1. Introduction

Bottom topography in the ocean is a determining factor in shelf jet dynamics. Topography guides the large-scale ocean circulation and is also important in creating small-scale motions. For example, coastal topography can induce wave breaking and thereby cascade energy to smaller length scales (Thorpe 2001).

Large-scale ocean models cannot accurately describe small-scale mixing as occurs in coastal regions since these phenomena are on the order of the grid size or smaller. Therefore parameterizations must be used to represent smaller scale mixing. Studies of small scale phenomena over continental shelfs are important since they help to improve our understanding of the physical processes and, perhaps, our parameterizations in the numerical models (Spall 2001; Samelson 1998). Some examples include the study of barotropic continental shelves in finescale models, which has been useful in predicting changes in surface elevations due to tides and storm surges (Heaps 1983; Davies and Xing 2001; Davies et al. 2003) and the analysis of the way in which the internal tide generated at a shelf edge diffuses into the oceanic interior (Munk and Wunsch 1998; Wunsch 2000; Davies and Xing 2001).

Another important process, which we focus on in this work, is that a change in topography may alter the stability characteristics of an oceanic jet. The presence of variable topography establishes a mean gradient in the background potential vorticity (PV) field. This provides a mechanism through which vortex tubes can be stretched or contracted (Pedlosky 1987) and, depending on the particular flow and topography, this can destabilize or stabilize the jet. Barotropic slope currents may be topographically steered from beneath in one of two directions. The current will be classified as prograde, if the cyclonic flank is in shallow water, or retrograde if the cyclonic flank is in deep water.

Typically, jets flowing over continental shelves have order-1 Rossby numbers and deformations in the layer on the same order as the mean depth of the layer itself. Some important examples of such flows are dense water flows over sills at the Denmark Strait overflow and the Strait of Gibraltar overflow from the Mediterranean. These flows can have Rossby numbers close to 1 and 0.5, respectively, which is why we cannot expect quasigeostrophy (QG) to describe their evolution. These currents are of fundamental importance as they introduce anomalous vorticity, temperature, salinity, and possibly nutrients into the ambient waters (Spall and Price 1998).

In the oceans, the highest shears in vorticity are located either above continental shelves or submerged ridges or seamounts. These two topographies are mathematically different in that the depth of the former is monotonically increasing away from the shore, whereas the gradient of the depth of the latter changes sign. An example of the latter is the South Pacific midocean ridge, as illustrated in Webb and The FRAM Group (1991) and Webb et al. (1991). Schmidt and Johnson (1997) modeled shear flow over ridges in the context of the barotropic shallow-water (SW) model. This model is appropriate in describing a homogeneous fluid that has a small aspect ratio. This allows us to assume, to a good order of approximation, that the dynamics is hydrostatic.

Schmidt and Johnson examined a current in both barotropic and continuously stratified fluid that travels along an infinitely long ridge or trench. For the barotropic case they analyze in detail the case where the topography is a top-hat profile. For the case most applicable to our work, the barotropic limit, they find that the flow is unstable for the relevant parameters that they consider. Their analysis indicated that the ridge topography destabilized the jet. Conversely, laboratory experiments of shear flow along a ridge, Hreinsson et al. (1997), indicated that this type of topography can stabilize the flow. Presently, however, there is no criterion to indicate which parameter regions induce what type of stability for troughs or shelves. Since shelves are more studied in the literature, we begin by analyzing the stability of this type of topography. In subsequent work, we will analyze the influence of troughs on the stability of jets.

The Middle Atlantic Bight (Beardsley et al. 1985), the Svalbard Bank (Johansen et al. 1989), and the deep western boundary current in the Atlantic (Warren 1981) are all examples of continental shelves that underlie oceanic jets. Analytical studies of the barotropic SW model in Pedlosky (1980), Collings (1986), Collings and Grimshaw (1980a, b), and Bidlot and Stern (1994) have found examples where coastal shelves destabilize the overlying shear flow. Barth (1989a, b) studied a simplified two-layer SW system under the geostrophic momentum approximation. The jet that he considered runs along steep retrograde topography. He determined that the topography was a stabilizing factor and that the steeper the topography, the greater the stabilizing effect. It will be shown that these results are consistent with our findings.

Li and McClimans (2000) analyzed the linear stability of barotropic jets that flow along a bottom shelf in the rigid-lid SW model. They were mostly concerned with the Svalbard Bank, but their analysis applies to other similar nonequatorial jets as well. In their analysis, they determined the neutral curves that lie in between the stable and unstable regions. Since these curves were symmetric for prograde and retrograde topographies, they concluded that both types of topographies had a stabilizing effect; observations of the retrograde current in the Svalbard Bank supported these conclusions. However, they observed that the prograde topography tended to have more unstable modes. This suggests that there may be an asymmetry in the growth rates for the two types of topography. To determine whether this is true, one needs to calculate the growth rates in the two unstable regions, which was not done in Li and McClimans.

Oshima (1987) studied, also in the context of the rigid-lid SW model, a jet traveling along piecewise linear topography. Their main focus was to determine the effect of the presence of a side boundary and linearly sloping topography. In their analysis they determine that retrograde flow (their case of Rossby number negative) stabilizes the flow. They do not address the issue of whether prograde topography stabilizes or destabilizes the jet. Instead they study how this type of topography alters the structure of the unstable modes, particularly the symmetry.

Narayanan and Webster (1987) investigated coastally trapped waves on a continental shelf within the SW model with a free surface. The linear stability problem that they solve is essentially the same as ours with the exception that they choose different profiles for the jet and topography. However, their work differs from ours since they study how the characteristics of the jet and topography affect the structure of the waves, whether it is more like a shelf wave or a shear wave, rather than the growth rate of the waves.

Davies et al. (2003) studied how the instability of a flow at the shelf edge is sensitive to open boundary conditions, inflow, and diffusion. Their model is different from ours in several ways. First, they include bottom friction. Second, they apply either Laplacian or biharmonic lateral friction. Third, they consider different types of inflow boundary conditions. In our model, we are trying to study the inviscid dynamics and include friction, not because of physical motivation, but because it is necessary for numerical stability. The particular structure of friction, Laplacian, is chosen over biharmonic because it is easier to implement in our finite-difference scheme. Also, since our model is located in a periodic channel, we do not need any inflow boundary conditions. We are studying the same physical mechanism, the instability of a jet over a shelf, but we investigate different aspects of the problem.

One of Li and McClimans’ assumptions, which we remove in what follows, is the rigid-lid approximation. This has the drawback that it requires numerical instead of analytical techniques to solve the linear stability problem. However, its advantage is that it allows for a wider range of motions by permitting the Froude number to be nonzero.

Lozier et al. (2002) studied the linear stability problem of a surface current over the Middle Atlantic Bight in the context of the stratified hydrostatic primitive equations. They determined that the shelfbreak front is unstable. They also calculated the temporal and spatial scales and the structure of the instability of this front. They considered different background density and velocity fields; however, they did not explore parameter space to observe whether there are instances where the jet can be stabilized. Their investigation therefore does not reveal the qualitative dependence of the instability on the relevant nondimensional parameters. We investigate the dependence of the stability of a jet on the Rossby and Froude numbers and topographic heights and slope in the context of the SW model, where the fluid is homogeneous and therefore easier to study.

The nonlinear evolution of unstable jets overlying continental shelves in the SW model has not been studied directly. The series of papers, Allen et al. (1990a, b) and Barth et al. (1990), studied the instability of a jet over a flat bottom. They observed the generation of a single row of vortices as opposed to a vortex street. Their motivation was not to study the physical problem but instead to compare the intermediate models between the QG and SW models. Other works that studied the instability of a jet overlying plane or barred beaches include Allen et al. (1996), Ozkan-Haller and Kirby (1999), and Slinn et al. (1998). These models are all in a nonrotating environment and far removed from our regions of interest. The nonlinear simulations in this article are generated with two goals in mind: to support the results from the linear analysis and to see for what topographic heights fluid is transported across the jet by eddy detachment. To begin, section 2 explains the governing equations and the particular jet and topographic profiles. Then, in section 3, we present the results of the linear stability analysis to learn how the instabilities are affected by variations in the Rossby number and topographic heights. Last, section 4 presents the numerical simulations that support the linear theory and illustrate how the fluid transport is directly related to the instability of the jet. We restrict our attention almost entirely to the case of Froude number equal to 0.1, which is appropriate for deep-water currents.

Both the linear and nonlinear calculations reveal an intricate dependence of growth rate on topography. The stability properties of the jet are asymmetric with respect to prograde and retrograde topography of the same size, unlike what is found in Li and McClimans (2000). It appears that retrograde topography is always stabilizing. For order-1 Rossby numbers and smaller, prograde topography is destabilizing. Therefore, we conclude that both the magnitude and orientation of topography are important factors in determining the stability of a jet.

2. The model

We aim to further our understanding of the stability of jets flowing along topographic edges near coastlines. Instead of dealing with the intricacies involved in realistic models, we consider an idealized setting that lets us focus on the physical processes at work. In particular, we assume that the geometry is a periodic channel with smooth, one-dimensional topography (depth of the fluid) of the form

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The coordinates x and y are directed in the across-shelf and along-shelf directions, respectively. The length scale L will be the jet width; H is the maximum resting fluid depth. Therefore, the nondimensional parameters α and β specify the relative width and amplitude of the topography.

a. Shallow-water equations

While much of the work on the instability of a barotropic jet has assumed the motion is QG, this restricts the fluid depth to have small-amplitude variation. We wish to allow for finite-amplitude topography and free- surface variations. Hence, QG is an inadequate model for our motions of interest. Instead, we assume the motion is governed by the fully nonlinear SW equations (Pedlosky 1987; Salmon 1998). In addition to allowing for large variations in the layer depth, this model also includes gravity waves into the system. The inviscid, reduced gravity, SW model in dimensional form is (Pedlosky 1987; Salmon 1998)

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and

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The parameters g′ and f are respectively the reduced gravity and the Coriolis parameter, which we take to be constant. We assume that f is positive, thereby situating the model in the Northern Hemisphere. In the Southern Hemisphere, with f < 0, the topographic Rossby waves travel in the opposite direction and therefore the definitions for prograde and retrograde are reversed, with shallow water on the left and right, respectively.

The variables u = (u, υ), h, and h_B denote the horizontal velocity field, layer thickness, and bottom topography, respectively. The free-surface deformation is then η = h − h_B. In addition, L and U are the scales of the horizontal motion and velocity based on the width and peak velocity of the jet. Also, H is the scale for the depth of fluid. The geometry is a periodic channel where the across- and along-channel coordinates are x and y, respectively, with rigid boundaries at x = ±1 and the channel ends y = ±1.

The two conventional nondimensional parameters are the Rossby number Ro = U/(fL) and rotational Froude number Fr = (fL)²/(g′H). Solving the problem numerically requires friction for stability; the dimensional friction parameter is ν. This introduces, as a third parameter, a numerical Reynolds number Re = UL/ν. We note that, because the numerical model also possesses implicit diffusion, this parameter is only an estimate of the diffusion in the system. The Reynolds number for our flows of interest are Re = Ro × 10⁴.

Since our primary concern is with the inviscid dynamics, we have not included bottom friction or surface stresses. The friction necessary to stabilize the numerical scheme will be discussed in section 2c.

We can interpret these equations three ways: as a homogeneous fluid over topography (g′ → g), as a moving layer below an infinitely thick layer at rest, or as a surface moving layer over an infinitely thick deep layer with a geostrophic flow (Dowling and Ingersoll 1989).

b. Basic state

The alongshore flow is a geostrophic Bickley jet that only varies on the across-shelf coordinate, x:

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The parameter Δη denotes the maximum amplitude of the surface deformations from the mean, which we take to be positive. Note that Eqs. (5) and (6) imply that the motion of the basic state is strictly alongshore. If β is positive or negative, the motion is prograde (with the shallow water on the right) or retrograde (with the shallow water on the left), respectively. It will be seen that this parameter is instrumental in determining the stability of the jet. Figure 1 illustrates the two possible topographic configurations.

c. Numerical method to solve the SW model

We study the nonlinear evolution of a jet by numerically integrating the SW equations. Instead of solving the equations in the canonical form of Eqs. (2) and (3), we formulate them in terms of the momentum transport functions, U = uh and V = υh. This form is preferred since the nonlinearities are in conservation form as are the friction terms. The equations are as follows:

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and

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The numerical model is the same as in Poulin and Flierl (2003) except that the x momentum transport equation has an additional term because of the variable topography beneath.

The equations are solved on an unstaggered grid with a finite-difference scheme. The advection and pressure terms are discretized by third-order upwinding and fourth-order center-differencing schemes, respectively. The equations are evolved forward in time by the third-order Adams–Bashford method (Fletcher 1991). The square domain is a 200 × 200 regular grid and the state variables are defined at each node of the grid. We have verified the results of particular non-QG simulations by doing similar calculations on a 400 × 400 grid to ensure these results are robust. There are many other possible methods that we could have used, including spectral, Godunov, essentially nonoscillatory (ENS), and flux- limited schemes. However, these are all more complicated to implement, and we have instead chosen to use the simpler method described above.

To guarantee numerical stability, two measures were taken. First, the time step was chosen so that the Courant–Friedrichs–Lewy condition was satisfied:

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Second, friction was introduced into the momentum transport equations as a Laplacian operator with a viscosity coefficient ν. Since we take ν = 10⁻⁵, the numerical Reynolds number is Ro × 10⁴. For certain simulations we have decreased and increased the viscosity coefficient by a factor of 10 and have found that, although the simulations may vary slightly, their essential behavior persists. Thus we do not believe that the introduction of friction alters the qualitative behavior of the instability.

Often, in spectral methods, one applies a hyperviscosity of the form ν(−1)ⁿ⁺¹∇²ⁿ for an integer value of n; this can retain a fairly inviscid character at eddy scales but dissipates near-grid-scale energy. Applying this type of diffusion is more problematic in finite-difference methods because of the large stencil it requires. This is why we have chosen the diffusion term to be a simple Laplacian.

The boundary conditions are no-normal flow and free slip in the x direction and periodic boundary conditions in the y direction. We have chosen free slip instead of no-slip because our grid is not fine enough to resolve the boundary layers that may develop. Even though the no-slip condition is valid at the microscopic scales, there is no reason to believe that it is the appropriate one when describing large-scale fluid dynamics, like the ones we are interested in (Pedlosky 1996).

This numerical code is used with two objectives in mind. We compare the growth rates from the simulations with those predicted by the linear stability analysis in order to determine whether linear theory predicts the initial development of the instability. Next, we investigate how the transport of fluid across the jet is affected by the various elevations of the bottom topography. We consider mainly monochromatic perturbations, that is, those containing only one wavelength in the along-channel direction, —because it presents a cleaner picture of the instability processes than a more realistic, polychromatic model. Our model illustrates what asymmetries develop in the absence of complicated vortex interactions.

3. Linear stability analysis

The governing equation of the linear perturbation dynamics is the same as in Poulin and Flierl (2003) except that :

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This is the Rayleigh equation for the SW model for a one-dimensional basic state in geostrophic balance overlying one-dimensional topography. The parameter k is the wavenumber and η̂(x), û(x), and υ̂(x) are the perturbation variables. Our numerical code solves this eigenvalue problem for the phase speed using a spectral scheme (Trefethen 2000). With the phase speeds, the growth rate for each mode is easily calculated since it is the wavenumber k multiplied by the imaginary part of the phase speed. Last, the growth rate for each wavenumber is the maximum over all of the modes. The minimum value of the growth rate is zero. Details on the computational method can be found in Poulin and Flierl (2003) and Poulin (2002).

The solid lines in Fig. 2 are contours of the relative growth rate (growth rate normalized by the Rossby number) for a range of Rossby numbers and topographic heights with Fr = 0.1 and α = 0.2. The two dashed lines denote the curves beneath which the gradient of the PV for the basic state does not change sign throughout the domain: a necessary condition to guarantee linearly stability known as Ripa’s first theorem (Ripa 1983; Poulin and Flierl 2003) or the Rayleigh criteria. These curves are approximately coincident with the zero contours and therefore serve to a good approximation as both necessary and sufficient conditions for instability. The zero contour is not plotted because numerical error makes it difficulty to calculate it precisely.

Figure 2 is divided into four sectors. The two outermost sectors, indicated by white space, are the regions in parameter space where the topography completely stabilizes the flow. The two sectors of the unstable region are separated by a dashed–dotted curve stemming out from the positive β axis with positive slope. This curve, which has been drawn in manually, defines the line of maximum relative growth. On both sides of this ray, the growth rates are monotonically decreasing in moving away from the maximal ray. We have observed that this division of Fig. 2 into these four different sectors is characteristic of this type of jet and topography.

The division of the plot into these sectors has important implications. First, retrograde topography (β < 0) always stabilizes the jet (lower growth rates than the flat bottom case) and, for small enough values of the Rossby number, the jet can be completely stabilized. Second, prograde topography (β > 0) can either stabilize or destabilize the jet (higher growth rates than the flat bottom case) depending on its height and the Rossby number. There are three cases. For large values of the Rossby number, the flow is always destabilized. However, for intermediate values of Ro, the flow is destabilized up to a critical value of β beyond which the growth rate decreases monotonically. In the third case, small Ro, there are regions of destabilization, stabilization and even a region where the flow is entirely stabilized. The calculations for Fig. 2 extended to β = 0.5 indicate that the line of maximum relative growth does not asymptote to a vertical line. Therefore, the magnitude of the nondimensional parameters isan important factor that determine the stability of a jet flowing along topography.

For a qualitatively similar jet and monotonic topography, Li and McClimans (2000) determined that the neutral curves are symmetric for prograde and retrograde topography. From this they deduced that the stability for both types of topography were symmetric and stabilizing. This is contradictory to what we observe in Fig. 2. The reason for the discrepancy is that they did not calculate the growth rates within the unstable regions. Indeed, we have computed the growth rates for precisely their situation and the results indicate that there is again an asymmetry for their topography: retrograde topography is stabilizing and prograde topography can be either stabilizing or destabilizing depending on the Rossby number and topographic slope. This conclusion is consistent with the fact that prograde topography tend to have more unstable modes, as shown in Li and McClimans (2000). They rationalized that the jet flowing above retrograde topography has more stable modes because topographic Rossby waves have a preferred direction of travel, with the shallow water on the right (Pedlosky 1987). In the case of retrograde topography, the topographic Rossby waves travel in the opposite direction as the jet, making it more difficult for the resonance to arise and therefore creating a more stable profile. For very steep prograde topography, the phase speeds of the waves are too fast to resonantly interact with the mean flow, which is why these particular mean flows are completely stable (Schmidt and Johnson 1997).

Figure 3 focuses on the QG reduction of our problem. Clearly, in QG a negative β stabilizes the flow and positive β may either destabilize or stabilize the jet. However, it need be remarked that the QG approximation only holds for small-amplitude topography, so that the prediction for large topographic slopes is extrapolating beyond its range of validity. In contrast, the SW model can describe these regimes without any difficulty.

The results of linear stability calculations for various heights and slopes are presented in Fig. 4 for Ro = 1.0 and Ro = 0.25. For a given value of β, an increase of slope corresponds to α decreasing. In the first case, for any particular slope the growth rate is an increasing function of the β except where β is large and the slope is small. Therefore, the tendency is for retrograde (prograde) topography to stabilize (destabilize) the jet as the topographic slope increases. However, when β is of small magnitude, the stability of the jet is nearly invariant with the slope since the topography is too shallow to have any effect, no matter what slope it has. The case of Ro = 0.25 is similar except that it has a larger stabilizing region of prograde topography. It is not shown here, but if the plot is extended to larger slopes we would discover a region where the flow is stabilized entirely. The QG limit is similar to the second plot of Fig. 4 for the case of small topographic heights. As is shown in the plot, in this regime there is very little evidence that prograde topography can be stabilizing.

Figure 4 also illustrates how the stability of a jet is altered by changing the ratio of the width of the topography versus the width of the jet: α. Since we keep the width of the jet constant in all of our calculations, an increase in the slope parameters corresponds to a decrease in α. The first plot indicates that decreasing this ratio causes prograde and retrograde topography tend to be destabilized and stabilized, respectively. The second plot shows similar results but, in addition, how increasing the ratio α for prograde topography can be either destabilizing or stabilizing depending on the particular values of the ratio and the topographic height.

The final plot of this section, Fig. 5, illustrates how the growth rate and wavenumber of the most unstable mode is affected by variations in the Froude number; the dashed line indicates the wavenumber k = 3π. Since this dependence was already studied in Poulin and Flierl (2003), we do not examine this in as great of detail as the other parameters. We do remark that, as the Froude number increases, the growth rate decreases and the most unstable mode moves to slightly smaller scales. The same qualitative behavior was determined in Poulin and Flierl (2003).

4. Numerical simulations

We explore the nonlinear dynamics to verify the predictions of the linear stability theory and to study the dependency of the across-shelf transport on the Rossby number and topographic parameter height. Moreover, these experiments will illustrate how various topographic heights alter the structure of the finite-amplitude waves and eddies. In all simulations we set α = 0.2 and Fr = 0.1. In the first set of experiments Ro = 1.0 and β increases from −0.4 to 0.4 in increments of 0.1. This range is sufficient to illustrate the effect of topography on the stability of a jet. The second set of simulations is exactly the same except that Ro = 0.25.

In the contour plots that follow, the two uppermost frames show the topographic profile. The first frame in each series of plots shows which side of the channel has deep and shallow topography, respectively. The across- channel and along-channel coordinates correspond to the x and y axes, respectively. The free surface decreases monotonically with increasing x. Unless otherwise stated, the frames show contour plots of the PV at six different instants in time during the destabilization of the jet. The solid and dashed lines denote relatively high and low PV, respectively. The contour intervals are different in each case since the extreme values of PV depend on the topography and Rossby number.

In the nonlinear calculations we choose to measure the growth rate based on changes in the layer depth h. We plot the log of the maximum difference of the perturbed height with that of the height of the basic state, choose the region that has a linear shape, where exponential growth is occurring, and find the best line fit in this area. The slope of this line gives us the growth rate of the perturbed nonlinear simulation.

A synopsis of the growth rates for both series of runs is presented in the two plots of Fig. 6. These graphs compare the growth rates predicted by the linear theory with those calculated in the nonlinear simulations. The qualitative behavior of the dependence of the growth rates on topographic height is clearly in agreement. The root-mean-square error in the cases of Ro = 0.25 and Ro = 1.0 are 0.020 and 0.018, respectively. The error is larger in the first case because the smaller numerical Reynolds number produced more diffusion. Notice in particular how the growth rate in the second figure is monotonic whereas in the first it is not. In the first plot, with Ro = 0.25, for β = 0.4 the growth rate is lower than the flat bottom equivalent with β = 0.0. This is unlike the other prograde cases where the growth rate is higher. This proves that, for small enough Rossby number, sufficiently large prograde topography can stabilize as well as destabilize a jet in relation to the flat bottom scenario.

5. Summary and conclusions

In this article, we have solved the linear stability problem of a barotropic jet in the SW model with a free surface. We also went beyond the linear theory and studied the nonlinear evolution of the instability through the use of numerical simulations. These analyses indicate, contrary to Li and McClimans (2000), that prograde and retrograde topography are asymmetric; the former and latter tend to destabilize and stabilize the jet, respectively. However, in the extraordinary case of small Rossby number and large prograde topography, the jet was stabilized by the topographic variations. The stabilizing effects of topography and Froude number both reduced the most unstable mode to smaller wavelengths. In the instances of strong stabilization, a barrier to transport was created that prevented any fluid from crossing the center of the jet. When this scenario occurs in coastal regions, this barrier would have a serious impact on the ambient biology that are dependent upon nutrients from the coastal waters.

Tidal forcing along coastline can create oscillatory jets that flow along topography; two examples are the Georges Bank (K. H. Brink 2001, personal communication) and the Cape Cod Bay (Poulin 2002). Our calculations indicate that the instabilities that develop are expected to grow most during the prograde phase of the period. Therefore, the instabilities that develop are most likely to be similar to the instabilities predicted by the prograde profiles.