Shallow Water Waves on the Rotating Sphere

Home , Homogeneity (physics), Shallow water equations

PHYSICAL RBVIEW E VOLUME SI, NUMBER S MAYl99S

Shallowwater waves on the rotating sphere

D. MUller andJ. J. O'Brien Centerfor Ocean-AtmosphericPrediction Studies,Florida State Univenity, TallahDSSee,Florida 32306 (Received8 September1994) Analytical solutions of the linear wave equation of shallow waters on the rotating, spherical surface have been found for standing as weD as for inertial waves. The spheroidal wave operator is shown to playa central role in this wave equation. Prolate spheroidal angular functions capture the latitude- longitude anisotropy, the east-westasymmetry, and the Yoshida inhomogeneity of wave propagation on the rotating spherical surface. The identification of the fundamental role of the spheroidal wave operator permits an overview over the system's wave number space. Three regimes can be distinguished. High-frequency gravity waves do not experience the Yoshida waveguide and exhibit Cariolis-induced east-westasymmetry. A second, low-frequency regime is exclusively populated by Rossby waves. The familiar .8-plane regime provides the appropriate approximation for equatorial, baroclinic, low- frequency waves. Gravity waves on the .8-plane exhibit an amplified Cariolis-induced east-westasymmetry. Validity and limitations of approximate dispersion relations are directly tested against numeri- caDy calculated solutions of the full eigenvalueproblem. PACSnumber(s): 47.32.-y. 47.3S.+i,92.~.Dj. 92.10.Hm

L INTRODUcnON [4] in the framework of the .B-pJaneapproximation. In this Cartesian approximation of the spherical geometry The first theoretical investigation of the large-scale near the equator Matsuno identified Rossby, mixed responseof the ocean-atmospheresystem to external tidal Rossby-gravity, as well as gravity waves, including the forces was conducted by Laplace [I] in terms of the KelVin wave. He moreover confirmed Yoshida's hy- linearized Euler equations on the rotating sphere. Since pothesis [5] of an equatorial waveguide. A subsequent then. these equations are known as "the tidal equations." numerical study of the spherical problem by Longuet- although their significance widely exceedsthe tidal prob- Higgins [6] provided a quantitative representation of the lem: the free. linearized Euler equations on the rotating eigenvalues and eigenfunctions of the tidal equation. sphere govern the basic small-amplitude dynamics of the These calculations demonstrated that Matsuno's theory shallow ftuid envelope of a rotating planet. As such they was qualitatively complete. However, in spite of its pose the fundamental linear problem of planetary circula- eleganceand fundamental role for the terrestrial climate tion theory with applications to the atmosphereas well as problem, the ,B-pJaneapproximation is not entirely free to the ocean. These applications include short term. from ambiguities. In contrast to Laplace's tidal equation large-scalephenomena like the tides in the narrow sense. it does not provide a physically meaningful nonrotating and long term. large-scalephenomena like the terrestrial limit and postulates, furthermore, an apparently paradox- climate problem and the circulation of the atmospheres ical east-west asymmetry of gravity of waves. For given of other planets. The generic dynamics of the mode number and frequency a westward traveling gravity overwhelming majority of contemporary numerical circu- wave has a different phase speed than its eastward coun- lation models are represented by Laplace's tidal equa- terpart. Such an asymmetry suggests a Doppler-type tions. effect due to a relative motion of medium and observer. In contrast to the nonrotating case. Laplace observed However, the "tidal" equations represent a situation that the special form of the rotation vector rendered the where both medium and observer are corotating and at three-dimensional (3D) spherical problem nonseparable. rest relative to each other. In the framework of Doppler With "the tidal equation" he established a 2D approxi- theory, these gravity waves should thus be symmetric. mation, which is separable in spherical coordinates. In Recently, exact, analytical solutions of the spherical ti- this framework the discussed what is now called the dal equation have been presented [7] for the special cases Lamb wave under the inftuence of external tidal forces. of standing waves and inertial waves. These results show Laplace was not able to evaluate the eigensolutionsof his that the eigenfunctions of the tidal equation are closely equation. A century later. Margules [2] identified asymp- related to spheroidal wave functions and underscore the totically two classesof solutions: gravity waves and the baroclinic, low-frequency nature of the ,B-pJaneapproxi- now called Rossby waves. An extensive investigation of mation. At high frequencies, the .B-pJaneapproximation these asymptotics by Hough [3] led to the notion of cannot appropriately accommodate the effectsof rotation "Hough functions" for the eigensolutionsof Laplace's tion the dispersion of gravity waves. In this regime, an al- dal equations. ternative approximation of the dispersion relation ap- The most comprehensivediscussion of the dynamics of plies. Presently, a complete analytical theory of the "tidal" problem to date has been given by Matsuno Laplace's tidal equations is still not available.

1063-6SlX/9S/S I(S)/4418(14)/506.00 n 4418 @ 1995 The American Physical Society n SHALLOW WATER WAVES ON nIB R.OT A TIN G SPHERE 4419

The "tidal" equations occupy their fundamental posi- tal dynamical nature of the large-scalecirculation. Verti- tion in planetary circulation theory essentially due to cal structure has to comply with covariance requirements their consistency with respect to geometry, dynamics, and for multilayer versions this is the case. Shallow wa- and inertial forces, i.e., their covariance with respect to ter theory with the covariant inclusion of vertical varia- the 2D non-Euclidean geometry of the spherical surface. bility as (prognostic or diagnostic) internal variability of a It is noteworthy t~t Laplace establishedhis form of the spatially 2D fluid representsthe dynamically appropriate equations some 60 years before Coriolis's work on framework for planetary circulation theory. Newton dynamics in rotating systems and-although The lack of a complete analytical theory of the linear spherical trigonometry was known at the time-some 75 shallow water equations on the rotating spheremore than years before Riemann's systematic discussion of non- two centuries after their formulation provides a measure Euclidean geometries. Laplace considered the diagnos- of their complexity. This is partly due to the necessarily tics of a barotropic gas. Structurally the same diagnos- curvilinear nature of the problem. Partial differential tics hold for shallow water theory in the vertically in- operators in curvilinear coordinates carry coordinate- tegrated ("transport") form. Linear, as well as nonlinear dependent coefficients, which complicate the derivation shallow water theories in transport form thus preserve of higher-order wave equations from the first order equa- the dynamically essential covariance of Laplace's tidal tions of motion. However, these problems are drastically equations. In view of their role for the large-scalecircu- simplified by utilizing tensor analysis in Riemann space, ~tion, the "tidal" equations are thus primarily the linear invoking in particular the concept of covariant shallow water equations on the rotating sphere. Howev- differentiation. In contrast to the partial derivative, the er, it has always been questioned whether shallow water covariant derivative "automatically" accounts for the theory can provide a sufficiently rich framework to cap- coefficient functions so that the usual differentiation rules ture the relevant features of the large-scalecirculation. for products and other combinations of tensors apply to Shallow water theory in its generic form has an unreal- covariant differentiation. The manipulation of istically trivial vertical structure. But, interestingly differential expressions then almost proceeds as in the enough, the incorporation of vertical structure, namely, caseof Cartesian coordinates. baroclinicity, into circulation models has been essentially The additional feature to be taken into account is the guided by the spirit of boundary layer theory with little non-Euclidean nature of the intrinsic geometry of the 2D regard for covariance requirements. The prototype of the spherical surface (it may be noted that the "3D spheri- resulting models is the set of the so called "primitive cal" geometry of the "primitive equations" with vertical equations," constructed from three basic ingredients: (1) coordinate z = r -a, where r is the radial coordinate of hydrostatics in the vertical, (2) generally a Boussinesq- 3D Euclidean, spherical geometry and a the Earth's ra- type formulation of stratification, and (3) Laplace's dius, is also non-Euclidean). For a non-Euclidean Coriolis parameter for the representation of inertial geometry the Riemann tensor no longer vanishes. This forces. As a mixture of 3D and 2D features, the "primi- fourth order tensor is a functional of the metric, measur- tive equations" are strictly noncovariant. The derivation ing the anticommutator of the covariant second order of physically meaningful results from these equations re- derivatives of a vector. As a consequenceof a nontrivial quires a sophisticated set of scaling conditions [8] and in Riemannian, the covariant second order derivatives of a the numerical approach a high degreeof sophistication in vector no longer commute. This characteristic feature of the selection of appropriate parameter ranges as well as tensor analysis in non-Euclidean Riemann spacebecomes in the evaluation of physically meaningful diagnostics relevant for the derivation of higher-order wave equa- from model results. Access to increasing computer time tions from the vectorial equations of motion. In the did not decreasethese problems. present paper different forms of the wave equation of Covariant shallow water theory, on the other hand, is shallow waters on the rotating sphere will be considered frequently utilized in numerical studies in the form of using tensor analysis. For these purposes index notation multilayer, reduced-gravity models [9]. This form elimi- is quite effective with indices m, n ,. . ., running from 1 to nates fast barotropic waves through the coding-effective 2, subscripts denoting covariant tensor components, su- assumption of a quiescent abyss. This is known to be perscripts contravariant components, and the semicolon fairly unrealistic and to "increase" relevance reduced- for covariant differentiation. &sentials of tensor analysis gravity models had to resort to the equally artificial, but can be found in a variety of textbooks [13,14] and the observationally more ambiguous, reference to an abyssal basic formulas for the 2D geometry of the spherical sur- "level of no motion." However, with accessto increasing face in geophysical coordinates (longitude A, latitude tp) computer' time the relevance issue ceased to be one of are given in [15] with additional details in the Appendix abyssaldogma and becamea matter of comparison to ob- to this paper. servations. It has now been demonstrated that covariant In this framework it has been shown [7] that the funda- shallow water models successfully simulate observable mental wave operator of shallow waters on the rotating large-scale features of the ocean circulation: in simple sphere is the 2D spheroidal wave equation. The separa- model versions on seasonal[10] and interannual [11] time tion of this wave equation results in ordinary difFerential scalesand in more elaborate multilayer versions even on equations with periodic coefficients. Formally similar decadal time scales [12]. This successindicates that shal- problems in other branches of fluid dynamics (stability of low water theory takes advantage of increasednumerical time periodic flow) or in electrodynamics (wave propaga- resolution becauseits covariance captures the fundamen- tion in an elliptical waveguide) lead to the related 4420 D. MULLER. AND J. J. O"BRIEN II

Mathieu equation. While Floquet theory provides a are given in the Appendix. As necessaryfor a covariant comprehensiveanalysis of Mathieu functions, a compara- system, the mass ftux density, defined in the divergence ble tool for spheroidal wave function does not exist. The term of the continuity equation, equals the momentum major problem stems from the irreguJar singularity of the density, defined in the prognostic term of the momentum spheroidal wave equation at infinity. This prevents the budget. With this identity, the vertically integrated form establishment of recurrence relations similar to those for (2.1) of a 3D, hydrostatic system satisfies Newton's first functions of the hypergeometric type [16]. Special case law in a spatially 2D sense. Correspondingly, Laplace's solutions, as obtained in [7], can thus not be generalized representation of the Coriolos force refers exclusively to a by meansof some algebra in wave number space. spatially 2D system. The identification of the prolate spheroidal wave For the potential vorticity z, defined by operator as a key element in the linear shallow water equations on the rotating spherical surface permits never- Rz=~vII;m-Ir/R , theless a qualitative overview of the system's wave num- the equations of motion (2.1) imply the relation ber space. The asymptotics of the spheroidal wave operator yield a set of approximations of the dispersion rela- Ro,z+l"v"=O, (2.2) tion for different regimes in wave number space, which essentially link the Margules limits with the {J plane. In where 1" is the contravariant gradient of the Conolis pa- the paper at hand, the validity and limitations of these rameter. The system's energy budget approximations are directly tested against numerical a,Re+s";" =0 computations of the eigenfrequenciesof rotating, spherical shallow waters. with total energy The code for the calculation of eigenvalueshas been / 2R kindly provided by P. Swarztrauber and is extensively Re=.!.Rvllv2 II +c2r2 discussedin [17]. In contrast to the numerical approach and Poynting vector $11=C2roll follows by the usual algo- of Longuet-Higgins [6] this code does not require the pri- rithm Newton dynamics from (2.1). or derivation of a wave equation. It rather takes direct One form of the linear wave equation of shallow waters advantage of the fact that shallow water theory on the rotating sphere is obtained by eliminating the represents a 2D vector field. Expanding the eigensolu- momentum vector from (2.1). To this end take the diver- tions in spherical vector harmonics, the first order equa- genceof (2.1b), tions of motion are transformed into an infinite, symmetric, homogeneous,pentadiagonallinear algebraic sys- (~-c2.6.)r-~II(fjm );11=0, tem. The eigenvaluesof this systemare readily computed numerically. Although the {J-plane approximation and where .6.denotes the 2D Laplacian of a scalar in spherical the numerical approach have been well establishedfor 30 coordinates as defined in the Appendix. With the repeat- ed use of (2.1b) and e"'1It"am= -8: the time derivative of years, they have-to our knowledge-never been com- pared in a common format. Besidesa direct validity test, this equation becomes such a comparison elucidates the characteristic dispersive [(0:-C2.6.)Ot +C2e"'lIf 110m)r=(f2jll);11 . features of the {J-plane regime within the entire wave number space. In conjunction with the quantitative rep- Evaluation of the right-hand side (RHS) with the help of resentation of the complete dispersion relation, the set of the massand potential vorticity budgets leadsto analytical approximations obtained here specifiesparticu- lar dispersion regimes in wave number space. The [(0:+ j2-c2.6.)Ot +C2~lIf 110m)r= -2R2fOtZ . (2.3) characteristics of these regimes differentiate the wide The wave operator on the LHS of (2.3) is the 2D prolate range of circulation phenomena, representedby shallow spheroidal wave operator. It is stressed that the waters on the rotating spherical surface. spheroidal character of this operator is here not a consequenceof underlying 3D spheroidal coordinates. The un- n. WAVE EQUATIONS derlying 3D coordinates are spherical and the spheroidal On the surface of the rotating sphere the linearized wave operator appears in (2.3) as a consequenceof the shallow water equations read nonvanishing Coriolis parameter. To eliminate the potential vorticity from (2.3) consider the operator c1tr+j";,,=O, (2.1a) d all=g allOt+e"" f

aJ. +E"'ftfj"'+c2a.r=o , (2.1b) with

dGlld f 2\~G where r denotes a perturbation of the equilibrium mass mil =(a2+t /Um . per unit area R and the covariant vector jll =Rv. =aR(vA. coSq>,v,,)the mass flux density near a Application of this operator to (2.1b) yields zero-velocity equilibrium. The constant c denotes the (a,z+ f2)jG= -c2dGllall' . (2.4) system'sintrinsic phase speedand the latitude-dependent scalar f = 20 sinq>is Laplace's Coriolis parameter. The After scalar multiplication with the covariant gradient of components of the antisymmetric Levi-Civita tensor E"," the Cariolis parameter this can be written with the help ~ SHALLOW WATER WAVES ON 'nIB ROTATING SPHERE 4421

of (2.2) as and (a2+/ 2)a z=c2dalll" a r R2 , , JG II . Af=f-;_=-20-1f. Hence, on multiplication of (2.3) with (a,z+/2) one ob- with the result tains f-Ai- = A(f-j. )-(Af)j-;- , (a,z+r)[(a,z+ r-c2A.)a, +c2E""'/ Nalll]r or with (2.1a)and (2.2) = -c2d-alllraNr. (2.5) f-Ai- =(Af)atr- R 2Aatz . This is the linear wave equation of shallow waters on the rotating sphere in tenns of the mass perturbation r. Hence, the scaJarproduct of (2.6b) with the gradient of f While this is not identically the prolate spheroidal wave becomes equation, it is nevertheless obvious that this operator R2[(a:+ f2-c2A)at+c2~f _am]atz= -c2(Af)a:r . plays a central role in (2.5). Alternatively, the wave equation may be written in terms of the momentum vector. Taking the time derivative of (2.lb), Together with (2.3) this equation forms a coupled set of prolate spheroidal wave equations for the mass perturba- a~jll-~mllf(~fj4+c2amr)+c2allatr=O , tion r and the potential vorticity z. Furthermore, if (2.7) is read in terms of Cartesian coordinates (x ,y) with and using ~~11111= -8: as well as the continuity equa- f=fJy, i.e., in particular, ~f=O, this equation reducesto tion in the last term. this becomes the farni1iar fJ-plane version of the Matsuno equation. (a~+ f2)jll-c2j4;4II-c2~mllfamr=O . (2.6a) Equation (2.7) is thus the spherical ~atsuno equation. Similarly, one obtains equations for the cross product To evaluate the gradient of the divergenceof the momen- of the gradient of f with the momentum vector tum Bux, usethe definition of potential vorticity Ru=~mfmj/l ~nJ.. =R2Z+ I"r s'" J' from (2.6b). Without derivation the equations and define the notation [(0:+ f2-c2~)cJ, +c2e1Jbfaab]r=2RcJ,u (2.8a) ~. ab. -2. ~1I=g JII;ab-a JII and so that formula (A2) of the Appendix assumesthe form [(~+ P-c2~>at+c2~fbaG]U j4;QII =Ajll +~lI4a4(R2z+ fr) . = - R [(~+ P-C2~>at +c2~f GabVz (2.gb) Inserting this into (2.6a) are mentioned here. Equations (2.3)-(2.8) demonstrate (a~+ r-c2~)jll-c2~II4(R2a4z+rf4)=O that, in spite' of the non-Euclidean geometry, covariant differentiation renders the evaluation of higher-order and dift"erentiatingagain with respect to time wave equations from the equations of motion (2.1) fairly (a~+ r-c2~>O,jll +c2~II4[a4(fbjb)+ f4jb;b]=O (2.6b) straightforward. On the spherical surface, waves propagate in a funda- yields a wave equation in terms of the momentum Bux mentally anisotropic environment: only in one direction vector. In contrast to the temporally fifth order, scalar is the propagation of waves possible, while standing equation (2.5) this is a temporally third order vector waves form in the perpendicular direction. Due to its to- equation. The prolate spheroidal vector operator is seen pology as an unbounded but finite domain, the spherical to playa prominent role in (2.6b). While both (2.5) and surface always acts as a waveguide. On the rotating (2.6b) exhibit some identifiable structure, they are never- sphere the propagation direction is the longitudinal direc- theless still fairly complex. A more symmetric form of tion and it can always be chosenas such in the nonrotat- the wave equation can be obtained with the help of cer- ing case. This suggeststhe dependent variables to be of tain derivatives of (2.6b). the form Consider first the scalar product of (2.6b) with the gradient of the Conolis parameter. To this end evaluate e -i(",f-MA.)F(y) where y =sintp and F(y) is a latitudinal eigenfunction. ~(f"j" )=gab(f"j" );4b To ensure sufficient dift"erentiability on the entire domain, =g~f"j,,;.. +2f";.j.;. +f.;..j. the zonal wave number M should be an integer. For functions of this form the scalar, prolate spheroidal wave using operator reducesto the prolate angular operator

/";m=-/G,,m M2 P=o (1- , 2)0 - _a2y2+v2 where G,.,.. is the Ricci tensor, defined in the Appendix, y , 1-,2 4422 D. MULLER. AND J. J. O'BRIEN 51 where v=aa>/c, while the Lamb parameter a=2afi/c A large part of the dynamical wealth of planetary cir- measuresthe ratio of the planet's angular velocity to the culations is due to the fact that the rotating spherical sur- propagation speed of effective pressure perturbations. face provides an anisotropic and inhomogeneousarena For small values of a, prolate angular functions resemble for the propagation of shallow water waves. Besidesthe associatedLegendre polynomials, while they behave like basic latitude-longitude anisotropy, rotation effectson the parabolic cylinder functions for large values of the Lamb spherical surface introduce an east-westasymmetry into parameter. The terrestrial weather and climate systems the propagation of waves. However, in a Doppler-type operate over a wide range of values of this parameter. framework gravity waves are expected to be symmetric, For atmospheric Lamb waves a~ 1, whereas a~S for as long as observer and medium are at rest relative to barotropicgravity wavesin the oceanand a ~ 300 for the each other. In addition to these asymmetries, Yoshida first baroclinic mode in the ocean. For the purposes of [5] has pointed out that the rotating spherical surface is planetary circulation theory, solutions at fixed values of a inhomogeneouswith respect to latitude. At low frequen- are therefore of limited interest and the solution of linear cies, latitudinal eigenfunctions behave oscillatorily only shallow waters on the rotating sphere has to be deter- in a narrow equatorial belt, the Yoshida waveguide. mined for the entire (real and positive) range of the Lamb The f -plane approximation, successfully employed in parameter. the study of internal gravity waves, does not admit Ross- With wave functions of the above form the coupled by waves, nor does it exhibit anisotropy or inhomogenei- system (2.3)and (2.7)becomes ty. In contrast, MarguIes's asymptotics [2] represent the anisotropic, globally homogeneouslimits of the system. (P-m )V= -2ayD , (2.9a) In the nonrotating case (.0.=0) one obtains from the (P+m )D=2ayV , (2.9b) wave equations of the previous section where D=r/R and V=aRz/c, while m=aM/v. For r=roPf(sintp)e-i8, (3.1a) Eqs. (2.8)one obtains correspondingly j"=-jod,,Pf(sinq»e-i8, (3.1b) (P-m)D=-2aU, with jo=iroc2/ll>, 6=ll>t-MA., and associatedLegendre (P+m)U=-(P-m)yV, polynominals pf. The dispersion relation of these long gravity waves with U=VA COSIp/c=au /2cfi. The system (2.9) can formally be diagonalized in different ways. For instance, ,,2=L(£+I), -L ~M~L , (P+m )y-l(P-m )V= -4a2yV , or alternatively in terms of the mode number No = L -1M I ~ 0, counting the zeros of the mass pertur- (P-m )y-l(P+m )D=-4a1yD , bation r in the open interval y E ( - 1,1 ), or ,,2=No(No+l)+(2No+l)IMI+M2, (3.1c) P(2ay+m )-lp( Y+D)=-(2ay-m )(Y+D) . is symmetric with respect to the zonal wave number M. P(2ay-m )-lp( Y-D)= -(2ay+m )(Y-D) . On the other hand, for nondivergent Bow the limit c 2-. 00 , r = constof the Matsunoequation (2.7) yields As in both casesthe RHS is not readily incorporated into a factorized fonD of the LHS. these equations do not sug- r=const. gest an obvious strategy to evaluate eigensolutions. Simi- lar expressionsare obtained from the U equations. Al- j/l=joEnmamp!.'(sinlp)e-i9, (3.2b) though the latitudinal eigenfunctions are generally not representing divergence-free Roesby waves with disper- just prolate angle functions, these equations indicate nev- sion relation erthelessthe fundamental role of the prolate angle operator. (J)=-2fiM/L(L+1), -L~Ma;-a2y2+v2]V=0. (3.4) spherical geometry in the vicinity of the equator, formu- lating the equations of motion (2.1) in these coordinates, This equation is exactly solved by and finally deriving the system's wave equation. Con- versely, it will here be shown that appropriate approxi- V=A(I-y2)I/2Si1(y;a) . mations of the spherical MatsUDoequation lead indeed to Here, A is a constant and SL-l(y;a) the prolate angular the same result on the .8 plane. The necessaryapproxi- function of degreeL ~ I and order ( -1 ), where the nega- mations define the conditions of validity of the .8-pIane tive value has been chosen for correspondencepurposes. approximation in physical and wave number space. From this expression, the full eigensolution for standing Assuming waveson the rotating sphericalsurface is found as V=( 1-)/2)IMI12p( 11) r=ro(a9'-~)Si1e-i< , (3.5a) with 11=)/V2a the spherical MatsUDO equation (3.3) jl=-jocoSf1fSi1e-ifBf, (3.5b) transforms to j2=ijoCiJSi1e-i<, (3.5c) [(2a-V)~ -2( IMI + 1)"a,,-fa112 + X]P with j 0 = a 2r0 and dispersion relation =2y-l[(2a-V>."o,,-IMlv+2aILm]F ,

yi:=£PA(L,I;a) , (3.5d) where X=v2-M2+m and Y=2ap. + 112. For large a» 1 and 2a »112, i.e., in the vicinity of the equator, where £pA(L,K;a)=£PA(L, -K;a) denotes the eigenval- )/2« 1, this is approximated by the equation for parabol- ue corresponding to the prolate angular function ic cylinder functions Sf(y;a). A closed expression for these eigenvaluesdoes not exist [16,18]. Furthermore, due to the lack of re- (a2-.l'rl2+" 4 ., N 2 +.l2 )P-O- , currence relations, (3.5) has to be given in terms of the which is the familiar .8-plane version of the Matsuno prolate angular function and its first derivatives. This equation with dispersion relation solution describes essentially standing gravity waves, where the /J-plane approximation denotes the case L = I .y3-[a(2N2+1)+M2]v-aM~O, (3.6) as standing mixed Rossby-gravity wave. In contrast to the nonrotating case, standing waves on the rotating where the mode number N 2 = L -1M I here counts the 4424 D . MULLER. AND J. J. O'BRIEN n

zeros of Y in the open interval ." E ( - Q), Q)). It may be The lack of recurrence relations leads again to a represen- noted that the spectrum of the parabolic cylinder opera- tation of (3.8) in terD1sof the prolate angular function tor is actually continuous. In the approximation of an and its first derivatives. This solution describes gravity operator with a discrete spectrum, the mode number wavesat the inertial frequency (J)= m, includingthe Kel- emerges here necessarily as an integer, thus selecting Vin wave for M = L + 1 > 0 and the gravity branch of the those parabolic cylinder functions which can be mixed Rossby-gravitywave for M = L > O. It accounts represented in terms of Hermite polynominals. In con- not only for the latitude-longitude anisotropy and latitu- trast, the conventional derivation of the Matsuno equa- dinal inhomogeneity, but also for the east-west asym- tion on the fJ plane has to confine itself quite arbitrarily metry. As indicated by (3.8d), v(M ):#:,.-(- M). to integer mode numbers. Moreover, this derivation of The cosine of latitude in the denominator of the the equatorial fJ plane can obviously not be generalizedto momentum flux of inertial waves raises the question of obtain a "midlatitude fJ plane." Besides formal regularity at the poles for this solution. Near the poles difficulties to define it as an approximation to the spheri- the prolate angular function behaveslike [16] cal problem, such a concept would necessarilyimply un- physical features like a "midlatitude waveguide," midlati- Sf-1(y;a)=(I-y2)IJlI12F(L,M-l;a) , tude mixed Rossby-gravity waves, and-independent of where F generally does not vanish. This is sufficient to en- coastal trapping-midlatitude Kelvin waves. The fJ sure the polar regularity of (3.8), except for M=O and 2. plane is thus the appropriate approximation for equatori- For M=2, tabulated values of the spheroidal wave func- al low-frequency waves, if the Lamb parameter is large. tion [16] show that Sl(y=:t:l;a)=O. However, it is not As an asymptotic expansion for large a , the fJ plane ex- obvious that this value is approached sufficiently fast for cludes a nonrotating limit and approximates the exterior all L and a. At M =0, (3.8) should not only be regular at of the Yoshida waveguide by an infinite plane. On Earth, the poles, but also coincide with (3.5) at v=a. With re- large a refer to baroclinic waves in the ocean. Baroclinic gard to the dispersion relations (3.Sd) and (3.8d) this low-frequency dynamics in the equatorial ocean are correspondence is obvious. Regularity and correspon- indeed successfully simulated by reduced-gravity models dence of the solutions can in this case even be checked in fJ-planegeometry [10,11]. directly, since S1;1(y;a2=EpA) is known in closed form A second exact solution in terms of prolate angular [16]. For even L functions is readily found from a low-order closure of (2.9b).Evaluation of the two-componentof (2.4) yields Si1(y;a2=EpA)= A sin(asintp)/COSip aY= _(u2_y2)-1[( l-y2)a,. +my]D , with constant A is a regular prolate angular function, if a is an even multiple of 'IT/2. A sinli1ar expression holds where u=v/a. With this representation of Y, Eq. (2.9b) for odd L [16]. Thus, if a is for instance an even multiple becomes of 'IT/2, the solutions (3.5) and (3.8) become

(p-m)D=-2(0'2_y2)-1[(l-y2)yo,+m0'2]D. (3.1a) r -cos(a sincp) , j 1- sintpsin( a sintp) , The structure of (3.1a) is similar to the spherical MatsUDO equation (3.3). A particularly simple form of (3.1a)is ob- j2-sin(asintp)/COSqJ. tained for inertial waves with 0' = l: This demonstrates that at M=O regular and common solutions (3.5) and (3.8) exact, although not for arbitrary ] D=O. (3.7b) [ (l-y2)a:-~i~*_a2y2+a2-M values of the Lamb parameter. Hence standing waves at l-y £l) = 2fi only occur if a assumescertain specific values. It This is exactly solved by is furthemlore noted that for M = t and t Eq. (3.7b) has exact solutions in terms of Mathieu functions. However, D=A.(l-y2)1/2sf-l(y;a) , at these values of M the full wave function is not continu- where A denotes a constant amplitude and Sr-1(y;a) ous for all longitudes AE[O,21T]. the prolate angular function of degree L ~ 0 and order Exact solutions in terms of prolate angular functions (M-l) with l-L ~M~L+l. The eigensolutions of capture the latitude-longitude anisotropy and the east- linear shallow waters for inertial waves on the rotating west asymmetry as well as the latitudinal inhomogeneity sphere are found from this expressionas of the rotating spherical surface. In contrast to associat- r=ro CO8'P8r-le -i9 , (3.8a) ed Legendre polynomials or parabolic cylinder functions they are capable of representing an equatorial waveguide on the finite spherical surface, where the inhomogeneity- jl =jo [~fI+~+COS'fJ ]sr-1e-i9, (3.8b) scale-i.e., the width of the Yoshida waveguide-is controlled by the Lamb parameter. In the corresponding j2 = -ijo(COS'fJ)-l[o" +(M -1)tantp ]Sr-1e-i9, (3.8c) limits the prolate angular functions reproduced Margules's asymptotics and the IJ-plane approximation, withjo=c2ro/20 and dispersion relation respectively. At sufficiently high frequencies, gravity waves are no longer confined by an equatorial waveguide, v"=a2=£PA(L,M-l;a)+M. (3.8d) but propagate on a globally homogeneous,though aniso- ~ SHALLOW WATER WAVPSON 11IE ROTATING SPHERE 442S

tropic spherical surface. From an alternative viewpoint, trary degreeof accuracy. For wave numbers M= -1,0, 1 the Lamb parameter appears as the remnant of a vertical Fig. 1 shows these eigenfrequenciesin comparison to nu- wave number in a vertically integrated, strictly horizontal merically computed eigenfrequenciesas a function of the systemand the dependenceof eigenfrequenciesand eigen- Lamb parameter a. Similar figures of numerical solu- functions on this "wave number" as a rotation-induced tions of the eigenvalue problem have been obtained by "vertical dispersion." In a two-layer model, the barotro- Longuet-Higgins [6], who shows u=v/a vs l/a. For all pic eigenfrequencylI)l(L,M;al) and the baroclinic eigen- three wave numbers the intersections of the dashed line frequency lI)2(L,M;a2) at the same L and M belong to v=a with the solid lines have been calculated with (3.8d), different eigenfunctions S!.'(y;al) and Sf!(y;a2)' respec- while the solid lines represent numerical computations. tively. In the nonrotating limit these eigenfunctions These computations use the Swarztrauber-K~~hara code reduce to the same associated Legendre polynominal [17] to solve the equations of motion (2.1) numerically. P!.'(y) and the system becomes strictly nondispersive. In Fig. 1{b), the solid lines represent in fact two calcula- There is, nevertheless,a pronounced difference between tions: numerical solutions of the eigenvalueproblem and the Lamb parameter and a wave number in the proper (3.5d). Both results obviously coincide. For a=O the nu- sense: for given a the prolate angular functions form a merical code [17] reproduces the Margules limits (3.1c) complete, orthonormal set of basis functions, while this is and (3.2c). not the case for given L and M. The Lamb parameter Figure 1(a) clearly shows low-frequency Rossby waves, provides the link between covariant, linear shallow wa- well separated by the mixed Rossby-gravity wave from ters and vertical variability that maintains consistency higher-frequency gr:avity modes. The standing waves in with respect to Laplace's representation of inertial forces Fig. 1{b)are gravity waves,where the fJ-planeapproxima- on the rotating spherical surface. Obviously, this vertical tion denotes the lowest mode as mixed Rossby-gravity variability has to appear as internal variability of a strict- wave. Figure l(c) depicts eastward traveling gravity ly 2D fluid. The representation of vertical variability in waves with the distinguished behavior of the Kelvin terms of a set of Lamb parameters and a vertical wave mode (No=O) for large values of a. In the fJ-plane ap- number is not only redundant, but will generally be in- proximation, the mode No = 1 is considered the gravity consistent. In practice, prolate angular functions are the branch of the mixed Rossby-gravity wave. Figure 1 indi- appropriate set of basis functions for a rotating, shallow cates a pronounced difference in the behavior of v for few-layer fluid on the global domain. small and large values of a, corresponding to the approximation of prolate angular functions in terms of associated IV. THE DISPERSION RELATION Legendre polynomials or parabolic cylinder functions. The transition region is approximately given by While exact eigensolutionscould here only be obtained 1 ~ a ~ 10, where these approximations converge poorly in special cases,the identification of the role of the pro- in comparison to the left and right boundary regions of late spheroidal wave operator is neverthelesssufficient for the diagrams. Under terrestrial conditions, this transi- a qualitative overview of the system's wave number tion range is occupied by Lamb waves, atmospheric grav- space. The asymptotics of this operator yield a set of ap- ity, and Rossby waves as well as barotropic waves in the proximations of the dispersion relation, which link the ocean. Margules limits with the .8 plane. These approximations For inertial waves (3.8), the east-west asymmetry can are here tested against numerical solutions of the full ei- be evaluated explicitly. According to (4.1b) the mode genvalue problem of shallow waters on the rotating number in this case is given as No = L -1M -11, count- spherical surface. ing the numbers of zeros of the mass perturbation r in the Benchmarks for these approximations are provided by open interval y e ( -1, 1). For given Lamb parameter the exact dispersion relations (3.5d) and (3.8d) as well as and mode number, (3.8d) has two solutions and with the Margules limits (3.1c) and (3.2c). The prolate angular eigenvaluesare explicitly known [16,18] in terms of the power series EpA(L,K;a)=A2+a2(2A2-2K2_1)/(4A2_3)+ . with A2=L(L+l), or for large a in terms of the asymptotic expansion

E"PA(L,K;a)=aq-2-~~ +3-4X2)

-(16a)-lq(p+3- 8K2)- (4.1b) FIG. 1. Nondimensional frequency V=QQ)/c versus Lamb parameter O.3 ~ a = 20 {} / c ~ 300 in logarithmic scalesfor mode with q=2N+l, p=N(N+l), and N=L-IKI. numbers O~No~S. Dashed line: v=a. (a) Zonal wave num- Higher-orderterms are given in [16,18]and the optimal ber M= -1. Solid lines: numerical computations. (b) Zonal matching of both expressionsdepends on L, K, and a. wave number M =0. Solid lines: numerical computations and The dispersion relations (3.Sd) with K = -1 and (3.8d) (3.Sd). (c) Zonal wave number M= 1. Solid lines: numerical with K=M-l can be evaluatedwith (4.1) to an arbi- computations. 4426 D. MULLER. AND J. J. O'BRIEN II

(4.tb) one obtains to O(~) server will find gravity waveswith Mg1w = t:t::¥a'l-aq+fp ,,=,../ L"[i"'+O - taM' such that i.e., the eastward propagation of gravity waves with M > 2v'L(L + 1)/a is inhibited, while stationary pertur- IMwl=M.-l physics of this asymmetry, assumea m.=2QMIL(L+l), O,O) (j)=2nM[I-L(L+I)]/L(L+I) . with constant U is an exact solution of the fully nonlinear These considerations demonstrate that the Lamb parame- shallow water equations on the rotating spherical surface. ter captures the Corlolis effect, as well as shear eft"ectsof It is uncritical in the present context that this solution stationary, zonal flows on the dispersion of shallow water generally requires some external time-independent forc- wav~ on the rotating spherical surface. Zonal currents ing for arbitrary values of U. This stationary ftow is non- on the surface of the sphere necessarilyexhibit shear. In divergent and exhibits the shear addition to the usual Doppler effect this shear introduces an east-w~t asymmetry into the propagation of waves UN;M=£- U sinII' . that is still seen by an observer moving with the flow. The east-w~t asymmetry introduced by the Cariolis term Linearization of the shallow water equations around this is physically closely related to this shear-inducedDoppler equilibrium leads to effect. (a, + uaA,)r+j";" =0 , The same east-westasymmetry is exhibited by gravity wav~ with v»a. In this frequency domain the denomi- (a,+ uaA)j.+E..Fjm+c2a.,=o , nator on the RHS of (3.7a) can be Taylor expanded. For (3.7a) this expansion yields an approximate prolate angu- where F=2(O+ U>siDf). On the spherical surface, the lar equation with the eigenvaluerelation stationary mean Bow is seen to introduce two effects in comparison to (2.1): fint, the partial time derivative is re- y3-EpA(L,M;a}v+aM ~O . placed by the substantial derivative following the mean The gravity modes which satisfy the defining inequality Bow. This is the Doppler effect in the fAmiliar sense. are thus given as the roots Secondly,the shear of the zonal Bow modifies the Coriolis parameter. As a stationary zonal Bow on a spherical sur- v,~2vocos'Y>a (4.3) face will general1ybe a shear Bow, this additional shear effect will always appear for a medium moving relative to of this dispersion relation with ~=EPA/3 and the observer. For the above system it is in particular ob- arccos(-aM /2~) . vious that the shear of a westward Bow with U = - 0 r = t (i.e., a medium at rest with respect to a nonrotating ob- The domain of validity of (4.3) in Fig. 1 is the upper left server) completely compensatesthe Coriolis effect. Con- comer above the dashed line. For a=S and mode num- versely, this implies that the effect of the Coriolis term on bers 2~No=L -1M! ~ 10 these gravity modesare shown the dispenion of gravity waves pn the rotating spherical in Fig. 2 as a function of the zonal wave number M in surface is physically equivalent to the shear effect of a comparison to numerically computed eigenfrequenciesof stationary, eastward zonal Bow on the dispenion of grav- the full system. For all mode numbers, (4.3) has beencal- ity waves on the nonrotating sphere, as seen by an ob- culated with the power series expansion (4.1a) for EPAto server moving with the Bow. O(~). Discrepancies appear at low No and small wave For general values of U the dispersion relation of the numbers M. For sufficiently high frequencies, however, above system (4.3) provides a good approximation to the eigenfrequencies of long gravity waves on the rotating spherical sur- OJ.=6J.(L,M;a. face. In agreement with (4.2) the dispersion diagram ex- has the same form as the dispersion relation of (2.1), hibits east-west asymmetry: westward branches are where 0). =0)- UM is the frequency seenby an observer steeper than their eastward counterparts. For a=O, (4.3) moving with the flow and a.=2a(O+ U)/c. In the case reduces to the Margules limit (3.1c) and the east- and of a westward current with U = - {} the corotating ob- westward branches become symmetrical with respect to H SHALLOW WATER WA vm ON THE ROTATING SPHER.E 4427

in Fig. l(a) is the part below a horlzontalline (not shown) v=l. For a=S and mode numbers 0~N2=L-IMI ~S the Rossby frequencies0' r = V r / a are shown in Fig. 3 as a function of M in comparison to numerically computed eigenfrequencies of the full system. For N 2 > 0 the dispersion relation (4.4) has been evaluated with the expansion (4.la) of EpAto O( a2), while for N 2=0 (the Ross- by branch of the mixed Rossby-gravity mode) the expan-

~ sion (4.lb) to O(ao) has been used. Except for small 1M I in the mixed Rossby-gravity mode, agreement is general- s ly good. Inspection of Fig. l(a) shows that for a=S the ( frequency of the mixed Rossby-gravity mode at M = -1 is larger than 1 and thus outside the domain of validity of (4.4). For c - ~, i.e., a -0, the eigenfrequencies (4.4) reduce to the Margules limit (3.2c). In the opposite extreme a» 1 they approach the familiar fJ-plane values. In contrast to the Margules limit and the {3 plane, the approximation (4.4) holds for Rossby waves at all values of the Lamb parameter and is not restricted to the vicinity of the equator. Since the regime-defining inequality involves ZONAL WAVE NUMBER .. - wave numbers and frequencies only, (4.4) applies in the FIG. 2. Normal east-westasymmetry of gravity waves:west- entire physical space. It includes in particular those ward branches are steeper than eastward branches. Nondimen- Rossby waves which are frequently representedin terms sional gravity frequencies v=a(j)/c versus zonal wave number of a "midlatitude {3 plane." As pointed out in the previ- M for modes 2~No~lO and Lamb parameter a=S. Solid lines: (4.3), dashedlines: numerical computations. ous section such a concept is formally as well as physically inconsistent, while the approximation (4.4) is free of these difficulties. Baroc1inic Rossby wavesin the midlati- M =0. For increasing a discrepanciesexpand to higher tude Pacific play an important role in the long-term mode numbers. However, with the inclusion of higher- aftereffectsof an E1Nino event [12]. order terms of (4.1a)it can always be achieved that agree- Linear fJ-plane dynamics approximate low-frequency waves in the equatorial waveguide for large values of the ment at v=a is fair and becomesrapidly better for v>a. Lamb parameter. The dispersive properties of gravity For M=O, (4.3) does not reduce exactly to (3..5d), al- and Rossby waves on the {3 plane are given by the roots though Fig. 2 indicates satisfactory quantitative agreement for sufficiently large v. For all values of a, gravity waves, which are well approximated by (4.3) do not experience an equatorial waveguide. These gravity waves gain large-scale relevance, if the Lamb parameter is small. On Earth this is the case for Lamb waves, atmospheric gravity waves, and barotropic gravity wavesin the ocean, where the latter are of particular significance for the ocean's responseto external tidal forces. On a slowly rotating planet like Venus, gravity waves (4.3) playa dominating role on large scales. .. >- A second regime in wave number space is defined by u the inequality M2»v2. In this case p.»l and the denominator on the RHS of the Matsuno equation (3.3) i can be Taylor expanded. This expansion leads to an approximation of the Matsuno equation in terms of a prolate angular equation with the eigenvaluerelation y3-E"PA(L,M;a)v-aM~O . The roots of this dispersion relation, which satisfy the defining inequality v,= -2vocos(y +t1r) < 1M I (4.4) ZONAL ,""... AU.""" . where v5=EPA/3 and FIG. 3. NondimensionalRossby frequenciesa=O)/2.o. versuszonal wavenumber M for modes0 ~ N 2~ S and Lamb r=tarccos(aM /2vb) parametera=5. Solid lines: (4.4), dashedlines: numerical represent Rossby waves. The ~omain of validity of (4.4) computations. 4428 D. MULLER. AND I. I. O'BRIEN II

of (3.6). For N2 >0, the low-frequency gravity waves are Margules limits for vanishing Lamb parameter. While representedby the root (3.6) does not exactly reduce to (3.8d) for strictly standing waves with M=O, Fig. 4 shows neverthelessthat quanti- Vg~2VOcosr (4.5a) tative agreement is satisfactory. In Fig. I the domain of while the Rossby modes are validity of the equatorial {J plane is the lower right of the diagrams, below the dashed line and to the right of a v,.~ -2VOCOS(r+t1T) (4.Sb) vertical line (not shown) a = 1. The {J plane thus excludes high-frequency gravity waves ( 'V~ a) and barotropic with ~=[a(2N2+1)+M2]/3 and Rossby waves (a ~ 1). r=tarccosME . root In obviouscontrast to (4.2) eastwardpropagating gravity waves have a higher phase speed than their westward Vrg~tM+tv~ (4.5c) counterparts at the samefrequency and mode number. It of (3.6) is taken, which combines the eastward gravity will here be demonstrated that this difference is in fact an branch for N2=O with the highest westward Rossby enhanced east-westasymmetry of the type (4.2). To this branch into the mixed Rossby-gravity mode. Finally, the end notice that the dispersion relations (3.1c), (3.8d), and KelVin wave is the trivial solution of the .B-planeversion (4.3) utilize a mode number No, while (3.2c), (3.Sd),(3.6), of the MatsUDo equation and its dispersion relation is and (4.4) organize modes with respect to N2. Both of consequently not included in (3.6). Rather, it has to be these numbers represent physically different quantities: reevaluated from the equations of motion as No is the number-of zeros of the mass perturbation r, while N 2 counts the zeros of the latitudinal momentum vk~Me(M) , (4.5d) component j2. To evaluate the role of these wave num- where e here denotesthe unit step function. For a= 100 bers consider the eigenfrequenciesof gravity waves in ei- and N 2 ~ 3 the dispersion relation (4.5) is shown in Fig. 4 ther representation. Figure 5 shows numerically comput- in comparison to numerically computed eigenfrequencies ed eigenfrequenciesof gravity waves for O~No ~ 10 at of the full problem. The agreement is generally good. Lamb parameters a=O,S,I00. Figure 6 depicts exactly For increasing a the agreement improves, while for the same eigenfrequencies,here, however, combined into smaller a or higher frequenciesdiscrepancies emerge. As modes according to constant N2. It is pointed out that a large-a approximation, (3.6) does not reduce to the Fig. 5 does not show the roots of (3.6). These roots exhibit additional quantitative discrepancieswith numerically calculated eigenvaluesdue to the fact that (3.6) involves only terms of O(ao) of (4.1b). These quantitative discrepancies can partly be decreased by including higher-order terms of (4.1b) into (3.6). The qualitative feature of interest here is the fact that the {J-plane ap-

..~ 5- f ~

FIG. 5. Enhancedeast-west asymmetry of low-frequency ZONAL WAVE NUMBER M gravity waves: westwardbranches of low gravity modesare FIG. 4. p-plane dispersion relation. Nondimensional fre- shifted upwards with increasinga. Numerically computed, quencies v=am/c versus zonal wave number M for modes nondimensionalgravity frequenciesfor O~No~ 10. (a) For N2 ~ 3 and Lamb parameter a= 100. Solid lines: (3.6), dashed Lamb parametera=O. (b) For Lamb parametera=5. (c) For lines: numerical computations. Lambparameter a= 100. a SHALLOW WATER.WAVES ON TIlE R.OTA11NGSPHERE 4429

to the discrete wave number M. the standard p-plane concept refers to a continuous wave number k = M / a so that in particular the group velocity u =(akO»N as the 2 speed of energy propagation is always well defined. In this framework. standing gravity waves on the {3 plane

~ (k =0) exhibit a finite group velocity. while westward t propagating gravity waves with wave number ka. defined I by u(ka)=O. are not associatedwith a flow of energy. In ! the N a representation. on the other hand. a group velocity of the above form is not even defined in this domain of wave number space. Figures 5 and 6 thus primarily indicate that neither the mode number N a nor N 2 is appropriate over the entire wave number space. The only parameters consistent in the entire wave number space are the zonal wave number M and the degree L of the prolate angular function. proximation necessarily provides the eigenfrequenciesin Under terrestrial conditions the dispersive properties the N 2 representation due to the very nature of the of the p-plane regime are those of equatorial. baroclinic MatsUDoequation. It is furthermore emphasizedthat on low-frequency waves in the ocean. These waves are of major significance in climate dynamics. The now widely the spherical surface only eigenfrequencies at integer known El Nino phenomenon. for instance. is essentially values of the zonal wave number M are physically meaningful. Lines in Figs. 5 and 6 have merely an auxiliary triggered by baroclinic KelVin waves. generated by function indicating which eigenfrequencies are con- changing winds in the western equatorial Pacific [11]. sidered as members of one mode. For increasing a, the No representation, Fig. 5, developsa jump betweenM =0 V. CONCLUSION and -1, while for frequenciesv ~ a the characteristics of With respect to dynamics as well as diagnostics, Fig. 2, namely, the east-westasymmetry corresponding to Laplace's tidal equations are the linear shallow water (4.2), are retained. The N 2 representation, on the other equations on the rotating spherical surface. These equa- hand, yields the familiar .B-planediagram, Fig. 5(c), for tions are a spatially 2D, covariant, and separableapprox- large a. With decreasinga, this representation develops imation to the covariant and nonseparable 3D problem. a minimum at M = - 1. In contrast to the implications of Covariant differentiation renders tensor analysis on the (3.6) this minimumdoes not approachthe axisM =0 nor non-Euclidean spherical surface straightforward. The does the steepnessof eastern and western branches de- prolate spheroidal wave operator plays a fundamental creasewith increasing N 2' role in the system's wave equation. Only in special cases The comparison of Figs. 5 and 6 demonstrates the can its latitudinal eigenfunctions be representedas simple modification of the dispersion of low-frequency gravity combinations of trigonometric functions and prolate an- waves with varying Lamb parameter: for larger a the gular functions. The form of the wave equations indi- westward branches not only steepen,but are translated to cates neverthelessthat these functions are a key element higher frequencies. Due to this translation eigenfrequen- of the general solution. cies with constant The rotating spherical surface provides an anisotropic and inhomogeneousarena for the propagation of Rossby N2=No-sgn(M) and long gravity waves. The fundamental latitude- can be combined into one mode. This relation between longitude anisotropy is due to its topology, while the No and N 2 ensures the same symmetry with respect to Coriolis effect introduces an additional east-west asym- the equator for the eigensolutions of all members of one metry, physically similar to the asymmetry induced by mode in both representations. The dift"erent east-west the shear of a stationary, zonal flow. The joint action of asymmetry of the P plane thus emerges in fact as an rotation and a finite intrinsic phase speedof shallow wa- enhanced east-west asymmetry of the type (4.2). The ters lead to a latitudinal inhomogeneity, which confines reason for the upward translation of westward low- low-frequency waves to an equatorial waveguide. Prolate frequency gravity branches is the emergence of Rossby angular functions represent this inhomogeneity on the waves in this part of wave number space: the uniqueness finite spherical surface, where the width of the Yoshida of a wave prohibits the intersection of the dispersion waveguide is controlled by the Lamb parameter. As this curves of Rossby and gravity waves. The amplified east- parameter distinguishes barotropic and baroclinic phase west asymmetry of gravity waves in the p-plane regime is speeds. it also permits the consistent incorporation of thus the necessarycondition for the coexistenceof Ross- vertical variability into rotating, spherical shallow waters by wavesand low-frequency gravity waves in the Yoshida as internal variability of an otherwise 2D fluid. For linearizations around a stationary, zonal flow, the Lamb waveguide. Although the labeling of gravity modes by the mode parameter captures the necessary shear effects of such number N 2 leads to good quantitative agreement in the flows on the spherical surface. .B-planeregime, it is not quite unproblematic. In contrast In the wave number spaceof shallow waters on the ro- 4430 D . MULLER. AND J. J. O'BR.IEN 51

r l - r l - tAnm tating spherical surface three regimes with distinct 12- 21---""" dispersion properties have been identified. For gravity 1 = sintp COSql waves at and above the inertial frequency the rotating rt . sphere is globally homogeneous and an equatorial The Ricci tensor is defined as a contraction of the waveguide does not exist. In addition to the latitude- Riemannian longitude anisotropy, these gravity waves exhibit Cariolis-induced east-westasymmetry with stepper west- Ginn =gabGambll ward branches of the dispersion diagram. Barotropic gravity waves in the ocean, generated by external tidal and has for the particular metric (AI) the form forces, propagate in this regime. A second, low- G - -2 mil -a gmll. frequency regime of Rossby waves can be distinguished for all valuesof the Lamb parameter. BesidesMargules's The Riemannian corresponding to (At) can be written as nondivergent Rossby waves and Rossby waves on the Gamb"=gabGm" -ganGmb . equatorial fJ plane, this regime includes midlatitude Ross- by waves, which are frequently representedin terms of a The covariant derivative of a covariant vector is "midlatitude fJ plane." The approximations obtained here avoid the inherent inconsistenciesof this concept. Final- vm;"=a,,vm -~8Va ly, for large values of the Lamb parameter, the fJ plane and of a contravariant vector admits gravity and Rossby waves with frequencies well Vm. = below the inertial frequency, coexisting in the Yoshida '8 a8 vm+rm-' Va waveguide. As a consequenceof the emergenceof Ross- For the covariant derivative of a mixed tensor one has by wavesin the westward part of wave number space,the east-west asymmetry of low-frequency gravity waves is Tm8;a=aaTm"-r:"Tmb+r:Tb,, . enhanced: westward gravity branches are not only steeper than their eastward counterparts, but are also The Laplacian of a scalar is translated to higher frequencies. Under terrestrial condi- (a"S)' =( + rm \:1"S M= 'II a" m,,/U tions the fJ-plane regime represents baroclinic low- =a -2 frequency waves in the equatorial ocean. These waves (COS-2a1;A.+a2 ..'-""-" -ta"",a )S are of major significance for the terrestrial climate prob- and its eigenfunctions are the spherical harmonics. The lem, as indicated by the crucial role of baroclinic Kelvin twofold covariant differentiation of a vector waves in the equatorial Pacific for the EI Nino phenomenon. Exact dispersion relations for strictly Va;m"=Va;/lm + Gbam"Vb standing wavesand for inertial waves, as well as the Mar- does not commute, since the Riemanni;ln does not van- gules limits, provide benchmarks for analytical approximations and numerical computations of the eigenfrequen- ish. The Laplacian of a covariant vector is cies of linear shallow water on the rotating spherical sur- gaby.",ab =(aa +rbba) aaV" -2ra bIIaby a face. +gab(r~r:+ r"abr: -abr:)V IN ACKNOWLEDGMENTS with longitudinal component The authors wish to thank the Physical Oceanography gabVl;ab =a -2(COS-2cpai+a~ +tan."a" + l)Y 1 Section of ONR for the basic support of COAPS, NASA for the support of D.M., and P. Swarztrauber for kindly -2a-~A.V2 providing the Swarztrauber-Kasahara code. We appreci- ate helpful discussionswith Dr. S. D. Meyers. and latitudinal component

g ab y . =a-2 ( cos-2_~+a2_+G :4 -tan2m)y APPENDIX 2 , ab Cf'U 1 f' -U'f'V f' T 2 +2(a COSfJ)-~lYl . For curvilinear coordinates A (longitude) and tp (latitude) on the surface of a sphere of radius a the covariant The gradient of a spherical harmonic metric is given as Ell =all Yf 2 COS2tp 0 is an eigenvector of the vector Laplacian with eigenvalue

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