## 1. Introduction

Considerable evidence has accumulated for more than a decade to support the notion of a continuous flow of continental shelf water along the east coast of North America from at least as far north as the northern Labrador shelf to Cape Hatteras (e.g., Chapman and Beardsley 1989; Loder et al. 1998; Khatiwala et al. 1999). Along substantial parts of this system, the relatively fresh and cold shelf waters are separated from the saltier and warmer offshore waters by a sharp density front, typically located near the shelf break, with a swift, surface-intensified alongfront jet. This shelfbreak front is particularly well established along the Labrador shelf and through the Middle Atlantic Bight (MAB), where it is persistent and robust despite large meanders and disturbances that temporarily change the location of the front and the direction of the alongfront jet, especially at the surface (e.g., Pickart et al. 1999).

Observations also show that this shelf circulation system loses an enormous fraction (about 90%) of its volume and freshwater transport during its transit from the northern Labrador shelf to the MAB (Loder et al. 1998). Where and how this happens remains a mystery. The loss could result from a gradual “leakage” of shelf water through the shelfbreak front. Another possibility, which is the subject of this paper, is that the loss could occur through advective offshore transport at specific locations where the topography turns abruptly—for example, at the southern tail of the Grand Banks—causing the frontal jet to leave (or separate from) the shelf. Of course, the loss could be a combination of both, or it could involve some other process altogether.

A related mystery on a somewhat smaller scale is the way in which the shelfbreak front negotiates the deep channels that cut across the entire shelf (e.g., the Laurentian and Northeast Channels). Observations show that the shelfbreak front sometimes follows the topography into the Northeast Channel, and at other times it crosses directly over the channel (e.g., Bisagni et al. 1996; Smith et al. 2003). The dynamics that determine its path are not understood.

Some features of a recent idealized model of the adjustment of a buoyant coastal current over a sloping bottom, developed by Chapman and Lentz (1994) and extended by Yankovsky and Chapman (1997) and Chapman (2000), have been linked to the dynamics of the shelfbreak front. In this model, a buoyant coastal source forms a surface-to-bottom front with a surface-intensified alongfront jet that is nearly geostrophic everywhere above the bottom boundary layer (Fig. 1). Downslope buoyancy advection in the bottom boundary layer moves the entire front offshore. As the front moves into deeper water, the vertical (thermal wind) shear in the alongfront velocity reduces the near-bottom alongfront and, hence, cross-front velocities. Eventually, the depth is great enough that the cross-front velocity near the foot of the front (where the front intersects the bottom) approaches zero, terminating the downslope Ekman transport. At this location, the bottom boundary layer detaches and flows upward along frontal isopycnals. Buoyancy advection can no longer move the front, so the front is trapped along this isobath, thus producing an advectively trapped buoyancy current (ATBC).

Lentz and Helfrich (2002) recently examined the ATBC (which they call a “slope-controlled” gravity current) in more detail and placed it in a more general context with other buoyant gravity currents. They show that an ATBC develops when the parameter *c*_{ω}/*c*_{α} ≫ 1, where *c*_{ω} is the nose propagation speed of a gravity current along a vertical wall and *c*_{α} is the nose propagation speed of a gravity current along a gently sloping bottom. In the present calculations, this ratio is always in the range of 3–18 in the region where the ATBC forms, so the dynamics described above dominate.

Strong evidence for the detachment of the bottom boundary layer at the foot of the MAB shelfbreak front has been reported recently by Houghton (1997) and Houghton and Visbeck (1998) using dye releases, Barth et al. (1998) from chlorophyll and suspended sediments, and Pickart (2000) from temperature variations along isopycnals. These studies suggest that the ATBC model may offer some insight into shelfbreak front behavior and may be a reasonable starting point for addressing the dynamics of a shelfbreak front encountering large changes in bottom topography. Therefore, the behavior of an ATBC (like that depicted in Fig. 1) at both a single sharp bathymetric bend and a cross-shelf channel are examined here using a primitive equation numerical model. As a preliminary step, some expectations are explored in section 2. The numerical model and configuration are described in section 3. The results are examined in section 4, followed by a discussion and summary in section 5.

## 2. Expectations

Of primary interest here is the degree to which an ATBC follows the bathymetry and the conditions, if any, under which an ATBC separates from the shelf. Before launching into the new numerical calculations, it is instructive to consider some dynamical ideas and previous results that may provide insight into the problem at hand.

The separation of a coastal buoyancy current traditionally has been studied using inviscid, reduced-gravity models of a current along a curved boundary (e.g., Klinger 1994). The current structure along a straight coast with a vertical wall provides the upstream flow condition (Fig. 2a). Conservation of potential vorticity is then used to find the flow structure for a specified coastline curvature. As the curvature increases, centrifugal upwelling raises the interface at the coast. The curvature eventually is great enough that the interface shoals to the surface, at which point the flow is assumed to separate from the coast.

There are several problems in applying the reduced-gravity results to an ATBC at a sharp change in bottom topography, that is, an ATBC with a radius of curvature comparable to the current width. First, changes in curvature are assumed to be gradual in the reduced-gravity models, and so the current can adjust slowly and continuously to the new curvature. This condition is basically violated by definition at a sharp change in bathymetry. Second, the reduced-gravity models usually assume an infinitely deep, quiescent lower layer, that is, no bottom topography. This assumption has been relaxed by adding a sloping bottom [Jiang (1995); see Garrett (1995) for a summary], although with qualitatively similar results. On the other hand, bottom topography is critical for the development of the ATBC. Third, all dissipation is neglected in the reduced-gravity models, whereas bottom friction plays a key role in the ATBC. Fourth, the interface shoals in the reduced-gravity models until it reaches the surface, whereas the trapping mechanism of the ATBC should inhibit large excursions of the foot of the front, perhaps preventing the front from shoaling at all. If so, separation of the ATBC must take a different form and involve different dynamics than the reduced-gravity models. Therefore, the reduced-gravity models are probably of limited use for understanding ATBC separation.

In regard to the fourth issue above, it is important to point out that, for the special case of a reduced-gravity current for which the upstream flow has a horizontal interface at the coast (i.e., zero velocity and zero relative vorticity there; Fig. 2b), an equally valid solution is one in which the upstream structure is placed an arbitrary distance offshore with no flow (zero velocity) in the region between the current and the coast (above the dashed interface in Fig. 2b). This idea will be useful later.

Bottom topography is expected to inhibit separation to some degree. For example, it is easy to show that unstratified, purely geostrophic flow must follow isobaths exactly. Ageostrophic processes (e.g., local accelerations, bottom friction, nonlinearities) break this constraint, allowing flow to cross isobaths. However, numerical calculations show that it is difficult to induce separation of large-scale, low-frequency unstratified flows with realistic scales and dynamical parameters because the topographic constraint is so powerful (e.g., Williams et al. 2001).

Stratified, purely geostrophic flow must also follow isobaths exactly, unless the velocity parallel to the isobaths vanishes at the bottom (e.g., Brink 1998). This condition suggests that an ATBC, which is nearly geostrophic in its fully adjusted, trapped state (Fig. 1), should flow along isobaths in its entirety (including the surface-intensified jet) as long as the alongfront velocity at the bottom is nonzero. If, however, the alongfront velocity at the foot of an ATBC vanishes or reverses, then the flow is free to cross isobaths and could separate from the topography. The conditions under which this situation happens are unclear.

*cross*-front velocity vanish at the foot of the front, but there is no guarantee that the

*along*front velocity will also vanish there. Recent calculations have shown that the alongfront velocity close to the bottom within the front is sensitive to the parameterization of vertical mixing (Chapman 2002); it reverses when constant vertical eddy viscosity is used, whereas it is weak and oscillatory when the vertical viscosity depends on the local Richardson number. Yet, the cross-front velocity vanishes and the front is trapped in both cases. Furthermore, Chapman and Lentz (1994) showed that the addition of a weak barotropic background current flowing in the direction of the frontal jet can produce an ATBC in which the alongfront velocity neither vanishes nor reverses anywhere. This result can be understood by considering the steady, linear alongshelf momentum balance appropriate for an ATBC near the bottom,

*u,*

*υ*) are the alongshelf (

*x*) and cross-shelf (

*y*) velocities, respectively;

*p*is pressure;

*f*is the Coriolis parameter;

*ρ*

_{0}is a reference density;

*A*

_{V}is the vertical eddy viscosity; and

*z*is the vertical coordinate pointing upward. Subscripts

*x*and

*z*denote partial differentiation. Assuming the pressure gradient to be nearly independent of

*z*near the bottom, (1) can be integrated from the bottom to the top of the bottom boundary layer, yielding

*δ*is the boundary layer thickness, the overbar represents the average over the boundary layer, and

*u*

_{b}is the alongshelf velocity at the bottom. There is no stress at the top of the bottom boundary layer, and a linear bottom stress condition has been applied, that is,

*A*

_{V}

*u*

_{z}=

*ru*at the bottom, where

*r*is a bottom friction coefficient.

In a fully developed ATBC with no background current, *p*_{x} = 0 and (1) becomes a simple Ekman balance. In this case, according to (2), *υ**u*_{b} also vanishes, so this ATBC must have *u*_{b} = 0 at the foot of the front in order to be trapped. This result implies that, if separation merely requires *u*_{b} = 0, then an ATBC with no background current could separate at *any* change in bathymetry, no matter how gentle it might be and regardless of the strength of the ATBC jet.

A background current in the direction of the frontal jet (+*x*) introduces a negative alongshelf pressure gradient, *p*_{x} < 0, so *υ**u*_{b} vanishing anywhere. This ATBC would not be expected to separate immediately where the bathymetry changes, but rather should follow the isobaths until the bottom velocity within and parallel to the front vanishes. Whether and where this happens depends on the dynamics of the flow at the change in bathymetry.

A weak background current is typically imposed to ensure that the entire buoyancy current flows in the Kelvin-wave direction, thereby eliminating any upstream influence [e.g., Yankovsky and Chapman (1997); Chapman (2000); Fong and Geyer (2002); see Garvine (2001) for a critical discussion of this approach]. The reasoning is that a weak background flow is often present in nature, and it probably has little effect on the frontal behavior. This may be true along a straight shelf, but, in light of the arguments just presented, the alteration of the bottom velocity may have an important effect on the behavior of an ATBC at a sharp change in bathymetry.

If an ATBC separates and remains nearly geostrophic, it could, in principle, continue indefinitely along its path at the separation point because a geostrophic jet in a flat-bottom ocean with constant Coriolis parameter “can maintain any circular path of constant curvature, including the limit of rectilinear flow” (Robinson and Niiler 1967, p. 274). Of course, instabilities may develop and ultimately alter the flow path and structure.

In summary, the alongfront jet in an ATBC is nearly geostrophic, and so it should tend to follow the local isobaths unless or until the alongfront bottom velocity within the front vanishes. At this point, the jet may separate from the shelf and flow into the deep ocean. The separation process and location should depend on the interplay among the buoyancy current, the bathymetry, and any additional background current. A numerical model is now used to examine the details of this scenario.

## 3. Numerical model and configuration

*s*-coordinate primitive equation model (SPEM 5.1), developed by D. Haidvogel's group at Rutgers University, is used to solve the following hydrostatic and Boussinesq momentum, density, and continuity equations

*u,*

*υ,*

*w*) are the (

*x,*

*y,*

*z*) velocity (

**u**) components,

*ρ*is the difference between the total density and a constant reference density

*ρ*

_{0},

*g*is gravitational acceleration,

*K*

_{V}is the vertical eddy diffusivity, and

*t*is time. Variables

*p,*

*f*, and

*A*

_{V}have been previously defined. Variables

*F*

_{u,υ,ρ}represent dissipative functions that are required for numerical stability. Subscripts

*x,*

*y,*

*z,*and

*t*denote partial differentiation.

*r*= 5 × 10

^{−4}m s

^{−1}is the bottom friction coefficient and

*h*(

*x,*

*y*) is the bottom depth.

The vertical viscosity and diffusivity are approximated using either the Mellor–Yamada level-2 turbulence closure scheme as applied in SPEM5.1 or a prescription based on the local Richardson number as in Yankovsky and Chapman (1997). In either case, the minimum background value of *A*_{V} and *K*_{V} is 10^{−5} m^{2} s^{−1} and a maximum limit of 10^{−3} m^{2} s^{−1} is imposed. Additional vertical mixing of density is applied in the form of instantaneous convective adjustment whenever the water column becomes statically unstable (i.e., when lighter water appears under heavier water). In the present calculations, this occurs only within the bottom boundary layer.

For numerical stability, Laplacian subgrid-scale mixing with constant mixing coefficients is applied along horizontal surfaces (i.e., *F*_{u,υ,ρ} = *ν*_{u,υ,ρ} ∇^{2}*u,* *υ,* *ρ*), using the smallest mixing coefficients that produce stable calculations: *ν*_{u,υ} = 20 m^{2} s^{−1} and *ν*_{ρ} = 5 m^{2} s^{−1}.

The model domain is shown in Fig. 3. The coast (denoted by the thick solid line) is located at *y* = 50 km for the first 100 km and then abruptly shifts to *y* = 0. The offshore boundary (*y* = 160 km) is a solid wall. The cross-shelf boundaries are open; variables are prescribed at *x* = 0, and an outgoing radiation condition is imposed at *x* = 250 km. As in Chapman (2000), the bottom is nearly flat at *x* = 0, rises rapidly to become uniform in *x* until *x* = 100 km. Here the isobaths make a sharp turn toward *y* = 0 and either remain parallel and close to the coast (Fig. 3a: a single bend) or return to their original offshore locations (Fig. 3b: a cross-shelf channel). The coastal depth is 50 m for all *x* > 50 km, and the deepest depth is 250 m. Some slight variations to these bathymetries are also used, but the basic structure is the same. These choices for bathymetry, though somewhat arbitrary, represent a compromise to accomodate dynamical, technical, and computational limitations of the model and available computers.

*f*= 10

^{−4}s

^{−1}. The horizontal grid is uniform, with Δ

*x*= 1.74 km and Δ

*y*= 1.67 km. Thirty vertical grid points are used, with a concentration near the bottom to resolve the bottom boundary layer. The

*s*-coordinate mapping is

*h*

_{c}= 50 m is the coastal depth and

*θ*= 2.5.

As in Chapman (2000), each calculation begins from rest. A buoyant geostrophic inflow with velocity *u*_{i} and density (*ρ*_{0} − Δ*ρ*) is prescribed at *t* = 0 in a triangular region at the upstream boundary (*x* = 0) with depth *h*_{i} = 100 m at the coast (*y* = 50 km) and extending a distance *y*_{i} = 20 km offshore (arrows in Fig. 3). The remainder of the fluid is at rest with constant density *ρ*_{0}. An additional weak spatially uniform background current *u*_{0} (with no density anomaly) is imposed at *x* = 0. All variables on the inflow boundary remain fixed at their initial values throughout the calculation. Each calculation is run for 200 days of simulation time, by which time the flow over the variable topography is essentially steady. The time step is 432 s.

## 4. Results

*h*

_{b}

*fT*

_{i}

*g*

^{1/2}

*T*

_{i}(=

*h*

_{i}

*y*

_{i}

*u*

_{i}/2 here) is the buoyant inflow transport and

*g*′ =

*g*Δ

*ρ*/

*ρ*

_{0}is reduced gravity. Chapman (2000) showed that (10) is a good predictor of the trapping depth for a wide range of inflow parameters, and so there is no need to explore parameter space fully here. Instead, the present calculations are designed to examine the effects of the buoyant inflow velocity

*u*

_{i}, the density anomaly Δ

*ρ,*and the imposed background current velocity

*u*

_{0}on the behavior of an ATBC encountering the variable bathymetry of Fig. 3. Therefore, for ease of comparison, the buoyant inflow parameters in all of the present calculations are chosen to make

*h*

_{b}= 100 m.

Figure 4 shows the surface and bottom density fields for three examples of an ATBC encountering a single bend. The background velocity is *u*_{0} = 0.025 m s^{−1}. In each case, the ATBC forms in the shelf region upstream of the bend, exactly as in Chapman (2000). The velocity fields (not shown) are like those plotted in Chapman's (2000) Fig. 4 and drawn here in Fig. 1: a narrow surface-intensified jet flowing along the density front, offshore flow in the bottom boundary layer shoreward of the front, and detachment of the bottom boundary layer at the foot of the front with upward flow along isopycnals. As the ATBC encounters the bathymetric bend, the response depends on the buoyant inflow velocity *u*_{i}. For a weak buoyant inflow (Figs. 4a,b), the density front follows the bathymetry fairly closely at both the surface and bottom. A stronger buoyant inflow (Figs. 4c,d) generates a surface front that begins to turn and follow the bend but then crosses the isobaths and leaves the shelf, after which it becomes unstable and develops large meanders. The strongest buoyant inflow (Figs. 4e,f) produces a surface front that turns slightly at the bend before shooting offshore over the deep region. In this case the instabilities are not as prevalent, presumably being swept downstream by the stronger jet. It is clear that the ATBCs in Figs. 4c and 4e separate from the shelf and, not surprising, do so more readily for a stronger buoyant inflow, that is, larger *u*_{i}.

Perhaps the most striking feature of Fig. 4 is that the foot of the density front continues to follow the isobaths fairly closely, shifting only slightly deeper than the trapping depth regardless of the buoyant inflow properties. This feature appears in all of the calculations that have been made for this study. Once the ATBC is established, the foot of the front hardly changes depth.

These density fields imply that the front must become enormously stretched in the horizontal direction when the ATBC separates from the shelf; that is, the foot of the front remains close to the coast while the surface expression is far offshore. Figure 5 demonstrates this stretching with vertical sections of *u* and *ρ* at *x* = 145 km, just downstream of the bend where the isobaths begin to parallel the coast. In each case, virtually all of the inflow transport is carried in the surface-intensified jet, located where the isopycnals shoal to the surface. In this part of the flow, both the front and jet are almost the same as they are upstream of the bend (not shown). In the cases of clear separation (Figs. 5b,c), the remainder of the front is nearly flat, extending from the sloping bottom to the edge of the jet. Once separated, the front and jet appear simply to continue unaltered as they move over the deeper water, having detached from the constraining topography. In Fig. 5b, an eddy is present between about *y* = 15 and 40 km, a consequence of the unstable region seen in Fig. 4c. For the strongest jet (Fig. 5c), the currents over the flat part of the front between the jet and the coast are close to zero. Apart from the eddy in Fig. 4b, both of the separated cases look very much like the schematic in Fig. 2b, indicating that this separation process is very different from that of the inviscid reduced-gravity models described in section 2.

A similar set of calculations, using the channel topography (Fig. 3b), is shown in Figs. 6 and 7. Again the background current is *u*_{0} = 0.025 m s^{−1}. The results are qualitatively like the single bend in Figs. 4 and 5. A stronger buoyant inflow produces separation farther offshore and reduces the penetration into the channel (Fig. 6). The foot of the front remains close to the trapping isobath, independent of the behavior of the frontal jet, leading to horizontal stretching of the front within the channel (Fig. 7) that looks much like Fig. 2b. For the strongest buoyant inflow (Figs. 6e,f and 7c), the frontal jet appears to cross directly over the channel with little or no influence of the bathymetry. One difference from the single bend is that the ATBC reforms on the shelf downstream of the channel where it remains stable even in the cases that are unstable downstream of the single bend. The detailed dynamics of this stabilization are not understood and are beyond the scope of this paper.

So far, the influence of the background current *u*_{0} is unclear. In the above cases, the value of *u*_{0} has been considerably smaller than the inflow velocity *u*_{i}, so its effect might be expected to be small. However, as was suggested in section 2, in the absence of a background current (i.e., *u*_{0} = 0), an ATBC might separate at a bend regardless of the strength of the buoyant inflow because the alongfront bottom velocity within the front would vanish before reaching the bend. To test this notion, the channel calculation in Figs. 6c,d has been repeated using different values of *u*_{0} (Fig. 8). The penetration of the front into the channel is clearly altered by the background current; a stronger background current (Fig. 8a) produces greater penetration, whereas a weaker background current (Fig. 8c) allows the front to cross directly over the channel much like the stronger inflow in Fig. 6e. Thus, even a weak background current can have a dominating effect on the separation process. Furthermore, notice that the front in Fig. 8c extends considerably farther offshore *upstream* of the channel (0 < *x* < 90 km), forming a “bulge” and suggesting that the front may be separating around the initial bend created by the rising topography at *x* < 40 km. If *u*_{0} is reduced further, the front generates a larger offshore bulge (not shown), and for *u*_{0} = 0 the bulge never stops growing. This result was also found by Chapman (2000; see his Figs. 3 and 6) but was not explained.

The examples presented above are consistent with the expectations in section 2 that separation should occur when the alongfront bottom velocity in the front vanishes. It was argued that the background current introduces a negative alongshelf pressure gradient [*p*_{x} < 0 in (2)] that allows the cross-shelf bottom velocity to vanish while the alongshelf bottom velocity remains positive, thereby preventing separation prior to the bathymetric bend. To demonstrate this, Fig. 9 shows the horizontal velocities at the bottom, along with the surface and bottom alongshelf pressure gradients (scaled by *ρ*_{0}*f* to produce velocities), measured on a cross-shelf transect upstream of the bend and averaged along the shelf in the range 60 < *x* < 90 km to remove oscillatory currents within the front (see Chapman 2002). The background current is *u*_{0} = 0.025 m s^{−1}, and the buoyant inflow velocity *u*_{i} is changed to produce different relative influences of *u*_{0}.

In each panel of Fig. 9, the cross-shelf velocity (*υ,* thin solid line) either nearly vanishes or reverses near *y* = 90 km, the location at which the foot of the front is trapped. The surface pressure gradient is basically a measure of the interior flow, which is essentially geostrophic. The interior flow is made up of two components: the background current, which flows onshore (*p*_{x} < 0), and the buoyant inflow, which contributes no pressure gradient if the front is perfectly parallel to the isobaths. In reality, the buoyant inflow makes a positive (negative) contribution to *p _{x}* if the front is directed slightly offshore (onshore). When the buoyant inflow is relatively weak (Fig. 9a), the background current contribution to

*p*

_{x}dominates, and so

*p*

_{x}is less than 0 everywhere, including at the bottom. The alongshelf velocity is positive across the front (

*u,*solid line with dots), consistent with (2). This flow penetrates well into the channel before separating (shown in Figs. 6a,b). For a slightly stronger buoyant inflow (Fig. 9b),

*p*

_{x}is again negative, but it is reduced so that

*u*almost goes to zero at the front. This flow penetrates slightly into the channel before separating (shown in Figs. 6c,d). For an even stronger buoyant inflow (Fig. 9c), the front is oriented slightly offshore, producing a large positive interior

*p*

_{x}that swamps the background current

*p*

_{x}at the bottom and generates a reversal of

*u*in the front. As a consequence, this flow easily crosses over the channel (not shown but similar to Figs. 6e,f).

ATBCs that do not separate immediately upon reaching the bend begin to turn the corner, where both the buoyant inflow and the background current experience a small nonlinear acceleration caused by the turning. In addition, the background current accelerates as the isobaths converge. These changes combine to generate increased alongfront velocities with enhanced offshore transport in the bottom Ekman layer. This leads to offshore buoyancy advection in the foot of the front, which moves the front into slightly deeper water, much like the adjustment of the ATBC upstream of the bend. The slight frontal movement can be seen in Figs. 4b,d,f and 6b,d,f. Thermal-wind shear reduces the alongfront velocity at the bottom in the deeper water until, at some point, the bottom alongfront velocity vanishes and the jet separates. This effect is illustrated by examining the bottom velocity parallel to the foot of the front and comparing the location at which it vanishes (i.e., the separation point) with the surface currents. Two examples are presented in Fig. 10; only the region of the model domain near the first bend is shown. The solid contours delineate the edges and the center of the front at the bottom. The vectors show the surface currents, and the solid circle indicates the location of the separation point. In both cases, the surface current diverges from the foot of the front precisely at the separation point, confirming the importance of the alongfront reversal in the bottom velocity. This behavior is true for all of the cases examined for this study.

Numerous calculations like those presented above have been made using a range of inflow velocities and background currents. In addition, calculations were repeated with different turbulence closure schemes (see section 3) and slightly different bathymetries. For each case, several diagnostics have been computed to generalize the results, if possible. Figure 11a shows the depth at which the front is trapped upstream of the bend (scaled by the theoretical trapping depth of *h*_{b} = 100 m) as a function of the ratio of the buoyant inflow velocity to the background current (*u*_{i}/*u*_{0}). Almost all of the trapping depths cluster around 0.95, indicating that the front is typically trapped slightly closer to the coast than *h*_{b}, as expected in the presence of a background current (Chapman and Lentz 1994). The points that fall close to 1 are all cases in which the background current is very small.

Figure 11b shows the offshore location of the separation point scaled by the offshore location of the *h*_{b} isobath upstream of the bend (i.e., *y* = 90 km). Smaller values indicate that the front moves farther around the bend (i.e., closer to the coast) before separating. A front that continues straight at the bend has a scaled separation point location of 1. The points collapse remarkably close to a single curve. The two points at *u*_{i}/*u*_{0} = 8 that fall below the curve are for a smoother bend for which the acceleration around the bend is less and so the front travels farther before the alongfront bottom velocity vanishes. This plot clearly shows that separation depends on the relative strength of the buoyant inflow and the background current. A relatively weak buoyant inflow follows the bathymetry, whereas a strong buoyant inflow separates at the start of the bend. The plot also shows that immediate separation occurs for any strength of buoyant inflow if the background current is absent (i.e., the separation point goes to 1 as *u*_{i}/*u*_{0} goes to ∞). Furthermore, the background current need not be terribly strong to influence separation; the influence is apparent in Fig. 11b when the background current is nearly an order of magnitude smaller than the buoyant inflow velocity.

As the current rounds the bend, the foot of the front gradually moves into deeper water before separating. Figure 11c shows the depth change that occurs during this process, scaled by the theoretical trapping depth *h*_{b}. The front exhibits a greater change in depth for cases with greater relative strength of the background current (i.e., smaller *u*_{i}/*u*_{0}). This behavior may be understood by noting that the frontal transport is now augmented to some extent by the background current in the buoyant inflow region, and so the front must move to water deeper than *h*_{b} to be trapped (i.e., the frontal transport is greater than *T*_{i}). The required depth change can be approximated by including the background transport in (10), that is, by replacing *T*_{i} with *T* = *h*_{i}*y*_{i}(*u*_{i} + *u*_{0})/2. This provides a fairly accurate estimate for the frontal transport that develops in the numerical model calculations (Fig. 12) and produces a scaled change in trapping depth of Δ*h* = (1 + *u*_{0}/*u*_{i})^{1/2} − 1, which is plotted as the solid curve in Fig. 11c. Although there is considerable scatter and most points fall below the curve, the trend is correct and suggests that the depth at the separation point is largely determined by the same mechanism that leads to the upstream trapping.

## 5. Discussion and summary

The response of an advectively trapped buoyancy current at a sharp change in bathymetry (either a single bend or a cross-shelf channel) has been examined here. An ATBC forms as follows. A buoyant coastal current generates a surface-to-bottom front with a nearly geostrophic surface-intensified alongfront jet. Buoyancy advection in the bottom boundary layer moves the front offshore. The front eventually reaches a depth at which the thermal-wind shear reduces the alongshelf bottom velocity enough that the cross-shelf velocity vanishes at the foot of the front, thereby eliminating the offshore buoyancy advection in the bottom boundary layer and trapping the front along this isobath (Fig. 1). All of the calculations presented here are in the parameter range that forms an ATBC according to the Lentz and Helfrich (2002) criterion.

The results show that the separation process of an ATBC is unique and surprising in several ways. For example, the front is trapped where the cross-front bottom velocity vanishes, but separation occurs where the alongfront bottom velocity vanishes within the front. In general, these two conditions may occur at different locations, depending crucially on any ambient or background currents that may be present. In the absence of background currents, the alongfront bottom velocity vanishes everywhere within the front of a fully developed ATBC, and so the ATBC separates immediately at the single bend or the channel, regardless of the strength of the frontal jet. Extended to general bathymetries, this result implies that an ATBC should separate at *any* bend, no matter its shape or size! A background current in the direction of the buoyant inflow imposes an alongfront pressure gradient that allows trapping without a vanishing alongfront bottom velocity, and so the ATBC must follow the isobaths to some degree at a bathymetric bend. As the current rounds the bend, local accelerations cause the flow to adjust such that the alongfront bottom velocity vanishes, producing separation at that point. Therefore, in some sense, the background current controls the separation process. However, the response is counterintuitive; stronger background currents inhibit separation while weaker background currents enhance separation. Furthermore, the background current need not be large to have a major impact. It can substantially alter separation even when it is almost an order of magnitude smaller than the buoyant inflow velocity.

Unlike most geophysical flows, separation of an ATBC is basically a linear process that does not require large nonlinear inertial contributions to the momentum balances. It is essentially controlled by the dynamics of the bottom boundary layer; these dynamics are crucial in both trapping the front and determining the path of the frontal jet.

Once separated, the surface-intensified jet continues relatively unimpeded into deeper water, at which point it may become unstable and form meanders and eddies. On the other hand, the foot of the front follows the original trapping isobath remarkably closely. Together these features lead to an enormously stretched and convoluted front, which is presumably more susceptible to mixing and destruction by other forcing mechanisms not included in the present model.

To the extent that an ATBC represents a shelfbreak front, the results suggest that shelfbreak fronts should tend to separate readily at bathymetric changes, unless ambient currents prevent separation. For example, it seems likely that a large part of the coastal current system from the Labrador Shelf to the MAB could easily be lost at the southern tail of the Grand Banks; not only is the bend sharp, but the strong ambient slope currents north of the Grand Banks should keep the buoyancy current attached to the topography, whereas the ambient currents south of the Grand Banks are likely to oppose the jet, certainly enhancing separation. At large channels, such as the Northeast Channel, background currents might originate offshore, in which case these ambient currents would probably control the penetration of the shelfbreak front into the channel. For instance, strong alongshelf currents originating offshore should force the front into the channel, and weak offshore currents should allow cross-overs. Furthermore, variations in the shelfbreak jet strength are probably relatively small as compared with variations in offshore currents (in terms of the ratio *u*_{i}/*u*_{0}), and so small changes in the ambient currents—for example, from 0 to 0.03 m s^{−1}—could be enough to change the path of the shelfbreak jet from crossing over the channel to penetrating far into the channel. The shelfbreak jet probably does not vary enough to accomplish the same changes, given fixed ambient currents. These speculations are consistent with the recent results of Smith et al. (2003), who conclude that forcing by offshore flows is the most likely cause of channel crossovers. Some of their drifter tracks look remarkably like the surface isopycnals in Fig. 8.

## Acknowledgments

I thank Larry Pratt, Ken Brink, Kipp Shearman, Steve Lentz, and Rich Garvine for their helpful discussions, comments, and suggestions. Financial support was provided by the Ocean Sciences Division of the National Science Foundation under Grant OCE-9809965 and is gratefully acknowledged.

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*Woods Hole Oceanographic Institution Contribution Number 10803.