The Quasi-Nondispersive Regimes of Long Extratropical Baroclinic Rossby Waves over (Slowly Varying) Topography

Rémi Tailleux Laboratoire de Météorologie Dynamique, Université Pierre et Marie Curie, Paris, France, and EGS-CSAG, University of Cape Town, Rondebosch, South Africa

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Abstract

Actual energy paths of long, extratropical baroclinic Rossby waves in the ocean are difficult to describe simply because they depend on the meridional-wavenumber-to-zonal-wavenumber ratio τ, a quantity that is difficult to estimate both observationally and theoretically. This paper shows, however, that this dependence is actually weak over any interval in which the zonal phase speed varies approximately linearly with τ, in which case the propagation becomes quasi-nondispersive (QND) and describable at leading order in terms of environmental conditions (i.e., topography and stratification) alone. As an example, the purely topographic case is shown to possess three main kinds of QND ray paths. The first is a topographic regime in which the rays follow approximately the contours f /hαc = a constant (αc is a near constant fixed by the strength of the stratification, f is the Coriolis parameter, and h is the ocean depth). The second and third are, respectively, “fast” and “slow” westward regimes little affected by topography and associated with the first and second bottom-pressure-compensated normal modes studied in previous work by Tailleux and McWilliams. Idealized examples show that actual rays can often be reproduced with reasonable accuracy by replacing the actual dispersion relation by its QND approximation. The topographic regime provides an upper bound (in general a large overestimate) of the maximum latitudinal excursions of actual rays. The method presented in this paper is interesting for enabling an optimal classification of purely azimuthally dispersive wave systems into simpler idealized QND wave regimes, which helps to rationalize previous empirical findings that the ray paths of long Rossby waves in the presence of mean flow and topography often seem to be independent of the wavenumber orientation. Two important side results are to establish that the baroclinic string function regime of Tyler and Käse is only valid over a tiny range of the topographic parameter and that long baroclinic Rossby waves propagating over topography do not obey any two-dimensional potential vorticity conservation principle. Given the importance of the latter principle in geophysical fluid dynamics, the lack of it in this case makes the concept of the QND regimes all the more important, for they are probably the only alternative to provide a simple and economical description of general purely azimuthally dispersive wave systems.

Corresponding author address: R. Tailleux, NCAS Centre for Global Atmospheric Modelling, Dept. of Meteorology, University of Reading, Earley Gate, P.O. Box 243, Reading RG6 6BB, United Kingdom. Email: r.g.j.tailleux@reading.ac.uk

Abstract

Actual energy paths of long, extratropical baroclinic Rossby waves in the ocean are difficult to describe simply because they depend on the meridional-wavenumber-to-zonal-wavenumber ratio τ, a quantity that is difficult to estimate both observationally and theoretically. This paper shows, however, that this dependence is actually weak over any interval in which the zonal phase speed varies approximately linearly with τ, in which case the propagation becomes quasi-nondispersive (QND) and describable at leading order in terms of environmental conditions (i.e., topography and stratification) alone. As an example, the purely topographic case is shown to possess three main kinds of QND ray paths. The first is a topographic regime in which the rays follow approximately the contours f /hαc = a constant (αc is a near constant fixed by the strength of the stratification, f is the Coriolis parameter, and h is the ocean depth). The second and third are, respectively, “fast” and “slow” westward regimes little affected by topography and associated with the first and second bottom-pressure-compensated normal modes studied in previous work by Tailleux and McWilliams. Idealized examples show that actual rays can often be reproduced with reasonable accuracy by replacing the actual dispersion relation by its QND approximation. The topographic regime provides an upper bound (in general a large overestimate) of the maximum latitudinal excursions of actual rays. The method presented in this paper is interesting for enabling an optimal classification of purely azimuthally dispersive wave systems into simpler idealized QND wave regimes, which helps to rationalize previous empirical findings that the ray paths of long Rossby waves in the presence of mean flow and topography often seem to be independent of the wavenumber orientation. Two important side results are to establish that the baroclinic string function regime of Tyler and Käse is only valid over a tiny range of the topographic parameter and that long baroclinic Rossby waves propagating over topography do not obey any two-dimensional potential vorticity conservation principle. Given the importance of the latter principle in geophysical fluid dynamics, the lack of it in this case makes the concept of the QND regimes all the more important, for they are probably the only alternative to provide a simple and economical description of general purely azimuthally dispersive wave systems.

Corresponding author address: R. Tailleux, NCAS Centre for Global Atmospheric Modelling, Dept. of Meteorology, University of Reading, Earley Gate, P.O. Box 243, Reading RG6 6BB, United Kingdom. Email: r.g.j.tailleux@reading.ac.uk

1. Introduction

In the standard linear theory (SLT; e.g., Gill 1982; Leblond and Mysak 1978) long baroclinic Rossby waves propagate westward nondispersively with the phase speed cR = −βc2/f2, where f is the Coriolis parameter, β is its latitudinal derivative, and c2 is the equivalent internal gravity wave speed squared of the mode considered. In the SLT, c2 is the solution of a classical Sturm–Liouville eigenvalue problem, whose properties depend upon the environmental conditions (i.e., the stratification and total ocean depth) only. Chelton and Schlax (1996) recently showed that the phase speeds predicted by SLT were in general too fast by a factor of up to 2–3 at mid- and high latitudes as compared with observations, using historical data for temperature and salinity to estimate c2 and hence cR.

Because the SLT discards the two important effects of bottom topography and background mean flow, most recent theoretical efforts have sought to understand how the latter may affect the phase speed of long extratropical baroclinic Rossby waves, either in isolation or in combination. To that end, Killworth and Blundell (2003a, b, together referred to as KB03 hereinafter) introduced a generalized long-wave model (GLWM) based on Wentzel–Kramers–Brillouin (WKB) theory to describe the propagation of long extratropical baroclinic Rossby waves in the presence of a slowly varying background mean flow v = (u, υ, w) and topography H(x, y). The vertical structure F (for vertical velocity) and zonal phase speed c = ω/kx of purely periodic waves are the eigensolutions of (see KB03 for details)
i1520-0485-36-1-104-e1
i1520-0485-36-1-104-e2
where (kx, ky) are the zonal and meridional wavenumber components, U = u + υky/kx, N2 is the squared buoyancy frequency, and μdim = ( f /β)(kyxH/kx − ∂yH). Special cases of the GLWM are the SLT (for μdim = v = 0); the flat-bottom mean flow theories of Killworth et al. (1997) and Colin de Verdière and Tailleux (2005) (for μdim = υ = 0); the topography-only studies of Killworth and Blundell (1999, hereinafter KB99), Tailleux (2004), and Tailleux (2003) (for v = 0); and the bottom-pressure compensation theory of Tailleux and McWilliams (2001) (for v = 0 and μdim = +∞).

Having obtained the dispersion relation from solving the GLWM, a basic fundamental issue is to understand the ray trajectories followed by the energy. For a realistically stratified ocean (e.g., KB99 and KB03), these trajectories are usually investigated empirically owing to the problem complexity. Despite this, the ray paths often seem to be insensitive to the wavenumber orientation, suggesting hidden simplifications. These simplifications would be welcome, because the complexity and range of possible solutions dramatically increase when dispersion, topography, and mean flow are all accounted for (Killworth and Blundell 2004), increasing the complexity of exposing the results intelligibly.

The main purpose of this paper is to show that simplifications can indeed be achieved owing to the dispersion relation in the present context being generically of the form
i1520-0485-36-1-104-e3
which stems from the fact that only the ratio of the meridional to zonal wavenumbers τ enters the GLWM [(1)(2)]. The second equality in (3) follows from the first one and is clarified below by (4). Equation (3) expresses that both the zonal phase speed c and the group velocity cg (and hence the rays) depend on the orientation of the wavenumber only (because they depend only on τ); that is, they are “azimuthally” dispersive. Although not a new result, its consequences for our understanding of ray properties have not yet been fully explored. Azimuthal dispersion is much less familiar than the dispersion of standard quasigeostrophic waves, dominated primarily by the wave vector’s norm rather than by its orientation (e.g., Straub 1994). Note, though, that azimuthal dispersion of the GLWM is only nontrivial when the meridional component of the velocity υ and/or zonal topographic variations Hx are retained, because otherwise the propagation is simply westward and nondispersive and is describable from environmental conditions alone. If so, τ must then be known at all points along a given ray path. The values of τ, however, are at present only poorly constrained, both observationally and theoretically. It is therefore theoretically important to determine whether there exist ways to approximate the ray paths in terms environmental conditions alone and, if so, when and under what assumptions.
To address these questions, let us mathematically quantify the sensitivity of the group velocity to the wavenumber orientation. From (3), we can compute cg = (∂kxω,∂kyω)=(cgx, cgy) and ∂τcg as follows:
i1520-0485-36-1-104-e4
i1520-0485-36-1-104-e5
so that, according to (5), the dependence of the group velocity upon the wavenumber orientation is minimized when the τ curvature of the zonal phase speed vanishes, that is, when c depends linearly on τ:
i1520-0485-36-1-104-e6
with C1 and C2 being functions of position only. In this case, we have simply cgx = C1 and cgy = C2 so that the rays obey the familiar autonomous ODE dy/dx = cgy(x, y)/cgx(x, y), which is integrable from the knowledge of environmental conditions alone.

In practice, ∂2c/∂τ2 usually differs from zero1 so that the best we can hope for is to approximate c to sufficient accuracy by its linear tangent over a sufficiently large interval of τ values [τmin, τmax]. If so, the problem reduces to identifying such intervals, each interval thus defining a dynamically distinct “quasi nondispersive” (QND) regime. In this paper, this idea is applied to the GLWM with no mean flow recently considered by KB99, Tailleux (2004), and Tailleux (2003), as a first step toward a more complete understanding of the full problem in (1) and (2), the justification being that the rays in KB03 differ little from those of KB99. In that case, this paper shows that τ can be subdivided into three–four subintervals defining as many QND regimes.

The organization of this paper is as follows: Ray theory is derived in section 2 and is simplified in section 3. The theory of QND regimes is given in section 4 and is tested on idealized examples in section 5. Section 6 discusses the effects of stratification. Section 7 summarizes and discusses the results.

2. Parameter dependences of the WKB Rossby wave phase speeds

a. The WKB eigenvalue problem for long Rossby waves over topography

Without mean flow, the dimensionless GLWM [(1)(2)] for purely periodic waves becomes
i1520-0485-36-1-104-e7
i1520-0485-36-1-104-e8
where
i1520-0485-36-1-104-e9
N2(z) = −−10dρ0/dz is the squared Brunt–Väisälä frequency, N0 = N(0), and H is the total ocean depth. The only restriction on N is that it depend upon depth only, with specific examples using an exponential buoyancy profile for which δ is taken as the e-folding scale. We also define
i1520-0485-36-1-104-e10
where kλ and kϕ are local angular wavenumbers defined from the phase function Σ by kλ = ∂Σ/∂Λ and kϕ = ∂Σ/∂Φ. Here, Σ = Σ(Λ, Φ) is assumed to be a function of the slow coordinates (Λ, Φ) = (ελ, εϕ), where ε is the assumed small classical WKB parameter and λ and ϕ denote longitude and latitude, respectively. The link with the dimensional wavenumbers kx and ky used in the introduction is given by
i1520-0485-36-1-104-e11
where R is the earth’s radius; Γ is the eigenvalue of (7)(8) and can be written
i1520-0485-36-1-104-e12
where ω = ω0R2Ω/(N20δ2) is the dimensionless counterpart of the dimensional frequency ω0 and Ω is the earth’s rotation rate. Equation (12) yields the dispersion relation of the system:
i1520-0485-36-1-104-e13
The parameter μ in (10) physically measures the ratio of the topographic β effect over the background planetary β effect (Rhines 1970) and critically controls the effect of topography on long Rossby waves. The parameter Z entering μ is a wavenumber ratio from which latitudinal effects have been removed. Indeed, over a flat bottom, assuming appropriate boundary conditions, it reduces to Z = 2(λλE), which is a function of longitude only, whereas τ = kϕ/kλ = 2(λλE)/tanϕ strongly depends on latitude (this definition of τ being the one appropriate in polar coordinates). The use of Z is shown below to simplify greatly the ray equations.
Equations (7)(8) show that μ and h are the two main parameters that control the spatial variations of the eigenvalue Γ and eigenmode F. In formal terms, let us write
i1520-0485-36-1-104-e14
For fixed μ and h, the number of possible solutions for Γ is infinite, as in the SLT. This is illustrated in Fig. 1, which depicts the quantity 1/Γ as a function of μ for h = 1 (deep stratification) and h = 5 (shallow stratification). In this paper, we shall focus only on the baroclinic (n = 1) mode. Tailleux (2003) discusses the pseudobarotropic mode (n = 0). The function Γ(μ, h) for the first baroclinic mode is depicted in Fig. 2. The salient feature of this function is an increased sensitivity to topographic variations in the deeply stratified case (low values of h) relative to that of the shallowly stratified case. This is manifest in the low relative variations of Γ for negative values of μ/h and values of h > 2. The same also occurs for positive values of μ, but less strongly. The last important feature is the rapid transition behavior occurring for small values of μ/h, which is apparent in Fig. 1 for all baroclinic modes.

b. Functional dependence of Γ upon h and μ

To establish how Γ varies with μ and h, we compute ∂Γ/∂μ and ∂Γ/∂h. Multiplying (7) by F and integrating by parts while accounting for (8) yields as a result
i1520-0485-36-1-104-e15
where
i1520-0485-36-1-104-e16
are two strictly positive functions of μ and h. In this section, a prime denotes differentiation with respect to σ and the index b denotes the bottom value: Fb = dF/(−h). Next, let us assume that all quantities (i.e., F, Γ, μ, and h) depend on some arbitrary parameter α. Thus, differentiating (7) and (8) with respect to α (in the same spirit as in KB99’s appendix A) yields
i1520-0485-36-1-104-e17
i1520-0485-36-1-104-e18
where νb = ν(−h) and ∂α = ∂/∂α. Now, multiplying (17) by F and integrating by parts over depth and accounting for (18) yields, after some algebra,
i1520-0485-36-1-104-e19
The sought-for result is obtained by successively replacing α by μ and h, which yields
i1520-0485-36-1-104-e20
Equation (20) establishes a universal characteristic of Γ—namely, to be an increasing function of both μ and h regardless of the stratification chosen. The functional dependence predicted by (20) is consistent with that illustrated in Figs. 1 and 2. The two main coefficients K2I and K2b, as well as K2r = K2b/K2I, are depicted as a function of μ/h, for two particular values of h, in Fig. 3. The coefficient K2I is seen to have a hyperbolic-tangent kind of profile, whereas K2b and K2r have a bell shape, reaching their maximum for the particular value μc discussed further in the text.

3. Reduction of the canonical ray equations

a. Standard canonical ray equations

The standard canonical ray equations (e.g., Lighthill 1978)
i1520-0485-36-1-104-e21
for the present problem were derived previously by KB99 and are given in appendix B for reference, where x = (λ, ϕ) and k = (kλ, kϕ). These equations can be further simplified thanks to the rays’ dispersion being controlled by Z only. To show this, we use (13) to rewrite the dispersion relation as follows:
i1520-0485-36-1-104-e22
From (22), the following expressions for the group velocity are obtained:
i1520-0485-36-1-104-e23
i1520-0485-36-1-104-e24
Equation (24) demonstrates clearly that the departure from purely zonal propagation is due to the azimuthal dispersion, for c = 0 if and only if c is independent of Z.

b. Evolution equation for Z

The importance of Z comes from the fact that the dispersion relation and Z’s definition allow us to write
i1520-0485-36-1-104-e25
Because the wave vector derives from a phase function Σ, it must satisfy the compatibility condition:
i1520-0485-36-1-104-e26
By inserting (25) into (26), some straightforward algebra yields the following evolution equation for Z:
i1520-0485-36-1-104-e27

c. Final reduced ray equations

Further reduction can be achieved by using λ instead of s as pseudo–time coordinate, since s does not enter the dispersion relation explicitly. This is simply achieved by dividing (24) and (27) by c:
i1520-0485-36-1-104-e28
Equations (28), supplied by suitable initial conditions along the eastern boundary, allow for the complete determination of the rays. Last, we relate cZ, cλ, and cϕ to the stratification and topography considered:
i1520-0485-36-1-104-e29
i1520-0485-36-1-104-e30
where D/ = ∂/∂λ + (c/c)∂/∂ϕ.

4. Quasi-nondispersive theory of baroclinic ray evolution

a. Qualitative results about ray propagation

Let us first show that a qualitative understanding of the ray trajectories can be readily achieved through a qualitative analysis of the evolution equation in (30). To that end, note that (30) can be rewritten as K2r tanϕϕhDϕ = K2r tanϕλhDλ or alternatively as = K2r tanϕ(∂ϕhDϕ + ∂λhDλ) = tanϕK2rDh. As a result, (30) can be rewritten in the two following equivalent ways:
i1520-0485-36-1-104-e31
A third useful form can be obtained by noting that λ does not enter these equations explicitly, so that the following also holds along the rays:
i1520-0485-36-1-104-e32
What (31) and (32) say is that it is a qualitative generic property of the rays to be systematically deflected equatorward (poleward) when propagating uphill (downhill). This makes propagation purely westward in the absence of topographic variations, as expected. Over variable topography, however, the rays will necessarily undergo latitudinal excursions and hence will undergo change in the Coriolis parameter, whose intensity is measured by the function αc(μ, h) = K2rh according to (32).

To understand the physical meaning of αc(μ, h), we note that (32) would immediately integrate exactly as D/ ln( f /hαc) = 0 if αc were constant. The particular case αc = 1 is recognized as classical barotropic propagation along f /h isocontours (e.g., Holland 1967). We may therefore classify the propagation as “subbarotropic” or “superbarotropic” depending on whether αc < 1 or αc > 1. For the exponential buoyancy profile, Fig. 4 shows that αc typically decays rapidly for increasing |μ| as O(1/μ2) (shown in appendix D), which, for a fixed h, reaches a maximum for μ = μc(h) such that |μc(h)/h| < 1 in general (Fig. 5). At first sight, αc appears to be essentially a function of μ/h, because its dependence upon h is weak; these slight variations upon h are nevertheless important to understand the behavior of specific ray examples given in section 5, as further discussed in section 6.

Somewhat unexpected and surprising is the possibility of superbarotropic behavior suggested by the existence of values αc > 1 in Fig. 4, which contrasts with the subbarotropic behavior of baroclinic rays obtained for a realistic stratification by KB99 and KB03. A closer examination of Fig. 4 shows, however, that values of αc > 1 only occur in the deeply stratified case (specifically for h < 2) and, furthermore, over a very narrow range of small values of μ/h. In fact, even in the case of a deeply stratified ocean, specific examples of rays shown in section 5 indicate that superbarotropic behavior occurs over too narrow of a range of μ/h values to be of practical significance.

The rapid decay of αc for large values of μ/h means that baroclinic rays are expected to be influenced significantly by topographic variations only for small values of μ/h and influenced very little for large values of μ/h. One therefore intuitively expects baroclinic rays over topography to fall mainly into three dynamically distinct regimes: one topographic regime for small values of μ/h and two essentially westward regimes for large negative and positive values of μ/h. The following section rationalizes this idea more quantitatively.

b. Theory of quasi-nondispersive regimes

Because αc is maximum for small μ/h but is very small for large |μ/h| (Fig. 4), the rays must be topographically altered in the former case but must remain mainly westward in the latter case. To formalize this idea, μ is in the present case a more natural parameter than τ to express azimuthal dispersion, so that the introduction’s results need to be adapted. First, we have from (B1) and (B2):
i1520-0485-36-1-104-e33
so that the condition for cg to be independent of μ is that Γ be a linear function of μ, that is, Γ ≈ An(h)μ + Bn(h). Over any interval in which this is approximately valid, one shows that
i1520-0485-36-1-104-e34
In this case, (31) becomes
i1520-0485-36-1-104-e35
which is integrable as follows:
i1520-0485-36-1-104-e36
with h0 being an arbitrary reference depth. Because Γ has a universal hyperbolic tangent–like shape, three straight lines are the minimum required to approximate it. Figure 6 (left panel) shows Γ approximated by its two asymptotes and the tangent line going through its finite inflection point μ = μc(h). In Fig. 6 (right panel), an additional fourth straight line tangent to Γ at μ = 0 is considered. Each straight line defines a nondispersive or quasi-nondispersive regime, which we describe below. The relevance of the QND regimes to interpret the behavior of actual rays is studied in Section 5.

c. Quasi-nondispersive topographic (QNDT) regime

The topographic QND regime is associated with the tangent going through Γ’s finite inflection point, which is also where K2r is maximum; see appendix C. The latter occurs at μ = μc(h), which is depicted in Fig. 5 for the exponential buoyancy profile. It follows that the QNDT regime provides an upper bound for the maximum latitudinal excursions of actual rays. The governing equation for QNDT rays is from (31):
i1520-0485-36-1-104-e37
which is exactly integrable as f /H(h) = a constant. The function αc[μc(h), h] is depicted in Fig. 7 (left panel). Of interest is that the function αc is seen to vary little with h, suggesting that a good approximation to (37) is f /hαc = a constant. The QNDT regime is superbarotropic for h < 1 and subbarotropic for h > 1, approximately.

d. Quasi-nondispersive westward regimes

The QND westward regimes are associated with the two asymptotes of Γ for large μ; see Fig. 6. Appendix D shows that Γ ≈ Γ0(h)[1 − C±0(h)]/μ for large μ, which implies Γμμ ≈ −2Γ0C±0(h)/μ3 → 0 when μ → ±∞. As shown in appendix D, K2rC±0(h)/μ2 for large μ, so that from (37) the propagation is purely westward at leading order, with the group velocity then being simply given by
i1520-0485-36-1-104-e38
and departures from zonality occurring only at O(1/μ2). Propagation “faster” and “slower” than standard occurs for positive and negative μ, the faster regime being that studied in Tailleux and McWilliams (2001). As seen in Fig. 2, cg as expressed by (38) depends sensitively on h only in the deeply stratified case (h < 1). Equation (38) is found to be an excellent approximation (with a relative error of less than 1%) of cg for values of μ/h ≥ 3, the approximation raising up to about 20% for μ/h ≈ 2. The coefficients C±0 (normalized by h) controlling the departure from the limit Γ(±∞, h) are depicted in Fig. 7 (right panel).

e. Behavior near μ = 0 and the string function of Tyler and Käse (2001)

Figure 8 shows the accuracy of the three-piecewise linear approximation to decrease for increasing h > 2 near μ ≈ 0, suggesting the need for a fourth QND regime associated with the linear tangent line of Γ at μ = 0. This four-piecewise linear approximation is depicted in Fig. 6 (right panel) for h = 5. From (20), we see that, at μ = 0, we have the interesting analytical property that Γμ(0, h) = Γh(0, h), so that (37) reduces to
i1520-0485-36-1-104-e39
which is immediately integrated as
i1520-0485-36-1-104-e40
Here, the quantity Γ(0, h)/f is recognized as the (baroclinic) string function of Tyler and Käse (2001). Figure 6 (right panel) shows, however, that the string function regime occurs only within a very small interval around μ = 0, and so side effects are likely to prevent it from being a truly well-defined nondispersive regime. This point will be further discussed in the following section. Note here that the above results establish rigorously the range of validity and existence conditions under which the string function can be defined for a continuously stratified fluid, which Tyler and Käse’s (2001) study did not address.

5. Test of the QND theory on an idealized case

To test the QND theory, we now compare rays obtained by solving the full ray equations with rays computed using the three- and four-piecewise linear approximation of the function Γ. The idealized examples presented use the exponential buoyancy profile and a Gaussian topography H varying with longitude only given by
i1520-0485-36-1-104-eq1
The ridge is centered in an ocean basin of zonal extent 2λ0 = 2π/3 rad = 120° and total mean depth Hmean = 4500 m. The ridge’s width is chosen as λT = 12π/225 rad = 9.6° to occupy the interval [40°, 80°]. The ray equations in (29) and (30) were integrated numerically with the initial conditions ϕ = π/4 and Z = 0 along the eastern boundary. Our comparison focuses here on 1) the ray latitude ϕ, 2) the normalized wavenumber ratio Z, 3) the topographic parameter μ, and 4) the wave speed Γ, depicted in Figs. 9 and 10 for a shallow stratification δ = 1000 m and a deep stratification δ = 5000 m. Both figures use ΔH = 1000 m, with the left (right) panels depicting the comparison with the three-piecewise (four-piecewise) linear approximation.

In the shallowly stratified case, Fig. 9a shows the “exact” ray to be strongly deflected southward on the ridge’s eastward side before resuming an essentially westward propagation on the ridge’s westward side along the latitude 37°, resulting in a total southward excursion of about 8°. In that case, the three-piecewise linear approximation greatly exaggerates this total southward deflection (by about 6°), this error being greatly reduced by the four-piecewise linear approximation (the error dropping to about 2°). On the other hand, both approximations correctly initiate the topographic and purely westward regimes at the right locations, that is, at the bottom of the ridge’s eastern flank and at the ridge’s top, respectively. Because the regime transitions are governed by the value of the topographic parameter μ, itself strongly correlated with the normalized ratio Z, it follows that both quantities are expected to be reproduced correctly by the three- and four-piecewise linear approximations. That this is so can be checked in Figs. 9b and 9c. In both cases, the discrepancy essentially arises because the topographic regime becomes very inaccurate quantitatively close to the ridge’s eastern side. With regard to Z and μ, Figs. 9b and 9c show that they are in general well predicted, especially μ, which is the most important quantity for determining the ray characteristics. Furthermore, Fig. 9c (left panel) shows that in the three-piecewise linear approximation μ follows initially the topographic regime until it enters the transition region that separates it from the slower westward regime close to the ridge’s top eastern side. It then rapidly enters the faster westward regime on the ridge’s eastward side, to resume its course in the topographic regime upon reentering the nearly flat part in the westernmost part of the basin. Essentially the same occurs for the four-piecewise linear approximation (Fig. 9c, right panel), except for the fact that the trajectories in the flat regions of the basin are now associated with the string function regime. In both cases (figure panels), Fig. 9c reveals that the error in the ray latitude evident in Fig. 9a can be explained by the fact that μ lies in the strongly dispersive transition region separating the topographic regime from the slow westward one close to the ridge’s top eastern side. Figure 9d displays the behavior of Γ along the rays. Because 3.5 ≤ h ≤ 4.5, we expect from Fig. 8 that the three-piecewise linear approximation will badly approximate Γ in the flat parts of the basin where μ ≈ 0. This is clearly verified in the left panel of Fig. 9d, the approximation being much better with the four-piecewise linear approximation (Fig. 9d, right panel).

In Fig. 10, 0.7 ≤ h ≤ 0.9 over the whole basin so that from Fig. 8 we expect little differences between the predictions of the two kinds of approximations for Γ, for then the topographic and string function regimes coincide. The left and right panels of Fig. 10 are therefore essentially identical, and only one panel need be commented upon. In contrast to the previous case, all the quantities appear to be approximated well by the QND theory. There are discrepancies, which occur mainly for Z, μ, and Γ (Figs. 10b–d) on the ridge’s western side, but these are largely acceptable in view of the more drastic approximations involved in WKB theory. Note that in this case the actual latitudinal ray excursion is smaller than in the previous case, being about 4° versus 8°. This behavior, which is not intuitive, is discussed in the next section.

6. Some remarks on the effects of stratification

To shed light on the combined effects of stratification and topography on Rossby wave propagation, this section seeks to clarify the role of stratification on the two following issues: 1) its effect on the speed of propagation along the rays and 2) its effect on the rays’ direction and latitudinal variations. For an exponential buoyancy profile, which possesses some of the salient features of realistic stratification, this amounts to quantifying the respective influence of the surface value N0, which measures the overall strength of the stratification, and of the e-folding scale δ, which measures the shallow or deep character of the stratification.

Given that the dimensional value of Γ is Γdim = N20δ2Γ(μ, h), clearly both N0 and δ are important in determining the speed along the rays. In the standard case (i.e., for μ = 0), Chelton et al. (1998) show that Γdim scales as the vertical integral of the buoyancy frequency as follows:
i1520-0485-36-1-104-eq2
with a proportionality factor that is a function of μ in general. In the exponential case, we have
i1520-0485-36-1-104-eq3
so that two dynamically distinct regimes arise: 1) the deeply stratified case h ≪ 1, for which 1 − ehh, and ΓdimN20H2, which is independent of δ so that Γ ∝ h2 is an increasing quadratic function of h,2 and 2) The shallowly stratified case h ≫ 1, for which ΓdimN20δ2 so that Γ ∝ 1 is independent of h, in agreement with the qualitative features of the function Γ(μ, h) depicted in Fig. 2 for fixed μ. It is obvious that μ is the second important parameter controlling Γdim, because it controls the abovementioned factor of proportionality, thereby modulating Γdim. As seen previously, Γ is an increasing function of μ so that its effect need not be discussed further. How the stratification controls μ is tackled below.
To address regime 2, a useful reference starting point is the two-layer model because its rays display the same qualitative features as in this study (e.g., see Tailleux and McWilliams 2002). In the two-layer model, the direction of the rays for a single “WKB” wave is that of the energy flux:
i1520-0485-36-1-104-e41
[see (2.15) of Tailleux and McWilliams 2002], where g′ is the reduced gravitational acceleration, p1 and p2 are the pressure perturbations in each layer, and H1 and H2 = HH1 are the upper- and lower-layer thicknesses respectively. Equation (41) says that the rays’ direction always lie between the purely westward contour lines of H1/f = a constant and the geostropic contour lines H2/f = a constant so that the latter is as an upper bound for the largest possible southward or northward latitudinal ray excursions and therefore is the two-layer analog of the QND topographic regime. The counterparts of N0 and δ in the two-layer model are g′ and the upper-layer thickness H1. According to (41), g′ affects the norm of F but not its direction. We therefore expect δ, rather than N0, to control the direction of the QND topographic rays, which would be consistent with the maximum topographic exponent being a function of h = H/δ only.

In the two-layer model, increasing H1 decreases H2 = HH1 and hence increases the maximum possible deflections of the contours H2/f = a constant. Therefore, decreasing δ (increasing h) is expected to decrease the maximum topographic exponent αc(μc). Figure 4 confirms this in the deeply stratified case for values h < 2–3, but the reverse occurs in the shallowly stratified case. The two-layer model accordingly appears to mimic the continuously stratified fluid only in the deeply stratified case, which was not obvious a priori. An analog for the shallowly stratified case remains to be found.

Is the above helpful to predict the corresponding response of actual rays?—Unfortunately not straightforwardly, as is apparent in Figs. 9 and 10, which provide examples of actual rays undergoing larger latitudinal excursions in the shallow δ = 1000 m case than in the deep δ = 5000 m one, in contrast with the QND topographic regime. Figure 11 illustrates this further by superimposing the actual ray behavior with the paths of the QND topographic regime, barotropic geostrophic contours f /h = a constant, and baroclinic string function Γ(0, h)/f of Tyler and Käse (2001), all originating from the same eastern boundary latitude. While the barotropic geostrophic contours f /h = a constant, given for reference, of course do not depend upon the stratification, both the string function and the QND topographic regime do, showing increased latitudinal variations as δ is increased. This confirms that an accurate prediction of actual ray latitudinal excursions requires the consideration of at least three–four QND regimes and that no single idealized QND regime can be expected to do it on its own; this obviously undermines the claim of Tyler and Käse (2001) that the string function alone can often reproduce ray behaviors.

From the above, it is clear that understanding actual ray behavior requires understanding the global behavior of the topographic exponent αc. This is difficult, because αc depends upon δ not only through h, but also through μ, whose dependence upon δ is a priori nonlocal because determining μ requires the integration of the ray equations. However, the function μ/h was empirically found to be largely insensitive to δ when it is negative—that is, in the topographic regime taking place on the ridge’s eastern flank—as illustrated in Fig. 12. The nonlocal behavior of μ can hence be ignored if αc is regarded as a function of μ/h and h rather than as a function of μ and h, as depicted in Fig. 4 (left panel). The differences between Figs. 9 and 10 can thus be explained by the fact that αc is larger for h = 1 than for h = 5 for negative μ/h, as shown in Fig. 4 (right panel). This is interesting, because at first sight αc(μ/h, h) appears to depend little upon h (Fig. 4, left panel), yet it appears that it is these slight variations upon h that determine the considerable differences in the latitudinal excursions between Figs. 9 and 10.

It is possible to trace the insensitivity of negative μ/h to changes in δ evidenced in Fig. 12 back to the properties of the Z equation in (29), provided one makes the approximation that αc(μ/h, h) is independent of h in the latter, as suggested by Fig. 4 (left panel). To see it, we rewrite the evolution equations for μ/h and Z for a purely zonal topography, using (10) and (29), as follows:
i1520-0485-36-1-104-e42
i1520-0485-36-1-104-e43
If αc(μ/h, h) ≈ αc(μ/h) in (43), then only the term depending on νb actually depends on δ. When this term can be neglected, both (42) and (43) depend formally only upon μ/h and no longer upon δ. The neglect of the νb term is justified for small values of |μ/h| but not for large ones; this precisely accounts for Fig. 12, because negative values of μ/h are much smaller than positive ones in absolute value.

In conclusion, it does not appear possible to predict accurately the total latitudinal variations of a given ray based on the knowledge of the stratification and topography alone, that is, without explicitly integrating the ray equations. Indeed, even in the simple exponential case, the differences between the two numerical experiments of Figs. 9 and 10 are caused by the very slight dependence of αc(μ/h, h) upon h, which is unlikely to be generic (unlike the dependence upon μ/h); inferences to more general stratifications are therefore not easily made, requiring studies on a case by case basis, as was done for instance by KB99 and KB03 and hence is not repeated here.

Instead, we examine the issue of whether the actual oceanic stratification is compatible with the exponential buoyancy model, as is often assumed. To that end, we estimate the maximum topographic exponent αc(μc) and the function C+0(h)/h, which describe respectively the quasi-nondispersive topographic and fast westward regimes, for realistic stratifications. Figures 13 and 14 show the values of αc(μc(h), h) and C+0(h)/h for the South Pacific, Atlantic, North Pacific, and Indian Oceans, as a function of latitude, computed from the 1° × 1° “Levitus” atlas dataset for temperature and salinity, each point representing a particular longitude. To be compatible with the exponential buoyancy model, αc should lie in the interval [0.8, 2] from Fig. 7 (left panel). In reality, αc is lower than unity almost everywhere, reaching values lower than 0.8 in a number of places, especially at low latitudes, suggesting that the exponential model is only appropriate at higher latitudes. An exception is the northwestern part of the Atlantic Ocean, where very high values of αc considerably larger than 2 are reached and where the exponential model is also likely to be invalid. With regard to the behavior of C+0/h, its values lie mostly within the interval [0.1, 0.3] and show a high degree of consistency between the different oceans. According to Fig. 7, such values would place actual oceans in the shallowly stratified category (large h). Of interest is that the same anomalous behavior is found in the abovementioned northwestern part of the Atlantic Ocean, confirming the analysis based on αc.

7. Summary and discussion

The energy trajectories of continuously stratified baroclinic long Rossby waves can be strongly deflected by variable topography when the wave fronts (i.e., lines of constant phase) and isobaths are nearly parallel (small |μ/h|), but these become nearly westward otherwise. The ray trajectories are a priori determined both by environmental conditions (topography and stratification) and by azimuthal dispersion, that is, the wave vector’s orientation. Yet, the propagation was found empirically by KB99 and KB03 to depend little on the wave-vector orientation. This is rationalized here by showing that weak sensitivity to the wave vector’s orientation, and hence quasi-nondispersive behavior, occur whenever the zonal phase speed varies locally approximately linearly with τ. Because the dependence of the zonal phase speed upon τ has a universal hyperbolic tangent–like shape, three such QND regimes are naturally defined, which corresponds to the topographic and two westward regimes, whose characteristics are primarily determined by environmental conditions alone.

The propagation along each particular QND topographic ray path approximately follows contours of the form f /hαc = a constant, where αc is a near constant that depends on the stratification and on the regime considered. The two (faster and slower than standard) westward regimes correspond to the first and second surface-intensified baroclinic modes discussed in Tailleux and McWilliams (2001). The agreement between rays computed with the full ray equations and those computed using a QND approximation of the dispersion relation is usually good, except in the shallowly stratified case in which a fourth QND regime associated with the string function of Tyler and Käse (2001) is needed to improve the comparison. Predicting accurately the ray latitudinal excursions requires solving the full ray equations: it is in general impossible for a single regime to provide a global approximation for the full ray equations, which contradicts Tyler and Käse’s (2001) claim that the baroclinic string function can often reproduce actual ray paths. At best, this can only be true over very limited areas.

We believe that the method discussed in this paper is important for allowing a systematic investigation of the underlying preferred directions of propagation (the hyperbolic structure) of the GLWM, which is potentially useful to understand dispersive waves also. Indeed, the situation discussed here is analogous to that of the barotropic case for which Holland (1967) finds the hyperbolic geostrophic contours f/H to remain useful for discussing nonhyperbolic short dispersive barotropic waves. A last point is that baroclinic Rossby wave propagation over topography is not easily discussed in terms of equivalent shallow-water dynamics and hence in terms of a 2D potential vorticity conservation principle. For lack of such a principle, the concepts of QND regimes may be the only alternative to a simple and economical description of purely azimuthally dispersive waves.

Acknowledgments

The author gratefully acknowledges Allan N. Kaufman and several anonymous reviewers who greatly helped to improve and clarify the manuscript both in presentation and content.

REFERENCES

  • Abramowitz, M., and I. A. Stegun, 1965: Handbook of Mathematical Functions. Dover, 1046 pp.

  • Chelton, D. B., and M. G. Schlax, 1996: Global observations of oceanic Rossby waves. Science, 272 , 234238.

  • Chelton, D. B., R. A. de Szoeke, M. G. Schlax, K. E. Naggar, and N. Siwertz, 1998: Geographical variability of the first baroclinic Rossby radius of deformation. J. Phys. Oceanogr., 28 , 433460.

    • Search Google Scholar
    • Export Citation
  • Colin de Verdière, A., and R. Tailleux, 2005: The interaction of a baroclinic mean flow with long Rossby waves. J. Phys. Oceanogr., 35 , 865879.

    • Search Google Scholar
    • Export Citation
  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Holland, W. R., 1967: On the wind-driven circulation in an ocean with bottom topography. Tellus, 19 , 582599.

  • Killworth, P. D., and J. Blundell, 1999: The effect of bottom topography on the speed of long extratropical planetary waves. J. Phys. Oceanogr., 29 , 26892710.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. Blundell, 2003a: Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part I: The local problem. J. Phys. Oceanogr., 33 , 784801.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. Blundell, 2003b: Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part II: Ray propagation and comparison with observations. J. Phys. Oceanogr., 33 , 802821.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. Blundell, 2004: The dispersion relation for planetary waves in the presence of mean flow and topography. Part I: Analytical theory and one-dimensional examples. J. Phys. Oceanogr., 34 , 26922711.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., D. B. Chelton, and R. deSzoeke, 1997: The speed of observed and theoretical long extratropical planetary waves. J. Phys. Oceanogr., 27 , 19461966.

    • Search Google Scholar
    • Export Citation
  • Leblond, P. H., and L. A. Mysak, 1978: Waves in the Ocean. Elsevier, 602 pp.

  • Lighthill, J., 1978: Waves in Fluids. Cambridge University Press, 504 pp.

  • Rhines, P. B., 1970: Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid Dyn., 1 , 273302.

  • Straub, D. N., 1994: Dispersion of Rossby waves in the presence of zonally varying topography. Geophys. Astrophys. Fluid Dyn., 75 , 107130.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., 2003: Comments on “The effect of bottom topography on the speed of long extratropical planetary waves.”. J. Phys. Oceanogr., 33 , 15361541.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., 2004: A WKB analysis of the surface signature and vertical structure of long extratropical baroclinic Rossby waves over topography. Ocean Modell., 6 , 191219.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., and J. C. McWilliams, 2001: The effect of bottom pressure decoupling on the speed of extratropical, baroclinic Rossby waves. J. Phys. Oceanogr., 31 , 14611476.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., and J. C. McWilliams, 2002: Energy propagation of long, extratropical Rossby waves over slowly varying zonal topography. J. Fluid Mech., 473 , 295319.

    • Search Google Scholar
    • Export Citation
  • Tyler, R. H., and R. Käse, 2001: A string function for describing the propagation of baroclinic anomalies in the ocean. J. Phys. Oceanogr., 31 , 765776.

    • Search Google Scholar
    • Export Citation

APPENDIX A

Eigensolutions for an Exponential Stratification

In the case in which the buoyancy profile is given by the exponential function N = N0ez/δ, the dimensionless eigenproblem can be put under the form
i1520-0485-36-1-104-ea1
with boundary conditions F(0) = 0 and F(−h) = μF′(−h). Its solution is classically expressed in terms of Bessel functions of zero order (e.g., Abramowitz and Stegun 1965; note that the latter use the notation ℓ2 where we use 1/Γ). The particular solution satisfying the upper boundary condition F0(0) = 0 is given by
i1520-0485-36-1-104-ea2
Defining F1(σ) = Y0−1/2)J1−1/2eσ) − J0−1/2)Y1−1/2eσ), we have F0(σ) = Γ−1/2eσF1(σ), yielding for the bottom boundary condition
i1520-0485-36-1-104-ea3
This equation provides an implicit equation for Γ−1/2 that in general must be solved numerically. To compute the coefficients K2b, K2r, and K2I, we use the analytical formula
i1520-0485-36-1-104-ea4
using formula (11.3.31) of Abramowitz and Stegun (1965, p. 484). This yields
i1520-0485-36-1-104-ea5
Then, from (15), one computes
i1520-0485-36-1-104-ea6

APPENDIX B

Standard Canonical Ray Equations

Standard algebra yields
i1520-0485-36-1-104-eb1
i1520-0485-36-1-104-eb2
i1520-0485-36-1-104-eb3
i1520-0485-36-1-104-eb4
where K2r = K2b/K2I = Fb2/∫0h F′(σ)2 . These equations are equivalent to those of KB99.

APPENDIX C

Inflection Point of Γ and Maximum of K2r

We want to prove that the inflection point of Γ occurring for finite μ = μc(h) coincides with the maximum of K2r. By construction, Γμμ[μc(h), h] = 0 so that by differentiating (20) with respect to μ it follows that
i1520-0485-36-1-104-ec1
Now, rewriting (15) under the form
i1520-0485-36-1-104-ec2
and differentiating this expression with respect to μ, accounting for (C1) and the fact that Γμ = Γ2K2b, yields
i1520-0485-36-1-104-ec3
The sought-for result is obtained by noting that K2r = (Γ2K2b)/(Γ2K2I). Differentiating the latter with respect to μ by accounting for (C1) and (C2) yields immediately ∂K2r/∂μ = 0, QED.

APPENDIX D

Behavior for Large μ

The behavior of baroclinic rays for large μ is obtained by expressing the eigensolutions of (7)(8) as a regular series expansion in powers of 1/μ as follows:
i1520-0485-36-1-104-ed1
The approach of inserting (D1) into (7)(8) and retaining the terms of equal powers of 1/μ yields a series of linear problems, which are omitted for brevity, with the leading-order problem between the eigenvalue problem for the surface-intensified vertical modes discussed in Tailleux and McWilliams (2001). Solving the first and second order of these problems yields
i1520-0485-36-1-104-ed2
i1520-0485-36-1-104-ed3
and hence
i1520-0485-36-1-104-ed4
Now, from (20), we have K2b = Γμ2, which yields
i1520-0485-36-1-104-ed5
By using (D4) and (D5) in combination with (15), we arrive at
i1520-0485-36-1-104-ed6

Fig. 1.
Fig. 1.

The eigenvalue 1/Γ as a function of μ/h for (left) h = 1 and (right) h = 5 for the exponential buoyancy profile described in appendix A. Each curve is numbered according to the flat-bottom modes classification (i.e., defined for μ = 0) such that n = 0 refers to the barotropic mode and n ≥ 1 to the baroclinic modes. Note that the barotropic curve n = 0 always become negative at μ/h = −1, regardless of h.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 2.
Fig. 2.

The function Γ(μ, h) for the first baroclinic mode (n = l), as a function of μ/h and h, for the exponential buoyancy profile.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 3.
Fig. 3.

The functions K2r(−) (solid line), K2b (dashed line), and K2I (dotted line) as a function of μ/h for the exponential buoyancy profile for (left) h = 1 and (right) h = 5.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 4.
Fig. 4.

(left) Contour plot of log10(αc) as a function of μ/h and h, for the exponential buoyancy profile (see text for explanation). The isovalue αc = 1 is contoured as the thick line, so that subbarotropic (superbarotropic) occurs for negative (positive) isovalues. (right) Selected examples of αc as a function of μ/h for the particular cases h = 1 (thick line) and h = 5 (dotted–dashed line).

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 5.
Fig. 5.

The critical value of μc(h)/h as a function of h at which the finite inflection point of Γ occurs.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 6.
Fig. 6.

The function Γ as a function of μ/h for h = 5 along with its piecewise linear approximations, using (left) three and (right) four nondispersive regimes.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 7.
Fig. 7.

(left) The maximum topographic exponent αc[μc(h), h] = hK2r[μc(h), h] as a function of h; superbarotropic and subbarotropic behaviors occur for αc greater than and lesser than 1, respectively. (right) The function C+0(h)/h (solid) and C0(h)/h (dashed) as a function of h.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 8.
Fig. 8.

The function Γ(0, h) (smooth curve) and its approximate value obtained from the three-piecewise linear approximation (upper curve). The discrepancy between the two curves becomes important for h > 2.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 9.
Fig. 9.

Comparison of ray quantities computed with the exact Γ (thick solid lines) vs computed with the piecewise linear approximation in terms of (left) three and (right) four nondispersive regimes (thin solid lines): (a) ray latitude ϕ, (b) normalized wavenumber ratio Z, (c) parameter μ along with the boundaries separating the different nondispersive regimes, and (d) the eigenvalue Γ. The topography is the Gaussian ridge varying only with λ, as described in the text, and the stratification is the exponential buoyancy profile. Parameter values are δ = 1000 m and ΔH = 1000 m.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 10.
Fig. 10.

Same as in Fig. 9 but with δ = 5000 m and ΔH = 1000 m.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 11.
Fig. 11.

(left) Comparison of the ray using the exact Γ of Fig. 9 (thick solid line) with the string function (dashed line), topographic regime (dotted line), and barotropic geostrophic f /h contours (thin solid line). (right) Same as in left panel but for the actual ray of Fig. 10 (thick solid line). Thin solid line is the barorotropic geostrophic contours f /h, and the dashed–dotted line is the superposition of the string function and topographic regime, which are indistinguishable in this case.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 12.
Fig. 12.

Evolution of the functions (left) μ/h and (right) μ as a function of longitude for various δ at fixed topography. The values of δ used are 1) δ = 100 m (for the curve having the weakest amplitude for positive μ/h; 2) δ = 500 m, 3) δ = 1000 m, 4) δ = 2500 m, 5) δ = 5000 m, and 6) δ = 10 000 m (for the curve having the largest amplitude for positive μ/h). The total topography is Hmean = 4500 m, and the topographic ridge height is ΔH = 1500 m.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 13.
Fig. 13.

The maximum topographic exponent αc(μc) = hK2r[μc(h), h] computed for the (a) South Pacific Ocean, (b) Atlantic Ocean, (c) North Pacific Ocean, and (d) Indian Ocean. Each point represents the value estimated from a 1° × 1° grid box.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

Fig. 14.
Fig. 14.

Values of the function C+0(h)/h for the (a) South Pacific Ocean; (b) Atlantic Ocean; (c) North Pacific Ocean; and (d) Indian Ocean. Each point represents the value estimated from a 1° × 1° grid box.

Citation: Journal of Physical Oceanography 36, 1; 10.1175/JPO2823.1

1

Except when υ = Hx = 0, in which case we have simply c = C1(x, y) = cgx(x, y), with the additional simplification C2 = cgy = 0, as mentioned earlier.

2

This case is actually similar to the constant-N case considered by Rhines (1970) and Straub (1994), which explains the independence upon δ.

Save
  • Abramowitz, M., and I. A. Stegun, 1965: Handbook of Mathematical Functions. Dover, 1046 pp.

  • Chelton, D. B., and M. G. Schlax, 1996: Global observations of oceanic Rossby waves. Science, 272 , 234238.

  • Chelton, D. B., R. A. de Szoeke, M. G. Schlax, K. E. Naggar, and N. Siwertz, 1998: Geographical variability of the first baroclinic Rossby radius of deformation. J. Phys. Oceanogr., 28 , 433460.

    • Search Google Scholar
    • Export Citation
  • Colin de Verdière, A., and R. Tailleux, 2005: The interaction of a baroclinic mean flow with long Rossby waves. J. Phys. Oceanogr., 35 , 865879.

    • Search Google Scholar
    • Export Citation
  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Holland, W. R., 1967: On the wind-driven circulation in an ocean with bottom topography. Tellus, 19 , 582599.

  • Killworth, P. D., and J. Blundell, 1999: The effect of bottom topography on the speed of long extratropical planetary waves. J. Phys. Oceanogr., 29 , 26892710.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. Blundell, 2003a: Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part I: The local problem. J. Phys. Oceanogr., 33 , 784801.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. Blundell, 2003b: Long extratropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography. Part II: Ray propagation and comparison with observations. J. Phys. Oceanogr., 33 , 802821.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., and J. Blundell, 2004: The dispersion relation for planetary waves in the presence of mean flow and topography. Part I: Analytical theory and one-dimensional examples. J. Phys. Oceanogr., 34 , 26922711.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., D. B. Chelton, and R. deSzoeke, 1997: The speed of observed and theoretical long extratropical planetary waves. J. Phys. Oceanogr., 27 , 19461966.

    • Search Google Scholar
    • Export Citation
  • Leblond, P. H., and L. A. Mysak, 1978: Waves in the Ocean. Elsevier, 602 pp.

  • Lighthill, J., 1978: Waves in Fluids. Cambridge University Press, 504 pp.

  • Rhines, P. B., 1970: Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid Dyn., 1 , 273302.

  • Straub, D. N., 1994: Dispersion of Rossby waves in the presence of zonally varying topography. Geophys. Astrophys. Fluid Dyn., 75 , 107130.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., 2003: Comments on “The effect of bottom topography on the speed of long extratropical planetary waves.”. J. Phys. Oceanogr., 33 , 15361541.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., 2004: A WKB analysis of the surface signature and vertical structure of long extratropical baroclinic Rossby waves over topography. Ocean Modell., 6 , 191219.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., and J. C. McWilliams, 2001: The effect of bottom pressure decoupling on the speed of extratropical, baroclinic Rossby waves. J. Phys. Oceanogr., 31 , 14611476.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R., and J. C. McWilliams, 2002: Energy propagation of long, extratropical Rossby waves over slowly varying zonal topography. J. Fluid Mech., 473 , 295319.

    • Search Google Scholar
    • Export Citation
  • Tyler, R. H., and R. Käse, 2001: A string function for describing the propagation of baroclinic anomalies in the ocean. J. Phys. Oceanogr., 31 , 765776.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    The eigenvalue 1/Γ as a function of μ/h for (left) h = 1 and (right) h = 5 for the exponential buoyancy profile described in appendix A. Each curve is numbered according to the flat-bottom modes classification (i.e., defined for μ = 0) such that n = 0 refers to the barotropic mode and n ≥ 1 to the baroclinic modes. Note that the barotropic curve n = 0 always become negative at μ/h = −1, regardless of h.

  • Fig. 2.

    The function Γ(μ, h) for the first baroclinic mode (n = l), as a function of μ/h and h, for the exponential buoyancy profile.

  • Fig. 3.

    The functions K2r(−) (solid line), K2b (dashed line), and K2I (dotted line) as a function of μ/h for the exponential buoyancy profile for (left) h = 1 and (right) h = 5.

  • Fig. 4.

    (left) Contour plot of log10(αc) as a function of μ/h and h, for the exponential buoyancy profile (see text for explanation). The isovalue αc = 1 is contoured as the thick line, so that subbarotropic (superbarotropic) occurs for negative (positive) isovalues. (right) Selected examples of αc as a function of μ/h for the particular cases h = 1 (thick line) and h = 5 (dotted–dashed line).

  • Fig. 5.

    The critical value of μc(h)/h as a function of h at which the finite inflection point of Γ occurs.

  • Fig. 6.

    The function Γ as a function of μ/h for h = 5 along with its piecewise linear approximations, using (left) three and (right) four nondispersive regimes.

  • Fig. 7.

    (left) The maximum topographic exponent αc[μc(h), h] = hK2r[μc(h), h] as a function of h; superbarotropic and subbarotropic behaviors occur for αc greater than and lesser than 1, respectively. (right) The function C+0(h)/h (solid) and C0(h)/h (dashed) as a function of h.

  • Fig. 8.

    The function Γ(0, h) (smooth curve) and its approximate value obtained from the three-piecewise linear approximation (upper curve). The discrepancy between the two curves becomes important for h > 2.

  • Fig. 9.

    Comparison of ray quantities computed with the exact Γ (thick solid lines) vs computed with the piecewise linear approximation in terms of (left) three and (right) four nondispersive regimes (thin solid lines): (a) ray latitude ϕ, (b) normalized wavenumber ratio Z, (c) parameter μ along with the boundaries separating the different nondispersive regimes, and (d) the eigenvalue Γ. The topography is the Gaussian ridge varying only with λ, as described in the text, and the stratification is the exponential buoyancy profile. Parameter values are δ = 1000 m and ΔH = 1000 m.

  • Fig. 10.

    Same as in Fig. 9 but with δ = 5000 m and ΔH = 1000 m.

  • Fig. 11.

    (left) Comparison of the ray using the exact Γ of Fig. 9 (thick solid line) with the string function (dashed line), topographic regime (dotted line), and barotropic geostrophic f /h contours (thin solid line). (right) Same as in left panel but for the actual ray of Fig. 10 (thick solid line). Thin solid line is the barorotropic geostrophic contours f /h, and the dashed–dotted line is the superposition of the string function and topographic regime, which are indistinguishable in this case.

  • Fig. 12.

    Evolution of the functions (left) μ/h and (right) μ as a function of longitude for various δ at fixed topography. The values of δ used are 1) δ = 100 m (for the curve having the weakest amplitude for positive μ/h; 2) δ = 500 m, 3) δ = 1000 m, 4) δ = 2500 m, 5) δ = 5000 m, and 6) δ = 10 000 m (for the curve having the largest amplitude for positive μ/h). The total topography is Hmean = 4500 m, and the topographic ridge height is ΔH = 1500 m.

  • Fig. 13.

    The maximum topographic exponent αc(μc) = hK2r[μc(h), h] computed for the (a) South Pacific Ocean, (b) Atlantic Ocean, (c) North Pacific Ocean, and (d) Indian Ocean. Each point represents the value estimated from a 1° × 1° grid box.

  • Fig. 14.

    Values of the function C+0(h)/h for the (a) South Pacific Ocean; (b) Atlantic Ocean; (c) North Pacific Ocean; and (d) Indian Ocean. Each point represents the value estimated from a 1° × 1° grid box.

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