## 1. Introduction

### a. Motivation and background

In layered models of the ocean the assumption of a deep resting layer is often made, motivated by the surface intensification of many phenomena. The propagation speed of first-mode, baroclinic Rossby waves in such models is always faster than in models with all the layers active. For example, in the so-called reduced-gravity or 1.5-layer model, baroclinic Rossby waves are faster than in the full two-layer model by the factor *H*/*H*_{2}, where *H*_{2} is the lower layer thickness and *H* the total ocean depth.

This behavior is qualitatively similar to the recent finding by Chelton and Schlax (1996) from TOPEX/Poseidon altimetry that observed extratropical, baroclinic Rossby waves are systematically faster than predicted by the standard linear theory. The discrepancy is illustrated in Fig. 1 (bottom), which displays the ratio of observed to theoretical phase speeds. This ratio is seen to increase poleward and can reach values as high as two to three at high latitudes. Given that the mixed layer and the pycnocline also increase with latitude on average, such a behavior is also expected for *H*/*H*_{2}.

The “Radon Transform” method used by Chelton and Schlax (1996) does not distinguish between low- and high-frequency motions within the altimetric dataset that resolves periods of months and longer; this method simply provides a global phase-speed estimate based on the assumption that the propagation is quasi-nondispersive. Subsequently, Zang and Wunsch (1999) computed frequency/wavenumber diagrams from TOPEX/Poseidon to determine an empirical dispersion relationship for the observed first-mode baroclinic Rossby waves. They found a discrepancy with the standard theory only for the higher frequency motions. This is in agreement with the study by Hong et al. (1998); they did not find enhanced phase speeds necessary to account for decadal sea-level variability in the North Atlantic subtropical gyre. Currently it is unclear what distinguishes these methods of analysis or lower and higher frequency Rossby waves, although the latter seem to have shorter zonal length scales in extratropical regions. In the present study, we focus on motions of subbasin length scale, though still in the nondispersive limit of Rossby wave propagation, hence we only attempt to compare our results with that of Chelton and Schlax (1996). Future studies should include dispersive and finite-domain effects as well to allow a more complete comparison with the frequency/wavenumber analysis of the altimetric data.

There have been several theoretical attempts to explain Chelton and Schlax's (1996) results. The most popular explanation to date is that mean-flow effects account for most of the discrepancy (e.g., Killworth et al. 1997; Dewar 1998; de Szoeke and Chelton 1999; Liu 1999). Estimates from this theory are shown in Fig. 1 (middle panel). They do better than the standard theory but are still systematically too low, suggesting that other effects might also be important.

Among the other theoretical possibilities, the speed doubling mechanism due to a periodic forcing proposed by White (1977) and Qiu et al. (1997) can be rejected because it is only a visual effect that is filtered out by the radon transform method used by Chelton and Schlax (1996). The radon transform is able to distinguish between individual propagating components so that it does not see the sumcos*ωt* + cos(*ωt* + *kx*) as a spatially modulated wave, 2 cos*kx* cos(*ωt* + *kx*/2), propagating at twice the standard-theory speed (as argued by White 1977), but as two waves yielding two peaks. The first has an infinite phase speed, and the second has the standard speed. This is easily verified by running the radon transform on the above expression.

An explanation involving the coupling with the atmosphere is investigated by White et al. (1997), but the analysis is not very convincing due to the coarse-resolution data and the limited time series used. The effect of a smooth topography in a continuously stratified fluid is investigated by Killworth and Blundell (1999) by means of a WKB theory, but the authors do not find a systematic phase-speed enhancement in this case. On the other hand, Tailleux and McWilliams (2000) find a systematic phase-speed enhancement over a steep topography in a two-layer model, a possibility previously raised by Rhines (1977) and Veronis (1981) and related to the faster phase speed of the reduced-gravity model.

*p*

_{2}is the bottom pressure,

*η*

_{2}the displacement of the layers interface,

*g*′ the reduced gravity of the density difference across the interface between the layers,

*H*

_{1}the upper layer thickness,

*f*the Coriolis parameter, and

*β*=

*df*/

*dy*its latitudinal derivative; see Tailleux and McWilliams (2000) for details about the derivation of (1). In the standard theory, the bottom pressure is proportional to the interface according to

*p*

_{2}= −

*H*

_{1}/

*H*

_{2}

*η*

_{2}so that the right-hand side contributes to the propagation to define an effective phase speed,

*c*= −

*g*′

*βH*

_{1}

*H*

_{2}/(

*f*). However, if we assume that for some reason ∂

^{2}H

_{x}p_{2}is uncorrelated to ∂

_{x}η_{2}, then the effective speed becomes

*c*= −

*g*′

*βH*

_{1}/

*f*

^{2}, which is faster by the factor

*H*/

*H*

_{2}. This condition is obviously realized in a reduced-gravity model that assumes

*p*

_{2}= 0.

In order to assess the relevance of this idea for the interpretation of Chelton and Schlax's (1996) results, we confront two main difficulties. First, we have to decide whether there is a reasonable basis for assuming either bottom pressure compensation or decoupling (i.e., decorrelation from pressure fluctuations in the upper ocean). Second, we need to determine how to estimate the analog of the enhancement factor *H*/*H*_{2} for a realistic stratified ocean.

### b. Basis for bottom pressure decoupling

In Tailleux and McWilliams (2000), the phase-speed enhancement occurs as the result of the decoupling of ∂_{x}p_{2} with ∂_{x}η_{2}. In this case, the responsible mechanism is the steep topography. Past studies investigating the effects of the topography on the baroclinic modes have shown the existence of top-trapped modes, but they generally failed to associate this feature with faster propagation (Rhines 1970; Straub 1994; Samelson 1992; Reznik and Tsybaneva 1994; Hallberg 1997).^{1}

The possibility of top-trapping does not appear to be uniquely related to the existence of topography. For instance, surface intensification occurs in the calculations of McWilliams et al. (1986) for the nonlinear propagation of a vortex over a flat bottom, with the resulting phase-speed enhancement noted by the authors. However, the physical mechanisms responsible for the surface intensification in these calculations are still largely unexplained. Certainly, the effects of nonlinearities, eddies, small-scale topography, and weak stratification of the deep ocean may all play a role in decoupling the abyss from the upper ocean. Potentially relevant are the concepts of arrested Ekman layers [e.g., see Garrett et al. (1993), Mellor and Wang (1996) and references therein], which can give rise to bottom pressure decoupling when isopycnals intersect topography. We also note that circumstances in which eddy–topography interactions are detuned in the deep ocean have been found in the numerical experiments of two-layer quasigeostrophic (QG) turbulence by Rhines (1975) and Rhines (1977).

The studies cited above compose the present theoretical evidence that bottom pressure compensation or decoupling and/or strong surface intensification frequently occur in the ocean, along with the accompanying phase-speed enhancement. In this paper, we propose a method to estimate the latter effect in a continuously stratified fluid. The pressure-decorrelation estimates are displayed in Fig. 1 (top), along with those from the mean-flow (middle) and standard (bottom) theories, and the pressure-decorrelation estimates are in reasonable agreement with the observational estimates.

### c. Quantification of the phase-speed enhancement factor

If one tries to estimate the value of the enhancement factor by using typical values for *H* and *H*_{2} based on the depth of the pycnocline (e.g., *H*_{2} = 4000 m, *H* = 5000 m), *H*/*H*_{2} appears to be quite small (1.25 in this example) compared to the observed phase-speed enhancement factor of 2 or more (Fig. 1, bottom). According to Flierl (1978), however, the value of *H*/*H*_{2} depends on which physical process one is interested in modeling accurately. For instance, Flierl (1978) finds that *H*_{2} = *H*/2 is required to model strong topographic effects accurately (thus implying *H*/*H*_{2} = 2), whereas the choice of a much thinner upper layer is better suited to modeling the response to wind forcing. In this paper, we propose a method that eliminates the concern for the right two-layer calibration by focusing on a continuously stratified fluid.

### d. Methodology and organization

*p*is a state vector of dimension

*N,*and

**A**an

*N*×

*N*square matrix. Within this simple framework, the standard theory is analogous to the case where

**A**is a diagonal matrix with constant coefficients. In this case, the modal amplitudes (the components

*p*of

_{i}*p*) are independent of each other, each obeying a scalar equation of the type ∂

*+*

_{t}p_{i}*c*∂

_{i}*= 0. In order to include realistic effects, like that of the mean flow (Killworth et al. 1997) or the topography (Killworth and Blundell 1999), the matrix*

_{x}p_{i}**A**must be transformed into one with variable coefficients, including nonzero off-diagonal ones. As a result, the

*p*become coupled. Under the assumption that there is a scale separation between the spatial variations of the environment and that of the waves, which justifies the use of WKB theory, this coupling can be interpreted in terms of anomalous propagation. In this case, the coupled modes and their phase speeds are simply determined by diagonalizing the matrix

_{i}**A**.

**A**is somehow perfectly known. As a result, the

*q*are expected to be well-behaved deterministic quantities, with well-defined spatial and temporal characteristics. However, this might not be the case. Indeed, one may easily conceive that the neglected nonlinearities, small scales, temporal variations, etc. if included, would have a significant impact on the dynamics of the system. In this paper, we assume that the main consequence of neglecting the above effects is to render aspects of the model dubious and inaccurate. Our strategy thus consists in modifying this dubious dynamics by an additional constraint that we believe may often be more accurate. In some ways this strategy is equivalent to making a closure assumption. In this paper, we assume that a particular combination of the

_{i}*p*that constitute the bottom pressure has, in fact, the behavior of a random variable. In other words, we add to the usual dynamics a constraint of the form

_{i}_{j=0}

^{N}

*s*=

_{j}p_{j}*X*(

*x, t*),

*X*a random process with unspecified characteristics and

*s*constants discussed later in the text. This constraint allows the expression of one of the

_{j}*p*,

_{i}*p*say, in terms of the others. After substitution of this constraint, our schematic problem (2) is transformed into one of the form,

_{N}*p** = (

*p*

_{1}, · · · ,

*p*

_{N−1}) has one component less than

*p*;

**A*** is a (

*N*− 1)-dimensional square matrix; and

**F**(

*x, t*) is a random vector. In this paper, we argue that the new wave properties are to be obtained by diagonalizing

**A***, not

**A**.

This paper is organized as follows: The coupled equations for the planetary geostrophic standard modes over topography that result from making the bottom pressure decoupling hypothesis are presented in section 2. The theory for the phase-speed enhancement resulting from bottom pressure decoupling and application to several illustrative examples are presented in section 3. The issue of the vertical structure of the modes is addressed in section 4. A comparison of our predictions with the observational estimates is presented in section 5. A discussion concludes the paper in section 6.

## 2. Planetary geostrophic equations

As stated in the introduction, we are interested in investigating how the coupling of the standard modes resulting from any departure from the idealized assumptions of the standard linear model—flat bottom, horizontally uniform stratification, free, inviscid, small perturbations to a state of rest, etc.—may modify the propagation properties of the planetary Rossby modes. Of course, all the effects neglected in the standard theory—forcing, dissipation, topography, nonlinearities, etc.—contribute to the coupling, so that the most general problem is quite complicated. For the reasons given in the introduction, we shall focus here on the topography effects only.

### a. Dynamical model

**u**= (

*u, υ*) denotes the horizontal velocity;

*w*the vertical velocity;

*p*=

*P*/

*ρ*

_{0}the pressure divided by a reference density

*ρ*

_{0};

*b*= −

*gρ*/

*ρ*

_{0}the buoyancy, with

*g*the gravitational acceleration and

*ρ*the density;

*N*

^{2}(

*z*) = −(

*g*/

*ρ*

_{0})

*d*

*ρ*

_{0}/

*dz*

*ρ*

_{0}(

*z*);

*f*the Coriolis parameter; and

*β*=

*df*/

*dy*the latitudinal derivative of

*f*. These equations are supplemented by the linearized boundary conditions,

*η*(

*x, y, t*) denotes the departure of the sea surface height from its resting position

*z*= 0, while

*z*= −

*H*(

*x, y*) denotes the position of the ocean bottom. From a mathematical viewpoint, the only departure from the standard linear theory lies in the bottom boundary condition (9) used in place of the standard (i.e., flat-bottom boundary condition)

*w*(−

*H*

_{0}) = 0.

### b. Projection onto the standard modes

*H*

_{0}[taken here as the maximum value of

*H*(

*x, y*)] and background stratification with buoyancy frequency

*N*(

*z*), the standard normal modes are defined as the eigenmodes of the following Sturm-Liouville problem:

*c*can be interpreted as the phase speed of the gravity wave modes of the system. The set of modes

_{k}^{′}

_{k}(

*z*)

_{k}(

*z*)

*p*by taking the vertical derivative of (13). Simplifying the result by means of (10) yields

*w*:

*x, y, t*) is defined by

_{k}at

*z*= −

*H*(

*x, y*), so it would vanish in the standard theory. It is therefore responsible for the modal coupling here. We can get a sense of the physical meaning of

_{j}(−

*H*) in a Taylor series,

_{j}

*H*

_{0}

_{j}

*H*

*H*

_{0}

*H*

^{′}

_{j}

*H*

_{j}(−

*H*) ≈ (

*H*

_{0}−

*H*)Φ

^{′}

_{j}(−

*H*) so that

*H*

_{0}−

*H*(

*x, y*) is the topographic anomaly and

**U**

_{b}the bottom velocity. In cases where the bottom velocity is in approximate geostrophic balance,

*p*the bottom pressure. Equation (20) shows that

_{b}*w*(0) =

*g*

^{−1}∂

*p*/∂

*t*(0). Using (13) and (16), this yields

*i*≠

*j.*As a result, the sought-for coupled modal equations are given by

*j*= 0), one can also neglect the time derivative in (27), which yields the classical Sverdrup balance. These classical approximations permit the simplification of (26) and (27) as follows:

*βc*

_{j}^{2}/

*f*

^{2}

*βR*

_{j}^{2},

*R*the

_{j}*j*th baroclinic Rossby radius of deformation.

### c. The hypothesis of bottom pressure decorrelation

*H*should be retained for modeling purposes. For this reason, we first use (29) to express

*H*

_{0}. Since we are much more confident about how to choose

*N*

^{2}(

*z*) and

*H*

_{0}than

*H*(

*x, y*) in the actual ocean, we believe that the equations constituting (32) have more informative content and a more deterministic character than (30) and (31).

*p*in terms of the normal modes. From (13), one has

_{b}^{′}

_{k}(−

*H*)

^{′}

_{k}(−

*H*

_{0})

*O*(

^{2})

^{″}

_{k}(−

*H*

_{0})

^{′}

_{k}(−

*H*)

^{′}

_{k}(−

*H*

_{0})

^{′}

_{k}(−

*H*)

^{′}

_{k}(−

*H*

_{0})

*k*= 0) since Φ

_{0}(

*z*) varies linearly with depth. Equation (33) allows the barotropic amplitude Π

_{0}to be expressed in terms of

*p*and the baroclinic amplitude Π

_{b}_{j},

*j*≥ 1. Inserting the result into (32) yields

As stated in the introduction, this paper argues that the randomness of the topography introduces a randomness in the system whose main consequence is to render (31) greatly uncertain and inaccurate. Since (31) and (35) are coupled, it follows that randomness must also be present in (35). On the physical grounds detailed in the introduction, which mainly stem from our previous analysis of the two-layer model solutions reported in Tailleux and McWilliams (2000), we believe that the main consequence of this randomness is that the right-hand side of (35) does not have any influence on propagating solutions of the system. It follows that the anomalous propagation due to the departure from the idealized assumptions of the standard linear model considered here can be understood by analyzing the propagation properties of (35) with the right-hand side taken as zero.

## 3. Propagation analysis

### a. General theory

*j*≥ 1, is an infinite-dimensional system. We analyze its properties by considering successive truncations. The truncated system of order

*n,*obtained by retaining only the terms

*k*= 1, · · · ,

*n*in the infinite series in (35), is of the following form:

*n*×

*n*matrix

**M**

_{n}being

*m*=

_{jk}*σ*+

_{jk}*c*

_{j}^{2}

*δ*,

_{jk}*j*= 1, · · · ,

*n,*

*k*= 1, · · · ,

*n*;

*δ*

_{jk}*δ*;

**Γ**the

*n*-dimensional vector with components Γ

_{j}/Γ

_{0},

*j*= 1, · · · ,

*n*; and

**Π**the

*n*-dimensional vector of components Π

_{j},

*j*= 1, · · · ,

*n.*As stated previously, we assume that the right-hand side of (37) does not contribute to the propagation of the wave modes. Thus, the propagation properties of (37) are obtained by diagonalizing the matrix

**M**

*, with the enhanced phase speed given by the greatest eigenvalue.*

_{n}#### 1) The coefficients *σ*_{jk}

*σ*

_{jk}**M**

*, the coefficients*

_{n}*σ*are needed. As seen in the previous section, those are given by the expression (36) whose definition involves the coefficients Γ

_{jk}_{j}(28). We first simplify the expression for the latter by establishing the two relations

_{j}over depth, in both cases accounting for the boundary conditions (11). By using these two expressions, (28) becomes

_{0}(

*z*). We estimate those by making the classical approximation consisting in regarding

^{′}

_{0}(

*z*)

^{′}

_{0}(

*z*) =

*B*

_{0}

*B*

_{0}a constant. As a result,

^{′}

_{0}(−

*H*

_{0})

*B*

_{0}and

^{0}

_{−H0}

^{′2}

_{0}(

*z*)

*dz*

*H*

_{0}

*B*

_{0}

^{2}

_{0}=

*g*/

*B*

_{0}, yielding

#### 2) Symmetrization of the problem

_{j},

*j*= 0, 1, · · · ,

*σ*in a different basis Φ

_{jk}_{j}such that Φ

_{j}=

*B*Φ̃

_{j}_{j},

*B*a constant, will be, according to (42),

_{j}*B*only affect the eigenvectors of

_{j}**M**

*, but not its eigenvalues. Since symmetric eigenvalue problems are the easiest to solve numerically, we therefore chose the*

_{n}*B*that render

_{j}**M**

_{n}symmetric. Thus, solving

*σ*=

_{jk}*σ*for

_{kj}*B*/

_{j}*B*yields

_{k}*H*

_{0}in (43) in the square roots of the denominator.

#### 3) Leading-order phase-speed enhancement

**M**

_{n}for

*n*large have to be estimated numerically. To shed some light on the problem, however, it is of interest to derive the formal anomalous phase speed at the lowest order of truncation. In this case,

*M*

_{1}is simply the scalar

*M*

_{1}

*c*

_{1}

^{2}

*σ*

_{11}

*σ*

_{11}/

*c*

_{1}

^{2}

*σ*

_{11}from (44) yields

^{′}

_{1}(

*z*),

*N*

^{2}, which allows WKB theory to be used to derive approximations to

*σ*, is considered. Third, the case of an exponential stratification, for which WKB theory fails, concludes our investigation.

_{jk}### b. Application 1: Two-layer stratification

*N*

^{2}

*z*

*g′δ*

*z*

*H*

_{1}

*g*′ is the reduced gravitational acceleration,

*H*

_{1}is the upper-layer thickness, and

*δ*the classical Dirac distribution. This singular profile admits only two eigenmodes. With the usual approximations, the barotropic mode takes the form Φ

_{0}(

*z*) =

*B*

_{0}(

*z*+

*H*

_{0}), while the baroclinic mode is given by Φ

_{1}(

*z*) =

*B*

_{1}(

*z*+

*H*

_{0}) (−

*H*

_{0}≤

*z*≤ −

*H*

_{1}), Φ

_{1}(

*z*) = −

*B*

_{1}

*H*

_{2}/

*H*

_{1}

*z*(−

*H*

_{1}≤

*z*≤ 0). These expressions for Φ

_{0}and Φ

_{1}are used to compute the coefficients Γ

_{0}and Γ

_{1},

*σ*

_{11},

### c. Application 2: Slowly varying stratification

*N*

^{2}is a slowly varying function of depth in the usual WKB sense, approximate expressions for the standard normal modes can be obtained by making use of WKB theory. Such a method was recently used with success by Chelton et al. (1998) to study the spatial variations of the first baroclinic modes in the ocean, thus indicating that WKB theory can be used for

*N*

^{2}profiles typical of the real ocean. When WKB theory is applicable, we show in the appendix that the approximate WKB expression for the

*σ*is given by

_{jk}*N*denotes the value of the Brunt–Vaisälä frequency near the ocean bottom and

_{b}*N*

*N*remains large enough since cases for which

_{b}*N*comes close to zero would correspond to a turning point situation where WKB theory is known to fail. One can intuitively understand that (50) breaks down for

*N*too small since from (10) one has

_{b}^{″}

_{m}(

*z*

*N*≪ 1, Φ

_{m}must be approximately linear in such regions. In such situation, Φ

_{m}behaves independently of the local values of

*N,*so one does not except Φ

_{1}(−

*H*

_{0}) to depend on

*N*As a result, (50) must greatly underestimate the phase speed enhancement if

_{b}.*N*

_{b}≈ 0. We shall further clarify this issue in the following section.

**M**

_{n}to be written under the particularly simple form

**M**

_{n}*c*

_{1}

^{2}

**I**

_{n}

*ξ*

**J**

_{n}

**I**

*the*

_{n}*n*×

*n*matrix of coefficients

*I*=

_{jk}*δ*/(

_{jk}*jk*) (

*δ*being the Kroenecker delta),

_{jk}**J**

*the matrix of coefficients*

_{n}*J*= 1/(

_{jk}*jk*), and

*ξ*= 2

*N*/

_{b}*N*

**I**

*+*

_{n}*ξ*

**J**

*It also shows that the amplification factor depends uniquely on the physical parameter*

_{n}.*ξ*for all

*n.*We computed the largest eigenvalue of

**I**

*+*

_{n}*ξ*

**J**

*for*

_{n}*n*varying between 2 and 300 by standard numerical methods, for

*ξ*regularly sampled between 0 and 2. In Fig. 2 we display the amplification factor as a function of

*ξ*for the three cases

*n*= 2,

*n*= 3, and

*n*= 300. For greater values of

*n,*the resulting curves are indistinguishable from that corresponding to

*n*= 300, indicating that convergence has been achieved. We note that all the curves differ little from each other for

*ξ*≤ 1, which corresponds to a ratio

*N*/

_{b}*N*

*ξ*and the asymptotic amplification factor occurs for

*ξ*= 2, where the latter reaches the value of 4, while the former is only 3. The value

*ξ*= 2 corresponds to the case

*N*=

_{b}*N*

*N*

^{2}= const.

### d. Application 3: Exponential stratification

*N*becoming too small, we analyze the idealized case of an exponential stratification,

_{b}*N*

*z*

*N*

_{0}

*e*

^{z/δ}

*N*

_{0}is the surface value and

*δ*the

*e*-folding scale. For large values of

*H*

_{0}/

*δ,*one verifies that (52) will in general violate the conditions (72) in the deep ocean. On the other hand, we expect the WKB expression (50) to remain valid for low values of

*H*

_{0}/

*δ.*In this paragraph, we seek to understand how (50) compares with the more accurate (46). For

*N*given by (52), (50) becomes

*H*

_{0}/

*δ*. We compare (53) with the more general (46),

^{′}

_{1}(

*z*)

*N*given by (52). The two formula were computed for

*H*

_{0}/

*δ*varying between 0.1 and 10. The result, depicted in Fig. 3, shows the expected difference for high values of

*H*

_{0}/

*δ*the two approaches becoming similar for

*H*

_{0}/

*δ*≈ 1. For comparison, we also added the curve corresponding to the “exact” amplification factor for this particular stratification, which would be obtained in the asymptotic limit of an infinite number of modes. A simple method to compute this limit is given at the end of the following section.

## 4. Interpretation of the nonstandard modes

A legitimate question is what is the vertical structure of the faster modes. The main result of this section is to link the present modes to the eigenmodes of the standard Sturm-Liouville problem for which the bottom boundary condition of vanishing vertical velocity is replaced by that of vanishing pressure. These modes correspond to the surface-intensified Rossby waves occurring over steep topography, which have been extensively discussed by Rhines (1970), Veronis (1981), Charney and Flierl (1981), and Straub (1994) among others. The following paragraphs present a demonstration of this result in three steps. First, we formally derive the theoretical series expansion for the vertical structure of the faster modes. Second, this series expansion is computed numerically for the case *N*^{2} = const and compared with the solution of the Sturm-Liouville problem with vanishing bottom pressure. Third, the result is demonstrated in the general case.

### a. Formal general solution

*p.*To that end, we first rewrite Eq. (37) more concisely as follows:

*ω.*Thus, if

*R*and

_{i}*μ*denote the

_{i}*i*th eigenvector and eigenvalue of

**M**

*, the general solution of (55) takes the form*

_{n}*s*is a constant,

_{l}*k*=

_{l}*ωf*

^{2}/(

*βμ*), and

_{l}**Π**

_{part}a particular solution satisfying the equation

*p*as the following series expansion:

*R*is the

_{j,l}*j*th component of the eigenvector

**R**

*After permuting the two sums, the above expression becomes*

_{l}._{l,new}as the vertical structure associated to the

*l*th anomalous mode (the case

*l*= 1 therefore corresponds to the structure of the “faster” first baroclinic mode), one needs to be aware that the term proportional to Π

_{0}might also contribute to it. Indeed, any term contained in Π

_{0}proportional to

*e*

^{i(klx+ωt)}

_{l,new}to give a different vertical structure. We note, however, that

^{′}

_{0}(

*z*)

^{′}

_{l,new}(

*z*).

^{′}

_{l,new}(

*z*),

### b. Application to the case N^{2} = const

^{′}

_{l,new}(

*z*)

*l*= 1, in the case of a constant

*N*

^{2}. After suitable normalization, a possible set of standard modes is given by

*σ*become

_{jk}*ξ*= 2

*N*/

_{b}*N*

*N*=

_{b}*N*when

*N*

^{2}is constant. The resulting coefficients for the matrix

**M**

*are*

_{n}*m*=

_{jk}*c*

_{1}

^{2}(2 +

*δ*

_{j,k}))/

*jk*

*δ*is the Kronecker

_{j,k}*δ.*If we denote by

*R*

_{1}the eigenvector of

**M**

*associated with the largest eigenvalue, the expression for*

_{n}^{′}

_{1,new}(

*z*)

*R*

_{j,1}were estimated numerically, as well as the sum (61). We find empirically that

*n*= 300 is more than sufficient for reaching numerical convergence. In this case, we find that the resulting vertical structure is indistinguishable from that given by the following analytical expression (up to an unimportant multiplicative constant):

_{1,new}(

*z*) = cos[

*π*(

*z*+

*H*

_{0})/2

*H*

_{0}] +

*z*/

*H*

_{0}. Interestingly, the function Φ

_{1,new}(

*z*) −

*z*/

*H*

_{0}= cos[

*π*(

*z*+

*H*

_{0})/2

*H*

_{0}] can be obtained from the standard Sturm-Liouville problem by changing the bottom boundary condition to Φ′(−

*H*

_{0}) = 0 instead of Φ(−

*H*

_{0}) = 0, that is, by replacing the condition of vanishing vertical velocity by that of vanishing bottom pressure.

### c. Eigenmode with zero bottom pressure

_{1,new}(

*z*) and the gravest eigenmode

*d*

*dz*(−

*H*

_{0}) = 0, such that

*A*is taken as

*α*, (66) is inserted into the Sturm-Liouville problem (63). This yields the following problem:

_{m}_{m}one obtains

*N*

^{2}Φ

_{n}= −

*c*

_{n}

^{2}Φ

^{″}

_{m}

_{j}(0) = 0,

*j*≥ 1, using the rigid-lid approximation], one realizes that (71) is identical to the expression for

*σ*(42) originally derived in section 3. This establishes that the anomalous phase speeds can, in fact, also be determined by solving the modified Sturm-Liouville problem (63). The main interest of this result is to offer a computationally efficient way to compute the anomalous phase speeds that is much cheaper than using the series expansion introduced in section 3. On the other hand, the latter series expansion remains the better way to get approximate theoretical results for the anomalous phase speeds. The reason is that WKB theory does not seem to work satisfactorily on the modified Sturm-Liouville problem for reasons that appear related to the bottom pressure boundary condition. Similar difficulties were encountered by Killworth and Blundell (1999) (Killworth 1999, personal communication) to compute approximate solutions for the WKB modes over topography.

_{jk}## 5. Interpretation of TOPEX/Poseidon measurements

We test the relevance of the pressure-decorrelation theory by comparing its predicted amplification factors with the ratios of observed to standard phase speed from Chelton and Schlax (1996), as well as with the ratio predicted by the mean-flow theory reported in Killworth et al. (1997). The extended Levitus dataset (Boyer and Levitus 1997) is used for the computation in the regions investigated by Chelton and Schlax (1996). The amplification factors predicted by WKB theory are compared with those predicted by solving the Sturm-Liouville problem with a vanishing bottom pressure instead of a vanishing vertical velocity.

### a. Predictions of WKB theory

Values of the ratio *N _{b}*/

*N*

*n*= 300 (the upper curve depicted in Fig. 2). The dataset used is the Levitus climatology extended to ¼° resolution by Boyer and Levitus (1997). The result, shown in Fig. 4, demonstrates that the amplification factor thus obtained in the midlatitudes has a value around 2. Values close to 3 are obtained in the regions of the Antarctic Circumpolar Current, as well as in the subpolar gyre in the North Atlantic, due to the weak stratification that exists there; this implies high values for the ratio

*N*/

_{b}*N*

*N*/

_{b}*N*

Estimates of the zonal phase speed from TOPEX/Poseidon altimetric data are generally determined from use of the radon transform at a given latitude over longitudinal bands spanning approximately 30°. Amplification factors are computed over the same longitudinal bands and displayed in Fig. 1 (top). For comparison, the results of the standard theory are displayed in Fig. 1 (bottom), and the results of the mean-flow theory are displayed in Fig. 1 (middle). A logarithmic scale is used so that overestimates are put on the same footing as underestimates. The standard theory clearly underestimates the observations. The mean-flow theory does better, but still underestimates observations. In contrast, the present theory overestimates observations, with discrepancy similar in magnitude to that of the mean-flow theory. Error bars are computed as one standard deviation of our estimates for the amplification factor over the aforementioned longitudinal bands.

### b. Predictions of exact asymptotic theory

The above comparison is repeated with the amplification factors computed from solving the Sturm-Liouville problem with a vanishing bottom pressure instead of a vanishing vertical velocity. This method was shown in the previous section to yield the asymptotic value of the anomalous phase speed due to bottom pressure decorrelation; therefore, its predictions are intrinsically more accurate than those of WKB theory. The modified Sturm-Liouville problem was solved by interpolating *N* on a regular vertical grid every Δ*z* = 20 m. Given the greatly increased computational cost of this approach compared to that of the WKB method, we solved the above eigenvalue problem on a 1° × 1° grid, obtained by averaging our ¼° × ¼° *N* vertical profiles dataset on one-degree square boxes. The result, shown in Fig. 5, shows significantly smoother patterns than Fig. 4, with values generally smaller than those predicted by WKB theory. For instance, amplification factors in the North Pacific are about 0.2 less than with the WKB method. However, both approaches appear consistent with each other, in the sense that they both show systematically higher amplification factors in the southern Hemisphere than in the Northern Hemisphere, in agreement with the Chelton and Schlax (1996) findings.

Although the two approaches yield somewhat different looking patterns on the global maps of Figs. 4 and 5, we find little differences between the two methods when restricting the comparison to the regions analyzed by Chelton and Schlax (1996), as depicted in Fig. 6 (note that the lower panel of Fig. 6 is the same as the upper panel of Fig. 1). Indeed, the main difference appears in the error bars, which are smaller for the asymptotic theory owing to lesser variance and resolution.

As an additional comparison between the two methods, we depicted in Fig. 7 a nonsmoothed longitudinal section of the deep Brunt–Väisälä frequency around 35°N (top panel), along with the theoretical phase-speed enhancement predicted by the asymptotic “exact” and zero-order WKB theories (bottom panel). The latter shows that WKB estimates are closely related to that of the deep Brunt–Väisälä frequency, as theory suggests, whereas such sensitivity is absent in the other theory. The two methods have both significant differences, but without any obvious bias between them, and many regions where they roughly coincide. Therefore, it appears difficult to say when WKB theory will overestimate the asymptotic “exact” theory and conversely. Nevertheless, even though WKB theory may not always be accurate, it does capture the phase-speed enhancement resulting from bottom pressure decorrelation at very little computational cost, with the additional advantage of relating the amplification factor to observable physical quantities in a simple way.

## 6. Discussion

The main result of this paper is to quantify the phase-speed enhancement that occurs when the bottom pressure fluctuations are decoupled from the overlaying ocean dynamics in a continuously stratified fluid. When WKB theory is applicable in the vertical direction, the amplification factor has a particularly simple form that depends only on the ratio of the deep Brunt–Väisälä frequency to its vertical average. Results are also established for more general conditions, but they are more difficult to interpret physically. In particular, we formally show that the amplification factor can be very generally related at leading order to the bottom value of the vertical modal function for the first baroclinic standard mode. We also show that the faster phase speeds can be obtained by solving the Sturm-Liouville problem for the standard modes by replacing the bottom condition of zero vertical velocity with that of vanishing pressure. Thus, the faster modes are related to the classical theoretical surface-intensified modes believed to occur over steep topography.

Although regarding the faster modes and phase speeds as solutions of the modified Sturm-Liouville problem may appear much simpler than regarding them as asymptotic solutions of a series expansion in terms of the standard modes, we believe this is only advantageous from a numerical viewpoint. Indeed, it turns out that the eigenmodes of the modified Sturm-Liouville problem are very difficult to study analytically, except in the very idealized case of a constant Brunt–Väisälä frequency. In particular, we failed to find approximate WKB solutions for realistic *N*^{2} profiles. Apparently, Killworth and Blundell (1999) encountered the same problem (Killworth 1999, personal communication) when attempting to find approximate WKB solutions to the vertical modes over slowly varying topography.

The physical mechanism responsible for the phase-speed enhancement is the dynamical coupling between the standard modes that can occur in a realistic ocean because of the violation of the basic assumptions made in the standard theory. In this paper we make the ad hoc assumptions that this coupling is independent of time and location and that the normal modes cancel each other exactly at the ocean bottom. In this respect, this paper extends to a continuously stratified fluid the well-known result that baroclinic Rossby waves are faster in a reduced-gravity, layered model than in the same model with all the layers active.

Of course, the assumption of a time- and spatially independent coupling between the standard modes is an idealization unlikely to occur in the real ocean. Still, we find evidence in the literature—albeit indirect and based on fragmentary results of numerical calculations, theory, and laboratory experiments—that it is not an unreasonable one. Our theoretical predictions outside the tropical band (10°S–10°N) show reasonable agreement with observational estimates. In contrast to the mean-flow theory whose estimates are generally systematically too low, ours are generally too high. On the other hand, since the assumption of total bottom pressure decorrelation is probably an exaggeration, then a more realistic, partial decorrelation will cause a somewhat smaller phase speed enhancement that is even closer to the observational estimates. Even so, the discrepancy with the observational estimates, assuming total decorrelation, is no greater than the discrepancy for the mean-flow theory.

As it now stands our theory remains speculative in part because the nature of the decorrelation is assumed rather than deduced directly from the equations of motion. In order to achieve further progress one needs to understand how other effects like nonlinearities, forcing, and friction affect the coupling among vertical modes. Further research on this topic might give a better understanding of the pressure decorrelation mechanisms, as well as of the limits of validity in assuming a deep resting layer in layered models of the ocean.

## Acknowledgments

We thank Dudley Chelton for kindly providing his data and for frequent discussions about phase-speed enhancement. We thank Peter Killworth for interesting exchanges and two anonymous reviewers for thorough comments and suggestions that helped improve the manuscript. This research was supported by the National Science Foundation through Grant OCE-9633681 and the National Aeronautics and Space Administration through Grant NAG5-3982.

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## APPENDIX

### WKB Approximation of *σ*_{jk}

_{jk}

*σ*requires an explicit knowledge of the standard baroclinic modes. Here, we use WKB theory to compute approximate expressions for the solution Φ

_{jk}_{m}(

*z*) of the Sturm-Liouville problem (10)–(11) by following the methodology recently used by Chelton et al. (1998). To be valid, WKB requires a scale separation between

*N*/

*c*and Φ

_{m}_{m}. Mathematically, this is generally verified if for all

*m,*

*c*decreases as the number of nodes of Φ

_{m}_{m}increases, the approximation is expected to improve as

*m*increases. In Chelton et al. (1998) the authors found that WKB theory gave satisfactory results for the first baroclinic mode, even though realistic

*N*

^{2}profiles rarely strictly comply with (A1), with a maximum error around 10%. The so-called physical optics approximation for Φ

_{m}is given by

*B*is an arbitrary constant. The corresponding approximation for

_{m}^{′}

_{m}(

*z*)

_{m}(0) = 0 for the baroclinic modes (

*m*≥ 1) (Gill 1982). From (A2), this imposes

*N*

*N.*

*σ*from (44), we use (A2) to first compute

_{jk}^{2}

*x*= (1 − cos2

*x*)/2, it comes

*uu*′ = (sin

*u*)′ so that

^{′}

_{m}(

*H*

_{0}) =

*B*

_{m}(

*N*/

_{b}*c*)

_{m}^{1/2}, where

*N*=

_{b}*N*(−

*H*

_{0}) denotes the bottom value of the buoyancy frequency. By inserting this result in combination with (A8) into (44), the resulting WKB approximation for

*σ*is given by

_{jk}*c*

_{1}=

*H*

_{0}

*N*

*π*and thus

*c*=

_{m}*c*

_{1}/

*m.*

The amplification factor as a function of 2*N*_{b}/*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

The amplification factor as a function of 2*N*_{b}/*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

The amplification factor as a function of 2*N*_{b}/*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Theoretical amplification factors for the exponential stratification profile as a function of the ratio *H*_{0}/*δ*: prediction of zero-order WKB theory (thick solid line) computed from Eq. (50), prediction of formal zero order theory (thin solid line) computed from Eq. (46), and prediction obtained by solving the modified Sturm-Liouville problem as explained at the end of section 4 (dotted–dashed line)

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Theoretical amplification factors for the exponential stratification profile as a function of the ratio *H*_{0}/*δ*: prediction of zero-order WKB theory (thick solid line) computed from Eq. (50), prediction of formal zero order theory (thin solid line) computed from Eq. (46), and prediction obtained by solving the modified Sturm-Liouville problem as explained at the end of section 4 (dotted–dashed line)

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Theoretical amplification factors for the exponential stratification profile as a function of the ratio *H*_{0}/*δ*: prediction of zero-order WKB theory (thick solid line) computed from Eq. (50), prediction of formal zero order theory (thin solid line) computed from Eq. (46), and prediction obtained by solving the modified Sturm-Liouville problem as explained at the end of section 4 (dotted–dashed line)

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Global map of the theoretical enhancement factor, 1 + 2*N*_{b}*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Global map of the theoretical enhancement factor, 1 + 2*N*_{b}*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Global map of the theoretical enhancement factor, 1 + 2*N*_{b}*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Same as in Fig. 4 but with the theoretical enhancement factor computed by solving the Sturm-Liouville problem with a bottom boundary condition of vanishing bottom pressure instead of vanishing vertical velocity. [The Levitus (1994) analysis grid is subsampled by a factor of four in each direction by averaging *N* profiles over 1° square boxes.] The contour intervals is the same as in the previous figure but the levels of gray are different. White areas denote regions with amplification factor greater than 2.2

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Same as in Fig. 4 but with the theoretical enhancement factor computed by solving the Sturm-Liouville problem with a bottom boundary condition of vanishing bottom pressure instead of vanishing vertical velocity. [The Levitus (1994) analysis grid is subsampled by a factor of four in each direction by averaging *N* profiles over 1° square boxes.] The contour intervals is the same as in the previous figure but the levels of gray are different. White areas denote regions with amplification factor greater than 2.2

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

Same as in Fig. 4 but with the theoretical enhancement factor computed by solving the Sturm-Liouville problem with a bottom boundary condition of vanishing bottom pressure instead of vanishing vertical velocity. [The Levitus (1994) analysis grid is subsampled by a factor of four in each direction by averaging *N* profiles over 1° square boxes.] The contour intervals is the same as in the previous figure but the levels of gray are different. White areas denote regions with amplification factor greater than 2.2

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

The ratio of observed to theoretical phase speed for the present theory computed from solving the (top) modified Sturm-Liouville problem and (bottom) by using WKB theory. Note that the bottom panel is the same as the upper panel of Fig. 1

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

The ratio of observed to theoretical phase speed for the present theory computed from solving the (top) modified Sturm-Liouville problem and (bottom) by using WKB theory. Note that the bottom panel is the same as the upper panel of Fig. 1

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

The ratio of observed to theoretical phase speed for the present theory computed from solving the (top) modified Sturm-Liouville problem and (bottom) by using WKB theory. Note that the bottom panel is the same as the upper panel of Fig. 1

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

(top) Zonal section of the deep Brunt–Väisälä frequency (in s^{−1}) at 35°N. The nonvanishing values correspond to the Mediterranean, Pacific, and Atlantic, respectively, from left to right. (bottom) Zonal section of the theoretical amplification factor at the same latitude: zero-order WKB theory, 1 + 2*N*_{b}/*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

(top) Zonal section of the deep Brunt–Väisälä frequency (in s^{−1}) at 35°N. The nonvanishing values correspond to the Mediterranean, Pacific, and Atlantic, respectively, from left to right. (bottom) Zonal section of the theoretical amplification factor at the same latitude: zero-order WKB theory, 1 + 2*N*_{b}/*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

(top) Zonal section of the deep Brunt–Väisälä frequency (in s^{−1}) at 35°N. The nonvanishing values correspond to the Mediterranean, Pacific, and Atlantic, respectively, from left to right. (bottom) Zonal section of the theoretical amplification factor at the same latitude: zero-order WKB theory, 1 + 2*N*_{b}/*N*

Citation: Journal of Physical Oceanography 31, 6; 10.1175/1520-0485(2001)031<1461:TEOBPD>2.0.CO;2

^{1}

However, Samelson (1992) points out that surface-trapped waves have higher frequencies by the ratio *H*/*H*_{2}, whose consequence is higher phase speeds.