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Equatorial Kelvin wave in finite difference models Ng, Max K.F.; Hsieh, William W. Jul 15, 1994

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JOURNAL  OF GEOPHYSICAL  RESEARCH,  VOL. 99, NO. C7, PAGES 14,173-14,185, JULY 15, 1994  The equatorial Kelvin wave in finite difference models Max K.F. Ng and William W. Hsieh Departmentof Oceanography, Universityof BritishColumbia,Vancouver,Canada  Abstract. Coarseresolutionoceanmodelstendto poorlyresolvemany smaller-scale phenomena,includingthe equatorialcurrentsnarrowlyconfinedaroundthe equator.We studythe freeequatorialKelvin wavein inviscidfinitedifferencemodelsusingthe AmkawaA, B, C, and E grids.Exactanalyticsolutionswith meridionalvelocityv = 0 are foundon the A, C, andE grids.As the assumption v = 0 is not valid on the B grid, the solutionis obtainednumerically by a "shooting" method.In all cases,thewaveremainsnondispersive, andthephasespeedis unchanged from thatin thecontinuumexceptin theB grid, whereit decreases with worsening resolution.The meanzonalheat transportby the Kelvin wave duringan El Nifio is comparedon the variousgrids.In termsof the currentsand sealevel displacements, the B grid bestmodelsthe equatorialKelvin wave undercoarseresolution,thoughin termsof zonalheattransportand phasevelocity,theC grid appearssuperior.The A andE gridsappearto havethemosttrouble.  Our theoretical predictions arecheckedexperimentally by generating equatorialKelvinwavesin linearshallow-waterequationmodelson thevariousgrids.Additionaleffectsof Rayleigh dampingandNewtoniancoolingarestudiedin theappendix. 1. Introduction  usedthe B grid; Bleck and Boudra [1981], Blumberg and  Distortions by finite difference effects are significant in coarse resolution ocean models, where the grid spacing is usually insufficient for properly resolving the internal Rossby radius. Nevertheless, there have only been a few studies of f'mite difference  effects in ocean models: Arakawa  and Lamb [1977] and Batteen and Han [1981] examined f'mite  differenceeffectson Poincar6waves,Henry [1981] andHsieh et al. [1983] on coastalKelvin waves, Wajsowicz [1986] on Rossby waves, Foreman [1987] on continental shelf waves, and O'Brien and Parham [ 1992] on equatorialKelvin waves. Global ocean climate models and coupled climate models tend to use grid spacingsof the orderof the internalequatorial Rossby radius (about 290 kin), which is inadequate for resolving in the meridional direction the equatorial Kelvin wave and the equatorial undercurrent,as both are trapped mainly within one Rossby radius of the equator. The zonal grid spacing,on the other hand, has a negligible effect, as the wavelength in the zonal direction is generally much larger than the Rossbyradius.Henceforthwe limit our studyto only finite  difference  effects in the meridional  direction.  Since the  equatorialKelvin wave has a centralrole in determiningthe equatorial climatology as well as the El Nifio-Southern Oscillation in climate models, the study of finite difference effects on the Kelvin wave is crucial to understanding biases in our climate  models.  In finite differenceoceanmodels,there are severalpossible arrangementsof the model variablesin the horizontal plane  (Figure1). Exceptfor therarelyusedArakawaA grid,thegrids  Melior [1983], Dietrich et al. [1987], and Haidvogel et al. [1991] used the C grid; while Maier-Reimer and Hasselmann [1987] used the E grid. The only previous study on finite difference effects on the equatorial Kelvin wave is a note by O'Brien and Parham [1992], examining a particularcase of the C grid. Our objective is to provide a comprehensivetreatment of the finite differenceeffectson the equatorialKelvin wave over the various grids. We presentexact analytical solutionsof the free equatorial Kelvin wave in the A, C, and E grids, and examine  how  resolution  affects  the wave  structure  and the  zonal heat transport. As the B grid does not lend itself to exact analytical treatment, a numerical treatment is used for the B grid. Comparisonsare made betweenthe variousgridsto determine which grid models a particular aspect of the equatorial Kelvin wave best. The outline of this paper is as follows: After laying down the governing equationsin section 2, we explore equatorial Kelvin waves on the C grid in section3, the E and A gridsin section4, and the B grid in section5. Zonal heat transports on the variousgrids are comparedin section6. Our theoretical predictions are tested experimentally in section 7 by generatingequatorialKelvin waves in shallow-waterequation models.Kelvin wavesunderRaleighdampingand Newtonian cooling in finite difference models are studied in the appendix.  2. Governing Equations  in Figure 1 are staggered,in that the variables are not all The unforced linearized shallow water equations on the located at the same site. The Geophysical Fluid Dynamics Laboratory(GFDL) modularoceanmodel and its predecessors equatorialfl planeare [e.g., Bryan and Cox, 1967] and the Oberhuber[1990] model  ut - •yv = -g'lix - eu,  vt +•Yu=-g'liy- •v,  Copyright1994 by the AmericanGeophysicalUnion.  lit + H(ux+ Vy) = - •rll,  Paper number94JC00473. 0148-0227/94/94JC-00473505.00  (1)  where u and v are the eastward and northward velocity 14,173  14,174  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DIH•ERENCE MODELS  ß  u,v,q  ß  u,v,q  u,v,q  U,V  U,V  x  u,v,q  u,v,q  l'  u,v,q  4  u,v,q  q  u,v,q  q  u  q  U,V  U,V  x  x  u,v,q  (a) A grid  T  x  (b) B grid  u  •  T  uv  U,V  v  rl  A  u  q  v  q  u  v  v  u  q  T•  q  u  U,V  q  A  q U,V  (c) C grid  T  U,V  (d) E grid  Figure 1. Orientationof the modelvariablesin the ArakawaA, B, C, andE grids,whereu, v representthe  eastward andnorthwar•d•yelocity components, andt/ thevertical displacement orthepressure fluctuation.  Tracerssuchastemperature or salinityarealsolocatedat the t/ sites.(The ill-behavedD grid resembles the C grid,but with the u andv pointsinterchanged). The meridionalgrid spacingA is nondimensionalized by the Rossbyradius.The zonalgridspacing is irrelevant, aswe will maketheassumption thatthe zonalwavelength is well resolvedby the zonalgrid spacing,thusignoringthe finite differenceeffectsin the zonaldirectionø components,q the vertical displacement,fl the northward [u,t/]: [u(y),tl(y)]exp[i(kxax)]. derivative of the Coriolis parameterat the equator, g' the reducedgravity, H the equivalentdepth,and • and y the Substituting(4) into (3) yields coefficientsfor mixing of momentumand heat respectively.  (4)  Equation (1) can representeither one of the internal modesin  (5)  r•=cu, yu=-CUy, c=a•lk=+l.  a continuously stratified fluid (McCreary, 1981], or the internal mode in a two-layer fluid (Gill, 1982], where q The equatoriallytrappedsolution,the equatorialKelvin wave, becomes the interface displacement.Arakawa and Lamb follows from choosingc=l; whence [1977] showedhow the shallowwaterequations are computed u(y)=uoex•pl-y 2/ 2), (6) on the variousfinite differencegrids. Restricting to zonal flows, we set v = 0. With the  with uo the wave amplitudeat the equfitor.The wave is nondispersive andeastwardpropagating, as the phasespeedc = 1. Equations(5) and (6) provide the continuumsolution, obtainthe nondimensionalized againstwhich the finite differencesolutionsof the following  horizontal length scale L the equatorial Rossby radius  •[c'lfi, timescale(c'fi)-1, andvertical length scale H,  wherec'=•[•'H, we equations  ut =-r G-t:u, yu=-rly,  th +% =-•a/,  (2)  sectionswill be compared.  where (u, v) is the nondimensionalizedvelocity transport (velocitymultipliedby the equivalentdepth),and •: andy are the nondimensionalized mixing coefficients. We begin with the undampedcase where the Rayleigh friction coefficientE and Newtoniancoolingcoefficienty are  3. Equatorial Kelvin Waves on the C Grid  zero, so (2) reducesto  have been ignored. A possiblearrangementof the C grid placesa u point on the equator(Figure 2a). Substituting(4) into the centerMifferenced form of the first andlastequations in (3) on this C1 grid yields  ut=-t/x,  yu=-•/•,  th=-u x.  (3)  Assuming a plane wave solution in the zonal direction, the dynamic variables take the form of  The finite differenceeffect on (3) is next investigated by looking at two different configurationsof the C grid (Figure 2), where the finite difference effects in the zonal direction  (Ouj= kT•j,  (01'•j = kuj,  (7)  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DIH•RENCE MODELS  C2  C1 U1ll 1 A  14,175  V1 A  V  Uo 1]o  EQUATOR  Uo TIo Vo  vo  u_1 •1_ 1  u_1 •1_ l  V_1  (a)  (b)  Figure2. Two possible configurations of theC gridabouttheequator,henceforth referredto astheC1 andC2 grids, respectively.  On the C2 grid (Figure2b), Equation(7) remainsvalid but withj the northwardindexdefinedby y = jA (]=0, +1,+2... ). The middleequationin (3) can be evaluatedin two possible Equations (Sa)and(Sb)arechanged, yielding,caseC2a, ways, dependingon how the Coriolis term is approximated: case Cla,  (j- «)A«(u•+u•_•)=-(• - •_• )/ A,  (8a)  +jA2/2cuJ -1'  (12)  +(j+«)A2/2cJ uj-l'  (13)  and caseC2b,  andcaseClb,  A«[juj+(j- 1)u j_1]=-(hi - nj-i )/ A,  (Sb) uj=  where in caseC l a, the Coriolisparameteris estimatedat a v point, then multiplied by the latitudinal averageof two u points;whereasin caseC lb, the averageof the Coriolis term yu at two u pointsis usedinstead.Thesetwo casescorrespond respectivelyto the potential-enstrophy-conserving scheme and the energy-conserving schemeby Sadourny[1975a,b], who found that the potential-enstrophy-conserving scheme (i.e., our caseC l a) to havesuperiorstability. From (7), we have  where the Coriolis terms are again treated in two ways, analogousto casesCla andC1b, respectively.Note that the phasespeedc of the equatorialKelvinwavein all fourC grid configurations is independent of the grid spacingA, a result analogousto the coastalKelvin wave on the C grid as found by Hsieh et al. [1983]. O'Brien and Parham [1992] examined caseC2a, and essentiallyderived (12). The meridional structure of u for the four cases at resolution  •lj = cuj,  c= +1,  (9)  wherefor the eastwardpropagating Kelvin wave,we choosec = 1. Hence,the Kelvinwavephasespeedis unaffected by the finite differenceeffectsin the C1 grid. Equations(9) and(Sa) yield the recurrencerelation  uj =  l+(j-«)A•/2c J"i-•'  (10)  caseCla, wheregiven,say,thevalueu0, all othervaluesuj are obtained.As in the caseof the continuumsolution(6), the finite differencesolutionis also symmetricaboutthe equator,  sinceu_j= uj. In theClb case, (9) and(8b)yield ui =  I+jA2/2c juJ -1'  which is alsosymmetricaboutthe equator.  (•)  A = 1 is shownin Figure 3, where the curveshave been normalizedto all have the samemeridionallyintegratedzonal masstransportas in the continuumsolution.In Figure3a, the value of u in caseCla is higher than that for the continuum solutionat the equator,butdeclinesawayfromtheequatorat a greater rate than the continuum solution, yielding a weak reverseflow by 3 Rossbyradii from the equator.The solution in case C lb is closer to that in the continuum.Similarly, Figure 3b showsthat the solutionin C2b is betterthan the solution in C2a' Hence for both the C 1 and C2 configurations,method b (i.e., latitudinally averagingthe Coriolis term yu from two u points)yields a more accurate equatorialKelvin wave than methoda (i.e., multiplyingthe Coriolis parameter at a v point to the averageof two u points), despite$adourny's [1975a,b] finding that methoda offered superiorstability. The percentageerror of the finite differencesolutionswith respectto the continuumsolutionat  A = 1 is shownin Figure4, whereresultsfrom othergrids (discussed in the followingsections)are alsoplottedto allow  14,176  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE D•tqSRENCE MODELS (a)  Co) 2.0  2.0  '  '  I  '  1.8  1  8  -  1  6  1.6  1  4  1.4  o x  C2a C2b  1.2 o  •I• 1.0 8  --  \  6 .4  .4  \  ß2 ........... - .2  0  .2  ----"-"•2.• .... • ,  I  .5  - ._-r F-- .- _--  ,  I  1.0  ,  I  1.5  i  I  2.0  ,  I  2.5  ,  I  3.0  ,  I  3.5  ,  4.0  -.2  ,  0  I  .5  ,  Distance fromEquator  I  ,  1.0  !  1.5  ,  I  ,  2.0  •  2.5  ,  ,  I  I  4.0  3.0  Distance fromEquator  Figure 3. The zonal transportu as a function of the distancefrom the equator (nondimensionalizedby the  equatorialRossbyradius)in (a) Cla (solidcurve),Clb (sh6rtdashcurve)andthecontinuum(longdashcurve), and (b) C2a (solid curve), C2b (shortdashcurve) and the continuum(long dashcurve). These curveshave all been normalized so that the total meridionallyintegratedu transport(i.e., area under the curve) is the samefor all the curves. The grid resolution A = 1 (where A is defined as the latitudinal grid spacingdivided by the Rossby radius). an intercomparisonamong the various grids. The changesin the zonally averagedrms error of u and h with changesin the grid resolutionA are shownin Figure5. For large enoughj, all four cases,i.e., (10) to (13), yield  uj •--uj_ 1, i.e., a grid-scale oscillation always occurs in the C grid when far enoughaway from the equator.How far away from the equatorwhen this oscillationoccursis controlledby  A. For large enoughA,-the grid-scaleoscillationsoccurright near the equator, analogous to the coastal Kelvin wave behavior found by Hsieh et al. [1983]. Fortunately, the Rossbyradius is so much larger in the equatorialregion than at midlatitudesthat this pathologicalequatorialKelvin wave is not likely to occur even in coarse resolution climate models.  Co)  (a) 4o  30 t ' [ ' [ ' I ' [ ' •' ] ' [ '  20  '  I  '  I  '  I  '  I  '  I  '  22  •Clb  lO  E1  -2½  -3½ -40  -50  -lOO  t .3  ,  [ .6  ,  [ .9  ,  • 1.2  ,  [ 1.5  ,  : 1.8  Distancefrom Equator  ,  [ 2.1  - ""•' 'B 2  -80  Cla  ,  -120 2.4  ,  0  ,  I  .5  ,  I  1.0  ,  I  1.5  i  I  I  2.0  I  2.5  Distancefrom Equator  Figure 4. The percentageerror in the f'mitedifferencesolutionof u with respectto the continuumsolutionfor cases(a) Cla (circles),Clb (crosses), E 1 (circledcrosses), andB 1 (triangles),and(b) for C2a (circles),C2b (crosses),and B2 (circledcrosses),with grid resolutionA =1.  3.2  14,177  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DIH•ERENCE MODELS (a) .6  I  '  I  '  I  '  I  •) '  I  '  I  I  '  .•  '  I  '  I  '  I  '  I  '  I  '  I  I I  .5  I  I  I  I  !  I  !  !  I  I  I  I  C2•  I  C2a  I  I  I  //// C1•  ///// C 1 • •,, // ,,xClb •  /  I  I  I  /  '  ,"  ø"•'  C2b  .1  ./  B1  •  ,,'.•./'" C2 b  .,./__••._..•øøøø ,,' _B2I ..... -•'--v'-"l 'e ,  •- -,•."•"•'-'Z .2  .4  .6  .8  1.0  1.2  1 .4  1.6  0  .2  .4  .6  .8  1.0  1.2  i  1 .4  Grid spacing A  Gridspacing A  Figure5. The zonallyaveraged Rmserrorfor (a) thezonaltransport u and(b) the sealeveldisplacement h of thefinitedifferenced equatorial Kelvinwave(withrespectto thatin theconQnuum) plottedasa functionof the  gridspacing A onthefourC grids,theEl, theB1, andB2 grids.  4. Equatorial Kelvin Waves on the E and A Grids When  uj+ 1+2JA2uj-uj_i =0, (j=O, +1,+2...).(16) c  finite  difference  effects in the zonal direction  are  ignored,the E and A gridsfrom Figure 1 collapseto the same grid shownin Figure 6. A u point is locatedon the equatorin the E1 grid, wherethe finite differenceformof (3) is givenby  rouj= k•/j, rm/j=kay, (•tj+•- z/j_•) jAuj =. 2A  (14)  In contrastto the first-order linear difference equations encounteredin the previoussectionon the C grid, Equation (16) is a second-orderlinear difference equation with a nonconstantcoefficient,asj is the independentvariable. A helpfulhint in solving(16) is providedby the recurrence relation of the modified Bessel functions [Abramowitz and $tegun, 1972]:  (15)  Fj+ 1(z)+2__j Fi(Z)-Fj_ 1(z)=0,  Equation(14) againleadsto (9), and the followingrecursion relation  is derive:  (17)  wherethe argumentZ may be complex.Whenj is an integer,  E1  E2  A/2  Uo v o 11o  EQUATOR  i Iuø vø  U_lv_•11_1  (a)  (b)  Figure6. Two possibleconfigurations of the E grid (or A grid) aboutthe equator;henceforth referredto as the E1 and E2 grids,respecQvely.  NG AND HSIEH: EQUATORIALKF.LVIN WAVE IN FINITE DIl•I•ERENCEMODELS  14,178  the two linearly independent solutions of (17) are the  appropriateto comparethe E grid of resolutionD with the B  •f•A. Sucha comparison in modified Bessel function ofthefirstkind, lj, and(-1)JKj, andC gridswithresolution with Kj themodifiedBesselfunction of thesecond kind. Figure5 wouldshowtheE1 gridto be competitivewith theC  Hence, Fj denotes /j, (-1)JKjoranylinear combination of grids, but still not with the B grids. thetwo.Bychoosing c=1andz= 1/A2, (16)and(17)are  identical; hence  (j=0,  +1,+2...),  With the E2 (and A2) grids, where the u point no longer lies on the equator(Figure6b), Equation(15) is repla• by  (j+«)Auj =- (rlj+l - rlj_l )  (lS)  (20)  2A  where A and B are arbitrary constants.Note that z is real in  leading to  Uj+ 1+--2c(j+•)A2u j -Uj_ 1=0, (j = 0,  thiscase.Figure7 shows at A =1, thebehavior of AIj in relation to the continuum solution (6) with both solutions normalized to having the samemeridionallyintegratedzonal  + 2... ).(2 l)  The recurrencerelation of the modified Besselfunctionsagain  transport. While AIj resembles the continuum solution, providesa hint. Consider  (-1)JKjdisplays wildlygrowing grid-scale oscillations away from the equator.Whetherthis extraoscillator),modeis a problem in numericalmodelsis investigatedin section7.  To obtain a bounded solution, theKj termisdropped in (18); hence  uj =Atj,  (j=0,  fj+l (Z)+  2(j+«)fj(z)_fj_i(z)=0, (j=0, ñ1,ñ2...),(22) z  where the argumentz may be complex. The two linearly independentsolutionsof (22) again involve the modified  The symmetryof the continuumsolution(6) aboutthe equator  Bessel functions, though offractional order, where /j+«and /_j_«arelinearly independent. Thegeneral solution of(21)  is also observedhere as uj-AIj--AI_j--u_j.  is  The  percentage error of AIj with respectto the continuum  f j =4•r/(2z) (AIj+« +BI_j_«),  (23)  solution is also shown in Figure 4a, which showsthat it is generallypoorerthan the C grid results,exceptwhenfar away from the equator.Similarly, Figure 5 showsthe E1 (and A1) grids to have larger zonally averagedrms errorsin u and h than the other grids.To be fair to the E grid, we needto point out that the E grid (with spacingA in bothdimensions)can be  spherical Bessel functions of the first and secondkinds, respectively. For a solution of (21) symmetric about the  regarded asa B grid(withspacing •A  equator, wechoose c= 1, z= 1/ A2 andA = B;hence  ) rotated by45ø  where A and B are arbitrary constants. The functions  4•r/ (2z)lj+«,%]•r / (2z)/_j_« areknown asthemodified  (Figure 1). Hence in Figures 4 and 5, it may be more  ,j =C(Ij+«+/_j_«), 2.0  '  I  '  I  1.8  '  I  '  I  '  I  '  i  '  I  1.00 E1  1.6  '  (24)  where the symmetry about the equator is easily seen from  u0=u_1=C(I«+/_«),u1=u_ 2=C(I•+/_•),etc.Figure 8 plotsthebehavior of uj ontheE2gridat A = 0.3, wherethe finite difference solution closely resembles the continuum solution near the equator, but develops large unbounded oscillations away from the equator. Reducing A does not eliminate the unbounded oscillations, but pushes their  1.4  appearance furtheraway from the equator.IncreasingA leads to theseunboundedoscillationsmoving closerto the equator,  %  such that at, for instance, A = 1, the solution is unreasonable  \ \  with oscillationsright near the equator.Even thoughthe C \ grid also has grid-scale oscillations(which do not grow .4 spatially)for large enoughA and far enoughaway from the equator,the situationfor the E2 caseis much worse,as the .2 oscillationsgrow rapidly spatiallyand are muchcloserto the equatorfor the sameA. Thus an equatoriallytrappedKelvin wave modeis not properlysupported in the E2 grid. For small -.2 , I , I , I , I , I , I , I , A, the theoretically predicted spatially growing oscillations 0 .5 1 .0 1 .5 2.0 2.5 3.0 3.5 4.0 probably occur too far away from the equatorto be a real Distancefrom Equator problem.Of all the grids, only the E2 and A2 have no grid Figure 7. The zonal transportu as a functionof the distance points on the equator at all, so for large enoughA, the from the equator in the E1 grid (solid curve) with the absenceof grids at the equator is likely to cause serious problems for modeling the equatorial Kelvin wave. This continnum solution shown as the dashed curve. The grid strangephenomenonis investigatedfurtherin section7. spacingA = 1. \  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DwH•RENCE MODELS 2.12  14,179  the breakdownof the v = 0 assumptionhere. Nonzero v terms need to be retainedin both the massandmomentumequations, since the simple balances of (7) and (14) are no longer possible with u and q located at different latitudes. After considerablealgebra, u andv can be eliminated,resultingin a recurrencerelation for q on the B 1 grid:  1.5-  1.0  ;/j+l + F;/j + G;/j_1= O,  (25a)  F = {2co2 (mt:2A 2+ 4co-2kA2- 2co3A 2) where + coA2 [j 2+(j- 1)2](4c02A 2-k2A2-4)  o  -4A4j(j - Ilk + coA2j(j - 1)]} /{co(k2A 2- 4)[c02 -(j- 1)2A2]}  G=(co 2-j2A2)/[co2- (j-1)2A2].  (25b)  •.5  0  1  2  3  4  5  6  7  8  Distancefrom Equator Figure 8. The zonal transportu as a functionof the distance from the equator in the E2 grid (solid curve) with the continuum solution shown as the dashed curve. The grid  spacingA = 0.3 (whichis muchsmallerthanthe A = 1 value used in Figures 3 and 7). Due to the divergence of the  The recurrencerelation for q is of secondorderas in the E grid case, but the additional v terms render (25) a much more complicated problem. As we could not find an analytic solutionfor (25), we useda numericalapproachbasedon the "shooting" method. Given o• and A, we integrate(25a) northwardstartingfrom  a point ;/-N severalRossbyradii southof the equator.Since  thesolution Canbe linearlyscaled,;/-N canbe arbitrarily  fixed. To integrate(25a) northward,we needto make an initial guessof two parameters,the wavenumber k and ;/-•v+l, i.e., meridionally integrated zonaltransport, the norm•alizationthe ;/ immediatelynorth of ;/-N- Both parameterscould be usedhere is as follows: the E2 solutionis chosento have the complex. Equation (25a) is integrated to the northernmost same u value as the continuum at the first point from the  point;/t•,anda costfunction isdefined asthesumof ;/2  equator.  over the few northernmostpoints. Since we are seeking an equatorially trapped solution, we minimize the cost function  5. Equatorial Kelvin Waveson the B Grid  by adjustingk and ;/-/½+1. Sincev was not eliminated,the  recurrencerelation (25) holds over the entire equatorialwave In contrastto the A, C, and E grids,the one-dimensionalB spectrum, which includes Rossby waves and gravity waves. grid (Figure9) hasthe variablesu andq at differentlatitudes. Hence, the frequency o) is deliberately set low to avoid The finite differenceequationsare againformulatedin two grid convergenceto the gravity waves. A wave period of 130 days is chosen. orientations,B 1 andB2, wherea u pointlies on the equatorin For the B 1 case, we found it necessaryto allow the B 1, but an q point lies on the equatorin B2. The major wavenumberto be complex.As A becomessmall, the ratio of differencefrom the previouscasesof the A, C, and E gridsis  B1  B2 UlV 1  A  T[0  uovn  A  Ul VI  EQUATOR  qo u0v 0  U. lV. 1  (a)  q-I  (b)  Figure9. Two possibleconfigurations of the B grid aboutthe equator,henceforthreferredto as the B 1 andB2 grids, respectively.  14,180  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DIH•ERENCE MODELS  !.0  increases(Figure 10), analogousto the behavior for coastal Kelvin waves on the B grid [Hsieh eta/., 1983]. Although the phase speed is dependent on A, the wave remains nondispersive• (i.e., o• o•k). For the wave structure, the latitudinal variation of u transportsis comparedto that of the continuum counterpart (Figure 11a). Even though the "shooting" method did not guarantee a zonally symmetric  g  solution, thenumerical solution didturnoutto besymmetric. In contrastto the C 1a and E1 grid cases,the wave in B1 is  seento be underestimated near the equator(Figure4a). At regionsaway from the equator,the wave is widened,similarto the B grid coastalKelvin wave [Hsieh eta/., 1983]. For the v transport,the ratio of v/u approaches zero as A tendsto zero, as expectedfrom the continuumtheory.In general,v/u is too small to be noticeablydifferent from zero. For the B2 case,eliminatingu andv leadsto  .7  (26a)  •j+l + Frlj + Grlj-1= 0 where  .6  I  0  I  ,  .4  I  ,  .$  i  ß  ,  1 .2  i  ,  1 .6  I  ,  2.0  2.4  Grid spacingA Figure 10. The phase speed c as a function of the grid resolution A for the B 1 grid (solidcurve)andfor the B2 grid (dashed curve). Observed pltase speed in shallow water equation models with the B2 grid (see section 7) are also plotted as the circled crossesfor comparisonwith the dashed  F= {2m2 (a•2A2+4m-2kA 2-2m3A 2)  +wA2 [(j_•)2+(j+•)2](4w2A2 _k2A2 -4) -4a4(j-•)(j +•)[k+•a 2(j-•)(j +•)]} /{•(k2a2_4)[•2- (j_ •)2A21} G=[•2 _(j +•)2a2]/[•2 _(j_ •)2a2].  (26b)  The propertiesof the solutionof (26) (Figures9, 10b, 4b, and the imaginary part to the real part of k convergesto zero, while the real part of k convergesto the wavenumberin the continuum solution, as expected. The imaginary part of k meansthat the Kelvin wave is zonally dampedon the B 1 grid. At A=I, the 130-dayperiod Kelvin wave hasa zonale-folding  5) are similarto thosefor the •1 case,exceptthat the wavenumber k now turns out to be real, meaning that the small zonal dampingfoundin the B 1 grid is not presentin the B2 grid. The decline in the phase speed with worsening resolution is more severe in the B2 grid than in B1 (Figure  damping scaleof 4.05x103 (Rossby radii),equivalent to  10). From Figure 5, the B2 grid outperformsB 1, with the B  about 29 times the Earth's circumference; hence the zonal  grids generallysuperiorto the A, C, and E grids in modelingu and q of the equatorialKelvin wave.  dampingis negligiblysmall.The phasespeedc decreases asA  Co)  (a) 1.2  '  I  '  I  '  I  '  i  '  I  '  I  '  I  '  I  '  I  '  1.2  '  I  '  I  '  I  '  I  '  I  '  I  '  i  '  I  '  I  I  '  I  '  oB2  oB1 1.0  1.0  '  A:  q.oo  .8  .8  dIIII ,  I  -4  ,  I  -3  ,  !  -2  ,  I  -1  ,  .2  I  0  ,  I  1  ,  J  2  ,  -.2 -6  I  -5  ,  I  -a  ,  I  -3  ,  I  -2  ,  !  -•  ,  I  z  ,  I  •  ,  I  2  ,  I  3  ,  I  ,  4  Distancefrom Equator Distancefrom Equator Figure 11.Thezonal transport uasafunction ofthedistance fromtheequator in'the(a)B1gridand(b)B2 grid, with the continnumsolutionshownas the dashedcurve. The grid spacingA = 1.  I  5  ,  NG AND HSIEH: EQUATORIAL KEI.VIN WAVE IN FINITE DIItqsRENCE MODELS Since (25a) and (26a) are secondorder differenceequations like (16), the B grid could also contain an extra spatially growing oscillatory mode like that in the E1 grid. Unfortunatelythe costfunctionin our shootingmethodwould only capturethe boundedsolutionsfor the B grid.  14,181 (34)  Mean ZHT is obtained from (32) and (34) where u(y)is computedfrom the continuumsolution(6) or from one of the C or E grid solutions.The integrationin (34) is computedto +8 Rossbyradii away from the equator.  For the B grid, since rI ½u, the integrals I2 in (32) are  6. Zona Heat Transport The thermoclinein the equatorialPacific is normally tilted  replacedby I3, where  in the zonal direction, from about 200 m on the western  I3=]yY•u•l dy, (I3)=«I•2u(y)•l(y) dy  Pacific to about $0 m in the easternPacific [Philander, 1990].  During an E1 Nifio, an equatorial Kelvin wave traverses eastward across the Pacific, transmitting a large heat flux, while causing a deepening of the thermocline and an anomalouswarming of the sea surface temperaturein the eastern Pacific. An immediate application of the present theory is to calculate the zonal heat flux transmittedby the equatorial Kelvin wave in the various grids and to compare with the correspondingvalue in the continuum. The meridionally and vertically integrated zonal heat transport(ZHT, measuredin J/s) is defined (in dimensional variables) as  ZHT =pCp I>•' [•DUrdz]dY  (35)  The model parametersusedto evaluatethe ZHT in (32) are  g'= 0.02 m s'2, H1= 200m, H2 = 4000m, T1= 303øK, and T2 = 290øK. The pre-E1Nifio situationin the equatorial Pacific has the interface on the western boundary lowered (i.e., a trough) and that at the eastern boundary raised (a "crest").During an E1 Nifio, the "trough"propagatesfrom the western boundaryto the easternboundaryin about2 months. This suggeststhat the E1 Nifio scenariocan be representedin our simple two-layer model by an equatorialKelvin wave with a wavelengthdoublethat of the zonal basinwidth and a period of about 4 months(we chose130 days). Figure 12 showsthe percentageerror in the ZHT on the four C grids, the El, the  C27) B1,  and the B2 grids with respectto the ZHT in the  continuum.The ZHT on the E2 grid is not consideredhere as  whereT is thetemperature, P andCp arethedensity and  this solution  specificheat of water, respectively,Yl and Y2 are latitudinal limits of integration, and D is the ocean depth. For simplicity, considera two-layermodel, with subscripts1 and 2 denoting the first and secondlayers. The vertical integral  Note that the ZHT valuesin Figure 12 were calculatedbasedon the assumptionthat the meridionally integratedu transports in the finite differencemodels were the same as the transport  term in (27) becomes  assumption,Clb, C2b, B1, andB2 all underestimated the ZHT, whileCla, C2a' andE1 overestimated theZHT relative  [vøutdz=uiti(Hi-h)+u2t2(H2 +h).  (28)  in  the  was found  continuum  to be unbounded  solution.  Under  in the last section.  this  normalization  20  where both temperatureand velocity are constantthroughout eachlayer, H i is the equilibriumwaterdepthin layer i, andh is the interface elevation. Substituting(28) into (27),  O  •,,  15  (29)  +T2H2I;•2 u2 dy+r2I;•2 u2 hdy]. These  dimensional  variables  are  related  to  the  nondimensionalvariablesu and •l by  u!=- H2 c'u,u2=H+H: H1 c'u, h=H•,  (30)  with H the equivalentdepthand c' thewave speeddefinedas •-10  H= HH: , + H: With • = u in the continuumas well as the A, C, and E grids, (29) can be expressedas  H2  -20  H2  1.6  ZHT =pCpLc'{-T1HI 1+T1 -•I 2+T2HI 1+T2•2-2 I2},(32) where L, the length scale,is the equatorialRossbyradius, and the nondimensionalintegralsare  ii =j•2udy, I2=lyY• u2dy.  (33)  Taking u = u(y)cos(kx- oat) and averagingover one wave period and one wavelength,we obtain  Grid spacingA  Figure 12. The percentage error in the mean zonal heat transportof the finite differencedequatorialKelvin wave (with respectto that in the continuum)plotted as a function of the  grid spacingA on the four C grids,the El, the B 1 and B2 grids.The frequencyand wavenumberof the Kelvin wave were chosen to be representativeof the Kelvin wave during a typical E1 Nifio.  14,182  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DIH•ERENCE MODELS  to the continuum solution. In general, the best ZHTs were found in C lb, C 2b, followed by B1 and B2, in decreasing orderof accuracy.It may seempuzzlingwhy the B grid, which models the currents and sea level displacementsbetter than the C grid, turns out to be less accuratein the ZHT. This may be explainedby the fact that for the C grid, u and11are located at the same latitude, while for the B grid, they are not. Since the productu•l is involved in calculatingthe ZHT, the B grid is at a disadvantagewhen its u and •l points are at different latitudes.Again as in Figure 5, the resultfor E 1 at resolution A should probably be compared with the other grids at  1.3  '  1.2  1.1  I  '  !  '  '  I  0  0  '  I  '  I  m  '  .  1.0  resolution %1• A, which gives E1agood accuracy forZHT. .7  7. Numerical Experiments To experimentally verify our theoretical predictions of finite differenceeffects,we ran shallow-waterequationmodels [Arakawa and Lamb, 1977] on the various grids, with absolutely no damping. Centered on the equator, the numericalmodelswere 12 Rossbyradii widemeridionally,and 38 Rossbyradii wide zonally (corresponding to about 11,000 km), with solid boundaries all around. The initial condition correspondedto having an equatorialKelvin wave (with half wavelength equal to the model zonal width). Zonally, the wave had a sinusoialshapewith a crestcenteredin the middle of the ocean.Our finite differencetheorypwvided the initial sea level displacementand currentsfor this wave. The wave was allowed to evolve without external forcing. As the wave propogatedeastward,we observedthe shapeand measuredthe propagation speed. Our theoreticalpredictionof decreasingphasespeedin the B grid with worseningresolutionwas indeedobserved.Figure 10 showssomeexperimentallymeasuredphasespeedsplotted againstthe theoreticalphasespeedon the B2 grid. Similarly, our predictionsthat the phasespeedshouldremain unchanged in the C grid with worsening resolutionwas also confirmed, with Figure 13 showingsomemeasuredphasespeedson the C2b grid. We also used the continuum solution (Section 2) as the initial condition on these grids. However, as the continuum solution is not the correct Kelvin wave mode on thesegrids, the initial wave soon began to disperseat a significantly faster rate than in the correspondingrun which used our theoretical  finite  difference  wave  solution  as the  initial  condition.  Our predictionsfor the A or E grids were also tested on shallow-water A grid models. As the resolutionworsens,the observedphasespeedon the A 1 grid becamegreaterthanour predictions (which was a constantindependentof the grid spacing) (Figure 13). The spatially growing grid-scale oscillatory mode permitted under (18) was not observed, indicatingthat the mode was not readily excited.However,our runs were only of 1-monthduration,which couldbe too short a time to excite such a mode. In practice,numericalmodels would have damping terms which would also control this oscillatory mode. For the A 2 case, where there is no true equatoriallytrappedKelvin mode, we couldonly run with the continuum  solution  as the  initial  condition.  At  coarse  resolution, the results were a significantly stronger dispersionof the original wave and a drasticdecline in the phase speed(shown as the circled crossesin Figure 13). The sharp decline in the phase speed was likely causedby the scatteringof the initial wave form into the other equatorial  .6  ^2  .5  .4  m  I  •  I  I  I  m  I  m  I  .4  Grid spacingA Figure 13. Observedphasespeedsin shallow water equation modelsas a functionof the grid spacingA for the A1 grid (diamondsymbols),the A2 grid (circledcwsses),andtheC2b grid (triangles).The theoreticalphasespeedis plotted as the dashed horizontal  line.  modes, such as the westward propagating Rossby modes, which wouldproducethe low observedphasespeed.For finer resolution,the growing oscillationsin our theory (Figure 8) occur too far off the equatorto be of real consequence in our numerical model, and the initial Kelvin wave propagated without  much deviation  from the continuum  solution.  8. Summary and Conclusion We have derived exact analytic solutions for the free equatorialKelvin wave on the ArakawaA, C, andE grids.By ignoring the finite difference effects in the zonal direction, the A grid solutionsare identicalto the E grid solutions.The v = 0 assumptionfor the Kelvin wave in the continuum also  holdson the A, C, and E grids.The B grid is very different from the A, C, andE gridsin that the u and•1 pointsdo not lie on the same latitude in the B grid, so that a direct relation between u and•1, suchas (7), is not possiblein the B grid, renderinga needfor a nonzerov to balancethe momentumand mass equations.Our main findings for the equatorialKelvin wave on the A, C, andE gridsare as follows. 1. The dispersionrelationsfor the finite differenceKelvin wave are identicalto the dispersionrelationin the continuum, for bothundampedanddampedwaves(seeappendix). 2. For the C grid, a better solution is obtained by calculatingthe Coriolis term in the v-transportequationby latitudinally averaging the yu term (methodb) than by multiplyingy at the v latitudeto the latitudinalaverageof two u points (methoda), thoughmethoda pwbably has superior stability [Sadourny, 1975a,b]. 3. The E grid with a u pointon theequator(i.e., E 1), allows an extra grid-scaleoscillatorymode besidesthe Kelvin mode. However,the grid-scalemodedoesnot appearreadilyexcited. The E grid withouta grid pointon the equator(i.e., E2) does  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DIH-ERENCE MODELS not pror•rly support an equatorially trapp•t Kelvin mode, thusenergywould leak to otherequatorialmodes. The main resultsfor the B grid are as follows. 1. Phase speed for the Kelvin wave decreases with worsening resolution (i.e., increasing A). However, the dispersionrelation remains nondispersive(i.e., (o= k).  2. On B 1 (with u point on the equator),there is a small zonal damping arising from the finite differenceeffect, which is absentin the B2 configuration. 3. Parameterv is generallynonzero,and only satisfiesv --• 0 as A --• 0. In general,v/u is negligiblysmall. 4. Among the A, B, C, and E grids, the B grid appearsto model the currents and sea level displacements of the equatorial Kelvin wave best under coarseresolution,with the B 2 configurationslightly better than B 1. However, in terms of zonal heat transportand phase speed,the C2b and C 1 b grids are better.  1 .05  !  14,183 !  .00  .95  .90  .85  .80  ..  •=5 T  .75  ø70  -  .65  '"  =0  Appendix: Damped Equatorial Kelvin Waves We now succinctly examine the equatorial Kelvin wave with damping terms in the governingequations.The zonal flow equations with damping are given by (2). Again assuminga plane wave form (4), it follows that  u(y)=u0exp[-•/( (o+i7)/((o+i•) y2/2]  5'  1'0  1'5 2'0 2'5 3'0 3'5 40 y (x10-')  Figure A1. The zonal phase speed miRe(k) plotted as a function of y for variousvaluesof •.  (A1)  and  •/(y)=Uo •[(m+i•)/ (m+iY)exp[-•[( m+ir)/(m+i•) y2/21 (^2)  withu0 andUo•](m+i•)l(m+iy)thewaveamplitudes foru and q, respectively, at the equator [see Yamagata and Philander, 1985]. They found that increasing• widens the Kelvin wave in the meridional direction, while in contrast,  increasing y concentratesthe wave around the equator. The dispersionrelation is  k2 = a•2+i(7+•)a•-• 7  ß600  (A3)  Thus Rayleigh friction and Newtonian cooling dispersethe wave. By assumingfree wave (i.e., 0• real), (A3) implies wave dampingin the zonal direction.With 0• set at a wave periodof 130 days, the zonal phasespeed,(o/Re(k), is shownin Figure A1, where an interestingcancellationeffect betweeny and• is observed.For instance,with • = 0, increasing¾leadsto a drop in the zonal phasespeed,as expected.However,if • is raised, it begins to cancel the slow-downeffect due to •, such that when e = •, the phasespeedhasreturnedto 1, as if bothy ande were zero. Further increasing e again leads to a drop in the phase speed, as illustrated by the curve labelled e = 5y. The damping parameterse and y also producephaseshifts in the meridional direction, as seen from (A1) and (A2). The constantphaselines in the x-y plane are describedby  -lm[4( m+iy)/( m+ie)]y2 /2+Re(k )x=const.(A4)  direction,as in the undampedcase.Also from (A1) and (A2), it is clear that for the particular case of g = y, the meridional structuresof u andq arereducedto the undampedsolution. For the C grid, we limit ourselvesto the C2b case,with the finite differenceform of (2) yielding  (o+ie)uj =kllj, «A2[(j+•)uj +(j-«)Uj_l]= •j-1-11j, ( •0+iy)•lj= kuj.  (A5)  The first and third equationsin (A5) combine to give the dispersionrelation, which is easily shown to be identical to the continuum dispersion relation (A3) and is therefore independentof A. The meridional structureis obtainedby combiningthe first two equationsin (A5):  uj =  1-(j-{)kA2/2(m+ie)] 1+(j+-'Jj.) kA 2/2( (.0+ ie)JUj-l' (A6)  case C2b. As u is now complex,there is an additionalphase shift in the meridional direction. We choose to plot the meridional profile of Re(u) in Figure A2 at resolutionA=I, with the curve again normalized to have the same meridionally integratedzonal transportas in the continuum case.The plot is generatedfor a waveperiodof 130 dayswith  (dimensionalized) e = 1/ 50 day'1andy = 1/ 10 day'l, typical valuesusedby YamagataandPhilander[1985]. A weakreverse flow is found at 2.5 Rossbyradius from the equator.The corresponding undamped solution (from Figure 3b) is superimposedas the short-dashed curvesfor comparison. The finite differencedequationswith respectto the E1 grid  To the order of O(rdo•,y/o•), the imaginarypart of the term in brackets in (A4) is directly proportionalto (y-g), thereby revealing the cancellationeffect between y and g. The phase lines (which were aligned in the meridional direction in the 2A (A7) undampedcase)may now slantbackwardsor forwardsfrom the ( m+ i7) •lj = kuj. equator,dependingon the relative size of g andy, as discussed by Yatnagataand Philander [1985]. For g = y or o• = 0, (A4) The dispersionrelation is again found to be identical to (A3), implies that the phase lines are aligned in the meridional while the meridionalstructureis describedby  (m+ie)uj =k•7j,jAuj =_(.•7j+l - •lj-1),  14,184  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DIH•ERENCE MODELS I  I  I  I  I  I  Uj+l +2jA2 (o.;+' k")')uj-uj_l=O, (j=0,ñl,:k2 .... ). (AS)  !  C2b  _  Equation(AS) resembles(16) in the undampedcaseexceptfor the dampingeffect iy appearingnow in the secondterm. The effect from œis exertedimplicitly in (AS) via 0) andk through the dispersionrelation (A3). Following the same line of argumentas in the undamped case, solutionsof (AS) can be written in termsof the modified Bessel functionsexcept that the function' s argumentis now complex. Hence,  \  k  uj(z)=Ali(z), .2  , z=  1  re+i?'--'l' A  (A9)  and from (A7),  iIj(z)= Aa}+ie=a•a}+ie m+iy  (A10)  The meridional structureof Re(u) in (A9) is shownin Figure A3 for A=I. z  .5  a.o  2.z  2.5  3.z  3.5  Distancefrom Equator Figure A2. Re(u) for the C2b grid plottedas a functionof the distance from the equator. The solid curve shows the C2b solution for nonzero 7 and • (as given in the text), with the correspondingcontinuumsolutionas the long-dashedcurve. The corresponding undam• (7 = • = 0) finite differenceand continuum solutions are also shown by the short-dashed curves.In the dampedcase,we have chosen7 > œ;hencethe damped continuum solution (long-dashed curve) is more concentratedaround the equator than the invisicid continuum solution (short-dashed curve). (If 7 < œ were chosen, the dampedcontinuumsolutionwould be wider thanthe undamped continuum solution.) 1.3.  ,  ,  1.2  E1 1.1  In summary,as •1 and v lie on the samelatitudein all the C and E grid configurations, their dispersion relations are identical to that for the continuum,i.e. (A3). For the B grids, one can derive analogousequationsto (25) and (26), though the functions F and G will be complex, and a numerical approachwill againbe needed. Acknowledgments. This researchwas supportedby the Canadian Natural Sciences and EngineeringResearchCouncil through the CanadianWOCE projectandthe researchgrantto W. Hsieh.  References  Abramowitz,M., andI. A. Stegun,Handbookof Mathematical Functions., 1046 pp., Dover, New York, 1972. Arakawa, A., and V. R. Lamb, ComputationalDesign of the Basic Dynamical Processes of the UCLA General Circulation Model., pp. 173-265, Academic,San Diego, Calif.,  1977.  Batteen, M. L., and Y. J. Han, On the computationalnoise of .9  finite difference schemesused in ocean models, Tellus, $$, 387-396, 1981.  Bleck, R., and D. B. Boudra, Initial testing of a numerical ocean circulation model using a hybrid (quasi-isopycnic) vertical coordinate,J. Phys. Oceanogr.,11(6), 755-770, 1981.  Blumberg,A. F., and G. L. Melior, Diagnosticand prognostic numerical circulationstudiesof the South Atlantic Bight, J. Geophys.Res., 88(C8), 4579-4592, 1983. Bryan, K., and M.D. Cox, A numericalinvestigationof the oceanicgeneralcirculation,Tellus, 19, 54-80, 1967. .3  Dietrich, D. E., M. G. Marietta, and P. J. Roache, An ocean  .2  modelling system with turbulent boundary layers and topography, Numerical description, lnt. J. Numer.  Methods Fluids, 7, 833-855, 1987. Foreman, M. G. G., An accuracy analysis of selectedfinite difference methods for shelf waves, Continental Shelf .'5 Res., 7(7), 773-803, 1987. Gill, A. E., Atmosphere-OceanDynamics,662 pp., Academic, San Diego, Calif., 1982. Figure A3. Re(u) for the E1 grid plottedas a functionof the Haidvogel, D. B., J. L. Wilkin, and R. Young, A semi-spectral distance from the equator. The solid curve shows the E 1 primitive equation ocean circulation model using vertical solution for nonzero 1, and œ (as given in the text), with the sigma and orthogonalcurvilinear horizontal coordinates, correspondingcontinuumsolutionas the long-dashedcurve. J. Cornput.Phys.,94(1), 151-185, 1991. The corresponding undamped(1'= œ= 0) finite differenceand Henry, R. F., Richardson-Sieleckischemesfor the shallowcontinuum solutions are also shown by the short-dashed water equations, with applicationsto Kelvin waves, J. curves. Cornput. Phys., 41(2), 389-406, 1981. .t  NG AND HSIEH: EQUATORIAL KELVIN WAVE IN FINITE DIH•ERENCE MODELS  14,185  Hsieh, W. W., M. K. Davey, and R. C. Wajsowicz, The free Sadourny,R., The dynamicsof finite differencemodelsof the Kelvin wave in finite differencenumericalmodels,J. Phys. shallow-water equations, J. Atmos. Sci., 32, 680-689, 1975a. Oceanogr., 13(8), 1383-1397, 1983. Maier-Reimer, E., and K. Hasselmann,Transportand storage Sadourny, R., Compressiblemodel flows on the sphere,J. Atmos. Sci., 32, 2103-2110, 1975b. of CO2 in the ocean-- An inorganic ocean-circulation carbon cycle model, Clim. Dyn., 2, 63-90, 1987. Wajsowicz, R. C., Free planetarywaves in finite difference McCreary, J., A linear stratified oceanmodel of the equatorial numerical models, J. Phys. Oceanogr., 16(4), 773-789, undercurrent, Philos. Trans. R. Soc. London A, 298, 603635, 1981. Oberhuber, J. M., Simulation of the Atlantic circulation with  a coupledsea ice-mixed layer-isopycnalgeneralcirculation model, Rep. ;59,Max-Planck-Inst. fur Meteorol., Hamburg, Germany, 1990. O'Brien, J. J., and F. Parham,EquatorialKelvin wavesdo not vanish, Mon. Weather. Rev., 120, 1764-1766, 1992. Philander, S. G., El Nitto, La Nitta, and the Southern  Oscillation., 289 pp., Academic, San Diego, Calif., 289 pp., 1990.  1986.  Yamagata, T., and S. G. H. Philander, The role of damped equatorial waves in the oceanic response to winds, J. Oceanogr.Soc. Jpn., 41(5), 345-357, 1985.  W. W. Hsieh and M. K. F. Ng, Departmentof Oceanography, University of British Columbia, Vancouver,B.C., CanadaV6T 1Z4.  (ReceivedSeptember3, 1992; revisedAugust 19, 1993; acceptedSeptember27, 1993.)  

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