2692 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34
The Dispersion Relation for Planetary Waves in the Presence of Mean Flow and Topography. Part I: Analytical Theory and One-Dimensional Examples
PETER D. KILLWORTH AND JEFFREY R. BLUNDELL Southampton Oceanography Centre, Empress Dock, Southampton, United Kingdom
(Manuscript received 14 October 2003, in ®nal form 4 May 2004)
ABSTRACT An eigenvalue problem for the dispersion relation for planetary waves in the presence of mean ¯ow and bottom topographic gradients is derived, under the Wentzel±Kramers±Brillouin±Jeffreys (WKBJ) assumption, for frequencies that are low when compared with the inertial frequency. Examples are given for the World Ocean that show a rich variety of behavior, including no frequency (or latitudinal) cutoff, solutions trapped at certain depths, coalescence of waves, and a lack of dispersion for most short waves.
1. Introduction Only one calculation has been made, to our knowl- edge, of the actual dispersion relationship for nonlong The discovery by Chelton and Schlax (1996) that planetary waves in the presence of mean ¯ow (but as- planetary waves propagated faster than traditional linear suming a ¯at bottom), by Fu and Chelton (2001), for a theory predicted has produced a large number of papers limited range of wavenumber and physical locations. examining possible explanations. These all involved de- These calculations showed the continued speedup of viations from the assumptions of small perturbations to planetary waves persisted at least some way into the an ocean at rest with a ¯at bottom. The most successful short-wave regime, but were not intended as a full dis- to date (though not completely so) is due to Killworth cussion of the problem. [Gnevyshev and Shrira (1989) et al. (1997) (and, much earlier, by Kang and Maagaard discuss short-wave theory in the quasigeostrophic limit 1979) who argued that background baroclinic mean ¯ow for an idealized problem.] modi®ed the potential vorticity distribution and thus the In this paper and its sequel, we calculate the disper- speed of long planetary waves. Many other papers ap- sion relationship for planetary waves in the presence of peared examining aspects of the problem. Most were slowly varying baroclinic mean ¯ow and bottom to- con®ned to two- or three-layer models for simplicity, pography. After formulating the problem (section 2), although such models appear to respond rather too some analysis is given for short waves (section 3). Sev- strongly to bottom topographic gradients, and can per- eral example dispersion curves (for east±west propa- mit wave speeds that could not occur in the continuously gation only) are discussed (section 4), showing a rich strati®ed equivalent (e.g., Liu 1999; see Killworth and variety of behavior. These are discussed in section 5. Blundell 2003a,bÐhenceforth KB03a,bÐfor a discus- The second part of the paper discusses the full two- sion). Few papers have examined the problem of wave dimensional dispersion relationship and gives global ex- propagation through a mean ¯ow in a non-¯at-bottom amples of this. ocean, save KB03a,b. In those papers, two critical as- sumptions were made: the Wentzel±Kramers±Brillouin± Jeffreys (WKBJ) assumption, invoking slowly varying 2. Formulation background ¯ow and topography, and the long-wave assumption. Although the WKBJ assumption is neces- Planetary waves are traditionally studied either within sary for the local problem to be meaningful, ray theory a quasigeostrophic model or by a WKBJ formulation calculations (KB03b; Killworth and Blundell 1999) combined with a long-wave assumption. We here relax show that waves become somewhat shorter as they prop- the latter assumption and show that, provided that the agate westward from an eastern boundary wavemaker. wave frequency remains small when compared with the absolute value of the Coriolis frequency f, the effects of non±long waves can be included. Corresponding author address: Dr. Peter Killworth, Process Mod- elling Group, James Rennell Division, Southampton Oceanography We assume that there is a background mean (denoted Centre, Empress Dock S014 3ZH, United Kingdom. by an overbar) ¯owu (x, y, z) ϭ (u , ,w ) relative to E-mail: [email protected] longitude , latitude , and depth z. The Coriolis fre-
᭧ 2004 American Meteorological Society
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quency f ϭ 2⍀ sin, where ⍀ is the rotation rate of layers. This immediately implies that all planetary wave the earth. This ¯ow varies on a horizontal length scale speeds are Doppler shifted by an amount proportional
Lbasin, assumed also to be the scale of topographic var- to the maximum value of Q in the ¯uid column (in the iations. The ¯ow is steady and geostrophic to leading purely east±west ¯ow to be considered later, this is order. Such ¯ows are not full solutions to the equations equivalent to a speed shift of the minimum value of u of motion, in general, and will require weak (steady) in the water column). Alternatively, there is a minimum forcing terms to exist. This is a standard assumption in possible frequency, that is, the maximum of Q in the perturbation and wave theory; Pedlosky (1987, pp. 567 water column, at which real waves may propagate. et seq., 617 et seq.) gives a useful discussion. Then The WKBJ formulation precludes abrupt topography ilM iRЈ ilM kRM or fronts (cf. Owen et al. 2002). The wave perturbations, uЈϭϪzzz Ϫ ϭϪ ϩ ϩsmall (2.8) denoted by primes, are small amplitude relative to the af f af af 2 cos mean and vary horizontally with a length scale Lwave K and Lbasin by assumption. The horizontal momentum equa- tions become essentially quasigeostrophic: ikMzzz iRuЈ ikM lRM Јϭ ϩ ϭ ϩ ϩsmall (2.9) 2 u M a cos facos af uЈϩ uЈϩ uЈϪ f ЈϭϪz ϩsmall (2.1) t a cos aacos de®ne uЈ and Ј in terms of M. The second terms are already order (/ f ) smaller than the ®rst (geostrophic) and and can thus be neglected when they occur in this form u M (but not when a divergence operator is used). The per- Јϩ Јϩ Јϩ fuЈϭϪz ϩsmall, (2.2) t a cos aa turbation vertical velocity can be obtained from the di- vergence equation where a is the earth's radius, t denotes time, and we 11 have de®ned Welander's (1959) function by w ϭϪ u Ϫ ( cos) (2.10) z a cos a cos pЈ M ϭ (2.3) z ik ilM kRM 0 ϭϪ Ϫzz ϩ for convenience in what follows. In (2.1) and (2.2), a cosaf af 2 cos ``small'' means either of order /| f | smaller or of order 1 ikMzz lRM cos Lwave/Lbasin. Ϫϩ Following normal WKBJ theory, we write a cosaf af 2 M ϭ F(z; , ) expi(k ϩ l Ϫ t), (2.4) 2 ikMzz iRM k where we assume that there is a rapid phase variation ϭϪ ϩl2 ϩ small (2.11) 2⍀a22sin af 222cos in the horizontal on the scale Lwave described by expi(k ϩ l Ϫ t), where k ϭ (k, l) is the local wavenumber, so that and the wave frequency. The vertical structure of M is described by F, which also varies slowly in the hor- ikM iRM k2 w ϭϪϩl2 izontal, on the scale L . The wavenumber k is de®ned 22 222 basin 2⍀a sin afcos in terms of longitude and latitude, and so is dimen- ikM sionless. The dimensional equivalent, that is, in units of ϭ (1 ϩ D), lengthϪ1,is 2⍀a22sin
kl where kdim ϭ , . (2.5) a cos a Rk2 D ϭϪ ϩl2 Ͻ 0 (2.12) Then we de®ne 2⍀kcos2 ku l Q ϭϩϭk ´ u and (2.6) as R Ͻ 0, k Ͻ 0. (The latter is by assumption since a cos a dim westward propagation is almost entirely what is ob- served in the ocean.) The term (1 ϩ D) is approximately R Q (2.7) ϭ Ϫ 1 for long waves (k → 0), but D can become important as in KB03a; R is (minus) the locally Doppler-shifted for nonlong waves. In particular, (1 ϩ D) can vanish, frequency. when Rk/ f is of order unity, or kഠO(| f |/) k 1. (For Precisely as in appendix A of KB03a, it can be shown an annual frequency, this would involve wavelengths of that R Ͻ 0 everywhere for real frequencies, that is, stable order 120 km.) waves (the frequency is assumed positive without loss The ®nal quantity to be expressed as a function of M of generality). In other words, there are no real critical is density, which, from the hydrostatic balance
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pЈϭϪz gЈ, (2.13) termwЈz is neglected against wЈ z on scaling grounds, assuming the perturbation ®elds have depth scales of becomes order H, the ocean depth. This will not always be the case for short waves. In fact, thewЈ term continues to ЈϭϪ 0 MЈ . (2.14) z g zz be negligible, and will be assumed so henceforth (ap- pendix A demonstrates this). We now consider substitution into the conservation Using the geostrophic and thermal wind relations for of density, which reads the horizontal mean ¯ows
(uЈϩuЈ )( ЈϩЈ ) 1 g Јϩt ϩ a cos a fu ϭϪ p and fuz ϭ and (2.16) a 00a ϩ wЈ ϩ wЈϭ0. (2.15) zz 1 g f ϭ p and f ϭϪ , (2.17) Immediately there is a potential dif®culty in that the last a cos z a cos term in (2.15) involves a third (vertical) derivative of 00 M, while the remainder of the equation is only second the phase relation (2.4) and substitution into (2.15) order. In previous long-wave work (e.g., KB03a,b) the gives
1 a cos f il i 000M ϩϪiku M ϪϪz M gazzcos[]gg zz2⍀a sin z 1 a f u ik ik ϩϪil 00M ϩ z M ϩ (1 ϩ D)M ϭ 0. (2.18) ag[]zz g2⍀a sin cos zaf2 sin z
Simplifying, we obtain linearly depends on frequency and contains an addi- tional term in Q. The mean ¯ow is assumed purely kuM l l Mku zz z zz z baroclinic; any weak barotropic ¯ow simply adds a ϪMzzϩϪM zϩϪM z a cos aaacos small Doppler shift to the frequency but has no other k impact. ϩ (1 ϩ D)MN 2 ϭ 0. (2.19) 2 The surface boundary condition remains a rigid lid, afsin that is, w ϭ 0atz ϭ 0, so that (1 ϩ D)F vanishes at Here N 2(z; , ) is the buoyancy frequency. Rewriting the surface. We shall use consistently the simpler con- and substituting in terms of F, dition F ϭ 0atz ϭ 0. (2.23) RFzzϪ RF z z ϩ SF ϭ 0, or, equivalently, (2.20) The possible additional solution D ϭϪ1atz ϭ 0is FS L(F) ϵϩz F ϭ 0, (2.21) discussed in some detail below, where it is shown to be 2 RRz related to the barotropic mode provided that the mean ¯ow satis®es a certain criterion (rare in the ocean). Since where the frequency for this solution can simply be written kN 2 down from the de®nition of D, we ignore this solution S ϭ (1 ϩ D), (2.22) af2 sin for the moment. The bottom boundary condition, at depth z ϭϪH, which can also be written is
kN22 Q k uЈ Ј S ϭ 1 Ϫϩl2 ϩ T, (2.22a) wЈϩ Hϩ H ϭ 0atz ϭϪH, af22sin[]2⍀kcos a cos a where which becomes
F(ϪH)(1 ϩ D) ϩ ␣Fz(ϪH) ϭ 0, (2.24) Nk22 T ϭϩl2 . where the ``gradient parameter'' af22cos 2 l Equation (2.20) or (2.21) is identical in form to that ␣ ϭ tan H Ϫ H (2.25) used by KB03a; the only differences are that S now k
Unauthenticated | Downloaded 09/28/21 08:30 PM UTC DECEMBER 2004 KILLWORTH AND BLUNDELL 2695 is the same as in KB03a. However, there is an extra in Part II, where it is noted that there is no natural mode term in D (which contains the frequency) in (2.24). This numbering when mean ¯ow is included. latter term plays no part if the bottom is locally ¯at. [There remains a debate about the bottom boundary 3. Asymptotics of the problem condition: we use here the usual one of no normal mass ¯ux into the system. See Tailleux and McWilliams It would appear natural to solve (2.20) for simple (2001) and KB03a for more details.] analytical forms of the mean ¯ow (constant shear, e.g.). The system (2.21), (2.23), and (2.24) can usefully be However, attempts at analytical solutions rapidly be- compared with (2.12), (2.16), and (2.17) of KB03a. The come tedious and unenlightening; constant shear and changes are that one extra term has been gained in the buoyancy frequency, for example, yields con¯uent hy- equation for F (in S) and one in the lower boundary pergeometric function solutions, while quadratic shear condition. Both of these extra terms are only important and constant buoyancy frequency gives a version of when the length scale is of order the deformation radius (3.16) below, which has no known analytical solutions. (in other words, D becomes order unity only in that Numerical solutions to simple cases yield no better in- case). The system includes several special cases. With sight than the solutions for real ocean locations in the no mean ¯ow and a ¯at bottom, following section. Accordingly we restrict attention here to the asymptotics of the problem, where considerable kN 22 k headway can be made. ϪF ϩ 1 ϩϩlF2 ϭ 0. zz af22sin[]2⍀kcos These asymptotics are related generically to the po- sition where Q k ´u reaches a maximum in depth. 2 2 ϭ dim This possesses solutions that satisfy Fzz ϭϪ(N /C )F, (For zero north±south wavenumber or zero north±south where C is an eigenvalue equal to the internal wave ¯ow, this simpli®es to the position whereu reaches a speed, yielding the dispersion relation minimum in depth.) The physical signi®cance of this 2⍀k location will be discussed later. ϭϪ , (2.26) kaf222 There are three possibilities: ϩ l2 ϩ cos22 C (i) Q is a maximum (an extremum) at the surface, (ii) Q has a local maximum within the ¯uid, or which is the familiar normal mode theoretical result (cf. (iii) Q is a maximum (an extremum) at the ¯oor. Gill 1982) rewritten into angular wavenumber terms. This has a maximum frequency given by The value of Q at the maximum will prove to play a strong role in the dispersion relationship by providing ⍀C ϭ cos. (2.27) a Doppler shift (recall that only real solutions are con- max af sidered in this paper). The distribution of the three pos- Alternatively, there is a maximum latitude for a given sibilities is not uniform in the World Ocean. It would frequency (Gill 1982) given by appear a priori that the orientation of the wave vector is needed before any distribution of the possibilities can C cot | | ϭ 2a. (2.28) be calculated. However, since in most of the ocean is If the bottom is not ¯at, but there is no ¯ow, the results small when compared with u (Killworth and Blundell of Rhines (1970) and later authors (e.g., Killworth and 2001) and observed wave vectors are oriented fairly Blundell 1999; Bobrovich and Reznik 1999) can be re- close to westward, the dependence on orientation is produced. If both mean ¯ow and bottom slopes are in- weak. For a purely westward orientation (l ϭ 0), Fig. cluded, but the waves are long, the theory of KB03a 1 shows where each possibility is located. Only 7% of appears. the ocean between 5Њ and 50Њ latitude has a maximum Other useful quantities can be determined from the Q at the surface, 58% has an internal maximum, and eigensolution (i.e., the dispersion relation for the sys- 35% has a maximum at the bottom. This latter tends to tem). These include the phase velocity occur at higher latitudes where the thermal wind rela- tionship has a poleward gradient of density over most ak cos al or all of the depth. Calculations with wave vectors at c p ϭ , (2.29) Ϯ45Њ to pure westward propagation yield similar maps KK22 and statistics identical to within a few percent of those and the group velocity above, though of course the depths of the middepth maximum depend on orientation. cg ϭ (a cos,a ). (2.30) kl Much of the asymptotics relates to short waves (K → Here K ϭ (k 2 ϩ l 2)1/2 is the amplitude of the wave vector, ϱ). In such cases, D is of order K 2, and so from (2.21) and (k, l) are the derivatives of frequency w.r.t. wave- the vertical length scale for the problem becomes small number. Both velocities are adjusted to give correct (either K Ϫ1 or K Ϫ2 depending on the problem). The speeds on the spherical earth. Details of the numerical reader unconcerned with asymptotics is advised to skip approach and computation of group velocity are given the remainder of this section.
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FIG. 1. Location in the World Ocean where the component of mean velocity along the wave vector, Q, reaches a maximum at (a) the surface (white), (b) the interior (medium gray), and (c) the bottom (light gray). The near-equator and shallow regions are masked in darker gray shades. A westward orientation of the wave vector is assumed; the diagram is very similar if the wave vector orientation is at Ϯ45Њ to westward. a. Q is a maximum (an extremum) at the surface 2⍀k ϭϪ . (3.3) k2 If Q is a maximum at the surface, then a trivial so- ϩ l2 lution satisfying the surface boundary condition is to cos2 require (1 ϩ D) ϭ 0, z ϭ 0. This speci®es the dispersion This is readily seen as the barotropic solution to the relationship, using (2.12), as planetary wave problem, in which w vanishes identically and there is no density perturbation (for no mean ¯ow). 2⍀k ϭ Qmax Ϫ , (3.1) For long waves, the frequency becomes much larger k2 than observed by remote sensing (and our assumptions ϩ l2 cos2 break down). When there is mean ¯ow, the solution is Doppler shifted by the surface velocity so that the fre- where Qmax will be consistent shorthand for the maxi- quency also becomes large for short waves. mum of Q. In the special case l ϭ 0 this reduces to As far as we are aware, this is the only solution per- 2⍀ cos2 ku mitted for reasonably short waves when there is a sur- ϭϪ ϩ min . (3.2) face maximum in Q (cf. the analysis in appendix B), kacos though there may be other solutions for long waves, but The structure of the solution merely requires that F sat- these do not exist over the full wavenumber range. Mean isfy (2.24) so that without loss of generality we can take ¯ows with a surface Q maximum usually satisfy the necessary conditions for baroclinic instabilities (cf. Gill F(ϪH) ϭ 1, derive Fz(ϪH) from (2.24), and then in- tegrate (2.21) upward to the surface. The choice that Q et al. 1974). is maximal at the surface ensures that R never vanishes (i.e., there is no critical layer) so that the solution is b. Q has a local maximum within the ¯uid well behaved everywhere. It is possible to solve the problem analytically for In the case of no mean ¯ow, the solution becomes short waves when Q has a maximum within the ¯uid.
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Its vertical extent also varies as K Ϫ1. This vertical struc- ture is given by an eigenvalue problem; because of the limited vertical extent involved, details of the bottom boundary condition (and, indeed, the surface condition) become irrelevant. The solutions are valid when K is large enough for the vertical scale to be small when compared with the depth but not so large as to render the neglect of vertical advection byw invalid. The wave-vector orientation is chosen to lie in the range /2 Յ Յ 3/2 so that k Ͻ 0. We have
u Q ϭ K cos ϩ sin ϵ Kq(z); a cos a k ´ u q(z) ϭ dim . (3.4) |kdim | The shape of this remains unchanged as K increases.
We assume that there is a value of z, z0 say, at which
Q is a maximum, again termed Qmax. We write this al- ternatively as FIG. 2. Eigenvalues of the nondimensional frequency correction  for short waves concentrated at middepth as a function of the con- Q ϭ Kq(z) ϭ Qmax ϩ KqÃ(z), qà Յ 0. (3.4a) trolling parameter ␥ (a measure of strati®cation and curvature of the mean ¯ow at its local minimum along the wave vector). For lower We seek a solution approximately Doppler shifted by values of ␥, the solution is well de®ned by  ഠ 0.68␥1.06; for larger Q so that values, it is well de®ned by  ഠ 0.96␥ Ϫ 1.53. max
ϭ Kqmax ϩ ␦. (3.5) Here ␦ is positive, so that no critical layer appears, The position of this maximum will depend on the ori- and is of the form entation of the wave vector, and such a maximum may disappear discontinuously as the orientation is changed. à ␦ ϭϾ0 (3.6) This problem was examined for a two-layer ¯uid by K Samelson (1989), who found a solution for short waves Doppler shifted by the upper-layer (westward) velocity. where KB03a note that care must be taken in interpretation of à Ͼ 0. layered results because such models permit solutions that would have critical layers if converted to a contin- Corrections to the theory will then be of order K Ϫ2.We uously strati®ed system; they showed that (real) critical note that layers cannot exist. (Indeed, Samelson's Fig. 2 shows à wave speeds that lie between the mean ¯ow in his two R ϭ Q Ϫ ϭ Kqà ϪϽ0, (3.7) layers, implying a critical layer in a more realistic con- K ®guration.) Nonetheless, these results suggest that a where qÃ(z 0) ϭ 0 and qà zz(z 0) Ͻ 0 as required for a max- Doppler shift by some aspect of the mean ¯ow may be imum of Q or qÃ. Near z ϭ z 0, generic in short-wave solutions. In the same year, Gnev- 1 yshev and Shrira (1989) examined the same problem in qà ഠ qà (z )(z Ϫ z )22ϭϪ(z Ϫ z ) , (3.8) a quasigeostrophic context, near a zonal ¯ow critical 2 zz 00 0 layer, avoiding the above dif®culties. They included hor- where izontal shear (excluded in our analysis below by the WKBJ assumption). Their main ®nding was that there 1 ϭϪ qà (z ) Ͼ 0 (3.9) is a natural tendency for a short-wave mode to occur at 2 zz 0 the level of the minimum zonal ¯ow (i.e., near a critical layer). The mode's vertical extent varies as K Ϫ1, and is is de®ned for convenience. The behavior of S is again Doppler shifted with the minimum of the zonal K cosNq22à Ϫ à /K cos ¯ow. S ϭ 1 Ϫ K ϩ sin2 , The nonquasigeostrophic analysis below con®rms af22sin []2⍀ cos cos their ®nding and demonstrates that a short-wave mode (3.10) occurs around a level at which the vector component of Ϫ1 the mean ¯ow along the wave vector is a local maxi- which is positive provided z is not within O(K )ofz0 mum, again in an approximate Doppler shift behavior. (because cos Ͻ 0) and varies as K 3 for large K. The
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FIG. 4. The lowest and second-lowest short-wave solutions, for the FIG. 3. Computed short-wave solutions at 20ЊS, 70ЊE using a ¯at- location in Fig. 3. The ®rm line shows the lowest mode; the dashed bottom boundary condition. (Solutions are almost identical if a non- line shows the second lowest. ¯at-bottom condition is used.) Solid line is k ϭϪ600, and the dashed line is k ϭϪ300; l ϭ 0 has been assumed. wavelengths short when compared with 600 km are re- R term in (2.20) varies only as K and is negative, and quired. Fu and Chelton's (2001) results show this to be so the solutions of the problem show strong exponential a realistic restriction. decay away from z . Substituting (3.11) and (3.14) into (2.20) gives, after 0 simpli®cation, the eigenvalue equation2 In the vicinity of z 0, 22 K cosN 2(z ) ( ϩ )FϪ 2F ϩ [␥ Ϫ ( ϩ )]F ϭ 0. S ϭ 0 (3.16) af2 sin Here the controlling parameter 1 cos2 2 2 2 ϫ 1 ϩϩsin ϩ Ã , cosN (z0) 2⍀ cos cos2 A ␥ ϭϪ Ͼ0 (3.17) [] af2 sin (3.11) and a nondimensional frequency correction where is a nondimensional coordinate given by AÃ  ϭϾ0 (3.18) z Ϫ z ϭ , and (3.12) 0 KA1/2 are de®ned, together with the boundary conditions 22 N (z0) cos F → 0, | | → ϱ. (3.19) A ϭϩsin2 (3.13) 22 2 af cos The boundary condition means that the details of the is usefully de®ned. Then bottom boundary condition become irrelevant in the limit of large K (if solutions exist). 1 We are not aware of any known solutions to this R ϭϪ 2 ϩ Ã . (3.14) KA problem. [In the quasigeostrophic limit, Gnevyshev and Shrira (1989) found the solutions to be parabolic cyl- The requirement that the vertical scale, O(a | f |/NK), inder functions.] Solutions were found numerically. The be small when compared with H yields1 lowest mode is symmetric about ϭ 0 and so has F(0) KNH ϭ 0. For large negative ,Fϳ exp(). The solutions k 1 (3.15) a| f | are found, for a given ␥, by integrating in to zero and varying  to locate a solution using a zero ®nder. (The or kdimNH/| f | k 1. With typical values, K k 50, or
2 The omission of advection by the mean vertical velocity is le- 1 This ensures that N can be considered constant in the expansion gitimate provided the wavelengths are not too small; see appendix A below, for example. for details.
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persive. This too totally contrasts with linear normal- mode theory where short waves are always dispersive. Since these short-wave solutions all possess the dis-
persion relation ϳ Qmax and, since any solution with no mean ¯ow would be strongly dispersive and have → 0 for large wavenumber, these solutions can be
regarded roughly as a simple Doppler shift by Qmax to the eigenfrequency for the problem without mean ¯ow.
c. Q is a maximum (an extremum) at the ¯oor For bottom slopes of a particular orientation, a bot- tom-trapped mode is permitted when Q has a maximum at the ¯oor. We again seek a solution whose vertical extent varies as K Ϫ1. We write again
Q ϭ Qmax ϩ KqÃ(z), qà Յ 0, (3.20) FIG. 5. The function K 0(x)/K 1(x) for positive x. where now qà Ϫ(z ϩ H) and ϭϪqà (ϪH) Ͼ 0. (3.21) next gravest mode is antisymmetric about ϭ 0, the ഠ z next symmetric, and so on. The number of solutions We seek a solution for frequency as depends on the value of ␥.) Q , (3.22) The lowest mode eigenvalues are shown in Fig. 2 (in ഠ max ϩ ␦ this short-wave limit there is again a natural mode num- which has the familiar Doppler shift as before but now bering). For lower values of ␥, the solution is well de- ␦ will be a constant as K varies, rather than decreasing ®ned by  ഠ 0.68␥1.06; for larger values, by  ഠ 0.96␥ as K Ϫ1. Ϫ 1.53. In all cases, ␥ Ϫ  K  in (3.21), which means Expanding in a vertically stretched coordinate
that S is negative at ϭ 0; that is, 1 ϩ D takes both 1/2 signs in the vertical. This analysis is con®rmed by actual ϭ KA (z ϩ H) (3.23) numerical solutions. In a region of steep topography [here A is de®ned as in (3.13), evaluated at the ¯oor] used by KB03a as an example (20ЊS, 70ЊE), ␥ is about and de®ning 500, giving a predicted value of  of 479. For k ϭ Ϫ600, l ϭ 0, the observed  (i.e., the correction to the ␦ ϭ Ä (3.24) 1/2 frequency from the simple Qmax) is 400. Doubling k to A Ϫ1200 gives  equal to 437. The error from the theory gives, after a little algebra, the equation has halved, so that the solution is indeed behaving as Ϫ1  ϭ 479 ϩ O(K ), the latter term being the next-order ( ϩ Ä )FϪ F Ϫ ( ϩ Ä )F ϭ 0. (3.25) correction. Figure 3 shows numerical solutions at this The boundary condition in the interior of the ¯uid is point for a ¯at-bottom boundary condition (those with simple: the actual bottom condition are similar except for nec- essary small changes near bottom). The similarity to the F → 0as → ϱ. (3.26) Gnevyshev and Shrira (1989) parabolic cylinder func- At the ¯oor, (2.24) must be evaluated. To leading order, tion solutions is evident. As noted, the eigenvalue problem can have several K cos2 D ϭϩsin2 ( ϩ Ä ) solutions, of higher and higher degree. As an example, 21/2 the k ϭϪ600 case has one more solution, shown in 2⍀ coscos A Fig. 4. When there are several solutions, the dispersion so that at the ¯oor (2.24) becomes relationships are all similar [varying as ϳ Qmax ϩ O(K Ϫ1)] with increasing number of zero crossings. In- 2⍀ cosA Ä F(0) ϩ ␣ F(0) ϭ 0. (3.27) terestingly, the gravest mode possesses the slowest cos2 group velocity. ϩ sin2 cos2 We stress that the vertical structure of the short wave modes is completely different from that for the long The solution of (3.25) that decays at in®nity is waves (see KB03a for examples of the latter). This is F ϭ ( ϩ Ä ) ( ϩ Ä ), (3.28) in contrast to traditional linear wave theory, where the K 1
vertical structure is always that of the linear mode. In where K 1 is the modi®ed Bessel function of order 1. This
addition, the existence of near-Doppler shift frequencies implies that F ϭϪ( ϩ Ä )K 0( ϩ Ä ), and the boundary means that short waves of this type are almost nondis- condition simpli®es to the dispersion relation
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FIG. 6. The dispersion relationship for zero north±south wavenumber at 45ЊN, 150ЊW when a ¯at-bottom condition has been imposed. The plots shown are frequency, group velocity, phase velocity, and the mean east±west ¯ow. In the ®rst three of these, a solid line shows solutions including mean ¯ow, and a dashed line shows solutions with no mean ¯ow (the dash±dotted line shows the barotropic mode, not shown henceforth). The ®rst three modes are shown (in the case of mean ¯ow, there is only one mode). A wavenumber of Ϫ600 corresponds to a wavelength of 47 km.
2⍀ cosA K 0(Ä ) accurate as wavenumber becomes larger. Formally these Ä ϭ ␣ , (3.29) would appear as zeroÄ in the foregoing analysis, and cos2 K 1(Ä ) ϩ sin2 the solution would have a singularity in the second de- cos2 rivative at the ¯oor. The solution (3.30) removes this where we note the presence of the negative factor cos singularity within a second boundary layer of width of order K Ϫ2. The details are given in appendix B and yield in the rhs. Figure 5 shows the behavior of K 0 (x)/K 1(x) for positive x. Because of the in®nite gradient of this a dispersion relation of the form function at the origin, there is precisely one solution to 1 Ϫ X Ä ϩ ␣A1/2(X ϩ Ä lnÄ Ϫ X Ä ) ϭ 0, (3.31) (3.29) when ␣ Ͻ 0 (making the multiplier on the rhs 12 3 positive), and no solution when ␣ Ͼ 0. If the north± where the Xi are positive nondimensional quantities. south wavenumber is zero, the condition for a solution Equation (3.31) may or may not possess solutions. For is that (tan)H Ͻ 0. zero bottom slope (␣ ϭ 0) there is a trivial solution, There are, however, other bottom-trapped solutions not Ä ϭϪ1/X1, which reduces to (3.1) in form again and covered by the above analysis. These solutions have remains essentially a barotropic-like solution whose ex- istence is not predicated on the assumption of short ␦ ഠ Ä (3.30) waves; the discussion in section 3a holds for this so- KA1/2 lution. If the bottom slope parameter ␣ is small, clearly so that the Doppler shift becomes progressively more there are solution with Ä ϭϪ1/X1 ϩ O(␣). For larger
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FIG. 7. The dispersion relationship for zero north±south wavenumber at 45ЊN, 150ЊW with the full bottom boundary condition. The plots shown are frequency, group velocity, phase velocity, and the mean east±west ¯ow. In the ®rst three of these, a solid line shows solutions including mean ¯ow; a dashed line shows solutions with no mean ¯ow. Only two modes are shown for
clarity. The frequency diagram is also annotated with a digit indicating the number of sign changes in the vertical of Fz, i.e., of the perturbation horizontal ¯ow ®eld. Other aspects of the behavior are also noted. A wavenumber of Ϫ600 corresponds to a wavelength of 47 km. The nondimensional value of the bottom slope parameter, ␣/H, is 0.33. values of ␣, nothing can be deduced in general about and Chelton 1999). In particular, the ®rst of these (Kill- existence of solutions. worth et al. 1997) studied ¯at-bottomed ¯ows where the Both of these bottom-trapped solutions owe their ex- north±south velocity and wavenumber were neglected. istence to the mean ¯ow and are quite dissimilar to the They found that in regions where the mean eastward
Rhines (1970) solutions for no mean ¯ow. Thus the velocity had a minimum (umin, say) somewhere within generic Doppler shifting by Qmax is likely to occur in the water column (i.e., a Qmax again), the long-wave most oceanic situations. phase velocity was given approximately by a simple Doppler offset of the velocity of the standard problem (without mean ¯ow) by u . d. Long waves min Although Killworth et al. (1997) could not formally In the limit of long waves (K → 0), D → 0 and (2.21), prove thisÐrelying on numerical solutions of simpli®ed (2.23), and (2.24) reduce to the problem described by problemsÐit is straightforward to extend these numer- KB03a, who discuss the global behavior of long-wave ical results to cases in which there is a mean northward solutions. Analysis of the effects of mean ¯ow on the velocity and a north±south wavenumber, using the def- speed of planetary waves has been given in several pa- inition of R. Clearly there must be at least a Doppler pers (e.g., Killworth et al. 1997; Dewar 1998; de Szoeke shift by Qmax to avoid a critical layer (which, recall,
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FIG. 8. The dispersion relationship for zero north±south wavenumber at 30ЊN, 30ЊW. Plots as in Fig. 7, showing up to four modes. A gap of twice 15 (the wavenumber interval in plotting) indicates where a solution may have become complex. A wavenumber of Ϫ600 corresponds to a wavelength of 58 km. The nondimensional value of the bottom slope parameter, ␣/H, is Ϫ0.83. cannot occur for real solutions), and so at the very least there is no simple rule of thumb for the speed of long there must be a change of frequency of Qmax or of phase waves. velocity of 4. Numerical examples aQmax k cos aQ max l , ϭ (aqmaxcos cos, aq max sin). The above analysis has indicated that the behavior of KK22 the dispersion relation is likely to be rich and varied. However, examination of the solutions of Killworth et Because for the most part there is likely to be a short- al. (1997) and KB03a shows that the additional speedup wave mode with a Doppler shift based on Qmax, there of long planetary waves is more complicated than this will in general be no latitudinal cutoff for waves of a simple shift. Indeed, if the east±west group velocity (in- given frequency such as the standard result (2.28). cluding mean ¯ow and topography) is written as To illustrate the variety of behavior, we consider so- lutions from various locations. The 1998 World Ocean cg ϭ u ϩ cg, min 0 Atlas (Antonov et al. 1998; Boyer et al. 1998) was used g wherec0 is the group velocity of long planetary waves for temperature and salinity data, and the ETOPO5 da- without mean ¯ow and over a ¯at bottom, the factor taset (National Geophysical Data Center 1988) for to- is found to vary between 0.1 and 2 or more (partly pography. Treatment to produce the baroclinic (i.e., no depending on the projection of the mean u ®eld onto net vertical integral) mean ¯ows and buoyancy fre- the second undisturbed vertical normal mode). Thus quency was as in KB03a.
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FIG. 9. The dispersion relationship for zero north±south wavenumber at 20ЊS, 70ЊE. Plots as in Fig. 8. A wavenumber of Ϫ600 corresponds to a wavelength of 63 km. The nondimensional value of the bottom slope parameter, ␣/H, is Ϫ1.00. a. 45ЊN, 150ЊW ¯ow possesses an internal minimum so that the short- wave theory in section 3b becomes valid. The long wave We begin with a pair of examples outlining the effects is increased in speed by an amount approximately u and differences produced by combinations of mean ¯ow min (cf. Killworth et al. 1997), a pattern that in fact holds and bottom topography.3 Figure 6 shows the dispersion approximately for all wavenumbers. There is no fre- relation at 45ЊN, 150ЊW, which would be obtained if the bottom were locally ¯at. The dash±dotted curve shows quency maximum, and at high wavenumber the solution the barotropic mode (which has no thermodynamic sig- asymptotes to a Doppler shift by umin. nature, and is thus henceforth omitted from future so- Adding in the bottom boundary condition (which will lutions with no mean ¯ow). The dashed curve shows be the case for all later examples) for the weak slope the standard no mean ¯ow theory (2.26) for the ®rst at this location (␣/H, a nondimensional measure of the two internal modes, demonstrating the frequency max- slope parameter, is 0.33) produces considerable change imum, strongly dispersive behavior at large wavenum- (Fig. 7). The solutions without mean ¯ow (dashed) show ber and nondispersive long-wave response. Adding the a bottom-trapped mode (Rhines 1970), whose horizontal mean ¯ow completely changes the solutions. The mean ¯ow perturbation does not change sign with depth, and which disappears when the wavenumber becomes too small in amplitude. A more traditional interior mode 3 The local hydrographic data are set, of course, by a mean ¯ow (with one sign change in depth) is also visible, though consistent with the local bottom slope, rather than a ¯at bottom; however, the slope is weak here so that the comparison is worth its quantitative details have been changed by the ad- making. dition of the bottom condition. For example, the long-
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FIG. 10. The dispersion relationship for zero north±south wavenumber at 45ЊS, 100ЊW. Plots as in Fig. 8. A wavenumber of Ϫ600 corresponds to a wavelength of 47 km. The nondimensional value of the bottom slope parameter, ␣/H, is 1.32. wave speed has been reduced to 81% of its ¯at bottom without mean ¯ow, but propagating faster westward. For value. Killworth and Blundell (1999) noted that there suf®ciently short waves the frequency of this mode are large local changes in wave speed induced by bottom reaches the value Qmax (i.e., a critical layer appears) and topography alone, but its effects tend to cancel over the wave ceases to exist for larger | k | . We have ex- wide basins. amined both analytically and numerically the problem Adding mean ¯ow induces further changes. For short in the vicinity of this value of k for complex solutions, waves, the mode trapped on the umin appears. As the but none were found; it appears that, when a critical wavenumber becomes smaller, the frequency begins to layer is reached, the solution simply terminates rather increase (after 1 ϩ D has changed sign at the ¯oor, than becoming complex. which changes the behavior of the equation for F and hence the number of sign changes). For suf®ciently high frequencies, the mean ¯ow becomes less relevant, and b. 30ЊN, 30ЊW the mode becomes indistinguishable from the Rhines bottom-trapped mode in the system without mean ¯ow. Figure 8 shows results for a region with a stronger There is also a second4 mode, qualitatively like that bottom slope (␣/H ϭϪ0.83), again with an interior umin. Without mean ¯ow, there is a similar picture with a Rhines bottom-trapped mode and a series of relatively 4 As noted, there is no natural numbering system for modes in the undisturbed internal modes. With mean ¯ow included, presence of mean ¯ow since modes can change their ordering by frequency, group, or phase velocity, or number of zero crossings in there is a short-wave mode showing a Doppler shift with the vertical as wavenumber is changed. We use an informal numbering umin as before, but this coalesces with the Rhines-like for the examples, which we hope is free from misinterpretation. mode at a wavenumber just less than Ϫ400. Here there
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FIG. 11. The dispersion relationship for zero north±south wavenumber at 45ЊN, 40ЊW. Plots as in Fig. 8. A wavenumber of Ϫ600 corresponds to a wavelength of 47 km. The nondimensional value of the bottom slope parameter, ␣/H, is 3.57. are two modes whose dispersion relations appear to be mode for longer waves, is now not the Doppler-shifted crossing, that is, close to possessing the same wave- mode at smaller wavelengths, its place being taken by number and frequency. This is the classical position for the ®rst internal mode. For long waves, this mode has linear resonance (or mode conversion) to occur. There a speed similar to the equivalent mode in the absence is an extensive literature in this ®eld (cf. Vanneste 2001 of mean ¯ow. There is a third mode that only exists for for a review), but applications to planetary wave theory long waves. have mainly been con®ned to two-layer cases without mean ¯ow (cf. Hallberg 1997; Kaufman et al. 1999; Tailleux and McWilliams 2002). Linear resonance im- c. 20ЊS, 70ЊE plies a potentially strong energy transfer between modes This region (Fig. 9) possesses strong bottom slope at such locations.5 In this case, no real modes corre- (␣/H ϭϪ1) and is the location used in Figs. 3 and 4 sponding to the two under consideration could be found showing vertical short-mode structure. As before, the near the apparent crossover point (though the third mode inclusion of mean ¯ow drastically changes the disper- exists, itself terminating almost immediately thereafter). sion relation. There is a Rhines-like trapped mode that The mode pair appears to have become weakly complex coalesces with one of the two internal modes (and which in this wavenumber range (``appears'' because we have may coalesce further with the second internal mode be- not solved here for any complex modes). The crossover fore ceasing to exist). These two internal modes both means that the wave, which becomes the Rhines trapped become Doppler shifted on umin for high wavenumber, corresponding to the ®rst two modes in section 3b. In- 5 Discussion of these issues is beyond the scope of this paper. terestingly, the smaller frequency solution has an almost
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FIG. 12. The dispersion relationship for zero north±south wavenumber at 45ЊN, 170ЊW. Plots as in Fig. 8. A wavenumber of Ϫ600 corresponds to a wavelength of 47 km. The nondimensional value of the bottom slope parameter, ␣/H, is Ϫ1.81. constant phase and group velocity across the entire e. 45ЊN, 40ЊW wavenumber range, corresponding to a solution with small  in that theory. Both modes show a small speedup This location (Fig. 11) has an extremum of umin at the above the no mean ¯ow case; this speedup increases for ¯oor and has the strongest bottom slope parameter of short waves. all those considered (␣/H ϭ 3.57) and is shown because it does not behave in the manner shown in section 3c. The mean ¯ow also has an internal minimum which is d. 45ЊS, 100ЊW almost, but not quite, the same size as the extremum.
We now examine the ®rst example with a bottom umin The vertical structure of the solutions shows a concen- (Fig. 10). The bottom slope is strong and positive (␣/ tration about the internal minimum, not the ¯oor. Pro- H ϭ 1.32), permitting a rapid long wave in the absence vided that the correction to the Doppler-shifted phase of mean ¯ow, discussed by Tailleux (2003) and Kill- velocity by the solutions in section 3b is large enough, worth and Blundell (2003c). Such a wave is dependent no critical layer occurs near the ¯oor, and the solutions on the topographic slope to survive and does not prop- are vanishingly small there. As the magnitude of the agate well. The mean ¯ow is negligible for this wave, wave vector increases, ®rst the second mode and (even- and so the mode persists when mean ¯ow is added. This tually) the ®rst mode have eigenfrequencies that gen- mode becomes the single bottom-trapped mode per- erate a near-critical layer at the ¯oor, and disappear for missible by the theory in section 3b, namely, the case larger wavenumber. (The values of wavenumber con-
ഠ Qmax ϩ constant/K. A second mode exists for suf- cerned are suf®ciently high that there is no physical ®ciently long waves, 50% faster than the equivalent interest in ®nding whether the mode transfers to the ¯oor mode without mean ¯ow. for larger wavenumber.) Thus a simple map of extreme
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FIG. 13. The dispersion relationship for zero north±south wavenumber at 13ЊN, 130ЊW. Plots as in Fig. 8. A wavenumber of Ϫ600 corresponds to a wavelength of 65 km. The nondimensional value of the bottom slope parameter, ␣/H, is Ϫ0.01. Note that there is a subsurface minimum in the mean u ®eld at a depth of about 30 m, which is not well resolved numerically by the 10-m grid employed. This leads to numerical errors in computing group velocity for the high-wavenumber cases with mean ¯ow; these have been replaced by a short dashed line.
values of umin and their location (e.g., Fig. 1) does not g. 13ЊN, 130ЊW tell the entire story. This case (Fig. 13) is again included to demonstrate that care must be taken with interpretation of solutions. f. 45ЊN, 170ЊW The mean ¯ow actually possesses a minimum at about We now show a case (Fig. 12) with an unambiguous 30 m, though this is hard to see on the scale of the diagram. Even this small depth is enough to produce bottom umin and with a negative slope parameter (␣/H ϭϪ1.81), which permits two bottom-trapped modes solutions trapped around the minimum; the pseudobar- (section 3b). The higher frequency of the pair becomes otropic solution (3.2) would possess a critical layer. The a Rhines mode at smaller wavenumber, similar to the mode produced (which has a very small correction to case without mean ¯ow. It tends to Qmax plus a constant the Doppler shift) exists over the entire range of wave- offset in frequency. The mode with the smaller fre- number, though the 10-m grid used in integration was quency (which appears as a normal internal mode for insuf®ciently ®ne for the group velocity to be computed long waves, though faster than the case of no mean accurately for high wavenumber; the short dashed line
¯ow), also tends to Qmax but with an offset decaying as shown is the obvious solution. Another Rhines-like 1/K. mode exists for a short range of wavenumber both with
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FIG. 14. The dispersion relationship for zero north±south wavenumber at 20ЊS, 0Њ. Plots as in Fig. 8. A wavenumber of Ϫ600 corresponds to a wavelength of 63 km. The nondimensional value of the bottom slope parameter, ␣/H, is Ϫ0.03. and without mean ¯ow; in the former case it ceases to rich variety of dispersion relationships possible in the exist when it meets the other solution. World Ocean, even for the restriction of zero north± south wavenumber. In no case does the relationship re- semble the traditional result (2.26), in terms of either h. 20ЊS, 0Њ the dispersion relationship or the vertical structure. For The ®nal example possesses a genuine minimum in long waves, the phase and group velocities are typically the u ®eld at the surface. With mean ¯ow, there are greater than the traditional theory (cf. KB03a), but not precisely three solutions (Fig. 14). The main one is the everywhere; there is at least a shift of frequency by pseudobarotropic mode (recall that this solution is spec- Qmax, however, to avoid a critical layer. For short waves, i®ed by umin). The second is a Rhines-like mode, which there may be no solutions; or there may be several so- ceases to exist for suf®ciently high wavenumber; the lutions, either trapped about an internal Qmax or trapped third is an internal mode, slower than its equivalent at a bottom extremum of Qmax. Solutions can cease to without mean ¯ow, which only exists in the long-wave exist abruptly [as with the Rhines (1970) solutions for end of the spectrum. no mean ¯ow], or can become complex, or can change their vertical structure qualitatively as well as quanti- tatively as the wavelength changes. There is usually no 5. Discussion frequency cutoff, with the Doppler shift at short waves These solutions show the planetary wave dispersion providing an asymptote until the assumption of small relation for a physically relevant, potentially observable frequency becomes invalid. wavenumber regime. These examples have shown the Many of the waves described here are trapped in the
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vertical, with a length scale varying as at least K Ϫ1 for wЈ w cot z ഠഠ Յ O(10Ϫ2 ) (A.1) short waves. Suf®ciently short waves trapped at mid- Ј Ha depth and at the ocean ¯oor would thus be invisible to t remote sensing devices, and are unlikely to be contrib- for any realistic values, where the linearized potential uting to the evidence for planetary wave speedups. The vorticity equation w z ϭ cot /a has been used to es- question of which of the modes presented here and in timatew . Thus the advection by the mean vertical ve- Part II would possess adequate surface signatures is im- locity can be neglected. portant, but will not be discussed further here. Despite the large variability in behavior, however, cer- b. Large wavenumber, R → 0 tain features persist: the requirement of at least a Dopp- When the wavenumber becomes large, the balance ler shift by Qmax to avoid critical layers; the lack of dispersion of short waves; the ability of waves to prop- (A.1) is disturbed both because R becomes small and agate at latitudes beyond their traditional cutoff values; because the vertical length scale of the perturbation be- and the disappearance or coalescence of modes, which comes small. This vertical length scale, from (3.13), is may be a source of modal energy or scattering. when is O(1) whether ␦ is large or small so that the We have not addressed why the location where Q dimensional vertical scale h ϭ 1/KA1/2 ϭ af/NK. The reaches a maximum should matter to the short-wave part ratio (A.1) then becomes of the spectrum. If this location is at middepth, so that wЈ w the maximum is a true maximum and not an extremum, z ഠ . RЈ Rh then kdim ´ u z ϭ 0 at this point. From thermal wind, this ١ ϭ 0 at such a point, so that k Now can be written k ϫ ١ are parallel. Thus particle trajectories, which lie and à af af  af33 af 2 along lines of constant phase, are normal to k and hence Rh Ն ഠഠ␥ ഠ lie along lines of constant density. To leading order, KNK NK2322 A N K NK then, the ¯ow is insensible to any degree of strati®cation. after simpli®cations and use of the term de®nitions. As the wavenumber (and frequency) rise, the dominant Thus term in (2.10) becomes the ®rst, implying R ϭ 0 at the maximum point. Because the last term in (2.10) is non- wNKH22 NK cot Յ w ഠ . zero, vertical adjustments have to be made as demon- Rh af222 a f strated by the asymptotic solution. It seems harder to make similar heuristic arguments for a bottom maximum This is much less than unity provided ١ are not now af here an extremum), because k and) K K ഠ 103 parallel, and the bottom slope plays an intrinsic role in (H N)1/2 the dynamics. for realistic values. This requires dimensional wave- Acknowledgments. This work is part of the James lengths much larger than 36 km. This is not a strong Rennell Division Core Project CSP1. Victor Shrira pro- restriction on wavelength, as other effects would come vided valuable discussions on short-wave theory, and into play at such very short wavelengths. comments by the referees, and by Bill Dewar and Roger Samelson, were most helpful. APPENDIX B
APPENDIX A The Second Bottom-Trapped Short-Wave Mode We pose (3.21) and (3.22), but now require the cor- The Neglect of Vertical Advection by the rection to the Doppler shifted frequency to be Mean Flow ␦ ϭ Ä . (B.1) 1/2 In (2.15), the termwЈz is neglected against wЈ z with- KA out proof. To demonstrate this, we consider two cases. The problem, to leading order in K Ϫ1 in terms of the stretched coordinate simply omits the terms in fre- a. R/ϭO(1) quency; that is, F Ϫ F Ϫ F ϭ 0 (B.2) Provided that the wavenumber is not too large (i.e., that the waves are not too short), then the term R is with solution of the same order as frequency, and the vertical scale F ϭ (). (B.3) for the perturbations is H. Then, in (2.15), considering K 1
the termwЈzt against the time derivative termЈ , we We de®ne a further stretched coordinate, describing a have layer within the (outer) layer, by
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plus the requirement of matching to the outer solution ϭ , (B.4) K (B.3) for large . (The value of 1 ϩ D can change sign within this inner layer, unlike the solution given pre- and, after algebra, the problem becomes viously.) This problem possesses a formally small last 1 N 2(ϪH) cos term in both equation and boundary condition. The ( ϩ Ä )F Ϫ F Ϫ Kaf2 2sin␣A 1/2 problem is solved either by use of matched asymptotic expansions, or more easily by inspection. To leading cos2 order, (B.5) requires ϩ sin2 2 cos ( ϩ Ä )F Ϫ F ഠ 0, (B.7) ϫ 1 ϩ ( ϩ ) F ϭ 0. (B.5) A1/22 cos ⍀ whose solution is The boundary condition becomes F ϭ B ϩ B ( ϩ Ä ),2 (B.8) 12 cos2 ϩ sin2 where B and B are constants. This must match with 2 1 2 cos the outer solution (B.3), which for small is unity to 21/2 1 ϩ F ϩ KA ␣F ϭ 0, A1/22⍀ cos leading order. Thus B1 ϭ 1, B 2 ϭ 0. The next order can then be evaluated by substituting F ϭ 1 in (B.5) and ϭ 0, (B.6) integrating, giving
cos2 ϩ sin2 2 1 cosN 2(ϪH) cos P( ϩ Ä ) F ϭϪ1 ϩ ( ϩ Ä ) ln( ϩ Ä ) Ϫ , (B.9) Kaf2 2sinAA 1/2 1/22⍀ cos K 2
where P is an unknown constant. Letting → ϱ, this where ␥ is Euler's constant 0.5771. From the de®nition must match with the limit of the outer problem to next of A, the ln terms match automatically, and P ϭ ␥ order, which is ϩ ln2. 1 The bottom boundary condition then yields the re- [Ϫ(␥ ϩ ln2) ϩ ln] K mainder of the dispersion relation as
cos2 cos2 ϩ sin2 ϩ sin2 2 2 cos cosN 2(ϪH) cos 1 ϩ Ä ϩ ␣A1/2 Ϫ1 ϩ Ä lnÄ Ϫ A1/2(␥ ϩ ln2)Ä ϭ 0. A1/22⍀ cos af 2sinAA 1/2 1/22⍀ cos (B.10)
This is of the form, recalling that cos Ͻ 0, strati®ed ocean of variable depth. Part 2: Continuously strati®ed ocean. J. Fluid Mech., 388, 147±169. 1/2 1 Ϫ X12Ä ϩ ␣A [X ϩ Ä lnÄ Ϫ (␥ ϩ ln2)Ä ] ϭ 0, Boyer, T. P., S. Levitus, J. Antonov, M. Conkright, T. O'Brien, and C. Stephens, 1998: Salinity of the Atlantic/Paci®c/Indian Ocean. Vols. (B.11) 4±6, World Ocean Atlas 1998, NOAA Atlas NESDIS 30, 166 pp. where X and X are positive constants. Chelton, D. B., and M. G. Schlax, 1996: Global observations of 1 2 oceanic Rossby waves. Science, 272, 234±238. de Szoeke, R. A., and D. B. Chelton, 1999: The modi®cation of long REFERENCES planetary waves by homogeneous potential vorticity layers. J. Phys. Oceanogr., 29, 500±511. Antonov, J., S. Levitus, T. P. Boyer, M. Conkright, T. O'Brien, and Dewar, W. K., 1998: On ``too fast'' baroclinic planetary waves in the C. Stephens, 1998: Temperature of the Atlantic/Paci®c/Indian general circulation. J. Phys. Oceanogr., 28, 1739±1758. Ocean. Vols. 1±3, World Ocean Atlas 1998, NOAA Atlas NES- Fu, L. L., and D. B. Chelton, 2001: Large-scale ocean circulation. DIS 27, 166 pp. Satellite Altimetry and Earth Sciences, L. L. Fu and A. Cazenave, Bobrovich, A. V., and G. M. Reznik, 1999: Planetary waves in a Eds., Academic Press, 133±169.
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