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  • Killworth, P. D., D. B. Chelton, and R. A. de Szoeke, 1997: The speed of observed and theoretical long extratropical planetary waves. J. Phys. Oceanogr.,27, 1946–1966.

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  • Rhines, P. B., 1970: Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid Dyn.,1, 273–302.

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  • Ripa, P., 1978: Normal Rossby modes of a closed basin with topography. J. Geophys. Res.,83, 1947–1957.

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  • Straub, D. N., 1990: Some effects of large scale topography in a baroclinic ocean. Ph.D. thesis, University of Washington, 164 pp.

  • ——, 1994: Dispersion of Rossby waves in the presence of zonally varying topography. Geophys. Astrophys. Fluid Dyn.,75, 107–130.

  • Suarez, A. A., 1971: The propagation and generation of topographic oscillations in the ocean. Ph.D. thesis, MIT/Woods Hole Oceanographic Institution, 195 pp.

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  • Welander, P., 1959: An advective model of the ocean thermocline. Tellus,11, 309–318.

  • White, W. B., 1977: Annual forcing of baroclinic long waves in the tropical North Pacific. J. Phys. Oceanogr.,7, 50–61.

  • View in gallery

    . 1. The buoyancy frequency N(z) typical of the North Pacific (s−1).

  • View in gallery

    (a) The bounds on phase speed ratio c/cflat for the five major ocean basins, implied by their basin-averaged stratification, as a function of depth. (b) The bounds computed for four subareas of the North Pacific, arranged in the SW, SE, NW, and NE of the basin. The longitude ranges are 140°–160°W (SW, NW) or 150°–130°E (SE, NE); the latitude ranges are 10°–20°N (SW, SE) or 30°–40°N (NW, NE).

  • View in gallery

    Contours of the speed ratio c/cflat for the case of topography varying with latitude only, as a function of tanθHθ and H, for the North Pacific.

  • View in gallery

    Results using ray theory for the case of a Gaussian hill of minimum depth 2000 m, centered on 20°N, 140°W with width 15° in both directions, with a background depth of 4000 m except for a slope near the eastern boundary. (a) The ray trajectories themselves, with rays initially spaced for clarity at 1/2° apart (a closer spacing of 1/8° was used to compute the data for all diagrams without rays). (b) The westward phase velocity (the westward group velocity is almost identical in most regions, and is not shown). (c) The phase speed ratio c/cflat, with contour interval 0.1; values less than unity are dashed; unity is dash–dotted. (d) The phase ϕ.

  • View in gallery

    As for Fig. 4 but for a hill of minimum depth 1500 m. A caustic is now formed on the poleward side of the hill. The location of the caustic is shown by letter C’s.

  • View in gallery

    As for Fig. 4 but for a depression with maximum depth 6000 m (the flat ocean depth remains 4000 m). (a) Westward phase velocity, (b) the ratio c/cflat (contours as Fig. 4), and (c) the phase ϕ.

  • View in gallery

    Results for the North Pacific. (a) Ray trajectories, using an initial spacing between rays of 1/4° with C’s marking caustics and D’s marking regions where rays enter depths less than 1000 m, beyond which a ray is not followed (again, a closer spacing of 1/8° was used to compute the data for all diagrams without rays); (b) the phase ϕ of the solution; (c) the computed westward phase speed; and (d) the ratio of this to the local flat-bottom phase speed (contours as in Fig. 4). (e) The smoothed topography used in the computations.

  • View in gallery

    Results for the South Pacific. (a) Ray trajectories, (b) the phase speed ratio (contours as in Fig. 4), and (c) the smoothed topography used.

  • View in gallery

    Results for the North Atlantic. Details as for Fig. 8.

  • View in gallery

    Results for the South Atlantic. Details as for Fig. 8.

  • View in gallery

    Results for the southern Indian. Details as for Fig. 8.

  • View in gallery

    The ratio of the values of the longitudinal averages of phase speed c and flat-bottom phase speed cflat as a function of latitude, over the five ocean basins considered. Values are not shown within 5° of the equator, where midlatitude planetary wave theory would not hold, or beyond 50°, where annual waves would not occur.

  • View in gallery

    The ratio of the values of the longitudinal averages of phase speed c and flat-bottom phase speed cflat, as a function of latitude, using the longitude bands defined by CS. Solid symbols show the ratios observed by CS, and open symbols show the topographic theory. Circles indicate Pacific values, squares Atlantic and Indian values.

  • View in gallery

    Values of surface height amplitude η0 for the five basins, with an arbitrary uniform initial amplitude at the eastern boundary: (a) the North Pacific, (b) the South Pacific, (c) the North Atlantic, (d) the South Atlantic, and (e) the southern Indian. Because of the wide variation, contour intervals are nonconstant. They are 0.2–2 by 0.2; 2.5–4 by 0.5; 5, 6, and 8; 10–40 by 5.

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The Effect of Bottom Topography on the Speed of Long Extratropical Planetary Waves

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  • 1 Southampton Oceanography Centre, Southampton, United Kingdom
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Abstract

This paper examines how slowly varying topography induces changes in all aspects of long planetary wave propagation, including phase speed and surface signature, through steering effects. The approach introduces a method for the exact solution of the vertical topographic eigenvalue problem for arbitrary realistic stratification and ray theory in the horizontal. It is shown that, for observed stratifications, first internal mode topographic waves have phase speeds between about 0.4 and twice the local flat-bottom phase speed. Increases occur on the western and equatorward sides of hills. Focusing of ray trajectories and caustics are common features of the solutions. Despite a bias between slowdown and speedup, on average there is little speedup except in high latitudes (where long-wave theory is less applicable). Calculations are performed for five main ocean basins, assuming waves are generated at the eastern coastline, using smoothed topography. These calculations confirm the above findings: there are significant local effects on wave speed, but these largely cancel over the basin scale. Thus, topographic effects cannot explain recent observations, which demonstrate long planetary waves propagating about twice as fast as linear theory. The presence of mean flow, which induces changes to the planetary vorticity gradient, remains the prime candidate for the observed speedup.

Corresponding author address: Dr. Peter D. Killworth, James Rennell Division for Ocean Circulation, Southampton Oceanography Centre, Empress Dock, Southampton SO14 3ZH, United Kingdom.

Email: P.Killworth@soc.soton.ac.uk

Abstract

This paper examines how slowly varying topography induces changes in all aspects of long planetary wave propagation, including phase speed and surface signature, through steering effects. The approach introduces a method for the exact solution of the vertical topographic eigenvalue problem for arbitrary realistic stratification and ray theory in the horizontal. It is shown that, for observed stratifications, first internal mode topographic waves have phase speeds between about 0.4 and twice the local flat-bottom phase speed. Increases occur on the western and equatorward sides of hills. Focusing of ray trajectories and caustics are common features of the solutions. Despite a bias between slowdown and speedup, on average there is little speedup except in high latitudes (where long-wave theory is less applicable). Calculations are performed for five main ocean basins, assuming waves are generated at the eastern coastline, using smoothed topography. These calculations confirm the above findings: there are significant local effects on wave speed, but these largely cancel over the basin scale. Thus, topographic effects cannot explain recent observations, which demonstrate long planetary waves propagating about twice as fast as linear theory. The presence of mean flow, which induces changes to the planetary vorticity gradient, remains the prime candidate for the observed speedup.

Corresponding author address: Dr. Peter D. Killworth, James Rennell Division for Ocean Circulation, Southampton Oceanography Centre, Empress Dock, Southampton SO14 3ZH, United Kingdom.

Email: P.Killworth@soc.soton.ac.uk

1. Introduction

The advent of satellite altimeters has had an immediate effect upon the theory of ocean wave propagation. Beginning with analysis by Chelton and Schlax (1996, hereafter CS), there have been many papers (e.g., Glazman et al. 1996; Cipollini et al. 1997, 1999) showing two distinct facts: first, that westward propagating baroclinic Rossby, or planetary, waves do exist and are observable from their surface signature and, second, that over significant fractions of ocean basins, the average phase speed of these waves is a factor of 2 or so larger than linear wave theory (Dickinson 1978; Gill 1982) would predict. Since these waves provide the main mechanism whereby the ocean responds at one location to changes elsewhere, it is important that we understand why this response occurs so much faster than classical theory would predict.

Killworth et al. (1997, henceforth KCS) discussed four mechanisms, all of which could be acting; in each mechanism, one or more of the assumptions of traditional linear theory has broken down. In their paper, they examined one of these mechanisms in detail, namely that the background state of the ocean involved existing flows, especially east–west, with speeds of the same order as those of the planetary waves themselves. These background flows induce a change to the potential vorticity gradient that can severely modify the wave propagation speed. KCS were able to show that this mechanism accounted successfully for observed phase speed changes in the Northern Hemisphere, but was only partly successful in the Southern. A three-layer model by Dewar (1998) has helped to explain why the predominant change is a speedup of the waves.

It is thus necessary to reexamine the other mechanisms to see which might be acting. The first mechanism KCS listed was that the response was for a forced, and not a free, system. This, as White (1977) and Meyers (1979) note, means that a wave of form sin(kxωt), with wavenumber k in the x direction and frequency ω in time t, can be forced by local winds with the same frequency. The response to such forcing, with a suitable boundary condition at the east coast, can be written as cos(kx/2 − ωt) sinkx/2, which appears to have a phase speed twice the original. This possibility has been discussed in detail, including the effects of decay terms, by Qiu et al. (1997). However, time–space diagrams of such solutions possess nodes at 2π/k intervals in x, whereas equivalent diagrams from satellite imagery consistently show propagation westward undisturbed by zeros; transform methods similar to those used on the data also fail to yield double phase speed estimates when applied to this simple model.

Another mechanism rejected by KCS was that of a nonlinear, rather than a linear, response. Indications from observations still fail to suggest any clear nonlinearity in the signal, so this mechanism can probably continue to be rejected.

This leaves the possibility that the varying ocean depth could act to speed up the planetary waves. KCS rejected this possibility, arguing that while wave steering by topography would certainly occur, it did not seem likely that such steering would preferentially act to speed up rather than slow down the wave propagation. However, despite the existence of solutions for special cases, there has been little work on the general problem, which is algebraically tedious, so that the possibility of phase speed increase by topographic steering remains open and will be investigated here.

Work on the steering and modifications to barotropic planetary waves has been extensive (e.g., Ripa 1978), and the key role played by the characteristics of the resulting equations, namely contours of depth divided by Coriolis parameter, are well understood. The modification of baroclinic planetary waves is understood only for a number of special cases, involving idealized stratification and topography. Rhines (1970) examined the dispersion relation for baroclinic planetary waves on a sloping bottom. This work was slightly extended by Suarez (1971) and Charney and Flierl (1981) to general topographic slopes, but concentrating on the dispersion relation and the vertical form of the baroclinic mode. Straub (1990, 1994) extended this work considerably, providing full details of vertical mode structure, as well as (1990) some discussion of ray paths for waves that were not necessarily long. There is evidence (Tokmakian and Challenor 1993) that wave speeds increased by 50% from east to west across the Mid-Atlantic Ridge, with a corresponding decrease in wavenumber by a factor of 3. Hallberg (1997) examined topographic coupling in a two-layer ocean and examined mode changes during the propagation of a wave.

In this paper we shall examine the propagation of unforced long baroclinic planetary waves, generated at an eastern boundary, as they move across a slowly varying topography. The work will frequently concentrate on phase velocity rather than group velocity, simply because this is the quantity most readily observed from altimetric data (although the westward components will turn out to be very similar). Hitherto, solutions for topographically steered planetary waves have involved simple vertical structures. Here we show how the problem may be solved semi-analytically in the vertical for arbitrary topography, producing a dispersion relation which can be used with the WKBJ, or ray, approximation in the horizontal (sections 2 to 5). An immediate finding is that the ratio of westward phase speed to the local flat-bottomed phase speed (e.g., that computed by Chelton et al. 1998 or KCS) lies between about 0.4 and 2. This immediately provides a potential bias in phase speed, in that the ratio can take values rather more above unity than it can below, giving an average that can lie above unity. (It will be shown, however, that this net speedup does not occur in most areas.) Caustics can appear in the ray theory used, beyond which ray theory fails. These caustics can be located numerically so that actual ocean topography (section 6) can be studied and maps of phase speed produced. Values are compared with observations in section 7.

The horizontal distribution of surface energy of waves generated at the eastern boundary is also of interest, and this is studied in section 8. We conclude with a discussion.

2. The model

We seek solutions for unforced long waves propagating in an ocean whose stratification varies vertically but not horizontally. This choice permits us to concentrate solely on the effects of topography variation for this paper (we show below that many topographic effects do not depend strongly on lateral stratification variation). Thus, the buoyancy frequency is N = N(z) only, where z is measured vertically upward, with the ocean surface at z = 0 and the floor at z = −H(λ, θ), where (λ, θ) are respectively longitude and latitude.

The longest scales present are assumed to be those of the topography and will implicitly be on the planetary scale. The scale of the waves will also be assumed long, in the combined sense that the waves are (i) long enough to be (planetary) geostrophic to leading order while (ii) much smaller than the topographic scale. This assumption will permit a WKBJ (or ray theory, or geometric optics) analysis of the wave propagation. Such assumptions are usually made for wave propagation in varying media, and the regions of their validity vary from case to case. In this case, clearly topography possesses variability on all possible horizontal length scales. We shall use a smoothed topographic representation in what follows and regard roughness on shorter length scales than the smoothing to be noise, while bearing in mind that the roughness may well be an efficient energy scatterer or, indeed, a source.

It should also be noted that WKBJ theory implicitly requires no scattering between propagating wave modes. Both Anderson and Killworth (1977) and Barnier (1988) found energy leakage from the baroclinic to the barotropic mode as a planetary wave passed over a ridge in a two-layer system, although in a wind-driven, and not free, configuration. Tailleux and McWilliams (1999, manuscript submitted to J. Phys. Oceanogr.; hereafter TM) extend these analyses to forced and unforced problems in a two-layer ocean and find results that are sometimes in conflict with those given here. However, as we shall note below, solutions using simplified vertical structures can give misleading results when compared with the exact continuously stratified problem, so comparison of the results is nontrivial.

The assumption of planetary geostrophy, while convenient, eliminates existing features such as turning latitudes that rely on waves shortening as they move poleward. Appendix B shows how one would relax the long-wave assumption, although the algebra would become distinctly more awkward. For the density structures considered here, appendix B also shows that turning latitudes occur at about 45°–48° (calculations follow CS in extending to 50° where possible) so that the long-wave assumption breaks down in subpolar regions even without the effects of topography.

It is convenient to use Welander’s (1959) formulation in terms of a single quantity, M; write
pρ0Mz
where p represents pressure and ρ the density with ρ0 a reference value. Then the geostrophic velocity field is given by
i1520-0485-29-10-2689-e2-2
where a is the radius of the earth, and f = 2Ω sinθ is the Coriolis parameter; Ω is the earth’s rotation rate. The density is given by
i1520-0485-29-10-2689-e2-5
and the (time dependent) equation for density conservation is a single equation for M:
i1520-0485-29-10-2689-e2-6
where an overbar denotes background values. This equation has two vertical boundary conditions. At the surface, neglecting Ekman pumping since we are interested in free waves,
wMz
while at the floor, no normal flow becomes
wuHzHMλθMHθMHλzH.
The set (2.6)–(2.8) forms a linear system for the time evolution of M, assuming the existence of suitable sidewall boundary conditions (themselves a difficult problem; cf. discussions by Huang 1986). Here we assume an oscillatory behavior at constant frequency ω so that
Miωt
To proceed, we appeal to WKBJ ideas and seek a solution like a first internal mode in the vertical (with a shape that varies slowly in the horizontal), plus a phase which varies rapidly. Explicitly, write
MFλ, θ, ziϕλ, θωt
where F is a shape function in the vertical with slow horizontal variation and ϕ is a phase function assumed rapidly varying compared with F; its gradients, however, change at the same rate as F. The problem will prove linear between ω and ϕ, in the sense that rescaling ω leads to an equal rescaling of ϕ due to the assumption of geostrophy. For definiteness, we choose a frequency of one cycle per year in what follows. Clearly higher frequencies, or correspondingly shorter waves, would eventually lead to a breakdown of these assumptions.
Substitution into (2.6) gives a second-order eigenvalue problem for F,
i1520-0485-29-10-2689-e2-11
plus boundary conditions
i1520-0485-29-10-2689-e2-12
Since ray theory will be used below, it is also convenient to define horizontal wavenumbers
kϕλlϕθ
in terms of which (2.11) and (2.13) become, respectively,
i1520-0485-29-10-2689-e2-15

The shape of (2.11) is that of the standard normal mode problem (e.g., Gill 1982), in which the east–west wavenumber ϕλ appears as an eigenvalue and plays the role more normally taken by the internal wave speed. In addition, it is of Sturm–Liouville form, a fact which will be crucial for the solution method in appendix A. For the ray theory that follows, a rapid and accurate solution of (2.11) to (2.13) is required to provide a dispersion relation relating frequency ω to the wavenumbers k, l, and H and its gradients. Two choices present themselves: The first is to use an approximate solution to (2.11) that possesses similar properties to the exact (and, in general, numerical) solution; the second is to use a numerical solution throughout. We shall make use of both in what follows.

Before proceeding to these solutions, we close the problem with an eastern boundary condition. For the free waves considered in this paper, we follow most authors and assume the waves are produced from a wavemaker on the eastern boundary, which permits waves to travel rapidly along the coast and so have almost constant phase there. Other boundary conditions could be employed but would need physical justification: for example, to start the problem in midbasin would require a specification of the phase there somehow. There may well be other sources for the waves (there is considerable evidence of amplification of at least the surface expression of waves across topography, possibly suggesting modal interactions such as those of TM; however, we note below that surface signatures are not necessarily well correlated with the wave energy). Whatever the boundary condition, coherent wave propagation is observed across entire basins from the eastern to the western boundary; compare Cipollini et al. (1996). Specifically, we set
ϕλλeθ
where λe(θ) is the longitude of the eastern boundary. We will usually take this to be where the depth reaches some given value, for simplicity. In this formulation, waves propagate westward with a phase velocity
i1520-0485-29-10-2689-e2-18
In order to find out how the westward phase speed c varies, properties of (2.11) to (2.13) must be determined. Now (2.15), together with (2.12) and an arbitrary scaling condition, such as
Fzz
has oscillatory solutions in z. [In fact, the full solution will have an amplitude a0(λ, θ), which will be derived from ray theory; discussion will be given later.] This means that as the quantity
i1520-0485-29-10-2689-e2-20
varies1 upward from zero, zeros of Fz(−H) and F(−H) interlace, beginning with the former. In particular, and of relevance here, F(−H) and Fz(−H) vary smoothly with μ, and there are (smallest possible) values of μ, namely μ1, μ2, μ3, such that
i1520-0485-29-10-2689-e2-21
Solutions for which μ1 < μ < μ3 can be termed the “first internal mode.” Their properties are discussed in the special case of constant N and topography varying north–south by Rhines (1970). Much of his work applies to the general case also. We first note that the boundary condition (2.16) requires a linear combination of F and Fz to vanish at the ocean floor. Whether or not (kHθlHλ) vanishes for μ in the range (μ1, μ3), it is straightforward to show that for some value of μ in this range, (2.16) is satisfied (and this is the lowest such μ possible).
If the ocean is locally flat, then this zero occurs at μ = μ2 by definition. In all other cases, μ is not μ2, but lies in the range (μ1, μ3). From (2.18) and (2.20), then, we have
i1520-0485-29-10-2689-e2-22
where cflat is the westward phase speed of a wave traveling across a locally flat ocean with the same depth. Thus,
i1520-0485-29-10-2689-e2-23
where the first factor is less than unity and the second is greater than unity. Both limits can be approached arbitrarily closely; examples will be given below. Note that the range depends only on H.

In general, the upper limit (μ22/μ21) is rather more above unity than the lower limit is below unity. Thus, if the topography in some region varies in a manner in which solutions filling the above range are achieved, then the average of the speed ratio c/cflat in that region will be above unity. In other words, the existence of topography can apparently act to speed up baroclinic planetary waves.

We now consider three approaches to the derivation of the dispersion relation: one simple approximation, one apparently better, and the exact solution.

a. The sinusoidal approximation

As an example of the range that can be achieved, we consider an approximate solution to (2.11). This uses a simplification of the WKBJ expansion of the vertical problem, taking account only of phase variations. Chelton et al. (1998) have shown this to give accurate eigenvalues—to within 5%—for the flat-bottomed problem. The full WKBJ-like solution adds an amplitude variation and will be dealt with next. We write, approximately,
i1520-0485-29-10-2689-e2-24
This will be termed the sinusoidal approximation. If the WKBJ assumptions were well satisfied in the vertical (i.e., if the solution oscillated rapidly), then this would be an excellent approximation (and indeed is for higher vertical modes). For the first mode, (2.24) is only completely accurate for constant N, a case discussed fully by Straub (1994). In particular, he discusses the change in vertical structure F with wavenumber size and orientation, a feature we shall almost entirely ignore henceforth. For other profiles, the error is small if, from substitution in (2.15),
i1520-0485-29-10-2689-eq1
where all quantities are evaluated at the floor. Thus, the approximate solution is valid provided, roughly, N is varying slowly at the ocean floor, which is frequently the case in the deep ocean. For typical stratification profiles, however, the condition is violated. Figure 1 shows a typical basin-averaged profile of buoyancy frequency (for the North Pacific). The changes in sign of Nz at depth may be induced by the wide spacing of the Levitus datasets (Levitus and Boyer 1994; Levitus et al. 1994.)
Although the approximate sinusoidal solution (2.24) may or may not be quantitatively accurate, it does share the behavior of the exact solution regarding the distribution of zeros. To see this, define a measure G of the speed of the first internal flat-bottom wave mode (whose speed in this approximation is G/π), where
i1520-0485-29-10-2689-e2-25
G typically takes values between 5 and 9 m s−1 as the depth H changes from 1000 to 4000 m.
Substitution from (2.16) and setting
qμG
gives a dispersion relationship
i1520-0485-29-10-2689-e2-27
When the bottom is flat, q = for the mth internal mode; we concentrate on the first mode only as this is the predominant mode observed in satellite data.

As Straub (1994) notes, if the rhs of (2.28) vanishes because the phase is aligned along constant depth (or constant G) contours, then q = π also, and the same dispersion relation occurs. However, the waves are then dispersive because ∂ω/∂kω/k.

For nonconstant topography, (2.28) permits multiple solutions. The types of solutions possible are discussed by Rhines (1970), Straub (1994), and Hallberg (1997). The lowest vertical mode—if it exists—exchanges sin for sinh in (2.24) and can take a form varying from bottom trapped through to near barotropic, depending upon the parameters. All other modes are baroclinic, and rely on the fact that tanq/q can take all values between ±∞. The solution corresponding to the first (i.e., lowest) internal mode would have q = π for a flat bottom, and F would vanish top and bottom as in normal theory. When there is topography, q moves away from π. But because tanq/q can take any value for q lying in the range (π/2, 3π/2), the first internal mode satisfies
i1520-0485-29-10-2689-e2-29
These limits immediately imply that
i1520-0485-29-10-2689-e2-30

The result (2.31), a particular example of (2.23), was deduced by Rhines (1977) for the special case of north–south topographic slope and constant N. He notes (as we shall see explicitly) that the value 4 is usually an overestimate. Nonetheless, (2.31), or more accurate versions below, implies an asymmetry in the observed phase speeds of planetary waves. For small amplitude topography, one might imagine that q would deviate by only small amounts from its flat-bottom value of π, so local estimates of phase velocity would themselves vary around the flat-bottom value without a bias toward higher or lower values. However, larger amplitude topography—or, equivalently, larger latitudes, increasing the rhs of (2.28)—could be expected to yield phase velocities biased toward values larger than the flat-bottom case. Indeed, G and its horizontal derivatives are in general of the same order so that tanq/q in (2.28) will take values of order unity or more, moving q a finite distance away from π.

b. The WKBJ approximation

The above approximation was convenient in that it shared the interlacing property of the zeros that the exact solution possesses. An apparently logical improvement is to take the leading order WKBJ solution by writing
i1520-0485-29-10-2689-e2-32
This produces an excellent approximation to the flat-bottom solution (Chelton et al. 1998), and it might be anticipated that this improvement would carry over to the case of varying bottom topography. However, it is simple to show that (2.32) loses the crucial property of interlacing zeros—and so fails to reproduce the qualitative behavior of the exact solution, let alone quantitative behavior—if Nz is negative at the ocean floor and sufficiently large. More precisely, if
GNzHNH
then the approximation to F in (2.32) possesses a zero of F at some μ2 as before, but has no zero of Fz for 0 < μ < μ2. For typical ocean profiles such as that in Fig. 1, this condition is satisfied at various depths within the water column, particularly between 3500 and 3800 m (although, as noted, this might be an artefact of the data spacing). Accordingly, (2.32) cannot be used as an approximate solution since its behavior differs qualitatively from the exact solution. For higher-order modes, the WKBJ solution recovers the interlacing property, but these are not the modes observed in data, and so are of less physical interest.

c. The exact problem

The values of μ1, μ2, and μ3 can be evaluated numerically for typical ocean profiles. Figure 2a shows the implied range of the phase speed ratio as a function of depth H for each of the five major ocean basins, using a horizontally averaged stratification. As noted (above, and by Rhines 1977), the approximate upper bound of 4 is an overestimate, with values between 1.75 and 2.25 being typical; the upper bound increases as the depth shallows. The lower bound, however, is usually close to the 4/9 of the approximate theory. The upper and lower bounds depend on the shape of the stratification, but not its absolute value (a rescaling of N implies a rescaling of k, but no change to the ratios considered here). Figure 2b demonstrates this clearly: the bounds on the phase speed ratio are largely independent of where within the North Pacific the stratification is computed, despite the known changes in stratification across midocean ridges, etc. This is especially true for deep water, and thus lends credence to our later use of basin-averaged stratification.

Accordingly, we might anticipate a speedup of waves over their flat-bottom speed, with values of the ratio depending on orientation and slope of the topography. Any average speedup (e.g., across a piece of topography) is weak for the deep ocean since the bias between speedup and slowdown is small in many of the basins, but would be noticeable in shallower parts of the ocean. (Note that the zeroth approximation above turns out to give excellent qualitative descriptions of many aspects of the solution, but it is inaccurate for speed estimates and so cannot be employed.)

Although analytical solutions for certain profiles of N(z) can be determined (e.g., when N is an exponential in depth), in general (2.15) must be solved numerically. This is time-consuming when ray tracing is involved since high accuracy is required. However, the Sturm–Liouville structure means that many of the difficulties can be circumvented and the problem solved both rapidly and accurately. Details are postponed to section 5.

3. Approximate solutions for one-dimensional topography

Two special cases can be solved straightforwardly, in which the topography varies only with latitude or with longitude. Both can be solved exactly using numerical solution methods, of course; here we merely indicate the behavior.

a. Topography varying only with latitude

Dispersion relations for uniform stratification and a small and uniform northward slope were discussed by Rhines (1970) as well as by later authors. The problem for each latitude becomes separate, with latitude only entering parametrically. We write the phase as
ϕ2θλλeψθ
so that the problem for F becomes
i1520-0485-29-10-2689-e3-2
Then, trivially,
i1520-0485-29-10-2689-e3-4
which can be computed as a function of depth H and topographic slope tanθHθ. Figure 3 shows this ratio computed using typical North Pacific stratification. When tanθHθ < 0 (the ocean shoals poleward), the discussion above shows that c/cflat > 1. This is connected with, but not identical to, the fact that the topographic component of the planetary vorticity f/H here increases the β effect, as Rhines (1970) notes. Conversely, when tanθHθ > 0 (the ocean deepens poleward), c/cflat < 1. The behavior of the phase speed ratio is nonlinear in tanθHθ, as Fig. 3 shows.

A change in depth of 2 km over 10° lat is equivalent to tanθHθ ≈ 6000 m at 30°N, so this has been adopted as a range. The contours are all nearly straight, suggesting that the phase speed ratio depends mainly on tanθHθ/H; it is possible to show that this is approximately the case by manipulation of (3.2) and (3.3). (Using the sinusoidal approximation, it is straightforward to show that the phase speed ratio depends solely on tanθGθ/G.) Both extreme values in (2.23) are easily achieved.

b. Topography varying only with longitude

A straightforward, but nonclosed, solution exists in this case when the eastern boundary is oriented north–south. By posing
ϕλ, θ2θχλ
both the F equation and its boundary conditions become dependent only on z and λ; derivatives of χ and H occur linearly together. However, the system is nonclosed because solutions of the F equation are required, and these are not available except in special cases or by approximations such as the sinusoidal solution above. Adopting the latter approach (not detailed here) we find that an asymmetry, similar to the case of latitudinal variation, is found between regions where the speed ratio is less than unity (on the eastern slopes of ridges, where Hλ is positive) and regions where the speed ratio is above unity (on the western slopes of ridges, where Hλ is negative). The speed ratio readily achieves its two bounds for realistic topographic gradients. The farther west the wave has propagated, the easier the bounds are achieved since the differing phase speeds at different latitudes induce a stronger degree of θ variation.

The same comments apply for the exact solution (obtained using ray methods discussed below). It is clear that east–west gradients in topography are more effective at changing the phase speed than are north–south gradients; we can thus anticipate that known features such as midocean ridges will play a role in changing phase speeds.

4. Small two-dimensional topography

For the general case of topography varying in both horizontal directions, little is known about propagation except for Straub’s (1994) discussions. The general problem requires, and is amenable to, ray theory (see below). There does not seem to be a straightforward baroclinic equivalent of the simple “characteristics lie along contours of f/H,” which holds in the barotropic case (e.g., Smith 1971). Nonetheless, we can anticipate that certain topographic regimes will cause difficulty for simple ray theory, just as closed contours of f/H create regions that rays for barotropic planetary waves cannot enter, and steep depth contours will produce caustics at which ray theory breaks down (Lighthill 1978).

When topographic gradients are small in some sense, (2.11)–(2.13) can be solved directly for the phase ϕ, essentially by regarding the problem as one for ϕλ and stepping the problem in longitude. Although restrictive, it is instructive to solve for the small gradient case to see an indication of how slopes in the two directions combine.

We write
i1520-0485-29-10-2689-e4-1
where ε is a small parameter, and expand to the first order in ε. We do not need to define ε here, as no asymptotics are involved; the results show that the expansion is valid provided |H/H| ≪ 1. At leading order the flat-bottom solution satisfies
i1520-0485-29-10-2689-e4-2
Here μ is defined as before, and μ0 is the value corresponding to the first zero of F (termed μ2 in section 2, but here subscripts denote the order of the expansion in ε). This gives, for later use,
ϕ02μ20a2ω2θλλeθ
As before, rays propagate due west for a flat bottom.
The first-order equation becomes
i1520-0485-29-10-2689-e4-4
Multiplying (4.4) by F0 and (4.2) by F1, subtracting, and integrating from z = −H0 to 0 gives
i1520-0485-29-10-2689-e4-6
after use of the surface boundary conditions. Using (4.5) to substitute for F1(−H0) gives
i1520-0485-29-10-2689-e4-7
Now the east–west phase speed is given by c = c0 + εc1 so that
i1520-0485-29-10-2689-e4-8
after use of (2.18). After simplification, we find the fractional change to c0 is
i1520-0485-29-10-2689-e4-9
There are thus four terms contributing to the local change in the phase speed. Neglecting that due to the orientation of the eastern boundary—although this can be important—the other three terms are readily identifiable from what has gone before. The term in H1 itself simply reflects a faster local velocity in deeper water. The term in tanθH1θ is precisely (the linearized version of) that occurring in the latitude-only case so that the phase speed again increases when the ocean shoals poleward. The term 2H1λ(λλe) predicts, again, an increase in phase speed when the topography is sloping up toward the east; this effect increases with distance westward, as mentioned for the longitude-only case earlier. For a sufficiently wide ocean, this increase would cause the linearization to break down.

Since the perturbation is small, all effects are additive. Thus, the largest increase in phase speed would be expected on the western and equatorward side of an isolated seamount, or the eastern and poleward side of an isolated depression. These effects will be seen to occur in regimes beyond the validity of this small perturbation theory. Note that some of these results are in conflict with those of TM, who use a two-layer model and discuss the importance of vertical mode coupling. It would be useful to continue their studies with a continuously stratified ocean, as we have seen that simplifications in the vertical can sometimes give misleading answers.

5. The ray theory approach

For more general topography, ray theory must be used. We now regard the dispersion relation implied by (2.15), (2.12), and (2.16) as being of the form ω = ω(λ, θ, k, l). If we follow a ray initially at the eastern boundary using a timelike coordinate s, then ray theory predicts (Lighthill 1978)
i1520-0485-29-10-2689-e5-1
so that rays move with the group velocity cg = (∂ω/∂k,ω/∂l) [here cg has units of inverse time and can be converted to an actual velocity by multiplying by (a cosθ, a) respectively]. The latter two of (5.1) ensure that /Ds = 0, that is, that frequency is conserved along a ray.

The apparent difficulty in employing ray theory is that we do not possess an analytical dispersion relation with which to compute (5.1). The existence of such a relation for simple problems permitted Straub’s (1994) analysis, for example. However, by using the Sturm–Liouville nature of the governing vertical equation, it becomes possible to compute the gradients of frequency in (5.1) without directly solving the vertical eigenvalue problem. Appendix A shows how this can be done efficiently. Then, for suitable initial conditions (also discussed in appendix A) (5.1) can be stepped westward. This stepping can continue indefinitely, provided only that the ray theory itself does not break down. Ray theory can lead to caustics in the solution; the determination of these is straightforward though algebraically tedious, and some formulas are provided in appendix A.

The simplest examples are those for topography varying in one or other of the two directions, and ray theory reproduces the results of section 3 in such cases (but for the exact rather than the sinusoidal solution).

The next simplest example, which serves as an instructive model problem for real topographies, is shown in Fig. 4. This is the solution for a Gaussian perturbation to H giving a hill of minimum depth 2000 m in the center of a simplified ocean of depth 4000 m (except for a narrow continental rise convenient for the boundary conditions; this produces the deviation at the eastern edge of the diagram). The hill is centered at 20°N, 140°W (a mean North Pacific stratification is assumed), with e-folding width of 15° in both latitude and longitude.

The rays (Fig. 4a) move predominantly westward, with a small southward deviation as the outer part of the hill is approached (rather like f/H characteristics in the barotropic case, but with far less deviation). On the northwestern side of the hill, there is a convergence of the rays that persists far to the west of the hill, but otherwise the rays appear straightforward. However, both westward phase and group velocities (which are almost identical; only the phase velocity is shown in Fig. 4b)2 show decreases on the eastern side of the hill and increases on the westward side, in rough agreement with the linear theory in section 4, save that the linear theory overestimates the reduction in phase speed. Figure 4c shows the phase speed ratio c/cflat and demonstrates the changes in phase speed clearly. Note that there is no change in the average of c/cflat over the entire basin. However, longitudinally averaged phase speed ratios show an increase over unity (up to 5%) in latitudes 5° to 30°N and a decrease below unity poleward of 30°N. The phase velocity varies strongly in direction (since north–south wavenumber vanishes at the coast, the phase velocity is directed northward there, for example). However, the group velocity is oriented everywhere within 10° of true west, save for two areas. The dynamically relevant area is in the convergence region to the northwest of the hill, where the group velocity turns northwestward (for a larger amplitude hill, as we shall see, a caustic forms in this area); the other, near the eastern boundary, occurs where the deviation is caused by the slope and initial conditions. The phase (Fig. 4d) is far from constant along depth contours—indeed, it is hard to distinguish where the topography is in Fig. 4d.

For a higher hill, a caustic appears to the north of the center of the hill: Fig. 5a shows the ray trajectories for a hill of minimum depth 1500 m, but otherwise the same configuration as for Fig. 4. The caustic runs northwest until about 37°N in this example, before being replaced by partial ray focusing. At the caustic, ray theory breaks down. There are methods for connecting rays across caustics (e.g., Lighthill 1978), but these are complicated here, and we shall not explore beyond the caustic. The focusing northward does not, however, mark a breakdown of ray theory in those areas, although it does indicate a convergence of wave energy (to which we shall return later), suggesting an increase in energy levels in such locations. The phase and group velocities remain similar (only the phase being shown) and again lie predominantly westward except near the caustic where by definition they become aligned northwestward along the caustic. (There is a rapid change in the north–south group velocity across the north–south axis of the hill, though the values on both sides remain small compared with the westward component.) There is again an increase in longitudinally averaged phase speed ratios between 5° and 25°N, by up to 6%, and a decrease poleward (some of which is due to the loss of the region of speedup removed by the caustic). The basin average shows a slight decrease, at 0.97.

Caustics tend to appear more easily where the phase ϕ changes rapidly, so their locations tend to be biased poleward and westward. This feature will be seen clearly in the oceanic examples to follow.

Figure 6 shows results for the opposite case, when the depression has a maximum depth of 6000 m, so that the amplitude is 2000 m as in Fig. 4. The rays (not shown) are almost completely oriented east–west, and it is hard to distinguish any topography from them. However, the other fields are similar to those in Fig. 4 (the less steep hill) but with opposite signs: the speedup of the phase and group velocity now occurring on the eastern side of the depression (i.e., again where the topography is rising toward the east). Averaging along an east–west line would produce little evidence of wave speed increase in this case, since the speedup and slowdown are of similar magnitude. North–south propagation remains minimal. Curiously, there is no tendency toward focusing in this case (whereas on a simple sign change argument one might have expected one), nor is there an equivalent turning of the group velocity where focusing might have been expected.

6. Oceanic examples

We now consider realistic topography, for five ocean basins: North and South Pacific, North and South Atlantic, and the southern Indian Oceans. The size of each basin is determined by the need for a well-defined eastern boundary to serve as the source of the rays, with sufficient width to develop the phase of the waves over at least several cycles. The Levitus datasets (Levitus and Boyer 1994; Levitus et al. 1994) were averaged horizontally over each region used to provide the buoyancy frequency as a function of depth. It is necessary to construct fields of H and all its derivatives up to third order, which are both continuous and consistent. Bicubic interpolation (including splines) introduces large oscillations in the fields on the scale of the data tabulation, leading to unacceptable H contours between data points. We therefore construct H and its derivatives via a Fourier representation. The ETOPO5 topography, resolution 1/12° (National Geophysical Data Center 1988), was averaged onto a ¼° grid. This topography was Fourier transformed in latitude and longitude, and a Lanczos sigma-factor filter (Lanczos 1957, 1966) was applied, truncating the spectral representation at a scale of 9° and attenuating the remaining high wavenumber components. The choice of an ideal smoothing length scale is unclear. With too short a scale, for example, 6°, the WKBJ assumption is likely to break down, and at all events the results are noticeably noisier; with too long a length scale, for example, 15°, important features such as midocean ridges are lost. The choice of 9° was a compromise between these extremes. The results depend quantitatively, but not qualitatively, on the smoothing length scale. We discuss the effects of less smooth topography below.

Both H and its derivatives were written out on the ¼° grid. (The choice of ¼° here was twofold: it was the finest resolution affordable and gave good conservation of frequency for all runs described here, to within a few parts in 104.) Bilinear interpolation of these tabulated fields was used to produce continuous values.

Rays were initiated at an eastern boundary defined by some suitable depth. Apart from surface elevation (see later) and minor details about ray paths, solutions do not depend noticeably on the choice of depth between 3000 and 3500 m. (Occasional geometric quirks meant that too deep a choice could induce unacceptable meanders in coastline shape.) For uniformity, an eastern boundary depth of 3500 m was used consistently. Rays are terminated when they leave the region, or reach a depth less than 1000 m.

We consider, for each basin in turn, both the solutions and, for later comparison with observations, the averaged phase speed ratio.3 Only long-wave calculations have been made so that the theory tends to become less relevant as one proceeds poleward, though Hallberg (1997) shows that horizontal WKBJ theory continues to yield good predictions.

a. The North Pacific

Figure 7 shows a selection of the results for the North Pacific. Rays (Fig. 7a) initially display the poleward tendency discussed in appendix A before becoming predominantly westward. In the subequatorial region the rays remain smooth across the basin, with minor deflections due to topography. Farther north caustics and shadow zones begin to appear, together with regions of ray concentration and divergence; the main shadow zone is at about 33°N. There are several examples of ray crossings. The depth becomes too shallow in the northwest corner to continue the integration. The phase ϕ shows a distorted version of what would occur with a flat bottom: phase lines oriented N–S near the coast and becoming more E–W oriented as the waves move west due to the poleward gradient of phase and group velocity. Indeed, the westward phase and group speeds are similar (Fig. 7c shows the phase speed). The N–S group velocity (not shown), by contrast, is much weaker, seldom reaching above 0.01 m s−1 except near the boundary. Thus, the direction of the group velocity is predominantly within a few degrees of westward except near the coast and where caustics turn at a high angle. This is in contrast to the direction of the phase velocity, which shows strong variation from a purely westward orientation.4

The flat-bottom phase speed cflat varies meridionally, dependent upon Coriolis parameter, but only weakly east–west. This is because the flat-bottom phase speed depends, roughly, on the integral of N(z) (Chelton et al. 1998) and N is small at large depth, so lessening the depth dependence. In the calculations of KCS, the much stronger east–west variation of phase speed was largely caused by horizontal variability of buoyancy frequency (removed here a priori). Figure 7d shows the phase speed ratio c/cflat. In high latitudes near the western boundary, the ratio becomes slightly noisy, but its behavior at more southern latitudes is as predicted, with bands of high ratio (up to 1.4) being followed, on the topographic scale, by bands of low ratio (around 0.5) along the Hawaiian chain, leading to a slight increase in zonally averaged speed. Indeed, longitudinal averages show phase speed ratios a few percent over unity between 14° and 19°N and well over unity (up to 38%) poleward of 41°N. Elsewhere, the effect of topography is to decrease phase speed so that, averaged over the entire basin, the phase speed ratio is 1.01. This weak effect is predominantly due to the deep North Pacific basin; recall that deep basins show little asymmetry in phase speeds.

The wavenumber k is of order 30–40 in much of the region, with higher values near the northern end. The N–S wavenumber l is small near the eastern boundary (it would be identically zero if the boundary were oriented N–S) but increases to values of order 100 south of about 34°N and with noisy values up to 200 north of there. Long-wave theory (see appendix B) requires wavenumbers small compared with 200, so our theory breaks down in such regions for the annual period—as it should since the turning latitude has been reached.

b. The South Pacific

Figure 8 shows selected diagrams for the South Pacific. Again, caustics and shadow zones appear (Fig. 8a), with a wide zone at 20°S and another developing at 37°S after the mid-Pacific rise has been crossed. Several apparent areas of focusing are visible. The westward phase and group velocities are extremely similar (a feature which applies throughout the basins). The westward phase speed ratio (Fig. 8b) is mostly under 1 east of the mid-Pacific rise but increases on its western flank as in the theoretical examples earlier, with coherent areas with ratios above 1.6. The phase speed ratio, averaged longitudinally, is above unity almost everywhere in the basin, with an average over the basin for the phase speed ratio of 1.08. Similar comments for the North Pacific apply to wavenumbers, phase, and direction of the group velocity.

c. The North Atlantic

Figure 9 shows results for the North Atlantic. Rays are effectively swept northward past the Bay of Biscay if the depth of the coastline is defined too shallow, with a caustic preventing rays leaving over a range of latitudes. Values presented here, as elsewhere, use a depth of 3500 m, although many caustics remain. Note for later discussion the focusing in a band at 36°N. As before, rays nearer the equator are relatively unmodified by topography, the effects of which are reduced by the tanθ factors in the theory. The phase speed peaks to the west and south of the Mid-Atlantic Ridge as suggested by our theory, causing the phase speed ratio (Fig. 9b) to show the now familiar strong increase in that region. This is a known feature in North Atlantic observations (CS; Cipollini et al. 1999) and is caused by changes in stratification and mean flow (KCS) as well as by the mechanism suggested here. The east–west wavenumber (not shown) reaches values of 100 at about 40°N, while N–S wavenumbers exceed the limits of the long-wave theory northward of about 30°N. Between 25° and 41°N, the phase speed ratio exceeds unity on average, although the basin average of the phase speed ratio is slightly under unity, at 0.97. There is evidence of a decrease in E–W wavenumber as the Mid-Atlantic Ridge is crossed, in qualitative agreement with the findings of Tokmakian and Challenor (1993).

d. The South Atlantic

The South Atlantic has a smaller latitudinal extent because of the extent of Africa, and relatively simple topography, so that the resulting ray diagram (Fig. 10a) is much cleaner than in the other basins. It shows almost complete westward orientation of the rays, with an extensive caustic to the south yielding a zone beyond which we cannot integrate. The N–S wavenumber reaches values of 300 near the caustic, demonstrating a breakdown of the long-wave theory. The phase speed ratio (Fig. 10b) shows two extensive regions of increase: the first is west of the mid-Atlantic rise as before, the second over a weak slope east of the meridian, largest at 26°S. The basin average of the speed ratio is 0.91 and is also under unity for each latitude.

e. The southern Indian

The topography of the southern Indian Ocean is complicated, even after smoothing (Fig. 11c). There is a near-shadow zone (associated with the almost east–west orientation near Indonesia) and several regions of focusing (Fig. 11a). Contours of the phase speed ratio form a collection of cells, with a basin average of 0.94; only between 30° and 23°S does the ratio exceed unity when averaged longitudinally.

f. Robustness to smaller-scale topography

The WKBJ theory requires that topographic variations occur on scales larger than the waves themselves. In the real ocean, topography varies on all possible scales. However, it is of interest to see how increasing topographic roughness modifies the solutions. We recomputed the solutions using a 6° rather than a 9° smoother. As might be expected, the solutions are somewhat noisier than before, with somewhat poorer frequency conservation along the rays. The ray diagrams show more areas of caustics but are mainly qualitatively similar to those shown here. Other quantities, for example, phase speeds, are quite robust to the change in topographic scale. The main differences are found in surface elevation (which is affected by ray divergences, which are sensitive to topographic changes); while some features of the surface elevation remain essentially unaltered, others are changed, so that results concerning surface elevation should be treated with caution.

7. Comparison with satellite observations

Because of the existence of satellite data, planetary wave theories can now be compared with observations. There are difficulties. The presence of mean flow cannot be removed from the data. Wave analyses from satellite observations still use large-scale averaging, whereas theories such as ours involve much detailed structure (although smoothing of topography has occurred).

The ocean basin results suggested little evidence for a net increase in basin-averaged westward phase speed due to the presence of topography. However, phase speed changes possess latitudinal structure. To show this, we construct longitudinal averages. Now the ocean basins chosen do not cover the entire ocean between (say) ±50° because of shallow areas, intrusion of western boundaries, shadow zones, and the need to start the integration with a well-defined eastern boundary condition. Nonetheless, they do form the majority of the World Ocean in this latitude range so that summaries of the results by averaging are meaningful. (Averaging over at least 5° is almost always involved in computations from satellite data anyway.)

First, consider the longitude-averaged speed ratio. To obtain this, all relevant points at any given latitude in the five basins of the previous section were used to compute the longitude-averaged value of phase speed c and flat-bottomed phase speed cflat. The ratio of these two quantities is shown as a function of latitude in Fig. 12. The ratio tends to be below unity in midlatitudes, save for a peak of 1.1 at 25°S, and rises toward higher values at higher latitudes, especially in the Northern Hemisphere. (The ratio is not shown near the equator since midlatitude theory would not apply there.)

A useful comparison is to compute the speed ratio for averages taken only over the longitude bands used by CS.5 This is shown in Fig. 13. The phase speed ratio predictions lie close to unity and show no clear pattern, unlike the CS ratios, which show the familiar poleward increase. Thus topographic variations cannot explain the phase speed increase in the observations, so the mean flow arguments of KCS remain the main explanation for such increases over wide areas.

8. Energy

Convergence or divergence along ray trajectories acts to change the amplitude of the signal (e.g., Lighthill 1978). The energy density E of the solution satisfies
i1520-0485-29-10-2689-e8-1
so that E increases rapidly in regions of convergence of the group velocity and decreases similarly in regions of divergence. For long waves, the energy is entirely potential, given by
i1520-0485-29-10-2689-e8-2
Defining the second integral in (8.2) as V (it is already tabulated for purposes of ray tracing), we obtain the amplitude of the wave as
i1520-0485-29-10-2689-e8-3
Satellite altimeters regularly measure the sea surface elevation η0, which is related to the amplitude a0 by
i1520-0485-29-10-2689-e8-4
so
i1520-0485-29-10-2689-e8-5
where
i1520-0485-29-10-2689-e8-6
is a nondimensional factor dependent only on μ (i.e., on how the solution varies along the ray) and H (i.e., on position). Computed values of η show that η(μi, H), i = 1, 2, 3 vary very weakly with H, and indeed η only varies between 0.045 and 0.07 across the permitted range of μ.

From an initial value for energy density E (here taken arbitrarily as a constant along the coastline, which implies the surface elevation η0 and amplitude a0 are also constant along the coastline), E can be integrated by (8.1) and converted through (8.5) and (8.6) to the surface expression of the waves for comparison with data. Like other computed quantities, the overall details of E and η0 do not depend on details of the computation. However, their maxima, which are frequently large, arise because of regions of convergence. These regions are strongly sensitive to details of coastline depth, etc. Thus Fig. 14, which shows contours of nondimensional values of η0 for the five basins, gives an accurate picture of the O(1) variation, but could mislead as to the size of the largest values by an order of magnitude. Of particular interest is the rapid change in magnitude of the elevation that occurs over a degree or two, despite the slowly varying nature of the problem. In the North Atlantic (14c) there is an intense band of high amplitude predicted at about 33°–35°N. Cipollini et al. (1997) find similar focusing in both sea surface height and sea surface temperature signals at 34°N. Detail is lacking in the model results due to a caustic and some refraction of the rays, but given the averaged topography and horizontally uniform stratification the agreement is as good as can be expected.

In general, the relation between η0 and the energy E of the vertical mode is nontrivial, depending as it does on E–W wavenumber in a nonlinear manner. Caution must therefore be used in attempting to infer modal energy from surface expressions. There are features in the surface elevation with no apparent expression in E:for example, η0 rises over the midocean ridge in the South Atlantic while the energy falls.

9. Conclusions

This paper has examined the propagation of long baroclinic planetary waves across slowly varying topography, for arbitrary (and physical) vertical stratification. Local phase speeds between about 0.4 and twice the flat-bottom speed are found, as suggested by Rhines (1977). The work has extended, and subsumed, previous work on special cases of such problems. We have shown that the full three-dimensional problem can be solved using ray theory in the horizontal without either a dispersion relation or specifically solving an eigenvalue problem at any location. Generic features, such as phase speed increases equatorward and westward of topographic hills, have been found.

In addition, this paper has returned to the intriguing question of accounting for the “too-rapid” propagation of planetary waves shown by remote sensing data. We note recent theory by Killworth et al. (1997), which showed how much of the discrepancy between observed and theoretical speeds could be explained by changes in planetary vorticity induced by the presence of background mean flows. However, that paper still predicted wave speeds in the midlatitude Southern Hemisphere that were consistently too slow. This paper has thus also examined a second possibility, that steering of long planetary waves by slowly varying topography can induce changes to wave speeds that result in an overall increase of speed.

Despite the bias between slowdown and speedup, oceanic calculations for five main ocean basins, assuming waves are generated at the eastern coastline, show little evidence for overall increases in speed of planetary waves by topography. By examining energy propagation, estimates for the changes in the amplitude of sea surface height variation could also be made, and these appear in qualitative agreement with the observations. (Since a basin-averaged stratification was used, the locations of features in our model will not coincide exactly with those seen in observations. Heavy averaging is also employed on satellite data analyses.)

Background mean flow remains the most likely contender to account for most of the observations of phase speed. Topographic steering will have a local effect and may interact with background mean flow in ways beyond the scope of this paper. To test this, several more calculations must be made. An extension of this paper would be to permit a buoyancy frequency that varies (smoothly) laterally as well as vertically, without inclusion of the concomitant mean flow. While not formally valid, it would permit a more detailed examination of the effects of local topography. This is straightforward using the sinusoidal approximation—which does not yield the correct phase speed—but far more tedious using any exact approach. Later papers should use simple and complex numerical models to see if these effects can be isolated in their results.

Acknowledgments

It is again a pleasure to note the help and enthusiasm of the Satellite Meteorology team at SOC. Dudley Chelton provided the satellite-derived phase speeds for comparison. George Nurser found an important and elusive algebraic error for which we are most grateful. A referee provided incisive comments leading to considerable improvements in the paper.

REFERENCES

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APPENDIX A

Ray Theory and Vertical Solution Details

To obtain the quantities in (5.1), the Sturm–Liouville structure of (2.15) is used. We let X denote any of (λ, θ, k, l) and differentiate (2.15) wrt X. This gives6
i1520-0485-29-10-2689-ea-1
where from (2.20), μ2 = (−k/2a2Ωω sin2θ) and
i1520-0485-29-10-2689-ea-2
with boundary conditions
i1520-0485-29-10-2689-ea-3
where
i1520-0485-29-10-2689-ea-5
(here all quantities are evaluated at z = −H) and
σkHθlHλ
is usefully defined.
Multiply (2.15) by FX, subtract (A.1) times F, and integrate from −H to 0. The top boundary conditions give no contribution, leaving merely
i1520-0485-29-10-2689-ea-7
We now substitute for F(−H) in terms of Fz(−H) from (2.16), so that the lhs of (A.7) becomes
i1520-0485-29-10-2689-eq3
From (A.4), this can be replaced with known quantities from the solution undifferentiated in X, involving R. After a little algebra, we have finally
i1520-0485-29-10-2689-ea-8
Thus, the differentials required for ray theory can be computed directly from the undifferentiated solution, requiring knowledge merely of F(−H), Fz(−H), and 0HN2F2 dz. One need merely solve (2.15) as an initial value problem, and store the values of F, Fz, and
i1520-0485-29-10-2689-eq1a
tabulated as functions of z and the parameter μ. For any given value of depth and μ, simple interpolation gives all quantities in (A.8) and integration can proceed.
The Eqs. (5.1) and (A.8) were solved with Runge–Kutta routines in the NAG library (Numerical Algorithms Group 1997). They also need initial conditions, at some eastern boundary. For idealized problems, the boundary can simply be defined, as in the main text, as λ = λe(θ). The wavenumber does not change along such a boundary so that
i1520-0485-29-10-2689-ea-9
If the ocean is assumed flat at the eastern boundary, then F(−H) = 0 there, and (A.9) together with (2.16), gives an implicit equation for k, given ω. Solution of this then implies both k and l, and the problem can be integrated.
For problems with real topography, the coastline must be defined in some manner. The obvious choice, by defining the coastline at the location of zero depth, gives singularities in the initial integration as well as almost certainly violating the vertical modal assumptions. Instead, we choose the boundary to occur at some constant depth along the eastern side of the ocean. The location of the boundary is then well defined. The boundary lies at an angle tan−1(−Hλ/Hθ) to the east since H is constant along the boundary. The wavenumber along the coastline must vanish so that
i1520-0485-29-10-2689-ea-10
This implies, from (2.16), that σ = kHθlHλ = 0 at the eastern boundary, so the ocean—as far as the dispersion relation is concerned—is essentially flat at the boundary [i.e., F(−H) = 0]. Substitution into (2.16) then gives an implicit equation for k given the frequency ω (this implicit equation need only be solved at boundary points, since wavenumber is thereafter predicted). Equation (A.10) then gives l, and the problem may be integrated.
For such a boundary, rays leave at an angle given by (5.1) as
i1520-0485-29-10-2689-ea-11
Inferences about this angle can be made for either the exact solution or the sinusoidal approximation, yielding similar results. For the exact solution, (A.8) and earlier give at the coastline
i1520-0485-29-10-2689-eq4
(recall that F(−H) vanishes here) so that
i1520-0485-29-10-2689-ea-12a
at the coastline, and
i1520-0485-29-10-2689-ea-12b
Without the first term on the rhs of (A.12b), the rays would follow the topography. The addition of the first term shows that for coastal topography oriented north–south, rays leave at angles that are almost westward near the equator and become progressively more poleward oriented as latitude increases; this effect is clearly visible in Fig. 4, for example. At the other extreme (coastal topography oriented east–west), the frequency of the waves becomes zero and waves cannot leave the coast. This latter case does not occur in the calculations presented in this paper, mainly as an accident of geometry but occasionally by selection of latitudes (e.g., the Gulf of Guinea).

Equations (A.12) show that the angle at which rays leave the “coast” depends on H and its gradients, that is, on the depth chosen to define the coastline. However, none of the solutions given vary noticeably with the chosen depth (we used depths of 3000 and 3500 m).

Caustic calculations

Computed solutions give no indication (except a visual suggestion, often hard to distinguish from focusing) when a caustic is reached, nor do internal consistency checks (e.g., conservation of frequency). It is therefore necessary to add additional checking for the existence of caustics along a ray.

Let τ be a measure of the distance along the eastern boundary that the trajectory starts from (τ can also be thought of as a label for the trajectory). A caustic appears when two neighboring trajectories, one at time s and one at time s + δs, reach the same location. Thus,
i1520-0485-29-10-2689-ea-13
or, expanding the rh sides,
i1520-0485-29-10-2689-ea-14
Eliminating the small quantities yields a caustic when
i1520-0485-29-10-2689-ea-15
To follow the values of this Jacobian requires differentials of the entire ray problem and its initial conditions wrt distance τ. This, in turn, requires second derivatives of ω wrt λ, θ, k, or l in all combinations. These can indeed be computed algebraically, but the details are exceptionally tedious; they required much independent checking and are not reproduced here; however, symmetries (e.g., between second derivatives of ω) have been used to verify the code. Worthy of note is that third derivatives of H are required for the caustic tracing. The algebra can be obtained from the authors electronically.

APPENDIX B

Short-Wave Modifications

The analysis in the text assumes that the planetary waves remain long; this need not be the case. This appendix discusses the modifications necessary for shorter waves and finds a condition on wavenumber for the applicability of long-wave theory.

Geostrophy is modified by the inclusion of time derivative terms, giving u and υ in terms of M as
i1520-0485-29-10-2689-eb1
to leading order, provided merely that ωf, certainly the case for annual or semiannual cycles. To leading order, mass conservation yields
i1520-0485-29-10-2689-eb3
after substitution for the wavenumbers. The linearized density equation is now modified by the presence of the latter term on the rhs of (B.3), giving the equivalent of (2.15) as
Fzzμ2N2F
where
i1520-0485-29-10-2689-eb5
replaces (2.20).
The bottom boundary condition, to leading order, only retains the geostrophic terms in the horizontal velocities but all terms in w. This becomes
i1520-0485-29-10-2689-eb6
where σ retains its original meaning (5.7). We can estimate how small the wavenumbers must be by using the sinusoidal approximation given by (2.24)–(2.26). Defining q′ = μG in a similar manner, the approximate dispersion relation becomes
i1520-0485-29-10-2689-eb7
while q′ is given by
i1520-0485-29-10-2689-eb8
With a flat bottom, (B.8) reduces to q′ = π again.

The condition for the wave to be long is thus that |k|G/afπ, or |k| ≪ 200 for typical midocean values. This is usually approximately satisfied in the solutions in this paper for the annual frequency except at high latitudes and forms part of the choice to confine calculations to ±50°. Note that because the problem is linear in ω and ϕ (and hence k and l), increasing the frequency implies increasing wavenumber, and so decreases the area for which long-wave theory is valid.

In the flat-bottom case, maximum frequency is found for l = 0, k = afπ cosθ/G. This frequency is G cotθ/2aπ. For the annual frequency, this implies a maximum latitude of 45° or 48° as G varies from 8 to 9, with higher maximum latitudes for lower frequencies, for example, half a period per year, as observed by CS in some areas.

i1520-0485-29-10-2689-f01

. 1. The buoyancy frequency N(z) typical of the North Pacific (s−1).

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 2.
Fig. 2.

(a) The bounds on phase speed ratio c/cflat for the five major ocean basins, implied by their basin-averaged stratification, as a function of depth. (b) The bounds computed for four subareas of the North Pacific, arranged in the SW, SE, NW, and NE of the basin. The longitude ranges are 140°–160°W (SW, NW) or 150°–130°E (SE, NE); the latitude ranges are 10°–20°N (SW, SE) or 30°–40°N (NW, NE).

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 3.
Fig. 3.

Contours of the speed ratio c/cflat for the case of topography varying with latitude only, as a function of tanθHθ and H, for the North Pacific.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 4.
Fig. 4.

Results using ray theory for the case of a Gaussian hill of minimum depth 2000 m, centered on 20°N, 140°W with width 15° in both directions, with a background depth of 4000 m except for a slope near the eastern boundary. (a) The ray trajectories themselves, with rays initially spaced for clarity at 1/2° apart (a closer spacing of 1/8° was used to compute the data for all diagrams without rays). (b) The westward phase velocity (the westward group velocity is almost identical in most regions, and is not shown). (c) The phase speed ratio c/cflat, with contour interval 0.1; values less than unity are dashed; unity is dash–dotted. (d) The phase ϕ.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 5.
Fig. 5.

As for Fig. 4 but for a hill of minimum depth 1500 m. A caustic is now formed on the poleward side of the hill. The location of the caustic is shown by letter C’s.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 6.
Fig. 6.

As for Fig. 4 but for a depression with maximum depth 6000 m (the flat ocean depth remains 4000 m). (a) Westward phase velocity, (b) the ratio c/cflat (contours as Fig. 4), and (c) the phase ϕ.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 7.
Fig. 7.

Results for the North Pacific. (a) Ray trajectories, using an initial spacing between rays of 1/4° with C’s marking caustics and D’s marking regions where rays enter depths less than 1000 m, beyond which a ray is not followed (again, a closer spacing of 1/8° was used to compute the data for all diagrams without rays); (b) the phase ϕ of the solution; (c) the computed westward phase speed; and (d) the ratio of this to the local flat-bottom phase speed (contours as in Fig. 4). (e) The smoothed topography used in the computations.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 8.
Fig. 8.

Results for the South Pacific. (a) Ray trajectories, (b) the phase speed ratio (contours as in Fig. 4), and (c) the smoothed topography used.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 9.
Fig. 9.

Results for the North Atlantic. Details as for Fig. 8.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 10.
Fig. 10.

Results for the South Atlantic. Details as for Fig. 8.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 11.
Fig. 11.

Results for the southern Indian. Details as for Fig. 8.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 12.
Fig. 12.

The ratio of the values of the longitudinal averages of phase speed c and flat-bottom phase speed cflat as a function of latitude, over the five ocean basins considered. Values are not shown within 5° of the equator, where midlatitude planetary wave theory would not hold, or beyond 50°, where annual waves would not occur.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 13.
Fig. 13.

The ratio of the values of the longitudinal averages of phase speed c and flat-bottom phase speed cflat, as a function of latitude, using the longitude bands defined by CS. Solid symbols show the ratios observed by CS, and open symbols show the topographic theory. Circles indicate Pacific values, squares Atlantic and Indian values.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

Fig. 14.
Fig. 14.

Values of surface height amplitude η0 for the five basins, with an arbitrary uniform initial amplitude at the eastern boundary: (a) the North Pacific, (b) the South Pacific, (c) the North Atlantic, (d) the South Atlantic, and (e) the southern Indian. Because of the wide variation, contour intervals are nonconstant. They are 0.2–2 by 0.2; 2.5–4 by 0.5; 5, 6, and 8; 10–40 by 5.

Citation: Journal of Physical Oceanography 29, 10; 10.1175/1520-0485(1999)029<2689:TEOBTO>2.0.CO;2

1

This assumes that the east–west wavenumber k is negative, that is, that the propagation is westward.

2

All quantities shown were computed by small-area averages over several rays, and many time steps, and then contoured.

3

In all cases, results (not shown) from the sinusoidal approximation are qualitatively, and frequently quantitatively, similar to the exact solutions, with two exceptions: the westward phase speed, where as we have seen the sinusoidal approximation yields consistent overestimates, and the surface elevation, discussed later.

4

It remains unclear whether analysis of satellite altimeter data yields phase or group velocity (D. Cromwell 1997, personal communication).

5

Not all of CS’s longitude ranges could be included because they sometimes included land or shallow water. Because a basin-averaged buoyancy frequency N(z) was used for this calculation, we have not shown here a direct comparison between actual phase speed values in observations and theory.

6

Further simplification can be made by redefining the latitudinal variable as sin2θ but, because we shall plot diagrams for real ocean basins, no change is made to the θ coordinate.

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