a. Quasigeostrophic equations

In this paper, we consider a two-layer quasigeostrophic (QG) model on the β plane and take into account the effect of a bottom topography with a constant slope (see Fig. 1). When the fluid is at rest, the first layer depth is H₁ and the second layer depth is H₂ − s_xx − s_yy, where (s_x, s_y) are the east–west and north–south slope components, respectively, and s = (s²_x + s²_y)^1/2 is the bottom slope. The densities are ρ₁ and ρ₂ respectively, the Coriolis frequency is f, and the gravitational acceleration is g.

The inviscid equations rely on the conservation of the total potential vorticity in each layer and can be written (see Pedlosky 1987, chapter 6)

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where

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is the potential vorticity anomaly in each layer and

_kπ_kβ_k,xxβ_k,yy(3)

is the total potential vorticity.

In these equations F₁ = f²/g′H₁, F₂ = f²/g′H₂ with g′ = g(ρ₂ − ρ₁)/ρ₁; β_{1, y} = β, β_1,x = 0; β_{2, y} = β + fs_y/H₂; β_2,x = fs_x/H₂, and β is the gradient of the planetary vorticity; β₂ = (β²_2,x + β²_2,y)^1/2 then measures the background potential vorticity gradient in the second layer and will be used to evaluate topographic dispersion. Also ψ_k is the streamfunction in layer k, J(A, B) = ∂_xA∂_yB − ∂_yA∂_xB is the Jacobian operator, and ζ = ∇²ψ is the relative vorticity.

b. Numerical model

The numerical model we use is a pseudospectral code in the horizontal (a biperiodic domain with a 128 × 128 horizontal grid; Orszag 1971). A biharmonic viscosity is used with a viscosity coefficient ν = 5 × 10⁷ to 5 × 10⁸ m⁴ s⁻¹ to avoid computational instability. The mesh size is dx = 20R/128 and depends on the vortex radius R (defined below). The time step is 2160 s, and the model is typically run for 150 days.

On the planetary beta plane, the vortex usually does not interact with the periodic continuation of the Rossby wave wake generated during its propagation within this time period. But for strong bottom slopes, the topographic Rossby waves rapidly crosses the domain and may interact with the vortex again. However, we believe that this does not alter our conclusions as different slope orientations do not seem to change the general vortex behavior.

c. Initial state, vortex structure

In this study we focus on vortices with a strong signature in the upper layer. We thus choose the initial streamfunctions as follows:

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where R is the vortex lengthscale and corresponds to the radius at which the velocity reaches its maximum, r is the distance from the vortex center, and Ω₁/2 its rotation rate near the center. As we only consider anticyclonic vortices, Ω₁ is positive. Notice that the initial minimum vorticity ζ₁ = ∇²ψ₁ is −Ω₁ (and not −Ω₁/2).

To test for the sensitivity of the processes to vortex structure, we have considered vortices for which ψ₂ = 0 initially (so that the lower layer was initially at rest) and vortices with no initial potential vorticity anomaly in the second layer, π₂ = 0. On the f or β plane, many studies have shown that these structures usually have a very different behavior (see Verron and Valcke 1994; Morel and McWilliams 1997). In addition to considering different initial deep flow, we have also tested for the sensitivity of our results to different slope orientations. The ψ₂ = 0 and π₂ = 0 cases have thus been systematically considered with different (s_x, s_y) couples (but the same bottom slope s) including east–west, north–south, or opposite sign slopes.

a. Process

As mentioned in the introduction, steep bottom topography is likely to maintain the lower-layer motion weak enough so that the upper-layer vortex evolves as in a reduced-gravity model. The bottom layer may however affect the general behavior and trajectory of the structure if either the lower-layer motion becomes strong enough so that the circulation it generates in the upper layer via stretching is comparable to the beta gyre or if the excitation of the topographic Rossby waves associated with the vortex displacement induces a strong energy leakage. If this latter and the second-layer motion are weak, the lower layer does not play a big role on the dynamics of the vortex, and it behaves as if the lower layer was at rest, or infinitely deep as in a reduced-gravity model.

In the following analysis, we address this issue assuming that the upper-layer vortex displacement is not affected by the topographic Rossby waves and behaves as if the lower layer was infinitely deep. We can then calculate the effect of the vortex displacement on the lower layer, and this yields an estimate of the lower-layer motion that can then be used to evaluate its influence on the development of the upper-layer beta gyre. We then seek a contradiction with the initial assumption (negligible effect of the lower layer) to derive an analytic criterion for the validity of the model.

b. Analysis

Let us develop Eq. (1) and derive a scaling for each term. The equation for the upper and lower layers can be written

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In (4a), if ψ₂ is weak enough, T2 and T4 are negligible and the equation is then equivalent to a reduced-gravity model on the β plane with an internal radius of deformation R₁ = 1/F^1/2₁. In this case, an intense upper-layer vortex keeps its axisymmetric part and translates essentially westward (see Sutyrin and Flierl 1994). This displacement is due to the development of a dipolar circulation component ψ^′₁ called beta gyre and can be evaluated together with the propagation speed C = R/τ₁ as T₁, T₃, and T₅ have the same order of magnitude (see McWilliams and Flierl 1979; Sutyrin and Flierl 1994; Reznik and Dewar 1994; Sutyrin and Morel 1997). At first, since T1 is associated with the axisymmetric part displacement, O(T1) ≈ CO(∇π₁). In the vortex core, both the stretching and relative vorticity terms have the same sign (negative for an anticyclone), and we can thus evaluate the order of magnitude of the potential vorticity anomaly π₁ by simply summing the order of magnitude of both terms. This yields O(T1) ≃ Ω₁(1 + ½F₁R²)/τ₁. (Notice the ½ coefficient for the order of magnitude of the stretching term. This coefficient depends, in fact, on the shape of the streamfunction but is usually close to ½ when R is defined as the radius where the velocity is maximum.) Concerning T3, as the Jacobian term is zero for a purely axisymmetric structure, we must take into account the beta gyre component ψ^′₁ to evaluate this nonlinear term. We get O(T3) ≈ ψ^′₁Ω₁/R². The last term represents advection of background potential vorticity by the vortex circulation and a rough estimate is O(T5) ≈ β₁RΩ₁/2. As each term has the same order of magnitude, we finally get

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Let us now consider (4b) and the lower-layer dynamics. As Rossby waves propagate along lines of constant total potential vorticity, dispersion occurs in this layer when potential vorticity contours extend to infinity. In our configuration, this happens when the order of magnitude of the background potential vorticity gradient β₂ = (β²_2, x + β²_2,y)^1/2 (≈fs/H₂ for a steep enough bottom topography) is greater than the potential vorticity anomaly gradient ‖∇π₂‖. This condition yields a constraint on the upper-layer vortex strength (considering that the order of magnitude of π₂ is mainly given by F₂ψ₁)

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This parameter controls dispersion at depth and, in practice, rapid scattering occurs for values of δ_disp above 1 (up to δ_disp ≈ 2 or so). In this case, the nonlinear terms T8 + T9 (associated with the second-layer potential vorticity gradient) can be neglected in comparison with T10 (associated with the background gradient of potential vorticity). On the f plane this sole criterion governs the vortex evolution: if it is satisfied, the vortex becomes compensated (see La Casce 1998). On the beta plane, the upper vortex propagates and this induces a motion at depth through term T7, which acts as a forcing in (4b). To evaluate the order of magnitude of the circulation in the second layer we assume that δ_disp is small enough to neglect T8 and T9 in (4b) (see appendix A). We find that ψ₂ can be decomposed into two terms evolving at two different timescales: a slow one, τ₁ = (2 + F₁R²)/β₁R, associated with the upper-layer forcing, and a rapid one, τ₂ = (2 + F₂R²)/β₂R, associated with the topographic scattering. Both parts have the same order of magnitude for the lower-layer vorticity (see appendix A); it can also be obtained if we notice that T10 and T7 must have the same order of magnitude (they are both prevailing terms for the slow part). We get

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When the vortex has an initial deep flow (for instance, when we choose π₂ = 0 initially), this scaling is, of course, not valid as long as the lower-layer motion has not been dissipated by the topographic Rossby waves. But this only lasts for a short time period of several τ₂.

c. Criteria for model validity

We now evaluate the influence of the lower-layer circulation on the upper-layer vortex evolution and look for a contradiction with the previous hypothesis (negligible influence). This circulation changes the first-layer velocity and potential vorticity fields via stretching. In (4a) the second term is mainly associated with the fast topographic wave part of ψ₂ (τ₂ ≪ τ₁) and represents the signature of the topographic Rossby waves in the first layer. Its ratio with T5 compares the topographic Rossby waves signature with the development of the beta gyre and will be called δ_TRW:

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The fourth term is associated with the changes in the potential vorticity distribution due to the lower-layer circulation and subsequent interface deviation. It is a nonlinear effect, and its ratio with T5 will be called δ_NL:

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The criteria for the validity of the reduced-gravity model are thus δ_disp < 1, δ_TRW ≪ 1 and δ_NL ≪ 1.

a. Dissipation

Let us imagine twin experiments with a vortex of the same initial shape in the upper layer, but a resting infinite lower layer on the one hand and a finite lower layer but with steep topography on the other hand. For the latter case, when δ_TRW and δ_NL are small, the topographic Rossby wave signature and the potential vorticity changes associated with the lower-layer motion (terms T2 and T4) can, a priori, be neglected in Eq. (4a) and the vortex in the first layer should evolve as if the lower layer was at rest. Thus, both experiments should yield the same evolution for the vortex in the first layer.

In fact, some discrepancies may appear because in the finite depth case, the upper-layer vortex displacement generates a motion in the lower layer that is associated with a significant energy loss on a long timescale. Thus some differences of the vortex strength and shape appear between both experiments after a while. To keep the same evolution, one must reinitialize the reduced-gravity experiment using the fields of the finite depth one with a period much shorter than the timescale associated with the decay due to the generation of bottom motion.

We have not been able to calculate the rate of decay (an estimation can however be given under some assumptions; see appendix B), but the numerical experiments seem to show that, in the reduced-gravity regime (δ_NL and δ_TRW small), it has little influence on the vortex evolution for the period of time considered in this paper (∼150 days).

b. Flat bottom

If there is no bottom topography, dispersion in the lower layer can only happen if δ_disp = F₂Ω₁R/β ⩽ 1, where β ≈ 2 × 10⁻¹¹ m⁻¹ s⁻¹ is the planetary beta. On the other hand, for the vortex in the first layer to resist dispersion and stay coherent, a similar scaling analysis can be performed, which yields δ^′_disp = F₁Ω₁R/β ≫ 1. So both criteria can only be satisfied if F₂/F₁ = H₁/H₂ ≪ 1.

This condition was first derived by Chassignet and Cushman-Roisin (1991), in the framework of a more general two-layer shallow-water model on the beta plane. They concluded that two conditions must be satisfied for the reduced-gravity model to be valid.

1. F₂/F₁ ≪ 1: the lower layer must be much bigger, by a factor of 10 or so, than the upper one.

2. min(F₁R²F₂R², F₂Ω₁R/β) ≪ 1: the vortex radius is either smaller than an internal deformation radius or smaller than a beta-based horizontal scale.

c. Sloping bottom case

When steep bottom topography is taken into account, the validity domain of the reduced-gravity regime is drastically changed. According to our scaling analysis, the bottom layer no longer needs to be much deeper than the upper one as steep topography is able to disperse the motion in the lower layer while keeping the upper-layer vortex coherent. Indeed, as β₂ depends on both the planetary beta and bottom slope, δ_disp can be greatly decreased if a steep slope is taken into account.

Let us first present an experiment with two layers of the same depth, for which Chassignet and Cushman-Roisin’s constraint is not satisfied. For this preliminary experiment, we have chosen H₁ = H₂ = 1500 m, f = 7 × 10⁻⁵ s⁻¹, β = 2 × 10⁻¹¹ m⁻¹ s⁻¹, g′ = 0.01, and a vortex with a maximum vorticity Ω₁ = 0.3f and a radius R = 40 km. The lower layer is initially at rest. Such a vortex is able to resist dispersion in the upper layer (δ^′_disp = F₁Ω₁R/β ≈ 15) while the lower layer is subject to dispersion when a bottom slope s = 0.01 is taken into account (δ_disp = F₂Ω₁RH₂/fs ≈ 0.8). In addition, in this parameter regime, the reduced-gravity model should be valid (we get δ_TRW ≈ 0.04 and δ_NL ≈ 0.1).

Figure 2 represents the vortex trajectories on the beta plane with and without bottom slope and compared to a reduced-gravity model (plain). The duration of each simulation is 150 days. Two different bottom slope orientations are presented: the dashed trajectory corresponds to the (s_x, s_y) = (0, 0.01) case (north–south slope) and circles to the (s_x, s_y) = (0.01, 0) case (east–west orientation). Initially, ψ₂ = 0 and the dotted line is associated with the flat bottom experiment. Even though some discrepancies appear after 80 days or so, the reduced-gravity model and the strong bottom slope cases yield comparable trajectories, whereas without topography the vortex displacement is quite different. Notice for instance how the trajectories are deviated westward in the reduced-gravity and strong slope cases. This is, in addition, independent of the slope orientation. The differences after 80 days are probably due to the energy loss associated with the development of motion in the lower layer for the finite depth case and we believe they can be avoided if the reduced-gravity model is reinitialized with the finite depth ψ₁ every 50 days in this case (the rate of decay calculated in appendix B yields T_decay ≈ 600 days).

a. Reduced-gravity regime

Let us now consider a more “realistic” two-layer configuration with the following characteristics: H₁ = 500 m, H₂ = 3500 m, g′ = 0.01 m s⁻², f = 7 × 10⁻⁵ s⁻¹ and β = 2 × 10⁻¹¹ m⁻¹ s⁻¹. The internal radius of deformation is R_d ≈ 30 km. We take a bottom slope s = 0.015 into account, the vortex maximum vorticity is Ω₁ = 0.3f and is held constant, but the radius R is now variable.

Figure 3 represents δ_disp as a function of the vortex radius. It shows that dispersion can occur in the lower layer for up to R = 100 km. Figure 4 shows δ_TRW (dashed) and δ_NL (plain) as a function of the vortex radius R. It shows that the reduced-gravity regime should be valid up to R ≈ 30 km or so (δ_TRW and δ_NL ⩽ 0.1). It is, in fact, rather difficult to estimate the largest value for which the reduced-gravity model and the finite depth/strong slope one give similar vortex evolution, as our criteria are based on a scaling analysis, not exact analytical calculations, and the discrepancies between both models cannot be evaluated. We have thus performed many experiments with different background stratification and found that, for the period of time considered here (150 days), both models roughly yield the same evolution if δ_TRW and δ_NL are both smaller than 0.3. Beyond this value, because of growing influence of the topographic Rossby waves in the upper layer, some discrepancies appear: the vortex shapes become different, and we observe a growing influence of the vortex initial deep flow (i.e., discrepancies appear between the ψ₂ = 0 and π₂ = 0 cases). The latter can, in fact, be explained if we consider the upper-layer potential vorticity. The ψ₂ = 0 and the π₂ = 0 initialization yields, indeed, very different Π₁ when the vortex radius becomes large. As this quantity is conserved and is important for the upper-vortex evolution, it is not really surprising to see growing influence of the initial deep flow when R is large.

It is however worth mentioning that as long as δ_NL ⩽ 1 or so, both the reduced-gravity model and the steep topography case yield similar evolutions in the sense that in both models the meridional displacement is weak and the trajectories are close, whereas in the finite depth/flat bottom case, the southward displacement is as large as the westward displacement and the translation speed is much more important. In addition, the vortex often exhibits several instability periods above a flat bottom but remains stable above a steep topography or in a reduced-gravity model. Thus, in this intermediate parameter regime (0.3 ⩽ δ_NL ⩽ 1), a vortex above a strong bottom slope appears to be trapped and to keep its signature in comparison with the flat bottom case (see Thierry 1996).

To illustrate these results, let us now compare the behavior of vortices in the reduced-gravity model to evolutions in the finite depth case with H₁ = 500 m, H₂ = 3500 m, g′ = 0.01 m s⁻², different bottom topographies, and initial lower-layer streamfunction.

Figures 5, 6, and 7 represent the trajectories of the vortices in the reduced-gravity model (plain), with a finite depth without topography and ψ₂ = 0 (dotted) initially and with a finite depth but a strong north–south slope (s_x, s_y) = (0, 0.015) and either ψ₂ = 0 (dashed) or π₂ = 0 (dash-dotted) initially. As mentioned previously, a different slope orientation (s_x, s_y) = (0.015, 0) was also considered with the ψ₂ = 0 initialization, and the corresponding trajectory is represented by circles. The results are given for three different vortex radii R = 30 km (Fig. 5), R = 40 km (Fig. 6), R = 80 km (Fig. 7) corresponding, respectively, to (δ_disp, δ_TRW, δ_NL) ≈ (0.3, 0.08, 0.1), (0.4, 0.18, 0.2), and (0.8, 0.5, 0.6). The trajectories of the vortex above an infinite resting lower layer and above a strong slope s = 0.015 are very close for R = 30 km and R = 40 km independently of the initial lower-layer streamfunction or slope orientation. They exhibit slight differences when R = 80 km (the final westward displacement are 235, 220, and 180 km for the reduced-gravity model, the finite depth/strong north–south slope with π₂ = 0 and ψ₂ = 0, respectively), but remain quite close in comparison with the flat bottom case. [Notice that, in Fig. 7, the total trajectory for the flat-bottom experiment did not fit in the plot and that the final vortex position was (x, y) = (−380, −430) km.] This result is somewhat surprising as for such radii, δ_TRW and δ_NL are not small and the topographic Rossby waves have a rather strong signature in the upper layer. We thus expected a non-negligible influence on the development of the beta gyre and a subsequent effect on the vortex propagation. That the topographic Rossby waves have no effect on the vortex displacement may be associated with their rather rapid propagation. Indeed, the beta gyre development timescale is τ₁, which is much bigger than τ₂, the timescale associated with the topographic Rossby waves. Thus, even though these latter have a strong instantaneous signal, their effect, when averaged over τ₁, can be quite small.

Figure 8 represents the upper-layer potential vorticity anomaly π₁ evolution at time t = 0, 50, 100, and 150 days for the reduced gravity case and R = 30 km. Notice the development of Rossby waves and the general westward displacement. Figure 9 compares the upper-layer potential vorticity anomaly after 150 days for a vortex with a radius R = 30 km in the reduced-gravity model (Fig. 9a), the finite depth/strong north–south slope, (s_x, s_y) = (0, 0.015), and either ψ₂ = 0 (Fig. 9b) or π₂ = 0 (Fig. 9c), and finally the finite depth/strong east–west slope, (s_x, s_y) = (0.015, 0), with ψ₂ = 0 (Fig. 9d). Each model yields rather close evolutions, whatever the initial lower-layer velocity field or slope orientation.

Figure 10 is the same as Fig. 9 except that the vortex radius is now R = 80 km and Fig. 10e represents the evolution above a flat bottom. Some discrepancies between the different models become clearer in the Rossby wave field and vortex structure but the vortex remains stable above an infinite lower layer or steep topography. This is not the case above a flat bottom where the vortex exhibits an unstable behavior and has a much larger propagation speed (see Fig. 10e).

b. Unstable regime

When δ_NL is increased beyond 1, the vortex above a strong slope is apparently unstable and dispersed by topographic Rossby waves.

Figure 11 illustrates the δ_NL ≥ 1 case. It represents π₁ after 300 days for an initial radius R = 120 km [(δ_disp, δ_TRW, δ_NL) ≈ (1.2, 0.7, 1.2)], in the reduced-gravity case (Fig. 11a), and the finite depth with a strong north–south (Fig. 11b) or east–west (Fig. 11c) slope and ψ₂ = 0 initially. The vortex remains stable above an infinite lower layer, but in the presence of a strong bottom slope it is unable to maintain its coherence in this case. This is quite clear with a north–south slope (Fig. 11b) where the vortex does not seem to move and is dispersed by topographic Rossby waves. The east–west slope case is closer to the reduced-gravity case, but some strong differences can be seen in the streamfunction field, which exhibits strong meanders and small-scale structures due to several instability period and subsequent scattering. As in this regime topographic Rossby waves have a strong signature in the upper layer (δ_TRW = 0.7), it is not surprising to see differences between two different slope orientations.

We have tested for the evolution of this vortex above a strong east–west or north–south slope and without planetary beta effect and found that it was unstable. As a compensated vortex above a slope is a steady state on the f plane, one can investigate the linear stability properties of the vortex and look for stable or unstable eigenmodes. But, to our knowledge, there does not exist any theory for the stability of an isolated compensated vortex over a slope in an unbounded domain, and previous studies have, as is the case in this one, relied on scaling analysis. For instance, La Casce (1998) based his analysis on the dispersion coefficient δ_disp. He found that when δ_disp ⩽ 0.5 or so, a compensated surface vortex is stabilized to weak perturbations above a strong bottom slope.