{"Bibliographic":{"Title":"Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop","Authors":"","Publication date":"2012","Publisher":""},"Administrative":{"Date created":"08-17-2023","Language":"English","Rights":"CC 0","Size":"0000734129"},"Pages":["NOAA Special Report\nPROCEEDINGS AND RESULTS OF THE 2011 NTHMP\nMODEL BENCHMARKING WORKSHOP\nNational Tsunami Hazard Mitigation Program\nHilo Bay\nHazard Assessment\nJuly 2012\nU.S. DEPARTMENT OF COMMERCE\nTexas A&M University at\nAM\nNOAA\nGalveston\nNational Oceanic and Atmospheric\nTEXAS A&M\nUNIVERSITY\nDEPARTMENT\nAdministration\nGALVESTON\nOF","NOAA Special Report\nPROCEEDINGS AND RESULTS OF THE 2011 NTHMP\nMODEL BENCHMARKING WORKSHOP\nNTHMP Model Benchmarking Workshop Contributors:\nStéphane Abadie (University of Rhode Island)\nFrank González (University of Washington)\nLIBRARY\nStephan Grilli (University of Rhode Island)\nJuan Horrillo (Texas A&M at Galveston)\nOCT 262012\nBill Knight (West Coast & Alaska Tsunami Warning Center)\nNational communic &\nDmitry Nicolsky (University of Alaska at Fairbanks)\nAtmospheric\nAdministration\nU.S. Dept. of Commerce\nVolker Roeber (University of Hawaii)\nFengyan Shi (University of Delaware)\nElena Tolkova (Joint Institute for the Study of the Atmosphere and Ocean, University of\nWashington/Pacific Marine Environmental Laboratory)\nYoshiki Yamazaki (University of Hawaii)\nJoseph Zhang (Oregon Health & Science University)\nEditor: Loren Pahlke (Cooperative Institute for Research in Environmental Sciences)\nNTHMP Mapping and Modeling Co-Chairs:\nMarie Eble (Pacific Marine Environmental Institute)\nGC\nBill Knight (West Coast & Alaska Tsunami Warning Center)\n223\nRick Wilson (California Geological Survey)\nN84\n2012\nJuly 2012\nATMOSPHER\nAND\nU.S. DEPARTMENT OF\nTexas A&M University\nSignature\nATM\nNOAA\nCOMMERCE\nat Galveston\nTEXAS A&M\nDEPARTMENT\nNational Oceanic and Atmospheric\nJNIVERSITY\nGALVESTON\nAdministration","National Tsunami Hazard Mitigation Program (NTHMP)\nii\nNOTICE\nMention of a commercial company or product does not constitute an endorsement by\nNOAA/NTHMP. Use of information from this publication concerning proprietary products or\nthe tests of such products for publicity or advertising purposes is not authorized.\nThis publication may be cited as:\n[NTHMP] National Tsunami Hazard Mitigation Program. 2012. Proceedings and Results of the\n2011 NTHMP Model Benchmarking Workshop. Boulder: U.S. Department of Commerce/\nNOAA/NTHMP; (NOAA Special Report). 436 p.\nA current PDF version of this document is maintained and available from the publications list at:\nhttp://nthmp.tsunami.gov","MODEL BENCHMARKING WORKSHOP AND RESULTS\niii\nContents\nList of Tables\nV\nList of Figures\nvii\n1\nNTHMP Model Benchmarking Workshop\n1\n1.1\nAcknowledgements\n1\n1.2\nExecutive summary\n1\n1.3\nIntroduction\n2\n1.4\nNTHMP models\n4\n1.5\nBenchmark tests\n13\n1.6\nWorkshop section summary\n20\n1.7\nRecommendations\n20\n1.8\nProposed benchmark tests and lessons learned\n21\n1.9\nModels comparison\n26\n1.10 References\n49\n2\nAlaska Tsunami Model\n56\n2.1\nIntroduction\n56\n2.2\nModel description\n56\n2.3\nBenchmark results\n58\n2.4\nLessons learned\n84\n2.5\nProposed benchmarks\n85\n2.6 References\n86\n3\nATFM (Alaska Tsunami Forecast Model)\n89\n3.1\nIntroduction\n89\n3.2\nModel description\n89\n3.3\nObservations and next steps\n103\n3.4 References\n104\n4\nFully Nonlinear Boussinesq Wave Model FUNWAVE-TVD, V. 1.0\n105\n4.1 Introduction\n105\n4.2\nModel description\n105\n4.3\nBasic hydrodynamic considerations\n110\n4.4\nAnalytical benchmarks\n111\n4.5\nLaboratory benchmarks\n120\n4.6 References\n133\n5\nGeoClaw Model\n135\n5.1 Introduction\n135\n5.2\nModel description\n135\n5.3\nBenchmark results\n139\n5.4\nFurther remarks and suggestions\n206\n5.5 References\n209","National Tsunami Hazard Mitigation Program (NTHMP)\niv\nMOST (Method Of Splitting Tsunamis) Numerical Model\n6\n212\n6.1\nModel description\n212\n6.2\nBenchmark problems\n216\nProposed benchmark problem: test for tolerance to depth discontinuities\n6.3\n233\n6.4 References\n237\n7\nNEOWAVE\n239\nModel description\n7.1\n239\nModel verification and validation\n7.2\n246\nProposed field benchmark problems\n7.3\n254\n7.4 Conclusions\n260\n7.5 References\n261\n7.6 Figures\n264\n8\nSELFE\n303\n8.1 Introduction\n303\nModel description\n8.2\n303\nBenchmark results\n8.3\n305\n8.4\nLessons learned\n334\n8.5 References\n335\n9\nTHETIS\n337\n9.1 Introduction\n337\nBP8: Three-dimensional landslide\n9.2\n338\n9.3\nReferences\n346\n10\nTSUNAMI3D\n348\n10.1 Introduction\n348\nTSUNAMI3D Governing Equations\n10.2\n349\n10.3 Lab experiments\n352\n10.4 Conclusions\n358\n10.5 References\n359\n11\nBOSZ\n361\n11.1 Introduction\n361\n11.2 Model description\n361\n11.3 Benchmark comparisons\n365\n11.4 Proposed benchmark problems\n369\n11.5 References\n374\n11.6 Figures\n376","MODEL BENCHMARKING WORKSHOP AND RESULTS\nV\nList of Tables\nTable 1-1: NTHMP model benchmarking workshop participants. Participant affiliation and\nmodel of expertise are provided.\n3\nTable 1-2: Summary of model characteristics (ALASKA, ATFM, BOSZ, FUNWAVE,\nGeoClaw)\n5\nTable 1-3: Summary of model characteristics (MOST, NEOWAVE, SELFE, THETIS,\nTSUNAMI3D)\n6\nTable 1-4: Current benchmark tests for model verification and validation\n14\nTable 1-5: Current allowable errors for model validation and verification, after Synolakis et\nal. (2007)\n28\nTable 1-6: Allowable errors for the main three categories used for benchmarking\n34\nTable 1-7: BP1: NTHMP models' errors with respect to the analytical solution for H =\n0.0185. a) surface profile errors at t = [35, 40, 45, 50, 55, 60, 65]. b) sea level time\nseries errors at X = 9.95 and X = 0.25. RMS: Normalized root mean square\ndeviation. MAX: Maximum amplitude or runup error\n38\nTable 1-8: BP4: NTHMP's models errors with respect to the lab experiment data. a) surface\nprofile errors for Case A, H = 0.0185. b) surface profile errors for Case B, H =\n0.30. RMS: Normalized root mean square deviations. MAX: Maximum amplitude\nor relative runup error\n41\nTable 1-9: BP6: Sea level time series NTHMP models' errors with respect to laboratory\nexperiment data. a) Case A, H = 0.045; b) Case B, H = 0.096; and c) Case C, H =\n0.181. RMS: Normalized root mean square deviation error. MAX: Maximum\namplitude or runup relative error\n46\nTable 1-10: - Runup NTHMP models' errors with respect to laboratory experiment data for\nCase A (H = 0.045), Case B (H = 0.096), and Case C (H = 0.181). RMS:\nNormalized root mean square deviation error. MAX: Maximum runup relative\n47\nerror.\nTable 1-11: BP9: NTHMP Models' relative error with respect to field measurement data,\nOkushiri Tsunami, 1993. a) Models' maximum amplitude error for Iwanai and\nEsashi gauges. b) Models' runup errors around Okushiri Island (see Figure 1-3).\nMAX: Maximum amplitude relative error. ERR: Runup relative error.\n49\nTable 2-1: Comparison between the numerically computed and measured runup at the\nvertical wall.\n70\nTable 4-1: Maximum runup for gauge 9 for different grid size\n111\nTable 4-2: Runup data from numerical calculations compared with runup law values\n113\nTable 4-3: Runup data from numerical calculations compared with runup law for N-wave\n116","National Tsunami Hazard Mitigation Program (NTHMP)\nvi\nTable 4-4: Maximum runup of solitary wave on composite beach compared to runup law\n118\n(29).\nTable 4-5: Percent error of predicted maximum runup calculated for each gauge in conical\nisland test.\n126\nTable 5-1: Runup values in mm. Lab results taken from Table 1 of Le Veque (2011). Two\ndifferent resolutions with 36 and 72 points in the y direction were compared, with\nmx = 4my points in the X direction\n150\nTable 7-1. Runup comparisons on vertical wall.\n248\nTable 7-2: Fault parameters of DCRC-17a (Takahashi et al., 1995)\n252\nTable 7-3: Recorded runup for the six trials from Matsuyama and Tanaka (2001).\n254\nTable 7-4: Fault parameters for the 29 September 2009 Mw = 8.1 2009 Samoa-Tonga\nEarthquake\n256\nTable 11-1: - Convergence test and computation time for BP4.\n366\n367\nTable 11-2: Runup on vertical wall in BP2.","MODEL BENCHMARKING WORKSHOP AND RESULTS\nvii\nList of Figures\nFigure 1-1: Definition Sketch for Canonical Bathymetry, i.e., Sloping Beach Connected to a\nConstant-depth Region.\n16\nFigure 1-2: Basin Geometry, coordinate system and location of gauges. Solid lines represent\napproximate basin and wavemaker surfaces. Circles along walls and dashed lines\nrepresent wave absorbing material. Note the gaps of approximately 0.38 m between\neach end of the wavemaker and the adjacent wall. (Figure courtesy of Frank\nGonzalez.)\n17\nFigure 1-3:a) Tide gauge locations at Iwanai and Esashi. b) Maximum runup measurements\naround Okushiri Island. Numbers in red indicate regions to determine maximum\nrunup. (Figure courtesy of Yoshiki Yamazaki)\n19\nFigure 1-4: Comparison of analytical solution (crosses) versus NTHMP's models surface\nprofiles (solid lines) during runup of a non-breaking wave of H = 0.0185 at t = [35,\n45,55,56]. The analytical solution can be found in Synolakis (1986)\n35\nFigure 1-5: Comparison of analytical solution (crosses) versus NTHMP's models surface\nprofiles (solid lines) during runup of a non-breaking solitary wave of H = 0.0185 at\n= [40, 50, 60]. The analytical solution can be found in Synolakis, (1986).\n36\nt\nFigure 1-6: Comparison between the analytical solution (crosses) versus NTHMP's models\n(solid lines) during the runup of a non-breaking solitary wave of H = 0.0185 on\n1:19.85 beach. The top and bottom plots represent comparisons at X = 9.95 and X =\n0.25, respectively. The analytical solution is taken from Synolakis (1986)\n37\nFigure 1-7: Comparison of experimental data (crosses) versus NTHMP's models surface\nprofiles (solid lines) during runup of a non-breaking wave (Case A, H = 0.0185) at t\n= [30, 40, 50, 60, 70].\n39\nFigure 1-8: Comparison of experimental data (crosses) versus NTHMP's models surface\nprofiles (solid lines) during runup of a breaking wave (Case B, H = 0.30) at t = [15,\n20, 25, 30]\n40\nFigure 1-9: Sea level time series comparison between experimental data (crosses) versus\nNTHMP's models results (solid lines) of a solitary wave of H = 0.045 (Case A) at\ngauges shown in Figure 1-2\n42\nFigure 1-10: - Sea level time series comparison between experimental data (crosses) versus\nNTHMP's models results (solid lines) of a solitary wave of H = 0.096 (Case B) at\ngauges shown in Figure 1-2\n43\nFigure 1 - 11: Sea level time series comparison between experimental data (crosses) versus\nNTHMP's models results (solid lines) of a solitary wave of H = 0.181 (Case C) at\ngauges shown in Figure 1-2\n44","National Tsunami Hazard Mitigation Program (NTHMP)\nviii\nFigure 1-12: Runup comparison around a conical island between experimental (crosses)\nversus NTHMP models' results (solid lines) for H = [0.045, 0.096, 0.181] (Cases A,\nB, and C). Briggs et al. (1995)\n45\nFigure 1-13: Sea level time series at two tide stations (Iwanai and Esashi) along the west\ncoast of Hokkaido island during 1993 Okushiri tsunami. NTHMP models' results\n(solid lines), observed water level (dashed line). Observations courtesy of Yeh et al.\n48\n(1996).\nFigure 2-1: Non-scaled sketch of a canonical beach with a wave climbing up.\n58\nFigure 2-2: Left plot: comparison between the analytically and numerically computed\nsolutions simulating runup of the non-breaking wave in the case of H/d = 0.019 on\nthe 1:19.85 beach. Right plot: an enlarged version of the left plot within the\nrectangle region. Two numerical solutions computed on grids with Ax = d/20 and\nAx = d/200 are shown at t = 55vd/g. The numerical solution is shown to be\nconverging to the analytical one as the spatial discretization is refined. The\nanalytical solution is according to Synolakis (1986)\n59\nFigure 2-3: Comparison between the analytical solution (hollow symbols) and the finite\ndifference solution (filled symbols) during the runup of the non-breaking solitary\nwave with H/d = 0.019 on 1:19.85 beach. The top and bottom plots represent\ncomparisons at X = 0.25d and X = 9.95d, respectively. The analytical solution is\naccording to Synolakis (1986)\n60\nFigure 2-4: Left, non-scaled sketch of the composite beach modeling Revere Beach,\nMassachusetts. Vertical lines mark the locations of gauges measuring the water\nlevel in laboratory experiments. Right, an incident wave recorded by Gauge 4. This\nrecord was used to set the water height h(Xb; t) at the inflow boundary condition.\n61\nFigure 2-5: Comparison between the analytically and numerically computed solutions at the\ngauge locations shown in Figure 2-4. Left plot: Case A, Right plot Case C\n62\nFigure 2-6: Comparison between the analytically and numerically computed solutions at\nseveral moments of time. LSWE and NLSWE stand for the numerical solutions\ncomputed with linear and non-linear assumptions, respectively. The analytical\nsolution is according to Liu et al. (2003)\n63\nFigure 2-7: Comparison of observed and simulated water profiles during runup of a non-\nbreaking wave in the case of H/d = 0.019. Observations are shown by dots. The\nanalytical predictions and numerical calculations are marked by hollow and filled\nsymbols, respectively. The measurements are provided courtesy of Synolakis\n(1986).\n64\nFigure 2-8: Laboratory measured and simulated waterfront path Xw of a solitary wave\nrunning up on a canonical beach. Measurements are represented by squares and","MODEL BENCHMARKING WORKSHOP AND RESULTS\nix\nnumerical simulations by a line. The measurements are provided courtesy of\nSynolakis (1986)\n65\nFigure 2-9: Comparison of measured and simulated water profiles during runup of a non-\nbreaking wave in the case of H/d = 0.04. Observations are shown by squares. The\nanalytical predictions and numerical calculations are marked by hollow and filled\nsymbols, respectively. The measurements are provided courtesy of Synolakis\n(1986).\n66\nFigure 2-10: Comparison of measured and simulated water profiles during runup of a non-\nbreaking wave in the case of H/d = 0.3. Observations are shown by squares. The\nanalytical predictions and numerical calculations are marked by hollow and filled\nsymbols, respectively. The measurements are provided courtesy of Synolakis\n(1986).\n67\nFigure 2-11: Non-dimensional maximum runup of solitary waves on the 1:19.85 sloping\nbeach versus the height of the initial wave. The measured runup values (Synolakis,\n1986) are marked by dots. The dashed line represents maximum runup values\ncomputed without an effect of bottom friction, i.e., V = 0. The solid lines represent\nmaximum runup values computed with the effect of bottom friction, i.e., V = 0.02\nand V = 0.04. The measurements are provided courtesy of Synolakis (1986)\n68\nFigure 2-12: Comparison between the numerically computed solutions and laboratory\nmeasurements at the gauge locations shown in Figure 2-4. Left plot: Case A, Right\nplot Case C.\n70\nFigure 2-13: Top-down non-scaled sketch of a conically-shaped island. The solid circles\nrepresent exterior and interior bases of the island. The dotted line shows an initial\nlocation of the shoreline. The dash-dotted line shows the extent of the high\nresolution computational grid. The dots mark gauge locations where laboratory\nmeasured water level is compared to numerical calculations. The in-flow boundary\ncondition is simulated on the segment AD, while the open boundary condition is\nmodeled on the segments AB, BC, and CD. Location of gauges 1, 2, 3, and 4 is\nschematic, while locations of the rest of gauges are precise.\n71\nFigure 2-14: Left plot: comparison between the computed and measured water level at\ngauges shown in Figure 2-13 for an incident solitary wave in the case of H/d =\n0.05. Right plot: comparison between computed and measured inundation zones.\nTop view of the island, with the lee side at 90°. The dotted line represents the\ninitial shoreline. The measurements are provided courtesy of Briggs et al. (1995)\n72\nFigure 2-15: Left plot: comparison between the computed and measured water level at\ngauges shown in Figure 2-13 for an incident solitary wave in the case of H/d = 0.2.\nRight plot: comparison between computed and measured inundation zones. Top\nview of the island, with the lee side at 90°. The dotted line represents the initial\nshoreline. The measurements are provided courtesy of Briggs et al. (1995).\n73","National Tsunami Hazard Mitigation Program (NTHMP)\nX\nFigure 2-16: The 3-D view of the computational domain and numerical solution at 12\nseconds. Locations of gauges, at which the modeled and measured water level\ndynamics are compared, are shown by arrows. Abbreviations Ch5, Ch7, and Ch9\nstand for Channel 5, 7, and 9, respectively. The inlet boundary is modeled at X = 0.\n74\nAt y = 0 and y = 3.4, the reflective boundary conditions are set\nFigure 2-17: Comparison of the computed water height with the laboratory measurements at\nwater gauges Ch5, Ch7, and Ch9. The measurements are provided courtesy of the\nThird International Workshop on Long-Wave Runup Models (Liu et al., 2007).\n75\nFigure 2-18: Left side: frames 10, 25, 40, 55, and 70 from the overhead movie of the\nlaboratory experiment. The time interval between frames is 0.5 seconds. The\ndashed yellow line shows the instantaneous location of the shoreline. Right side:\nsnapshots of the numerical solution at the time intervals corresponding to the movie\nframes. The blue shaded area corresponds to the water domain and is considered to\nbe wet. The frames are provided courtesy of the Third International Workshop on\n76\nLong-Wave Runup Models (Liu et al., 2007).\nFigure 2-19: Schematic of the experimental setup. Locations of the water gauges are marked\nby green dots. The profiles along which the runup is measured are shown by red\n77\nlines\nFigure 2-20: Time histories of the block motion for the submerged case with A = -0.025 m\n78\nand A = -0.1 m.\nFigure 2-21: The top plots show the comparison between the computed and measured water\nlevel dynamics at two gauges, shown in Figure 2-19. The bottom plots show the\ncomparison between the computed and measured runup along two profiles, shown\n79\nin Figure 2-19.\nFigure 2-22: The computational domain used to simulate 1993 Okushiri tsunami. The\ntriangles mark the locations of the tide gauge stations that observed water levels to\nwhich we compare model dynamics. The contours mark the seafloor displacement\ncaused by the Hokkaido-Nansei-Oki earthquake (Takahashi et al., 1995).\n80\nFigure 2-23: Computed and observed water levels at two tide stations located along the west\ncoast of Hokkaido island during 1993 Okushiri tsunami. The observations are\n81\nprovided courtesy of Yeh et al. (1996)\nFigure 2-24: The computed and observed runup in meters at 19 locations along the coast of\nOkushiri island after the 1993 Okushiri tsunami. The observations are provided\n82\ncourtesy of Kato and Tsuji (1994)\nFigure 2-25: Numerical modeling of a tsunami wave overflowing the Aonae peninsula,\nviewed from above. The dashed black and solid red contours represent the water\nlevel and land elevation, respectively. The upper left plot shows an approaching 5\nmeter high wave. As the wave approaches, it steepens and overtops the peninsula as","MODEL BENCHMARKING WORKSHOP AND RESULTS\nxi\nillustrated by the upper right plot. In the lower left plot, the wavefront bends around\nthe peninsula and propagates in the direction of Hamatsumae. In the lower right\nplot, the water retreats and the seabed becomes partially dry.\n83\nFigure 2-26: The computed and observed runup in the vicinity of the Aonae peninsula after\n1993 Okushiri tsunami. The triangles mark the locations where the observations\nwere conducted. The computed runup distribution has a local maximum near\nHamatsumae, as observed by eyewitnesses. The observations are provided courtesy\nof Kato and Tsuji (1994).\n84\nFigure 2-27: Water height profiles 5(x, 0, t) for numerical (solid) and analytical (hollow)\nsolutions at t = 3T, 3.1T, 3.2T, 3.3T, 3.4T, and 3.5T, where T is the period of the\ncorresponding oscillatory mode.\n86\nFigure 3-1: Initial condition for BP1. Note figure is not drawn to scale. The right vertical\nscale shows the range of sea level used in the benchmark, and the left vertical scale\nshows the ratio of maximum heights (H) to depth (D)\n91\nFigure 3-2: Non-dimensional sea level profiles as a function of non-dimensional distance for\nBP1. Dashed red lines are the analytic result while the solid blue are modeled\nresults for dimensionless times T = 35 to 45\n92\nFigure 3-3: Non-dimensional sea level profiles as a function of non-dimensional distance for\nBP1. Dashed red lines are the analytic result while the solid blue are the modeled\nresults for dimensionless times T = 55 to 70\n92\nFigure 3-4: Non-dimensional sea level as a function of non-dimensional time at two\npositions for BP1. Dashed red lines are the analytic result while the solid blue lines\nare model results\n93\nFigure 3-5: Comparison of non-dimensional model runup (blue line) to the analytical result\n(red line) with increasing mesh resolution. All other computations for this\nbenchmark were computed with D/DX = 20\n94\nFigure 3-6: Non-dimensional sea level VS. non-dimensional distance for several non-\ndimensional times. Model results and experimental data for H/D = 0.0185. Model\nresults in black, experimental data in red.\n95\nFigure 3-7: Non-dimensional sea level VS. non-dimensional distance for several non-\ndimensional times. Model results and experimental data for H/D = 0.3. Model\nresults in black, experimental data in red.\n95\nFigure 3-8: Hydrostatic model results and experimental data for H/D = 0.3. Model results in\nblack, experimental data in red\n96\nFigure 3-9: Model results for R/D0 for increasing H/D with various values of a and U.\n96\nFigure 3-10: Run C model results for BP6 in one second steps from time t = 29 to 34\nseconds.\n97","National Tsunami Hazard Mitigation Program (NTHMP)\nxii\n98\nFigure 3-11: Plan view sketch of BP6 domain, sub-mesh, and gauge locations.\nFigure 3-12: Surface elevation VS. time at four gauge locations for BP6, Cases A, B, and C.\n99\nModel results are in black, and the experimental data are in red.\nFigure 3-13: Comparison of model runup results with experimental runups for BP6: cases A,\n100\nB, and C.\nFigure 3-14: Comparison of model results with tide gauge data from the Iwanai and Esashi\ngauges. Model results are in black, and the measured data are in red\n101\nFigure 3-15: Model inundation sequence along the Aonae Peninsula from the 1993 Okushiri\n102\nIsland Tsunami.\nFigure 3-16: Maximum model runups and tsunami amplitudes. Magenta is 14 m, red is\n12 m, and orange is 10 m. The original coastline is drawn in blue. The arrow\n102\nlocates the region near Hamatsumae.\nFigure 3-17: Comparison of model inundation forecasts with observations for the 1993\nOkushiri Island Tsunami. The spread in model results is from a series of runups at\n103\ngrid points near the test point.\nFigure 4-1: Definition sketch for simple beach bathymetry (from Synolakis et al. (2007,\n112\nFigure A1))\nFigure 4-2: Numerical simulation data for maximum runup of nonbreaking waves climbing\n114\nup different beach slopes. Solid line represents the runup law (25)\nFigure 4-3: The water level profiles during runup of the non-breaking wave in the case of\nH/d = 0.019 on a 1:19.85 beach. The solid blue line represents the analytical\nsolution according to Synolakis (1986), and the dashed red line represents the\n115\nresults of numerical simulation.\nFigure 4-4: The water level dynamics at two locations X/d = 0.25 and X/d = 9.95. Solid blue\nline represents the analytical solution in according to Synolakis (1986), and dashed\n116\nred line represents the numerical simulation\nFigure 4-5: Numerical simulation data for the maximum runup of N-waves climbing up\ndifferent beach slopes. The solid line represents the runup law (28).\n117\nFigure 4-6: Definition sketch for Revere Beach (from Synolakis et al. (2007, Figure A7))\n118\nFigure 4-7: Time evolution of nonbreaking H/d = 0.0378 wave on composite beach. The red\nline shows the numerical solution and the blue line represents the analytic solution. 119\nFigure 4-8: Time evolution of breaking H/d = 0.2578 initial wave on composite beach. The\nred line shows the numerical solution and the blue line represents the analytic\n119\nsolution.\nFigure 4-9: Time evolution of breaking H/d = 0.6404 initial wave on composite beach. The\nred line shows the numerical solution and the blue line represents the analytic\n120\nsolution.","MODEL BENCHMARKING WORKSHOP AND RESULTS\nxiii\nFigure 4-10: Time evolution of nonbreaking H/d = 0.0185 initial wave. The solid line shows\nthe numerical solution and the dots represent the laboratory data.\n121\nFigure 4-11: Time evolution of breaking H/d = 0.3 initial wave. The solid line shows the\nnumerical solution and the dots represent the laboratory data.\n122\nFigure 4-12: Time evolution of nonbreaking H/d = 0.0378 initial wave on composite beach.\nThe red line shows the numerical solution and the blue line represents the\nlaboratory data\n123\nFigure 4-13: Time evolution of breaking H/d = 0.2578 initial wave on composite beach. The\nred line shows the numerical solution and the blue line represents the laboratory\ndata.\n123\nFigure 4-14: Time evolution of breaking H/d = 0.6404 initial wave on composite beach. The\nred line shows the numerical solution and the blue line represents the laboratory\ndata.\n124\nFigure 4-15: View of conical island (top) and basin (bottom) (from Synolakis et al. (2007,\nFigure A16)).\n125\nFigure 4-16: Definition sketch for conical island. All dimensions are in cm (from Synolakis\net al. (2007, Figure A17))\n125\nFigure 4-17: Schematic gauge locations around the conical island. From Synolakis et al.\n(2007, Figure A18).\n126\nFigure 4-18: Comparison of computed and measured time series of free surface for H/d =\n0.045. Solid lines: measured, Dashed lines: Computed.\n127\nFigure 4-19: Comparison of computed and measured time series of free surface for H/d =\n0.091. Solid lines: measured, Dashed lines: Computed.\n127\nFigure 4-20: Comparison of computed and measured time series of free surface for H/d =\n0.181. Solid lines: measured, Dashed lines: Computed.\n128\nFigure 4-21: Bathymetric profile for experimental setup for Monai Valley experiment (2007,\nFigure A24))\n129\nFigure 4-22: Initial wave profile for Monai Valley experiment (2007, Figure A25))\n129\nFigure 4-23: Computational area for Monai Valley experiment (2007, Figure A26)).\n130\nFigure 4-24: Computational area for Monai Valley numerical simulation\n130\nFigure 4-25: Comparison of computed and measured time series of free surface. Dashed\nlines: Computed, Solid lines: Measured\n131\nFigure 4-26: Comparison between extracted movie frames from the overhead movie of the\nlaboratory experiment (left) (from http://burn.giseis.alaska.edu/file\ndoed/Dmitry/BM7_description.zip) and numerical simulation (right)\n132\nFigure 5-1: Sketch of canonical beach and approaching wave\n140","National Tsunami Hazard Mitigation Program (NTHMP)\nxiv\nFigure 5-2: Profile plots for the times specified in Task 2. For each pair of plots at a\nparticular time, the top frame provides a full view of the incoming wave and the\nbottom frame provides an expanded view of the inundation area. In some regions,\n141\nthe analytic and GeoClaw solutions lie atop one another\nFigure 5-3: Left column: Water level time series at location x/d = 9. 95. Right column:\n142\nWater level time series at location x/d = 0.25\n142\nFigure 5-4: Runup on canonical beach as a function of time.\n144\nFigure 5-5: Case A\n145\nFigure 5-6: Case B\n146\nFigure 5-7: Case C\n147\nFigure 5-8: Convergence Plot for Gauge 4 in Case C\nFigure 5-9: Sample results for d = 0.061. The water surface n(x, y, t) (colors with dark red\n+0.02 m and dark blue -0.02 m) and bathymetry (0.01 m contour levels). Only a\nportion of the computational domain is shown. Grid resolution: Ax = Ay = 0.025 m\non the full domain, with refinement to Ax = Ay = 0.0025 m in the nearshore region\nin the rectangular box. The full domain goes to X = 6.2 and to y = 1.8\n149\nFigure 5-10: Gauge and runup results for d = 0.061. Three different resolutions with my =\n18, 36, and 72 points in the y direction were compared, with mx = 4my points in\n152\nthe X direction.\n153\nFigure 5-11: Gauge and runup results for d = 0.08\n154\nFigure 5-12: Gauge and runup results for d = 0.1\n155\nFigure 5-13: Gauge and runup results for d = 0.12\n156\nFigure 5-14: Gauge and runup results for d = 0.14\n157\nFigure 5-15: Gauge and runup results for d = 0.149\n158\nFigure 5-16: Gauge and runup results for d = 0.189\n159\nFigure 5-17: Schematic of computational domain\nFigure 5-18: Runup computations and lab measurements for the low amplitude case. In the\npaired plots for each time value, the bottom frame provides a zoomed view of the\ninundation area for the incident wave presented in the top frame\n161\nFigure 5-19: Runup computations and lab measurements for the high amplitude case. In the\npaired plots for each time value, the bottom frame provides a zoomed view of the\ninundation area for the incident wave presented in the top frame\n162\nFigure 5-20: Maximum runup estimate of 0.085 cm for the low amplitude case, occurring at\n163\n55 seconds of the computation.\nFigure 5-21: Maximum runup estimate of 0.42 cm for the high amplitude case, occurring at\n163\n40 seconds of the computation.","MODEL BENCHMARKING WORKSHOP AND RESULTS\nXV\nFigure 5-22: Scatter plot of nondimensional maximum runup, R/d, versus nondimensional\nincident wave height, H/d, resulting from a total of more than 40 experiments\nconducted by Y. Joseph Zhan and described at Le Veque (2011). Red dots indicate\nthe numerical results\n164\nFigure 5-23: Case A\n166\nFigure 5-24: Case B\n167\nFigure 5-25: Case C\n168\nFigure 5-26: Convergence plot for Gauge 4 in Case C\n169\nFigure 5-27: Basin geometry and coordinate system. Solid lines represent approximate basin\nand wavemaker vertical surfaces. Circles along walls and dashed lines represent\nrolls of wave absorbing material. Note the gaps of approximately 0.38 m between\neach end of the wavemaker and the adjacent wall. Gauge positions are given in\nFigure 5-28.\n172\nFigure 5-28: Coordinates of laboratory gauges 1, 2, 3, 4, 6, 9, 16, and 22\n173\nFigure 5-29: Animation snapshots of Case A for the 12.5 cm resolution computational grid,\n174\nFigure 5-30: Animation snapshots of Case C for the 12.5 cm resolution computational grid.\n175\nFigure 5-31: Comparison of laboratory gauge and GeoClaw time series for the Case C, 100\ncm resolution grid\n176\nFigure\n5-32: Comparison of laboratory gauge and GeoClaw time series for the Case C, 50\ncm resolution computational grid\n177\nFigure 5-33: Comparison of laboratory gauge and GeoClaw time series for the Case C, 12.5\ncm resolution computational grid\n178\nFigure 5-34: Comparison of laboratory gauge and GeoClaw time series for the Case A, 12.5\ncm resolution computational grid\n179\nFigure 5-35: Island runup for Case A, using a 12.5 cm resolution computational grid\n180\nFigure 5-36: Island runup for Case C, using a 12.5 cm resolution computational grid\n181\nFigure 5-37: Island runup for Case C on a 12.5 cm grid, for differing values of Manning's\nfriction coefficient, M, and the 'Dry Cell Depth', DCD, threshold.\n182\nFigure 5-38: Left column: on 423 X 243 grid (same as given bathymetry). Right column: 211\nX 121 grid\n185\nFigure 5-39: Left column: Frames 10, 25, and 40 from the movie. Right column: Zoomed\nview of computation.\n186\nFigure 5-40: Left column: Frames 55 and 70 from the movie. Right column: Zoomed view\nof computation\n187\nFigure 5-41: Maximum runup relative to observed location (white dot)\n188","National Tsunami Hazard Mitigation Program (NTHMP)\nxvi\nFigure 5-42: Single grid 140 X 40 GeoClaw simulation of Case 1 to illustrate moving\nbathymetry and gauge locations.\n191\nFigure 5-43: Left column: Time histories of the surface elevation with respect to still water\nlevel for case 1. Right column: Time histories of the runup measurements with\nrespect to still water level for case 1, at Runup gauges 2 and 3. Note: runup values\nare negated in this figure for both GeoClaw and lab data due to a programming\n192\nglitch\nFigure 5-44: Left column: Time histories of the surface elevation with respect to still water\nlevel for case 2. Right column: Time histories of the runup measurements with\nrespect to still water level for case 2, at Runup gauges 2 and 3. Note: runup values\nare negated in this figure for both GeoClaw and lab data due to a programming\nglitch.\n193\nFigure 5-45: Full computational domain for one simulation, in which AMR grids are\nfocused near the Aonae peninsula at the south of Okushiri Island\n195\nFigure 5-46: Zoom on Okushiri Island.\n197\nFigure 5-47: Zoom on the Aonae peninsula showing the first wave arriving from the west\nand the second from the east. Color map shows elevation of sea surface. 4-meter\ncontours of bathymetry and topography are shown.\n198\nFigure 5-48: Inundation map of the Aonae peninsula. Color map shows maximum fluid\ndepth over entire computation at each point. 4-meter contours of bathymetry and\ntopography are shown\n199\nFigure 5-49: Photographs of the Aonae peninsula taken shortly after the event. Left: From\nhttp://www.usc.edu/dept/tsunamis/hokkaido/aonae.html, Right: From\nhttp://nctr.pmel.noaa.gov/okushiri_devastation.html credited to Y. Tsuji.\n199\nFigure 5-50: Inundation map of the Hamatsumae neighborhood just east of the Aonae\npeninsula. Color map shows maximum fluid depth over entire computation at each\npoint, with the same color scale as Figure 5-48. 4-meter contours of bathymetry and\ntopography are shown\n200\nFigure 5-51: Top: Locations of field observations by three independent field survey teams,\nrelative to the computational bathy/topo grid system. Only the observations of Tsuji\n(left figure) were used in this study due to misregistration of the other two data sets.\nBottom: Measured and computed runup at 21 points around Okushiri Island where\nmeasured by the Tsuji team. Red circles are measurements; green diamonds are\nestimated from the computation. Two red circles at the same point represent\nestimates of minimum and maximum inundation observed near the point. Two\ngreen diamonds at the same point represent values estimated when the model was\nrun with and without bottom friction (Manning coefficient 0.025). The runup\ncomputed with bottom friction is the smaller value.\n201","MODEL BENCHMARKING WORKSHOP AND RESULTS\nxvii\nFigure 5-52: Scatter plot illustrating the correlation between measured and computed values\nfor the values shown in Figure 5-51.\n202\nFigure 5-53: Inundation map of the valley north of Monai. Color map shows maximum fluid\ndepth over entire computation at each point. 4-meter contours of bathymetry and\ntopography are shown. Compare to Figure 5-41 showing the related wave tank\nsimulation.\n203\nFigure 5-54: Analog tide gauge records at Iwanai and Esashi\n204\nFigure 5-55: Iwanai digitized tide gauge record (black line) and GeoClaw (blue line) time\nseries.\n205\nFigure 5-56: Esashi digitized tide gauge record (black line) and GeoClaw (blue line) time\nseries.\n206\nFigure 6-1: Time histories at locations x/d = 0.25 (left) and x/d = 9.95 (right), analytical\nNSW solution (black), and numerical solution (red), in dimensionless units\n217\nFigure 6-2: Water level profiles at t/T = 35, 45, 55, and 65: analytical NSW solution (black),\nand numerical solution in grid nodes (red dots), in dimensionless units\n217\nFigure 6-3: Water level profiles for H/d = 0.0185 at t/t = 30, 40, 50, 60, and 70:\nmeasurements (black dots) and numerical solution in grid nodes (red dots), in\ndimensionless units\n218\nFigure 6-4: Water level profiles for H/d = 0.3 at t/t = 15, 20, 25, and 30: measurements\n(black dots) and numerical solution in grid nodes (red dots), in dimensionless units.. 218\nFigure 6-5: Time histories at gauges 4-10 (top to bottom), provided by measurements\n(black), analytical LSW solution (green), and numerical solution (red). Case A.\n219\nFigure 6-6: Time histories at gauges 4-10 (top to bottom), provided by measurements\n(black), analytical LSW solution (green), and numerical solution (red). Case B\n220\nFigure 6-7: Time histories at gauges 4-10 (top to bottom), provided by measurements\n(black), analytical LSW solution (green), and numerical solution (red). Case C\n221\nFigure 6-8: Time histories at the wall, provided by analytical LSW solution (green) and\nnumerical solution (red). Left to right: cases A, B, and C\n221\nFigure 6-9: Time histories at gauges 1-4, 6, 9, 16, and 22: simulated with MOST (red) and\nlaboratory measurements (black). Case A\n223\nFigure 6-10: Time histories at gauges 1-4, 6, 9, 16, and 22: simulated with MOST (red) and\nlaboratory measurements (black). Case B.\n224\nFigure 6-11: Time histories at gauges 1-4, 6, 9, 16, and 22: simulated with MOST (red) and\nlaboratory measurements (black). Case C.\n224","xviii\nNational Tsunami Hazard Mitigation Program (NTHMP)\nFigure 6-12: Maximum runup height around the island: computed with MOST (red) at dx =\n5 cm (top) and dx = 2.5 cm (bottom), and laboratory measurements (black\ntriangles). Case A.\n225\nFigure 6-13: Maximum runup height around the island: computed with MOST (red) at dx =\n5 cm (top) and dx = 2.5 cm (bottom), and laboratory measurements (black\ntriangles). Case B\n226\nFigure 6-14: Maximum runup height around the island: computed with MOST (red) at dx =\n5 cm (top) and dx = 2.5 cm (bottom), and laboratory measurements (black\ntriangles). Case C\n227\nFigure 6-15: Time histories at gauges 5, 7, and 9: simulated with MOST (red) and laboratory\n228\nmeasurements.\nFigure 6-16: Snapshots of computed water surface, to be compared with video frames 10,\n25, 40, 55, and 70. The time when each snapshot is taken is shown in the plot.\nWater elevation is shown with respect to still surface (sea) or with respect to\ninundated land level. Color scale - cm.\n229\nFigure 6-17: Top left: grid A with grid B contour shown in red. Bottom left: grid B with grid\nC contour shown in red. Right: grid C with contours of grids Da and Dm shown in\nred. Color scale - m.\n230\nFigure 6-18: Top: simulated time history at the southern tip of Okushiri Island; red dots\nmark wave phases at which snapshots below were taken. Bottom: snapshot of the\nsea surface displacement / water height over inundated land (color) and the velocity\nfield (arrows), 5 min (left) and 14 min (right) after the earthquake. Color scale - m 231\nFigure 6-19: Computed water level at Iwanai and Esashi tide gauges (red) VS. measurements\n(black circles).\n232\nFigure 6-20: Maximum modeled runup distribution around Okushiri Island (center pane),\nAonae area (bottom), Monai area (left). Red dots show locations of field\nmeasurements (0.01° was subtracted from the latitude vector). Color scale - m.\n233\nFigure 6-21: Pulse trajectory in x-t plane.\n234\nFigure 6-22: Corresponding pulse series (14) whose propagation over the constant depth d in\nthe flat basin was modeled by MOST.\n236\nFigure 6-23: Time histories at x = 0 of the evolution of a single pulse in a basin (13) for d1 =\nd2 = 0.8d, 0.6d, 0.4d, 0.2d (red) and that of the corresponding pulse series over the\nflat bottom (black)\n237\nFigure 7-1: Schematic of free-surface flow generated by seafloor deformation.\n264\nFigure 7-2: Linear dispersion relation.\nAiry wave theory;\n, classical Boussinesq\nequations of Peregrine (1967);\n(red), depth-integrated, non-hydrostatic\nequations\n265","MODEL BENCHMARKING WORKSHOP AND RESULTS\nxix\nFigure 7-3: Definition sketch of the discretization scheme.\n266\nFigure 7-4: Schematic of a two-level nested grid system.\n267\nFigure 7-5: Schematic of two-way grid-nesting and time-integration schemes.\n268\nFigure 7-6: Definition sketch of solitary wave on compound slope. o, gauge locations.\n268\nFigure 7-7: Time series of surface elevation at gauges on compound slope with A/h = 0.039.\n(black), analytical solution of Kânoglu and Synolakis (1998);\n(red),\ncomputed data\n269\nFigure 7-8: Time series of surface elevation at gauges on compound slope with A/h = 0.264.\n(black), analytical solution of Kânoglu and Synolakis (1998);\n(red),\ncomputed data\n270\nFigure 7-9: Time series of surface elevation at gauges on compound slope with A/h = 0.696.\n(black), analytical solution of Kânoglu and Synolakis (1998);\n(red),\ncomputed data\n271\nFigure 7-10: Time series of surface elevation at gauges on compound slope with A/h =\n(black), laboratory data of Briggs et al. (1996);\n0.039.\n(red), solution\nwith NH-Hybrid scheme;\n(blue), solution without NH-Hybrid scheme.\n272\nFigure 7-11: Time series of surface elevation at gauges on compound slope with A/h =\n0.264.\n(black), laboratory data of Briggs et al. (1996);\n(red), solution\nwith NH-Hybrid scheme;\n(blue), solution without NH-Hybrid scheme.\n273\nFigure 7-12: Time series of surface elevation at gauges on compound slope with A/h =\n0.696.\n(black), laboratory data of Briggs et al. (1996);\n(red), solution\nwith NH-Hybrid scheme;\n(blue), solution without NH-Hybrid scheme.\n274\nFigure 7-13: Definition sketch of solitary wave runup on plane beach\n275\nFigure 7-14: Transformation and runup of a solitary wave on a 1:19.85 plane beach with\nA/h = 0.019. o, analytical solution of Synolakis (1987);\n(red), computed data. 275\nFigure 7-15: Time series of surface elevation at x/h = -9.95 and x/h = -0.25 on a 1:19.85\nplane beach with A/h = 0.019. o, analytical solution of Synolakis (1987);\n(red), computed data\n276\nFigure 7-16: - Convergence of numerical solution on a 1:19.85 plane beach with A/h = 0.019.\nanalytical solution of Synolakis (1987);\n(red), computed data with Ax/h\n,\n= 0.025;\n(magenta), computed data with Ax/h = 0.0625;\n(cyan),\ncomputed data with Ax/h = 0.125;\n(red), computed data with Ax/h = 0.25.\n276\nFigure 7-17: Transformation and runup of a solitary wave on a 1:19.85 plane beach with\nA/h = 0.0185. o, laboratory data of Synolakis (1987);\n(red), computed data.\n277","National Tsunami Hazard Mitigation Program (NTHMP)\nXX\nFigure 7-18: Transformation and runup of a solitary wave on a 1:19.85 plane beach with\nA/h = 0.03. o, laboratory data of Synolakis (1987);\n(red), non-hydrostatic\nsolution with MCA;\n(blue), hydrostatic solution without MCA.\n278\nFigure 7-19: Transformation and runup of a solitary wave on a 1:19.85 plane beach with\nA/h = 0.03. o, laboratory data of Synolakis (1987);\n(red), non-hydrostatic\nsolution with MCA;\n(blue), hydrostatic solution with MCA.\n279\nFigure 7-20: Schematic sketch of the conical island experiment. (a) Perspective view. (b)\nSide view (center cross section). o, gauge locations\n280\nFigure 7-21: Time series of surface elevation at gauges around a conical island. (a) A/h =\n0.045. (b) A/h = 0.096. (c) A/h = 0.181. o, laboratory data from Briggs et al.\n(1995);\n(red), computed data.\n281\nFigure 7-22: Inundation and runup around a conical island. (a) A/h = 0.045. (b) A/h = 0.096.\n(c) A/h = 0.181. o, laboratory data from Briggs et al. (1995);\n(red), computed\ndata.\n281\nFigure 7-23: Schematic sketch of the complex reef system experiment in ISEC BP1. (a)\nPlain view. (b) Side view (center cross section)\n282\nFigure 7-24: Schematic sketch of the complex reef system experiment in ISEC BP2. (a)\nPlain view. (b) Side view (center cross section)\n283\nFigure 7-25: Geometry of reef system and wave gauge and ADV locations for ISEC BP1.\nO\n(white), wave gauge; o (red), wave gauge and ADV\n284\nFigure 7-26: Geometry of reef system and wave gauge and ADV locations for ISEC BP2.\nO\n(white), wave gauge; O (red), wave gauge and ADV\n284\nFigure 7-27: Time series of surface elevation and velocity at gauges for ISEC BP1. (a)\nSurface elevations at y = 0 m. (b) Surface elevations at y = 5 m. (c) Surface\nelevations at X = 25 m. (d) Horizontal velocity comparison at ADV.\n(black),\nlaboratory data of Swigler and Lynett (2011);\n(red), solution with NH-Hybrid\nscheme;\n(blue), solution without NH-Hybrid scheme\n285\nFigure 7-28: Time series of surface elevation and velocity at gauges for ISEC BP2. (a)\nSurface elevations at y = 0 m. (b) Surface elevations at y = 5 m. (c) Surface\nelevations at X = 25 m. (d) Horizontal velocity comparison at ADV.\n(black),\nlaboratory data of Swigler and Lynett (2011);\n(red), solution with NH-Hybrid\nscheme;\n(blue), solution without NH-Hybrid scheme\n286\nFigure 7-29: Bathymetry and topography in the model region for the 1993 Hokkaido Nansei-\nOki tsunami. (a) level-1 computational domain, (b) level-3 domain. o (red),\nepicenter; O (white), tide gauge stations\n287\nFigure 7-30: Schematic of planar fault model.\n287","MODEL BENCHMARKING WORKSHOP AND RESULTS\nxxi\nFigure 7-31: Initial sea surface elevation for the 1993 Hokkaido Nansei-Oki tsunami. Solid\nrectangles indicate subfaults;\nuplift contours at 0.5-m intervals;\n,\nsubsidence contours at 0.2-m intervals; O (red), epicenter; O (white), tide gauge\nstations\n288\nFigure 7-32: Time series of surface elevation at tide gauges.\n(black), recorded data;\n(red), computed data.\n288\nFigure 7-33: Runup around Okushiri Island. O (white): recorded runup;\n(red),\ncomputed runup.\n289\nFigure 7-34: Input data for Monai Valley experiment. (a) Computational domain. (b) Initial\nwave profile. o, gauge locations;\n(black), topography contours at 0.0125-m\nintervals;\n(grey), bathymetry contours at 0.0125-m intervals;\n, initial\nprofile.\n290\nFigure 7-35: Time series of surface elevation at gauges in Monai Valley experiment.\n(black), laboratory data of Matsuyama and Tanaka (2001);\n(red), computed\ndata.\n291\nFigure 7-36: Runup and inundation comparisons. (a) Runup, (b) Inundation. o, laboratory\ndata of Matsuyama and Tanaka (2001);\n(red), computed data;\n(black),\ntopography contours at 0.0125-m intervals;\n(grey), bathymetry contours at\n0.0125-m intervals.\n292\nFigure 7-37: Bathymetry and topography in the model region for the 2009 Samoa Tsunami.\n(a) level-1 computational domain, (b) close-up view of epicenter and Samoa\nIslands. o (red), epicenter; O (white), water-level stations\n293\nFigure 7-38: Original bathymetry and topography in the model region for the 2009 Samoa\ntsunami. (a) close-up view of faults and Samoa Islands in the level-1 computational\ndomain, (b) level-2 domain, (c) level-3 domain , (d) level-4 domain. O (red),\nepicenter; O (white), water-level stations.\n294\nFigure 7-39: Smoothed bathymetry and topography with the depth-dependent Gaussian\nfunction in the model region for the 2009 Samoa tsunami. (a) close-up view of\nfaults and Samoa Islands in the level-1 computational domain, (b) the level-2\ndomain, (c) the level-3 domain , (d) the level-4 domain. O (red), epicenter;\nO\n(white), water-level stations.\n295\nFigure 7-40: Time series and spectra of surface elevations at water level stations. (black),\nrecorded data; (red), computed data.\n296\nFigure 7-41: Runup and inundation at inner Pago Pago Harbor.\n(white), recorded\ninundation; o(white): recorded runup; (blue): recorded flow depth plus land\nelevation; (red), computed data; (black), coastline; (grey), depth contours at 10-m\nintervals.\n297","National Tsunami Hazard Mitigation Program (NTHMP)\nxxii\nFigure 7-42: Original bathymetry and topography at south shore of Oahu, Hawaii and Kilo\nNalu Nearshore Reef Observatory. O (white), water level and velocity measurement\n297\npoints.\nFigure 7-43: Original bathymetry and topography in the model region. (a) level-1\ncomputational domains in the Pacific. (b) level-2 domain over Hawaiian Islands,\n(c) level-3 domain of Oahu. o (red), epicenter\n298\nFigure 7-44: Smoothed bathymetry and topography with the depth-dependent Gaussian\nfunction. (a) level-2 domain over Hawaiian Islands, (b) level-3 domain of Oahu, (c)\nlevel-4 domain of the south shore of Oahu.\n299\nFigure 7-45: Slip distribution and initial sea surface deformation. (a) USGS finite fault\nsolution for the 2006 Kuril earthquake, (b) Sea surface deformation for the 2006\nKuril tsunami, (c) USGS finite fault solution for the 2010 Chile earthquake, (d) Sea\nsurface deformation for the 2010 Chile tsunami. O (red), epicenter.\n300\nFigure 7-46: Time series and spectra of surface elevation and velocity at Honolulu tide\ngauge and Kilo Nalu ADCP for the 2006 Kuril tsunami.\n(black), recorded\ndata;\n(red), computed data.\n301\nFigure 7-47: Time series and spectra of surface elevation and velocity at Honolulu tide\ngauge and Kilo Nalu ADCP for the 2010 Chile tsunami.\n(black), recorded\n302\ndata;\n(red), computed data.\nFigure 8-1: Inundation algorithm in SELFE. The orange line is the shoreline from the\nprevious time step, and the cyan lines are corrections made to obtain the shoreline\nat the new time step because points A is wetted and B is dried.\n305\n306\nFigure 8-2: Domain sketch\nFigure 8-3: Comparison of surface profiles at various times for the non-breaking wave case\nA (H/d = 0.0185). All variables have been non-dimensionalized. The RMSE at t =\n60 (near the maximum runup) is 0.001 and the Willmott skill is 0.998 (we have\nrestricted the calculation of errors to x<2 to remove the uninteresting part of the\nsolution offshore)\n307\nFigure 8-4: Comparison of elevation time series at two locations. The RMSE at the two\nstations are 0.0012 and 0.0008, and the Willmott skill are 0.994 and 0.999\nrespectively\n308\nFigure 8-5: Scaling effects through comparison of surface profiles between model results\nusing two different values of d. Note that the corresponding dimensional times are\ndifferent due to the different scaling.\n309\nFigure 8-6: Convergence test for case A at a time near the maximum runup (t = 60).\n310\nFigure 8-7: Schematics of the composite beach and locations of gauges\n312","MODEL BENCHMARKING WORKSHOP AND RESULTS\nxxiii\nFigure 8-8: Comparison of elevation time history at 8 stations for case A. Gauge 4 is located\nat the domain boundary and serves as a check for the imposed boundary condition;\ngauge 11 is at the vertical wall, where the RMSE is 1.5 mm and the Willmott skill\nis 0.96.\n312\nFigure 8-9: Comparison of surface profiles for case A. The Manning friction coefficient is\nno = 0.016. The error for the maximum elevation at t = 60 (near the maximum\nrunup ) is 2%.\n314\nFigure 8-10: Comparison of surface profiles for breaking-wave case C (H/d = 0.3). Model\nresults with two choices of bottom friction are shown\n315\nFigure 8-11: Runups as a function of the incident wave height. The maximum runup error\nfor non-breaking waves is 1.9%.\n316\nFigure 8-12: Comparison of elevation time series for case A.\n317\nFigure 8-13: Comparison of elevation time series for case B\n318\nFigure 8-14: Comparison of elevation time series for case C\n319\nFigure 8-15: Comparison of elevation time series at 4 gauges for 3 cases. (a) shows the\nexperimental setup\n320\nFigure 8-16: Nodes in the unstructured grid\n321\nFigure 8-17: Comparison of runups around the conical island for the 3 cases. The errors at\nthe back of the island are: 2.6%, 4.3% and 14% for the 3 cases.\n322\nFigure 8-18: Comparison of runups around the conical island for the 3 cases, in spatial form. 323\nFigure 8-19: Comparison of elevations at 3 gauges in front of the valley as shown in (a).\n324\nFigure 8-20: Inundation sequence near the narrow valley; t = 16.6 is close to the maximum\nrunup, which is ~9 cm\n325\nFigure 8-21: (a) Bathymetry as embedded in DEMs; (b) unstructured grid and (c) zoom-in\naround the Okushiri Island. The white arrow in (a) indicates a mismatch of\nbathymetry from multiple DEM sources\n326\nFigure 8-22: Comparison of elevations at 2 tide gauges\n327\nFigure 8-23: Comparison of runups around the island. Red and green numbers are from the\nmodel, with the green numbers indicating larger errors\n328\nFigure 8-24: Arrival of 2 waves at Aonae. The 1st wave came from the west while the 2nd\nwave attacked from the east.\n329\nFigure 8-25: Domain sketch for N-wave runup.\n330\nFigure 8-26: Comparison of elevation (top) and velocity (bottom) at 3 times. The error in the\nrunup is 5%\n330\nFigure 8-27: Comparison of time history of the shoreline position (top) and velocity\n(bottom).\n331","National Tsunami Hazard Mitigation Program (NTHMP)\nxxiv\nFigure 8-28: Grid used in the 1964 event, with multiple zooms. The insert histogram shows\nthe distribution of equivalent radii of all elements; over 90% of the elements have a\n333\nradius of 40 m or less\nFigure 8-29: Comparison at two tide gauges; the model results are from (a) static tides; and\n(b) dynamic tide (black) or superposition of tides on top of static tide results\n333\n(green)\nFigure 8-30: Comparison of inundation at Cannon Beach. The black dots are field estimates\nfrom Witter (2008) and the red dots are from the model, (a) with static tides, and\n(b) with dynamic tides.\n334\nFigure 9-1: Simulation of Heinrich's experiment (Heinrich, 1992) using THETIS. Top panel:\ndensity contours and flow streamlines at time t = (a) 0.5 S (b) 1 S, (c): Time\nevolution of vertical slide displacement. (): Heinrich's (1992) experiments, (-):\nTHETIS, (d): Free surface deformation at t = 0.5 S, (): Heinrich's (1992)\nexperiments, dot filled contour: THETIS\n338\nFigure 9-2: Sketch of BP8 (after Liu et al., 2005)\n340\nFigure 9-3: Snapshots of slide/water and water/air interfaces at different times for grid 2\nwith 170x100x120 cells, using the setup sketched in Figure 9-2. Slide initial\nsubmergence is D = -0.1 m, slide density is 2.14. (a) t = 0.7 S, (b) t = 1.4 S, (c)\nt\n=\n2.1 s, (d) = 3.5 S.\n341\nFigure 9-4: Time evolution of slide center of mass, using the setup sketched in Figure 9-2.\nSolid lines: numerical results with (a) 62x76x24 cells and (b) 170x100x120 cells\n(): experimental data. Initial slide submergence is A = -0.1 m, slide density is 2.14. 341\nFigure 9-5: Comparison between numerical results (solid lines) and experimental data (.) for\nthe time histories of free surface elevations at wave gauge 1 (top figure) and wave\ngauge 2 (bottom figure), using the setup sketched in Figure 9-2. Initial slide\nsubmergence is A = -0.1 m, slide density is 2.14, (a) 62x76x24 cells, (b)\n170x100x120 cells, and (red curve) 260x200x120 cells\n343\nFigure 9-6: Comparison between numerical results (solid lines) and experimental data () for\nthe time histories of runup at gauge 2 (top) and gauge 3 (bottom) ), using the setup\nsketched in Figure 9-2. Initial submergence is A = -0.1 m, slide density is 2.14. (a)\n62x76x24 cells, (b) 170x100x120 cells\n344\nFigure 9-7: Time evolution of slide center of mass. Initial slide submergence is A = -\n0.025 m, slide density is 2.79. Solid line: numerical results with 170 X 100 X 120\ncells; () experimental data\n345\nFigure 9-8: Comparison between numerical results (solid lines) and experimental data () for\nthe time histories of free surface elevations at wave gauge 2 (Figure 9-2). Initial\nslide submergence is A = -0.025 m, slide density is 2.79. 170 X 100 X 120 cells.\n346","MODEL BENCHMARKING WORKSHOP AND RESULTS\nXXV\nFigure 10-1: Experiment of a solid wedge induced waves. Tsunami generation and runup\ndue to three-dimensional landslide (Synolakis et al., 2007)\n353\nFigure 10-2: TSUNAMI3D snapshots of the solid wedge sliding down the slope and induced\nwaves at time = [ 1.0, 1.5, 2.0 and 2.5] sec after slide initiation.\n354\nFigure 10-3: Comparison of TSUNAMI3D numerical result (black broken line) against\nexperiment (blue solid line), case A = 0.025 m. Red line is the normalized error\nplotted in time\n355\nFigure 10-4: Comparison of TSUNAMI3D numerical result (black broken line) against\nexperiment (blue solid line), case A = 0.010 m. Red line is the normalized error\nplotted in time\n356\nFigure 10-5: Comparison of the TSUNAMI3D model's numerical result against the\nanalytical solution, nonlinear and linear shallow water approximation. Benchmark\nTsunami generation and runup due to two-dimensional landslide, Synolakis et al.,\n(2007), OAR PMEL-135.\n357\nFigure 10-6: Comparison of the TSUNAMI3D model's numerical result (thick broken line)\nagainst the analytical solution. the nonlinear and linear shallow water\napproximation. Tsunami generation and runup due to two-dimensional landslide,\nSynolakis et al., (2007), OAR PMEL-135.\n358\nFigure 11-1: Linear dispersion properties. (blue), za = -0.5208132 from BOSZ; (red),\nza = -0.5375 from Nwogu (1993); (black), za = -0.42265 equivalent to Peregrine\n(1967).\n376\nFigure 11-2: Definition sketch of BP4: Solitary wave runup on a simple beach.\n376\nFigure 11-3: Free surface profiles of solitary wave transformation on a 1:19.85 simple beach\nwith A/h = 0.3 and Ax/h = 0.125. Solid lines and circles denote BOSZ and\nmeasured data.\n377\nFigure 11-4: Solitary wave runup on a simple beach. (a) 1:19.85 (Synolakis, 1987). (b) 1:15\n(Li and Raichlen, 2002). (c) 1:5.67 (Hall and Watts, 1953). Solid lines (blue), and\ncircles denote computed and measured data.\n378\nFigure 11-5: Definition sketch of BP2: Solitary wave on a composite beach. Circles denote\nwave gauge locations\n379\nFigure 11-6: Solitary wave on a composite beach, BP2, case A. (blue) denotes\nsolution\nfrom BOSZ. (black) indicates laboratory data from Briggs et al. (1996)\n379\nFigure 11-7: Solitary wave on a composite beach, BP2, case B. (blue) denotes solution\nfrom BOSZ. (black) indicates laboratory data from Briggs et al. (1996)\n380\nFigure 11-8: Solitary wave on a composite beach, BP2, case C. (blue) denotes the solution\nfrom BOSZ. (black) indicates laboratory data from Briggs et al. (1996)\n381","National Tsunami Hazard Mitigation Program (NTHMP)\nxxvi\nFigure 11-9: Schematics of the conical island laboratory experiment, BP6. (Top) Perspective\nview. (Bottom) Cross-sectional view along centerline. Circles denote gauge\n382\nlocations\nFigure 11-10: Wave transformation around the conical island, BP6 for A/h = 0.181\n383\nFigure 11-11: Free surface profiles of wave transformation around the conical island, BP6.\n(a) A/h = 0.045; (b) A/h = 0.096; (c) A/h = 0.181. Solid lines and circles denote\n384\ncomputed and measured data\nFigure 11-12: Maximum inundation around the conical island, BP6. (a) A/h = 0.045; (b)\nA/h = 0.096; (c) A/h = 0.181. Solid lines and circles denote computed and\n384\nmeasured data.\nFigure 11-13: Bathymetry of the Monai valley experiment, BP7. White circles denote wave\ngauge locations. The gauge at the lower boundary is used as a control for the wave\n385\ninput (see Figure 11-14 below)\nFigure 11-14: Initial N-wave profile near the left boundary, BP7. (blue) denotes solution\nfrom BOSZ. (black) indicates initial wave profile from Matsuyama and Tanaka\n385\n(2001).\nFigure 11-15: Time series of free surface elevation in Monai valley experiment, BP7. (blue)\ndenotes solution from BOSZ. (black) indicates experimental data from Matsuyama\n386\nand Tanaka (2001)\nFigure 11-16: Maximum inundation in Monai valley experiment, BP7. (Blue) denotes\nsolution from BOSZ. (Red) circles indicate observed inundation from Matsuyama\nand Tanaka (2001). Black lines are topographic contours at 1.25 cm intervals\n387\nFigure 11-17: Maximum inundation in Monai valley experiment, top view, BP7. (Red)\n387\ncircles denote locations of observed inundation from Figure 11-16.\nFigure 11-18: ISEC BM1 test. Two-dimensional reef model of 1:12 slope in the Large Wave\nFlume at Oregon State University (folded into two rows at the 28.25 m\nmeasurement point for better visibility). Circles and vertical lines indicate wave\ngauge locations. The water level in the Benchmark test case is 2.5 m resulting in a\nwater depth of 0.136 m over the reef flat. The reef crest (red trapezoid) is 1.25 m\nlong and rises 0.201 m above the reef flat (exposed by 0.065 m), with a 1:12\noffshore and a 1:15 onshore slope. The plywood reef crest and the finished concrete\ntank are described by the same Manning coefficient of n = 0.014 m/s1/3. The tank\nis 3.66 m wide. A two-dimensional bathymetry file is provided in the benchmark\n388\npackage.\nFigure 11-19: Snapshots of free surface profiles for propagation of solitary wave with A/h =\n0.3 over 1:12 slope and exposed reef crest. Solid lines and circles denote computed\nand measured data. The experimental wave overturns describing a large air cavity.","MODEL BENCHMARKING WORKSHOP AND RESULTS\nxxvii\nThe depth-integrated numerical formulation approximates the 3-D process as a 2-D\ndiscontinuity with high accuracy\n389\nFigure 11-20: Time series of free surface at all wave gauges for propagation of solitary wave\nwith A/h = 0.3 over 1:12 slope and exposed reef crest. Solid lines and circles\ndenote computed and measured data. Notice the dispersive waves over the slope\n(30 - 50 m) generated from the reflected bore.\n390\nFigure 11-21: Close-up of time series of free surface at X = 35.9 m at beginning of 1:12\nslope. Solid lines and circles denote computed and measured data. Signal showing\ndispersive waves generated from the reflected bore that propagated over the slope\ninto the direction of the wavemaker.\n391\nFigure 11-22: Time series of x-directed velocity at X = 54.4 m close to the reef crest. Solid\nlines and circles denote computed and measured data. The laboratory wave already\noverturned at this location causing strong supercritical flow\n391\nFigure 11-23: Time series of free surface at all wave gauges with different grid resolution\nand identical model setup. (blue), BOSZ with Ax = 0.05 m (black), BOSZ with\nAx = 0.101 m (red), BOSZ with Ax = 0.20 m\n392\nFigure 11-24: Close-up of time series of free surface at X = 35.9 m close to the reef crest\nwith different grid resolution and identical model setup. (blue), BOSZ with Ax =\n0.05 m (black), BOSZ with Ax = 0.10 m (red), BOSZ with Ax = 0.20 m\n393\nFigure 11-25: Snapshots of free surface profiles for propagation of solitary wave with A/h =\n0.3 over 1:12 slope and exposed reef crest. (blue), BOSZ dispersive, (red), BOSZ\nhydrostatic (nonlinear shallow water equations) after tg/h = 64.8\n394\nFigure 11-26: Time series of free surface at all wave gauges. Solutions from BOSZ with\nidentical numerical scheme and input parameters for (blue), BOSZ dispersive,\n(red), BOSZ hydrostatic (nonlinear shallow water equations) after tg/h = 64.8\n395\nFigure 11-27: Close-up of time series of free surface at X = 35.9 m. Solutions from BOSZ\nwith identical numerical scheme and input parameters for (blue), BOSZ dispersive,\n(red), BOSZ hydrostatic (nonlinear shallow water equations) after tg/h = 64.8\n396\nFigure 11-28: Time series of x-directed velocity at X = 54.4 m. Solutions from BOSZ with\nidentical numerical scheme and input parameters for (blue), BOSZ dispersive,\n(red), BOSZ hydrostatic (nonlinear shallow water equations) after tg/h = 64.8\n396\nFigure 11-29: Perspective view of ISEC BM1 bathymetry from laser scan at original\nresolution of X = 5 cm. Circles denote wave gauge locations at free surface and\ncorresponding location on tank bottom. Red crosses at gauges 3, 6, and 13 indicate\npositions of acoustic Doppler velocimeters for velocity measurements.\n397\nFigure 11-30: Snapshots of solitary wave transformation in ISEC BM1\n398","National Tsunami Hazard Mitigation Program (NTHMP)\nxxviii\nFigure 11-31: - Time series of free surface profiles along basin centerline in ISEC BM1. Solid\n399\nlines and circles denote computed and measured data.\nFigure 11-32: - Time series of free surface profiles along transect at X = -5 m in ISEC BM1.\nSolid lines and circles denote computed and measured data\n400\nFigure 11-33: - Time series of free surface profiles along alongshore transect in ISEC BM1.\nSolid lines and circles denote computed and measured data\n401\nFigure 1-34: - Time series of velocity in ISEC BM1. (a) Cross-shore component. (b)\nAlongshore component. Solid lines and circles denote computed and measured\n402\ndata.\nFigure 11-35: Perspective view of ISEC BM2 bathymetry from laser scan at original\nresolution of. Ax = 5 cm. Conical island was added after laser scan. Circles denote\nwave gauge locations at free surface and corresponding location on tank bottom.\nRed crosses at gauges 2, 3, and 10 indicate positions of acoustic Doppler\n403\nvelocimeters for velocity measurements.\nFigure 11-36: Snapshots of solitary wave transformation in ISEC BM2\n404\nFigure 11-37: Time series of free surface profiles at wave gauges in ISEC BM2. Solid lines\n405\nand circles denote computed and measured data\nFigure 11-38: Time series of velocity in ISEC BM2. (a) Cross-shore component. (b)\nAlongshore component. Solid lines and circles denote computed and measured\n406\ndata.","MODEL BENCHMARKING WORKSHOP AND RESULTS\nxxix","National Tsunami Hazard Mitigation Program (NTHMP)","1\nNTHMP Model Benchmarking Workshop\nNTHMP Mapping and Modeling Subcommittee\n1.1\nAcknowledgements\nThe authors and participants in the National Tsunami Hazard Mitigation Program (NTHMP)\nModel Benchmarking Workshop wish to thank the NTHMP and the National Oceanic and\nAtmospheric Administration (NOAA) for providing the funding for the modeling activities\nassociated with this work, and Texas A&M University at Galveston for hosting the workshop\nitself. In addition, we thank the collaborators and reviewers, especially those at the NOAA\nNational Geophysical Data Center, who supported this work and provided invaluable discussion\nand review.\n1.2\nExecutive summary\nModel validation is essential to the production of accurate hazard mitigation and public\nsafety products. Held March 31 st to April 2nd. 2011 at Texas A&M University at Galveston under\nthe auspices of the NTHMP Mapping and Modeling Subcommittee (MMS), the Model\nBenchmarking workshop participants were tasked with developing and implementing a strategy\nfor the validation of tsunami inundation models. As stated in the NTHMP Strategic Plan, \"All\nNTHMP-funded models will meet established standards by 2012.\" Accordingly, during this\nworkshop, participants presented the results of applying their tsunami propagation models to a\nseries of pre-established benchmark tests. The participants also began the process of clearly\ndefining the validation procedure that all such models will be required to follow in order to\nobtain NTHMP funding for activities involving the development of public safety products such\nas inundation and evacuation maps. Specifically, participants, most representing nationally\nrenowned tsunami modeling organizations, decided that a model will be deemed validated when\nit successfully simulates a series of tsunami benchmark tests covering all relevant tsunami\nprocesses targeted by a specific study. According to the analyses and model comparisons\nundertaken by the workshop participants, all the presented models tested as being capable of\npredicting propagation and runup of tsunami waves in most geophysical conditions.\nThe initial list of NTHMP benchmark tests, accessible at http://nctr.pmel.noaa.gov/\nbenchmark, was established based on the OAR-PMEL-135 report (Synolakis et al., 2007).\nBecause an additional important goal of the workshop was to revise the list of benchmark tests,\nparticipants proposed and discussed new benchmark tests, particularly in relation to landslide\ntsunamis and recent large co-seismic tsunami events.\nThe workshop results presented and described in these proceedings represent an important\nstep in attaining consistency in tsunami inundation modeling and mapping among federal\nagencies, states, and communities.","National Tsunami Hazard Mitigation Program (NTHMP)\n2\n1.3\nIntroduction\nThe coastal states and territories of the United States are vulnerable to devastating tsunamis\nsimilar to the 2004 Indian Ocean and 2011 Tohoku-oki events (Dunbar and Weaver, 2008). Over\nthe past several decades, these states and territories have been developing tsunami inundation\nmaps to form the basis of community evacuation plans. This map-making process drastically\naccelerated in 2005, following the catastrophic 2004 Indian Ocean tsunami, when the NTHMP\nreceived an increase of funding from the Tsunami Warning and Education Act for community-\nlevel tsunami preparedness activities, including inundation modeling and mapping. At the same\ntime, the NTHMP was mandated by the National Science and Technology Council to implement\na set of tsunami hazard mitigation recommendations. Because, at the time these\nrecommendations were made, there were no commonly accepted tools for simulating potential\ntsunami inundation, the council recommended that NTHMP \"develop standardized and\ncoordinated tsunami hazard and risk assessment methodologies for all coastal regions of the U.S.\nand its territories.\" The standards were to be developed to ensure sufficient quality of the tsunami\ninundation maps and to ensure a basic level of consistency between efforts in terms of products.\nTo this end, Synolakis et al. (2007) proposed a set of benchmark tests of tsunami computer\nmodels, aimed at ensuring that all models were vetted through a benchmarking process.\nDuring the 2011 NTHMP Model Benchmarking Workshop, participants, as identified in\nTable 1-1, presented the hydrodynamic models currently in use by states and territories to\nproduce inundation and evacuation products. The models ranged from a full application of the 3-\nD Navier-Stokes equations and Boussinesq approach to the depth-averaged non-linear shallow\nwater equations (SWEs). Each model was put through a benchmarking process in which\nnumerical model results were analyzed and compared with:\nThe analytical solution for a specific benchmark test,\nThe results of laboratory experiments,\nThe observed field data for the July 1993 Okushiri Island event.\nIt should be noted that not all models were applied to all benchmark tests. Models were applied\nto only those tests corresponding to tsunami processes taken into account for developing\ninundation maps in the modelers' own geographic areas.\nResults of systematic model testing exposed the limitations and clearly identified the\nattributes of each model. This led to collegial discussions, constructive criticisms, and proposed\ncollaboration among participants to address outstanding issues. According to these analyses and\ncomparisons, all the presented models were deemed capable of predicting the propagation and\nrunup of tsunami waves in most of the geophysical conditions covered by the applicable\nbenchmark tests. Brief descriptions of each model presented at the workshop and validated\nthrough the benchmarking process are given in the following section.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n3\nTable 1-1: NTHMP model benchmarking workshop participants. Participant affiliation and model of\nexpertise are provided.\nName\nAffiliation\nModel\nUniversity of Rhode\nStephane Abadie\nTHETIS\nIsland (URI), East Coast\nAggeliki\nUniversity of Southern\nBarberopoulou\nCalifornia (USC), CA\nNational Geophysical\nBarry Eakins\nData Center (NGDC),\nNOAA\nUniversity of Washington\nFrank Gonzalez\nGeoClaw\n(UW), WA\nStephan Grilli\nURI, East Coast\nFUNWAVE / THETIS\nUniversity of Alaska\nRoger Hansen\nFairbanks (UAF), AK\nJeff Harris\nURI, East Coast\nTexas A&M at Galveston\nConference\nJuan Horrillo\n(TAMUG),\nsponsor &\nTSUNAMI3D\nGulf of Mexico Coast\nhost\nPuerto Rico Seismic\nVictor Huerfano\nNetwork (PRSN), PR\nWest Coast & Alaska\nMMS co-\nBill Knight\nTsunami Warning Center\nchair\nATFM\n(WC/ATWC), NOAA\nUAF, AK\nDmitry Nicolsky\nALASKA GI-T / GI-L\nUniversity of Delaware\nFengyan Shi\nFUNWAVE\n(UD), East Coast\nUW, Joint Institute for the\nStudy of the Atmosphere\nElena Tolkova\nand Oceans/Pacific\nMOST\nMarine Environmental\nLaboratory, NOAA\nUniversity of Hawaii\nVolker Roeber\nBOSZ\n(UH), HI\nCalifornia Geological\nMMS co-\nRick Wilson\nSurvey, CA\nchair\nYoshiki Yamazaki\nUH, HI\nNEOWAVE\nOregon Health & Science\nJoseph Zhang\nSELFE\nUniversity (OHSU), OR","National Tsunami Hazard Mitigation Program (NTHMP)\n4\n1.4\nNTHMP models\nOver the past few decades, a variety of tsunami propagation models have been developed,\nbased on a variety of governing equations, numerical methods, spatial and temporal discret-\nization techniques, and wetting-drying algorithms used to predict tsunami runup.\nFinite difference methods (FDM) were initially developed for solving linear shallow water\nwave equations (LSWEs), based on the work of Hansen (1956) and Fisher (1959). A detailed\nreview of these methods can be found in Kowalik and Murty (1993a) and Imamura (1996).\nBased on these initial FDM approaches, the tsunami propagation model referred to as TUNAMI\n(Tohoku University's Numerical Analysis Model for Investigation; Imamura, 1995) was\ndeveloped. In this work, the water level dynamics near the shoreline are computed by\nparameterizing a water flux quantity, the so-called \"discharge\" (Imamura, 1996), and nonlinear\nshallow water equations (NSWEs) are formulated in a flux-conservative form, which helps\npreserve water mass throughout the computations. A similar numerical model, also extensively\nused for tsunami modeling, is COMCOT (COrnell Multigrid COupled Tsunami; Liu et al.,\n1998), where a moving boundary algorithm is used to find the shoreline location. During the\nsame period, Titov and Synolakis (1995) presented the VTCS model, now known as MOST\n(Method Of Splitting Tsunami), which can compute runup without adding an artificial viscosity\nor friction factor. An innovative idea of the MOST model is its ability to track the shoreline by\nadding new temporal grid points.\nWhile the earlier models were hydrostatic and hence non-dispersive, more recently, long\nwave models have been developed on the basis of Boussinesq equations and applied to\nsimulating tsunami propagation. While these equations were initially both weakly nonlinear and\ndispersive (Peregrine, 1967), fully nonlinear approximations with extended dispersion properties\nwere developed (e.g., Wei et al., 1995; Kirby et al., 1998; Lynett et al., 2002). In shallow water,\nthese fully nonlinear Boussinesq models (FNBMs) extend NSWEs to include non-hydrostatic\ndispersive effects. FNBMs have led to operational tsunami models (e.g., FUNWAVE), which\nwere applied to simulating some of the recent significant tsunami events (e.g., Watts et al., 2003;\nIoualalen et al., 2007; Tappin et al., 2008). An alternative governing equation describing a\ndispersive wave through non-hydrostatic pressure has been proposed by Stelling and Zijlema\n(2003). The models based on their formulations provide a slightly better dispersive property than\nPeregrine's (1967) depth-integrated model, and are comparative to FNBMs using a multiple-\nlayer model. In total, more than ten tsunami simulation models were proposed over the past two\ndecades or so, including, in addition to the earlier references, Mader and Lukas (1984), Kowalik\nand Murty (1993b), George and eVeque (2006), and Zhang and Baptista (2008).\nThe next section briefly describes each of the numerical models presented and discussed\nduring the 2011 NTHMP Model Benchmarking Workshop. Summaries of the main\ncharacteristics and capabilities of each model are given in Table 1-2 and Table 1-3, followed by\na short write-up of each of the ten models run through the testing process.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n5\nTable 1-2: Summary of model characteristics (ALASKA, ATFM, BOSZ, FUNWAVE, GeoClaw)\nModel Features\nALASKA\nATFM\nBOSZ\nFUNWAVE\nGeoClaw\nApproximation\nShallow Water\nShallow Water\nBoussinesq\nBoussinesq\nShallow Water\nWave\nNo\nYes (Optional)\nYes\nYes\nNo\ndispersion\nTwo-way\nTwo-way\nGrid nesting\nTwo-way\nNo\nOne-Way\n(submeshes)\nAMR\nCoordinate\nCartesian/\nCartesian/\nCartesian/\nCartesian/\nCartesian\nSpherical\nSpherical\nSpherical\nSpherical\nsystem\nHybrid finite\nNumerical\nFinite\nFinite\nFinite volume\nvol./finite\nFinite volume\nscheme\ndifference\ndifference\ndifference\nCo-seismic+\nTsunami\nCo-seismic +\nLandslide\nCo-seismic+\nCo-seismic\nCo-seismic\nLandslide\n(Initiate\nLandslide\nsource\nTHETIS)\nShock-\nShock-\nRunup\nMoving\nVolume of\ncapturing/\ncapturing/\nSlot technique\napproach\nboundary\nFluid\nRiemann\nRiemann\nsolution\nsolution\nCo-array\nParallelization\nMPI\nOpenMP\nMPI\nOpenMP\nFORTRAN\nDocumentation\nLimited\nLimited\nLimited\nYes\nYes\nCommand line/\nGraphics\nCommand line\nCommand line\nExecution\nCommand line\nGraphics\ninterface\ninterface","6\nNational Tsunami Hazard Mitigation Program (NTHMP)\nTable 1-3: Summary of model characteristics (MOST, NEOWAVE, SELFE, THETIS, TSUNAMI3D)\nModel Features\nMOST\nNEOWAVE\nSELFE\nTHETIS\nTSUNAMI3D\nHydro./\nHydro./\n3-D Navier-\n3-D Navier-\nApproximation\nShallow Water\nNonhydro.\nNonhydro.\nStokes\nStokes\nWave\nYes\nYes\nYes (Optional)\nYes\nYes (Optional)\ndispersion\n(Numerical)\nTwo-way\nStructured.\nStructured.\nGrid nesting\nOne-way\nTwo-way\nUnstructured\nVariable mesh\nVariable mesh\nmesh\nCoordinate\nCartesian/\nCartesian/\nCartesian/\nCartesian/\nCartesian\nSpherical\nSpherical\nSpherical\nCylindrical\nsystem\nNumerical\nFinite\nFinite\nFinite element/\nFinite volume\nFinite volume\nscheme\ndifference\ndifference\nFinite volume\nCo-seismic+\nLandslide\nLandslide\nTsunami\nLandslide\nCo-seismic\nCo-seismic\n(Coupled to\n(Coupled to\n(Initiate\nsource\nFUNWAVE)\nNEOWAVE)\nTSUNAMI3D)\nRunup\nHorizontal\nHorizontal\nIterative\nVolume of\nVolume of\napproach\nprojection\nprojection\nprojection\nFluid\nFluid\nParallelization\nOpenMP\nNo\nMPI\nMPI\nOpenMP/MPI\nDocumentation\nLimited\nYes\nYes\nYes\nLimited\nGraphics\nExecution\nCommand line\nCommand line\nCommand line\nCommand line\ninterface\n1.4.1 Alaska Geophysical Institute Parallel Robust Inundation Modeling Environment-Tectonic\n(GI'-T)\nThe ALASKA GI'-T is a numerical code that simulates the propagation and runup of co-\nseismic tsunami waves in the framework of NSWE theory. The numerical code adopts a\nstaggered leapfrog FDM scheme to solve the shallow water equations formulated for depth-\naveraged water fluxes in spherical coordinates on Arakawa C-grid (Arakawa and Lamb, 1977).\nThe spatial derivatives are discretized by central difference and upwind difference schemes\n(Fletcher, 1991). The friction term is discretized by a semi-implicit scheme according to Goto et\nal. (1997). A temporal position of the shoreline is calculated using a free-surface moving\nboundary algorithm, validated by Nicolsky et al. (2010). The FDM scheme is coded in\nFORTRAN using the Portable, Extensible Toolkit for Scientific computations (Balay et al.,\n2004) and the MPI standard. For large scale problems, the developed algorithm employs two-\nway nested grids.\nThe model can be accessed through a wizard-style internet-based interface that guides users\nthrough setup, execution, and retrieval of tsunami modeling results. The interface is based on\nGoogle Maps API, which simplifies interactions with the geospatial database of hypothetical,\nhistorical sources and elevation datasets needed to simulate propagation and runup of the\ntsunami.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n7\nFor submarine landslide modeling the University of Alaska Fairbanks has developed the\nALASKA GI'-L (Geophysical Institute Parallel Robust Inundation Modeling Environment -\nLandslide) code. The ALASKA GI'-L is a numerical code that simulates propagation and runup\nof landslide-generated tsunami waves in the framework of the long-wave approximation to water\ndynamics. The slide is an incompressible, isotropic 3-D viscous flow, propagating over realistic\nbathymetry. It is assumed that the horizontal velocities of the slide have a parabolic vertical\nprofile. Equations for slide and water waves are two-way coupled. This coupling was initially\nproposed by Jiang and LeBlond (1992), and later improved by Fine et al. (1998). The numerical\ncode adopts a staggered leapfrog finite difference scheme to solve the governing equations\nformulated for slide velocities and depth-averaged water fluxes in Cartesian coordinates. A\ntemporal position of the shoreline is calculated using the same algorithm as in the GI'-T model.\nThe finite difference scheme is coded in FORTRAN and can be implicitly parallelized.\nThis model was successfully applied to simulate a tsunami event in fjords near Seward,\nWhittier, and Skagway harbors, where tsunami waves were generated by submarine landslides in\n1964 and 1994 (Fine et al., 1998; Thomson et al., 2001; Suleimani et al., 2009; Nicolsky et al.,\n2010).\n1.4.2 The Alaska Tsunami Forecast Model (ATFM)\nThe Alaska Tsunami Forecast Model (ATFM) began as a collaborative effort between two\nresearchers (Zygmunt Kowalik and Paul Whitmore) and became operational at the West Coast\nAlaska Tsunami Warning Center (WCATWC) in 1997 (Kowalik and Whitmore, 1991;\nWhitmore and Sokolowski, 1996). This is known as the \"classic\" model currently used in\nWCATWC operations. From 2004 until the present time, the model has been substantially\nreworked into a second forecasting model which is called ATFMv2. The benchmark challenge\nproblems were computed with this newer model.\nThe ATFM solves the NSWEs. Two equations of motion and one continuity equation are\nformulated in spherical coordinates and solved on structured, nested meshes. The two horizontal\ncomponents of velocity (u and v) are depth-averaged. The vertical component of velocity is not\nconsidered in this hydrostatic formulation. The solution technique for u and V is based on a\ndifferencing method described in Kowalik and Murty (1993a), and the sea level is computed\nwith a second-order accurate, upwind scheme that conserves mass to machine accuracy (van\nLeer, 1977). The run-up / run-down method is based on the VOF approach pioneered by Nichols\nand Hirt (1980), and Hirt and Nichols (1981). There is no explicit dispersion in the model,\nalthough a non-hydrostatic addition is in the testing phase and was used for benchmarks BP4 and\nBP6 (Walters, 2005; Yamazaki et al., 2009). Submeshes are nested within parent meshes to\nincrease spatial resolution where needed. Information is passed both from low to high-resolution\nmeshes and back, based on a mass conserving interconnecting scheme (Berger and Leveque,\n1998). Discretization for the three field variables uses the staggered \"C grid\" layout.\nThe model is coded in FORTRAN 90 and in Co-array FORTRAN. It has been run on PCs, a\nCray X1, and a Penguin Computing cluster comprised of Opteron processors.\n1.4.3 Boussinesq Model for Ocean and Surf Zones (BOSZ)\nBOSZ is a numerical model for propagation, transformation, breaking, and runup of water\nwaves. BOSZ was specifically designed for nearshore wave processes in the presence of fringing\nreefs in tropical and sub-tropical regions around the world. BOSZ was developed with the goal to\nobtain reliable and robust results in addressing the complementary but somewhat opposing","National Tsunami Hazard Mitigation Program (NTHMP)\n8\nphysical processes of flux and dispersion throughout a single numerical model. BOSZ was\nintentionally kept as simple as possible and yet containing the main features to accurately\ndescribe nearshore wave processes. BOSZ combines the weakly dispersive properties of a\nBoussinesq-type model with the shock capturing capabilities of the conservative form of the\nNSWEs. BOSZ allows the simulation of dispersive waves up to medium order as well as\nsupercritical flows with discontinuities. The depth-integrated governing equations are derived\nfrom Nwogu (1993), an extended (with respect to linear dispersion) Boussinesq approach with\nconserved variables that satisfy conservation of mass and momentum for Fr > 1. The continuity\nand momentum equations contain the conservative form of the nonlinear shallow-water\nequations to capture shock-related hydraulic processes.\nThe governing equations are solved with a conservative finite volume Godunov-type\nscheme. The finite volume method benefits from a fifth-order Total Variation Diminishing\n(TVD) reconstruction procedure that evaluates the inter-cell variables. The exact Riemann solver\nof Wu and Cheung (2009) supplies the inter-cell flux and bathymetry source terms. The well-\nbalanced scheme of Liang and Marche (2009) eliminates depth interpolation errors in the domain\nand preserves continuity at moving boundaries over irregular topography. The moving waterline\nis part of the Riemann solution and does not require additional treatments. A fourth-order\nexplicit Adam-Bashforth-Moulton scheme integrates the governing equations in time and\nevaluates the conserved variables.\nBOSZ is primarily used for modeling surf-zone and swash processes of swell and wind\nwaves. The model can be applied to near-field tsunami scenarios. However, to date BOSZ is not\nbased on spherical coordinates nor does it support nested grids. The code is written in MATLAB\nwith most of its processing in embedded pre-compiled C++ MEX subroutines. This combines\nfast computations with a user-friendly code development interface for debugging, modifying,\nand demonstration. The current version of BOSZ runs on a serial processor.\n1.4.4 FUNWAVE-TVD model\nThe FUNWAVE tsunami propagation and runup model is based on fully nonlinear and\ndispersive Boussinesq equations, retaining information to leading order in frequency dispersion\nO[(kh)2] and to all orders in nonlinearity (Wei and Kirby, 1995; Wei et al., 1995). Instead of\ntracking the moving boundary during wave run-up/run-down on the beach or coastlines,\nFUNWAVE treats the entire computational domain as an active fluid domain by employing an\nimproved version of the slot or permeable-seabed technique, i.e., the moving shoreline algorithm\nproposed by Chen et al. (2000) and Kennedy et al. (2000) for the runup simulation. The basic\nidea behind this technique is to replace the solid bottom where there is little or no water covering\nthe land with a porous seabed or to assume that the solid bottom contains narrow slots. This is\nincorporated in terms of mass flux and free surface elevation in order to conserve mass in the\npresence of slots. The model includes bottom friction, energy dissipation to account for the wave\nbreaking and a subgrid turbulence scheme too. The bottom friction is modeled by the use of the\nquadratic law with bottom friction coefficient. The subgrid turbulence is modeled in terms of\nSmagorinsky-subgrid turbulent mixing type to account for the effect of the underlying current\nfield. The energy dissipation due to wave breaking in shallow water is treated by the use of\nmomentum mixing terms. The associated eddy viscosity is essentially proportional to the\ngradient of the horizontal velocity, which is strongly localized on the front face of the breaking\nwave.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n9\nFUNWAVE-TVD is an extension of FUNWAVE, formulated in both Cartesian coordinates\n(Shi et al., 2012) and in spherical coordinates with Coriolis effects (Kirby et al., 2009; 2012) for\napplication to ocean basin scale problems. This new model uses a hybrid finite-volume and\nFDM-MUSCL-TVD scheme. As in FUNWAVE, improved linear dispersive properties are\nachieved, up to the deep water limit, by expressing the BM equations in terms of the horizontal\nvelocity vector at 0.531 times the local depth, as in Nwogu (1993). Additionally, wave breaking\nis more accurately modeled by switching from the Boussinesq equations to the NSWE, when the\nlocal height to depth ratio exceeds 0.8. FUNWAVE-TVD's latest implementation is fully\nparallelized using MPI-FORTRAN, for efficient use on distributed memory clusters. One-way\ngrid nesting was implemented to allow for grid refinement near tsunami sources and near the\ncoast. This latest version was used for running the tsunami benchmarks.\nFUNWAVE-TVD has been used to model landslide or co-seismic tsunamis. A pre-processor\nallows the user to specify the initial tsunami source condition in terms of a hot start, either from\nthe underwater landslide (slides or slumps) solution of Grilli et al. (2002), Grilli and Watts\n(2005) and Watts et al. (2005), or for co-seismic tsunamis based on the standard Okada (1985)\nsolution. More recently, both landslide and co-seismic tsunamis have also been dynamically\ngenerated (as a space and time-varying bottom boundary condition), using the non-hydrostatic,\nsigma coordinate model NHWAVE (Ma et al., 2012), whose solution is then interpolated into\nFUNWAVE's Cartesian or spherical grid.\n1.4.5 GeoClaw model\nThe GeoClaw model is based on the NSWEs and uses a finite volume method on adaptively\nrefined rectangular grids (Cartesian or lat-long). The method exactly conserves mass (except\nnear the shoreline when refining or de-refining grids) and conserves momentum over a flat\nbottom. This method is based on Godunov's method: at each cell interface a one-dimensional\nRiemann problem is solved normal to the edge, which reduces to a one-dimensional shallow\nwater model with piecewise constant initial data, with left and right values given by the cell\naverages on each side. The jump in bathymetry between the cells is incorporated into the\nRiemann solution in a manner that makes the method \"well balanced\": the steady state of the\nocean at rest is exactly maintained. The shoreline is handled by allowing dry cells to have depth\n0 and to dynamically change between wet and dry. The method is second order accurate in\nsmooth regions but nonlinear limiters are used to create \"shock-capturing\" methods (LeVeque,\n2002) that maintain sharp non-oscillatory solutions and non-negative depth even in the nonlinear\nregime. The method is stable to Courant number 1 and very robust. The Manning friction term is\nincorporated using a fractional step method.\nAdaptive mesh refinement to several nested levels is allowed, with arbitrary refinement\nratios at each level. Refinement is done by flagging cells for refinement (based on wave height\nand specification of the areas of interest). The flagged cells at each level are clustered into\nrectangular patches for refinement to the next level, as described in detail in Berger and LeVeque\n(1998). The high-resolution methods and adaptive refinement algorithms have been extensively\ntested in the Clawpack software that has been in development since 1994. GeoClaw includes\nspecial techniques to deal with bathymetry data, well-balancing, and wetting/drying, and is an\noutgrowth of the TsunamiClaw software developed in George (2006). The algorithms and\nsoftware are described in more detail in Berger et al. (2011) and Veque et al. (2011).","10\nNational Tsunami Hazard Mitigation Program (NTHMP)\nFor modeling earthquake-generated tsunamis, the co-seismic seafloor motion is modeled by\nadjusting the bathymetry dynamically each time step. An Okada (1985) model can be used to\ntranslate fault models to seafloor motion. For modeling landslide-generated tsunamis, the\nseafloor motion is modeled by adjusting the bathymetry dynamically each time step. The\nlandslide motion is generally computed first, and GeoClaw has been used with a Savage-Hutter\nmodel to simulate the motion of the landslide itself, in current work by Ph.D. student Jihwan\nKim. This has been compared with two-layer fully coupled models and found adequate for\nlandslides in sufficiently deep water.\nThe main code is written in Fortran, with a Python user interface and plotting modules. All\nof the code is open source, hosted at https://github.com/organizations/clawpack. Additional\ndocumentation is available at http://www.clawpack.org/geoclaw.\n1.4.6 Method Of Splitting Tsunamis (MOST)\nThe MOST model simulates propagation and runup of gravity waves according to depth-\nintegrated NSWEs. The algorithm is based on the method of fractional steps which reduces the\n2-D problem to a 1-D problem in each direction. To progress the solution through a time step,\ntwo 1-D problems are solved sequentially. Each 1-D problem is formulated in terms of Riemann\ninvariants. MOST's computational algorithm uses a forward difference scheme in time and\ncentered differences for spatial derivatives (Titov and Synolakis, 1998; Burwell et al., 2007).\nFriction is represented by a Manning term. The model operates on structured grids given in\nCartesian or spherical coordinates. The algorithm is coded in Fortran 95 and parallelized using\nOpenMP.\nMOST's inundation algorithm is a 1-D algorithm that uses horizontal projection of the water\nlevel in the last wet node onto the beach to move the instantaneous shoreline position (Titov and\nSynolakis, 1995). The simulation can be initiated given initial seafloor deformation or by\nproviding lateral boundary conditions. The latter facilitates grid nesting with one-way coupling.\nThe operational version of the MOST model determines source parameters for the tsunami\nwave itself by incorporating observations into forecast methodology. Just as hurricane forecasts\nrely on observations (radar, aircraft, satellite, ocean systems) to forecast the path of a hurricane\nfollowing generation, the operational forecast relies on deep-ocean bottom pressure observations\nof the tsunami waves after generation. The specific operational procedure is hard-coded for a\nthree-nested-grid configuration forced through the boundary of the outer grid. The boundary\ninput is supplied by the database of an ocean-wide 24-hour-long simulation of tsunami wave\npropagation, for numerous tsunamis generated by hypothetical Mw 7.5 earthquakes covering\nworldwide subduction zones (Gica et al., 2008). These data sets are linearly combined to imitate\nan arbitrary tsunami scenario in the deep ocean. Access to the operational version is offered via\ninternet-enabled interface (ComMIT), which allows for the selection of model input data, use of\nshared databases, display of model output through a graphical user interface (GUI), and sharing\nsimulation results.\n1.4.7 Non-hydrostatic Evolution of Ocean WAVEs (NEOWAVE)\nThe Non-hydrostatic Evolution of Ocean WAVEs (NEOWAVE) model is a shock-\ncapturing, dispersive model in a spherical coordinate system for basin-wide evolution and coastal\nrunup of tsunamis using two-way nested computational grids (Yamazaki et al., 2011). This\ndepth-integrated model describes dispersive waves through the non-hydrostatic pressure and\nvertical velocity (Stelling and Zijlema, 2003, and Yamazaki et al., 2009). The vertical velocity","MODEL BENCHMARKING WORKSHOP AND RESULTS\n11\nterm also facilitates modeling of tsunami generation from seafloor deformation to account for the\ntime-sequence of the earthquake rupture process (Yamazaki et al., 2011). The semi-implicit,\nstaggered finite difference model captures flow discontinuities associated with bores or hydraulic\njumps through the momentum conserved advection (MCA) scheme, which embeds the upwind\nflux approximation of Mader (1988) in the shock-capturing scheme of Stelling and Duinmeijer\n(2003).\nNEOWAVE builds on the nonlinear shallow-water model of Kowalik et al. (2005) with the\nnon-hydrostatic terms and the momentum-conserved advection scheme (Yamazaki et al., 2009).\nThe grid refinement scheme is implemented in the model to capture tsunami physics in adequate\ngrid resolution. To ensure propagation of dispersive waves and discontinuities across\ncomputational grids of different resolution, a two-way grid-nesting scheme utilizes the Dirichlet\ncondition of the non-hydrostatic pressure and both the horizontal velocity and surface elevation\nat the inter-grid boundary (Yamazaki et al., 2011). The present model tracks the wet/dry\ninterface using the approach based on Kowalik and Murty (1993a) to compute the runup and\ninundation. The wet/dry interface is predicted by horizontal projection of sea level at the adjacent\nwet cell, and obtained through integration of the momentum and continuity equations (Yamazaki\net al., 2009).\n1.4.8 Semi-implicit Eulerian-Lagrangian Finite Elements (SELFE)\nThe tsunami propagation and inundation model SELFE (Zhang and Baptista, 2008a) was\nenvisioned at its inception to be an open source, community supported, 3-D hydrodynamic/\nhydraulic model. Originally developed to address the challenging 3-D baroclinic circulation in\nthe Columba River estuary, the SELFE model has since been adopted by 100+ groups (most\nrecent web count) around the world. It has evolved into a comprehensive modeling system\nencompassing such physical/biology processes as general circulation, tsunami and hurricane\nstorm surge inundation, ecology and water quality, sediment transport, wave-current interaction\nand oil spill. A central web site dedicated to this model, http://www.stccmop.org/\nCORIE/modeling/selfe/, is maintained, along with a user mailing list and mail archive system,\nand an annual user group meeting has been organized since 2004, along with online training\ncourses which are occasionally conducted for users.\nSELFE combines numerical accuracy with efficiency and robustness. It is based on the 3-D\nNSWEs. The time stepping in SELFE is done semi-implicitly for the momentum and continuity\nequations, and together with the Eulerian-Lagrangian method for the treatment of the advection,\nthe stringent CFL stability condition is bypassed. The use of unstructured grids in the horizontal\ndimension further enhances the model efficiency and flexibility due to their superior capability in\nfitting complex coastal boundary and resolving bathymetric and topographic features as well\ncoastal structures. The model can be configured in multiple ways (e.g., hydrostatic or non-\nhydrostatic options, etc.), but in tsunami applications the 2-D depth-averaged hydrostatic\nconfiguration is typically applied for maximum efficiency. Since 2007, all components of the\nSELFE modeling system have been fully parallelized using domain decomposition and Message\npassing Protocol (MPI). The inundation algorithm in SELFE uses a simple iterative procedure to\ncapture the moving shoreline (Zhang and Baptista, 2008b).\nThe model has been successfully applied in the recent simulation of the impact of the 1964\nAlaska event on the US west coast (Zhang et al., 2011), and in the study of paleo-tsunamis near\nan Oregon town (Priest et al., 2010; Witter et al., 2011).","National Tsunami Hazard Mitigation Program (NTHMP)\n12\n1.4.9 THETIS\nTHETIS is a multi-fluid Navier-Stokes (NS) solver developed by the TREFLE CNRS\nlaboratory at the University of Bordeaux I. It is a multipurpose CFD code, freely available to\nresearchers (http://thetis.enscbp.fr) and fully parallelized. For tsunami modeling, THETIS has\nbeen applied to tsunami generated by subaerial landslides (Abadie et al., 2010). In this case, the\nmodel solves the incompressible NS equations for water, air and the slide. Basically, at any time,\nthe computational domain is considered as being filled by one \"equivalent\" fluid, whose physical\nproperties (namely density and viscosity) vary with space. Subgrid turbulent dissipation is\nmodeled based on a Large Eddy Simulation approach, using a mixed scale subgrid model (Lubin\net al., 2006). The governing equations (i.e., conservation of mass and momentum) are discretized\non a fixed mesh, which may be Cartesian, cylindrical or curvilinear, using the finite volume\nmethod. These governing equations are exact, except for interfacial meshes, where momentum\nfluxes are only approximated, due to the presence of several fluids. NS equations are solved\nusing a two-step projection method. Fluid-fluid interfaces are tracked using the VOF method.\nFor most flows, the PLIC algorithm (e.g., Abadie et al., 1998) allows ensuring an accurate\ntracking while keeping the interface discontinuous. However, for very violent flows with fast\ndroplet ejection, the PLIC method may cause divergence of the projection algorithm. In such\ncases, the interface is smoothed either by allowing a slight diffusion process, after each PLIC\niteration, or by using a TVD scheme solving Eulerian advection equations for the interfaces.\nTHETIS has been extensively validated for many theoretical and experimental flow cases.\nHence, each new version of THETIS has to successfully solve more than 50 validation cases\nwithin a certain expected accuracy, before being released.\n1.4.10 Tsunami Solution Using Navier-Stokes Algorithm with Multiple Interfaces\n(TSUNAMI3D)\nThe TSUNAMI3D model was developed by the University of Alaska Fairbanks (Horrillo,\n2006) and Texas A&M University at Galveston (TAMUG). The TSUNAMI3D code solves\ntransient fluid flows with free surface boundaries, based on the concept of the fractional VOF.\nThe code uses an Eulerian mesh of rectangular cells having variable sizes. The fluid equations\nsolved are the FDM approximation of the Navier-Stokes and the continuity equations. The basic\nmode of operation is for single fluid calculation having multiple free surfaces. However,\nTSUNAMI3D can also be used for calculations involving two fluids separated by a sharp or non-\nsharp (diffusive) interface, for instance, water and mud. In either case, the fluids may be treated\nas incompressible. Internal obstacles or topography are defined by blocking out fully or partially\nany desired combination of cells in the mesh.\nThe code is based on the developments originated in Los Alamos National Laboratory\n(LANL) during the 1970s, Hirt and Nichols (1981). In particular, the VOF algorithm for tracking\nthe movement of a free surface interface between two fluids or fluid-void has been simplified\nspecifically for the 3-D mode of operation, to account for the horizontal distortion of the\ncomputational cells with respect to the vertical scale that is proper in the construction of efficient\n3-D grids for tsunami calculations. In addition, the pressure term has been split into two\ncomponents, hydrostatic and non-hydrostatic. The splitting of the pressure term facilitates the\nhydrostatic solution by merely switching off the non-hydrostatic pressure term. Therefore,\nTSUNAMI3D can be used to separate non-hydrostatic effects from the full solution while\nkeeping the 3-D structure. The TSUNAMI3D model is suitable for complex tsunami generation","MODEL BENCHMARKING WORKSHOP AND RESULTS\n13\nbecause it has the capability to consider moving or deformable objects, subaerial/subsea\nlandslide sources, soil rheology, and complex vertical or lateral bottom deformation.\n1.5\nBenchmark tests\nAs per NTHMP rules, it is mandatory that all numerical models used in inundation mapping\nbe validated and verified by 2012. Although not required, the same should apply to models used\nin tsunami warning or emergency planning. This is best done by subjecting each model to a\nseries of benchmark tests commonly accepted by the community. The three usual categories of\nreference data used for defining benchmark tests for tsunami numerical model validation and\nverification are: (i) analytical solutions; (ii) laboratory experiments; and (iii) field measurements.\nVarious benchmark tests defined in these categories test some features, but not all, of the tsunami\nmodels. For instance, some benchmark tests are focused on validating and verifying model\nsimulations of co-seismic tsunamis sources, while others are developed for landslide sources.\nSimilarly, some analytic solutions solve NSWEs, which do not feature dispersion and hence do\nnot include the more complete physics included in Boussinesq, non-hydrostatic, or NS models.\nThe validation of numerical models is a continuous process. Even proven numerical models\nmust be subjected to additional testing as new knowledge/methods or better data are obtained.\nNew benchmark tests must also be defined to address new tsunami source characteristics or\ncomplex coastal impact. Therefore all existing NTHMP numerical models, according to their\ncapability, have to be tested regularly against a selected set of benchmark tests, for validation\nand verification. The official suite of benchmark tests was originally assembled based on the\nrecommendations of Synolakis et al. (2007). This official set of benchmark tests has its origin in\npast tsunami model validation workshops that were organized to verify the performance of\ntsunami models according to the state of knowledge and new data obtained from recent tsunamis\nat the time, see Table 1-4. These were the 1995 Long-Wave Run-up Models Workshop held in\nFriday Harbor, Washington and the 2004 Workshop held in Catalina Island, California. A short\ndescription of each of the benchmark tests with their main intent can be found in Synolakis et al.\n(2007), and the complete suite of benchmark tests and related data are available\nat\nhttp://nctr.pmel.noaa.gov/benchmark/ and at the University of Washington (UW) Wiki,\nhttp://depts.washington.edu/clawpack/links/noaa-tsunami-benchmarks/.","National Tsunami Hazard Mitigation Program (NTHMP)\n14\nTable 1-4: Current benchmark tests for model verification and validation\nBenchmark\nCategory\nDescription\nTest\nSingle Wave on a Simple Beach\nBP1*\nAnalytical\nSolitary Wave on a Composite Beach\nBP2\nSolution\nSub-aerial Landslide on Simple Beach (2-D Landslide)\nBP3\nSolitary Wave on a Simple Beach\nBP4*\nSolitary Wave on a Composite Beach\nBP5\nSolitary Wave on a Conical Island\nBP6*\nLaboratory\nTsunami Runup onto a Complex Three-Dimensional Beach.\nExperiment\nBP7\nMonai Valley\nTsunami Generation and Runup Due to Three-Dimensional\nBP8\nLandslide\nOkushiri Island Tsunami\nBP9*\nField\nMeasurements\nRat Island Tsunami\nBP10\n* Benchmark test used for NTHMP's model comparison\nDuring the detailed analysis of modeling results during the 2011 NTHMP workshop, it\nbecame apparent that some of the analytical benchmarks were formulated under certain\nconditions (i.e., to solve certain classes of equations such as NSWE) that prohibit the direct use\nof the derived analytical solution for accurate benchmarking of some tsunami models. For\nexample:\nBP3 proposes to test the generation and propagation of the sub-aerial landslide-generated\ntsunami against an analytical solution that is obtained using overly simplified governing\nequations [i.e., linear shallow water (LSW) that do not feature dispersion]. In addition,\nthe sliding mass in this problem is not conservative (the mass changes with time). During\nthe workshop closing session, it was suggested that this benchmark be replaced in the\nfuture by a benchmark test that is similar in nature but presenting more realistic physics.\nBP5 (Solitary Wave in a Composite Beach) earned a similar recommendation from the\nworkshop participants. This benchmark tests numerical models in extreme conditions that\nare not typical for geophysical tsunamis and can produce incongruent numerical results,\ne.g., the so-called \"splash-type\" runup at the vertical wall.\nBP8 (Tsunami Generation and Runup Due to Three-Dimensional Landslide) can only be\nrealistically solved using a full 3-D Navier-Stokes model, which is not one of the\npreferred approximations used in most tsunami numerical models (e.g., NSWEs or\nBoussinesq). Models using NSWE approximation show numerical instabilities in this\ncase that are due to a shock formation at the edge of the triangular wedge of sliding\nmaterial. It is still unknown whether a Boussinesq approximation would help eliminate\nthe shock formation and produce a good agreement with laboratory measurements.\nTherefore, it was suggested that this benchmark be replaced by one that can be applied to","MODEL BENCHMARKING WORKSHOP AND RESULTS\n15\na broader spectrum of numerical models or approximations, by changing the geometry of\nthe sliding material.\nAlthough each presented model was tested against nearly every benchmark test listed in\nSynolakis et al. (2007), for the sake of brevity, only a few benchmark tests have been selected for\ndoing both the NTHMP model verification and a cross-model comparison. These are the\nbenchmark tests which are the most applicable to testing geophysical tsunamis in realistic\nconditions: BP1, BP4, BP6 and BP9 (see Table 1-4). Following is a brief description of each of\nthese tests.\n1.5.1 BP1 analytical: Solitary wave on a simple beach\n-\nIn this test, the bathymetry consists of a channel of constant depth d, connected to a plane\nsloping beach of angle = cot (19.85) = 2.88° A sketch (with distorted scale) of the canonical\nbeach is displayed in Figure -1. The x coordinate increases monotonically seaward, x = 0 is\nthe initial shore location, and the toe of the beach is located at x = X = cot(B) The wave of\nheight H is initially centered at x = X, at t=0. This benchmark test is focused on modeling\nthe runup of an incident non-breaking solitary wave such that H = H d = 0.0185 . H is the\ndimensionless wave height. The initial wave profile is given by,\nn(x,0) = Hsech 2 (r(x-x))\nWhere y = V3H/4. The initial wave-particle velocity in the numerical experiments is set,\nfollowing Titov and Synolakis (1995) as:\nu(x,0)== =\nThe recommended procedure is to set a non-reflective boundary condition at the left side of the\ncomputational domain and then check that the computed non-dimensional variables such as n/d,\nu/ do not depend on the value of d.\nTo perform this benchmark, it is necessary to compare numerically and analytically\ncomputed water level profiles at certain times, and then to compare the numerically and\nanalytically computed water level dynamics at two locations along the beach during propagation\nand reflection of the wave. The analytical solution is derived based on NSWEs. All other\nrequirements to satisfy BP1 can be found at https://github.com/rjleveque/nthmp-benchmark-\nproblems.","National Tsunami Hazard Mitigation Program (NTHMP)\n16\nH\nR\nd\nx=0\nInitial Shoreline\nPosition\nX\nXs\nFigure 1-1: Definition Sketch for Canonical Bathymetry, i.e., Sloping Beach Connected to a Constant-\ndepth Region.\n1.5.2 BP4 laboratory experiment: Solitary wave on a simple beach\nThis benchmark is the laboratory experiment counterpart of BP1 and requires comparison of\nthe model solution to laboratory measurements collected during experiments in a 32-meter long\nwave tank at the California Institute of Technology. The geometry of the tank and laboratory\nequipment used to generate long-waves is described by Hammack (1972), Goring (1978) and\nSynolakis (1986).\nMore than 40 experiments with solitary waves of varying heights were performed\n(Synolakis, 1986). The height-to-depth ratio H = H d in these experiments ranged from 0.021\nto 0.626. The water level profiles at several times were measured for waves with H = 0.0185\n,\nH = 0.045 , and H = 0.3 According to observations, the solitary wave breaking occurs when\nH > 0.045 The wave with H = 0.0185 did not break in the laboratory experiments,\nsimulating a realistic tsunami, and hence this case is used to compare numerical results to\nlaboratory data in this study. The initial and boundary conditions for this benchmark test are\nexactly the same as in the case of BP1. The choice for d is somewhat arbitrary, but the depth\nused in the laboratory experiments was approximately 0.3 m.\nTo perform this BP, it is first necessary to compare numerically computed water surface\nprofiles to the laboratory data in the case of H = 0.0185 at certain given times. In addition, it is\nrequired to compare numerical model results and laboratory data in the case of H = 0.3 which\ncorresponds to a breaking wave. All others requirements to satisfy BP4 can be found at\nhttps://github.com/rjleveque/nthmp-benchmark-problems\n1.5.3 BP6 laboratory experiment: Solitary wave on a conical island\nTo validate tsunami propagation models for a 3-D case, Synolakis et al. (2007) proposed the\nuse of a laboratory experiment developed to study the inundation of Babi Island by the\nDecember 12, 1992 tsunami (Yeh et al., 1994). The tsunami attacked this conically shaped island\nfrom the north, but an extremely high inundation was observed in the south. A model of the\nconical island was constructed in a wave tank at the US Army Engineer Waterways\nExperimental Station (Briggs et al., 1995).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n17\nY\n-25\nd = 32 cm\n0°\n20\n(12.96,13.80)\n15\n22\n270°\n90°\n16\n10\n25 m\n180°\n23 m\n5\n0.38 in\n20\n5\n15\n10\nX\n29.3 m\nFigure 1-2: Basin Geometry, coordinate system and location of gauges. Solid lines represent\napproximate basin and wavemaker surfaces. Circles along walls and dashed lines represent wave\nabsorbing material. Note the gaps of approximately 0.38 m between each end of the wavemaker and\nthe adjacent wall. (Figure courtesy of Frank Gonzalez.)\nFigure 1-2 was developed based on personal communications with Mike Briggs. The basin\nwall dimensions were 29.3 X 30 m. An absorbing material was installed to define a smaller, 25 X\n28.2 m wave basin. The absorbing material used was synthetic horsehair about 2 inches thick,\nrolled into cylinders approximately 0.9 m in diameter, and characterized by a reflection\ncoefficient that varied somewhat with wave frequency, but was of the order of 12%. The length\nof the wavemaker was 27.4 m. A few differences between Figure 1-2 and previously published\nfigures of the wave basin stem primarily from the fact that (a) the wavemaker face extended\nabout 2 m into the wave basin and (b) a gap of approximately 0.38 m was present between each\nend of the wavemaker and the bottom and top walls.","National Tsunami Hazard Mitigation Program (NTHMP)\n18\nThe island had the shape of a truncated, right circular cone with diameters of 7.2 m at the toe\nand 2.2 m at the crest. The vertical height of the island was approximately 0.625 m, with 1V on\n4H beach face (i.e., B = 14°). The water depth was set at 0.32 m in the basin. The interested\nreader is referred to\nhttp://chl.erdc.usace.army.mil/chl.aspx?p=s&a=Projects:35or\nhttp://nctr.pmel.noaa.gov/benchmark/Laboratory/Laboratory_ConicalIsland/index.html\nfor detailed descriptions of laboratory experiments and data files. All requirements to satisfy this\nbenchmark test can be found at https://github.com/rjleveque/nthmp-benchmark-problems.\nTo perform this benchmark, it is necessary to demonstrate that two modeled wave fronts\nsplit in front of the island and collide behind it (as edge waves), while comparing computed\nwater level and runup with the laboratory data at gauges.\n1.5.4 BP9 field measurements: Okushiri Island Tsunami\nOn July 12, 1993, the Mw 7.8 Hokkaido-Nansei-Oki earthquake generated a tsunami that\nseverely inundated coastal areas in northern Japan. Most of the damage was concentrated around\nOkushiri Island located west of Hokkaido. The tsunami runup around Okushiri Island was\nmeasured by the Hokkaido Tsunami Survey Group (1993), which reported up to 31.7 m runup\nnear Monai village. The detailed runup measurements, together with high-resolution bathymetric\nsurveys, done before and after the earthquake, allow for testing of numerical methods and\nvalidation of the shallow water approximation to simulate real tsunamis.\nOne of the difficulties in modeling a geophysical tsunami lies in specifying the initial\nconditions of the water surface displacement and velocities. In the absence of detailed earthquake\nmodels, the deformation in the Earth's crust is commonly computed by analytical formulae\n(Okada, 1985), which assumes a simple dislocation along an inclined plane in a homogeneous\ninfinite half space. Owing to the near incompressibility of water and small rise times, the initial\nwater surface displacement is typically set equal to the crustal surface displacement, while the\ninitial water velocity is assumed to be zero. In the case of the Hokkaido-Nansei-Oki earthquake,\nseveral earth crustal deformation models were proposed. The interested reader is referred to a list\nof the deformation models for this earthquake in Yeh et al. (1996).\nThe original Kansai University bathymetry/topography Digital Elevation Models (DEMs)\nand the tectonic source were developed for the 1995 Friday Harbor Long Wave Runup Model\nworkshop (Takahashi, 1996), and are available at the NOAA/PMEL Okushiri Benchmark test\nwebsite http://nctr.pmel.noaa.gov/benchmark/Field/Field_Okushiri/index.html. Unfortunately,\nthe developed Digital Elevation Models suffer from the apparent addition of rows and columns\nof so-called \"ghost cells\" to accommodate requirements of certain numerical models, and from\nsignificant horizontal and vertical misalignments of neighboring or embedded grids. The nesting\nbetween the Digital Elevation Models was attempted to be restored by finding an optimal\nalignment of the bathymetry/topography contours across boundaries of the fine-coarse grid.\nDespite all efforts, the final set of Digital Elevation Models still suffers from a lateral shift, due\nto conversion errors between the old Tokyo Datum and WGS84 datum.\nTo perform BP9, it is necessary to compute water level dynamics at Iwanai and Esashi tide\ngauges on Hokkaido Island, Figure 1-3-a; and the runup distribution around Okushiri Island at\nthe regions enumerated in Figure 1-3-b. Other requirements to satisfy this benchmark test as well","MODEL BENCHMARKING WORKSHOP AND RESULTS\n19\nas recommendations for potential improvements can be found at https://github.com/rjleveque/\ninthmp-benchmark-problems,\na)\n43°N\nIwanai/\nHokkaido\nJapan\nSea\n42°N\nEsashi\n15\n10\n5\n41°N\n0\n42.25°N\nb)\nInaho\n17\n42.20°N\n15\n16\n2\nOkushiri\n19\n42.15°N\nIsland\n5\n18\nMonai\n42.10°N\n14\nAonae\n10\n42.05°N\n35 30 25 20 15 10 5\nOo\n0 5 10 15\n(m)\n(m)\n5\n10\n(u)\n15\n20\n25\n139.40 E 139.45° E 139.50 E 139.55° E\nFigure 1-3: a) Tide gauge locations at Iwanai and Esashi. b) Maximum runup measurements around\nOkushiri Island. Numbers in red indicate regions to determine maximum runup. (Figure courtesy of\nYoshiki Yamazaki).","National Tsunami Hazard Mitigation Program (NTHMP)\n20\n1.6\nWorkshop section summary\nThe workshop comprised two parts. The first part was devoted to presentations by individual\nmodelers of their benchmarking results, with each presentation followed by a general discussion\nwith the NTHMP Mapping and Modeling Subcommittee members in attendance. The second\npart was focused entirely on crafting recommendations to the NTHMP Coordinating Committee,\nto critiquing model results, and setting goals to consolidate the validation process.\nMost of the workshop attendees presented their model results for the existing benchmark\ntests, as defined in the report OAR-PMEL-135 (Synolakis et al., 2007). In view of the presented\nresults, it became clear that some of these benchmark tests were not well enough defined and that\nothers were missing supporting data, or at least some of the data were difficult to locate. The\nrelevance of current benchmarks, how to conduct the required peer review of models and\nbenchmarking results, how to develop \"pass/fail\" criteria for the benchmarks, and how to submit\nmodel results, were additional discussion topics. Several long-term recommendations for\nproposal to the NTHMP were identified as the result of these discussions. The long-term\nrecommendations, along with short-term recommendations of note are summarized in the next\nsection.\nAn additional topic focused on what mechanism should be employed for future validation of\nmodels. In general, future models may be verified by comparison with models having already\npassed benchmarking. A presentation of results showing new comparisons should be made to the\nMapping and Modeling Subcommittee in order for newly tested models to be considered verified\nor validated.\n1.7\nRecommendations\nDiscussions that occurred during the NTHMP Model Benchmarking workshop led to a\nseries of both short- and long-term recommendations. These are presented here for further\nconsideration.\n1.7.1 Short-term recommendations\n1. Benchmarks shall include analytical, laboratory experiment, and field measurements\nbenchmark tests.\n2. Existing benchmark tests in OAR-PMEL-135 will be retained with the exception of\nBP3, to be replaced by laboratory experiments performed for an underwater rigid\nslide with Gaussian shape (Enet and Grilli, 2007) and BP8 (3-D slide), which will be\nreplaced with a more carefully documented challenge problem. Case A in BP6, case\nB in BP4, as well as cases B and C in BP5, will be optional.\n3. The UW github site (https://github.com/rjleveque/nthmp-benchmark-problems) will\nbe the temporary repository for benchmark tests and model results.\n4. A pass / fail criterion will be developed by consensus of NTHMP-MMS members, in\nconsultation with state modelers, prior to reviewing model results deposited on the\n\"results\" directory of the UW github site.\n1.7.2 Long-term recommendations\n1. Establish and maintain a benchmark test repository, perhaps under NTHMP, NGDC\nor PMEL. This will require a partially funded position. Material accumulated on the\nUW site will eventually be transferred to the repository.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n21\n2. Conduct periodic reviews of current benchmarks with consideration of new proposed\nbenchmarks.\n3. Adopt additional BPs proposed for future use including: new field BPs for Samoa\n2009, Chile 2010, Japan 2011; a sub-aerial slide laboratory BP; two submarine slide\nlab benchmark tests; a submarine slide field benchmark test (such as the PNG event\nproposed by Stephan Grilli); a seiche laboratory benchmark test (proposed by\nAggeliki Barberopoulou); an analytical seiche benchmark test (proposed by Bill\nKnight); a grid alignment sensitivity benchmark test such as that briefly described in\nthe GeoClaw Results Report and documented in Berger et al., 2011 (proposed by UW\nGeoClaw Tsunami Modeling Group). A proposal was made to create a folder on the\nUW repository to collect this information on new benchmark tests.\n4.\nProvide financial support to develop and incorporate new, standardized Digital\nElevation Models for the field benchmark tests; this is critical to the quality and\ncredibility of benchmark test simulation results.\n5. Provide modest financial support to individuals that agree to develop the necessary\ninformation and database for each benchmark test; this is sorely needed to improve\nthe quality of these important resources and to minimize the time-consuming\ndifficulties that beset modelers that attempt these benchmark tests.\n6. Continue use of existing benchmarks for the immediate future, although cases well\noutside the shallow-water approximation should be omitted.\n7. Immediately replace BP3 and BP8 (sub-aerial landslide on a simple beach, and 3-D\nslide) with similar, but more carefully documented and comprehensive, benchmark\ntests.\n8. As agreed to in earlier MMS meeting discussions, models may be validated for either\nco-seismic or slide sources (or both). Slide source validation specifically requires\npassing the slide benchmarks. (NOTE: In the separate landslide workshop that\nfollowed the NTHMP benchmarking workshop in Galveston, there were some\ndiscussions about using justifiable initial conditions for a landslide source with\nvalidated, generic wave propagation models as an alternative for the completion of\nthe slide benchmarks.)\n1.8\nProposed benchmark tests and lessons learned\n1.8.1\nProposed benchmark tests\n1.8.1.1 Analytical: Convergence studies\nBy: Frank González et al.\nThe one-dimensional test problems currently involve exact solutions that are themselves\ndifficult to calculate numerically, e.g. requiring numerical quadrature of Bessel functions. It is\nvery useful that tabulated values of these solutions have been provided. However, rather than\nusing limited tests for which such \"exact\" solutions are known, it might be preferable to carefully\ntest a 1-D numerical model and show that it converges, and then use this with very fine grids to\ngenerate reference solutions. Fully converged solutions could be provided in tabulated form as\nwell and could be as accurate as needed for a given class of equations. It would then be possible\nto generate a much wider variety of test problems. In particular, more realistic bathymetry could","National Tsunami Hazard Mitigation Program (NTHMP)\n22\nbe used, for example on the scale of the ocean, a continental shelf and beach, rather than\nmodeling only a beach.\n1.8.1.2 Analytical: Sloshing in a parabolic basin\nBy: Dmitry Nicolsky et al., Bill Knight, and Frank González et al.\nWe propose to test the model against the analytical solution in the cases of frictionless water\nflow in 2-D parabolic basins that can model fjord-type settings typical for the Alaska coast. The\nanalytical solution to this problem is described as nonlinear normal mode oscillations of water\n(Thacker, 1981). This is a good test of wetting and drying as well as conservation. See Gallardo\net al. (2007) and the test problem in GeoClaw: http://www.clawpack.org/clawpack-4.x/apps/\ntsunami/bowl-slosh/README.html.\n1.8.1.3 Analytical: Symmetry preservation and grid alignment sensitivity studies\nBy: Frank González et al.\nHigh-accuracy one-dimensional reference solutions can also be used to test a full two\ndimensional code, by creating bathymetry that varies in only one direction at some angle to the\ntwo-dimensional grid. A plane wave approaching such a planar beach would ideally remain one-\ndimensional, but at an angle to the grid this would test the two-dimensional inundation\nalgorithms.\nThis idea can be extended to consider radially symmetric problems, such as a radially\nsymmetric ocean with a Gaussian initial perturbation at the center. The waves generated should\nreach the shore at the same time in all directions, but the shore will be at different angles to the\ngrid in different locations and it is valuable to compare the accuracy in different locations. The\ntwo-dimensional equations can be reformulated as a one-dimensional equation in the radial\ndirection (with geometric source terms) and a very fine grid solution to this problem can be used\nas a reference solution. Features could also be added at one point along the shore and this\nlocation rotated to test the ability of the code to give orientation-independent results. Some\nGeoClaw results of this nature are presented in Berger et al. (2011) and LeVeque et al. (2011).\n1.8.1.4 Analytical: Test for tolerance to depth discontinuities\nBy: Elena Tolkova\nThe appropriateness of representing differential equations with various difference schemes\nis based on the assumption that the physical variables do not change significantly within the\nsampling intervals in space and time. However, in tsunami propagation applications, the ocean\ndepth supplied by Digital Elevation Models can undergo large changes between neighboring\nnodes. The suggested exercise aims to verify the correct operation of the tsunami model in the\nbasin with abrupt depth changes.\n1.8.1.5 Laboratory Experiment: Solitary wave over 2-D reef system\nBy: Volker Roeber and Kwok Fai Cheung\nWe conducted two series of laboratory experiments at Oregon State University in 2007 and\n2009 that included 198 tests with 10 two-dimensional reef configurations at a range of water\ndepths. Each test included a series of incident solitary wave heights. These test cases are a logical\nextension of the current benchmark for validation of inundation models. Though the laboratory\nexperiments focus on shock-related hydraulic processes such as wave breaking and bore\nformation, the collected data allow examination of shoaling, reflection, wave breaking, and","23\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nswashing dynamics. We propose the results from one of the test configurations with H/d = 0.3 as\na future NTHMP benchmark test and provide a detailed description of hydraulic processes with\nadditional data from BOSZ.\n1.8.1.6 Laboratory Experiment: Solitary wave over 3-D reef system\nBy: Volker Roeber and Kwok Fai Cheung\nThe National Science Foundation funded a workshop and a benchmarking exercise for\ninundation models at Oregon State University in 2009. The organizer provided two benchmark\ntest cases with data from laboratory experiments at the Tsunami Wave Basin. Swigler and Lynett\n(2011) provided a detailed description of the experiments, instrumentation, and data post-\nprocessing. These test cases, which involve wave transformation over three-dimensional reef\nconfigurations, are logical extensions of the two-dimensional reef experiments from Roeber\n(2010). The laboratory data allow validation of models in handling dispersion and flux-\ndominated processes simultaneously.\n1.8.1.7 Laboratory Experiment: Shoaling waves and runup along a wide continental shelf\nBy: Dmitry Nicolsky\nIt is noted in BP6 that the distance between the wave-maker and island is short and the wave\ndispersion effect does not have enough time to modify the wave and thus the runup of non-\nhydrostatic and hydrostatic models are more or less the same. A new benchmark test should be\nsimilar in nature to BP3, where the vertical wall is replaced by a sloping beach. The focus of this\nbenchmark test will be on modeling propagation and shoaling of the wave across the continental\nshape and its runup at the beach.\n1.8.1.8 Laboratory Experiment: The saucer slide\nBy: F. Enet and S. Grilli\nExperiments were performed in the 3.7 m wide, 1.8 m deep, and 30 m long wave tank of the\nOcean Engineering Department at the University of Rhode Island, Grilli and Watts (2001) and\nGrilli et al. (2002). The experimental setup was designed to be as simple as possible to build,\nwhile allowing one to illustrate and quantify the key physical phenomena occurring during\nlandslide tsunami generation. In each experiment, a smooth and streamlined rigid body slides\ndown a plane slope, starting from different initial submergence depths, and generates surface\nwaves. Different conditions of wave nonlinearity and dispersion are generated by varying the\nmodel slide initial submergence depth. Surface elevations are measured with capacitance gauges.\nRunup is measured at the tank axis using a video camera. Landslide acceleration is measured\nwith a micro accelerometer embedded within the model slide, and its time of passage is further\nrecorded at three locations down the slope. The repeatability of experiments is very good.\nLandslide kinematics is inferred from these measurements and an analytic law of motion is\nderived, based on which the slide added mass and drag coefficients are computed. Characteristic\ndistance and time of slide motion, as well as a characteristic tsunami wavelength, are parameters\nderived from these analyses. Measured wave elevations yield characteristic tsunami amplitudes,\nwhich are found to be well predicted by empirical equations derived in earlier work, based on\ntwo-dimensional numerical computations. The strongly dispersive nature and directionality of\ntsunamis generated by underwater landslides is confirmed by wave measurements at gauges.\nMeasured coastal runup is analyzed and found to correlate well with initial slide submergence\ndepth or characteristic tsunami amplitude.","24\nNational Tsunami Hazard Mitigation Program (NTHMP)\n1.8.1.9 Field Measurements: The 1964 Alaska tsunami\nBy: Joseph Zhang\nThe March 28, 1964 Prince William Sound (Alaska) earthquake produced a mega\ntransoceanic tsunami that represented the largest tsunami that impacted the US and Canadian\nwest coast on record. There is a wealth of field records for this event from tide gauges and\neyewitness reports. Therefore we have been using this event as a representative of remote\nsources in our mitigation studies in Oregon.\n1.8.1.10 Field Measurements: The Papua New Guinea (PNG) landslide-generated tsunami\nBy: Stephan Grilli\nThe tsunami that struck New Guinea on July 17, 1998 was the most devastating tsunami (in\nterms of casualties) generated by a landslide. The high reported runups, source definition and\ngood runup data collected by tsunami survey teams makes this benchmark test a good candidate\nfor benchmarking.\n1.8.1.11 Field Measurements: Tsunami propagation for the 2004 Indian Ocean Tsunami\nBy: Stephan Grilli et al.\nThe Mw 9.3 earthquake on 26 December, 2004, ruptured over 1200 km along the Andaman-\nSunda trench from Sumatra to the Nicobar and Andaman Islands, generating destructive tsunami\naffecting the coastal communities throughout the Indian Ocean. The 2004 Indian Ocean tsunami\nwas recorded by satellite radar altimeters and many tide gauges. Three satellite radar altimeters,\nTOPEX/Poseidon, Jason-1, and Envisat, recorded the 2004 Indian Ocean tsunami propagating in\ndeep water across the Indian Ocean and many nearshore tide gauges also measured the tsunami\nsignal, both in far- and near-field. These are valuable data to validate tsunami models for open\nocean propagation as well as for coastal impact.\n1.8.1.12 Field Measurements: Propagation and runup for the 2009 South Pacific Tsunami\nBy: Yoshiki Yamazaki et al.\nThe South Pacific Tsunami on 29 September 2009 generated by Mw 8.1 earthquake south of\nthe Samoan Islands struck Tutuila Island, American Samoa. Tsunami was recorded at DART\nbuoys and Pago Pago Harbor at Tutuila. Tsunami arrived at mid tide and produced maximum\nrunup of 17.6 m with detrimental impact on Tutuila. Several field survey teams measured runup\nand inundation around Tutuila. The high quality surface elevation data and runup/inundation\nmeasurements with available high resolution DEM data from NOAA NGDC provide the field\nbenchmark to validate tsunami model to simulate propagation and runup.\n1.8.1.13 Field Measurements: Tsunami induced currents for the 2006 Kuril, and 2010 Chile\ntsunamis\nBy: Yoshiki Yamazaki et al.\nThe Mw 8.3 earthquake on 15 November 2006 at the Kuril Islands and the Mw 8.8\nearthquake on February 27, 2010 in Chile generated tsunamis that reached the Hawaiian Islands.\nAlthough Tsunami impacts due to these events were not severe along the Hawaii coastline, both\ntsunamis caused rapid changes in water level and unusual currents around the Hawaiian Islands.\nThe Kilo Nalu Nearshore Reef Observatory located south of Oahu, Hawaii recorded clear signals\nof the flow velocity and surface elevation associated with the 2006 Kuril and 2010 Chile","MODEL BENCHMARKING WORKSHOP AND RESULTS\n25\ntsunamis. Velocity measurements made during each of these tsunamis provide the opportunity to\nvalidate a tsunami model's capability to estimate tsunami induced currents for tsunamis arriving\nat the Islands from different directions.\n1.8.1.14 Field Measurements: Distant and near-field tsunami impacts for the 2011 Tohoku\nTsunamis\nBy: Many authors\nThe 2011 Tohoku earthquake of Mw 9.0 generated a massive tsunami that devastated the\nentire northeastern Japan coasts and damaged coastal infrastructure across the Pacific. The\nextensive the dense geodetic instruments, the numerous water level stations, post-event surveys\nacross the Pacific provide the best quality datasets and coverage of any tsunami to date for model\nvalidation.\n1.8.2 Lessons learned\nAn analytical solution in BP3 is derived under certain circumstances that prohibit any direct\nemployment of the analytical solution in accurate benchmarking of the tsunami models.\nTherefore, it is suggested that this benchmark should be replaced by a problem that is similar in\nnature.\nIn BP8, the numerical results derived within the NSWE approximation show numerical\ninstabilities that are due to a shock formation at the edge of the triangular wedge modeling\nsliding material. It is still unknown whether a Boussinesq approximation would help to eliminate\nthe shock formation and produce better agreement with the laboratory measurements. However,\n3-D full Navier-Stokes solutions do fairly well at reproducing BP8's free surface time series and\nrunup. Anyway, we emphasize the rigid sliding block used in this benchmark test has an overly\nsimplified geometry to represent an actual landslide. Therefore, it is suggested that this\nbenchmark should be replaced by a problem with a more realistic geometry of the sliding\nmaterial.\nIn BP9, the computed runup around Okushiri Island is within the variability of field\nobservations. The computer simulation of the 1993 Okushiri tsunami captures the overland flow\nat the cape Aonae, where the maximum destruction was reported. However, the local extreme\nrunup, e.g. in the narrow gully near the village of Monai, is sensitive to the near shore\ninterpolation of bathymetry and topography.\nDuring the benchmarking exercise, one of the most taxing problems we encountered was\nrelated to the incomplete information regarding field tests. We had to spend a considerable\namount of time gleaning files from various sources. For the field test (Okushiri, etc.) some\ncritical pieces of information (such as the horizontal datums of the Digital Elevation Models) are\nstill missing. Perhaps the most serious problem with field tests is uncertainty about the geometry\nof the earthquake source. This issue causes serious errors in simulations for areas proximal to the\nsource (e.g., poor match of simulated runup on the east coast of Okushiri Island).\nThe set of benchmark tests proposed in OAR-PMEL-135 was found to be mostly\nappropriate except for a few extreme cases (e.g., the larger wave breaking case C in the\ncomposite beach case). The combination of analytical, lab and field tests adequately tests the\nperformance of models.\nGoing forward, we suggest that each model be validated by comparing it with recent well-\ndocumented tsunami events, such as 2004 Indian Ocean, 2010 Chile, and 2011 Tohoku. These","National Tsunami Hazard Mitigation Program (NTHMP)\n26\nevents have the advantage of being well documented in terms of witness accounts and a wealth\nof observations (e.g., satellite imagery, amateur and professional videos, and DART® deep-\nocean bottom pressure) not previously captured on a wide scale. The synoptic look at these\nevents makes them suitable for tsunami propagation model validation.\n1.9\nModels comparison\n1.9.1 Summary\nAs stated in the NTHMP Strategic Plan, all numerical models used by NTHMP program\npartners for preparation of tsunami inundation and evacuation maps and associated products are\nrequired to meet standard criteria by the end of 2012. Accordingly, the primary objective of the\nNTHMP model benchmarking workshop held in Galveston, Texas in March 2012 was to ensure\nthat each candidate model successfully simulate a series of pre-defined and accepted tsunami\nbenchmark tests. Model results were compared against the relative reference solution for each\nbenchmark test (e.g. analytic solutions, lab experiment, and tsunami field data for the case of the\nJuly 1993 Okushiri Island, Japan tsunami.) A subset of the benchmark tests listed in OAR-\nPMEL-135 was selected as benchmark priorities (BP) to adequately test the performance of each\ncandidate NTHMP numerical model under the conditions of intended use. Results of each model\nwere compared under the same data format and error formulation. Note that although each\nmodeling group used benchmark tests in addition to those specified, the model comparisons\npresented here are limited in scope to the tests specified as part of the subset. The benchmark test\npriorities are as follows:\nSingle wave on simple beach - analytical solution (BP1)\nSingle wave on simple beach - laboratory experiment (BP4)\nSolitary wave on conical island - laboratory experiment (BP6)\nOkushiri Island - field measurements (BP9)\n1.9.2 Error definition to measure model accuracy\nTo determine NTHMP models performance, three error formulae were selected and tailored\nto each of the four benchmark tests. These are:\nThe normalized Root Mean Square deviations\nThe Relative Error for maximum wave height or runup\nThe Relative Error for multiple runup values recorded in a specific region\nThese three errors are described below.\nRMS: The normalized Root Mean Square deviations error is applied within a space segment\nor time period to all observed data points. The error is defined as\n(1)\nRMS n\n=\nThe RMS is a frequently used measure of the differences between values predicted by a model (\n5m) and values obtained/observed either from an analytical solution, laboratory experiments, or\nfield measurements. The obtained/observed values are assumed as the benchmark test reference","MODEL BENCHMARKING WORKSHOP AND RESULTS\n27\ndata and hence assumed as true, although errors in their estimation are probable, especially for\nlaboratory experiment and field measurements where systematic errors might exit. The RMS is\nnormalized with respect to the magnitude between the maximum and minimum observed values,\ni.e., wave height (Semax Semin), or the maximum wave amplitude or runup Semax The maximum\nwave height usually corresponds to the first or second wave in a tsunami wave train, measured\nfrom crest to trough and n is the number of observed points obtained within an arbitrary space\nsegment or time period. This RMS error is thus a time or space-dependent error, based on model\nprediction, which are interpolated at the location/time of each observed point. Hence RMS error\nis sensitive to phase lags in the predicted values and its main use is only to assess the accuracy of\nthe model in predicting the entire set of observed data (overall model performance).\nMAX: To quantify each model's predictive accuracy for the maximum wave amplitude or\nrunup regardless of the location where, or time this maximum occurs, the following formulation\nis used to measure differences between maximum predicted (5m) and observed (5e) values.\nMAX =\n(2)\nNote that the expression used for MAX is a relative error based on the maximum magnitude of\nthe observed values.\nERR: Another error used to measure the model performance is the relative error for multiple\nvalues that are collected in one specific location or region. This error is practically the same as\nMAX but it has been defined to determine model accuracy in predicting runup against multiple\nvalues that have been recorded by a tsunami survey team in a region with similar inundation\ncharacteristics or geomorphology. The regional data set is reduced to three values, minimum,\nmaximum, and average, that represents the inundation at a specific location. The error is then\ndefined as:\n0\nERR =\n(3)\nOtherwise\nWhere the denominator D is one of the following values ]. D is selected\naccording to the numerator minimum value. For example, if is the minimum\nvalue in the numerator, then the denominator = Seave \"\nThe MAX and ERR errors are the only formulation used for verification and validation with\nobserved data in the model comparison and discussion. However, RMS errors further help\nto\nassess the accuracy of the model in predicting the entire set of observed data (with the correct\nmagnitude and phase); in other words, it is a good metric tool for assessing overall model\nperformance.\n1.9.3 Models comparison and discussion\nTo verify suitability of the NTHMP numerical models for the tsunami problem, the Mapping\nand Modeling Subcommittee decided, as a first attempt, to use as a threshold the allowable errors\nstated in the standard OAR-PMEL-135 for each benchmark test (Synolakis et al., 2007). For the","National Tsunami Hazard Mitigation Program (NTHMP)\n28\ninter-model comparison, a subset of four benchmark tests that almost all models could simulate\nand which was deemed to adequately test the performance of each model was selected from the\navailable list,. The selected benchmark tests represent all the categories of reference data used to\nassess each model, namely: a) analytical solution, b) laboratory measurements and c) field\nmeasurements. The allowable errors vary according to these categories, the tested parameter\n(e.g., maximum amplitude/runup), and wave condition (e.g., breaking, non-breaking). Model\naccuracy is quantified by way of the error formulae previously defined, i.e., RMS, MAX, and\nERR. According to OAR-PMEL-135, the models' maximum predicted values should not differ\nby more than the pre-established error threshold indicated in Table 1-5.\nTable 1-5: Current allowable errors for model validation and verification, after Synolakis et al. (2007)\nOAR-PMEL-135\nBenchmark\nTested Quantity and Wave Condition\nCategory\nAllowable Error\nTest\nBP1\nAnalytical\nRunup/amplitude (non-breaking wave)\n<5%\nSolution\nRunup/amplitude (non-breaking wave)\n<5%\nBP4\nLaboratory\nRunup/amplitude (breaking wave)\n<10%\nMeasurements\n<20%\nBP6\nLaboratory\nRunup\nMeasurements\n<20%\nBP9\nField\nRunup\nMeasurements\n1.9.4 Summary of comparisons for the four selected benchmark tests\nThe following sections summarize NTHMP model comparisons for the four selected\nbenchmark tests.\n1.9.4.1 BP1 Analytical Solution: Single wave on simple beach\nLet us define the following non-dimensional variables: H =\nA focus in developing a tsunami modeling algorithm is to simulate extreme positions of the\nshoreline - the maximum runup and rundown. Figure 1-4 and Figure 1-5 show comparisons of\nanalytical (i.e., NSW) and computed water surface profiles during runup and rundown of a\nsolitary wave of initial height H = 0.0185 over a beach of slope 1:19.85 cot B = 19.85) (see\nFigure 1-1), at t = 35 to 65 by At = 5 increments. The maximum runup and rundown occur in the\nnumerical simulation at t = 55 and 70, respectively. Note, the t = 70 plot is not shown in the\nfigures. Results indicate that all the numerical solutions match very well the analytical solution\nduring runup, while the maximum rundown shows some variability between the computed water\nsurface profiles. This is likely due to the use of different shoreline tracking (i.e., wetting-drying)\nalgorithms, different model schemes, and different spatial and temporal discretizations. In future\nbenchmark validations, it is recommended that the numerical solutions be computed to show\nconvergence to the analytical prediction for maximum rundown, i.e., here at t = 70. It is\nnoteworthy to mention that for BP1, dispersive models such as SELFE, NEOWAVE, and\nFUNWAVE were restricted to run in a non-dispersive mode, because the validation data are\nbased on an analytical solution of the non-dispersive NSW equations","MODEL BENCHMARKING WORKSHOP AND RESULTS\n29\nTable 1-7-a (case H = 0.0185, non-breaking wave) shows, for all NTHMP models, the mean\nRMS errors (for computed surface profiles) range between 4% and mean MAX errors (for\nmaximum wave amplitude or runup, regardless of location) range between 1 - 5%. Based on\nthese results, the overall performance of all the models is deemed very good (RMS < 4% and for\nMAX, the model errors are kept below 5%). According to Synolakis et al. (2007), \"any well-\nbenchmarked code should produce results within 5% of the calculated value from the analytical\nsolution.\"\nFigure 1-6 compares numerical and analytical solutions of water level dynamics at locations\nx 0.25 (near the initial shoreline) and 9.95 (between the beach toe and initial wave crest) during\npropagation and reflection for the case H = 0.0185. During rundown, both numerical and\nanalytical solutions predict a water backwash between t = 67 and 82, with the location x = 0.25\nbecoming temporarily dry, while X = 9.95 remains wet throughout the entire duration of the\nexperiment. Comparison of the analytical and numerical solutions at these two locations reveals\nthat the computational error (RMS) is typically less that 3% (see Table 1-7-b), and the agreement\nis quite good for all NTHMP models. For MAX (maximum wave amplitude), the models' mean\nerrors range between 0 - 2%. Therefore, one can conclude that all NTHMP models predicted\nextremely well the analytical solution of BP1. The models' MAX errors are kept below the\nallowable threshold of 5%, although some small differences between models occur, presumably\ndue to the variety of numerical schemes used and the different space and time discretization\nmethods adopted for the solution.\n1.9.4.2 BP4 Laboratory Experiment: Single wave on simple beach\nThe time evolution of the free surface profile for an initial wave of height H = 0.0185 (Case\nA) shoaling over a sloping beach with 1:19.85 slope, is shown in Figure 1-7, between t = 30 and\n70 by At = 10 increments. Laboratory measurements of the surface elevation at these times are\ncompared to the NTHMP models solutions. Overall, we see a good agreement between the\nnumerical solutions and the laboratory measurements for all times corresponding to the\npropagation and runup of the wave. One noticeable observation is that differences among\nnumerical solutions are smaller than differences between any of the model solutions and the\nlaboratory data.\nAt time t = 50, which is near the maximum runup (t = 55), two models based on the NSWE\n(ALASKA and GEOCLAW) predict a slightly higher runup than the other models and the\nexperimental data. These differences could be attributed in part to the zero friction assumption\nand fine spatial discretization used in the numerical solution. However, other NSWE models\n(MOST and ATFM) captured the runup somewhat better. In the case of ATFM, the improvement\nis likely due to the addition of the non-hydrostatic component. It is noticeable from Figure 1-7\nthat the models that include dispersive effects produce better results at t = 50 and 60 than non-\ndispersive models. The numerical solutions vary the most at t = 70, which corresponds to the\nmaximum rundown. There, we see that the non-dispersive models (MOST, ALASKA and\nGEOCLAW) compute larger rundowns than the dispersive models.\nThe estimation of model errors are shown in Figure 1-6-a. The mean RMS errors range\nbetween 7 - 11% and, for the maximum wave amplitude or runup (MAX), regardless of its\nlocation, mean errors range between 2 - 10%. Hence, model errors are kept below 10%, the\nthreshold agreed upon for laboratory measurement benchmarks during the Catalina Island model\nvalidation workshop, in 2004.","National Tsunami Hazard Mitigation Program (NTHMP)\n30\nFigure 1-8 similarly presents the time evolution of free surface profiles for an initial wave of\nheight =0.30 (Case B) between t = 15 and 30 by At = 5 increments, computed by NTHMP\nmodels, as compared to laboratory experiments. This is a very challenging case, in which the\nwave breaks during runup in laboratory experiments. For non-dispersive or NSWE models, this\ncase becomes even more challenging. For instance, NSWE models predict that the leading front\nof the solitary wave will steepen and become singular shortly after the initiation of the\ncomputations. The numerical singularity propagates towards the beach until it meets the\nshoreline where the singularity dissipates. The existence of strong wave breaking does prevent a\ngood agreement of the NSWE solutions with the laboratory measurements (for details on a non-\ndispersive solution see each individual reports that use NSWE). However, since numerical\ndispersion can compensate for the absence of physical dispersion in NSWE models (e.g., by\nadjusting the spatial discretization as the MOST model does), the effective simulation of\nbreaking waves in NSWE models is still an active area of research. Results show that the\ninclusion of wave dispersion in the models allows the wave to initially steepen-up without\nbreaking, between t = 15 and 20, which results in a good match with laboratory measurements.\nTable 1-6-b indicates for Case B, mean RMS errors ranging from 5 - 8% and mean MAX errors\nfor the maximum wave amplitude ranging from 5% to 12%. Although, for some models, the\nlatter errors are slightly larger than the accepted threshold for laboratory measurements (<10%),\nthese errors are still deemed more than acceptable, considering the difficulty in reproducing this\nbenchmark test that features breaking waves.\n1.9.4.3 BP6 Laboratory Experiment: Solitary wave on conical island\nIn BP6, experiments with different wave heights were conducted in the large-scale wave\ntank at Coastal Engineering Research Center, Vicksburg, Mississippi. The time series of surface\nelevation comparison and associated errors at gauges 6, 9, 16, and 22 for a variety of incident\nsolitary waves impinging into a conical island are shown in Figure 1-9 through Figure 1-12, and\nin Table 1-9 and Table 1-10. Three different wave heights or cases were selected from the\nlaboratory experiment to validate the numerical models, i.e., Case A, H = 0.045 ; Case B,\nH = 0.096 and Case C, H = 0.181\nFigure 1-9, Figure 1-10, and Figure 1-11 show the computed and observed sea levels at the\nfour gauges around the island, for the various cases of incident waves. The model comparison\ntimes have been selected to avoid the first reflection of the wave. One of the challenges in\nmodeling the observed waves was the application of appropriate boundary conditions and\ngenerating the wave in the numerical models. Although various techniques were employed to\naddress these challenges, all computed solutions matched well the observed water level\ndynamics at the given locations or gauges. For all SWE models, such as ALASKA, GEOCLAW\nand MOST, the simulated waves are seen to steepen faster than in laboratory experiments (e.g.,\nin Case C; H = 0.181) This is a well-known effect of the shallow water approximation, where\nthe lack of dispersive terms yields so-called \"shallow water steepening\" of waves. Visual\nexamination of the models' results reveals that the dispersive models, such as BOSZ,\nFUNWAVE, NEOWAVE, and SELFE, capture the water level dynamics slightly better than the\nnon-dispersive models. However, while models based on the Boussinesq-type or non-hydrostatic\napproximations feature wave dispersion effects, they did not show an appreciable improvement\nover the SWE models in matching this particular set of laboratory observations. This can also be\ndeduced from Table 1-9, where the models' mean RMS errors for cases A, B and C have a narrow\nrange of variation, i.e., between 7% and 10%. For maximum wave amplitude at gauges,","MODEL BENCHMARKING WORKSHOP AND RESULTS\n31\nregardless of the location the maxima occurred, the mean models' errors (MAX) range between\n4% and 19%. These models' errors have a wider variation, and there is a clear trend that\ndispersive models perform slightly better. One non-dispersive model, ALASKA, was the\nexception, with its performance for this specific BP case being quite good (perhaps due to\nnumerical dispersion). All models' errors for BP6 are below the 20% threshold recommended in\nOAR-PMEL-135 (see Table 1-5).\nFigure 1-12 shows the modeled maximum runup distribution versus laboratory experiments,\naround the conical island, for all three wave height cases. We again observe that, in general,\ndispersive models reproduce the maximum runup slightly better. The models' RMS errors for\nmaximum runup around the conical island range from 12% to 22%, and the mean relative errors\nMAX, range between 3% and 10%. As they do not differ by more than 20% from the laboratory\nmeasurements, maximum runup values predicted by all the models are considered to be fairly\ngood.\n1.9.4.4 BP9 Field Measurement: Okushiri Island\nBP9 provides model validation and as such is an important check of the ability of the\npresented models to simulate realistic tsunami events. The sea level dynamic results modeled\nwith various NTHMP models are compared to the tidal gauge records taken during the first hour\nfollowing the earthquake. Figure 1-13 shows the computed and observed water level dynamics at\nIwanai and Esashi, respectively. For the Esashi tide gauge, the arrival time of each computed\nwave matches well the arrival of the recorded leading tsunami wave. The correlation of positive\nand negative phases between the computed and observed waves is also rather good, although the\ncomputed waves at both locations have a larger range and frequency of variability than the\nobserved waves. For the Iwanai case, the time shift discrepancies between the measured and\nobserved waves could be explained by the lack of detailed bathymetry or coastal geomorphology\nnear tide stations, inaccuracies in the specified initial conditions, and a potential delayed\nresponse of the tide gauge hardware (Yeh et al., 1996). The models' relative errors on maximum\namplitude (MAX) at Iwanai and Esashi are shown in Table 1-11-a. In the Esashi case, the ATFM,\nFUNWAVE and NEOWAVE models predicted the free surface time series and the maximum\nwave amplitude with maximum amplitude relative errors in the range of 10%-19%, below the\n20% error threshold recommended in OAR-PMEL-135. Other models did not perform as well,\nbut during the benchmarking exercise, many modelers struggled with incomplete information\nregarding this benchmark. They reported having to spend considerable time collecting files from\nvarious sources, which instead of helping, actually increased the uncertainties. Some critical\npieces of information, such as the horizontal datum of the DEMs, are still missing and the\ngeometry of the earthquake source is doubtful too. These information gaps caused serious errors\nin the simulations for areas both near to and far from the source. Therefore, the workshop\nparticipants recommend that BP9 be revised to address these issues, if possible, and that\nadditional historic events be simulated for model validation.\nIn the Iwanai case, the errors on maximum amplitude modeled are very large. In view of the\nclose similarity among model results and the large discrepancy with respect to measurements, it\nis quite clear that the Iwanai tide record is too uncertain and likely not adequate to be used as a\nvalidation case. After a closer examination of the Iwanai gauge location, it seems that the gauge\nis surrounded by a complex port layout, which is not resolved well in the current grid resolution\nused in the models. Therefore, it is suggested that Iwanai's tide records be replaced with another","National Tsunami Hazard Mitigation Program (NTHMP)\n32\n(nearby) tide gauge for which adequate surrounding bathymetry and coastal geomorphology data\nexist.\nFigure 1-3-b shows the locations (black circles) on Okushiri island where the runup was\nmeasured shortly after the 1993 tsunami. These measurements were obtained from Imamura-\nShuto's group (data researched by Yoshiki Yamazaki) (regional data are enumerated in red).\nFrom Table 1-11-b, based on the models' runup relative errors ERR, it can be seen that all\nNTHMP's models obtained mean runup relative error below the permissible 20%. A closer\nexamination of the modelers' individual reports, however, shows that there are several\nexceptions where the modeled runup underestimates the observations by a larger amount. For\nexample, the modeled runup in the narrow gully near the village of Monai is underestimated in\nmost of the models. The discrepancy between the measured and computed runup values there\nmight be explained by the lack of accurate bathymetry and topography data near Monai,\nuncertainties in the initial water surface displacement, or by limitations of the current models\nwhen high vertical acceleration occurs. Almost all presented models capture a sequence of events\nrelated to the inundation of the city of Aonae (model runup relative ERR = 0% at regional\nlocation 9 on Table 1-11-b). In computer experiments, it is easy to observe an approximately 5 m\nhigh wave approaching the Aonae peninsula from the west. The wave drastically steepens over\nthe shallow areas, runs up on the western side on the Aonae peninsula, and then sweeps across\nthe tip of the peninsula. Then due to the shallow depth around the peninsula, the wave front\nbows, bends around the Aonae peninsula, and subsequently hits the town of Hamatsumae.\nAlthough some computed results show discrepancies with measured runup regional data, all the\npresented models reproduced this overall pattern quite well. Probably, the largest source of\ndiscrepancy is associated with the interpretation of incomplete information regarding the field\ndata, horizontal datums of the DEM, and uncertainties in the sea floor deformation.\n1.9.5 Benchmarking results for landslide generated tsunami models\nTo benchmark numerical models to simulate landslide-generated tsunamis, Synolakis et al.\n(2007) propose two benchmark problems, namely BP3 and BP8. The first BP is based on an\nanalytical solution, while the other is a laboratory experiment. The analytical solution in BP3 is\nderived under certain conditions that prohibit any direct use of the solution to accurately\nbenchmark some tsunami models (i.e., dispersive/non-hydrostatic models). All models that\nattempted to predict the landslide-generated wave according to BP3 have a good qualitative\nagreement with the analytical solution provided in Liu et al. (2003), although quantitative\nagreement may be lacking for the reason discussed before. Therefore, it is suggested that this\nbenchmark be replaced by a problem that is similar in nature.\nOnly two SWE models (GEOCLAW and ALASKA) and two Navier-Stokes models\n(THETIS and TSUNAMI3D) were used to compare the simulated water dynamics to laboratory\nobservations in BP8. Before comparing the numerical results to the laboratory observations, it\nwas observed, that for this particular BP, the SWEs are not applicable to simulate 3-D flows,\nespecially around the sliding wedge. Away from the shore, the water depth is comparable with or\ngreater to the observed wavelength and thus the shallow water assumption is no longer valid.\nThe SWE models produce computational results that exhibit severe numerical oscillations\ndue to the formation of a shock wave (see individual reports). The Navier-Stokes models achieve\na much better comparison of numerical results with laboratory observations. The THETIS model\nshows that it is possible both to predict a slide location and wave dynamics at the same time. The","MODEL BENCHMARKING WORKSHOP AND RESULTS\n33\nTSUNAMI3D model also shows a good match of the computed water level to the laboratory\nobservations. Unfortunately, the computational resources required for meaningfully running\nNavier-Stokes models are still prohibitive. However, the coupling of Navier-Stokes model\n(landslide tsunami generation region) and SWEs model (tsunami propagation and inundation) for\npractical tsunami application is possible. The coupling of the two approaches has been applied\nsuccessfully for construction of inundation map in the Gulf of Mexico, Horrillo et al. (2009) and\nrecently for the US east coast inundation assessment, Kirby and Grilli (2011). Moreover, a closer\nexamination of BP8 reveals that this BP does not constitute a good test for the tsunami models,\ndue to an unrealistic landslide profile - a rigid wedge (with vertical front and side walls) sliding\ninto the water. Therefore, it is suggested that BP8 be replaced by a problem with a more realistic\ngeometry of the sliding material. In Section 1.8, we briefly mention some proposed benchmark\nproblems for the future validation and verification of landslide tsunami models.\n1.9.6 Conclusions\nWorkshop participants recommend continuing to use the existing benchmark tests listed in\nOAR-PMEL-125 in the immediate future, although the cases well outside the shallow-water\napproximation can be omitted. In particular, BP3 should be replaced with a newly proposed\nlaboratory benchmark for an underwater landslide with a Gaussian shape, and BP8 should be\nreplaced by a more carefully documented challenge problem; in BP4, Case B should be optional;\nin BP5, Cases B and C should be optional, and in BP6, Case A should be optional. Historic\nevents in addition to the 1993 Hokkaido-Nansei-Oki (Okushiri Island, Japan) should be added to\nthe suite of benchmark tests to further validate all models.\nThe 1993 Hokkaido-Nansei-Oki Tsunami event remains one of the most important and\nthoroughly documented cases of extensive tsunami runup available to the tsunami community.\nBut the bathymetric and topographic computational grids are flawed by severe horizontal and\nvertical misalignment errors. It is necessary to build improved, nested coastal digital elevation\nmodels (DEMs) of the Okushiri, Japan area to replace the 8 existing DEMs that have large,\nknown inaccuracies that negatively impact tsunami modeling results. The best publicly available\nbathymetry and topographic data should be obtained through collaboration with Japanese\ninstitutions and both structured and unstructured nested DEMs should be developed to support\nmultiple tsunami modeling codes. It was evident after examination of numerical models results\nthat the Iwanai tide record is faulty too. The workshop participants suggest that this tide record\nbe replaced by another (nearby) tide gauge record where surrounding bathymetry and coastal\ngeomorphology data are better suited for unambiguous model validation.\nIn BP1, all NTHMP's models predicted very well the analytical solution of the wave\nevolution (H = 0.0185) at different times, with errors less than the allowable 5%. In BP4, for\nCase A (H = 0.0185), all mean model errors were kept below 10% (which the 2004 Catalina\nIsland model validation workshop suggested is an allowable error for laboratory measurements\nbenchmark tests). For Case B H = 0.30, breaking wave), the maximum wave amplitude model\nerrors range from 5% to 12%. Although, the mean amplitude model errors are slightly larger than\nthe accepted standard error for laboratory measurements benchmark tests (<10%), the workshop\nparticipants concluded that these errors are more than acceptable, considering the difficulty in\nreproducing this benchmark test, which features breaking waves. In BP6, Case A (H = 0.045)\nCase B H = 0.096), and Case C H = 0.181), a clear trend was observed showing that\ndispersive models perform slightly better than non-dispersive models. However, one non-","National Tsunami Hazard Mitigation Program (NTHMP)\n34\ndispersive model, ALASKA, was the exception, with its performance for this specific test case\nbeing quite good. For BP6, model errors are kept below the 20% recommended in OAR-PMEL-\n135 (see Table 1-5). For BP9, runup around Okushiri Island, participants concluded that,\nalthough some computed results show discrepancies with measured runup regional data, all the\npresented models reproduced the overall observed pattern of wave arrival quite well. All\nNTHMP models obtained mean runup relative errors below the acceptable 20%.\nThe workshop participants represent nationally renowned tsunami modeling organizations\nand, therefore, are well qualified for validating each numerical model. A pass / fail criteria for\nthe verification and validation based on comparison with observations of each numerical model\nwas developed by consensus of NTHMP MMS members during the benchmarking workshop, in\nconsultation with state modelers and based on the review of modeling results. Future model\nvalidation testing and acceptance will follow the existing criteria, and will also be reviewed by\nthe MMS members and their modeling experts.\nThe following table summarizes the allowable errors for the main three categories used for\nbenchmarking. These errors are quite similar to those recommended in OAR-PMEL-135,\nSynolakis et al. (2007). The only difference is in the allowable error for the laboratory\nmeasurement category. For the sake of simplicity, the error criterion of 10% is applied to both\nbreaking and non-breaking waves.\nTable 1-6: Allowable errors for the main three categories used for benchmarking\nAllowable Errors MAX /ERR/ RMS\nCategory\nAnalytical\n<5%\nSolution\nLaboratory\n<10%\nMeasurements\nField\n<20%\nMeasurements","MODEL BENCHMARKING WORKSHOP AND RESULTS\n35\n1.9.7 Model comparisons: Summary figures and tables\n1.9.7.1 Single Wave on a Simple Beach - Analytical Solutions (BP1)\nBP1 (H=0.0185)\n0.03\nt =35\n0.02\nALASKA\nATFM\n0.01\nBOSZ\nFUNWAVE\n0\nGEOCLAW\nMOST\nNEOWAVE\n0.04\nt =45\nSELFE\n0.03\nAnalytical data\n+\n0.02\n0.01\n0\nt=55\n0.08\n0.06\n0.04\n0.02\n0\n0.02\n==65\n0.01\n0\n-2\n-1\n0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\nX\nFigure 1-4: Comparison of analytical solution (crosses) versus NTHMP's models surface profiles (solid\nlines) during runup of a non-breaking wave of H = 0.0185 at t = [35, 45, 55, 56]. The analytical solution\ncan be found in Synolakis (1986).","36\nNational Tsunami Hazard Mitigation Program (NTHMP)\nBP1 (H=0.0185)\n0.04\nt=40\n0.02\nALASKA\nATFM\nBOSZ\nFUNWAVE\n0\nGEOCLAW\nMOST\nNEOWAVE\nSELFE\n0.08\nAnalytical data\nt =50\n0.06\n0.04\n0.02\n0\n0.08\nt\n=60\n0.06\n0.04\n0.02\n0\n-2\n-1\n0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\nX\nFigure 1-5: Comparison of analytical solution (crosses) versus NTHMP's models surface profiles (solid\nlines) during runup of a non-breaking solitary wave of H = 0.0185 at t = [40, 50, 60]. The analytical\nsolution can be found in Synolakis, (1986).","37\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nBP1 (H=0.0185)\n0.03\nLocation: X = 9.95\n0.02\n0.01\n0\nALASKA\n-0.01\nATFM\nBOSZ\nFUNWAVE\n-0.02\nGEOCLAW\nMOST\nNEOWAVE\nSELFE\nAnalytical data\n+\n0.05\nLocation: X = 0.25\n0.04\n0.03\n0.02\n0.01\n0\n-0.01\n-0.02\n120\n60\n70\n80\n90\n100\n110\n0\n10\n20\n30\n40\n50\nt\nFigure 1-6: Comparison between the analytical solution (crosses) versus NTHMP's models (solid lines)\nduring the runup of a non-breaking solitary wave of H = 0.0185 on 1:19.85 beach. The top and bottom\nplots represent comparisons at X = 9.95 and X = 0.25, respectively. The analytical solution is taken\nfrom Synolakis (1986).","38\nNational Tsunami Hazard Mitigation Program (NTHMP)\nTable 1-7: BP1: NTHMP models' errors with respect to the analytical solution for H = 0.0185. a) surface\nprofile errors at t = [35, 40, 45, 50, 55, 60, 65]. b) sea level time series errors at X = 9.95 and X = 0.25.\nRMS: Normalized root mean square deviation. MAX: Maximum amplitude or runup error.\na)\nMODEL ERROR for CASE H=0.0185\nt=35\nt 40\nt 45\nt = 50\nt 55\nt = 60\nt 65\nMean\nMODEL\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMax\nALASKA\n1%\n1%\n1%\n0%\n1%\n0%\n0%\n2%\n0%\n0%\n0%\n2%\n2%\n0%\n1%\n1%\nATFM\n1%\n1%\n1%\n0%\n1%\n1%\n0%\n1%\n0%\n1%\n0%\n2%\n2%\n0%\n1%\n1%\nBOSZ\n2%\n1%\n2%\n1%\n1%\n1%\n1%\n2%\n0%\n0%\n1%\n0%\n6%\n7%\n2%\n2%\nFUNWAVE\n3%\n1%\n2%\n2%\n2%\n3%\n1%\n4%\n1%\n2%\n0%\n2%\n4%\n2%\n2%\n2%\nGEOCLAW\n1%\n2%\n1%\n2%\n1%\n2%\n0%\n2%\n0%\n0%\n1%\n1%\n5%\n1%\n1%\n1%\nMOST\n6%\n5%\n5%\n5%\n4%\n7%\n2%\n2%\n1%\n0%\n0%\n0%\n10%\n13%\n4%\n5%\nNEOWAVE\n1%\n0%\n1%\n0%\n1%\n1%\n0%\n2%\n0%\n0%\n0%\n1%\n2%\n1%\n1%\n1%\nSELFE\n3%\n4%\n3%\n4%\n2%\n3%\n1%\n2%\n1%\n0%\n1%\n0%\n5%\n1%\n2%\n2%\nMean\n2%\n2%\n2%\n2%\n2%\n2%\n1%\n2%\n0%\n0%\n0%\n1%\n5%\n3%\nb)\nMODEL ERROR\nx = 9.95\nx 0.25\nMean\nMODEL\nRMS\nMAX\nRMS\nMAX\nRMS\nMax\nALASKA\n1%\n1%\n1%\n1%\n1%\n1%\nATFM\n2%\n2%\n2%\n1%\n2%\n2%\nBOSZ\n3%\n0%\n1%\n2%\n2%\n1%\nFUNWAVE\n3%\n0%\n2%\n0%\n3%\n0%\nGEOCLAW\n2%\n1%\n2%\n1%\n2%\n1%\nMOST\n3%\n2%\n3%\n1%\n3%\n2%\nNEOWAVE\n2%\n1%\n2%\n0%\n2%\n1%\nSELFE\n2%\n2%\n1%\n1%\n2%\n2%\nMean\n2%\n1%\n2%\n1%","MODEL BENCHMARKING WORKSHOP AND RESULTS\n39\n1.9.7.2 Single Wave on a Simple Beach - Laboratory Experiment (BP4)\nBP4 (H=0.0185)\n0.03\nt =30\n0.02\n0.01\n0\n0.04\n=40\n0.03\n0.02\nALASKA\n0.01\nATFM\nBOSZ\n0\nFUNWAVE\nGEOCLAW\n0.08\nMOST\nt==00\nNEOWAVE\n0,06\nSELFE\n0.04\nExperiment data\n+\n0.02\n0\nt=60\n0.06\n0.04\n0.02\n0\n0.02\nt =70\n0.01\n0\n-0.01\n-0.02\n-0.03\n-0.04\n6\n7\n8\n9\n10\n-2\n-1\n0\n1\n2\n3\n4\n5\nX\nFigure 1-7: Comparison of experimental data (crosses) versus NTHMP's models surface profiles (solid\nlines) during runup of a non-breaking wave (Case A, H = 0.0185) at t = [30, 40, 50, 60, 70].","40\nNational Tsunami Hazard Mitigation Program (NTHMP)\nBP4 (H=0.30)\n0.4\nt=15\n0.3\n0.2\n0.1\n0\nATFM\nBOSZ\nFUNWAVE\nNEOWAVE\nExperiment data\n+\n0.4\nt=20\n0.3\n0.2\n0.1\n0\nIII\nIIII\n0.2\nt=25\n0.1\n0\n0.4\nt=30\n0.3\n0.2\n0.1\n0\n-10\n-8\n-6\n-4\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\nX\nFigure 1-8: Comparison of experimental data (crosses) versus NTHMP's models surface profiles (solid\nlines) during runup of a breaking wave (Case B, H = 0.30) at t = [15, 20, 25, 30].","41\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nTable 1-8: BP4: NTHMP's models errors with respect to the lab experiment data. a) surface profile\nerrors for Case A, H = 0.0185. b) surface profile errors for Case B, H = 0.30. RMS: Normalized root\nmean square deviations. MAX: Maximum amplitude or relative runup error.\nMODEL ERROR for CASE H=0.0185\na)\nt 60\nt 70\nMean\nt 30\nt 40\nt 50\nRMS\nMax\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nMODEL\nRMS\nMAX\n16%\n11%\n10%\n17%\n4%\n0%\n24%\n9%\n11%\n6%\n7%\nALASKA\n10%\n10%\n9%\n7%\n5%\n4%\n2%\n5%\n8%\nATFM\n10%\n1%\n8%\n3%\n2%\n0%\n10%\n3%\n8%\n2%\n4%\n0%\n4%\n13%\n3%\n7%\nBOSZ\n8%\n5%\n6%\n5%\n14%\n12%\n6%\n4%\n3%\n4%\nFUNWAVE\n11%\n2%\n19%\n11%\n10%\n16%\n5%\n4%\n22%\n6%\n11%\n3%\n7%\nGEOCLAW\n11%\n19%\n9%\n9%\n4%\n5%\n6%\n4%\n0%\n10%\n4%\n6%\n0%\nMOST\n3%\n3%\n12%\n3%\n8%\n5%\n8%\n1%\n5%\n4%\n4%\nNEOWAVE\n10%\n8%\n2%\n4%\n3%\n14%\n2%\n10%\n3%\n9%\n1%\n5%\n3%\nSELFE\n9%\n7%\n5%\n3%\n16%\n4%\n8%\n3%\n5%\nMean\n11%\nMODEL ERROR for CASE H=0.30\nb)\nt 30\nt 15\nt = 20\nt 25\nMean\nMAX\nRMS\nMax\nMAX\nRMS\nMAX\nRMS\nMODEL\nRMS\nMAX\nRMS\nNA\nNA\nNA\nNA\nNA\nNA\nALASKA\nNA\nNA\nNA\nNA\n5%\n9%\n3%\n4%\n7%\n9%\n4%\n11%\n1%\n6%\nATFM\n3%\n4%\n5%\n9%\n7%\n14%\n7%\n13%\nBOSZ\n4%\n5%\n6%\n5%\n12%\n5%\n15%\n6%\n16%\n5%\n11%\n4%\nFUNWAVE\nNA\nNA\nNA\nNA\nNA\nNA\nGEOCLAW\nNA\nNA\nNA\nNA\nNA\nNA\nNA\nNA\nNA\nMOST\nNA\nNA\nNA\nNA\nNA\n4%\n8%\n8%\n6%\n13%\n11%\n7%\n5%\nNEOWAVE\n9%\n1%\nNA\nNA\nNA\nNA\nNA\nNA\nNA\nSELFE\nNA\nNA\nNA\n6%\n10%\n4%\n6%\nMean\n7%\n6%\n9%\n11%","42\nNational Tsunami Hazard Mitigation Program (NTHMP)\n1.9.7.3 Solitary Wave on Conical Island - Laboratory Experiment (BP6)\nBP6 (H=0.045)\n2\nGAUGE 6\n1\n0\n-1\nALASKA\n-2\nATFM\nBOSZ\nFUNWAVE\nGEOCLAW\n3\nGAUGE 9\nMOST\n2\nNEOWAVE\nSELFE\n1\nExperiment data\n+\n0\n-1\n-2\n3\nGAUGE 16\n2\n1\n0\n-1\n-2\n3\nGAUGE 22\n2\n1\n0\n-1\n-2\n25\n30\n35\n40\nt (sec)\nFigure 1-9: Sea level time series comparison between experimental data (crosses) versus NTHMP's\nmodels results (solid lines) of a solitary wave of H = 0.045 (Case A) at gauges shown in Figure 1-2.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n43\nBP6 (H=0.096)\n4\nGAUGE 6\n3\n2\n1\n0\n-1\n-2\nALASKA\n-3\nATFM\nBOSZ\nFUNWAVE\n6\nGEOCLAW\nGAUGE 9\n5\nMOST\n4\nNEOWAVE\n3\nSELFE\n2\nExperiment data\n1\n+\n0\n-1\n-2\n-3\n-4\n5\nGAUGE 16\n4\n3\n2\n1\n0\n-1\n-2\n6\nGAUGE 22\n5\n4\n3\n2\n1\n0\n-1\n-2\n-3\n25\n30\n35\n40\nt (sec)\nFigure 1-10: Sea level time series comparison between experimental data (crosses) versus NTHMP's\nmodels results (solid lines) of a solitary wave of H = 0.096 (Case B) at gauges shown in Figure 1-2.","National Tsunami Hazard Mitigation Program (NTHMP)\n44\nBP6 (H=0.181)\n8\nGAUGE 6\n6\n4\n2\n0\n-2\n-4\n8\nGAUGE 9\n6\n4\n2\n0\n-2\n-4\nALASKA\n-6\nATFM\n-8\nBOSZ\nFUNWAVE\nGEOCLAW\n8\nMOST\nGAUGE 16\nNEOWAVE\n6\nSELFE\n4\nExperiment data\n2\n0\n-2\n-4\n12\nGAUGE 22\n10\n8\n6\n4\n2\n0\n-2\n-4\n40\n35\n30\n25\nt (sec)\nFigure 1-11: Sea level time series comparison between experimental data (crosses) versus NTHMP's\nmodels results (solid lines) of a solitary wave of H = 0.181 (Case C) at gauges shown in Figure 1-2.","MODEL BENCHMARKING WORKSHOP AND RESULTS\nBP6\n5\nCASE (A) H=0.045\n4.5\n4\n3.5\n3\n2.5\n2\nALASKA\n1.5\n+\nATFM\n++\n1\nBOSZ\n0.5\nFUNWAVE\n0\nGEOCLAW\nMOST\nNEOWAVE\nSELFE\n+\nExperiment data\n10\nCASE (B) H=0.096\n9\n#\n8\n+\n7\n6\n5\n+\n4\n+\n+\n+\n3\n2\n20\nCASE (C) H=0.181\n17.5\n15\n12.5\n10\n7.5\n5\n2.5\n0\n60\n120\n180\n240\n300\n360\nDirection (°)\nFigure 1-12: Runup comparison around a conical island between experimental (crosses) versus NTH\nmodels' results (solid lines) for H = [0.045, 0.096, 0.181] (Cases A, B, and C). Briggs et al. (1995)","National Tsunami Hazard Mitigation Program (NTHMP)\n46\nTable 1-9: BP6: Sea level time series NTHMP models' errors with respect to laboratory experiment\ndata. a) Case A, H = 0.045; b) Case B, H = 0.096; and c) Case C, H = 0.181. RMS: Normalized root mean\nsquare deviation error. MAX: Maximum amplitude or runup relative error.\nSEA LEVEL MODEL ERROR for CASE (A) H=0.045\na)\nGauge #6\nGauge #9\nGauge #16\nGauge #22\nMean\nRMS\nMax\nMODEL\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\n7%\n9%\nALASKA\n6%\n0%\n6%\n6%\n10%\n15%\n7%\n16%\n8%\n4%\nATFM\n9%\n2%\n8%\n0%\n8%\n13%\n8%\n2%\n9%\n13%\nBOSZ\n9%\n12%\n9%\n4%\n8%\n14%\n8%\n23%\n10%\n10%\nFUNWAVE\n10%\n5%\n10%\n14%\n10%\n2%\n10%\n20%\n8%\n16%\nGEOCLAW\n6%\n12%\n8%\n19%\n10%\n5%\n9%\n27%\n8%\n19%\nMOST\n6%\n14%\n8%\n18%\n9%\n11%\n7%\n31%\n7%\n14%\nNEOWAVE\n6%\n14%\n7%\n11%\n9%\n15%\n6%\n15%\n7%\n8%\nSELFE\n6%\n2%\n5%\n4%\n8%\n18%\n8%\n7%\nMean\n7%\n8%\n8%\n10%\n9%\n12%\n8%\n18%\nSEA LEVEL MODEL ERROR for CASE (B) H=0.096\nb)\nGauge #6\nGauge #9\nGauge #16\nGauge #22\nMean\nMODEL\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMax\nALASKA\n8%\n6%\n9%\n1%\n6%\n9%\n5%\n21%\n7%\n9%\nATFM\n8%\n3%\n8%\n1%\n6%\n15%\n9%\n8%\n8%\n7%\nBOSZ\n8%\n12%\n7%\n13%\n6%\n1%\n8%\n19%\n7%\n11%\nFUNWAVE\n9%\n2%\n9%\n17%\n7%\n8%\n12%\n17%\n9%\n11%\nGEOCLAW\n9%\n4%\n9%\n6%\n9%\n10%\n13%\n50%\n10%\n18%\nMOST\n7%\n8%\n8%\n13%\n6%\n2%\n8%\n49%\n7%\n18%\nNEOWAVE\n6%\n15%\n7%\n15%\n5%\n0%\n10%\n26%\n7%\n14%\nSELFE\n6%\n1%\n7%\n3%\n7%\n10%\n9%\n28%\n7%\n11%\nMean\n8%\n6%\n8%\n9%\n7%\n7%\n9%\n27%\nSEA LEVEL MODEL ERROR for CASE (C) H=0.181\nc)\nGauge #6\nGauge #9\nGauge #16\nGauge #22\nMean\nMODEL\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMax\n10%\n10%\nALASKA\n13%\n16%\n13%\n2%\n10%\n1%\n5%\n20%\nATFM\n8%\n2%\n11%\n15%\n6%\n4%\n8%\n2%\n8%\n6%\nBOSZ\n7%\n7%\n11%\n18%\n7%\n5%\n8%\n3%\n8%\n8%\nFUNWAVE\n8%\n1%\n11%\n16%\n8%\n1%\n11%\n8%\n10%\n7%\nGEOCLAW\n10%\n6%\n11%\n6%\n9%\n6%\n8%\n15%\n10%\n8%\nMOST\n8%\n1%\n9%\n18%\n7%\n1%\n6%\n20%\n8%\n10%\nNEOWAVE\n5%\n2%\n8%\n16%\n6%\n4%\n10%\n26%\n7%\n12%\nSELFE\n6%\n3%\n12%\n13%\n8%\n5%\n8%\n22%\n9%\n11%\nMean\n8%\n5%\n11%\n13%\n8%\n3%\n8%\n15%","MODEL BENCHMARKING WORKSHOP AND RESULTS\n47\nTable 1-10: Runup NTHMP models' errors with respect to laboratory experiment data for Case A (H =\n0.045), Case B (H = 0.096), and Case C (H = 0.181). RMS: Normalized root mean square deviation error.\nMAX: Maximum runup relative error.\nRUNUP MODEL ERROR\nCASE (A) H=0.045\nCASE (B) H=0.096\nCASE (C) H=0.181\nMean\nMODEL\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nRMS\nMAX\nALASKA\n17%\n25%\n25%\n2%\n8%\n3%\n17%\n10%\nATFM\n13%\n2%\n22%\n8%\n15%\n3%\n17%\n4%\nBOSZ\n17%\n0%\n13%\n7%\n5%\n6%\n12%\n4%\nFUNWAVE\n17%\n2%\n16%\n10%\n10%\n3%\n14%\n5%\nGEOCLAW\n24%\n9%\n24%\n2%\n17%\n3%\n22%\n5%\nMOST\n14%\n1%\n13%\n2%\n11%\n11%\n13%\n5%\nNEOWAVE\n24%\n11%\n18%\n4%\n11%\n9%\n18%\n8%\nSELFE\n14%\n4%\n11%\n4%\n10%\n2%\n12%\n3%\nMean\n18%\n7%\n18%\n5%\n11%\n5%","National Tsunami Hazard Mitigation Program (NTHMP)\n48\n1.9.7.4 Models Comparison: Okushiri Island - Field Measurement (BP9)\nBP9\n4\nIWANAI\n3\n2\n1\n0\n-1\nALASKA\n-2\nATFM\n-3\nBOSZ\n-4\nFUNWAVE\nGEOCLAW\nMOST\nNEOWAVE\n4\nSELFE\nESASHI\n3\nField data\n2\n1\n0\n-1\n-2\n-3\n-4\n50\n60\n20\n30\n40\n0\n10\nt (min)\nFigure 1-13: Sea level time series at two tide stations (Iwanai and Esashi) along the west coast of\nHokkaido island during 1993 Okushiri tsunami. NTHMP models' results (solid lines), observed water\nlevel (dashed line). Observations courtesy of Yeh et al. (1996).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n49\nTable 1-11: BP9: NTHMP Models' relative error with respect to field measurement data, Okushiri\nTsunami, 1993. a) Models' maximum amplitude error for Iwanai and Esashi gauges. b) Models' runup\nerrors around Okushiri Island (see Figure 1-3). MAX: Maximum amplitude relative error. ERR: Runup\nrelative error.\na)\nSEALEVEL MODEL ERROR\nIWANAI\nESASHI\nMODEL\nMAX\nMAX\nMean\nALASKA\n57%\n80%\n69%\nATFM\n19%\n8%\n14%\nFUNWAVE\n27%\n10%\n19%\nGEOCLAW\n59%\n99%\n79%\nMOST\n23%\n42%\n33%\nNEOWAVE\n6%\n14%\n10%\nSELFE\n59%\n45%\n52%\nMean\n36%\n43%\nb)\nRUNUP MODEL ERROR\nALASKA\nATFM\nFUNWAVE\nGEOCLAW\nMOST\nNEOWAVE\nSELFE\nRegion\nLongitude\nLatitude\nERR\nERR\nERR\nERR\nERR\nERR\nERR\nMean\n1\n139.4292117\n42.18818149\n8%\n0%\n16%\n0%\n0%\n12%\n0%\n5%\n2\n139.4111857\n42.16276287\n5%\n5%\n8%\n0%\n9%\n21%\n11%\n8%\n3\n139.4182612\n42.13740439\n14%\n23%\n25%\n41%\n4%\n25%\n58%\n27%\n4\n139.4280358\n42.09301238\n1%\n4%\n1%\n16%\n6%\n12%\n7%\n5\n139.4262450\n42.11655479\n0%\n2%\n4%\n7%\n11%\n6%\n14%\n6%\n6\n139.4237147\n42.10041415\n0%\n0%\n0%\n0%\n13%\n30%\n0%\n6%\n7\n139.4289018\n42.07663658\n22%\n22%\n8%\n4%\n29%\n18%\n5%\n15%\n8\n139.4278534\n42.06546152\n28%\n0%\n0%\n0%\n44%\n0%\n0%\n10%\n9\n139.4515399\n42.04469655\n0%\n0%\n0%\n0%\n0%\n0%\n0%\n0%\n10\n139.4565284\n42.05169226\n0%\n16%\n37%\n0%\n0%\n0%\n0%\n8%\n11\n139.4720138\n42.05808988\n11%\n0%\n0%\n0%\n0%\n0%\n0%\n2%\n12\n139.5150461\n42.21524909\n0%\n13%\n0%\n12%\n17%\n11%\n14%\n10%\n13\n139.5545494\n42.22698164\n15%\n43%\n17%\n0%\n21%\n0%\n0%\n14%\n14\n139.4934307\n42.06450128\n24%\n71%\n16%\n161%\n82%\n95%\n71%\n74%\n15\n139.5474599\n42.18744879\n10%\n7%\n36%\n0%\n5%\n27%\n10%\n14%\n16\n139.5258982\n42.17101221\n0%\n17%\n8%\n8%\n34%\n0%\n13%\n11%\n17\n139.5625242\n42.21198369\n4%\n23%\n11%\n23%\n1%\n3%\n41%\n15%\n18\n139.5190997\n42.11305805\n10%\n27%\n16%\n54%\n53%\n59%\n20%\n34%\n19\n139.5210766\n42.15137635\n3%\n9%\n14%\n21%\n24%\n6%\n58%\n19%\nMean\n9%\n15%\n12%\n17%\n19%\n17%\n17%\n1.10\nReferences\nAbadie S, Caltagirone JP, Watremez P. 1998. Mecanisme de generation du jet secondaire\nascendant dans un deferlement plongeant. C. R. Mecanique, 326:553-559.\nAbadie S, Morichon D, Grilli S, Glockner S. 2010. A three fluid model to simulate waves\ngenerated by subaerial landslides. Coastal Engineering, 57, 9, 779-794.\nArakawa A, Lamb V. 1977. Computational design of the basic dynamical processes of the\nUCLA general circulation model. In: Methods in Computational Physics. Vol. 17.\nAcademic Press, 174-267.","National Tsunami Hazard Mitigation Program (NTHMP)\n50\nBalay S, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, Curfman McInnes L,\nSmith B, Zhang H. 2004. PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 2.1.5,\nArgonne National Laboratory.\nBerger MJ, LeVeque RJ. 1998. Adaptive mesh refinement using wave-propagation algorithms\nfor hyperbolic systems. 1998. SIAM J. Numer. Anal. 35, 2298-2316.\nBerger MJ, George DL, LeVeque RJ, Mandli KT. 2011. The GeoClaw software for depth-\naveraged flows with adaptive refinement. Adv. Water Res. 34, 1195-1206.\nBriggs MJ, Synolakis CE, Harkins GS, Green D. 1995 Laboratory experiments of tsunami runup\non a circular island. Pure Appl. Geophys., 144, 569-593.\nBurwell D, Tolkova E, Chawla A. 2007. Diffusion and Dispersion Characterization of a\nNumerical Tsunami Model. Ocean Modelling, Vol.19/1-2, pp. 10-30.\nloi:10.1016/j.ocemod.2007.05.003\nChen Q, Kirby JT, Dalrymple RA, Kennedy AB, Chawla A. 2000. Boussinesq modeling of wave\ntransformation, breaking, and runup. II: 2D. J Wtrwy Port Coast and Oc Engrg ASCE\n126(1):48-56.\nDunbar PK, Weaver CS. 2008. U.S. States and Territories National Tsunami Hazard\nAssessment: Historical Record and Sources for Waves. Prepared for the National\nTsunami Hazard Mitigation Program by NOAA and USGS. 59 pp.\nEnet F, Grilli ST. 2007. Experimental Study of Tsunami Generation by Three-dimensional Rigid\nUnderwater Landslides. Journal of Waterway Port Coastal and Ocean Engineering,\n133(6), 442-454.\nFine IV, Rabinovich AB, Kulikov EA, Thomson RE, Bornhold BD. 1998. Numerical Modelling\nof Landslide-Generated Tsunamis with Application to the Skagway Harbor Tsunami of\nNovember 3, 1994. In Proceedings, International Conference on Tsunamis (Paris, 1998)\npp. 211-223.\nFischer G. 1959. Ein numerisches verfahren zur errechnung von windstau und gezeiten in\nrandmeeren. Tellus 11, 60-76.\nFletcher C. 1991. Computational Techniques for Fluid Dynamics 1. Springer-Verlag, 401 pp.\nGallardo JM, Parés C, Castro M. 2007. On a well-balanced high-order finite volume scheme for\nshallow water equations with topography and dry areas. J. Comput. Phys. 227, 574-601.\nGeorge DL. 2006 Finite Volume Methods and Adaptive Refinement for Tsunami Propagation\nand Inundation, PhD Dissertation, University of Washington.\nGeorge DL, Leveque RJ. 2006. Finite volume methods and adaptive refinement for global\ntsunami propagation and inundation. Science of Tsunami Hazards 24(5), 319-328.\nGica E, Spillane MC, Titov VV, Chamberlin CD, Newman JC. 2008. Development of the\nforecast propagation database for NOAA's Short-Term Inundation Forecast for Tsunamis\n(SIFT). NOAA Tech. Memo. OAR PMEL-139, 89 pp.\nGoring DG. 1978. Tsunamis-the propagation of long waves onto a shelf. WM. Keck\nLaboratory of Hydraulics and Water Resources, California Institute of Technology,\nReport No. KH-R-38.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n51\nGoto C, Ogawa Y, Shuto N, Imamura F. 1997. Numerical method of tsunami simulation with the\nleap-frog scheme. Manuals and Guides 35, UNESCO: IUGG/IOC TIME Project.\nGrilli ST, Watts P. 2001. Modeling of tsunami generation by an underwater landslide in a 3-D\nnumerical wave tank. Proc. of the 11th Offshore and Polar Engrg. Conf., ISOPE01,\nStavanger, Norway, 3, 132-139.\nGrilli ST, Vogelmann S, Watts P. 2002. Development of a 3-D Numerical Wave Tank for\nmodeling tsunami generation by underwater landslides. Engineering Analysis with\nBoundary Elements, 26(4), 301-313\nGrilli ST, Watts P. 2005. Tsunami generation by submarine mass failure Part I : Modeling,\nexperimental validation, and sensitivity analysis. J. Waterway Port Coastal and Ocean\nEngng., 131(6), 283-297.\nHammack JL. 1972. Tsunamis-A model for their generation and propagation. WM. Keck\nLaboratory of Hydraulics and Water Resources, California Institute of Technology,\nReport No. KH-R-28.\nHansen W, 1956. Theorie zur errechnung des wasserstands und derstromungen in randemeeren.\nTellus 8, 287-300.\nHirt CW, Nichols BD. , 1981. Volume of Fluid (VOF) Method for the dynamics of free\nboundaries. J. of Computational Physics, 39, pp. 201-225.\nHokkaido Tsunami Survey Group, 1993. Tsunami devastates Japanese coastal regions. EOS,\nTransactions AGU 74 (37), 417-432.\nHorrillo J. 2006. Numerical Method for Tsunami Calculation Using Full Navier-Stokes\nEquations and Volume of Fluid Method. Thesis dissertation presented to the University\nof Alaska Fairbanks.\nHorrillo J, Wood AL, Williams C, Parambath A, Kim G-B. 2009. Construction of Tsunami\nInundation Maps in the Gulf of Mexico. A report to the National Tsunami Hazard\nMitigation Program, NOAA.\nImamura F. 1995. Tsunami numerical simulation with the staggered leap-frog scheme. Tech.\nrep., School Disaster Control Research Center, Tohoku University, Manuscript for\nTUNAMI code, 33 pp.\nImamura F. 1996. Review of tsunami simulation with a finite difference method. In: Yeh H, Liu\nP, Synolakis C. (Eds.), Long-Wave Runup Models. World Scientific, pp. 25-42.\nIoualalen M, Asavanant J, Kaewbanjak N, Grilli ST, Kirby JT, Watts P. 2007. Modeling the 26\nDecember 2004 Indian Ocean tsunami: Case study of impact in Thailand. J. of Geophys.\nRes., 112, C-07024.\nJiang L, LeBlond PH. (1992). The coupling of a submarine slide and the surface waves which it\ngenerates. J. Geoph. Res., 97(C8), 12731-12744.\nKennedy AB, Chen Q, Kirby JT, Dalrymple RA. 2000 Boussinesq modeling of wave\ntransformation, breaking, and run-up. I: 1D. J Wtrwy Port Coast and Oc Engrg ASCE\n126(1):39-47.","52\nNational Tsunami Hazard Mitigation Program (NTHMP)\nKirby J, Grilli S. 2011. Modeling Tsunami Inundation and Assessing Tsunami Hazards for the U.\nS. East Coast. NTHMP Semi-Annual Report March 29, 2011. Award Number:\nNA10NWS4670010.\nKirby JT, Pophet N, Shi F, Grilli ST. 2009. Basin scale tsunami propagation modeling using\nboussinesq models: Parallel implementation in spherical coordinates. In Proc. WCCE-\nECCE-TCCE Joint Conf. on Earthquake and Tsunami (Istanbul, Turkey, June 22-24),\npaper 100:(published on CD).\nKirby JT, Shi F, Harris JC, Grilli ST. 2012. Sensitivity analysis of trans-oceanic tsunami\npropagation to dispersive and Coriolis effects. Ocean Modeling, (in preparation):42 pp.\nKirby J, Wei G, Chen Q, Kennedy A, Dalrymple R. 1998. FUNWAVE 1.0, fully nonlinear\nboussinesq wave model documentation and users manual. Tech. Rep. Research Report\nNo. CACR-98-06, Center for Applied Coastal Research, University of Delaware.\nKowalik Z, Murty TS. 1993a. Numerical modeling of ocean dynamics. World Scientific Publ.,\n481 pp\nKowalik Z, Murty TS. 1993b. Numerical simulation of two-dimensional tsunami runup. Marine\nGeodesy 16, 87-100.\nKowalik Z, Whitmore PM. 1991. An investigation of two tsunamis recorded at Adak, Alaska.\nScience of Tsunami Hazards, 9, 67-83.\nKowalik Z, Knight W, Logan T, Whitmore P. 2005. Numerical modeling of the global tsunami:\nIndonesian Tsunami of 26 December 2004. Science of Tsunami Hazards, 23(1), 40-56.\nLeVeque RJ. 2002. Finite Volume Methods for Hyperbolic Problems, Cambridge University\nPress.\nLeVeque RJ, George DL, Berger MJ. 2011. Tsunami modeling with adaptively refined finite\nvolume methods. Acta Numerica 2011 211-289.\nLiang Q, Marche F. 2009. Numerical resolution of well-balanced shallow water equations with\ncomplex source terms. Advances in Water Resources 32(6):873-884.\nLiu PL-F, Woo S, Cho Y. 1998. Computer programs for tsunami propagation and inundation.\nTech. rep., Cornell University, 104 pp.\nLiu PL-F, Lynett P, Synolakis CE. (2003): Analytical solutions for forced long waves on a\nsloping beach. J. Fluid Mech., 478, 101-109.\nLubin P, Vincent S, Abadie S, Caltagirone JP. 2006. Three-dimensional Large Eddy Simulation\nof air entrainment under plunging breaking waves. Coastal Engineering, Volume 53,\nissue 8, .631-655.\nLynett P, Wu T-R., Liu PL-F. 2002. Modeling wave runup with depth-integrated equations.\nCoastal Engineering 46(2), 89-107.\nMa G, Shi F, Kirby JT. 2012. Shock-capturing non-hydrostatic model for fully dispersive surface\nwave processes. Ocean Modeling, 43-44:22-35.\nMader CL. 1988. Numerical Modeling of water waves. University of California Press, Berkeley,\nCalifornia.\nMader C, Lukas S. 1984. SWAN-A Shallow Water, Long Wave Code. Tech. Rep. HIG-84-4,\nHawaii Institute of Geophysics, University of Hawaii.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n53\nNichols BD, Hirt CW, Hotchkiss RS. 1980. SOLA-VOF: a solution algorithm for transient fluid\nflow with multiple boundaries. Los Alamos Scientific Laboratory Report, LA-8355.\nNicolsky DJ, Suleimani EN, Hansen RA. 2010. Validation and Verification of a Numerical\nModel for Tsunami Propagation and Runup. Pure Appl. Geophys. 168 (2011), 1199-\n1222.\nNwogu O. 1993. Alternative form of Boussinesq equations for near shore wave propagation. J.\nWtrwy., Port, Coast., and Oc. Engrg., ASCE, 119(6), 618-638.\nOkada Y. 1985. Surface deformation due to shear and tensile faults in a half-space. Bulletin of\nthe Seismological Society of America 75, 1135-1154.\nPeregrine D. 1967. Long waves on a beach. Journal of Fluid Mechanics 27(4), 815-827.\nPriest GR, Goldfinger C, Wang K, Witter RC, Zhang Y, Baptista AM. 2010 Confidence levels\nfor tsunami-inundation limits in northern Oregon inferred from a 10,000-year history of\ngreat earthquakes at the Cascadia subduction zone. Natural Hazards, 54(1), 27-73.\nRoeber V. 2010. Boussinesq-type model for nearshore wave processes in fringing reef\nenvironment. PhD Dissertation, University of Hawaii, Honolulu.\nShi F, Kirby JT, Harris JC, Geiman JD, Grilli ST. 2012. A highorder adaptive time-stepping\nTVD solver for Boussinesq modeling of breaking waves and coastal inundation. Ocean\nModeling, 43-44:36-51.\nStelling GS, Duinmeijer SPA, 2003. A staggered conservative scheme for every Froude number\nin rapidly varied shallow water flows. Int. J. Numer. Meth. Fluids; 43:1329-1354 (DOI:\n10.1002/fld.537)\nStelling GS, Zijlema M. 2003. An accurate and efficient finite-difference algorithm for non-\nhydrostatic free-surface flow with application to wave propagation. International Journal\nfor Numerical Methods in Fluids, 43(1), 1-23.\nSuleimani E, Hansen R, Haeussler P. 2009. Numerical study of tsunami generated by multiple\nsubmarine slope failures in Resurrection Bay, Alaska, during the M9.2 1964 earthquake.\nPure appl. geophys., V. 166, 131-152, doi: 10.1007/s00024-004-0430-31\nSwigler DT, Lynett P. (2011) Laboratory study of the three-dimensional turbulence and\nkinematic properties associated with a solitary wave traveling over an alongshore-\nvariable, shallow shelf. In review.\nSynolakis CE. 1986 The Runup of Long Waves. Ph.D. Thesis, California Institute of\nTechnology, Pasadena, California, 91125, 228 pp.\nSynolakis CE, Bernard EN, Titov VV, Kanoglu U, González FI. 2007. OAR PMEL-135\nStandards, criteria, and procedures for NOAA evaluation of tsunami numerical models.\nTechnical report, NOAA Tech. Memo. OAR PMEL-135, NOAA/Pacific Marine\nEnvironmental Laboratory, Seattle, WA.\nTakahashi T. 1996. Benchmark problem 4; the 1993 Okushiri tsunami-Data, conditions and\nphenomena. In Long-Wave Runup Models, World Scientific, 384-403.\nTappin DR, Watts P, Grilli ST. 2008. The Papua New Guinea tsunami of 17 July 1998: anatomy\nof a catastrophic event. Nat. Hazards Earth Syst. Sci. 8, 243-266.","54\nNational Tsunami Hazard Mitigation Program (NTHMP)\nThacker C-T. 1981. Some exact solutions to the nonlinear shallow-water wave equations.\nJ.Fluide Mech., vol. 107, pp.499-508, 1981.\nThomson RE, Rabinovich AB, Kulikov EA, Fine IV, Bornhold BD. 2001. On numerical\nsimulation of the landslide-generated tsunami of November 3, 1994 in Skagway Harbor,\nAlaska. In: Hebenstreit G. (Ed.), Tsunami Research at the End of a Critical Decade.\nKluwer, Dorderecht, pp. 243-282.\nTitov V, Synolakis C. 1995. Evolution and runup of breaking and nonbreaking waves using\nVTSC2. Journal of Waterway, Port, Coastal and Ocean Engineering 121 (6), 308-316.\nTitov VV, Synolakis CE, 1998. Numerical modeling of tidal wave runup. J. Waterw. Port Coast.\nOcean Eng., 124(4), 157-171.\nvan Leer B. 1977, Towards the ultimate conservative difference scheme III. Upstream-centered\nfinite-difference schemes for ideal compressible flow. J. Comp. Phys. 23 (3): 263-275\nWalters RA. 2005. A semi-implicit finite element model for non-hydrostatic (dispersive) surface\nwaves. International Journal for Numerical Methods in Fluids 49(7):721-737.\nWatts P, Grilli ST, Tappin D, Fryer GJ. 2005. Tsunami generation by submarine mass failure\nPart II: Predictive Equations and case studies. J. Waterway Port Coastal and Ocean\nEngng., 131(6), 298-310\nWei G, Kirby JT. 1995 Time-dependent numerical code for extended Boussinesq equations. J\nWtrwy Port Coast and Oc Engrg ASCE 121(5):251-261\nWei G, Kirby JT, Grilli ST, Subramanya R. 1995 A fully nonlinear Boussinesq model for free\nsurface waves. Part 1: highly nonlinear unsteady waves. J Fluid Mech 294:71-92\nWhitmore PM, Sokolowski TJ. (1996). Predicting tsunami amplitudes along the North American\ncoast from tsunamis generated in the northwest Pacific Ocean during tsunami warnings.\nScience of Tsunami Hazards, 14, 147-166.\nWitter R, Zhang Y, Wang K, Priest G, Goldfinger C, Stimely L, English J, Ferro P. 2011.\nSimulating tsunami inundation at Bandon, Coos County, Oregon, using hypothetical\nCascadia and Alaska earthquake scenarios. Special paper 43, Oregon Department of\nGeology and Mineral Industries.\nWu Y, Cheung KF. 2008. Explicit solution to the exact Riemann problem and application in\nnonlinear shallow-water equations. International Journal for Numerical Methods in\nFluids, V. 57, pp 1649-1668.\nYamazaki Y, Kowalik Z, Cheung KF. 2009. Depth-integrated, nonhydrostatic model for wave\nbreaking and run-up. Int. J. Numer. Methods Fluids, 61(5), 473-497.\nYamazaki Y, Cheung KF, Kowalik Z. 2011. Depth-integrated, non-hydrostatic model with grid\nnesting for tsunami generation, propagation, and run-up. Int. J. Numerical Methods\nFluids, 67(12), 2081-2107.\nYeh H, Liu PL-F, Briggs M, Synolakis CE. 1994. Tsunami catastrophe in Babi Island. Nature,\n372, 6503-6508.\nYeh H, Liu PL-F, Synolakis CE. 1996. Long-Wave Runup Models. World Scientific, 403 pp.\nZhang Y, Baptista AM. 2008a SELFE: A semi-implicit Eulerian-Lagrangian finite-element\nmodel for cross-scale ocean circulation. Ocean Modelling, 21(3-4), 71-96.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n55\nZhang Y, Baptista AM. 2008b An Efficient and Robust Tsunami Model on Unstructured Grids.\nPart I: Inundation Benchmarks. Pure and Applied Geophysics, 165, 2229-2248.\nZhang Y, Witter RW, Priest GP. 2011 Nonlinear Tsunami-Tide Interaction in 1964 Prince\nWilliam Sound Tsunami. Ocean Modelling (submitted).","56\nNational Tsunami Hazard Mitigation Program (NTHMP)\n2 Alaska Tsunami Model\nDmitry Nicolsky\nGeophysical Institute of the University of Alaska Fairbanks and the State of Alaska Division of\nGeological & Geophysical Surveys\n2.1\nIntroduction\nNTHMP-funded efforts in Alaska are focused on improvement of the emergency response to\nthe tsunami hazards in coastal communities. Typical tsunami-related hazards originate from\nvertical and horizontal coseismic tectonic displacements and the failure of unconsolidated\nmaterials from above and below the sea surface.\nTo help mitigate the tsunami hazards, we subject our tsunami model, developed at the\nGeophysical Institute, University of Alaska Fairbanks, to a series of benchmark problems\nproposed by Synolakis et al., (2007). The benchmark problems deal with validating and verifying\nthe ability of the model to accurately simulate a tsunami caused by vertical coseismic tectonic\ndisplacement or by the failure of unconsolidated material. There is, however, no benchmark\nproblem focused on validating the model's ability to simulate seiche waves caused by lateral\ncoseismic displacements. In certain Alaska fjords, lateral tectonic displacements might produce\na\nseiche wave comparable in size with a landslide-generated tsunami (Plafker et al., 1969).\nIn this report, we provide a brief description of three tsunami models used to predict a\npotential inundation. Each model addresses one of the above-mentioned tsunami hazards.\n2.2\nModel description\nThe suit of numerical models employed by the State of Alaska for the tsunami inundation\nmapping project comprises:\nTectonic tsunami model\nLandslide-generated tsunami model\nSeiche tsunami model\n2.2.1 Tectonic tsunami model\nThis is a numerical model that has been described and tested through a set of analytical,\nlaboratory, and field benchmark problems (Nicolsky et al., 2011). This model solves nonlinear\nshallow water equations, commonly used to predict the propagation of long waves in the ocean\nand the inundation of coastal areas (Synolakis and Bernard, 2006).\nThe water depth, n, and the horizontal water velocity, u, in the ocean are described in the\nspherical coordinates by the mass and linear momentum conservation principles:\non\n(1)","MODEL BENCHMARKING WORKSHOP AND RESULTS\n57\n(2)\n=\nat\nHere,\n=h+n\n(3)\nis the water level, h is the bathymetry, g is the acceleration of gravity, f is the Coriolis parameter,\nerand e, is the outward unit normal vector on the sphere. The system of equations (1) and (2) is\napproximated in spherical coordinates by finite differences on Arakawa C-grid (Arakawa and\nLamb, 1977). The spatial derivatives are discretized by central difference and upwind difference\nschemes (Fletcher, 1991). The friction term T is discretized by a semi-implicit scheme according\nto Goto et al. (1997). Equations (1) and (2) are solved semi-implicitly in time using a first order\nscheme (Kowalik and Murty, 1993). The finite difference scheme is coded in FORTRAN using\nthe Portable, Extensible Toolkit for Scientific computations (Balay et al., 2004) and the MPI\nstandard (Gropp et al., 1999).\nThe initial water deformation is assumed to be equal to a coseismic uplift and subsidence of\nthe sea floor. The vertical displacement can be set arbitrarily or computed by Okada (1985)\nformulae, requiring the epicenter location, area, dip, rake, strike, and amount of slip on the fault.\n2.2.2 Landslide-generated tsunami model\nTo simulate tsunamis produced by underwater slope failures, we use a numerical model of a\nviscous underwater slide with full interactions between the deforming slide and the water waves\nthat it generates. The shallow viscous slide equations are coupled with shallow water equations\n(1) and (2) by substituting equation (3) with\nwhere S is the landslide thickness. The thickness is computed according to the model initially\nproposed by Jiang and LeBlond (1992), improved by Fine et al. (1998). The full system of\nequations is provided by Suleimani et al. (2011), who successfully used it to model landslide-\ngenerated tsunami in Resurrection Bay, Alaska. The Fine model's assumptions and applicability\nto simulating underwater mudflows are discussed by Jiang and LeBlond (1992, 1994) in their\nformulation of the viscous slide model. The model uses long-wave approximation for water\nwaves and the deforming slide, which means that the wavelength is much greater than the local\nwater depth, and the slide thickness is much smaller than the characteristic length of the slide\nalong the slope (Jiang and LeBlond, 1994).\n2.2.3 Seiche tsunami model\nNicolsky and others, (2010) considered a fixed coordinate system to model runup of the\nseiche tsunami by considering motion of the reservoir by solving equations (1) and (2), while\ntaking into account that h = h(x,y,t). If there is no vertical displacement and the lateral ground\nvelocity, Ug, is known then\noh\n(4)\nTo facilitate computations, it is convenient to simulate water dynamics in the reference frame\nmoving with the land. The change of reference coordinate systems, i.e., from a fixed system to a\nmoving one, introduces new terms into the original mass and momentum conservation principles,","National Tsunami Hazard Mitigation Program (NTHMP)\n58\ni.e., equations (1) and (2), respectively. At the same time, equation (4) reduces to h = const and\nthus the final system of equations is\n(1a)\n(2a)\nat\nEquations (1a) and (2a) are discretized in Cartesian coordinates by finite differences on Arakawa\nC-grid (Arakawa and Lamb, 1977). The spatial derivatives are discretized by central difference\nand upwind difference schemes (Fletcher, 1991). The runup is modeled by the method employed\nto simulate the runup of tectonic waves (Nicolsky, 2011).\n2.3\nBenchmark results\nBP1: Solitary wave on simple beach - analytical\n2.3.1\nWe verify our numerical method by comparing numerical and analytical solutions that\ndescribe 1-D solitary wave runup. The analytical solution to a specific solitary wave runup on the\nsloping beach was derived by Synolakis (1986). In this problem, the wave of height H is initially\ncentered at distance L from the beach toe and is schematically shown in Figure 2-1.\nR\nH\nd\nL\nX0\nFigure 2-1: Non-scaled sketch of a canonical beach with a wave climbing up.\nThe value of L = arccosh (V20)/y is the half-length of the solitary wave, and the initial depth\nprofile is given by\nwhere X1 = X + L, and y=V3H/4d = The initial wave-particle velocity in the computer\nexperiments is set, following Titov and Synolakis (1995), as:\n=\nFirst, we check the ability of the method to model runup on a beach by simulating the runup\nof a solitary wave when d=1, = 100, and 500 m, and then comparing the numerical and analytical\nsolutions. In these numerical experiments, the 1-D domain, with total length 400d, is discretized\nwith spacing Ax = d/20. The computational time step At = 10-3. vd/g satisfies the Courant-\nFriedrichs-Levy stability criterion (Courant et al., 1928). The results suggest that the computed\nnondimensional variables such as n/d, v/Vgd do not depend on the value of d, and further that the\nnumerical predictions are in good agreement with analytical solutions for H/d = 0.019. We","MODEL BENCHMARKING WORKSHOP AND RESULTS\n59\ndiscuss comparison of numerical and analytical solutions for H/d = 12 0.019 in greater detail\nlater in this section. For error analysis, the water mass before and after wave reflection from the\nbeach is calculated, finding a total mass decrease of less than 0.01% in each case. This negligibly\nsmall error in the mass conservation is well within established criteria (Synolakis et al., 2007).\n0.10\n0.10\nAnalytical solution\n1/2\nt=40(d/g)¹\nNumerical solution, Ax=d/20\n0.08\nt=55(d/g)1(2)\nNumerical solution, Ax=d/200\nt=70(d/g)1(2)\n0.06\n0.08\n0.04\n0.02\n0.00\n0.06\n-0.02\nHollow Analytical solution\nSolid Numerical solution, Ax=d/20\n-0.04\n0\n5\n10\n15\n20\n-2.0\n-1.5\n-1.0\n-0.5\n0.0\nx/d\nx/d\nFigure 2-2: Left plot: comparison between the analytically and numerically computed solutions\nsimulating runup of the non-breaking wave in the case of H/d = 0.019 on the 1:19.85 beach. Right plot:\nan enlarged version of the left plot within the rectangle region. Two numerical solutions computed on\ngrids with Ax = d/20 and Ax = d/200 are shown at t = 55vd/g. The numerical solution is shown to be\nconverging to the analytical one as the spatial discretization is refined. The analytical solution is\naccording to Synolakis (1986).\nA focus in developing a tsunami modeling algorithm is to simulate extreme positions of the\nshoreline-the maximum runup and rundown. Figure 2-2 shows computed water surface profiles\nat the maximum runup and rundown of a solitary wave in the case of H/d = 0.019. The maximum\nrunup in the numerical simulation occurs at t ~55vd/g and this solution has a 15% error with\nrespect to the derived analytical solution. After refining the computational grid from Ax = d/20 to\nAx = d/200, the analytically and numerically computed maximum runup values differ by less\nthan 2%, which is within the recommended criteria (Synolakis et al., 2008). We additionally\nchecked convergence of numerically computed maximum rundown to its analytical prediction.\nFor the computational grid with Ax = d/20, the difference between the numerical and analytical\nrundown values is at most 16%. After the grid refinement to Ax = d/200, the difference is less\nthan 3%. The results show that the numerical solution converges to the analytical prediction at\nthe extreme locations of the shoreline, and that the recommended 5% error in numerical solution\nis achieved.","National Tsunami Hazard Mitigation Program (NTHMP)\n60\n0.06\nx/d=0.25\nAnalytical solution\nNumerical solution\n0.04\n0.02\n0.00\n80\n100\n0\n20\n40\n60\n1/2\nt(g/d)1\nx/d=9.95\n0.02\n0.01\n0.00\nAnalytical solution\n-0.01\nNumerical solution\n0\n20\n40\n60\n80\n100\nt(g/d)1/2\n1/2\nFigure 2-3: Comparison between the analytical solution (hollow symbols) and the finite difference\nsolution (filled symbols) during the runup of the non-breaking solitary wave with H/d = 0.019 on\n1:19.85 beach. The top and bottom plots represent comparisons at X = 0.25d and X = 9.95d,\nrespectively. The analytical solution is according to Synolakis (1986).\nFigure 2-3 shows numerically and analytically computed water level dynamics at locations\nx/d = 0.25 (near the initial shoreline) and x/d = 9.95 (between the beach toe and initial wave\ncrest) during propagation and reflection in the case H/d = 0.019. During rundown, both\nnumerical and analytical solutions show that water retreats from t = 67vd/g to t = 82vd/g, and the\npoint x/d = 0.25 temporally becomes dry, while the point x/d = 9.95 remains wet throughout the\nentire length of the computer experiment. Comparison of the analytical and numerical solutions\nat these two points reveals that the computational error is typically less that 2% for Ax = d/20,\nand that the agreement between the two solutions is quite good even on the coarse grid.\n2.3.2 BP2. Solitary wave on composite beach - analytical\nTypically, a real-life beach has an irregular bathymetry, which is much more complicated\nthan the one described in the previous benchmark problem. One of the simplest approximations\nto the irregular bathymetry is obtained by utilizing piece-wise linear functions. Kânoglu and\nSynolakis (1998) developed an exact analytical solution to the linearized shallow water equations\n(1-2) in order to predict propagation and runup of wave over the piece-wise linear beach. In this\nbenchmark, we compare our finite difference solution to the analytical solution (Kânoglu and\nSynolakis, 1998) in the case of waves propagating over a composite beach.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n61\nA composite beach simulating the geometrical dimensions of Revere Beach, Massachusetts,\nwas built in a U.S. Army Corps of Engineers, Coastal Engineering Research Center in\nVicksburg, Mississippi. The beach consists of three piece-wise linear segments and a vertical\nwall, shown in Figure 2-4.\nX=X, G4\nG5\nG6\nG7\nG8\nG9 G10\n0.04\nGauge\n0.02\n1/13\nd=0.218m\n1/150\n1/53\n0.9m\n5.04m\n4.36m\n2.93m\n0.00\n5\n10\n15\n20\n25\nt (g/d)\nFigure 2-4: Left, non-scaled sketch of the composite beach modeling Revere Beach, Massachusetts.\nVertical lines mark the locations of gauges measuring the water level in laboratory experiments. Right,\nan incident wave recorded by Gauge 4. This record was used to set the water height h(Xb; t) at the\ninflow boundary condition.\nThe laboratory equipment and the beach profile are described in Yeh, et al. (1996) and\nKânoglu and Synolakis (1998). The slopes of the segments, starting from the wall, are 1 = 13,\n1 = 150, and 1 = 53, respectively. At the beginning and in the middle of each sloping segment a\nwater gauge measuring time dynamics of the water surface height was installed\nWe compare our numerical solution to the analytical solution modeling propagation of\nwaves over the composite beach. We note that in the numerical experiments, we neglect all non-\nlinearities in (1-2) and solve the linear shallow water equations. In Figure 2-5, we show the\ncomparison between the analytical solution (5) and the obtained numerical solution at the\nlocations where the gauges were installed. The numerical solution was computed using the grid\nsize Ax=d/20, where the quantity d stands for the still water equal to 0:218 m at Gauge 4.\nAnalyzing the results, we observe that the numerical solution closely matches the analytical one,\nand the difference between them is typically less than 5%.","National Tsunami Hazard Mitigation Program (NTHMP)\n62\nCase A\nCase C\n10\n20\nE 0.2\nE\nObservations\nGauge4\nLSWE Numerical solution\nAnalytical solution\nObservations\nGauge4\nLSWE Numerical solution\nAnalytical solution\nnot\n10\n01\n0\n-\n25\n5\n10\n15\n20\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\nE\n02\nGauge5\nGauge5\nand\n10\n01\n0\n0\n5\n10\n15\n20\n25\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\nE\n02\nGauge6\nGauge6\n10\n0\n0\n0\n10\n15\n20\n25\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\nE\n0.2\nGauge7\nGauge7\n10\n0.1\n0\n0\n5\n10\n15\n20\n25\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\n0.2\nGauge8\nGauge8\nAnd\n10\n01\n0\nve\n5\n10\n15\n20\n25\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\nE\n0.2\nGauge9\nGauge9\nMAA\n10\n0.1\n0\n0\n5\n10\n15\n20\n25\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\nE\n02\nGauge 10\nMAA\nGauge1 10\n10\n0\n0\n0\n5\n10\n15\n20\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n0.03\n0.4\nWall\nWall\n0.02\n02\n0.01\n0\n0\nwe\n5\n10\n15\n20\n25\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\nTime\nTime\nFigure 2-5: Comparison between the analytically and numerically computed solutions at the gauge\nlocations shown in Figure 2-4. Left plot: Case A, Right plot Case C.\n2.3.3 BP3: Subaerial landslide on simple beach - analytical\nLiu et al. (2003) considered tsunami generation by a moving slide on a uniformly sloping\nbeach, using the forced linear shallow water-wave equation as in Tuck and Hwang (1972);\n02 E at2\nx0,\nwhere the quantities and specify the beach slope. Following Liu et al. (2003), Synolakis et al.\n(2007) consider a translating Gaussian shaped mass, described by s(x,t) = exp(-(-t)2) with 52 2 =\n4ux/tan B. The analytical solution for this translating Gaussian shape is given by Liu et al. (2003)\nThe goal of this benchmark problem is to compare the numerical predictions with the\nanalytical solution. Figure 2-6 presents a comparison between the analytical and numerical\nsolutions to equations (1) and (2). Note that the analytical solutions and the numerical solutions\nto the linear shallow water equations are derived using two different boundary conditions.\nTherefore, the discrepancy between these solutions is rather large.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n63\nu=0.1, B=5\nTime 0.50 seconds.\nTime = 1.00 seconds.\n0.5\n0.8\n0.4\n0.6\n0.3\n0.4\n0.2\n0.2\n0.1\n0\n0\nNLSWE, Numerical solution\n-0.1\nLSWE, Numerical solution\n-0.2\nFBnd LSWE, Analytical\n-0.2\nSlide profile\n-0,4\n0\n0.5\n1\n1.5\n0\n0.5\n1\n1.5\n2\n2.5\n3\nTime = 2.50 seconds.\nTime = 4.50 seconds.\n2\n1\n1.5\n1\n0.5\n0.5\n0\n0\n-0.5\n-0.5\n-1\n0\n1\n2\n3\n4\n5\n0\n2\n4\n6\n8\n10\nFigure 2-6: Comparison between the analytically and numerically computed solutions at several\nmoments of time. LSWE and NLSWE stand for the numerical solutions computed with linear and non-\nlinear assumptions, respectively. The analytical solution is according to Liu et al. (2003).\n2.3.4 BP4: Solitary wave on simple beach - laboratory\nMore than 40 laboratory experiments were conducted in the wave tank by Synolakis (1986).\nIn this benchmark problem, we perform numerical modeling of the water dynamics observed\nduring these experiments. In the computer experiment, we assume that the wave tank is 400d in\nlength and discretized by a uniformly spaced grid with Ax = d/200. To model common\ngeophysical conditions, we assume that d = 500 m, although scalability shows that appropriately\nscaled results do not depend on d. Additionally, we assume that there is no bottom friction, i.e.,\nV = 0. We analyze the effects of bottom friction on water dynamics later in this sub-section.\nIn the first series of laboratory experiments, the runup of a non-breaking solitary wave with\nH/d = 0.019 is studied. We plot laboratory measured water level by black rectangles in Figure\n2-7. In the same Figure, the numerical and analytical solutions are plotted by lines with solid and\nhollow triangles, respectively. Agreement between analytical and numerical computed solutions\nis more than sufficient for all snapshots; the discrepancy between the solutions is much smaller\nthan the discrepancy between any one of the solutions and the laboratory data.","National Tsunami Hazard Mitigation Program (NTHMP)\n64\n0.04\nt=30(d/g)1/2\n1/2\n0.02\n0.00\n0\n5\n10\n15\n20\n0.04\nt=40(d/g)1/2\n1/2\n0.02\n0.00\n0\n5\n10\n15\n20\n0.08\nt=50(d/g)1/2,\n0.04\n0.00\n0\n5\n10\n15\n20\n0.08\nt=60(d/g)1/2.\n0.04\n0.00\n0\n5\n10\n15\n20\n0.04\nt=70(d/g)1\"\n1/2\n0.00\nMeasured data\nNumerical solution\n-0.04\nAnalytical solution\n0\n5\n10\n15\n20\nx/d\nFigure 2-7: Comparison of observed and simulated water profiles during runup of a non-breaking\nwave in the case of H/d = 0.019. Observations are shown by dots The analytical predictions and\nnumerical calculations are marked by hollow and filled symbols, respectively. The measurements are\nprovided courtesy of Synolakis (1986).\nThe computed solutions have slightly higher runup than observations, and the computed\nmaximum runup also exceeds the physical measurements, visible in Figure 2-8 where we show\nthe numerically modeled and observed waterfront path Xw(t). Here, the measured data are plotted\nby rectangles while the computed path is plotted by a line with hollow triangles. For a detailed\nanalysis of the discrepancy between the analytical solution and laboratory results, which we\nprescribe to the zero friction assumption in the computer modeling, we refer interested readers to\nSynolakis (1986).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n65\n100\n75\nMeasured\nNumerical solution\n50\n25\n0\n0\n4\n8\n12\n16\n20\n/d\nX\nFigure 2-8: Laboratory measured and simulated waterfront path Xw of a solitary wave running up on a\ncanonical beach. Measurements are represented by squares and numerical simulations by a line. The\nmeasurements are provided courtesy of Synolakis (1986).\nIn the second series of laboratory experiments, a solitary wave with the initial amplitude\nH/d = 0.04 propagates and inundates the sloping beach. We display laboratory data by black\nrectangles and the results of numerical modeling by lines with solid triangles in Figure 2-9.\nAccording to analytical predictions derived using the zero bottom friction assumption, the\nsolitary wave breaks only if its initial height satisfies H/d > 0.029 (Synolakis, 1987). The ratio of\nH/d = 0.04 for which the laboratory data are collected satisfies this condition, SO a breaking wave\nis expected. Laboratory experiments, however, show that this wave does not break. This lack of\nbreaking in the laboratory experiment is explained by bottom friction and dispersion effects on\nwave dynamics. Further, the numerically simulated wave also fails to break, but is on the verge\nof breaking at t = 38vd/g during runup and at t = 62vd/g during rundown. This behavior in the\nnumerical solution is explained by numerical dispersion and dissipation, introduced by the finite\ndifference discretization of the partial derivatives. The slight numerical dissipation brings\nstability into the calculations and produces computational results that are in good agreement with\nlaboratory measurements.","National Tsunami Hazard Mitigation Program (NTHMP)\n66\n0.15\nt=32(d/g)1/2.\n0.10\n0.05\n0.00\n-0.05\n-5\n0\n5\n10\n15\n20\n0.15\nt=38(d/g)1/2.\n0.10\n0.05\n0.00\n-0.05\n-5\n0\n5\n10\n15\n20\n0.20\nt=44(d/g)1/2.\n0.10\n0.00\n-5\n0\n5\n10\n15\n20\n0.20\nt=50(d/g)1/2-\n0.10\n0.00\n-5\n0\n5\n10\n15\n20\nt=56(d/g)1/2\n1/2\n0.20\n0.10\n0.00\n-5\n0\n5\n10\n15\n20\n0.15\nt=62(d/g)1/2\nMeasured data\n0.10\nNumerical solution\n0.05\n0.00\n-0.05\n-5\n0\n5\n10\n15\n20\nx/d\nFigure 2-9: Comparison of measured and simulated water profiles during runup of a non-breaking\nwave in the case of H/d = 0.04. Observations are shown by squares. The analytical predictions and\nnumerical calculations are marked by hollow and filled symbols, respectively. The measurements are\nprovided courtesy of Synolakis (1986).\nFinally, in the third series of laboratory experiments, the runup in the H/d = 0.3 case is\nstudied. Both in computer and in laboratory experiments, the wave severely breaks. The leading\nfront of the solitary wave steepens and becomes singular shortly after the beginning of\ncomputations. The numerical singularity propagates towards the beach until it meets the","MODEL BENCHMARKING WORKSHOP AND RESULTS\n67\nshoreline where the singularity dissipates. Figure 2-10 shows our numerical solution, plotted by a\nline with solid triangles. The existence of strong wave breaking prevents a good agreement of\nour solution with the laboratory measurements. We observe that between moments t = 15/d/g\nand t = 20/d/g, the computed wave propagates faster than the measured wave, because the\nnumerical solution is computed using the primitive shallow water approximation (2-3) where\ndispersive terms are neglected. Inclusion of the wave dispersion leads to Boussinesq-type\nequations.\nIn all previous computations, for both breaking and non-breaking waves, the computed\nmaximum runup, denoted by R, is found to be higher than its laboratory measured value. One\npossible explanation for this discrepancy is the assumption of zero bottom friction in the model.\nIn the following sensitivity study, we examine whether the bottom friction can effectively\nparameterize wave breaking and eddy viscosity to accurately predict the maximum runup height\nR both for breaking and non-breaking waves. We begin with a discussion of non-breaking waves.\n0.50\n1/2\nt=15(d/g)\n0.25\n0.00\n-10\n-5\n0\n5\n10\n15\n20\nt=20(d/g)\n0.4\n0.2\n0.0\n-10\n-5\n0\n5\n10\n15\n20\n1/2\nt=25(d/g)\n0.4\nSolution of Zelt (1991)\n0.2\n0.0\n-10\n0\n10\n20\n0.5\nt=30(d/g)1/2\nMeasured data\nNumerical solution\n0.0\n-10\n-5\n0\n5\n10\n15\n20\nx/d\nFigure 2-10: Comparison of measured and simulated water profiles during runup of a non-breaking\nwave in the case of H/d = 0.3. Observations are shown by squares. The analytical predictions and\nnumerical calculations are marked by hollow and filled symbols, respectively. The measurements are\nprovided courtesy of Synolakis (1986).\nIn several series of computer experiments, we model inundation of the sloping beach by\nwaves with different H/d ratios. In each series, the bottom water friction is parameterized by the\nManning friction coefficient V, a certain fixed number. In Figure 2-11, we plot the computed\nmaximum runup R Id versus H/d for V = 0, V = 0.02, and V = 0.04. In the same plot, we also","National Tsunami Hazard Mitigation Program (NTHMP)\n68\ndisplay laboratory measurements (Synolakis, 1987). We observe that for small non-breaking\nwaves with H/d < 0.01, numerically simulated maximum runup heights do not depend on V and\nare in good agreement with laboratory data. For intermediate non-breaking waves\n0.01 0.03, the maximum runup height strongly depends on V because a wave\nbecomes a thin layer of liquid traveling up the slope after breaking, and friction is inversely\nproportionally to the water depth. Analyzing our computer experiments, we conclude that the\nmeasured runup height can be well approximated for the case in which the Manning friction\ncoefficient V = 0.03.\n1\n0.1\nMeasurements\nCalculations, No friction\nCalculations, v=0.02\nCalculations, v=0.04\n0.01\n0.1\n1\nH/d\nFigure 2-11: Non-dimensional maximum runup of solitary waves on the 1:19.85 sloping beach versus\nthe height of the initial wave. The measured runup values (Synolakis, 1986) are marked by dots. The\ndashed line represents maximum runup values computed without an effect of bottom friction, i.e.,\nv=0. The solid lines represent maximum runup values computed with the effect of bottom friction,\ni.e., V = 0.02 and V = 0.04. The measurements are provided courtesy of Synolakis (1986).\n2.3.5 BP5: Solitary wave on composite beach - laboratory\nIn a series of laboratory experiments, solitary waves of various heights H were generated in\nthe tank by paddle motion. Their water surface height dynamics were recorded by ten\ncapacitance gauges. Additionally, the maximum runup on the vertical wall was visually\nmeasured. Records of propagation of solitary waves with various ratios of Hld, where d is the\nstill water depth, and values of the maximum runup can be found in Briggs et al. (1995). In Yeh\net al. (1996) computational results from several numerical models are compared to certain","MODEL BENCHMARKING WORKSHOP AND RESULTS\n69\nlaboratory records, namely cases A, B, and C related to H/d = 0.038, H/d = 0.259, and H/d =\n0.681, respectively.\nTo simulate the laboratory recorded wave dynamics, we discretize the computational\ndomain shown in Figure 2-4 into grid cell of the length Ax-d/20 and use the time step At =\n0.05vd/g. To model the incident solitary wave, we specify the water surface height at the inflow\nboundary located at Gauge 4. Once the wave was generated, the boundary condition is modified\nto simulate the non-reflective behavior.\nIn Figure 2-12, we display the laboratory measured water level in Cases A and C by a line\nwith hollow circles. In Case A, the generated wave has the ratio H/d = 0.038 and it does not\nbreak in either the laboratory or computer experiments. The computational results in the case of\nnon-linear shallow water equations are shown by lines with triangles in Figure 2-12. The\ncomparison of the numerical results in the case of linear shallow water equations is provided in\nFigure 2-5.\nWe note that the wave simulated by the linearized equations does not break while shoring,\nwhereas the solution to non-linear equations indicates that the wave front sharpens and finally\nthe wave breaks after reflection from the vertical wall. We conclude that the linear solution\nreproduces the observations satisfactorily, while the non-linear solution clearly indicates\nshortcomings of the primitive shallow water approximation, due to negligence of the dispersion.\nIn Case C, the wave front of the numerical non-linear solution sharpens and approaches the\nlimiting case of a vertical leading edge as the wave approaches the vertical wall. There is also a\ntiming error with the predicted wave traveling =5% faster than the observed wave. The\nsharpening of the wave front and the timing error are probably due to the dispersive terms being\nneglected in the primitive shallow water approximation.","National Tsunami Hazard Mitigation Program (NTHMP)\n70\nCase\nCase C\n10\n20\nE 02\nNL SWE Numerical solution\nAnalytical solution\nObservations\nGauge4\nNLSWE Numerical solution\nAnalytical solution\nObservations\nGauge4\n10\n01\n0\n0\n10\n15\n20\n25\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\nE 0.2\nGauge5\nGauge5\n10\n0.1\n0\n0\n10\n15\n20\n25\n2\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\n0.2\nGauge6\nGauge6\n10\n0.1\n0\n0\n5\n10\n15\n20\n25\n2\n6\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\n0.2\nGauge7\nGauge7\n10\n0\n0\n0\n10\n2\n8\n10\n12\n14\n16\n18\n20\n10\n20\n0.2\nGauge8\nGauge8\nMAS\n10\n0\n0\n0\n10\n15\n20\n2\n4\n6\nB\n10\n12\n14\n16\n18\n20\n22\n10\n20\n0.2\nGauge9\nGauge9\nAA\n10\n0\n0\n0\n15\n20\n2\n8\n10\n12\n14\n16\n18\n20\n22\n10\n20\n0.2\nGauge 10\nGauge 10\nAM\n10\n0\n0\n20\n2\n10\n12\n14\n16\n18\n20\n22\n0.03\n04\nWall\nWall\n0 02\n0.2\n0.01\n0\n0\n-\n10\n12\n5\n10\n15\n20\n2\n4\n6\n8\n14\n16\n18\n20\n22\nTime\nTime\n$\nFigure 2-12: Comparison between the numerically computed solutions and laboratory measurements\nat the gauge locations shown in Figure 2-4. Left plot: Case A, Right plot Case C.\nIn Table 2-1, we list values of the maximal computed runup at the vertical wall versus the\nobserved values. For case A, the predicted runup is under-predicted by 10%. For cases B and C,\nthe error is significantly larger because when the breaking wave collapses into the vertical wall,\nthe shallow water theory is no longer applicable and more complicated models are necessary to\nmodel the observed runup more precisely.\nTable 2-1: Comparison between the numerically computed and measured runup at the vertical wall.\nMeasured\nComputed\nR, cm\nR/d\nR\nR/d\nCase A\n2.74\n0.13\n2.18\n0.1\nCase B\n45.72\n2.10\n8.81\n0.4\nCase C\n27.43\n1.26\n12.91\n0.6\n2.3.6 BP6: Solitary wave on a conical island - laboratory\nWe simulate propagation and runup of a solitary wave on a conically shaped island. To\nvalidate our numerical method, we use a laboratory experiment focused on studying inundation","71\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nof the Babi island by the 12 December 1992 tsunami. The tsunami attacked the conically shaped\nBabi island from the north, but extremely high inundation was observed in the south. A model of\nthe conical island was constructed in a wave tank at the US Army Engineer Waterways\nExperimental Station (Briggs et al., 1995). Figure 2-13 shows a sketch of the conical island and\nthe location of several sensors that recorded the water level dynamics. Along one side of the\ntank, a wave generator directed plane solitary waves toward the island. Interested readers are\nreferred to Liu et al. (1995) where the laboratory experiments and measured data are described in\ndetail.\nC\nB\n22\n16\n6\ni\n2\n3\n4\nD\nA\nFigure 2-13: Top-down non-scaled sketch of a conically-shaped island. The solid circles represent\nexterior and interior bases of the island. The dotted line shows an initial location of the shoreline. The\ndash-dotted line shows the extent of the high resolution computational grid. The dots mark gauge\nlocations where laboratory measured water level is compared to numerical calculations. The in-flow\nboundary condition is simulated on the segment AD, while the open boundary condition is modeled\non the segments AB, BC, and CD. Location of gauges 1, 2, 3, and 4 is schematic, while locations of the\nrest of gauges are precise.\nExperiments with different wave heights, H, were conducted in the wave tank. A goal of the\nexperiments was to demonstrate that after the tsunami hits the island, it splits into two waves\ntraveling with their crests perpendicular to the shoreline. Once these waves meet behind the\nisland, they collide and produce a local extremum in runup. In this work, we model the highest\ngenerated wave. This is a formidable test of the numerical algorithm because the modeled wave\nis steeper than most realistic tsunamis (Titov and Synolakis, 1998).\nIn our computer experiment, we discretize the entire basin with a coarse resolution grid with\nspacing Ax/d = Ay/d = 5/32, where the undisturbed water depth d = 0.32 m. However, in the\nvicinity of the conical island, we also construct a fine resolution grid with a cell size Ax/d =\nAy/d = 1/32 to include the entire island and its exterior base. To couple these two grids, we use\nan algorithm described by Kowalik and Murty (1993) and Goto et al. (1997) in which the water\nflux from the coarse resolution grid is passed to the fine resolution grid, and the water level n\nfrom the fine resolution grid is returned back to the coarse grid at each time step. To simulate the\nincident wave, the water level at the boundary AD is set according to measurements at gauges 1 -\n4, instead of modeling the action of generator paddles. On all other sides of the computational\ndomain, we define open boundary conditions. The choice of boundary conditions in the model\ndiffers from conditions imposed by horsehair-type absorbers along the tank wall, SO the","72\nNational Tsunami Hazard Mitigation Program (NTHMP)\ncomputed water dynamics cannot model laboratory data after the time T when wave crest reaches\nBC. Therefore, the computer simulation is terminated at T.\nOn the left plots in Figure 2-14 and Figure 2-15, we compare computed and measured water\nlevel dynamics in the case of H/d = 0.05 and H/d = 0.2, respectively. The computations are\nterminated after the first reflection of the wave from the island. The simulated wave breaks and\nsteepens faster than in laboratory measurements, a well-known effect of the shallow water\napproximation in which dispersive terms are neglected. Despite the extensive wave breaking, the\ncomputed runup is in good agreement with laboratory data, as shown on the right plots in Figure\n2-14 and Figure 2-15. The error between the measured and simulated runup everywhere around\nthe island, except the lee side, is within 10%, below suggested errors (Synolakis et al., 2008).\nThe simulated runup does not match the measured runup at the lee side, due to low order wave\ntheory used to simulate the vertical velocity in the computer experiment. Still, the error between\nthe measured and simulated runup at the lee side of the island is less than 20% and is within\nacceptable criteria (Synolakis et al., 2007).\n0.04\nMeasured\nComputed\nGage 9\n0.02\n90\n3\n0.00\n120\n60\nCase A\n-0.02\n2\n30\n40\n50\n150\n30\nMeasured\nComputed\n0.02\nGage 16\n1\n0.01\nE\n0.00\n0\n180\n0\n-0.01\n30\n40\n50\n1\n0.02\nMeasured\nComputed\nGage 22\n210\n0.01\n330\n2\n0.00\n-0.01\n240\n300\n3\n30\n40\n50\n270\nMeasured run-up\nComputed run-up\nInitial shoreline\nCase A\nTime,\nFigure 2-14: Left plot: comparison between the computed and measured water level at gauges shown\nin Figure 2-13 for an incident solitary wave in the case of H/d = 0.05. Right plot: comparison between\ncomputed and measured inundation zones. Top view of the island, with the lee side at 90°. The\ndotted line represents the initial shoreline. The measurements are provided courtesy of Briggs et al.\n(1995).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n73\n0.10\nMeasured\nComputed\n90\nGage 9\nCase C\n0.05\n3\n120\n60\n0.00\n2\n-0.05\n150\n30\n26\n28\n30\n32\n34\n36\n38\n40\n0.10\n1\nMeasured\nComputed\nGage 16\n0.05\n0\n180\n0\n0.00\n-0.05\n26\n28\n30\n32\n34\n36\n38\n40\n1\n0.10\nMeasured\nComputed\n330\nGage 22\n210\n0.05\n2\n0.00\n240\n300\n-0.05\n270\n26\n28\n30\n32\n34\n36\n38\n40\nMeasured run-up\nComputed run-up\nInitial shoreline\nTime, S\nCase C\nFigure 2-15: Left plot: comparison between the computed and measured water level at gauges shown\nin Figure 2-13 for an incident solitary wave in the case of H/d = 0.2. Right plot: comparison between\ncomputed and measured inundation zones. Top view of the island, with the lee side at 90°. The\ndotted line represents the initial shoreline. The measurements are provided courtesy of Briggs et al.\n(1995).\n2.3.7 BP7: Tsunami runup onto a complex three-dimensional model of the Monai Valley\nbeach - laboratory\nA laboratory experiment, using a large-scale tank at the Central Research Institute for\nElectric Power Industry, was focused on modeling runup of a long wave on a complex beach\nnear the village of Monai (Liu et al., 2007). The beach in the laboratory wave tank was a 1:400\nscale model of the bathymetry and topography around a very narrow gully, where extreme runup\nwas measured. The incoming wave in the experiment was created by wave paddles located away\nfrom the shoreline, and the induced water level dynamics were recorded at several locations by\ngauges. Figure 2-16 shows a snapshot of the simulated water level and the relative location of the\ngauges with respect to the shoreline.","74\nNational Tsunami Hazard Mitigation Program (NTHMP)\nComputational time = 12.0 seconds\nCh 9\nCh 7\nCh 5\n12\n15\n11\n05\n0\n05\n11\n15\n3\n2.5\n2\n1.5\n1\n0.5\n1\ny m\n0\n0\nx\nFigure 2-16: The 3-D view of the computational domain and numerical solution at 12 seconds.\nLocations of gauges, at which the modeled and measured water level dynamics are compared, are\nshown by arrows. Abbreviations Ch5, Ch7, and Ch9 stand for Channel 5, 7, and 9, respectively. The\ninlet boundary is modeled at X = 0. At y =0 and y = 3.4, the reflective boundary conditions are set.\nThe computational domain represents a 5.5 by 3.4 m portion of the wave tank near the shore\nand is divided into 0.014 X 0.014 m grid cells. The incident wave is prescribed at X = 0 for the\nfirst 22.5 seconds, after which a non-reflective boundary condition is set at x = 0. The boundary\nconditions along segments y = 0, y = 3.4, and x = 5.5 are set to be totally reflective. To model the\nbottom friction, we select V = 0.01, which is the closest value of the Manning's coefficient for the\nsmooth bottom material of the wave tank. The time step is set to 5.10-4 seconds to satisfy the\nstability condition.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n75\n0.04\nChannel 5\nMeasured\nComputed\n0.02\n0.00\n-0.02\n0\n5\n10\n15\n20\n25\n0.04\nMeasured\nComputed\nChannel 7\n0.02\n0.00\n0\n5\n10\n15\n20\n25\n0.04\nMeasured\nComputed\nChannel 9\n0.02\n0.00\n0\n5\n10\n15\n20\n25\nTime, S\nFigure 2-17: Comparison of the computed water height with the laboratory measurements at water\ngauges Ch5, Ch7, , and Ch9. The measurements are provided courtesy of the Third International\nWorkshop on Long-Wave Runup Models (Liu et al., 2007).\nFigure 2-17 shows plots of the computed and measured water surface dynamics by lines\nmarked with triangles and rectangles, respectively. The water level dynamics are shown at\nchannels 5, 7, and 9 for the first 25 seconds, during which the maximum runup occurs. As noted\nby (Zhang and Baptista, 2008), the observed water elevation for the first 10 seconds cannot be\naccurately modeled due to the existence of initial water disturbances in the wave tank. In the\ncomputer experiment, the positive wave arrives at the gauges less than 0.3 seconds after the\nmeasured wave. Also, the maximum computed water level at each gauge is less than the\nmeasurements by less than 5%. Therefore, we conclude that despite minor inconsistencies, the\nnumerical solution matches well with observations at each gauge.","National Tsunami Hazard Mitigation Program (NTHMP)\n76\nFigure 2-18: Left side: frames 10, 25, 40, 55, and 70 from the overhead movie of the laboratory\nexperiment. The time interval between frames is 0.5 seconds. The dashed yellow line shows the\ninstantaneous location of the shoreline. Right side: snapshots of the numerical solution at the time\nintervals corresponding to the movie frames. The blue shaded area corresponds to the water domain\nand is considered to be wet. The frames are provided courtesy of the Third International Workshop on\nLong-Wave Runup Models (Liu et al., 2007).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n77\nIn addition to point-wise comparison at the gauges, we plot snapshots of the computed and\nobserved water height of the whole domain. On the left side in Figure 2-18, we display five\nframes extracted from a video taken during the laboratory experiment. These frames are 0.5\nseconds apart and are focused on the narrow gully where the highest runup is observed. On the\nright side in Figure 2-18, we show snapshots of the numerically computed water level at times\nsynchronous with those of the video frames. Side-by-side comparison of these series of frames\nreveals a good agreement of the numerical solution to the observations throughout the domain,\nwhere the maximum runup occurred. Furthermore, Figure 2-18 shows that the numerical method\nis able to capture a rapid sequence of runup and rundown.\n2.3.8 BP8: Three-dimensional landslide - laboratory\nLarge-scale experiments have been conducted in a wave tank with a plane slope (1:2)\nlocated at one end of the tank. Detailed description of the tank geometry and experimental\nequipment can be found in Liu et al. (2005). A solid wedge was used to model the landslide. The\nhorizontal surface of the wedge was initially positioned either a small distance, A, above or\nbelow the still water level to reproduce a subaerial or submarine landslide.\n0.91\nD\nh\n[SIDE VIEW]\n2h\n1.2446\n[PLANE VIEW]\nwave gauge 2\n1.83\nRunup Gauge 3\n0.635\n0.61\nRunup Gauge 2\n0.305\nwave gauge 1\nRunup Gauge 1\n0.91\nFigure 2-19: Schematic of the experimental setup. Locations of the water gauges are marked by green\ndots. The profiles along which the runup is measured are shown by red lines.","National Tsunami Hazard Mitigation Program (NTHMP)\n78\nThe wedge was released from rest, abruptly moving downslope under gravity. The wedge\nwas instrumented with a position indicator to determine the velocity and position time histories.\nA = -0.1 m\nA = -0.025m\n500\n500\n250\n250\n0\n0\n2\n3\n4\n3\n4\n0\n1\n0\n1\n2\nt (sec)\nt (sec)\nFigure 2-20: Time histories of the block motion for the submerged case with A = -0.025 m and A =\n-0.1 m.\nThe goal of this benchmark problem is to compare the numerical predictions with the\nlaboratory measurements. The domain, shown in Figure 2-19, was discretized with Ax = Ay =\n5.10-3 meter spatial resolution and a time step of 101 5 seconds. Location of the wedge was\nprescribed according to the time histories of the block motion, shown in Figure 2-20. Figure 2-21\ndisplays the comparison between the measurements and the numerical solution.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n79\n\"Submerged\"\nslide,\n1=-0.1\n\"Subaerial\"\nslide,\nA=-0.025\nWater level gauge 1\nWater level gauge 1\n0.1\n0.15\nComputed\nObserved\nComputed\nObserved\n0.1\n0.05\nE\n0.05\n0\n0\n-0.05\n-0.05\n-0.1\n-0\n0\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\n4\n4.5\n5\n0\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\n4\n4.5\n5\nTime sec\nTime sec\nWater level gauge 2\nWater level gauge 2\n0.05\nComputed\nObserved\nComputed\nObserved\n0.04\nE\n0.02\n0\n0\n-0.02\n-0.04\n-0.05\n-0.06\n0\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\n4\n4.5\n5\n0\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\n4\n4.5\n5\nTime sec\nTime sec\nRunup gauge 2\nRunup gauge 2\n0.05\n0.1\nComputed\nObserved\nComputed\nObserved\n0.05\n0\n0\n-0.05\n-0.05\n-0.1\n0\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\n4\n4.5\n5\n0\n0.5\n1\n15\n2\n25\n3\n3.5\n4\n4.5\n5\nTime sec\nTime sec\nRunup gauge 3\nRunup gauge 3\n0.05\n0.1\nComputed\nObserved\nComputed\nObserved\n0.05\nE\n0\n0\n-0.05\n-0.05\n-0.1\n0\n05\n1\n1.5\n2\n2.5\n3\n3.5\n4\n4.5\n5\n0\n0.5\n1\n1.5\n2\n25\n3\n3.5\n4\n4.5\n5\nTime sec\nTime sec\nFigure 2-21: The top plots show the comparison between the computed and measured water level\ndynamics at two gauges, shown in Figure 2-19. The bottom plots show the comparison between the\ncomputed and measured runup along two profiles, shown in Figure 2-19.\nWe note that the numerical solution to the non-linear shallow water equations has some\nnumerical instability due to the formation of the shock wave.\n2.3.9 BP9: Okushiri Island tsunami - field\nThe bathymetry/topography digital elevation model (DEM) for Okushiri island was\nprovided by the Disaster Control Research Center (DCRC) at Tohoku University, Japan. The\ndata consist of several nested grids of increasing spatial resolution ranging from 450 m to 5 m.\nThe grids are focused on the Monai and Aonae regions where the maximum runup and\ndevastation was reported in 1993.\nWe began to analyze computational results by comparing the numerically computed water\nlevel dynamics to tidal gauge records of the first hour after the earthquake. Figure 2-23 shows\nthe computed and observed water level dynamics at the stations, marked in Figure 2-22 by\ntriangles. The arrival time of the computed wave matches well with the arrival of the leading\ntsunami wave. The correlation of positive and negative phases between the computed and\nobserved waves is rather good, although the computed wave at both locations has a larger range","National Tsunami Hazard Mitigation Program (NTHMP)\n80\nand frequency of variability than the observed wave. The discrepancies between the measured\nand observed waves can be explained by the lack of detailed bathymetry near tide stations,\nlimitations of the shallow water approximation model, and inaccuracy of the specified initial\nconditions.\n0\n43'00'N\nIwanai\n2\n42'30'N\n1\n0\nOkushin Island\nMonai\nHamatsumae\nHokkaido Island\n42'00'N\n2\nonae\n3\nEsashi\n1\n0\n41*30'N\n139°00'E\n139'30'E\n140*00'E\n140°30'E\nFigure 2-22: The computational domain used to simulate 1993 Okushiri tsunami. The triangles mark\nthe locations of the tide gauge stations that observed water levels to which we compare model\ndynamics. The contours mark the seafloor displacement caused by the Hokkaido-Nansei-Oki\nearthquake (Takahashi et al., 1995).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n81\nEsashi tidal station\n4\nObserved\nComputed\n2\n0\n-2\n-4\nIwanai tidal station\n4\nObserved\nComputed\n2\n0\n-2\n-4\n0\n15\n30\n45\n60\nTime since the earthquake, S\nFigure 2-23: Computed and observed water levels at two tide stations located along the west coast of\nHokkaido island during 1993 Okushiri tsunami. The observations are provided courtesy of Yeh et al.\n(1996).\nFigure 2-24 shows the locations on Okushiri island where the runup was measured shortly\nafter the 1993 tsunami. To compare the computed and observed runup, we discretize the\nshoreline into several regions, with each region enclosing the part of the shoreline lying closer to\na certain observation point than to any other. Within each region, we compute the maximum and\nminimum values of the simulated runup and compare this variability to the observations at each\npoint. We note that almost everywhere around the island, the observed values lie within the\nmodeled range of variability. There are, however, several exceptions where the modeled runup\nunderestimates the observations. For example, the modeled runup in the narrow gully near the\nvillage of Monai is underestimated partially because of the reasons discussed below.","National Tsunami Hazard Mitigation Program (NTHMP)\n82\nObserved 8.7 11.1\nComputed 3.9 8.3\nObserved 5.7 9.6\nObserved 5.0\nComputed 4.3 - 87\nComputed 2.6 4.9\nObserved 5.6 9.0\nObserved 3.3\nComputed 4.5 9.0\nComputed 2.7 5.5\nObserved 3.4\nComputed 4.7 9.9\nObserved 5.1 7.0\nComputed 6.1 9.7\nObserved 6.8 7.7\nComputed 3.0 8.2\nObserved 13\nComputed 7.3 - 12.9\nObserved 16.3\nObserved 3.4\nComputed 9.9 14.6\nComputed 4.3 9.6\nObserved 30.6\nComputed 17.2 21.5\no\nObserved 12.6\nComputed 9.5 15.7\nObserved 18.7\nComputed 8.9 15.1\nObserved 22.8\nObserved 4.8\nComputed 3.3 - 15.7\nComputed 4.7 - 17.7\nObserved 8.3 13.2\nComputed 6.3 - 18.0\nObserved 3.2 10.2\nComputed 4.9 8.8\nObserved 12.4\nComputed 5.1 19.5\nFigure 2-24: The computed and observed runup in meters at 19 locations along the coast of Okushiri\nisland after the 1993 Okushiri tsunami. The observations are provided courtesy of Kato and Tsuji\n(1994).\nWe recall that near Monai village the runup is modeled using 5 meter resolution\ncomputational grids. The bathymetry/topography data within the 5 meter grid are based either on\nthe DEM provided by the DCRC, or on the DEM used to construct a wave tank in the laboratory\nexperiment discussed in the previous sub-section. In both DEMs, the narrow gully is identical,\nbut there is a small difference in elevation near the shoreline. The numerical computations using\nthe DCRC DEM show that the computed maximum runup in the narrow gully is 17.2 m. By\ncomparison, utilizing the wave tank DEM, the resulting runup in the gully is 21.5 m. We\nemphasize that model parameters as well as bathymetry/topography in computational grids\ncoarser than 5 m are the same in both simulations. The difference between the maximum runup\nvalues in these two simulations reveals the high sensitivity of the runup to nearshore\nbathymetry/topography, and underlines the importance of the near-shore bathymetry data for\naccurate runup predictions. Therefore, the discrepancy between the measured and computed\nrunup values may be explained by the lack of accurate bathymetry/topography data near Monai,","MODEL BENCHMARKING WORKSHOP AND RESULTS\n83\nuncertainties in the initial water surface displacement, or finally by limitations of the shallow\nwater approximation to model 3-D flows.\nIn Figure 2-25, we show a sequence of snapshots depicting the simulated waves inundating\nthe city of Aonae. The 0, 5, and 10 meter ground elevations contours are shown by thick lines.\nThe first snapshot corresponds to 280 seconds after the earthquake, and each snapshot is 60\nsecond after the previous one. In the first snapshot, it is easy to observe the approaching 5 meter\nhigh wave via water level contours shown by the dashed lines. While the wave approaches the\nAonae peninsula, it drastically steepens over the shallow areas as shown in the second snapshot,\nshown upper right. The wave runs-up on the western side on the Aonae peninsula and reaches\nthe 10 meter high mark. In the third snapshot, shown lower left, the wave sweeps across the\npeninsula. The speed of the water traveling across the tip of the peninsula, where the greatest\ndestruction occurred, is numerically estimated at up to 12 m/s, which is in good agreement with\nobservations. In the last snapshot, shown lower right, we show the Aonae peninsula after the\nretreat of the computed wave.\n12\n12\n10\n10\n8\n8\n6\n6\n4\n4\n2\n2\n0\n0\n-2\n-2\n4\n-4\n12\n12\n10\n10\n8\n8\n6\n4\n2\n2\n0\n0\n7.\n-2\n-2\n4\n4\nFigure 2-25: Numerical modeling of a tsunami wave overflowing the Aonae peninsula, viewed from\nabove. The dashed black and solid red contours represent the water level and land elevation,\nrespectively. The upper left plot shows an approaching 5 meter high wave. As the wave approaches, it\nsteepens and overtops the peninsula as illustrated by the upper right plot. In the lower left plot, the\nwavefront bends around the peninsula and propagates in the direction of Hamatsumae. In the lower\nright plot, the water retreats and the seabed becomes partially dry.\nWe note that due to the shallow depth around the peninsula, the simulation reveals that the\nwavefront bows, then bends around the Aonae peninsula, and subsequently hits the town of\nHamatsumae. The computed runup at Hamatsumae reaches 15 m and matches well with field\nobservations. Numerical modeling shows that during the reflection of the first wave that hit\nHamatsumae, a wave traveling toward the Aonae peninsula has formed. Both in the computer","84\nNational Tsunami Hazard Mitigation Program (NTHMP)\nexperiment and in eyewitness reports, this second wave hits the Aonae peninsula from the south-\neast direction approximately 10 minutes after the first wave. The damage due to the second wave\nis localized on the eastern side as reported by eyewitnesses. In Figure 2-26, we provide the\ncontours of the maximum computed runup around the Aonae peninsula. This computer\nexperiment shows that the numerical algorithm is stable, successfully models the overland flow,\nand captures the runup of reflected waves.\nObserved 8.3 - 13.2m\nObserved 3.2 - 10.2m\nObserved: 12.4m\nComputed 12-16m\nComputed: 8-12m\nComputed: 4-8m\nFigure 2-26: The computed and observed runup in the vicinity of the Aonae peninsula after 1993\nOkushiri tsunami. The triangles mark the locations where the observations were conducted. The\ncomputed runup distribution has a local maximum near Hamatsumae, as observed by eyewitnesses.\nThe observations are provided courtesy of Kato and Tsuji (1994).\n2.4\nLessons learned\nA numerical model for the simulation of tsunami propagation and runup is verified and\nvalidated using NOAA standards and criteria. In computer experiments modeling the\npropagation and runup of tsunami waves, specified by BP1, BP2, BP4, BP6 and BP7, numerical\ncalculations are within the established errors proposed by Synolakis et al., (2008).\nThe analytical solution in BP3 is derived under circumstances that prohibit any direct\nemployment of the analytical solution in accurate benchmarking of the tsunami models.\nTherefore, we suggest that this benchmark should be replaced by a problem that is similar in\nnature.\nThe numerical results agree with the runup measurements in Case A of BP5, but do not\npredict the runup in Case B and Case C, if the water dynamics is modeled by using the non-","MODEL BENCHMARKING WORKSHOP AND RESULTS\n85\nlinear shallow water equations. It seems that in these lab experiments, a wave hitting the vertical\nwall creates air bubbles that produce a splash that is actually measured in Case B and Case C.\nIn BP8, the numerical results derived within the non-linear shallow water approximation\nshow numerical instabilities that are due to a shock formation at the edge of the triangular wedge\nmodeling sliding material. It is still unknown whether a Boussinesq approximation would help to\neliminate the shock formation and produce better agreement with the laboratory measurements.\nAnyway, we emphasize that the geometry of the rigid sliding block used in this BP is too simple\nto represent an actual landslide. Therefore, we suggest that this benchmark should be replaced by\na problem with a more realistic geometry of the sliding material.\nIn BP9, the computed runup around Okushiri island is within the variability of field\nobservations. However, the local extreme runup, e.g., in the narrow gully near the village of\nMonai, is sensitive to the near shore interpolation of bathymetry/topography. The computer\nsimulation of the 1993 Okushiri tsunami also captures the overland flow at the Aonae peninsula,\nwhere the maximum destruction was reported.\n2.5\nProposed benchmarks\nBenchmarking of the seiche tsunami model (1a)-(2a) that shares the same runup algorithm\nas the tectonic tsunami model is partially addressed by the list of considered BPs. Unfortunately,\nit is hard to derive an analytical solution for the full system (1a)-(2a). However, it is possible to\nmodel a physical effect of these terms as follows. The ground velocity Ug has non-zero values\nonly for a few seconds, i.e., over the period when the horizontal tectonic displacement occurs.\nWithin this time period the water velocity, u, can be considered small and then the water surface\ndisplacement can be approximated by Vh.u In the case of the parabolic water basin, this initial\ndisturbance is a plane profile. Therefore, we propose to test the model against the analytical\nsolution in the cases of frictionless water flow in 2-D parabolic basins that can model fjord-type\nsettings typical of the Alaska coast. The analytical solution to this problem is described as\nnonlinear normal mode oscillations of water (Thacker, 1981). We assume that the bathymetry is\ngiven by\nh=ho(x2+y2-12)\nThen, an analytical solution to (1)-(2) is described by oscillations such that the water surface\nremains planar:\nS =\nwhere a is the amplitude of the motion, 2w' =f+Jw,and\nTo conduct numerical experiments, we set L = 500, g 9.8, 10, a = 1, and f = 0.01. To\nestimate accuracy of the numerical scheme we compute the numerical solution on the series of\ngrids (Ax = 5, Ax = 2.5, and Ax = 1.25) and compare it to the analytical ones. In all\ncomputations, the time step At is fixed and is equal to 10 3T, where the constant T is a period of\noscillations, i.e., T = 2n/w'. The numerical water surfaces after the third revolution are shown in\nFigure 2-27.","86\nNational Tsunami Hazard Mitigation Program (NTHMP)\n0\nSolid\n- Numerical solution\n-2\nHollow - Analytical solution\nt=3.0T\n-4\nt=3.1T\nt=3.2T\n-6\nt=3.3T\nt=3.4T\nt=3.5T,\n-8\n-10\n-400\n-200\n0\n200\n400\nx-axis, m\nFigure 2-27: Water height profiles E(x, 0, t) for numerical (solid) and analytical (hollow) solutions at t =\n3T, 3.1T, 3.2T, 3.3T, 3.4T, and 3.5T, where T is the period of the corresponding oscillatory mode.\n2.6\nReferences\nArakawa A, Lamb V. 1977. Computational design of the basic dynamical processes of the\nUCLA general circulation model. In: Methods in Computational Physics. Vol. 17.\nAcademic Press, pp. 174-267.\nBalay S, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith\nBF, Zhang H. 2004. PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 2.1.5,\nArgonne National Laboratory.\nBriggs M, Synolakis C, Harkins G, Green D. 1995. Laboratory experiments of tsunami runup on\na circular island. Pure and Applied Geophysics 144, pp. 569-593.\nCourant R, Friedrichs K, Lewy H. 1928. Uber die partiellen differenzengleichungen der\nmathematischen physic. Mathematische Annalen 100, pp. 32-74.\nFine I, Rabinovich A, Kulikov E, Thomson R, Bornhold B. 1998. Numerical modeling of\nlandslide-generated tsunamis with application to the Skagway Harbor tsunami of","MODEL BENCHMARKING WORKSHOP AND RESULTS\n87\nNovember 3, 1994. In Proceedings of international conference on tsunamis: Paris, pp.\n211-223.\nFletcher C. 1991. Computational Techniques for Fluid Dynamics 1. Springer-Verlag, 401 pp.\nGoto C, Ogawa Y, Shuto N, Imamura F. 1997. Numerical method of tsunami simulation with the\nleap-frog scheme. Manuals and Guides 35, UNESCO: IUGG/IOC TIME Project.\nGropp W, Lusk E, Skjellum A. 1999. Using MPI: Portable Parallel Programming with the\nMessage-Passing Interface. The MIT press, 406 pp.\nJiang L, LeBlond P. 1992. The coupling of a submarine slide and the surface waves which it\ngenerates: Journal of Geophysical Research, 97(C8), 12731-12744.\nJiang L, LeBlond P. 1994. Three-dimensional modeling of tsunami generation due to a\nsubmarine mudslide. Journal of Physical Oceanography, 24(3):559-572.\nKânoglu U, Synolakis C. 1998. Long wave runup on piecewise linear topographies. Journal of\nFluid Mechanics 374, 1-28.\nKato K, Tsuji Y. 1994. Estimation of fault parameters of the 1993 Hokkaido-Nansei-Oki\nearthquake and tsunami characteristics. Bulletin of the Earthquake Research Institute 69,\n39-66, University of Tokyo.\nKowalik Z, Murty T. 1993. Numerical modeling of ocean dynamics. World Scientific, 481 pp.\nLiu PL-F, Cho YS, Briggs MJ, Kânoglu U, Synolakis CE. 1995. Runup of solitary waves on a\ncircular island. J. Fluid Mech, 302, 259-285.\nLiu PL-F, Lynett P, Synolakis CE. 2003. Analytical solutions for forced long waves on a sloping\nbeach. J. Fluid Mech, 478, 101-109.\nLiu PL-F, Wu T-R, Raichlen F, Synolakis CE, Borrero J. 2005. Runup and rundown generated\nby three-dimensional sliding masses. J. Fluid Mech, 536, 107-144.\nLiu PL-F, Yeh H, Synolakis C. 2007. Advanced Numerical Models for Simulating Tsunami\nWaves and Runup. Vol. 10 of Advances in Coastal and Ocean Engineering. World\nScientific, Proceedings of the Third International Workshop on Long-Wave Runup\nModels, Catalina, (2004) Benchmark problems, pp. 223-230.\nLynett P, Wu T-R, Liu PL-F. 2002. Modeling wave runup with depth-integrated equations.\nCoastal Engineering 46 (2): 89-107.\nNicolsky DJ, Suleimani EN, Hansen RA. 2010. Numerical modeling of the 1964 Alaska tsunami\nin western Passage Canal and Whittier, Alaska. Natural Hazards and Earth System\nSciences, 10, 2489-2505, doi:10.5194/nhess-10-1-2010.\nNicolsky DJ, Suleimani E, Hansen R. 2011. Validation and verification of a numerical model for\ntsunami propagation and runup. Pure and Applied Geophysics, 168:1199-1222, doi\n10.1007/s00024-010-0231-9.\nOkada Y. 1985. Surface deformation due to shear and tensile faults in a half-space. Bulletin of\nthe Seismological Society of America 75, 1135-1154.\nPlafker G, Kachadoorian R, Eckel E, Mayo L. 1969. Effects of the Earthquake of March 27,\n1964 on various communities: U.S. Geological Survey Professional Paper 542-G, 50 p.","88\nNational Tsunami Hazard Mitigation Program (NTHMP)\nSuleimani E, Nicolsky D, Haeussler P, Hansen R. 2011. Combined effects of tectonic and\nlandslide-generated tsunami runup at Seward, Alaska during the Mw9.2 1964 earthquake,\nPure Appl. Geophys 168, pp. 1053--1074, doi: 10.1007/s00024-010-0228-4.\nSynolakis C. 1986. The Runup of Long Waves. Ph.D. thesis, California Institute of Technology,\nPasadena, California, 228 pp..\nSynolakis C, Bernard E. 2006. Tsunami science before and beyond Boxing Day 2004:\nPhilosophical Transactions of the Royal Society A, V. 364, n. 1845, p. 2231-2265.\nSynolakis C, Bernard E, Titov V, Kânoglu U, González F. 2007. Standards, criteria, and\nprocedures for NOAA evaluation of tsunami numerical models. OAR PMEL-135 Special\nReport, NOAA/OAR/PMEL, Seattle, Washington, 55 pp.\nSynolakis C, Bernard E, Titov V, Kânoglu U, González F. 2008. Validation and verification of\ntsunami numerical models. Pure and Applied Geophysics 165, 2197-2228.\nTakahashi T, Takahashi T, Shuto N, Imamura F, Ortiz M. 1995. Source models for the 1993\nHokkaido-Nansei-Oki earthquake tsunami. Pure and Applied Geophysics 144, 747-768.\nThacker W. 1981. Some exact solutions to the nonlinear shallow-water wave equations. Journal\nof Fluid Mechanics 107, 499-508.\nTitov V, Synolakis C. 1995. Evolution and runup of breaking and nonbreaking waves using\nVTSC2. Journal of Waterway, Port, Coastal and Ocean Engineering 121 (6): 308-316.\nTitov V, Synolakis C. 1998. Numerical modeling of tidal wave runup. Journal of Waterway,\nPort, Coastal and Ocean Engineering 124, 157-171.\nTuck EO, Hwang LS. 1972. Long wave generation on a sloping beach. J. Fluid Mech. 51, 449-\n461.\nYeh H, Liu PL-F, Synolakis C. 1996. Long-Wave Runup Models. World Scientific, 403 pp.\nZhang Y, Baptista A. 2008. An efficient and robust tsunami model on unstructured grids. Part I:\nInundation benchmarks. Pure and Applied Geophysics 165, 2229-2248.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n89\n3\nATFM (Alaska Tsunami Forecast Model)\nBill Knight\nZygmunt Kowalik: University of Alaska, Fairbanks- Institute of Marine Sciences (UAF-IMS)\nPaul Whitmore: NOAA/NWS/West Coast and Alaska Tsunami Warning Center (WCATWC),\nPalmer, AK\nBill Knight: WCATWC, Palmer AK, UAF-IMS, Fairbanks, AK\n3.1\nIntroduction\nThe Alaska Tsunami Forecast Model (ATFM) began as a collaborative effort between two\nof the members (Kowalik and Whitmore) and became operational at the WCATWC in 1997\n(Kowalik and Whitmore, 1991; Whitmore and Sokolowski, 1996). This is known as the \"classic\"\nmodel currently used in WCATWC operations. This model has been tested against the analytical\nsolutions of Carrier and Greenspan (1958) and Thacker (1981). From 2004 on, the model has\nbeen substantially reworked into a second forecast model, which is called ATFMv2. The\nbenchmark challenge problems were computed with this newer model.\nThe design philosophy behind both models is to focus on tectonic sources and to use the\nmodels to pre-compute hundreds of tsunami \"scenarios\". Because of the tectonic-only usage, a\nsubset of the benchmarks proposed in Synolakis et al. (2007) has been validated here.\n3.2\nModel description\nThe ATFM solves the non-linear shallow water equations. Two equations of motion and one\ncontinuity equation are formulated in spherical coordinates and solved on structured, nested\nmeshes. The two horizontal components of velocity (U and V) are depth averaged. The vertical\ncomponent of velocity (W) is not considered in the hydrostatic formulation.\nThe solution technique for U and V is based on a differencing method described in Kowalik\nand Murty (1993), and the sea level (n) is computed with a second-order accurate, upwind\nscheme that conserves mass to machine accuracy (Van Leer, 1977). The runup / run-down\nmethod is based on the VOF approach pioneered by Nichols and Hirt (1980), and Hirt and\nNichols (1981). There is no explicit dispersion in the model, although a non-hydrostatic addition\nis under development and is in the testing phase (Walters, 2005, Yamazaki et al. 2009). Sub-\nmeshes are nested within parent meshes to increase spatial resolution where needed. Information\nis passed both from low to high resolution meshes and back, based on a mass conserving\ninterconnect scheme (Berger and Leveque, 1998). Discretization for the 3 field variables (U, V,\nand n) uses the staggered \"C grid\" layout.\nThe model is coded in FORTRAN 90 and in Co-array FORTRAN. It has been run on PCs, a\nCray X1, and a Penguin Computing cluster comprised of Opteron processors.","National Tsunami Hazard Mitigation Program (NTHMP)\n90\nSome notation:\nRe = radius of earth (assumed uniform)\nS = earth's rotation rate\na = longitude, Q = latitude\nU and V are the east, and north pointing velocity components\nn = sea surface elevation\nh = still water depth\nus = ocean bottom uplift\na = dimensionless bottom friction parameter\nG = gravity acceleration\nThe equations solved are:\nauthorization = 20V\n(1)\n(2)\nhc)0\n0\n(3)\n=\nThe UV and U2 terms in the equations of motion (1 and 2) are small and are neglected in the\nATFM. Note that for benchmark problems 1, 4, and 6, the equations were also re-formulated in\nsimpler 2-D Cartesian coordinates.\n3.2.1 BP1: Solitary wave on a simple beach - analytical\nThe model solution is compared to an analytical solution of the one dimensional, hydrostatic\nshallow water equations. With that comparison in mind, the bottom friction parameter has been\nset to zero. Numerical dispersion is also near zero through the choice of high spatial resolution.\nThe problem set up is shown in Figure 3-1.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n91\n1\n0.02\n0.75\n0.015\nBathymetry\nSea Level\n0.5\n0.01\nH\n0.25\n0.005\n0\n0\n-0.25\n-0.005\nD\n-0.5\n-0.01\n-0.75\n-0.015\n-1\n-0.02\n-10\n0\n10\n20\n30\n40\n50\n60\n70\n80\n90\n100\nX/D\nFigure 3-1: Initial condition for BP1. Note figure is not drawn to scale. The right vertical scale shows\nthe range of sea level used in the benchmark, and the left vertical scale shows the ratio of maximum\nheights (H) to depth (D).\nThe runup (R) in this and in later benchmarks is defined as maximum height reached by the\nwater on dry land. In this benchmark, runup (R), sea level (n), and time (t) are replaced with\ndimensionless counterparts t * going The beach\ntheir\nslope is fixed at 2.88 degrees and\nthe initial non-dimensional amplitude (H/D) is set to 0.019. At to, the solitary wave is positioned\nsuch that the sea level equals e -3 of the maximum height H at the toe of the slope (X/D = 20). The\n\"beach\" is the origin of the coordinate X, with x increasing towards deep water. A uniform mesh\nwith resolution D/20 was used to compute sea level profiles at six dimensionless times. The time\n1\nto insure the Courant-Friedrichs-Lewy (CFL) condition was met. Results\nstep was set at\n200\ng\nare displayed in Figure 3-2 and Figure 3-3.","National Tsunami Hazard Mitigation Program (NTHMP)\n92\n0.1\n0.075\nt=55\n0.05\n0.025\nt=65\n0\nT=70\n-0.025\n-0.05\n13\n18\n-2\n3\n8\nX/D\nFigure 3-2: Non-dimensional sea level profiles as a function of non-dimensional distance for BP1.\nDashed red lines are the analytic result while the solid blue are modeled results for dimensionless\ntimes T = 35 to 45.\n0.075\n0.05\nt=45\nT=35\n0.025\nt=40\n0\n-0.025\n-2\n3\n8\n13\n18\nX/D\nFigure 3-3: Non-dimensional sea level profiles as a function of non-dimensional distance for BP1.\nDashed red lines are the analytic result while the solid blue are the modeled results for dimensionless\ntimes T = 55 to 70.\nThe maximum predicted runup (R/D) is 0.0901 at T = 55.4. This is within 2% of the\ntheoretical runup value. The total system mass, normalized to the initial solitary wave mass,\nvaries by only 1.0x10 -6 during the model run. Additional plots of sea surface elevation VS. time at\ntwo positions x/D = 0.25 (near the initial shoreline) and x/D = 9.95 (between beach and initial","MODEL BENCHMARKING WORKSHOP AND RESULTS\n93\nsolitary wave crest are displayed in Figure 3-4. The data gap at x/D = 0.25 for dimensionless\ntimes 67 5.598\nThe value 5.598 12 1.5/ tan(0) is determined by the fact that the water has a depth of 1.5 m on\nthe flat portion and the beach slope is 15° SO 0 = 15/180 12 0.2618.","National Tsunami Hazard Mitigation Program (NTHMP)\n148\nAt time t = 0 the sliding mass is located at X = x0, determined by the initial depth d\naccording to\nAt time t the sliding mass is centered at xc = xo + s(t) cos(A) where s(t) is the function\ndiscretized in the data file kinematics.txt. However, this is only true for small t. After some time\nthe mass hits the horizontal bottom of the tank. According to paper by Enet and Grilli (2003) and\ncommunication with Stephan Grilli, the mass stops at this point. This is not made clear in the\nproblem specification (LeVeque, 2011).\nTo determine (x,y,t) for each finite volume grid cell with center (xi,yj) the value 5 must\nbe found SO that\n& sin(A) = Xi\nwhere &c = xcl cos(A) + s(t) is the & location of the center of mass at this time. Determining\n& requires solving the nonlinear equation\ncos sin(A) - Xi = 0.\nIn our Fortran code this equation is solved using the library routine zeroin, available from Netlib\n(http://www.netlib.org/go/zeroin.f).\nOnce & has been found, the bathymetry is\nB(xi,yj,t) tan(0) +\n5.3.3.3 What we did\nThe moving bathymetry is handled by recomputing Bn = B(xi,Yj,tn) in each time step\nat the center of each finite volume grid cell, by solving a nonlinear equation as described\nabove. This is the standard approach for handling moving bathymetry in GeoClaw: the\nvalue is adjusted but the fluid depth hij remains the same, SO that the water column is\nsimply displaced vertically in any cell where the bathymetry changes. For bathymetry\nthat is smoothly varying in space and time, as in this problem, this is considered a\nreasonable approach. Note, however, that no momentum is directly imparted to the water\nby the moving bathymetry.\nThe problem was solved using a fixed grid with 72n X 18n grid cells on the domain\n-1 x 6.2 and 0 y < 1.8 m. Three resolutions corresponding to n = 1,2,4\nwere used to test convergence.\nA second level of refined grid was used in the region -0.1 X 0.1 and y\n0.1 surrounding the point x 22 0,y = 0 on the shoreline where the runup Ru must be\ncalculated. In each case this grid was 10 times finer in each direction than the base grid.\nAdaptive mesh refinement (with moving grids) was not used.\nThe problem was solved on Osys 1.8 with solid wall boundary conditions at y = 0.\nThis gives the correct solution in this domain and the solution in the other half of the\nwave tank -1.8 y 0 is easily constructed by symmetry if desired.\nSolid wall boundary conditions were also used at y = 1.8. At x = 1 the boundary\ncondition doesn't matter since this region is always dry, and at x = 6.2 outflow\nboundary conditions were used. Zero-order extrapolation, which generally gives a very","MODEL BENCHMARKING WORKSHOP AND RESULTS\n149\ngood approximation to non-reflecting boundary conditions as described in Section\n5.2.3.1. Solid wall boundary conditions are implemented as described in Section 5.2.3.2.\nSee Section 5.2.3 for more information about these boundary conditions.\n5.3.3.4 Numerical simulations\nFigure 5-9 shows two frames from a sample computation for the case d = 0.061. Colors\nindicate the surface elevation and contours show the bathymetry with the upper half of\nthe sliding mass.\n5.3.3.5 Gauge comparisons\nSimulated gauges were placed at the 4 locations that match the wave tank measurements, as\nindicated in Figure 5-9.\nCase 1: surface at 0.00 seconds\n1.0\n0.8\nU.6\n0.4\n0.2\n0.0\n-1.0\n-0.5\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\nCase 1: surface at 1.50 seconds\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0\n-1.0\n-0.5\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\nFigure 5-9: Sample results for d = 0.061. The water surface n(x, y, t) (colors with dark red +0.02 m and\ndark blue -0.02 m) and bathymetry (0.01 m contour levels). Only a portion of the computational\ndomain is shown. Grid resolution: Ax = Ay = 0.025 m on the full domain, with refinement to Ax = Ay =\n0.0025 m in the nearshore region in the rectangular box. The full domain goes to X = 6.2 and to y = 1.8.\nThe surface elevation n(t) at each gauge was recorded every time step. These results are\nshown in Figure 5-10 through Figure 5-16 for the 7 test cases.\nReasonable agreement is generally seen for the initial peak and trough at Gauges 1, 2, and 4.\nOn the other hand, Gauge 3, located along the centerline, shows quite different results than the\nmeasurements and generally exhibits a steeper dip in n as the mass passes this point. The\nmeasurements also show an oscillatory wave train behind the initial peak and trough that is not\ncaptured in the simulations obtained with the shallow water equations. This is consistent with\nclaims in LeVeque (2011) and Enet and Grilli (2003) that dispersive effects are important for\nthese short wavelength waves that cannot be captured by the non-dispersive shallow water","National Tsunami Hazard Mitigation Program (NTHMP)\n150\nequations. By contrast, the Boussinesq model used in Fuhrman and Madsen (2009) does display\nthese dispersive ripples.\n5.3.3.6 Runup measurements\nThe runup is measured near y = 0 by keeping track of the approximate shoreline position in\nthe first row of grid cells j = 1, whose centers lie at y = Ay/2. In each time step, we loop over\nall cells i = 1,2, and look for the first cell for which hij > f, where f = 0.001 (1 mm) was\nchosen as a depth below which the cell is considered dry. The value Xs = iAx, the right edge of\nthis finite volume cell, was then used as the shoreline location at this time. The runup at each\ntime is then computed as Xs tan(A), and this value was output for later plotting, and for\ncomputing the maximum runup Ru required for the benchmark.\nFigure 5-10 through Figure 5-16 show the runup as functions of time for each test case.\nSome of these plots exhibit strange behavior for later times. This was due to the fact that we used\na limited domain and also that we used a refined grid over only a fairly small region near the\norigin.\nApproximate maximum runup values are tabulated in Table 5-1. These values are based on\nthe minimum values seen in the figures for early times. It is not clear if these are correct in all\ncases. Also these grids are fairly coarse. But because the gauge data do not match particularly\nwell and we do not believe shallow water is a suitable model for this problem, we did not pursue\nthis further.\nTable 5-1: Runup values in mm. Lab results taken from Table 1 of LeVeque (2011). Two different\nresolutions with 36 and 72 points in the y direction were compared, with mx = 4my points in the X\ndirection.\nD\nLab\nMy 36\nMy = 72\n0.061\n6.2\n8.0\n8.7\n0.080\n5.7\n5.4\n6.0\n0.100\n4.4\n2.7\n4.3\n0.120\n3.4\n2.7\n3.4\n0.140\n2.3\n2.7\n2.7\n0.149\n2.7\n2.7\n2.7\n0.189\n2.0\n1.4\n2.0\n5.3.3.7 Lessons learned and suggestions for improvement\nIt might be useful to other groups doing this problem in the future if the bathymetry\nZ = Bo(x,y) were tabulated, corresponding to the mass centered at x0 = 0 on the slope\nwith 0 = 15°. From this, the bathymetry at later times could be interpolated by shifting\nby Xc. Computing Bo(x,y) from the given 3(5,n) requires solving a nonlinear equation\nat each (x,y) as outlined in Section 5.3.3.3.\nIt is stated in LeVeque (2011) that 3(5,n) represents a \"Gaussian mass\" but this is not a\nGaussian function.\nThe non-dispersive shallow water equations do not appear adequate to model the\noscillatory wave train observed in the laboratory. The shallow water equations may still","MODEL BENCHMARKING WORKSHOP AND RESULTS\n151\nbe useful for modeling landslides of this nature because the initial peak amplitude and run\nup values are in the right ballpark, but comparison with laboratory measurements is not a\nsuitable means of judging convergence or accuracy of the numerical method. For this\nreason, it would be valuable if the community could agree on what the \"correct\"\nconverged solution to the shallow water equations is for this problem, and if this solution\n(or at least the values at the gauges) were tabulated for comparison in future benchmark\nstudies.\nThe runup results in the laboratory might be affected by the rail along which the mass\nslides, which is visible in Figure 1 of LeVeque (2011) and is along y = 0, the point\nwhere it is stated that the runup should be measured. In fact, the runup must have been\nmeasure slightly above this point, as indicated in Figure 9 of Enet and Grilli (2003). The\nrail appears to be several mm high and should affect the fluid dynamics. This rail could\neasily be added to the bathymetry if its dimensions were known.","152\nNational Tsunami Hazard Mitigation Program (NTHMP)\nGauge 1 for d = 0.061\nGauge 2 for d = 0.061\n0.005\n0.02\n0.000\n0.01\n-0.005\n0.00\n-0.010\n-0.01\n-0.015\nmy 18\nmy 18\n-0.02\n-0.070\nmy 36\nmy 36\nmy 72\nmy 72\nLab data\nLab data\n-0.025\n-0.03\n0\n1\n2\n3\n4\n5\n0\n1\n2\n,\n4\n5\nGauge 3 for d = 0.061\nGauge 4 for d = 0.061\n0.015\n0.010\n0.00\n0.005\n0.000\n-0.05\n-0.005\n-0.010\nmy 18\n-0.10\nmy 18\nmy 36\nmy 36\n-0.015\nmy 72\nmy 72\nLab data\nLab data\n-0.020,\n0\n1\n2\n4\n5\n1\n2\n3\n4\n5\nRunup for d = 0.061\n0.015\nmy 18\nmy 36\n0.010\nmy 72\n0.005\n0.000\n-0.005\n-0.010\n-0.015\n1\n2\n0\n3\n4\n5\nFigure 5-10: Gauge and runup results for d = 0.061. Three different resolutions with my = 18, 36, and\n72 points in the y direction were compared, with mx = 4my points in the X direction.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n153\nGauge 1 for d = 0.080\nGauge 2 for d = 0.080\n0.005\n0.015\n0.010\n0.000\n0.005\n-0.005\n0.000\n-0.005\n-0.010\n-0.010\nmy 18\nmy 18\n-0,015\nmy 36\nmy 36\n-0.015\nmy 72\nmy 72\nLab data\nLab data\n-0.020\n-0.020\n2\nD\n1\n2\n3\n4\nI\n3\n4\nGauge 3 for d = 0.080\nGauge 4 for d = 0.080\n0.02\n0.010\n0.00\n0.005\n-0.02\n0.000\n-0.04\n-0.005\nmy 18\nmy 18\n-0.06\n-0,010\nmy 36\nmy 36\nmy 72\nmy 72\nLab data\nLab data\n-0.05\n-0.015\n2\n1\n2\n3\n4\n0\n3\n3\n4\n5\nRunup for d = 0.080\n0.015\nmy 18\nmy 36\n0.010\nmy 72\n0.005\n0.000\n-0.005\n-0.010\n-0.015\n1\n2\n0\n3\n4\n5\nFigure 5-11: Gauge and runup results for d = 0.08.","154\nNational Tsunami Hazard Mitigation Program (NTHMP)\nGauge 1 for d = 0.100\nGauge 2 for d = 0.100\n0.006\n0.010\n0.004\n0.005\n0.002\n0.000\n0.000\n-0.002\n-0.005\n-0.004\n-0.006\n-0.010\nmy 18\nmy 18\n-0.008\nmy 36\nmy 36\n-0.015\n-0.010\nmy 72\nmy 72\nLab data\nLab data\n-0.012\n-0.020\n1\n2\n3\n4\n5\n0\n1\n2\n3\n4\n5\nGauge 3 for d = 0.100\nGauge 4 for d = 0.100\n0.02\n0.010\n0.00\n0.005\n-0.02\n0.000\n-0.04\n-0.005\n-0.06\nmy 18\n-0.010\nmy 18\nmy 36\n-0.00\nmy 36\nmy 72\nmy 72\nLab data\n-0.10\n-0.015\nI\n2\n3\n4\n5\n3\n2\n3\n4\nRunup for d = 0.100\n0.015\nmy 18\nmy 36\n0.010\nmy 72\n0.005\n0.000\n-0.005\n-0.010\n-0.015\n1\n0\n2\n3\na\n5\nFigure 5-12: Gauge and runup results for d = 0.1.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n155\nGauge 1 for d = 0.120\nGauge 2 for d = 0.120\n0.006\n0.010\n0.004\n0.005\n0.002\n0.000\n0.000\n-0.002\n-0.004\n-0.005\n-0.006\n-0.008\nmy 18\nmy 18\n-0.010\nmy 36\nmy 36\n-0.010\nmy 72\nmy 72\nLab data\nLab data\n-0.012\n-0.015\n1\n2\n3\n4\n5\n1\n2\n3\n5\nGauge 3 for d = 0.120\nGauge 4 for d = 0.120\n0.008\n0.006\n0.00\n0.004\n0.002\n-0.02\n0.000\n-0.002\n-0.04\n-0.004\nmy 18\n-0.006\nmy 18\nmy 36\n-0.06\nmy 36\n-0.006\nmy 72\nmy 72\nLab data\n-0.010\n0\n1\n2\n3\nA\n1\n2\n0\n3\n4\nRunup for d = 0.120\n0.015\nmy 18\nmy 36\n0.010\nmy 72\n0.005\n0.000\n-0.005\n-0.010\n-0.015\n0\n1\n2\n3\n4\n5\nFigure 5-13: Gauge and runup results for d = 0.12.","National Tsunami Hazard Mitigation Program (NTHMP)\n156\nGauge 2 for d = 0.140\nGauge 1 for d = 0.140\n0.006\n0.006\n0.004\n0.004\n0.002\n0.002\n0.000\n0.000\n-0.002\n-0.002\n-0.004\n-0.004\n-0.006\n-0.006\nmy 18\nmy 18\n-0.008\n-0.008\nmy 36\nmy 36\nmy 72\nmy 72\n-0.010\n-0.010\nLab data\nLab data\n-0.012\n-0.012\nI\n2\n3\n4\n1\n2\n3\n4\nGauge 4 for d = 0.140\nGauge 3 for d = 0.140\n0.006\n0.02\n0.01\n0.004\n0.00\n0.002\n-0.01\n0.000\n-0.02\n-0.002\n-0,03\n-0.004\n-0.04\nmy 18\nmy 18\nmy 36\nmy 36\n-0 006\nmy 72\n-0.05\nmy 72\nLab data\nLab data\n-0.00B\n-0.06\n1\n2\n3\n4\n1\n2\n3\n4\nRunup for d = 0.140\n0.015\nmy 18\nmy 36\n0.010\nmy 72\n0.005\n0.000\n-0.005\n-0.010\n-0.015\n1\n2\n3\n4\n5\n0\nFigure 5-14: Gauge and runup results for d = 0.14.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n157\nGauge 1 for d = 0.149\nGauge 2 for d = 0.149\n0.006\n0.004\n0.004\n0.002\n0.002\n0.000\n0.000\n-0.002\n-0.002\n-0.004\n-0.004\n-0.006\n-0.006\nmy 18\nmy 18\nmy 36\nmy 36\n-0.008\n-0.008\nmy 72\nmy 72\nLab data\nLab data\n-0.010\n-0.010\n1\n2\n3\n4\n5\n0\n1\n2\n3\n4\n5\nGauge 3 for d = 0.149\nGauge 4 for d = 0.149\n0.02\n0.006\n0.01\n0.004\n0.00\n0.002\n-0.01\n0.000\n-0.02\n-0.002\n-0.03\n-0 004\nmy 18\nmy 18\nmy 36\n-0.04\n-0.006\nmy 36\nmy 72\nmy 72\nLab data\n-0.05,\n-0.00B\n0\n1\n2\n3\n5\n1\n2\n3\n4\nRunup for d = 0.149\n0015\nmy 18\nmy 36\n0.010\nmy 72\n0.005\n0.000\n-0.005\n-0.010\n-0.015\n0\n1\n2\n3\n4\n5\nFigure 5-15: Gauge and runup results for d = 0.149.","National Tsunami Hazard Mitigation Program (NTHMP)\n158\nGauge 2 for d = 0.189\nGauge 1 for d = 0.189\n0.006\n0.002\n0.004\n0.000\n0.002\n0.000\n-0.002\n-0.002\n-0.004\n-0.004\nmy 18\nmy 18\nmy 36\nmy 36\n-0.006\nmy 72\nmy 72\n-0.006\nLab data\nLab data\n-0.006\n4\n0\n1\n2\n3\n4\n5\n1\n2\n3\nGauge 4 for d = 0.189\nGauge 3 for d = 0.189\n0.004\n0.01\n0.003\n0.00\n0.002\n0.001\n-0.01\n0.000\n-0.001\n-0.02\n-0.002\nmy 18\nmy 18\n-0.003\n-0.03\nmy 36\nmy 36\nmy 72\nmy 72\n-0.004\nLab data\nLab data\n-0.005\n-0.04\n2\n1\n4\n5\n1\n2\n3\n4\n5\n0\nRunup for d = 0.189\n0.015\nmy 18\nmy 36\n0.010\nmy 72\n0.005\n0.000\n-0.005\n-0.010\n-0.015\n1\n2\n3\n4\n5\n0\nFigure 5-16: Gauge and runup results for d = 0.189.\n5.3.4 BP4: Single wave on simple beach - laboratory\nPMEL-135, pp. 5 & 30-33\nProblem description provided by Y. Joseph Zhan, at LeVeque (2011).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n159\n5.3.4.1 Description\nThis benchmark is the laboratory counterpart to BP1 (Single wave on a simple beach:\nAnalytic). A wave tank at the California Institute of Technology in Pasadena was used. The tank\nwas 31.73 m long, 60.96 cm deep, and 39.97 cm wide; the bottom of the tank consisted of\npainted stainless steel plates. An instrument carriage was mounted on rails that ran along the\nentire length of the tank, permitting the arbitrary positioning of measurement sites. A ramp was\ninstalled at one end of the tank to model the bathymetry of the canonical beach configuration -\ni.e., a constant-depth region adjoining a sloping beach. The beach ramp was sealed to the tank\nside walls and the beach slope corresponded to angle = arccot(19.85). Figure 5-17 presents\nthe computational domain used in this test.\n%\nR\nH\nd\nL\nX0\nFigure 5-17: Schematic of computational domain.\n5.3.4.2 Tasks\na. Compare numerically calculated surface profiles at t/T = 30:10:70 for the non-\nbreaking case H/d = 0.0185 with the lab data.\nb. Compare numerically calculated surface profiles at t/T = 15:5:30 for the breaking\ncase H/d = 0.3 with the lab data\nc. (Optional) Demonstrate the scalability of the code by using different d\nd. Compute maximum runups for at least one non-breaking and one breaking wave case.\n5.3.4.3 Problems encountered\nProblems that prevented completion of the benchmark were not encountered.\n5.3.4.4 What we did\nUsed g = 1 and no friction.\nThe bathymetry consisted of a deep plateau of constant depth d connected to a sloping\nbeach of angle B = arccot(19.85). Note that the toe of the beach was located at\nx = X0 = d cot\nThe initial waveform of the wave was given by\nn(x,0) = Hsech2(y(x - X1)/d)\nwhere L = arccosh( / (20))/y, X1 = X0 + L, and Y =\n(3H/4d).\nThe\nspeed\nof","National Tsunami Hazard Mitigation Program (NTHMP)\n160\nthe wave is given by:\nu(x,0) = - Jg/dn(x,0) = -\nFor the low amplitude case, we set d = 1 cm, H = 0.0185 cm, and ran the\ncomputations on an 800 X 2 grid, where the x domain spanned from x = -10 to x\n=\n60.\nFor the high amplitude case, we set d = 1 cm, H = 0.3 cm, and ran the computations on\na 1200 X 2 grid, where the X domain spanned from x = -10 to x = 60.\nWe allowed variable time stepping based on a CFL number of 0.9\n5.3.4.5 Results\nTasks a and b in Figure 5-18 and Figure 5-19 present the computed and measured surface\nprofiles for the low and high amplitude cases, respectively. Correspondence is excellent\nin the low amplitude case. In the high amplitude case the computed amplitude is smaller\nand the steepness greater than that of the measured wave - a consequence of the fact that\nthe experimental parameters violate the shallow water wave assumptions.\nTask c: This optional task was not addressed.\nTask d: Figure 5-20 and Figure 5-21 present the results for maximum runup computations\nfor the low amplitude and high amplitude wave cases. The results can be expressed as the\nnon-dimensional data pairs (H/d, R/d) = (0.0185, 0.085) and (0.3, 0.42) for the high and\nlow amplitude cases, respectively. The low amplitude result falls well within the scatter\nplot results of Zhan (LeVeque, 2011) presented in Figure 5-22, while the high amplitude\nresult falls somewhat below, as might be expected in light of the comments made in the\nTask a and b discussion, above.\n5.3.4.6 Lessons learned\nFor test cases in which amplitudes are SO large that the shallow water wave assumptions are\nviolated, it can be expected that computed and observed wave height and runup will not agree as\nwell as in cases characterized by amplitudes for which the shallow water wave assumptions are\nvalid.\nFor the amplitude H/d = 0.3 case, our observed runup of 0.42 agrees well with the\nexperimental results shown in Figure 5-22.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n161\nSurface at time t If 30.00000000\nSurface at time t If 40.00000000\n0.10\n0.10\n0.08\n0.08\n0.06\n0.06\n0.04\n0.04\n0.02\n0.02\n0.00\n0.00\n-0.02\n-0.02\n-0.04\n-0.04\n-5\n0\n5\n10\n15\n20\n-5\no\n5\n10\n15\n20\nSurface at time t\n30.00000000\nSurface at time t\n40.00000000\nif\n=\n0.12\n0.12\n0.10\n0.10\n0.08\n0.08\n0.06\n0.06\n0.04\n0.04\n0.02\n0.02\n0.00\n0.00\n-0.02\n-0.02\n-0.04\n-0.04\n-1.0\n-0.5\n0.0\n0.5\nI.D\n1.5\n2.0\n-1.0\n-0.5\n0.0\n0.5\n10\n1.5\n2.0\nSurface at time t = 50.00000000\nSurface at time t = 60.00000000\n0.10\n0.10\n0.08\n0.08\n0.06\n0.06\n0.04\n0.04\n0.02\n0.02\n0.00\n0.00\n-0.02\n-0.02\n-0.04\n-0.04\n0\n5\n10\n15\n20\n-5\n0\n5\n10\n15\n20\nSurface at time t\n50.00000000\nSurface at time t\n=\n60.00000000\n=\n0.12\n0.12\n0.10\n0.10\n0.06\n0.08\n0.06\n0.06\n0.04\n0.04\n0.02\n0.02\n0.00\n0.00\n-0.02\n-0.02\n-0.04\n-0.04\n-1.0\n-0.5\n0.0\n0.5\n10\n1.5\n2.0\n-1.0\n-0.5\n0.0\n0.5\n10\n1.5\n2.0\nSurface at time t = 70.00000000\n0.10\n0.08\n0.06\n0.04\n0.02\n0.00\n-0.02\n-0.04\n-5\n0\n5\n10\n15\n20\nSurface at time t\n70.00000000\n=\n0.12\n0.10\n0.08\n0.06\n0.04\n0.02\n0.00\n-0.02\n-0.04\n-1.0\n-0.5\n0.0\n0.5\n10\n1.5\n2.0\nFigure 5-18: Runup computations and lab measurements for the low amplitude case. In the paired\nplots for each time value, the bottom frame provides a zoomed view of the inundation area for the\nincident wave presented in the top frame.","National Tsunami Hazard Mitigation Program (NTHMP)\n162\nSurface at time t If 20.00000000\nSurface at time t = 15.00000000\n0.25\n0.25\n0.20\n0.20\n0.15\n0.15\n0.10\n0.10\n0.05\n0.05\n0.00\n0.00\n5\n10\n15\n20\n20\no\n-5\n0\n5\n10\n15\nSurface at time t = 20.00000000\nSurface at time t =\n15.00000000\n0.25\n0.25\n0.20\n0.20\n0.15\n0.15\n0.10\n0.10\n0.05\n0.05\n0.00\n0.00\n0\n1\n-4\n2\n-3\n-2\n-1\n3\n4\n-3\n-2\n-1\n0\n1\n2\n3\nSurface at time t = 30.00000000\nSurface at time t = 25.00000000\n0.25\n0.25\n0.20\n0.20\n0.15\n0.15\n0.10\n0.10\n0.05\n0.05\n0.00\n0.00\n5\n10\n15\n20\n0\n5\n10\n15\n20\n-5\n0\nSurface at time t = 30.00000000\nSurface at time t =\n25.00000000\n0,25\n0.25\n0.20\n0.20\n0.15\n0.15\n0.10\n0.10\n0.05\n0.05\n0.00\n0.00\n1\n4\n-3\n-2\n-1\n0\n1\n2\n3\n-4\n-3\n-2\n-1\nO\n2\n3\nFigure 5-19: Runup computations and lab measurements for the high amplitude case. In the paired\nplots for each time value, the bottom frame provides a zoomed view of the inundation area for the\nincident wave presented in the top frame.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n163\nMaximum Geoclaw Runup as a Function of Time\n0.10\nMaximum runup at (55,0.085)\n0.08\n0.06\n0.04\n0.02\n0.00\n-0.02\n-0.04\n0\n10\n20\n30\n40\n50\n60\n70\nTime(s)\nFigure 5-20: Maximum runup estimate of 0.085 cm for the low amplitude case, occurring at 55\nseconds of the computation.\nMaximum Geoclaw Runup as a Function of Time\n0.45\nMaximum runup at (40,0,42)\n0.40\n0.35\n0.30\n0.25\n0.20\n0.15\n0.10\n0.05\n0.00\n0\n20\n40\n60\nBO\n100\n120\nTime(s)\nFigure 5-21: Maximum runup estimate of 0.42 cm for the high amplitude case, occurring at 40 seconds\nof the computation.","National Tsunami Hazard Mitigation Program (NTHMP)\n164\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0\n0.0\n0.2\n0.4\n0.6\n0.8\nH/d\nScatter plot of nondimensional total of more maximum than 40 runup, experiments R/d, results. versus conducted by Y. Joseph Zhan\nnondimensional\nFigure wave described height, 5-22: H/d, at LeVeque resulting (2011). from Red a dots indicate the numerical\nincident\nand\n5.3.5 BP5: Solitary wave on composite beach - laboratory\n5.3.5.1 Problem specification\nPMEL- Problem description 37-39. provided beach by Elena laboratory/BP5 Tolkova at description.pdf LeVeque n.d.). (2011): BP05-ElenaT-\n- 135, pp. 6 &\nSolitary Hydraulics wave on composite Laboratory Problem using Description the nonlinear (Briggs, shallow linear water equations. equations and\n-\ncomparing This Coastal is to the laboratory same problem data rather as in than BP2, to but the analytic solution of the","MODEL BENCHMARKING WORKSHOP AND RESULTS\n165\n5.3.5.2 What we did\nWe solved the shallow water wave equation in Cartesian coordinates with g = 9.81 and\nno friction.\nTo specify the incoming wave from the left boundary of our computational domain we\nused the first ten seconds of measurements taken at Gauge 4. After ten seconds the left\nboundary switched to be a non-reflecting boundary. This boundary is selected because the\nend of our computational domain is not the end of the physical wave tank. The\nimplementation of these boundary conditions is described in Section 5.2.3.\nWe solved on a 600 X 2 grid with no adaptive mesh refinement.\n5.3.5.3 Gauge comparisons\nThe results for cases A, B, and C are shown in Figure 5-23, Figure 5-24, and Figure 5-25\nrespectively, where Gauge 11 is placed at the vertical wall.\n5.3.5.4 Convergence Study\nWe performed a test to see how well Clawpack converged to the gauge measurements as we\nincreased the number of grid cells in our computational domain. We found that as the number of\ngrid cells was increased that the computed solution converged and had a shock in approximately\nthe same location as in the gauge data. The results are shown in Figure 5-26.\n5.3.5.5 Lessons learned\nIn this benchmark problem, we found that using the measured data from Gauge 4 as\nboundary conditions on a shorter domain, starting at this gauge, provided more accurate results\nthan using the wavemaker position and a longer domain to model the entire tank. It appears that a\nsimilar assumption is made in the provided analytic solutions, as they match up nearly perfectly\nwith the lab data for the first ten seconds.\nOverall this benchmark problem is a good test for one-dimensional codes. Case C exhibits\ndispersion in the laboratory results not seen with the nonlinear shallow water equations.\nThe benchmark problem specifications could be improved by specifying the computational\ndomain and the specific data source that should be used to model the incoming wave.","166\nNational Tsunami Hazard Mitigation Program (NTHMP)\nSurface at gauge 4\nSurface at gauge s\nSurface at gauge 6\nGeoClaw\nGeoClaw\nGeoClaw\nLab Observation\nLab Observation\nLab Observation\n0.015\n0.015\n0.015\n0010\n0.010\n0.010\n0.005\n5.885\nsand\n0.000\n0.000\n0.000\nmar\nMC\n2\n30\n10\n20\n25\n:\n30\nJO\n29\ntime\nSurface at gauge ,\nSurface at gauge 8\nSurface at gauge 9\nGeoClaw\nGeoClaw\nGeoClaw\nLab Observation\nLab Observation\nLab Observation\n0015\nCOIS\n0015\n3910\n0.010\n0.010\n0.005\ncaus\n9.665\n0.000\n0.000\n0.000\n- with\n1\n10\n15\n20\n25\n1\n10\n20\n2\n:\n10\n70\n2\ntime\n-\nSurface at gauge 10\nSurface at gauge 11\nGeoClaw\nGeoClaw\nLab Observation\nCOIN\n0.015\n0.010\n0.010\n2.865\nand\nwas\n10\n1\n10\n15\nX\n15\n25\n1\na\n25\ntime\nTime\nFigure 5-23: Case A","MODEL BENCHMARKING WORKSHOP AND RESULTS\n167\nSurface at gauge 4\nSurface at gauge 5\nSurface at gauge 6\nGeol law\nGeoClaw\nGeol law\n0.01\n8.81\n0.07\nLab Data\nLab Data\nLab Data\n0.34\n5 04\n0.00\n0.05\nG.05\n0.05\n0.84\nC.D4\n0.04\nas\n0.03\nGDA\n0.07\n0.07\n0.00\n0.01\n0.01\n0.01\n0.00\n0.00\n0.00\n10\n24\n15\n10\n15\nA\ntime\nSurface at gauge ,\nSurface at gauge #\nSurface at gauge y\nGeoClaw\nGeoClaw\nGeoClaw\n001\n0.07\n0.07\nLab Data\nLab Data\nLab Data\n5.00\n8.00\n0.00\n0.01\n0.05\nues\n0.04\n0.04\n0.04\n0.07\nGUT\n0.01\nHOJ\n007\n0.07\n0.01\n0.01\n00\nDo\n0.00\n0.00\n0.00\nI\n30\nIS\nX\n29\n2\n10\na\n25\ntime\ntime\nSurface at gauge 10\nSurface of gauge 11\nGeoClaw\nGeoClaw\n0.07\nLab Data\n0.15\nDIDE\n0.05\n0.10\n0.04\nCLASS\n0.0%\n0.02\n001\n0.00\n0.00\n1\n10\n25\n:\n10\n15\nVO\n2\ntime\nFigure 5-24: Case B","168\nNational Tsunami Hazard Mitigation Program (NTHMP)\nSurface at gauge 4\nSurface at gauge 5\nSurface at gauge 6\nGeoClaw\nGeol law\nGeoClaw\nLab Data\nLab Data\nLab Data\n0.15\n0.15\n0.15\n0.30\n6.10\n0.10\non\nam\na.m.\n0.00\ncan\n0.00\n16\nB\nto\n25\nto\nis\n1\nX\nH\n3\ntime\ntime\ntime\nSurface at gauge ,\nSurface at gauge 8\nSurface at gauge 9\nGeoClaw\nGeoC law\nGeoClaw\nLab Data\nLab Data\nLab Data\n0.15\n0.15\n0.15\n0.10\n0.10\n0.10\non\num\non\n0.00\n0.00\n9.00\n5\n10\nSurface at gauge 10\nSurface at gauge 11\n0.20\nGeoClaw\nGeoc law\nLab Data\n0.15\n015\n0.10\n610\nour\n0.05\n0.00\n#:00\n20\n10\n15\n70\n29\ntime\nFigure 5-25: Case C","169\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nSurface at gauge 4\n0.20\nGeoClaw 200 Grid Points\nGeoClaw 400 Grid Points\nGeoClaw 600 Grid Points\nLab Data\n0.15\n0.10\n0,05\n0.00\n15\n20\n25\n5\n10\ntime\nFigure 5-26: Convergence plot for Gauge 4 in Case C\n5.3.6 BP6: Solitary wave on a conical island - laboratory\nThe Corps of Engineers website is the primary documentation for this benchmark\nproblem: http://chl.erdc.usace.army.mil/chl.aspx?p=s&a=Projects:35\nA problem description is also provided by Frank González at LeVeque (2011): BP06-\nFrankG-Solitary wave on a conical island/Description.pdf\nNumerous other publications also describe this experiment, in varying detail: (Synolakis\net al., 2007; Briggs et al., 1994; Liu et al., 1994; Briggs et al., 1995; Liu et al., 1995;\nBriggs et al., 1996; Fujima et al., 2000)\n5.3.6.1 Description\nThe goal of this benchmark problem (BP) is to compare computed model results with\nlaboratory measurements obtained during a physical modeling experiment conducted at the\nCoastal and Hydraulic Laboratory, Engineering Research and Development Center of the U.S.\nArmy Corps of Engineers. The laboratory physical model was constructed as an idealized\nrepresentation of Babi Island in the Flores Sea, Indonesia, to compare with Babi Island runup\nmeasured shortly after the 12 December 1992 Flores Island tsunami (Yeh et al., 1994).\n5.3.6.2 Problems encountered\nDetails of the laboratory setup and, therefore, of the computational domain could not be\ndetermined by the available documentation (above). The version of the domain used in\nthis report is presented in Figure 5-27; this specification of the domain was developed\nafter personal communication with Michael Briggs, U.S. Army Corps of Engineers, who","170\nNational Tsunami Hazard Mitigation Program (NTHMP)\nprovided additional information on physical details of the laboratory experiment.\nUnfortunately, it is not certain that accurate specification of details of the laboratory setup\nhave been resolved. In particular, the following items were not well documented and\nremain open to question: (a) the distance from the wavemaker face to the island center\nand (b) open gaps at each end of the wavemaker.\nErroneous entries were found in data files ts2a.txt, ts2b.txt and ts2cnewl.txt. Several\nentries of the letter 'M' triggered read-in error messages; they were replaced by linear\ninterpolation or extrapolation of neighboring values.\nInitial values for some laboratory data were non-zero, rather than the zero values\nexpected for initial wave basin conditions corresponding to still water.\n5.3.6.3 What we did\nUsed g = 9.81 and no friction.\nUsed the computational domain presented in Figure 5-27.\nUsed open boundary conditions for the top, bottom and right walls, and for the gaps\nbetween the ends of the wavemaker and the top and bottom walls.\nUsed inflow boundary conditions for the face of the wavemaker, as described in the\nModel Description section of this report.\nSimulated Cases A and C, each with three different grid sizes and resolution, to\ndemonstrate convergence: 28 X 24 (100 cm), 56 X 47 (50 cm) and 223 X 185 (12.5 cm)\nSimulated optional Case B; the results are not presented here, but they were submitted for\nanalysis and inclusion in the workshop summary report.\nAn additional computational experiment was conducted to document the effect of varying\ntwo model parameters on the results - the Manning coefficient of friction (M) and the\nDry Cell Depth (DCD) threshold. Several values of each parameter were used in this\nexperiment.\n5.3.6.4 Results\nRequirements of this benchmark test were to:\nDemonstrate that two wave fronts split in front of the island and collide behind it.\nDemonstrate convergence of the solution as the computational grid is refined.\nCompare computed water level with laboratory data for Cases A and C at gauges 1, 2, 3,\n4, 6, 9, 16, and 22 (files ts2a.txt, ts2cnewl.txt).\nCompare computed island runup with laboratory gauge data (files run2a.txt, run2c.txt)\nThe first benchmark requirement was satisfied, as seen in Figure 5-29 and Figure 5-30. Thus,\nfor Cases A and C we see in frames t = 30 to t = 36 seconds that the initial wave splits into two\nwave fronts in front of the island, which then collide behind the island.\nThe second benchmark requirement was satisfied, as seen in Figure 5-31, Figure 5-32, and\nFigure 5-33. The agreement between Lab and GeoClaw time series is seen to improve as the\ncomputational grid resolution is decreased from 100 to 12.5 cm. The most obvious manifestation\nof this convergence is the improved value of the first wave amplitude.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n171\nThe third benchmark requirement is satisfied by the comparisons presented in Figure 5-34\nand Figure 5-33. Good agreement is seen overall and, in particular, between computed and\nmeasured time series for the first wave. The agreement for later wave details becomes\nprogressively worse, as multiple reflections and refraction occur at the basin boundaries, the\nwavemaker face, and the island. Note that in some cases, the laboratory gauge data are\ncharacterized by non-zero initial values, which would be expected in the case of an initial\ncondition corresponding to still water in the wave basin (see, e.g., gauge 2 for Cases A and C).\nThe final benchmark requirement is satisfied by the runup values presented in Figure 5-35\nand Figure 5-36, in which good agreement is seen between the computed and measured runup on\nthe conical island.\n5.3.6.5 Sensitivity of runup to friction and 'Dry Cell Depth' parameters\nNine additional simulations of Case C were run on the 12.5 cm grid to test the sensitivity of\ncomputed runup values to variations in Manning's friction coefficient and the threshold depth for\nwhich a \"Dry Cell\" is identified by GeoClaw. The results are presented in Figure 5-37. We see\nthat runup estimates can be significantly affected by changes in the value of each parameter.\nBecause the friction term is a function of water depth, we also see that these effects vary spatially\nover the computational domain; for example, the frame (DCD, M) = (0.01, 0.0) provides the best\nfit for inundation values on the lower side of the conical island, but increasing the friction\ndegrades this fit and improves the fit to runup measurements directly behind the conical island -\nsee frames (DCD, M) = (0.01, 0.012) and (0.01, 0.025). Similar frictional effects are seen in the\nOkushiri Island field benchmark problem, in which runup computations with M = 0.0 and M =\n0.25 are compared with field observations (see Figure 5-51).\n5.3.6.6 Lessons learned\nAccurate specification of the computational domain is essential, and every effort should\nbe made to acquire this information.\nResults demonstrate that the long wave equations are adequate to describe the major\nfeatures of propagation, refraction, and runup observed in the laboratory experiment.\nEven with the unresolved details of the computational domain and lab data (i.e., non-zero\ninitial values) the available data still provide a good benchmark test.\nBoth the friction and the dry cell depth parameters have a significant, spatially variable,\neffect on runup computations.","National Tsunami Hazard Mitigation Program (NTHMP)\n172\nY\n25\n-\nd = 32 cm\n20\n-\n(12.96, 13.80)\n15\n+\n10\n-\n25 m\n23 m\n5\n0.38 m\n20\n15\n10\n5\nI\nI\nX\n29.3 m\nFigure 5-27: Basin geometry and coordinate system. Solid lines represent approximate basin and\nwavemaker vertical surfaces. Circles along walls and dashed lines represent rolls of wave absorbing\nmaterial. Note the gaps of approximately 0.38 m between each end of the wavemaker and the\nadjacent wall. Gauge positions are given in Figure 5-28.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n173\nGage\nX, m\nY, m\nZ, cm\nComment\nID\n1\n16.05\n32.0\nA: 5.76\n2\n14.55\n32.0\nB: 6.82\nIncident gage\n3\n13.05\n32.0\nC: 7.56\n4\n11.55\n32.0\n6\n9.36\n13.80\n31.7\n270 deg transect\n9\n10.36\n13.80\n8.2\n16\n12.96\n11.22\n7.9\n0 deg transect\n22\n15.56\n13.80\n8.3\n90 deg transect\nFigure 5-28: Coordinates of laboratory gauges 1, 2, 3, 4, 6, 9, 16, and 22.","National Tsunami Hazard Mitigation Program (NTHMP)\n174\n0.030\n0.030\nCase A. 12.5 cm grid. Surface at time t\n28.0000000\nCase A. 12.5 cm grid. Surface at time t\n30.00000000\n=\n=\n0.024\n0.024\n0.018\n0.018\n20\n20\n0.012\n0.012\n22\n22\n15\n15\n0.006\n0.006\n16\n16\n0.000\n0.000\n10\n10\n-0.006\n-0.006\n1234\n1234\n-0.012\n-0.012\n5\n5\n-0.018\n-0.018\n100\n101\n102\n100\n101\n102\n0\n-0.024\n0\n-0.024\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n-0.030\n-0.030\n0.030\n0.030\nCase A. 12.5 cm grid. Surface at time t\n32.0000000\nCase A. 12.5 cm grid. Surface at time t =\n34.00000000\n=\n0.024\n0.024\n0.018\n0.018\n20\n20\n0.012\n0.012\n15\n15\n0.006\n0.006\n16\n16\n0.000\n0.000\n10\n10\n-0.006\n-0.006\n1234\n1234\n-0.012\n-0.012\n5\n5\n-0.018\n-0.018\n100\n101\n102\n100\n101\n102\n0\n-0.024\n0\n-0.024\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n-0.030\n-0.030\n0.030\n0.030\nCase A. 12.5 cm grid. Surface at time t =\n36.0000000\nCase A. 12.5 cm grid. Surface at time t\n38.0000000\n=\n0.024\n0.024\n0.018\n0.018\n20\n20\n0.012\n0.012\nX\n22\n15\n15\n0.006\n0.006\n16\n16\n0.000\n0.000\n10\n10\n-0.000\n-0.000\n1234\n4234\n-0.012\n-0.012\n5\n5\n-0.018\n-0.018\n100\n101\n102\n100\n101\n102\n0\n-0.024\n0\n-0.024\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n-0.030\n-0.030\nFigure 5-29: Animation snapshots of Case A for the 12.5 cm resolution computational grid.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n175\n0.10\n0.10\nCase C. 12.5 cm grid. Surface at time t\n28.0000000\nCase C. 12.5 cm grid. Surface at time t\n=\n30.00000000\n=\n0.08\n0.08\n20\n0.06\n20\n0.06\n0.04\n0.04\n22\n22\n15\n15\n0.02\n0.02\n16\n16\n0.00\n0.00\n10\n10\n-0.02\n-0.02\n1234\n1234\n-0.04\n-0.04\n5\n5\n-0.06\n-0.06\n100\n101\n102\n100\n101\n102\n0\n-0.08\n0\n-0.08\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n-0.10\n-0.10\n0.10\n0.10\nCase C, 12.5 cm grid. Surface at time t =\n32.00000000\nCase C. 12.5 cm grid. Surface at time t\n34.0000000\n=\n0.08\n0.08\n20\n0.06\n20\n0.06\n0.04\n0.04\n15\n15\n0.02\n0.02\n16\n16\n0.00\n0.00\n10\n10\n-0.02\n-0.02\n. 1234\n1234\n-0.04\n-0.04\n5\n5\n-0.06\n-0.06\n100\n101\n102\n100\n101\n102\n0\n-0.05\n0\n-0.08\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n-0.10\n-0.10\n0.10\n0.10\nCase C. 12.5 cm grid. Surface at time t =\n36.0000000\nCase C. 12.5 cm grid. Surface at time t =\n38.0000000\n0.08\n0.08\n20\n0.06\n20\n0.06\n0.04\n0.04\n27\n22\n15\n15\n0.02\n0.02\n16\n16\n0.00\n0.00\n10\n10\n-0.02\n-0.02\n1234\n1234\n-0.04\n-0.04\n5\n5\n-0.06\n-0.06\n100\n101\n102\n100\n101\n102\n0\n-0.08\n0\n-0.08\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n-0.10\n-0.10\nFigure 5-30: Animation snapshots of Case C for the 12.5 cm resolution computational grid.","National Tsunami Hazard Mitigation Program (NTHMP)\n176\nCase C with 100 cm grid cells Surface at gauge 2\nCase C with 100 cm grid cells Surface at gauge\n1\n010\n1110\n0.05\n0.05\n0.00\n0.00\n-0.05\n-0.05\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-7 10\n-0.10\n40\nSE\nBD\n20\n30\n40\n50\n-\nTHE\nBOU\n20\n$\n75\ntarte\nbrite\nCase C with 100 cm grid cells Surface at gauge 4\nCase C with 100 cm gnd cells Surface at gauge 3\n0.10\n0.10\nass\nass\n0,00\n0.00\n-0.05\n4F05\nGeoClaw\nGeoClaw\nLab Date\nLab Data\nat 10.\n- 10\n20\n3D\n40\n50\nso\nno\n30\n20\n30\n40\n50\n00\nTO\nao\ntime\ntime\nCase C with 100 cm grid cells Surface at gauge o\nCase C with 100 cm grid cells Surface at gauge 9\n010\n010\n0.0%\n0.05\nthat\n000\n-0.05\n-0.05\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-0.10\n-0.10.\n20\n20\nat\n40\n50\n30\n40\n20\nfii\n60\nThe\nBO\n60\n80\nname\ntime\nCase C with 100 cm grid cells Surface at gauge 22\nCase C with 100 cm grid cells Surface at gauge 16\n0.10\n0.10\nnos\nILDS\n0.00\n0.00\n-0.05\n-0.05\nGeoClaw\nGeoClaw\nLAD Data\nLab Data\n-if 10\nat 10.\n30\n30\n40\n50\n60\nTO\nso\n20\n40\n50\nso\nTO\n80\n20\nBirther\nFigure 5-31: Comparison of laboratory gauge and GeoClaw time series for the Case C, 100 cm\nresolution grid.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n177\nCase C with 50 cm grid cells. Surface at gauge 1\nCase C with 50 cm grid cells Surface at gauge 2\n0.30\n0.10\n0.05\n0.03\nB.net\n0.00\n-005\n-1k05\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-0.10\n-0.10\n20\n30\n4D\n500\nDO\nTO\n30\n20\n30\n4D\nNO\neo\nTO\n30\ntime\ntime\nCase C with 50 cm grid cells, Surface at gauge 3\nCase C with 50 cm grid cells. Surface at gauge 4\n010\n0.10\n1\n0.03\n0.05\n0.00\n0.00\n-0.05\n-0.05\nGeoClaw\nGeoClaw\nLab Data\nLab Date\n-0.10\n0.10\n20\n30\n40\n50\n100\nTO\nall\n20\n30\n40\n30\nBC\nTO\nas\ntime\ntime\nCase C with 50 cm grid cells. Surface at gauge 6\nCase C with so cm grid cells. Surface at gauge 9\n0.30\n010\nI\n0,05\n0.05\none\n-0.05\n-0.05\nGeoClaw\nGepClaw\nLab Data\nLab Data\n-0.00\n30.\n20\n30\n40\n50\nSO\n60\nBK\n20\n30\n4b\n50\n60\n70\nBO\ntime\ntime\nCase C with 50 cm grid cells Surface at gauge in\nCase C with 50 cm grid cells Surface at gauge 22\nn\nID\nDUE\n0.05\n(kas\n0.00\n0.00\n-0.05\n-0.05\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-0.10\n-0.10\n2n\n30\n40\n30\nso\nTO\nand\n20\n30\n417\n50\nNO\nTO\nas\ntime\nUNIVERSITY\nFigure 5-32: Comparison of laboratory gauge and GeoClaw time series for the Case C, 50 cm resolution\ncomputational grid.","National Tsunami Hazard Mitigation Program (NTHMP)\n178\nCase C with 12. 5 cm grid cells Surface at gauge\n2\nCase C with 12.5 cm grid cells Surface at gauge 1\n010\n0.10\n0.05\n0.05\n0.00\n0.00\n-0.05\n-0.05\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-0.10\n-0.10\n40\nso\nIND\n70\nan\nTO\nand\n20\n30\n20\n30\n40\n50\nED\ntime\ntime\nCase C with 12 5 cm grid cells Surface at gauge 4\nCase C with 12.5 cm grid cells Surface at gauge 3\n0.10\n0.10\n0,05\n0.05\n0.00\n0.00\n-0.05\n-0.05\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-D 102\n-0 103\n4D\n50\n20\n20\n30\n60\n70\nBO\n30\n4D\n90\nBIO\n70\nBO\ntarrue\ntames\nCase C with 12.5 cm grid cells Surface at gauge 9\nCase C with 12.5 cm grid cells Surface at gauge 5\n0.10\n0.10\n0.03\n0.05\n0.00\n0.00\n-0.05\n-0.05\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n- 10\n-0.10\nTO\nab\n#\n30\n40\n50\nINC\n20\n30\n413\nNO\nAC\n70\n20\ntime\nLIMITAL\nCase C with 12.5 cm grid cells. Surface at gauge 22\nCase C with 12.5 cm grid cells, Surface at gauge 16\n0.10\n0.10\n0.05\n0.05\n0.00\n0.00\n-D.05\n-0.05\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-D\n111\n-0 10\n30\n4D\nso\n60\nto\n80\n40\nSo\n60\n70\n40\n20\n30\n20\ntime\ntime\nFigure 5-33: Comparison of laboratory gauge and GeoClaw time series for the Case C, 12.5 cm\nresolution computational grid.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n179\nCase A with 12.5 cm grid cells. Surface at gauga 1\nCase A with 12.5 cm grid cells. Surface at gauge 2\n0.03\n0.03\n0.02\n0.02\n0.01\n001\n0.150\n11,450\n-0.01\n-0.01\n-0.02\n-0.02\nGeoClaw\nGeoClaw\nLab Date\nLab Data\n-0.03\n-001\n20\n30\n4D\nso\neo\n70\nno\n20\n30\n40\nto\nSO\n70\nBO\ntime\ntime\nCase A with 12.5 cm grid cells Surface at gauge 3\nCase A with 12.5 cm grid cells Surface at gauge 4\n003\nDD\n0.02\n0.02\n0.01\n0.01\n0.00\n0.00\n-0.01\n-0.01\n-0.02\n-0.02\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-001\n-0.03\n20\n30\n40\n503\n80\n70\nAND\n20\n30\n413\nso\nN\nTO\nas\ntime\ncames\nCase A with 12.5 cm grid cells Surface at gauge 6\nCase A with 12 5 cm grid tells Surface at gauge 9\n0.03\n0.03\n6.02\nwas\n0.01\nno\n0.00\n0.00\n-0.01\n-0.01\n-0.02\n-D 02\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-0.03\n-001\n30\n40\nNO\n40\n50\n20\n60\n70\n#O\n20\n30\nRIO\n70\nBO\ntime\ntime\nCase A with 12 5 cm grid cells Surface at gauge 16\nCase A with 12.5 cm grid cells Surface at gauge 22\n0.00\n0.00\n0.02\n0.02\n0,01\n0.03\n0.00\n0.00\n-0.01\n-0.01\n-0.02\n-0.02\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n-0.035\n0.01\n2n\n30\n417\n50\never\nTO\nin\n20\n30\n413\n30\nSCD\nF\nan\ntime\ntime\nFigure 5-34: Comparison of laboratory gauge and GeoClaw time series for the Case A, 12.5 cm\nresolution computational grid.","National Tsunami Hazard Mitigation Program (NTHMP)\n180\nCase A, 12.5 cm grid: Island topography and dry zone\n16\n15\n14\n13\n12\n11\n10\nLab Data\nGeoClaw\n9\n10\n11\n12\n13\n14\n15\n16\n17\nFigure 5-35: Island runup for Case A, using a 12.5 cm resolution computational grid.","181\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nCase C, 12.5 cm grid: Island topography and dry zone\n16\n15\n14\n13\n12\n11\n10\nLab Data\nGeoClaw\n9\n13\n14\n15\n16\n17\n10\n11\n12\nFigure 5-36: Island runup for Case C, using a 12.5 cm resolution computational grid.","National Tsunami Hazard Mitigation Program (NTHMP)\n182\n0.0\n0.001\n0.01\nDry cell depth, m\nFigure 5-37: Island runup for Case C on a 12.5 cm grid, for differing values of Manning's friction\ncoefficient, M, and the 'Dry Cell Depth', DCD, threshold.\n5.3.7 BP7: Monai valley beach - laboratory\n5.3.7.1 Problem specification\nPMEL-135, pp. 6 & 45-46.\nProblem description provided by Dmitry Nicolsky, at LeVeque (2011): BP07-DmitryN-\nMonai valley beach/description.pdf\nThe original experiment is fully described by Matsuyama and Tanaka (2001).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n183\n5.3.7.2 What we did\nWe solved the nonlinear shallow water equations in Cartesian coordinates with g = 9.81\nand no friction.\nWe used the given initial wave to specify a boundary condition at the left boundary up to\ntime 20. This was done by filling ghost cells each time step at the left edge of the\ncomputational domain with depth values interpolated from the given time trace at x =\n0.\nMomentum values were determined as described in Section 5.2.3.3.\nAfter time 20, the boundary condition procedure switched to non-reflecting boundary\nconditions (see Section 5.2.3.1) at the left boundary, SO reflected waves exit. (Note that\nthe wave tank was much longer than the specified computational domain.)\nWe solved on 423 X 243 grid (same as bathymetry), with no adaptive mesh refinement.\nSolid wall boundary conditions were used at the top and bottom.\nWe also solved on 211 X 121 grid, coarser by roughly a factor of 2, for comparison as a\ntest of convergence.\n5.3.7.3 Gauge comparisons\nFigure 5-38 shows a comparison of the GeoClaw results with the laboratory values at the\nthree gauges requested, with both grid resolutions. The two resolutions give very comparable\nresults, indicating that the solution presented is close to a converged solution of the shallow\nwater equations. The results are, in general, a good match to the laboratory measurements.\n5.3.7.4 Frame comparisons\nSee Figure 5-39 and Figure 5-40 for comparisons of the Frames 10, 25, 40, 55, and 70 from\nthe overhead movie with GeoClaw results at roughly corresponding times. These results are from\nthe 423 X 243 grid (same as given bathymetry).\nThe movie had a rate of 30 fps, SO the frames are 0.5 seconds apart. However, it is not clear\nwhat the starting time was for Frame 1 relative to the simulation time. In the Benchmark\nDescription (LeVeque, 2011), it is stated that \"frame 10 approximately occurs at 15.3 seconds,\"\nbut then later \"it is recommend that each modeler find times of the snapshots that best fit the\ndata.\" We found reasonably good agreement starting at 15.0 seconds for Frame 1 and then taking\n0.5 second increments, as shown in Figure 5-39 and Figure 5-40.\nThe yellow dashed lines on the frames from the movie show the approximate shoreline, and\nwere provided as part of the benchmark specification (LeVeque, 2011). The actual shoreline\nlocation is, of course, somewhat ambiguous in the movie, and also in the computation. The\nfigures of the Geo-Claw computation show the shoreline two different ways:\nThe cells colored blue are finite volume cells where the fluid depth is greater than 0.0001\nm. Those colored green have less fluid or are dry.\nThe black dashed line is a contour line where depth = 0.002 m, which agrees better with\nthe movie frames and might be a depth that can actually be detected in the movie frames.\n5.3.7.5 Runup in the valley\nThe file DBS RUNUP.tx from the benchmark specification contains the runup at 3\nlocations as observed in 6 runs of the same wavetank experiment. The relevant location for runup","184\nNational Tsunami Hazard Mitigation Program (NTHMP)\nin the valley is the first point at x = 5.1575, y = 1.88 m. The six values given are 0.0875, 0.09,\n0.08, 0.09, 0.1, 0.09, with an average value of approximately 0.09.\nIn the computation, the maximum runup was observed at time t 2 16.5. This frame is\nshown in Figure 5-41 with a white dot at the location x = 5.1575,y = 1.88 and several contour\nlevels marked. The contour lines are at levels 0.01 m apart. The maximum runup appears to be\naround 0.08 to 0.10 m depending on what water depth is used to judge.\n5.3.7.6 Lessons learned\nThis problem has data that are fairly well specified, and has wave tank geometry that\nscales up to a reasonable physical tsunami problem (because it was designed by scaling\ndown a physical problem).\nSolutions to the shallow water equations fit the data quite well, as found both in our\nexperiments and by other modelers. This gives a reassuring test of the validity of shallow\nwater equations for real tsunamis.\nThis benchmark problem appears to be a good test for tsunami models. It has been widely\nused and many models have been shown to give results that agree quite well with the\nlaboratory measurements.\nThe laboratory test also appears to agree very well with the actual tsunami it was\ndesigned to model. Compare Figure 5-41 to Figure 5-53.\nThe benchmark problem specification could be improved by specifying the computational\ngrids that are to be used. We show results for a grid that matches the resolution of the\nbathymetry provided and a second computation at half the resolution, but this should be\nspecified as part of the problem.\nThe input data only go out to 20 seconds. The first waves are modeled well, but later\nwaves in the laboratory data (not shown here) are not seen in the computation. If a longer\ntime history were provided for the input data, it might be possible to match later waves\nbetter. Note that the computational domain is only part of the wave tank, which was\n205 m long (Matsuyama and Tanaka, 2001). Presumably it is impossible to obtain more\ndata at this point.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n185\nSurface at gauge 5\nSurface at gauge 5\n0.05\n0.05\nGeoClaw\nGeoClaw\ntopography\ntopography\n0.04\n0.04\nsea level\nsea level\nlab data\nlab data\n0.03\n0.03\n0.07\n0.02\n0.01\n001\n0,00\n0.00\n-0.01\n-0.01\n-0.02\n+0.02\n5\n10\n15\n20\n25\n70\n25\n0\n5\n10\n15\ntime\ntime\nSurface at gauge 7\nSurface at gauge 7\n0.05\n0.05\nGeoClaw\nGeoClaw\ntopography\ntopography\n0.04\n0.04\nsea level\nsea level\nlab data\nlab data\n0.03\n0.03\n0.07\n0.02\naol\n0.01\n0.00\n0.00\n-0.01\n-0.01\n-0.02\n-0.02\n15\n20\n25\n10\n15\n20\n25\n5\n10\n0\n5\ntime\ntime\nSurface at gauge 9\nSurface at gauge 9\n0.05\n0.05\nGeoClaw\nGeoClaw\ntopography\ntopography\n0.04\n0.04\nsea level\nsea level\nlab data\nlab data\n0.03\n0.03\n0.07\n0.07\n001\n001\n0.00\n0.00\n-0.01\n-001\n-0.02\n-0.02\n5\n10\n15\n20\n25\n0\n5\n10\n15\n70\n25\ntime\ntime\nFigure 5-38: Left column: on 423 X 243 grid (same as given bathymetry). Right column: 211 X 121 grid.","National Tsunami Hazard Mitigation Program (NTHMP)\n186\nMonai Valley at time t\n15.00000000\n=\n2.2\n2.1\n2.0\n1.9\n1.8\n1.7\n1.6\n187\n4.8\n4.9\n5.0\n5.1\n5.2\nMonai Valley at time t =\n15.50000000\n2.2\n2.1\n2.0\n1.9\n1.8\n1.7\n1.6\n117\n4.8\n4.9\n5.0\n5.1\n5.2\nMonai Valley at time t =\n16.00000000\n2.2\n2.1\n2.0\n1.9\n1.8\n1.7\n1.6\n1/7/\n4.8\n4.9\n5.0\n5.1\n5.2\nFigure 5-39: Left column: Frames 10, 25, and 40 from the movie. Right column: Zoomed view of\ncomputation.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n187\nMonai Valley at time t\n16.50000000\n=\n2.2\n2.1\n2.0\n1.9\n1.8\n1.7\n1.6\n117\n4.8\n4.9\n5.0\n5.1\n5.2\nMonai Valley at time t a\n17.00000000\n2.2\n2.1\n2.0\n1.9\n1.8\n1.7\n1.6\n1.4.7\n4.8\n4.9\n5.0\n5.1\n5.2\nFigure 5-40: Left column: Frames 55 and 70 from the movie. Right column: Zoomed view of\ncomputation.","National Tsunami Hazard Mitigation Program (NTHMP)\n188\nMonai Valley at time t =\n16.50000000\n2.2\n2=0.12\nz=0\n2.1\nz=0.01\n2.0\n1.9\n1.8\n1.7\n1.6\n1.5\n5.0\n5.1\n5.2\n4.7\n4.8\n4.9\nFigure 5-41: Maximum runup relative to observed location (white dot).\n5.3.8 BP8a: Old 3-D landslide - laboratory\nThere are plans to replace this benchmark problem with a new one. This has not yet\nhappened. This old benchmark problem consists of a wedge sliding on a plane beach. See Figure\n5-42.\n5.3.8.1 Problem specification\nPMEL-135, pp. 7 & 47-48 (Synolakis et al., 2007).\nThe original experiment is fully described on NOAA's benchmarking website which can\nbe found at http://nctr.pmel.noaa.gov/benchmark/Laboratory/","MODEL BENCHMARKING WORKSHOP AND RESULTS\n189\n5.3.8.2 What we did\nWe solved the nonlinear shallow water equations in Cartesian coordinates with g = 9.81\nand no friction.\nWe used the given laboratory data and problem setup to create our initial topography and\nbathymetry. While there were data provided up to time 20 S, we only conducted\nsimulations up to time 10 S, as was done on NOAA's benchmarking website. We\nspecified the movement of the wedge by using the time histories of the block motion\nprovided for the problem. In order to implement this, we adjusted the bathymetry every\ntime step to capture the wedge sliding down the linear beach. The slope of this linear\nbeach was 12. Due to the symmetry of the problem, we simplified the problem to half of\nthe domain of the tank, specifying an outflow or non-reflecting boundary condition at the\nright boundary SO reflected waves exit. We also specified a solid wall boundary condition\nat all other boundaries. (Note that the wave tank was much longer than the specified\ncomputational domain.) We used zero-order extrapolation, which generally gives a very\ngood approximation to non-reflecting boundary conditions as described in Section\n5.2.3.1. Solid wall boundary conditions are implemented as described in Section 5.2.3.2.\nSee Section 5.2.3 for more information on how these boundary conditions were specified.\nThe moving bathymetry is handled by recomputing = B(x),Yj,tn) in each time step,\nat the center of each finite volume grid cell, based on the specified bathymetry. This is the\nstandard approach for handling moving bathymetry in GeoClaw: the value is adjusted\nbut the fluid depth huj remains the same, SO that the water column is simply displaced\nvertically in any cell where the bathymetry changes. For bathymetry that is smoothly\nvarying in space and time this is considered a reasonable approach (see Section 5.3.3, for\nexample). Note, however, that no momentum is directly imparted to the water by the\nmoving bathymetry.\nFor this problem, the vertical face of the wedge makes this approach inadequate. The\ndiscontinuity in the moving bathymetry means that in each time step the bathymetry near\nthe face will gain an increment of 0.455 m, lifting the water in this cell by this amount.\nThis is not at all physical. Instead, the moving face should impart horizontal momentum\nto the water.\nGiven this inaccuracy and the full three-dimensional nature of the physical flow, we do\nnot expect to obtain very good comparisons computationally.\nWe solved on a 35 X 10 grid with 3 levels of adaptive mesh refinement. We refined in the\nx-and y-directions by a factor of 6 from levels 1 to 2 and levels 2 to 3. We refined in time\nby a factor of 3. We specified level 3 refinement on a rectangle with x values of [-0.4, 2]\nand y values of [0, 1].\nWe compared the simulated gauge data with the laboratory gauge data to determine\nGeoClaw's accuracy on this problem.\n5.3.8.3 Gauge comparisons\nFigure 5-43 shows a comparison of the GeoClaw results with the laboratory values at the\ntwo wave gauges and two runup gauges requested for case 1. The gauge data for gauge 1 is\ninitially very \"noisy\" but the overall behavior seems to be captured well. We suspect that\nbecause gauge 1 was in the wedge's path of travel and because the wedge was specified as part","National Tsunami Hazard Mitigation Program (NTHMP)\n190\nof our bathymetry, this created strong oscillations in our wave formations and an overshoot\nrelative to the lab results.\nFigure 5-44 shows a comparison of the GeoClaw results with the laboratory values at the\ntwo wave gauges and two runup gauges requested for case 2. As for case 1, the gauge data for\ngauge 1 is initially very \"noisy\" but the overall behavior seems to be roughly consistent with the\nlab results.\n5.3.8.4 Lessons learned\nIt is not clear that the shallow water equations are a good model for this problem. The\nflow should be fully three-dimensional around this sliding wedge and it is not clear that\nany depth-averaged model will do well.\nAt some distance away from the shore, the depth will be greater than wave length and the\nshallow water equations will no longer be valid.\nThe vertical face causes numerical difficulties.\nOverall, GeoClaw seems to model the surface elevations with respect to still water level\nwell for both cases. However, gauge 1 seems to have issues from shortly after the start of\nthe simulation to about 2 seconds. As mentioned earlier, it seems that this phenomenon is\nmore of a result of how the bathymetry is specified than GeoClaw's ability to model this\nlandslide. To smooth the data, one could try interpolating the data SO that the moving\nbathymetry is smooth instead of piecewise. This should greatly reduce the heavy\noscillations. Another approach would be to add a slope to the leading face of the wedge.\nThis would ensure a more gradual drop in bathymetry as the wedge propagates across the\nlinear beach.\nThis benchmark problem does not appear to be a good test for tsunami models. The\ndimensions do not seem to be reasonable relative to true submarine landslide problems.\nThe vertical face does not seem realistic and causes numerical difficulties.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n191\nCase 1: surface at 0.00 seconds\n1.5\n1.0\n2\n0.5\n1\n0.0\n-1\n0\n1\n2\n3\n4\n5\nCase 1: surface at 1.00 seconds\n1.5\n1.0\n0.5\n0.0\n-1\n0\n1\n2\n3\n4\n5\nCase 1: surface at 2.00 seconds\n1.5\n1.0\n2\n0.5\n0.0\n-1\no\n1\n2\n3\n4\n5\nFigure 5-42: Single grid 140 X 40 GeoClaw simulation of Case 1 to illustrate moving bathymetry and\ngauge locations.","National Tsunami Hazard Mitigation Program (NTHMP)\n192\nSurface at gauge 1\n0.2\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n0.0\n0.05\n-02\n0.00\n-0.4\n-0.05\n-0.6\n2\n4\n6\n8\n10\n2\n4\n6\n8\n10\n0\ntime\nSurface at gauge 2\nGeoClaw\nGeoClaw\nLab Data\nLab Data\n0.15\n0.05\n0.10\n0.05\n0.00\n0.00\n-0.05\n-0.05\n-0.10\n-0.15\n-0.10\n-0.20\n6\n8\n10\n2\n4\n6\nB\n2\n4\ntime\nFigure 5-43: Left column: Time histories of the surface elevation with respect to still water level for\ncase 1. Right column: Time histories of the runup measurements with respect to still water level for\ncase 1, at Runup gauges 2 and 3. Note: runup values are negated in this figure for both GeoClaw and\nlab data due to a programming glitch.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n193\nSurface at gauge 1\n0.2\nGeoClaw\nGeoClaw\n0.1\nLab Data\nLab Data\n0.06\n0.0\n0.04\n-0.1\n0.02\n-0.2\n0.00\n-0.3\n-0.02\n-0.4\n-0.04\n-0.5\n-0.06\n-0.6\n2\n4\n6\n8\n10\n4\n6\n8\n10\ntime\nSurface at gauge 2\nGeoClaw\nGeoClaw\nLab Data\n0.10\nLab Data\n0.04\n0.02\n0.05\n0.00\n0.00\n-0.02\n-0.05\n-0.04\n-0.10\n-0.06\n2\n4\n6\n8\n10\n0\n2\n4\n6\n8\n10\ntime\nFigure 5-44: Left column: Time histories of the surface elevation with respect to still water level for\ncase 2. Right column: Time histories of the runup measurements with respect to still water level for\ncase 2, at Runup gauges 2 and 3. Note: runup values are negated in this figure for both GeoClaw and\nlab data due to a programming glitch.\n5.3.9 BP9: Okushiri Island - field\nPMEL-135, pp. 8 & 48-53 (Synolakis et al., 2007).\nA problem description is provided by Frank González at LeVeque (2011) BP09-FrankG-\nOkushiri island/Description.pdf\nNumerous other publications also describe this event, in varying detail: (DCRC, 1994;\nHTSG, 1993; Kato and Tsuji, 1994; Takahashi et al., 1995; Yeh et al., 1994).\n5.3.9.1 Description\nThe goal of this benchmark problem (BP) is to compare computed model results with field\nobservations of the 1993 Okushiri Island tsunami.\n5.3.9.2 Problems encountered\nTwo basic problems were encountered:\nPoor quality of the computational bathymetric/topographic grids.","National Tsunami Hazard Mitigation Program (NTHMP)\n194\nInaccurate spatial registration of field observational data with the model grids.\n5.3.9.3 What we did\nUsed g = 9.81 and Manning coefficient 0.025 for the friction term. We also ran many\nof the tests with no friction for comparison.\nUsed bathy/topo grids and source grid for the Disaster Control Research Center solution\nDCRC17a. Dmitry Nicolsky provided improved versions of the originals developed by\nKansai University, in which severe misalignments in the original data were reduced (but\nnot eliminated).\n5.3.9.4 Problem Requirements\nRequirements of this benchmark test were to compute:\n1. Runup around Aonae.\n2. Arrival of the first wave to Aonae.\n3. Two waves at Aonae approximately 10 min apart; the first wave from the west, the\nsecond from the east.\n4. Water level at Iwanai and Esashi tide gauges.\n5. Maximum modeled runup distribution around Okushiri Island.\n6. Modeled runup height at Hamatsumae.\n7. Modeled runup height at a valley north of Monai.\n5.3.9.5 Results\nFigure 5-45 through Figure 5-48 show results of one computation where AMR is used to\nconcentrate grid points near the southern Aonae peninsula and (Requirements 1, 2, 3). The\nrectangular boxes show regions of refinement. The coarsest grid is a 60 X 60 grid on a 1-degree\nsquare as shown in Figure 5-45. Five levels of refinement are used going down by factors 2, 4, 4,\nand 6 from each level to the next. In this computation, Level 4 is only allowed on the southern\nhalf of Okushiri Island and Level 5 only around the Aonae peninsula.\nFigure 5-46 shows a zoom on the island and Figure 5-47 a further zoom on the peninsula.\nArrival of the first wave at Aonae (Requirement 2) is seen from the west at about t = 5 minutes.\nThe second major wave arrives from the east at about 10 minutes.\nFigure 5-48 shows the inundation level on the peninsula. The color scale indicates the\nmaximum depth of water recorded at each point on a fixed grid that is placed around this region.\nThis can be compared to the photographs shown in Figure 5-49.\nA slight modification of this run was done to focus on the Hamatsumae region just east of\nthe peninsula. Figure 5-50 shows the maximum inundation in this region.\nThe bottom panel of Figure 5-51 shows the runup at various other points around the island\nas measured by the team of Y. Tsuji (top panel), along with values computed using GeoClaw.\nFigure 5-52 shows a scatter plot of the correlation between the observations and the computed\nvalues. The GeoClaw values were obtained by placing a small fixed grid around each\nobservation point and recording the maximum water depth at each point on this grid at each\ntimestep of the computation, using the built-in feature of GeoClaw. The maximum depth over\ntime can also be accumulated at these points and updated each time step. Plots of the maxima","MODEL BENCHMARKING WORKSHOP AND RESULTS\n195\nover these grids give a visualization of the maximum extent of inundation. Such plots are shown\nin Figure 5-48 and Figure 5-53, with 4-meter contours. For most other observation points\ncontours of topography at 2-meter increments were plotted in order to better estimate the\nmaximum runup in a small region centered about each observation point.\nFigure 5-53 shows the inundation map for the Valley north of Monai, with 4-meter contour\nlines (Requirement 7). Inundation to roughly 32 m is observed, in accordance with observations.\nFor this run a finer grid was used in the region around the value (refining by a factor 24 rather\nthan 6 in the level 5 grid), and the finer scale bathymetry provided in this region was used.\nRequirement 4 is the comparison of observed and computed water levels at the Iwanai and\nEsashi tide gauge stations; the analog records are shown in Figure 5-54, taken from Takahashi et\nal. (1995). The digitized tide gauge data are compared with the GeoClaw time series in Figure\n5-55 and Figure 5-56. At Iwanai, the digitized tide gauge record is clearly undersampled\n(compare Figure 5-54 and Figure 5-55), but does capture the peaks and troughs of the analog\nrecord. We see that the first wave arrival time and the overall wave amplitudes are comparable,\nbut that the GeoClaw tsunami waves are about half the period of the waves recorded by the\nIwanai tide gauge. Considering the regularity of the long train of waves in the tide gauge record,\nit is probable that longer period resonance modes at Iwanai were excited by the incident tsunami;\nif so, then higher resolution bathy/topo grids would be required to capture these resonant\noscillations in a numerical simulation. At Esashi, it appears that the digitized tide record reflects\nthe main features of the analog record (compare Figure 5-54 and Figure 5-56). However, the\nstrange shape of the analog wave form makes it likely that there are problems with the tide gauge\nrecord; the record suffers from either mechanical/electronic, or damage by debris, or simply a\ndamped or mismatched response function in the tsunami frequency band. In spite of this, the first\narrival and timing of the first two tsunami waves are in good correspondence, though the\namplitude and individual wave forms are not.\nSurface at 3.00 minutes\nSurface at 6.00 minutes\n42.4\n42.4\n42.2\n42.2\n42.0\n42.0\n41.8\n41.8\n41.6\n41.6\n139.2\nFigure 5-45: Full computational domain for one simulation, in which AMR grids are focused near the\nAonae peninsula at the south of Okushiri Island.","National Tsunami Hazard Mitigation Program (NTHMP)\n196\n5.3.9.6 Lessons learned\nThis benchmark problem requires much more work to qualify as a credible test of tsunami\ninundation models. We have little confidence in:\nThe quality of the bathy/topo computational grids. A number of mismatches and\ndiscontinuities still exist in the system of grids.\nThe accuracy of the geospatial registration of observational data with model latitude and\nlongitude positions. Figure 5-51 presents the observation locations of each of three field\nsurvey teams - Professor Yoshinobu Tsuji, Tokyo University (Tsuji), the United States-\nJapan Cooperative Program on Natural Resources (UJNR) and the Tohoku University\n(Tohoku) teams. The bathy/topo computational grids were adjusted to match the positions\nof the Tsuji observations, but it is clear that this created a systematic error in the\nregistration of the grids with the Tohoku field observations and, in all likelihood, the\nUJNR field observations. As another example, there appear to be discrepancies in the\nseveral field team reports of the latitude and longitude of the highest runup observed, i.e.,\nthe value of over 30 m in a\n\"\nsmall valley north of Monai\n\"\nSuch positioning errors\ncan be critical with respect to accurate comparisons of observed and computed runup.\n5.3.9.7 Recommendations\nThe Okushiri event tsunami runup and eyewitness reports remains one of the most valuable\ndatasets for model comparisons in existence, but the quality of this dataset must be improved to\nqualify as a credible benchmark problem. We recommend that an effort be supported to\nDevelop a high quality bathy/topo grid system.\nResolve ambiguities and discrepancies currently found in the various team data reports,\nand improve the geospatial registration of observed and modeled values.\nProvide adequate documentation of the resulting benchmark problem dataset.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n197\nSurface at 3.00 minutes\nSurface at 4.00 minutes\nSurface at 5.00 minutes\n42.2\n42.2\n2340\n2396\n1390\nSurface at 6.00 minutes\nSurface at 7.00 minutes\nSurface at 8.00 minutes\n42.2\n42.2\n2394\nSurface at 9.00 minutes\nSurface at 10.00 minutes\nSurface at 11.00 minutes\n42.2\n42.2\n42.2\nFigure 5-46: Zoom on Okushiri Island.","National Tsunami Hazard Mitigation Program (NTHMP)\n198\nEta on FG 1 at time = 5.28 minutes\nEta on FG 1 at time = 9.67 minutes\n42.055\n20\n42.055\n20\n16\n16\n12\n12\n42.05\n42.05\n8\n8\n4\n4\n42.045\n0\n42.045\n0\n-4\n-4\n-8\n-8\n42.04\n42.04\n-12\n-12\n-16\n-16\n42.035\n-20\n42.035\n-20\n139.45\n139.44\n455\nFigure 5-47: Zoom on the Aonae peninsula showing the first wave arriving from the west and the\nsecond from the east. Color map shows elevation of sea surface. 4-meter contours of bathymetry and\ntopography are shown.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n199\nInundated region for t <= 12.59 minutes\n42.055\n10\no\n9\n8\n42.05\n7\n6\n42.045\n5\n4\n3\n42.04\n2\n1\n42.035\n0\n139.44\n139.445\n139.45\n139,455\n139.46\nFigure 5-48: Inundation map of the Aonae peninsula. Color map shows maximum fluid depth over\nentire computation at each point. 4-meter contours of bathymetry and topography are shown.\nFigure 5-49: Photographs of the Aonae peninsula taken shortly after the event.\nLeft: From http://www.usc.edu/dept/tsunamis/hokkaido/aonae.html\nRight: From hhttp://nctr.pmel.noaa.gov/okushiri_devastation.html, credited to Y. Tsuji.","200\nNational Tsunami Hazard Mitigation Program (NTHMP)\nInundated region for t <= 14.00 minutes\n42.065\n42.06\n42.055\n42.05\n139.455\n139.485\n139.49\nFigure 5-50: Inundation map of the Hamatsumae neighborhood just east of the Aonae peninsula.\nColor map shows maximum fluid depth over entire computation at each point, with the same color\nscale as Figure 5-48. 4-meter contours of bathymetry and topography are shown.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n201\nUJNR Stat ions\nTsuji Stations\nTohoku Stations\n42.25\n<<<\ndos\non\n42.2\nit\n(210)\n3%\nas\n279\n42.15\n(213\n(250\n432\n420\n1226\n400\nd07\n#12\nThe\nand\n<06\n42.1\ndes\n011\nJage\nd12\n80\nd22\n42.05\n42\n139.4\n139.45\n139.5\n139.55\n139.55\n139.4\n139.45\n139.5\n139.55\n139.4\n139.45\n139.5\nLongi tude\nLongi tude\nLongi tude\n30 meters\n20 meters\n10 meters\n0 meters\nFigure 5-51: Top: Locations of field observations by three independent field survey teams, relative to\nthe computational bathy/topo grid system. Only the observations of Tsuji (left figure) were used in\nthis study due to misregistration of the other two data sets. Bottom: Measured and computed runup\nat 21 points around Okushiri Island where measured by the Tsuji team. Red circles are measurements;\ngreen diamonds are estimated from the computation. Two red circles at the same point represent\nestimates of minimum and maximum inundation observed near the point. Two green diamonds at the\nsame point represent values estimated when the model was run with and without bottom friction\n(Manning coefficient 0.025). The runup computed with bottom friction is the smaller value.","202\nNational Tsunami Hazard Mitigation Program (NTHMP)\nScatter plot of observed vs. computed\n35\n30\n25\n20\n15\n10\n5\n0\n0\n5\n10\n15\n20\n25\n30\n35\nGeoClaw runup (m)\nFigure 5-52: Scatter plot illustrating the correlation between measured and computed values for the\nvalues shown in Figure 5-51.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n203\nInundated region for t V =\n270.000\n42.101\n10\n9\n42.1005\n8\n42.1\n7\n42.0995\n6\n42.099\n5\n4\n42.0985\n3\n42.098\n2\n42.0975\n1\n42.097\n0\n139.4255\nFigure 5-53: Inundation map of the valley north of Monai. Color map shows maximum fluid depth\nover entire computation at each point. 4-meter contours of bathymetry and topography are shown.\nCompare to Figure 5-41 showing the related wave tank simulation.","National Tsunami Hazard Mitigation Program (NTHMP)\n204\nIwanai\n1\n0\n-1\n0\n60\n120\n180\n240\n300\n360\nTime (min)\nEsashi\n1\n0\n-1\n40\n0\n10\n20\n30\nTime (min)\nFigure 5-54: Analog tide gauge records at Iwanai and Esashi.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n205\nSurface at gauge 900\n4\n3\n2\n1\n0\n-1\n-\n-2\n-3\nsurface\ntopography\n-4\n0\n900\n1800\n2700\n3600\ntime\nFigure 5-55: Iwanai digitized tide gauge record (black line) and GeoClaw (blue line) time series.","National Tsunami Hazard Mitigation Program (NTHMP)\n206\nSurface at gauge 901\n4\n3\n2\n1\n0\n-1\n-2\n-3\nsurface\ntopography\n-4\n1800\n2700\n3600\n0\n900\ntime\nFigure 5-56: Esashi digitized tide gauge record (black line) and GeoClaw (blue line) time series.\nFurther remarks and suggestions\n5.4\nComments on the current benchmark problems\n5.4.1\nThe current set of benchmark problems do a good job of testing some aspects of a tsunami\nsimulation code. However, there are some shortcomings that have become apparent to us in the\ncourse of working through these problems and that could be addressed in the future.\nSeveral of the problems are not well specified in terms of the data provided. These\ndifficulties have been noted in our discussion of the individual problems.\nIn some problems there is not a clear description of how the simulation is supposed to be\nset up, or how the accuracy of the solution should be quantified. Allowing flexibility is\nperhaps necessary to allow for differences in capabilities of existing simulation codes, but\nwe feel this could be better constrained. In particular, there is no indication in the\nproblem descriptions of what grid resolution should be used. There are requirements to\n\"demonstrate convergence,\" but for practical applications it is important to know that\nadequately accurate results can be obtained on grids with a reasonable resolution in terms\nof computing time constraints.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n207\nFriction parameter values significantly affect model runup computations at both\nlaboratory and field experiment scales (see Figure 5-37 and Figure 5-51). Different\nmodels may use different formulations of the friction terms and some consideration\nshould be given to testing and reporting on the effect of different values of the friction\nparameter when conducting benchmark problem simulations.\nRunup computational algorithms frequently employ a parameter that sets a threshold\nlevel below which a cell is declared dry, i.e., the Dry Cell Depth. As in the case of\nfriction, the value of this parameter can also affect the resulting runup computation (see\nFigure 5-37), and some consideration should be given to testing and reporting on how\nvariations in such model parameters can affect runup computation results for benchmark\nproblems.\nCurrently there is no requirement to report CPU time required to solve each problem.\nThis would be interesting information to have when comparing different approaches, and\nwould be a necessary component if the benchmark did require a particular grid resolution,\nbecause grid resolution alone is not necessarily a good indication of computational effort\nneeded.\nSome of the benchmark problems compare numerical results to the exact solutions of the\nlinear or nonlinear shallow water equations. For these problems, any code that solves the\nequations in question should converge to the correct solution, but it may also be of\ninterest to know how rapidly the error goes to zero, and how good the solution is on\nunder-resolved grids that may be more representative of what would be used in actual\ntsunami simulations.\nOther benchmark problems require comparison with wave tank experiments. In some\ncases (e.g., with breaking waves) it cannot be expected that the code converges to the\nexperimental results because the equations used in tsunami modeling are only\napproximations. Different codes may use different approximations and SO this\ncomparison may be valuable, but because many codes use the same shallow water\napproximations, for these problems it would be valuable to have some agreement as to\nwhat a \"converged solution\" of the shallow water equations looks like.\nBP8a, studied in Section 5.3.8, does not seem to scale well as a model of a real landslide,\nand has difficulties associated with the vertical face that are not likely to be seen in real\nlandslides, where momentum transfer is probably secondary to the vertical displacement\nof the water column in creating a tsunami. The short wavelength waves generated by the\ndiscontinuity in this problem also accentuate the need to use dispersive corrections in\norder to obtain reasonable approximations. While dispersive terms may be very important\nfor some submarine landslide generated tsunamis, there may be other cases where they\nare less important and the ability to model such events with shallow water equations is\nimportant because these equations can be solved with explicit methods that are often\norders of magnitude faster than implicit dispersive solvers. (This may be particularly\nimportant in doing probabilistic hazard assessment requiring a large number of\nscenarios.) We believe it would be valuable to develop landslide benchmarks that model\nevents such as a large mass failure on the continental slope, which the current\nbenchmarks do not address.","National Tsunami Hazard Mitigation Program (NTHMP)\n208\nThe Okushiri Island BP9 requires comparison to field observations. Beyond the technical\ndifficulties with datasets for this problem, there are also questions regarding (a) the\naccuracy of the earthquake source being used, (b) the accuracy of some of the field\nobservations and tide gauges. This makes it difficult to assess the accuracy of a\nsimulation code. This will always be a problem in comparing with actual events, but our\nfeeling is that to form a meaningful benchmark there should be some agreement in the\ncommunity regarding how large the deviation between the computed solutions and the\nobservations are expected to be, rather than an expectation that results converge to\nobservations as the grid is refined.\n5.4.2 Suggestions for future benchmark problems\nWe believe there are other possible benchmark problems that should be considered by the\ncommunity in order to better test tsunami simulation codes.\nThe one-dimensional test problems currently involve exact solutions that are themselves\ndifficult to calculate numerically, e.g., requiring numerical quadrature of Bessel\nfunctions. It is very useful that tabulated values of these solutions have been provided.\nHowever, rather than using limited tests for which such \"exact\" solutions are known, it\nmight be preferable to carefully test a 1-D numerical model and show that it converges,\nand then use this with very fine grids to generate reference solutions. Fully converged\nsolutions could be provided in tabulated form and could be as accurate as needed. It\nwould then be possible to generate a much wider variety of test problems. In particular,\nmore realistic bathymetry could be used, for example on the scale of the ocean, a\ncontinental shelf and beach, rather than modeling only a beach.\nHigh-accuracy one-dimensional reference solutions can also be used to test a full two\ndimensional code, by creating bathymetry that varies in only one direction at some angle\nto the two-dimensional grid. A plane wave approaching such a planar beach would\nideally remain one-dimensional, but at an angle to the grid this would test the two-\ndimensional inundation algorithms.\nThis idea can be extended to consider radially symmetric problems, such as a radially\nsymmetric ocean with a Gaussian initial perturbation at the center. The waves generated\nshould reach the shore at the same time in all directions, but the shore will be at different\nangles to the grid in different locations and it is valuable to compare the accuracy in\ndifferent locations. The two-dimensional equations can be reformulated as a one-\ndimensional equation in the radial direction (with geometric source terms) and a very fine\ngrid solution to this problem can be used as a reference solution.\nFeatures could also be added at one point along the shore and this location rotated to test\nthe ability of the code to give orientation-independent results. Some GeoClaw results of\nthis nature are presented in Berger et al. (2011) and LeVeque et al. (2011).\nA very simple exact solution is known for water in a parabolic bowl, in which the water\nsurface is linear at all times but the water sloshes around in a circular motion. This is a\ngood test of wetting and drying as well as conservation. See for example Gallardo et al.\n(2007), Thacker (1981), and the test problem in GeoClaw: http://www.clawpack.org/\nclawpack-4.x/apps/tsunami/bowl-slosh/README.html","MODEL BENCHMARKING WORKSHOP AND RESULTS\n209\nExtensive observations are available for recent events such as Chile 2010 or Tohoku\n2011, including DART buoys, tide gauges, and field observations of inundation and\nrunup. It would be valuable to develop new benchmark problems based on specific data\nsets, including specified bathymetry and earthquake source (or seafloor displacement).\n5.5\nReferences\nBale DS, LeVeque RJ, Mitran S, Rossmanith JA. 2002. A wave propagation method for\nconservation laws and balance laws with spatially varying flux functions. SIAM J. Sci.\nComput., 24:955-978.\nBehrens J, LeVeque RJ. 2011. Modeling and simulating tsunamis with an eye to hazard\nmitigation. SIAM News, 44(4), May.\nBerger MJ, Calhoun DA, Helzel C, LeVeque RJ. 2009. Logically rectangular finite volume\nmethods with adaptive refinement on the sphere. Phil. Trans. R. Soc. A, 367:4483-4496.\nBerger MJ, Colella P. 1989. Local adaptive mesh refinement for shock hydrodynamics. J.\nComput. Phys., 82:64-84.\nBerger MJ, George DL, LeVeque RJ, Mandli KT. 2011. The GeoClaw software for depth-\naveraged flows with adaptive refinement. Adv. Water Res.\nBerger MJ, eVeque RJ. 1998. Adaptive mesh refinement using wave-propagation algorithms\nfor hyperbolic systems. SIAM J. Numer. Anal., 35:2298-2316.\nBerger MJ, Oliger J. 1984. Adaptive mesh refinement for hyperbolic partial differential\nequations. J. Comput. Phys., 53:484-512.\nBriggs M. No date (n.d.) Runup of solitary waves on a vertical wall. Coastal Hydraulics\nLaboratory http://chl.erdc.usace.army.mil/chl.aspx?p=s&a=Projects:36. Accessed n.d.\nBriggs MJ, Synolakis CE, Harkins GS. 1994. Tsunami runup on a conical island. In Proc. of\nWaves T Physical and Numerical Modelling, 21-24 August 1994, Vancouver, Canada,\npages 446-455, August (1994).\nBriggs MJ, Synolakis CE, Harkins GS, Green D. 1995. Laboratory experiments of tsunami runup\non a circular island. Pure Appl. Geophys., 144:569-593.\nBriggs MJ, Synolakis CE, Harkins GS, Green DR. 1996. Runup of solitary waves on a circular\nisland. In Proceedings of the Second International Long-Wave Runup Models, Friday\nHarbor, Washington, 12-16 September (1995), pages 363-374.\nDCRC (Disaster Control Research Center). 1994. Tsunami Engineering Technical Report No.\n11. Tohoku University, Tohoku, Japan.\nEnet F, Grilli ST. 2003. Experimental study of tsunami generation by three-dimensional rigid\nunderwater landslides. Journal of Waterway, Port, Coastal, and Ocean Engineering,\n133:442-454.\nFuhrman DR, Madsen PA. 2009. Tsunami generation, propagation, and run-up with a high-order\nboussinesq model. Coastal Eng., 56:747-758.\nFujima K, Briggs MJ, Yuliadi D. 2000. Runup of tsunamis with transient wave profiles incident\non a conical island. Coastal Engineering Journal, 42:175-195.\nGallardo JM, Parés C, Castro M. 2007. On a well-balanced high-order finite volume scheme for\nshallow water equations with topography and dry areas. J. Comput. Phys., 227:574-601.","National Tsunami Hazard Mitigation Program (NTHMP)\n210\nGeoClaw documentation. 2011. http://www.clawpack.org/geoclaw.\nGeorge DL. 2006. Finite Volume Methods and Adaptive Refinement for Tsunami Propagation\nand Inundation. PhD thesis, University of Washington.\nGeorge DL. 2008. Augmented Riemann solvers for the shallow water equations over variable\ntopography with steady states and inundation. J. Comput. Phys., 227(6):3089-3113,\nMarch.\nGeorge D. 2010. Adaptive finite volume methods with well-balanced Riemann solvers for\nmodeling floods in rugged terrain: Application to the Malpasset dam-break flood (France,\n1959). Int. J. Numer. Meth. Fluids.\nGeorge DL, Iverson RM. 2010. A two-phase debris-flow model that includes coupled evolution\nof volume fractions, granular dilatency, and pore-flud pressure. Submitted to Italian\nJournal of Engineering, Geology and Environment.\nGeorge DL, LeVeque RJ. 2006. Finite volume methods and adaptive refinement for global\ntsunami propagation and local inundation. Science of Tsunami Hazards, 24:319-328.\nHTSG (Hokkaido Tsunami Survey Group). 1993. Tsunami devastates japanese coastal region.\nEos Trans. Am. Geophys. Union, 74:417, 432.\nKato K, Tsuji Y. 1994. Estimation of fault parameters of the 1993 hokkaido-nansei-oki\nearthquake and tsunami characteristics. Bull. Earthq. Res. Inst., Univ. Tokyo, 69:39-66.\nLangseth JO, LeVeque RJ. 2000. A wave-propagation method for three-dimensional hyperbolic\nconservation laws. J. Comput. Phys., 165:126-166.\nLeVeque RJ. 1996. High-resolution conservative algorithms for advection in incompressible\nflow. SIAM J. Numer. Anal., 33:627-665.\nLeVeque RJ. 1997. Wave propagation algorithms for multi-dimensional hyperbolic systems.\nJournal of Computational Physics, 131:327-335.\nLeVeque RJ. 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge University\nPress.\nLeVeque RJ. 2007. Finite Difference Methods for Ordinary and Partial Differential Equations,\nSteady State and Time Dependent Problems. SIAM, Philadelphia.\nLeVeque RJ. 2010. A well-balanced path-integral f-wave method for hyperbolic problems with\nsource terms. J. Sci. Comput. To appear, www.clawpack.org/links/wbfwave10.\nLeVeque RJ. 2011. rjleveque / inthmp-benchmark-problems https://github.com/rjleveque/nthmp-\nbenchmark-problems. Accessed n.d.\nLeVeque RJ, Berger MJ, et al. No date (n.d.) Clawpack. http://www.clawpack.org. Accessed n.d.\nLeVeque RJ, George DL. 2004.. High-resolution finite volume methods for the shallow water\nequations with bathymetry and dry states. In PL-F. Liu, H. Yeh, and C. Synolakis,\neditors, Proceedings of Long-Wave Workshop, Catalina, volume 10, pages 43-73. World\nScientific. http://www.amath.washington.edu/~rjl/pubs/catalina04/.\nLeVeque RJ, George DL, Berger MJ. 2011. Tsunami modeling with adaptively refined finite\nvolume methods. Acta Numerica, pages 211-289.\nLiu PL-F, Cho Y-S, Briggs MJ, Kânoglu U, Synolakis CE. 1995. Runup of solitary waves on a\ncircular island. J. Fluid Mech., 302:259-285.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n211\nLiu PL-F, Cho Y-S, Fujima K. 1994. Numerical solutions of three-dimensional run-up on a\ncircular island. In Proceedings of the International Symposium: Waves - Physical and\nNumerical Modeling, 21-24 August 1994, Vancouver, Canada, volume 2, pages 1031-\n1040.\nMandli KT. 2011. Finite Volume Methods for the Multilayer Shallow Water Equations with\nApplications to Storm Surges. PhD thesis, University of Washington.\nMatsuyama M, Tanaka H. 2001. An experimental study of the highest run-up height in the 1993\nHokkaido Nansei-oki earthquake tsunami. ITS Proceedings, pages 879-889.\nRoe PL. 1981. Approximate Riemann solvers, parameter vectors, and difference schemes. J.\nComput. Phys., 43:357-372.\nSynolakis CE, Bernard EN, Titov VV, Kanoglu U, Gonzáles F. 2007. Standards, criteria, and\nprocedures for NOAA evaluation of tsunami numerical models. NOAA Tech. Memo.\nOAR PMEL-135, http://nctr.pmel.noaa.gov/benchmark/SP_3053.pdf\nTakahashi Tomo., Takahashi Take., Shuto N, Imamura F, Ortiz M. 1995. Source models for the\n1993 hokkaido nansei-oki earthquake tsunami. Pure and Applied Geophysics, 144:74-\n767.\nThacker WC. 1981. Some exact solutions to the nonlinear shallow water wave equations. J.\nFluid Mech., 107:499-508.\nYeh H, Liu P, Briggs M, Synolakis C. 1994. Propagation and amplification of tsunamis at coastal\nboundaries: Nature, 372:353-355.\nYeh H, Liu P, Synolakis C, editors. 1996. Benchmark Problem 4. The 1993 Okushiri Data,\nConditions and Phenomena. Singapore: World Scientific Publishing Co. Pte. Ltd..\nZhang S, Yuen DA, Zhu A, Song S. 2011. Use of many-core architectures for high-resolution\nsimulation of Tohoku 2011 Tsunami waves. Submitted to Supercomputing 2011\nconference.","212\nNational Tsunami Hazard Mitigation Program (NTHMP)\n6\nMOST (Method Of Splitting Tsunamis) Numerical Model\nElena Tolkova\nJoint Institute for the Study of the Atmosphere and Ocean, University of Washington/PME\netolkova@u.washington.edu\n6.1\nModel description\nThe Method Of Splitting Tsunamis (MOST) numerical model (Titov and Synolakis, 1998) is\na set of code developed to simulate three processes of tsunami evolution: generation by an\nearthquake, transoceanic propagation, and inundation of dry land. MOST has been used by the\nNOAA Center for Tsunami Research (NCTR) in the development of tsunami inundation forecast\nmodels. These forecast models are supported by an ocean-wide database of 24-hour-long\ntsunami wave propagation simulations of numerous tsunami scenarios, each generated by\nhypothetical earthquakes from unit sources covering worldwide subduction zones (Gica et al.,\n2008). As a tsunami wave propagates across the ocean and reaches tsunameter observation sites,\nthe forecasting system uses a data inversion technique coupled with these pre-computed tsunami\ngeneration scenarios to deduce the tsunami source, in terms of the earthquake unit sources\nidentified in the database (Percival et al., 2009). A linear combination of the pre-computed\ntsunami scenarios is then used to determine the offshore tsunami waves and to incorporate\nsynthetic boundary conditions of water elevation and flow velocities into site-specific forecast\nmodels. The main objective of the forecast model is to provide an accurate estimate of wave\narrival time, wave height, and inundation extent at a particular location within minutes of the\nearthquake, in advance of wave arrival. Previous and present development of forecast models in\nthe Pacific (Titov and González, 1997; Wei et al., 2008; Titov, 2009; Tang et al., 2009) have\nvalidated the accuracy and efficiency of each forecast model currently implemented in the real-\ntime tsunami forecast system.\n6.1.1 Governing equations\nThe MOST model is intended to solve a system of depth-averaged continuity and\nmomentum equations:\n(1a)\n(1b)\n=\nwhere +d(x,,22) and n and d refer to the free surface displacement and\nundisturbed water depth, respectively, V(x1,x2,t) is the depth-averaged velocity vector in the\nhorizontal plane, g is the acceleration due to gravity, and y=n2 = is the Manning friction\ncoefficient. The solution is based on the method of fractional steps (Yanenko, 1971; Durran,\n1999) which reduces the 2-D problem to two 1-D problems by setting either spatial derivative to","MODEL BENCHMARKING WORKSHOP AND RESULTS\n213\nzero. The majority of the benchmark problems below were simulated in the Cartesian\ncoordinates (x1,x2) with friction set to zero. In those settings, the two 1-D problems yielded by\nsplitting are:\n(2a)\n(2b)\n8/24/2012-0\n(2c)\nand\n(3a)\n(3b)\n(3c)\nThe MOST method is a numerical technique of solving system (1) by computing solutions\nfor the next time step of the simplified systems (2) and (3) sequentially.\nThe first two equations in (2) (and in (3) analogically) can be re-written in terms of Riemann\ninvariants\nand eigenvalues V1 I gh\n(4a)\n(4b)\nAdding equations (4) together simplifies to (2b). Subtracting the two equations simplifies to\n(2a). Equations (4) and (2c) are the set of equations that are solved in MOST to propagate the\nsolution in the x1 -direction. The corresponding primitive variables V1 and h are then obtained\nfrom the characteristics p and q as\nV1 =(p+q)/2,h=(p-q)2/16g, =\n(5)\nand propagation in the X2 -direction is computed according to (3), with the first two equations\nbeing solved in terms of\n=","National Tsunami Hazard Mitigation Program (NTHMP)\n214\nThe numerical scheme for solving the characteristic equations (4) in one dimension, with\n(optionally) varying space step followed by runup on a beach, has been given in Titov and\nSynolakis (1995). Details of the MOST numerical scheme can also be found in Titov and\nSynolakis (1998) and Burwell et al. (2007). The MOST model solver is second-order accurate in\nAx, except on boundaries.\nTwo-dimensional extension of the 1-D model in either Cartesian or spherical coordinates is\ndescribed in Titov and Synolakis (1998) and Titov and González (1997). MOST adaptation to an\narbitrary orthogonal curvilinear coordinate system is developed in Tolkova (2008).\n6.1.2 Wetting/drying interface\nThe MOST model inundation algorithm uses a horizontal projection of the water level in the\nlast wet node onto the beach to adjust the length of the last wet cell, as discussed in Titov and\nSynolakis (1998). As the moving shoreline position steps over a fixed grid node, the algorithm\nwould add or exclude a wet node SO the wet area would expand at runup and shrink at rundown.\nA node is considered dry when the water height at this node is less than a threshold.\n6.1.3 Boundary conditions\nTwo types of boundary conditions are implemented in MOST: totally reflective and totally\ntransparent, as discussed in detail in Titov and Synolakis (1998).\nAt either boundary, the outgoing Riemann invariant is advanced for the next time step tn\nusing a 1st order difference scheme applied to the solution at tn-1, while the incoming invariant is\nspecified according to the type of the boundary. It is assumed that the flow is subcritical, and\ntherefore 2, > 0 and the p-invariant propagates to the right, while A2 < 0 and the q-invariant\npropagates to the left.\nOn the reflecting left boundary, the p-invariant at the 1st node is set to -q at each time step.\nThis procedure implies V1 = 0 at the 1st node, which sets the reflective wall immediately behind\nthe 1st node.\nOn the transparent left boundary, the p-invariant is assigned to\np1(t), =\n(6)\nwhere di is depth at the 1st node, and u, and n, represent external forcing through given\nvelocity and elevation.\nIn general, values u, and n, would not coincide with the solution in the 1st node, except\nwhen they are consistent with the value of the outgoing Riemann invariant, that is, satisfying\nu,(tn) -2/g(d,+n,(1,))=q,(1,) -\n(7)\nThe last equation reflects the fact that, for an initial-boundary-value problem in the quadrant\nx>0,1>0, only one boundary condition (either velocity or elevation) is needed on the t-axis\n(Mei, 1983).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n215\nIn particular, if near the left boundary at tn there is no wave inside the domain propagating\ntoward the boundary, then q,(t,)=-2/gd, and condition (7) defines the velocity of the\nincoming wave, given its elevation (or vice versa) as:\n(8)\nu,\nIn the absence of any inflow through the left boundary, u, and n, are set to zero.\nLikewise, in the N-th (last) node, the p-invariant at tn is computed with the solution at tn-1 in\nthe last two nodes, while the q-invariant at tn is set to either -Pn (t) (reflective right boundary),\nor\n(9)\n-\n(transparent right boundary), where dn is depth at the last node, and Ur and N, represent forcing\nthrough the right boundary. As long as no wave escapes through the right boundary, velocity and\nelevation in a wave entering the domain from the right satisfy\n(10)\n=\n6.1.4 Numerical dispersion and dissipation in MOST\nDesigned to solve non-dispersive shallow-water equations, the MOST model nevertheless\npossesses significant numerical dispersion and, in some cases, dissipation properties for the\nhigher wave numbers. The numerical dispersion can be used to mimic physical dispersion. As\nshown in Burwell et al. (2007), MOST dispersive and dissipative properties are determined by a\nsingle parameter B=VO/C, also referred to as the Courant number, where o=Vgd and\nC = Ax/At\nMOST yields the non-dispersive solution traveling at exactly long-wave velocity V only\nwhen B=1. This is also the edge case, because at B>1 the numerical scheme becomes\nunstable.\nWhen 1/52B<1, the harmonic component with the highest wave number of r / Ax\nalways travels at phase speed C . At the same time, in this range of B , the components with\nhigher wave numbers are subject to numerical dissipation, which is greater the closer B is to\n1/52\nWhen B<1/J2, , the shortest wave of 24x-length always travels with zero phase speed.\nNumerical dissipation also drops down and remains lower, the smaller the value of B. For\nB < 0.3, MOST phase velocities (function of wave number k) are very close to the limit\n(11)","216\nNational Tsunami Hazard Mitigation Program (NTHMP)\nFor well-resolved waves (kAx<<1), MOST velocities coincide with Linear Wave Theory\n(LWT) phase velocities\ntanh kd\n(12)\nwhen Ax = d , whereas the time step should be just small enough to provide for low B.\nMaintaining a specific Ax and Courant number in a basin with variable depth is straight-\nforward in the 1-D case. The grid spacing should vary as Vd and the Courant number should be\nas close as possible to 1, to keep the solution stable, if the objective is to approach a nonlinear\nshallow water (NSW) solution over the wide range of wave numbers. If, however, the wave is\nwell resolved (containing only low wave numbers), then keeping B close to 1 is not essential.\nThe grid spacing is set equal to depth while the Courant number is kept low, whenever a\nbenchmark test calls for matching laboratory measurements taken in dispersive settings.\nEmulating LWT dispersion is not possible with finer (spacing less than depth) grids.\n6.2\nBenchmark problems\nThe MOST model has been benchmarked against benchmark problems 1, 2, 4-7, and 9,\nwhich focus on verifying the simulations of wave propagation and subsequent inundation with\nlaboratory, analytical, and field data. The results are presented below.\n6.2.1 BP1: Solitary wave on a simple beach - analytical\nThe solitary wave on a simple beach problem for BP1 is focused on modeling runup of a\nnon-breaking solitary wave of height H, normally incident to a plane sloping beach. The\nsimulation was performed using a 384-node grid, which encompassed a 45d-long segment of\nconstant depth d connected to a 50d-long slope of angle = arccot(19.85). In simulations, the\ndepth of the flat part of the basin was d=1 m and the initial wave height was H = 0.0185d. The\ngrid spacing was set to 1 m (depth) over the flat segment, then varied as , but not less than\n0.1d. The time increment At = 0.03 S provided for = 0.1. Results with dimension of length are\nexpressed in units of depth; time is expressed in units of T = Jd/g.\nComputed time histories of surface elevation at x/d=0.25 and x / d = 9.95 VS. the\nanalytical solution are shown in Figure 6-1. Water level profiles at t /t = 35, 45, 55, and 65 are\nshown in Figure 6-2. Maximum computed runup is 0.08d.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n217\n0.05\n0.025\n/d=0.25\nx/d=9.95\n0.02\n0.04\n0.015\n0.03\n0.01\n0.02\n0.005\n0.01\n0\n0\n-0.005\n-0.01\n-0.01\n-0.015\n-0.02\n20\n40\n60\n80\n100\n120\n20\n40\n60\n80\n100\n120\nT\nT\nFigure 6-1: Time histories at locations x/d = 0.25 (left) and x/d = 9.95 (right), analytical NSW solution\n(black), and numerical solution (red), in dimensionless units.\n0.1\n35\n0.05\n0\n-0.05\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n0.1\n45\n0.05\n0\n-0.05\n14\n16\n-2\n0\n2\n4\n6\n8\n10\n12\n0.1\n55\n0.05\n0\n-0.05\n10\n-2\n0\n2\n4\n6\n8\n12\n14\n16\n0.1\n65\n0.05\n0\n-0.05\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\nx/d\nFigure 6-2: Water level profiles at t/T = 35, 45, 55, and 65: analytical NSW solution (black), and\nnumerical solution in grid nodes (red dots), in dimensionless units.\n6.2.2 BP4: Solitary wave on a simple beach - laboratory\nThis problem is a laboratory counterpart to BP1. It was simulated under the same settings as\nBP1 for initial wave heights H = 0.0185₫ and H = 0.3d.\nComputed water level profiles are shown in Figure 6-3 and Figure 6-4. Maximum computed\nrunup is 0.08d for H/d = 0.0185, and 0.265d for H/d= 0.3.","218\nNational Tsunami Hazard Mitigation Program (NTHMP)\n0.1\n30\n0.05\n0\n-0.05\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n0.1\n40\n0.05\n0\n-0.05\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n0.1\n50\n0.05\n0\n-0.05\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n0.1\n60\n0.05\n0\n-0.05\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n0.1\n70\n0.05\n0\n-0.05\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\nx/d\nFigure 6-3: Water level profiles for H/d = 0.0185 at t/T = 30, 40, 50, 60, and 70: measurements (black\ndots) and numerical solution in grid nodes (red dots), in dimensionless units.\n0.4\n15\n0.2\n0\n-0.2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n0.4\n20\n0.2\n0\n-0.2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n0.4\n25\n0.2\n0\n-0.2\n-4\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n0.4\n30\n0.2\n0\n-0.2\n-8\n-6\n-4\n-2\n0\n2\n4\n6\n8\n10\n12\nx/d\nFigure 6-4: Water level profiles for H/d = 0.3 at t/T = 15, 20, 25, and 30: measurements (black dots)\nand numerical solution in grid nodes (red dots), in dimensionless units.\n6.2.3 BP2/5: Solitary wave on a composite beach - analytical, laboratory\nThe objective of this benchmark is modeling propagation of incident and reflected solitary\nwaves of different heights (cases A, B, and C) in a basin of complex geometry.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n219\nThe experiment was modeled as a 1-D problem. The computational domain starts at gauge 4\n(whose position X4 is different among the three cases A, B, and C). The simulation is initiated\nwith the boundary input of no(t) and vo(t) at X = X4, where n is taken from an actual record of a\ndirect pulse (prior to 275 s) at the incident gauge. Given n, velocity V is computed according to\nequation (8).\nThe grid parameters were selected with the intent to use numerics to mimic physical\ndispersion; that is, the grid spacing dx equals depth (but is no smaller than 0.05 m), starting with\nd = 0.218 m on the deep end. The resulting grid size was 72 nodes (case A), 66 nodes (case B),\nand 64 nodes (case C). The time increment was 0.02 S (A), 0.003 S (B), and 0.01 S (C), which\nkept the Courant number under 0.3 everywhere.\nNo attempt was made to match the analytical LSW solution because, as can be seen from the\nfigures, nonlinearity affects propagation speed even for the lowest pulse (case A). For this\nproblem, the MOST model does not have a \"linear\" mode for meaningful comparisons with the\nLSW solution.\nComputed time histories of surface elevation at gauges 4-10 VS. the lab measurements and\nanalytical solutions for the three heights of the incident pulse are shown in Figure 6-5 through\nFigure 6-7; time histories by the wall are shown in Figure 6-8. The maximum simulated runup at\nthe wall was 2.24 cm / 0.1d (case A), 18.2 cm / 0.84d (case B), and 22.4 cm / 1.03d (case C).\n7.8\n7.2\n6.6\n6\n5.4\n4.8\n4.2\n3.6\n3\n2.4\n1.8\n1.2\n0.6\n0\n271\n273\n275\n277\n279\n281\n283\n285\n287\n289\nsec\nFigure 6-5: Time histories at gauges 4-10 (top to bottom), provided by measurements (black),\nanalytical LSW solution (green), and numerical solution (red). Case A.","National Tsunami Hazard Mitigation Program (NTHMP)\n220\n65\n60\n55\nM\n50\nManco\n45\n40\n35\ncm\n30\n25\n20\n15\n10\n5\n0\n270\n272\n274\n276\n278\n280\n282\n284\n286\nsec\nFigure 6-6: Time histories at gauges 4-10 (top to bottom), provided by measurements (black),\nanalytical LSW solution (green), and numerical solution (red). Case B.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n221\nAnd\n130\n120\nAND\n110\n100\n90\n80\nM\n70\ncm\n60\nArea\n50\n40\nMhat\n30\n20\n10\n0\n268\n270\n272\n274\n276\n278\n280\n282\n284\nsec\nFigure 6-7: Time histories at gauges 4-10 (top to bottom), provided by measurements (black),\nanalytical LSW solution (green), and numerical solution (red). Case C.\n2.5\n20\n50\n2\n40\n15\n1.5\n30\n10\n1\n20\n5\n0.5\n10\n0\n0\n0\n-0.5\n-5\n-10\n275\n280\n285\n290\n295\n275\n280\n285\n290\n265\n270\n275\n280\n285\nsec\nsec\nsec\nFigure 6-8: Time histories at the wall, provided by analytical LSW solution (green) and numerical\nsolution (red). Left to right: cases A, B, and C.","National Tsunami Hazard Mitigation Program (NTHMP)\n222\n6.2.4 BP6: Conical island\nThis benchmark problem was simulated using two nested grids. The outer grid enclosed the\nentire basin area starting with the wavemaker at x = 0. Radiative boundary conditions were\napplied on the other three sides (basin walls). The size of the grid was 25 m X 30 m, or 84 X 101\nnodes at dx = 30 cm spacing. The time increment of dt = 0.01 S with the depth-equal spacing\nprovided for imitating physical dispersion as the pulse propagates from the wavemaker toward\nthe island.\nBoundary input to the domain was provided according to the paddle velocities (computed by\ndifferentiating the given paddle trajectory x(t), with some smoothing applied) complemented\nwith elevation computed according to equation (8). The boundary conditions were applied at x =\n0. Because the paddle stroke was under 30 cm, that is, under the grid cell size in all the cases, no\nactual reduction in the size of the basin was accounted for. After the stroke was completed (in\n158 S in case A, 123 S in case B, and 65 S in case C), reflective boundary conditions were applied\nat x = 0.\nRunup onto the island was simulated within a finer 10 m X 10 m grid enclosing the island.\nBoundary input on the four sides was provided from the outer grid. Two resolutions for the inner\ngrid were considered: 201 X 201 nodes at dx = 5 cm spacing and dt = 0.02 S time increment, and\n401 X 401 nodes at dx = 2.5 cm spacing and dt = 0.01 S time increment.\nSimulated time histories at eight gauges VS. the laboratory measurements for the three cases\nare shown in Figure 6-9 through Figure 6-11. Time histories at the incident gauges 1-4 were\nobtained in the outer grid, while those at the gauges around the island (6, 9, 16, and 22) were\ncomputed in the inner grid with dx = 5 cm resolution. The boundary of the inundated area around\nthe island, computed at 5 and 2.5 cm grid spacing for each of the three cases, is shown in Figure\n6-12 through Figure 6-14.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n223\n2\ngage 1\n2\ngage 6\ncm\n0\n0\n-2\n-2\n25\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\n2\ngage 2\n2\ngage 9\ncm\n0\n0\n-2\n-2\n25\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\n2\ngage 3\n2\ngage 16\n0\n0\n-2\n-2\n25\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\n2\ngage 4\n2\ngage 22\ncm\n0\n0\n-2\n-2\n25\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\nsec\nsec\nFigure 6-9: Time histories at gauges 1-4, 6, 9, 16, and 22: simulated with MOST (red) and laboratory\nmeasurements (black). Case A.","National Tsunami Hazard Mitigation Program (NTHMP)\n224\n6\n6\n4\n4\ngage 6\ngage 1\n2\n2\n0\n0\n-2\n-2\n-4\n-4\n40\n45\n50\n25\n30\n35\n40\n45\n50\n25\n30\n35\n6\n6\n4\n4\ngage 9\ngage 2\n2\n2\n0\n0\n-2\n-2\n-4\n-4\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\n25\n30\n6\n6\n4\n4\ngage 16\ngage 3\n2\n2\n0\n0\n-2\n-2\n-4\n-4\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\n25\n6\n6\n4\n4\ngage 22\ngage 4\n2\n2\n0\n0\n-2\n-2\n-4\n-4\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\n25\nsec\nsec\nFigure 6-10: Time histories at gauges 1-4, 6, 9, 16, and 22: simulated with MOST (red) and laboratory\nmeasurements (black). Case B.\n10\n10\ngage 6\ngage 1\n5\n5\n0\n0\n-5\n-5\n25\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\n10\n10\ngage 2\ngage 9\n5\n5\n0\n0\n-5\n-5\n40\n45\n50\n25\n30\n35\n40\n45\n50\n25\n30\n35\n10\n10\ngage 3\ngage 16\n5\n5\n0\n0\n-5\n-5\n50\n25\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n10\n10\ngage 22\ngage 4\n5\n5\n0\n0\n-5\n-5\n25\n30\n35\n40\n45\n50\n25\n30\n35\n40\n45\n50\nsec\nsec\nFigure 6-11: Time histories at gauges 1-4, 6, 9, 16, and 22: simulated with MOST (red) and laboratory\nmeasurements (black). Case C.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n225\nMaximal Run-up Height, case A\n5\n16\n4\n15\n3\n14\ncm\nu\nX\n2\n13\ny\n1\n12\n0\n11\n12\n13\n14\n15\n0\n60\n120\n180\n240\n300\n360\ndegree\nm\nMaximal Run-up Height, case A fine\n5\n16\n4\n15\n3\n14\ncm\nX\n2\n13\ny\n1\n12\n0\n11\n12\n13\n14\n15\n0\n60\n120\n180\n240\n300\n360\ndegree\nm\nFigure 6-12: Maximum runup height around the island: computed with MOST (red) at dx = 5 cm (top)\nand dx = 2.5 cm (bottom), and laboratory measurements (black triangles). Case A.","National Tsunami Hazard Mitigation Program (NTHMP)\n226\nMaximal Run-up Height, case B\n10\n16\n8\n15\n6\n14\ncm\nE\nA\nX\n4\n13\ny\n2\n12\n0\n240\n300\n360\n11\n12\n13\n14\n15\n0\n60\n120\n180\ndegree\nm\nMaximal Run-up Height, case B fine\n10\n16\n8\n15\n6\n14\n4\nE\nA\nA\nX\n4\n13\ny\n2\n12\n0\n11\n12\n13\n14\n15\n0\n60\n120\n180\n240\n300\n360\ndegree\nm\nFigure 6-13: Maximum runup height around the island: computed with MOST (red) at dx = 5 cm (top)\nand dx = 2.5 cm (bottom), and laboratory measurements (black triangles). Case B.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n227\nMaximal Run-up Height, case C\n20\n16\n15\n15\n14\nE\n10\nX\n13\ny\n5\n12\n0\n360\n11\n12\n13\n14\n15\n0\n60\n120\n180\n240\n300\ndegree\nm\nMaximal Run-up - Height, case C fine\n20\n16\n15\n15\n14\n104\nu\nX\n13\ny\n5\n12\n0\n300\n360\n11\n12\n13\n14\n15\n0\n60\n120\n180\n240\ndegree\nm\nFigure 6-14: Maximum runup height around the island: computed with MOST (red) at dx = 5 cm (top)\nand dx = 2.5 cm (bottom), and laboratory measurements (black triangles). Case C.\n6.2.5 BP7: Runup onto a laboratory model of Monai Beach\nComputations were performed with 0.01 S time step using the original grid of 393 X 244\nnodes at 1.4 cm spacing, covering the region of 5.488 m X 3.402 m. Reflective walls were\nimposed at two sides, y = 0 and y = 3.402. Boundary input into the domain was provided\naccording to the given elevation time history at x = 0, complemented with x-direction velocity\ncomputed according to (8).\nShown in Figure 6-15 and Figure 6-16 are time histories at the gauges and snapshots of the\nsimulation to be compared with the laboratory data and recordings.\nThe area shown in the snapshots is 4.6 m L/2}\n(13)","234\nNational Tsunami Hazard Mitigation Program (NTHMP)\nInitially, the pulse no(x) is localized in the central segment. Its evolution in the central segment\ncan be described as a sequence of partial reflections at X = + L/2 (see Figure 6-21). Thus the\nsolution in the central segment is identical to the solution for a series no(x) of pulses centered at\nX = 0,+1,+2L,+3L, ... propagating over the constant depth d:\n(14)\nk1\nwhere R1 and R2 are the amplitude reflection coefficients on the left and right discontinuity. The\ncorresponding time history at x = 0, for a symmetric pulse, is:\n- -\n(15)\nwhere r=1/T,T=L/Jgd,g being the gravity acceleration.\nIn the shallow-water approximation, the reflection coefficients are:\n(16)\nt\nx R1\nx R2\n3T\n2T\nR1*R2*R1\nR2*R1\nR1\nR2\nR1*R2\nR2*R1*R2\n-3L\n-2L\n-L\no\nL\n2L\n3L\nx\nFigure 6-21: Pulse trajectory in x-t plane.\n6.3.3 Procedure\nFor a given pulse no(x) localized within the central segment, model its evolution over the\ndepth (13).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n235\nUnder the same computational settings, model the evolution of (14) over the constant\ndepth d.\nCompare the solutions in the central segment, in particular the time histories at X = 0.\nComment: comparing the model outputs as above, rather than comparing a model output to\nthe analytical solution (15), ensures that the discrepancies between the solutions are only due to\nreflections in the first solution. Other numerical artifacts, if any, contribute equally to both model\noutputs.\n6.3.4 MOST example\nThe next example illustrates the suggested test performed with the MOST model, with d1 set\nequal to d2, and, consequently, R1 = R2 = R. Four cases were considered, representing different\nratios of the outer depths (d1) to the depth of the central segment (d) in the segmented basin:\ndid = 0.8, 0.6, 0.4, 0.2. Figure 6-22 shows the corresponding pulse series (14) whose\npropagation over the constant depth d in the flat basin is to be modeled. Ideally, within the\ncentral segment, the solution for a single pulse in the segmented basin (13) should coincide with\nthe solution for the corresponding pulse series in the flat basin.\nAs seen in Figure 6-23, in the first three cases, the MOST model reproduced the theoretical\nreflection very closely. In the last case, representing the largest change in depth, the MOST\nmodel over-estimated the amplitude of the reflected wave.","National Tsunami Hazard Mitigation Program (NTHMP)\n236\n1\n1\n0.6 d\n0.8 d\n0.8\n0.8\nR= 0.13\nR= 0.06\n0.6\n0.6\n0.4\n0.4\n0.2\n0.2\n0\n0\n-4\n-3\n-2\n-1\n0\n1\n2\n3\n4\n5\n-5\n-4\n-3\n-2\n-1\n0\n1\n2\n3\n4\n5\n-5\n1\n1\n0.2 d\n0.4 d\n0.8\n0.8\nR= 0.38\nR= 0.23\n0.6\n0.6\n0.4\n0.4\n0.2\n0.2\n0\n0\n-1\n0\n1\n2\n3\n4\n5\n-5\n-4\n-3\n-2\n-1\n0\n1\n2\n3\n4\n5\n-5\n-4\n-3\n-2\nx/L\nx/L\nFigure 6-22: Corresponding pulse series (14) whose propagation over the constant depth d in the flat\nbasin was modeled by MOST.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n237\n1\n1\n0.8 d\n0.6 d\n0.8\n0.8\n0.6\n0.6\n0.4\n0.4\n0.2\n0.2\n0\n0\n0\n1\n2\n3\n4\n5\n0\n1\n2\n3\n4\n5\n1\n1\n0.4 d\n0.2 d\n0.8\n0.8\n0.6\n0.6\n0.4\n0.4\n0.2\n0.2\n0\n0\n0\n1\n2\n3\n4\n5\n0\n1\n2\n3\n4\n5\nt/T\nt / T\nFigure 6-23: Time histories at x = 0 of the evolution of a single pulse in a basin (13) for d1 = d2 = 0.8d,\n0.6d, 0.4d, 0.2d (red) and that of the corresponding pulse series over the flat bottom (black).\n6.4\nReferences\nBurwell D, Tolkova E, Chawla A. 2007. Diffusion and dispersion characterization of a numerical\ntsunami model. Ocean Modelling, 19(1-2), 10-30,doi:10.1016/j.ocemod.2007.05.003\nDurran\nDR. 1999. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics.\nSpringer-Verlag, New York, Berlin, Heidelberg. 465 pp.\nGica E, Spillane MC, Titov VV, Chamberlin CD, Newman JC. 2008. Development of the\nforecast propagation database for NOAA's Short-Term Inundation Forecast for Tsunamis\n(SIFT), NOAA Tech. Memo. OAR PMEL-139, 89 pp.\nPercival DB, Arcas D, Denbo DW, Eble MC, Gica E, Mofjeld HO, Spillane MC, Tang L, Titov\nVV. 2009. Extracting tsunami source parameters via inversion of DART buoy data.\nR\nNOAA Tech. Memo. OAR PMEL-144, 22 pp.\nMei CC. 1983. The Applied Dynamics of Ocean Surface Waves. John Wiley & Sons, New York,\nChichester, Brisbane, Toronto, Singapore. 740 pp.\nTang L, Titov VV, Chamberlin CD. 2009. Development, testing, and applications of site-specific\ntsunami inundation models for real-time forecasting. J. Geophys. Res., 114, C12025,\ndoi: 10.1029/2009JC005476.","National Tsunami Hazard Mitigation Program (NTHMP)\n238\nTitov VV. 2009. Tsunami forecasting. Chapter 12 in The Sea, Volume 15: Tsunamis. Harvard\nUniversity Press, Cambridge, MA and London, England, 371-400.\nTitov VV, Synolakis CE. 1995. Modeling of breaking and nonbreaking long-wave evolution and\nrunup using VTCS-2. J. Waterw. Port Coast. Ocean Eng., 121(6), 308-316.\nTitov VV, González FI. 1997. Implementation and testing of the Method of Splitting Tsunami\n(MOST) model. NOAA Tech. Memo. ERL PMEL-112, NTIS: PB98-122773,\nNOAA/Pacific Marine Environmental Laboratory, Seattle, WA, 11 pp.\nTitov VV, Synolakis CE. 1998. Numerical modeling of tidal wave runup. J. Waterw. Port Coast.\nOcean Eng., 124(4), 157-171.\nTolkova E. 2008. Curvilinear MOST and its first application: Regional Forecast version 2. In:\nBurwell D and Tolkova E, Curvilinear version of the MOST model with application to\nthe coast-wide tsunami forecast. NOAA Tech. Memo. OAR PMEL-142, Part 2, 28 pp.\nWei Y, Bernard E, Tang L, Weiss R, Titov V, Moore C, Spillane M, Hopkins M, Kanoglu U.\n2008. Real-time experimental forecast of the Peruvian tsunami of August 2007 for U.S.\ncoastlines. Geophys. Res. Lett., 35, L04609, doi: 10.1029/2007GL032250.\nYanenko NN. 1971. The Method of Fractional Steps. Translated from Russian by M. Holt.\nSpringer, New York, Berlin, Heidelberg.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n239\n7 NEOWAVE\nYoshiki Yamazaki, Kwok Fai Cheung, Zygmunt Kowalik, Thorne Lay, and Geno Pawlak\nYoshiki Yamazaki, Kwok Fai Cheung, Geno Pawlak: Department of Ocean and Resources\nEngineering, University of Hawaii at Manoa, Honolulu, Hawaii\nZygmunt Kowalik: Institute of Marine Science, University of Alaska, Fairbanks, Alaska\nThorne Lay: Department of Earth and Planetary Sciences, University of California Santa Cruz,\nSanta Cruz, California\n7.1\nModel description\nNEOWAVE (Non-hydrostatic Evolution of Ocean WAVEs) is a shock-capturing, dispersive\nmodel in the spherical coordinate system for tsunami generation, basin-wide evolution and\ncoastal inundation and runup using two-way nested computational grids (Yamazaki et al., 2011).\nThe depth-integrated model describes dispersive waves through the non-hydrostatic pressure and\nvertical velocity, which also account for tsunami generation from kinematic seafloor\ndeformation. The semi-implicit, staggered finite difference model captures flow discontinuities\nassociated with bores or hydraulic jumps through the momentum-conserved advection scheme\nwith an upwind flux approximation (Yamazaki et al., 2009). A two-way grid-nesting scheme\nutilizes the Dirichlet condition of the non-hydrostatic pressure and both the horizontal velocity\nand surface elevation at the inter-grid boundary to ensure propagation of dispersive waves and\ndiscontinuities across computational grids of different resolution.\n7.1.1 Theoretical formulation\nNEOWAVE utilizes a spherical coordinates system (2, , z), in which a is longitude, is\nlatitude, and Z denotes normal distance from the ocean surface, to model basin-wide propagation\nand coastal inundation of tsunamis. Figure 7-1 provides a schematic of the free-surface flow\ngenerated by seafloor deformation. The boundary conditions at the free surface and seafloor\nfacilitate the depth-integration of the Euler equations. The flow depth D is the distance between\nthe two boundaries defined as\nD=5+(h-n)\n(1)\nwhere 5 is the surface elevation from the still-water level, h is the water depth, and n is the\nseafloor displacement.\nThe depth-integrated, non-hydrostatic governing equations, which include the a, , and Z\nmomentum equations as well as the continuity equation, are\nau at Rcoso U au an an","National Tsunami Hazard Mitigation Program (NTHMP)\n240\n(2)\n2\n(3)\n2 R 20 DR\ndo\nD\naw_q\n(4)\nat D\n=0\n(5)\nwhere U, V, and W are depth-averaged velocity components in the a, , and Z directions; q is the\nnon-hydrostatic pressure at the seafloor; g is acceleration due to gravity; and n is Manning's\ncoefficient accounting for surface roughness. Because of the assumption of a linear distribution,\nW is the average value of the vertical velocity at the free surface and seafloor given by the\nsimplified kinematic boundary conditions:\nit z = S\n(6)\natz==hhn\n(7)\nRcos\non\nR\nao\nEquations (6) and (7) describe the average vertical velocity W as a function of U, V, S, and n to\nclose the depth-integrated governing equations for non-hydrostatic flows.\nThe depth-integrated non-hydrostatic equations describe wave dispersion through the non-\nhydrostatic pressure and vertical velocity. Although the basic assumptions are consistent with\nthose in the classical Boussinesq equations of Peregrine (1967), the dispersion characteristics\ndiffer due to the variables employed and truncation of terms in the derivation of the governing\nequations. The linearized, depth-integrated, non-hydrostatic governing equations in the x\ndirection read:\n(8)\n(9)\n(10)\nwhere Wis and W-h denote the vertical velocity at the free surface and the seafloor. The one-\ndimensional kinematic conditions on the free surface and the seafloor are given by\n(11)","MODEL BENCHMARKING WORKSHOP AND RESULTS\n241\n(12)\nThe non-hydrostatic pressure q in terms of S and U can be obtained from the vertical momentum\nequation (8) by substitution of the kinematic boundary conditions (11) and (12). The resulting\ngoverning equations for constant water depth become\n(13)\n(14)\nFollowing Madsen et al. (1991) and Nwogu (1993), the linear dispersive relation is derived by\nconsidering a system of small amplitude periodic waves in the form:\n(15)\n(16)\nwhere k and C denote the wave number and celerity. Substitution of (15) and (16) into the\nlinearized depth-integrated, non-hydrostatic equations (13) and (14) yields the dispersion\nrelation:\nthe\n(17)\nwhere k is wave number and kh is the water depth parameter. Figure 7-2 compares the linear\ndispersion relations of the depth-integrated, non-hydrostatic equation (17) and the classical\nBoussinesq equations of Peregrine (1967) with the exact relation from Airy wave theory. The\napplicable range of a model is up to an error of 5% in the linear\ndispersion relation according to Madsen et al. (1991). Within the intermediate water depth of\n/10 < kh < TT, the dispersion relation of the depth-integrated non-hydrostatic equations has an\nerror less than 5%. The classical Boussinesq equation of Peregrine (1967) has an error of 5% at\nkh = 1.35 and a maximum of 20% within the intermediate depth range. This provides a\ntheoretical proof of the observations by Stelling and Zijlema (2003), Walters (2005), and\nYamazaki et al., (2009) that their non-hydrostatic models produce better dispersion\ncharacteristics than the classical Boussinesq equations.\n7.1.2 Numerical formulation\nThe numerical formulation includes the solution schemes for the hydrostatic and non-\nhydrostatic components of the governing equations. Figure 7-3 shows the space-staggered grid\nfor the computation. The model calculates the horizontal velocity components U and V at the cell\ninterface, and the free surface elevation S, the non-hydrostatic pressure q, and the vertical\nvelocity W at the cell center, where the water depth h is defined.\nIntegration of the continuity equation (5) provides an update of the surface elevation at the\ncenter of cell (j, k) in terms of the fluxes, FLX and FLY along the longitude and latitude, at the\ncell interfaces as","National Tsunami Hazard Mitigation Program (NTHMP)\n242\nLYkco(+2)-FY(+2)\n(18)\nRAocosok\nwhere m denotes the time step, At is the time step size, and A a and Ao are the respective grid\nsizes. The flux terms at (j, k) are estimated using the upwind flux approximation of Mader (1988)\nas\nFLX,U\n(19a)\n2\n(19b)\n2\nin which\n(20)\nThe horizontal momentum equations provide the velocity components U and V at (m+1) in (19)\nfor the update of the surface elevation in (18). Integration of the momentum equations (2) and (3)\nis performed in two steps. With the non-hydrostatic terms omitted, an explicit hydrostatic\nsolution is given by\nAt\nRANCOSO\n(21)\nsin( Ao/2)\nR2) At At\n(22)\nRAO","243\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nwhere the subscripts p and n indicates upwind and downwind approximations of the advective\nspeeds and the over bar denotes average quantities.\nStelling and Duinmeijer (2003) proposed an alternate shock-capturing scheme for finite-\ndifference and finite volume solutions of the nonlinear shallow-water equations. The\ndiscretization scheme of the advective speeds in the horizontal momentum equations\napproximate breaking waves as bores or hydraulic jumps and conserves flow volume across\ndiscontinuities. NEOWAVE utilizes the momentum conserved advection (MCA) scheme of\nYamazaki et al. (2009 and 2011), which embeds the upwind flux approximation of Mader (1988)\nin the shock-capturing scheme of Stelling and Duinmeijer (2003) to handle flow discontinuity\nassociated with tsunami wave breaking and bore as part of the hydrostatic solution.\nThe numerical scheme for the non-hydrostatic solution is implicit. Integration of the non-\nhydrostatic terms in the horizontal momentum equations completes the update of the horizontal\nvelocity from (21) and (22)\n(23)\nRANcoso,\n2\nRad At\n(24)\nwhere\n(25a)\nAj.k =\n(25b)\nB j.\nDiscretization of the vertical momentum equation (4) gives the vertical velocity at the free\nsurface as\nm+1\n(26)\nThe vertical velocity on the seafloor is evaluated from the boundary condition (7) as\n2ioi\nRANCOSOK\nRANCOSOK\ni\n(27)\nRAO\nRAO\nin which\n(28)\n2\nwhere","244\nNational Tsunami Hazard Mitigation Program (NTHMP)\n(29)\nThe horizontal velocity components and the vertical velocity at the free surface are now\nexpressed in terms of the non-hydrostatic pressure.\nThe non-hydrostatic pressure is calculated implicitly using the three-dimensional\ncontinuity equation discretized in the form\n(30)\nRAd cos o k\nSubstitution of (23), (24), and (26) into the continuity equation (30) gives a linear system of\nPoisson-type equations\n[P]{q}={{}}\n(31)\nwhich provides the non-hydrostatic pressure at each time step. The matrix equation (31), in\nwhich [P] is non-symmetric, can be solved by the strongly implicit procedure (SIP) of Stone\n(1968). At each time step, the computation starts with the calculation of the hydrostatic solution\nof the horizontal velocity using (21) and (22). The non-hydrostatic pressure is then calculated\nusing (31) and the horizontal velocity is updated with (23) and (24) to account for non-\nhydrostatic effects. The computation for the non-hydrostatic solution is complete with the\ncalculation of the free surface elevation as well as the free surface and the bottom vertical\nvelocities from (18), (26), and (27), respectively.\nThe governing equations in the spherical coordinate system (2, 0) can be transformed into\nthe Cartesian system (x, y) for modeling laboratory and analytical numerical experiments. The\ngrid spacing in the x and y directions becomes\nAx = RAN cos Ay = RAO\n(32)\nWith = 0, the earth curvature effects and Coriolis terms in the horizontal momentum vanish.\nThese modifications will recover the numerical formulation in the Cartesian coordinate system as\nderived by Yamazaki et al. (2009).\n7.1.3 Wet-dry moving boundary condition\nFor inundation or runup calculations, special numerical treatments are necessary to describe\nthe moving waterline in the swash zone. The present model tracks the interface between wet and\ndry cells using the approach of Kowalik and Murty (1993). The basic idea is to extrapolate the\nnumerical solution from the wet region onto the beach. The non-hydrostatic pressure is set to be\nzero at the wet cells along the wet-dry interface to conform to the physical problem and to\nimprove stability of the scheme.\nThe moving waterline scheme provides an update of the wet-dry interface as well as the\nassociated flow depth and velocity at the beginning of every time step. A marker CELL\".\" first\nupdates the wet-dry status of each cell based on the flow depth and surface elevation. If the flow\ndepth D is positive, the cell is under water and CELL\" =1, = and if D is zero or negative, the\ncell is dry and CELL\" =0. This captures the retreat of the waterline in an ebb flow. The surface","245\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nelevation along the interface then determines any advancement of the waterline. For flows in the\npositive a or X direction, if CELL is dry and is wet, CELL\" is reevaluated as\nCELL\"1(wet)\nIf CELL\" becomes wet, the scheme assigns the flow depth and velocity at the cell as\nThe marker CELL\" is then updated for flows in the negative direction. The same procedures are\nimplemented in the or y direction to complete the wet-dry status of the cell. If water flows into\na new wet cell from multiple directions, the flow depth is averaged.\nOnce the wet-dry cell interface is open by setting CELL\" =1, = the flow depth D and\nvelocity (U, are assigned to the new wet cell to complete the update of the wet-dry\ninterface at time step m. The surface elevation 5mth and the flow velocity Umt Vmt) over the\ncomputational domain are obtained from integration of the momentum and continuity equations\nalong with the implicit solution of the non-hydrostatic pressure as outlined in Section 7.1.2. The\nmoving waterline scheme is then repeated to update the wet-dry interface at the beginning of the\n(m+1) time step. This approach remains stable and robust for the non-hydrostatic flows without\nartificial dissipation mechanisms.\n7.1.4 Grid refinement scheme (two-way, coupled grid nesting scheme)\nMost grid-nesting schemes for tsunami models use a system of rectangular computational\ndomains with the fluxes as input to a fine inner grid and the surface elevation as output to the\nouter grid (Liu et al., 1995b, Goto et al., 1997, and Wei et al., 2003). A non-hydrostatic model\nneeds to input the horizontal velocity, surface elevation as well as the non-hydrostatic pressure to\nensure propagation of breaking and dispersive waves across inter-grid boundaries. Figure 7-4\nillustrates the grid setup and data transfer protocol. The outer and inner grids align at the cell\ncenters, where they share the same water depth. The surface elevation and non-hydrostatic\npressure are input at the cell centers as indicated by the red dots. While the tangential component\nof the velocity is applied along the inter-grid boundary, the normal component is applied on the\ninboard side of the boundary cells as indicated by the red dash. Existing grid-nesting schemes,\nwhich only input the normal velocity, cannot handle propagation of discontinuities through MCA\nscheme and Coriolis effects across inter-grid boundaries.\nFigure 7-5 shows a schematic of the solution procedure with the two-way grid-nesting\nscheme. The time step of the outer grid must be divisible by the inner-grid time step 2. The\ncalculation begins at time t with the complete solutions at both the outer and inner grids. The\ntime-integration procedure provides the solution over the outer grid at (t + At1). Linear\ninterpolation of the horizontal velocity, surface elevation, and non-hydrostatic pressure in time\nand space provides the input boundary conditions to the inner grid. The time-integration\nprocedure then computes the inner grid solution at 2 increments until (t + - At1). The hydrostatic\nsolution is computed explicitly from the input surface elevation and horizontal velocity. The non-","246\nNational Tsunami Hazard Mitigation Program (NTHMP)\nhydrostatic solution is implicit and requires reorganization of the matrix equation (31) for the\ninput non-hydrostatic pressure from the outer grid. After the computation in the inner grid\nreaches (t + 11, the surface elevation at the outer grid is then updated with the average value\nfrom the overlapping inner grid cells to complete the procedure. This feedback mechanism is\nsimilar to the grid-nesting scheme of Goto et al. (1997).\n7.1.5 Non-hydrostatic/hydrostatic hybrid (NH-Hybrid) scheme\nDispersive wave models often introduce numerical artifacts when waves break on a beach\n(Zelt, 1991). A narrow peak appears at the leading edge, and the peak grows narrower and higher\nduring shoaling and runup processes leading to instability. This problem is observed only for\nhigh amplitude waves on a gentle slope. NEOWAVE utilizes the non-hydrostatic/hydrostatic\nhybrid (NH-Hybrid) scheme to solve this overshoot instability problem associated with energetic\nwave breaking. When steep waves approach the shore with energetic breaking, nonlinearity is\npredominant and dispersion becomes secondary (Tonelli and Petti, 2010). The dispersion terms,\nwhich cause the overshoot instability, can be disabled prior to wave breaking. This will allow the\nmomentum conserved advection scheme to model the breaking waves as bores or hydraulic jump\nwithout artificial dissipation.\nA marker CELL B is used to track the breaking area. The cells with the non-hydrostatic\nsolution are set to CELL B = 0. We use the breaking initiation criterion\n(33)\nto disable the dispersive terms locally and the breaking termination criterion\n(34)\nto reactivate the dispersion terms. These criteria are checked every time step for every individual\ngrid cell, and there is no need to track the wave direction and breaking duration. Once a cell hit\nthe breaking initiation criterion (33), CELL B = is assigned to the cell. With the dispersion\nterms disabled locally, the non-hydrostatic pressure q = 0 is specified at all cells with\nCELL - B = 1, and Dirichlet B.C. is applied at cell interface between CELL B = 1 and\nCELL - B = 0. The status remains until the cell reaches the breaking termination criteria (34).\nThe marker becomes CELL B 0 and the cell includes computation of the non-hydrostatic\npressure.\nWe implemented the NH-Hybrid scheme with the wave breaking criteria, (33) and (34), to\nmodel energetic breaking events. Tonelli and Petti (2010) and Xiao et al. (2010) implemented the\nsimilar schemes with different breaking criteria. Tonelli and Petti (2010) used the ratio of wave\namplitude and water depth, while Xiao et al. (2010) used the surface elevation gradient as wave\nbreaking criteria. The proposed breaking criteria are verified with laboratory experiments.\n7.2\nModel verification and validation\nThis section provides verification and validation of NEOWAVE with analytical solutions,\nlaboratory data, and field measurements in five test cases. The first test case on solitary wave\ntransformation over a compound slope provides an assessment of the basic hydrodynamic\nprocesses of wave propagation, shoaling, breaking, and reflection. The test cases on solitary","MODEL BENCHMARKING WORKSHOP AND RESULTS\n247\nwave transformation on a plane beach, a conical island, and a reef system allow examination of\nthe numerical schemes for wave breaking and swashing. The fifth test case involving laboratory\nand field data of the 1993 Hokkaido Nansei-Oki earthquake tsunami validates the model for\npractical application.\n7.2.1 BP4: Solitary wave on compound slope\nBriggs et al. (1996) conducted a laboratory experiment to examine solitary wave\npropagation and transformation over a compound slope and the subsequent runup on a vertical\nwall. Figure 7-6 shows the experiment setup in a 23.2-m long, 0.45-m wide flume at the U.S.\nAmy Engineer Waterways Experiment Station, Vicksburg, MS. The water is 0.218 m deep over\nthe 15.05 m long flat bed in front of the compound slope of 1:53, 1:150, and 1:13. Capacitance\nwave gauges record surface elevation at ten locations along the flume. Runup on the vertical wall\nis measured visually at every run. A computer-controlled hydraulic wavemaker generates solitary\nwaves toward the slope and wall complex. Gauge 4 with a variable location at half wavelength\nfrom the toe of the compound slope recorded incident wave heights of A/h = 0.039, 0.264 and\n0.696 for the three cases A, B, and C. The approximate wavelength of a solitary wave is given by\n(35)\nin which the wave number k = 33/4h (Synolakis, 1987).\nWe first verify the linear shallow-water component of NEOWAVE with the analytical\nsolution of Kânoglu and Synolakis (1998). The analytical solution describes linear wave\ntransformation and reflection from the compound slope beginning with the recorded surface\nelevation at gauge 4. The computational domain covers the flume between gauge 4 and the\nvertical wall. The time series of the recorded solitary wave is input at the left boundary, which\nturns into an open boundary afterward. The computation uses Ax = Ay = 1 cm for the one-\ndimensional problem and At = 0.002 S to achieve a Courant number of Cr = 0.41. A Manning's\nroughness coefficient n = 0.0 is used according to the linear solution of Kânoglu and Synolakis\n(1998). Figure 7-7, Figure 7-8, and Figure 7-9 compare the computed and analytical solutions at\ngauges 4 to 10 in the experiment of Briggs et al. (1996) as well as an artificial gauge 11 at the\nvertical wall. The excellent agreement for all for three cases verifies the linear shallow-water\nsolver of NEOWAVE to appropriately model wave propagation, shoaling, and reflection. The\nsmall hump toward the end of the computation in cases B and C are reflection from the open\nboundary because of the steep nonlinear input profile.\nThe laboratory data include a wide range of wave amplitudes including non-breaking and\nbreaking events for validation of the wave breaking criteria, (33) and (34) and the NH-hybrid\nscheme for modeling the wave breaking processes. The computational domain covers the entire\nlength of the flume with Ax = Ay = 1 cm and a Manning's roughness coefficient of n = 0.01 for\nthe smooth glass beach. The solitary wave is initiated at 4 m from the left boundary with the\ninitial wave heights of A/h = 0.039, 0.264 and 0.8716, which provide best fits with the recorded\nwave heights at gauge 4. An extremely high amplitude of A/h = 0.8716 is used in case C to\nreproduce the recorded wave amplitude at gauge 4 because the present model cannot accurately\ndescribe the dispersion and nonlinearity of such high amplitude solitary waves. The recorded\ndata at gauge 4 are also used as a reference for adjustment of the timing of the computed\nwaveforms with and without the NH-hybrid scheme.","248\nNational Tsunami Hazard Mitigation Program (NTHMP)\nFigure 7-10 and Figure 7-11 compare the two solutions with the measurements at gauges 4\nto 10 for case A and case B, respectively. Case A is a non-breaking event, while case B produced\na wave breaker at or near the wall. Both solutions show very good agreement of the amplitude\nand waveform with the measurements, despite slight phase lags in the reflected wave. The\nsolutions with and without the NH-Hybrid scheme are identical in case A confirming that the\nhybrid scheme does not modify the solution in non-breaking cases. The hybrid scheme avoids\novershoot from dispersion when wave breaking occurs and produces a smaller reflected wave\namplitude in case B. Figure 7-12 shows the comparison for case C, in which the wave breaks\nbetween gauges 7 and 8 and then re-forms and shoals to the vertical wall without further\nbreaking. The NH-Hybrid scheme shows very good agreement of the amplitude and waveform\nwith measurements. Without it, the solution shows a strong artificial peak and the wave\npropagates faster due to the artificially high amplitude.\nTable 7-1. Runup comparisons on vertical wall.\nCase\nMeasured\nComputed runup with\nComputed runup without\nNo.\nrunup (cm)\nhybrid scheme (cm)\nhybrid scheme (cm)\nA\n2.74\n2.22\n2.22\nB\n46.72\n19.53\n22.98\nC\n27.43\n24.11\n58.99\nTable 7-1 provides a comparison of the measured runup at the wall and the solutions with\nand without the NH-Hybrid scheme. The NH-Hybrid scheme provides good predictions of the\nrunup for case A and case C, but significantly underestimates the runup in case B. Because the\nwave broke near the wall, the depth integrated model cannot reproduce the splash from the jet.\nThe computed results with and without the NH-Hybrid scheme are the same in case A, showing\nthat the hybrid scheme does not alter the solution in non-breaking waves. On the other hand, the\ncomputed runup without the NH-Hybrid scheme in case B and case C is much higher due to\nartificial overshoot. The results verify the proposed wave breaking criteria (33) and (34) and the\nNH-Hybrid scheme for computation of wave transformation and runup over a wide range of\nwave amplitudes.\n7.2.2 BP1/5: Solitary wave on a plane beach\nSynolakis (1987) derived an analytical solution from the non-linear shallow water equations\nfor solitary wave transformation and runup on a plane beach. Figure 7-13 provides a schematic\nof the physical problem with A indicating the incident wave height, the beach slope, and R the\nrunup. The solitary wave is initiated at a half wavelength from the toe of the beach. The\nanalytical solution allows verification of the non-linear shallow water solution and the moving\nboundary condition of NEOWAVE. We consider a beach slope of 1:19.85 and a wave height of\nA/h = 0.019 and ignore surface roughness as in the analytical solution. The grid and time step\nsize of Ax/h = 0.025 and At(g/h)\" = 0.025 gives a Courant number of Cr = 0.19. Figure 7-14 and\nFigure 7-15 show excellent agreement between the computed and analytical surface profiles\nalong the model domain and at x/h = -9.95 and -0.25. We recomputed the surface profiles at\nt(g/h)' = 50 and 70 with additional grid resolutions of Ax/h = 0.0625, 0.125, and 0.25. As\nobserved in Figure 7-16, the higher the grid resolution, the better the convergence is to the\nanalytical solution. These agreements of computed and analytical solutions verify the\nNEOWAVE model's moving boundary condition and the convergence of the model's solution.","249\nMODEL BENCHMARKING WORKSHOP AND RESULTS\nSynolakis (1987) also conducted a series of laboratory experiments that have become the\nstandard for validation of runup models. In the numerical model, we use Ax/h = 0.125 and a\nCourant number of Cr = 0.2. Surface roughness becomes important for runup over gentle slopes\nand a Manning's coefficient n = 0.01 describes the surface condition of the smooth glass beach\nin the laboratory experiments (Chaudhry, 1993). Synolakis (1997) provided a series of surface\nprofiles with a beach slope of 1:19.85 and solitary wave heights of A/h = 0.0185 and A/h = 0.3.\nFigure 7-17 shows a comparison of the measured profiles with the computed results for the non-\nbreaking conditions with A/h = 0.0185. The computed results, with and without the MCA shock\ncapturing scheme, provide identical solutions in the non-breaking case. The model reproduces\nthe runup and drawdown processes and gives very good agreement with the laboratory data.\nThe MCA scheme is instrumental in modeling the breaking conditions for A/h = 0.3. Figure\n7-18 and Figure 7-19 compare the laboratory data, the non-hydrostatic solution with the MCA\nscheme, and the hydrostatic solutions with and without the MCA scheme. The laboratory data\nshow wave breaking between t(g/h)'s = 20 and 25 as the solitary wave reaches the beach and\ndevelopment of a hydraulic jump at t(g/h) = 50 when the water recedes from the beach. The\nnon-hydrostatic solution with MCA scheme reproduces wave breaking without the use of\npredefined criteria and matches the surface elevation and runup on the beach. Both hydrostatic\nsolutions fail to reproduce the surface profile at t(g/h)' = 25 immediately after wave breaking.\nThe solution without the MCA scheme shows significant volume loss and underestimates the\nsurface elevation on the beach and eventually the runup. The MCA scheme conserves the water\nvolume and provides good agreement with laboratory data in the subsequent runup and\ndrawdown process. A minor discrepancy on the location of the hydraulic jump occurs around the\npeak of the return flow at t(g/h)'s = 55. The finite volume model with Reimann solver of Wei et\nal. (2006) also produces a similar discrepancy with the laboratory data. This may be attributed to\nthe three-dimensional flow structure that is not amenable to depth-integrated solutions. The\ncomparisons show that the MCA scheme can effectively capture discontinuous flows associated\nwith energetic wave breaking and can describe the subsequent runup on the beach without an\nempirical dissipation term in both the hydrostatic and non-hydrostatic solutions.\n7.2.3 BP6: Solitary wave transformation on conical island\nBriggs et al. (1995) conducted a large-scale laboratory experiment to investigate solitary\nwave runup on a conical island at the U.S. Army Engineer Waterways Experiment Station,\nVicksburg, MS. The collected data have become a standard for validation of runup models (Liu\net al., 1995a; Titov and Synolakis, 1998; Chen et al., 2000; Lynett et al., 2002; and Wei et al.,\n2006; and Yamazaki et al. 2009). Figure 7-20 shows a schematic of the experiment. The basin is\n25 m by 30 m. The circular island has the shape of a truncated cone with diameters of 7.2 m at\nthe base and 2.2 m at the crest. The island is 0.625 m high and has a side slope of 1:4. The\nsurface of the island and basin has a smooth concrete finish. A 27.4-m long directional spectral\nwavemaker, which consists of 61 paddles, generates solitary waves for the experiment. Wave\nabsorbers at the three sidewalls reduce reflection in the basin.\nThe experiment covers the water depths h = 0.32 and 0.42 m and the solitary wave heights\nA/h = 0.05, 0.1 and 0.2. The present study considers the smaller water depth h = 0.32 m, which\nprovides a more critical test case for the non-hydrostatic model. In the computation, the solitary\nwave is generated from the left boundary with the measured initial wave heights of A/h = 0.045,\n0.096, and 0.181 at gauge 2. These measured wave heights, instead of the target wave heights\nA/h = 0.05, 0.1, and 0.2 in the laboratory experiment, better represent the recorded data and thus","250\nNational Tsunami Hazard Mitigation Program (NTHMP)\nthe incident wave conditions to the conical island. The open boundary condition is imposed at\nthe lateral boundaries to model the effects of the wave absorbers. We use Ax = Ay = 5 cm, At =\n0.01 sec, and a Manning's roughness coefficient n = 0.012 for the smooth concrete finish\naccording to Chaudhry (1993).\nA number of gauges recorded the transformation of the solitary wave around the conical\nisland. Figure 7-21 shows the time series of the measured and computed free surface elevations\nat selected gauges. With reference to Figure 7-20, gauges 2 and 6 are located in front of the\nisland and 9, 16, and 22 are placed just outside the still waterline around the island. These gauges\nprovide sufficient coverage of the representative wave conditions in the experiment. The\nmeasured data at gauge 2 provide a reference for adjustment of the timing of the computed\nwaveforms. The computed result shows excellent agreement with the measured time series,\nincluding the depression following the leading wave that was not adequately reproduced in\nprevious studies. The present model well describes the phase of the peak, but slightly\noverestimates the leading wave amplitude at gauges 9 and 22 as the wave height increases.\nFigure 7-22 shows good agreement of the measured and computed inundation and runup around\nthe conical island.\nMost of the previous studies neglected friction in this numerical experiment (Liu et al.,\n1995a; Titov and Synolakis, 1998; Chen et al., 2000; and Lynett et al., 2002). A test of the model\nwith n = 0.0 gave very similar results as n = 0.012. As pointed out in Liu et al. (1995a), the\ncomputed results are not sensitive to the surface roughness coefficient due to the steep 1:4 slope\nof the conical island. The computed results are comparable or slightly better than the extended\nBoussinesq solutions of Chen et al. (2000) and Lynett et al. (2002) that use empirical relations\nwith adjustable coefficients to describe wave breaking. The overall agreement between the\ncomputed results and laboratory data indicates the capability of the present model to estimate\nwave transformation, breaking, and inundation in the two horizontal dimensions.\n7.2.4 ISEC BM1/2: Solitary wave transformation over complex reef system\nNEOWAVE entered the 2009 Benchmark Challenge at the Inundation Science and\nEngineering Cooperative (ISEC) Community Workshop, Corvallis, Oregon (http://isec.nacse.\norg/workshop/2009_isec/). The two benchmarks are based on the laboratory experiment of\nSwigler and Lynett (2011) at the large wave basin in the O.H. Hinsdale Wave Research\nLaboratory, Oregon State University. The basin is 48.8 m long, 26.5 m wide, and 2.1 m deep.\nThe 26.5-m long piston-type wavemaker consists of 29 paddles. Swigler and Lynett (2011)\nconstructed the 43.5 m long, 26.5 m wide experiment setup of a complex reef system shown in\nFigure 7-23. That figure shows the setup of ISEC BM1 with a triangular reef on a compound\nslope of 1:15.5, 1:31.25, and 1:550. The reef is flat with a gentle slope of 1:620. The top flat\nportion of the compound slope is 0.69 m high where it begins underwater and reaches 0.95 m\nhigh on shore. The reef has a maximum slope of 1:3.5 at the apex that gradually transitions to\n1:31.25 on the compound slope. ISEC BM2 includes a cone near the apex of the reef as shown in\nFigure 7-24. The cone has a 0.45 m height and 3 m radius at water level.\nFigure 7-25 and Figure 7-26 show laser scans of the finished reef-slope configurations and\nthe instrument layouts. Resistance-type wire and ultrasonic wave gauges were used to measure\nsurface elevation and the Norteck Vectrino 3D acoustic-Doppler velocimeters (ADV) were used\nto record velocity. In the numerical model, we use Ax = Ay = 5 cm and At = 0.002 S for a\nCourant number of Cr = 0.16. The computational grid captures the surface pattern and a","MODEL BENCHMARKING WORKSHOP AND RESULTS\n251\nManning coefficient of n = 0.012 accounts for the subgrid roughness of the finished concrete\nsurface. Both benchmarks have the same water depth of 0.78 m and the incident solitary wave\nheight of A/h = 0.5. We examine the model results with and without the NH-Hybrid scheme,\nbecause energetic wave breaking occurred during the laboratory experiment.\nFigure 7-27(a) and (b) show the time series of the two solutions and the measured free\nsurface elevations at the gauges along the centerline and at 5 m offset. The measured data at\n(x,y) = (7.5 m, 0 m) provides a reference for adjustment of the timing of the computed\nwaveforms. Wave breaking occurs near (15 m, 0 m) at the center and (17 m, 5 m) when the wave\napproaches the edge of the shallow reef flat. Figure 7-27(c) indicates the subsequent bore\npropagates across the reef flat to the row of gauges near the waterline. The solution without the\nNH-Hybrid scheme produces an anomalous peak at the gauges in the surf zone. On the other\nhand, the solution with the NH-Hybrid scheme reproduces the amplitude and phase of the wave\nbreaker. The subsequent reflected waves show clear phase lags from the measurement because\nthe horizontal movement of the wavemaker, which is about 1 m, is not considered in\ncomputation. Figure 7-27(d) shows the velocity comparison at the three ADVs. The computed\nresults agree with the first wave very well.\nThe addition of the cone in ISEC BM2 modifies the wave processes on the reef flat. Figure\n7-28 shows comparisons of the two solutions with the measurements of the free surface elevation\nand velocity at the gauges. The model with the NH-Hybrid scheme successfully reproduces the\nwave transformation and energetic breaking over the complex reef system. There are some\ndiscrepancies in both the surface elevation and velocity comparisons, while the overall\nagreement is very good considering the strong nonlinearity and energetic wave breaking. These\nnumerical experiments have validated the capability of NEOWAVE in handling wave processes\nbeyond what are required of tsunami modeling.\n7.2.5 BP7/9: The 1993 Hokkaido Nansei-Oki Earthquake Tsunami\nAn Ms 7.8 earthquake occurred west of Hokkaido, Japan on July 12, 1993 at 13:17 GMT.\nThe subsequent tsunami hit the west coast of Hokkaido and devastated an entire coastal\ncommunity on Okushiri Island along its way. The runup reached 31.7 m in a small valley near\nMonai on the west side of the island. Shuto and Matsutomi (1995) and Takahashi et al. (1995)\nconducted a field survey and a numerical model study after the event. The Hokkaido Tsunami\nSurvey Group studied the impacts and characteristics of the earthquake and tsunami, including\nrunup, ground deformation, arrival time, and structural damage (Shuto and Matsutomi, 1995).\nThe Disaster Control Research Center (DCRC) in Tohoku University investigated the focal\nmechanism and derived a set of fault parameters (DCRC-17a) summarized in Table 7-2 that\nproduces overall good agreement with recorded runup and bottom deformation (Takahashi et al.,\n1995). DCRC provided the measured runup, recorded tide gauge data, initial deformation, and\ndigitized bathymetry from nautical charts as a field benchmark for tsunami models (Takahashi,\n1996).","National Tsunami Hazard Mitigation Program (NTHMP)\n252\nTable 7-2: Fault parameters of DCRC-17a (Takahashi et al., 1995).\nLatitude\nLongitude\nLength\nWidth\nStrike\nDip\nRake\nDislocation\nDepth\nNo\n(km)\n(°N)\n(E)\nangle\nangle\n(Slip) (m)\n(km)\n(km)\nangle\n(°)\n(°)\n(°)\n43.130*\n139.400\n105*\n5.71\n10\n1\n90\n25\n188\n35\n139.250\n175\n60\n80*\n2.50\n5\n42.340\n2\n30\n25\n5\n42.100\n139.300\n60\n80*\n12.00\n3\n24.5\n25\n163\nModifications based on the initial deformation profile of Takahashi et al. (1995).\n*\nThe present study uses three levels of nested grids to model the tsunami runup on Okushiri\nIsland due to the 1993 Hokkaido Nansei-Oki earthquake. Figure 7-29(a) shows the level-1 grid at\n450 m resolution off western Hokkaido and the coverage of the level-2 and level-3 grids. An\nopen boundary condition allows radiation of tsunami waves away from the domain. The level-2\ngrid resolves the shelf and slope off Hokkaido at 150 m resolution and provides a transition to\nthe level-3 grid at 50 m resolution around Okushiri Island as shown in Figure 7-29(b). The\ndigital elevation model is derived from the 450-m, 150-m, and 50-m gridded datasets from\nDCRC as well as the JMA 20-sec bathymetry data. The Generic Mapping Tools (GMT)\ninterpolates the data to produce the nested computational grids. The computation with the 3\nlevels of nested-grids covers 1 hour of elapsed time, and uses At = 0.5 S (Cr = 0.3) for the level-1\ngrid, At = 0.125 S (Cr = 0.2) for the level-2 grid, and At = 0.125 S (Cr = 0.2) for the level-3 grid,\nwith a Manning's coefficient of n = 0.025 for the ocean bottom.\nThe planar fault model of Okada (1985) describes the earth surface deformation in terms of\nthe depth, orientation, and slip of a rectangular fault as shown in Figure 7-30. The deformation is\na linear function of the slip and dimensions of the fault. Superposition of the planar fault\nsolutions from the subfaults gives vertical displacement of the seafloor as a function of the\nlongitude and latitude (a, 0):\n(36)\nwhere n = 3 is the number of subfaults for DCRC-17a model, ni is the vertical earth surface\ndeformation associated with rupture of subfault i from Okada (1985). Figure 7-31 shows initial\nsea surface deformation, which is obtained from (36) with the fault parameters in Table 7-2.\nFigure 7-32 shows the computed and recorded surface elevations at the Esashi and Iwanai\ntide gauges on the west coast of Hokkaido. The computed results with the 150-m grid at Esashi\nshow good agreement of the arrival time and amplitude, but do not reproduce the waveform. The\ncomputed waves at Iwanai in the 450-m grid arrive about 5 min earlier than the recorded data.\nEsashi and Iwanai Bays are surrounded by multiple breakwaters and the available tide gauge\ncoordinates are accurate to the arcmin. The bathymetry data at both tide gauges come from the\n450-m grid, which does not show breakwaters and indicates the tide gauges are on land. The\ncomputed surface elevations are obtained at the nearest locations with different water depths.\nThe discrepancies between the computed and recorded surface elevations come from multiple\nfactors.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n253\nFigure 7-33 shows the computed and measured runup around Okushiri. The present model\nproduces good agreement with measurements around most of the island, but underestimates the\nrunup on the southwest coast from Monai to Aonae, consistent with previous studies (e.g.,\nTakahashi et al., 1995, and Tanioka and Satake, 1996). Takahashi et al. (1995) derived the fault\nparameters to reproduce the overall tsunami runup around the entire island as well as the vertical\nground deformation over the Island. The parameters do not reflect either the 20-m runup at\nHamatsuma (due to lack of detailed topography data) or the largest recorded runup of 31.7 m in\nMoani Valley (due to the low-resolution computation). Shuto and Matsutomi (1995) suggested\nthe fault parameters and the 45-m grid should produce 10-25 m of runup near the Moani valley.\nWe computed the results with and without dispersion and shock capturing in this benchmark and\nobtained identical solutions meaning that this tsunami is fully hydrostatic and does not involve\nenergetic wave breaking. Even though the computed results show discrepancies with the\nrecorded tide gauge and runup data, the present model reproduces the overall patterns and very\nsimilar results as the previous studies. This validates that NEOWAVE is capable modeling field\ntsunamis with at least the same level of accuracy as the existing tsunami models in fully\nhydrostatic and non-breaking events.\nMatsuyama and Tanaka (2001) conducted a laboratory study at the Central Research\nInstitute for Electric Power Industry (CRIEPI) to investigate the runup in Monai Valley during\nthe 1993 Hokkaido Nansei-Oki earthquake tsunami. Eyewitnesses reported an initial withdrawal\nof the water followed by a large tsunami wave at the coast, causing the 31.7-m runup. Such\nobservations fit the description of the leading-depression N-wave, which is often used in\nlaboratory studies of tsunami impact close to the source. The CRIEPI wave flume is 205 m long,\n3.4 m wide, and 6 m high with a hydraulic, piston-type wavemaker capable of generating N-\nwaves. The 1:400-scale coastal relief model around Monai Valley was constructed of painted\nplywood and installed approximately 140 m from the wavemaker. The initial N-wave was\nrelatively long with a very gentle profile and dispersed into a series of short-period waves over\nthe coastal relief model in the experiment. Figure 7-34(a) shows the setup of the numerical\nexperiment over the 5.475-m-long and 3.4-m-wide relief model. We use Ax = Ay = 1.25 cm (5\nm\nfull scale) and At = 0.0025 sec to achieve a Courant number of Cr = 0.23. A Manning's\ncoefficient n = 0.012 describes the surface roughness of the painted plywood relief model\n(Chaudhry, 1993). Figure 7-34(b) shows the input N-wave profile at the left boundary with 0.135\nm (54 m full scale) water depth. The incident waves shoal over a plane slope before refracting\nand diffracting around a small island on a shallow bank. Figure 7-35 shows very good agreement\nof the computed surface elevations with the laboratory measurements at three gauges behind the\nisland. The model reproduces the small-amplitude dispersive waves generated by reflection from\nthe coast. The minor phase lag in the dispersive waves is likely due to errors in the small-scale\nlaboratory experiment.","National Tsunami Hazard Mitigation Program (NTHMP)\n254\nTable 7-3: Recorded runup for the six trials from Matsuyama and Tanaka (2001).\ny =2.32 m\nTrial No.\nMaximum, Rmax (cm)\ny = 2.2062 m\nR (cm) (Full scale in m)\nR (cm) (Full scale in m)\n(Full scale in m)\n5.25 (21.0)\n5.25 (21.0)\n209 105\n8.75 (35.0)\n5.50 (22.0)\n209_106\n9.00 (36.0)\n5.75 (23.0)\n5.50 (22.0)\n209_107\n8.00 (32.0)\n5.50 (22.0)\n210_101\n9.00 (36.0)\n6.50 (26.0)\n5.75 (23.0)\n6.75 (27.0)\n5.75 (23.0)\n210_102\n10.00 (40.0)\n6.50 (26.0)\n5.75 (23.0)\n210 103\n9.00 (36.0)\n5.58 (22.3)\nMean\n8.97 (35.8)\n6.04 (24.2)\nTable 7-3 lists the runup measurements along transects at y = 2.2062 and 2.32 m as well as\nthe maximum value inside Monai Valley from a series of tests in the laboratory experiment of\nMatsuyama and Tanaka (2001). Figure 7-36 shows the computed runup and inundation with the\nrange and mean value of the recorded runup. The computed results show good agreement with\nthe measured data despite the uncertainty in the laboratory experiment. Comparisons of the\nsurface elevation, runup, and inundation from the Monai Valley experiment validate the model's\nability to handle dispersive wave and runup processes over complex nearshore bathymetry and\ntopography.\nProposed field benchmark problems\n7.3\n29 September 2009 South Pacific tsunami: water level records and runup\n7.3.1\n7.3.1.1 Introduction\nThe Samoa-Tonga earthquake occurred near the Tonga trench on 29 September 2009 at\n17:48:10 UTC. The US Geological Survey (USGS) determined the epicenter at 15.509°S\n172.034°W and estimated the moment magnitude Mw of 8.1. Figure 7-37 shows the epicenter\nand the locations of the water-level stations in the region. The main energy of the resulting\ntsunami propagated toward Tonga and American Samoa. The tide gauge in Pago Pago Harbor,\nTutuila (American Samoa) and the DART buoys 51425, 51426, and 54401 surrounding the\nrupture area recorded clear signals of the tsunami. The destructive waves arrived at mid tide and\nproduced maximum runup of 17.6 m with detrimental impact on Tutuila. Several field survey\nteams recorded and documented the tsunami runup and inundation around Tutuila and provided\nuseful data for model validation (Jaffe et al., 2010, Koshimura et al., 2009, and Okal et al.,\n2010).\nThe source mechanism is still an on-going research topic, but the data are available to\nreproduce the tsunami with reasonable accuracy. The high resolution bathymetry and topography\ndata around American Samoa and the well-recorded water level and runup will be one of the best\ndatasets for tsunami model validation. This report provides a description of the field benchmark\nand demonstrates its use in the validation of NEOWAVE (Yamazaki et al., 2009; 2011).\n7.3.1.2 Bathymetry and topography data\nThe digital elevation model is derived from a blended dataset of multiple sources. The 0.5-\narcmin (2 900-m) General Bathymetric Chart of the Oceans (GEBCO) from the British","MODEL BENCHMARKING WORKSHOP AND RESULTS\n255\nOceanographic Data Centre (BODC) provides the bathymetry for the Pacific Ocean. The Coastal\nRelief Model from the National Geophysical Data Center (NGDC) covers the American Samoa\nregion and Tutuila at 3 and 0.3333 arcsec (~90 and 10 m) resolution respectively. Embedded in\nthe NGDC dataset are multibeam and satellite measurements around Tutuila and high-resolution\nLiDAR (Light Detection and Ranging) survey data at Pago Pago Harbor. We have converted the\ndatasets to reference the WGS 84 datum and the mean-sea level (MSL). The Generic Mapping\nTools (GMT) interpolates the data to produce computational grids.\nFour levels of nested grids are used to reconstruct the 2009 South Pacific Tsunami from its\ngeneration at the earthquake source to runup at Pago Pago Harbor. Figure 7-37(a) shows the\ncoverage of the level-1 grid, which extends across the south-central Pacific at 1-arc min (~ 1800-\nm) resolution. An open boundary condition in the level-1 grid allows radiation of tsunami waves\naway from the domain. Figure 7-38(a) shows a close-up view around the source and the outlines\nof the grids at the next three levels. The level-2 grid in Figure 7-38(b) covers American Samoa\nand the surrounding seabed at 7.5-arcsec 225-m) resolution to capture wave transformation\naround the island group. The level-3 grid in Figure 7-38(c) resolves the insular shelf and slope of\nTutuila at 1.5 arcsec (~45 m). The rugged, volcanic island sits on a shallow shelf of less than\n100 m depth covered by mesophotic corals (Bare et al., 2010). The insular slope is steep with\ngradients up to 1:2 on the west side and drops off abruptly to over 3000 m depth in the\nsurrounding ocean. The level-4 grid in Figure 7-38(d) covers Pago Pago Harbor at 0.3 arcsec (~9\nm) resolution for computation of inundation as well as tide gauge signals. This nested-grid\nsystem describes wave dynamics at resolutions compatible with the physical process and spatial\nscale for optimization of computational resources.\nDispersive wave models including non-hydrostatic and Boussinesq-type models are prone to\ninstability over localized, steep bottom gradients at high-resolution computations (Horrillo et al.,\n2006; Lovholt and Pederson, 2008). The depth-dependent Gaussian function of Yamazaki et al.\n(2011) smooths localized bottom features in relation to the water depth while retaining the\nbathymetry important for modeling of tsunami transformation and runup. Figure 7-39 shows the\nsmoothed bathymetry and topography using the depth-dependent Gaussian function. Modelers\ncan use either the original or smoothed dataset depending on the model being validated.\n7.3.1.3 Fault parameters and initial condition\nGreat earthquakes typically involve rupture of a single fault or a fault network with an\noverall distribution of coseismic motion that can be characterized by a dominant geometry (e.g.,\nthrust faulting, normal faulting, or strike-slip faulting). However, when stress from one rupture\nimmediately activates faulting on a very distinct fault orientation, a non-double-couple point-\nsource mechanism may be obtained by seismic wave analysis, and the precise orientation of any\nof the activated fault geometries may not be directly inferred from the point-source\nrepresentation. This was the case for the 29 September 2009 Mw = 8.1 Samoa-Tonga earthquake\n(17:48:11 UTC, 15.51°S, 172.03°W) located near the northern Tonga subduction zone (Figure\n7-37). The global centroid-moment tensor analysis (Ekström G, GCMT Project.\nhttp://www.globalcmt.org/, 2009) obtained an unusual non-double couple moment tensor for this\ngreat event.\nSubsequent detailed analysis of seismic waves ranging from 1-s period body waves to\n~3000-s normal modes demonstrated that this event actually involved two great earthquake\nruptures with distinct faulting geometries. The main earthquake resulted from extensional","National Tsunami Hazard Mitigation Program (NTHMP)\n256\nintraplate faulting with Mw = 8.1 in the shallow Pacific plate near the outer Tonga trench slope\nand followed within 40 S by compressional thrust faulting with Mw ~ 8.0, 50-100 km to the\nsouthwest along the subduction zone megathrust fault between the underthrusting Pacific plate\nand the overriding Tonga Block (Lay et al., 2010). Campaign GPS measurements on the Samoa\nislands and Niuatoputapu, and the computed tsunami signals at DART buoys 51425, 51426, and\n54401 also suggested that composite extensional and compressional faulting had occurred\n(Beavan et al., 2010). The latter study noted that the polarities and waveforms for the tsunami\nrecordings could not be matched by the extensional faulting alone, and even use of the full non-\ndouble couple point-source mechanism could not match the signals. Similarly, the NOAA PMEL\nassumed thrust faulting for a Tonga subduction zone event of this size could not match all\npolarities of the three nearest DART buoys (e.g., NOAA Center for Tsunami Research. Tsunami\nevent-September 29, 2009 Samoa. http://nctr.pmel.noaa.gov/samoa20090929/, 2009).\nThis great Samoa-Tonga earthquake doublet is certainly unusual, but presents an\nopportunity to assess the superposition of tsunami signals from the two great earthquakes with\ndifferent geometries. The relatively slow phase speed of the tsunami waves allow refinements of\naspects of the doublet source model that are not well-resolved by seismic wave analysis due to\nthe strong interference of signals from the two events. Table 7-4 lists the fault parameters\ndeducted from Lay et al. (2010) and Beavan et al. (2010) through a sensitivity analysis of\ntsunami signals at the Pago Pago tide gauge and DART 51425, 51426, and 54401 using\nNEOWAVE. The event includes a normal fault and two thrust faults. The sensitivity analysis\nover strike angles from 320° to 350°, dip angles from 15° to 35°, fault lengths from 91 to 120\nkm, and fault widths from 15 km to 35 km covers the NE dipping normal fault of Lay et al.\n(2010) and the USGS finite fault solution used in Yamazaki et al. (2011). The computed surface\nelevations obtained from the NE dipping (shallow dipping) normal fault plane provide better\nagreement with recorded data than the SW dipping (steeply dipping) normal fault in Lay et al.\n(2010). This is consistent with tsunami modeling results and GPS measurements in Beavan et al.\n(2010) as well as tsunami modeling results in Yamazaki et al. (2011).\nTable 7-4: Fault parameters for the 29 September 2009 Mw = 8.1 2009 Samoa-Tonga Earthquake\nStrike\nDip\nRake\nSlip\nDepth\nLatitude\nLongitude\nRise\nRupture\nNo\nLength\nWidth\ntime\ninitiation\nangle\nangle\nangle\n(W)\n(s)\ntime (sec)\n(km)\n(km)\n(°)\n(°)\n(°)\n(m)\n(km)\n(°N)\n171.9693\n41\n0\n1\n110\n35\n340\n35\n265\n6.60\n5.2\n16.0151\n40\n49\n2\n50\n75\n175\n20\n90\n4.62\n5\n15.5260\n172.3703\n20\n90\n4.71\n5\n16.0402\n172.3250\n40\n90\n3\n50\n75\n180\nThe planar fault model of Okada (1985) describes the earth surface deformation in terms of\nthe depth, orientation, and slip of a rectangular fault. The deformation is a linear function of the\nslip and dimensions of the fault. Let t denote time and (A, 0) the longitude and latitude.\nSuperposition of the planar fault solutions from the subfaults gives the time sequence of the\nvertical displacement of the seafloor:","MODEL BENCHMARKING WORKSHOP AND RESULTS\n257\n(37)\nin which\n0\n(38)\nwhere n = 3 is the number of subfaults for the 2009 Samoa-Tonga Earthquake, ni is the vertical\nearth surface deformation associated with rupture of subfault i from Okada (1985), and ti and Ti\nare the corresponding rupture initiation time and rise time. The source time function (38) defines\na linear motion of the slip at each subfault to approximate the rupture process in the absence of\ndetailed source time information (Irikura, 1983).\n7.3.1.4 Results and discussion\nThe 2009 South Pacific Tsunami is reconstructed from its generation at the earthquake\nsource to runup at Pago Pago Harbor with the four levels of nested grids and the smoothed\nbathymetry data in Figure 7-39. A Manning's coefficient n = 0.035 represents the surface\nroughness of the coral reefs in the near-shore seabed according to Bretschneider et al. (1986).\nThe momentum-conserved advection scheme is used at the level-4 grid to describe flow\ndiscontinuities associated with wave breaking and bore formation.\nTsunami energy propagation is directional with the majority perpendicular to the fault-line.\nThe main energy propagates toward Tonga and American Samoa, where the tide gauge in Pago\nPago Harbor recorded a strong signal of the tsunami. The three DART buoys, which are located\nroughly along the strike direction off the main energy beams, also recorded clear signals. Figure\n7-40 shows a comparison of the recorded and computed waveforms and spectra at the Pago Pago\ntide gauge and at the three DART buoys. The computed results show good agreement of the\narrival time, amplitude, and frequency content with the measurements. The model reproduces the\ninitial negative wave at the tide gauge and captures the distinct 11- and 18-min oscillations at\nPago Pago Harbor. The computed surface elevations at the three DART buoys show very good\nagreement with the recorded data. NEOWAVE, with the fault parameters in Table 7-4,\nsuccessfully reproduces the arrival time, amplitude, and polarity of the recorded signals at the\nthree DART buoys, which could not be achieved with a single normal fault (Yamazaki et al.,\n2011). DART 51426 and 54401 recorded the high-amplitude reflected waves from Tutuila\narriving at 1.6 and 3.2 hours after the earthquake. The computed results at DART 51426 cannot\nreproduce the amplitudes, but reasonably predicts the reflected wave's arrival time. The\ncomputed results at DART 54401 show excellent agreement with recorded data, including the\nreflected waves.\nFigure 7-41 compares the computed runup and inundation around Pago Pago Harbor with\nthe measurements from Koshimura et al. (2009) and Okal et al. (2010). The model reproduces\nthe overall pattern and height of the runup along the coast. The minor discrepancies between the\ncomputed results and the measurements are primarily due to the difficulties in modeling\ninundation in the built environment around Pago Pago Harbor. The large runup at the tip and low\nvalues in the outer harbor are results of resonance. Such oscillations, which extend over the","National Tsunami Hazard Mitigation Program (NTHMP)\n258\nentire insular shelf and slope, provide an explanation for the local amplification and the disparate\nproperty damage along the coastlines of Tutuila (Roeber et al., 2010; Yamazaki et al., 2011).\nNEOWAVE reproduces the surface elevations at the Pago Pago tide gauge and the DART buoys\nas well as the runup around the harbor. The proposed field benchmark problem demonstrates the\nmodel capability in describing generation, propagation, and runup of tsunamis.\nThe 2009 South Pacific Tsunami is an excellent benchmark with high quality bathymetric\nand topographic data, water level records, and runup and inundation measurements. This is the\nfirst complete dataset of a significant event that has both water level as well as runup and\ninundation records at the same location. On the other hand, the initial condition is still an\nongoing research topic. Modelers may use any suitable fault parameters in the validation.\n7.3.2 2006 Kuril and 2010 Chile tsunamis: nearshore surface elevation and current\n7.3.2.1 Introduction\nConcurrent surface elevation and flow measurements of tsunamis in nearshore waters have\nbeen rare. The Kilo Nalu Observatory, which is a cabled coastal monitoring station at the south\nshore of Oahu, shown in Figure 7-42, represents an important asset in tsunami research. The\nfacility is managed by the Department of Ocean and Resources Engineering, School of Ocean\nand Earth Science and Technology (SOEST), University of Hawaii (UH) to provide real-time\nhydrodynamic and geochemical data since 2004 (Pawlak et al., 2009). The observatory extends\nfrom 12 to 20 m water depth on an open coast. An acoustic Doppler current profiler (ADCP) was\ndeployed at the observatory in 2006 to measure real-time velocity and pressure at 12 m depth and\n400 m from the shore (21.2885°N, 157.8648°W). The ADCP has detected several tsunami events\nsince its deployment and recorded clear signals of the 2006 Kuril and 2010 Chile tsunamis.\nThe Honolulu tide gauge (21.3033°N, 157.8645°W) adjacent to the Kilo Nalu Observatory\nalso recorded clear signals of the 2006 Kuril and 2010 Chile tsunamis. The recorded data reflect\nthe flows in a basin through narrow harbor channels. In contrast, the surface elevation and\nvelocity data at the Kilo Nalu ADCP represents nearshore wave processes on an open coast. The\nrecorded tsunami data and high-resolution bathymetry in the region provide a valuable field\nbenchmark to examine model capabilities in describing detailed nearshore flow conditions and\ntheir transformation into harbor basins. The use of the USGS finite fault solutions of the 2006\nKuril and 2010 Chile earthquakes as initial conditions for tsunami modeling have been validated\nwith DART and tide gauge data by Munger and Cheung (2006) and Yamazaki and Cheung\n(2011). We demonstrate the implementation of this benchmark with the non-hydrostatic free\nsurface flow model NEOWAVE (Yamazaki et al., 2009; 2011).\n7.3.2.2 Bathymetry and topography data\nThe digital elevation model is derived from a blended dataset from multiple bathymetry and\ntopography data sources. The 1-arcmin (2 1800-m) ETOPO1 from National Geophysical Data\nCenter (NGDC) provides the bathymetry for the Pacific Ocean. The 1.5-arcsec ~ 50-m) SOEST\nmutibeam data covers the Hawaiian Islands bathymetry and the gaps are filled with the 5-arcsec\n(~ 150-m) Japan Marine Science and Technology Center (JAMSTEC) and US Geological\nSurvey (USGS) data. The 4-m Scanning Hydrographic Operational Airborne LiDAR Survey\n(SHOALS) data from U.S. Army Corps of Engineers (USACE) covers the nearshore bathymetry","MODEL BENCHMARKING WORKSHOP AND RESULTS\n259\nfrom 40 m water depth to the coast, while the 1-m LiDAR topography from the Federal\nEmergency Management Agency (FEMA) and the U.S. Army Corps of Engineers (USACE)\ncovers the coastal area up to 15 m elevation. The areas above the 15-m contour are filled with the\n30-m National Elevation Dataset (NED) from USGS. The multiple datasets were merged with\nArcGIS, converted to reference the WGS 84 datum and the mean-sea level (MSL). The Generic\nMapping Tools (GMT) interpolates the data to produce nested computational grids.\nFour levels of nested grids are used to reconstruct the 2006 Kuril and 2010 Chile tsunamis\nfrom the respective source to the south shore of Oahu. Figure 7-43(a) shows the coverage of the\nlevel-1 grids for the 2006 Kuril and 2010 Chile tsunamis over the north and southeast Pacific at\n2-arc min 4000-m) resolution. An open boundary condition allows radiation of tsunami waves\naway from the computational domain. The level-2 grid in Figure 7-43(b) covers the Hawaiian\nIslands at 15-arcsec (2 500-m) resolution to capture wave transformation around the island\ngroup. The level-3 grid in Figure 7-43(c) resolves the insular shelf and slope of Oahu at 3 arcsec\n(~ 100 m) and provides a transition to the level-4 grid, which covers the south shore of Oahu at\n0.3 arcsec (~10 m) resolution for computation of tsunami signals at the Kilo Nalu ADCP and at\nthe Honolulu tide gauge, as shown in Figure 7-42. This nested-grid system describes wave\ndynamics at resolutions compatible with the physical process and at spatial scales that optimize\ncomputational resources.\nDispersive wave models, including the present depth-integrated, non-hydrostatic models and\nBoussinesq-type models, are prone to instability over localized, steep bottom gradients at high-\nresolution computations (Horrillo et al., 2006; Lovholt and Pederson, 2008). In this study, the\ndepth-dependent Gaussian function of Yamazaki et al. (2011) is used to smooth bathymetric\nfeatures smaller than the water depth to resolve the stability issues without significantly\nmodifying the nearshore wave transformation shown in Figure 7-44. Modelers can use either the\noriginal or smoothed datasets depending on the type of models being examined.\n7.3.2.3 Finite fault models and initial conditions\nThe Mw 8.3 Kuril earthquake occurred along Kuril Trench 495 km SSW of Severo-Kuril'sk,\non 15 November 2006 at 11:14:17 UTC or 1:14:17 a.m. Hawaii Standard Time (HST). The\nresulting tsunami traveled across the Pacific reaching Hawaii at 7:20 a.m. (HST), 6 hours after\nthe earthquake. The Mw 8.8 Chile earthquake occurred along the Peru-Chile Trench offshore of\nMaule, Chile (105 km NNE of Concepcion) on 27 February 2010 at 6:34:14 UTC (8:34:14 p.m.\nHST), and generated a destructive tsunami in the near field that resulted in warnings across the\nPacific. The tsunami reached Hawaii at 11:19 a.m., 14.5 hours after the earthquake. Neither\ntsunami caused major damage in Hawaii, but prolonged oscillations and strong currents\nassociated with tsunamis were observed at harbors and embayments around the Hawaiian Islands\n(Bricker et al., 2007, and Munger and Cheung, 2008).\nUSGS analyzed the 2006 Kuril and the 2010 Chile earthquake using the finite fault\nalgorithm of Ji et al. (2002). For the 2006 Kuril earthquake, the finite fault solution utilizes a 400\nkm long and 137.5 km wide fault, which is sub-divided into 220 subfaults of 20 km long and\n12.5 km wide, each with strike angle = 220° and dip angle = 14.89°, and hypocenter, 46.616°N,\n153.224°E, at 26.7-km depth. The resulting finite fault solution, shown in Figure 7-45(a),\nestimates seismic moment of 3.9 1021 N.m (Mw 8.3). (Ji C, 2010, http://earthquake.usgs.gov/\nearthquakes/eqinthenews/2006/usvcam/finite fault.php). For the 2010 Chile earthquake, the","National Tsunami Hazard Mitigation Program (NTHMP)\n260\nsolution utilized a fault of 540 km long and 200 km wide subdivided into 180 subfaults of 30 km\nby 20 km, with strike angle = 16.13 and dip angle = 14.84°, and hypocenter, 35.83°S, 72.67°W,\nat 35-km depth (Hayes G, 2010, http://earthquake.usgs.gov/earthquakes/eqinthenews\n/2010/us2010tfan/finite fault1.php). The resulting finite fault solution in Figure 7-45(c)\nestimates seismic moment of 2.0 x 1022 N.m (Mw 8.8). The USGS finite fault solutions of the\n2006 Kuril and 2010 Chile earthquakes in Figure 7-45(a) and (c) provide the sea surface\ndeformations in Figure 7-45(b) and (d) through (36) as the initial conditions of the two\nbenchmarks.\n7.3.2.4 Results and discussion\nThe 2006 Kuril and 2010 Chile tsunamis are reconstructed from the generation at the\nearthquake sources to the nearshore transformation at the south shore of Oahu, using the\nsmoothed bathymetry. A Manning's coefficient n = 0.035 represents the surface roughness of the\ncoral reefs in the near-shore seabed according to Bretschneider et al. (1986). The momentum-\nconserved advection scheme is used at the level-4 grid to describe flow discontinuities associated\nwith wave breaking and bore formation.\nFigure 7-46 shows a comparison of the recorded and computed data at the Honolulu tide\ngauge and at Kilo Nalu for the 2006 Kuril tsunami. The computed surface elevations show very\ngood agreement of the arrival time, amplitude, and frequency content with the measurements.\nHowever, the computed velocity component, u, along the longitude cannot capture the first\ndistinct signal and underestimates the overall amplitude. The computed velocity component, v,\nalong the latitude shows very good agreement of the arrival time and amplitude, of the first two\nwaves especially, but it still underestimates the overall amplitudes.\nFigure 7-47 compares the recorded and computed data at the Honolulu tide gauge and at\nKilo Nalu for the 2010 Chile tsunami. The model reproduces the measured surface elevations at\nboth the Honolulu tide gauge and at the Kilo Nalu ADCP well, but slightly overestimates the\nsubsequent waves at the tide gauge. On the other hand, the computed subsequent waves at the\nKilo Nalu ADCP show very good agreement of the arrival time, amplitude, and frequency\ncontent. The measured velocity show a strong noise level compared to the surface elevation.\nBoth computed velocity components show overall good agreement of the arrival time and\namplitude with the measurements.\nThe computed results at the Honolulu tide gauge for both the 2006 Kuril and 2010 Chile\ntsunamis shows slight overestimations of the amplitude at subsequent waves. The 10-m grid may\nnot be able to describe detailed features of the wharves in the harbor to appropriately model\nharbor oscillation. Inaccurate harbor oscillation prediction could lead the growing amplitude in\nthe computed surface elevation at the Honolulu tide gauge. Considering the long distance\npropagation across the Pacific, scattering around the many islets and seamounts, and the effects\nof background currents, NEOWAVE successfully reproduces the Honolulu tide gauge records\nand the surface elevation and velocity at the Kilo Nalu ADCP. This validates its capability to\npredict surface elevations and currents for distant tsunamis.\n7.4\nConclusions\nThis report has demonstrated the versatility and robustness of the shock-capturing,\ndispersive wave model, NEOWAVE, for tsunami propagation, transformation, and runup. The\nformulation in the spherical coordinate system accounts for the earth's curvature in basin-wide","MODEL BENCHMARKING WORKSHOP AND RESULTS\n261\ntsunami propagation and yet is flexible enough to model inundation at a regional scale through a\nsystem of two-way nested grids. Implementation of an upwind flux approximation in the\nmomentum-conserved advection scheme and a mixed non-hydrostatic and hydrostatic hybrid\nscheme allows modeling of flow discontinuities associated with bores and hydraulic jumps\nduring the runup and drawdown process. A series of numerical experiments with analytical\nsolutions, laboratory data, and field data have verified and validated NEOWAVE as a tool for\ntsunami inundation mapping.\n7.5\nReferences\nBare AY, Grimshaw KL, Rooney JJ, Sabater MG, Fenner D, Carroll B. 2010. Mesophotic\ncommunities of the insular shelf at Tutuila, American Samoa, Coral Reefs, 29(2), 369-\n377.\nBeavan J, Wang X, Holden,C, Wilson K, Power W, Prasetya G, Bevis M, Kautoke R. 2010.\nNear-simultaneous great earthquakes at Tongan megathrust and outer rise in September\n2009, Nature, 466(7309), 959-963.\nBretschneider CL, Krock HJ, Nakazaki E, Casciano FM. 1986. Roughness of Typical Hawaiian\nTerrain for Tsunami Run-up Calculations: A User's Manual. J.K.K. Look Laboratory\nReport, University of Hawaii, Honolulu, Hawaii.\nBricker JD, Munger S, Pequignet C, Wells JR, Pawlak G, Cheung KF. 2007. ADCP\nobservations of edge waves off Oahu in the wake of the November 2006 Kuril Island\nTsunami. Geophysical Research Letters, 34(23), L23617, doi: 10.1029/2007GL032015\nBriggs MJ, Synolakis CE, Harkins GS, Green DR. 1995. Laboratory experiments of tsunami\nrunup on a circular island. Pure and Applied Geophysics, 144(3/4), 569-593.\nBriggs MJ, Synolakis CE, Kanoglu U, Green DR. 1996. Benchmark Problem 3: runup of solitary\nwaves on a vertical wall. In Long-Wave Runup Models, Yeh H, Liu PL-F, Synolakis C.\n(eds). World Scientific, Singapore, pp. 375-383.\nChaudhry MH. 1993. Open-Channel Flow. Prentice-Hall Inc., Englewood Cliffs, New Jersey,\n483 p.\nChen Q, Kirby JT, Dalrymple RA, Kennedy AB, Chawla A. 2000. Boussinesq modeling of wave\ntransformation, breaking, and runup. II: 2D. Journal of Waterway, Port, Coastal, and\nOcean Engineering, 126(1), 48-56.\nGoto C, Ogawa Y, Shuto N, Imamura F. 1997. IUGG/IOC Time Project: Numerical Method of\nTsunami Simulation with the Leap-Frog Scheme, Manuals and Guides No. 35,\nIntergovernmental Oceanographic Commission of UNESCO, Paris.\nHorrillo JJ, Kowalik Z, Shigihara Y. 2006. Wave dispersion study in the Indian Ocean tsunami\nof December 26, 2004. Marine Geodesy, 29(1), 149-166.\nIrikura K. 1983. Semi-empirical estimation of strong ground motions during large earthquakes,\nBulletin of the Disaster Prevention Research Institute, 33-2(298), 63-104.\nJaffe BE, Gelfenbaum G, Buckley ML, Watt S, Apotsos A, Stevens AW, Richmond BM. 2010.\nThe limit of inundation of the September 29, 2009, tsunami on Tutuila, American Samoa.\nUS Geological Survey Open-File Report 2010-1018, U.S. Geological Survey.","262\nNational Tsunami Hazard Mitigation Program (NTHMP)\nJi C, Wald DJ, Helmberger DV. 2002. Source description of the 1999 Hector Mine, California,\nEarthquake, Part I: Wavelet domain inversion theory and resolution analysis. Bulletin of\nthe Seismological Society of America, 92(4), 1192-1207.\nKânoglu U, Synolakis CE. 1998. Long wave runup on piecewise linear topographies. Journal of\nFluid Mechanics, 374, 1-28.\nKoshimura S, Nishimura Y, Nakamura Y, Namegaya Y, Fryer GJ, Akapo A, Kong LS, Vargo D.\n2009. Field survey of the 2009 tsunami in American Samoa. EOS Transactions of the\nAmerican Geophysical Union, 90(52), Fall Meeting Supplemental Abstract U23F-07.\nKowalik Z, Murty TS. 1993. Numerical Modeling of Ocean Dynamics. World Scientific,\nSingapore, 481 p.\nLay T, Ammon CJ, Kanamori H, Rivera L, Koper K, Hutko A. 2010. The 2009 Samoa-Tonga\ngreat earthquake triggered doublet. Nature, 466(7309), 964-968.\nLiu PL-F, Cho YS, Briggs MJ, Kânoglu U, Synolakis CE. 1995a. Runup of solitary wave on a\ncircular island. Journal of Fluid Mechanics, 302, 259-285.\nLiu PL-F, Cho YS, Yoon SB, Seo SN. 1995b. Numerical simulations of the 1960 Chilean\nTsunami propagation and inundation at Hilo, Hawaii. In Tsunami: Progress in Prediction,\nDisaster Prevention and Warning. Tsuchiya Y, Shuto, N. (eds). Kluwer Academic\nPublishers, Netherlands, pp. 99-116.\nLovholt F, Pederson G. 2008. Instabilities of Boussinesq models in non-uniform depth.\nInternational Journal for Numerical Methods in Fluids, 61(6), 606-637.\nLynett PJ, Wu TR, Liu PL-F. 2002. Modeling wave runup with depth-integrated equations.\nCoastal Engineering, 46(2), 89-107.\nMader CL. 1988. Numerical Modeling of Water Waves. University of California Press,\nCalifornia, 206 p.\nMadsen PA, Murray R, Sorensen OR. 1991. A new form of the boussinesq equations with\nimproved linear dispersion characteristics. Coastal Engineering, 15(4), 371-388.\nMatsuyama M, Tanaka H. 2001. An experimental study of the highest run-up height in the 1993\nHokkaido Nansei-oki earthquake tsunami. In Proceedings of the International Tsunami\nSymposium 2001, Seattle, Washington, pp. 879-889.\nMunger S, Cheung KF. 2008. Resonance in Hawaii waters from the 2006 Kuril Islands Tsunami.\nGeophysical Research Letters, 35(7), L07605, doi: 10.1029/2007 GL032843.\nNwogu O. 1993. Alternative form of Boussinesq equations for nearshore wave propagation.\nJournal of Waterway, Port, Coastal, and Ocean Engineering, 119(6), 618-638.\nOkada Y. 1985. Surface deformation due to shear and tensile faults in a half space. Bulletin of\nthe Seismological Society of America, 75(4), 1135-1154.\nOkal EA, Fritz HM, Synolakis CE, Borrero JC, Weiss R, Lynett PJ, Titov VV, Foteinis S, Jaffe\nBE, Liu PL-F, Chan IC. 2010. Field Survey of the Samoa Tsunami on 29 September\n2009. Seismological Society of America, 81(4), 577-591.\nPawlak G, De Carlo EH, Fram JP, Hebert AB, Jones CS, McLaughlin BE, McManus MA,\nMillikan KS, Sansone FJ, Stanton TP, Wells JR. 2009. Development, deployment, and","263\nMODEL BENCHMARKING WORKSHOP AND RESULTS\noperation of Kilo Nalu nearshore cabled observatory. In Proceedings of the IEEE\nOCEANS 2009, Bremen, Germany, May 2009, pp 1-10.\nPeregrine DH. 1967. Long waves on a beach. Journal of Fluid Mechanics, 27(4), 815-827.\nRoeber V, Yamazaki Y, Cheung KF. 2010. Resonance and impact of the 2009 Samoa tsunami\naround Tutuila, American Samoa, Geophysical Research Letters, 37(21), L21604, doi:\n10.1029/2010GL044419.\nShuto N, Matsutomi H. 1995. Field survey of the 1993 Hokkaido Nansei-oki earhtuquake\ntsunami. Pure and Applied Geophysics, 144(3/4), 649-663.\nStelling GS, Duinmeijer SP.A. 2003. A staggered conservative scheme for every Froude number\nin rapidly varied shallow water flows. International Journal for Numerical Methods in\nFluids, 43(12), 1329-1354.\nStelling GS, Zijlema M. 2003. An accurate and efficient finite-difference algorithm for non-\nhydrostatic free-surface flow with application to wave propagation. International Journal\nfor Numerical Methods in Fluids, 43(1), 1-23.\nStone HL. 1968. Iterative solution of implicit approximations of multidimensional partial\ndifferential equations. SIAM (Society for Industrial and Applied mathematics) Journal\non Numerical Analysis, 5(3), 530-558.\nSwigler DT, Lynett P. 2011. Laboratory study of the three-dimensional turbulence and kinematic\nproperties associated with a solitary wave traveling over an alongshore-variable, shallow\nshelf. in review.\nSynolakis CE. 1987. The runup of solitary waves. Journal of Fluid Mechanics, 185, 523-545.\nTakahashi T. 1996. Benchmark problem 4; the 1993 Okushiri tsunami-Data, conditions and\nphenomena. In Long-Wave Runup Models, World Scientific, 384-403.\nTakahashi T, Takahashi T, Shuto N, Imamura F, Ortiz M. 1995. Source models for the 1993\nHokkaido Nansei-oki earthuquake tsunami. Pure and Applied Geophysics, 144(3/4), 747-\n767.\nTanioka Y, Satake K. 1996. Tsunami runup on Okushiri Island. In Long-Wave Runup Models,\nYeh H, Liu PL-F, Synolakis C. (eds). World Scientific, Singapore, pp. 249-257.\nTitov VV, Synolakis CE. 1998. Numerical modeling of tidal wave runup. Journal of Waterway,\nPort, Coastal, and Ocean Engineering, 124(4), 157-171.\nTonelli M, Petti M. 2010. Finite volume scheme for the solution of 2D extended Boussinesq\nequations in the surf zone. Ocean Engineering, 37(7), 567-582.\nWalters RA. 2005. A semi-implicit finite element model for non-hydrostatic (dispersive) surface\nwaves. International Journal for Numerical Methods in Fluids, 49(7), 721-737.\nWei Y, Cheung KF, Curtis GD, McCreery CS. 2003. Inverse algorithm for tsunami forecasts.\nJournal of Waterway, Port, Coastal, and Ocean Engineering, 129(2), 60-69.\nXiao H, Young YL, Prévost JH. 2010. Hydro- and morpho-dynamic modeling of breaking\nsolitary waves over a fine sand beach. Part II: Numerical simulation. Marine Geology,\n269(3-4), 119-131.","264\nNational Tsunami Hazard Mitigation Program (NTHMP)\nYamazaki Y, Cheung KF. 2011. Shelf resonance and impact of near-field tsunami generated by\nthe 2010 Chile earthquake. Geophysical Research Letters, 38(12), L12605, doi:\n10.1029/2011GL047508.\nYamazaki Y, Cheung KF, Kowalik Z. 2011. Depth-integrated, non-hydrostatic model with grid\nnesting for tsunami generation, propagation, and run-up. International Journal for\nNumerical Methods in Fluids, 67(12), 2081-2107.\nYamazaki Y, Kowalik Z, Cheung KF. 2009. Depth-integrated, non-hydrostatic model for wave\nbreaking and runup. International Journal for Numerical Methods in Fluids, 61(5), 473-\n497.\nZelt JA. 1991. The run-up of nonbreaking and breaking solitary waves. Coastal Engineering,\n15(3), 205-246.\n7.6\nFigures\nZ\na,\nW (w)\nh\nU, V\n(u, v)\nn\nFigure 7-1: Schematic of free-surface flow generated by seafloor deformation.","265\nMODEL BENCHMARKING WORKSHOP AND RESULTS\n1\n0.9\n0.8\n0.7\n0.6\n0.5\n0.4\n0.3\n4.0\n5.0\n0\n1.0\n2.0\n3.0\nkh\nclassical Boussinesq equations of\nFigure 7-2: Linear dispersion relation.\nAiry wave theory;\n,\n(red), depth-integrated, non-hydrostatic equations.\nPeregrine (1967);","266\nNational Tsunami Hazard Mitigation Program (NTHMP)\nk+1\nVj.k\nUj.k\nk\nSik\nk-1\nj-1\nj\nj+1\na\nFigure 7-3: Definition sketch of the discretization scheme.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n267\n(y)\na (x)\nInter grid\nboundary\nInter grid\nboundary\nFigure 7-4: Schematic of a two-level nested grid system.","National Tsunami Hazard Mitigation Program (NTHMP)\n268\nLevel-1 Grid\nSTEP. 1\nSTEP 2\nSTEP 6\nt+12\n1+2A12\nSTEP 3\nSTEP 5\nLevel-2 Grid\nSTEP 4\nFigure 7-5: Schematic of two-way grid-nesting and time-integration schemes.\nVertical Wall\n23.23 m\nIncident wave\n4\n8\n9\n10\nA\nB C 5\n6\n7\nWave\n1/13\n21.8 cm\nMarker\n1/150\n1/53\n*\n15.04 m\n4.36 m\n2.93 m 0.9 m\nFigure 7-6: Definition sketch of solitary wave on compound slope. o, gauge locations.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n269\n2\nGauge NO 4\n1\n0\n2\nGauge NO 5\n1\n0\n2\nGauge NO 6\n1\n0\n2\nGauge NO 7\n1\n0\n2\nGauge NO 8\n1\n0\n2\nGauge NO 9\n1\n0\n2\nGauge NO10\n1\n0\n2\nGauge NO11\n1\n0\n268\n273\n278\n283\n288\n293\nt (sec)\nFigure 7-7: Time series of surface elevation at gauges on compound slope with A/h = 0.039.\n(black), analytical solution of Kânoglu and Synolakis (1998);\n(red), computed data.","National Tsunami Hazard Mitigation Program (NTHMP)\n270\n10\nGauge NO 4\n5\n0\n10\nGauge NO 5\n5\n0\n10\nGauge NO 6\n5\n0\n10\nGauge NO 7\n5\n0\n10\nGauge NO 8\n5\n0\n10\nGauge NO 9\n5\n0\n10\nGauge NO10\n5\n0\n20\nGauge NO11\n10\n0\n282\n287\n267\n272\n277\nt (sec)\nFigure 7-8: Time series of surface elevation at gauges on compound slope with A/h = 0.264.\n(black), analytical solution of Kânoglu and Synolakis (1998);\n(red), computed data.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n271\n20\nGauge NO 4\n15\n10\n5\n0\n20\nGauge NO 5\n15\n10\n5\n0\n20\nGauge NO 6\n15\n10\n5\n0\n20\nGauge NO 7\n15\n10\n5\n0\n20\nGauge NO 8\n15\n10\n5\n0\n20\nGauge NO 9\n15\n10\n5\n0\n20\nGauge NO10\n15\n10\n5\n0\n50\nGauge NO11\n40\n30\n20\n10\n0\n-10\n266\n271\n276\n281\n286\nt (sec)\nFigure 7-9: Time series of surface elevation at gauges on compound slope with A/h = 0.696.\n(black), analytical solution of Kânoglu and Synolakis (1998);\n(red), computed data.","National Tsunami Hazard Mitigation Program (NTHMP)\n272\n2\nGauge NO 4\n1\n0\n2\nGauge NO 5\n1\n0\n2\nGauge NO 6\n1\n0\n2\nGauge NO 7\n1\n0\n2\nGauge NO 8\n1\n0\n2\nGauge NO 9\n1\n0\n2\nGauge NO10\n1\n0\n295\n265\n270\n275\n280\n285\n290\nt (sec)\nFigure 7-10: Time series of surface elevation at gauges on compound slope with A/h = 0.039.\n(black), laboratory data of Briggs et al. (1996);\n(red), solution with NH-Hybrid scheme;\n(blue), solution without NH-Hybrid scheme.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n273\n10\nGauge NO 4\n5\n0\n10\nGauge NO 5\n5\n0\n10\nGauge NO 6\n5\n0\n10\nGauge NO 7\n5\n0\n10\nGauge NO 8\n5\n0\n10\nGauge NO 9\n5\n0\n10\nGauge NO10\n5\n0\n265\n270\n275\n280\n285\n290\n295\nt (sec)\nFigure 7-11: Time series of surface elevation at gauges on compound slope with A/h = 0.264.\n(black), laboratory data of Briggs et al. (1996);\n(red), solution with NH-Hybrid scheme;\n(blue), solution without NH-Hybrid scheme.","National Tsunami Hazard Mitigation Program (NTHMP)\n274\n20\nGauge NO 4\n15\n10\n5\n0\n20\nGauge NO 5\n15\n10\n5\n0\n20\nGauge NO 6\n15\n10\n5\n0\n20\nGauge NO 7\n15\n10\n5\n0\n20\nGauge NO 8\n15\n10\n5\n0\n20\nGauge NO 9\n15\n10\n5\n0\n20\nGauge NO10\n15\n10\n5\n0\n290\n295\n265\n270\n275\n280\n285\nt (sec)\nFigure 7-12: Time series of surface elevation at gauges on compound slope with A/h = 0.696.\n(black), laboratory data of Briggs et al. (1996);\n(red), solution with NH-Hybrid scheme;\n(blue), solution without NH-Hybrid scheme.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n275\nR\nA\nh\nL / 2\nFigure 7-13: Definition sketch of solitary wave runup on plane beach.\n0.10\nt (g/h) 1/2 =35\nt (g/h) 1/2 =55\n0.05\n0.00\n-0.05\n0.10\nt (g/h) 1/2 =60\nt (g/h) 1/2 =40\n0.05\n0.00\n-0.05\n0.10\nt (g/h) 1/2 =45\nt (g/h) 1/2 =65\n0.05\n0.00\n-0.05\n0.10\nt (g/h) 1/2 =70\nt (g/h) 1/2 =50\n0.05\n0.00\n-0.05\n0\n-20\n-15\n-10\n-5\n0\n-20\n-15\n-10\n-5\nx/h\nx/h\nFigure 7-14: Transformation and runup of a solitary wave on a 1:19.85 plane beach with A/h = 0.019.\no, analytical solution of Synolakis (1987);\n(red), computed data.","276\nNational Tsunami Hazard Mitigation Program (NTHMP)\n0.05\nX / h = -9.95\nX / h =-0.25\n0.04\n0.03\n0.02\n0.01\n0.00\n-0.01\n-0.02\n0\n20\n40\n60\n80\n100\n120 0\n20\n40\n60\n80\n100\n120\nt (g /h) 1/2\nt (g /h)¹/2\nFigure 7-15: Time series of surface elevation at x/h = -9.95 and x/h = -0.25 on a 1:19.85 plane beach\nwith A/h = 0.019. o, analytical solution of Synolakis (1987);\n(red), computed data.\n0.10\n0.08\nt (g/h)¹ 1/2 = 55\nt (g/h)¹/2 = 70\n0.06\n0.08\n0.04\n0.02\n0.06\n0\n-0.02\n0.04\n-0.04\n-0.06\n0.02\n-0.08\n0\n1\n2\n-4\n-3\n-2\n-1\n0\nx/h\nx/h\nFigure 7-16: Convergence of numerical solution on a 1:19.85 plane beach with A/h = 0.019.\nanalytical solution of Synolakis (1987);\n(red), computed data with Ax/h = 0.025;\n(magenta),\ncomputed data with Ax/h = 0.0625;\n(cyan), computed data with Ax/h = 0.125;\n(red),\ncomputed data with Ax/h = 0.25.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n277\n0.075\nt (g/h)¹/2 = 30\n0.050\n0.025\n0.000\n-0.025\n0.075\nt (g/h) 1/2 = 40\n0.050\n0.025\n0.000\n-0.025\n0.075\nt (g/h)¹/2 = 50\n0.050\n0.025\n0.000\n-0.025\n0.075\nt (g/h) 1/2 = 60\n0.050\n0.025\n0.000\n-0.025\n0.075\nt (g/h)¹/2 = 70\n0.050\n0.025\n0.000\n-0.025\n0\n2\n-20\n-18\n-16\n-14\n-12\n-10\n-8\n-6\n-4\n-2\nX /h\nFigure 7-17: Transformation and runup of a solitary wave on a 1:19.85 plane beach with A/h = 0.0185.\no, laboratory data of Synolakis (1987);\n(red), computed data.","278\nNational Tsunami Hazard Mitigation Program (NTHMP)\nt (g/h) 1/2 = 10\nt (g/h) 1/2 = 40\n0.4\n0.2\n0.0\n(g/h)¹/2 = 15\n(g/h)¹/2 = 45\n0.4\n0.2\n0.0\nt (g/h)¹ 1/2 = 20\nt (g/h) 1/2 = 50\n0.4\n0.2\n0.0\nt (g/h)¹/2 1/2 = 25\nt (g/h)¹ 1/2 = 55\n0.4\n0.2\n0.0\nt (g/h) 1/2 = 30\nt (g/h) 1/2 = 60\n0.4\n0.2\n0.0\nt (g/h) 1/2 = 35\nt (g/h)¹ 1/2 = 65\n0.4\n0.2\n0.0\n-20\n-15\n-10\n-5\n0\n5\n10\n-20\n-15\n-10\n-5\n0\n5\n10\nx / h\nX / h\nFigure 7-18: Transformation and runup of a solitary wave on a 1:19.85 plane beach with A/h = 0.03. o,\nlaboratory data of Synolakis (1987);\n(red), non-hydrostatic solution with MCA;\n(blue),\nhydrostatic solution without MCA.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n279\nt (g/h) 1/2 = 40\n(g/h) 1/2 = 10\n0.4\n0.2\n0.0\n(g/h)¹/2 = 45\nt (g/h)¹/2 = 15\n0.4\n0.2\n0.0\nt (g/h)¹ 1/2 = 50\nt (g/h) 1/2 = 20\n0.4\n0.2\n0.0\nt (g/h)¹/2 = 55\nt (g/h)1/ 1/2 = 25\n0.4\n0.2\n0.0\n(g/h) 1/2 = 60\nt (g/h) 1/2 = 30\n0.4\n0.2\n0.0\nt (g/h)¹ 1/2 = 65\nt (g/h) 1/2 = 35\n0.4\n0.2\n0.0\n-20\n-15\n-10\n-5\n0\n5\n10\n-20\n-15\n-10\n-5\n0\n5\n10\nx / h\nx/h\nFigure 7-19: Transformation and runup of a solitary wave on a 1:19.85 plane beach with A/h = 0.03. o,\n(blue),\nlaboratory data of Synolakis (1987);\n(red), non-hydrostatic solution with MCA;\nhydrostatic solution with MCA.","280\nNational Tsunami Hazard Mitigation Program (NTHMP)\n90°\n22\n0°\n16\n9\nB\nC\n6,\n2\n180°\nB\n:\nA.,\n270°\n25m\n30m\nWave Maker\n(a).\n2\n3.6m\n1.1 m\nIncident wave\nA B C\n6\n9\n22\n30.5 cm\no\n32.0 cm\n(b).\nFigure 7-20: Schematic sketch of the conical island experiment. (a) Perspective view. (b) Side view\n(center cross section). o, gauge locations.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n281\n(c)\n(a)\n(b)\n5\n5\n10\ngauge 2\ngauge 2\ngauge 2\n5\n0\n0\n0\n-5\n-5\n-5\n-10\n5\n5\n10\ngauge 6\ngauge 6\ngauge 6\n5\n0\n0\n0\n-5\n-5\n-5\n-10\n5\n5\n10\ngauge 9\ngauge 9\ngauge 9\n5\n0\n0\n0\n-5\n-5\n-5\n-10\n5\n5\n10\ngauge 16\ngauge 16\ngauge 16\n5\n0\n0\n0\n-5\n-5\n-5\n-10\n5\n15\ngauge 22\ngauge 22\ngauge 22\n5\n10\n0\n5\n0\n0\n-5\n-5\n25\n30\n35\n40\n25\n30\n35\n40\n25\n30\n35\n40\nTime (sec)\nTime (sec)\nTime (sec)\nFigure 7-21: Time series of surface elevation at gauges around a conical island. (a) A/h = 0.045. (b)\nA/h = 0.096. (c) A/h = 0.181. o, laboratory data from Briggs et al. (1995);\n(red), computed data.\n(a)\n(b)\n(c)\n90°\n90°\n90°\n135°\n45°\n135°\n45°\n135°\n45°\n0°\n180°\n0°\n180°\n0°\n180°\n315°\n225°\n315°\n225°\n315°\n225°\n270°\n270°\n270°\n4\n4\n4\n3\n3\n3\n2\n2\n2\n1\n1\n1\n00\n0\n0\n0\n0\n30\n60\n90\n120\n150\n180\n60\n90\n120\n150\n180\n0\n30\n60\n90\n120\n150\n180\n0\n30\nDirection (°)\nDirection (°)\nDirection (°)\nFigure 7-22: Inundation and runup around a conical island. (a) A/h = 0.045. (b) A/h = 0.096. (c) A/h =\n0.181. o, laboratory data from Briggs et al. (1995);\n(red), computed data.","282\nNational Tsunami Hazard Mitigation Program (NTHMP)\n43.5m\n27.5m\n16.0 m\n10.2m\n7.3m\n10,0 fo\n12.6m\n12.4 m\n5.0 m\n11.0m\n26.5 m\n(a). Overview\nIncidentwave\n1/620\n17 cm\n1/550\n7 cm\nI\n15 cm\n9 bm\n31 cm\n1/31.25\n78 cm\n1/3.5\n1/15.5\n(b). Side view\nFigure 7-23: Schematic sketch of the complex reef system experiment in ISEC BP1. (a) Plain view. (b)\nSide view (center cross section).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n283\n43.5m\n27.5m\n16.0 m\n10.2 m\n7.3m\n10.0 m\n17.1\n12.6m\n12.4 m\n5.0m\n11.0m\n26.5 m\nF-3 m\n(a). Overview\nIncidentwave\n1/620\n17 cm\n7 cm\n1/550\n45\ncm\nI\n15 cm\n9 cm\n31 cm\n1/31.25\n78 cm\n1/3.5\n1/15.5\n(b). Side view\nFigure 7-24: Schematic sketch of the complex reef system experiment in ISEC BP2. (a) Plain view. (b)\nSide view (center cross section).","National Tsunami Hazard Mitigation Program (NTHMP)\n284\nWavemarker\n0 : wave gauge\n: wave gauge and ADV\nFigure 7-25: Geometry of reef system and wave gauge and ADV locations for ISEC BP1. O (white), wave\ngauge; o (red), wave gauge and ADV.\nWavemarker\n0\n: wave gauge\n: wave gauge and ADV\nFigure 7-26: Geometry of reef system and wave gauge and ADV locations for ISEC BP2. o (white), wave\ngauge; O (red), wave gauge and ADV.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n285\ny 0.0 m\ny 5.0 m\n40\nx =7.5 m\n40\n=7.5 m\n20\n20\n0\n0\n40\n40\n=11.5 m\nx=11.5m\n20\n20\n0\n0\n40\n40\n=13.0 m\n=13.0 m\n20\n20\n0\n0\n40\n40\nx =15.0 m\n=15.0\n20\n20\n0\n0\n40\nx =17.0 m\n40\n=17.0\n20\n20\n0\n0\n40\n40\n=21.0 m\nx=21.0\n20\n20\n0\n0\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\nt (sec)\nt (sec)\n(a).\n(b).\nx 25.0 m\n300\n40\nx =0.0 m\n200\nx =13.0 m. y =0.0 m\n100\n20\n0\n0\n-100\n-200\n300\n40\nx =2.0 m\n200\n=21.0 m y =0.0 m\n100\n20\n0\n0\n-100\n-200\n300\n40\n=5.0 m\n200\n=21.0 m y =-5.0 m\n100\n20\n0\n-100\n0\n-200\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n40\n=7.0 m\n50\nx =13.0 m. y =0.0 m\n20\n25\n0\n0\n-25\n-50\n40\n=10.0 m\n50\nx =21.0 m y =0.0 m\n20\n25\n0\n0\n-25\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n-50\nt (sec)\n50\nx=21.0m.y=-5.0m\n25\n0\n-25\n-50\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\nt (sec)\n(c).\n(d).\nFigure 7-27: Time series of surface elevation and velocity at gauges for ISEC BP1. (a) Surface elevations\nat y = 0 m. (b) Surface elevations at y = 5 m. (c) Surface elevations at X = 25 m. (d) Horizontal velocity\ncomparison at ADV.\n(black), laboratory data of Swigler and Lynett (2011);\n(red), solution\nwith NH-Hybrid scheme;\n(blue), solution without NH-Hybrid scheme.","National Tsunami Hazard Mitigation Program (NTHMP)\n286\ny 0.0\n300\n40\n=7.5 m\n200\nX =13.0m, y =0.0 m\n20\n100\n0\n0\n40\nx =13.0 m\n-100\n20\n-200\n0\n300\n200\nX =21.0 m, y =0.0 m\n40\nx =21.0\n100\n20\n0\n0\n-100\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n(a)\nt (sec)\n-200\n300\n5.0 m\n200\nX =21.0 m, y =-5.0 m\n40\nx =7.5 m\n100\n20\n0\n0\n-100\n40\nx =13.0 m\n-200\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n20\nt (sec)\n0\n50\nx=13.0m, y =0.0 m\n40\nX =21.0\n25\n20\n0\n0\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n-25\nt (sec)\n(b).\n-50\n50\nX =25.0m\nX =21.0 m, y =0.0 m\n40\ny =0.0 m\n25\n20\n0\n0\n-25\n40\ny =5.0 m\n-50\n20\n50\nX =21.0 m, y =-5.0 m\n0\n25\n40\ny =10.0\n0\n20\n-25\n0\n-50\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\nt (sec)\n(c).\n(d).\nt (sec)\nFigure 7-28: Time series of surface elevation and velocity at gauges for ISEC BP2. (a) Surface elevations\nat y=0m. (b) Surface elevations at y = 5 m. (c) Surface elevations at X = 25 m. (d) Horizontal velocity\n(black), laboratory data of Swigler and Lynett (2011);\n(red), solution\ncomparison at ADV.\n(blue), solution without NH-Hybrid scheme.\nwith NH-Hybrid scheme;","MODEL BENCHMARKING WORKSHOP AND RESULTS\n287\n43°N\nIwanai\n42.2°N\nHokkaido\nOkushiri\nJapan\nIsland\nSea\n42°N\nMonai\n42.1°N\nEsashio\n(a)\n(b)\n41°N\nFigure 7-29: Bathymetry and topography in the model region for the 1993 Hokkaido Nansei-Oki\ntsunami. (a) level-1 computational domain, (b) level-3 domain. o (red), epicenter; o (white), tide\ngauge stations.\nNorth Pole\nSea bottom\nStrike\nStrike direction\nangle\nReference\npoint\nReference\nLength\ndepth\nHanging\nWidth\nwall\nFoot wall\nSlip\nDip\nangle\nRake angle\nFigure 7-30: Schematic of planar fault model.","National Tsunami Hazard Mitigation Program (NTHMP)\n288\n43°N\nIwanai\nHokkaido\nJapan\nSea\n42°N\nEsashio\n41°N\nFigure 7-31: Initial sea surface elevation for the 1993 Hokkaido Nansei-Oki tsunami. Solid rectangles\nuplift contours at 0.5-m intervals;\nsubsidence contours at 0.2-m\nindicate subfaults;\n-,\n,\nintervals; O (red), epicenter; o (white), tide gauge stations.\n3.0\nEsashi\n2.0\n1.0\n0.0\nS\n-1.0\n-2.0\n-3.0\n3.0\nIwanai\n2.0\n1.0\n0.0\n-1.0\n-2.0\n-3.0\n0\n5\n10\n15\n20\n25\n30\n35\n40\n45\n50\n55\n60\nElapse Time (min)\n(red),\nFigure 7-32: Time series of surface elevation at tide gauges.\n(black), recorded data;\ncomputed data.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n289\n15\n10\n(u)\n5\n0\n42.25°N\n8\nInaho\n8\n42,20°N\nOkushiri\n42.15°N\nIsland\no\nO\nMonai\n42.10°N\nAonae\n42.05°N\n35 30 25 20 15 10 5\n°0\n0 5 10 15\n(m)\n(m)\n5\n10\n(ul)\n15\n20\n25\n139.40° E 139.45° E 139.50 E 139.55° E\nFigure 7-33: Runup around Okushiri Island. O (white): recorded runup;\n(red), computed runup.","National Tsunami Hazard Mitigation Program (NTHMP)\n290\n3\n9\n2\n7\n5\n1\n0\n0\n1\n2\n3\n4\n5\n(a)\nX (m)\n2\n1\n0\n-1\n-2\n20\n0\n5\n10\n15\n(b)\nTime (sec)\nFigure 7-34: Input data for Monai Valley experiment. (a) Computational domain. (b) Initial wave\n(grey),\nprofile. o, gauge locations;\n(black), topography contours at 0.0125-m intervals;\nbathymetry contours at 0.0125-m intervals;\n, initial profile.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n291\n6\ngauge 5\n4\n2\n0\n-2\n6\ngauge 7\n4\n2\n0\n-2\n6\ngauge 9\n4\n2\n0\n-2\n0\n10\n20\n30\nTime (sec)\nFigure 7-35: Time series of surface elevation at gauges in Monai Valley experiment.\n(black),\nlaboratory data of Matsuyama and Tanaka (2001);\n(red), computed data.","292\nNational Tsunami Hazard Mitigation Program (NTHMP)\n2.8\n2.8\n2.6\n2.6\n2.4\n2.4\n10t\n2.2\n2.2\n2\n2.0\n1.8\n1.8\n1.6\n1.6\n1.4\n1.4\n0.1\n0.075\n0.05\n0.025\n0\n4.4\n4.6\n4.8\n5\n5.2\n(40)\n(30)\n(20)\n(10)\nRunup (m)\nX (m)\n(a)\n(b)\nFigure 7-36: Runup and inundation comparisons. (a) Runup, (b) Inundation. o, laboratory data of\nMatsuyama and Tanaka (2001);\n(red), computed data;\n(black), topography contours at\n0.0125-m intervals;\n(grey), bathymetry contours at 0.0125-m intervals.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n293\nSamoa Islands\n13°S\nWestern Samoa\n10°S\nDART51425\nAmerican Samoa\nUpolu\nSamoa Islands\n14°S\nTutuila\nTau\n15°S\nPago Pago\nPago Pago\n20°S\nDART51426\n25°S\n16°S\nSouth Pacific Ocean\n30°S\n17°S\nDART54401\n35°S\n(a)\nFigure 7-37: Bathymetry and topography in the model region for the 2009 Samoa Tsunami. (a) level-1\ncomputational domain, (b) close-up view of epicenter and Samoa Islands. o (red), epicenter; O (white),\nwater-level stations.","National Tsunami Hazard Mitigation Program (NTHMP)\n294\nWestern Samoa\n(m)\nAmerican Samoa\n2000\nAmerican Samoa\n14.0°S\n1000\nSavaii\nUpolu\n0\nTau\n-1000\nTutuila\n-2000\n15°S\nTau\n-3000\n14.5°S\nTutuila\n-4000\n-5000\n16°S\n-6000\n-7000\n15.0°S\n-8000\n(b)\na\n-9000\n14.2°S\n4.27°S\nPago\"\n14.28\n14.3°S\nTutuila Island\n(d)\nFigure 7-38: Original bathymetry and topography in the model region for the 2009 Samoa tsunami. (a)\nclose-up view of faults and Samoa Islands in the level-1 computational domain, (b) level-2 domain, (c)\nlevel-3 domain, (d) level-4 domain. O (red), epicenter; O (white), water-level stations.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n295\nWestern Samoa\n(m)\nAmerican Samoa\n2000\nAmerican Samoa\n1000\n14.0°S\n14°S\nSavaii\nUpolu\n0\nTau\n-1000\nTutuila\n-2000\n15°S\nTau\n-3000\n14.5°S\nTutuila\n-4000\n-5000\n16°S\n-6000\n-7000\n18881\nM.0011\n15.0°S\n-8000\n698\n(b)\n(a)\n-9000\n14.2°S\n14.27°S\nPago\n14.28\n14.3°S\nTutuila Island\n1006\n1005\n14.4°S\n(d)\n(c)\nFigure 7-39: Smoothed bathymetry and topography with the depth-dependent Gaussian function in\nthe model region for the 2009 Samoa tsunami. (a) close-up view of faults and Samoa Islands in the\nlevel-1 computational domain, (b) the level-2 domain, (c) the level-3 domain , (d) the level-4 domain.\no (red), epicenter; O (white), water-level stations.","296\nNational Tsunami Hazard Mitigation Program (NTHMP)\nSurface Elevation (cm)\nAmplitude Spectrum (cm-s)\n400\n50\nPago Pago\n300\n40\n200\n20 30 10\n100\n0\n-100\n-200\n0\n10.0\n0,6\nDART 51425\n0.5\n5.0\n0.4\n0.0\n0.3\n0.2\n-5.0\n0.1\n-10.0\n0.0\n10.0\n1.2\nDART 51426\n1,0\n5.0\nAlina\n0.8\n0.0\n0.5\n0.4\n-5.0\n0.2\n-10.0\n0.0\n10.0\n0.6\nDART 54401\n0.5\n5.0\n0.4\n0.0\n0.3\n0.2\n-5.0\n0.1\n-10.0\n0.0\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\n4.0 10°\n10\n102\nElapse Time (hours)\nPeriod (min)\nFigure 7-40: Time series and spectra of surface elevations at water level stations. (black), recorded\ndata; (red), computed data.","297\nMODEL BENCHMARKING WORKSHOP AND RESULTS\n10\n8\n6\n4\n2\n0\n14.270\n14.275\n14.280\n0 2 4 6 8 10\n10 8 6 4 20\n(m)\n(m)\n2\n4\n(u)\n6\n8\n10\n170.67\n170.66\n170.69\n170.68\n170.71\n170.70\nLongitude (W)\n(white), recorded inundation;\nFigure 7-41: Runup and inundation at inner Pago Pago Harbor.\no(white): recorded runup; o(blue): recorded flow depth plus land elevation; (red), computed data;\n(black), coastline; (grey), depth contours at 10-m intervals.\n21.32°N\nHonolulu\nInternational Airport\nSand Island\nHonolulu\n21.30N\nOF\ntide gauge\nAla Moana\nKilo Nalu\nADCP\n21,28°N\nWaikiki\nKilo Nalu\nNearshore Reef\nObservatory\nFigure 7-42: Original bathymetry and topography at south shore of Oahu, Hawaii and Kilo Nalu\nNearshore Reef Observatory. o (white), water level and velocity measurement points.","National Tsunami Hazard Mitigation Program (NTHMP)\n298\n60°N\n40N\nLevel-1 grid for\nthe 2006 Kuril tsunami\n20°S\nLevel-1 grid for\n40°S\nthe 2010 Chile tsunami\n(a)\n60°S\n21.6 N\n21.0°N\n20.0°N\n19.0°N\n(b)\n(c)\nFigure 7-43: Original bathymetry and topography in the model region. (a) level-1 computational\ndomains in the Pacific. (b) level-2 domain over Hawaiian Islands, (c) level-3 domain of Oahu. o (red),\nepicenter.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n299\n21.6 N\n21.0°N\n21.4°N\n20.0°N\n19.0°N\n(a)\n(b)\n21.32°N\n21.30N\n21.28°N\nFigure 7-44: Smoothed bathymetry and topography with the depth-dependent Gaussian function. (a)\nlevel-2 domain over Hawaiian Islands, (b) level-3 domain of Oahu, (c) level-4 domain of the south\nshore of Oahu.","National Tsunami Hazard Mitigation Program (NTHMP)\n300\n50°N\n50°N\n48°N\n48°N\n46°N\n44°N\n44°N\n0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n-1\n0\n1\n2\n3\n(b)\nseafloor vertical displacement (m)\n(a)\nslip (m)\n32°S\n32°S\n34°S\n34°S\n36°S\n36°S\n38°S\n38°S\n-1 0 1 2 3 4 5\n0\n5\n10\n15\n-2\nslip (m)\nseafloor vertical displacement (m)\n(c)\n(d)\nFigure 7-45: Slip distribution and initial sea surface deformation. (a) USGS finite fault solution for the\n2006 Kuril earthquake, (b) Sea surface deformation for the 2006 Kuril tsunami, (c) USGS finite fault\nsolution for the 2010 Chile earthquake, (d) Sea surface deformation for the 2010 Chile tsunami.\n(red), epicenter.\nO","MODEL BENCHMARKING WORKSHOP AND RESULTS\n301\nHonolulu Tide Gauge\nAmplitude Spectrum (cm.s)\n30\n2.0\n20\n10\n0\n1.0\n-10\n-20\n-30\n0.0\nKilo Nalu ADCP\n20\n2.0\n10\nmark\n0\n1.0\n-10\n-20\n0.0\n10\n0.5\n0.4\n5\n0.3\n0\n0.2\n-5\n0.1\n-10\n0.0\n10\n0,5\n0.4\n5\n0.3\n0\n0.2\n-5\n0.1\n-10\n0.0\n5\n6\n7\n8\n9\n10 10°\n10\n102\nElapse Time (hours)\nPeriod (min)\nFigure 7-46: Time series and spectra of surface elevation and velocity at Honolulu tide gauge and Kilo\nNalu ADCP for the 2006 Kuril tsunami.\n(black), recorded data;\n(red), computed data.","National Tsunami Hazard Mitigation Program (NTHMP)\n302\nHonolulu Tide Gauge\nAmplitude Spectrum (cm.s)\n1.4\n50.0\n1.2\n25.0\n1.0\n0.8\n0.0\n0.6\n0.4\n-25.0\n0.2\n0.0\n-50.0\nKilo Nalu ADCP\n1.4\n40.0\n1.2\n20.0\n1.0\n0.8\n0.0\n0.6\n0.4\n-20.0\n0.2\n0.0\n-40.0\n1.0\n10.0\n5.0\n0.5\n0.0\n-5.0\n0.0\n-10.0\n0.2\n10.0\n5.0\n0.0\n0.1\n-5.0\n0.0\n-10.0\n13\n14\n15\n16\n17\n18\n19\n20\n10°\n10\n102\n10\n11\n12\nPeriod (min)\nElapse Time (hours)\nFigure 7-47: Time series and spectra of surface elevation and velocity at Honolulu tide gauge and Kilo\n(black), recorded data;\n(red), computed data.\nNalu ADCP for the 2010 Chile tsunami.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n303\n8 SELFE\nY. Joseph Zhang (OHSU)\nOregon Health & Science University (OHSU) and Oregon Department of Geology & Mineral\nIndustries (DOGAMI)\n8.1\nIntroduction\nThe state of Oregon is part of the US Pacific Northwest (PNW) region with extensive\ncoastline and vibrant coastal communities. The primary tsunami hazards posed to these\ncommunities come from the seismic sources including both local (Cascadia Subduction Zone -\nCSZ) and remote sources, with landslides being of relatively minor concern. We (OHSU) have\nbeen working in partnership with DOGAMI in the preparation and dissemination of inundation\nmaps since the 1990s, first under Oregon Senate Bill 379. Current funding from NTHMP has\nenabled us to apply state-of-the-art technology, i.e., new geological and computational models,\nthe latter based on unstructured grids, to remap the entire OR coastal counties for both local and\nremote sources. Because we are primarily focusing on seismic tsunamis, the tsunami propagation\nand inundation model we used has been carefully benchmarked for this purpose. We have\nhowever, also preliminarily benchmarked the non-hydrostatic version of the model for landslide\nproblems and we may consider formally doing SO in the future.\n8.2\nModel description\nThe fault dislocation model we use for the CSZ earthquakes is based on Okada's point\nsource model but with extensive geophysical constraints (Priest et al., 2009; Witter et al., in\nreview). We have developed 15 full-margin rupture models for Cascadia subduction zone\nearthquakes that define vertical seafloor deformation used to simulate tsunami inundation at\nBandon, Oregon. Rupture models include slip partitioned to a splay fault in the accretionary\nwedge and models that vary the updip limit of slip on a buried megathrust fault. Coseismic slip is\nestimated from turbidite paleoseismic records (Goldfinger et al., in press) and constraints from\ntsunami simulations at Bradley Lake (Witter et al., in review). Alternative earthquake source\nscenarios are evaluated using a logic tree that ranks model performance based on geophysical\nand geological data. Scenario weights at the branch ends of the logic tree are the products of the\nweights of the two parameters, earthquake size and fault geometry, and represent the relative\nconfidence that a particular model represents a reasonable rupture scenario based on geological\nand geophysical data, theoretical models and the judgment of the scientific team. The basal (and\nmost important) branch of the logic tree, estimated fault slip from recurrence interval, is based on\nthe frequency of turbidite interevent times over the last 10,000 years. The second branch of the\nlogic tree evaluates three rupture geometries that vary the distribution of slip on the updip end\nof\nthe locked zone: (1) activation of a shallow splay fault in the accretionary wedge; (2) shallow\nburied rupture that tapers slip to zero at the deformation front; and (3) a deeper buried rupture","National Tsunami Hazard Mitigation Program (NTHMP)\n304\nthat tapers slip to zero beneath a sharp break in the slope of the accretionary wedge offshore\nWashington and northern Oregon.\nFor remote sources we have limited ourselves to the Alaska-Aleutian subduction zone as\nhistorically it poses the most severe threat to the Oregon coast. We have considered two\nscenarios: the 1964 Great Alaska earthquake and a hypothetical worst case scenario. The source\nmodel for the 1964 event came from Johnson et al. (1996) while that for the worst case scenario\nwas originally used for a pilot study at Seaside, OR (TPSWG, 2006). Both scenarios involve Mw\n~9.2 earthquakes near the Gulf of Alaska. Results of simulations for the 1964 tsunami are\nchecked against historical observations of water levels and wave runup along the Oregon coast,\nallowing verification of the hydrodynamic model.\nThe worst case Gulf of Alaska earthquake scenario, identified as \"Source 3\" in Table 1 of\nGonzalez et al. (2009), has uniform slip on 12 subfaults with each subfault assigned an\nindividual slip value of 15, 20, 25 or 30 m. These extreme parameters result in maximum\nseafloor uplift nearly twice as large as the uplift produced by the 1964 earthquake (as estimated\nby Johnson et al. (1996)). Analyses of the maximum tsunami amplitude simulated for this source\nshow beams of higher energy directed toward the Oregon coast compared with other Alaska-\nAleutian subduction zone sources (TPSWG, 2006). Because of its precedent use for the Seaside\ntsunami study by TPSWG (2006), we consider the hypothetical Gulf of Alaska scenario as a\nmaximum distant tsunami source; however, testing the geological plausibility of the scenario was\nbeyond the scope of this study.\nThe tsunami propagation and inundation model we use is SELFE (Zhang and Baptista\n2008a), which was envisioned at its inception to be an open-source community-supported 3-D\nhydrodynamic/hydraulic model. This philosophy remains the corner stone of the model to this\nday. Originally developed to address the challenging 3-D baroclinic circulation in the Columba\nRiver estuary, it has since been adopted by 65+ groups around the world and evolved into a\ncomprehensive modeling system encompassing such physical/biology processes as general\ncirculation (Burla et al. 2010), tsunami and hurricane storm surge inundation (as in the on-going\nIOOS sponsored SURA project), ecology and water quality (Rodrigues et al. 2009ab), sediment\ntransport (Pinto et al. 2011), wave-current interaction (Roland et al. 2011) and oil spill (Azvedo\net al. 2009). Currently we maintain a central web site dedicated to this model\n(http://www.stccmop.org/CORIE/modeling/selfe/), maintain a user mailing list and mail archive\nsystem, organize annual user group meetings (since 2004), and occasionally conduct online\ntraining courses for users.\nThe rapid growth of the SELFE user community owes a great deal to the numerical scheme\nused in SELFE, which combines numerical accuracy with efficiency and robustness; the last two\nmodel traits are indispensable for large-scale practical applications as commonly found in\ntsunami hazard mitigation studies. The time stepping in SELFE is done semi-implicitly for the\nmomentum and continuity equations, and together with the Eulerian-Lagrangian method (ELM)\nfor the treatment of the advection, the stringent CFL stability condition is bypassed. The\nremaining stability conditions are related to the horizontal viscosity and baroclinic gradient\nterms, which are very mild (in the case of tsunami applications, these conditions are absent). The\nuse of unstructured (triangular) grids in the horizontal dimension further enhances the model\nefficiency and flexibility due to their superior capability in fitting complex coastal boundary and\nresolving bathymetric features, topographic features, and coastal structures. The vertical grid\nused in SELFE (for 3-D model) uses hybrid terrain-following coordinates (generalized o, or so-","MODEL BENCHMARKING WORKSHOP AND RESULTS\n305\ncalled \"S\"; Song and Haidvogel 1994) and Z coordinates. The model can be configured in\nmultiple ways: 2-D or 3-D; Cartesian (i.e., map projection) or spherical (latitude/longitude);\nhydrostatic or non-hydrostatic options, etc. In tsunami applications, we typically apply the 2-D\nhydrostatic configuration for maximum efficiency. An exception is for the land-slide generated\ntsunamis, to which we have applied the 3-D non-hydrostatic SELFE. For seismic tsunamis, we\nalso explicitly model the earthquake stage (i.e., with moving bed) in order to obtain accurate\ninitial acceleration (Zhang and Baptista 2008b).\nThe inundation algorithm in SELFE uses a simple iterative procedure to capture the moving\nshoreline as shown in Figure 8-1. Because a semi-implicit scheme is used, which enables\nexceptionally large time step, the iterative procedure allows wetting and drying over multiple\nlayers of elements over a single time step (i.e., with local CFL number >1). This simple\nprocedure has led to accurate and stable results even near the wet/dry front where supercritical\nflow is not uncommon.\nThe version used in this study (v3.1g) does not explicitly model wave breaking effects, and\ntherefore breaking waves are often represented as shock fronts in the model results. This\napproach is consistent with the conservatism we attempt to build in for inundation maps.\nHowever, conceptually it is not difficult to include the breaking effects in the model in the future.\ndry\nA\nTn\nB\nwet\nFigure 8-1: Inundation algorithm in SELFE. The orange line is the shoreline from the previous time\nstep, and the cyan lines are corrections made to obtain the shoreline at the new time step because\npoints A is wetted and B is dried.\nSince 2007, all components of the SELFE modeling system have been fully parallelized\nusing domain decomposition and Message Passing Protocol (MPI). This has further enhanced\nefficiency. For example, in the recent simulation of the impact of the 1964 Alaska event on the\nUS west coast, we used a large grid (with 2.9 million nodes and 6 million elements) to resolve 12\nmajor estuaries and rivers in the PNW (Zhang et al. 2011). With 256 CPUs on NASA's Pleiades\ncluster, the 6-hour simulation took only 2.25 hours wall-clock time. The largest grid we have\nimplemented on Pleiades for SELFE SO far has over 10 million nodes in the horizontal and 26\nlevels in the vertical.\n8.3\nBenchmark results\nAs mentioned before, here we will only report results for those benchmark problems\nrelevant to seismic sources.","National Tsunami Hazard Mitigation Program (NTHMP)\n306\n8.3.1 BP1: Solitary wave on simple beach - analytical\n%\nR\nH\nd\nL\nX0\nFigure 8-2: Domain sketch.\nThis canonical problem deals with a single solitary wave propagating along a constant depth\nand then over a sloping beach (Figure 8-2). The problem is completely defined by 3 parameters:\nd (offshore depth), (beach slope) and H (height of the solitary wave over constant depth), and\nall variables are non-dimensionalized with respect to d. The goal is to validate the model for both\npropagation and inundation on the beach.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n307\n0.04\nt=35\nSELFÉ - NRMSD=3%\n0.02\nERR Max Wave Amp=4%\nSLOPE\n0\nSELFE\n-0.02\nANALYTICAL\n0.04\nt=40\nSELFÉ - NRMSD=3%\n0.02\nERR Max Wave Amp=4%\n0\n-0.02\n0.06\nt=45\n0.04\nSELFE - NRMSD-2%\nERR Max Wave.Amp =3%\n0.02\n0\n-0.02\nn\n0.08\nt=50\n0.06\nSELFE - NRMSD= 1%\n0.04\n- ERR Max Wave Amp =2%\n0.02\n0\n-0.02\n0.1\nt=55\n0.08\nSELFE -NRMSD=1%\n0.06\n0.04\nERR- Max Wave Amp =0%\n0.02\n0\n-0.02\n0.08\nt=60\n0.06\nSELFE - NRMSD=1%\n0.04\nERR Max Wave Amp =0%\n-\n0.02\n0\n-0.02\n0.04\nt=65\nSELFE - NRMSD=5%\n0.02\nERR Max Wave Amp =1%\n0\n-0.02\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\nX\nFigure 8-3: Comparison of surface profiles at various times for the non-breaking wave case A (H/d =\n0.0185). All variables have been non-dimensionalized. The RMSE at t=60 (near the maximum runup)\nis 0.001 and the Willmott skill is 0.998 (we have restricted the calculation of errors to x<2 to remove\nthe uninteresting part of the solution offshore).","National Tsunami Hazard Mitigation Program (NTHMP)\n308\n0.03\nx/h=9.95\nSELFE\n0.02\nSELFE - NRMSD=2%\nANALYTICAL\nERR Max Runup Amp 2%\n0.01\n0\n-0.01\n-0.02\n0.05\nx/h=0.25\n0.04\nSELFE - NRMSD=1%\n0.03\nERR Max Runup Amp = 1%\n0.02\nn\n0.01\n0\n-0.01\n-0.02\n100\n120\n0\n20\n40\n60\n80\nt\nFigure 8-4: Comparison of elevation time series at two locations. The RMSE at the two stations are\n0.0012 and 0.0008, and the Willmott skill are 0.994 and 0.999 respectively.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n309\n0.02\n0.01\n0.00\nd=9.81m, t=70s\n-0.01\nd=14.1264m, t=84s\n-0.02\n-0.03\nn\n0.001\nDifference\n0.000\n-0.001\n0\n2\n4\n6\n8\nX\nFigure 8-5: Scaling effects through comparison of surface profiles between model results using two\ndifferent values of d. Note that the corresponding dimensional times are different due to the different\nscaling.","National Tsunami Hazard Mitigation Program (NTHMP)\n310\n1-60\n0.08\nAnalytical\nAx=0.1*d, At=0.05s\n0.075\nAx=0.05*d, At=0.025s\nAx=0.025*d, At=0.0125s\n0.07\nAx=0.0125*d, At=0.00625\n0.065\nn\n0.06\n0.055\n0.05\n0.045\n0.04\n-1.5\n-1.4\n-1.3\n-1.2\n-1.1\n-1\n-0.9\n-0.8\n-0.7\n-0.6\n-0.5\nX\nFigure 8-6: Convergence test for case A at a time near the maximum runup (t = 60).\nFor convenience, we chose d = 9.81 m in the model simulations, although scalability of the\nmodel is also tested below (Figure 8-5). The model grid covers - 98m< x 686.7 m and is 2\nelements wide in the y direction. A \"uniform\" grid with Ax = Ay = 0.1*d was used with each\nsquare divided into 2 triangular elements, and therefore there are altogether 2403 nodes in the\ngrid. The time step was set at At = 0.05 S, and the implicitness factor at 0.6 (the maximum runup\nis somewhat sensitive to this factor; as explained in Zhang and Baptista (2008b), the best\naccuracy is achieved near 0.5). On the left boundary, a Flather type open boundary condition was\nimposed to minimize reflection there. We used frictionless bottom and nonlinear shallow-water\nequation (SWE) in order to be consistent with the assumptions made in the analytical solution.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n311\nThe comparisons were made in several ways. First the surface profiles at multiple times are\ncompared in Figure 8-3. The model is able to accurately simulate the entire runup and run-down\nprocess. In addition to the conventional root-mean-square error (RMSE) we also used the\nWillmott skill number which is defined as:\n(1)\nwhere m and o are model and data respectively, and O is the data mean. This number combines\nthe contributions from both RMSE and the correlation (Willmott et al. 1985); a Willmott number\nof 1 indicates a perfect skill and the model is more skilled with a higher W.\nSecondly, the time series at two stations were compared (Figure 8-4). Note that at one\nstation (x = 0.25) wetting and drying occurred as indicated in both the model results and the\nanalytical solution. Again the model skill is high in this aspect.\nThirdly, the scaling effects in the model were also examined. As mentioned before, there is\nonly 1 parameter used in scaling all variables: the offshore depth d. Figure 8-5 shows model\nresults from using different values of d and it can be seen that the difference in the elevations is\nvery small, even near the instantaneous shoreline.\nThe convergence of the model is illustrated by using successively finer grid size and time\nstep; the grid size and time step were varied in such a way that the CFL number remains constant\nas required by the model (Zhang and Baptista 2008a). The most challenging aspect of this test is\nto obtain the convergence for the maximum runup. As shown in Figure 8-6, the modeled runup\nindeed converges to the analytical value; at the finest resolution the error for the runup value is\n3%.\nDue to the small grid size used in this benchmark problem, the CPU time is modest. For\nexample, the 100-time-unit run shown in Figure 8-4 took 1 min wall-clock time to complete with\n1 CPU (all tests shown in this report, unless otherwise noted, were conducted on an Intel cluster\nrunning on Rocks version 5.0 with CPU clock speed of 2.66 GHz and gigabit copper Ethernet\nnetwork connection).\n8.3.2 BP2: Solitary wave on composite beach - analytical\nThis problem was modeled against the Revere Beach located approximately 6 miles\nnortheast of Boston in the City of Revere, Massachusetts. A physical model was constructed at\nthe Coastal Engineering Laboratory of the U.S. Army Corps of Engineers, Vicksburg,\nMississippi facility. The model beach consists of three piecewise-linear slopes of 1:53, 1:150,\nand 1:13 from seaward to shoreward with a vertical wall at the shoreline (Figure 8-7). In the\nexperiments, the wavemaker was moved to different locations for each of the 3 cases: A, B and\nC. Here we simply used the measured time series at gauge 4 as the boundary condition for the\nmodel. For comparison with the analytical solution, which was derived from the linearized SWE,\nwe will only consider case A for the reason stated below.","National Tsunami Hazard Mitigation Program (NTHMP)\n312\nG4\nG5\nG6\nG7\nG8\nG9 G10\n1/13\nd=0.218m\n1/150\n1/53\nLB4\nL45\n0.9m\n4.36m\n2.93m\n45\nCase A : L45=2.40m\nLB4 is the distance between\nCase B : L45=0.98m\nthe boundary and G4\nCase C : L45=0.64m\nFigure 8-7: Schematics of the composite beach and locations of gauges.\nGauge4\nGauge5\nGauge6\n0.01\n0.01\n0.01\n0\n0\n0\n-0.01\n-0.01\n-0.01\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\nTime (sec)\nGauge7\nGauge8\nGauge9\n0.01\n0.01\n0.02\n0\n0\n0\n-0.01\n-0.01\n-0.02\n15\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n5\n10\n20\n25\nGauge10\nGauge11\n0.02\n0.05\n0\n0\nAnalytical\nModel\n-0.02\n-0.05\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\nFigure 8-8: Comparison of elevation time history at 8 stations for case A. Gauge 4 is located at the\ndomain boundary and serves as a check for the imposed boundary condition; gauge 11 is at the\nvertical wall, where the RMSE is 1.5 mm and the Willmott skill is 0.96.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n313\nIn the model, a rectangular domain with variable grid size (Ay = 2.5 mm and Ax varies from\n1\ncm at the left boundary to 1 mm at the vertical wall, in order to capture the large runup there)\nwas used to cover 12.64m0.045. The modeled runups are accurate up to this limit and\nare less accurate beyond (Figure 8-11). We expect better results to be obtained when we\nincorporate the wave breaking effects.","National Tsunami Hazard Mitigation Program (NTHMP)\n314\nt=30\n0.1\nLab\nModel\n0.05\n0\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\nt=40\n0.1\n0.05\n0\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\nt=50\n0.1\n0.05\n0\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\nt=60\n0.1\n0.05\n0\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\nt=70\n0.1\n0.05\n0\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\nX\nFigure 8-9: Comparison of surface profiles for case A. The Manning friction coefficient is no = 0.016.\nThe error for the maximum elevation at t = 60 (near the maximum runup ) is 2%.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n315\nt=15\nLab\n0.4\nModel (no=0.016)\nModel (n.=0)\n0.2\n0\n-10\n-5\n0\n5\n10\n15\n20\nt=20\n0.4\n0.2\n0\n-10\n-5\n0\n5\n10\n15\n20\nn\nt=25\n0.4\n0.2\n0\n-10\n-5\n0\n5\n10\n15\n20\nt=30\n0.4\n0.2\n0\n-10\n-5\n0\n5\n10\n15\n20\nX\nFigure 8-10: Comparison of surface profiles for breaking-wave case C (H/d = 0.3). Model results with\ntwo choices of bottom friction are shown.","National Tsunami Hazard Mitigation Program (NTHMP)\n316\n0.6\n0.4\nBreaking\nLab data\nModel\nR/d\n0.2\n0.0\n0.315\n0.180\n0.225\n0.270\n0.000\n0.045\n0.090\n0.135\nH/d\nFigure 8-11: Runups as a function of the incident wave height. The maximum runup error for non-\nbreaking waves is 1.9%.\n8.3.4 BP5: Solitary wave on composite beach - laboratory\nThe set-up of the experiments has been described in Section 8.3.2. The model uses the\nnonlinear SWE instead of the linear SWE as in Section 8.3.2. We have also explored variation of\nthe Manning coefficient (no) but found little sensitivity in the results to this parameter. The\nresults presented below were obtained using no = 0.\nThe elevation time series for all 3 cases (A, B, and C) are presented in Figure 8-12 through\nFigure 8-14. The mismatches in mean water level at gauge 7 in Figure 8-12, etc., suggest\nproblems in the lab data. In general, the model results compare reasonably well with the lab data,\nexcept for the largest wave case C (Figure 8-14).\nFurthermore, the modeled runups at the vertical wall are also in reasonable agreement with\nthe data for cases A and C. For case A, the modeled and lab measured runups are 2.3 cm and\n2.7 cm respectively. For case C, they are 24.9 cm and 27.4 cm. For case B, the large waves\ncolliding with the wall generated very high splash which explains the largest runup measured","MODEL BENCHMARKING WORKSHOP AND RESULTS\n317\namong the 3 cases (45.7 cm). Because the model does not explicitly simulate this process, the\nmodeled runup was substantially lower (19.1 cm). In all three cases, the model runs were stable\nwith no sign of instability.\n-3\n-3\n-3\nGauge4\nGauge5\nGauge6\nx10\nx10\nX 10\n10\n10\n10\n5\n5\n5\n0\n0\n0\n-5\n-5\n-5\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\nTime (sec)\n-3\n-3\nGauge7\nGauge8\nGauge9\nX 10\nx 10\n10\n10\n0.02\n5\n5\n0.01\n0\n0\n0\n-5\n-5\n-0.01\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\nGauge10\n0.02\n0.01\nLab\nModel\n0\n-0.01\n0\n5\n10\n15\n20\n25\nFigure 8-12: Comparison of elevation time series for case A.","National Tsunami Hazard Mitigation Program (NTHMP)\n318\nGauge6\nGauge4\nGauge5\n0.1\n0.1\n0.1\n0\n0\n0\n-0.1\n-0.1\n-0.1\n0\n5\n10\n15\n20\n25\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n5\nTime (sec)\nGauge9\nGauge7\nGauge8\n0.1\n0.1\n0.1\n0\n0\n0\n-0.1\n-0.1\n-0.1\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\nGauge10\n0.1\n0\nLab\nModel\n-0.1\n0\n5\n10\n15\n20\n25\nFigure 8-13: Comparison of elevation time series for case B.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n319\nGauge4\nGauge5\nGauge6\n0.2\n0.2\n0.5\n0\n0\n0\n-0.2\n-0.2\n-0.5\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\nTime (sec)\nGauge7\nGauge8\nGauge9\n0.2\n0.2\n0.2\n0\n0\n0\n-0.2\n-0.2\n-0,2\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\n0\n5\n10\n15\n20\n25\nGauge10\n0.2\nLab\nModel\n0\n-0.2\n0\n5\n10\n15\n20\n25\nFigure 8-14: Comparison of elevation time series for case C.\n8.3.5 BP6: Solitary wave on a conical island\nTo examine the model performance in more than one dimension, the experiment of solitary\nwave around a conical island (Figure 8-15a) is simulated numerically here. The lab data include\ntime series at various gauges in the tank as well as runup measurements around the perimeter of\nthe conical island.\nWe did not model the wavemaker in the experiment but instead used the time series at a\ngauge close to the wavemaker as the boundary condition. An unstructured grid was generated for\nthis problem to better resolve the downwave side of the island where large runups due to\ncollision of two waves were expected. A 10 cm grid size was used at the outer tank boundaries,\n1 cm at the boundary of the flat top of the cone (which has a diameter of 2.2 m), and 5 cm in the\nupwave half of the circle that defines the toe of the conical island, and 2 cm for the downwave\nhalf (Figure 8-16). The total number of the nodes was 259231. The time step was set at 0.02 S,\nand the Manning's no = 0.01. The total simulation time was 60 S, which took 38 min wall-clock\ntime on 24 CPUs.","National Tsunami Hazard Mitigation Program (NTHMP)\n320\nFigure 8-15 shows the comparison of elevation time series at 4 gauges for 3 cases, with a\nprogressively larger incident wave amplitude. The modeled elevations are in good agreement\nwith the lab data and the model was stable for all cases.\nThe modeled runup distribution around the island is compared to the measured values in\nFigure 8-17. Figure 8-18 is a spatial representation of the same information as in Figure 8-17.\nThe agreement is excellent for cases A and B; the maximum errors were below 5%. For case C,\nthe model results exhibit a slight asymmetry (Figure 8-18) and an under-estimation of\nthe\nmaximum runup at the back of the island. The wave breaking that occurred in this case cannot be\naccurately modeled by SELFE.\n(b) Case A\nGauge 6\n0.05\n0\nData\n-0.05\nModel\n20\n5\n10\n15\nGauge 9\n0.05\n0\n(a)\n-0.05\n5\n10\n15\n20\n0.05\nGauge 16\n0\n-0.05\n20\n5\nGauge 22 10\n15\n0.05\n0\n-0.05\n20\n5\n10\nTime (sec) 15\n(d) Case C\n(c) Case B\n0.1\n0.05\n0\n0\n-0.1\n-0.05\n20\n5\n10\n15\n5\n10\n15\n20\n0.1\n0.05\n0\n0\n-0.1\n-0.05\n20\n5\n10\n15\n5\n10\n15\n20\n0.1\n0.05\n0\n0\n-0.1\n-0.05\n15\n20\n20\n5\n10\n5\n10\n15\n0.1\n0.05\n0\n0\n-0.1\n-0.05\n15\n20\n20\n5\n10\n5\n10\n15\nFigure 8-15: Comparison of elevation time series at 4 gauges for 3 cases. (a) shows the experimental\nsetup.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n321\nToe of island\nFigure 8-16: Nodes in the unstructured grid.","National Tsunami Hazard Mitigation Program (NTHMP)\n322\nCase A\nData\n0.04\nModel\n0.02\n0\n300\n350\n400\n150\n200\n250\n0\n50\n100\nCase B\n0.1\n0.05\n0\n350\n400\n0\n50\n100\n150\n200\n250\n300\nCase C\n0.2\n0.1\n0\n150\n200\n250\n300\n350\n400\n0\n50\n100\nAzimuth (degrees)\nFigure 8-17: Comparison of runups around the conical island for the 3 cases. The errors at the back of\nthe island are: 2.6%, 4.3% and 14% for the 3 cases.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n323\n2\n2\n1.5\n1.5\n1\n1\n0.5\nDATA\n0.5\n0\nModel\nCase A\n0\nCase B\n-0.5\n-0.5\n-1\n-1\n-1.5\n-1.5\n-2\n-2\n-2\n-1.5\n-1\n-0.5\n0\n0.5\n1\n1.5\n2\n-2\n-1.5\n-1\n-0.5\n0\n0.5\n1\n1.5\n2\n2\n1.5\n1\n0.5\n0\nCase C\n-0.5\n-1\n-1.5\n-2\n-2\n-1.5\n-1\n-0.5\n0\n0.5\n1\n1.5\n2\nFigure 8-18: Comparison of runups around the conical island for the 3 cases, in spatial form.\n8.3.6 BP7: Wave runup on Monai Valley\nThis test considers the wave tank experiment that models the 1993 Okushiri Island tsunami.\nThe lab data include time series at 3 gauges (Figure 8-19a) as well as video images that illustrate\nthe runup sequence in a narrow valley near Monai.\nThe model used a uniform grid resolution of 1.4 cm and a time step of 0.01 S . Therefore\nthere were altogether 95892 nodes in the grid. We have tested the sensitivity to the Manning's no\nand found little influence from this parameter. The results below were obtained using no = 0. The\ntotal simulation time was 22.5 S, which took 9 min wall-clock time with 20 CPUs.\nThe comparison of time series at the 3 gauges in front of the valley is shown in Figure 8-19.\nThe model was able to capture the arrival time and the amplitude of the first waves well. The\ninundation sequence in the narrow valley is shown in Figure 8-20, which agrees qualitatively","National Tsunami Hazard Mitigation Program (NTHMP)\n324\nwith the lab observation: the modeled maximum runup of 9 cm is close to the observed mean\nvalue of 10 cm.\n(b)\n(a)\n4\n(m)\ny\n3\nOkuahiri\nGauge 5\nIsland\n2\nGauge 9\n1\nGauge 7\nLab data\nMonai\nValley\n0\nSELFE\nGauge\n5\n5\n-1\n3\nGauge 7\n05\n1\n:\n1)\nD's\nlife\n23\nat\nx (m)\n-1\n5\nGauge 9\n3\n1\n-1\n10\n12\n14\n16\n18\n20\n22\nTime (s)\nFigure 8-19: Comparison of elevations at 3 gauges in front of the valley as shown in (a).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n325\nt=16\nt=16.6 S\nS\nt=17.2\nFigure 8-20: Inundation sequence near the narrow valley;. t = 16.6 is close to the maximum runup,\nwhich is ~9 cm .\n8.3.7 BP9: Field - Okushiri Island\nThe first field test conducted was the 1993 Hokkaido-Nansei-Ok tsunami that impacted the\nJapanese island of Okushiri. The historical data were originally collected by the post-tsunami\nsurvey group and passed onto the author by Dr. Tomo Takahashi; they include the pre- and post-\nevent bathymetry survey data, the estimated source information, tide gauge records at 2 gauges,\nand estimated runup distribution around the island. However, the horizontal datum used in the\nfiles was later found to contain errors when overlaid with modern maps. The files used in this\nstudy were corrected by Dr. Dmitry Nicolsky (U. Alaska Fairbanks). As shown in Figure 8-21a,\nthere still appear to be mismatches among the various files that define the DEMs.","National Tsunami Hazard Mitigation Program (NTHMP)\n326\n(c)\n(b)\n(a)\n0\n260\n520\n780\n1040\n1300\n1560\n1820\n2080\n2340\n2600\n2860\n3120\n3380\n3640\n3900\nFigure 8-21: (a) Bathymetry as embedded in DEMs; (b) unstructured grid and (c) zoom-in around the\nOkushiri Island. The white arrow in (a) indicates a mismatch of bathymetry from multiple DEM\nsources.\nTo better capture the runup distribution around the island we used variable resolution around\nthe island, with 30 m resolution at the shoreline, and ~5 m around the narrow Tsuji Valley where\nthe maximum runup (~31 m) was observed (Figure 8-21). At the outer ocean boundary we used a\ncoarse resolution of 7 km. The total number of nodes in the grid was 1,617,562 with over 95%\nspent around the island. A time step of 0.5 S was used to carry out the 1 hour simulation, which\ntook 4.75 hours on 40 CPUs.\nThe comparison at the 2 tide gauges is shown in Figure 8-22. While the modeled arrival time\napproximately matched the gauge record at Esashi, it was too early at Iwanai, suggesting errors\nin the source information; note that a similar mismatch was also observed by Kato and Tsuji\n(1994).\nOn the other hand, the agreement between the model and data for the distribution of the\nrunups around the island was much more reasonable (Figure 8-23). Note that the data were\ndigitized from a figure in Kato and Tsuji (1994) and therefore the precise locations of\nobservation were unknown. Nevertheless, the model seems to have captured well the variation\nalong the west, north, and south coast, with large errors along the east coast, where the errors in\nthe source information may be more pronounced. The model adequately simulated the large\nrunups around the Tsuji Valley (with 20% error) on the west coast, and around Aonae-\nHamatsumae on the south coast. The two waves reported in various post-tsunami surveys as\narriving ~10 min apart devastated the town of Aonae; these were correctly simulated by the\nmodel (Figure 8-24).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n327\n3\nEsashi\n2\nModel\nData\n1\n0\n1\n2\n0\n10\n20\n30\n40\n3\n2\nIwanai\n1\n0\n1\n2\n3\n0\n10\n20\n30\n40\nFigure 8-22: Comparison of elevations at 2 tide gauges.","National Tsunami Hazard Mitigation Program (NTHMP)\n328\nObserved : 8.7 - 11.1\n4.9\n6.8\nObserved : 9.6 - 5.7\n2.3\nObserved : 5.0\n5.1\nObserved : 5.6-9.0 -\nObserved : 3.3\n2.7\n8.6\nObserved : 3.4\nObserved : 5.1 - 7.0 7.0\n5.2\nObserved : 6.8 - 7.7\nObserved : 13 10.1\n13.9\nObserved : 16.3\n6.5\nObserved : 3.4\n24.3 (20%\nObserved : 30.6\n14.2\nObserved : 12.6\nObserved : 18.7\n18.3\nObserved : 22.8\n17.2-22\n11.8\nObserved : 4.8\nObserved : 8.3-13.2 - 9.0\nObserved : 3.2 - 10.2 7.1\nObserved : 12.4\n12.6\nFigure 8-23: Comparison of runups around the island. Red and green numbers are from the model,\nwith the green numbers indicating larger errors.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n329\n20\nTime=4.8333\n15\n10\nnae\n5\n0\n-5\n2U\nTime=13\n15\n10\n5\n0\n-5\nFigure 8-24: Arrival of 2 waves at Aonae. The 1st wave came from the west while the 2nd wave\nattacked from the east.\n8.3.8 N-Wave runup on a beach\nThis test was originally proposed at the 3rd Workshop on Longwave Runup. The 1-D\nproblem considers an initial N wave on a sloping beach (Figure 8-25) and has a nonlinear","330\nNational Tsunami Hazard Mitigation Program (NTHMP)\nanalytical solution. Note that this test is more nonlinear than that for the solitary wave case\n(Section 8.3.1) and is therefore a good test for the nonlinear SWE solvers.\nThe model grid used a uniform resolution of 5 m in both X and y directions, a time step of\n0.1 S and a frictionless bottom. The total number of nodes in the grid was 176011. With 25\nCPUs, the 240 s-run took 24 min to complete.\nComparison of the along-slope elevation and velocity profiles at 3 times is shown in Figure\n8-26; the errors for elevation were smaller than for the velocity, and the error for the runup was\n5%. The model accurately simulated the entire runup and rundown phases.\nMSL\nFigure 8-25: Domain sketch for N-wave runup.\n30\nAnalytical\n20\nModel\nt=220s\n10\n0\nt=175s\n160s\n-10\n-20\n-30\n-40\n-300-200-100\n0\n100\n200\n300\n400\n500\n600\n700\n800\n8\nt=160s\n4\n=220s\n0\n-4\n-8\n-12\nt=175s\n-16\n300 - 200-100\n0\n100\n200\n300\n400\n500\n600\n700\n800\nAlong beach direction (m)\nFigure 8-26: Comparison of elevation (top) and velocity (bottom) at 3 times. The error in the runup is\n5%.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n331\nComparison of the time history of shoreline position and velocity is shown in Figure 8-27.\nIn general, larger errors were observed during the runup phase than in the rundown phase; in\naddition there appeared to be some minor instability during the runup phase as can be seen from\nthe modeled shoreline velocity. Overall the model was able to adequately capture the shoreline\nevolution.\n300\nAnalytical\n200\nModel\n100\n0\n100\n200\n10\n5\n0\n-5\n-10\n15\n20\n0\n100\n200\n300\nTime (sec)\nFigure 8-27: Comparison of time history of the shoreline position (top) and velocity (bottom).\n8.3.9 1964 Alaska - field\nThe March 28, 1964 Prince William Sound (Alaska) earthquake produced a mega trans-\noceanic tsunami that represented the largest tsunami that impacted US and Canadian west coast\non record. There is a wealth of field records for this event from tide gauges and eyewitness\nreports. Therefore, we have been using this event as a representative of remote sources in our\nmitigation studies in Oregon.\nWe have recently conducted a detailed propagation and inundation study for this event for\nthe Pacific Northwest coast (WA, OR and CA), and detailed model results are being reviewed as\na journal publication (Zhang et al. 2011). Therefore we only present some highlights from this\nstudy below.","National Tsunami Hazard Mitigation Program (NTHMP)\n332\nUsing a large unstructured grid (with over 2.9 million nodes) to cover the Pacific from\nPrince William Sound to the west coast, we were able to resolve 12 major estuaries and rivers in\nWashington and Oregon (Figure 8-28). The resolution at the shoreline was 80 m in general, and\n5-30 m in the 12 estuaries and rivers. The Manning no = 0.025. The time step was set at 1 S\nA key question we strived to answer from this study was the role of tides in inundation. To\nthis end, we have conducted two sets of simulation, one with static tides (with the vertical datum\nfixed at local MHW) and the other with dynamic tides. For the latter, we conducted a tidal\nsimulation for 7.15 days before the event in order to reach a dynamic equilibrium.\nFigure 8-29 shows the comparison of elevation time series at two tide gauges, one of which\nis located deep inside the Columbia River estuary, from the 2 simulations mentioned above. The\nmain finding was that the linear superposition of tides on top of the static tide results led to an\nunder-estimation of the wave amplitude as compared to the data due to significant nonlinear\ninteraction between tides and tsunamis. The results with dynamic tides rectified this under-\nestimation (Figure 8-29).\nThe nonlinear interaction was even more pronounced in the maximum inundation extent as\nin the city of Cannon Beach, Oregon (Figure 8-30). With the static tides, the inundation extent\nwas severely under-estimated, even though the starting water level was higher (MHW as\nopposed to MSL used in the dynamic tide simulations). The inundation line predicted by\nincluding dynamic tides significantly improved the model results (Figure 8-30b). Spectral\nanalysis was conducted to verify that the nonlinear interaction occurred along the narrow Ecola\nCreek (not shown).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n333\n30\n+\n500m\nAC\n25\n20\n15\nGrays Harbor\n10\n5\nColumbia River\n500h\n0\nNC\n0\n10\n20\n30\n40\n50\n60\n70\n80\n90\n100\nEquivalent radius (m)\n50km\n0m\n500m\nCoos Bay\nFigure 8-28: Grid used in the 1964 event, with multiple zooms. The insert histogram shows the\ndistribution of equivalent radii of all elements; over 90% of the elements have a radius of 40 m or less.\n2.0\nStatic tides\n0.2\nDynamic tides\n1.0\n0.0\n-0.2\n0.0\nRUN40e\nDATA\n-0.4\n-1.0\nAstoria\n-0.6\n-2.0\n-0.8\n0.0\n2.0\n0.0\n1.0\n2.0\n3.0\n4.0\n6.0\n4.0\n5.0\n8.0\n6.0\n3.0\n2.0\nTide+tsunami\n2.0\n1.0\nDATA\nTsunami alone + tidal result\n1.0\n0.0\n0.0\n-1.0\n-1.0\n-2.0\nCrescent\n-2.0\n-3.0\n0.0\n1.0\n2.0\n3.0\n4.0\n5.0\n0.0\n6.0\n2.0\n4.0\n6.0\n8.0\nTime (hours)\nTime (hours)\nFigure 8-29: Comparison at two tide gauges; the model results are from (a) static tides; and (b)\ndynamic tide (black) or superposition of tides on top of static tide results (green).","National Tsunami Hazard Mitigation Program (NTHMP)\n334\n(a) Tsunami alone\n(b) Tsunami+tide\nLeidelHouse\nSteudel House\nBell harbor\nElk Creek Bridge\nData\nTsunami+ Tide\nSt.\nCannon Beach\nSt.\nI 1 ft\nMeters\nMeters\n0 10\n0 10\nFigure 8-30: Comparison of inundation at Cannon Beach. The black dots are field estimates from\nWitter (2008) and the red dots are from the model, (a) with static tides, and (b) with dynamic tides.\nThe simulations were conducted at NASA's Pleiades cluster, and the 6-hour static tide\nsimulation took 2.25 hours wall-clock time with 256 CPUs. The model efficiency has enabled us\nto effectively conduct simulations with even larger grids (~10 million nodes).\nLessons learned\n8.4\nDuring the benchmarking exercise, one of the most perplexing problems we have\nencountered was related to the incomplete information regarding those tests. We had to spend a\nconsiderable amount of time gleaning files from various sources. For the field tests (Okushiri,\netc.) some critical pieces of information (such as the horizontal datums of the DEMs) are still\nmissing. Perhaps the most serious problem with field tests is uncertainty about the geometry of\nthe earthquake source. This issue causes serious errors in simulations for areas proximal to the\nsource (e.g., poor match of simulated runup on the east coast of Okushiri Island).\nThe set of benchmark problems proposed in OAR-PMEL-135 was found to be mostly\nappropriate except for a few extreme cases (e.g., the larger wave breaking case C in the\ncomposite beach case). The combination of analytical, lab, and field tests adequately tests the\nperformance of models.\nGoing forward we suggest inclusion of more recent field cases. These events have the\nadvantage of better documentation and of the availability of new types of observation (satellite","MODEL BENCHMARKING WORKSHOP AND RESULTS\n335\nimagery, amateur and professional videos, and DART buoy records, etc.), and therefore serve as\nbetter validation tools for models.\n8.5\nReferences\nAzvedo A, Oliveira A, Fortunato AB, Bertin X. 2009. Application of an Eulerian-Lagrangian oil\nspill modeling system to the Prestige accident: trajectory analysis. J. Coastal. Res., 56,\n777-781.\nBurla M, Baptista AM, Zhang Y, Frolov S. 2010. Seasonal and inter-annual variability of the\nColumbia River plume: a perspective enabled by multi-year simulation databases.\nJournal of Geophysical Research, 115, C00B16.\nGoldfinger C, Nelson CH, Johnson JE, Morey AE, Gutiérrez-Pastor J, Karabanov E, Eriksson\nAT, Gràcia E, Dunhill G, Patton J, Enkin R, Dallimore A, Vallier T, and the Shipboard\nScientific Parties. 2010. Turbidite Event History: Methods and Implications for Holocene\nPaleoseismicity of the Cascadia Subduction Zone: U.S. Geological Survey Professional\nPaper 1661-F (in press).\nGonzalez FI., et al. 2009. Probabilistic tsunami hazard assessment at Seaside, Oregon, for near-\nand far-field seismic sources. J. Geophys. Res., 114, C11023.\nJohnson J, Satake K, Holdahl S, Sauber J. 1996. The 1964 Prince William Sound earthquake:\njoint inversion of tsunami and geodetic data. J. Geophys. Res., 101(B1), 523-532.\nKato K, Tsuji Y. 1994. Estimation of fault parameters of the 1993 Hokkaido-Nansei-Oki\nearthquake and tsunami characteristics. Bull. Earthq. Res. Inst., Univ. of Tokyo, vol. 69,\n39-66.\nPinto L, Fortunato AB, Zhang Y, Oliveira A, Sancho FE.P. Development and validation of a\nthree-dimensional morphodynamic modelling system, Ocean Modelling (submitted).\nPriest GR, Goldfinger C, Wang K, Witter RC, Zhang YJ, Baptista AM. 2009. Confidence levels\nfor tsunami-inundation limits in northern Oregon inferred from a 10,000-year history of\ngreat earthquakes at the Cascadia subduction zone: Natural Hazards, DOI\n10.1007/s11069-009-9453-5.\nRodrigues M, Oliveira A, Queiroga H, Fortunato AB, Zhang Y. 2009a. Three-Dimensional\nModeling of the Lower Trophic Levels in the Ria de Aveiro (Portugal), Ecological\nModelling, 220(9-10), 1274-1290.\nRodrigues M, Oliveira A, Costa M, Fortunato AB, Zhang Y. 2009b. Sensitivity analysis of an\necological model applied to the Ria de Aveiro. Journal of Coastal Research, SI56, 448-\n452.\nRoland A, Zhang Y, Wang HV, Maderich V, Brovchenko I. 2011. A fully coupled wave-current\nmodel on unstructured grids, Compt. & Geosciences (submitted).\nSong Y, Haidvogel DB. 1994. A semi-implicit ocean circulation model using a generalized\ntopography-following coordinate system. J. Comp. Phys., 115(1), 228-244.\nTPSWG (Tsunami Pilot Study Working Group). 2006. Seaside, Oregon tsunami pilot study-\nmodernization of FEMA flood hazard maps. National Oceanic and Atmospheric\nAdministration OAR Special Report, NOAA/OAR/PMEL, Seattle, WA.","National Tsunami Hazard Mitigation Program (NTHMP)\n336\nWillmott CJ; Ackleson SG; Davis RE; Feddema JJ; Klink KM; Legates DR; O'Donnell J; Rowe\nCM. 1985. Statistics for the Evaluation and Comparison of Models, J. Geophys. Res., Vol.\n90, No. C5, pp. 8995-9005\nWitter RC. 2008. Prehistoric Cascadia tsunami inundation and runup at Cannon Beach, Clatsop\nCounty, Oregon, Oregon Department of Geology and Mineral Industries, Open-File\nReport O-08-12, 36 p., appendices.\nWitter RC, Zhang JY, Wang K, Priest GR, Goldfinger C, Stimely L, Ferro P, (in review)\nSimulating tsunami inundation at bandon, oregon using hypothetical Cascadia and Alaska\nearthquake scenarios: Oregon Department of Geology and Mineral Industries Special\nPaper.\nZhang Y, Baptista AM. 2008a. SELFE: A semi-implicit Eulerian-Lagrangian finite-element\nmodel for cross-scale ocean circulation. Ocean Modelling, 21 (3-4), 71-96.\nZhang Y, Baptista AM. 2008b. An Efficient and Robust Tsunami Model on Unstructured Grids.\nPart I: Inundation Benchmarks. Pure and Applied Geophysics, 165, 2229-2248.\nZhang Y, Witter RW, Priest GP. 2011. Nonlinear Tsunami-Tide Interaction in 1964 Prince\nWilliam Sound Tsunami, Ocean Modelling (submitted).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n337\n9 THETIS\nStéphane Abadie\nLaboratoire SIAME, Université de Pau et des Pays de l'Adour, Anglet, France, and Fulbright\nScholar at Department of Ocean Engineering, University of Rhode Island, Narragansett, USA\n9.1\nIntroduction\nTHETIS is a multi-fluid Navier-Stokes (NS) solver developed by the TREFLE CNRS\nlaboratory at the University of Bordeaux I. It is a multi purpose CFD code, freely available to\nresearchers (http://thetis.enscbp.fr) and fully parallelized. THETIS solves the incompressible NS\nequations for water, air, and the slide. Basically, at any time, the computational domain is\nconsidered as being filled by one \"equivalent\" fluid, whose physical properties (namely density\nand viscosity) vary with space. Subgrid turbulent dissipation is modeled based on a Large Eddy\nSimulation approach, using a mixed scale subgrid model (Lubin et al., 2006). The governing\nequations (i.e., conservation of mass and momentum) are discretized on a fixed mesh, which may\nbe Cartesian, cylindrical or curvilinear, using the finite volume method. These governing\nequations are exact, except for interfacial meshes, where momentum fluxes are only\napproximated, due to the presence of several fluids. NS equations are solved using a two-step\nprojection method. Fluid-fluid interfaces are tracked using the VOF method. For most flows, the\nPLIC algorithm (e.g., Abadie et al., 1998) enables accurate tracking while keeping the interface\ndiscontinuous. However, for very violent flows with fast droplet ejection, the PLIC method may\ncause divergence of the projection algorithm. In such cases, the interface is smoothed either by\nallowing a slight diffusion process, after each PLIC iteration, or using a TVD scheme solving\nEulerian advection equations for the interfaces.\nTHETIS has been extensively validated for many theoretical and experimental flow cases.\nHence, each new version of THETIS has to successfully solve more than 50 validation cases\nwithin a certain expected accuracy, before being released.\nAbadie et al. (2006, 2008, 2010) simulated classical landslide tsunami benchmarks\ninvolving rigid bodies using THETIS. In these papers, rigid slides are simulated as a Newtonian\nfluid, for which deformation is prevented by specifying an infinite viscosity. With the use of this\nso-called penalty method, the slide displacement is computed implicitly and thus no longer\nprescribed as in most other similar studies. Abadie et al. (2010) performed convergence studies\nand showed that result accuracy on the free surface is satisfactory as long as enough grid cells\nare used to ensure that the slide motion is correctly reproduced. Figure 9-1 shows an example of\nTHETIS' results for the 2-D experiment of Heinrich (1992). In the latter experiment, waves were\ngenerated by a triangular rigid wedge sliding down a 45° slope. Figure 9-1a and Figure 9-1b\nshow two snapshots of model results with interface contours and flow streamlines. The flow is\nsolved within the wedge as well, where the very large viscosity, as expected, yields rectilinear\nstreamlines (confirming that the rigid slide moves as a whole). Figure 9-1c shows the slide law","National Tsunami Hazard Mitigation Program (NTHMP)\n338\nof motion simulated, as compared to experimental data, and Figure 9-1d shows the subsequent\nfree surface deformation.\nThis penalty method is an improvement that may help NS models to become more suitable\nfor more realistic applications or case studies, in which, of course, slide motion is always\nunknown. However, real slides are far more complex than solid bodies. Our next step is thus to\nvalidate the model for waves generated by deformable slides. For that purpose, we have\nimplemented the generalized non Newtonian fluid model (i.e., a Herschel-Bulkley fluid)\nin\nTHETIS and are currently performing tests using this constitutive law.\nBP8: Three-dimensional landslide\n9.2\nNavier-Stokes multi-fluid models such as THETIS are adapted to wave generation by\nviolent subaerial landslides, as they can deal with multiple interface reconnections (i.e.,\noverturning/breaking waves). They may also be used for shallow water submarine landslide\ncases, provided that the ratio of slide thickness to depth is large enough, as the whole domain has\nto be meshed. For larger depth however, models relating pressure to free surface elevation will\nbe less time consuming, for a comparable accuracy and thus more adapted.\n(b)\n(a)\n0.10\n(d)\n(c)\n0.05\n0.4\n0.00\n-0.05\n0.3\n-0.10\n0.2\n-0.15\n-0.20\n0.1\n-0.25\n0.0\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\n4.0\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nx(m)\nt(8)\nFigure 9-1: Simulation of Heinrich's experiment (Heinrich, 1992) using THETIS. Top panel: density\ncontours and flow streamlines at time t =: (a) 0.5 S (b) 1 S, (c): Time evolution of vertical slide\ndisplacement. (): Heinrich's (1992) experiments, (-): THETIS, (d): Free surface deformation at t =\n0.5 S, (): Heinrich's (1992) experiments, dot filled contour: THETIS.\nIn the NTHMP work along the East Coast, THETIS is used only to simulate the tsunami\nwave source in the potential case of Cumbre Vieja Volcano flank collapse (La Palma, Canary\nIslands), featuring a large subaerial landslide. For that reason, THETIS has only been tested","MODEL BENCHMARKING WORKSHOP AND RESULTS\n339\nagainst BP8, which features a rigid body close to the free surface. Note also, that for the case of\nLa Palma, runup is not studied with THETIS but instead, through a model coupling approach\nemploying THETIS to compute the tsunami generation and later the long-wave propagation and\ncoastal impact model FUNWAVE to study propagation and runup. For this reason, the THETIS\nmodel's results on runup in BP8 are only indicative.\nIn BP8, instead of prescribing slide motion (as in Liu et al., 2005, for instance), we solved\nfor the full coupling between slide and water and hence calculated slide motion as part of the\nsolution, where the rigid wedge is modeled as a Newtonian fluid of large viscosity (105 Pa.s).\nHere, we present in detail the first benchmark case in which the slide initial submergence is\nA = -0.1 m and slide density ratio is 2.14; x' is in the slope direction, z' is perpendicular to the\nslope plane (gravity being in this case inclined by 26.56° with respect to z'), y = y' is in the\nlongshore direction. Due to symmetry with respect to the middle vertical plane, the\ncomputational domain only represents half the experimental flume (including runup gauge 1 and\nwave gauge 1). Two numerical grids were tested to assess discretization effects on slide motion\nand, subsequently, on free surface deformation. Both grids were irregularly distributed over x'\nand z' to account for the need for finer resolution close to the generation zone. The first grid\n(mesh 1, with 62 X 76 X 24 cells) is comparable to the mesh used in Liu et al. (2005). The finest\ngrid cell size is Ax' = 0.039 m, Az' = 0.0196 m, Ay' being constant and equal to 0.077 m.\nFor the second mesh used (mesh 2, with 170 X 100 X 120 cells), the finest grid cell size is\nAx' = Az' = 0.015 m, Ay' was constant and also equal to 0.015 m. The nondimensional finest\ngrid sizes are respectively Ax'/L = Az'/L = 3.8% for mesh 1 and 1.4% for mesh 2 (with\nL = 1.017 m). High resolution 3-D computations are very computationally intensive and\ntherefore simulations were run using the parallel version of our model.","National Tsunami Hazard Mitigation Program (NTHMP)\n340\nI\n091\nD\nh\n[SIDE VICW]\n2h\n1.2446\n[PLANE VIEW]\nWave gauge 2\n1.83\nRun-up gauge 3\n0.635\n0.61\nRun-up gauge 2\n0.305\nWave gauge 1\n0.91\nRun-up gauge 1\nFigure 9-2: Sketch of BP8 (after Liu et al., 2005)\nFigure 9-3 shows snapshots of water/air and slide/water interfaces computed at four\ndifferent time steps for the finest mesh, using the NS-PLIC model. In the last image at t = 3.5 S,\nthe slide has stopped on the horizontal bottom of the experimental flume, which is represented in\nsimulations but not visible on Figure 9-3. In this snapshot sequence, we observe the generation\nof a wave train, and its subsequent propagation and reflection from the vertical sidewalls of the\nflume. Maximum runup, which occurs at the plane of symmetry on the flume axis, is seen to also\npropagate toward and reflect off the sidewalls.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n341\n(a)\n(b)\n(c)\n(d)\nFigure 9-3: Snapshots of slide/water and water/air interfaces at different times for grid 2 with\n170x100x120 cells, using the setup sketched in Figure 9-2. Slide initial submergence is D = -0.1 m, slide\ndensity is 2.14. (a) t = 0.7 S, (b) t = 1.4 s S, (c) t = 2.1 S, (d) t = 3.5 S.\n4.5\n(b)\n4.0\n3.5\n(a)\n3.0\n2.5\n2.0\n4\n1.5\n1.0\n0.5\n0.Q\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\nt(s)\nFigure 9-4: Time evolution of slide center of mass, using the setup sketched in Figure 9-2. Solid lines:\nnumerical results with (a) 62x76x24 cells and (b) 170x100x120 cells ; (): experimental data. Initial\nslide submergence is A = -0.1 m, slide density is 2.14.","342\nNational Tsunami Hazard Mitigation Program (NTHMP)\nFigure 9-4 shows the simulated slide center of mass motion as a function of time, as\ncompared to experimental data. The measured slide motion is well reproduced in the finer mesh\ngrid, with a RMS deviation of 0.16 m. In the coarser grid, the slide is much slower than in\nexperiments (larger RMS deviation of 0.80 m). A computation performed using an intermediate\ngrid size (not presented here) yielded a slide motion curve in between the two presented curves,\nindicating a consistent behavior of the model. Other computations for different initial\nsubmergence values also matched experimental data well, provided that a fine enough grid (i.e.,\nwith Ax/L~1%) is used. A closer inspection of our results indicates that resolving the coupling\nbetween slide motion and water flow in the numerical model is achievable with good accuracy,\nbut requires an overall grid about 20 times larger than when slide motion is a priori specified, as\nin Liu et al. (2005).\nFigure 9-5 compares the experimental data to the surface elevations simulated at wave\ngauges 1 and 2, both in the generation area (see Figure 9-2). At gauge 1, the first elevation wave\nand trough are both well modeled in the finer grid 2 (respectively 17% and 5% of relative errors),\nwhereas the second wave is much higher (162%) than measured. This was also observed by Wu\n(2004) for this gauge location and a slide initial submergence D = -0.05. Our last attempt\nusing a finer grid step (mesh: 260 X 200 X 120) helps, reducing the error from 162% to 114% for\nthis second wave.\nWave phase and, hence, celerity, are correctly predicted in the finer grid 2, whereas the\nsecond wave is too slow in the coarser grid 1. Note that wave heights are also under-predicted in\nthe latter grid, which is consistent with a slower slide. At gauge 2, the lateral spreading of the\nwave is also well simulated using grid 2; the simulated wave elevations are close to the\nexperimental results (17%, 10%, 30% respectively for wave crest 1, trough 1, wave crest 2), even\nthough waves seem to be a bit slower. In grid 1 again, the slower slide generates both smaller\nand slower waves as compared to experimental data.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n343\n0.10\n(b)\n0.05\n(a)\n0.00\n-0.05\n-0.10\n0.10\n0.05\n(b)\n0.00\n(a)\n-0.05\n0.10\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\n4.0\nt(s)\nFigure 9-5: Comparison between numerical results (solid lines) and experimental data () for the time\nhistories of free surface elevations at wave gauge 1 (top figure) and wave gauge 2 (bottom figure),\nusing the setup sketched in Figure 9-2. Initial slide submergence is A = -0.1 m, slide density is 2.14, (a)\n62x76x24 cells, (b) 170x100x120 cells, and (red curve) 260x200x120 cells.\nFigure 9-6, similarly, compares runup simulated at runup gauges 2 and 3 (see Figure 9-2) to\nexperimental data. At gauge 2, in the finer grid, numerical results closely match experiments,\nexcept during the first runup phase where the model overestimates the recorded runup value and\ngenerates a quicker runup motion. In the coarser grid, run-down and runup values are both\noverestimated. A similar behavior is observed at gauge 3. Wu (2004) also reported such\ndiscrepancies between runup data and numerical result with an initial submergence A = -0.05.\nNote that we used here a free slip boundary condition on the bottom, which may explain the\noverestimation of the runup results. The model accuracy may be improved by using a no-slip\ncondition.","National Tsunami Hazard Mitigation Program (NTHMP)\n344\n0.10\n(b)\n(a)\n0.05\n0.00\n-0.05\n-0.10\n0\n1\n2\n3\n4\n5\n6\nt(s)\n0.10\n(a)\n(b)\n0.05\n0.00\n-0.05\n-0.10\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\n4.0\nt(s)\nFigure 9-6: Comparison between numerical results (solid lines) and experimental data () for the time\nhistories of runup at gauge 2 (top) and gauge 3 (bottom) ), using the setup sketched in Figure 9-2.\nInitial submergence is A = -0.1 m, slide density is 2.14. (a) 62x76x24 cells, (b) 170x100x120 cells.\nFinally, we also simulated the second benchmark case featuring the same rigid body, but\nwith a lower submergence A = -0.025 m and a higher slide density ratio of 2.79. In the finer\nmesh, the only results shown here, slide motion is accurately predicted (Figure 9-7). Also a\ncomparison of free surface elevation at wave gauge 2, presented in Figure 9-8, shows the\naccuracy that may be expected by using THETIS in this kind of problem.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n345\n4.5\n4.0\n3.5\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\nexperimental data Liu et al. (2005)\nnumerical results 170x100x120\n0.87\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\ntime(s)\nFigure 9-7: Time evolution of slide center of mass. Initial slide submergence is A = -0.025 m, slide\ndensity is 2.79. Solid line: numerical results with 170 X 100 X 120 cells; (): experimental data.","National Tsunami Hazard Mitigation Program (NTHMP)\n346\n0.10\n0.05\n0.00\n-0.05\n-0.10\nexperimental data Liu et al. (2005)\nnumerical results 170x100x120\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\n4.0\ntime(s)\nFigure 9-8: Comparison between numerical results (solid lines) and experimental data () for the time\nhistories of free surface elevations at wave gauge 2 (Figure 9-2). Initial slide submergence is\nA = -0.025 m, slide density is 2.79. 170 X 100 X 120 cells.\n9.3 References\nAbadie S, Caltagirone JP, Watremez P. 1998. Splash-up generation in a plunging breaker\n.\nComptes Rendus de l'Académie des Sciences -Series IIB -Mechanics-Physics-Astronomy,\nVolume 326, Issue 9, Pages 553-559\nAbadie S, Grilli S, Glockner S. 2006. A coupled numerical model for tsunami generated by\nsubaerial and submarine mass failures, in Proc. 30th Intl. Conf. Coastal Engng., San\nDiego, California, USA. 1420-1431.\nAbadie, S, Morichon, D, Grilli, S, Glockner, S. 2008. VOF/Navier-Stokes numerical modeling of\nsurface waves generated by subaerial landslides, La Houille Blanche, 1,21-26.\nAbadie S, Morichon D, Grilli S, Glockner S. 2010 A three fluid model to simulate waves\ngenerated by subaerial landslides. Coastal Engineering, 57, 9, 779-794.\nHeinrich P. 1992. Nonlinear water waves generated by submarine and aerial landslides. J.\nWtrwy, Port, Coast, and Oc. Engrg., ASCE, 118(3), 249-266.\nLiu PL-F, Wu TR, Raichlen F, Synolakis CE, Borrero JC. 2005. Runup and rundown generated\nby three-dimensional sliding masses. J. Fluid Mech., 536, 107-144.\nLubin P, Vincent S, Abadie S, Caltagirone JP. 2006. Three-dimensional Large Eddy Simulation\nof air entrainment under plunging breaking waves, Coastal Engineering, Volume 53,\nissue 8, .631-655.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n347\nWu TR. 2004. A Numerical Study of Three-Dimensional Breaking Waves and Turbulence\nEffects. PhD Study. Cornell University.","National Tsunami Hazard Mitigation Program (NTHMP)\n348\n10 TSUNAMI3D\nJuan J. Horrillo, Amanda L. Wood, Gyeong-Bo Kim, Ashwin Parambath, Zygmunt Kowalik\nMaritime Systems Engineering, Texas A&M University at Galveston, Texas; US\nOcean Engineering, Texas A&M University at College Station, Texas, US\nInstitute of Marine Science, University of Alaska Fairbanks, Alaska, US\n10.1 Introduction\nIn accordance with National Tsunami Hazard Mitigation Program (NTHMP) guidance,\ntsunami numerical models need to be tested against the NOAA PMEL 135 validation\nbenchmarks before receiving approval to be used as calculation tools for inundation mapping and\nmitigation. In this section, two experimental benchmarking problems have been selected to\nvalidate a proposed 3-D numerical model for tsunami generation by sub-aerial and submarine\nlandslides. The 3-D model, dubbed TSUNAMI3D (for Tsunami Solution Using Navier-Stokes\nAlgorithm with Multiple Interfaces), uses the Navier-Stokes approach and the Volume of Fluid\nMethod (VOF) to track the fluid free surface. The model has been developed by the University\nof Alaska Fairbanks (UAF) and by Texas A&M University at Galveston (TAMUG).\nTSUNAMI3D solves transient fluid flow with free surface boundaries, based on the concept\nof the fractional VOF. The code uses an Eulerian mesh of rectangular cells having variable sizes.\nThe fluid equations solved are the finite difference approximation of the Navier-Stokes and the\ncontinuity equations. The basic mode of operation is for single fluid calculation having multiple\nfree surfaces. However, TSUNAMI3D can also be used for calculations involving two fluids\nseparated by either a sharp or indefinite (diffusive) interface, for instance, water and mud. In\neither case, the fluids may be treated as incompressible. Internal obstacles or topography are\ndefined by blocking out fully or partly any desired combination of cells in the mesh.\nThe TSUNAMI3D code was exclusively developed for tsunami calculations. The first\nversion of the code was developed under the direction of Professor Zygmunt Kowalik, UAF.\nOther important contributions to the code are due to William Knight, West Coast & Alaska\nTsunami Warning Center (WC/ATWC) and Ed Kornkven, Arctic Region Supercomputing\nCenter (ARSC). The code is based on development done at the Los Alamos National Laboratory\n(LANL) during the 70's, following the work done by C. W. Hirt and a group of researchers,\nincluding, among others, A. A. Amsden, T. D. Butler, L. D. Cloutman, B. J. Daly, R. S.\nHotchkiss, C. Mader, R. C. Mjolsness, B. D. Nichols, H. M. Ruppel, M. D. Torrey and\nKothe.\nThe current TSUNAMI3D code has undergone dramatic changes from its original\nconception in Horrillo's thesis (2006). In particular, the VOF algorithm for tracking the\nmovement of a free surface interface between two fluids or between a fluid and a void has been\nsimplified especially for the 3-D mode of operation. The simplification accounts for the\nhorizontal distortion of the computational cells with respect to the vertical scale that is proper in","MODEL BENCHMARKING WORKSHOP AND RESULTS\n349\nthe construction of efficient 3-D grids for tsunami calculations. In addition, the pressure term has\nbeen split into two components, hydrostatic and non-hydrostatic. The splitting of the pressure\nterm allows users of the model to obtain a hydrostatic solution by merely switching off the non-\nhydrostatic pressure term. Therefore, TSUNAMI3D can be used to separate out non-hydrostatic\neffects from the full solution while keeping the three dimensional structure. TSUNAMI3D is\nsuitable for complex tsunami generation because it is capable of modeling: (1) moving or\ndeformable objects, (2) subaerial/subsea landslide sources, (3) soil rheology, and (4) complex\nvertical or lateral bottom deformation.\nTSUNAMI3D is in constant development and it requires FORTRAN and MATLAB (post-\nprocessing) programming skills to put it to work. Usually, a 3-D simulation requires a large\namount of computer memory and CPU-wall-time to obtain the solution. A few subroutines, the\nmost computational demanding, are parallelized using OPENMP directives. In the 2-D mode of\noperation, the code uses the PETSC-MPI library to solve the pressure field.\n10.2 TSUNAMI3D Governing Equations\nThe governing equations to describe the flow of two incompressible Newtonian fluids (e.g.,\nwater and mud) in a domain S((t), are given by the equation of conservation of mass\n(1)\ndx oy oz\nand the conservative equation of momentum given by:\nA-water:\n(2)\nOx\n(3)","National Tsunami Hazard Mitigation Program (NTHMP)\n350\n(4)\nwhere, u(x,y,z.)), v(x,y,z,t) and w(x,y,z,t) are the velocity components along the\ncoordinate axes of the fluid at any point x x xi + yj + zk at time t, n, (x,y,t) is the water-surface\nelevation measured from a vertical datum, P1 is the density of the fluid, and is the non-\nhydrostatic pressure. The kinematic viscosity H,/p, can be related to the water eddy viscosity and\ng is the acceleration due to gravity. The total pressure, p=Phyd = +q, has been divided into the\nhydrostatic pressure Phyd = pig(n1 such that - and the dynamic pressure q.\nHerez is the elevation measured from a vertical datum to the cell center. The velocities u,v\nand W associated with a computational cell are located at the right, back and the top face of the\ncell respectively. The non-hydrostatic pressure q(x,y,z,t) is located at the cell center as the\nhydrostatic pressure is.\nFor the simulation of landslide induced-tsunami waves, an additional set of equations are\nincluded for the second fluid (mud) with density P2. Here, for this particular set of equations, the\nmud is considered as a Newtonian fluid (mud rheology is not considered). Then, the set of\ngoverning equations for the momentum in the second layer (mud) are given by:\nB-mud:\n(5)\n(6)","MODEL BENCHMARKING WORKSHOP AND RESULTS\n351\n(7)\nAgain, here u(x,y,z.t), v(x,y,z,1) and w(x,y,z,t) are the velocity components of the\nfluid along the coordinate axes at time t.N2 (x,y,t) is the mud-surface elevation measured from a\nvertical datum; P2 is the mud density and a is the fluid-mud density ratio given by P1P2 The\nkinematic viscosity H2/P2 can be related to a constitutive model for mud rheology for a non-\nNewtonian fluid (not discussed here). The total pressure, p = Phyd + q, has been divided into the\nhydrostatic pressure = + P2(n2 - z)] and the dynamic pressure q.\nBoth interfaces, water surface elevation (water-void) and mud surface elevation (fluid/void-\nmud), are traced using a simplified VOF method based on the donor-acceptor algorithm of Hirt\nand Nichols (1981). The simplified VOF method features a scalar function F1.2 (x,y,z,t) to\ndefine the water/mud region in space and time. Here, sub-indices 1 and 2 represent water and\nmud respectively. The F function accounts for the fractional volume of fluid/mud contained in\nthe cell (fluid concentration in the computational cell). A unit value for F corresponds to a fluid\ncell totally filled with water/mud, while a value of zero indicates an empty cell. Therefore, a cell\nwith an F value between zero and one is a surface cell or a water-mud interface cell. The\nequation describing the F function is given by\n(8)\nwhich states that F1.2 propagates with the fluid velocities u , V and W. Physical properties in\neach cell element, i.e., the density and eddy viscosity, can be weighted in terms of the\nF1.2 (x,y,z,t) function. For example, a general expression for density is determined by the\nfollowing equation and condition,\np(x,y,z,t)\n(9)\nF1 >= F2\nEq. 9 indicates that advection or transportation of the second fluid (mud) requires the\nexistence of the first fluid (water) in the cell such that F1 >= F, (saturated condition). For\ninstance, in a mud parcel that is isolated from the atmosphere / void (the subaerial landslide\ncase), the initial value of F1 always equals F2 in the control volume cell (F2 = F ) This technique\ngreatly simplifies calculations of both free surfaces, because the advection algorithm for the\nsecond fluid (mud) is an external procedure that is completed once the advection of the first fluid\n(water) is done.\nThe scalar function F is located at the cell center as the hydrostatic (Phvd ) and non-\nhydrostatic (q) pressures. Eq. 8 is only solved in the water and mud domains. The water and\nmud surface elevations N1.2 (x,y,1) are a mere byproduct of F1.2 and they are calculated by\nlocating the water/void or mud/water interfaces along the water/mud column at each (x,y)\nlocation in time. This implies that breaking waves are not allowed, because just one value of n1\nand N2 are kept for each (x,y) location. This assertion is valid for cells with a large distortion","National Tsunami Hazard Mitigation Program (NTHMP)\n352\nratio (horizontal/vertical scale ratio) much greater than two (>> 2) which is a common case in\nmesh generation for practical tsunami calculations.\nThe code uses an Eulerian mesh of rectangular cells having variable sizes. The governing\nequations are solved by using the standard volume difference scheme starting with field variables\nsuch as u , V W, q and F. which are known at time t = 0 Notice that n is a function of F and\nis known once F is determined. All variable are treated explicitly with the exception of the non-\nhydrostatic pressure field q which is treated implicitly. The governing equations are solved by\ndiscretizing field variables spatially and temporally in the domain to obtain new field variables at\nany required time. Nonlinear terms are approximated using an up-wind down-wind approach up\nto the third order. The hydrodynamic pressure field q is calculated through the Poisson's\nequation by using the incomplete Choleski conjugate gradient method to solve the resulting\nsystem of linear equations.\nTurbulence closure approximations are not considered in the model solution. Instead, the\nscale of turbulence is mainly accomplished using a simplified general eddy viscous formulation\ndescribed by two phase fluids, water and mud (Direct Numerical Simulations (DNS)). The DNS\napproximation is presumably correct when the computational domain is relatively well resolved.\nIn addition, for rapid landslide tsunami generation (very steep slope), the energy transfer\nmechanism is mainly the pressure. Therefore, energy loss due to turbulence is expected to be\nsmall and comparable to other numerical assumptions and physical processes. Even though the\nturbulence mechanisms are solved in a very simple manner, the final solution is expected to be\nadequate. The simplified general eddy viscous formulation indicated previously in the\nmomentum equations is discussed in more detail in the following lines.\nThe friction term can be tuned for internal friction within the water body by means of the\nwater eddy viscosity. For instances, in practical tsunami calculations, a value for H,/p, typically\nranges from 10-6 12/sec to 10-5 m²/sec. For mud, on the other hand, a typical value for H2/P2\nranges from 10 1 /sec to 103 m²/sec. At the water-mud interfaces, the eddy viscosity is\ncalculated by a weighting function similar to the expression and condition indicated for density\nexpression in Eq. 9. The eddy viscosity expression for water-mud interface cells is\n(10)\nP\nFor a well resolved domain (very high resolution), additional friction mechanisms can be\nimplemented, for instance, the no-slip condition. The no-slip condition is enforced at all\ncomputational cells that are in contact with the sea-bottom or walls, i.e., duloz=0 In addition,\nto mimic further the bottom friction, an exponential function is provided to increase the fluid\neddy viscosity to one or several orders of magnitude at computational fluid cells located at some\nshort distance from the sea-bottom or walls.\n10.3\nLab experiments\nIt is important for any tsunami numerical model to be evaluated against standard\nbenchmarking cases suggested by National Tsunami Hazard Mitigation Program (NTHMP),\nNational Oceanic and Atmospheric Administration (NOAA). In this section, two experimental","MODEL BENCHMARKING WORKSHOP AND RESULTS\n353\nlandslide cases have been chosen to validate the TSUNAMI3D code for tsunami generation\ncaused by sub-aerial and sub-sea landslides. The two test cases are:\nTsunami generation and runup due to 3-D landslide\nTsunami generation and runup due to 2-D landslide\n10.3.1 BP8: Tsunami generation and runup due to 3-D landslide\nThis test compares TSUNAMI3D numerical results with laboratory data obtained in a series\nof 3-D experiments carried out at Oregon State University (OSU) by Raichlen et al., (2003) and\nSynolakis et al., (2003).\nIn the 3-D lab experiments, a solid wedge was used in a large wave tank to represent a\nlandslide inducing tsunami waves, see Figure 10-1.\nFigure 10-1: Experiment of a solid wedge induced waves. Tsunami generation and runup due to three-\ndimensional landslide (Synolakis et al., 2007).\nThe solid wedge has a triangular section with a horizontal length of 91 cm, a height of 45.5\ncm, and a width of 61 cm. The wedge rests in a sloping bottom of the wave tank and is released\nfrom repose. The horizontal surface of the wedge was positioned in two different small distances\nbelow the still water level (case A = 0.025 m and case A = 0.10 m) to generate waves with\ndifferent energies and characteristics. Detailed information and access to the data are found in","National Tsunami Hazard Mitigation Program (NTHMP)\n354\nthe report, Tsunami Generation and Runup Due to Three dimensional Landslide, Synolakis et al.,\n(2007), OAR PMEL 135.\nTSUNAMI3D was tested against the experimental results for cases A = 0.025 m and\nA = 0.10 m. Figure 10-2 shows a set of snapshots for the case A = 0.025 , taken from the 3-D\nnumerical model result. Domain dimensions, free surface elevation, and velocity vectors\nprojected at plane y = 0 for time 1.0, 1.5, 2.0 and 2.5 sec are displayed in the figure. For\nnumerical efficiency, the domain is reduced in half by cutting it through its plane of symmetry at\ny = 0, , see Figure 10-2. The dimension of the computational box in the X, y and Z direction is\n6.10 m X 1.85 m X 3.05 m, respectively; total number of computational cells is 4.6 million (246\nX\n76 X 246); space step 0.025 m X 0.025 mx 0.0125 m; and maximum time step of 0.001 sec. The\nmotion of the solid block was prescribed in the 3-D model according to a given wedge location\ntime series. The computation time for 4 sec of simulation took 4 hours using a PC with 8 CPUs.\nb)\na)\n0.3\n0.3\n8\n8,\n-0.5\n-0.5\n4\n4\n-1.5\n-1.5\n0.04\nt=1.5s\nt=1.0s\n-2\n-2\n0.035\n-2.5\n0.03\n-2.5\n-2.75\n-2.75\n0.025\n-0.6\n-0.6\n0.02\n0.015\n1.85\n1.5\n0.01\n0.5\n0.005\n5.5\n0\n(M)\nd)\nc)\n-0.005\n0.01\n0.3\n0.3\n%\n-0.015\n8,\n-0.02\n-0.5\n-0.5\n-0.025\n-1\n-1\n-0.03\n(m)\n0.035\n-1.5\n-1.5\nt=2.5s\nt=2.0s\n-2\n2\n-2.5\n-2.5\n-2.75\n-2.75\n-0.6\n-0.6\n1.85\n1.85\n1.5\n1.5\n1\n3.5\n0.5\n0.5\n4.5\n5.5\nFigure 10-2: TSUNAMI3D snapshots of the solid wedge sliding down the slope and induced waves at\ntime = [ 1.0, 1.5, 2.0 and 2.5] sec after slide initiation.\nThe TSUNAMI3D model's results are portrayed against the experimental results in Figure\n10-3 and Figure 10-4. Overall numerical results agree fairly well with the experimental results.\nSome discrepancies in timing are evident, especially in the runup results. The rebound wave\n(second wave recorded in case A = 0.025 m by gauge #1 resulting from the drag of the wedge) is\nslightly overestimated by the model. Similar overestimation in the rebound wave has been","MODEL BENCHMARKING WORKSHOP AND RESULTS\n355\nreported by other authors, e.g., Abadie et al., (2010). The model has been tuned for internal\nfriction by means of the eddy viscosity value of 10 -6 m²/sec. The no slip condition is used for\nsea-bottom friction. In addition, an exponential function increases the internal friction by one or\nseveral orders of magnitude at all computational fluid cells in contact with walls or sea-bottom.\nThe small discrepancy observed in the model's results is mainly attributable to energy dissipation\ndue to turbulence processes. As previously indicated, a turbulence mechanism is not incorporated\nin the model solution, SO the scale of turbulence is calculated by using the DNS approximation.\nTherefore, energy loss due to turbulence is expected to be small and comparable to other\nnumerical assumptions and physical processes.\nThe normalized error (ERR) shown in Figure 10-3 and Figure 10-4 is used to measure model\naccuracy in time. The ERR is defined as\nn(t)\n1\n(11)\ni=1\n=\nn(t)\n-\nmin\nmax\nERR(t) is the accumulative error between the values predicted by the model (5m) and the values\nobserved in the physical experiment (5.). The error is normalized with respect to the distance\nbetween the maximum and minimum values obtained in the lab experiment (Semar 50min),\nwhich usually corresponds to the first or second wave height. n(t) is the number of observed\npoints at a given time, SO errors can be calculated as a function of time. This method permits\nmeasuring model accuracy for the first, second and subsequent waves. For instance, At Gage 1\nfor the lab experiment with A = 0.025 m, the accumulative model error for predicting the wave\nheight at t = [0.95, 2.25, 3.85] sec. are ERR(t) = [1.5%, 7.6%, 6%] respectively. Notice that the\nselected times are associated with the first, second and third waves respectively.\n0.05\n8\nGage 1\nRunup Gage 2\n0.04\n20\nresults ERR= 9.7053%\n0\n6\nresults ERR =6.1356%\n0\n(un)\n-0.05\n10\n-0.1\nEXPERIMENT\n2\n-0.04\nresults\nERR(1)%\n-0.15\n4°\n0\n1\n2\n3\n0\n1\n2\n3\nt(sec)\nt(sec)\n0.05\n6\nGage 2\nRunup Gage 3\nresults ERR= 18.2675% 20\nresults ERR= 4.6656%\n0.04\n0\n15\n(L)\n0\n10\n-0.05\n-0.04\n5\n-0.1\nO\n0\n1\n2\n3\n0\n1\n2\n3\n4\nt(sec)\nt(sec)\nFigure 10-3: Comparison of TSUNAMI3D numerical result (black broken line) against experiment (blue\nsolid line), case A = 0.025 m. Red line is the normalized error plotted in time.","National Tsunami Hazard Mitigation Program (NTHMP)\n356\n0.05\n20\n10\nGage 1\nRunup Gage 2\n0.04\nresults ERR =9.5688%\nresults ERR =8.6992%\n0\n(\n(\n10\n0\nF\n-0.05\nEXPERIMENT\nresults\n-0.04\nERR(t)%\n0\n-0.1\n0\n1\n2\n3\n0\n1\n2\n3\n4\nt(sec)\nt(sec)\n0.05\n20\nGage 2\nRunup Gage 3\n0.04\nresults ERR =5.4214%\nresults ERR =10.4749%\n(\n(I))\n0\n0\n-10\nOF\nE\n2\n-0.04\n-0.05\no\n10\n0\n1\n2\n3\n4\n0\n1\n2\n3\n4\nt(sec)\nt(sec)\nFigure 10-4: Comparison of TSUNAMI3D numerical result (black broken line) against experiment (blue\nsolid line), case A = 0.010 m. Red line is the normalized error plotted in time.\n10.3.2 BP3: Tsunami generation and runup due to 2-D landslide\nThe second experiment aims to predict the free surface elevation and runup associated with\na 1-D translating Gaussian shaped mass which is initially at the shoreline (Synolakis et al. 2007).\nIn dimensional form, the seafloor deformation can be described by\nn(x,t)=H(x)+n.(x,t)\n(12)\n(13)\n(14)\nn.\nwhere n. is the slide thickness, 8 is the maximum slide thickness, =SL = is the thickness slide\nlength ratio, B is the slope angle, and L is the slide length. After it begins to move, the landslide\nmass moves at constant acceleration. The experimental case and its analytical solution are\ndescribed in great detail in Liu et al., (2003).\nFor this experiment, only the benchmark Case B has been chosen to test TSUNAMI3D\nbecause Case B offers higher nonlinearity than its counterpart, Case A. The experimental setup\ninformation for Case B follows.\nCase B: tan Blu = 1, , B = 5.7 ° 8=1 m, u = 0.1; and snapshots of the free surface are\ndetermined at non-dimensional time: 4.5\nTo solve Case B numerically, an additional 1-D hydrostatic numerical model has been\nconsidered to compare with the analytical linear-solution and visualize model differences and","MODEL BENCHMARKING WORKSHOP AND RESULTS\n357\nbehaviors with respect to TSUNAMI3D (non-hydrostatic). The 1-D hydrostatic model is based\non the linear and non-linear shallow water approximation, LSW / NLSW respectively. It is\nexpected that the LSW numerical solution will match the analytical linear-solution because both\nuse the linear approach. On the other hand, if nonlinear effects are important, the NLSW results\nshould depart from the linear solution and the free surface profile might be comparable to the\nTSUNAMI3D results, which are non-linear too. It is important to mention that TSUNAMI3D\nincludes the vertical component of velocity and acceleration (non-hydrostatic), SO dispersive\neffects might cause the solution to differ from shallow water (SW) solutions, which are, in\nprinciple, depth integrated methods. If non-hydrostatic effects are weak, the NLSW result will be\ncomparable to the full solution of TSUNAMI3D.\nThe finite difference solution of momentum equation for the LSW or NLSW approximation\nis solved on a staggered grid using the original depth integrated code (an early version of\nNEOWAVE, Yamazaki et al., 2008), Kowalik and Murty, 1993. The nonlinear term for the\nNLSW method has second order approximation in space and first order in time.\nThe TSUNAMI3D model has been extended to deal with deformable moving objects in the\ncomputational domain, SO the Gaussian slide deformation and motion is prescribed according to\nEq. 12. As can be gleaned from Figure 10-5 and Figure 10-6, the TSUNAMI3D results agreed\nvery well with the NLSW results. However, some differences are evident, such as the wave\nskewness and the location of the shoreline. It is important to mention that TSUNAMI3D solves\nthe full Navier-Stokes equations with the vertical and horizontal velocities being variables along\nthe water column, while the shallow water (SW) approximation assumes constant horizontal\nvelocity and no vertical velocity. All SW numerical results, including the analytical solution,\nmissed the effects caused by the vertical acceleration or the non-hydrostatic behavior of the\ngenerated impulsive wave.\n2\nSolid lines: NLSW\nThin dashed lines: LSW\nThick dash lines: TSUNAMI3D\nCircles: Analytical solution\n1.5\nt=4.5\n1\nt=2.5\nt=1.0\n0.5\nAt=0.5\n0\n-0.5\n-1\n0\n2\n4\n6\n8\n10\n12\n14\nx (Dimensionless)\nFigure 10-5: Comparison of the TSUNAMI3D model's numerical result against the analytical solution,\nnonlinear and linear shallow water approximation. Benchmark Tsunami generation and runup due to\ntwo-dimensional landslide, Synolakis et al., (2007), OAR PMEL-135.","National Tsunami Hazard Mitigation Program (NTHMP)\n358\nt=0.5\nt=1.0\n0.6\nAnalytical\n0.6\nLSW\n0.5\nNLSW\n0.4\n0.4\nTSUNAMI3D\n0.3\n0.2\n0.2\n0\n0.1\n0\n-0.2\n-0.1\n-0.4\n-0.2\n0\n0.5\n1\n1.5\n0\n0.5\n1\n1.5\n2\nt=2.5\n0.4\nt=4.5\n1\n0.2\n0.8\n0.6\n0\n0.4\n-0.2\n0.2\n-0,4\n0\n-0.6\n-0.2\n-0.8\n-0.4\n0\n0.5\n1\n1.5\n2\n2.5\n3\n0\n1\n2\n3\n4\nx\nx\nFigure 10-6: Comparison of the TSUNAMI3D model's numerical result (thick broken line) against the\nanalytical solution. the nonlinear and linear shallow water approximation. Tsunami generation and\nrunup due to two-dimensional landslide, Synolakis et al., (2007), OAR PMEL-135.\nIt is believed, that the full solution is ideally suited for cases where relatively high vertical\nvelocity/acceleration occurs. In this particular case, as dispersion effects develop, TSUNAMI3D\npredicts a more elongated, skewed and less tall wave than the NLSW approach does for later\ntimes in the model run, i.e.,\nWhen nonlinearity is not strong, all\nnumerical approximation agreed very well with the analytical solution, especially at earlier times\nin the model run, i.e.,\n10.4 Conclusions\nIn accordance with National Tsunami Hazard Mitigation Program (NTHMP) guidance,\ntsunami numerical models must be tested against NOAA PMEL 135 benchmark standards for\ntheir approval. Therefore, TSUNAMI3D, a model solving Navier-Stokes equations for two fluids\n(water, mud and void), with a Volume of Fluid (VOF) algorithm to track fluid interfaces has\nbeen applied to the benchmark landslide problems. For the 3-D tsunami calculation, the\nalgorithm has been simplified to account for the horizontal distortion of the computational cells\nwith respect to the vertical scale. The pressure term has been split into two components,\nhydrostatic and non-hydrostatic. The splitting of the pressure term permits the determination of\nthe hydrostatic solution by merely switching off the non-hydrostatic pressure terms. The\nturbulence process is solved in a very simple manner using Direct Numerical Simulations (DNS).\nTSUNAMI3D is validated against a set of established benchmarking problems for underwater\nand subaerial landslides. They are: a) Tsunami generation and runup due to 3-D landslide; b)\nTsunami generation and runup due to 2-D landslide.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n359\nBoth cases are simulated with a prescribed slide motion. Overall numerical results agree\nfairly well with the experimental data.\nFor the solid wedge experiment (3-D case), some small discrepancies in timing are evident,\nespecially in the runup results. The rebound wave resulting from the drag of the landslide wedge\nis slightly overestimated by the model. The small difference is mainly attributed to the\nsimplification of the model to account for energy loss due to the turbulence process. Even though\nthe turbulence mechanism is solved in a very simple manner using DNS, the overall solution was\nadequate. The DNS assumption for this particular test correctly predicted the impulsive waves\ngenerated by the submarine solid slides, from generation to the propagation stages.\nTSUNAMI3D code performed efficiently for the landslide simulation domain, with sizes on\nthe order of 5 million cells. The computer time required to simulate 4 sec of landslide simulation\nwas approximately 4 hours using a PC with 8 CPUs.\nWith respect to the 2-D analytical (linear-solution) experiment, all the models matched the\nanalytical results at the early stages of the wave generation when nonlinearity effects are not\nstrong. However, at later stages of wave generation, when nonlinearity and dispersive effects are\nimportant, the NLSW and TSUNAMI3D results are comparable, although some differences are\nvisible due to the dispersive nature of the resulting wave. All SW numerical results, including the\nanalytical solution, missed the complicated wave configuration caused by the vertical\nacceleration or the non-hydrostatic effects during the later stages of wave evolution.\n10.5\nReferences\nAbadie SD, Morichon SD, Grilli S, Glockner S. 2010. Numerical simulation of waves generated\nby landslides using a multiple-fluid Navier-Stokes model. Coastal Engineering, 57:779-\n794.\nHirt CW, Nichols BD. 1981. Volume of Fluid Method for the Dynamics of Free Boundaries. J.\nComp. Phys., 39:201-225, 1981.\nHorrillo J. 2006. Numerical Method for Tsunami Calculation Using Full Navier-Stokes Equations and\nVolume of Fluid Method. Thesis dissertation presented to the University of Alaska Fairbanks.\nKowalik Z, Murty TS. 1993. Numerical Simulation of Two-Dimensional Tsunami Runup.\nMarine Geodesy, 16:87-100.\nLiu PL-F, Lynett P, Synolakis CE. 2003. Analytical solution for forced long waves on a sloping\nbeach. J. Fluid Mech., 478:101-109.\nRaichlen F, Synolakis CE. 2003. Runup from three dimensional sliding mass. In Briggs, M.\nCoutitas, Ch., editors, in Proceedings of the long wave Symposium 2003, pages 247-256,\nXXX IAHR Congress Proceedings, ISBN-960-243-593-3.\nSynolakis CE, Bernard EN, Titov VV, Kanoglu U, González FI. 2007. OAR PMEL-135\nStandards, criteria, and procedures for NOAA evaluation of tsunami numerical models.\nTechnical report, NOAA Tech. Memo. OAR PMEL-135, NOAA/Pacific Marine\nEnvironmental Laboratory, Seattle, WA.\nSynolakis CE, Raichlen F. 2003. In Submarine Mass Movements and Their Consequences.\nWaves and Runup generated by a three-dimensional sliding mass, Advances in Natural\nHazards. Locat, J. and Mienert, J.(Kluwer Academic publishers, Dordrect).","National Tsunami Hazard Mitigation Program (NTHMP)\n360\nYamazaki Y, Kowalik Z, Kwok Fai Cheung. 2008. Depth-integrated, non-hydrostaticmodel\nforwave breaking and run-up, Int. J. Numer. Meth. Fluids, DOI: 10.1002/fld.1952.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n361\n11 BOSZ\nVolker Roeber, Kwok Fai Cheung\nDepartment of Ocean and Resources Engineering, University of Hawaii at Manoa\n11.1\nIntroduction\nBOSZ (Boussinesq model for Ocean & Surf Zones) is a numerical model for propagation,\ntransformation, breaking, and runup of water waves. BOSZ was specifically designed for\nnearshore wave processes in the presence of fringing reefs in tropical and sub-tropical regions\naround the world. The extended lagoons and steep flanks of most reef systems produce unique\ncoastal wave processes that are challenging to numerical models used in coastal engineering\ndesign and flood hazard assessment. Typical nearshore processes that numerical models have to\ndeal with are abrupt transitions from dispersion-dominated to flux-dominated flow through a\nwide range of wave breaking conditions. The formation, propagation, and runup of breaking\nwaves or bores involve shock-related hydraulic processes in which additional treatments are\nnecessary to enforce conservation laws across flow discontinuities. A numerical model designed\nfor such extreme conditions provides a robust platform for a wide range of nearshore wave\nconditions elsewhere around the world. BOSZ was developed with the goal of obtaining reliable\nand robust results in addressing the complementary but somewhat opposing physical processes\nof flux and dispersion throughout a single numerical model. BOSZ was intentionally kept as\nsimple as possible and yet containing the main features to accurately describe nearshore wave\nprocess. Often more sophisticated governing equations or numerical schemes encounter\nproblems such as instabilities before they can utilize their theoretical advantages over a simpler\nsolution.\nBOSZ is primarily used for modeling nearshore and surf-zone processes of swell and wind\nwaves. The model can be applied to near-field tsunami scenarios. However, to this date BOSZ is\nnot based on spherical coordinates nor does it support nested grids.\n11.2\nModel description\n11.2.1 Governing equations and numerical scheme\nBOSZ combines the dispersive properties of a Boussinesq-type model with the shock-\ncapturing capabilities of the conservative form of the nonlinear shallow-water equations. BOSZ\nallows the simulation of dispersive waves up to medium order as well as supercritical flows with\ndiscontinuities. The depth-integrated governing equations are derived from Nwogu's (1993)\nextended Boussinesq approach with conserved variables that satisfy conservation of mass and\nmomentum for Fr>1. The continuity and momentum equations contain the conservative form of\nthe nonlinear shallow-water equations to capture shock-related hydraulic processes. In contrast,\nmost depth-integrated formulations utilize physical variables and require additional treatments","362\nNational Tsunami Hazard Mitigation Program (NTHMP)\nfor momentum conservation under supercritical flow conditions associated with breaking waves,\nbores or hydraulic jumps. The employed Boussinesq-type equations are given as\n,+(Hu)x+(Hv),\n(1)\n+\n=0\n(2)\n+\n(3)\nwhere H is total water depth, h is still water level, n is the free surface elevation, u and V are the\nhorizontal flow velocities, g is gravitational acceleration and Za is the reference depth at which\nthe velocity is evaluated.\nThe governing equations are solved with a conservative finite volume Godunov-type\nscheme. The finite volume method benefits from a two-dimensional third-order TVD (Total\nVariation Diminishing) reconstruction procedure (Kim and Kim, 2005) that evaluates the inter-\ncell variables. The user can chose between the exact Riemann solver of Wu and Cheung (2008)\nand the approximate HLL Riemann solver for the solution of the inter-cell flux and bathymetry\nsource terms. The flux and bathymetry source terms are well-balanced and the numerical scheme\npreserves continuity at moving boundaries over irregular topography. The moving waterline is\npart of the Riemann solution and does not require additional treatments. A second-order explicit\nRunge-Kutta scheme integrates the governing equations in time and evaluates the conserved\nvariables, which in turn provide the two horizontal velocity components through the solution of","MODEL BENCHMARKING WORKSHOP AND RESULTS\n363\nseparate tri-diagonal matrices. The mixed spatial and temporal derivatives are split up throughout\nthe solution procedure SO that the two-dimensional equations can be treated as a series of one-\ndimensional problems in the X and y directions that can easily be processed on a parallel\ncomputer architecture. The solution procedure reduces numerical diffusion and provides accurate\ndescriptions of the flow and runup over relatively coarse grids and with large time steps for\nefficient computation over large geographic areas.\n11.2.2 Dispersion properties\nIn addition to the nonlinear shallow water equations, the governing Boussinesq-type\nequations contain parabolic terms to describe frequency dispersion. These terms involve the\nreference depth Za, which can be used to optimize the dispersion properties. Comparing the\nlinearized form of the governing equations with Airy wave theory leads to an optimum value of\nZa over a range of water depth parameter kh. Moving the value of Za toward the free surface gear\nthe equations toward shorter period components, and vice versa. The optimum value is\ndetermined from an error analysis over the range of 0 < kh < 2 but with stronger emphasis on kh\n< TT. Figure 11-1 illustrates the dispersion relation from the linearized version of the governing\nequations and Airy wave theory. The present value of Za = -0.5208132 results in less than 1%\nerror in celerity for kh < IT and less than 4% at kh = 5. This means, that the model accounts\nreasonably well for wavelengths almost as short as the local water depth.\n11.2.3 Wave breaking\nDepth-integrated models do not describe overturning of the free surface and thus cannot\nfully reproduce the wave breaking processes. However, the use of conserved variables in the\npresent governing equations allows approximation of breaking waves as discontinuous flows.\nThe flow is flux dominated along the breaking wave front and non-hydrostatic effects are\nnegligible. The Boussinesq-type equations locally reduce to the nonlinear shallow-water\nequations for the shock-related hydraulic processes, while the waves in the rest of the domain\nremain dispersive. Even though the flow is flux-dominated, the governing equations balance\namplitude dispersion of the shock with frequency dispersion. This might lead to a local anomaly\nat the breaking location that results in numerical instability depending on the order of the\ndispersion terms, the numerical scheme, and most important, the grid size. This problem can be\nobserved in a wide range of dispersive depth-integrated equations. Hence, the present solution to\nthis issue can be applied to other numerical models in the same way.\nBOSZ uses an approach to locally and momentarily deactivate the dispersion terms for the\nRiemann solver to describe the breaking wave as a bore or hydraulic jump. We utilize the local\nmomentum gradients, (HU)x and (HV), as the indicators consistent with the flux-based\nformulation in the governing equations. Dispersion terms are deactivated in cells where\n0.5[|(HU)xl +(HU)x]>BBgH\n(4)\nand\n0.5[|(HV)y/+(HV)y]z BgH\n(5)\nwith B=0.5 denoting the onset criterion of wave breaking. Even though (HU)x and (HU)y, in\na\nstrict sense, are not velocity terms, calibration with laboratory data would define their relations to\nthe velocity for the wave-breaking criterion. BOSZ has been calibrated with laboratory data from","National Tsunami Hazard Mitigation Program (NTHMP)\n364\nTing and Kirby (1994) as well as a wide range of wave breaking scenarios in the benchmark test\ncases considered here.\nThe form of Eqs. (4) and (5) accounts for the direction of the flow and rules out wave\nbreaking behind the crest, where large momentum gradients might occur in short period waves.\nThis criterion focuses at the potential source of instability typically at the steep wave front\ninstead of the wave crest and allows dispersion to take effects over most of the surf zone. Since\nthe Riemann solver used in BOSZ is exact, the solution over the breaking cells satisfies exact\nconservation laws of mass and momentum. This provides a robust solution over an area that is\noften the source of instabilities. The approach determines the breaking threshold using only one\ncoefficient without the need for a wave front tracking mechanism. This method eliminates a\nprimary source of numerical instability while enabling the numerical solution to account for\nenergy dissipation associated with wave breaking. For many practical applications, only a very\nlimited number of cells represent the breaking wave front. The dispersion terms will hardly\napproach critical value and as a consequence, the threshold in Eqs. (4) and (5) will mostly not be\nexceeded even though the flow is locally supercritical. This means that if there is simply no\npotential instability arising along the wave front, the model run remains stable. Excessive\nsuppression of dispersion based on the water depth to wave height criterion of Tonelli and Petti\n(2010) is not necessary. Independently of Eqs. (4) and (5), the governing equations in BOSZ\nautomatically account for momentum conservation if Fr > 1 and therefore, no additional terms\nare needed nor does the numerical scheme require modifications under wave breaking\nconditions.\n11.2.4 Model extensions\nBOSZ has been greatly extended over the past months. The model package now gives the\nuser the option to choose from two higher-order formulations:\nThe conservative formulation of the extended Boussinesq equations of Wei. et al. (1995),\nsimilar to the governing equation of FUNWAVE-TVD\nThe conservative formulation of the set fourth-order Boussinesq-type equations by\nMadsen and Schäffer (1998)\nThe higher-order formulations are based on the same numerical scheme as the standard\ngoverning equations with an identical treatment of wave breaking and moving boundary (runup).\nThe different formulations are compatible and the model is arranged in modular form, thus\ngiving the option to compute parts of the numerical grid with governing equations of different\norders depending on the practical problem. The dispersion properties of the higher-order\nformulation based on the equations by Madsen and Schäffer (1998) show almost perfect\nagreement with Airy wave theory up to kh = 2n. However, satisfactory results can be achieved\nfor most of the following benchmark problems with low order dispersion properties. Thus, all\ntests were conducted using the governing equations described in Section 11.2.1.\n11.2.5 System requirements\nThe model has about 100 variables in matrix format and several other scalar variables that\nrequire roughly 750 MB of computer memory for a grid of 1000 by 1000 cells. The code is\nwritten in MATLAB with most of its processing in embedded pre-compiled C MEX subroutines.\nThis combines fast computation with a user-friendly code development interface for debugging,\nmodifying, and demonstration. BOSZ can be run on multiple processors through OMP parallel","MODEL BENCHMARKING WORKSHOP AND RESULTS\n365\nprocessing of the C MEX routines. Parallel processing also includes the solutions of the linear\nsystems of equations that are formed by the two momentum equations. The post-processing and\nplotting can also be done in MATLAB and useful scripts are provided with the model package.\nSoftware requirements include a MATLAB license as well as a compatible C compiler. The code\nhas been successfully run on Mac os, Windows, and Linux cluster platforms with MATLAB\n2007 and 2010 in combination with GNU C and Microsoft Visual 2010 Express compilers.\n11.3\nBenchmark comparisons\n11.3.1 BP4: Single wave on a simple beach\nSolitary wave runup on a plane slope is one of the most intensively studied problems in\nlong-wave modeling. In particular, the laboratory experiments of Hall and Watts (1953) and\nSynolakis (1987) have provided frequently used data for validation of wave breaking and runup\nmodels. Figure 11-2 shows a schematic of the experiment with A indicating the initial solitary\nwave height, R the runup, and the beach slope. We first focus on case C from Synolakis (1987)\nwith A/h = 0.3 that involves wave breaking in front of the slope. This simple two-dimensional\ngeometry allows for testing of grid dependency and computation time. The computational\ndomain is set up with a length of 25 m. The governing equations contain derivatives of up to the\nthird order that need at least 6 cells in each horizontal direction in the computation. The model\nwas tested with four different grids with identical model parameters. A Courant number of C1 =\n0.45 ensures the stability of the time integration and a Manning coefficient of n = 0.01 s/m ¹/3\ndefines the surface roughness of the smooth glass beach in the laboratory experiment. The initial\nsolitary wave is located at x/h = -20 from the beach toe, allowing the initial wave profile to adjust\nto the governing equations before reaching the slope.\nThe runup on a beach with = 19.85 and a wave height of A/h = 0.3 is first examined.\nFigure 11-3 compares the measured surface profiles and the model results. Different from\nnonlinear shallow-water models, which lack dispersion terms, the present model reproduces the\nshoaling process up to 1g/g = 20 at the onset of a plunging breaker as observed in the\nlaboratory experiment. With the conserved variables, the model mimics the three-dimensional\nbreaking process as a collapsing bore and conserves the flow volume and momentum during the\nprocess. The resulting surge reaches the maximum elevation of R/h = 0.55 around tvg/h = 40.\nThe model shows a minor discrepancy with the laboratory data starting around Wg/h = 50,\nwhen a hydraulic jump begins to develop from the drawdown. Because the computed results\nagree with the measured data toward the end, the local disagreement might be due to a\ncombination of instrumentation and model errors. The drawdown process introduces air\nentrainment in the water column and splashes at the surface that are difficult to measure by any\ninstrument. The depth-integrated model also cannot fully capture the complex two-dimensional\nflow structure in the vertical plane.","National Tsunami Hazard Mitigation Program (NTHMP)\n366\nTable 11-1: Convergence test and computation time for BP4.\nComputation time\nNumber of\nRunup\nGrid Size\n1 Intel Nehalem i7 CPU 2.66 GHz\n[cm]\nAx/h\nGrid cells\n18.2\n32 sec\n0.25\n1500\n01 min 20 sec\n18.9\n0.125\n3000\n04 min 18 sec\n0.0625\n6000\n19.0\n21 min 35 sec\n0.025\n15000\n18.9\nTable 11-1 shows the CPU time and computed runup for a range of grid resolutions for 20\nsec of simulation time. The CPU time increases approximately as a quadratic function of grid\nresolution. A grid size of Ax/h = 0.125 is sufficient to reproduce the runup. The computed runup\nremains very similar at finer resolution, indicating convergence of the solution. This can be\nattributed to the low numerical diffusion of the finite volume scheme.\nFigure 11-4 plots the measured and computed runup as a function of the initial solitary wave\nheight A/h for beach slopes of 1:19.85, 1:15, and 1:5.67. The data show good agreement over a\nwide range of breaking and non-breaking events characterized by a bilinear distribution with a\ndistinct transition. For the 1:19.85 and 1:15 slopes, the data to the right of the transition represent\nplunging breakers, whereas the relatively steep slope of 1:5.67 produces surging waves without\nflow discontinuities or breaking. The model is able to simulate the runup for incident wave\nheights of up to A/h = 0.7, which is beyond the model's range of nonlinearity. The tests prove the\nvalidity of the use of the Riemann solver in combination with local deactivation of dispersion\nterms for wave breaking in depth-integrated dispersive models. The model runs stably and the\nresults are not grid sensitive as indicated in the table above.\n11.3.2 BP2: Solitary wave on a composite slope\nBriggs et al. (1996) conducted a laboratory experiment at the U.S. Army Corps of Engineers\nWaterways Experiment Station, Vicksburg, MS, to examine propagation and transformation of a\nsolitary wave over a compound slope as well as the subsequent runup on a vertical wall. Figure\n11-5 illustrates the test setup, which comprises a flat section of 15.05 m length and 0.218 m\ndepth and a compound slope of 1:53, 1:150, and 1:13. Three cases, A, B, and C, were\ninvestigated with solitary wave heights of A/h = 0.039, 0.264 and 0.696 measured at gauge 4.\nThe solitons then transform over the three slopes before being reflected from the right wall. Case\nA is a small-amplitude non-breaking solitary wave. Case B is a medium size soliton that\nundergoes significant shoaling over the slopes. The wave breaks over the 1:150 slope with the\nwave face slamming onto the wall causing a very high splash-up. Case C uses a large initial\nsolitary wave with very high nonlinearity and dispersion that are beyond the range of\napplicability of the governing equation in Section 11.2.1. With intense breaking over the first\nslope, bore formation and subsequent transition of the bore into dispersive waves, this test is an\ninteresting validation that pushes depth-integrated formulations to their limit. In all three cases\nthe waves are initiated at the left boundary. Timing is adjusted by aligning the numerical solution\nwith the measured peak at wave gauge 4. We conducted a convergence test similar to the one\ndescribed in Section 11.3.1 and found that the numerical solutions agree well for grid sizes\nbelow Ax = 1.5 cm. All cases are run with Cr = 0.45 and a Manning coefficient n = 0.01","MODEL BENCHMARKING WORKSHOP AND RESULTS\n367\nFigure 11-6, Figure 11-7, and Figure 11-8 show comparisons between the computed and\nlaboratory data. In Figure 11-6, the discrepancy between the numerical solution and the\nlaboratory data can probably be attributed to the poorly defined initial solitary wave in the\nexperimental study. The recorded profile is slightly narrower than the computed profile at gauge\n4. The wavemaker is probably not calibrated to generate such a small amplitude wave. The initial\nwave profile is better defined for case B in Figure 11-7. The model well captures the shoaling\nand breaking processes over the slope as well as the subsequent reflection of the dispersive\nwaves. A small discrepancy can be observed around gauge 10, where the model cannot fully\naccount for the breaking wave with overturning of the free surface, resulting in larger\namplitudes. The results of case C are shown in Figure 11-8. BOSZ is able to propagate a highly\nnonlinear and dispersive solitary over the flat bed. Although discrepancies arise in the shoaling\nand breaking processes, the amplitude and phase of the reflected dispersive waves match the\nmeasured data fairly well. This demonstrates that the wave breaking technique accurately\nreproduces transitions between flux and dispersion dominated flows even for conditions beyond\nthe applicable range of the model. It should be noted that for case C the threshold for\ndeactivation of the dispersion terms had to be elevated to B = 0.6 (see Eqs. (4) and (5)) to avoid\nanticipated negligence of dispersion due to very high flux gradients at the steep flanks of the\ninitial solitary wave. This is of minor concern because the model was not primarily designed for\npropagation of highly nonlinear waves such as the one in this test case.\nTable 11-2 compares the recorded and computed runup on the wall. The computed runup in\ncases A and C is about 80% of the measured values, which can be considered satisfactory for a\ndepth-integrated model without an explicit description of the vertical motion of water particles.\nThe large discrepancy in case B can probably be attributed to strong splash-up and to vertical\nsheet flow from the impinging wave that are beyond the scope of this model.\nTable 11-2: Runup on vertical wall in BP2.\nCase\nExperimental\nComputed runup from\nNo.\nrunup [cm]\nBOSZ [cm]\nA\n2.74\n2.20\nB\n46.72\n16.40\nC\n27.43\n22.10\n11.3.3 BP6: Solitary wave on conical island\nTransformation of long waves around an island has attracted a lot of attention in the research\ncommunity. Refraction and diffraction of long waves may result in significant inundation on the\nlee side and trapping of energy around an island. A common observation is that waves refract\naround an island on the two sides, collide in the back with additional energy from the diffracted\nwaves, and continue to wrap around the island. Briggs et al. (1995) conducted a large-scale\nlaboratory experiment to investigate solitary wave transformation around a conical island. Figure\n11-9 shows schematics of the laboratory experiment. The basin is 25 m by 30 m. The circular\nisland has the shape of a truncated cone constructed of concrete with diameters of 7.2 m at the\nbottom and 2.2 m at the top. The island is 0.625 m high and has a side slope of 1:4. A 27.4-m\nlong directional wavemaker consisting of 61 paddles generates the input solitary waves for the\nlaboratory test. Wave absorbers at the three remaining sidewalls reduce reflection in the basin.","National Tsunami Hazard Mitigation Program (NTHMP)\n368\nThe experiment covers the water depths h = 0.32 and 0.42 m and the solitary wave heights\nA/h = 0.05, 0.1 and 0.2. The present study considers the smaller water depth h = 0.32 m, which\nprovides a more challenging test case for dispersive wave models. In the computation, the\nsolitary wave is generated from the left boundary with the measured incident wave heights of\nA/h = 0.045, 0.096, and 0.181. These measured wave heights, instead of the target wave heights\nA/h = 0.05, 0.1, and 0.2 in the laboratory experiment, better represent the recorded data at gauge\n2 and thus the incident wave conditions to the conical island. A solid boundary condition is\nimposed at the lateral boundaries. The wave absorbers, which are not effective for long waves,\nare not considered. The model is set up with a grid of Ax = Ay = 5 cm. A Manning's roughness\ncoefficient n = 0.014 s/m 1/3 accounts for the smooth concrete finish according to Chaudhry\n(1993). The Courant number is set to Cr = 0.45. Figure 1-10 shows a series of snapshots as the\nsolitary wave with A/h = 0.181 propagates around the island. The result shows the maximum\nrunup at the front and refraction and trapping of the solitary wave over the island slope. The\ntrapped waves from the two sides superpose with the diffracted wave on the leeside of the island.\nWave breaking occurs around the island according to Titov and Synolakis (1998) and reduces the\nrunup particularly on the leeside of the cone. After the solitary wave passes the island, the\ntrapped waves continue to wrap around to the front.\nA number of gauges recorded the transformation of the solitary wave around the conical\nisland. Figure 11-11 shows the time series of the solutions and the measured free surface\nelevations at selected gauges. With reference to Figure 11-9, gauges 2 and 6 are located in front\nof the island and 9, 16, and 22 are placed close to the still waterline at 0°, 270°, and 180° around\nthe island. These gauges provide sufficient coverage of the wave conditions important to the\nexperiment. The measured data at gauge 2 provides a reference for adjustment of the timing of\nthe computed waveforms. The results show the incident wave profile and reflection from the\nisland. With higher nonlinearity, the crest of the solitary wave is narrower and the reflection is\nmore distinct. Wave breaking was observed at gauge 9 for A/h = 0.181 and 0.096. The model\naccurately describes the subsequent drawdown in all three cases, but just like most published\nmodels, does not reproduce the small amplitude short period waves afterwards. A careful\ninspection of the laboratory setup shows the wavemaker is 2.6 m shorter than the basin width and\ndiffraction of the incident wave at the two ends of the wavemaker likely generated these short\nperiod waves. For A/h = 0.181, the wave breaks as it wraps around the island and the model\napproximates the wave breaking process as bores resulting in steep front faces. The waves\nwrapped around from the two sides superpose and break symmetrically as if it is a standing\nwave. This is a classic example of the two-shock interaction in the Riemann solver used in the\npresent model. Most empirical wave breaking mechanisms cater to typical spilling and plunging\nwave breakers such as those examined by Ting and Kirby (1994) and might not correctly account\nfor these conditions. Figure 11-12 compares the measured data and the computed inundation in\nits original resolution around the conical island. The computed results show good agreement with\nthe laboratory data and are symmetric about the wave propagation direction despite the use of a\nCartesian grid to describe curved surfaces. Especially the inundation at the lee side of the island\nis captured very well. Doubling the grid spacing to Ax = Ay = 10 cm hardly affects the quality of\nthe results. However, the maximum runup outline appears less detailed due to the coarser grid.\n11.3.4 BP7: Runup at Monai valley\nThe 1993 Hokkaido Nansei-Oki tsunami is a well-studied event. In particular, Matsuyama\nand Tanaka (2001) conducted laboratory experiments at the Central Research Institute for","MODEL BENCHMARKING WORKSHOP AND RESULTS\n369\nElectric Power Industry (CRIEPI) to examine the extreme runup of over 30 m at Monai Valley\nlocated between two headlands and sheltered by the small Muen Island. The CRIEPI wave flume\nis 205 m long, 3.4 m wide, and 6 m high with a hydraulic wavemaker capable of generating\nleading depression N-waves, which are typical for near-field tsunamis. The area around Monai\nValley was reconstructed in the tank at a 1:400 scale based on bathymetric data as shown in\nFigure 11-13. A wave gauge near the wavemaker recorded the initial low amplitude N-wave as\nshown in Figure 11-14 and is used in BOSZ as boundary input with the free surface elevation\ninterpolated from the data according to the model time step. The computational domain is\ndiscretized with Ax = Ay = 2 cm. As in the other benchmark cases, the Courant number is kept\nconsistent at Cr = 0.45. A Manning coefficient of n = 0.014 s/m 1/3 describes the surface\nroughness of the plywood model (Chaudhry, 1993). Figure 11-15 shows the comparison between\nthe computed and recorded data at the wave gauges located between Muen Island and Monai\nValley in the experiment. The model cannot fully resolve all the details associated with the\ninteractions between refracted and diffracted waves around Muen Island and the drawdown from\nMonai Valley. The scattered waves are not highly dispersive. The short period waves in the\nrecord probably result from scattering over the physical model terrain at small water depth.\nHowever, the model agrees well with the overall wave field. It should be noted that very similar\nresults were obtained from non-dispersive shallow-water models, because dispersion has only\nmarginal influence on this experiment.\nThe maximum inundation limit can be seen in Figure 11-16 and Figure 11-17. The well-\nbalanced finite volume scheme reproduces the inundation at three monitored locations including\none in the narrow Monai Valley without requiring an excessively fine grid resolution.\n11.4\nProposed benchmark problems\n11.4.1 Solitary wave over 2-D reef\nWe conducted two series of laboratory experiments at Oregon State University in 2007 and\n2009 that included 198 tests with 10 two-dimensional reef configurations at a range of water\ndepths. Each test included a series of incident solitary wave heights. These test cases are a logical\nextension of the current benchmark for validation of inundation models. Though the laboratory\nexperiments focuses on shock-related hydraulic processes such as wave breaking and bore\nformation, the collected data allow examination of shoaling, reflection, wave breaking, and\nswashing dynamics. We propose the results from one of the test configurations with A/h = 0.3 as\na future NTHMP benchmark problem and provide a detailed description of hydraulic processes\nwith additional data from BOSZ. Additional experimental data for 0.05 < A/h < 0.5 can be\nrequested and we can also provide data for different bathymetries.\nAs we will demonstrate, this test is suitable for dispersive and non-dispersive models.\nThough the test is essentially two-dimensional, it is recommended to compute the numerical\nsolution over the x and y directions by simply rotating the input grid. The solutions should be\nidentical to ensure that the model works flawlessly and the computer code is free of errors. The\nproposed test case examines many aspects of numerical models that are important in tsunami\ninundation modeling:\nWave breaking and bore formation\nTransitions between dispersion and flux-dominated flows\nTransitions between sub and supercritical flows","National Tsunami Hazard Mitigation Program (NTHMP)\n370\nMass and momentum conservation for Fr 1\nMass and momentum conservation at moving waterlines\nWave shoaling and reflection\n11.4.1.1 Experiment setup\nThe test was conducted in the Large Wave Flume (LWF) at the Hinsdale Wave Research\nLaboratory at Oregon State University, Corvallis, OR. The LWF has a length of 104 m, a width\nof 3.66 m, and a height of 4.57 m. The wavemaker was installed in early 2009 prior to the series\nof experiments. A hydraulic actuator moves a single piston, which in turn drives a 4.57-m high\nwaveboard at a maximum speed of 4.0 m/s. This facility can accommodate tests with a scale up\nto 2.5 times that in the Tsunami Wave Basin at the same laboratory. Figure 11-18 shows a\nschematic of the experiment setup. The effective length of the flume is 83.7 m with a 1:12 slope\nstarting at 25.9 m from the wavemaker. The reef flat is at an elevation of 2.361 m from the tank\nfloor. Because most fringing reefs have a sheltered lagoon, a reef crest composed of marine\nplywood is attached to the reef edge and smoothly transitions the reef slope to 2.565 m elevation.\nThe reef crest is 1.25 m long and the offshore and onshore slopes are both 1:12. With a water\nlevel of 2.5 m, the lagoon has a shallow depth 0.136 m while the reef crest remains exposed by\n0.065 m. Resistance-type and ultrasonic-type wave gauges recorded the free surface elevation\nalong the tank. Two Acoustic Doppler velocimeters (ADV) recorded the three-dimensional flow\nvelocity at 54.41 m and 58.07 m from the wavemaker. However, the particular location at 58.07\nm was subject to very intensive wave breaking and air entrainment and the recorded ADV data\nshow insufficient quality for this study and thus are omitted. More details about the experimental\nprocedure can be found in Roeber et al. (2010) and Roeber (2010).\n11.4.1.2 Hydraulic processes and model - data comparison\nThe hydraulic processes can be best explained by a combination of model and laboratory\ndata. The computational domain uses a grid size of Ax = Ay = 0.05 m with a Courant number of\nCr = 0.45. A Manning coefficient of =0.014 s/m 1/ 1/3 from Chaudhry (1993) describes the\nsmooth, finished concrete and plywood surfaces of the flume. Figure 11-19 shows the computed\nand measured free surface profiles along the flume. The 0.75-m solitary wave gives a\ndimensionless wave height of A/h = 0.3. After being generated at the left boundary from an\nanalytical solution, the solitary wave shoals over the relatively gentle slope. The profile becomes\nnear vertical and the wave begins to break around 1/g/h = Observations during the\nlaboratory experiment indicate subsequent overturning of the free surface and the development\nof a plunging breaker impinging on the reef crest with a large air cavity and subsequent splash-\nup around 1/g/h BOSZ mimics the plunging breaker as a collapsing bore and correctly\ndescribes the free surface profile during the entire process. The flow transitions to advection- or\nflux-dominated over the reef flat such that the conservative form of the governing equations\nbecomes instrumental in capturing the pertinent hydraulic processes. Around 1/g/h = 70, the\nbroken wave begins to travel down the back slope of the reef crest and generates a supercritical\nflow displacing the initially still water in the lagoon. The flow generates a hydraulic jump off the\nback reef and a propagating bore downstream. Laboratory observations indicate overturning of\nthe free surface at the hydraulic jump as the supercritical flow transfers volume and momentum\nto the subcritical flow that fuels a propagating bore at the front. The hydraulic jump initially\nmoves downstream with the strong supercritical flow. Around 1g/h = 80, the momentum flux","MODEL BENCHMARKING WORKSHOP AND RESULTS\n371\nbalances at the flow discontinuity and the hydraulic jump becomes stationary momentarily. The\npresent model detects the breaking at the hydraulic jump during the process through the\nmomentum gradient in the Riemann solver approach that would otherwise not be accounted for\nby conventional methods based on free surface motion. The hydraulic jump subsequently\ndiminishes with the flow and moves back to the reef crest as a bore. In the meanwhile, the\npropagating bore shows a gradual reduction in amplitude and continues to propagate\ndownstream.\nThe end wall of the flume reflects the bore back to the lagoon that in turn overtops the reef\ncrest as sheet flow generating a hydraulic jump on the fore reef. The reflected bore, which has\nlower Froude number, produces a series of dispersive waves that warrant a closer examination.\nFigure 11-20 compares the computed and recorded surface elevation time series. The wave\ngauge right next to the wall at x = 80 m shows superposition of the approaching and reflected\nbores propagating in the opposite directions. The time series at x = 65.2 on the reef flat shows the\napproaching bore and the reflected bore from the end wall as well as its reflection from the reef\ncrest. The process continues with the subsequent reflection from the end wall. The steep wave\nfronts demonstrate the shock-related hydraulic processes in the flux-dominated flow. The time\nseries at x = 57.9 m shows overtopping at the reef crest. As the water rushes down the fore reef,\nthe flow transitions to dispersion-dominated through a hydraulic jump. Observations during the\nexperiments confirm an overturning free surface with air entrainment near x = 54.4 m. The\nhydraulic jump initially generates an offshore propagating bore, which transforms into a train of\nwaves over the increasing water depth at the fore reef for x < 50.4 m. The resulting undulations\nintensify as higher harmonics are released from the wave packet. At the same time, a long period\nreflected wave propagates in onshore direction and superposes with the released higher\nharmonics from the offshore propagating bore. Wave gauges located near the toe of the slope\nrecord highly dispersive waves of kh > 15. Figure 11-21 shows a close up view of the free\nsurface at the wave gauge at x = 35.9 m in about 1.7 m water depth. Though the wavelengths of\nthese harmonics are below the applicable range of the model, the waveforms are well captured\nwith good agreement of the phase and only slight over-predictions in the amplitude. Figure 11-22\nshows the velocity in the x direction at a location in front of the reef crest, where the laboratory\nwave already overturns. Despite its depth-integrated structure, the model captures the velocity of\nthe initial strong supercritical flow conditions 3.5) and subsequent dispersive waves.\nThe model reproduces the long and intermediate-period oscillations even after a long\nsimulation involving a series of wave breaking and reflection in the flume. The conservative\nstructure of the model allows description of the transition between super and subcritical flows\nand the present wave-breaking model reproduces the surging and plunging waves over the reef.\nThis test also demonstrates the applicability of the Riemann solver model for wave breaking. The\nlocal deactivation of dispersion terms efficiently eliminates potential instabilities, and at the\nsame time, does not alter the dispersion properties of subsequent wave transformation processes\nsuch as the release of higher order dispersive waves from a decaying bore.\n11.4.1.3 Grid dependency and dispersion\nFigure 11-23 and Figure 11-24 illustrate the model grid dependency and resolution of shock\nand dispersive waves. Despite the fact that the tank geometry cannot be perfectly resolved with\nAx = 0.1 and 0.2 m, the model shows good agreement with the measurements. The model is able\nto resolve most of the dispersive waves at x = 35.9 m with Ax = 0.10 m and only a small phase\nshift can be observed. This might be due to a small offset of the wall location in the coarse grid","National Tsunami Hazard Mitigation Program (NTHMP)\n372\nthat alters the timing of the reflection. Even with a grid of Ax = 0.20 m the model is able to\nresolve the main features of the recorded waveform including the leading dispersive waves. The\nresults for the coarser grids are very satisfying considering the much shorter computation time\nand smaller output files.\n11.4.1.4 Comparison between dispersive and hydrostatic solutions\nBOSZ has an option to use the embedded nonlinear shallow-water equations by turning off\nthe dispersion terms in the computation. We recomputed the bore propagation over the reef flat\nusing the same discretization, Courant number, and Manning's coefficient without the dispersion\nterm. The computation starts off with the full model up to Wg/h 64.8, which allows the\nsolitary wave to propagate up to its breaking point without any difference. The dispersion terms\nare turned off in the entire domain afterward and the computation continues based on the\nnonlinear shallow-water equations describing only hydrostatic effects. Figure 11-25 through\nFigure 11-28 show comparisons of the solutions. Despite the fundamental differences in the\ngoverning equations, as the flow becomes completely flux-dominated, both solutions become\nalmost identical. The subsequent bore formation does not depend on dispersion, and as long as\nthe governing equations conserve momentum, the shock front moves at the correct speed and\nheight. The small difference in phase speed results from the small changes in the remaining wave\nfield to the left of the reef crest that fuel the bore. Especially as the bore decays along the flume,\nits Froude number decreases and the bore approaches the stage toward an undular bore. As the\nhydrostatic solution continues to dissipate energy through the hydraulic jump, the Boussinesq\nsolution conserves energy in the vicinity of the weakening shock and correctly reproduces its\nspeed. After the wave gets reflected from the wall and propagates over the reef crest back into\nthe deeper part of the tank, the shallow-water solution fails to resolve the dispersive wave train,\nwhich develops from the decaying bore along the offshore slope. Instead, the shallow-water\nsolution describes the wave train as a decaying discontinuity. The lack of dispersive effects can\nalso be seen in the velocity profile that essentially follows the trend of the free surface profile.\nThis example has demonstrated the key features for a tsunami inundation model. The results\nfrom the nonlinear shallow-water equations can give very satisfying predictions of the flux-\ndominated processes. We would like to encourage modelers using nonlinear shallow-water\nmodels to undertake this test case by simply initializing the calculation with the initial solitary\nwave placed at the first wave gauge instead of generating the input from the left boundary.\n11.4.2 ISEC BMI, ISEC BM2: Solitary wave over 3-D reefsystem\nThe National Science Foundation funded a workshop and a benchmarking exercise for\ninundation models at Oregon State University in 2009. The organizer provided two benchmark\ntest cases with data from laboratory experiments at the Tsunami Wave Basin. Swigler and Lynett\n(2011) provided a detailed description of the experiments, instrumentation, and data post-\nprocessing. These test cases, which involve wave transformation over three-dimensional reef\nconfigurations, are logical extensions of the two-dimensional reef experiments from Section\n11.4.1. The laboratory data allows validation of models in handling dispersion and flux-\ndominated processes simultaneously. Information about the test cases can be found at\nhttp://isec.nacse.org/workshop/2009_isec/benchmarks.html\nFigure 11-29 shows the reef configuration determined from a laser scan and the setup of the\ninstrumentation for ISEC BM1. The neutral position of the wavemaker is at x = 0. The main","MODEL BENCHMARKING WORKSHOP AND RESULTS\n373\nfeature in the experiment is a triangular reef flat submerged between 7.5 and 9 cm below the still\nwater level. The reef sits on top of a bilinear background profile extending from x = 10.2 to\n17.7 m at a slope of 1:16 and from x = 17.7 to 32.4 m at 1:32. The slope of the reef is 1:3.5 at the\napex and flares to 1:16 over a distance of 9 m on either side to converge the wave energy. The\nwater depth in front of the bilinear profile varies slightly around 0.78 m. The top of the relief\nmodel has an elevation between 0.16 to 0.13 m above the still water level with a mild grade to\nthe back of the basin. Time series of the water surface elevation were recorded along transects at\nthe centerline from gauges 1 to 7, at 5 m offset from gauges 8 to 13, and at the edge of the reef\nflat from gauges 7 and 14 to 17. Velocity measurements are available at gauges 3, 6, and 13.\nBecause of the limited supply of instruments, the data were recorded over several weeks from\na\nnumber of repetitions of the same test conditions with the instrument array repositioned along the\nbasin.\nThe model is set up with a grid of Ax = Ay = 0.1 m, a Courant number of Cr = 0.45, and a\nManning roughness of n = 0.014 s/m 1/3 as in the two-dimensional reef experiments. On a single\nCPU, the model takes almost 2 hours of runtime for 90 sec of simulation. With parallel\ncomputation, the computation time reduces almost linearly; the simulation finishes in around 16\nminutes on 8 CPUs. The incident solitary wave has a height of 0.39 m giving rise to strongly\nnonlinear conditions with A/h = 0.5. Figure 11-30 shows a series of snapshots as the solitary\nwave transforms over the reef and slope complex. In the laboratory experiment, spilling at the\ncrest occurred locally at t = 5 sec, when the solitary wave reached the apex of the reef with little\nshoaling over the steep slope. With shoaling of the wave along the sides, plunging waves\nsubsequently developed along the entire length of reef edge at t = 8 sec. The model reproduces\nthe breaking process as a collapsing bore spreading across the triangular reef flat. The flow\ntransitions into a surge moving up the initially dry slope and overtops the reef and slope\ncomplex. At t = 21 sec, drawdown of the water has already occurred on the slope, while the sheet\nflow on the top continues to move forward. As observed in the laboratory experiment, the upper\nslope is mostly dry with water trickling down the streaks of the concrete surface and the sheet\nflow at the top has been reflected from the back wall as a bore over the impounded water by t\n=\n35 sec. The panel at t = 48 sec shows the second reflection from the wavemaker and sloshing of\nthe impounded water at the top separated by the dry upper slope. Though the test uses a high\nsolitary wave as input that itself poses challenging dispersive conditions, the flow stays mainly\nflux-dominated throughout the entire experiment after wave breaking occurs. For this part,\nnonlinear shallow-water models can give satisfying results if the initial solitary wave is placed at\nabout 6 m from the left boundary. Dispersion, however, is responsible for many of the flow\ndetails in Figure 11-30.\nThe data from the laboratory experiment allow a quantitative comparison with the model\nprediction. Figure 11-31 and Figure 11-32 show good agreement of the computed and recorded\nsurface elevations during the initial steepening and breaking of the solitary wave along the two\ncross-shore transects of the basin. Figure 11-33 shows good agreement of the bore propagation in\nthe alongshore direction. The timing of the first and second reflection from the wavemaker\nmainly depends on the water depth over the reef. Figure 11-34 compares the recorded and\ncomputed flow velocity components at three of the gauges in the x and y directions. At the apex\nof the shelf, the model reproduces the entire recorded time series of the x component of the\nvelocity. The recorded data are not continuous at the two other locations at the reef flat but\ngenerally agree with the model output in the x and y directions, albeit with minor phase shift of","374\nNational Tsunami Hazard Mitigation Program (NTHMP)\nthe reflection. We noticed a slight decrease of the phase lag of the reflection through shortening\nthe domain by 0.5 m at the offshore boundary and increasing the water depth by ~1.5 cm, that\ncorrespond to the final wavemaker position and the equivalent volume displaced. However, this\nadjustment does not account for the dynamics that is induced by the forward motion of the\nwavemaker as it generates the initial wave.\nISEC BM2 utilizes the same relief model but with a concrete cone of 6 m diameter and 0.45\nm height fitted to the apex of the reef between x = 14 and 20 m. Figure 11-35 shows the test\nconfiguration and instrumentation layout. The presence of the cone modifies the hydraulic\nprocesses over the reef flat and provides even more complex wave dynamics for model\nvalidation. The model setup is similar to that for ISEC BM1 with a grid of Ax = Ay = 0.10 m, a\nCourant number of Cr = 0.45, and a Manning roughness of n = 0.014 Figure 11-36 shows\nsnapshots of the computed free surface elevation in the basin. The solitary wave breaks at the\napex of the reef flat at t = 5.1 sec and the resulting surge completely overtops the cone at t =\n6.6 sec. The refracted waves from the two sides of the cone and the diffracted waves converge in\nthe back at t = 8.6 sec. While the refracted waves continue to wrap around as trapped waves, the\ndiffracted waves radiate from the back of the cone. During this process, the flux gradients in the\nx and y directions trigger the threshold to deactivate dispersion along the breaking wave front\nand the model is stable during this critical episode of the simulation. The diffracted wave on the\nleeside of the cone propagates up the slope reinforcing the refracted waves from the reef edge.\nThe drawdown of the diffracted wave generates a bore, which collides with the reflection from\nthe wavemaker over the reef flat around t = 17 sec and part of which is trapped around the cone\nas shown in the panel at t = 21.2 sec. After about 45 sec, and not shown in the snapshots, small\nvortices are generated in the vicinity of the reef edge and are moved around the conical island.\nFigure 11-37 compares the computed and recorded surface elevations. The model reproduces the\nrecorded surface elevations in front of the cone and the collapse of the bore behind the cone. The\nmodel matches the x component of the velocity at gauge 3 reasonably well as shown in Figure\n11-38. The recorded data immediately behind the cone at gauge 6 missed the initial wave\nprobably due to the turbulence and air entrainment, which could be the reason for the small\ndiscrepancies in the alongshore velocity component. Gauge 10 recorded most of the initial wave\nand gives good agreement with the computed data. The y component of the velocity is only a\nfraction of the x component with distinct secondary flow features of which the overall trend is\naccounted for in the model. Simulations with a coarser grid of Ax = 0.20 m, which takes only 10\nmin of computation time, still account for the main flow structure and provide very reasonable\nagreement with the laboratory data - especially along the bore.\nThe overall agreement between the computed and recorded data demonstrates the validity of\nBOSZ in handling multiple hydraulic processes, transitions from flux to dispersion-dominated\nflows as well as a variety of wave breakers in the two-dimensional horizontal plane with a\nmoving boundary.\n11.5 References\nBriggs MJ, Synolakis CE, Harkins GS, Green DR. 1995. Laboratory experiments of tsunami\nrunup on a circular island. Pure and Applied Geophysics, 144(3/4), 569-593.\nBriggs MJ, Synolakis CE, Kanoglu U, Green DR. 1996. Benchmark Problem 3: runup of solitary\nwaves on a vertical wall. In Long-Wave Runup Models, Yeh, H, Liu PL-F., and\nSynolakis, C. (eds). World Scientific, Singapore, pp. 375-383.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n375\nChaudhry MH. 1993. Open-Channel Flow. Prentice-Hall Inc., Englewood Cliffs, New Jersey,\n483 p.\nHall JV, Watts JW. 1953. Laboratory investigation of the vertical rise of solitary waves on\nimpermeable slopes. Technical Memo No. 33, Beach Erosion Board, US Army Corps of\nEngineers.\nKim KH, Kim C. 2005. Accurate, efficient and monotonic numerical methods for\nmultidimensional compressible flows. Part II: Multi-dimensional limiting process.\nJournal of Computational Physics 208, 570-615.\nMaden PA, Schäffer, HA. 1998. Higher-order Boussinesq-type equations for surface gravity\nwaves: Derivation and Analysis. Phil. Trans.: Math., Phys. and Eng. Sciences, Vol. 356\n(1749), 3123-3184.\nMatsuyama M, Tanaka H. 2001. An experimental study of the highest run-up height in the 1993\nHokkaido Nansei-oki earthquake tsunami. In Proceedings of the International Tsunami\nSymposium 2001, Seattle, Washington, pp. 879-889.\nNwogu O. 1993. An alternative form of the Boussinesq equations for nearshore wave\npropagation. Journal of Waterway, Port, Coastal, and Ocean Engineering, 119 (6), 618-\n638.\nRoeber V. 2010. Boussinesq-type model for nearshore wave processes in fringing reef\nenvironment. PhD Dissertation, University of Hawaii, Honolulu.\nRoeber V, Cheung KF, Kobayashi MH. 2010. Shock-capturing Boussinesq-type model for\nnearshore wave processes, Coastal Engineering, 57 (4), 407-423.\nSwigler DT, Lynett P. 2011. Laboratory study of the three-dimensional turbulence and kinematic\nproperties associated with a solitary wave traveling over an alongshore-variable, shallow\nshelf. In review.\nSynolakis CE. 1987. The runup of solitary waves. Journal of Fluid Mechanics, 185, 523-545.\nTing FC.K, Kirby JT. 1994. Observation of undertow and turbulence in a laboratory surf zone.\nCoastal Engineering 24 (3-4), 51-80.\nTitov VV, Synolakis CE, 1998. Numerical modeling of tidal wave runup. J. Waterw. Port Coast.\nOcean Eng., 124(4), 157-171.\nTonelli M, Petti M. 2010. Finite volume scheme for the solution of 2D extended Boussinesq\nequations in the surf zone. Ocean Engineering, 37(7), 567-582.\nWei G, Kirby JT, Grilli ST, Subramanya R. 1995. A fully nonlinear Boussinesq model for\nsurface waves: Part I. Highly nonlinear unsteady waves, J. Fluid Mech., 294, 71-92.\nWu Y, Cheung KF. 2008. Explicit solution to the exact Riemann problem and application in\nnonlinear shallow-water equations. International Journal for Numerical Methods in\nFluids 57 (11), 1649-1668.","National Tsunami Hazard Mitigation Program (NTHMP)\n376\n11.6 Figures\n1.20\n1.15\n1.10\n1.05\n1\n0.95\n0.90\n0.85\n0.80\n0\n1\n2\n3\n4\n5\n6\n7\n8\nkh\nFigure 11-1: Linear dispersion properties. (blue), za = -0.5208132 from BOSZ; (red), za = -0.5375 from\nNwogu (1993); (black), za = -0.42265 equivalent to Peregrine (1967).\nR\nA\nV\nh\nX\nX\nFigure 11-2: Definition sketch of BP4: Solitary wave runup on a simple beach.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n377\nt(g/h)0.5 = 10\nt(g/h)0.5 = 40\n0.4\n0.2\n0\nt(g/h)0.5 = 15\nt(g/h)0.5 = 45\n0.4\n0.2\n0\n0.6\nt(g/h)0.5 =20\nt(g/h)0.5 = 50\n0.4\n0.2\n0\n0.6\nt(g/h)0.5 = 25\nt(g/h)0.5 0.5 = 55\n0.4\n0.2\n0\n0.6\nt(g/h)0.. 0.5 = 30\nt(g/h)° 0.5 = 60\n0.4\n0.2\n0\n0.6\nt(g/h)0.5 = 35\nt(g/h)0.5 = 65\n0.4\n0.2\n0\n5\n10\n15\n20\n25\n30\n5\n10\n15\n20\n25\n30\nX /h\nx / h\nFigure 11-3: Free surface profiles of solitary wave transformation on a 1:19.85 simple beach with\nA/h = 0.3 and Ax/h = 0.125. Solid lines and circles denote BOSZ and measured data.","378\nNational Tsunami Hazard Mitigation Program (NTHMP)\n10°\n1\n10\n10-2\n10-3\n-2\n10-1\n1\n0\n10°\n10\n10\n1\n10\n10\n10-3\n-2\n10-1\n10°\n10\n1\n10\n10°\n10\n10\n10-2\n10-1\n10°\nA/h\nFigure 11-4: Solitary wave runup on a simple beach. (a) 1:19.85 (Synolakis, 1987). (b) 1:15\n(Li\nand\nRaichlen, 2002). (c) 1:5.67 (Hall and Watts, 1953). Solid lines (blue), and circles denote computed and\nmeasured data.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n379\n0.2\n0.1\n4\nA BC: 5\n6\n7\n8\n9\n10\n0\n1:13\n-0.1\n1:150\n-0.2\n1:53\n0\n5\n10\n15\n20\nTank Extension [m]\nFigure 11-5: Definition sketch of BP2: Solitary wave on a composite beach. Circles denote wave gauge\nlocations.\n2.0\nGauge 4\n1.0\n0\n2.0\nGauge 5\n1.0\n0\n2.0\nGauge 6\n1.0\n0\n2.0\nGauge 7\n1.0\n0\n2.0\nGauge 8\n1.0\n0\n2.0\nGauge 9\n1.0\n0\n2.0\nGauge 10\n1.0\n0\n265\n270\n275\n280\n285\n290\n295\nTIME [sec]\nFigure 11-6: Solitary wave on a composite beach, BP2, case A. (blue) denotes solution from BOSZ.\n(black) indicates laboratory data from Briggs et al. (1996).","National Tsunami Hazard Mitigation Program (NTHMP)\n380\n10\nGauge 4\n5\n0\n10\nGauge 5\n5\n0\n10\nGauge 6\n5\n0\n10\nGauge 7\n5\n0\n10\nGauge 8\n5\n0\n10\nGauge 9\n5\n0\n10\nGauge 10\n5\n0\n290\n295\n275\n280\n285\n265\n270\nTIME [sec]\nFigure 11-7: Solitary wave on a composite beach, BP2, case B. (blue) denotes solution from BOSZ.\n(black) indicates laboratory data from Briggs et al. (1996).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n381\n20\nGauge 4\n15\n10\n5\n0\n20\nGauge 5\n15\n10\n5\n0\n20\nGauge 6\n15\n10\n5\n0\n20\nGauge 7\n15\n10\n5\n0\n20\nGauge 8\n15\n10\n5\n0\n20\nGauge 9\n15\n10\n5\n0\n20\nGauge 10\n15\n10\n5\n0\n265\n270\n275\n280\n285\n290\n295\nTIME [sec]\nFigure 11-8: Solitary wave on a composite beach, BP2, case C. (blue) denotes the solution from BOSZ.\n(black) indicates laboratory data from Briggs et al. (1996).","National Tsunami Hazard Mitigation Program (NTHMP)\n382\n180\n22\n2\n9\n16\n25m\nLength [m]\n20\n25\n0\n5\n10\n15\n3.6 m\n0.305\n2\nA B C\n6\n9\n16\n22\no\n-0.320\nFigure 11-9: Schematics of the conical island laboratory experiment, BP6. (Top) Perspective view.\n(Bottom) Cross-sectional view along centerline. Circles denote gauge locations.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n383\nTime: 26.8 sec\nTime: 31.4 sec\nTime 29.3 sec\nTime 33.3 sec\nTime: 29.7 sec\nTime 34.0 sec\nTime: 30.6 sec\nTime 35.0 sec\nFigure 11-10: Wave transformation around the conical island, BP6 for A/h = 0.181.","National Tsunami Hazard Mitigation Program (NTHMP)\n384\n(c)\n(a)\n(b)\n5\n5\n10\ngage 2\ngage 2\ngage 2\n5\n0\n0\n0\n-5\n-5\n.5\n10\n5\n5\ngage 6\ngage 6\ngage 6\n5\n0\n0\n0\n.5\n-5\n-5\n5\n5\n10\ngage 9\ngage 9\ngage 9\n5\n0\n0\n10000\n0\nO\n-5\n-5\n-5\n5\n5\n10\ngage 16\ngage 16\ngage 16\n5\n0\n0\n0\n-5\n-5\n-5\n5\n5\n10\ngage 22\ngage 22\ngage 22\n5\n0\n0\n0\n.5\n-5\n-5\n35\n40\n25\n30\n35\n40\n25\n30\n35\n40\n25\n30\nTime [sec]\nTime [sec]\nTime [sec]\nFigure 11-11: Free surface profiles of wave transformation around the conical island, BP6. (a) A/h =\n0.045; (b) A/h = 0.096; (c) A/h = 0.181. Solid lines and circles denote computed and measured data.\n(c)\n(a)\n(b)\nFigure 11-12: Maximum inundation around the conical island, BP6. (a) A/h = 0.045; (b) A/h = 0.096; (c)\nA/h = 0.181. Solid lines and circles denote computed and measured data.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n385\n0.1\n0\n-0.1\n5.0\n4.0\n3.0\n2.0\n0\n0.5\n1.0\n1.0\n1.5\n2.0\n2.5\n3.0\n0 3.5\nFigure 11-13: Bathymetry of the Monai valley experiment, BP7. White circles denote wave gauge\nlocations. The gauge at the lower boundary is used as a control for the wave input (see Figure 11-14\nbelow).\n6\nWave Input\n4\n2\n0\n-2\n0\n10\n20\nTime [sec]\nFigure 11-14: Initial N-wave profile near the left boundary, BP7. (blue) denotes solution from BOSZ.\n(black) indicates initial wave profile from Matsuyama and Tanaka (2001).","National Tsunami Hazard Mitigation Program (NTHMP)\n386\n6\ngauge 5\n4\n2\n0\n-2\n6\ngauge 7\n4\n2\n0\n-2\n6\ngauge 9\n4\n2\n0\n-2\n10\n20\n0\nTime [sec]\nFigure 11-15: Time series of free surface elevation in Monai valley experiment, BP7. (blue) denotes\nsolution from BOSZ. (black) indicates experimental data from Matsuyama and Tanaka (2001).","MODEL BENCHMARKING WORKSHOP AND RESULTS\n387\n2.8\n2.6\n2.4\n2.2\n2\n1.8\n1.6\n1.4\n4.4\n4.6\n4.8\n5\n5.2\nFigure 11-16: Maximum inundation in Monai valley experiment, BP7. (Blue) denotes solution from\nBOSZ. (Red) circles indicate observed inundation from Matsuyama and Tanaka (2001). Black lines are\ntopographic contours at 1.25 cm intervals.\n3.5\n3\n2.5\n2\n[m]\n1.5\n1\n0.5\n0\n0\n1\n2\n3\n4\n5\n[m]\nFigure 11-17: Maximum inundation in Monai valley experiment, top view, BP7. (Red) circles denote\nlocations of observed inundation from Figure 11-16.","National Tsunami Hazard Mitigation Program (NTHMP)\n388\n28.25m\n25.9m\n104.0m\nx-037m\nrigid\n29.55m\nResistance Wave Gauge (14) (wg)\nFigure 11-18: ISEC BM1 test. Two-dimensional reef model of 1:12 slope in the Large Wave Flume at\nOregon State University (folded into two rows at the 28.25 m measurement point for better visibility).\nCircles and vertical lines indicate wave gauge locations. The water level in the Benchmark test case is\n2.5 m resulting in a water depth of 0.136 m over the reef flat. The reef crest (red trapezoid) is 1.25 m\nlong and rises 0.201 m above the reef flat (exposed by 0.065 m), with a 1:12 offshore and a 1:15\nonshore slope. The plywood reef crest and the finished concrete tank are described by the same\nManning coefficient of n = 0.014 m/s1/3. The tank is 3.66 m wide. A two-dimensional bathymetry file\nis provided in the benchmark package.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n389\n0.4\nt(g/h) 0.5 =55\nn/h 0.2\n0\n-0.2\n-0-2\n0.4\nt(g/h) 0.5 =63\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h) 0.5 = 65.9\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h) 0.5 = 68.2\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h) 0.5 = 69\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h) 0.5 =70.8\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h)' 0.5 = 72.3\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h)' 0.5 = 76.2\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h) 0.5 = 80.2\nn/h 0.2\n0\n-0.2\n8.4\nt(g/h) 0.5 = 98.5\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h) 0.5 = 112.2\nn/h 0.2\n0\n-0.2\n0.4\nt(g/h) 0.5 = 130\nn/h 0.2\n0\n-0.2\n0\n-1\n-2\n0\n10\n20\n30\n40\n50\n60\n70\n80\nTank Extension [m]\nFigure 11-19: Snapshots of free surface profiles for propagation of solitary wave with A/h = 0.3 over\n1:12 slope and exposed reef crest. Solid lines and circles denote computed and measured data. The\nexperimental wave overturns describing a large air cavity. The depth-integrated numerical\nformulation approximates the 3-D process as a 2-D discontinuity with high accuracy.","National Tsunami Hazard Mitigation Program (NTHMP)\n390\n0.4\nX = 48.2 m\nX = 80 m\nn/h\n0.2\n0\n0.4\nX = 46.1 m\nX = 72.7 m\nn/h\n0.2\n0\n0.4\nX = 44.3 m\nX = 65.2 m\nn/h\n0.2\n0\n0.4\n8\nX = 40.6 m\nX = 61.5 m\nn/h\n0.2\n0\n0.4\nX = 35.9 m\nX = 57.9 m\nn/h\n0.2\n0\n0.4\nX = 28.6\nX = 54.4 m\nn/h\n0.2\n0\n0.4\nX = 17.6m\nX = 50.4 m\nn/h\n0.2\n0\n200\n50\n100\n150\n200\n50\n100\n150\nt (g/h)0 0.5\nt (g/h) 0.5\nX\nx\nx\nx\nx\n0\n-1\n-2\n0\n10\n20\n30\n40\n50\n60\n70\n80\nTank Extension [m]\nFigure 11-20: Time series of free surface at all wave gauges for propagation of solitary wave with\nA/h = 0.3 over 1:12 slope and exposed reef crest. Solid lines and circles denote computed and\nmeasured data. Notice the dispersive waves over the slope (30 - 50 m) generated from the reflected\nbore.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n391\n0.1\nX = 35.9 m\nn/h 0.05\n0\n110\n115\n120\n125\n130\n135\n140\n145\n150\n155\n160\nt (g/h) 0.5\nFigure 11-21: Close-up of time series of free surface at X = 35.9 m at beginning of 1:12 slope. Solid lines\nand circles denote computed and measured data. Signal showing dispersive waves generated from\nthe reflected bore that propagated over the slope into the direction of the wavemaker.\n4\nX = 54.4 m\n2\n0\n-2\n60\n80\n100\n120\n140\n160\n180\n200\n(g/h) 0.5\n0.5\n0.0\n-0.5\n-1.0\n-1.5\n-2.0\n-2.5\n25\n30\n35\n40\n45\n50\n55\n60\n65\n70\nTank Extension [m]\nFigure 11-22: Time series of x-directed velocity at X = 54.4 m close to the reef crest. Solid lines and\ncircles denote computed and measured data. The laboratory wave already overturned at this location\ncausing strong supercritical flow.","National Tsunami Hazard Mitigation Program (NTHMP)\n392\n0.4\nX = 48.2 m\nX = 80 m\nn/h\n0.2\n0\n0.4\nX = 72.7 m\nX = 46.1 m\nn/h\n0.2\n0\n0.4\nX = 65.2 m\nX = 44.3 m\nn/h\n0.2\n0\n0.4\n8\nX = 61.5 m\nX = 40.6 m\nn/h\n0.2\n0\n0.4\nX = 57.9 m\nX = 35.9 m\nn/h\n0.2\n0\n0.4\nX = 54.4 m\nX = 28.6 m\nn/h\n0.2\n0\n0.4\nX = 50.4 m\nX = 17.6 m\nn/h\n0,2\n0\n50\n100\n150\n200\n50\n100\nt (g/h) 0.5 150\n200\nt (g/h) 0.5\nx\n0\n-1\n-2\n0\n10\n20\n30\n40\n50\n60\n70\n80\nTank Extension [m]\nFigure 11-23: Time series of free surface at all wave gauges with different grid resolution and identical\nmodel setup.\n(blue), BOSZ with Ax = 0.05 m\n(black), BOSZ with Ax = 0.10 m\n(red),BOSZ with Ax = 0.20 m","MODEL BENCHMARKING WORKSHOP AND RESULTS\n393\n0.1\nX = 35.9 m\nn/h 0.05\n0\n110\n115\n120\n125\n130\n135\n140\n145\n150\n155\n160\nt (g/h) 0.5\nFigure 11-24: Close-up of time series of free surface at X = 35.9 m close to the reef crest with different\ngrid resolution and identical model setup.\n(blue), BOSZ with Ax = 0.05 m\n(black), BOSZ with Ax = 0.10 m\n(red),BOSZ with Ax = 0.20 m","National Tsunami Hazard Mitigation Program (NTHMP)\n394\nt(g/h)° 55\nn/h 0.2\n0\n-0.2\nt(g/h) 0.5 = 63\nn/h 0.2\n0\n-0.2\nt(g/h) 0.5 = 65.9\nn/h 0.2\n0\n-0.2\nt(g/h)° 0.5 = 68.2\nn/h 0.2\n0\n-0.2\nt(g/h) 0.5 = 69\nn/h 0.2\n0\n-0.2\nt(g/h) 0.5 = 70.8\nn/h 0.2\n0\n-0.2\nt(g/h) 0.5 =72.3\nn/h 0.2\n0\n-0.2\nt(g/h)° 0.5 = 76.3\nn/h 0.2\n0\n-0.2\nt(g/h) 0.5 = 80.2\nn/h 0.2\n0\n-0.2\nt(g/h)° 0.5 = 98.5\nn/h 0.2\n0\n-0.2\nt(g/h)° 0.5 = 112.2\nn/h 0.2\n0\n-0.2\nt(g/h) 0.5 = 130\nn/h 0.2\n0\n-0.2\n0\n-1\n-2\n0\n10\n20\n30\n40\n50\n60\n70\n80\nTank Extension [m]\nFigure 11-25: Snapshots of free surface profiles for propagation of solitary wave with A/h = 0.3 over\n1:12 slope and exposed reef crest.\n(blue), BOSZ dispersive,\n(red), BOSZ hydrostatic (nonlinear shallow water equations)\nafter tg/h = 64.8.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n395\n0.4\nX = 80 m\nX = 48.2 m\n0.2\n0\n0.4\nX = 72.7 m\nX = 46.1 m\nn/h o.2\n0\n0.4\nX = 65.2 m\nX = 44.3 m\nn/h o.2\n0\n0.4\n8\nX = 61.5 m\nX = 40.6 m\nn/h o.2\n0\n0.4\nX = 57.9 m\nX = 35.9 m\nn/h o.2\n0\n0.4\nX = 54.4 m\nX = 28.6 m\nn/h o.2\n0\n0.4\nX = 50.4 m\nX = 17.6 m\nn/h\n0.2\n0\n50\n100\n150\n200\n50\n100\n150\n200\nt (g/h) 0.5\nt (g/h)0.5\nX\nX\n0\n-1\n-2\n0\n10\n20\n30\n40\n50\n60\n70\n80\nTank Extension [m]\nFigure 11-26: Time series of free surface at all wave gauges. Solutions from BOSZ with identical\nnumerical scheme and input parameters for (blue), BOSZ dispersive, (red), BOSZ hydrostatic\n(nonlinear shallow water equations) after t/g/h=64.8 =","National Tsunami Hazard Mitigation Program (NTHMP)\n396\nX = 35.9 m\n0.1\nn/h 0.05\n0\n110\n115\n120\n125\n130\n135\n140\n145\n150\n155\n160\nt (g/h) 0.5\nFigure 11-27: Close-up of time series of free surface at X = 35.9 m. Solutions from BOSZ with identical\nnumerical scheme and input parameters for\n(blue), BOSZ dispersive,\n(red), BOSZ hydrostatic (nonlinear shallow water equations)\nafter tg/h = 64.8.\n4\nX = 54.4 m\n2\n0\n-2\n60\n80\n100\n120\n140\n160\n180\n200\nt (g/h)0.5\n0.5\n0.0\n-0.5\n-1.0\n-1.5\n-2.0\n-2.5\n25\n30\n35\n40\n45\n50\n55\n60\n65\n70\nTank Extension [m]\nFigure 11-28: Time series of x-directed velocity at X = 54.4 m. Solutions from BOSZ with identical\nnumerical scheme and input parameters for\n(blue), BOSZ dispersive,\n(red), BOSZ hydrostatic (nonlinear shallow water equations)\nafter t/g/h = 64.8.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n397\n17\nDepth\n16\n15\n0.5\n14\n97\n0 [m]\n6\n13\ne\n-0.5\n12\n1\n11\n10\n9\n-1.0\n8\n40\n10\n30\n5\n20\n0 [m]\nWidth\nLength\n-5\n10\n-10\n0 [m]\nFigure 11-29: Perspective view of ISEC BM1 bathymetry from laser scan at original resolution of X = 5\ncm. Circles denote wave gauge locations at free surface and corresponding location on tank bottom.\nRed crosses at gauges 3, 6, and 13 indicate positions of acoustic Doppler velocimeters for velocity\nmeasurements.","National Tsunami Hazard Mitigation Program (NTHMP)\n398\nTime 3.0 sec\nTime 16.0 sec\n0.5\n0.5\n0\n0\n-0.5\n-0.5\n10.0\n10.0\n5.0\n5.0\n0\n0.0\n0\n0.0\n5\n5\n10\n10\n15\n-5.0\n15\n-5.0\n20\n20\n25\n25\n30\n-10.0\n-10.0\n30\n35\n35\n40\n40\nTime 5.0 sec\nTime 21.0 sec\n0.5\n0.5\n0\n0\n-0.5\n-0.5\n10.0\n10.0\n5.0\n5.0\n0\n0.0\n0\n0.0\n5\n5\n10\n10\n15\n-5.0\n15\n-5.0\n20\n20\n25\n25\n30\n-10.0\n30\n-10.0\n35\n35\n40\n40\nTime 35.0 sec\nTime 8.0 sec\n0.5\n0.5\n0\n0\n-0.5\n-0.5\n10.0\n10.0\n5.0\n5.0\n0\n0\n00\n0.0\n5\n5\n10\n10\n15\n-5.0\n15\n-5.0\n20\n20\n25\n25\n-10.0\n30\n-10.0\n30\n35\n35\n40\n40\nTime 11.0 sec\nTime 45.0 sec\n0.5\n0.5\n0\no\n-0.5\n-0.5\n10.0\n10.0\n5.0\n5.0\n0\n0\n0.0\n0.0\n5\n5\n10\n10\n15\n-5.0\n15\n-5.0\n20\n20\n25\n25\n30\n-10.0\n30\n-10.0\n35\n35\n40\n40\nFigure 11-30: Snapshots of solitary wave transformation in ISEC BM1.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n399\n0.6\n0.4\nGauge 1\nn/h\n0.2\n0.0\n0.6\n0.4\nGauge 2\n0.2\n0.0\n0.6\nGauge 3\n0.4\nn/H\n0.2\n0.0\n0.6\nGauge 4\nn/hp.4\n0.2\n0.0\n0.6\nGauge 5\nn/hp.4\n0.2\n0.0\n0.6\nGauge 6\nn/hp.4\n0.2\n0.0\n0.6\nGauge 7\nn/hp.4\n0.2\n0.0\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\n220\n240\n260\n280\nt (g/h) 0.5\nFigure 11-31: Time series of free surface profiles along basin centerline in ISEC BM1. Solid lines and\ncircles denote computed and measured data.","National Tsunami Hazard Mitigation Program (NTHMP)\n400\n0.6\nGauge 8\nn/hp.4\n0.2\n0.0\n0.6\nGauge 9\nn/h.4\n0.2\n0.0\n0.6\nGauge 10\nn/hp.4\n0.2\n0.0\n0.6\nGauge 11\nn/hp.4\n0.2\n0.0\n0.6\nGauge 12\nn/hp.4\n0.2\n0.0\n0.6\nGauge 13\nn/hp.4\n0.2\n0.0\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\n220\n240\n260\n280\nt (g/h) 0.5\nFigure 11-32: Time series of free surface profiles along transect at x = -5 m in ISEC BM1. Solid lines and\ncircles denote computed and measured data.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n401\n0.6\nn/hp.4\nGauge 14\n0.2\n0.0\n0.6\nGauge 15\nn/hp.4\n0.2\n0.0\n0.6\nGauge 16\nn/hp.4\n0.2\n0.0\n0.6\nn/hp.4\nGauge 17\n0.2\n0.0\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\n220\n240\n260\n280\nt (g/h) 0.5\nFigure 11-33: Time series of free surface profiles along alongshore transect in ISEC BM1. Solid lines\nand circles denote computed and measured data.","National Tsunami Hazard Mitigation Program (NTHMP)\n402\n3.0\n(a)\n2.0\nGauge 3\n1.0\n0\n-1.0\n3.0\n2.0\nGauge 6\n1.0\n0\n-1.0\n3.0\n2.0\nGauge 13\n1.0\n0\n-1.0\n1.0\nGauge 3\n(b)\n0\n-1.0\n1.0\nGauge 6\n0\n-1.0\n1.0\nGauge 13\n0\n-1.0\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\n220\n240\n260\n280\nt (g/h)0.5 0.5\nFigure 11-34: Time series of velocity in ISEC BM1. (a) Cross-shore component. (b) Alongshore\ncomponent. Solid lines and circles denote computed and measured data.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n403\nDepth\n9\n8\n0.5\n7\n04\n3\n6\n0 [m]\n5\n10\n2\n1\n-0.5\n-1.0\n40\n10\n30\n5\n20\n0 [m]\nLength\nWidth\n-5\n10\n-10\n0 [m]\nFigure 11-35: Perspective view of ISEC BM2 bathymetry from laser scan at original resolution of. Ax =\n5 cm. Conical island was added after laser scan. Circles denote wave gauge locations at free surface\nand corresponding location on tank bottom. Red crosses at gauges 2, 3, and 10 indicate positions of\nacoustic Doppler velocimeters for velocity measurements.","National Tsunami Hazard Mitigation Program (NTHMP)\n404\nTime 5.1 sec\nTime 12.2 sec\nTime 15.2 sec\nTime: 6.6 sec\nTime 8.6 sec\nTime 17.2 sec\nTime 21.2 sec\nTime 9.2sec\nFigure 11-36: Snapshots of solitary wave transformation in ISEC BM2.","MODEL BENCHMARKING WORKSHOP AND RESULTS\n405\n0.6\nGauge 1\n0.4\nn/h\n0.2\n0.0\n0.6\nGauge 2\n0.4\nn/h\n0.2\n0.0\n0.6\nGauge 3\n0.4\nn/h\n0.2\n0.0\n0.6\nGauge 4\n0.4\nn/h\n0.2\n0.0\n0.6\nGauge 5\nn/hp.4\n0.2\n0.0\n0.6\nGauge 6\nn/hp.4\n0.2\n0.0\n0.6\nGauge 7\n0.4\n0.2\n0.0\n0.6\nGauge 8\n0.4\nn/h\n0.2\n0.0\n0.6\nGauge 9\n0.4\nn/h\n0.2\n0.0\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\n220\n240\n260\n280\nt (g/h) 0.5\nFigure 11-37: Time series of free surface profiles at wave gauges in ISEC BM2. Solid lines and circles\ndenote computed and measured data.","406\nNational Tsunami Hazard Mitigation Program (NTHMP)\n(a)\n3.0\n2.0\nGauge 2\n1.0\n0\n-1.0\n3.0\n2.0\nGauge 3\n1.0\n0\n-1.0\n3.0\n2.0\nGauge 10\n1.0\n0\n-1.0\n(b)\n1.0\nGauge 2\n0\n-1.0\n1.0\nGauge 3\n0\n-1.0\n1.0\nGauge 10\n0\n-1.0\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\n220\n240\n260\n280\nt (g/h)0.5 0.5\nFigure 11-38: Time series of velocity in ISEC BM2. (a) Cross-shore component. (b) Alongshore\ncomponent. Solid lines and circles denote computed and measured data."]}