{"Bibliographic":{"Title":"Addition of orography to the semi-implicit version of the Shuman-Hovermale model","Authors":"","Publication date":"1978","Publisher":""},"Administrative":{"Date created":"08-20-2023","Language":"English","Rights":"CC 0","Size":"0000022716"},"Pages":["H\nQC\n851\nU6\nN5\nno.62\nOF\nCOMMUNITY\n*\n*\nNOAA Technical Memorandum NWS NMC-62\nWITH\nAustralia\nSTATES\nOF\nADDITION OF OROGRAPHY TO THE SEMI-IMPLICIT\nVERSION OF THE SHUMAN-HOVERMALE MODEL\nNational Meteorological Center\nWashington, D. C.\nApril 1978\nnoaa\nNATIONAL OCEANIC AND\nNational Weather\nATMOSPHERIC ADMINISTRATION\nService","NOAA TECHNICAL MEMORANDONS\nNational Meteorological Center\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 Memorandums 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 Memorandums 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. 22161. Prices vary for\npaper copies; $3.00 microfiche. Order by accession number, when given, in parentheses.\nWeather Bureau Technical Notes\nTN\n22\nNMC 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 35 Saturation 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 37 NMC 36 Summary of Verification of Numerical Operational Tropical Cyclone Forecast Tracks for\n1965. March 1966. (PB-170-410)\nTN 40 NMC 37 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\nNMC 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\n45\nOn a Certain Type of Integration Error in Numerical Weather Prediction Models. Hans\nOkland, September 1969. (PB-187-795)\nWBTM NMC 46 Noise Analysis of a Limited-Area Fine-Mesh Prediction Model. Joseph P. Gerrity, Jr., and\nRonald D. McPherson, February 1970. (PB-191-188)\nNMC 47 The National Air Pollution Potential Forecast Program. Edward Gross, May 1970. (PB-192-\nWBTM\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)","HC\n851\nU6N5\n110.62\nNOAA Technical Memorandum NWS NMC-62\nADDITION OF OROGRAPHY TO THE SEMI-IMPLICIT\nVERSION OF THE SHUMAN-HOVERMALE MODEL\nKenneth A. Campana\nKenneth Campan\nCENTRAL\nLIBRARY\nAUG 07 1978\nNational Meteorological Center\nN.O.A.A.\nWashington, D. C.\nU.S. Dept. of Commerce\nApril 1978\nAND ATMOSPHERIC\nNOAA\nUNITED STATES\nNATIONAL OCEANIC AND\nNational Weather\nService\nDEPARTMENT OF COMMERCE\nATMOSPHERIC ADMINISTRATION\nGeorge P. Cressman, Director\nJuanita M. Kreps, Secretary\nRichard A. Frank, Administrator\nS DEPARTMENT OF\n78\n2827","CONTENTS\nAbstract\n1\n1. Introduction\n1\n2. Semi-implicit transformation\n3\n3. Orography\n6\nfitness\n4. Changes to model equations\n11\n14\nAcknowledgments\n14\nReferences\nii","ADDITION OF OROGRAPHY TO THE SEMI-IMPLICIT\nVERSION OF THE SHUMAN-HOVERMALE MODEL\nKenneth A. Campana\nNational Meteorological Center, NWS, NOAA\nWashington, D. C. 20233\nABSTRACT. The semi-implicit version of the Shuman-\nHovermale model now includes orography. The incor-\nporation of mountains into the model was made with\ngreat difficulty and is documented in this report.\nCare must be taken when splitting the pressure\ngradient term in the equations of motion into im-\nplicit and explicit parts. The orographic effect\non the pressure gradient term is exhibited in two\nways--as the gradient of surface geopotential and\nas the gradient of surface pressure. These two\neffects must be calculated together as either im-\nplicit or explicit parts in order that the long\ntime step will not cause serious orographic-related\ntruncation errors. Necessary changes to model\nequations are documented.\n1. INTRODUCTION\nThe semi-implicit time integration scheme has been reported in the lit-\nerature for several years and is used in a number of multilaver numerical\nweather prediction models around the globe. The implicit treatment","permits a long time step to be used in a forecast model because it time-\naverages terms in the equations which govern the fastest moving gravity\nwaves 1 A11 other terms are treated in the normal explicit sense. The\ncomputation time savings resulting from the long time step make the semi-\nimplicit technique particularly attractive for numerical models that are\nbeing used in an operational forecasting environment. Because a set of\nHelmholtz equations must be solved during each time step, the savings\nfrom the semi-implicit method is not so great as would be expected from\nthe longer time step. However, computation time savings of four to one\nare reported for a six to one ratio of time step intervals in semi-\nimplicit versus explicit runs (Kwizak and Robert, 1971).\nA semi-implicit version of the Shuman-Hovermale 6-1ayer primitive equa-\ntion model (6L PE) has been developed at the National Meteorological Center\n(NMC) by Gerrity et al. (1973), and early experimental results without\norography have been published by Campana (1974). It is a simplified re-\nsearch model patterned after the 6L PE, but uses none of its physical\nparameterizations and has twice its grid length. Tests that included\norography were initially unsuccessful, and it was with great difficulty\nthat mountains were incorporated into the model. The purpose of this re- -\nport is to document the solution to the mountain problem in the semi- -\nimplicit model. The first section will briefly describe the splitting of\nthe equations into implicit and explicit parts. The next section will\ndiscuss the mountain problem and its solution. The final section will\n1Pressure gradient term in the equations of motion and divergence term in\nthe continuity equation.\n2","present the model equations which must be adjusted to fit the above\nsolution. The actual model is not discussed in great detail, so the\nreader is referred to Gerrity (1973) for all the particulars. In this\nreport, the terminology used by Gerrity (1973) will be used where\nappropriate.\n2. SEMI-IMPLICIT TRANSFORMATION\nIn order to more easily discuss the mountain problem in the next section,\na brief description of the transformation of the equations of motion to\nsemi-implicit time differencing is helpful. The equation of motion for the\nV component of the wind is used for this discussion:\nicto av av +\n(1)\nwhere\nt = time,\nu = horizontal wind component in the x-direction,\nV = horizontal wind component in the y-direction,\nof = vertical wind component in the o-direction,\nP = pressure,\na = specific volume,\n= geopotential,\n£ = Coriolis and map factor terms = am ,\nm = map factor, and\nf = Coriolis force.\nNote that, unlike other models at NMC, this one uses temperature and\npressure as the thermodynamic variables.\n3","First, eq. (1) is simplified by employing a linearization procedure. Each\nvariable, X, is assumed to be composed of a basic state, X, varying only\nwith o, and a deviation from this basic state, X',\n(2)\nX = X + X ' .\nImplicit calculations are done only on the resulting linear terms. Basic\nstate values for the thermodynamic variables are obtained from the U.S.\nStandard Atmosphere (1962) using \"representative\" o-layer pressures. The\nbasic state wind field is one of no motion\nTaking the a agr ap term in eq. (1) and defining\na=a+a\nand\n=p+p'\nP\n0,\nfunction\nof\nsince\nis\n=\na\no only; one obtains\nthe\n(3)\ndy\nIn the semi-implicit treatment of eq. (1), terms on the left side are time-\naveraged (implicit calculation). Rewriting eq. (1) using the linearization\nprocess for all terms except one obtains :\nat m - - + Friction (4)\nNote that the a-ap term is nonlinear and is calculated on the explicit\ndy\n(non-time averaged) side.\nLetting superscripts T-1, T, and T+1 denote quantities evaluated explic-\nitly at past, present, and future time levels, the following definitions\nof the time average, x2t, and the time derivative, ax are useful when\n4","transforming eq. (4) to its semi-implicit counterpart,\n-1\n=\n.\nImplicit treatment of the left side of eq. (4) and dropping the primes from\nall variables leaves the following:\nT\n(5)\n,\n(.....) T represents all terms on the right side of eq.(4).\nwhere\nIn the actual model equations, o-layer pressure thicknesses, ap/do, are\nused in the pressure gradient term, rather than pressure itself. Further,\nin order to close the system of equations, T2t is transformed into implicit\nterms involving ppt and =2t and into other terms, R, calculated explicitly.\nReplacing T2t by these terms in eq. (5), and using the actual model variables,\none obtains:\n+ At(.....)k (6)\nk = vertical index,\nwhere aRK 8k,j hk,j\nand the matrices and all result from the transformation\nof o2 (section 4 in Gerrity, By solving a set of Helmholtz equa-\ntions, obtains the three ap2t and the four 72 which are needed to\none\ncompute -2t from eq. (6).\nThe preceding general description of the semi-implicit transformation\nnow allows one to proceed to a discussion of the orographic problem.\n5","3. OROGRAPHY\nSemi-implicit model experiments without orography were quite successful\nusing a time step of 1 hour. When mountains were introduced, however,\nerroneous orographic scale features developed over large mountain masses\nand were amplified with time. An example of this problem over the\nRockies and Himalayas is shown in figure 1. Tests with lower mountain\nelevations only lessened the real difficulty. When the model was run in\nan entirely explicit mode (and thus a shorter time step) the problem dis-\nappeared (fig. 2). . Further tests with the semi-implicit version, using a\ntime step as short as the explicit mode above, also yielded trouble-free\nforecasts. There appeared to be severe time truncation errors near oro-\ngraphy when using a long time step.\nAfter a great deal of reflection and experimentation, the problem\nappeared to be related to the implicit/explicit splitting of the pressure\ngradient term2 in the tropospheric sigma domain. Recalling eq. (1), , the\npressure gradient near mountains is made up of two relatively large terms\nhaving opposite signs\nThrough the semi-implicit trans-\nformation on this equation, these terms are further broken into implicit\nand explicit parts. Close examination shows that these two parts also can\nbe large terms of opposite sign in the vicinity of mountains. Since the\nbasic state pressure, p, is not a function of (x,y), gradients of pressure\nnear orography remain in the deviation part, p\". Thus a good portion of\nap\nterm near mountains remains on the implicit side of eq. (4)\nthe large\na\ndy\n2 In the equations of motion.\n6","001\n06\nof\n10\n001\n06\n021\ngo\nD9\n09\ng\nThe\n88\n50\n1\n558\n80\n588\nH\nYou\n20\n<90\n80\n10\nFigure 1. .--Semi-implicit 500-mb heights (dekameters), time step = 3600 s, 11-hr\nforecast from 0000 GMT 24 August 1972. Contour interval 6 dekameters.\n7","001\nOE\ngo\n889\n001\nozl\nOR\nOS\n20\n40\n7\nI\n534\n558\n70\n80\n588\nH\nso\n60\n20\n90\n10\n0000 2. GMT -Explicit 24 August 500-mb 1972. heights, Contour time interval step = 6 dekameters. 600 s, 12-hr forecast from\nFigure\n8","Later, however, the process of transforming Tot leaves the gradient\nin\nof ground elevation on the explicit side of eq. (6) imbedded in the\nJRK\nterm. Examination of the two parts of the pressure gradient term\ndy\nat a grid point near steep mountains shows they both are larger than any\nother term in eq . (6). Table 1 displays the size of these pressure-gradient\nparts and their sum in the lowest tropospheric layer of the model during a\nsemi-implicit forecast (1-hour time step). The sum amplifies with time\nand the implicit part seems to cause most of the increase, In only 11\nforecast hours, negative model pressures appear over the mountains and\nproduce a model failure.\nSince the gradient of model ground height and the gradient of model sur- -\nface pressure are of opposite sign, the magnitude of both parts of the\npressure-gradient term can be reduced by calculating both of them on the\nsame side of the equation, either explicitly or implicitly, rather than\nseparately. In order to disrupt the model formulated by Gerrity (1973)\nas little as possible, a redefinition of the pressure deviation, p\", is\nmade in the troposphere:\n(7)\n,\nwhere p is a surface pressure at the top of the model mountains obtained\nfrom the U.S. Standard Atmosphere. A11 parts of eq. (7) are functions of\n(x,y) and in is time invariant. Now redefine the implicit term, ap\nin\neq. (4):\n(8)\nMoving the time invariant quantity to the explicit side of eq. (4)\n9","Table 1. -- Implicit and explicit parts of pressure gradient term in\neq. (6) at one grid point (k = 6) (units in m/s)\nexpressed as effect on v2t/m.\nForecast\nPressure\nPressure\nTotal\nhour\ngradient\ngradient\npressure\n(implicit)\n(explicit)\ngradient\n+ 1.3\n1\n+ 14.5\n- 13.2\n2\n+ 14.7\n- 13.2\n+ 1.5\n+ 1.7\n3\n+ 14.8\n- 13.1\n4\n+ 14.9\n- 13.2\n+ 1.7\n5\n+ 15.0\n- 13.1\n+ 1.9\n6\n+ 15.0\n- 13.1\n+ 1.9\n7\n+ 15.2\n- 13.0\n+ 2.2\n8\n+ 15.4\n- 13.0\n+ 2.4\n9\n+ 15.8\n- 12.9\n+ 2.9\n+ 3.6\n10\n+ 16.3\n- 12.7\n11\n+ 16.9\n- 12.6\n+ 4.3\n10","one obtains\n(9)\ndy\nIn essence, the tropospheric basic state pressure is adjusted to\ndp\naccount for orography. Recalling that o-layer pressure thickness, do'\nis used rather than pressure, P, in the actual model, eq. (6) in the\ntroposphere (k = 4,5,6,7) becomes:\n(10)\nOf course, in a like manner there is a\nterm in the u-equation of motion.\nThis redefinition of the deviation part of the pressure variable and its\nproper splitting into implicit and explicit parts removed the amplifying\nmountain features. Successful semi-implicit forecasts using an hour time\nstep have been made beyond 48 hours. Examination of the two parts of the\npressure gradient term at one grid point in table 2 shows them to be an\norder of magnitude smaller with the above modification than with the old\nformulation (table 1). The implicit part, which seemed responsible for the\namplification, is now under control.\n4. CHANGES TO MODEL EQUATIONS\nThis section documents changes to the actual semi-implicit model equations\nthat are necessary to remove the mountain problem. Gerrity (1973) denotes\nsigma domain pressure thicknesses as II, so eq.(7) becomes\n(11)\n,\n11","Table 2. -- Implicit and explicit parts of pressure gradient term in\neq. (10) at one grid point (k = 6) (units in m/s)\nexpressed as effect on v2t /m.\nTotal\nPressure\nPressure\nForecast\nhour\ngradient\ngradient\npressure\n(implicit)\n(explicit)\ngradient\n+ .4\n- .7\n+ 1.1\n1\n+ .3\n- .8\n+ 1.1\n2\n+ .3\n.8\n+ 1.1\n3\n-\n+ .4\n- .7\n+ 1.1\n4\n+ .6\n- .6\n+ 1.2\n5\n+ .6\n.6\n+ 1.2\n6\n+ .6\n.6\n+ 1.2\n7\n+ .4\n.7\n+ 1.1\n8\n+ .4\n.7\n+ 1.1\n9\n+ .4\n- .7\n+ 1.1\n10\n+ .4\n.7\n+ 1.1\n11\n12","where k = 3 is the tropospheric sigma domain. Notice that (i) refers to\nbasic state variables, that the primes on the deviation parts are dropped\nand that the p notation for the standard atmosphere surface\npressure at the mountain tops is retained. Changed model equations are\npresented below, where equation numbers noted are those from Gerrity\n(1973) :\n1. (83) becomes\n>T-1\nThe -\n2. Eq. (116) becomes:\nT-1\nChanges must also be made to other equations that contain :\n3. Eq. (107) becomes:\nP -\n4. Eq. (112) becomes:\nKk3k -\n5. Eq. (114) becomes:\n6. Eq.(125) becomes:\n-\n13","Changes also have to be made to the Helmholtz equations, since the\ntropospheric pressure thickness, \"3\" on the implicit side of the equations\nhas been changed to \"3\" through eq. (11).\n7. Eq. (236) becomes:\n22, -2t W1 - -2t W2 , -2t W3 , -2t w4}\nACKNOWLEDGMENTS\nThe author wishes to thank Joseph Gerrity of the National Meteorological\nCenter for his invaluable contributions during this \"agonizing\" research.\nSpecial thanks to Mary Daigle for typing the report and to Tom Krzenski\nfor drafting the figures.\nREFERENCES\nCampana, K. A., , 1974: \"Status Report on a Semi-Implicit Version of the\nShuman-Hovermale Model.\" NOAA Technical Memorandum NWS-NMC-54.\nU.S. Dept. of Commerce, Washington, D.C., 22 pp.\nGerrity, J. P., , R. D. McPherson, and S. H. Scolnik, 1973: \"A Semi-Implicit\nVersion of the Shuman-Hovermale Model.\" NOAA Technical Memorandum\nNWS-NMC-53, U.S. Dept. of Commerce, Washington, D.C., 44 pp.\nKwizak, M. and A. J. Robert, 1971: \"A Semi-Implicit Scheme for Grid Point\nAtmospheric Models of the Primitive Equations.\" Monthly Weather\nReview, 99, pp. 32-36.\n14","(Continued from inside front cover)\nNOAA Technical Memorandums\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. (COM-71-00595)\nUpdating Asynoptic Data for Use in Objective Analysis. Armand J. Desmarais, December\nNWS NMC 51\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)\nNWS NMC 54\nStatus Report on a Semi-Implicit Version of the Shuman-Hovermale Model. Kenneth Campana,\nMarch 1974. (COM-74-11096/AS)\nNWS NMC 55\nAn Evaluation of the National Meteorological Center's Experimental Boundary Layer model.\nPaul D. Polger, December 1974. (COM-75-10267/AS)\nNWS NMC 56\nTheoretical and Experimental Comparison of Selected Time Integration Methods Applied to\nFour-Dimensional Data Assimilation. Ronald D. McPherson and Robert E. Kistler, April\n1975. (COM-75-10882/AS)\nNWS NMC 57\nA Test of the Impact of NOAA-2 VTPR Soundings on Operational Analyses and Forecasts.\nWilliam D. Bonner, Paul L. Lemar, Robert J. Van Haaren, Armand J. Desmarais, and Hugh M.\nO'Neil, February 1976. (PB-256-075)\nNWS NMC 58\nOperational-Type Analyses Derived Without Radiosonde Data from NIMBUS 5 and NOAA 2 Temp-\nerature Soundings. William D. Bonner, Robert van Haaren, and Christopher M. Hayden, March\n1976. (PB-256-099)\nNWS NMC 59\nDecomposition of a Wind Field on the Sphere. Clifford H. Dey and John A. Brown, Jr.\nApril 1976. (PB-265-422)\nNWS NMC 60\nThe LFM Model 1976: A Documentation. Joseph P. Gerrity, Jr., December 1977.\nNWS NMC 61\nSemi-Implicit Higher Order Version of the Shuman-Hovermale Model. Kenneth A. Campana,\nApril 1978.","NOAA CENTRAL LIBRARY\nCIRC QC851.U6 N5 no.62\nCampana, Ken Addition of orography to t\n3 8398 0002 4760 5\nNOAA SCIENTIFIC AND TECHNICAL PUBLICATIONS\nThe National Oceanic and Atmospheric Administration was established as part of the Department of\nCommerce on October 3, 1970. 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