{"Bibliographic":{"Title":"Semi-implicit higher order version of the Shuman-Hovermale model","Authors":"","Publication date":"1978","Publisher":""},"Administrative":{"Date created":"08-20-2023","Language":"English","Rights":"CC 0","Size":"0000056165"},"Pages":["85\nH\nQC\n851\nOF\nCOMMUNITY\nU6\nN5\nno.61\nNOAA Technical Memorandum NWS NMC-61\n*\n*\nSTATES\nOF\nSEMI-IMPLICIT - HIGHER ORDER VERSION OF\nTHE SHUMAN-HOVERMALE MODEL\nNational Meteorological Center\nWashington, D. C.\nApril 1978\nnoaa\nNATIONAL OCEANIC AND\nNational Weather\nATMOSPHERIC ADMINISTRATION\nService","National 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 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 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\n37\nNMC\n36\nSummary of Verification of Numerical Operational Tropical Cyclone Forecast Tracks for\n1965. March 1966. (PB-170-410)\nTN\n40 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. E. 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\n43 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 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 NMC 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)","H\nPC\n851\nU6\nN5\nno.61\nNOAA Technical Memorandum NWS NMC-61\nSEMI-IMPLICIT - HIGHER ORDER VERSION OF\nTHE SHUMAN-HOVERMALE MODEL\nCENTRAL\nLIBRARY\nKenneth A. Campana\nAUG 071978\nN.O.A.A.\nS. Dept. of Commerce\nNational Meteorological Center\nWashington, D. C.\nApril 1978\nAND ATMOSPHERIC\nAMOUNT NOAA\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 COMMUNITY\nOF\n78\n2829","","CONTENTS\n1\nAbstract\n1\n1. Introduction\n2\n2. Initial data\n5\n3. Model physics\n8\n4. Model stability\n9\n5. Higher order finite differencing\n11\n6. Results\n14\n7. Conclusions\n15\nAcknowledgments\n16\nReferences\nFIGURES\n1. -Location of forecast variables on the horizontal grid.\n2. ---Vertical structure of the model.\n3. - --Semi-implicit 500-mb heights, 48 hr from 0000 GMT Feb. 17,\n1977, with regular smoothing--contour - - interval 6 decameters.\n4. Semi-implicit 500-mb heights, 48 hr from 0000 GMT Feb. 17,\n1977, with one-half the regular smoothing - - contour interval\n6 decameters.\n5. Semi-implicit 500-mb heights, 84 hr from 0000 GMT Feb. 17,\n1977, with regular smoothing--contour interval 6 decameters.\n6. Semi-implicit 500-mb heights, 84 hr from 0000 GMT Feb. 17,\n1977, with one-half the regular smoothing - contour interval\n6 decameters.\n-Semi-implicit, fourth order, 500-mb heights, 168 hr from\n7.\n0000 GMT Feb. 17, 1977--contour interval 6 decameters.\niii","8. --E2 model, sea level pressure, 24 hr from 0000 GMT Oct. 8,\n976--contour interval 4 mb.\n9. --E4 model, sea level pressure, 24 hr from 0000 GMT Oct. 8,\n1976--contour interval 4 mb.\n10. --E2 model, 500-mb heights, 24 hr from 0000 GMT Oct. 8, 1976--\ncontour interval 6 decameters.\n11. --E4 model, 500-mb heights, 24 hr from 0000 GMT Oct. 8, 1976--\ncontour interval 6 decameters.\n12.--E2 model, sea level pressure, 48 hr--contour interval 4 mb.\n13.--E4 model, sea level pressure, 48 hr--contour interval 4 mb.\n14.--E2 model, 500-mb heights, 48 hr--contour interval 6 decameters.\n15.--E4 model, 500-mb heights, 48 hr--contour interval 6 decameters.\n16.--S4 model, sea level pressure, 24 hr from 0000 GMT Oct. 8, 1976--\ncontour interval 4 mb.\n17. --6L PE model, sea level pressure, 24 hr from 0000 GMT Oct. 8,\n6--contour interval 4 mb.\n18. - -NMC sea level pressure analysis, 0000 GMT Oct. 9, 1976,\n24-hr verification--contour interval 4 mb.\n19. --S4 model, 500-mb height, 24 hr from 0000 GMT Oct. 8,\n1976--contour interval 6 decameters.\n20.--6L PE model, 500-mb height, 24 hr from 0000 GMT Oct. 8,\n76--contour interval 6 decameters.\n21. -NMC 500-mb height analysis, 0000 GMT Oct. 9, 1976, 24-hr\nverification--contour interval 6 decameters.\n22.--S4 model, sea level pressure, 48 hr--contour interval 4 mb.\n23.--6L PE model, sea level pressure, 48 hr--contour interval\n4 mb.\n24. .--NMC sea level pressure analysis, 0000 GMT Oct. 10, 1976,\n48-hr verification--contour interval 4 mb.\n25. - -S4 model, 500-mb height, 48 hr--contour interval 6 decameters.\n26.--6L PE model, 500-mb height, 48 hr--contour interval 6\ndecameters.\niv","27. - -NMC 500-mb height analysis, 0000 GMT Oct. 10, 1976,\n48-hr verification--contour interval 6 decameters.\n28. - E4 model, 36-48 hr accumulated precipitation from 0000 GMT\nOct. 8, 1976--contour interval 1 centimeter.\n29. --S4 model, 36-48 hr accumulated precipitation from\n0000 GMT Oct. 8, 1976--contour interval 1 centimeter.\n30. --6L PE model, 36-48 hr accumulated precipitation from\n0000 GMT Oct. 8, 1976--contour interval 1 centimeter.\n31.--E4 and S4 models S1 scores (500 mb) at 24 hr and 48 hr for\n6 cases in October 1976. Each data point is from one of\nthree NMC verification areas in the Northern Hemisphere\n(North America, Europe, and Asia).\n32. -6L PE and E2 models S1 scores (sea level pressure) at 24 hr\nand 48 hr for 6 cases in October 1976.\n33. --6L PE and E2 models S1 scores (500 mb) at 24 hr and 48 hr\nfor 6 cases in October 1976.\n34. --6L PE and S4 models S1 scores (sea level pressure) at 24 hr\nand 48 hr for 6 cases in October 1976.\n35. --6L PE and S4 models S1 scores (500 mb) at 24 hr and 48 hr\nfor 6 cases in October 1976.\nV","SEMI-IMPLICIT HIGHER ORDER VERSION OF THE SHUMAN-HOVERMALE MODEL\nKenneth A. Campana\nDevelopment Division\nNational Meteorological Center, NWS, NOAA\nCamp Springs, Md.\nABSTRACT. The atmospheric numerical model\ndeveloped by Shuman and Hovermale at the National\nMeteorological Center has been reformulated for semi-\nimplicit integration. Orography, moisture, friction,\nand diabatic effects have been incorporated into this\nnew model. Higher order horizontal finite differencing\nis used in the advective terms of the model equations\nto reduce truncation error. Forecasts have been made\nfor a number of real data cases using a time step up\nto 45 minutes, and the semi-implicit technique has\nproven stable to at least 7 days. The model is com-\npetitive with the operational model at the National\nMeteorological Center in terms of both forecast accuracv\nand computation time efficiency.\n1. INTRODUCTION\nSemi-implicit integration schemes have been used in atmospheric models\nfor a number of years. The implicit treatment is a time-averaging\nprocess and is applied to the terms in the model equations governing\nthe fastest moving gravity waves. This acts to slow down these waves\nand permits a much longer time step to be used than is allowed in a\nfully explicit model. The computation time savings associated with\nthe longer time step makes the semi-implicit method attractive for\natmospheric models being used in operational forecasting environments.\nThe Shuman-Hovermale (1968) six-layer primitive equation model (6L PE)\nhas been used as an operational forecasting tool for several years\nat the National Meteorological Center (NMC). With the economic potential\nof the semi-implicit technique in mind, NMC began to experiment with the\n6L PE using this integration scheme. Formulation of the semi-implicit\nversion of the Shuman-Hovermale model was made by Gerritv, McPherson,\nand Scolnik (1973) and early testing with real data was reported by\nCampana (1974). The early experiments were feasibility studies con-\nducted with a simplified research model patterned after the 6L PE, but\nusing none of its physical parameterizations and having twice its grid\nspacing. Since that time, a great deal of effort has been expended in\nconstructing a version of the semi-implicit model which contains all\nthe features of the 6L PE.","The purpose of this report is to document the additions and changes\nmade to the original semi-implicit research model in order to bring it\nto its new state. No attempt is made to reproduce the multitude of\nequations and fine details found in the papers of Cerrity et al. (1973)\nand Campana (1974). Some of the model changes involve only computer\nprogramming techniques and are noted here simply for information\npurposes. A device that allows one to use \"variables\" rather than\nnumbers in FORTRAN dimension statements makes it easy to change arrav,\nand thus grid, sizes. Modular programming, using many small\nsubroutines, simplifies experimentation with various forms of\nthe physical parameterizations. Higher order horizontal finite\ndifferencing is feasible because the required number of data rows are\nalways in central memory of the computer. Finally, the ability to\nswitch the model from a semi-implicit mode to an explicit one, which\nwas part of the earlier research model, has been retained.\nThe details of the NMC semi-implicit model, which have not been dis-\ncussed elsewhere, will be described in the following sections. Section\n2 contains a description of the current method used to obtain initial\ndata for the model. Section 3 discusses the addition of the 6L PE\nphysical processes to the model, but will primarily concentrate on the\nmoisture equation. Section 4 shows the techniques emploved to produce\nstable forecasts beyond the 2-day range. Section 5 describes the\nhigher-order finite differencing on advection processes in the model.\nFinally, in section 6, some experimental forecasts are presented.\n2. INITIAL DATA\nModel variables are located on a forecast grid having the same\nhorizontal 1 and vertical resolution as the 6L PE. However, by horizon-\ntally staggering the wind components (u,v) with respect to the other\nmodel variables (fig. 1) the effective grid mesh length is reduced.\nIn fig. 2 the vertical dimension is shown--p is pressure, T is absolute\ntemperature, is the vector wind, is geopotential, q is specific\nhumidity, and & is vertical motion. The model's vertical structure is\na o system composed of four domains (of, 's' 'I'' OB) each of which\nvaries between zero and unity. The history variables are the pressure\nthickness of each o-domain, and the temperature, wind, and specific\nhumidity defined at the midpoint of each o-layer (specific humidity\nonly in the lowest four layers). Diagnostic variables such as and :\nare defined at the interfaces between o-layers. In fig. 2, PA is the\npressure thickness of the computational cap (k=1), PS is the pressure\nthickness of the stratosphere (k=2,3), and PT is the pressure thickness\nof the troposphere (k=4,5,6,7). The boundary layer (k=7) pressure thick-\nness, PC, is a constant 50 mbs.\n381 kilometers at 60°N on a polar stereographic map projection.\n2","Initial data for the model is obtained from the NMC analyses bv using\nthe current 6L PE procedure. Winds in o layers are obtained from a\nvertical interpolation of the NMC constant pressure wind analyses rather\nthan the Balance Equation, but otherwise the technique is as described\nby Shuman and Hovermale (1968). Problems arise when deriving temperatures\nfor the semi-implicit model via this 6L PE initialization scheme, because\nabsolute temperature (T), rather than potential temperature (0), is the\nforecast variable. The 6L PE hvdrostatic equation is\n(1)\nwhile for the semi-implicit model it is\nad - a a\n(2)\nwhere the specific volume a = (RT/p); IT is the Exner function,\n(p/1000)R/Cp; R is the gas constant; and cp is the specific heat at\nconstant pressure (the geopotential ( o ) is defined at o-interfaces).\nEqn. (2) is rewritten as\ndo\n2(en p) = = RT\n(3)\nor using\n(4)\nT=TT\nit becomes\ndo\n(5)\n= - RTO,\n(ln p)\nwhere T is the Exner function defined in o-layers. Both models should\nhave the same initial distribution of pressure and geopotential\n(and thus temperature), but 0 solutions to the finite-\ndifference forms of eqns. (1) and (5) will differ markedly in the\nupper atmosphere above 500 mb. To assure the same thermodynamic\nstructure in both the 6L PE and semi-implicit models, the definition\nof TT in eqns. (4) and (5) is changed. Solving eqns. (1) and (5) for\nTT one obtains (in finite-difference form):\n(6)\nwhere A denotes a vertical difference. Note that TT on the right side\nof eqn. (6) is still the Exner function at o-interfaces. This new\ndefinition of TT is then used with eqn. (4) whenever there is any\nconversion between absolute temperature and potential temperature.\n3","The method used to obtain initial data for the computational cap has\nalso been changed. In the operational 6L PE scheme, a complex pro-\ncedure is employed which insures an initial balance between the wind\nand mass fields in this topmost model layer. The initial wind field\nis one of no motion and the procedure produces a pressure and geo-\npotential distribution which has a zero pressure gradient everywhere;\n(7)\nThe potential temperature, cap, is a constant in both space and time.\nThe computational cap pressure gradient in the semi-implicit model,\nhowever, is\n(8)\nThe cap pressure thickness is set to a constant 100 mb at all grid\npoints, thus making the second term in eqn. (8) initially zero.\nTherefore, to obtain an initial balance between wind and mass fields,\none need only look at the geopotential term. Defining acap to be\nconstant in space and time and using eqn. (2), one sees that the\nterm is identical at top and bottom of the cap. Since the initial\nwinds at 100 mb are constrained to be in at least geostrophic balance\nby the NMC analysis scheme, use of these winds as initial data for the\ncap should result in a balanced state. Thus, for the cap (k = 1 in\nfig. 2)\nPe = 100 mb\nU1 = u at 100 mb\nV1 = V at 100 mb.\nThe constant specific volume, acap' is defined by the Equation of State\nbelow:\nacap = RT/P\nwhere T = 250°K1 and P = 50 mb.\n1A temperature below 200°K produced a model failure during a forecast while\na temperature over 400°K produced a large temporal oscillation in the model\nStratosphere.\n4","Several other changes have been made to the 6L PE initialization\nscheme and they are noted below:\n1. The tropopause (o-interface above layer 4 in fig. 2) is redefined\nso that its pressure is never less than 200 mb. This insures an\ninitial stratospheric pressure thickness of at least 100 mb. During\nthe forecast cycle this constraint is removed.\n2. No initial moisture is available for the topmost tropo-\nspheric layer, so none is defined (q1 III 0). Any moisture accumulating\nthere during a forecast comes from vertical advection processes.\n3. The NMC wind analyses produce u, components at grid point\nlocations, whereas the semi-implicit model needs them at grid box\nlocations ; (see fig. 1). Wind data are interpolated horizontally from\nthe grid points using a biquadratic scheme--a Bessel interpolation\nformula with second-order terms (Saucier 1955, p. 121).\n4. Orography is now part of the semi-implicit model. Necessary\nchanges to the original method of splitting the orographic terms into\nimplicit and explicit parts have been made by Campana (1978). The 6L PE\nmountain field is used, but it is filtered horizontally using Shuman's\n(1957) nine-point smoother. Unfiltered mountains produced model\nfailures in both semi-implicit and explicit versions of the new model\nafter 1 1/2 forecast days. These difficulties were noise problems and were\nnot related to those documented by Campana (1978).\n3. MODEL PHYSICS\nAll physical processes contained in the 6L PE are included in the\nsemi-implicit model. Since they have been discussed by Shuman and\nHovermale (1968), fine details are omitted from this\nreport. Modifications are made to the semi-implicit model in order to\nuse existing FORTRAN coded algorithms from the 6L PE model. Absolute\ntemperature is converted to potential temperature, via eans. (4) and (6),\nin order to use the existing dry and moist convective adjustment schemes.\nAfter the adjustment, the reverse conversion occurs, from 0 to T.\nEqns. (4) and (6) are also used to obtain long-wave cooling rates for\nthe new model that are equivalent to those of the 6L PE.\nThe primary purpose of this section is to document the addition of a\nmoisture variable to the semi-implicit model. Specific humidity, a,\nis used and is defined in the four o-layers below the model\ntropopause (fig. 2). In order to use the precipitation, latent\nheating, and solar absorption (due to moisture) algorithms from the\n6L PE, specific humidity is converted temporarily to precipitable water,\nPWAT:\nPWAT = do do ,\n(9)\nwhere g is gravity.\n5","The moisture equation (10) and its finite difference forms are pre-\nsented below in order to complete the set of model equations given bv\nGerrity et al. (1973) :\n(10)\n+ CE,\nwhere the symbols are:\nt = time\nx,y = horizontal coordinates\no = vertical coordinate\nu, = horizontal wind components\n& = vertical wind component\nm = map factor\nCE = influence of condensation and evaporation (as in 6L PE)\nSince moisture influences the model through latent heating from the\nprecipitation algorithm, which in turn is part of the explicitly cal-\nculated terms in the thermodynamic equation, eqn. (10) is completelv\nexplicit (i.e., no semi-implicit parts). Initial fears concerning\nexcess precipitation associated with the long time step in the semi-\nimplicit model have proven unfounded.\nThe most difficult term to compute in eqn. (10) is the vertical\nadvection term, & aq/do. The evaluation of this term at\nthe top of the boundary layer is quite complex. A formulation exactly\nthe same as described in Gerrity et al. (1973, section 3.5) is used,\nwhere the 2q/20 term at this interface is defined below for both the\ntroposphere (subscript T) and the boundary laver (subscript B).\n(11)\n(12)\n20B\nPc\n(13)\n6","Note that PT and Pc are the tropospheric and boundary layer pressure\nthicknesses, respectively ; [see Gerrity et al. 1973, eqns. (129) and\n(130)]. The following equation is useful when discussing the specifics\nof the moisture equation. It relates the two definitions of & at the\ntop of the boundary layer which are valid either in the troposphere\n(87) or in the boundary layer (8B)--see eqn. (134) Gerrity et al. (1973) :\n(14)\nThe moisture equation in the troposphere (k = 1,2,3 for a) becomes\n(15)\nM1 = 3/2 05(9)- 9, =\n(16)\n(17)\n+ 6\n(18)\nSince the vertical advection, & 2q/20, is computed at o-interfaces,\nthe \"1\"2\" in the Mk term comes from vertically averaging & 2q/20 to the\no-layers.\nIn the boundary layer (k = 4 for q), eqn. (10) is:\n(19)\nCE\n(20)\nwhere OB is defined in eqn. (14) and\n(21)\nThe finite difference forms of the horizontal and vertical advection\nterms in eqns. (15) and (19) are consistent with the other model\nequations. The reader's attention is drawn to Shuman and Hovermale\n(1968) where the finite-difference notation used below is defined.\nNoting that overbars, (T), refer to averaging processes along\n7","specified grid axes, representative terms are calculated as follows:\nxy\ndq\n-X\n(22)\nIx\nthe\n+\nu\nV\nxy\nM1 3/2 (92-91)\n(23)\nCareful consideration of where each variable is located on the grid\nshows that the above finite differences are valid at the same locations\nas q.\n4. MODEL STABILITY\nTechniques used to control computational noise that develops during\na 6L PE forecast are also included in the semi-implicit model. They\nare quite effective in damping noise and allowing the forecast to\nproceed reliably beyond 48 hours.\nThe first tool employed is a time filter of the form used by Asselin\n(1972) which controls high frequency noise developing during a model\nforecast. The filter is shown in eqn. (24) and is applied at each time\nstep to the u and V wind components.\n(24)\n=\nwhere E = 0.3 and T-1,T, T+1 superscripts refer to past, present, and\nfuture time levels. After obtaining T+1 variables, eqn. (24) is solved\nfor the time filtered variable, ( ) *. This new quantity is used during\nthe succeeding forecast cycle when it becomes the T-1 variable.\nLimited testing with a larger E, meaning less filtering, allowed too\nmuch computational noise to develop in the wind fields.\nThe second technique controls high wave number noise in space. It\nis a Shuman (1957) smoother-desmoother and is applied to pressure,\ntemperature, and moisture variables at each time step. These spatially\nsmoothed quantities become the T-1 variables in the succeeding forecast\nstep. In the 6L PE it is applied to only pressure and temperature;\nthe smoothing coefficient, v, has a value of .022 with a 10-minute step.\nNoting that the strength of the filter depends on the number of times\nthat it is applied, as well as the size of V, the semi-implicit model\ncoefficient is approximately 21/2 times stronger than that used in the\n1\n6L PE.\n1 The coefficient, v, would be equivalent to 0.058 in the 6L PE.\n8","An experiment with a filter half as strong, and thus similar to\nthe 6L PE, produced results in which meteorological features differed\nlittle from the heavier filter test. At 48 hours, the 500-mb\nheight forecasts are virtually identical (figs. 3 and 4). By 84 hours,\nhowever, there are some small differences in the intensity of\nmeteorological features, but more importantly, computational noise\nappears in the weaker filter test (figs. 5 and 6) Thus, the heavier\nsmoother is beneficial to the model without adversely affecting its\nforecasts of meteorological scale features.\nAnother device is used to control problems in the model stratosphere.\nThe semi-implicit model constrains the stratosphere to be at least\n50 mb thick during a forecast by \"adding mass\" from the model tropo-\nsphere at offending grid points. Whenever this occurs the wind field\nhas become noisy, so the winds are mixed vertically and horizontally.\nNoting that the u,v wind components are staggered with respect to\nthe other variables, the four u,v winds surrounding the offending\nstratospheric pressure are averaged to the grid point and mixed verti-\ncally as in the 6L PE. The vertically mixed wind is inserted back\ninto the four surrounding u,v locations in each o-layer which partici-\npated in the vertical mixing. The 6L PE uses 6 mb as its limiting\npressure thickness, but by the time the stratosphere has become that\nthin, all other variables must be rather nonmeteorological. Reflecting\nthe stability of the semi-implicit model, this stratospheric device\nis rarely needed during a forecast.\nBoth the time and space filters act to reduce the maximum allowable\ntime step of the model, but they have helped create a stable\nforecast model. 1 In the only attempt at \"long-range\" forecasting with\nthe semi-implicit model, forecasts were made out to 168 hours from 0000 GMT\n17 February 1977. Although necessarily smooth, the large-scale\nmeteorological features appear quite realistic (fig. 7). No verifi-\ncation is presented for this forecast, however.\n5. HIGHER ORDER FINITE DIFFERENCING\nAlthough not related to the semi-implicit method, higher order\n(horizontal) finite differencing has been quite successful in the new\nmodel. Thus it is deemed to be of sufficient interest to be included\nin this report. Its use in the model is based on the paper by Gerrity,\nMcPherson, and Polger (1972) There the conclusion was that higher\norder, more accurate, forms of Shuman's (1968) semimomentum finite-\ndifferencing for the advective processes would reduce truncation error\nin a grid point model while requiring only a slight decrease in time\nstep length. The reduction in truncation error would result in\n1A backward implicit time differencing technique used in conjunction\nwith the time filter is an alternative method for controlling non-\nmeteorological noise in the model. Little experimentation has been\ndone with this technique.\n9","improved translation speeds of meteorological-scale features. Often\nthe major difference between the 6L PE and its finer mesh counterpart,\nthe LFM¹, is the speed with which major features are moved during a\nforecast--the LFM being more accurate. Higher order differencing\ncould make the 6L PE more competitive with the LFM in these cases. Of\ncourse, in situations where the resolution of the finer mesh is\nnecessary to delineate important small-scale features in the initial\ndata and during the forecast, the LFM probably will be superior.\nA paper by Gerrity (1973) shows the higher-order scheme that is used\nwith the semi-implicit model. These fourth-order2 differences are\napplied to the advective terms in all model equations; although near\nmodel boundaries, where fewer grid points are available, second-order\napproximations are used. The derivation of the higher-order finite-\ndifference forms is not presented here; however, Gerrity's (1973)\neqns. (14) and (15) are shown below. Fourth-order averaging in one\ndimension is\nh9/8)- 1/8)3x\n(25)\n,\nwhile fourth-order differencing is\n(26)\nwhere T.()x Shuman's regular second-order approximations and\n(33x, ( ) 3x are derived from them.\nThe fourth-order semi-implicit model's finite difference form for\nthe advection terms are shown below. Eqn. (22) from section 3 becomes\n(27)\nThe advection terms in the equations of motion are handled somewhat\ndifferently due to the staggering of variables\n(28)\nLimited-area fine-mesh model-- distance of 190.5 km at 60°N on\na polar stereographic map projection.\n2The order of the scheme refers to the exponential power of the grid\ninterval in the leading term of those omitted from a series expansion\nof the finite-difference form--the higher the order, the more terms are\nretained.\n10","Note that the advecting wind in eqn. (28) contains second-order\naveraging. Although all results documented in this report use this\nsecond-order approximation on the advecting wind, a fourth-order form\nis more consistent with the finite differencing. 1 For future testing\nof this model, changes have been made that use fourth-order averaging\non the advecting wind.\n6. RESULTS\nA number of experiments have been made using the semi-implicit\nversion of the Shuman-Hovermale model at NMC. Some of the results\nfrom six case studies made from initial data for October 1976 are\npresented in this section. Pressure level data for the semi-implicit\nmodel are interpolated from o data in exactly the same manner as for\nthe 6L PE. This has necessitated an averaging of the u,v winds to the\ngrid points and a resultant small loss in amplitude for any quantity\nderived from these velocity fields. In order to keep the various\nversions of the new model from confusing the reader, the following\nabbreviations are defined:\nS4 - semi-implicit,\nfourth-order finite differencing\nS2 - semi-implicit,\nsecond-order finite differencing\nE4 - explicit version, fourth-order finite differencing\nE2 - explicit version, second-order finite differencing\nThe main advantage of the semi-implicit method is the longer time\nstep permitted and the associated savings in computation time. This\nsaving is partially offset by having to solve a set of Helmholtz\nequations for each time step. Using an iterative process described by\nSela and Scolnik (1972), these equations are transformed so that each\none can be solved independently by a two-dimensional Liebmann relaxation\nscheme. When using this method on the IBM 360/195 system, double pre-\ncision arithmetic (64-bit word) is required for the transformation\nprocess and the relaxation scheme. Each independent equation provides\none of the model's pressure thicknesses (P0, PS' PT) or one of the non-\nzero vertical motions (8) Each equation has a separate over-relaxation\ncoefficient in order to speed up the computations. The convergence\ncriteria are for the transformed variables and translate into approxi-\nmately 10-2 mb for the pressure thicknesses and 10-9 mb/sec for & .\nIn practice, the relaxation calculations account for about 10% of the\ntotal computation time (the transformation process takes an additional\n5%).\n1 Several comparisons of 48-hr forecasts made using second or\nfourth-order averaging showed little difference.\n11","No experiments have been made to determine the maximum allowable\ntime step for either the S2 or S4 models, but the S2 model has safely\nused a 45-minute step. During about half the calendar year, a 45-\nminute step has been used with the S4 model; however, in winter months\nwhen the wind speeds are higher a shorter step is needed. For convenience,\n30 minutes is used. Table 1 shows some approximate computation times\nfor 48-hour forecasts using different versions of the new model. This\ndoes not include the \"wait\" time involved with reading from or writing\nto disk.\nTable 1. Approximate computation times for 48-hour forecasts.\nTime for 48-hr\nTime\nForecast (min)\nStep\nModel\n45\nS2\n9.25\n45\nS4\n10.00\n30\nS4\n13.75\n7.5\nE4\n36.00\nNote from the first two entries in the table that the more complex\ncalculations used in the higher order finite differences for the S4\nmodel increase the computation time by about 10%. Comparing the E4\nmodel with its time step of 7.5 minutes and the S4 model with a time\nstep of 45 minutes, there is almost a 4:1 ratio in computation time.\nIf the more expensive radiation calculations were made every time\nstep in the E4 model, as is done in the S4 model, then a 4:1 ratio\nwould easily be attained.\nSince the higher order finite difference scheme is chosen to be used\nwith the semi-implicit model, it is instructive to show some results\nthat led to this decision. Sea level pressure and 500-mb height fore-\ncasts for 24 and 48 hours are shown in figs. 8 to 15. The comparisons are\nbetween the E2 model and the more accurate E4 model for a case (0000 GMT\nOctober 8, 1976) in which the 6L PE produced a poor 48-hour forecast\nover North America. At 24 hours (figs. 8 to 11), there is little\ndifference between the E2 and E4 models; however, by 48 hours (figs. 12 to\n15) the fourth-order scheme has resulted in faster motion for the\nmeteorological features primarily at 500 mb. The faster movement of\nthe ridge-trough pattern over the United States at 48 hours verifies\nbetter against the analysis (fig. 27) than the E2 model forecast. While\nother test cases are not described in this report, indications from them\nare that the faster translational speeds due to the higher order scheme\nare more pronounced at higher atmospheric levels where wind speeds are\nstronger.\nIt seems that in every paper on the semi-implicit method in the\nliterature, the point is made that there is little difference between\nexplicit and semi-implicit versions of the same model. In keeping with\n12","tradition, this author will do likewise. Comparison is made between\nthe fourth-order models S4 and E4 for the October 8, 1976 case. Figures\n13 and 22 show 48-hour forecasts of sea level pressure; figs. 15 and\n25 show 500-mb height forecasts; and figs. 28 and 29 show 36-48 hour\naccumulated precipitation (centimeters) for the E4 and S4 models,\nrespectively. The explicit version is somewhat noisier in the tropical\nregions and its stronger vertical motion probably accounts for the\nprecipitation differences, but overall the forecasts for the two models\nare virtually identical. This is brought out more dramatically in\nfig. 31. It is a graph of S1 scores at 500 mb for the six test cases\nin October 1976. Twenty-four and 48-hour scores in three NMC\nverification areas over the Northern Hemisphere are shown. Each point\non the graph gives the appropriate S1 score for the E4 (abscissa) and\nS4\n(ordinate) models. The models are equivalent in S1 score if the\npoints fall on the 45° diagonal. As can be seen, the models are\nvirtually identical at 500 mb for both forecast hours.\nOne might speculate how the new model compares with the 6L PE.\nSince the 6L PE has second-order finite differencing, the fairest com-\nparison is with the E2 model. However, one should expect some difference\nsimply because the effective grid length of the E2 model is smaller due\nto the staggering of forecast variables. The time step for the\nE2 model is only 7.5 minutes compared with 10 minutes for the\n6L PE. Differences in orography, initial winds, and smoothing coefficients\nwill also contribute to differences in the results. Forecasts for the\nOctober 8 case are shown in figs. 8, 10, 12, and 14 for the E2 model and\nin figs. 17, 20, 23, and 26 for the 6L PE. The reader will be able to\nmake more comparisons than will be put into this report, but several\nconclusions may be drawn. First, the E2 forecasts appear smoother than\nthe 6L PE--compare 24-hour sea level forecasts in the Gulf of Alaska in\nfigs. 8 and 17. Second, at 48 hours the E2 forecasts have moved some\nfeatures faster (and more correctly) than the 6L PE--compare the sea-\nlevel low pressure north of the Caspian Sea in figs. 12 and 23, and\ncompare the lower part of the 500-mb trough over southeastern United\nStates in figs. 14 and 26. Model comparisons using graphs of S1 scores\nat sea level and 500 mb are shown in figs. 32 and 33. At sea level\nthere may be a slight advantage given to the E2 model by 48 hours;\nhowever, at 500 mb the E2 model is clearly superior to the 6L PE. These\nresults may be due, in part, to the smoother E2 forecasts.\nThe most competitive model with respect to the 6L PE is the S4 version--\nboth in terms of computational speed (semi-implicit) and in terms of\naccuracy (fourth-order differencing). Forecasts for the S4 model are\nshown in figs. 16, 19, 22, and 25, and they can be compared with those\nof the 6L PE (figs. 17, 20, 23, and 26) or the E2 model (figs. 8, 10,\n12, and 14). Verification charts are shown in figs. 18, 21, 24, and 27,\nwhile the 12-hour accumulated precipitation for the 36- to 48-hour period\n13","is shown in fig. 29 for the S4 model and in fig. 30 for the 6L PE.\nThe same conclusions can be drawn about the comparison between the\ntwo models as was done above for the E2 model. The S4 forecasts appear\nsmoother than those of the 6L PE, and translation speeds of meteoro-\nlogical systems are faster. The potential for better numerical fore-\ncasts using a more accurate higher order scheme is shown in the 48-hour\n500-mb map over the eastern United States (compare figs. 25 and 26 with\nfig. 27). While some of this good forecast can be attributed to the\nnew model (E2 shows improvement over the 6L PE), much of it results\nfrom the higher order scheme (compare fig. 25 with fig. 14). Graphs\nof S1 scores for the S4 model and the 6L PE are shown in figs. 34 and\n35. While at sea level there is no clear victor, the S4 model is\nsuperior at 500 mb in most instances.\n7. CONCLUSIONS\nThe Shuman-Hovermale numerical model at NMC has been reformulated\nsuccessfully for semi-implicit time integration. Tests have been made\nwith various versions of the new model and comparisons have been made\nusing forecast results. Several conclusions are noted below:\n1) Semi-implicit and explicit versions of the new model produce\nvirtually identical forecasts. The vertical motion in the semi-implicit\ntests has less amplitude than in the explicit runs and this appears\nas less noise on pressure level maps. This does not make the comparable\nforecasts any less identical.\n2) The new model in its second-order form is superior to the 6L PE.\nSince both models have virtually the same initial data and produce\npressure level forecast maps in an identical manner, the superiority\nof the new model seems due its reduction of effective grid length by\nstaggering variables horizontally. However, there may be some loss\nof detail in the new model due to the heavier smoothing.\n3) Higher-order finite differencing on the horizontal advection\nterms in the model has increased the translation speeds of some\nmeteorological-scale features--usually in a more correct sense. The effect\nseems more pronounced at higher atmospheric levels where wind speeds are\nstronger, and at longer forecast times when the cumulative effect of\nthe greater accuracy becomes more apparent.\nNo plans have been made to use the new model as an operational\ntool, even though it produces forecasts that are generally superior\nto those of the 6L PE. It requires about the same amount of computer\nresources as the operational model, because it uses the same horizontal\ngrid mesh; however, its computation efficiency is not impressive when\ncompared with the highly optimized 6L PE. of course, the semi-implicit\n14","version of the new model is quite efficient when compared with its\nexplicit counterpart (table 1), but the crucial computation time\nin the NMC operational environment is the one which includes \"wait\"\ntime associated with input/output operations. This \"wall\" time\nis approximately 24 minutes for a 48-hour 6L PE forecast. Some\nwork has been done to optimize the new model, but the best wall\ntimes for the semi-implicit versions are approximately twice those\ntimes listed in table 1--that is, about 19 minutes for the S2 model\nand about 28 minutes for the S4 model (30-minute time step). The\nquestion of computational efficiency (i.e., wall time) will need to\nbe resolved if the new model is to become useful operationally.\nIf the S4 model with its more accurate finite differencing\nproduces forecasts that are competitive with the finer mesh version\nof the 6L PE currently being tested, 1 then this question of efficiency\nrecedes into the background.\nACKNOWLEDGMENTS\nThe author wishes to thank Joseph Gerrity for his important contri-\nbution toward incorporating orography into the model; also Bill Collins\nfor his much needed assistance in obtaining initial temperature data\nin the model that is equivalent to the 6L PE. Finally, special thanks\nto Mary Daigle for typing the report, to Bob van Haaren for producing\nthe S1 scores, and to Tom Krzenski for drafting some of the figures.\n1 Wall time over an hour for a 48-hour forecast.\n15","REFERENCES\nAsselin, Richard A., \"Frequency filter for time integrations,\"\n\"\nMonthly Weather Review, 100, 1972, pp. 487-490.\nCampana, Kenneth A., \"Status report on a semi-implicit version of\nthe Shuman-Hovermale model,\" NOAA Technical Memorandum NWS-NMC 54,\n1974, 22 pp.\nCampana, Kenneth A., \"Addition of orography to the semi-implicit\nversion of the Shuman-Hovermale model,\" to be published as a NOAA\nTechnical Memorandum in 1978.\nGerrity, Joseph P. McPherson, Ronald D., and Polger, Paul D., \"On\nthe efficient reduction of truncation error in numerical weather\nprediction models,\" Monthly Weather Review, 100, 1972, pp. 637-643.\nGerrity, Joseph P., \"Numerical advection experiments with higher order,\naccurate, semi-momentum approximations,\" Monthly Weather Review,\n101, 1973, pp. 231-234.\nGerrity, Joseph P., , McPherson, Ronald D., and Scolnik, Stephen, \"A\nsemi-implicit version of the Shuman-Hovermale model,\" NOAA Technical\nMemorandum NWS-NMC 53, 1973, 44 pp.\nSaucier, Walter J., , Principles of meteorological analysis, University\nof Chicago Press, 1955, 438 pp.\nSela, Joseph, and Scolnik, Stephen, \"Method for solving simultaneous\nHelmholtz equations,\" Monthly Weather Review, 100, 1972, pp. 644-645.\nShuman, Frederick G., \"Numerical methods in weather prediction,\nII. Smoothing and filtering,\" Monthly Weather Review, 85, 1957,\npp. 357-361.\nShuman, Frederick G., and Hovermale, John B. , \"An operational six-layer\nprimitive equation model,\" Journal of Applied Meteorology, 7, 1968,\npp. 525-547.\n16","x\nu,v\nP,O,T,a\np,o,\nFigure 1. -- Location of forecast variables on the horizontal grid.\nk\n81=0,01\nCAP\nVI,T1\n1\nto\n=\n2\nV2, I2\ns = =\n03033\n3\n's = 3/4\n3 T 3\nSTRATOSPHERE\n04=0,04\n= Ps\n4\nV4,4,4,\n859 $5\nT = 1/3\nTROPOSHERE\n°T=\n5\n069 $6\nOT = 213\nT6,93\n079°7\nOT=5%\nThe\nFigure 2. Vertical structure of the model.\n17","110\n9/0\n1203\n80\nW\n130\n70\n140\n60\n0\n50\n150\nx\n40\n160\n30\n170\n20\n180\n2\n498\nWr\n534\n10\n170\n510\n0\n160\n492\n582\n10\n150\nL\n522\n540\n20\n140\n588\n570\n30\n130\n582\nx\n120\n40\n110\n50\n100\n60\n90\n70\n80\nFigure 3. . --Semi-implicit - 500-mb heights, 48 hr from 0000 GMT Feb. 17, 1977,\nwith regular smoothing--contour interval 6 decameters.\n18","100\n10\n90\n120\n80\n&\n130\n70\n140\n60\n150\n50\n160\nor\n40\n170\n30\n180\n20\n2\n498\nWILL\n534\n170\n510t\n10\n160\n0\nx\n492\n10\n582\n150\n10\n522\n140\nD\n20\n558 E\n130\n58.2\n30\n588\n120\n40\n110\n50\n100\n60\n90\n70\n80\nFigure 4. . - --Semi-implicit 500-mb heights, 48 hr from 0000 GMT Feb. 17, 1977,\nwith one-half the regular smoothing--contour interval 6 decameters.\n19","110\n9/0\n:\n120\n&0\n&\n70\n130\n140\n60\n0\n150\n50\n160\n40\n70\n30\n180\n498\n20\nwith\na\n510\nx\n170\n10\n522\n.\n522\n3\nx\n160\n0\n498\n588\n150\n10\n522\nx\na\n140\nQ\n20\n552\nx\n130\n30\n576\nx\n120\n40\n110\n50\n100\n60\n9/0\n70\n80\nT\nFigure 5. . --Semi-implicit - 500-mb heights, 84 hr from 0000 GMT Feb. 17, 1977,\nwith regular smoothing--contour -- interval 6 decameters c\n20","100\n90\n120\n80\n70\nL\n1\n0\n60\n150\n50\nx\n0\n160\n40\n0\n120\n30\n180\n498\n20\nO\n510\na\nO\n170\n10\n522\n522\n160\n0\n504\n150\n10\n522\nMA\n594\nx\n140\n20\n570\n130\n30\nx\n582\no:\n20\n40\n110\n50\n100\n60\n9/0\n70\n80\nFigure 6.--Semi-implicit 500-mb heights, , 84 hr from 0000 GMT Feb. 17, , 1977,\nwith one-half the regular smoothing--contour interval 6 decameters,\n21","G\n110\n9/0\n:\n80\n120s\nW\n70\n130\n60\n140\n50\n150\n40\n160\n30\n170\n582\n510\n20\n180\n504\n10\n170\no\n8\n0\n510\n160\n0\n10\n150\n528\nx\nXX\n&\n20\nx\n140\n588\nx\n30\n130\n582\nx\n40\n120\n50\n110\n60\n100\n70\n9/0\n80\nFigure 7. --Semi-implicit, fourth order, , 500-mb heights, 168 hr from 0000 GMT\nFeb. 17, 1977-- contour interval 6 decameters\n22","110\n90\n120\n80\n130\n70\n140\n60\n150\n50\n160\n40\n032\n170\n30\n180\n20\n000\na\n020\n170\n10\n000\nn\n988\n020\nTOUTER\n160\n0\na\n008\n150\n10\ns\n012\nath\n140\n20\n020\n024\nK\n130\n30\n0\n120\n40\n012\n110\n50\n90\n100\n60\n90\n70\n80\nFigure 8. --E2 -- model, , sea level pressure, 24 hr from 0000 GMT Oct. 8, 1976--\ncontour interval 4 mb.\n23","100\n110\n90\n80\n120\n70\n130\nother\n60\n140\n50\n0\n150\n40\nor\n160\n032\n30\n170\nas\n20\n180\n000\na\n020\n10\n170\no\n000\n5\nwas\n988\nx\n020\nYOUTH\n0\n160\n004\nx\no\n008\nV\n&\n10\n150\n012\nfor\n20\n140\n020\n024\n30\n130\n0\n40\n120\n012\n50\n110\n100\n60\n70\n90\n80\n1\nFigure 9. - -E4 model, sea level pressure, 24 hr from 0000 GMT Oct. . 8, 1976--\ncontour interval 4 mb.\n24","100\n110\n90\n120\n&0\n130\n70\n60\n140\n588\n50\n150\n40\n160\n30\n170\n&\n20\n180\n516\nIt\n540\n4\n534\n170\n10\nD\n4\n504\n160\n0\n564\n540\n150\n10\n588\n588\n570\n20\n140\n30\n130\n0\n120\n40\n110\n50\nor\n100\n60\n90\n70\n80\nFigure 10. -- E2 model, 500-mb heights, 24 hr from 0000 GMT Oct. 8, 1976--\ncontour interval 6 decameters,\n25","100\n110\n90\n1203\n0\n70\n130\n60\n140\n588\n50\n150\n40\n160\n30\n170\n20\n180\n516\n540\nn\n534\n10\n170\no\n504\n402\n0\n160\n$564\n540\n10\n150\n588\n588\n20\n140\n570\n30\n130\n0\n40\n120\n50\n110\non\n100\n60\n90\n70\n80\nFigure 11. . - - E4 model, 500-mb heights, 24 hr from 0000 GMT Oct. 8, 1976--\ncontour interval 6 decameters.\n26","100\n1\n10\n90\n120\n80\n130\n70\n60\n140\n150\n50\n40\n160\n032\n008\n30\n170\na\n996\n016\n20\n180\n020\n020\na\n10\n170\nDO\n0\n100\n988\n0\n160\n5\n000\nC\n008\n020\n012\nm\n10\n150\n008\n€\n020\n020\n20\n140\n0\n130\n40\n120\n50\n110\nor\n100\n60\n90\n70\n80\n1\nFigure 12.--E2 model, , sea level pressure, 48 hr--contour interval 4 mb .\n27","100\n10\n90\n80\n120\n70\n130\n60\n140\n50\n150\n0\n40\nor\n160\n028\n30\n170\n008\n996\n016\n20\n180\n020\n016\n020\n10\n170\n**\n0\n0\n996\n988\n0\n160\na\nB\n008\n012\n020\n10\n150\n008)\n020\n20\n140\n020\n012\n0\n30\n130\n40\n120\n50\n110\n100\n60\n90\n70\n80\n7\nFigure 13. - E4 model, , sea level pressure, 48 hr--contour interval 4 mb.\n28","100\n110\n90\n120\n80\nW\n130\n70\n140\n60\n588\n150\n0\n50\n160\n40\n30\n170\na\n180\n20\n534\n516\n170\n10\n546\n26\nb\n510\n&\nx\n160\n510\n0\n588\n540\n150\n10\n564\n140\n20\n588\n130\n30\n0\n120\n40\n110\n50\n100\n60\n90\n70\n80\nFigure 14. . - E2 model, 500-mb heights, 48 hr--contour interval 6 decameters.\n29","100\n110\n90\n80\n120\n70\n130\n60\n140\n50\n588\n150\n40\n160\n30\n170\n20\n180\n534\n516\nof\n10\n170\n546\n&\nb\n510\n510\n0\n160\no\n588\n540\n10\n150\n20\n140\n564\n30\n130\n582\n0\n40\n120\n50\n110\n100\n60\n90\n70\n80\nFigure 15. -- E4 model, , 500-mb heights, 48 hr--contour interval 6 decameters.\n30","100\n110\n90\n120\n80\n130\n70\n140\n60\n150\n50\n160\n40\n032\n170\n30\n020\n180\n20\n000\n020\n170\n10\n1/\n000\nand\n1988\n004\n160\n0\n&\n008\n,\n016\n$\n150\n10\n012\n140\n20\n024\n020\n012\n130\n30\n0\n120\n40\n012\n110\n50\n100\n60\n90\n70\n80\nFigure 16. . - - S4 model, sea level pressure, 24 hr from 0000 GMT Oct. 8, 1976--\ncontour interval 4 mb.\n31","10\n910\n:\n8/0\n1203\n8\n70\n130\n60\n140\n50\n150\n0\n&\n012\n160\nIs\na\n036\n30\n170\n008\n20\n180\n1:\n024\n10\n170\n.\n020\n000 4\n1\n992\n008\n0\n160\nX\n004\n008\n&\n10\n150\nD\n020\n20\n140\nf\n024\n012\n30\n30\nx\n40\n120\n110\n50\n100\n6NO\n9/0\n70\n80\nFigure 17. . -- --6L PE model , sea level pressure, 24 hr from 0000 GMT Oct. . 8,1976--\ncontour interval 4 mb. a\n32","00\n110\n9/0\n:\n120\n80\nTO\n130\n70\n140\n60\nx\n150\n50\n160\n40\n036\n170\n30\nin\nD\n024\n180\n20\nC\n004\n028\nA\n&\na\n020\n0\n008\n170\n10\n020\n004\n984\nDOT\n160\n0\n008\n012\nL\n020\n150\n$\n10\n0\n028\n25\n140\n20\n020\n130\n30\n0\n008\n120\n40\n110\n50\n100\n60\n9/0\n70\n80\n,\nFigure 18 . - - NMC sea level pressure analysis, , 0000 GMT Oct. 9, , 1976, 24-hr\nverification - - contour interval 6 decameters.\n33","0\n90\n110\n80\n120\n70\n130\n60\n140\n588\n50\n150\n40\n160\n30\n170\n26\n180\n516\n54 08\nE\n534\n170\nL\n0\n504\nwith\n0\n160\n564\nx\n540\n10\n150\n&\n588\n588\n20\n140\n570\n30\n130\n0\n40\n120\n50\n110\n60\n100\n90\n70\n80\nFigure 19 . - - S4 model, 500-mb height, 24 hr from 0000 GMT Oct. o 8, 1976--\ncontour interval 6 decameters.\n34","10\n9/0\n:\n120\n80\n8\n130\n70\n/\n140\n60\nx\n150\n50\n582\n40\n160\n&\n170\n30\n180\n20\n516\n546\n534\n170\n10\n.\n504\nx\n160\n0\n564\n540\n150\n10\nx\n588\n588\n140\n570\n20\n130\n30\nx\n120\n40\nMO\n50\n100\n60\n9/0\nX0\n80\nFigure 20. . -- 6L PE model, 500-mb height, 24 hr from 0000 GMT Oct. 8, 1976--\ncontour interval 6 decameters.\n35","no\n10\n90\n8/0\n120\nto\n70\n130\n60\n140\n582\n50\n150\n40\n160\n30\n170\n20\n180\n510\n4\n540\n:\n528\n10\n170\n.\nM\nour\n0\n516\n160\n564\n540\n10\nla\n150\nLK\n588\n588\n570\n20\n140\n30\n130\nx\n40\n120\n50\n110\na\n100\n60\n90\n70\n80\nFigure 21. . - - -NMC 500-mb height analysis, 0000 GMT Oct. 9, 1976, 24-hr\nverification - - contour interval 6 decameters.\n36","110\n90\n120\n80\n70\n130\n60\n140\n50\n150\n0\n40\n160\n008\n028\n170\n996\n020\n20\n180\n020\n020\nl\na\n10\n170\nas\n996\na\n0\n988\n160\n0\n0\n020\n008\n012\n150\n10\n008\n020\n020\n140\n20\n130\n30\n0\n120\n40\n110\n50\nor\n100\n60\n90\n70\n80\nFigure 22. . - -S4 model, , sea level pressure, 48 hr--contour interval 4 mb e\n37","o\na\n110\n9/0\n:\n1203\n80\nto\n130\n70\n60\n140\nx\n150\n40\n160\n008\n036\n30\n170\n024\n996\na\n20\n180\n020\n024\n170\n10\nX\no\n992\n988\n024\n160\n008\n0\nX\n6\n012\n150\n10\n020\n004\nx\n140\n20\n020\nx\n130\n30\n0\nx\n120\n40\n110\n50\n100\n60\n9/0\n70\n80\nFigure 23. . -- -6L PE model, , sea level pressure, 48 hr--contour interval 4 mb. 3\n38","10\n90\n:\n1203\n8/0\nto\n130\n70\n140\n60\nx\n150\n50\n012\n160\n40\na\n036\n170\n30\n-\n992\n024\n&\n180\n20\na\n028\n020\n170\n10\n992\n996\n024\nDOFT\n160\n980\n0\n0\n020\n150\n10\nx\n024\n996\n016\n140\n2\n012\n20\n130\n30\nx\n120\n40\n110\n50\n100\n60\n9/0\n70\n80\nFigure 24. . - --NMC sea level pressure analysis, 0000 GMT Oct. 10, , 1976, 48-hr\nverification--contour interval 4 mb.\n39","100\n110\n90\n80\n120\n70\n130\n60\n140\n50\n150\n588\n40\nor\n160\n30\n170\n534\n20\n180\n516\n546\n10\n170\n510\nx\nof\nS\n516\n0\n160\n&\n540\n10\n150\n582\n20\n140\n564\n588\n30\n130\n0\n40\n120\n50\n110\niV\n100\n60\n90\n70\n80\nFigure 25. -- S4 model, , 500-mb height, 48 hr -contour interval 6 decameters.\n40","110\n90\n120s\n8/0\n&\n130\n7/0\neth\n140\n60\n150\n50\n582\n160\n40\nx\n170\n30\n534\n180\nto\n522\n20\n546\na\n170\n10\n504\n160\n516\n0\nC\nis\n540\n151\n10\n582\n588\nna\n564\n140\n0\n20\n130\n30\nx\n120\n40\n110\n50\n100\n60\n9/0\n70\n80\nFigure 26. . -- -6L PE model, , 500-mb height, 48 hr--contour interval 6 decameters.\n41","0\n90\n80\n120\n5\n70\n130\n60\n140\n50\n150\n582\n160\n30\n170\n588\n534\n516\n20\n180\n10\n170\n546\no\n&\n504\n0\n160\n516\n564\n540\na\n10\n150\n564\na\n588\n588\n20\n140\nx\n30\n130\nx\n40\n120\n5\n0\n110\na\n60\n100\n90\n70\n80\nFigure 27 . -- --NMC 500-mb height analysis, 0000 GMT Oct. . 10, , 1976, 48-hr\nverification--contour interval 6 decameters,\n42","10\n:\n9/0\n1203\n8/0\nto\n130\n70\n140\n60\n150\n50\nx\n160\n40\n6\n170\n30\n1\n180\n0\n20\nO\n170\n10\n2\n0\n160\n0\n150\n0\n10\n1\n140\nx\n22\n20\n3\n130\n30\nx\n120\n40\n110\n50\n100\n60\n9/0\n70\n80\nFigure 28. -- E4 model, 36-48 hr accumulated precipitation from 0000 GMT Oct. 8,\n1976--contour interval 1 centimeter,\n43","10\n90\n8/0\n120\nTO\n70\n130\n60\n140\nx\n50\n150\n40\n160\n4\n30\n170\n20\n180\n0\n10\n170\n2+1\nK\nm\n(w/o\nx\n^\n1\n0\n0\n160\n1\n150\n10\n22\n140\n20\nx\n130\n30\nx\n120\n40\n110\n50\n100\n60\n9/0\n70\n80\nFigure 29 . -S4 model, 36-48 hr accumulated precipitation from 0000 GMT Oct. 8,\n1976--contour interval 1 centimeter.\n44","00\n110\n9/0\n:\n1203\n80\n&\n130\n70\n60\n140\nD\n0\n50\n150\n40\n160\nX\nx\n(5\nx\n30\n170\n08\n20\n180\n2\n0\n170\n10\n3\n17\nx\n7\n0\n160\n0\n1\nx\n00\n0\nx\nto\n1\n150\n0\n10\n2\nx\n140\n20\n1\n2\nx\n130\n30\nx\n120\n40\n110\n5\n100\n60\n9\n/0\n70\n80\n30. . -- 6L PE model, , 36-48 hr accumulated precipitation from 0000 GMT\nFigure\nOct. 8, 1976-- -- contour interval 1 centimeter\n45","50\n45\nE 4\nBETTER S I SCORE\n40\n35\nS4\nS4\nBETTER S I SCORE\n30\n24 HR\n25\n48 HR\nMEAN\n20\n25\n30\n35\n40\n50\n45\nE4\nFigure 3. - - E4 and S4 models S1 scores (500 mb) at 24 hr and 48 hr for 6 cases\nin October 1976. Each data point is from one of three NMC veri-\nfication areas in the Northern Hemisphere (North America, Europe,\nand Asia).\n46","80\n70\nE 2\nBETTER S I SCORE\n60\n50\n6L-PE -\n40\nBETTER SI SCORE\n6L-PE\n24 HR\n30\n48 HR\nMEAN\n20\n30\n40\n50\n60\n70\n80\nE2\nFigure 32. - - 6L PE and E2 models S1 scores (sea level pressure) at 24 hr and\n48 hr for 6 cases in October 1976.\n47","51\nE2\n47\nBETTER S I SCORE\n43\n39\n35\n6L-PE\n6L-PE\nBETTER S SCORE\n31\n24 HR\n48 HR\n27\nMEAN\n23\n31\n35\n39\n43\n47\n51\n27\nE2\nFigure 33. -- 6L PE and E2 models S1 scores (500 mb) at 24 hr and 48 hr for 6\ncases in October 1976.\n48","80\nS4\nBETTER S I SCORE\n70\n60\n50\n6L-PE\n6L - PE\n40\nBETTER SI SCORE\n30\n24 HR.\n48 HR.\nMEAN\n20\n20\n30\n40\n5Q\n60\n70\n80\nS4\nFigure 34. - - 6L PE and S4 models S1 scores (sea level pressure) at 24 hr and\n48 hr for 6 cases in October 1976.\n49","50\nS4\nBETTER SI SCORE\n45\n40\n35\n6L - PE\n30\n(\n6L- PE\nBETTER SI SCORE\n25\n24 HR\n48 HR\nMEAN\n20\n20\n25\n30\n35\n40\n45\n50\nS4\nFigure 35. - - 6L PE and S4 models S1 scores (500 mb) at 24 hr and 48 hr for\n6 cases in October 1976.\n50","(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)\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)\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)\nA Test of the Impact of NOAA-2 VTPR Soundings on Operational Analyses and Forecasts.\nNWS NMC 57\nWilliam D. Bonner, Paul L. Lemar, Robert J. Van Haaren, Armand J. Desmarais, and Hugh M.\nO'Neil, February 1976. (PB-256075)\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-256099)\nNWS NMC 59\nDecomposition of a Wind Field on the Sphere. Clifford H. Dey and John A. Brown, Jr.\nApril 1976. (PB-265422)\nNWS NMC 60\nThe LFM Model 1976: A Documentation. Joseph P. Cerrity, Jr., December 1977."]}