## 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 *c _{R}* = −

*βc*

^{2}/

*f*

^{2}, where

*f*is the Coriolis parameter,

*β*is its latitudinal derivative, and

*c*

^{2}is the equivalent internal gravity wave speed squared of the mode considered. In the SLT,

*c*

^{2}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

*c*

^{2}and hence

*c*.

_{R}**= (**v

*,*u

*υ*

*) and topography*w

*H*(

*x*,

*y*). The vertical structure

*F*(for vertical velocity) and zonal phase speed

*c*=

*ω*/

*k*of purely periodic waves are the eigensolutions of (see KB03 for details)

_{x}*k*,

_{x}*k*) are the zonal and meridional wavenumber components,

_{y}*U*=

*+*u

*/*υ k

_{y}*k*,

_{x}*N*

^{2}is the squared buoyancy frequency, and

*μ*

_{dim}= (

*f*/

*β*)(

*k*∂

_{y}*/*

_{x}H*k*− ∂

_{x}*). Special cases of the GLWM are the SLT (for*

_{y}H*μ*

_{dim}=

**= 0); the flat-bottom mean flow theories of Killworth et al. (1997) and Colin de Verdière and Tailleux (2005) (for**v

*μ*

_{dim}=

*υ*

**= 0); and the bottom-pressure compensation theory of Tailleux and McWilliams (2001) (for**v

**= 0 and**v

*μ*

_{dim}= +∞).

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.

*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

**c**

*(and hence the rays) depend on the orientation of the wavenumber only (because they depend only on*

_{g}*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*

*H*are retained, because otherwise the propagation is simply westward and nondispersive and is describable from environmental conditions alone. If so,

_{x}*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.

**c**

_{g}= (∂

*,∂*

_{kx}ω*)=(*

_{ky}ω*c*,

_{gx}*c*) and ∂

_{gy}

_{τ}**c**

*as follows:*

_{g}*τ*curvature of the zonal phase speed vanishes, that is, when

*c*depends linearly on

*τ:*

*C*

_{1}and

*C*

_{2}being functions of position only. In this case, we have simply

*c*=

_{gx}*C*

_{1}and

*c*=

_{gy}*C*

_{2}so that the rays obey the familiar autonomous ODE

*dy*/

*dx*=

*c*(

_{gy}*x*,

*y*)/

*c*(

_{gx}*x*,

*y*), which is integrable from the knowledge of environmental conditions alone.

In practice, ∂^{2}*c*/∂*τ* ^{2} usually differs from zero^{1} 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 [*τ*_{min}, *τ*_{max}]. 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

*N*

^{2}(

*z*) = −

*gρ*

^{−1}

_{0}d

*ρ*

_{0}/

*dz*is the squared Brunt–Väisälä frequency,

*N*

_{0}=

*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

*k*and

_{λ}*k*are local angular wavenumbers defined from the phase function Σ by

_{ϕ}*k*= ∂Σ/∂Λ and

_{λ}*k*= ∂Σ/∂Φ. Here, Σ = Σ(Λ, Φ) is assumed to be a function of the slow coordinates (Λ, Φ) = (

_{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

*k*and

_{x}*k*used in the introduction is given by

_{y}*R*is the earth’s radius; Γ is the eigenvalue of (7)–(8) and can be written

*ω*=

*ω*

_{0}

*R*

^{2}Ω/(

*N*

^{2}

_{0}

*δ*

^{2}) is the dimensionless counterpart of the dimensional frequency

*ω*

_{0}and Ω is the earth’s rotation rate. Equation (12) yields the dispersion relation of the system:

*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(

*λ*−

*λ*), which is a function of longitude only, whereas

_{E}*τ*=

*k*/

_{ϕ}*k*= 2(

_{λ}*λ*−

*λ*)/tan

_{E}*ϕ*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.

*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

*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

*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

*μ*and

*h*. In this section, a prime denotes differentiation with respect to

*σ*and the index

*b*denotes the bottom value:

*F*′

_{b}=

*dF*/

*dσ*(−

*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

*ν*=

_{b}*ν*(−

*h*) and ∂

*= ∂/∂*

_{α}*α*. Now, multiplying (17) by

*F*and integrating by parts over depth and accounting for (18) yields, after some algebra,

*α*by

*μ*and

*h*, which yields

*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

*K*

^{2}

_{I}and

*K*

^{2}

_{b}, as well as

*K*

^{2}

_{r}=

*K*

^{2}

_{b}/

*K*

^{2}

_{I}, are depicted as a function of

*μ*/

*h*, for two particular values of

*h*, in Fig. 3. The coefficient

*K*

^{2}

_{I}is seen to have a hyperbolic-tangent kind of profile, whereas

*K*

^{2}

_{b}and

*K*

^{2}

_{r}have a bell shape, reaching their maximum for the particular value

*μ*discussed further in the text.

_{c}## 3. Reduction of the canonical ray equations

### a. Standard canonical ray equations

**x**= (

*λ*,

*ϕ*) and

**k**= (

*k*,

_{λ}*k*). These equations can be further simplified thanks to the rays’ dispersion being controlled by

_{ϕ}*Z*only. To show this, we use (13) to rewrite the dispersion relation as follows:

*c*= 0 if and only if

_{gϕ}*c*is independent of

*Z*.

### b. Evolution equation for Z

*Z*comes from the fact that the dispersion relation and

*Z*’s definition allow us to write

*Z*:

### c. Final reduced ray equations

*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*:

_{gλ}*c*,

_{Z}*c*, and

_{λ}*c*to the stratification and topography considered:

_{ϕ}*D*/

*Dλ*= ∂/∂

*λ*+ (

*c*/

_{gϕ}*c*)∂/∂

_{gλ}*ϕ*.

## 4. Quasi-nondispersive theory of baroclinic ray evolution

### a. Qualitative results about ray propagation

*Dϕ*−

*K*

^{2}

_{r}tan

*ϕ*∂

*=*

_{ϕ}hDϕ*K*

^{2}

_{r}tan

*ϕ*∂

*or alternatively as*

_{λ}hDλ*Dϕ*=

*K*

^{2}

_{r}tan

*ϕ*(∂

*+ ∂*

_{ϕ}hDϕ*) = tan*

_{λ}hDλ*ϕK*

^{2}

*. As a result, (30) can be rewritten in the two following equivalent ways:*

_{r}Dh*λ*does not enter these equations explicitly, so that the following also holds along the rays:

*α*(

_{c}*μ*,

*h*) =

*K*

^{2}

*according to (32).*

_{r}hTo understand the physical meaning of *α _{c}*(

*μ*,

*h*), we note that (32) would immediately integrate exactly as

*D*/

*Dλ*ln(

*f*/

*h*) = 0 if

^{αc}*α*were constant. The particular case

_{c}*α*= 1 is recognized as classical barotropic propagation along

_{c}*f*/

*h*isocontours (e.g., Holland 1967). We may therefore classify the propagation as “subbarotropic” or “superbarotropic” depending on whether

*α*< 1 or

_{c}*α*> 1. For the exponential buoyancy profile, Fig. 4 shows that

_{c}*α*typically decays rapidly for increasing |

_{c}*μ*| as

*O*(1/

*μ*

^{2}) (shown in appendix D), which, for a fixed

*h*, reaches a maximum for

*μ*=

*μ*(

_{c}*h*) such that |

*μ*(

_{c}*h*)/

*h*| < 1 in general (Fig. 5). At first sight,

*α*appears to be essentially a function of

_{c}*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 *α _{c}* > 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

*α*> 1 only occur in the deeply stratified case (specifically for

_{c}*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 *α _{c}* 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

*α*is maximum for small

_{c}*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):

**c**

*to be independent of*

_{g}*μ*is that Γ be a linear function of

*μ*, that is, Γ ≈

*A*(

_{n}*h*)

*μ*+

*B*(

_{n}*h*). Over any interval in which this is approximately valid, one shows that

*h*

_{0}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*=

*μ*(

_{c}*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

*K*

^{2}

_{r}is maximum; see appendix C. The latter occurs at

*μ*=

*μ*(

_{c}*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):

*f*/H(

*h*) = a constant. The function

*α*[

_{c}*μ*(

_{c}*h*),

*h*] is depicted in Fig. 7 (left panel). Of interest is that the function

*α*is seen to vary little with

_{c}*h*, suggesting that a good approximation to (37) is

*f*/

*h*= a constant. The QNDT regime is superbarotropic for

^{αc}*h*< 1 and subbarotropic for

*h*> 1, approximately.

### d. Quasi-nondispersive westward regimes

*μ*; see Fig. 6. Appendix D shows that Γ ≈ Γ

_{0}(

*h*)[1 −

*C*

^{±}

_{0}(

*h*)]/

*μ*for large

*μ*, which implies Γ

*≈ −2Γ*

_{μμ}_{0}

*C*

^{±}

_{0}(

*h*)/

*μ*

^{3}→ 0 when

*μ*→ ±∞. As shown in appendix D,

*K*

^{2}

_{r}≈

*C*

^{±}

_{0}(

*h*)/

*μ*

^{2}for large

*μ*, so that from (37) the propagation is purely westward at leading order, with the group velocity then being simply given by

*O*(1/

*μ*

^{2}). 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,

**c**

*as expressed by (38) depends sensitively on*

_{g}*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

**c**

*for values of*

_{g}*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)

*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 Γ

*(0,*

_{μ}*h*) = Γ

*(0,*

_{h}*h*), so that (37) reduces to

*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

*H*varying with longitude only given by

*λ*

_{0}= 2

*π*/3 rad = 120° and total mean depth

*H*

_{mean}= 4500 m. The ridge’s width is chosen as

*λ*= 12

_{T}*π*/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 Δ

*H*= 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 *N*_{0}, 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.

_{dim}=

*N*

^{2}

_{0}

*δ*

^{2}Γ(

*μ*,

*h*), clearly both

*N*

_{0}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 Γ

_{dim}scales as the vertical integral of the buoyancy frequency as follows:

*μ*in general. In the exponential case, we have

*h*≪ 1, for which 1 −

*e*

^{−h}≈

*h*, and Γ

_{dim}∝

*N*

^{2}

_{0}

*H*

^{2}, which is independent of

*δ*so that Γ ∝

*h*

^{2}is an increasing quadratic function of

*h*,

^{2}and 2) The shallowly stratified case

*h*≫ 1, for which Γ

_{dim}∝

*N*

^{2}

_{0}

*δ*

^{2}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 Γ

_{dim}, because it controls the abovementioned factor of proportionality, thereby modulating Γ

_{dim}. 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.

*g*′ is the reduced gravitational acceleration,

*p*

_{1}and

*p*

_{2}are the pressure perturbations in each layer, and

*H*

_{1}and

*H*

_{2}=

*H*−

*H*

_{1}are the upper- and lower-layer thicknesses respectively. Equation (41) says that the rays’ direction always lie between the purely westward contour lines of

*H*

_{1}/

*f*= a constant and the geostropic contour lines

*H*

_{2}/

*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

*N*

_{0}and

*δ*in the two-layer model are

*g*′ and the upper-layer thickness

*H*

_{1}. According to (41),

*g*′ affects the norm of F but not its direction. We therefore expect

*δ*, rather than

*N*

_{0}, 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 *H*_{1} decreases *H*_{2} = *H* − *H*_{1} and hence increases the maximum possible deflections of the contours *H*_{2}/*f* = a constant. Therefore, decreasing *δ* (increasing *h)* is expected to decrease the maximum topographic exponent *α _{c}*(

*μ*). Figure 4 confirms this in the deeply stratified case for values

_{c}*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 *α _{c}.* This is difficult, because

*α*depends upon

_{c}*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

*α*is regarded as a function of

_{c}*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

*α*is larger for

_{c}*h*= 1 than for

*h*= 5 for negative

*μ/h*, as shown in Fig. 4 (right panel). This is interesting, because at first sight

*α*(

_{c}*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.

*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

*α*(

_{c}*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:

*α*(

_{c}*μ*/

*h*,

*h*) ≈

*α*(

_{c}*μ*/

*h*) in (43), then only the term depending on

*ν*actually depends on

_{b}*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

*ν*term is justified for small values of |

_{b}*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 *α _{c}*(

*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 *α _{c}*(

*μ*) and the function

_{c}*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

*α*(

_{c}*μ*(

_{c}*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,

*α*should lie in the interval [0.8, 2] from Fig. 7 (left panel). In reality,

_{c}*α*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

_{c}*α*considerably larger than 2 are reached and where the exponential model is also likely to be invalid. With regard to the behavior of

_{c}*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

*α*

_{c}.## 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* /*h ^{αc}* = a constant, where

*α*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.

_{c}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****,**234–238.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****,**433–460.Colin de Verdière, A., and R. Tailleux, 2005: The interaction of a baroclinic mean flow with long Rossby waves.

,*J. Phys. Oceanogr.***35****,**865–879.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****,**582–599.Killworth, P. D., and J. Blundell, 1999: The effect of bottom topography on the speed of long extratropical planetary waves.

,*J. Phys. Oceanogr.***29****,**2689–2710.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****,**784–801.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****,**802–821.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****,**2692–2711.Killworth, P. D., D. B. Chelton, and R. deSzoeke, 1997: The speed of observed and theoretical long extratropical planetary waves.

,*J. Phys. Oceanogr.***27****,**1946–1966.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****,**273–302.Straub, D. N., 1994: Dispersion of Rossby waves in the presence of zonally varying topography.

,*Geophys. Astrophys. Fluid Dyn.***75****,**107–130.Tailleux, R., 2003: Comments on “The effect of bottom topography on the speed of long extratropical planetary waves.”.

,*J. Phys. Oceanogr.***33****,**1536–1541.Tailleux, R., 2004: A WKB analysis of the surface signature and vertical structure of long extratropical baroclinic Rossby waves over topography.

,*Ocean Modell.***6****,**191–219.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****,**1461–1476.Tailleux, R., and J. C. McWilliams, 2002: Energy propagation of long, extratropical Rossby waves over slowly varying zonal topography.

,*J. Fluid Mech.***473****,**295–319.Tyler, R. H., and R. Käse, 2001: A string function for describing the propagation of baroclinic anomalies in the ocean.

,*J. Phys. Oceanogr.***31****,**765–776.

## APPENDIX A

### Eigensolutions for an Exponential Stratification

*N*=

*N*

_{0}

*e*

^{z/δ}, the dimensionless eigenproblem can be put under the form

*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 ℓ

^{2}where we use 1/Γ). The particular solution satisfying the upper boundary condition

*F*

_{0}(0) = 0 is given by

*F*

_{1}(

*σ*) =

*Y*

_{0}(Γ

^{−1/2})

*J*

_{1}(Γ

^{−1/2}

*e*) −

^{σ}*J*

_{0}(Γ

^{−1/2})

*Y*

_{1}(Γ

^{−1/2}

*e*), we have

^{σ}*F*′

_{0}(

*σ*) =

*−*Γ

^{−1/2}

*e*

^{σ}F_{1}(

*σ*), yielding for the bottom boundary condition

^{−1/2}that in general must be solved numerically. To compute the coefficients

*K*

^{2}

_{b},

*K*

^{2}

_{r}, and

*K*

^{2}

_{I}, we use the analytical formula

## APPENDIX B

### Standard Canonical Ray Equations

*K*

^{2}

_{r}=

*K*

^{2}

_{b}/

*K*

^{2}

_{I}=

*F*′

_{b}^{2}/∫

^{0}

_{−h}

*F*′(

*σ*)

^{2}

*dσ*. These equations are equivalent to those of KB99.

## APPENDIX C

### Inflection Point of Γ and Maximum of K2r

*μ*=

*μ*(

_{c}*h*) coincides with the maximum of

*K*

^{2}

_{r}. By construction, Γ

*[*

_{μμ}*μ*(

_{c}*h*),

*h*] = 0 so that by differentiating (20) with respect to

*μ*it follows that

*μ*, accounting for (C1) and the fact that Γ

*= Γ*

_{μ}^{2}

*K*

^{2}

_{b}, yields

*K*

^{2}

_{r}= (Γ

^{2}

*K*

^{2}

_{b})/(Γ

^{2}

*K*

^{2}

_{I}). Differentiating the latter with respect to

*μ*by accounting for (C1) and (C2) yields immediately ∂

*K*

^{2}

_{r}/∂

*μ*= 0, QED.

## APPENDIX D

### Behavior for Large μ

*μ*is obtained by expressing the eigensolutions of (7)–(8) as a regular series expansion in powers of 1/

*μ*as follows:

*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

*K*

^{2}

_{b}= Γ

*/Γ*

_{μ}^{2}, which yields

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

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

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

The functions *K*^{2}_{r}(−) (solid line), *K*^{2}_{b} (dashed line), and *K*^{2}_{I} (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

The functions *K*^{2}_{r}(−) (solid line), *K*^{2}_{b} (dashed line), and *K*^{2}_{I} (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

The functions *K*^{2}_{r}(−) (solid line), *K*^{2}_{b} (dashed line), and *K*^{2}_{I} (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

(left) Contour plot of log_{10}(*α _{c}*) as a function of

*μ*/

*h*and

*h*, for the exponential buoyancy profile (see text for explanation). The isovalue

*α*= 1 is contoured as the thick line, so that subbarotropic (superbarotropic) occurs for negative (positive) isovalues. (right) Selected examples of

_{c}*α*as a function of

_{c}*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

(left) Contour plot of log_{10}(*α _{c}*) as a function of

*μ*/

*h*and

*h*, for the exponential buoyancy profile (see text for explanation). The isovalue

*α*= 1 is contoured as the thick line, so that subbarotropic (superbarotropic) occurs for negative (positive) isovalues. (right) Selected examples of

_{c}*α*as a function of

_{c}*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

(left) Contour plot of log_{10}(*α _{c}*) as a function of

*μ*/

*h*and

*h*, for the exponential buoyancy profile (see text for explanation). The isovalue

*α*= 1 is contoured as the thick line, so that subbarotropic (superbarotropic) occurs for negative (positive) isovalues. (right) Selected examples of

_{c}*α*as a function of

_{c}*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

The critical value of *μ _{c}*(

*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

The critical value of *μ _{c}*(

*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

The critical value of *μ _{c}*(

*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

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

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

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

(left) The maximum topographic exponent *α _{c}*[

*μ*(

_{c}*h*),

*h*] =

*hK*

^{2}

_{r}[

*μ*(

_{c}*h*),

*h*] as a function of

*h*; superbarotropic and subbarotropic behaviors occur for

*α*greater than and lesser than 1, respectively. (right) The function

_{c}*C*

^{+}

_{0}(

*h*)/

*h*(solid) and

*C*

^{−}

_{0}(

*h*)/

*h*(dashed) as a function of

*h*.

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

(left) The maximum topographic exponent *α _{c}*[

*μ*(

_{c}*h*),

*h*] =

*hK*

^{2}

_{r}[

*μ*(

_{c}*h*),

*h*] as a function of

*h*; superbarotropic and subbarotropic behaviors occur for

*α*greater than and lesser than 1, respectively. (right) The function

_{c}*C*

^{+}

_{0}(

*h*)/

*h*(solid) and

*C*

^{−}

_{0}(

*h*)/

*h*(dashed) as a function of

*h*.

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

(left) The maximum topographic exponent *α _{c}*[

*μ*(

_{c}*h*),

*h*] =

*hK*

^{2}

_{r}[

*μ*(

_{c}*h*),

*h*] as a function of

*h*; superbarotropic and subbarotropic behaviors occur for

*α*greater than and lesser than 1, respectively. (right) The function

_{c}*C*

^{+}

_{0}(

*h*)/

*h*(solid) and

*C*

^{−}

_{0}(

*h*)/

*h*(dashed) as a function of

*h*.

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

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

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

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

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 Δ*H* = 1000 m.

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

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 Δ*H* = 1000 m.

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

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 Δ*H* = 1000 m.

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

Same as in Fig. 9 but with *δ* = 5000 m and Δ*H* = 1000 m.

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

Same as in Fig. 9 but with *δ* = 5000 m and Δ*H* = 1000 m.

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

Same as in Fig. 9 but with *δ* = 5000 m and Δ*H* = 1000 m.

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

(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

(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

(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

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 *H*_{mean} = 4500 m, and the topographic ridge height is Δ*H* = 1500 m.

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

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 *H*_{mean} = 4500 m, and the topographic ridge height is Δ*H* = 1500 m.

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

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 *H*_{mean} = 4500 m, and the topographic ridge height is Δ*H* = 1500 m.

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

The maximum topographic exponent *α _{c}*(

*μ*) =

_{c}*hK*

^{2}

_{r}[

*μ*(

_{c}*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

The maximum topographic exponent *α _{c}*(

*μ*) =

_{c}*hK*

^{2}

_{r}[

*μ*(

_{c}*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

The maximum topographic exponent *α _{c}*(

*μ*) =

_{c}*hK*

^{2}

_{r}[

*μ*(

_{c}*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

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

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

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**H _{x}* = 0, in which case we have simply

*c*=

*C*

_{1}(

*x*,

*y*) =

*c*(

_{gx}*x*,

*y*), with the additional simplification

*C*

_{2}=

*c*= 0, as mentioned earlier.

_{gy}^{2}

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