{"Bibliographic":{"Title":"Status report on a semi-implicit version of the Shuman-Hovermale model","Authors":"","Publication date":"1974","Publisher":""},"Administrative":{"Date created":"08-20-2023","Language":"English","Rights":"CC 0","Size":"0000033208"},"Pages":["A\nQC\n851\nOF\nU6N5\nno.54\nc.1\nA Technical Memorandum NWS NMC 54\n*\n*\nANGREST\nSTATES\nOF\nSTATUS REPORT ON A SEMI-IMPLICIT VERSION\nOF THE SHUMAN-HOVERMALE MODEL\nKenneth Campana\nNational Meteorological Center\nWashington, D.C.\nMarch 1974\nnoaa\nNATIONAL OCEANIC AND\nNATIONAL WEATHER\nATMOSPHERIC ADMINISTRATION\nSERVICE","Na\nNational Weather Service, National Meterological Center Series\nThe National Meteorological Center (NMC) of the National Weather Service (NWS) produces weather anal-\nyses and forecasts for the Northern Hemisphere. Areal coverage is being expanded to include the entire\nglobe. The Center conducts research and development to improve the accuracy of forecasts, to provide\ninformation in the most useful form, and to present data as automatically as practicable.\nNOAA Technical Memoranda in the NWS NMC series facilitate rapid dissemination of material of general\ninterest which may be preliminary in nature and which may be published formally elsewhere at a later\ndate. Publications 34 through 37 are in the former series, Weather Bureau Technical Notes (TN), Na-\ntional Meterological Center Technical Memoranda; publications 38 through 48 are in the former series\nESSA Technical Memoranda, Weather Bureau Technical Memoranda (WBTM). Beginning with 49, publications\nare now part of the series, NOAA Technical Memoranda NWS.\nPublications listed below are available from the National Technical Information Service (NTIS), U.S.\nDepartment of Commerce, Sills Bldg., 5285 Port Royal Road, Springfield, Va. 22151. Price: $3.00 paper\ncopy; $1.45 microfiche. Order by accession number, when given, in parentheses.\nWeather Bureau Technical Notes\nTN 22 NMC 34 Tropospheric Heating and Cooling for Selected Days and Locations over the United States\nDuring Winter 1960 and Spring 1962. Philip F. Clapp and Francis J. Winninghoff, 1965.\n(PB-170-584)\nTN\n30\nNMC\n35\nSaturation Thickness Tables for the Dry Adiabatic, Pseudo-adiabatic, and Standard Atmo-\nspheres. Jerrold A. LaRue and Russell J. Younkin, January 1966. (PB-169-382)\nTN\n37\nNMC\n36 Summary of Verification of Numerical Operational Tropical Cyclone Forecast Tracks for\n1965. March 1966. (PB-170-410)\nTN\n40\nNMC\n37 Catalog of 5-Day Mean 700-mb. Height Anomaly Centers 1947-1963 and Suggested Applica-\ntions. J. F. O'Connor, April 1966. (PB-170-376)\nESSA Technical Memoranda\nWBTM NMC 38 A Summary of the First-Guess Fields Used for Operational Analyses. J. F. McDonell, Feb-\nruary 1967. (AD-810-279)\nWBTM NMC 39 Objective Numerical Prediction Out to Six Days Using the Primitive Equation Model--A Test\nCase. A. J. Wagner, May 1967. (PB-174-920)\nWBTM NMC 40 A Snow Index. R. J. Younkin, June 1967. (PB-175-641)\nWBTM NMC 41 Detailed Sounding Analysis and Computer Forecasts of the Lifted Index. John D. Stackpole,\nAugust 1967. (PB-175-928)\nWBTM NMC 42 On Analysis and Initialization for the Primitive Forecast Equations. Takashi Nitta and\nJohn B. Hovermale, October 1967. (PB-176-510)\nWBTM NMC 43 The Air Pollution Potential Forecast Program. John D. Stackpole, November 1967. (PB-176-\n949)\nWBTM NMC 44 Northern Hemisphere Cloud Cover for Selected Late Fall Seasons Using TIROS Nephanalyses.\nPhilip F. Clapp, December 1968. (PB-186-392)\nWBTM\nNMC 45 On a Certain Type of Integration Error in Numerical Weather Prediction Models. Hans\nOkland, September 1969. (PB-187-795)\nWBTM\nNMC 46 Noise Analysis of a Limited-Area Fine-Mesh Prediction Model. Joseph P. Gerrity, Jr., and\nRonald D. McPherson, February 1970. (PB-191-188)\nWBTM\nNMC 47 The National Air Pollution Potential Forecast Program. Edward Gross, May 1970. (PB-192-\n324)\nWBTM NMC 48 Recent Studies of Computational Stability. Joseph P. Gerrity, Jr., and Ronald D. McPher-\nson, May 1970. (PB-192-979)\n(Continued on inside back cover)","A\nac\n851\nU6N5\nno.54\nC.I\nNOAA Technical Memorandum NWS NMC 54\nSTATUS REPORT ON A SEMI-IMPLICIT VERSION\nOF THE SHUMAN-HOVERMALE MODEL\nKenneth Campana\nNational Meteorological Center\nWashington, D.C.\nMarch 1974\nATMOSPHERIC SCIENCES\nLIBRARY\nMAY 23 1974\nN.O.A.A.\nU. S. Dept. of Commerce\nAND ATMOSPHERIC\nMONTH NOAA\nUNITED STATES\nNATIONAL OCEANIC AND\nNATIONAL WEATHER\nDEPARTMENT OF COMMERCE\nATMOSPHERIC ADMINISTRATION\nSERVICE\nFrederick B. Dent, Secretary\nRobert M. White, Administrator\nGeorge P. Cressman, Director\nS DEPARTMENT OF country\n74\n2258","Abstract\n1. Introduction\n2. Initialization\n3. Forecast section\n4. Boundary conditions\n5. Summary, results, and future plans\nAcknowledgments\nReferences\nAppendix A. High frequency computational mode control\nN\nAppendix B. Restoration boundary condition\nYRARSIL\nMPTES YAM","STATUS REPORT ON A SEMI-IMPLICIT VERSION OF THE SHUMAN-HOVERMALE MODEL\nKenneth A. Campana\nNational Meteorological Center\nABSTRACT. The numerical weather prediction model\ndeveloped by Shuman and Hovermale has been re-\nformulated for semi-implicit integration. Early\nresults of real data experimentation with the new\nmodel are presented in this paper. Orography,\nfriction, and diabatic effects are not treated in\nthis model. Successful forecasts have been made\nto 48 hours that compare favorably with results\nfrom a parallel explicit integration.\n1. INTRODUCTION\nThis report documents the current state of developmental testing of\nthe semi-implicit version of the Shuman-Hovermale six-layer primitive\nequation model. A description of the operational (OPNL) model at the\nNational Meteorological Center (NMC) is given by Shuman and Hovermale\n(1968), while the semi-implicit (SIMPL) model is described by Gerrity,\nMcPherson, and Scolnik (1973). The principal advantage of semi-implicit\nintegration is the longer time step allowed in comparison with the ex-\nplicit leap-frog scheme. The possible use of a semi-implicit scheme in\nfuture fine-mesh and hurricane models at NMC has been a prime reason for\nreal data experimentation with the SIMPL model.\nAlthough construction of the SIMPL model closely paralleled that of\nthe OPNL model, some features of the new model are different. Of course,\nthe major difference is the method of time integration used by each\nmodel--but there are other important ones, which are briefly noted here.\nThe grid of the new model is staggered in space--having pressure,\ntemperature, and vertical motion on one set of grid points and the u, V\nvelocity components on the other (fig. 1). It uses absolute temperature\nrather than potential temperature as the thermodynamic variable. For\nthe sake of economy and ease of experimentation, the grid spacing has\nbeen set to 762 km (twice that of the OPNL) and orography, friction,\nand diabatic effects have not been employed. The model is flexible so\nthat it will be relatively easy to perform numerical experiments with\nhigher-order differencing, explicit integration, etc.\nThis report describes the problems encountered when making real data\nforecasts with the SIMPL model and'some \"solutions\" to them. The diffi-\nculties involved in obtaining initial data for the model are discussed\nin section 2. Sections 3 and 4 contain discussion of features that have\nbeen added to the model in order to produce stable forecasts. Finally,\ncomparison is made between 48-hour forecasts of the SIMPL and OPNL\n1","models. Although problems still exist with initialization and boundary\nconditions and a more complex version of the model (including orography,\nmoisture, etc.) awaits development, the results indicate that the Shuman-\nHovermale model has been adapted successfully for semi-implicit integra-\ntion.\n2. INITIALIZATION\nRather than develop a new initialization scheme for the SIMPL model,\nit was decided to employ the current operational procedure--wit modifi-\ncations. The OPNL scheme described by Shuman and Hovermale (1968) con-\nsists of partitioning the vertical coordinate into six \"sigma\" layers\n(and a computational cap), and the interpolation of data from the ten\nNMC mandatory-level pressure analyses to this sigma coordinate system.\nVertical consistency between temperature and pressure is assured through\nuse of the hydrostatic equation. Winds in sigma layers are no longer\nobtained from the Balance Equation, as described by Shuman (1968), but\nresult from a vertical interpolation of the NMC constant pressure wind\nanalyses (TPB #65, 1971). Divergence is removed from these sigma layer\nwinds.\nProblems arise when using this initialization procedure to provide data\nfor the SIMPL model, because different thermodynamic variables are used\nin the two models. The form of the hydrostatic equation differs in each,\nand this difference was found to be important in early experiments with\nthe new model. Therefore, the OPNL initialization has been restructured\nto use temperature T rather than potential temperature 0, with equations\n(2) and (3) used in place of the OPNL hydrostatic equation (1).\n(1)\ndo/dt - cp6\n(2)\n20/2p = - a\n(3)\n= RT/p\na\nwhere is geopotential at sigma surfaces, a is specific volume in sigma\nlayers, R is the gas constant, cp is the specific heat at constant\npressure, and TT is the Exner Function, (p/1000)\nOther changes have been made in the initialization and are noted here.\nThe two models differ in the method of vertical interpolation of initial\ndata and in the definition of the middle of a sigma layer. The OPNL model\ninterpolates with respect to ln p and uses the pressure at the mean ln p,\nwhereas the SIMPL model interpolates with respect to pressure, P, and uses\nthe mean pressure itself. Additionally, the complex OPNL technique for\ndefining the computational cap is no longer valid. In the new model, the\npressure thickness of the cap simply is set at 100 mb and the constant\npotential temperature therein is obtained by extrapolating temperature\n2","upward from below at each grid point and averaging over the entire grid.\nThus, unlike the OPNL model, there is no initial balance between mass and\nwind fields in the computational cap. This has caused trouble and a\ntemporary \"solution\" is discussed in section 3.\nAppropriate sigma layer data for the SIMPL model now exist, but more\nproblems arise when extracting data for the coarser grid. Since the new\ngrid spacing is twice as large as that of the OPNL model, data are used\nfrom every other grid point. Small-scale features that are of larger\nscale in the OPNL model are thus introduced into the new model. Also, the\nwind field has acquired divergence during the horizontal interpolation to\nthe coarser grid. Since early tests using this data were unsuccessful,\nsome adjustment to these initial data was obviously needed. Even though\nthe data looked good to the \"naked eye\", geostrophic winds computed at\nthe initial hour were contaminated with small-scale features which, when\ncompared to the relatively smooth \"real\" winds, implied a poor initial\nbalance between the wind and mass fields. One solution to this problem\nwas to horizontally smooth all data using Shuman's (1957) nine-point\nfilter in order to remove the noise in the data and hopefully create a\npartial balance. The filter consists of a nine-point smoothing pass\nfollowed by a nine-point desmoothing pass (the order is immaterial),\nwith the data being found at its original location at the completion of\neach step (either grid points or grid boxes). All two-grid features are\nremoved, whereas the larger scales are hardly affected. Since divergence\nmust be removed from the wind field, a variation of this filtering process\nis used on the velocity components. The u,v winds are first desmoothed\nin grid boxes and then averaged to grid points. The divergence is\nremoved at these grid points and the resulting winds are averaged back to\ngrid boxes. The two \"averagings\" are equivalent to a nine-point smoothing\npass. Although some divergence is created by the final averaging, it is\nstill several orders of magnitude smaller than that generated during the\nforecast and does not seem to be important.\nBy restructuring the OPNL initialization and horizontally filtering the\ninterpolated data, a fairly consistent set of initial data for the SIMPL\nmodel now exists. Experimentation has shown that much of the initialization\nproblem has been solved by this technique. The next section discusses the\nforecast part of the model and additions that have been made to it.\n3. FORECAST SECTION\nThe details of the semi-implicit modification to the OPNL model and the\nassociated forecast equations are fully described by Gerrity et al. (1973).\nSeveral features have been added to the SIMPL model which were not reported\npreviously, and they have been successful in promoting a longer, more stable\nforecast by controlling the nonmeteorological noise that develops during\nthe integration.\n3","First, a dry convective adjustment is being used. The forecast tempera-\nture lapse rates are checked for stability and, if unstable, are adjusted\nto fit the dry adiabatic lapse rate. An adjustment procedure exactly the\nsame as that used in the OPNL model is employed. First, all temperatures,\nT, are converted to potential temperatures, 0, by using eq. (4):\n(4)\n0 = T/W\nThen at each grid point, temperatures are checked to see that 20/2p 0\neverywhere in the vertical. If 20/20 > 0, then an iterative procedure is\nused to adjust the unstable temperatures to a neutral lapse rate\n(dd/dd = 0) Most convective adjustments have occurred in the two layers\nastride the internal material surface in our experiments1. This technique\nhas prevented instabilities, which had appeared in previous tests, from\ndeveloping in the model stratosphere.\nA second feature used in the new model is a time filter of the form used\nby Asselin (1972). High frequency computational modes have developed in\nexperimental forecasts and in order to control some of this noise, the\ntime filter, shown in eq. (5), is applied to all variables:\n1-a\n(5)\nwhere a = 0.3, and T-1, T, T+1 refer to the past, present, and future\ntime levels. The starred ) * quantity on the left is the time-smoothed\nvariable and is valid at the T-1 time level during the succeeding forecast\ncycle. The OPNL model uses a time filter on the winds only (TPB #31,\n1969), but a similar test on the SIMPL model was not so successful as\nfiltering all variables.\nAnother addition to the forecast equations concerns treatment of the\ncomputational cap. As noted in the section on initialization, there is no\ninitial balance in the cap--with the result that strong winds quickly\ndevelop there during a forecast and harm variables in the stratospheric\nlayers. Further research will be devoted to the initialization procedure\nfor the cap; but we have chosen to avoid the problem, at present, by using\nthe following technique. The u and V components of the wind have been\nconstrained to change very slowly in the cap, eq. (6), as though there\nwere a balance everywhere on the grid.\n1This is probably due to the fact that the vertical temperature gradient\nacross the material surface does not explicitly enter the forecast equations.\n4","vt11\n(6)\nY II .25\nItl\nT+1\nu*\nand\nV*\nare the values to which the T+1 winds are constrained.\nNote that the actual forecast value of the winds would be used if Y = 1.\nSince the winds were the most adversely affected variables in the cap, they\nare the only ones constrained and the pressure and temperature are allowed\nto vary as the equations dictate. This device does keep the cap and strato-\nspheric variables under control during the forecast, although the difficulty\nreturns eventually.\nTwo-grid noise occurring around the lateral boundaries of the model and\npropagating into the forecast region remains a serious problem in forecasts\nfor periods longer than 2 days. Since the SIMPL model has a very coarse\nmesh, the active meteorological region (poleward from 30°N) is only a few\ngrid points from the lateral boundaries and this noise problem has a harm-\nful effect on the forecast. Application of Shuman's space filter to all\nvariables at specific time intervals has controlled this noise and allowed\nthe model to run successfully past 4 days. It is assumed that this device\nwill not be needed when the boundary problem is solved. Experiments with\nvarious boundary conditions and their effect on this problem are presented\nin the next section of this report.\nTwo other techniques have been employed in order to control the high\nfrequency computational modes; however, they have not been entirely success-\nful and are not used in the current version of the model. One involves the\nfinite difference form of the Coriolis term in the equations of motion, and\nthe other is the backward implicit time differencing scheme. Both experi-\nments are documented in appendix A.\n4. BOUNDARY CONDITIONS\nAs implied in the previous section, the most serious problem with the\nSIMPL model is the propagation of high wavenumber noise into the forecast\nregion from the lateral boundaries. The original formulation of the model\nrequired all variables to be held constant at their initial values on the\nboundary during a forecast. Although the velocity components are in grid\nboxes, winds must be supplied at boundary grid points for each time step\nand it is these winds that were held constant. Early tests showed a great\ndeal of noise being generated at the lateral boundaries, so experiments\nwere run in which boundary variables were allowed to vary in time.\nCurrently, a method is being used in which the boundary grid point winds\nare allowed to change while satisfying the OPNL model's free-slip wall\ncondition--no normal wind at the wall and no gradient of tangential wind\nacross the wall (Shuman and Hovermale 1968). The wall is defined in grid\nboxes next to the boundary grid points--an example of which is shown in\n5","figure 2. In order to impose the boundary conditions on the winds, which\nare carried in grid boxes, values for them must be obtained at grid points.\nFirst, u and V velocity components are averaged to the interior grid point\nnext to the boundary (fig. 3); then the boundary condition shown in figure\n2 is applied in order to obtain winds on the boundary (fig. 3), and finally\nthe velocity components at grid points are averaged back to the first\ninterior grid box (fig. 4). Imposition of the boundary conditions thus\nmeans that forecast winds in the first xy interior box are not used. Spatially\naveraged quantities are denoted by ( (fig. 5). This technique greatly\nimproved the forecast by controlling some of the noise near the boundaries;\nhowever, the other variables were kept constant on the lateral boundaries\nand harmful noise still occurred in forecasts past 2 days. In an attempt\nto control the boundary noise for longer periods of time, the variables\nnear the edges of the grid were constrained to remain near their initially\nsmooth values. This technique created additional problems and is not\nbeing used. It is presented in appendix B for documentation purposes only.\nFurther testing of boundary conditions uncovered a method that is\ncurrently producing the most successful SIMPL forecasts. The OPNL free\nslip wall condition is now being used for all variables in the SIMPL model.\nThe condition on pressure, temperature, and vertical velocity is one of no\ngradient across the wall, in which the boundary value is set equal to that\nof the first interior grid point. At each time step during the forecast,\na set of Helmholtz equations are solved for time-averaged values of pressure\nand vertical velocity (Gerrity et al. 1973). In the original formulation\nof the model, the relaxation procedure that was used to solve the equations\nemployed a Dirichlet boundary condition1; now the relaxation has been\nadjusted to use the Neumann boundary condition noted above. This condition\nis satisfied by changing the finite difference form of the Laplacian\noperator next to the boundary to that shown in figure 7--compare with the\nDirichlet condition in figure 6.\nIncorporation of the full set of OPNL boundary conditions into the SIMPL\nmodel has produced the best results so far; the boundaries are quieter and\nthe level of forecast noise is comparable to that of the OPNL forecast\nthrough 48 hours. These results are discussed in the next section.\n5. SUMMARY, RESULTS, AND FUTURE PLANS\nIn summary, the semi-implicit version of the Shuman-Hovermale six-layer\nprimitive equation model has been run successfully with real data. It is\nless complex than the OPNL model, having twice as large a grid spacing\nand having no orography, friction, or diabatic effects. Initial data are\nobtained from the OPNL initialization scheme, which has been adjusted for\nthe SIMPL model, extracted from every other grid point, and filtered hori-\nzontally. A convective adjustment, time filter, and constraint on the cap\nwinds, all have been added to the forecast equations. The model has been\nreformulated to use the operational free-slip wall boundary condition.\n1This fit the constant boundary condition then being used.\n6","Successful forecasts, for 0000 GMT August 24, 1972, have been made beyond\n48 hours with the semi-implicit model. For comparison, parallel 48-hour\nforecasts have been made with the OPNL model and with the explicit version\nof the new model (15-minute time step). Orography and diabatic effects\nhave been removed from the OPNL model to make it more comparable with the\nnew model. Figures 8 and 9 show the initial data used for the SIMPL model\n(and its explicit version) at 1000 mb and 500 mb. Figures 10, 11, and 12\nshow 1000-mb 48-hour forecasts for the three test runs and figures 13, 14,\nand 15 show the 500-mb results. None of the figures have been filtered for\noutput purposes, so some noise is seen around the boundaries. The semi-\nimplicit and explicit versions of the new model have produced very similar\nforecasts at both 1000 mb and 500 mb. The forecast using the explicit\nmethod took approximately 21/2 times longer to run on the computer, even\nthough it used 4 times as many time steps as the semi-implicit test. The\nnecessity of solving a system of Helmholtz equations each time step in the\nsemi-implicit version reduces the apparent four-to-one advantage implied by\ntime step length.\nWhen comparison is made between the semi-implicit scheme on a coarse grid\n(figs. 10 and 13) and the OPNL scheme on a finer grid (figs. 12 and 15),\nwe see that the large-scale features have been similarly forecast by both\nmodels. of course, smaller scale features and fine details have been lost\nin the SIMPL model which has one-quarter the number of grid points. These\nresults indicate that the six-layer OPNL model, without orography, friction,\nor diabatic effects, has been successfully adapted for semi-implicit inte-\ngration.\nFurther research is planned for the new model--but, in the operational\nenvironment at NMC, priorities change and of necessity so will the aims\nof the development effort being made on the SIMPL model. At present, NMC\nenvisions the potential use of the semi-implicit technique for fine-mesh\nregional models and for the planned hurricane model. Certainly, the\nlateral boundary problems that are evident in the current hemispheric\nversion of the model will become more crucial to the finer-mesh regional\nones, so research must continue regarding the specification of model\nvariables on the boundary. Incorporation of orography, moisture, radiation,\nand other physical effects into the SIMPL model is presently being planned.\nInitial tests with coarse grid orography have been unsuccessful and a\nmethod of filtering this field may have to be employed. Higher-order\ndifferencing tests and explicit vs. implicit comparisons can be made on\nthe current version of the model. Other tests will be made reducing grid\nsize. A future report will be made documenting progress in these areas.\nACKNOWLEDGMENTS\nThe author would like to thank Joseph Gerrity and Ronald McPherson for\ntheir important contributions to this project, also Mary Daigle for typing\nthe report and Tom Krzenski for drafting the figures.\n7","REFERENCES\nAsselin, Richard A., \"Frequency filter for time integrations,\" \" Monthly\nWeather Review, 100, 1972, pp. 487-490.\nGerrity, Joseph P., , and McPherson, Ronald D. \"On an efficient scheme for\nthe numerical integration of a primitive equation barotropic model, \"\nJournal of Applied Meteorology, 10, 1971, pp. 353-363.\nGerrity, Joseph P. , McPherson, Ronald D., and Scolnik, Stephen, \"A semi-\nimplicit version of the Shuman-Hovermale model,\" NOAA Technical Memo-\nrandum NWS-NMC 53, 1973, 44 pp.\nKesel, Philip G., and Winninghoff, Francis J., \"The Fleet Numerical\nWeather Central operational primitive equation model,\" 11 Monthly Weather\nReview, 100, 1972, pp. 360-373.\nShuman, Frederick G., \"Numerical methods in weather prediction, II.\nSmoothing and filtering,\" Monthly Weather Review, 85, 1957, pp. 357-361.\nShuman, Frederick G., and Hovermale, John B. \"An operational six-layer\nprimitive equation model,\" Journal of Applied Meteorology, 7, 1968, pp.\n525-547.\nTechnical Procedures Bulletin No. 31: \"Further changes in the NMC six-\nlayer model,\" \" Sept. 9, 1969.\nTechnical Procedures Bulletin No. 65: \"New initialization procedure for\nthe six-layer (PE) numerical prediction model,\" Aug. 12, 1971.\n8","APPENDIX A. HIGH FREQUENCY COMPUTATIONAL MODE CONTROL\nTwo experimental changes made to the SIMPL model in order to control\nhigh frequency computational modes have not been completely successful.\nThey are presented here for documentation purposes only.\nThe first test involves the finite-difference form of the Coriolis term\nin the equations of motion. As presented by Gerrity, McPherson, and\nScolnik (1973) in their equation 281, this term contains a great deal of\nhorizontal averaging. Noting that the Coriolis parameter, f, is defined\nat grid points and the V component of the wind and map factor, m, is\ndefined in grid boxes, the Coriolis term for the u equation of motion is\nshown below:\nxy\n]\nxy\nf\nxy\nThe\naveraging is defined in figure 5. Pressure, temperature, and\ngeopotential are defined at grid points, so the horizontal differencing\ninvolved in the pressure gradient term of the equation brings it to the\ngrid boxes without any additional averaging. Since the Coriolis and\npressure gradient terms are related geostrophically, excess averaging on\none of them might upset a balance in the forecast equations. Gerrity and\nMcPherson (1971) had successfully used unaveraged velocity components in\nthe Coriolis term to preserve a reasonable geostrophic balance, so it was\nattempted in the SIMPL model. Defining the Coriolis parameter, f, in grid\nboxes (as well as grid points), the finite difference form of the Coriolis\nterm reduces to f without any horizontal averaging. The technique\nreduced a great deal of the \"noise\" in the forecast vertical motion fields\nearly in the forecast, perhaps implying a better balance; however, later\nin the forecast, difficulty arose in all variables near one corner of the\ngrid and quickly spread. We have been unsuccessful in determining the\ncause of the problem, so the tentative conclusion is that the extra aver-\naging on the Coriolis term may be beneficial.\nThe other test attempted was to use the backward implicit time differ-\nencing scheme. A feature of this technique is that it damps the physical\ngravitational mode, and in conjunction with the time filter (which damps\nthe computational mode) is a potential method of controlling non-\nmeteorological noise in the model. In test forecasts, the backward\nimplicit method has proven somewhat successful--noise is reduced, forecast\nvariables are smoother, etc. However, the technique is incompatible for\nlong forecasts with the cap constraints that were described in section 3.\nUntil further research is done on the cap problem, these constraints will\nbe retained and the backward implicit method will be held in reserve.\n9","APPENDIX B. RESTORATION BOUNDARY CONDITION\nA method of controlling high-frequency noise near the boundary, which is\nemployed by the Navy in a primitive equation model (Kesel and Winninghoff\n1972), has been used with the SIMPL model. Experiments have shown that it\nis successful at quieting the boundaries, but the forecasts on the coarse\ngrid of the model are not satisfactory. It is presented here only for\ndocumentation purposes.\nIn section 4 of this report, tests with different boundary conditions\nwere discussed. The experiment that kept all boundary variables constant,\nbut allowed the winds to change while satisfying the OPNL boundary condition,\ngenerated a great deal of noise. In order to forecast for periods beyond\n2 days, an artificial device was used around the boundaries in which all\nvariables at T+1 were restored to their T values in a latitudinally depend-\nent zone:\n$ 0-4\nAO\n00=20°N =\nAo = 10°\nwhere ( )* t+1 is the value that the forecast variable acquires. Near the\nboundary (where B = 0), each variable is completely restored to its T\nvalue. This device did quiet the boundary region, but seemed to generate\nnoise at the edge of the restoration zone. On a grid with many more\npoints, the boundary is more removed from the active meteorological region,\nand this restoration technique would be more successful (a related type\nrestoration zone is used in the OPNL model). The restoration zone has to\nbe several grid lengths wide1, , so in the SIMPL model it penetrates to 30°N.\nAlthough longer forecasts could be produced by this technique, it had a\nharmful effect on meteorological systems--compare a 48-hour 500-mb forecast\nusing this method (fig. 16) with the OPNL forecast (fig. 15). A more\nsatisfactory result has been produced by using OPNL boundary conditions on\nall variables (fig. 13).\n1 Tests with do = 15° and A = 5° were not very successful.\n10","Po,I,:\nPo,I,O\ngrid point\ngrid box\nu, V\nFigure 1.\nwall\ngrid box\ngrid point\nsh\nN\nVD n = 0\nav\nS = 0\ndn\nFigure 2. -- Free-slip wall condition at left lateral boundary.\n11","wall\nu,v\nS\ngrid point\n- ,\nV\n,\ngrid box\nu,v\nFigure 3. -- Averaging wind components to grid points, then imposing wall\ncondition (left lateral boundary). .\nwall\n, very V1\n, , VAL\nu,v\n-uxy VEY ,\nuxy\nFigure 4. . -- Averaging wind components back to boundary grid boxes after\nimposing the wall condition.\n12","grid point\ngrid box\nV2\nV1\nxy\nvx = 1/4 1 IN V1\nV\ni=1\nV4\nV3\nFigure 5. .\nwall\nFigure 6. -- Grid points used to form the Laplacian operator -- -Dirichlet\ncondition.\nwall\nFigure 7. - - Grid points used to form the Laplacian operator wall condition.\n13","001\n06\nfor\n08\nOE\nI\n09\n001\n011\n06\n02\n08\nOL\nI\n08\nOL\n20\nI\nx\n8\nto\n$\n27\nI\n60\n50\n12\n0\n20\n$00\n60\n50\n70-\n80\n30\n60\n20\nL\n-90\n70\n80\n10\nFigure 8. . - - 1000-mb heights (meters) 0000 GMT Aug. 24, 1972.\n14","001\n06\nfor\nOB\nOL\nOE\n2\n0\n.\n001\no\nON\n06\nOA\nOR\nOL\n09\nnss\n04\nI\n2\n+\nto\n528\n540\n552\n4564\n576\n50\n594\nH\n10\n60\n20\n90\n80\n10\nFigure 9. -500-mb heights (decameters) 0000 GMT Aug. 24, 1972.\n15","00\nOE\nR\n16","011\nto\nH\n80\nL\n120\n120\nFigure 11. -Explicit version of semi-implicit 1000-mb heights,\n48-hr.\n17","001\n06\nfor\n08\n08\n0g\n0\nOF\n011\n001\n06\nor\nI\n10\n6th\n60\n60\n50-\n60\nA\n80 120\n130\nI\nH\n30\n180\n60\n100\n20\n10\nJ\nFigure 12. - -OPNL six-layer (explicit) 1000-mb heights, 48-hr.\n18","OH\n021\n80\n594\nH\nheights,\n48-hr.\nFigure\n13.\n19","001\nfor\nOE\nO\n001\nou\n06\n021\n0g\n09\n20\n7\n50-\n70-\n80\nH\n80\nFigure 14. -Explicit version of semi-implicit 500-mb heights, 48-hr.\n20","IV'I\n001\n06\nH\n001\n021\nQS\n08\n20\n10\n0522\n534\n552\n57\n60\n70\n80\n588\n3.\nH\n30\n20\nFigure 15. . -- -OPNL six-layer (explicit) 500-mb heights, 48-hr.\n21","001\n06\n08\nor\n7.0\nOE\n889\n9LS\nH\nour\n05\n04\nOG\nOL\n795\n09\n2299\n20\n570\n20\n576\n582\na\n588\n80\n594\n30\nH\n110\n100\n20\n90\n10\nL\n110\nFigure 16. -- Semi-implicit 500-mb heights, 48-hr restoration boundary\ncondition.\n22","(Continued from inside front cover)\nNOAA Technical Memoranda\nNWS NMC 49\nA Study of Non-Linear Computational Instability for a Two-Dimensional Model. Paul D.\nPolger, February 1971. (COM-71-00246)\nNWS NMC 50\nRecent Research in Numerical Methods at the National Meteorological Center. Ronald D.\nMcPherson, April 1971.\nNWS NMC 51\nUpdating Asynoptic Data for Use in Objective Analysis. Armand J. Desmarais, December\n1972. (COM-73-10078)\nNWS NMC 52\nToward Developing a Quality Control System for Rawinsonde Reports. Frederick G. Finger\nand Arthur R. Thomas, February 1973. (COM-73-10673)\nNWS NMC 53\nA Semi-Implicit Version of the Shuman-Hovermale Model. Joseph P. Gerrity, Jr., Ronald D.\nMcPherson, and Stephen Scolnik. July 1973. (COM-73-11323)"]}