## 1. Introduction

We investigate the effects of bottom topography on the creation, propagation, amplification, depletion, and distortion of baroclinic Rossby waves. Our work is motivated in part by the Chelton and Schlax (1996) analysis of TOPEX/Poseidon altimetric measurements showing that steep topographic features significantly alter the phase speed and amplitude of low-frequency signals associated with baroclinic Rossby waves. For instance, Fig. 2 in Chelton and Schlax (1996) demonstrates that propagating patterns in the sea surface height are larger in amplitude west of major topographic features, such as the Emperor Seamounts at 39°N, 170°E, the southeast flank of Hess Rise at 32°N, 175°E, and the Hawaiian Ridge at 21°N, 155°W; similar results are found for the North Atlantic (Tokmakian and Challenor 1993; Schlax and Chelton 1994). In some instances, the waves seem to amplify over the midocean topography, while in others they seem to be created. In the North Atlantic the phase speed of baroclinic Rossby waves appears to change abruptly over the midocean ridge.

Also shown by Chelton and Schlax is that the baroclinic Rossby waves propagate faster than predicted by the standard linear flat-bottom theory, outside the tropical band (10°S–10°N), with an amplification factor seemingly increasing poleward. Killworth et al. (1997) argues that the modification of the background planetary potential vorticity field by the mean, large-scale circulation is the most likely cause of the difference. In principle, a mean flow can speed up, slow down, or even reverse the propagation of Rossby waves. According to Dewar (1998) and de Szoeke and Chelton (1999), the reason for speedup might be due to to the existence of homogeneous potential vorticity regions below the thermocline associated with the vertical shear. However, despite providing some improvement over the standard theory, the theory of Killworth et al. (1997, their Fig. 1) still predicts speeds that are systematically too low, especially in the Southern Hemisphere. It also provides no improvement in the band (10°S–10°N), where these wave theories predict systematically high speeds. This may be due to the neglect of the meridional shear of mean zonal currents, which reduces the potential vorticity gradient, hence the effective *β* value and Rossby wave speed (Philander 1979; Chang and Philander 1989;Zheng et al. 1994). Other mechanisms related to wave nonlinearity, forcing, and dissipation are rejected in Killworth et al. (1997).

Topography deserves attention as an influence because previous theories suggest that it can systematically enhance the phase speed of first-baroclinic Rossby waves. In a two-layer model, Rhines (1977) and Veronis (1981) use a WKB approach to show that the propagation of long Rossby waves over a steep topography remains westward and nondispersive, as in the flat-bottom case, but faster by the factor *H*/*H*_{2} > 1 [i.e., *c* = *g*′*βH*_{1}/*f*^{2} instead of the standard *c* = *g*′*βH*_{1}(*H* − *H*_{1})/*f*^{2}*H,* where *H*_{1} is upper layer thickness, *H* is total depth, *f* is Coriolis frequency, *β* = *df*/*dy, g*′ = *g*Δ*ρ*/*ρ* is the reduced gravitational acceleration, and Δ*ρ* is the density difference across the layer interface]. In this steep limit, achieved when the topographic *β* effect *f*‖ ∇ *H*‖/*H* is greater than the planetary *β,* the waves are trapped in the upper layer and propagate as if unaffected by the topography. The existence of top-trapped, propagating modes is a well-known results of WKB theory for baroclinic Rossby waves over topography (e.g., Samelson 1992; Reznik and Tsybaneva 1994; Straub 1994;Hallberg 1997). In this case, baroclinic Rossby waves have the same phase speed as in a 1.5-layer model (i.e., a two-layer model with a resting deep layer, for which decoupling with the topography occurs trivially).

The use of a 1.5-layer model is common among physical oceanographers. One rationalization stems from the observation that velocities and pressure gradients are generally larger near the ocean surface than near the bottom, a situation sometimes called “pressure compensation” (e.g., Mellor and Wang 1996). Pressure compensation sometimes develops in the absence of topography. It occurs in wind-forced gyre spinup (Anderson and Gill 1975; Anderson and Killworth 1977), regardless of the nature of the topography. It also occurs for a baroclinic eddy propagating over a flat bottom (McWilliams et al. 1986), with an accompanying speed enhancement. From these examples it is tempting to hypothesize that pressure compensation is widespread, as would then be its associated phase speed enhancement. This hypothesis is an attractive alternative or complement to the mean-flow theory in interpreting the results of Chelton and Schlax (1996). As yet, the speed enhancement by compensation has not been estimated for a continuously stratified fluid.

How are baroclinic Rossby waves generated, and where are they amplified and depleted? In the theory of White (1977), a time-varying, zonally uniform wind generates zonally propagating waves at the eastern boundary in order to assure no normal flow. The numerical solutions in Barnier (1988) show that free baroclinic Rossby waves can be generated in the ocean interior by the wind or by an ocean ridge. Once the waves are generated, they undergo changes in shape by the action of the wind forcing and, for short wavelengths, by dispersive effects. Viscous damping and bottom drag dissipate the waves (e.g., see Qiu et al. 1997). In Sakamoto and Yamagata (1997), a baroclinic eddy is depleted while propagating over a ridge as a result of topographic coupling with the barotropic mode, which the authors interpret in terms of JEBAR (joint effect of baroclinicity and relief). In Barnier (1988), the creation of baroclinic Rossby waves over a ridge indicates the possibility of energy transfer from the barotropic mode to the baroclinic one. In Sakamoto and Yamagata (1997), the depletion of a baroclinic eddy over a ridge suggests that the energy transfer can also occur in the opposite direction.

One interpretive issue related to JEBAR is how topography couples the standard modes to produce new wave modes in the absence of forcing and dissipation. Another issue is whether these new modes are coupled or independent. The first issue is addressed with WKB theory in, for example, Rhines (1970), Charney and Flierl (1981), Straub (1994), Samelson (1992), and Reznik and Tsybaneva (1994). These studies mainly focus on the effect of topography in the locally valid dispersion relation and the vertical modal structure. Analyses of the phase speed over a latitudinally varying topography in a continuously stratified fluid are given in Rhines (1970) and for an arbitrary (but smooth) topography in the long-wave limit in Killworth and Blundell (1999). These papers predict that the ratio of wave speeds with and without topography should lie within the interval (4/9 to 4); however, by calculating rays emanating from eastern boundaries over realistic but highly smoothed topography, Killworth and Blundell (1999) concludes against a net enhancement of phase speed, in contrast with the findings of Rhines (1977) and Veronis (1981) for a steep topography [a condition rarely encountered in Killworth and Blundell (1999) where only very smooth topography is considered].

The second JEBAR issue, related to the coupling of WKB modes, has not received much attention so far. The main reason is that an absence of coupling between wave modes is usually assumed a priori in the WKB approach. The single-mode assumption is indeed essential to the success of action&ndash℅nservation theorems for instance. Such an assumption implicitly requires that the WKB approximation remains uniformly valid throughout the domain considered. However, this needs not be the case. Evidence that the WKB modes are, in fact, unavoidably coupled is implicitly contained in the numerical simulations by Barnier (1988), Sakamoto and Yamagata (1997), or in the ones presented in this paper. In these papers, one usually starts from a situation with a single standard wave mode in the flat eastern part of the basin and ends up with two distinct standard wave modes in the flat western part, the middle part of the basin being occupied by some topographic features over which wave modes are not as unambiguously defined as over the flat-bottom parts. If there was no mode coupling over the topographic features, only one single mode would be obtained in the western part by the action conservation principle. From a theoretical viewpoint, the coupling of WKB modes must be intrinsically linked to the breakdown of WKB theory. We are not aware, however, of previous works describing the general circumstances of this link. A first step toward a theoretical understanding of WKB-mode coupling is achieved by Hallberg (1997) who suggests that the breakdown of WKB theory is likely to occur near turning points. Such points correspond to a region in wavenumber space where the dispersion curve *ω* = const of the mode considered exhibits a sharp gradient. The physical interpretation given by Hallberg is that the Rossby wave energy cannot stay within a single wave mode at a turning point without violating mass conservation. On this basis, he is able to derive heuristic estimates for the amplitude of the additional mode required. In this respect, this case is different from the previously known case of coastal waves where the switch of one type of mode to a different type occurs with the energy staying within a single mode (Allen and Romea 1980). In contrast, Killworth and Blundell (1999) assumes explicitly that no energy scattering between WKB modes occurs and makes ray calculations for the world oceans that exhibit caustics, but no turning points. In this case, the absence of turning points seems to be due to the lack of dispersive effects (P. D. Killworth 1999, personal communication). Since caustics are also known to be associated with a local breakdown of WKB theory, they may also be linked to WKB-mode coupling, although the generally accepted idea is that they correspond to a transition region between a one-mode solution to a no-mode solution (e.g., Lighthill 1978).

Since the above remarks suggest that even the WKB modes are not uncoupled over topography, one may as well deal with the standard modes whose definition has the advantage to be independent of position. For this reason, we shall deal in this paper only with the standard baroclinic and barotropic modes, not with their generalized WKB counterparts. A simple theoretical model is constructed to address the above JEBAR issues for forced waves. We seek insight into the mechanisms associated with the generation, propagation, and decay of baroclinic Rossby waves that are strongly affected by steep topography. Since a WKB theory applies to unforced waves in a slowly varying medium, we rely primarily on numerical solutions of planetary geostrophic equations appropriate to large-scale waves; not surprisingly, though, our solutions show some similarities to those derived by a WKB approach. The material is organized as follows: the model equations are in section 2, theoretical analyses are in sections 3 and 4, illustrative numerical solutions are in section 5, and concluding remarks are in section 6.

## 2. Model formulation

*ρ*

_{1}overlying a deep layer of greater density

*ρ*

_{2}(Fig. 1). The unperturbed thicknesses of these layers are denoted by

*H*

_{1}and

*H*

_{2}=

*H*−

*H*

_{1}respectively, with

*H*(

*x,y*) the total ocean depth. We denote the perturbation elevations of the surface and interface by

*η*

_{1}and

*η*

_{2}, respectively. Thus, the total layer thicknesses are

*h*

_{1}=

*H*

_{1}+

*η*

_{1}−

*η*

_{2}and

*h*

_{2}=

*H*

_{2}+

*η*

_{2}. For small disturbances to a stratified state of rest, we use the linearized primitive equations for the layer horizontal transports

*U*

_{i}=

*h*

_{i}

**u**

_{i}≈

*H*

_{i}

**u**

_{i}:

*p*

_{1}=

*gη*

_{1}and

*p*

_{2}=

*g*(

*η*

_{1}+

*ϵη*

_{2});

*r*is a damping rate representing viscous processes.

_{t}

*η*

_{1}in (2), are neglected: see Colin de Verdière 1988]. As a result of the rigid-lid approximation, the vertically integrated horizontal transport is divergenceless, and hence can be expressed in terms of a streamfunction by

**U**=

**U**

_{1}+

**U**

_{2}=

**ẑ**× ∇Ψ. The vertically integrated momentum equations can be expressed in terms of Ψ as follows:

*f*

*H*

_{1}

*p*

_{1}

*H*

_{2}

*p*

_{2}

*τ.*

*p̃*=

*p*

_{2}−

*p*

_{1}=

*g*′

*η*

_{2}only can be obtained, given by

*H.*Equation (7) is the evolution equation for the baroclinic pressure, obtained from (4) by using

**U**

_{2}= (

*H*

_{2}/

*H*)

**ẑ**× ∇Ψ + (

*H*

_{2}/

*H*)

**ẑ**×

*τ*/

*f*+

*H*

_{1}

*H*

_{2}/(

*fH*)

**ẑ**× ∇

*p̃*that results from eliminating

*p*

_{1}between (1) and (3). In the flat-bottom case, Eqs. (7) and (6) are uncoupled, so Ψ and

*p̃*are related to the standard barotropic and baroclinic modes. Alternatively, Eq. (7) can be combined with (6) to be rewritten under the form

*H*curl(

*τ*/

*H*) = curl

*τ*+ (

**ẑ**×

*τ*)·∇

*H*/

*H,*curl(

*H*

_{2}

*τ*/

*Hf*) = (

*H*

_{2}/

*H*)

*w*

_{E}+

*H*

_{1}(

**ẑ**×

*τ*)·∇

*H*/

*fH*

^{2}, and

*fJ*(

*H,*Ψ)/

*H*

^{2}=

*J*(Ψ,

*f*/

*H*) − (

*β*/

*H*)∂

_{x}Ψ;

*w*

_{E}= curl(

*τ*/

*f*) is the classical Ekman pumping velocity. An equation for the surface pressure in terms of

*p̃*can be obtained by taking the curl of (5) divided by

*f,*yielding

*p*

_{2}by using

*p*

_{1}=

*p*

_{2}−

*p̃.*Another possibility to determine

*p*

_{1}and

*p*

_{2}as a function of

*p̃*and Ψ would be to integrate (5) from the eastern boundary.

It would be a mistake to regard Eqs. (7) and (8) as forced wave equations&mdash∞ the first case with characteristics *H*_{1}*H*_{2}/*fH* = const., in the second case with zonal characteristics&mdashcause the signal propagation is determined by the coupling between Eqs. (6) and (7), as analyzed below. Such a mistake would be similar to the one often made by regarding JEBAR as a forcing term in the equation for the barotropic mode, as a result of assuming the stratification known when in fact the latter is to be determined as part of the solution.

## 3. Effects of topography on the baroclinic mode

### a. Topography varying only with latitude

*H*′(

*y*) =

*dH*/

*dy.*We introduce the dimensionless parameter

*μ*=

*fH*′(

*y*)/

*βH*to characterize the steepness of the topography relative to that of the Coriolis frequency. As a result, inserting (10) into (8) and using

*d*/

*dy*(

*f*/

*H*) = (

*β*/

*H*)(1 −

*μ*) yields

*c*=

*g*′

*βH*

_{1}

*H*

_{2}/

*f*

^{2}

*H.*Equation (11) has exactly the same structure as the equation for the baroclinic mode in the flat-bottom case. The only difference is that both the phase speed and the forcing term are multiplied by the same factor 1 +

*μH*

_{1}/(

*μ*− 1)

*H*

_{2}. We are interested here in the asymptotic behavior of Eq. (11) in the limit of a steep topography, that is the case

*μ*≫ 1. Formally, when

*μ*increases indefinitely, this factor tends to

*H*/

*H*

_{2}> 1, and Eq. (11) tends to

*x*-independent Ekman pumping (hence also approximately with an

*x*dependence on a scale large compared to the baroclinic wavelength), the baroclinic response in

*p̃*is amplified by the same factor

*H*/

*H*

_{2}since this factor multiplies the wind forcing. Furthermore, since the baroclinic contribution to

*p*

_{1}is −(

*H*

_{2}/

*H*)

*p̃*over a flat bottom [from Eq. (9)] while

*p*

_{1}≈ −

*p̃*in a pressure compensated situation (as occurs for

*μ*≫ 1; see below), the ratio |

*p*

_{1}/

*p̃*| also is amplified by the factor

*H*/

*H*

_{2}, and overall the Rossby wave sea-level response is amplified by the factor (

*H*/

*H*

_{2})

^{2}compared to the flat-bottom response. For small-scale

*x*dependence in

*w*

_{E}, the possibility of resonances makes the degree of topographic amplification more dependent upon the particular situation.

*H*=

*H*(

*y*), Eq. (9) becomes

*p*

_{2}=

*p*

_{1}+

*p̃.*Here

*F*(

*y,t*) is the eastern boundary value for the quantity (

*d*/

*dy*)(

*H*/

*f*)

*p*

_{1}+ (

*d*/

*dy*)(

*H*

_{2}/

*f*)

*p̃.*For large

*μ,*the asymptotic expressions for

*p*

_{1}and

*p*

_{2}become

*μ,*indicative of pressure compensation.

### b. Topography varying only with longitude

*x*=

*x*

_{f}<

*x*

_{E}. The resulting barotropic characteristics, Π ≡

*f*/

*H*= const, are illustrated in Fig. 2 along with other notation. Sometimes, for analytic tractability, the continuous topographic contours are approximated as straight lines (right panel of Fig. 2). In this case Eq. (6) becomes

*dx*and

*dy*are related by

_{x}

*dx*

_{y}

*dy*

*dx*and using (19) yields

_{y}=

*β*/

*H*and

*H*curl(

*τ*/

*H*) = curl

*τ*−

*τ*

_{y}

*H*′(

*x*)/

*H*(

*x*) when

*H*varies only with longitude, Eq. (20) can be rewritten as

*x,y*) and (

*x*

_{f},

*y*

_{f}), belonging to the same characteristic, where the latter point is taken at the boundary between the flat-bottom region and the region with topographic variations. The result is

*x, y*

*x*

*y*

*x*

_{f}

*y*

_{f}

*y*′ and

*y*

_{f}can be regarded as function of

*x.*By differentiating (23) with respect to

*x,*the following expressions for

*dy*′/

*dx*and

*dy*

_{f}/

*dx*are obtained:

*μ*is the dimensionless topographic steepness parameter defined by

*μ*=

*fH*′(

*x*)/

*βH*(

*x*). By keeping in mind (24), as well as that

*H*(

*x*′)/

*f*(

*y*′) =

*H*(

*x*)/

*f*(

*y*) along a characteristic, we can take the derivative of (22) with respect to

*x*to obtain

*β*∂

_{x}Ψ − curl

*τ,*which also appears in (8) in a way that distinguishes in (25) between the wind forcing and the baroclinic response functionally related to

*p̃.*A provisional decomposition is

*p̃*) denotes “functional of

*p̃,*” defined by

*p̃.*The left side of this equation has the classical form for zonal propagation at the standard long Rossby wave speed,

*c,*while the right side contains both local and nonlocal wind forcing and an unfamiliar integral operator, ℒ. In some special cases below, we will show how ℒ(

*p̃*) contributes both to an enhancement of the propagation speed and to an energy exchange with the barotropic mode (see appendix D).

#### 1) Solution in the flat-bottom region: *x*_{f} ≤ *x* ≤ *x*_{E}

*t**(

*x,t*) =

*t*+ (

*x*−

*x*

_{E})/

*c.*

#### 2) Solution over a linear slope

*H*(

*x*), is a linear function of longitude (see the right side of Fig. 2) and the wind stress,

*τ*(

*y,t*), is independent of longitude. In this case, we show in appendix B that ℱ takes the following form:

*y** is such that

*y*≤

*y** ≤

*y*

_{f}. As can be seen, ℱ comprises two terms, one proportional to

*μ*(i.e., increasing with topographic steepness) and one that is order unity (i.e., of the same order of magnitude as the baroclinic forcing in the flat-bottom case). Thus, over steep topography, ℱ is dominated by the first term and can become arbitrarily large.

*x*

_{0},

*y*

_{0}) on the characteristic joining (

*x,y*) to (

*x*

_{f},

*y*

_{f}), we show in appendix A how to perform a local analysis of the operator ℒ(

*p̃*) in terms of the following quantities:

*p̂*(

*k,l*) is the Fourier spectrum of

*p̃*with longitudinal and latitudinal wavenumbers

*k*and

*l*respectively, 𝒟

_{ε}(

*x*

_{0},

*y*

_{0}) is the strip of the wavenumber space defined by −ε ≤

*k*+

*lμ*(

*x*

_{0},

*y*

_{0}) ≤ ε with ε ≪ 1, and 𝒟

_{T}(

*x*

_{0},

*y*

_{0}) is the total wavenumber space minus 𝒟

_{ε}(

*x*

_{0},

*y*

_{0}). The domains 𝒟

_{ε}(

*x*

_{0},

*y*

_{0}) and 𝒟

_{T}(

*x*

_{0},

*y*

_{0}) form a partition of the total wavenumber space so that

*p̃*=

*p̃*

_{ε}+

*p̃*

_{T}. Since the operator ℒ is linear, it follows that

*p̃*

*p̃*

_{ε}

*p̃*

_{T}

*p̃*

_{ε}) and ℒ(

*p̃*

_{T}) in the limit of large

*μ*are shown in appendix A to be given by

*p̃*) is dominated either by ℒ(

*p̃*

_{T}) or ℒ(

*p̃*

_{ϵ}) with the corresponding forms for Eq. (29) as follows:

*p̃*locally parallel to

*f*/

*H,*ℒ(

*p̃*

_{ϵ}) ≫ ℒ(

*p̃*

_{T}):

*x,y*) close to (

*x*

_{0},

*y*

_{0}), we have by construction

*J*(

*p̃*

_{ϵ},

*f*/

*H*) ≈ 0. In the case where

*p̃*

_{ϵ}is dominant, we shall have

*J*(

*p̃,f*/

*H*) ≈ 0 so that

*p̃*must appear locally parallel to the

*f*/

*H*contours, hence the denomination for the regime (41). On the other hand, note that it is not mathematically accurate to say that the first regime (40) should imply for

*p̃*to be locally nonparallel to the

*f*/

*H*contours. The reason is that there are some instances where

*J*(

*p̃*

_{T},

*f*/

*H*) vanishes even if

*p̃*

_{T}differs from zero, for example, at a saddle point or a local extremum. This subtlety, however, should not be important in practice because such cases are generally very isolated.

Outside the two above regimes, the equation must be a linear combination of (41) and (40) in proportion to the contribution of *p̃*_{ϵ} and *p̃*_{T} to the total sum *p̃.* In general, however, since the measure of 𝒟_{ε} tends toward zero as ε goes to zero, *p̃*_{ε} should be close to zero if *p̂*(*k,l*) is finite over 𝒟_{ε}, so the regime (40) should be the one most often encountered.

Equations (40) and (41) are one of the main theoretical results of this paper. In the main regime [i.e., Eq. (40)], as in the case of a topography varying in latitude only, there is an increase in phase speed by the factor *H*/*H*_{2}. The baroclinic forcing term, on the other hand, is much different since it increases with the topographic steepness to become much larger than over a flat bottom. This property is interpreted in the following sections as the means of creation of free baroclinic Rossby waves by an ocean ridge, which occurs through the conversion of energy from wind-driven barotropic motions. In contrast, in the second regime for which *p̃* is locally parallel to *f*/*H* contours [i.e., Eq. (41)], a diffusion-like term occurs that does not exist in the flat-bottom case. The associated baroclinic energy dissipation is a result of conversion from the baroclinic to the barotropic mode.

Is there a phase speed enhancement associated with pressure compensation in this case? An explicit mathematical answer to this question—as for the topography varying only with latitude—could not be obtained. From our numerical solutions below, we learn that in some instances enhanced speed is observed, even where pressure compensation does not occur. This indicates that pressure compensation is actually not a necessary condition for enhanced propagation to occur, as is further discussed at the end of section 5.

## 4. Theory for the generation of Rossby waves by a ridge

*μ*≫ 1) affects the character of wind-forced, baroclinic Rossby waves in three main ways: it systematically enhances their phase speed by the ratio

*H*/

*H*

_{2}, it introduces a dissipative term due to energy conversion to the barotropic mode where

*p̃*is aligned with

*f*/

*H,*and it increases the effectiveness of the wind forcing by

*O*(

*μ*) through energy conversion from the barotropic mode. Of these three effects, only the last is likely to be important in the generation and/or amplification of baroclinic Rossby waves by an ocean ridge. In this section, we investigate the dynamics of a simple model that attempts to describe how the third effect works. If we neglect the terms causing the phase speed amplification and the dissipation, Eq. (40) reduces to the simple forced-wave equation,

*g*′

*H*

_{2}

*w*

_{E}/

*H*+ ℱ* with ℱ* from (40). As shown above, the latter is

*O*(

*μ*) times greater over a steep topography than over a flat bottom. To gain insight into the dynamics of (42), we replace ℱ by idealized forms that match its qualitative character;specifically, we consider the following two types of “ridge forcing”:

*l*centered on

*x*

_{0}, representing the effect of the ridge.

*k*

_{T}=

*π*/

*l*is an approximate wavenumber for the ridge and

*k*=

*ω*/

*c*is the wavenumber of the Rossby waves. In each case the amplitude of the free waves generated by the ocean ridge is proportional to the amplitude of the ridge forcing, as well as its temporal period. The amplitude of the free waves generated also depends on the ratio of the wave wavelength to the width of the topographic region, or equivalently

*k*/

*k*

_{T}, with the functional dependences shown in Fig. 4. There is a notable difference in the responses to the two types of forcing. For the smooth forcing, the amplitude function (45) has a global maximum at

*k*/

*k*

_{T}≈ 0.83, vanishes for

*k*/

*k*

_{T}= 2, and then becomes very small for values

*k*/

*k*

_{T}≥ 2. Thus, free waves can be efficiently generated only if the width of the ridge forcing is comparable with the wavelength of the Rossby waves. On the other hand, free waves with maximum amplitude 2

*μw*

_{1}/

*ω*can be generated by abrupt ridge forcing even if the width of the ridge is not comparable with the wavelength, whenever

*k*/

*k*

_{T}takes on the discrete values

*n*+ ½, with

*n*any integer. This limitation is not likely to be important in nature, however, since the frequency spectrum of the wind is broadband so that Rossby waves with wavelengths satisfying the required constraint should always exist.

## 5. Numerical solutions

### a. Cases

Our previous analytical results are extended by three cases of idealized numerical solutions where the topography is a meridional midocean ridge of fixed longitudinal width but with several different heights, *δH* (Fig. 5). First, we investigate the propagation of a freely propagating baroclinic eddy initially east of the ridge, similar to Sakamoto and Yamagata (1997), to illustrate the enhancement of the phase speed over the ridge. Second, we consider the case of the baroclinic “wavemaker” that excites time-periodic motions at the eastern boundary (e.g., emission due to scattering by poleward-propagating coastal waves (Milliff and McWilliams 1994)]: without additional forcing, baroclinic Rossby waves are strongly attenuated west of the ridge through coupling with the barotropic mode. In contrast, amplification is predicted for the same problem with WKB theory in Killworth and Blundell (1999). Third, baroclinic Rossby waves are forced by zonally uniform Ekman pumping to show that in this case it is possible for large-amplitude baroclinic Rossby waves to be generated west of the ridge. In all cases we chose for the density stratification *ϵ* = (*ρ*_{2} − *ρ*_{1})/*ρ*_{0} = 0.002, *H*_{1} = 1200 m, and *H*_{2} = 2800 m (i.e., *H* = 4000 m) away from the ridge. The somehow unrealistic values for the ridge and stratification are dictated by the following considerations: 1) the ridge be wide enough for propagation features to be easily detected in longitude–time plots and 2) the amplification factor *H*/*H*_{2} be important enough for the enhanced and flat-bottom phase speeds to be clearly distinct.

### b. Numerical method

*f*; this allows us to use

*p*

_{2}in place of Ψ. Equation (48) is Eq. (4), using the geostrophic approximation for

**u**

_{2}and a rewriting of Eq. (47) as follows:

*p*

_{2}vanishes. The energy equation corresponding for this system is shown in appendix D to be

_{E}at the boundary determines where the energy enters and leaves the domain. Because the flux is sign-definite with respect to the

*p*

_{i}, we can deduce the necessary boundary information: with respect to the first term on eastern and western boundaries, the former site is an energy source (where

*p*

_{1}must be specified) and the latter a sink (where

*p*

_{1}is freely evolving); with respect to the second term,

*p*

_{2}must be specified where there is an energy source, namely, where (

**ẑ**× ∇

*H*

_{2}/

*f*)·

**n̂**> 0 with

**n̂**the outward normal direction.

For all cases the numerical domain is 120° wide in longitude and 50° wide in latitude with the southern boundary at 10°N. Thus, our boundary conditions are *p*_{1} and *p*_{2} specified at the eastern boundary and *p*_{2} = 0 (i.e., no energy flux) where its value is required on the northern and southern boundaries. For time integrations we use second-order, centered finite difference operators, with a spatial resolution of 200 points in longitude × 800 points in latitude (in order to resolve the characteristics accurately and limit erroneous numerical dispersion when the ridge is steep). A spatially implicit scheme is used to integrate (47) for *p*_{2}, and a Runge– Kutta scheme is used to step (48) for *η*_{2} forward in time, with a time step Δ*t* = 1.25 × 10^{4} s.

*p*

_{2}

*p̂*

_{2}

*e*

^{iωt}

*η*

_{2}

*η̂*

_{2}

*e*

^{iωt}

*η*

_{1}

*η̂*

_{1}

*e*

^{iωt}

*p̂*

_{2}and

*η̂*

_{2}

*p̂*

_{2}alone can be obtained by integrating (54) in longitude, which yields

*η̂*

_{2}

*p̂*

_{2}:

*c*

_{1}=

*g*′

*βH*

_{1}/

*f*

^{2}and

*k*

_{1}=

*ω*/

*c*

_{1}. Inserting (55) into (53) then yields

*H*

_{2}/

*f*= const, which is solved numerically by a method of characteristics using spline interpolation under tension (using the FITPACK package with a tension parameter value of unity). Once

*p̂*

_{2}is obtained,

*η̂*

_{2}

*η̂*

_{1}

### c. Results

#### 1) Freely propagating baroclinic eddy

In this case an initial Gaussian eddy propagates over a ridge located at the center of the computational domain. Figure 6 shows sea level and interface height at several times for the case *δH* = 1000 m. Figure 7 is the same but for the case *δH* = 2500 m. In general, the eddy shape appears to be less distorted by the steeper topography in the second case, where apart from signals propagating meridionally along the flanks of the ridge, the primary pattern change is a zonal elongation due to the Rossby wave speed decreasing with latitude ∝*β*/*f*_{2} (as also occurs with a flat bottom). This is an indication that the system becomes less dispersive as the steepness of the topography increases. We examine the propagation with Hovmöller diagrams (i.e., longitude–time plots) in Figs. 8 and 9 for an eddy core at latitude 40°N. For comparison we also draw a solid line to represent the flat-bottom speed, *c* = *g*′*βH*_{1}*H*_{2}/*f*^{2}*H,* and a dashed line to represent propagation enhanced by the factor *H*/*H*_{2} discussed above. The speed enhancement in the case with *δH* = 1000 m is modest, but with *δH* = 2500 m it is substantial and close to the predicted factor, *H*/*H*_{2}, over a wide longitude band. Again, dispersive effects appear weaker with the steeper topography, where the spread of *η*_{i} contours is much smaller west of the ridge.

Figure 10 displays the temporal decay of baroclinic energy, ∫∫½*η*^{2}_{2}

#### 2) Eastern wavemaker

*η*

_{1}and −

*η*

_{2}(

*ϵH*

_{2}/

*H*) at an annual frequency,

*ω*= 2

*π*/

*T, T*= 1 yr, and with a sinusoidal shape in latitude,

*y*

_{S}is the latitude of the southern boundary and

*D*the latitudinal extent of the basin; this forcing creates no barotropic energy there. We choose

*A*= 10 m. Also, no forcing is imposed in the interior, so the only source of energy is at the eastern boundary. We set

*ŵ*

_{E}= 0. For comparison, the solutions for

*η̂*

_{1}

*η̂*

_{2}

*k*=

*ω*/

*c*;for each quantity the root-mean-square variability,

*η̂*

_{1}

*η̂*

_{2}

*δH*= 1000 m,

*δH*= 1500 m,

*δH*= 2000 m, and

*δH*= 2500 m, respectively. In each figure the eastern part of the domain is flat, so the solutions are the same as described above. In the steeper case (Fig. 15), the coupling along the

*f*/

*H*contours is quite strong, so most of the baroclinic energy is converted to barotropic energy that then propagates out of the domain to the south. As a result, the amount of baroclinic energy present in the western part of the domain is quite small; this behavior is also evident in the sea level. With a less steep ridge (Fig. 14), less energy is lost through the southern boundary because the

*f*/

*H*contours are then more zonally aligned. As a result, barotropic energy is seen to appear in the northwestern part of the domain in the sea level, suggesting that it was provided by conversion of baroclinic energy from the northeastern part of the domain. Although the amount of baroclinic energy in the western part has somewhat increased in comparison to the previous case, it still remains considerably reduced in comparison to its level in the eastern part. As the ridge height further decreases, Figs. 12 and 13, the sea level variability in the western part of the domain increases much more rapidly than the interface (i.e., baroclinic) variability. Indeed, in some locations, the variability of the sea level west of the ridge has become of the same order of magnitude, sometimes even larger, than in the eastern part of the domain, whereas the baroclinic variability is uniformly much lower than in the east. These results demonstrate clearly that the coupling with the barotropic mode cannot be neglected.

#### 3) Wind forcing

*ŵ*

_{E}

*y*

*w*

_{0}

*π*

*y*

*y*

_{S}

*D*

*w*

_{0}= 10

^{−6}m s

^{−1}. For comparison, the solutions for

*η̂*

_{1}

*η̂*

_{2}

*H*= 2500 m, but in this case a significant amount of energy leaves the domain along strongly nonzonal characteristics (cf. the wavemaker case). In general, the largest baroclinic energy occurs in the southwestern part of the domain (as also occurs in the wavemaker case for the baroclinic waves that cross the ridge). The maximum value of the baroclinic mode in the eastern part of the domain is 7 m, while in the west it can reach values as high as 30 m in the δ

*H*= 2000 m solution, with typical values well above 10 m. These results demonstrate that wind forcing is associated with the creation of baroclinic energy west of the ridge, in agreement with our theoretical prediction.

### d. Enhanced propagation speed and baroclinic compensation

_{x}

*η*

_{2}every one degree in latitude for the steep topographic case

*δH*= 2000 m &lsqb e zonal derivative is used to filter out the zonally independent component that is part of the directly forced response (White 1977)]. Two main regimes arise over the ridge (Fig. 20, right panel): an ill-defined propagation south of 30°N approximately, and a well-defined, enhanced propagation speed north of 30°N. As seen in Fig. 21, the first regime is achieved for values of the steepness parameter

*μ*below or around unity, while the second regime is achieved for values of

*μ*greater than unity. From the theoretical considerations in the introduction, we might expect that pressure compensation would be found in the northern part of the domain, but actually it is not. To understand this behavior, let us recall the equation for the baroclinic mode (48), which is valid over topography as well as over a flat-bottom. The way it is written suggests free waves with a phase speed of

*g*′

*βH*

_{1}/

*f*

^{2}, but actually this depends on whether ∂

_{x}

*p*

_{2}is correlated to ∂

_{x}

*η*

_{2}. In the flat-bottom case, for instance, the two quantities are related by

*η*

_{2}is

*g*′

*βH*

_{1}

*H*

_{2}/

*f*

^{2}

*H.*For a zonally uniform Ekman pumping, such a dependency would result in a straight line in a plot showing ∂

_{x}

*p*

_{2}versus ∂

_{x}

*η*

_{2}. Such a plot is Fig. 22 for the same cases depicted in Fig. 20. In the southern part of the domain, ∂

_{x}

*p*

_{2}unexpectedly is strongly correlated to ∂

_{x}

*η*

_{2}with the approximate dependence

_{x}

*η*

_{2}and ∂

_{x}

*p*

_{2}in Eq. (48) approximately cancel each other so that the resulting equation is no longer a wave equation, explaining why propagation is not apparent in the southern part of the basin. This seemingly well-defined correlation for the regime

*μ*=

*O*(1) is as yet unexplained. This correlation between ∂

_{x}

*p*

_{2}and ∂

_{x}

*η*

_{2}decreases as the latitude increases, as indicated by the seemingly random pattern in Fig. 22 (the three bottom panels). Propagation becomes well defined, and the phase speed is close to

*g*′

*βH*

_{1}/

*f*

^{2}. From this example, we conclude that the general condition for enhanced propagation speed is not pressure compensation (i.e.,

*p*

_{2}≈ 0) but a lack of correlation between ∂

_{x}

*p*

_{2}and ∂

_{x}

*η*

_{2}.

### e. Barotropic–baroclinic coupling

*H*

_{1}/

*H*

^{2}

*J*(

*H,*Ψ). In order to gain further insight into the coupling of the two modes, we seek to understand when the latter term acts as a source or sink of energy for the baroclinic mode. After multiplying Eq. (7) by

*η*

_{2}, a baroclinic energy equation is obtained in which this term implies a possible energy conversion:

*η*

_{2}are nearly parallel to

*f*/

*H*contours, as anticipated in section 3b(2). Overall, the sink is greater than the source, so the coupling term contributes to dissipate the baroclinic mode. In the wind-forced case the coupling term is dominantly a source of energy, localized mainly in the southwestern part of the ridge. Its role as a sink appears sparse and negligible, though the negative values do occur mostly to the east of the ridge. Therefore, it contributes mainly to increase the energy of the baroclinic mode. In these two cases the source and sink regions roughly coincide but their strengths differ.

### f. Implications for sea level observations

In Chelton and Schlax (1996) sea level is low-pass filtered to remove the steric effect (i.e., the dominant zonally quasi-uniform effect due to seasonal heating and cooling). This filtering also reduces the impact of the barotropic mode and wind forced response whose zonal large-scale signatures may visually alter propagating signals in the same way as the steric effect. An alternative to the low-pass filter is to take the *x* derivative of the signal. Fig. 20 compares Hovmöller diagrams of ∂_{x}*η*_{1} (left panel) versus ∂_{x}*η*_{2} (right panel) depicted every 10° of latitude starting from 15°N (from top to bottom) in the case *δH* = 2000 m. In the flat-bottom eastern and western regions, where propagation is well defined, the visual appearance between ∂_{x}*η*_{1} and ∂_{x}*η*_{2} is identical at all latitudes, with the propagation as predicted by the standard theory (indicated by the dashed line). This is not the case over the ridge for the plots at 15° and 25°N where propagation is ill-defined and coincides with small values of the steepness parameter. Well-defined propagation over the ridge is observed only north of 25°N, mostly on the ridge eastern flank where the steepness parameter is greater than unity. In these cases, ∂_{x}*η*_{1} and ∂_{x}*η*_{2} indicate similar propagation characteristics, with a phase speed intermediate between the elevated value *g*′*βH*_{1}/*f*^{2} discussed in this paper (depicted as a solid line) and twice the standard speed (depicted as a dot&ndash↓shed line). From these examples, we conclude that it is well justified to regard the zonally low-pass filtered sea level as mimicking the behavior of first-mode baroclinic Rossby waves, although this does not seem to be true in regions where *μ* = *O*(1), a case that remains to be investigated.

## 6. Concluding remarks

### a. Summary

Chelton and Schlax (1996) shows that major topographic features in the ocean influence the propagation speed and amplitude of baroclinic Rossby waves. We present here theoretical and numerical results from a two-layer model with wind-forced waves over strong topography. Our main theoretical result is an explicit expression for the evolution equation of the baroclinic mode in the special case where the topography varies linearly in longitude and the wind stress depends on latitude only. We also obtain an analytical expression in the case where the topography varies only with latitude for an arbitrary wind stress. These analytic results and additional numerical solutions with a midocean ridge indicate that steep topography alters the character of baroclinic Rossby waves in three primary ways.

#### 1) Phase speed enhancement

Steep topography enhances systematically the phase speed of baroclinic Rossby waves by the factor *H*/*H*_{2} > 1, compared to the standard flat-bottom speed, while hardly changing the westwardly directed, nondispersive behavior. In contrast, the propagation of barotropic Rossby waves can become strongly nonzonal as a result of topographic steering along the characteristics *f*/*H* = const. These results are consistent with those in Rhines (1977) and Veronis (1981) for free waves. The phase speed amplification factor, *H*/*H*_{2}, is the same as the one arising in a pressure-compensated model, but it can occur in more general situations. Our numerical solutions reveal that pressure compensation does not always accompany enhanced propagation speed for a topography varying with longitude only, while we have shown analytically that it does in the case of a steep topography varying only with latitude. A more general condition for speed enhancement is that the lower-layer pressure zonal variations be uncorrelated to those of the layer interface.

#### 2) Rossby wave generation

Baroclinic Rossby waves are generated over a ridge in the case with wind forcing, while strongly dissipated in the case without it. Analysis of the coupling between the baroclinic and barotropic modes shows it to be a source of baroclinic energy in the former case and a sink in the latter case. In the forced case, the transfer of barotropic energy to the baroclinic mode occurs mainly in the southwestern part of the ridge. A simple analytical model mimicking the behavior of the baroclinic evolution equation over the ridge illustrates the wave creation.

#### 3) Rossby wave dissipation

Spatial variations in the bottom topography introduce a dissipation-like term in the evolution equation for the baroclinic mode, which is important only when the interfacial height contours are locally parallel to the contours *f*/*H* = const. Analysis of the coupling between the barotropic and baroclinic mode confirms this view. In the wavemaker case, for instance, the coupling acts as a sink of baroclinic energy, on the eastward side of the ridge where the wave pattern is strongly correlated with the *f*/*H* contours. This term is interpreted as being responsible for the conversion of energy from the baroclinic to barotropic mode: it is dominant in the absence of external forcing and a strong barotropic mode and it causes the decrease of baroclinic amplitude across the ridge in our propagating eddy and baroclinic-wavemaker cases, as well as in the solutions of Sakamoto and Yamagata (1997). Topographic dissipation through mode coupling is an important additional mechanism to viscous decay (e.g., Qiu et al. 1997) for the decay of baroclinic Rossby waves in the ocean, and it does not rely on a parameter as poorly constrained as eddy viscosity.

### b. Discussion

#### 1) Pattern distortion

The primary cause of pattern distortion is due to the decrease with latitude of propagation speed of baroclinic Rossby waves. This is well illustrated in Fig. 4 of Chelton and Schlax (1996). There is a further cause from the spatially nonuniform conversion of energy to the barotropic mode, as well from dispersive effects [i.e., unequal phase and group velocities of baroclinic waves over topography (Straub 1994)], although those are weak over a steep topography (Rhines 1977; Veronis 1981).

#### 2) Comparison with WKB solutions

Killworth and Blundell (1999) addresses issues similar to those in this paper but in a completely different framework. It focuses on free waves in a continuously stratified ocean while we address wind-forced waves in a two-layer model, and it uses a WKB theory where we use direct integration. Our approach allows for energy scattering between modes and both steep and slowly varying topography. In our solutions the energy transfer between the barotropic and baroclinic modes is a primary influence on the behavior of baroclinic Rossby waves over topography: baroclinic waves would not be strongly dissipated during propagation over a ridge, and baroclinic Rossby waves would not be created west of ridge. In contrast, a WKB theory assumes that mode scattering is negligible, and Killworth and Blundell (1999) concludes that the topography has little effect on baroclinic Rossby waves.

#### 3) Prospects

A wind-forced, two-layer model seems to account, at least qualitatively, for enhanced phase speed and amplitude variations in the vicinity of major topographic features observed in the altimetric measurements of sea surface height. Still, a two-layer model is a crude representation of the actual stratification, and its predictions are quantitatively doubtful. In this respect the phase speed prediction is probably the more sensitive one since its enhancement ratio is *H*/*H*_{2}, whose continuously stratified analog is uncertain. On the other hand, the free-wave generation and dissipation rely on the form of the evolution equation for the first baroclinic mode, which is similar in two-layer and continuously stratified fluids. Our results allow some insight into the nature of the coupling between the barotropic and baroclinic modes over a variable topography, by providing some explicit expressions for the coupling in terms of localized dissipation and forcing. However, we believe that a wider range of topographic and forcing patterns should be examined beyond the simple configurations considered here (we hope to report on this later).

## Acknowledgments

We gratefully acknowledge fruitful discussions with Bob Hallberg, Dudley Chelton, and Roland deSzoeke. A reviewer’s comments greatly helped us clarify some of the issues. This work 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 A

### Approximation of ℒ(*p̃*)

*p̃*) is defined in section 3b by

*y*′(

*x*′) such that Π(

*x*′,

*y*′(

*x*′)) = const. The integral is decomposed as follows:

*μ*=

*fH*′(

*x*)/

*βH*(

*x*), we have ∂

*μ*/∂

*y*=

*H*′(

*x*)/

*H*(

*x*) =

*βμ*/

*f.*From scaling ∂/∂

*y*by 1/

*λ,*the second term of (A2) is

*O*(

*fλ*/

*β*) compared to the first term, hence can be neglected assuming

*λ*small compared to the planetary scale

*f*/

*β.*

*p̃*in terms of its Fourier integral

*x*′,

*y*′(

*x*′)) = const, it comes

*dy*′/

*dx*′ = −∂

_{x}Π/∂

_{y}Π =

*fH*′(

*x*)/(

*βH*) =

*μ*; hence

*d*(

*kx*′ +

*ly*′) = (

*k*+

*lμ*)

*dx*′ so that

*dμ*/

*dx*′)

_{Π&equals℅nst}= [

*f*/(

*βH*)]∂

^{2}

*H*/∂

*x*

^{2}and assuming

*β*constant (i.e., the

*β*-plane approximation). Equation (A6) is estimated by approximating the characteristic curves by piecewise linear straight lines along which

*μ*is constant. As a result, the path integral over the interval [

*x,x*

_{f}] is split into

*N*− 1 path integrals of the form:

*μ*≫ 1, two cases arise: case I,

*k*+

*lμ*

_{n}≠ 0: expanding the following term as a series in inverse powers of

*μ*

_{n}

*k*+

*lμ*

_{n}≈ 0: for a straight characteristic, we have

*kx*′ +

*ly*′ =

*k*(

*x*′ −

*x*) +

*l*(

*y*′ −

*y*) +

*kx*+

*ly*= (

*k*+

*lμ*)(

*x*′ −

*x*) +

*kx*+

*ly*=

*kx*+

*ly*; using

*l*≈ −

*k*/

*μ*

_{n}, it follows that

_{ε}defined by −ε ≤

*k*+

*lμ*≤ ε, ε ≪ 1, and 𝒟

_{T}its complement. As a result, using (A9) and (A10), noting that (∂

_{x}+

*μ*∂

_{y})

*p̃*

_{ε}= 0 and hence that

*p̃*

*p̃*

_{ε}

*p̃*

_{T}

## APPENDIX B

### Approximation of ℱ

_{1}+ ℱ

_{2}with

_{1}and ℱ

_{2}are estimated assuming 1) the topography varies linearly so that

*μ*is constant along a characteristic and 2) the wind stress components depend on latitude only so that curl

*τ*= −

*dτ*

_{x}/

*dy*and hence ∂

_{y}curl

*τ*= −

*d*

^{2}

*τ*

_{x}/

*dy*

^{2}. Integrate by parts the integral term in ℱ

_{1}, namely,

*g*(

*y*) we have [

*d*/

*dx*′]

*g*(

*y*′) = [

*dg*/

*dy*](

*y*′)

*dy*′/

*dx*′ =

*μ*[

*dg*/

*dy*](

*y*′). It follows that

_{1}becomes

_{2}, we first write

*g*(

*y*). To estimate the last term in (B6), note that ∂

*μ*/∂

*y*=

*H*′(

*x*)/

*H*(

*x*) =

*βμ*/

*f.*Since

*βμ*/

*f*is constant in sign along the integral path, a classical theorem in integration theory allows us to write

*y** an undetermined latitude comprised between

*y*

_{f}and

*y.*As a result, ℱ

_{2}becomes

## APPENDIX C

### Analytical Solutions of the Forced Wave Equation

#### Solution for smooth forcing

#### Solution for abrupt forcing

#### Amount of free waves generated

*μw*

_{1}in

*p̃*

_{I}(

*x,t*). Their amplitude is given by

*A*

_{smooth}and

*A*

_{abrupt}are depicted in Fig. 4.

## APPENDIX D

Left panel: Smooth characteristics, *f*/*H* = const. Right panel: Same characteristics approximated by piecewise straight lines

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Left panel: Smooth characteristics, *f*/*H* = const. Right panel: Same characteristics approximated by piecewise straight lines

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Left panel: Smooth characteristics, *f*/*H* = const. Right panel: Same characteristics approximated by piecewise straight lines

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Idealized forcing distributions: abrupt (solid line) and smooth (dashed line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Idealized forcing distributions: abrupt (solid line) and smooth (dashed line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Idealized forcing distributions: abrupt (solid line) and smooth (dashed line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Amplitude of the free baroclinic waves generated west of the idealized forcing distributions of Fig. 3 as a function of *k*/*k*_{T}: solid line is for the abrupt forcing; dashed line is for the smooth forcing

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Amplitude of the free baroclinic waves generated west of the idealized forcing distributions of Fig. 3 as a function of *k*/*k*_{T}: solid line is for the abrupt forcing; dashed line is for the smooth forcing

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Amplitude of the free baroclinic waves generated west of the idealized forcing distributions of Fig. 3 as a function of *k*/*k*_{T}: solid line is for the abrupt forcing; dashed line is for the smooth forcing

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Top: The idealized ridge for the four cases *δH* = 1000 m (solid), *δH* = 1500 m (dotted), *δH* = 2000 m (dotted– dashed), and *δH* = 2500 m (dashed). The horizontal solid line indicates the position of the layer interface. Middle: dimensionless contours *f*/*H* = const for *δH* = 1000 m. Bottom: As above but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Top: The idealized ridge for the four cases *δH* = 1000 m (solid), *δH* = 1500 m (dotted), *δH* = 2000 m (dotted– dashed), and *δH* = 2500 m (dashed). The horizontal solid line indicates the position of the layer interface. Middle: dimensionless contours *f*/*H* = const for *δH* = 1000 m. Bottom: As above but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Top: The idealized ridge for the four cases *δH* = 1000 m (solid), *δH* = 1500 m (dotted), *δH* = 2000 m (dotted– dashed), and *δH* = 2500 m (dashed). The horizontal solid line indicates the position of the layer interface. Middle: dimensionless contours *f*/*H* = const for *δH* = 1000 m. Bottom: As above but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Instantaneous values of the sea level (left panel) and layer interface (right panel) for the times *t* = 0, *t* = 578, *t* = 1157, and *t* = 1735 (in days), from top to bottom, in the free propagating eddy experiment with *δH* = 1000 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Instantaneous values of the sea level (left panel) and layer interface (right panel) for the times *t* = 0, *t* = 578, *t* = 1157, and *t* = 1735 (in days), from top to bottom, in the free propagating eddy experiment with *δH* = 1000 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Instantaneous values of the sea level (left panel) and layer interface (right panel) for the times *t* = 0, *t* = 578, *t* = 1157, and *t* = 1735 (in days), from top to bottom, in the free propagating eddy experiment with *δH* = 1000 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 6 but with *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 6 but with *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 6 but with *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Hovmöller diagrams at 40°N latitude for the sea level (top) and layer interface (bottom) in the free propagating eddy experiment with *δH* = 1000 m. Superimposed are straight lines for the standard phase speed (solid) and elevated phase speed *c*_{1} = *g*′*βH*_{1}/*f*^{2} discussed in the text (dashed)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Hovmöller diagrams at 40°N latitude for the sea level (top) and layer interface (bottom) in the free propagating eddy experiment with *δH* = 1000 m. Superimposed are straight lines for the standard phase speed (solid) and elevated phase speed *c*_{1} = *g*′*βH*_{1}/*f*^{2} discussed in the text (dashed)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Hovmöller diagrams at 40°N latitude for the sea level (top) and layer interface (bottom) in the free propagating eddy experiment with *δH* = 1000 m. Superimposed are straight lines for the standard phase speed (solid) and elevated phase speed *c*_{1} = *g*′*βH*_{1}/*f*^{2} discussed in the text (dashed)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 8 but with *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 8 but with *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 8 but with *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Temporal decay of the baroclinic energy (basin integral of *η*^{2}_{2}*δH* = 1000 m (dashed line) and *δH* = 2500 m (solid line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Temporal decay of the baroclinic energy (basin integral of *η*^{2}_{2}*δH* = 1000 m (dashed line) and *δH* = 2500 m (solid line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Temporal decay of the baroclinic energy (basin integral of *η*^{2}_{2}*δH* = 1000 m (dashed line) and *δH* = 2500 m (solid line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Left panel: Energy fluxes *dE*/*dt* estimated from the curves depicted in Fig. 10 (dashed line) superimposed with those estimated from the flux formula given in appendix D (soline line). Right panel: The difference between the two curves, which gives an indication of the spurious numerical energy sources and sinks

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Left panel: Energy fluxes *dE*/*dt* estimated from the curves depicted in Fig. 10 (dashed line) superimposed with those estimated from the flux formula given in appendix D (soline line). Right panel: The difference between the two curves, which gives an indication of the spurious numerical energy sources and sinks

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Left panel: Energy fluxes *dE*/*dt* estimated from the curves depicted in Fig. 10 (dashed line) superimposed with those estimated from the flux formula given in appendix D (soline line). Right panel: The difference between the two curves, which gives an indication of the spurious numerical energy sources and sinks

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Root-mean-square time variability of the sea level (top) and layer interface (bottom) in the wavemaker experiment with *δH* = 1000. Units are in millimeters for the sea level and meters for the layer interface.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Root-mean-square time variability of the sea level (top) and layer interface (bottom) in the wavemaker experiment with *δH* = 1000. Units are in millimeters for the sea level and meters for the layer interface.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Root-mean-square time variability of the sea level (top) and layer interface (bottom) in the wavemaker experiment with *δH* = 1000. Units are in millimeters for the sea level and meters for the layer interface.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 1500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 1500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 1500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 2000 m.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 2000 m.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 2000 m.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 12 but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Root-mean-square time variability of the sea level (top) and layer interface (bottom) in the wind forced experiment with *δH* = 1000 m. Units are millimeters for the sea level and meters for the layer interface.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Root-mean-square time variability of the sea level (top) and layer interface (bottom) in the wind forced experiment with *δH* = 1000 m. Units are millimeters for the sea level and meters for the layer interface.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Root-mean-square time variability of the sea level (top) and layer interface (bottom) in the wind forced experiment with *δH* = 1000 m. Units are millimeters for the sea level and meters for the layer interface.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 1500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 1500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 1500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 2000 m.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 2000 m.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 2000 m.

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

As in Fig. 16 but for *δH* = 2500 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Hovmöller diagrams for ∂_{x}*η*_{1} (left) and ∂_{x}*η*_{2} (right) every 10° latitude from 15°N (top) to 55°N (bottom), in the wind forced expt. with *δH* = 2000 m. Straight lines indicate propagation at the standard phase speed (dashed line), enhanced phase speed discussed in the text (solid line) and twice the standard phase speed (dotted&ndash↓shed line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Hovmöller diagrams for ∂_{x}*η*_{1} (left) and ∂_{x}*η*_{2} (right) every 10° latitude from 15°N (top) to 55°N (bottom), in the wind forced expt. with *δH* = 2000 m. Straight lines indicate propagation at the standard phase speed (dashed line), enhanced phase speed discussed in the text (solid line) and twice the standard phase speed (dotted&ndash↓shed line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Hovmöller diagrams for ∂_{x}*η*_{1} (left) and ∂_{x}*η*_{2} (right) every 10° latitude from 15°N (top) to 55°N (bottom), in the wind forced expt. with *δH* = 2000 m. Straight lines indicate propagation at the standard phase speed (dashed line), enhanced phase speed discussed in the text (solid line) and twice the standard phase speed (dotted&ndash↓shed line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Dimensionless parameter *μ* = *fH*′(*x*)/(*βH*) characterizing the steepness of the topography in the case *δH* = 2000 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Dimensionless parameter *μ* = *fH*′(*x*)/(*βH*) characterizing the steepness of the topography in the case *δH* = 2000 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Dimensionless parameter *μ* = *fH*′(*x*)/(*βH*) characterizing the steepness of the topography in the case *δH* = 2000 m

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Term [*βH*_{1}/*f*^{2}]∂*p*_{2}/∂*x* vs −[*g*′*βH*_{1}/*f*^{2}]∂*η*_{2}/∂*x.* The two straight lines superimposed indicate the linear relations ∂*p*_{2}/∂*x* = [*g*′∂*η*_{2}/∂*x* (the less horizontal line) and ∂*p*_{2}/∂*x* = [*g*′(*H*_{1}/*H*)]∂*η*_{2}/∂*x* (the most horizontal line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Term [*βH*_{1}/*f*^{2}]∂*p*_{2}/∂*x* vs −[*g*′*βH*_{1}/*f*^{2}]∂*η*_{2}/∂*x.* The two straight lines superimposed indicate the linear relations ∂*p*_{2}/∂*x* = [*g*′∂*η*_{2}/∂*x* (the less horizontal line) and ∂*p*_{2}/∂*x* = [*g*′(*H*_{1}/*H*)]∂*η*_{2}/∂*x* (the most horizontal line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Term [*βH*_{1}/*f*^{2}]∂*p*_{2}/∂*x* vs −[*g*′*βH*_{1}/*f*^{2}]∂*η*_{2}/∂*x.* The two straight lines superimposed indicate the linear relations ∂*p*_{2}/∂*x* = [*g*′∂*η*_{2}/∂*x* (the less horizontal line) and ∂*p*_{2}/∂*x* = [*g*′(*H*_{1}/*H*)]∂*η*_{2}/∂*x* (the most horizontal line)

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Contribution of the coupling term (*H*_{1}/*H*^{2})*J*(*H,*Ψ)*δH* = 1000 m, (b) *δH* = 1500 m, (c) *δH* = 2000 m, (d) *δH* = 2500 m. Negative values, indicating a sink of baroclinic energy, are shaded

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Contribution of the coupling term (*H*_{1}/*H*^{2})*J*(*H,*Ψ)*δH* = 1000 m, (b) *δH* = 1500 m, (c) *δH* = 2000 m, (d) *δH* = 2500 m. Negative values, indicating a sink of baroclinic energy, are shaded

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2

Contribution of the coupling term (*H*_{1}/*H*^{2})*J*(*H,*Ψ)*δH* = 1000 m, (b) *δH* = 1500 m, (c) *δH* = 2000 m, (d) *δH* = 2500 m. Negative values, indicating a sink of baroclinic energy, are shaded

Citation: Journal of Physical Oceanography 30, 9; 10.1175/1520-0485(2000)030<2186:ACADOW>2.0.CO;2