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 v 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) 5 (u ,y ,w ) relative to E-mail: [email protected] longitude l, latitude u, and depth z. The Coriolis fre- q 2004 American Meteorological Society Unauthenticated | Downloaded 09/28/21 08:30 PM UTC DECEMBER 2004 KILLWORTH AND BLUNDELL 2693 quency f 5 2V sinu, where V 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 iRy9 ilM kRM or fronts (cf. Owen et al. 2002). The wave perturbations, u952zzz 2 52 1 1small (2.8) denoted by primes, are small amplitude relative to the af f af af 2 cosu mean and vary horizontally with a length scale Lwave K and Lbasin by assumption. The horizontal momentum equa- tions become essentially quasigeostrophic: ikMzzz iRu9 ikM lRM y95 1 5 1 1small (2.9) 2 u y M a cosu facosu af u91 u91 u92 f y952zl 1small (2.1) t a cosu luaacosu de®ne u9 and y9 in terms of M. The second terms are already order (v/ f ) smaller than the ®rst (geostrophic) and and can thus be neglected when they occur in this form u y M (but not when a divergence operator is used). The per- y91 y91 y91 fu952zu 1small, (2.2) t a cosu luaa 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 52 u 2 (y cosu) (2.10) z a cosu lua cosu p9 M 5 (2.3) z r ik ilM kRM 0 52 2zz 1 for convenience in what follows. In (2.1) and (2.2), a cosu12af af 2 cosu ``small'' means either of order v/| f | smaller or of order 1 ikMzz lRM cosu Lwave/Lbasin. 21 Following normal WKBJ theory, we write a cosu12af af 2 u M 5 F(z; l, u) expi(kl 1 lu 2 vt), (2.4) 2 ikMzz iRM k where we assume that there is a rapid phase variation 52 1l2 1 small (2.11) 2Va22sin u af 22212cos u in the horizontal on the scale Lwave described by expi(kl 1 lu 2 vt), where k 5 (k, l) is the local wavenumber, so that and v the wave frequency. The vertical structure of M is described by F, which also varies slowly in the hor- ikM iRM k2 w 521l2 izontal, on the scale L . The wavenumber k is de®ned 22 222 basin 2Va sin u af12cos u in terms of longitude and latitude, and so is dimen- ikM sionless. The dimensional equivalent, that is, in units of 5 (1 1 D), length21,is 2Va22sin u kl where kdim 5 , . (2.5) 12a cosu a Rk2 D 52 1l2 , 0 (2.12) Then we de®ne 2Vk12cos2u ku ly Q 515k ´ u and (2.6) as R , 0, k , 0. (The latter is by assumption since a cosu a dim westward propagation is almost entirely what is ob- served in the ocean.) The term (1 1 D) is approximately R Q (2.7) 5 2 v 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 1 D) can vanish, frequency. when Rk/ f is of order unity, or køO(| f |/v) k 1. (For Precisely as in appendix A of KB03a, it can be shown an annual frequency, this would involve wavelengths of that R , 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 Unauthenticated | Downloaded 09/28/21 08:30 PM UTC 2694 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 p952z gr9, (2.13) termwr9z is neglected against w9r z on scaling grounds, assuming the perturbation ®elds have depth scales of becomes order H, the ocean depth. This will not always be the r case for short waves. In fact, thewr9 term continues to r952 0 M9 . (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 (urll91u9r )(yr uu91y9r ) 1 g r91t 1 a cosu a fu 52 pu and fuz 5 ru and (2.16) ar 00ar 1 w9r 1 wr950. (2.15) zz 1 g f y 5 p and f y 52 r , (2.17) Immediately there is a potential dif®culty in that the last a cosur l z a cosur l 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.