OCTOBER 1982 TROND IVERSEN AND THOR ERIK NORDENG 1347 A Convergent Method for Solving the Balance Equation TROND IVERSEN Norwegian Institute for Air Research, N-2001 Lillestrom, Norway THOR ERIK NORDENG The Norwegian Meteorological Institute, Blindern, Oslo 3, Norway (Manuscript received 26 February 1982, in final form 25 June 1982) ABSTRACT A well known method for the solution of the balance equation is analyzed. It is shown that the methodis convergent, provided that under-relaxation is used and that the approximate solutions satisfy the ellipticitycriteria of the equation. The technique on which the presented analysis is based can be utilized for other nonlinear iterationproblems.1. Introduction The balance equation is deduced from the divergence equation by omitting the local time change ofthe divergence. This is sufficient to eliminate inertiagravity waves from the solution of the equations ofmotion. If, in addition, all terms that are connectedwith the divergent part of the wind are omitted, weobtain the balance equation given by Charney (1955),from which the mass field may be found, when thehorizontal streamfunction is given, or the streamfunction may be found when the mass field is given.The latter approach has been the method in commonuse, though it leaves the problem of solving a nonlinear equation which generally is elliptic and hyperbolic in different regions. With atmospheric valuesfor the coeffcients, however, the equation is mainlyelliptic and is solved as a boundary-value problem.As for the geostrophic approximation, the wind component obtained from this version of the balanceequation is uniquely determined by the geopotential.The balance equation has been used to get the solenoidal wind component in initialization for primitiveequation models and in balanced models. In thehigher-order balanced systems (Pedersen and Gronskei, 1969), the divergent part of the wind is not removed from the equation, and the balanced state hasto be found by an iteration between all the equationsconsidered. The balance equation is still solved forthe streamfunction. The coefficients are, however,somewhat altered. Recently, the nonlinear normal mode initialization(Machenhauer, 1977) has been presented and utilizedsuccessfully and extensively, and more or less has superseded the use of the balance equation for initial0027-0644/82/101347-07505.75c 1982 American Meteorological Societyization. However, even in the nonlinear normal modeapproach, there are convergence problems, and someof these are shown to be related to the problem ofnon-existence of solutions to the classical balanceequation (Daley, 1978; Tribbia, 1981). Various iterative schemes have been proposed tosolve the finite difference analogues to the' balanceequation. In this paper, we will give a closer examination of the method given by Miyakoda (1956) andShuman (1957). Paegle and Tomlinson (1.975) foundthat in some cases, the method seemed to converge,but subsequently developed an oscillation which prevented further convergence. They described a slightmodification of the iteration technique which seemedto be convergent. The test cases that originally diverged, reached convergence. They did not give anyconvergence analysis of the method. Although thismethod seems to be the most common for solvingthe balance equation, we have not been able to findan analytic investigation of the convergence properties in the literature. Since the method of Paegle andTomlinson is fast and accurate and seems to be convergent, we have decided to make a 'closer examination. The solution method and a discussion of the ellip' ticity criteria of the equation are described in Section2. The convergence analysis is found in Section 3,while data experiments are given in Section 4.2. Eilipticity criteria and solution method We will study the equation~-2 + 2f~' + 2Vf. V~k - 2V2q- I D - 2a[J(~, ~) + J(~,~, co)] = 0,(1)1348 MONTHLY WEATHER REVIEW VOLUME 110where ~' is the vertical component of the relative vorticity, b the divergence and D the deformation of thehorizontal motion, w is the vertical motion in pressure coordinates (w = dp/dt), f the Coriolis parameter,~b the streamfunction, ~o the geopotential, V the horizontal gradient operator at constant pressure levels,and a is a tracer (to be described below). Eq. (1) is obtained by omitting the local timechange of the divergence from the divergence equation. This is sufficient to filter out inertia-gravitywaves. In addition, the horizontal velocity has beensplit into divergent and rotational parts, and all termsthat depend only upon the divergent part of the velocity have been dropped. This simplification is justified by scale analysis (see, e.g., Haltiner, 1971) orenergy considerations (Lorenz, 1960):D = (~ - ~yy)2 + 4~xy2 - a[4~,xv(xxx - Xyy) - 4x~,v(~',~ - ~byy)], (2) ~- A2 ,+ B2 + aF. (3) Subscripts x, y or p denote partial differentiationwith respect to the specified variable, D is a simplification of the total deformation, and x is the velocitypotential. Because of the omission of the (X, x)-terms,D may, in extreme cases, become negative. When a = 0 we have the nonlinear balance equation in its.usual form. For t/= I the coupling termsbetween the streamfunction and the velocity potentialare retained. The last two terms of (1) have a simplephysical meaning. They are equivalent to 2J(-, 5)+ (O/Op)J(f, w), and describe change of divergencedue to advection with the horizontal rotational windand to the vertical convergence of vertical velocityadvection. The balance equation is of the MongeAmpere type, which is .discussed by Courant andHilbert (1962). The condition for ellipticity, hereafter referred toas the Rellich condition (Courant and Hilbert, 1962,p. 324), is ~72~0f ~[G(Xxx-Xyy)2''I-4Xxy2]J,_ ~- >- 1 +35-a - , (4)where G = -2Vf. V~ + 2a[J(~, b) + J(~., ~0)1.From (1) and (4) we obtain > ~xy+a~ (Sa) For large-sc~e motion in extratropical re~ons ofthe No.hem Hemispher6, we may assume that theabsolute vorticity is positive. Thus, we have to choosethe solution where ~,~ + ~ 4- ax~, > 0 '1 (Sb) ~y + ~ - axxy > 0 Eq. (1) is solved with respect to ~' and the solutionwith positive absolute vorticity is chosen: ~ = _] +-(f2 + 2V2~ + D + G)TM. (6)For a solution to exist, the .radic~ must be positive,i.e., (7)The criteria (4) and (7) coincide when the defomation field is zero and x = 0. The condition (4) is usually more restrictive than (7). This means that, ingeneral, it is not sufficient to fome the radic~ of (6)to be positive. The equation may still be h~erbolicin that re,on. For fixed w and X, Eq. (6) is solved by the iteration(1 + y)~2~.+~ = ~V2~ _ f + ~2 + 2V2q + D~ + G~)~/2, (8)where ~ is the relaxation parameter introduced byPae~e and Tomlinson (1975). When conve~ence isachieved, it is seen that the two sides of (8) coincidefor all values of ~. To avoid a three-dimensional rel~ation, the tern J(~v, w) was kept constant duringthe .iteration. Guess v~ues of ~ were used. The ellipticity criteria (5) ~e required to be fulfilledin all points during the iteration [as Aragon (1958)emphasized]. The finite difference approximmion of (8) w~token from Miyakoda (1960), scheme B, ~th someextensions in the approximation of the defomationtern. To cflculate the Jacobian, we use the ~mephysical considerations as Miyakoda in his B-scheme,i.e., J(p, q) means the difference of inflow and omflowof a quantity q, ca~ed by the streamfunction p ina small area ~ound the ~d point under consideration. This Jacobian operator conse~es enstrophy(Arakawa, 1966). To solve the Poisson equation, ~veml methodsmay be utilized. We have cho~n an effident combination between Fourier expansion and Gau~ elimination (see Potter, 1973). Since ~1 combinationsbetween wavenumber and coordinates ~e known(a finite number, 39 X 37 = 1443 in our ~d) thetrigonometric c~culations are peffomed one, andthe values tabulated. As bounda~ condition for ~,we chooseOCTOBER 1982 TROND IVERSEN AND THOR ERIK NORDENG 1349where as is a line element along the boundary (Bolin,1956).3. Convergence analysis We will restrict this analysis to the simple nonlinearbalance equation. With a = O, (8) may be squaredto give[~v+l .~ ~/(~v+l -- ~v)]2 -- f2 -- (a42 .J_ B2)~ - 2V2~0 + 2V/. V~bv = 0, (9)Where )/ = V2~ + f and D = A2 + B2. Replacing vby v - 1 and subtracting, we find0'(1 + ~,)v2e+~ = n%,v2e + (~ - ~y/)(~=v - ~y/)where e' = ~' - ~-i. We have assumed that changes in ~ from one iteration step to the next are sm~l. Consequently, second-order te~s in e have been omi~ed. To an~yze(10), we employ ~most the ~me procedure ~ described by Aragon (1958). The inte~afion ~ea is~sumed to be recmngul~, and is divided into ~d~ints equally spaced Mth mesh size d. Eq. (10) may~ simplified by ten~fively treating the quantities f,~=~, ~yy~ and ~y' ~ constant. The ~fferential equation ~ then redumd to one ~ constant coe~cients.Under the~ conffitions, we ~11 fu~her assume thate may be extended to a periodic function and expandit in a finite Fourier ~fies. By ~suming the secondorder derivatives of ~ to ~ consents, we actu~lyapproximme ~ by a polynomi~ of secon&order. Thissimplification is justified ~ long m the hofizon~v~ation of vo~icity and defo~mion over one ~dlength d is sm~l. We then ob~n tK-I M-I ( 2~ .2~my~) ~k=0 m=0where x and y only can take discrete values. (x = 0,K- 1), (y = 0, M- 1) and K-I M-, [ .2~rkx 2~y) 1 2 2 e~exp[-/~-i- . A,~,m = ~ x=0 y=0 - Since (iO) is regarded as a linear equation withconstant coe~cients, it is su~cient to examine oneFou~er component. Derivatives may be obt~ned byfinite differences. This leads to ~ + I k,m jv Aa.~ = ~ + B~ ~ ~k,m ~ 1+~whereB~,m = (~k~x - ~by~)tcos'-~- - cos -2f;y sin ~? sin 2-~] x cos~+cos~-2 n~ .When k = m = 0, the denominator in B is zero. However, for these infinite long waves, ~ is constant and(10) is always fulfilled. If we assume thin the approximate solution mtisfies the ellipticity criteria (5),is always ~eater than zero. For convergence, a sufficient condition is that IA~,ml/l&~lIf IB~,~l < l, this criterion is fulfilled for any y - 0.~t us ~sume thin ~ = 0 (no under-relaxmion). Thenthe conditions for convergence are =k =m(2~ + f) sin~ ~ + (2~; + f) sin2 M 2=k. 2=m + ~y sin ~ sin ~ > 0, (12a) ~m ~k(2~g + f) sin 2=k. 2=m - ~~ sin ~ sin ~ > 0. (12b)If we put 2~[~ + f = & and 2~~ + f = b2, the criteria(5) Mth a = 0, ~ves (Arnason, 1958) & > 0,Thus we see that - 2 =k =m 2=k 2=mfil s~n ~ + [ 2=mI - 2=k ~m 1 sin~sin > fi~ sin X ~1112621/2 = bl1/2 Sin ~ -- ~21/2 sin ( =k =m 1[ 2=k + 2sin~sin~-5 sin~sin X ~11/2~21/2. 'The expression inside the last parentheses is alwaysgreater than or equal to zero for the wavenumbersconsidered, and (12a) is always fulfilled. A similarargument may be used to verify (12b). IB~,/,I is, thus, always < 1, and the solution methodshould be convergent, whether under-relaxation is1350 MONTHLY WEATHER REVIEW VOLUME I10 FIG. 1. The effect of underrelaxation in the balance equation.Y as a function of X for different values of the relaxation parameter %utilized or not, provided the approximate solutions~b~ satisfy the ellipticity criteria of the balance equation. From experiments we know, however, that themethod may fail, and it is obvious that we have 10stinformation during the deduction of (I 2). We havetreated f as a constant. This simplification removesthe term Vf. V~~, which primarily carries information.about the long waves of the error spectrum (Asselin,1967). Another simplification is the linearizing ofX72ev. When oscillations occur, (Ve~)2 may not at allbe small compared to n'V2e~, especially in regionswhere n~ is small. We may get some insight into the nonlinear effectsby the use of a method described by Eliassen et al.(1964). This method does not assume constant coefficients either. Eq. (8) is written as (1 + 3,)n~+l = 3'n~ + R% (13)or equivalently (1 -- 3')~v = 3'nv--I __ R~-~, (14)where R is the square root in (8). We subtract (14)from (13) and divide by (~ - ~'-~):~+~-~_ 3' _ +~1 D~-/y'-I~_ ~-i 1 + 3' 1 + 3' n'- ~-~ X [~v+l __ ~v __ ~t(~v+l _ nv-i)]-l, (15)where Dr = (A2 + B2)~. The term ~Tf. Vev has againbeen omitted 0c constant). In the case of oscillations,the term (n~*~ - n~-l) is small, compared to(n~*~ + n~) and may be omitted. Eq. (I 5) then becomes 1 + x, Y- 1+3' 1--~ ~+~ - ~ Dr -- Dv-I 1Fig. 1 shows Y = Y(X) for different values of 3'.Paegle and Tomlinson (1975) found 3' = % to beoptimal. For convergence, Y has to be less than unity when -we increase v. The figure clearly shows that the convergence interval is extended when 3' is increased.The extension is, however, only for negative valuesof X where Y is predominately negative. A negativeYmeans that (n~ - n~-~) and (n~+~ - ~) have differentsigns. Hence, 3' seems to have no impact with themonotonic mode. However, in this case, the iterationsteps may be chosen small enough (by an increaseof 3') for the linear convergence discussion to be valid.Therefore, by a sufficiently large 3' the iteration stepsare ensured to be small enough for' convergence tobe obtained, both in the oscillating and the monotonic case. Too-large values of 3' will, however, increase the computer time due to these non-oscillatingmodes. We examine the criterion X > -X0, where X0 is apositive constant and assume that D~ - D~-~ > O,while n~ - ~v-1 ~ 0. ThusD~ - D~-1 < 2~(~/~-I - ~/~)Xo, ~ = V2(*/~+~ + ~).We divide by/5 (a non-negative quantity) and obtain D~ - D~-~ 2~ (n~_1 _ ngXo --- F(n, D, Xo). (16)The left-hand side of (16) is the relative change of thedeformation field. The inequality gives an upperbound on this change for convergence to be reached. From the ellipticity criterion (5a) with a = 0, it iseasily verified that D < 72 or equivalently 1/D > 1/*/2.We may now estimate a lower bound for the functionF. We have 2~ (,F_~ ~-x _ ~ e ~-- -~ - l.ff)x0 ~ 2x0 -~ ~-- mmin.If the relative change of the deformation field is notpermitted to be greater than Fmin, X is always greaterthan.-X0. To secure convergence, relative change indeformation must be less than 2Xo times the corresponding relative change in ~vorticity. When 3' = 0(no under-relaxation) this multiplier is 2, while3' = ~ gives 2X0 -- ~0fi. A sufficiently large 3' willprovide iteration steps small enough to obtain convergence. Too-large values of 3' will, however, increase the computer time due to non-oscillatingmodes. It is therefore important to find an optimalvalue of 3' which both secures convergence and minimizes computer time. These results may also be applied to the full balanceequation (a = 1), and other similar iteration problems. Of course, it is not generally possible to get agood relationship between the variables involved (asbetween deformation and vorticity in this case), butthe qualitative results are the same. Under-relaxationOCTOBER 1982TROND IVERSEN ANDhelps when oscillations occur and permits largerchanges in the right-hand side (rhs) of the equationthan without under-relaxation.4. Numerical experimentsA number of experiments on actual data were runto test the method. We used Cartesian coordinateson a polar-stcrcographic map-projection. The griddistance was 300 km at 60-N and the grid contained1443 (=39 x 37) points. The height field for the testcases were obtained from the routine analysis of theNorwegian Meteorological Institute. The method ofanalysis is an itcrativc method described by Biorheim(1979). It consists of a relatively simple two-dimcnsional objective analysis scheme.The tests wcrc performed at the 300 mb level. Inadvance, the height field was modified to satisfy theRellich condition (4), which was simplified to ~72(~ ~ -f-+ > x,By introducing/5 = 1 - (2F/f), we get T+/5 >0. ~Ve used the values/5 = 0.8 and /5 = 1.0 in theexperiments. The -ormcr means that the F is estimated to be greater than -f/10, while the latterclaims F to be positive. The method of correction isa simplification of an iterative technique given byShuman (1957). When the height is adjusted in a gridpoint, there is no compensation for this correctionto the adjacent points. Since an exact value of/5 isimpossible to estimate in advance, ellipticity is notsecured after the Rellich correction. This is a problemparticularly if a large /5 is chosen in order not tochange the observed height field more than necessary.However, since the Rellich criterion (4) and the criteria for ellipticity (5) are merely two ways of describing the same thing, remaining hyperbolic areas TABLE 1. The number of iterations required for convergence forthe nonlinear balance equation (a = 0) as a function of the underrelaxation parameter ~ and Rellich correction parameter ~.Month/Day Time1979 (GMT) 0 1/3 0 1/3 2/301/03 0000 10 13 13 13 1401/04 0000 11 13 * 14 1501/05 0000 10 9 * * 1604/10 0000 14 17 * 17 1004/14 1200 8 8 12 8 1004/15 1200 8 8 * * 1004/16 1200 9 8 23 8 200.8 ~ = 1.0* Divergence.THOR ERIK NORDENG1351 10 ~, A ~I~ I 1 / I II / I ., ,'ix ,,x ;,:; i/,q .._., . o .. .. -" i I I ('!~ I\ -" I ~ : I I ~l ~ I I I / - I /I II I,, I .......... ~':1/3 g=2/3 Fro. 2. Maximum change in ~ from one iteration step to thenext for different values of the relaxation parameter ~ versus iteration order ~, The case ~s 0000 GMT 5 January ]979 at 300 mb.-10are given a special treatment during the relaxationprocedure: Preliminary values of ~p are checked, andeventually changed just enough for (5) to be fulfilledbefore computing the rhs of the balance equation (seethe Appendix for details). Negative values of the radical will then never occur (as long as D is non-negative). Obviously, a reduction of the 300 mb height causesa lowering of the mean temperature of the modelatmosphere. For the test cases, the maximum heightcorrection at 300 mb varies from 54 to 136 m when/5 = 0.8, and from 43 to 79 m when/5 = 1.0. However,the areas of strong anticyclonic vorticity are seldomlimited to a single layer. They are connected to deepsynoptic features. Therefore, the thickness reductionbetween 500 and 300 mb, which also is a measureof the mean temperature reduction in the layer, isless than the maximum height reduction (usually 1020%). For/5 = 0.8, this causes a mean layer temperature reduction in the range 2-8-C for an air columnin the areas of large Rellich-correction. Results from the experiments for the nonlinearbalance equation (a = 0) are summarized in Table1. In these tests, the solution of the linear balanceequation is the first guess. When the Rellich criterionis strong (/5 = 0.8), convergence is achieved for allthe test cases. Under-relaxation is seen to slow theconvergence rate for three situations. Here, the slowest convergence is found for a non-oscillating mode.This is in contrast to the cases characterized by oscillations (e.g., the cases 0000 GMT 5 January 1979and 1200 GMT 16 April 1979). When/5 = 1.0 in theRellich condition, i.e., geostrophic vorticity must be1352 MONTHLY WEATHER REVIEW VOLUME ll0 TABLE 2. The number of iterations required for convergence forthe complete balance equation (a = 1) as a function of relaxationparameter % In the last column the computations are performedwithout an ellipticity correction during the iteration.Month/Day Time1979 (GMT) 0 1/3 2/3 2/301/03 0000 18 12 14 1501/04 0000 * 11 13 *01/05 0000 * * 14 *04/10 0000 * 19 21 2404/14 '1200 * 8 10 *04/15 1200 * * 10 1004/16 1200 * 7 9 9* Divergence.greater than -f/2, the method diverges for some situations. However, when 7 = 2/3, we reach convergencefor these cases too. The effect of under-relaxation is clearly demonstrated for the case 0000 GMT 5 January 1979, whichdiverges for -y = 0 and 3~ = 1/3, but converges for7 = 2/3 (Fig. 2). These results are in full accordancewith those given by Paegle and Tomlinson (1975),and with the analytic discussion in Section 3. The complete balance equation (a = 1) has beentested on the same set of situations as the nonlinearequation (a = 0). Results are found in Table 2. Asinitial values for ~k, o~ and x the solutions of the linearbalance equation, the quasi-geostrophic omega equation and the continuity equation were utilized. Without under-relaxation, the method fails in all but onecase. Under-relaxation is, however, seen to help inthis case as it did for the nonlinear equation. If the radical becomes negative (only possible fora = 1), it has to be given a small positive value. Witha strong Rellich condition (/~ = 0.8), the radical remains positive during the iterations. However, a weakRellich condition (/5 = 1.0) gives a negative radicalin several cases. A characteristic feature in all ourtests is that the radical is positive initially. After somescan, usually 5 to 6, it becomes negative. This is typically for the divergent cases, though it may becomenegative even if the method converges. By forcing theradical to be positive, or adjusting the preliminaryvalues of~b to satisfy (5), the solenoidal wind obtainedwill not be in exact balance with the geopotential inthe influenced area. This means that the Rellich correction made in advance is not sufficient. For theconvergence analysis to be valid, the conditions forellipticity (5) have to be fulfilled for all the approximate solutions in the iteration procedure. When theserestrictions' are relaxed, we are no longer ensured ofconvergence. This is clearly demonstrated in the lastcolumn of Table 2. When the comput~ations are carded out without this ellipticity correction, three caseswhich originally converged now diverge..5. Conclusion The Miyakoda method for solving the balanceequation has been in use for decades. However, asfar as we know, there has been no attempt, in theliterature, to give any convergence analysis of themethod. We therefore hope that this paper will contribute to the understanding of the solution methodfor the equation. A drawback when solving the balance equation isthat the height field has to be modified to get an elliptic equation. Some of these modifications, but notall, may be explained by poor objective analysis techniques or scarce observations. However, these experiments also indicate that the needed changes of thegiven height field are reduced when strong under-relaxation is utilized. Finally, it should be mentioned that the convergence analysis used in this paper is straightforwardand that the technique can be applied to other meteorological problems. Acknowledgments. This paper is a part of the authors' joint doctoral thesis at the University of Oslo,for which Professor Kaare Pedersen has been the scientific adviser. Our appreciation goes to him for his guidance andhelp during that period and to Professor Arnt Eliassenfor his critical and thorough review of the manuscript,which resulted in a reorganization of some sectionsof the paper. The Norwegian Meteorological Institutepartly sponsored the computer costs, and The Norwegian Council for Science and Humanities (NAVF)contributed a fellowship for Trond Iversen (Proj. No.D. 10-09-018). APPENDIXDefineThe Ellipticity Criteria rl = ~xx + ~ + aXxy, r2 = ~kyy + ~ - axxy.If r~ or r2 is < O, the correction ~j = ~ij + d~b whered~ = (d2/m2)(~ - e) and ~ = min(r~, r2), ensures thinr~ and r2 are - e (=10-7 s-~); d is the grid distanceand m is the map factor. Ifr3 = (~ + ~ + axxy)(~yy + ~-- axxy) (~x~+ a ~is less than zero, the solution of m \2 m2 ~-~d~ / - r/~-~ d~ + r3 - a = O,OCTOBER 1982 TROND IVERSEN AND THOR ERIK NORDENG 1353ensures that r3 becomes equal to a (= 10-13 S-2). Inorder not to destroy the correction obtained, the solution with the negative sign in front of the radicalmust be used, i.e., d2 df = 2m---5 7 - [72 - 4(r3 - a)m.The radical is positive since r3 ~< 0 and a > 0. REFERENCESArakawa, A., 1966: Computational design for long-term numerical - integration of the equations of fluid motion: Two-dimensionalincompressible flow, Part 1. J. Comput, Phys., 1, 119-143.Arnason, G., 1958: A convergent method for solving the balance equation. J. Meteor., 15, 220-225.Asselin, R., 1967: The operational solution of the balance equation. Tellus, 19, 24-31.Bjorheim, K., 1979: The objective analysis scheme for operational use at the Norwegian Meteorological Institute. Tech. Rep. No. 40, The Norwegian Meteorological Institute, Oslo, 44 pp.Bolin, B., 1956: An improved barotropic model and some aspects of using the balance equation for three-dimensional flow. Tellus, 8, 61-73.Charney, J. G., 1955: The use of primitive equations of motion in numerical prediction. Tellus, 7, 22-26.Courant, R., and D. Hilbert, 1962: Methods of Mathematical Phys ics, Vol. 2. Interscience, 324-326.Daley, R. W:, 1978: Variational nonlinear normal mode initial ization. Tellus, 30, 201-218.Elliassen, A., A. Grammeltvedt and O. Bremnes, 1964: Studies in numerical weather prediction and the dynamics of fronts. Final Report, A. Eliassen, Ed., Inst. Theor. Meteor., University of Oslo, 26-29.Haltiner, G. J., 1971: Numerical Weather Prediction. Wiley, 317 pp.Lorenz, E., 1960: Energy and numerical weather prediction. Tellus, 12, 364-373.Machenhauer, B., 1977: On the dynamics of gravity oscillations in a shallow-water model, with application to normal mode initialization. Beitr. Phys. Atmos., 50, 253-271.Miyakoda, K., 1956: On a method of solving the balance equation. J. Meteor. Soc. Japan, 34, 364-367. , 1960: Numerical solution of the balance equation. Coll. Meteor. Pap., 10, Nos. 1-2, Geophys. Inst., Tokyo University.Paegle, J., and E. M. Tomlinson, 1975: Solution of the balance equation by Fourier transform and Gauss elimination. Mon. Wea. Rev., 103, 528-535.Pedersen, K., and K. E. Gronskei, 1969: A method of initialization for dynamical weather forecasting and a balanced model. Geo phys. Norv., 27, No. 7. Potter, D., 1973: Computational Physics. Wiley, 304 pp. Shuman, F. G., 1957: Numerical methods in weather prediction:I. The balance equation. Mon. Wea. Rev., 85, 329-332.Tribbia, J. J., 1981: Nonlinear normal-mode balancing and the ellipticity condition. Mon. Wea. Rev., 109, 1751-1761.