{"Bibliographic":{"Title":"Theoretical and experimental comparison of selected time integration methods applied to four-dimensional assimilation","Authors":"","Publication date":"1975","Publisher":""},"Administrative":{"Date created":"08-20-2023","Language":"English","Rights":"CC 0","Size":"0000088703"},"Pages":["A\nD823\nQC\nOF COMMUNITY\n851\nU6N5\nno.56\nical Memorandum NWS NMC-56\n*\n*\nwith STATES OF Avenue\nc.l\nTHEORETICAL AND EXPERIMENTAL COMPARISON\nOF SELECTED TIME INTEGRATION METHODS\nAPPLIED TO FOUR - DIMENSIONAL DATA ASSIMILATION\nRonald D. McPherson and Robert E. Kistler\nNational Meteorological Center\nWashington, D.C.\nApril 1975\nREVOLUTION\n1776-1976\nnoaa\nNATIONAL OCEANIC AND\nNational Weather\nATMOSPHERIC ADMINISTRATION\nService","NOAA TECHNICAL MEMORANDA\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 Memoranda in the NWS NMC series facilitate rapid dissemination of material of general\ninterest which may be preliminary in nature and which may be published formally elsewhere at a later\ndate. Publications 34 through 37 are in the former series, Weather Bureau Technical Notes (TN), Na-\ntional Meterological Center Technical Memoranda; publications 38 through 48 are in the former series\nESSA Technical Memoranda, Weather Bureau Technical Memoranda (WBTM). Beginning with 49, publications\nare now part of the series, NOAA Technical Memoranda NWS.\nPublications listed below are available from the National Technical Information Service (NTIS), U.S.\nDepartment of Commerce, Sills Bldg., 5285 Port Royal Road, Springfield, Va. 22151. Price: $3.00 paper\ncopy; $1.45 microfiche. Order by accession number, when given, in parentheses.\nWeather Bureau Technical Notes\n22 NMC 34 Tropospheric Heating and Cooling for Selected Days and Locations over the United States\nTN\nDuring Winter 1960 and Spring 1962. Philip F. Clapp and Francis J. Winninghoff, 1965.\n(PB-170-584)\nNMC 35 Saturation Thickness Tables for the Dry Adiabatic, Pseudo-adiabatic, and Standard Atmo-\nTN\n30\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)\nCatalog of 5-Day Mean 700-mb. Height Anomaly Centers 1947-1963 and Suggested Applica-\nTN\n40\nNMC\n37\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 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)\nNMC 45 On a Certain Type of Integration Error in Numerical Weather Prediction Models. Hans\nWBTM\nOkland, September 1969. (PB-187-795)\nNoise Analysis of a Limited-Area Fine-Mesh Prediction Model. Joseph P. Gerrity, Jr., and\nWBTM\nNMC\n46\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)","A\nQC\n851\nU6N5\nno.56\nC./\nNOAA Technical Memorandum NWS NMC-56\nTHEORETICAL AND EXPERIMENTAL COMPARISON\nOF SELECTED TIME INTEGRATION METHODS\nAPPLIED TO FOUR-DIMENSIONAL DATA ASSIMILATION\nRonald D. McPherson and Robert E. Kistler\nNational Meteorological Center\nWashington, D.C.\nApril 1975\nATMOSPHERIC SCIENCES\nLIBRARY\nJUN 5 1975\nN.O.A.A.\nU. S. Dept. of Commerce\nAND ATMOSPHERIC\nNOAA\nUNITED STATES\nNational Weather\nNATIONAL OCEANIC AND\nDEPARTMENT OF COMMERCE\nATMOSPHERIC ADMINISTRATION\nService\nFrederick B. Dent, Secretary\nGeorge P. Cressman Director\nRobert M. White, Administrator\nSS OF communication\n1994\n15","CONTENTS\n1\nAbstract\n1\n1. Introduction\n4\n2. Theoretical analyses\n6\nA. Euler-backward\n8\nB. Explicit-centered\n10\nC. Semi-implicit-centered\n12\nD. Semi-implicit-backward\nE. Viscous method\n14\n20\nF. Discussion\n3. Experimental integrations\n22\n4. Application to four-dimensional data assimilation\n27\n5. Summary\n31\n6. Acknowledgments\n32\n7. References\n33\n8. Tables\n36\n9. List of figures\n49","THEORETICAL AND EXPERIMENTAL COMPARISON OF SELECTED\nTIME INTEGRATION METHODS APPLIED TO FOUR-DIMENSIONAL DATA ASSIMILATION\nRonald D. McPherson and Robert E. Kistler\nData Assimilation Branch, Development Division\nNational Meteorological Center, NOAA, Washington, D. C.\nABSTRACT. The characteristics of several damping\ntime integration methods, which have been used in\nfour-dimensional data assimilation, are examined\ntheoretically and experimentally. Included are two\nexplicit methods, two semi-implicit methods, and a\nviscous method. The results indicate that, although\nthere are differences in capability to damp gravita-\ntional noise, all the methods performed equally well\nin an experimental assimilation of real data.\n1. INTRODUCTION\nThe principal element of dynamic four-dimensional data assimilation\nis a numerical prediction model which serves as an integrator of\nobservations distributed in space and time. At intervals of a few\ntime steps, the marching process is interrupted and the current model\nrepresentation of the atmosphere is adjusted in one or more subareas\nof the grid lattice, in accordance with timely observations available\nin those areas. A primitive equation model tends to interpret the\nadjustments as local imbalances; this is commonly referred to as","\"shocking\" the model, which responds by producing gravitational oscilla-\ntions. Each adjusted subarea thus acts as a source of gravity wave\nnoise. In an assimilation integration the model is shocked frequently,\nwith the result that there is continual generation of noise. Ordinary\nfinite-difference methods are thus subjected to a special stress.\nMinimizing the effect of the model shock while simultaneously ensuring\nretention of the beneficial effects of the inserted data is a major\nproblem in the design of a four-dimensional data assimilation system.\nPrevious experiments have suggested several procedures for gingerly\ninserting the observations into the model. Bengtsson and Gustavsson\n(1972) used a local objective analysis method to interpolate the obser-\nvations to nearby grid points in a fashion which reduces the effect of\nobservational error. Morel, Rabreau, and Lefevre (1971) have indicated\nthe desirability of repeated use of each observation in order to\nreinforce the model's \"memory\" of the inserted data. Hayden (1973)\nused both of these and, in addition, a local balancing of the motion\nfield from observations of the mass field. These procedures ameliorate\nthe noise problem, but do not eliminate it. Therefore, most investigators\nhave found it necessary to also use a damping finite-difference system.\nOur purpose in this paper is to examine the properties of several\nintegration methods, most of which have been used in published assimila-\ntion experiments, in order to determine if any advantage is obtained by\nselecting one in preference to another. The methods investigated include\ntwo explicit and two semi-implicit damping time integration techniques,\nand a viscous method. of the explicit methods, the first is the Euler-\n2","backward, used in assimilation experiments by Talagrand (1972) and by\nHayden, among others. The second is the centered-difference, or \"leapfrog\"\nmethod, coupled with a rather powerful time filter devised by Robert (1966).\nThis method is incorporated in the National Meteorological Center 8-1evel\nglobal model and will be used in forthcoming data assimilation experiments.\nIt has been used, without the time filter, in a recent study by Halberstam\n(1974) comparing three finite-difference systems applied to four-dimensional\nassimilation. The first of the semi-implicit methods is a centered-\ndifference formulation devised by Kwizak and Robert (1971) and used,\nwith the Robert time filter, in an assimilation experiment by Rutherford\nand Asselin (1972). The second is the semi-implicit-backward method\n(Kurihara 1965), also coupled with the Robert time filter. To our\nknowledge, it has not been used in previous data assimilation experiments\nbut appears to be suitable for this application. Finally, the viscous\nmethod appends special viscosity terms, which affect only the irrotational\ncomponent of the wind, to the equations of motion. This idea has been\nused by Shuman and Stackpole (1969) in numerical stability studies, and\nby Morel and Talagrand (1974) in data assimilation experiments. Sadourny\n(1973) has also studied the method.\nIn the following section, we present theoretical investigations of\nthe several methods. Much of the information has been published previously,\nnotably by Kurihara and by Asselin (1972). We have followed the pattern\nestablished by those authors in extending their analyses to include the\nsemi-implicit-backward and the viscous methods with the Robert time filter.\n3","For convenience of comparison, we have also included, in abbreviated form,\nKurihara's analysis of the Euler-backward and Asselin's treatment of the\nexplicit and semi-implicit centered-difference forms with the time filter.\nThe third section summarizes a series of integrations from unbalanced\ninitial data using the different methods, designed to subject the findings\nof the linear analyses to experimental confirmation. In the last section,\nwe apply the methods in an experiment in which real asynoptic observa-\ntions are assimilated within the framework of a primitive-equation\nbarotropic model.\n2. THEORETICAL ANALYSES\nThe analyses in this section are patterned after those of Kurihara.\nThe system of linear differential equations may be written as\n(1)\nav\nav\nU\nfu\n= 0\n(2)\nat\nU\nfUv\n-\n(3)\nIn these equations, U is a constant zonal current, H is the mean depth of\nthe fluid varying only in the y-direction such that\n(4)\nand u, V, and 0 are perturbation wind components and geopotential.\nWe assume solutions of trigonometric form,\nuo\nu\n(x-ct)\nV\n=\nVo\ne\n(5)\n,\n4","where C is the phase speed, k is the wavenumber given by 2w/L, L is the\nwavelength, and , Vo' and 00 are unknown amplitudes. By substituting\n(5) into (1-3) and imposing the requirement that the determinant of the\nresulting system of linear algebraic equations must vanish, we obtain an\nequation in the phase speed c,\n(U-c)3- (gH f2/k2)(U-c) + £2/k2U = 0.\n(6)\nThe solution of (6) yields three real roots,\n(7)\n(8)\n(9)\na = - (gH + f2/k2) and b = Uf2/k2 and l =\nwhere\nWith these values for the Cj' the amplitudes and Vo may be determined\nin terms of Po, and the solutions to the system (1-3) may then be written\nas\n(10)\n(11)\nand\n(12)\n5","It may be demonstrated that with substitution of the solutions (10-12)\ninto the original system, any of (1), (2), or (3) may be written in the\nform\ndh\nanit\n(13)\n=\nwhere hj represents Uj, Vj, or j. The second term on the left may be\nidentified with advective processes and the third term with oscillatory\nprocesses. These two terms are treated differently in the semi-implicit\nmethods.\nThe form of (13) is of considerable convenience, for it permits an\nadvantageous economy of algebraic manipulations in the following analyses:\neach integration method (excluding the viscous method) may be applied\nto a single equation of the form of (13) instead of to the original system\nof three equations. The result is at most a quadratic equation with\ncomplex coefficients, which may be solved by standard methods. If each\nintegration method is applied to the original system of three equations,\nthe result is at most a sixth-order equation with complex coefficients,\nwhich must be solved numerically. A demonstration of this is presented\nin the analysis of the viscous method, which cannot be written in the form\nof (13). For the first four analyses, however, we will avail ourselves of\nthe proffered economy and apply each method to (13).\nA. Euler-backward -\nIn the case of both explicit methods, the second and third terms of (13)\nare evaluated at the same time level, and so (13) reduces to\ndh\n+\n(14)\n6","The Euler-backward method applied to (14) is written as\n(15)\n(16)\nCombining (15) and (16) leads to\nhjn+1 [1- ikcjAt - =\n(17)\nWe now represent the time-dependent part of the solutions (5) as\n(18)\nIII\n*\nwhere cj* denotes the numerical phase velocity. Substitution of\n(18) into (17) yields\n(19)\nSj=1-\nThe complex variable 5j may be expressed in the polar form 5j = 15y/e16 =\nwhere 15jl represents the magnitude of the time-dependent part of the\nsolution: /5j/<1 indicates damping. The phase angle oj may be identified\nthrough (18) as - kcj*st.\nFrom (19)\n(kcj&t)2)4,\n(20)\nand\n(21)\n= =\n7","It is convenient to express the phase response in terms of the ratio, Tj'\nof the numerical phase speed to the analytic phase speed, given by\ntj==\n(22)\nEquations (20) and (22) have been evaluated for a range of kcjAt, and\nthe results presented in column 2 of Tables 1, 3, 5, and 7, and in Figures\n1 and 5. The amplitude response is less than unity for WAT < 1 (w III kc);\nfor larger values, the method is amplifying. Maximum damping is about\n13 percent per time step at wst = 0.7, with no discrimination between\nmeteorological and gravitational modes of the same frequency.\nOur primary interest in this paper is in the amplitude response of the\nvarious methods. For completeness, however, we have included the phase\nresponse. Column 2, Table 3, shows that the numerical phase speed for\n0 < WAt < 1 is greater than the analytic phase speed. A spurious\nacceleration is therefore a result of using the Euler-backward. In practice,\nthis is counteracted by using space-differencing systems which under-\nestimate the phase speed.\nB. Explicit-centered\nIn this method, the time derivative in (14) is replaced by centered-\ndifference approximation\n(23)\nwhere the bar notation denotes a quantity which has been affected by\nthe time filter, written in the form\n(24)\n8","The coefficient a ranges from zero to unity. If (23) and (24) are\nwritten for the next time level as well, the set of four equations may\nbe manipulated to eliminate the unfiltered quantities in favor of\nfiltered ones. The result is\nht21o\n(25)\nAgain making use of (18), we obtain a quadratic equation in 5j,\n(26)\n- =\nwhich has the form\n(27)\n52 + + iB)5 + (C + iD) = 0.\nThe solutions of (27) may be expressed as\n(28)\nwhere R = A2 - - B2 - 40, and S = 2AB - 4D. The positive sign\ncorresponds to the physical mode, and the negative sign to the compu-\ntational mode. With\nA = a - 1\nB=2kcjAt\n(29)\nC==\nD = - kcjst (1-a),\nwe have calculated\n(30)\n=\n9","and\n(31)\n=\nfor a range of values of kcjAt. The results are presented in columns 3-6\nof Tables 1-8, and in Figures 1 and 5.\nWith a = 1, the amplitude response is unity for WAt < 1. For wAt > 1,\nthe physical mode is damped and the computational mode amplified. The\namplitude response decreases as a decreases; for a = 0.3, the physical\nmode is damped 44 percent per time step, at WAt = 0.7. For\nsmall WAt, the computational mode is severely damped, but is neutral at\nWAt = 0.7 and amplifying for larger values. Numerical stability therefore\nrequires a 30-percent reduction in the allowable time step if a is\ndecreased from unity to 0.3. As with the Euler-backward method, the\nexplicit-centered with the time filter is not capable of discriminating\nbetween meteorological and gravitational modes of the same frequency.\nTable 3 and Figure 5 indicate the spurious acceleration associated with\nthis method. The effect of the time filter is to exaggerate the acceleration.\nC. Semi-implicit-centered\nIn this formulation, the second and third terms of (13) are treated\ndifferently. The second is evaluated explicitly, that is, at the current\ntime level, but the third is evaluated by a time average centered on the\ncurrent time:\nhjn+1 - -\n(32)\n10","where a = kUAt and = The bar notation denotes a\nquantity affected by the time filter, given by (24). As in the explicit- -\ncentered case, eqn. (32) may be written for the next time level, and the\ntwo filter equations (eqn. (24) and its analogue for the (n+1) time level)\nmay be manipulated to eliminate all reference to unfiltered quantities.\nThe result is\n(33)\n-\nIt may be observed that for cj = U, bj = 0, and (33) reduces to (25), as\nit should.\nSubstitution of (18) into (33) results in a quadratic equation in Tj of\nthe form of (27), with\n(34)\nc - a tab2 - ab(1-a)\nD = 2ab - all-a)\nColumns 7-10 of Tables 1-8 and Figures 2-5 give the results of numerical\nevaluation of (28), using the definitions (34), in terms of 15jl and Tj.\n11","The amplitude response for a = 1 is neutral for meteorological modes\n(Figure 2a) provided wAt < 1. Gravitational modes (Figure 3) are neutral\nfor WAt much larger than unity. It is this property which renders semi-\nimplicit methods attractive, for it permits the use of much larger time\nsteps. Stability with WAt > 1 is obtained by retarding the motion of the\ngravity waves. Column 7 of Table 7 illustrates the spurious deceleration\nof the gravitational modes. Meteorological modes (column 7, Table 3) are\nsubject to slightly greater acceleration than is the case for the explicit\nmethods.\nAs the filter coefficient decreases from unity, the meteorological modes\nare damped somewhat more strongly by the present method than by the\nexplicit-centered, and the accelerative tendency is increased. Gravitational\nmodes are strongly damped for WAt > 1. The time filter tends to counteract\nthe reduction of numerical phase velocity.\nThese results indicate that the semi-implicit-centered method discriminates\nbetween meteorological and gravitational modes, in contrast to either of the\nexplicit methods. The use of the time filter enables the exploitation of\nthat capability, resulting in a system in which gravitational noise is\nstrongly damped, but with much less effect on the meteorological modes.\nD. Semi-implicit-backward\nThis method is a variation of the semi-implicit centered method. Here,\nthe third term of (13) is evaluated at the future time level rather than by\nan average centered at t = nst. The expression corresponding to (32) is\n(35)\n- - -\n12","where the barred quantity is defined by (24). Eliminating the unbarred\nquantities from (24), (35), and their analogues for the subsequent time\nstep results in an equation corresponding to (33),\n(1-a)b\n1 + 4b2\n-\n(36)\n+i\n-\nAgain making use of (18) in (36), we obtain a quadratic in Tj of the form\nof (27), with\n+ + bb2\nB = 2a + b 1 - a\n(37)\nc = - a - 2ab(1-a)\nD = 2ab - a(1-a)\nD\n1 + 4b2\nWe have used (37) to evaluate the solutions from (28). The results are\ngiven in terms of 15jl and Tj in columns 11-14 of Tables 1-8, and in\nFigures 2-5.\nKurihara found that this method, without the time filter, is strongly\ndamping with respect to gravity modes, but slightly amplifies.meteorological\nmodes. His results are confirmed in column 11 of Tables 1 and 5 and in\nFigure 2a. This distinction, while not altogether advantageous, nevertheless\ndemonstrates an excellent ability to discriminate between gravitational and\nmeteorological modes.\n13","Including the time filter eliminates the amplification of the meteorological\nmode for wat < 0.9 and reinforces the damping of the gravity modes. For\na = 0.9, the meteorological mode is slightly damped, retaining more than\n99 percent of its amplitude for wAt < 0.9, while the maximum damping of\nthe gravity mode is more than 70 percent.\nThe phase response is similar to that of the previous method: meteoro-\nlogical modes are accelerated spuriously and gravitational modes retarded.\nE. Viscous Method\nShuman and Stackpole suggested damping terms in the equations of motion\nof the form\ndu\n(38)\n+\nv\nat\ndv\n(39)\n+\n,\nat\nwhere n denotes vorticity and D divergence. If the vorticity and divergence\nequations are formed from (38) and (39), ,\nan\nV.V2n\n(40)\n0\n+\n=\n-\nat\naD\n+\n-\n(41)\nat\nIt may be seen that the term with coefficient V in (38) and (39) affects\nonly the rotational component of the wind, a fact pointed out by Lamb (1928)\nLikewise, the term with coefficient u affects only the divergent component\nof the wind. Thus, by setting v = 0, the divergence can be damped without\ndirect impact on the vorticity. Since gravity modes are mainly associated\nwith the divergent wind component, and meteorological modes with the rotational\n14","wind, this device offers the possibility of selective damping of the\ngravitational noise during numerical integration.\nThe system of equations (1-3) permits no variation in the y-direction.\nHence, the divergence D becomes and the viscous term in (38) reduces\nto 324 Eqn. (1) thus becomes\n(42)\n+\nand (2) and (3) are unaltered. Substitution of (5) into the system\n(2, 3, and 42) results in an equation identical to (6),\n(43)\n- = 0\nexcept for the second-power term. The solutions of (43) are considerably\nmore complicated than (7-9). We may write them symbolically as\n(44)\n, = A + B\nwhere\n=-(A+ B)+i(A-B)3]\n(45)\nA =\nB =\na = -\nb = +\n15","An attempt to write the Xj in more readily interpretable forms proved\nneither profitable nor instructive, and so numerical evaluation was\nnecessary. The calculations were carried out with U = 50 m sec- ,\ngH = 8 X 104m2sec-2, f = f45' and = 107m2sech and 108m2sec-1 The\nwavelength L was varied from 103 km to 20 x 10 3 km. For each mode, the\nphase speed Cj, its real and imaginary parts cj (r) and cj(i), and the\nkcj(i)\ndeviation from unity of the amplification factor per unit time, e\n1,\nare given in Table 9. The imaginary parts of both gravity mode phase\nspeeds are negative for the entire range of wavelength, and are greatest\nin absolute value for short waves. The amplification factor is, therefore,\nless than unity; i.e., , the gravity modes are damped. The damping is\ngreatest for short waves, least for long waves. This behavior is illustrated\nin Figure 6.\nThe imaginary part of the meteorological phase speed is negative for very\nshort waves, but changes sign at approximately 2000 km. Very short\nmeteorological modes are therefore damped, but waves longer than 2000 km\nare amplified, albeit very slightly. Figure 7 shows the amplification of\nthe meteorological mode as a function of wavelength. The maximum growth\nrate is of the order of 10-8 per unit time, which would require more than\na year for the amplitude of the meteorological mode to increase by a factor\nof e.\nEven so, the presence of an amplifying mode suggests that the Robert time\nfilter might be useful in conjunction with this method. The following\nanalysis of the viscous method with the explicit-centered differencing\nmethod includes the time filter.\n16","The pattern of the previous four analyses is not applicable here, for the\nsystem (2, 3, and 42) and its solutions cannot be written in the form of\n(13). Accordingly, it is necessary to resort to a less elegant method.\nWe apply explicit-centered differences to the system (2, 3, and 42) and\nobtain\nun+1 un-1 + 2ikAtUu + 2k2Atuun-1 - 2Atfvn + 2ikAt = 0\n(46)\nvn+1 vn-1 + 2ikAtUvn + 2Atfun = 0\n(47)\nand\non+1 - on-1 + 2ikAtU - 2AtfUvn + 2ikAtgHu = 0.\n(48)\nThe equations for the time-filtered quantities are\nun-1)\n(49)\n(50)\n=\n(51)\nThe six equations (46-51) may be also written for the subsequent time\nlevel, and the combined system of 12 equations may be manipulated to produce\na system of three equations in the time-filtered quantities:\nunti + [(a-1) (1-kc) + 2ia]u\" - [a:1-2c) - + ia (1-a) ]un-1\n-\n+ [a 1 + 2ia]vn + ]vn-1 + = (53)\n+ [2ibvgH]u\" - [ibvgH(1-a)] ]un-1 = 0\n(54)\n17","where\na = kUAt\nb = kvgH At\nc=k2ust\nd = fAt .\nWe assume solutions of a form analogous to (18),\n(un)\nand substitute these into (52-54). The result is a system of three linear\nequations,\n(52 + [(a-1)(1-1c) + 2ia]s - [a(1-2c) + ia (1-a)])u\n- [2dg - d(1-a)]vo + - [2ikAtt - ikAt(1-a) 100 0\n(55)\n[2d5 + 2ia]s - [a + ia (1-a)])v = 0\n(56)\n[ - -\n(57)\n+ [a-1 -\nthe determinant of which must vanish. The imposition of this requirement\nleads to a sixth-order polynomial in 5, with complex coefficients. Lacking\na general method for finding the roots of such an expression, we have\ninstead sought approximate solutions by numerical means.\nAn iterative procedure was devised to locate the zeroes of the determinant\nof (55-57) in the complex plane, as functions of kcjAt, where cj was\n18","calculated by (44) and (45). The calculations were performed with the\nextended precision arithmetic of the IBM 360/195. The accuracy of the\nprocedure was checked by obtaining the approximate solutions for u = 0,\na = 1 (the explicit-centered, without filter) for which analytic solutions\nare available. For kcAt < 1, the analytic zeroes all lie on the unit\ncircle in the complex plane, so that /5/ = 1. The approximate solutions\nobtained differed from unity only in the fifteenth decimal place.\nValues of 5 and T were calculated from the zeroes of the determinant,\nand are presented in Tables 10-17. The amplitude responses for the\nmeteorological modes (physical and computational) are given in Tables 10\nand 11 for = 0, 107m2sec-1, and 108m2sec-1. For u = 0, the numerical\nsystem reduces to the explicit-centered method, so the entries in columns\n2-5 of Tables 10 and 11 should agree with those in columns 3-6 of Tables 1\nand 2.\nComparison of Tables 10 and 11 with Tables 1 and 2 indicates that the\nmeteorological modes are affected only very slightly by the viscous term.\nSlight amplification is indeed evident from columns 6 and 10 of Table 10.\nHowever, the time filter effectively suppresses the weak instability, as\nwas the case with the semi-implicit-backward system. Columns 2, 6, and 10\nof Tables 12 and 13 show that the viscous term has a small tendency to\nreduce the spurious acceleration resulting from centered differences.\nThe amplitudes of the gravity modes are significantly reduced, especially\nfor large . For wat = 0.5 and a = 1, the reduction is 1.2 percent per time\nstep for u = 107m2sech but nearly 15 percent for u = 108m2sec-1 The time\n19","filter further reduces the amplitudes, and the combined effect is large.\nWith WAt = 0.6, a = 0.3, and = the physical gravity mode loses\nnearly 50 percent of its amplitude each time step. In contrast to the phase\nresponse of the meteorological mode, the viscous term exaggerates the spurious\nacceleration of the gravity modes.\nF. Discussion\nThe principal performance factor governing the selection of a time integration\nmethod for four-dimensional assimilation is the ability to exhibit maximum\ndamping of gravitational noise with minimum effect on the meteorological modes.\nFrom the preceding analyses, we may examine the relative merit of each method\nfrom this point of view.\nOf the explicit methods, including the viscous method, all damp the\ngravitational mode except for the explicit-centered with a = 1. For example,\nconsider the amplitude responses for WAt = 0.5 from columns 2-6 in Table 5\nand columns 2-9 in Table 14. Maximum damping of the gravity mode is\nobtained by the viscous method with = 108m2sech with a = 0.3. Even\nwithout the time filter, the viscous method has a greater effect on the\ngravity mode than either the Euler-backward or the explicit-centered. With\nrespect to the meteorological mode, the viscous method has the least impact\nand the Euler-backward the greatest. In practice, however, this distinction\nis almost negligible. For numerical stability, the time step must be\nsufficiently small that the fastest allowable mode cannot move more than one\ngrid interval per time step. In explicit methods, the fastest mode is the\nexternal gravity mode. Since the meteorological mode has a much lower\nfrequency, the associated Wm At is much less than unity. It may be seen from\nthe tables that the damping of the physical meteorological mode for\nwmAt << 1 is quite small.\n20","For the semi-implicit methods, the gravity mode is stable for very large\ntime steps, and in practice the time step is chosen such that WmAt is\napproximately unity for the meteorological mode. Columns 9, 10, 13, and 14\nof Table 1 indicate that the use of the time filter under these circumstances\ncan have a significant damping effect on the meteorological mode. If the\ntime filter is to be used in conjunction with a semi-implicit method, the\nfilter coefficient a should not depart greatly from unity. Table 5, columns\n7-14, present the amplitude responses for the eastward-moving physical\ngravity mode. Greater damping is exhibited by the semi-implicit-backward\nmethod for all values of WAt. The use of the time filter with this method\ndoes not augment the damping to any great degree, and in fact reduces the\ndamping for large WAt. In practice, wgAt will be large. For example, if\n-1\nthe phase speeds of the meteorological mode and gravity mode are 50 m sec\nand 300 m sec-1 respectively, and At is chosen such that WmAt = 0.6 for the\nmeteorological mode, then wgAt = 3.6 for the gravity mode. The damping\nresponse at this value is very large; a gravity wave would lose 60-65 percent\nof its amplitude in one time step. Even for \"gAt < 1, the response is\nsuperior to all of the explicit methods except the viscous method with large\n.\nThe semi-implicit-centered method derives all of its damping response from\nthe time filter and its stability at large wgAt. For a practical value of\nwgAt = 3.6 and a = 0.3, the damping is much stronger than with any of the\nexplicit methods. As noted above, however, such a value of the filter\ncoefficient a may result in undesirable damping of the meteorological mode.\n21","In summary, none of the explicit methods have a significant effect on\nthe amplitude of the meteorological mode, and the maximum effect on the\ngravity mode amplitude is obtained by the viscous method. The semi-\nimplicit-backward formulation exhibits greater damping of the gravity mode\nand less effect on the meteorological mode than the semi-implicit-centered.\nWe next examine the amplitude responses of the several methods within\nthe context of integration from real and unbalanced initial data.\n3. EXPERIMENTAL INTEGRATIONS\nA primitive-equation barotropic model was integrated from real initial\ndata to 36 hours. Based on the preceding analysis, five integration methods\nwere investigated: two explicit methods, the Euler-backward and the\nexplicit-centered with the heavy (a = 0.3) time filter; the viscous method\nwith = 107m2sech and both with a light (a = 0.9) time filter;\nand two semi-implicit methods, the centered version with the heavy (a = 0.3)\ntime filter, and the backward formulation with the light (a = 0.9) time\nfilter. For reference, an integration of the explicit-centered without the\ntime filter is included.\nThe domain of the model is a quasi-hemispheric rectangle on a polar stereo-\ngraphic projection of the Northern Hemisphere. The grid lattice is a 27 by\n29 array of points separated by 762 km at 6ON. Wind components, u and V, are\ndefined at the centers of grid boxes, while the geopotential is defined at\ngrid points. This arrangement of history variables has previously been used\nby McPherson (1971). Lateral boundary specifications are such that geo -\npotentials are invariant with time at the outer row of grid points, and the\nvelocity components are invariant with time at the outer row of grid boxes.\n22","Initial data were the same for all experiments. Operational National\nMeteorological Center analyses of 500-mb heights and wind components were\nobtained for 0000 GMT, 30 March 1973, and extracted to the grid lattice\ndescribed above. No attempt was made to reduce the rather large imbalances\npresent initially.\nThe finite-difference equations have the form\n(58)\n= U\n(59)\n=\n-\nxy\n=\n(60)\nwhere the velocity components are scaled by the map factor m, and\n5 =\nK 1/[(mu)2+ (mv)2] =\nfo = Coriolis parameter at 45N,\nf = deviation of Coriolis parameter,\nC = gH, H = mean depth of the fluid = 5572 m.\n12 234 m sec-1\nThe averaging and differencing notation is standard:\nwhere 4 is an arbitrary variable. The right-hand sides of (58-60) are\nevaluated identically in each integration, using values at the current\ntime, t = nAt. The difference between explicit and semi-implicit methods\n23","lies in the treatment of the second term on the left sides of (58-60).\nSpecifically, the forms of each method are:\nExplicit-centered\n(61)\n(62)\n(63)\nEuler-backward\n(64)\n(65)\n*om\n(66)\n(67)\n=\n(68)\n=\npnt1\n(69)\nViscous method\n(70)\nn-1\n(71)\n(72)\n24","Semi-implicit centered\nU\n(73)\n=\nV\n(74)\ne =\n(75)\nSemi-implicit backward\nTHE\nU\n(76)\n=\nV\n(77)\nZ\n(78)\n=\nFor those methods in which the Robert time filter was employed, the form\nof (24) was applied to all three history variables. The asterisk\nnotation in (67-69) indicates that the calculations are performed using\nthe variables obtained in (64-66). A11 explicit methods used a 10-minute\ntime step, while a 60-minute time step was used in both semi-implicit\nmethods.\nThe relative damping properties of the several methods are illustrated\nin Figures 8-10 by means of the time variation of height departure from\nits initial value of an individual grid point. In each of the three\ndiagrams, the solid curve represents the reference experiment, the\nexplicit-centered without the time filter. Two short-period oscillations\nare prominent: one with a period of 2-3 hours and large amplitude, and\nanother of larger period (12-14 hours) and smaller amplitude. These\noscillations were also characteristic of several other grid points examined.\n25","In Figure 8, the two explicit methods are compared. Both effectively\nremove the higher frequency oscillation, although the Euler-backward\nmethod has perhaps slightly more effect in the initial six hours. The\nlower requency oscillation is damped but not eliminated by both methods.\nFigure 9 displays the viscous method with two values of the coefficient\n, and the reference experiment. With the lesser value of u, the method is\nrelatively ineffective against either prominent oscillation, requiring\nmuch more time than either nonviscous explicit method to remove the higher-\nfrequency wave. However, for u = 108m2sec-1 the method is quite effective\nagainst both oscillations. Its performance is slightly superior to the\ntwo nonviscous explicit methods initially, but beyond 6 hours the\ndifference is very small.\nThe semi-implicit methods are shown in Figure 10. As suggested by the\nlinear analyses, both methods are superior to any of the explicit methods\nin quickly damping both oscillations. of the two, the backward formulation\nis the most effective.\nIn summary, all of the methods tested proved quite effective in damping\nthe short-period oscillation. All except the viscous method with small u\nmanage to greatly reduce the amplitude in the first few hours. A distinc-\ntion between the methods is more noticeable with respect to the longer-\nperiod wave. The semi-implicit methods, especially the backward version,\nare the most effective, followed by the viscous method with large U.\nWe now describe the performance of these five methods in an experiment\nin four-dimensional data assimilation.\n26","4. APPLICATION TO FOUR-DIMENSIONAL DATA ASSIMILATION\nFor this experiment, the initial data were NMC operational analyses\nof 500-mb heights and winds for 1200 GMT, 8 April 1973, as extracted\nto the grid lattice previously described. Initialization consisted\nof integrating the semi-implicit backward method forward and backward\nover the 12-hour period centered on 1200 GMT, i.e., from the initial\nstate at 1200 GMT the model marched forward to 1800 GMT, then backward\nto 0600 GMT, then forward again to 1800 GMT, for the equivalent of\n5.25 days. The final 12 hours of initialization (i.e., 0600 GMT-1800\nGMT) did not exhibit noticeable gravitational oscillations and were\nstored as a \"reference\" state. The reference heights and winds at\n1200 GMT were taken as the initial conditions for each assimilation\nexperiment.\nValues of 500-mb height as calculated from Vertical Temperature\nProfile Radiometer (VTPR) soundings during the period 0600-1800 GMT,\n8 April 1973, were taken as the data to be inserted into each assimilation\nintegration. The data were stratified by time into 2-hour blocks\ncentered on odd hours; for example, observations between 1200 and 1400\nGMT were assumed synoptic at 1300 GMT. This procedure effectively groups\nthe observations on an orbit-by-orbit basis. No observations south of 20N\nwere used. We did not perform any checking on the accuracy of the\nobservations, beyond an examination for obviously erroneous values.\nThe number of observations per insertion is given in the table below.\n27","Number of VTPR reports per insertion.\nNumber of\nInsertion Time\nObservations\n0700 GMT\n16\n11\n0900\n14\n\"\n1100\n32\n11\n1300\n18\n\"\n1500\n3\n\"\n1700\n5\nTotal\n88\nThe \"repeated insertion\" technique of Morel et al. (1971) was employed\nin the experiment. Each integration proceeded forward and backward\nbetween 0600 and 1800 GMT, as in the initialization, for the\nequivalent of 10.5 days. The individual sets of observations were\ninserted into the integrations repeatedly during this process, according\nto the following procedures:\nAt each insertion time, the available observations were interpolated by\na successive-approximation objective analysis method (Cressman 1959) only\nto nearby grid points; the objective analysis was \"local\" in that sense.\nFour scans through each data set were made, with scan radii of 2.375, 1.8,\n1.1, and 0.9 grid increments respectively. The first guess for the\nobjective analysis was the current predicted height fields. After analysis,\nthe resulting correction to the first guess was smoothed and desmoothed\ntwice, using a filter designed by Shuman (1957), with coefficients of 1/2\nand - 1/3 for smoothing and desmoothing, respectively. The analysis and\nfiltering insures smoothness of the inserted fields along the boundaries of\nthe local objective analysis area.\n28","Following the local objective analysis and filtering procedures,\ncorrections to the wind components were calculated according to the\nlocal balancing procedure of Hayden. The adjusted wind may be expressed\nas\n(79)\nwhere Va is the adjusted wind, VLF is the predicted wind, gf is the\npredicted geostrophic component, and V ga is the geostrophic component as\ncalculated from the objectively-analyzed heights. This amounts to a one-\nfor-one replacement of the forecast geostrophic wind by the \"observed\"\ngeostrophic wind.\nThe height and wind corrections were inserted directly into the model\nat both current (t = nAt) and immediate past (n-1) times in all except\nthe Euler-backward method. This procedure was found to be beneficial in\ninhibiting the development of a temporal computational mode.\nAt each insertion time, the root-mean-square (RMS) difference, Ef,\nbetween the observations and the current prediction interpolated to the\nobservation locations, was calculated. After each analysis, a similar\nRMS difference, Ea' between the interpolated analysis and the observations\nwas formed. Following the completion of each half-cycle (0600-1800 GMT\nor 1800-0600 GMT), \"pooled\" values and Ea were formed from all six\ninsertions. These numbers form the basis for evaluating the experiment,\nand are tabulated in Table 18. The experiments are numbered as follows:\n1. Euler-backward\n2. Explicit-centered, a = 0.3\n3. Viscous method, u = 108m2sec-1 a = 0.9\n4. Semi-implicit centered, a = 0.3\n5. Semi-implicit backward, a=0.9.\n29","A measure of the degree to which the data have been assimilated may be\nobtained from the behavior of the quantity\ngk=\n(80)\nwhere the superscript indicates the cycle number. Perfect assimilation\nof error-free data would be indicated by Ea and S = 0: the model\nwould return to the same perfect state on cycle (k+1) as it had on the\n(k) th cycle. In practice, imperfect data prevents E = 0, and the insertion\nof other data sets during the cycle precludes s = 0. More reasonable\ncriteria for successful data assimilation are that and E should decline\nduring the cycling process, approaching asymptotically a level depending\nupon the error characteristics of the data, and that S should likewise\nbecome small.\nFigure 11 displays E F as a function of the number of cycles for each of\nthe five experiments. The values plotted for cycle 0.25, the first 6 hours\nfrom 1200 GMT to 1800 GMT, include three data sets with only 26 observations.\nBy the end of the first backward cycle (1800 GMT-0600 GMT), all of the\ndata sets have been inserted at least once, and E f for all experiments\nare greater than 30 m. However, when all the data sets have been inserted\nat least twice (0600 GMT-1800 GMT), the values of E f drop sharply, and\ndecline slowly in subsequent cycles.\nFigure 12 shows the behavior of S during the cycling process. A minimum\nis reached in all five experiments by the end of the first forward cycle\n(1.0). Afterward, there is a distinct difference between the forward and\nbackward half cycles which is attributable to the inhomogeneous distribution\nof data within the interval.\n30","Both illustrations show that successful assimilation, according to our\ncriteria, has been achieved by all of the integration methods, and that\nmost of the assimilation has been accomplished when each data set has been\ninserted twice. Experiments reported elsewhere by Kistler and McPherson\n(1974) suggest that the rapidity of this adjustment is due in large\nmeasure to the technique of locally balancing the winds at insertion times.\nIntracomparison of Ef and s for the five integration methods reveals\nlittle difference between them. Orbit by orbit and cycle by cycle, the\nmethods are very similar in performance. For example, at the end of cycle\n1, the greatest difference in E f between any of the five is only 1.3 m.\nBy cycle 2, this has been reduced to 0.7 m, and to 0.4 m by cycle 4.\nTherefore, the differing damping characteristics of these five methods\nwas not an overriding factor with regard to these four-dimensional data\nassimilation experiments.\n5. SUMMARY\nSeveral methods of damping gravitational oscillations, such as might be\nintroduced during the assimilation of asynoptic data, have been examined\nboth theoretically and experimentally. Although there are differences in\ndamping characteristics, suggesting that one method or another might be\npreferable, their application to the assimilation of real data indicates\nthat these differences are not of major importance.\nThe selection of a damping integration method for use in four-dimensional\ndata assimilation may then be made on some other basis, such as computa- -\ntional economics or performance criteria other than the damping of gravi-\ntational oscillations. The semi-implicit methods offer the greatest\npotential for economical use in data assimilation.\n31","of the explicit methods examined here, the Euler-backward is the least\neconomical. Both the viscous method and the explicit-centered method\nwith the time filter require approximately the same amount of computer\ntime, and both performed equally well in the experiments reported here.\nHowever, there is evidence (McPherson 1973) that unfortunate side effects,\nincluding suppression of precipitation, and unrealistic amplification of\nflow patterns near mountains, may result from using the viscous method in\na multilevel model. A prudent selection among the explicit methods would\ntherefore be the explicit-centered scheme in conjunction with the time\nfilter.\nACKNOWLEDGMENTS\nThe authors are indebted to J. A. Brown, W. D. Bonner, C. H. Dey,\nJ. P. Gerrity, and J. B. Hovermale, of NMC, for their comments and\nsuggestions during the course of the research and on the manuscript.\nW. G. Collins and P. D. Polger, also of NMC, kindly allowed us to\nuse certain of their computer programs. Mrs. Doris Gordon assisted\nwith the programming. The manuscript was typed by Mrs. Mary Daigle\nand Mrs. Kay Weinstein, and the illustrations were drafted by Eugene\nBrown.\n32","REFERENCES\nAsselin, Richard, \"Frequency filter for time integrations,\" Monthly\nWeather Review, vol. 100, no. 6, June 1972, pp. 487-490.\nBengtsson, Lennart, and Gustavsson, Nils, \"Assimilation of non-synoptic\nobservations,\" Tellus, vol. 24, no. 5, Stockholm, Sweden, 1972,\npp. 383-399.\nCressman, George, \"An operational objective weather analysis system,\"\nMonthly Weather Review, vol. 87, no. 10, October 1959, pp. 367-374.\nHalberstam, Isidore, \"A study of three finite-difference schemes and\ntheir role in asynoptic meteorological data assimilation,\" Journal\nof the Atmospheric Sciences, vol. 31, no. 8, November 1974, pp. 1964-\n1973.\nHayden, Christopher, \"Experiments in the four-dimensional assimilation\nof Nimbus 4 SIRS data,\" Journal of Applied Meteorology, vol. 12, no. 3,\nApril 1973, pp. 425-436.\nKistler, Robert, and McPherson, Ronald, \"On the use of local balancing\nin four-dimensional data assimilation,\" submitted for publication in\nMonthly Weather Review, 1974.\nKurihara, Yoshio, \"On the use of implicit and iterative methods for\nthe time integration of the wave equation,\" Monthly Weather Review,\nvol. 93, no. 1, January 1965, pp. 33-46.\nKwizak, Michael, and Robert, Andre, \"A semi-implicit scheme for grid\npoint atmospheric models of the primitive equations,\" Monthly Weather\nReview, vol. 99, no. 1, January 1971, pp. 32-36.\n33","Lamb, Horace, Hydrodynamics, 6th Edition, Dover Publications, New York,\nN. Y., 1932, 738 pp.\nMcPherson, Ronald, \"Note on the semi-implicit integration of a fine mesh\nlimited-area prediction model on an offset grid,\" Monthly Weather Review,\nvol. 99, no. 3, March 1971, pp. 242-246.\nMcPherson, Ronald (National Meteorological Center, NOAA, Suitland, Md.) ,\n\"Noise suppression in primitive equation models,\" paper presented at\nthe Second Conference on Numerical Prediction, 1-4 October 1973,\nMonterey, California, 1973.\nMorel, Pierre, Lefevre, G., and Rabreau, G., \"On initialization and non-\nsynoptic data assimilation, Tellus, vol. 23, no. 3, Stockholm, Sweden,\n1971, pp. 197-206.\nMorel, Pierre, and Talagrand, Olivier, \"The dynamic approach to meteoro-\nlogical data assimilation,\" Tellus, vol. 26, no. 3, Stockholm, Sweden,\n1974, pp. 334-344.\nRobert, André, \"The integration of a low order spectral form of the\nprimitive meteorological equations,\" Journal of the Meteorological\nSociety of Japan, ser. 2, vol. 44, no. 5, Tokyo, 1966.\nRutherford, Ian, and Asselin, Richard, \"Adjustment of the wind field to\ngeopotential data in a primitive equations model,\" Journal of the\nAtmospheric Sciences, vol. 29, no. 6, September 1972, pp. 1059-1063.\nSadourny, Robert, \"Forced geostrophic adjustment in large scale flows,\"\nunpublished manuscript, Laboratoire de Meteorologie Dynamique du CNRS,\n92--Meudon-Bellevue, France, 1973.\n34","Shuman, Frederick, \"Numerical methods in weather prediction: II.\nSmoothing and filtering, Monthly Weather Review, vol. 85, no. 11,\nNovember 1957, pp. 357-361.\nShuman, Frederick, and Stackpole, John, \"The currently operational NMC\nmodel, and results of a recent numerical experiment,\" Proceedings WMO/\nIUGG Symposium of Numerical Weather Prediction, Tokyo, Japan, November 26-\nDecember 4, 1968, II-85,98, Japan Meteorological Agency, Tokyo, 1969.\nTalagrand, Olivier, \"On the damping of high-frequency motions in four-\ndimensional assimilation of meteorological data,\" Journal of the\nAtmospheric Sciences, vol. 29, no. 8, November 1972, pp. 1571-1574.\n35","a=0.3\n.3091\n.3354\n.3771\n.4333\n.5057\n.6070\n1.0538\n1.3417\n1.5906\n1.8243\n2.2700\na=0.3\n.9974\n.9895\n.9753\n.9531\n.9186\n.8578\n.5505\n.4784\n.4434\n.4221\n.3974\n(14)\n(14)\nSEMI-IMPLICIT BACKWARD\nSEMI-IMPLICIT BACKWARD\na=0.5\n.5033\n.5134\n.5303\n.5544\n.5867\n.6297\n.6905\n1.1470\n1.4527\n1.7130\n2.1868\na=0.5\n.9985\n.9937\n.9855\n.9731\n.9553\n.9292\n.8876\n.5611\n.4657\n.4154\n.3598\n(13)\n(13)\nTABLE 2. - AMPLITUDE RESPONSE FOR METEOROLOGICAL (COMPUTATIONAL) MODE, EXPLICIT AND SEMI-IMPLICIT METHODS\nAMPLITUDE RESPONSE FOR METEOROLOGICAL (PHYSICAL) MODE, EXPLICIT AND SEMI-IMPLICIT METHODS\na=0.9\n.9002\n.9007\n.9015\n.9027\n.9042\n.9060\n.9078\n.9087\n.8995\n1.3690\n1.9754\na=0.9\n.9999\n.9995\n.9988\n.9980\n.9968\n.9955\n.9943\n.9942\n.6613\n1.0054\n.4595\n(12)\n(12)\n.9999\n.9994\n.9987\n.9976\n.9960\n.9938\n.9905\n.9851\n.9733\n1.1961\n1.9079\n.5239\n1.0001\n1.0006\n1.0013\n1.0024\n1.0039\n1,0061\n1.0094\n1.0149\n1.0271\n.8357\na=1.0\na=1.0\n(11)\n(11)\na=0.3\n.3092\n.3359\n.3784\n.4357\n.5103\n.6185\n1.0455\n1.3356\n1,5849\n1,8187\n2,2642\na=0.3\n.9973\n.8452\n.9889\n.9739\n.9504\n.9134\n.5572\n.4828\n.4471\n.4255\n.4006\n(10)\n(10)\nSEMI-IMPLICIT CENTERED\nSEMI-IMPLICIT CENTERED\n.7094\n,9211\n.5912\n.6376\n1.1392\n1.4475\n1.7083\n2,1824\na=0.5\n.9983\n.9932\n.9842\n.9706\n.9508\n.8680\n.5680\n.4703\n.4195\n.3635\na=0.5\n.5035\n.5139\n.5316\n.5569\n(9)\n(9)\na=0.9\n.9003\n.9012\n.9028\n.9051\n.9083\n.9124\n.9178\n.9253\n.9377\n1.3651\n1.9744\na=0.9\n.9997\n.9989\n.9976\n.9956\n.9928\n.9892\n.9843\n.9775\n.9659\n.6644\n.4610\n(8)\n(8)\n1.0000\n1.0000\n1,0000\n1,0000\n1,0000\n1,0000\n1.0000\n1.0000\n1,0000\n1.1775\n1,9079\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1,0000\n.8492\n.5240\na=1.0\na=1.0\n(7)\n(7)\na=0.3\n.3089\n.3348\n.3760\n.4317\n.5043\n.6083\n1.0216\n1.3138\n1.5622\n1.7944\n2.2363\na=0.3\n.9973\n.9740\n.9889\n.9506\n.9142\n.8485\n.5624\n.4835\n.4467\n.4244\n.3989\n(6)\n(6)\nEXPLICIT CENTERED\nEXPLICIT CENTERED\na=0.5\n.5033\n.5134\n.5304\n.5548\n.5878\n.6325\n.7003\n1.1070\n1.4197\n1.6801\n2.1513\na=0.5\n.9983\n.9932\n.9842\n.9707\n.9511\n.9220\n.8716\n.5784\n.4738\n.4209\n.3630\n(5)\n(5)\n.9352\n1.3132\n1,9338\n.9989\na=0.9\n.9003\n.9012\n.9027\n.9049\na=0.9\n.9997\n.9976\n.9956\n.9929\n.9893\n.9845\n.9779\n.9672\n.6896\n.4695\n.9079\n.9118\n.9169\n.9240\n(4)\n(4)\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n.5367\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.8633\na=1.0\na=1.0\n(3)\n(3)\nTABLE 1.\nEULER\nEULER\n.9950\n.9806\n.9582\n.9304\n.9014\n.8773\n.8661\n.8773\n.9198\n1.0000\n1.2781\n(2)\n(2)\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.2\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.2\n(1)\nWAt\n0.1\n(1)","a=0.3\n1.005\n1.019\n1.046\n1.090\n1.166\n1.334\n1.281\n0.947\n0.740\n0.597\n0.414\na=0.3\n-3.354\n-3.255\n-3.129\n-3.011\n-2.930\n-2.280\n-1.713\n-1.598\n-1.466\n-1.347\n-1.151\n(14)\n(14)\nTABLE 4 - - RATIO OF NUMERICAL TO ANALYTIC PHASE SPEEDS FOR THE METEOROLOGICAL (COMPUTATIONAL) MODE, EXPLICIT AND SEMI-IMPLICIT METHODS\nTABLE 3. -RATIO OF NUMERICAL TO ANALYTIC PHASE SPEEDS FOR THE METEOROLOGICAL (PHYSICAL) MODE, EXPLICIT AND SEMI-IMPLICIT METHODS\nSEMI-IMPLICIT BACKWARD\nSEMI-IMPLICIT BACKWARD\na=0.5\n1.004\n1.014\n1.034\n1.065\n1.112\n1.191\n1.360\n1.409\n1.126\n0.673\na=0.5\n-2.040\n-2.041\n-2.045\n-2.054\n-2.077\n(13)\n0.937\n-2.129\n-2.219\n-1.639\n-1.516\n-1.391\n-1.183\n(13)\na=0.9\n1.002\n1.008\n1.019\n1.035\n1.058\n1.090\n1.226\na=0.9\n(12)\n1.137\n1.209\n1.358\n1.520\n-1.141\n-1.148\n-1.158\n-1.175\n-1.197\n-1.229\n-1.276\n-1.349\n-1.497\n-1.482\n-1.253\n(12)\na=1.0\n1.002\n1.007\n1.016\n1.030\n1.049\n1.076\n1.113\n1.168\n1.261\n1.482\n1.272\na=1.0\n-1.029\n-1.034\n-1.043\n(11)\n-1,057\n-1.076\n(11)\n-1.103\n-1.140\n-1.195\n-1,288\n-1.508\n-1.299\na=0.3\n1.005\n1.163\n1.325\n1.260\n0.410\na=0.3\n-3.352\n-3.252\n(10)\n1.019\n1.045\n1.088\n0.936\n0.733\n0.592\n-3.124\n-3.003\n-2.919\n-2,298\n(10)\n-1.743\n-1,619\n-1.484\n-1.364\n-1.166\nSEMI-IMPLICIT CENTERED\nSEMI-IMPLICIT CENTERED\na=0.5\n1.004\n1.014\n1.033\n1.064\n1.111\n1.188\n1.353\n1.380\n1.109\n0.919\n0.664\na=0.5\n-2.040\n-2.040\n-2.043\n-2,052\n-2,073\n-2.122\n-2,231\n-1.673\n-1.539\n-1.410\n-1,200\n(9)\n(9)\na=0.9\n1.002\n1.008\n1.019\n1.035\n1.058\n1.090\n1.136\n1.209\n1.359\n1.482\n1.205\na=0.9\n-1.141\n-1.148\n-1.158\n-1.174\n-1.197\n-1.229\n-1,275\n-1,348\n-1.498\n-1,521\n-1.274\n(8)\n(8)\na=1.0\n1.002\n1.007\n1.016\n1.030\n1.049\n1.076\n1.113\n1.168\n1.262\n1.557\n1.296\na=1.0\n-1.029\n-1.034\n-1.043\n-1.057\n-1.076\n-1,103\n-1,140\n-1.195\n-1.289\n-1.557\n-1.415\n(7)\n(7)\na=0.3\n1.004\n1.018\n1.044\n1.086\n1.157\n1.309\n1.284\n0.952\n0.745\n0.601\n0.417\na=0.3\n-3.279\n-3.201\n-3.079\n-2.963\n-2,882\n-2,343\n-1.744\n-1.626\n-1.494\n-1.374\n-1.178\n(6)\n(6)\nEXPLICIT CENTERED\nEXPLICIT CENTERED\na=0.5\n1.003\n1.014\n1.032\n1.062\n1.107\n1.181\n1.332\n1.405\n1.128\n0.935\n0.676\na=0.5\n-2.000\n-2.001\n-2.004\n-2.013\n-2.034\n-2.082\n-2.205\n-1.679\n-1.549\n-1.421\n-1.212\n(5)\n(5)\na=0.9\n1.002\n1.008\n1.018\n1.034\n1.056\n1.086\n1.130\n1.198\n1.329\n1.498\n1.220\na=0.9\n-1.113\n-1.119\n-1.129\n-1.145\n-1.167\n-1.197\n-1.244\n-1.308\n-1.440\n-1.533\n-1.287\n(4)\n(4)\na=1.0\n1.002\n1.007\n1.016\n1.029\n1.047\n1.073\n1.108\n1.159\n1.244\n1.571\n1.309\na=1.0\n-1.002\n-1.007\n-1.016\n-1.029\n-1.047\n-1.073\n-1.108\n-1.159\n-1.244\n-1.571\n-1.309\n(3)\n(3)\nEULER\n1.007\nEULER\n1.027\n1.062\n1.111\n1.176\n1.255\n1.345\n1.435\n1.514\n1.571\n-1.016\n(2)\n(2)\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n(1)\n0.6\n0.7\n0.8\n0.9\n1.0\n1.2\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n(1)\n0.7\n0.8\n0.9\n1.0\n1.2","a=0.3\n.9890\n.9589\n.9158\n.5803\n.5429\n.5135\n.4900\n(14)\n.8670\n.8177\n.7712\n.7288\n.6911\n.6578\n.6285\n.4711\n.4556\n.4427\n.4320\n.4230\n.4153\n.4088\n.4031\n.3982\n.3939\n.3902\n.3869\n.3840\n.3814\n.3790\nSEMI-IMPLICIT BACKWARD\na=0.5\n.9900\n.9625\n.9229\n.8776\n.8412\n.7867\n.7455\n.7081\n.6745\n.6444\n.5932\n.5519\n.5180\n.4898\n.4661\n.4459\n(13)\n.4285\n.4133\n.4001\n.3884\n.3780\n.3688\n.3605\n.3530\n.3463\n.3401\n.3346\n.3295\n.3248\nAMPLITUDE RESPONSE FOR EASTWARD-MOVING GRAVITY (PHYSICAL) MODE, EXPLICIT AND SEMI-IMPLICIT METHODS\na=0.9\n.9914\n.9326\n.9674\n.8921\n.8498\n.8084\n.7693\n.7332\n.5371\n(12)\n.7001\n.6700\n.6176\n.5739\n.5057\n.4787\n.4551\n.4343\n.4158\n.3992\n.3843\n.3707\n.3583\n.3470\n.3366\n.3269\n.3179\n.3096\n.3018\n.2944\na=1.0\n.9917\n.9684\n.8533\n.8125\n.6226\n.5787\n.5417\n.5100\n.4826\n.4586\n(11)\n.9345\n.8948\n.7739\n.7381\n.7052\n.6751\n.4375\n.4187\n.4018\n.3866\n.3728\n.3602\n.3486\n.3379\n.3280\n.3189\n.3103\n.3023\n.2948\na=0.3\n.9973\n.9895\n.9770\n.9606\n.9412\n(10)\n.9197\n.8970\n.8738\n.8507\n8281\n.7856\n.7477\n.7145\n.6858\n.6610\n.6397\n.6213\n.6053\n,5914\n.5792\n.5684\n.5589\n.5505\n.5429\n.5361\n.5300\n.5245\n.5194\n.5148\nSEMI-IMPLICIT CENTERED\na=0.5\n.9983\n.9858\n.9935\n.9757\n.9638\n.9506\n.9368\n.9227\n.9088\n.8952\n.8697\n.8470\n.8272\n.8101\n.7954\n.7827\n.7716\n.7620\n.7536\n.7462\n.7396\n.7338\n.7286\n.7239\n.7197\n.7158\n.7123\n.7092\n.7062\n(9)\na=0,9\n.9997\n.9990\n.9978\n.9962\n.9943\n.9878\n.9922\n.9900\n.9855\n.9834\n.9793\n.9756\n.9724\n.9650\n.9696\n.9671\n.9631\n.9615\n,9601\n.9588\n.9576\n.9566\n.9557\n,9549\n.9541\n.9534\n.9528\n.9522\n.9517\n(8)\na=1.0\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1,0000\n1,0000\n1,0000\n1.0000\n1.0000\n1,0000\n1.0000\n1.0000\n1.0000\n1,0000\n1.0000\n1.0000\n1,0000\n1,0000\n1.0000\n1,0000\n1.0000\n1.0000\n1,0000\n1.0000\n1.0000\n(7)\na=0.3\n.9973\n.9889\n.9740\n.9506\n.9142\n.8485\n.5624\n.4835\n.4467\n.4244\n.3989\n(6)\nEXPLICIT CENTERED\na=0,5\n.9983\n.9932\n.9842\n.9707\n.9511\n.9220\n.8716\n.5874\n.4738\n.4209\n.3630\n(5)\na=0.9\n.9997\n.9989\n.9976\n.9956\n.9929\n.9893\n.9845\n.9779\n.9672\n.6896\n.4695\n(4)\na=1.0\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n.5367\n(3)\nTABLE 5\nEULER\n.9950\n.9806\n.9582\n.9304\n.9014\n.8773\n.8661\n.8773\n.9198\n1.0000\n1.2781\n(2)\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n1.4\n(1)\n0.9\n1.0\n1.2\n1.6\n1.8\n2.0\n2.2\n2.4\n2.6\n2.8\n3.0\n3.2\n3.4\n3.6\n3.8\n4.0\n4.2\n4.4\n4.6\n4.8","METHODS\na=0.3\n.2992\n(14)\n.2969\n.2933\n.2888\n.2836\n.2781\n.2724\n.2668\n.2613\n.2561\n.2464\n.2378\n.2306\n.2242\n.2188\n.2141\n.2100\n.2065\n.2035\n.2008\n.1985\n.1964\n.1946\n.1930\n.1916\n.1904\n.1892\n.1882\n.1873\nSEMI-IMPLICIT\nSEMI-IMPLICIT BACKWARD\na=0.5\n(13)\n.4979\n.4920\n.4830\n.4718\n.4594\n.4465\n.4336\n.4210\n.4090\n.3975\n.3766\n.3582\n.3422\n.3280\n.3156\n.3046\n.2948\n.2861\n.2782\n.2712\n.2648\n.2590\n.2537\n.2489\n.2445\n.2404\n.2367\n.2333\n.2301\nTABLE 6.--AMPLITUDE RESPONSE FOR THE COMPUTATIONAL MODE CORRESPONDING TO THE EASTWARD-MOVING GRAVITY MODE, EXPLICIT AND\na=0.9\n.8949\n.8807\n.8595\n(12)\n.8341\n.8067\n.7791\n.7522\n.7268\n.7029\n.6807\n.6411\n.6072\n.5780\n.5526\n.5304\n.5108\n.4933\n.4777\n.4636\n.4508\n.4391\n.4285\n.4186\n.4095\n.4011\n.3932\n.3859\n.3798\n.3726\na=1.0\n(11)\n.9941\n.9776\n.9531\n.9239\n.8926\n.8612\n.8308\n.8021\n.7753\n.7505\n.7064\n.6688\n.6365\n.6086\n.5842\n.5627\n.5437\n.5266\n.5112\n.4973\n.4846\n.4729\n.4622\n.4523\n.4431\n.4346\n.4267\n.4192\n.4122\na=0.3\n(10)\n.3001\n.3004\n.3009\n.3016\n.3024\n.3034\n.3046\n.3059\n.3073\n.3088\n.3120\n.3137\n.3190\n.3225\n.3260\n.3295\n.3328\n.3359\n.3390\n.3419\n.3446\n.3472\n.3497\n.3520\n.3542\n.3563\n.3583\n,3602\n.3619\nSEMI-IMPLICIT CENTERED\na=0.5\n.5003\n.5010\n.5022\n.5039\n.5059\n.5083\n.5109\n.5137\n.5166\n.5197\n.5259\n.5320\n.5380\n.5436\n.5488\n.5623\n.5728\n.5536\n.5581\n.5661\n.5696\n.5758\n.5786\n.5812\n.5835\n.5857\n.5878\n.5897\n.5915\n(9)\na=0.9\n.9001\n.9004\n.9010\n.9017\n.9025\n.9034\n.9045\n.9056\n.9067\n.9078\n.9099\n.9120\n.9139\n.9156\n.9171\n.9185\n.9198\n.9209\n.9219\n.9229\n.9237\n.9245\n.9252\n.9258\n.9264\n.9269\n.9274\n.9279\n.9283\n(8)\na=1.0\n1.0000\n1.0000\n1.0000\n1,0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1,0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n(7)\na=0.3\n.3089\n.3348\n.3760\n.4317\n.5043\n.6083\n1.0216\n1.3138\n1.5622\n1.7944\n2.2363\n(6)\nEXPLICIT CENTERED\na=0.5\n.5033\n.5134\n.5304\n.5548\n.5878\n.6325\n.7003\n1.1070\n1.4197\n1.6801\n2.1513\n(5)\na=0.9\n.9003\n.9012\n.9027\n.9049\n.9079\n.9118\n.9169\n.9240\n.9352\n1.3132\n1.9338\n(4)\na=1.0\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.8633\n(3)\nEULER\n(2)\n0.2\n0.3\n0.4\n0.5\n(1)\nWAt\n0.1\n0.6\n0.7\n0.8\n0.9\n1.0\n1.2\n1.4\n1.6\n1.8\n2.0\n2.2\n2.4\n2.6\n2.8\n3.0\n3.2\n3.4\n3.6\n3.8\n4.0\n4.2\n4.4\n4.6\n4.8",".188\n.117\n.105\n.095\n.086\n.079\n.072\n.066\n.061\n.056\n.052\n.398\n.336\n.287\n.247\n.214\n.165\n.146\n.131\n.826\n.758\n.692\n.630\n.573\n.521\n.475\nTABLE 7 RATIO OF NUMERICAL TO ANALYTIC PHASE SPEED FOR THE EASTWARD-MOVING GRAVITY (PHYSICAL) MODE, EXPLICIT AND SEMI-IMPLICIT METHODS\na=0.3\n.986\n.948\n.892\n(14)\nSEMI-IMPLICIT BACKWARD\n.280\n.252\n.228\n.208\n.190\n.174\n.161\n.148\n.138\n.128\n.119\n.111\n.104\n.098\n.092\na=0.5\n.988\n.954\n.904\n.846\n.786\n.726\n.671\n.619\n.573\n.531\n.459\n.401\n.353\n.313\n(13)\n.264\n.205\n.535\n.482\n.438\n.400\n.369\n.342\n.318\n.298\n.280\n.249\n.236\n.225\n.214\n.196\n.188\n.180\n.174\na=0.9\n.990\n.962\n.921\n.872\n.821\n.770\n.722\n.678\n.637\n.600\n(12)\n.190\n.280\n.266\n.253\n.241\n.231\n.221\n.212\n.204\n.197\na=1.0\n.549\n.496\n.453\n.416\n.385\n.358\n.334\n.314\n.296\n.990\n.963\n.924\n.877\n.827\n.778\n.732\n.688\n.648\n.612\n(11)\n.473\n.494\n.453\n.434\n.410\n.384\n.362\n.342\n.323\n.307\na=0.3\n.998\n.991\n.981\n.967\n.950\n.931\n.911\n.888\n.866\n.843\n.797\n.752\n.710\n.671\n.635\n.602\n.572\n.544\n.518\n(10)\nSEMI-IMPLICIT CENTERED\n.784\na=0.5\n.998\n.737\n.693\n.653\n.615\n.581\n.550\n.552\n.496\n.473\n.451\n.431\n.413\n.396\n.380\n.366\n.352\n,340\n.326\n.991\n.980\n.965\n.947\n.927\n.904\n.881\n.857\n.832\n(9)\n.770\n.722\n.677\n.456\n.435\n.415\n.397\n.381\n.366\n.351\n.338\n.326\n.315\n.636\n.598\n.564\n.533\n.505\n.480\na=0.9\n.998\n.990\n.978\n.962\n.943\n.921\n.897\n.872\n.846\n.821\n(8)\na=1.0\n.768\n.433\n.413\n.395\n.379\n.364\n.349\n.336\n.324\n.313\n.997\n.990\n.978\n.961\n.942\n.920\n.896\n.871\n.845\n.819\n.719\n.674\n.633\n.596\n.562\n.531\n.503\n.477\n.454\n(7)\na=0.3\n1.004\n1.018\n1.044\n1.086\n1.157\n1.309\n1.284\n0.952\n0.745\n0.601\n0.417\n(6)\nEXPLICIT CENTERED\na=0.5\n1.003\n1.014\n1.032\n1.062\n1.107\n1.181\n1.332\n1.405\n1.128\n0.935\n0.676\n(5)\n1.220\na=0.9\n1.002\n1.008\n1.018\n1.034\n1.056\n1.086\n1.130\n1.198\n1.329\n1.498\n(4)\na=1.0\n1.002\n1.007\n1.016\n1.029\n1.047\n1.073\n1.108\n1.159\n1.244\n1.571\n1.309\n(3)\nEULER\n1.007\n1.027\n1.062\n1.111\n1.176\n1.255\n1.345\n1.435\n1.514\n1.571\n-1.016\n(2)\n3.2\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.2\n1.4\n1.6\n1.8\n2.0\n2.2\n2.4\n2.6\n2.8\n3.0\n3.4\n3.6\n3.8\n4.0\n4.2\n4.4\n4.6\n4.8\n(1)\n40","(14)\na=0.3\n.351\n.345\n.334\n.322\n.307\n.291\n.275\n.259\n.244\n.229\n.202\n.178\n.157\n.037\n.139\n.124\n.111\n.099\n.089\n.081\n.073\n.066\n.061\n.055\n.051\n.047\n.043\n.040\n.034\nTABLE 8 RATIO OF NUMERICAL TO ANALYTIC PHASE SPEED FOR THE COMPUTATIONAL MODE CORRESPONDING TO THE EASTWARD-MOVING GRAVITY MODE,\nSEMI-IMPLICIT BACKWARD\na=0.5\n.548\n(13)\n.537\n.520\n.499\n.475\n.451\n.427\n.404\n.382\n.361\n.238\n.323\n.291\n.262\n.217\n.199\n.183\n.168\n.156\n.144\n.134\n.125\n.117\n.109\n.103\n.096\n.091\n.086\n.081\n(12)\na=0.9\n.679\n.662\n.636\n.606\n.572\n.539\n.507\n.477\n.449\n.423\n.378\n.340\n.308\n.280\n.257\n.237\n.220\n.204\n.191\n.179\n.168\n.159\n.150\n.142\n.135\n.128\n.122\n.117\n.112\na=1.0\n(11)\n.695\n.677\n.650\n.617\n.583\n.548\n.515\n.483\n.454\n.428\n.380\n.342\n.309\n.282\n.258\n.238\n.220\n.205\n.191\n.179\n.169\n.159\n.150\n.142\n.135\n.129\n.123\n.117\n.112\na=0.3\n.353\n.352\n.351\n.349\n.346\n.343\n.340\n.336\n.332\n(10)\n.328\n.319\n.250\n.309\n.299\n.289\n.279\n.269\n.259\n.241\n.233\n.225\n.218\n.211\n.204\n.198\n.191\n.186\n.180\n.175\nSEMI-IMPLICIT CENTERED\na=0.5\n.552\n.463\n.549\n.545\n.539\n.532\n.524\n.515\n.505\n.495\n.484\n.441\n420\n.399\n.380\n.362\n.346\n.330\n,316\n.302\n.290\n.278\n,268\n.258\n.248\n.239\n.231\n.224\n.216\n(9)\na=0.9\n.683\n.679\n.672\n.663\n.652\n.639\n.625\n.611\n.595\n.579\n.548\n.518\n.489\n.462\n.437\n.414\n.393\n.374\n.356\n.340\n.325\n.311\n.298\n.286\n.276\n.265\n.256\n.247\n.239\n(8)\na=1.0\n.688\n.700\n.696\n.679\n.667\n.654\n.639\n.623\n.607\n.591\n.558\n.527\n.497\n.469\n.444\n.420\n.399\n.379\n.361\n.344\n,329\n.315\n.302\n.290\n.279\n.267\n.259\n.250\n.242\n(7)\na=0.3\n-3.297\n-3.201\n-3.079\n-2.963\n-2.882\n-2.343\n-1,744\n-1.626\n-1.494\n-1.374\n-1.178\n(6)\nEXPLICIT AND SEMI-IMPLICIT METHODS\nEXPLICIT CENTERED\na=0.5\n-2.000\n-2.001\n-2.004\n-2.013\n-2.034\n-2.082\n-2.205\n-1.679\n-1.549\n-1.421\n-1.212\n(5)\na=0.9\n-1.113\n-1.119\n-1.129\n-1.145\n-1.167\n-1.197\n-1.244\n-1.308\n-1.440\n-1.533\n-1.287\n(4)\na=1.0\n-1.002\n-1.007\n-1.016\n-1.029\n-1.047\n-1.073\n-1.108\n-1.159\n-1.244\n-1.571\n-1.309\n(3)\nEULER\n(2)\n(1)\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.2\n1.4\n1.6\n1.8\n2.0\n2.2\n2.4\n2.6\n2.8\n3.0\n3.2\n3.4\n3.6\n3.8\n4.0\n4.2\n4.4\n4.6\n4.8","-1.44 X 10-9\n-1.12 X 10-3\n-2.87 X 10-3\n-0.12 X 10-9\n-4.94 X 10-4\n-4.93 X 10-4\n0.56 X 10-9\n-2.19 X 10-4\n-2.19 X 10-4\n1.22 X 10-9\n-1.23 X 10-4\n-1.23 X 10-4\n1.97 X 10-9\n-0.79 X 10-4\n-0.79 X 10-4\n6.84 X 10-9\n-0.20 X 10-4\n-0.20 X 10-4\n14.90 X 10-9\n-0.50 X 10-5\n-0.48 X 10-5\nC.\n- EFFECT OF THE VISCOUS TERM ON THE PHASE SPEEDS. C is the real part and Ci is the imaginary part of the phase speed\n-2.29 X 10-4\n-0.37 X 10-4\n2.66 X 10-4\n7.79 X 10-4\n15.72 X 10-4\n108.86 X 10-4\n486.48 X 10-4\nH = 108 m2 sec-1\n-450.64\n-177.67\n-157.14\n-157.02\n-104.79\n-104.64\n-78.64\n-78.44\n-62.95\n-62.72\n-31.63\n-31.21\n-15.04\n-15.66\n49.96\n50.07\n49.97\n49.83\n285.87\n-185.70\n49.62\n-213.70\n314.08\n49.34\n-223.36\n324.02\n48.97\n-228.29\n329.23\n46.12\n-240.86\n344.74\n36.92\n-271.98\n385.06\ncr\n49.96\n184.59\n453.41\n49.83\n243.26\n326.16\n49.62\n238.02\n331.05\n49.34\n236.80\n333.39\n48.97\n236.81\n335.24\n46.12\n242.93\n346.15\n36.92\n272.43\n385.35\nc\n-0.18 X 10-9\n-0.20 X 10-3\n-0.20 X 10-3\n-.01 X 10-9\n-0.49 X 10-4\n-0.49 X 10-4\n.06 X 10-9\n-0.22 X 10-4\n-0.22 X 10-4\n0.12 X 10-9\n-0.12 X 10-4\n-0.12 X 10-4\n0.20 X 10-9\n-0.79 X 10-5\n-0.79 X 10-5\n0.68 X 10-9\n-0.20 X 10-5\n-0.20 X 10-5\n1.47 X 10-9\n-0.50 X 10-6\n-0.48 X 10-6\nkci-1\nr\n-0.29 X 10-4\n-0.04 X 10-4\n0.26 X 10-4\n0.78 X 10-4\n1.57 X 10-4\n10.89 X 10-4\n46.90 X 10-4\nH = 107 m² sec-1\n-31.42\n-31.41\n-15.71\n-15.70\n-10.48\n-10.47\n-7.86\n-7.84\n-6.29\n-6.27\n-3.16\n-3.12\n-1.60\n-1.54\n49.96\n-231.19\n331.23\n49.83\n-232.80\n332.96\n49.62\n-233.53\n333.91\n49.34\n-234.30\n334.96\n48.97\n-235.22\n336.25\n46.12\n-242.52\n346.41\n37.39\n-270.50\n383.11\nCI\n49.96\n233.32\n332.27\n49.83\n233.33\n333.34\n49.62\n233.76\n334.07\n49.34\n234.43\n335.05\n48.97\n235.30\n336.30\n46.11\n242.54\n346.42\n37.39\n270.50\n383.11\n/c\nMODE\n1\n2\n3\n1\n2\n3\n1\n2\n3\n1\n2\n3\n1\n2\n3\n1\n2\n3\n1\n2\n3\nTABLE 9\nWAVELENGTH\n(KM)\n1000\n2000\n3000\n4000\n5000\n10000\n20000","0.99728917\n0.98890407\n0.97398548\n0.95060748\n0.91419230\n0.84854319\n0.56238931\n0.48354489\n0.30889681\n0.33477543\n0.37598016\n0.43169101\n0.50424905\n0.60826972\n1.02161807\n1.31383690\na=0.3\n(13)\na=0.3\n(13)\n0.99832591\n0.99319014\n0.98422666\n0.97068362\n0.95106497\n0.92196591\n0.87158545\n0.57845049\n0.47383963\n0.50333814\n0.51340154\n0.53038627\n0.55478728\n0.58778818\n0.63245735\n0.70026150\n1.10696296\n1.41966350\na=0.5\n(12)\na=0.5\n(12)\nH = 108 m² sec-1\n0.99973776\n0.99893826\n0.99757153\n0.99558127\n0.99287189\n0.98928021\n0.98450530\n0.97788729\n0.96718615\n0.68961532\n0.90029476\n0.90118526\n0.90270141\n0.90489943\n0.90787485\n0.91179071\n0.91694778\n0.92400578\n0.93520452\n1.31315188\n(11)\na=0.9\na=0.9\n(11)\n=\nTABLE 11. AMPLITUDE RESPONSE FOR THE METEOROLOGICAL (COMPUTATIONAL) MODE, VISCOUS METHOD\n10. AMPLITUDE RESPONSE FOR THE METEOROLOGICAL (PHYSICAL) MODE, VISCOUS METHOD\n1.00000173\n1.00000347\n1.00000520\n1.00000695\n1.00000868\n1.00001042\n1.00001217\n1.00001391\n1.00001566\n1.00001741\n0.64178427\n1.00000173\n1.00000347\n1.00000520\n1.00000694\n1.00000868\n1.00001042\n1.00001217\n1.00001391\n1.00001566\n1.00001741\n1.55821569\na=1.0\n(10)\na=1.0\n(10)\n0.99728802\n0.98890172\n0.97398177\n0.95060213\n0.91418456\n0.84852964\n0.56237939\n0.48353932\n0.30889626\n0.33477452\n0.37597924\n0.43169059\n0.50424999\n0.60827559\n1.02163002\n1.31384473\na=0.3\na=0,3\n(9)\n(9)\n0.99832460\n0.99318749\n0.98422257\n0.97067794\n0.95105736\n0.92195559\n0.87156889\n0.57843018\n0.47382758\n0.50333724\n0.51339983\n0.53038392\n0.55478455\n0.58778554\n0.63245571\n0.70026452\n1.10698495\n1.41967750\na=0.5\na=0.5\nu = 107 m² sec-1\n(8)\n= 107 m2 sec-1\n(8)\n0.99973624\n0.99893521\n0.99756694\n0.99557511\n0.99286413\n0.98927078\n0.98449409\n0.97787403\n0.96716975\n0.68957024\n0.90029332\n0.90118239\n0.90269711\n0.90489375\n0.90786780\n0.91178237\n0.91693826\n0.92399554\n0.93519426\n1.31319706\na=0.9\na=0.9\n(7)\n(7)\n1.00000017\n1.00000035\n1.00000052\n1.00000069\n1.00000087\n1.00000104\n1.00000121\n1.00000139\n1.00000156\n1.00000173\n0.64174661\n1.00000017\n1.00000035\n1.00000052\n1.00000069\n1.00000087\n1.00000104\n1.00000121\n1.00000139\n1.00000156\n1.00000173\n1.55825335\na=1.0\na=1,0\n(6)\n(6)\n0.99728789\n0.98890146\n0.97398136\n0.95060154\n0.91418370\n0.84852815\n0.56237830\n0.48353871\n0.30889620\n0.33477441\n0.37597913\n0.43169054\n0.50425009\n0.60827624\n1.02163134\n1.31384559\na=0.3\na=0.3\n(5)\n(5)\n0.99832446\n0.99318720\n0.98422212\n0.97067731\n0.95105652\n0.92195445\n0.87156706\n0.57842794\n0.47382625\n0.50333714\n0.51339964\n0.53038366\n0.55478424\n0.58778525\n0.63245553\n0.70026485\n1.10698737\n1.41967904\na=0.5\na=0.5\n(4)\n(4)\nTABLE\n=0\n=0\n0.99973607\n0.99893487\n0.99756643\n0.99557443\n0.99286327\n0.98926973\n0.98442846\n0.97787257\n0.96716793\n0.68956526\n0.90029316\n0.90118207\n0.90269664\n0.90489312\n0.90786702\n0.91178144\n0.91693720\n0.92399942\n0.93519313\n1.31320205\na=0.9\na=0.9\n(3)\n(3)\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n0.64174246\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.00000000\n1.55825751\na=1.0\na=1.0\n(2)\n(2)\n0.1\n0.2\n0.3\n0.5\n0.6\n0.7\n0.8\n(1)\nWAt\n0.4\n0.9\n1.0\n1.1\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.1\n(1)\nWAt","-3.296718\n-3.201367\n-3.079417\n-2.963080\n-2.881715\n-2.342780\n-1.744445\n-1.626425\n1.004406\n1.018245\n1.043680\n1.085775\n1.157392\n1.308979\n1.284385\n0.951846\na=0.3\na=0.3\n(13)\n(13)\n-2.000043\n-2.000760\n-2.003970\n-2.013097\n-2.034421\n-2.081714\n-2.204656\n-1.678727\n-1.548635\n1.003360\n1.013789\n1.032456\n1.061843\n1.107140\n1.181030\n1.332207\n1.404855\n1.127802\na=0.5\na=0.5\n(12)\n(12)\n11= 108 m2 sec-1 =\nu = 108 m2 sec-1 =\nTABLE 13. - - RATIO OF NUMERICAL TO ANALYTIC PHASE SPEED FOR THE METEOROLOGICAL (COMPUTATIONAL) MODE, VISCOUS METHOD\nTABLE 12. RATIO OF NUMERICAL TO ANALYTIC PHASE SPEED FOR THE METEOROLOGICAL (PHYSICAL) MODE, VISCOUS METHOD\n1.001938\n1.007874\n-1.113044\n-1.118966\n-1.129263\n-1.144673\n-1.166493\n-1.197034\n-1.240805\n-1.308421\n-1.439487\n-1.532709\n1.018194\n1,033637\n1.055498\n1.086089\n1.129920\n1.197604\n1.328747\n1.498230\na=0.9\na=0.9\n(11)\n(11)\n-1.001672\n-1.006786\n-1.015637\n-1.028784\n-1.047188\n-1.072489\n-1.107694\n-1.159096\n-1.244152\n-1.564893\n-1.427997\n1.001672\n1.006786\n1.015637\n1.028784\n1.047188\n1,072489\n1.107694\n1.159096\n1,244152\n1.564893\n1.427997\na=1.0\na=1.0\n(10)\n(10)\n-3.296726\n-3.201381\n-3.079434\n-2.963100\n-2.881737\n-2.342750\n-1.744462\n-1.626433\n1.004407\n1.018247\n1.043683\n1.085779\n1.157399\n1,308995\n1.284355\n0.951826\na=0.3\na=0.3\n(9)\n(9)\n-2.000047\n-2.000768\n-2.003982\n-2.013113\n-2.034442\n-2.081740\n-2.204697\n-1.678740\n-1.548640\n1.003361\n1.013791\n1.032460\n1,061848\n1.107148\n1.181042\n1.332233\n1,404826\n1.127781\na=0.5\na=0.5\np = 107 m2 sec-1\nH = 107 m² sec-1\n(8)\n(8)\n-1.113046\n-1.118970\n-1.129269\n-1.144681\n-1.166503\n-1.197047\n-1.240823\n-1.308446\n-1.439535\n-1.532712\n1.001939\n1.007878\n1.018199\n1,033644\n1,055506\n1,086101\n1.129936\n1,197627\n1,328792\n1.498224\na=0.9\na=0.9\n(7)\n(7)\n-1.001674\n-1.006789\n-1.015642\n-1.028791\n-1.047196\n-1.072501\n-1.107709\n-1.159117\n-1.244185\n-1.570796\n-1.427997\n1.001674\n1.006789\n1.015642\n1,028791\n1.047196\n1.072501\n1.107709\n1.159117\n1.244185\n1.570796\n1.427997\na=1.0\na=1.0\n(6)\n(6)\n-3.296727\n-3.201382\n1.004407\n1.018247\n1.043683\n-3.079436\n-2.963102\n-2.881740\n-2.342746\n-1.744464\n-1.626434\n1.085779\n1.157400\n1.308997\n1.284352\n0.951823\na=0.3\na=0.3\n(5)\n(5)\n-2.000048\n-2.000769\n-2.003983\n-2.013114\n-2.034444\n-2.081743\n-2.204702\n-1.678741\n-1.548641\n1.003361\n1.013792\n1.032460\n1.061848\n1.107149\n1.181044\n1.332224\n1.404823\n1.127778\na=0.5\na=0.5\n(4)\n(4)\n= 0\nU= 0\n1.001940\n1.007878\n1.018200\n1.033644\n1.055508\n1.086102\n1.129937\n1.197630\n1.328797\n1.498223\n-1.113046\n-1.118971\n-1.129270\n-1.144682\n-1.166504\n-1.197049\n-1.240825\n-1.308449\n-1.439540\n-1.532712\na=0.9\na=0.9\n(3)\n(3)\n1.001674\n1.006790\n1.015642\n1.028792\n1.047198\n1.072502\n1.107711\n1.159119\n1.244188\n1.570617\n1.427997\n-1.001674\n-1.006790\n-1.015642\n-1.028792\n-1.047198\n-1.072502\n-1.107711\n-1.159119\n-1.244188\n-1.570617\n-1.427997\na=1.0\na=1.0\n(2)\n(2)\nWAt\n0.1\n0.2\n0.3\n0.4\n0,5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.1\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.1\n(1)\n(1)",".97946847\n.95121423\n.91249472\n.85697560\n.76268488\n.53404473\n.44212854\n.40505238\n.30017417\n.32010336\n.36149537\n.42832223\n.54137896\n.87040114\n1.17847429\n1.43257547\nTABLE 15. - - AMPLITUDE RESPONSE FOR THE COMPUTATIONAL MODE CORRESPONDING TO THE EASTWARD-MOVING GRAVITY MODE,\na=0.3\na=0.3\n(9)\n(9)\nTABLE 14 AMPLITUDE RESPONSE FOR THE EASTWARD-MOVING GRAVITY (PHYSICAL) MODE, VISCOUS METHOD\n.97794894\n.95027299\n.91523184\n.86939553\n.80407084\n.67153500\n.42382682\n.34289889\n.48921616\n.48525518\n.49013372\n.50750694\n.54635990\n.65959526\n1.06650882\n1.35980977\na=0.5\na=0.5\nu = 108 m2 sec-1\n(8)\nu = 108 m2 sec-1\n(8)\n.97586850\n.94895932\n.91866064\n.88380356\n.84287744\n.79245720\n.71936317\n.37675350\n.22538409\n.87764260\n.85388858\n.82848714\n.80117269\n.77177578\n.74084662\n.71709223\n1.15049667\n1.47886491\na=0.9\na=0.9\n(7)\n(7)\n.97547947\n.94871425\n.91920697\n.88626494\n.84887892\n.80548816\n.75347056\n.44212476\n.24818319\n.97547947\n.94871425\n.91920697\n.88626494\n.84887892\n.80548816\n.75347056\n1.07011767\n1.43572220\na=1.0\na=1.0\n(6)\n(6)\n.99555888\n.98534702\n.96837957\n.94248172\n.90235828\n.82714268\n.54795225\n.47527613\n.44036615\n.30805565\n.33338897\n.37457350\n.43103184\n.50577126\n.61805744\n1.03912588\n1.32548793\n1.57217608\na=0.3\na=0.3\n(5)\n(5)\n.99635795\n.98918384\n.97803608\n.96203844\n.93943429\n.90605080\n.84486000\n.54943339\n.45599232\n.50198613\n.51081140\n.52680790\n.55064164\n.58378690\n.63009577\n.70629429\n1.13840468\n1.44034524\na=0.5\na=0.5\nu = 107 m² sec-1\n(4)\nu = 107 m² sec-1\n(4)\n.99744598\n.99432730\n.99060244\n.98619983\n.98100083\n.97480176\n.96721477\n.95731734\n.94084796\n.62710142\n.89812349\n.89683589\n.89611794\n.89626099\n.89709542\n.89899163\n.90230327\n.90795370\n.92020832\n1.37529766\na=0.9\na=0.9\n(3)\n(3)\nVISCOUS METHOD\n.98805628\n.99764937\n.99528002\n.99289160\n.99048301\n.98560867\n.98314062\n.98065175\n.97814170\n.78152122\n.99764937\n.99528002\n.99289160\n.99048301\n.98805628\n.98560867\n.98314062\n.98065175\n.97814170\n1.21790035\na=1.0\na=1.0\n(2)\n(2)\n1.1\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.1\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n(1)\n(1)\n45","-1.914340\n-3.420237\n-3.431713\n-3.385994\n-3.324153\n-2.978291\n-2.129046\n-1.728224\n0.811318\n1.002451\n1.028843\n1.071309\n1.140565\n1.267262\n1.212576\n0.533992\na=0.3\na=0.3\n(9)\nTABLE 16.--RATIO OF NUMERICAL TO ANALYTIC PHASE SPEED FOR THE EASTWARD-MOVING GRAVITY (PHYSICAL) MODE,\n(9)\nRATIO OF NUMERICAL TO ANALYTIC PHASE SPEED FOR THE COMPUTATIONAL MODE CORRESPONDING TO THE\n-2.053340\n-2.127670\n-2.213239\n-2.316696\n-2.457628\n-2.520938\n-2.001265\n-1.792142\n1.220463\n0.848617\n1.000762\n1.029260\n1.071569\n1.135025\n1.242781\n1.471926\na=0.5\na=0.5\nu = 108 m2 sec-1\n(8)\nu = 108 m2 sec-1\n(8)\n-1.114455\n-1.152182\n-1.201648\n-1.510290\n-1.268219\n-1.362479\n-1.823191\n-1.928878\n-1.719491\n0.997121\n1.028260\n1.069822\n1.126672\n1.208584\n1.340002\n1.629698\n1.768245\n1.472544\na=0.9\na=0.9\n(7)\n(7)\n1.124668\n-0.996266\n-1.027870\n-1.069231\n-1.124668\n-1.202475\n-1,321802\n-1.549521\n-1.963495\n-1,745329\n0.996266\n1.027870\n1.069231\n1.202475\n1.321802\n1.549521\n1.963495\n1.745329\na=1.0\na=1.0\n(6)\n(6)\nEASTWARD-MOVING GRAVITY MODE, VISCOUS METHOD\n-3.308272\n-3.221895\n-3.105696\n-2,993078\n-2.915734\n-2.295472\n-1.767930\n-1.638063\n-1.502384\n1.005387\n1.020461\n1.047451\n1.091816\n1.167627\n1.333474\n1.239375\n0.920045\n0.717913\na=0.3\na=0.3\n(5)\n(5)\n1.004773\n1.016928\n1.037631\n1.069660\n1.118869\n1.199994\n1.376048\n1.363379\n1.094736\n-2.006184\n-2.013097\n-2.022371\n-2.037573\n-2.065571\n-2.122194\n-2.216487\n-1.696229\n-1.557184\na=0.5\na=0.5\nu = 107 m² sec-1\n(4)\nu = 107 m2 sec-1\n(4)\n-1.115562\n-1.124371\n-1.137777\n-1.156667\n-1.182578\n-1.218273\n-1.269293\n-1.349503\n-1.523135\n1.003925\n1.012216\n1.025107\n1.043483\n1.068882\n1.104067\n1.154581\n1.234287\n1.407417\n1.489354\n-1.536024\na=0.9\nVISCOUS METHOD\na=0.9\n(3)\n(3)\n1.003766\n1.011347\n1.022856\n1.038976\n1.060854\n1.090868\n1.131514\n1.192005\n1.297435\n1.570796\n-1.003766\n-1.011347\n-1.022856\n-1.038976\n-1.060854\n-1.090468\n-1.131514\n-1.192005\n-1.297435\n-1.570796\na=1.0\na=1.0\n(2)\n(2)\nTABLE 17.\nWAt\nWAt\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.1\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.1\n(1)\n(1)","20.2\n20.3\n20.0\n20.0\n19.9\n15.9\n15.8\n16.0\n16.3\n16.6\n12.1\n12.1\n12.1\n12.2\n10.7\n10.7\n10.7\n10.7\n11.0\n9.8\n9.8\n9.8\n9.8\n9.7\nEa\nPOOLED\n37.3\n37.4\n37.5\n37.0\n37.3\n33.0\n32.9\n34.1\n34.0\n34.9\n16.6\n16.7\n17.1\n16.4\n14.7\n15.0\n15.1\n14.4\n15.8\n12.1\n12.2\n12.6\n11.9\n12.0\nEf\nTABLE 18. RMS HEIGHT DIFFERENCE (EF) IN METERS BETWEEN FIRST GUESS (INTERPOLATED TO OBSERVATION LOCATION) AND OBSERVATION,\n15.6\n15.6\n15.8\n15.5\n15.6\n8.2\n8.6\n7.7\n7.8\n6.9\n3.0\n2.6\n2.7\n2.9\n1.5\n1.3\n1.4\n1.4\n1.9\n1.0\n0.8\n1.1\n0.7\n1.4\nEa\n17Z\n27.2\n27.2\n27.6\n27.0\n27.0\n14.0\n14.8\n13.5\n14.6\n12.9\n9.8\n7.7\n5.7\n6.8\n3.1\n2.7\n2.6\n3.2\n4.4\n2.8\n2.4\n1.6\n1.9\n2.6\nE f\n17.9\n17.9\n18.6\n15.4\n17.0\n6.6\n6.9\n4.6\n6.1\n3.8\n3.1\n3.2\n3.2\n2.6\n1.4\n1.5\n0.7\n1.2\n0.7\n0.9\n1.0\n1.1\n0.7\n2.0\nEa\n15Z\nAND THE RMS HEIGHT DIFFERENCE (Ea) BETWEEN INTERPOLATED ANALYSIS AND OBSERVATION\n40.8\n41.1\n42.4\n34.4\n38.2\n14.1\n14.9\n9.9\n13.2\n8.4\n6.6\n6.4\n7.5\n5.6\n2.7\n2.9\n2.8\n2.2\n1.6\n1.3\n1.7\n2.2\n1.1\n3.9\nEf\n21.6\n21.7\n21.2\n21.7\n21.4\n17.0\n17.0\n17.2\n17.7\n18.3\n15.8\n15.8\n15.8\n16.1\n13.8\n13.8\n13.9\n14.0\n14.5\n13.1\n13.2\n13.2\n13.2\n13.1\nEa\n13Z\n39.1\n39.2\n39.0\n39.7\n39.5\n22.4\n22.3\n22.5\n24.2\n26.1\n18.9\n19.0\n19.4\n19.2\n15.8\n15.9\n16.1\n16.1\n17.1\n15.4\n15.5\n15.6\n15.5\n15.7\nEf\n16.1\n16.1\n16.5\n16.3\n16.6\n11.8\n11.9\n11.8\n11.8\n10.8\n10.8\n10.9\n10.8\n11.1\n9.7\n9.7\n9.6\n9.6\n9.6\nEa\n11Z\n33.9\n33.6\n38.1\n34.9\n36.3\n15.6\n15.8\n17.2\n15.4\n13.7\n13.8\n14.8\n13.6\n14.4\n11.5\n11.6\n12.8\n11.3\n11.3\nEf\n12.5\n12.4\n13.0\n13.0\n13.4\n10.9\n10.9\n9.0\n11.3\n11.0\n9.0\n9.1\n8.9\n9.0\n8.7\n8.7\n8.7\n8.7\n8.4\nEa\n09Z\n20.9\n20.7\n22.1\n23.2\n24.4\n16.8\n17.0\n17.7\n16.6\n11.8\n12.0\n12.5\n11.7\n12.3\n13.6\n14.1\n13.5\n13.0\n13.0\nEf\n19.0\n19.2\n18.8\n20.0\n20.4\n11.8\n11.6\n11.1\n11.7\n10.2\n10.3\n10.2\n10.3\n10.4\n8.7\n8.7\n8.8\n8.9\n9.0\nEa\n07Z\n51.4\n51.4\n49.3\n51.8\n52.3\n18.6\n18.9\n17.2\n18.4\n20.1\n21.3\n19.4\n19.1\n22.2\n10.1\n10.2\n10.4\n10.2\n10.7\nEf\nEXPERIMENT\n1\n2\n3\n4\n5\n1\n2\n3\n4\n5\n1\n2\n3\n4\n5\n1\n2\n3\n4\n5\n1\n2\n3\n4\n5\nCYCLE\n0.25\n0.5\n1.0\n1.5\n2.0","9.3\n9.4\n9.3\n9.3\n9.5\n8.8\n9.1\n8.8\n8.7\n8.7\n8.5\n8.6\n8.5\n8.5\n8.7\n8.1\n8.2\n8.1\n8.1\n8.1\n6.6\n6.6\n6.5\n6.5\n6.6\nEa\na\nPOOLED\n12.7\n13.0\n13.0\n12.6\n13.3\n10.9\n11.1\n11.3\n10.8\n11.1\n11.8\n12.1\n12.0\n11.7\n12.4\n10.3\n10.4\n10.6\n10.2\n10.6\n8.7\n8.8\n9.1\n8.7\n9.1\nEf\nTABLE 18. RMS HEIGHT DIFFERENCE (Ef) IN METERS BETWEEN FIRST GUESS (INTERPOLATED TO OBSERVATION LOCATION) AND OBSERVATION,\n0.6\n0.5\n0.7\n0.3\n1.1\n0.5\n0.5\n0.8\n0.4\n1.0\n0.3\n0.2\n0.5\n0.3\n1.0\n0.2\n0.3\n0.7\n0.5\n0.9\n0.2\n0.1\n0.5\n0.8\n0.8\nEa\n17Z\n1.1\n1.0\n1.0\n0.8\n2.6\n1.3\n1.4\n1.7\n0.8\n2.1\n0.5\n0.5\n0.8\n0.5\n2.2\n0.7\n0.9\n2.0\n0.7\n1.8\n0.3\n0.1\n1.8\n1.0\n1.5\nEf\n0.7\n0.7\n0.9\n0.6\n1.4\n0.5\n0.6\n0.7\n0.5\n1.6\n0.4\n0.4\n0.9\n0.4\n1.5\n0.3\n0.3\n0.6\n0.4\n1.6\n0.1\n0.1\n0.4\n0.2\n1.2\nEa\n15Z\nAND THE RMS HEIGHT DIFFERENCE (Ea) BETWEEN INTERPOLATED ANALYSIS AND OBSERVATION\n0.8\n1.1\n2.4\n0.7\n3.0\n0.8\n0.9\n1.3\n0.6\n3.1\n0.5\n0.6\n2.7\n0.5\n3.1\n0.6\n0.5\n1.1\n0.5\n2.9\n0.1\n0.1\n0.8\n0.2\n2.3\nEf\n11.6\n11.6\n11.7\n11.6\n12.0\n11.4\n11.4\n11.6\n11.5\n11.5\n10.1\n10.1\n10.2\n10.1\n10.4\n10.3\n10.3\n10.4\n10.3\n10.4\n7.7\n7.6\n7.8\n7.6\n7.9\nEa\n13Z\n13.2\n13.3\n13.4\n13.3\n14.0\n13.7\n13.8\n13.9\n13.8\n14.2\n11.6\n11.6\n11.7\n11.6\n12.1\n12.7\n12.7\n12.9\n12.8\n13.3\n10.5\n10.4\n10.5\n10.6\n11.3\nEf\n9.4\n9.5\n9.6\n9.4\n9.8\n8.8\n8.8\n8.7\n8.6\n8.6\n8.8\n8.8\n8.9\n8.7\n9.2\n8.2\n8.3\n8.2\n8.2\n8.1\n7.0\n7.0\n7.0\n6.9\n6.8\nEa\n11Z\n12.0\n12.1\n12.8\n11.9\n12.7\n10.3\n10.4\n11.3\n10.1\n10.0\n11.3\n11.4\n12.2\n11.3\n12.2\n9.8\n9.9\n10.6\n9.6\n9.4\n8.7\n8.7\n9.6\n8.6\n8.4\nEf\n7.7\n7.8\n7.6\n7.7\n7.7\n7.7\n7.7\n7.5\n7.7\n7.5\n7.0\n7.0\n6.8\n7.0\n6.9\n7.1\n7.1\n6.8\n7.2\n7.0\n5.6\n5,6\n5.3\n5.8\n5.6\nEa\n09Z\n10.0\n10.2\n10.0\n9.9\n10.2\n12.5\n13.0\n12.2\n12.1\n12.4\n9.2\n9.5\n9.0\n9.2\n9.5\n11.8\n12.3\n11.4\n11.5\n11.9\n9.8\n10.1\n9.4\n9.7\n10.2\nEf\n9.8\n9.9\n9.5\n9.7\n9.4\n8.2\n8.4\n8.3\n8.3\n8.5\n9.3\n9.6\n9.0\n9.2\n9.0\n7.9\n8.0\n7.9\n7.9\n8.2\n6.7\n6.8\n6.7\n6.7\n7.1\nEa\n07Z\n17.8\n18.6\n17.4\n17.3\n18.4\n9.7\n9.9\n9.9\n9.6\n10.5\n17.0\n17.8\n16.4\n16.7\n17.4\n9.3\n9.5\n9.4\n9.2\n10.4\n7.8\n8.0\n8.0\n7.7\n9.2\nEf\nEXPERIMENT\n1\n2\n3\n4\n5\n1\n2\n3\n4\n5\n1\n2\n3\n4\n5\n1\n2\n3\n4\n5\n1\n2\n3\n4\n5\nCYCLE\n2.5\n3.0\n3.5\n4.0\n10.0\n88","LIST OF FIGURES\nFigure 1.--Amplitude response of the explicit methods as a function of\nWAt: (a) physical modes; (b) computational modes. The response is\nidentical for both meteorological and gravitational modes.\nFigure 2.-Amplitude response of semi-implicit methods as a function\nof WAt for meteorological modes: (a) physical; (b) computational.\nOnly curves for a=1.0 are plotted in (b) ; curves for other values\nare similar to those of Figure 1b. Note the expanded scale of the\nordinates as compared with those of Figure 1.\nFigure 3.--Amplitude response of the semi-implicit methods as a\nfunction of WAt for eastward-moving gravity (physical) modes.\nOnly the curve for a=1.0 is plotted for the semi-implicit backward\nmethod; see Table 5 for remaining values.\nFigure 4.--Same as Figure 3, but for computational mode corresponding\nto the eastward-moving gravity wave.\nFigure 5.--Ratio of numerical to analytic phase speed for the eastward-\nmoving gravity (physical) mode as a function of wAt for selected\nintegration methods.\nFigure 6.--Amplification factor per unit time as a function of wave-\nlength for the eastward-moving gravity mode and for two values of\nthe coefficient of the viscous term.\nFigure 7.--Same as Figure 6, for the meteorological mode.\nFigure 3.--Height change from the initial value at an individual grid\npoint for integrations of the explicit methods from unbalanced initial\ndata.\n49","Figure 9.--Same as Figure 8, for the viscous method.\nFigure 10.--Same as Figure 8, for the semi-implicit methods.\nFigure 11.--Behavior of Ef as a function of cycle number for each of\nthe five experiments.\nFigure 12. --Behavior of S as a function of cycle number for each of\nthe five experiments.\n50","1.2\na. PHYSICAL MODES\n1.1\nEULER BACKWARD\n1.0\nEXPLICIT CENTERED, a=1.0\nAAA\nEXPLICIT CENTERED, a=0.9\n0.9\nEXPLICIT CENTERED, a=0.5\n0.8\nEXPLICIT CENTERED, a=0.3\nx\n0.7\n0.6\nR\n0.5\nx\nx\n0.4\n0.3\n0.2\n0.1\n0\n1.8\n0\n.1\n.2\n.3\n.5\n.9\n1.0\n1.2\n1.4\n1.6\n.4\n.6\n.7\n.8\nwat\n1.2\n1.1\nb. COMPUTATIONAL MODES\n1.0\n0.9\n0.8\nx\n0.7\n0.6\nR\n0.5\nx\n0.4\n0.3\n0.2\n0.1\n0\n1.0\n1.2\n1.4\n1.6\n1.8\n0\n.1\n.2\n.3\n.4\n.5\n.6\n.7\n.8\n.9\nwat\nFigure 1. - -Amplitude response of the explicit methods as a function of WAt:\n(a) physical modes; (b) computational modes. The response is identical\nfor both meteorological and gravitational modes.\n51","1.02\n1.01\n1.0\n.99\n.98\n.97\n.96\nR\na. PHYSICAL MODES\n.95\nIMPLICIT CENTERED, a=1.0\n.94\nIMPLICIT CENTERED, a=0.9\nIMPLICIT CENTERED, a=0.5\n.93\nIMPLICIT CENTERED, a=0.3\n.92\nIMPLICIT BACKWARD, a=1.0\nIMPLICIT BACKWARD, a=0.9\n.91\n.90\n0\n.1\n.2\n.3\n.4\n.5\n.6\n.7\n.8\n.9\n1.0\n1.2\n1.4\n1.6\n1.8\nwat\n1.02\n1.01\nb.\nCOMPUTATIONAL MODES\n1.0\n.99\n.98\n.97\nR\n.96\n.95\n.94\n.93\n.92\n.91\n.90\n0\n.1\n.2\n.3\n.4\n.5\n.6\n.7\n.8\n.9\n1.0\n1.2\n1.4\n1.6\n1.8\nwat\nFigure 2. - -Amplitude response of semi-implicit methods as a function of WAt\nfor meteorological modes: (a) physical; (b) computational. Only curves\nfor a=1.0 are plotted in (b) ; curves for other values are similar to\nthose of Figure 1b. Note the expanded scale of the ordinates as compared\nwith those of Figure 1.\n52","1.0 1.2 1.4 1.6 1,8 2.0 2,2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3,8 4.0 4.2 4.4 4.6 4.8 5.0\nresponse of the semi-implicit methods as a function of WAt for eastward-moving backward\nFigure 3. --Amplitude (physical) modes. Only the curve for a=1.0 is plotted for the semi-implicit\nwat\nmethod; see Table 5 for remaining values.\nIMPLICIT CENTERED, a=1.0\nIMPLICIT CENTERED, a=0.9\nIMPLICIT CENTERED, a=0.5\nIMPLICIT CENTERED, a=0.3\nIMPLICIT BACKWARD, a=1.0\n.8\n.6\nA\ngravity\n.4\n.2\n0\n1.1\n1.0\n0.9\n0.8\n0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0\nR","5.0\nIMPLICIT CENTERED, CENTERED, a=1.0 a=0.9 a=0.5\nIMPLICIT CENTERED, a=0.3\nIMPLICIT IMPLICIT CENTERED, a=1.0\na=0.9\n4.8\nIMPLICIT IMPLICIT BACKWARD,\n4.6\nADD\n000\n3, but for computational gravity wave. mode\n1.0 1.2 1.4 1.6 1.8 2.0 2.2 wat 2.4 2.6 2.8 3.0 3.2 3.4 3.6\nFigure corresponding 4. - -Same as to Figure the eastward-moving\n0\n1.2\n1.1\n1.0\n0.9\n0.8\n0.3\n0.7\nR 0.6\n0.5\n0.4\n0.2\n0.1\n0","1.07\nx\nEXPLICIT CENTERED, a=1.0\n1.06\n6\nx\nEULER BACKWARD\n1.05\nx\nEXPLICIT CENTERED, a=0.3\nx\nIMPLICIT CENTERED, a=0.3\n1.04\n1.4\nIMPLICIT BACKWARD, a=0.9\n1.03\n1.3\n1.02\n1.2\n1.01\n1.1\n1.0\n0.2\n0.4\n0.6\n0.8\n1.0\n1.2\n1.4\n1.6\n1.8\n2.0\n2.2\n2.4\n2.6\n2.8\nx\nwat\n.99\n0.9\n.98\n0.8\nx\n.97\n0.7\nx\nx\nx\n.96\n0.6\n.95\n0.5\n0.4\n.94\n.93\n0.3\n.92\n0.2\nFigure 5. -- Ratio of numerical to analytic phase speed for the eastward-\nmoving gravity (physical) mode as a function of WAt for selected\nintegration methods.\n55","10\n5\nR=0\n7\nR=10\n0\n-5\n8\nR=108\n-10\n-15\n-20\n-25\n-30\n-35\n-40\n-45\ne\n-50\n22\n24\n26\n16\n18\n20\n2\n4\n6\n8\n10\n12\n14\nWAVELENGTH L (103 KM)\nFigure 6 . -- Amplification factor per unit time as a function of wavelength\nfor the eastward-moving gravity mode and for two values of the coefficient\nof the viscous term.\n56","18\n16\n14\n12\nu=108\n10\n8\n6\n4\nu=107\n=0\n2\n0\n-2\n0\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\nWAVELENGTH L (103 KM)\nFigure 7. -- Same as Figure 6, for the meteorological mode.\n57","","","60","5\n4\n10.0\n3\n2\n1\nFigure 11. Behavior of Ef as a function of cycle number for each of the five experiments.\n5\n4\n3.0\n3\n2\n1\n5\n4\n2.5\n3\n2\n1\n5\n4\n2.0\n3\nCYCLE\n2\n1\n5\n4\n1.5\n3\n2\n1\n5\n4\n1.0\n3\n2\n1\n5\n3 4\n0.5\n2\n1\n5\n4\n0.25\n3\n2\n1\n40\n30\n10\n(m) 20\n0\nEf","5\n4\nFigure 12. Behavior of S as a function of cycle number for each of the five experiments.\n4.0\n3\n2\n1\n5\n4\n3.5\n3\n2\n1\n5\n4\n3.0\n3\n2\n1\n5\nCYCLE\n4\n2.5\n3\n2\n1\n5\n4\n2.0\n3\n2\n1\n5\n4\n1.5\n3\n2\n1\n4 5\n1.0\n3\n2\n1\n5\n4\n0.5\n3\n2\n1\n-5\n15\n10\n20\n0\n5\n(m)\nS","(Continued from inside front cover)\nNOAA Technical Memoranda\nNWS NMC 49\nA Study of Non-Linear Computational Instability for a Two-Dimensional Model. Paul\nD.\nPolger, February 1971. (COM-71-00246)\nNWS NMC 50\nRecent Research in Numerical Methods at the National Meteorological Center. Ronald D.\nMcPherson, April 1971.\nNWS NMC 51\nUpdating Asynoptic Data for Use in Objective Analysis. Armand J. Desmarais, December\n1972. (COM-73-10078)\nNWS NMC 52\nToward Developing a Quality Control System for Rawinsonde Reports. Frederick G. Finger\nand Arthur R. Thomas, February 1973. (COM-73-10673)\nNWS NMC 53\nA Semi-Implicit Version of the Shuman-Hovermale Model. Joseph P. Gerrity, Jr., Ronald D.\nMcPherson, and Stephen Scolnik. July 1973. (COM-73-11323)\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)"]}