{"Bibliographic":{"Title":"A semi-implicit version of the Shuman-Hovermale model","Authors":"","Publication date":"1973","Publisher":""},"Administrative":{"Date created":"08-20-2023","Language":"English","Rights":"CC 0","Size":"0000031389"},"Pages":["A\nNOAA TM NWS NMC-53\nQC\n851\nU6N5\nNOAA Technical Memorandum NWS NMC-53\nCOM\nno.53\nPUBL\nc.1\nU.S. DEPARTMENT OF COMMERCE\nNational Oceanic and Atmospheric Administration\nNational Weather Service\nwith\navenue\nSTATES\nOF\nA Semi-Implicit Version of\nThe Shuman-Hovermale Model\nJOSEPH P. GERRITY, JR.\nRONALD D. McPHERSON\nSTEPHEN SCOLNIK\nNational\nMeteorological\nCenter\nWASHINGTON, D.C.\nJuly 1973","A\nQC\n851\nU6N5\nno.53\nc./\nU.S. DEPARTMENT OF COMMERCE\nNational Oceanic and Atmospheric Administration\nNational Weather Service\nNOAA Technical Memorandum NWS NMC-53\nA SEMI-IMPLICIT VERSION OF\nTHE SHUMAN-HOVERMALE MODEL\nJoseph P / Gerrity, Jr.\nRonald D. McPherson\nStephen Scolnik\nATMOSPHERIC SCIENCES\nLIBRARY,\nSEP 06 1973\nN.O.A.A.\nU. S. Dept. of Commerce\nNational Meteorological Center\nWASHINGTON, D.C.\nJuly 1973\n73 5319","UDC 551.509.313\n551.5\nMeteorology\nSynoptic analysis and forecasting\n.509\nNumerical prediction models\n.313\n01834980MT/\nV8A9913\nTo JoeO .2 .U\nii","CONTENTS\nAbstract\n1\n1.0\nIntroduction\n1\n2.0\nThe general equations\n2\n2.1 The definition of a domains\n4\n2.2 The isentropic cap equations\n5\n2.3 The stratospheric domain equations\n5\n2.4 The boundary layer equations\n7\n2.5 The tropospheric domain equations\n8\n2.6 A simplifying manipulation\n9\n3.0 The finite difference equations\n12\n3.1 The isentropic cap\n13\n3.2 The stratosphere\n14\n3.3\nThe troposphere\n16\n3.4\nThe boundary layer\n20\n3.5 Formulas for the evaluation of vertical advection\nat the top of the boundary layer\n22\n4.0 The reduction of the simultaneous equations\n23\n4.1 Manipulation of the thermodynamic equations\n24\n4.2 Combination of the ideal gas and thermodynamic equations 25\n4.3 The hydrostatic equations\n28\n4.4\nThe equations of motion\n31\n4.5 The continuity equations\n34\n5.0 Method for solving the simultaneous equations\n37\niii","Contents (continued)\n40\n6.0 The horizontal finite differences\n40\n6.1 Notation for finite differences\n40\n6.2 The finite difference expressions\n43\n6.3 Lateral boundary conditions\n43\n7.0 Acknowledgments\n44\n8.0 References\niv","A SEMI-IMPLICIT VERSION OF\nTHE SHUMAN-HOVERMALE MODEL\nJoseph P. Gerrity, Jr., Ronald D. McPherson,\nand Stephen H. Scolnik\nDevelopment Division\nNational Meteorological Center\nABSTRACT. The numerical weather prediction model,\ndeveloped by Shuman and Hovermale, has been reformulated\nto accommodate the use of a semi-implicit integration\nmethod. The derivation of the finite difference\nequations for the new model is given in this paper. A\ntechnique is presented for solving the system of simul-\ntaneous equations that arises from the derivation.\n1. INTRODUCTION\nThe linear, computational stability criterion that must be satisfied\nfor an explicit integration of the equations governing a numerical\nweather prediction (NWP) model is especially severe when the horizontal\nmesh size is reduced. In anticipation of the implementation of finer\nmesh models at the National Meteorological Center (NMC) over the next\nfew years, an effort is being made to explore the potential of a new\nscheme for integrating the model equations.\nThe semi-implicit technique, originally advocated by Marchuk (1964)\nfor use in NWP models and subsequently used by Robert and Kwizak (1971),\nhas a much relaxed linear computational stability criterion. Experience\ngained with a two-layer model (Gerrity and McPherson 1971) suggested\nthat one may achieve a factor of four computational speed advantage\nsolely through the use of this technique.\nIn this report, the equations for a semi-implicit form of the NMC\nmultilayer primitive equation model (cf. Shuman and Hovermale 1968) are\nderived. The finite difference methods employed for spatial derivatives\nin the new model have been kept as close to those schemes employed in\nthe operational model as was feasible. Certain departures were necessary,\nhowever, in order to implement the quasi-linearization required by the\nsemi-implicit technique.\nIt is presently planned to prepare a second report in which the results\nobtained in developmental testing now in progress will be documented.","2\n2.0 THE GENERAL EQUATIONS\nThe equations governing large-scale (quasi-static) meteorological phenomena\nmay be expressed using Cartesian coordinates on a polar stereographic pro-\njection of the Northern Hemisphere and a general o vertical coordinate, as\nfollows:\n(1)\n(2)\n-\n(3)\n(4)\npa = RT\n(5)\n(6)\n^\n(7)\nf = 28sind -\nThe map coordinates are defined in terms of the latitude, $, and the\nlongitude, 1, on an equivalent, spherical earth of radius, a, by\nX = a cos così m(0)\ny = a cos sin m(d)\nThe map factor m(d) that accounts for the variation of distance on the earth\ncorresponding to unit distance on the map may be expressed as\nsing","3\nin which $ o is the latitude at which the map distance corresponds exactly\nto that on the earth. To calculate m(d) from X and y, one may use the\nfollowing formula for sing,\nsing sing = 1 - RR = RR\nRR\nin which\nRR III [ (1 + 1 sind T2 (x2 + y2)\nTable 1. -Symbolic notation\nt\ntime\nCartesian map coordinates\nX , y\ngeneral vertical coordinate\no\na\npartial derivative with respect to time\nat\ncomponent of the horizontal wind in the X direction\nu\n11\n\"\n11\n\"\n11\nV\n11\n\"\n\"\ny\ngeopotential\nspecific volume\na\nP\npressure\nSo\nangular velocity of the Earth\n:\nindividual time derivative of vertical position\nFx,Fy\nunspecified sources of momentum per unit mass\nR\nthe gas constant for pure dry air\nT\ntemperature\nthe specific heat at constant pressure for pure dry air\ncp\nQ\nunspecified source of thermal energy per unit mass\nThe symbolism used in the equations is defined in table 1. Eqs. (1) and\n(2) are the Eulerían equations governing the horizontal velocity. Eq. (3)","4\nexpresses the quasi-static approximation between gravity and the static air\npressure. Eq. (4) is the ideal gas law for dry air. Eq. (5) is the thermo-\ndynamic energy equation. Eq. (6) is the form taken by the continuity\nequation for a quasi-static atmosphere. Eq. (7) defines a symbolic combi-\nnation of the Coriolis acceleration and the influence of the variation of\nthe map factor.\n2.1 The Definition of a-Domains\nThe prediction model developed by Shuman and Hovermale (1968) utilizes four\nseparate definitions for the vertical coordinate, O. Taken together, the\nfour o-domains encompass the entire model atmosphere. The four domains are\nidentified by the terms: boundary layer, troposphere, stratosphere, and\nisentropic cap.\nIf P stands for atmospheric pressure, one may define Q within each domain\nas follows:\nBoundary layer\nTroposphere\nPc\nStratosphere\n°S\n=\n=\nIsentropic Cap\no\n0\nThe parameters, PC' \"1\" \"2 and TT 3 have the dimensions of pressure and are\nindependent of O. In each domain, the relevant o varies between zero and\nunity.\nThe boundary surfaces separating the isentropic cap from the stratosphere,\nand the stratosphere from the troposphere, are treated as material surfaces.\nBy virtue of the definitions of these surfaces and the quasi-static approxi-\nmation, one need not explicitly calculate the deformation of the surfaces,\nas would generally be necessary.\nOne may next enumerate the form taken by the general equations within each\no domain.","5\n2.2 The Isentropic Cap Equations\nAs suggested by its name, this layer will be treated as an autobaro-\ntropic layer. The thermodynamic energy equation is written for isen-\ntropic physical processes (i.e., Q=0 0), and the stratification is\nassumed to be neutral (dry adiabatic lapse rate of temperature). The\npotential temperature will be chosen to be homogeneous throughout the\nlayer initially and will remain so throughout all time.\nThe equations for the layer may be expressed:\n(8)\n=\ndy\n(9)\n= -\nand\n(10)\n(11)\n(12)\n(13)\n+ = -\nse=0 =\n(14)\nThe result expressed in eq. (14) follows from the autobarotropy of\nthe layer and the boundary conditions, viz., 08 vanishes at the\nmaterial surface, separating the isentropic layer from the strato-\nsphere, and also at the top of the model atmosphere (of==)) =\nThere are no frictional terms in the equations of motion.\n2.3 The Stratospheric Domain Equations\nThe model stratosphere is bounded, above and below, by material surfaces\nat which &S vanishes. This assumption is used to derive two forms for","6\nthe continuity equation. The full set of equations in this domain are:\n(15)\n+ +\nav\n(16)\n=\n(17)\n= 0\n(18)\n- any =\n- +\n(19)\n(20)\n(21)\n+\n(22)\n(23)\n(24)\nEq. (21), with the definitions (22) and (23), is obtained by integrating\nthe continuity equation with respect to os and by using the boundary con-\nditions on s at 's = 0 and as = 1. Eq. (24) is derived by differentiating\nthe continuity equation with respect to °S and recalling that 2 and 11 are\nnot dependent upon O.","7\n2.4 The Boundary Layer Equations\nThe model treats the boundary layer as a domain of constant mass. This\nis accomplished by treating the parameter Pc as a fixed constant. The\nsurface of the earth forms the lower boundary of this layer, consequently\nOB = 0 when OB = 1. This defines the pressure at the ground to be the\nIt is only in this layer that the frictional terms will be kept in the\nequations of motion. They shall be calculated in keeping with the method\nused in the NMC model (Shuman and Hovermale 1968).\nThe upper boundary of this layer is a nonmaterial interface with the\ntropospheric domain. The pressure, wind, and temperature are assumed to\nbe continuously varying functions through the interface. This assertion\nshall be used subsequently, in the formulation of the finite difference\nequations.\nThe equations may be written for the boundary layer as follows:\nFx\n(25)\n=\n(26)\n=\n(27)\n= RT\n(28)\nP + -\nma + +\n(30)\n(31)\nEq. (30) follows from the continuity equation through the use of the facts\nthat the parameter Pc is a fixed constant and that OB = 0 at OB = 1.","8\n2.5 The Troposphere Domain Equations\nFinally, we come to the equations for the tropospheric domain which is\nbounded, above, by a material surface, and below, by a porous interface with\nthe boundary layer. The equations become:\n(32)\n=\nav\n(33)\n(34)\n+ a (73 - Pc) =\n0\n=\n+ =\n(35)\n+\n+\nQT\n(36)\n+\n(37)\n(38)\n(39a)\n(39b)\n(40a)","9\nvB=\n(40b)\n(41)\n(42)\nThe form of the vertically integrated, continuity equation, (37), follows\nfrom the use of eq. (38). The latter equation follows in turn from the\nassumption that the variables are continuous through the interface between\nthe troposphere and the boundary layer.\n2.6 A Simplifying Manipulation\nIn the application of the semi-implicit method, it is inconvenient to deal\nwith the two time derivatives that appear in the thermodynamic energy\nequation. If the vertically integrated continuity equations are used, one\nmay rewrite the thermodynamic energy equations in the following forms:\nThe isentropic cap:\n(43)\nA subscript 0 was inserted on u and V to avoid confusion in subsequent\nequations. Since the horizontal wind doesn't vary with a in the isentropic\ncap, the 0 simply denotes the wind at any vertical level within the cap.\nThe stratospheric domain:\n+\n+\n+","10\nThe tropospheric domain:\nP +\n+ + ue +\nm\n+ mama +\n+ ma/(u-us)anz + (v-vs)anz)\n(45)\n+ + + OT","11\n06=0\np=0\n.5\np=.5TT1\nos=0;06=1.\n82-0,42\nP=T11\nS = . 25\n2,T2,A\np=1+.25,\n2\nS = . 5\n,,\np=1+.5TT,\nV3,T3,a3\n06=.75\nP=T1+.75T,\n=0;.\n4=0,44\nP=1+2\n=1/6\n°T=1/3\n8570,05,T5\n=1/2\n5,T5,A5\n=2/3\n=5/6\n,\n1\n87#0,47,I7\n°B=.5\nV7,I7,A7\nPc\nOB=1\n,\np=T1++3\nFigure 1. - Schematic of the vertical structure of the finite difference\nequations.","12\nThe boundary layer:\n+\n+\n(46)\n+\n+\n+\n3.0 THE FINITE DIFFERENCE EQUATIONS\nThe equations given in section 2 will now be replaced by a set of finite\ndifference equations. The schematic of the model's vertical structure is\ngiven in figure 1.\ns\nIn writing the equations, we shall leave open for the present the form to\nbe used for horizontal derivatives. It is convenient to use a vector notation\nin which the presence or absence of the map scale factor is denoted as follows:\n(47)\n(48)\n(49)\nV.V III","1.3\n(50)\n(51)\nIt will be assumed that the thermodynamic variables are composed of a\nbasic state value, denoted by a circumflex; and a deviation, denoted by a\nprime, e.g., P = p + p. = The basic state values are taken to vary only\nwith o and not with time or horizontal position.\nFinally, we shall employ a superscript, T, to denote quantities evaluated\nexplicitly at time, TAt. A superscript overbar 2t will denote a time\naverage, e.g.,\n(52)\nIn terms of this last convention, a centered difference second order\napproximation for the time derivative is\n(53)\nin which At is the time interval between consecutive values of the parameters\nin the course of the integration.\n3.1 The Isentropic Cap\nThe equations of motion:\n(54)\nwith\n(55)\n(56)\ntk is a unit vector directed toward the local zenith; it serves to rotate\nthe horizontal wind vector.","14\nThe continuity equation:\n(57)\n+\nP\n(58)\nThe thermodynamic equation:\n(59)\n(60)\nThe hydrostatic equation:\n(61)\n(62)\nThe ideal gas law:\n(63)\n-\n(64)\n3.2 The Stratosphere\nThere are two sublayers within the model stratosphere. The simple structure\nof the layer lends itself to a compact enumeration of the equations. To\nallow this, we use the definitions\n1/4, k = 2\n°k = 3/4, k = 3\n(65)","15\nThe equations of motion: (k = 2, 3)\nAt +\n(66)\n5 +T-1 - (67)\n(68)\n(69)\nThe thermodynamic equations : (k = 2,3)\n(70)\n(71)\n+\n(72)\n(73)","16\nThe hydrostatic equation: (k ==2,3) =\n(74)\n(75)\nThe ideal gas equation\n(76)\n+\n(77)\nThe continuity equation\n-2t\n(78)\nP +\n(79)\nThe \"sigma-dot\" equation\n(80)\n(81)\n3.3 The Troposphere\nThe model troposphere is divided into three layers. . The equations may\nonce more be enumerated in a compact form. The vertical advection terms\nare however awkward to handle, and for this reason are defined separately.\n(There may be some advantage in looking ahead to section 3.5) We define\n04 = 1/6, °5 = 1/2, and 6 = 5/6.","LI\nJO\n+ to +\n(78)\n(88)\n+\n(78)\n(98)\n(98)\n(L8)\n(888)\n(988)\n.","18\nThe thermodynamic energy equations: =4,5,6 =\n(89)\n(90)\n+\n(91)\n(92)\n(93)\n(94)\n(95)\n(96)\n(97)\n(98)","19\nRT*(73-Pc)\n(99)\n-\n(100)\n(101)\n(102)\n(103)\nRT*T(- cc)\n(104)\n-\n(105)\nThe continuity equation\n12th + At +\n(106)\n(107)\nAtv.","20\nThe \"sigma-dot\" equations\n(108)\n-2t\n-2t\n(109)\nS7\n(110)\n(k = 4, 5, 6,6 6)\nThe hydrostatic equation:\n(111)\n+\n(112)\n.\n(k = 4, 5, 6)\nThe ideal gas equation:\n(113)\n+\n(114)\n3.4 The Boundary Layer\nThe boundary layer is modeled by a single layer within which the frictional\ninfluence is calculated.\nThe equations of motion:\n-2t\n(115)","21\n(116)\n(117)\n(118)\nThe frictional term F will be calculated using the same method employed\nin the NMC model (Shuman and Hovermale 1968).\nThe thermodynamic energy equation\nT2 At D. + + -2t\n(119)\n+\n+\n- At\n+\n(120)\nTHE\n(121)","22\nThe hydrostatic equation\n(122)\n(123)\nThe ideal gas equation\n-2t -2t + -\n(124)\nI7(a-7 -\n(125)\n3.5 Formulas for the Evaluation of Vertical Advection\nat the Top of the Boundary Layer\nIn the tropospheric equations and in the boundary layer equations, it was\nnecessary to use an approximation for the evaluation of vertical advection\nof temperature and wind at the interface. How the approximations were\nformulated is described below.\nIt is sufficient to show the process for just one parameter, say u; the\nother parameters are treated analogously.\nThe fundamental assumption is that the parameter varies linearly with\npressure through the region. If we let P6 be the pressure at the midpoint\nof the lowest tropospheric layer, P7 the pressure at the middle of the\nboundary layer, and p* the pressure at the interface, then the value of u\nat the interface, u*, is given by\n(126)\nu*u6(1-)+u7\nwith\n(127)\nPc\n(128)","23\nTo evaluate derivatives near the interface, we use\n(129)\n(130)\nAnother relation valid at the interface involves the value of & there.\nDepending upon whether one views 8 from the troposphere or from the\nboundary layer, one has (cf. Shuman and Hovermale 1968, eq. 6.3)\n(131)\nNow from the boundary layer continuity equation, one has\n(132)\nConsequently, one may write at OT=1\n(133)\nOnce = is known, °B at °B = 0 may be expressed as\n(134)\nThese relations were used in sections 3.3 and 3.4. At the interface,\nfollows from this approximation scheme.\n4.0 THE REDUCTION OF THE SIMULTANEOUS EQUATIONS\nThe finite difference equations developed above may be reduced to a set\nof seven equations in the seven quantities,\n= 1,2,3) and ok (k = 4,5,6,7).\nIn this section, we will show the process of elimination by which the\nreduction has been accomplished.","24\n4.1 Manipulation of the Thermodynamic Equations\nAs they were written, the thermodynamic equations involve the horizontal\nwinds,\nWe may use the continuity equations to replace the wind in the thermo-\ndynamic equations. The result of this manipulation is given here:\n(135)\n(136)\n2,3)\n(137)\n+ + = 4,5,6)\n(138)\n=\nThe right hand sides of these equations are defined as,\n(139)\n(140)\n(k = 2,3)\n(141)\n(k\n4,5,6)\n=\n(142)","25\n4.2 Combination of the Ideal Gas and Thermodynamic Equations\nThe Ideal Gas equations may be rewritten in the form,\n(143)\n(k = 2,3),\n(144)\n=\n(k = 4,5,6)(145)\nP\n(146)\n+\nThe parameters are defined to be,\n(k = 2,3),\n(147)\n-\n= 4,5,6),\n(148)\nPc\n(149)\nequations obtained in section 4.1 used to replace T2t in\nWhen the are\nthe ideal gas equations, one obtains:\n+\n(150)","26\n2\n(151)\n+ (k = 2,3)\n(152)\n+ ; (k = 4,5,6)\na\n(153)\nThese equations may be written more compactly by defining the following\nparameters:\n(154)\n(155)\n(k = = 2,3)\n(156)\n(k = 4,5,6)\n7,1273a(-1)\n(157)\nkg (k = 2,3)\n(158)\n(159)\n= =\n(160)\n= 0","27\n= 0 ; ; =\n(161)\n(162)\nand finally,\n,I\nH1 = 2 RKT I I I\n(163)\nk=2,...,6\n(164)\n,T\n(165)\nThe reduced equations are then,\n(166)\n+ (k = 2,3)\n(167)\n=\n+\n(168)\n<=4,5,6)\n+ 77,7 87 -2t ,T\n(169)","28\n4.3 The Hydrostatic Equations\nThe hydrostatic equations may be written\n(170)\nP\n(171)\n4,5,6)\n(k\nk\n2,3)\n=\n(172)\n(173)\nThe reduced expressions for 22t , derived in the preceding section,\nmay be introduced into the equations above.\n(174)\n+\n(175)\n+ =\n+\n(176)\n+ = 2,3)\n2\n(177)","29\nUsing these expressions, one may derive the following equations for\nthe geopotential at each level:\n673a1\n(178)\nRT\n(179)\n(180)\n7\n+\n+\nk=6\n(181)\na\n+\n+\n(182)\n-2t\n+\nk=5\n+\n(183)\na\n-2t\n(184)\n+\n+\n+\n+","30\n(185)\n(186)\n-2t\n-2t\n+ 7 + + 1667787 + (bang 6,6\n-2t\n+ + +\n(187)\n(188)\nk=2\n+ (b7,7 + 166,7)87\n+\n-2t\n+ + + (b5,5 + + +\n(189)\n(190)\nk=2\n+ + + 166,7)87\n+ (b6,6 + 05,6) + (b5,5 + + (b3,3 + b2,3) +\n(191)","31\n4.4 The Equations of Motion\nOne may now use the results of the previous section to eliminate the\ngeopotential from the equations of motion. The equations can be put\ninto the compact form\n-2t\n(192)\nThe definition of the new parameters is as follows:\n(193)\n(194)\n(195)\n(196)\n(197)\n(198)\n1,1 =\n1,2\n61.4\n(199)\n=","32\n(200)\n(201)\n-2t\n(202)\nwith\n(203)\n(204)\n(205)\n(206)\n82,3 - 81,3\n(207)\n(208)\n(209)\n(210)\n83,3 - 81,3","33\n63,1 - 41,1 = 1,2 ; 1,3\n(211)\n(212)\n84,2\n(213)\n(214)\nh4,2 = h1,2 4,3 = h1,3\n(215)\nh4,4 = 0\n(216)\n(217)\n+\n(218)\n; ; h5,3 1/25,55 h5,4 = = 0 (219)","34\n(220)\n(221)\n(222)\n(223)\nh6,3 = h6,4 -\n(224)\n(225)\n(226)\n(227)\n=\n7,3\n4.5 The Continuity Equations\nThe equations of motion have been manipulated so that only The and\n-2t\nappear. The continuity equations may now be used in conjunction with\nthe equations of motion to obtain a simultaneous set of equations only in\nterms of et and 2t. We may recall the continuity and \"sigma-dot\"\nequations using W in place of :\n(228)\n(229)\n(230)\nD.","35\n(231)\n3 +\n(232)\n(233)\nD.\n(234)\nIf we use the symbol i2 to denote the operator v.G(), the result of\nsubstituting from the equations of motion into the continuity equations\nmay be written,\nm² G2 CP + EP = F\n(235)\nC and E are 7x 7 square matrices; and P and F are the column\nvectors with transforms,\nP22\n(236)\n=\n(237)\nThe elements of the matrices C and E, and of the vector F, are\n(238)\n(239)\n(240)\n=\n(241)","36\nC2,j = At 72 (82,j + 83,j), 1 j j 3 3\n(242)\nC2,j = At 7/2 (,(j-3) + 63,(j-3)} 4 s j 5 7\n(243)\n(244)\n=0,35js7\n(245)\n+ # 55,j + (246)\n(247)\n+\n(248)\ne3,1e3,20;e3,3-1; =\n(249)\n(250)\n(251)\n4 s j 5 7\n(252)\n(253)\n(254)\n(255)\n4 s j s 7","37\ne5,4-3,e5,56,5,6-3\ne5,7= 0\n(256)\n(257)\n1 4 j 5 3\n(258)\nC6,js,()4()47\n(259)\n(260)\n(261)\n15js3\n(262)\n(263)\n€7,7 = + 4\n(264)\n(265)\n5.0 METHOD FOR SOLVING THE SIMULTANEOUS EQUATIONS\nEquations (235) in section 4.5 may be solved in a straightforward manner,\nbut the process can be shortened if the equations are first transformed by\nmatrix operations.\nThe matrix C is nonsingular, therefore one may rewrite the equation as,\n22p1E=C1\n(266)","38\nin which c-1 is the inverse of C. The coefficient matrix, C-1E, may be\ntransformed to a lower triangular matrix H by use of a method given by\nParlett (1968, pp. 116-117). By that iterative process (Sela and\nScolnik 1972), one may determine a nonsingular upper triangular matrix\nB, such that\n(267)\nH E B C - E B\nis lower triangular. Multiplication of (266) by B gives\nm² D2HB1 = X\n(268)\nin which V is the transformed variable\n= B P\n(269)\nThe solution of (268) may be calculated without any requirement for\niterative cycling among the several scalar component equations.\nSince the matrices C and E depend only upon the basic state and the\ngrid intervals, Ax and At, one may determine the B and H matrices once,\nfor all.\nThe method used for solving each of the scalar components of (268) is\nthe Liebmann relaxation technique. A first guess for the dependent\nvariable must be prescribed and so also must a criterion for estimating\nconvergence of the relaxation process.\nSince the basic dependent variable is 1, the initial guess is most\nreadily given in terms of P, say . The initial guess for V must\ntherefore be constructed by means of (269):\n(270)\nSimilarly, the convergence criterion for the relaxation is readily\nstated in terms of the dependent variable P. One must, however, manipu-\nlate that statement in such a way that a suitable criterion is developed\nfor the variable, V. Suppose, for example, that the convergence criteria\non P states that P should not vary by more than D between consecutive\niterations of the relaxation process, i.e.,\n(271)\nNow the relation between PA and V may be used in (271), ,\n(272)\nor\n(273)","39\nIf the difference between vtl and is E, then using the fact that\nB-1 and B are upper triangular matrices, one may assert that (273) will\nbe satisfied if\nm = 1,7\n(274)\ngiven that bi\nm,n n, where (bi)m,n are the elements of B-1.\nOne of the properties of B-1 is that\n(bi)m,m 1\n(275)\nA sufficient condition for satisfying (274) is\nm = 1,7.\n(276)\nExpanding (274) and using the properties of (bi), , the following\nconditions may be shown to be sufficient for satisfaction of (274);\nviz.,\n<\n(277)\nin which min { a, b, c} means that the smallest element of the\nlist a, C is to be taken.\nBy calculation of (277) given D and B-1, the appropriate convergence\ncriteria for solution of (268) may be established, once for all.","40\n6.0 THE HORIZONTAL FINITE DIFFERENCES\nThe horizontal coordinates are made discrete using equally spaced co-\nordinate lines. There results a lattice of squares with equal map area.\nThe intersections formed by the coordinate lines are referred to as grid\npoints. The midpoint of each square will be referred to as a cell.\nThe semi-implicit linearization scheme identifies the naturally\nstaggered relation between the horizontal velocity components and the\nthermodynamic parameters and vertical velocity. There are a number of\nways in which the natural lattice structure may be exploited in the design\nof the finite difference equations. The method used in the present model\nwas selected because of its ready adaptation to the use of higher order\napproximations of nonlinear advection and the successful use of a similar\nmethod by Kwizak and Robert (1971)\nThe method may be described briefly by stating that the thermodynamical\nparameters and the vertical velocity are carried at gridpoints, whereas\nthe horizontal wind vector is carried in the cells.\nThe use of a polar-stereographic map as the coordinate base requires\nthat one consider the statement of boundary conditions. For the present,\nwe shall only remark that the boundary surface is assumed to coincide with\nthe set of grid points forming a rectangular enclosure of the integration\ndomain. This decision places the boundary at the locations occupied by\nthe thermodynamic quantities and the vertical velocity. The specification\nof these parameters is necessary and sufficient for the proper statement\nof the linear gravitational oscillations permitted by the model.\n6.1 Notation for Finite Differences\nOur use of a staggered arrangement of the horizontal components of the\nvelocity with respect to the other parameters might lead to a cumbersome\nnotation for the difference operations. The simplest course is to assume\nthat the reader is familiar with the notation introduced by Shuman (1962)\nand used in Shuman and Hovermale (1968). In particular, attention is\ndrawn to the definition of higher order approximation formulas given in\nGerrity (1973) The questions that may arise regarding the location at\nwhich an approximation is applicable can be resolved by some careful\nreflection more readily than by a tedious enumeration of formulae.\n6.2 The Finite Difference Expressions\nIn order to be as concise as possible, only certain typical approxi-\nmations will be enumerated here. It is not difficult to change finite\ndifference schemes within computer codes and the methods described below\nare not regarded as fixed. It would be improper, however, to fail to\nindicate the outlines of the approximation approach being followed in this\nwork.","41\nConsider first the basic system of simultaneous equations given as\neq. (235). That equation may be expressed as\n(280)\n+ EP = F\na\nThe parameters W and TT constituting the elements of P are defined at\ngrid points. The coefficient C is a constant, consequently the operator\nreturns a value at the grid points. This operator has the property of\nadmitting two independent solutions on alternate gridpoints. One may\nfilter the solution to suppress such a lattice separation if it is found\nto be desirable. The map factor, m, is located at the middle of the grid\nsquares and the \"bar x,y\" returns a value at a grid point.\nTo discuss the approximation of the \"forcing functions\" in the several\nequations, we shall have recourse to the tropospheric equations outlined\nin section 3.3.\nFor the equations of motion, eq. (83) gives the forcing function in\nvector form. For simplicity, we shall consider only the X scalar component\nof vk denoted here as Uk.\nxy +\n(281)\n.\nThe parameter fk is calculated as\n(281a)\n.\nThe vertical advection term BK has the form typified by\nxy\n(282)\nWe next consider the forcing terms in the thermodynamic equation,\ngiven as eq. (90). Rather than stringing out a long equation, we may\nsimply give the approximations used for representative terms in the\nexpression for the forcing function\n(283)","42\n(284)\nxy\nI\n(285)\nTHE\n(286)\nh h\nT\n(287)\nIt may be observed that some of these terms are very nonlinear and may\nconstitute a source of difficulty. The approximations given above may\nnot be the best choice; experiments with alternative approximations are\nanticipated.\nThe forcing function appearing in the continuity equation (107) is\napproximated by\nxy\n(288)\nThe forcing functions in the \"sigma-dot\" equations are approximated\nin the form typified by the following in which the y dependence of 3\nhas been omitted.\nxy\n(289)\nThis enumeration should be sufficient to enable the reader to re-\nconstruct the remaining details.","43\n6.3 Lateral Boundary Conditions\nThe lattice structure adopted for the present model places the lateral\nboundary at the gridpoints. This is a convenient arrangement since it\nleads to a Dirichlet boundary value problem for the simultaneous equation\n(235). The boundary conditions on j2t, 2t, and 2t must be specified.\nIt is also necessary to specify uxy and at the boundary gridpoints.\nIn our preliminary experiments on a hemispheric domain, we have not\nexpended much effort in detailed diagnosis of the impact of the specifi-\ncation of these boundary conditions. The ultimate aim of this development\neffort is to apply the model to a limited area. Special problems arise in\nthat case (cf. Davies 1973) which must be addressed prior to making a final\ndetermination of the questions arising in the treatment of lateral boundary\nconditions.\n7.0 ACKNOWLEDGMENTS\nSeveral members of the staff of the National Meteorological Center have\nlent support to this project. Particularly significant contributions have\nbeen made by Joseph Sela, Kenneth Campana, and Paul Polger.\nAndré Robert and Richard Asselin of the Dynamic Prediction Research\nDivision of the Department of the Environment, Canada, have been most\ngenerous in sharing with us their experience with implicit modeling\ntechniques.","44\n8.0 REFERENCES\nDavies, H. C., , \"On the lateral boundary conditions for the primitive\nequations, Journal of Atmospheric Sciences, 30, 1973, 147-150.\nGerrity, J. P., , and McPherson, R. D., \"A semi-implicit integration\nscheme for baroclinic models,\" \" NMC Office Note 65, National\nMeteorological Center, Suitland, Maryland, 1971.\nGerrity, J. P., \"Numerical advection experiments with higher order\naccurate, semimomentum approximations,\" Monthly Weather Review,\n101, 1973, (in press).\nKwizak, M., and Robert, A. J., \"Implicit integration of a grid point\nmodel, \" Monthly Weather Review, 99, 1, Jan. 1971, pp. 32-36.\nMarchuk, G. I., \"A new approach to the numerical solution of differen-\ntial equations of atmospheric processes,\" WMO Tech. Note 66,\nUNIPUB, N.Y., 1964, pp. 212-226.\nParlett, B. N. , \"The LU and QR algorithms, Mathematical Methods for\nDigital Computers, Vol. 11, Ralston, A., and Wilf, H. S., Editors,\nJohn Wiley & Sons, N. Y., 1968, pp. 116-130.\nSela, J., , and Scolnik, S., \"Method for solving simultaneous Helmholtz\nequations,\" Monthly Weather Review, 100, 1972, 644-645.\nShuman, F. G., \"Numerical experiments with the primitive equations,\"\nProceedings of the International Symposium on Numerical Weather\nPrediction, Tokyo, Japan, Meteorol. Soc. of Japan, Tokyo, 1962,\npp. 85-107.\nShuman, F. G., and Hovermale, J. B., \"An operational six-layer primitive\nequation model,\" \" Journal of Applied Meteorology, 7, 1968, 525-547."]}