a. Model equations and solution method

The model used in this study consists of a steady background current flowing along the slope of idealized bathymetry (Fig. 1). The model uses a Cartesian coordinate system, where x is the offshore axis (positive offshore), y is the alongshore axis (positive upstream), and z is the vertical axis (positive upward). Note that this coordinate system differs from that presented in the last section, in which the system used by Pedlosky (1964, 1979) was preserved. For this formulation of the shelfbreak jet, our preference is to orient the y axis in the alongshore direction. Here, water depth is given by h(x) and the alongshore background flow is given by V(x, z). The background flow is in thermal wind balance with a mean density field ρ(x, z) according to f₀V_z = B_x, where f₀ is a constant Coriolis parameter, B is the mean buoyancy defined by B = −gρ(x, z)/ρ₀, with ρ₀ being the reference density and g the gravitational acceleration. To assess the stability of this basic state, three-dimensional velocity and density perturbations are superposed onto a two-dimensional background velocity and density field. The evolution of the perturbations is governed by the hydrostatic primitive equations, linearized about a geostrophic background state:

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The Cartesian components of the perturbation velocity are (u, υ, w); π is perturbation pressure divided by ρ₀, and b is the perturbation buoyancy. Equations (3.1)–(3.3) are the equations of motion with no frictional or external forces, and Eqs. (3.4) and (3.5), derived from the conservation of mass, represent the continuity equation for an incompressible fluid and the conservation of density, respectively. Boundary conditions stipulate that normal flow across solid boundaries is zero [u = 0 at x = 0 and w = −uh_x at z = −h(x)] and that disturbances vanish at distances far from the coast. In addition, the rigid-lid approximation (w = 0 at z = 0) is imposed. [Note: For our model, we set z = 0 at the sea surface, which differs from the formulation used in section 2 in which z = 0 defines the bottom boundary. We chose to follow the Pedlosky (1979) formulation in section 2.]

To transform the irregular grid created by the bathymetry into a regular rectangular grid (in the x and z directions), a mapping is applied to Eqs. (3.1)–(3.5). Such a mapping facilitates the use of basis functions, which are described below. The mapping, applied according to ζ = 1 + z/h(x), yields a new vertical velocity ω given by ω = w − (ζ − 1)uh_x (Xue and Mellor 1993). With this mapping it is convenient to define transport variables: (u*, υ*, b*) = (uh, υh, bh). In addition, all variables are cast in dimensionless form using the following scaling: x′ = x/L₀, y′ = y/L₀, ζ′=ζ, t′ = tf₀, u′ = u*/u₀, υ′ = υ*/u₀, ω′ = (L₀/u₀)ω, h′ = h/H₀, π′ = πH₀/(u₀f₀L₀), b′ = b*H₀/(u₀f₀L₀), B′ = B/(H₀N²₀), and V′ = V/V₀, where L₀ and H₀ are the horizontal and vertical length scales, respectively; u₀ is the typical perturbation velocity times H₀; N₀ is the Brunt–Väisälä frequency; and V₀ is the maximum of the background velocity V(x, z). For all model runs L₀ = 100 km, H₀ = 200 m, and f₀ = 9.37 × 10⁻⁵ s⁻¹ (corresponding to a latitude of 40°N). Solutions for the three-dimensional perturbations are sought in the form:

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where σ is the complex frequency (σ = σ_r + iσ_i), l is the alongfront wavenumber, the starred variables are the structure functions in x and z, and Re{} denotes the real part of the expression inside the curly braces. Substitution of Eqs. (3.6) and (3.7) into Eqs. (3.1)–(3.5) yields

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where R₀ = V₀/(f₀L₀) is the Rossby number for the background flow and S is the Burger number, defined as [N₀H₀/(f₀L₀)]². For the numerical calculation of Eqs. (3.8)–(3.12), the background flow and the perturbations are confined to a domain bounded by x = 0, x = 100 km, the sea surface (ζ = 1), and the seafloor (ζ = 0) (Fig. 1). The size of the domain was chosen to balance the need for computational affordability with the need for all perturbations to be vanishingly small at the channel walls. In addition, to provide for no normal flow, u* is set to 0 at x = 100 km. With these boundary conditions, Eqs. (3.8)–(3.12) form an eigenvalue problem with a complex frequency given by σ and eigenmodes given by Eqs. (3.6) and (3.7). The solution of this eigenvalue problem is achieved with a combination of Galerkin and Fourier collocation schemes. To obtain a solution, structure functions [e.g., u*(x, z)] are spectrally decomposed into orthogonal sets of trigonometric basis functions in the vertical (ζ) and cross-shelf (x) directions, as detailed in Xue and Mellor (1993). For numerical solution these expansions are truncated at a finite number of spectral modes M. These expansions are substituted into Eqs. (3.8)–(3.12) and manipulated to produce a standard matrix eigenvalue equation, which is solved by standard linear algebra methods for a specified background field, truncation level, and wavenumber. All model runs for this study used M = 44. Readers are referred to LRG for a discussion of model convergence. Model solutions yield growth rate (−σ_i), phase speed (−σ_r/l), and modal structure of the instabilities at each wavenumber.

b. Jet velocity profiles and bathymetry

The background velocity field, modeled as a cross-shelf Gaussian waveform that exponentially decays with depth, is expressed as

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where σ_x [=4 ln(V₀/0.1 m s⁻¹)/x²_d] and σ_z [=ln(V₀/0.1 m s⁻¹)/z_d] are formulated such that the 10 cm s⁻¹ isotach will fall at distances x_d/2 from x₀ and z_d from z₀. For all model runs, the center of the jet is placed at x = 50 km (x₀ = 50 km) and the maximum jet velocity is at the surface (z₀ = 0 m). The width and depth of the 10 cm s⁻¹ isotach are specified for each model run by the variables x_d and z_d, respectively. The background density field B(x, z) is set by the velocity field and the specification of density at the offshore boundary and the condition of thermal wind balance. At the offshore boundary, the stratification is assumed to be linear (B_z is constant) for all model runs, with the deepest isopycnal set at ρ = 1.0275 g cm⁻³ (Linder 1996). To create changes in the background stratification, the uppermost isopycnal ρ_shallow is varied at the eastern boundary. For brevity, values of ρ_shallow will be expressed using sigma-t units.

Using these parameters, two qualitatively different types of jets were produced: an idealized Middle Atlantic Bight frontal jet with strong vertical shear and one with very weak vertical shear, such that it could be characterized as a weakly baroclinic jet. The former was studied extensively in LRG, where it was shown that the modal structures associated with this jet are often surface-trapped, except for the very lowest of wave numbers. To probe the interaction of unstable modes and the bottom slope, the latter jet was also chosen for this study. The same functional form, as given above, was used for both jets. They differ only in their value of σ_z [Eq. (3.13)], which controls the strength of the vertical shear.

The bathymetry used in our model study was determined from the fit of a hyperbolic tangent functional form (Xue and Mellor 1993) to the bathymetric measurements in the Nantucket Shoals region (Linder 1996). This fit is given by

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where H_s (shelf depth) = 60 m, H_d (maximum domain depth) = 200 m, x_m (location of maximum slope) = 50 km, and α (lateral extent of slope) = 15 km. The offshore asymptotic depth in the Middle Atlantic Bight is on the order of 2000 m, but we have used 200 m in our model study to reduce the computational grid. A sensitivity study to offshore depth found negligible differences between model runs using 200 m as the offshore depth and those using depths in excess of 200 m. To study model sensitivity to bottom slope and to isolate quantities that are significant along the seafloor, the bathymetry of the model was modified by varying α in Eq. (3.14) above. The range of bathymetric slopes, produced by varying α from 5 km (steep) to a value of 25 km (less steep), is shown in Fig. 2. It is important to note that for our study, as for LRG, we have chosen to keep the frontal jet’s centerline position fixed, to maintain constant stratification at the offshore boundary and to maintain symmetrical horizontal velocity shear. Each of these constraints is made for the purpose of simplification and could be relaxed in future studies of this frontal system.

c. Prograde and retrograde jets

To understand how flow direction relative to the bottom slope affects flow stability, the velocity and density fields of the retrograde jets were constructed to be mirror images to the prograde fields, with the only exception being the orientation of the isopycnals to the bottom slope. Given the structure of the density field imposed by the thermal wind balance, isopycnals associated with a retrograde surface-intensified jet slope in the opposite direction from the bathymetric slope (Fig. 2) while the isopycnals associated with a surface-intensified prograde jet slope in the same direction as the bathymetric slope. To create equivalent interior fields, the velocity field was simply reversed while the following procedure was used for the density fields. The density field for the retrograde jet was computed by specifying the boundary conditions along the eastern border and then integrating westward using the thermal wind relation. The density values obtained along the western boundary for the retrograde jet were then specified as the eastern boundary conditions for the prograde jet. A westward integration from these boundary conditions yielded a “mirror image” of the retrograde density field.

a. Weakly baroclinic jet

A stability analysis on a weakly stratified flow (ρ_shallow = 27.25) was conducted using three different topographies. In all three cases, the structure of the jet remained unchanged; however, either the underlying bathymetry was changed or the jet’s flow direction relative to the bathymetry was changed. In one case the jet was retrograde, in another case it was prograde, and in another case the bottom was flat. For all three cases, the jet is unstable over a range of wavenumbers (Fig. 3), with the maximum growth rate occurring between wavenumbers 11 and 13, corresponding to approximately 50 km. (For all of the weakly baroclinic jets, significant growth rates were restricted to the wavenumber range of 0–30.) Of note is the distinct difference in the magnitude of the growth rates for each of the three cases. For this weakly stratified jet, the retrograde jet is more unstable than that for the flat-bottom case, which is more unstable than the prograde jet. It is apparent that, for weakly stratified jets, topography destabilizes retrograde jets and stabilizes prograde jets. As seen from Eq. (4.1), downgradient perturbation fluxes are consistent with the growth of the perturbations (∂ /∂t > 0) for both the flat-bottom retrograde and prograde jets. With the addition of topography (∂h_B/∂x < 0), the perturbation flux for the retrograde jet becomes more negative; thus the downgradient flux is increased, resulting in a more unstable jet as evidenced in Fig. 3. For the prograde jet, however, the addition of a topographic slope, where |(∂z/∂x)_ρ| < |∂h_B/∂x|, creates upgradient fluxes, which act to stabilize the flow, as seen in Fig. 3. Calculation of the kinetic and potential energy exchange between the mean and perturbations fields shows that the addition of topography increases the conversion of mean potential energy to eddy potential energy for the retrograde jet, yet for the prograde jet the topographic addition creates a transfer from the eddy potential energy to the mean potential energy, consistent with the upgradient fluxes discussed above. The difference in the conversion between the mean and eddy potential energies is reflected in the growth-rate differences for these three cases.

The difference in the topographic effect on the retrograde and prograde jet is illustrated qualitatively in Fig. 4. Because there is no flow normal to the boundary, the perturbation flow is constrained to lie parallel with the topographic slope. Thus, the degree to which velocity perturbations can create perturbation density fluxes is strongly dependent upon the slope of the isopycnals. For instance, when the isopycnal slope matches the topographic slope, the perturbation density flux will tend to zero because the perturbation velocity is essentially along a density surface. However, as the “wedge” between the bottom and the isopycnal “opens,” the perturbation density fluxes become nonnegligible because the perturbation velocity at the boundary now crosses a density surface. In Fig. 4, the upper (lower) line represents a sloping isopycnal for a retrograde (prograde) jet. When topography is added, it is apparent that downgradient fluxes for the retrograde jet will increase and those for the prograde jet will decrease. The effect of this differing geometry is apparent in Fig. 5 where the perturbation density fluxes (averaged over one wavelength) associated with the fastest-growing wave for the retrograde and prograde jets shown in Fig. 3 are presented. The retrograde jet is characterized by bottom-intensified downgradient fluxes ( < 0), and the prograde jet is characterized by relatively weak upgradient fluxes (also < 0 and also bottom-intensified). These fields are consistent with the relationship expressed in Eq. (4.1).

Based on the arguments above, a steepening of the bottom slope is expected to further destabilize a retrograde jet while further stabilizing a prograde jet. To test this expectation, flow characteristics for both retrograde and prograde jets were held constant while the bottom slope was changed. Shown in Fig. 6 are the results of a study in which α was varied to create more and less steep slopes. As evident from Fig. 6a, a steepening of the slope destabilized the retrograde jet; an inspection of Fig. 6b shows the reverse: increasing bottom slopes are associated with decreasing growth rates for prograde jets. It is important to keep in mind that the perturbations are of mixed nature; that is, energy is gained through baroclinic/barotropic instabilities. In both the retrograde and prograde cases, an increasing bottom slope should lead to the stabilization of the barotropic instability. Given that the retrograde jet is destabilized, we surmise that the strong downgradient density fluxes that would increase with increasing slope are sufficient to offset the barotropic tendency for stabilization. Overall, it is important to note that a relatively small change in the bottom slope has a larger impact on the prograde jet than on the retrograde jet, perhaps because the barotropic and baroclinic instabilities both act to stabilize the flow.

Using the geometric argument discussed above, it is readily apparent how a change in the background stratification will affect the stability of these jets. Given that the slope of the isopycnal is determined by

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an increase in the stratification will act to reduce the magnitude of the isopycnal slope for both the retrograde and prograde jet, as shown in Fig. 4. An increase in stratification essentially flattens the isopycnals for both types of jets, such that a convergence of the stability characteristics at high stratification is expected. To check the validity of this statement, the background stratification was increased for the retrograde and prograde jets used to produce Fig. 3. As seen in Fig. 7a, the retrograde (prograde) jet is indeed stabilized (destabilized) as the stratification is increased.¹ The growth rates of the two jets converge for strong stratification, when both jets have essentially the same geometric configuration for the isopycnals and bottom slope. This result implies that the barotropic instability for these retrograde and prograde jets is equally affected by the topography, consistent with the Li and McClimans (2000) study of a purely barotropic shelfbreak jet. Though not shown here, when the highly stratified retrograde and prograde jets are run with a flat bottom, the growth rate increases, reinforcing the result that topography stabilizes a barotropic instability. Of interest is that even though these jets have strong horizontal shear (R₀ = 0.72), their growth rates are affected by relatively small changes in the orientation of the isopycnals relative to the bottom slope. It is clear that the release of potential energy by the perturbation fluxes at the bottom has a significant effect, though in no cases was the flow completely stabilized by the addition of topography. However, growth rates can vary by more than a factor of 2 on the basis of bottom isopycnal orientation alone (Fig. 7a). Thus, even for weakly baroclinic jets, a small change in the background isopycnal slope can produce growth-rate changes that are effectively distinct (<1 day as compared with 3 days; Fig. 7a). In fact, such changes in the growth rate are comparable in magnitude to those when the horizontal shear of the background flow is doubled (LRG). Thus, relatively small temporal and/or downstream changes in the background structure of the shelfbreak jet could have large consequences for the resultant variability. Of importance is that these changes would depend upon whether the jet was prograde or retrograde.

A calculation of the energy conversions for each of the model runs confirms the expectation that changes in the orientation of the isopycnals relative to the bathymetry affect the energy conversions, in turn affecting the perturbation growth rate. Though the available potential energy is small relative to the mean kinetic energy for the weakly baroclinic jets, it is not zero, and, as evidenced by the changes in the growth rates when the mean kinetic energy is held constant, its release is significant to the resultant instability. A change in the isopycnal slope relative to the bathymetric slope produces significant changes in the horizontal heat flux term (indicating the transfer of mean potential energy to eddy potential energy). Changes in this energy conversion are reflected in the growth-rate changes.

To test the intuitive understanding of the effect of stratification changes on flow stability, illustrated schematically in Fig. 4 and explained above, the right-hand side of Eq. (4.1) is evaluated as a function of cross-stream distance for each of the model jet configurations shown in Fig. 7a. As seen in Fig. 7b, the magnitude of the perturbation density fluxes is maximized near the jet center, primarily because the bottom slope is maximized there, but also because the isopycnal slopes are also maximized there. For the retrograde jet, the downgradient perturbation flux ( < 0) decreases in magnitude as stratification is increased, thus stabilizing the flow. An increase in stratification for the prograde jet, however, increases the magnitude of the perturbation flux, destabilizing the flow. The convergence, as noted in the growth-rate curves, is also apparent for the perturbation fluxes at strong stratification (Fig. 7b). However, we note that while the increase in stratification for the prograde jet essentially opens the wedge between the bottom slope and the isopycnals, thus creating a larger magnitude for the perturbation density flux, the direction of the fluxes are upgradient.

Because baroclinic instability is characterized by energy exchanges in which the mean available potential energy is lost to the eddy potential energy, which is then transferred to the eddy kinetic energy, it is instructive to examine the structure of the cross-shelf momentum fluxes, uυ, for these unstable jets. Shown in Fig. 8 are the cross-shelf momentum fluxes associated with the fastest-growing wave for some of the weakly and strongly stratified retrograde and prograde jets studied in Fig. 7. The flux fields exhibit a signature of downgradient behavior: the two lobes, split at the V_max location, are of opposite sign, because the mean velocity gradient changes sign at V_max. Such a structure implies the importance of the barotropic instability for this high-Rossby-number flow (R₀ = 0.72). For the weakly stratified jet, the fluxes are nearly uniform in the vertical direction, with a slight surface intensification. The retrograde jet, which has been destabilized by topography, has a relatively stronger flux field than the prograde jet, which was stabilized by topography. Of importance is that the fluxes for these weakly stratified jets are present at the bottom, in contrast to the fluxes for the strongly stratified retrograde and prograde jet. For these jets, the same doubled-lobed structure, centered on V_max, is present; however, the flux field is strongly surface intensified such that there is no signature at the bottom. In effect, stratification has insulated the perturbation from the influence of the underlying bathymetry. Such behavior is reminiscent of baroclinic (Eady) modes in which the depth scale of the perturbation is essentially the Rossby height H_R, given by f₀/(N₀l) (Gill 1982). As the stratification increases, the Rossby height, where the steering level of the perturbation is located, decreases and the perturbation “moves off” the bottom (Barth 1994). (For the dependence of H_R on l, the alongfront wavenumber, refer to LRG.) It is noted also that the perturbation momentum flux is greater for the retrograde jet than for the prograde jet, perhaps an indication of the stronger perturbation density flux that results for that jet. Also, the offshore intensification of the perturbation momentum flux for the highly stratified jets suggests the possibility of an asymmetry in cross-frontal exchange.

In summary, topography stabilizes a weakly baroclinic prograde jet and destabilizes a weakly baroclinic retrograde jet. However, as stratification is increased, the prograde jet is destabilized and the retrograde jet is stabilized, resulting in no difference in the growth rates between these jet types. These results essentially show that the geometry of the isopycnal relative to the bottom slope either promotes or stifles perturbation density fluxes that feed the growth of the instability. This result, predicted for quasigeostrophic dynamics, is shown to hold for a range of stratifications and bottom slopes for retrograde and prograde jets governed by primitive equation dynamics. It is interesting that the bottom density fluxes have an impact on the flow stability despite the fact that the flow is strongly barotropic. Different runs in which the background Rossby number varied gave qualitatively the same results; that is, the retrograde/prograde jet dependence on topography and stratification was preserved. Also, the effect of the Rossby number on the stability characteristics was unchanged from that reported in LRG; that is, topography was not a modifier of this behavior.

b. Strongly baroclinic jet

The effect of topography on flow instability was tested for a model jet with considerable baroclinic structure. Unlike the jets studied in the previous section, these baroclinic jets are characterized by strong horizontal and vertical shear. We chose to study the standard jet configuration used in LRG’s parameter study: a jet with a depth of 70 m, V_max of 60 cm s⁻¹, and R₀ of 0.72, for both weak and strong stratification. As shown in Fig. 9, the growth-rate curves for the flat-bottom case, the retrograde case, and the prograde case exhibit only small differences—particularly so for the weakly stratified jet. As was demonstrated in LRG, the perturbation modes for these baroclinic jets are surface trapped, concentrated where there is appreciable vertical shear (according to their Rossby height, as discussed earlier). Because the perturbation velocities near the sloping boundary would be negligible, it follows that any perturbation density flux would be too weak to affect the overall stability of the jet. It is interesting to note that the effect of increasing stratification for each of these baroclinic jets is to stabilize the flow (see the parameter study conducted in LRG), unlike the cases discussed above in which increasing stratification produced a more (less) stable retrograde (prograde) jet. Because a flattening of the isopycnals generally reduces the potential energy available to feed an instability, reduced growth rates are expected. This expectation, however, is altered when considering the perturbation fluxes near the bottom and their dependence on the orientation of the isopycnal slope relative to the bottom slope, as explained in the previous section.

In addition to being insensitive to the bottom slope, the growth-rate curves for these strongly baroclinic jets differ from those shown in Figs. 3 and 6 in that the growth rate increases with increasing wavenumber and, especially for the low-stratification runs, the maximum growth rate lies beyond the l = 30 cutoff. As demonstrated by LRG and Barth (1994), the instability of jets with appreciable vertical shear is dominated by a frontal instability mode characterized by smaller wavelengths and smaller vertical scale than the traditional baroclinic (Eady) mode (Moore and Peltier 1989). In all model runs for both the retrograde and the prograde jets, this frontal instability mode was insensitive to the presence of the bottom slope and to changes in the orientation of the background isopycnals to the bottom slope. As noted in section 2, Barth (1994) also noted this insensitivity in his study of coastal upwelling jets. Furthermore, in all parameter studies run in LRG, the frontal instability mode dominated—that is, it had a larger growth rate than the baroclinic mode. Thus, we are left to conclude that for those jets with relatively strong vertical shear, such that the Rossby height ≪ water depth, the stability of the jet will be insensitive to the bottom. Our results have shown that the baroclinic mode is sensitive to the bottom configuration, but only for some jet/slope configurations. Topography has been shown to influence the stability of weakly baroclinic jets, characterized by weak stratification. An increase in the stratification reduces, and eventually eliminates, this dependence. In addition, an increase in the vertical velocity shear such that the jet is more surface intensified reduces and then eliminates the effect of topography on stratification. In both cases, it is the contribution of perturbation density fluxes near the boundary to the instability that has been altered. The degree to which the stratification needs to be increased to eliminate any topographic influence depends strongly upon the strength of the flow field near the bottom boundary. This dependence is shown in Fig. 10 in which the difference in growth rate between a prograde jet and a retrograde jet is plotted as a function of z_d (the depth of the 10 cm s⁻¹ isotach) for varying background stratifications at the eastern boundary. [Note that for those cases in which z_d exceeds the water depth, the implication is that 10 cm s⁻¹ would have fallen at z_d given the structure of V(z).] For weakly stratified flows (ρ_shallow = 27.25 and 26.0), the effect of topography is seen to increase as z_d is increased. In other words, the geometry of the sloping isopycnals at the bottom boundary influences the growth rate for those jets with appreciable velocity shear near the bottom. On the other hand, strongly stratified jets (ρ_shallow = 24.0, 22.0, and 20.0) are not affected by the underlying topography over a range of depths. For these cases, the Rossby height is sufficiently removed from the bottom such that the perturbation is insulated from any topographic effects.