A Technique of Objective Analysis and Initialization for the Primitive Forecast Equations

652 MONTHLY WEATHERREVIEW vel. 91, No. 9

UDC 551.609.313 A TECHNIQUE OF OBJECTIVEANALYSIS AND INITIALIZATION FOR THE PRIMITIVEFORECAST EQUATIONS TAKASHINITTA and JOHN B. HOVERMALE *

National Meteorological Center, Weather Bureau, ESSA, Washington D.C. ABSTRACT A technique .of initialization for the primitive forecast equations is presented. The method consists of a marching prediction scheme performed in such a manner that the large-scale solution remains approximately steady, and high-frequency modes created in the adjustment process are damped selectively by time differencing with the Euler- backward method. The scheme places no restriction on the wind divergence field and ensures truncation error consistency between initialization and forecast equations.

1. INTRODUCTION. ~ -~ ~ initialthe data also causes excitation of high-frequency modes and that a good estimate of the divergent charac- Themutual adjustment process between massand teristics could beobtained through use of the quasi- under the' influence Of the and geostrophic o-equation. Later, several authors(&&el- gravity of theearth is one of themost interesting physical mann, 1961 ; Miyakoda, 1963; and and characteristics Of motion in the atmosphere and the Baumhefner, 1966) included the terms of second-order Oceans (Rossby, 1937-1938a1 1937-1938b; Cahnl 1945; approximationin the filtered equations and utilized 1953; Obukhov, 1949; and Phillips, 1963)* This approaches involving simultaneous solution of the balance process disperses the energy Of the deviations from the ando-equations. All the abovehave had mathe- equilibrium state between the pressureand Coriolis matical limitations concerning the existence and unique- andl in the case of the atmosphere,maintains ness of solution under a given boundarycondition, i.e., the quasi-geostrophic state of the large-scale motion that the so-called ellipticity condition, but Fjortoft (1962) is observed on weathercharts. In addition, it seems to has proposed a method of solution applicable in elliptic thepresent authors that dissipative forces inthe free as well as hyperbolic areas. atmosphere act to suppress the high-frequency The mathematical and physical approachestaken in sound-gravity-inertia motions andthus contribute tothe initializations with the equations give, in all

state' The lower frequency modes Of these cases, inexactconditions of balance when applied to the motions are not as responsive to viscous forces as those primitive equations. The refined second-order equa- Of high and have a life Of hours tions may approach the balanced state supposed by the (e.g., tidal motions). primitiveequations, but a perfectagreement on the It is well known that observational data inthe atmos- state of mutual adjustment can be attained. men phere contain excessive imbalances between the Coriolis one adds to this problem the complexities~ introduced and pressure forces and that these deviationslead to when secondary physical effects are considered (e.g., large-amplitude gravity-inertia waves if time integrations flow Over mountains, radiation, friction, sensible heating, are attempted with the primitive equations (Richardson, convection, etc.), the task of maintaining balance con- 1922). The sources of these imbalances may lie in analyses sistent with the primitive forecast equations becomes as much as in observational errors. Although actual im- even difficult. balance do.es occur in the atmosphere, it is doubtful that The first departure from the early con- much of this is accurately revealed in the data. Perhaps ventional approach of initialization \vas taken by Miya- areas Of imbalanceare e.g.7 those of cyclostrophic koda and Moyer (1968). The design of their scheme makes change, and frontal and convective activity, but accurate it more applicable for use with the primitive equations representation of the excess is impossible. and, in particular, well suited for the addition of heating, A general be friction, and convectionsimulationswithin the dynrimic withdiagnostic forms of the second-order filtered equa- frame\vork. The behind this has The scheme avoids the problem of the ellipticity con- gradually since Hinkelmann (l951) proposed geostrophic ditionpresent in previousapproaches but retains a balancing. The lack of flow curvature influences in the restriction on the horizontal divergence (the first and geostrophic method was improved in the balance approach second individual time derivatives of divergence are suggested CharneY (1955); but the use Of the ''balanCe assumed to be zero) normally employed in conventional equation''gave only rotational characteristics of the initializations. Admittedly, this approximation is quite flow*It was Pointed out by Hinkelmann (1959) good on thesynoptic scale at middlelatitudes. The and (lg60) that a lack Of a divergence in application to othersituations is, however, quite limited.

1 Present affiliation: Japan MeteorologicalAgency, Tokyo Finally,theMiyakoda-Moyer scheme cannot ensure 2 Present affiliation: The Pennsylvania State University, University Park a truncation error consistency between the initialization 3 Balance of forces in the atmosphere on the earth should (strictly speaking) include consideration of cyclostrophic effects as well as the effect of large-scale divirgence. equations and the forecast equations. The unique char-

Unauthenticated | Downloaded 09/27/21 09:15 AM UTC Sepfernber 1969 Takashi Nitta and John B. Hovermale 653 acter of the numerical initialization produces truncation behavior of the scheme has been shown to depend on inconsistencies that can lead to imbalances in the interior the time step, as well as the frequency of oscillations. of the grid and particularly at the boundaries when the A simple form of a prognostic equation, forecast is begun. Any imbalance due to truncation differences can be expected to lead to fictitiousgravity butat= F(u), (1) waves when the forecast is begun. in the Euler-backward time difference may be expressed 2. INITIALIZATION BASED ON THEPRIMITIVE as follows : FORECASTEQUATIONS u*=u'+F(u')At The initializationprocedure presented here utilizes and (2) u'+l- "u7 +F(u*)At. the primitive forecast equations directly. It departs from theMiyakoda-Moyer scheme in two ways. First, no The symbol At represents thetime increment over explicit restriction on the horizontal divergence is re- which the extrapolation is made. If a completely reversible quired.Second, although the same time-differencing differential equationis treated andintegrated forward scheme (the"Euler-backward" method) is employed, it and thenbackward in time, the original solution will is applied in a different manner; i.e., an actual iteration of be obtained. However, if the Euler-backward method is forward and backwardforecasts are performed around employed with a finite At, a damping effect is introduced theinitial time. The method corresponds to anormal, on the high-frequency physical modes. After one forward- marching prediction scheme where the large-scale field is backward forecast cycle, a wave of angular frequency w quasi-steady, and only the high-frequency modes change experiences a reduction rate in amplitude given by with time. The possible advantages to be gained with this approach (l+iwAt-~~ht~)(l-i~At-w~At~)=l-u~At~+~~At~(3) are 1) it eliminates the truncation inconsistency problem discussed above and 2) it may have application in a wider where i=,/y. For waveswhere oAtforce and the Coriolis issomewhat arbitrary.Furthermore, it is not assured force dominatethe process. Recently,Blumen, 1967, that along with the elimination of noise in the calculation discussed the nonlinear effect in the geostrophic adjust- a meteorologically significant balance may be realized. ment process.) Theboundary conditionsand error due These uncertainties will be investigated in two ways. to discreterepresentation of thecontinuous fields may First, an analysis of a simple linear system will be per- excite gravity-inertiawaves unique to the particular formed relatingto a problem of barotropic, frictionless model. And afterward,the dispersive effects would dis- flow. Secondly,nonlineara experiment is presented sipatethe resultant noise ina manner peculiar to the based on numericalsimulations of a simple two-layer mathematicalformulation. It would seem that, under model. theseconditions, mathematical consistency down to truncation equivalence should be considered in the design of initialization forecast schemes. 3. INITIALIZATION OF A SIMPLE LINEARIZED SYSTEM Inthe approachto a balanced state in the present We shall consider the linearized equations of motion scheme, data errors, numerical errors, etc. can be expected which govern an incompressible homogeneous atmosphere to lead to fictitious noise just as with the forecast equa- over the rotating flatdomain, tions. Since theseexcitations areat the source com- putationally(as opposed to physically) generated, it (dV/dt)+jkXV=-v4 (41 does not seem inconsistent to removethis noise by a and computationalartifact. A time-integration scheme that &$/at= -HVa V. (5 1 accomplishes thisand, to some extent,simulates the viscous effect of damping high-frequency osdlationsis We transformequations (4) and (5) into a forward dif- the Euler-backwardmethod or its modified version ference form, i.e., (Matsuno, 1966a, 1966b; Kurihara, 1965).4 The damping V(.+l)_V(T)=_gkXV(r)--V~(l) (6) 4 Through personal communication one of the authors has found that Matsuno (1967) and and Gamho and Okamura (1967) have also experimented with thisscheme in initialization @(l+l)-+(T)=-.yHV.V(T) and forecast problems in a manner similar to that discussed here. (7 1

Unauthenticated | Downloaded 09/27/21 09:15 AM UTC 654 MONTHLY WEATHERREVIEW Vol. 97, No. 9 FORWARD FORECAST A*=C(v)-~D(y)+~VzG(‘), (17) B*=D‘~’+~C‘y’+yVuG(y), (18) T=O T = At 4*=G(Y)+~H(VZC(Y)+VuD(Y)), (19)

A(Y+I) = C‘d- BB*+TVZd*, (20 1 BACKWARD FORECAST B‘’+”=D(u)+PA*+yVu~*, (21) and ~(’+l)=~=G(v)+yH(V,A*+VuB*). (221

RECOVERY OF MASS FIELD OR VELOCITY FIELD During a single cycle of the forward and backward forecasts, A and B are unknown variables, and the time-dif- ferenced formulas give rise to two equations in A a.nd B. The final solution, for the case where sinusoidal spatial variation of quantities is assumed, turns out to be

A=-m+#J, B=rk/B)d). (231 FORWARD FORECAST FORWARD FORECAST Here, k=2i sin(mAs) and 1=2i sin(nAs) where m=2~/D, and n=2r/L. The symbols L and D represent the wave T = -At T=O T = At lengths of harmonicsin the z- and y-directions, respectively. This solution shows the geostrophic relation. The OR WIND FIELD differences between the final solution, A and B, and the vth iterated solution, A(’)and B(Y),are denoted ALI(~) and AB(Y),i.e., BACKWARD FORECAS d(v)=A-A(v) (24 1 and AB(v)=B-B(v). (25) FIGURE1.-Schematic representation of iterationmethods for init.ialization with the primitive forecast equations. Defining X as a convergentrate, we can write AA(v)=XY, (26 1

From equations (11) through (22), the quadratic equation for X is obtained. Two solutions for X indicate that the present scheme is conditionally convergent. The criterion for the convergence is At

where B=H[sin2 (mAs) +sin2(nAs)]/As2.

For a sinusoidal wave with a unit amplitude,

u=sin m(z-ct) (28 1

wherem=2n/L and cis a phase speed, the Euler-backward scheme equation (2) turns out to be

UT+ 1 =RUT (29) where R isan amplification rate (Kurihara, 1965). For equations (4) and (5), R=l-G (kcAt)-(kcAt)2. (30) If we denote an angular frequencyw=kc and i=-, the amplification factor RF for the forward forecast becomes RF= 1 “iwAt-WW2At2. (31) For the backward forecast, RB= 1 +iwAt- dAt2. (32)

Unauthenticated | Downloaded 09/27/21 09:15 AM UTC September 1969 Takashi Nitta and John B. Hovermale 655 An operation of a single cycle of the forward and backward forecast results inthe amplification rate RFB as follows (see fig. 2): I RFB=RFRB=(1+iwAt-wzAt2) (l"iwAt-~~At~) = 1 -w2At2+ w4At4. (33) I 1.0 ;- EUm-BACKWABD If we adoptthe modified Euler-backwarditeration \ (Kurihara, 1965; Matsuno, 1966b), RFB becomes \ \ RFB= 1%w6At6.--'At2+ (34) 0.5 '. \ \ \ If we choose aproper value of At that satisfies the MODIplW EU~-BACKWAQD \- computationalstability condition, waves of wAt

The rateThe of damping is higher for the modified Euler- uA t scheme than that for the Original FIGURE2.-Amplification rate of Euler-backwardmethod with backward iteration. backward respect to w A t.

4. INITIALIZATION OF A NONLINEAR SYSTEM Anumerical experiment of the proposed scheme was performed witha two-level primitiveequation model (see Appendix). The model offered a fine opportunity to perform controlexperiments with a minimum of con- taminating influences. Idealized atmospheric data were constructed from an extended forecast, and modifications were introduced at any earlier day to simulate the cir- cumstances of an initialization.Problems involving observationalerror and inconsistencies between model andatmosphere were avertedby this generation of fictitious data. In the case presented here, geopotentials from the 17th FIGURE3.-Initialized divergence in the upper layer. of the two- day of the model atmosphere were used with the corres- layer experimental model. ponding rotationalwind components as first estimates for the initialization scheme. The process outlined above Although the process is still converging at 150 iterations, was carried out for almost 150 cycles. The resultant wind the rate is discouragingly slow. divergence was compared to the actual divergence field The question remains unanswered whether or not the originally developed bythe model. The two solutions technique in its present design is accurate enough to be (original and initialized) maybe comparedin figures 3 of practical use. One need maybe a modified compu- and 4. tational technique to speed convergence. Perhaps, further We see that, from an initial guess of no divergence, diagnosis of thefeatures involved inthe nonconverged the large-scale divergence field is recovered rather accu- portion of the solution (such as large-scale or small-scale rately. The close agreement of the zero line in the figures featuresand rotational or irrotationalcomponents) will is most remarkable. The most evident faultof the method lend some insight in this direction. seems to be an inability to reproduce all of the amplitude of theirrotational components. Although the patterns 5. FEASIBILITY OF APPLICATION OF INITIALIZATION are not shown, thesituation is essentially the samein METHODS TO OBJECTIVE ANALYSIS the lower layer of the model. The correspondingvorticity fields in the upper layer The attainment of a balanced state between mass and are shown in figures 5 and 6. We note only that the ro- velocity fields isa basic prerequisite of initial data for tational fields, as we canexpect, are slightlydamped forecasts with the primitive equations, but in addition it during the iterations of forward and backward forecasts. is one of the basic goals in objective analysis. At present The inability of themethod to recover t,he exact the simplestbalance formulation, that based on the solution is indicated in figure 7 showing the root-mean- geostrophic relation, is widely used in objective analysis. square errorin the wind withrespect to theiterations Sasaki (1958) andWashington (1964) have suggested performed. This shows t'hat, even with the close approxi- sophisticated generalizations based on the balance equa- mation suggested in the divergence and vorticity fields, tion proposed by Charney (1955). Bedient, McDonell, somewhat less than half the error of the nondivergent and Shimomura (1967) are performing experiments also wind was removed through theiteration technique. aimed at more generalized balanced analysis.

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FIGURE4.--Observed divergence in the upper layerof the two-layer FIGURE6.-Observed vorticity in the upper layer of the two-layer experimental model. experimental model.

of forcescannot be assumed in the mesoscale range, it seems possible that the proposed method could converge to the controlled state with acceleration terms consistent with the mesoscale (Yeh and Li, 1964). Mutual adjustment may be simulated in addition to the idealized conditions of perfect wind-to-height or height-to-wind adjustment. Such-options may be important in application to a global analysisinitialization scheme where variation in physical modes and observationalerror may influence therelative importance of winds and heights. In middle latitudes where heights are more reliable, wind-to-height adjustment would be desir- FIGURE5.--Initialized vorticity in the upper layer of the two-layer able. On the other hand, in theTropics where wind obser- experimental model. vations become more meaningful, some procedure empha- sizing .the role of the wind field should beattempted, The initialization method presentedin thepresent paper eitherthrough a height-to-wind process or a mutual may also have application in objective analysis and could adjustment technique. However, it is not a simple matter be particularly effective if employed witha scheme to of data reliability that dictates the directionof adjustment. extrapolateinformation through time. For example, in Fundamental physical principles are involved. For smaller operational forecasts, data could be carried along in time scales (Rossby number well above unity), adjustment is in the coordinate system of the forecast equations (e.g., dominated by the wind field. Scales where theEossby on the a-surface presently employed inmost forecast’ number is below unity involve adjustment where heights models with the primitive equations). Then, observations tend to be dominating. Thus, the mere fact that winds could beintroduced through conventional first-guess in the Tropics are more reliable than the heights is no interpolation methods directly on the coordinate surfaces. guarantee of successful height-to-wind adjustment,and Finally, the finer details of adjustment could be attained this facet of the proposed initialization must be studied through use of an initializationapproach of thetype more thoroughly. suggested here. In the proposed scheme, one can visualize amore It seems to the authors that theapproach outlined above general analysis initialization which is both mathemati- has special advantages for the .-type coordinate systems. cally and physically consistent with the primitive forecast It bypasses the need for an error-producing vertical inter- equations and able to account to some ‘extent for varia- polation of analyses from the more conventional p-system tions inrelative data accuracy as well asthe many and combines analyses and initialization into onestep. physical modes encountered on different scales. The latter characteristic, of course, is desirable inany coordinate system. ~~~~~~!~ The generality of the approach mentioned above should The two-layer primitive equation model used .to manu- make it useful in a wide range of physical problems. It facture data for the initialization study is similar to the requires no constraint on the divergence and thus should model developed byMintz (1964) at theUniversity of have a definite advantage over conventional methods in California, Los Angeles. The present model departs in the scales of motion where divergent modes are important. following features: In particular, the very long waves of hemispheric scale 1) A smaller area is used as the forecast domain. should be handled more correctly; and, at the other end 2) No energy sources or sinks are included. of thespectrum, mesoscale data initializationmight be 3) No orographic effect is included. performedwith more correct treatment of thegravity- 4) Potential temperature is carried as a history variable inertia wave influence. Although an approximate balance instead of temperature.

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NUMBER OF ITERATIONS FIGURE7.-Root-mean-square wind error as a function of iteration number.

b=O 15 a N. D=1 ,, ,,,,, ,, ,, , , , 111- rr,rrrP=P, FIGURE5.-Horizontal grid domain of two-layer model. FIGURE9.-Vertical grid domain of two-layer model.

Cr, is set as zero at the surface (ps=surface pressure) and 5) A slightly different, finite space differencing in the at 200 mb (p=pT). continuity equation is used. Data for the reference atmosphere was created by a 6) The dissipational effect is considered throughskin numerical integration of the model out to 26 days. Initial frictionand lateral diffusion inthe freeatmosphere. data for the integration consisted of a single long wave, However, in some cases, the diffusion coefficients are 12000 km in length (at the latitude where the map factor reduced in magnitude. equals l), embedded in a zonal currentwith a baro- The domain of integration is resolved with 20 by 10 clinically unstable vertical wind shear. During the fore- grid net with a grid interval of 600 km. A 15-min time cast, the axis of the initial long wave begins to take on step is employed. A rigid, slippery wall conditionis an upstream tilt with height. After about 10 days, waves imposed at thenorth-south boundaries of the domain, of higher numberbegan to predominate.At around 15 at 65" N. and 15" N., respectively, on aMercator pro- days, the reference atmosphere took on the appearance of jection (see fig. 8). Cyclic conditions are assumed at the atypical weather map. Additional baroclinic develop- east-west boundaries. ments occurred. as the calculation wa&\extended to 26 The vertical dimension is resolved in two layers in a days. The "observed" data for the initl'alization experi- a-coordinatesystem. Placement of variables inthe ment were extracted from 'the referen\ce atmosphere vertical may be seen in figure 9. The vertical u-velocity, during the period of realistic appearance.

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ACKNOWLEDGMENTS Matsuno, T., “NumericalIntegration of thePrimitive Equations The authors wish to expresstheir thanks to Dr. Frederick G. by a SimulatedBackward Differencc Method,” Journal of the Shuman.The present work wasinitiated by hissuggestion and MeteorologicalSociety of Japan, Ser. 2, Vol. 44, No. 1, Feb. has been accomplished with his encouragement. 1966a, pp. 76-84. The authors are grateful toDr. Kikuro Miyakoda for stimulating Matsuno, T., “A Finite Differencc Scheme for Time Integrations of discussions and valuable suggestions and express their gratitude for OscillatoryEquations With Second OrderAccuracy and Sharp the helpfd comments of Dr. Syuknro Manabe and Professors Taro Cut-Off forHigh Frequencies,” Journal of theMeteorological Matsuno, Akio Arakawa, and Norman Phillips. Society of Japan, Ser. 2, Vol. 44, No. 1, Feb. 1966b, pp. 85-88. The authors are very much indebted to Mr. Allen E.Brinkley Matsuno, T., Department of Physics, Kyushu University, Fukuoka, for his assistance in preparation of the figures and to Mrs. Judith Japan, June 1967, (personal communication). Mintz, Y., “Very Long-Term Global Integration of the Primitive Turner and Mrs. Judy Rippel for their help in preparation of the manlwxipt. Equations of AtmosphericMotion,” WMO-I UGGSymposium on Research and Development rlspects of Long-Range Forecasting, REFERENCES Boulder,Colorado, 1964, WorldMeteorological Organization Bedient, H. A., McDonell, J. E., andShimomura, D. S., “An Technical Note No. 66, Geneva, 1966, pp. 141-167. Objective Analysis Procedure toImprove the Use of Winds,” Miyakoda, K., “Some Characteristic Features of Winter Circulation paperpresented at the 48th Annual Meeting of the American inthe Troposphere and the Lower Stratosphere,” Technical Geophysical Union, Washington, D.C., Apr. 17-20, 1967. Report No. 14, Department of Geophysical Sciences, The Univers- Blumen, W.,“On Nonlinear Geostrophic Adjustment,” Journal of ity of Chicago, Dec. 1963, 93 pp. the Atmospheric Sciences, Vol. 24, No. 4, July 1967, pp. 325-332. Miyakoda, K., and Moyer, R. W., “A Method of Initialization for Bolin, B., “TheAdjustment of a Non-Balanced Velocity Field DynamicalWeather Forecasting,” Tellus, Vol. 20, No. 1, 1968, Towards Geostrophic Equilibrium in a Stratified Fluid,” Tellus, pp. 115-128. Vol. 5, NO.3, Aug. 1953, pp. 373-385. Obukhov, A., “Kvoprosu o geostrofichcskomvetre,” (On the Cahn, A., “An Investigation of the Free Oscillations of a Simple Question of GeostrophicWinds), SeryaGeofzika i Geologya, Current System,” Journal of Meteorology, Vol. 2, No. 2, June 1945, Vol. 13, No. 4, Izvestiia,Akademiia Nauk, SSSR, July- pp. 113-119. Aug. 1949, pp. 281-306. Charney, J. G., “The Use of the Primitive Equations of Motion in Phillips, N. A., “On the Problem of Initial Data for the Primitive Numerical Prediction,” Tellus, Vol. 7, No. 1, Feb. 1955, pp. 22-26. Equations,” Tellus, Vol. 12, No. 2, May 1960, pp. 121-126. Fjortoft, R., “ANumerical Method of Solving CertainPartial Phillips,N. A., “Geostrophic Motion,” Reviews of Geophysics, DifferentialEquations of Second Order,” GeofysiskePublika- Vol. 1, No. 2, iMay 1963, pp. 123-176. sjoner, Vol. 24, Oslo, 1962, pp. 229-239. Richardson, L. F., Weather Prediction by Numerical Process, Cam- Gambo, K., andOkamura, Y., ElectricComputation Center, bridge University Press, London, 1922, 236 pp. JapanMeteorological Agency, Tokyo,June 1967, (personal Rossby, C.-G., “Onthe Mutual Adjustment of Pressureand communication). Velocity Distributions in CertainSimple Current Systems,” Hinkelmann, K., “Der Mechanism des Meteorologischen LLrmes,” Journal of Marine Research, Vol. 1, No. 1, 1937-B938a, pp. 15-28. (Mechanism of Meteorological“Noise”), Tellus, Vol. 3, No. 4, Rossby, C.-G.,“On theMutual Adjustment of Pressureand NOV.1951, pp. 283-296. VelocityDistributions in CertainSimple Current Systems, 11,” Hinkelmann, “Ein numerisches Experiment mit den primitiven K., Journal of MarineResearch, Vol. 1, No. 3, 1937-19386, pp. Gleichungen,” (A NumericalExperiment With the Primitive 239-263. Equation), TheAtmosphere and Sea in Motion, Rockefeller Sasaki, Y., “AnObjective Analysis Based on theVariational Institute Press, New York., 1959, pp. 486-500. Method,” Journal of the Meteorological Society of Japan, Ser. 2, Hinkelmann, K., “Non-CharacteristicFiltering of Meteorological VO~.36, NO. 3, June 1958, p1J. 77-88. Noise Waves,”Deutscher Wetterdienst, West Germany, 1961, 19 pp., (unpublished manuscript). Washington, W. M., “Initialization of Primitive Equation Models Krishnamurti, T. N., and Baumhefner, D., “Structure of a Tropical for NumericalWeather Prediction,” FinalReport (Part I), DisturbanceBased on Solutions of a Multilevel Baroclinic Contract No. AF19 (604)-7261, The Pennsylvania State Univer- Model,” Journal of Applied Meteorology, Vol. 5, No. 4, Aug. 1966, sity, University Park, Dec. 1964, 92 pp. pp. 396-406. Yeh, Tu-Cheng, and Li, Mai-Tsun, “The Adaptation Between the Kurihara, Y., “On the Use of Implicit and Iterative Methods for Pressure and the Wind Field in the Meso and Smallscale Motion,” the Time Integration of the Wave Equation,” Monthly Weather ActaMeteorologica Sinica, Vol. 34, No. 4, Peking,Nov. 1964. Review, Vol. 93, No. 1, Jan. 1965, pp. 33-46. pp. 409-423.

[Received November 15, 1968; Tewised April 1, 18691

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