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The adjustment of a shelfbreak jet to cross-shelf topography
Deep-Sea Research II 48 (2001) 373}393
The adjustment of a shelfbreak jet to cross-shelf topography夽 William J. Williams*, Glen G. Gawarkiewicz, Robert C. Beardsley Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Received 7 October 1998; received in revised form 14 July 1999; accepted 10 December 1999
Abstract A rigid-lid, "nite-di!erence numerical model is used to study the adjustment of inviscid, along-shelf, barotropic shelfbreak jets to cross-shelf, channel topography. The channel is embedded in the shelf topography perpendicularly to the shelfbreak, the shelfbreak jet #ows with the direction of propagation of long-wavelength, topographic Rossby waves, and the coast is su$ciently distant so as not to a!ect the #ow. Three models are used that vary the strength of the channel topography S"(f*hD/;h, where f is the Coriolis parameter, *h is the di!erence between the shelf depth and the channel depth, D is the width of the slope into the channel, ; is the maximum speed of the jet at the in#ow, and h is the depth of the shelf. Estimation of the path of the jet from the in#ow parameters and the geometry of the channel is possible in some cases. Generally, for large S the #ow follows the topography of the channel and for small S the #ow crosses the channel. The motivation for this study is the episodic #ow of Scotian Shelf water from the Scotian Shelf across the Northeast Channel to Georges Bank. The steady, inviscid, non-linear, barotropic dynamics studied here do not allow such a #ow for a channel of similar dimensions to the Northeast Channel and for #ow speeds within the oceanographic range. Other factors such as strati"cation, wind stress and time variability need to be introduced to account for this phenomenon. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Scotian Shelf water episodically #ows from the Scotian Shelf across the Northeast Channel to Georges Bank (Bisagni et al., 1996). Georges Bank is on the western edge of the North Atlantic Ocean at approximately 663W 423N (Fig. 1). To the northeast of Georges Bank lies the Northeast Channel which separates Georges Bank from the Scotian Shelf. The channel is 45-km wide and has 夽
Paper published in December 2000. * Corresponding author; Present address: Institute of Marine Sciences, University of Alaska, P.O. Box 757220, Fairbanks, AK 99775-7220, USA. Fax: #1-907-474-7204. E-mail address: [email protected] (W.J. Williams). 0967-0645/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 4 5 ( 0 0 ) 0 0 0 8 5 - 0
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Fig. 1. A map of the Northeast Channel area showing the mean circulation (solid arrows), and the episodic #ow across the Northeast Channel (dashed arrow). The Northeast Channel is marked by NEC and the x- and y-axis are marked in degrees longitude and degrees latitude, respectively.
a sill depth of 240 m. Typical shelf depths in this region are +100 m, and so the channel forms a relatively deep, wide gap in the otherwise continuous shelf topography. The channel is also the deepest route for water to #ow between the Atlantic Ocean and the Gulf of Maine. Surface circulation over the shelf in this region is generally from the northeast to the southwest (Chapman et al., 1986; Butman and Beardsley, 1987) (Fig. 1), with a component of the #ow being close to the shelfbreak (Smith and Petrie, 1983; Smith, 1983). This southwestward #ow is complicated by the presence of the Northeast Channel and the Gulf of Maine. The typical route for the #ow over the Scotian Shelf is for it to turn to the right, before reaching the Northeast Channel, and #ow past Cape Sable into the Gulf of Maine (Smith and Petrie, 1983; Smith, 1983). Only after #owing around the Gulf of Maine does the water reach Georges Bank and continue southwestward (Butman and Beardsley, 1987). This circuitous route is due to the presence of the Northeast Channel. A barotropic, geostrophic #ow is constrained to follow isobaths (conserving potential vorticity) so such a #ow over the Scotian Shelf #owing towards the Northeast Channel would be required to turn and #ow into the Gulf of Maine rather than crossing the channel. Episodic #ows of Scotian Shelf water across the Northeast Channel have been observed many times (Bisagni et al., 1996). Observations of these are clearest in late-winter/early-spring when the
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di!erence in temperature and salinity between Scotian Shelf water and the water over Georges Bank is the greatest. Perhaps the best set of observations was collected in 1992 and is described in Bisagni et al. (1996). During February of that year, a plume of cold, fresh Scotian Shelf water crossed the Northeast Channel to the southern #ank of Georges Bank. The plume was clearly visible from AVHRR imagery and persisted into June. Flows that cross the isobaths of the Northeast Channel cannot be considered simply barotropic, geostrophically balanced currents. Factors that could allow episodic #ow from the Scotian Shelf to Georges Bank are strati"cation, wind stress, variable in#ow, and inertia. To begin to study these factors, the possible role of inertia is discussed here by considering homogeneous #ows in the presence of topography similar to the Northeast Channel. Simple scalings of the shallow-water potential vorticity equations are developed in the next section. These scalings give an estimate of the dynamical size of the channel-like topography in comparison to the shelfbreak jet. Three model domains are then described which allow exploration of the adjustment of a shelfbreak jet to strong, weak, and wide channel topography. A numerical model, used to solve for the #ow "elds in each domain, is then introduced and described in Section 3. Next, we present the results from using the numerical model in each model domain, and then consider the irreversibility of the #ow "elds generated by the numerical model, the stability of the #ow and directions for future research.
2. Models Isobaths near the shelfbreak of the southwestern Scotian Shelf turn sharply 903 to the right at the Northeast Channel with a radius of curvature r of about 5 km. The Rossby number R of a #ow M following these isobaths is ;/fr, where ; is the #ow speed scale and f"10\ s\ is the Coriolis parameter. For a #ow with ;'0.2 m s\, R '0.4. R is not small in this case and the dynamical M M importance of #ow inertia needs to be considered. The simplest equations of motion that can be used to consider inertial e!ects are the inviscid, shallow water equations on an f-plane with a rigid lid 1 *u #(u ) )u#f;u"! P, o *t
) (hu)"0,
(1) (2)
where u is the velocity vector, t is time, o "1000 kg m\ is the density, P is the pressure on the rigid lid, and h is the depth. Density is assumed constant throughout the domain. The rigid-lid approximation is expected to be very good since the barotropic Rossby radius is much larger than the width of the Northeast Channel and the depth variations are large. In the models presented here, a geostrophic jet, centered close to the shelfbreak and running parallel to the shelfbreak, impinges on channel-like topography running perpendicular to the shelfbreak (for example, see Fig. 3). The orientation of the jet and channel is such that it mimics the #ow over the shelfbreak of the Scotian Shelf towards the Northeast Channel. Ageostrophic adjustments of the model jet to the topography are of interest here and will contain some departure of the #ow from its initial depth. An estimate of the scale of this departure can be obtained from the
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potential-vorticity equation
f#u *u #u ) u! u ) h"0, h *t
(3)
where u is the relative vorticity de"ned by *v/*x!*u/*y. If the #ow is a balance between the non-linear advection of relative vorticity u ) u and the stretching of vortex columns due to the topography (( f#u)/h)u ) h, then the length scale for the #ow used to estimate u must be
; . (4) b 2 ; is again the #ow speed scale and b "f*h/Dh is the topographic b-e!ect, where *h is a scale for 2 the change in height of the topography, D a scale for the width of the topography, and h a scale for the typical depth. If L is compared with the scale for the topography D, a measure of the strength of the topography is obtained
L"
L f*hD S" " . ;h D
(5)
Three models are used in this paper: the corner model (Fig. 2), the narrow-channel model (Fig. 3), and the wide-channel model (Fig. 4). They consider the e!ect on the #ow of varying the strength of the topographic parameter S. The corner model (Fig. 2) contains only a shelf with a sharp right hand turn in the shelfbreak, which is meant to mimic the southwestern Scotian Shelf and its border with the Northeast Channel. For jet speeds within the oceanographic range (0.1';'2.0 m s\), the topography of the slope is strong (20'S'4) and the jet is expected to follow the topography. The manner of adjustment of the jet to the corner is of interest. Klinger (1993) studied the adjustment of a barotropic coastal jet to a sharp bend in the coastline and topography. Due to its inertia, this coastal jet overshoots the corner in the coastline and a stationary eddy forms between the jet and the coast immediately downstream from the corner, the Rossby number of the #ow there being O(1). In the corner model used here, the coast is su$ciently distant so as to play no role in the adjustment of the jet to the sharp corner in the shelf topography, and so it is not required that the jet overshoot at the corner. The narrow-channel model (Fig. 3) features a long channel embedded in the shelf with arbitrarily weak topography (small S) so that it is possible to have #ow crossing the topography. This provides a route for #ow to cross the channel or #ow around the edge of the channel. The shelf-width and channel-length are as large as possible so as to closely simulate the adjustment of the #ow at the shelfbreak to an in"nitely long channel. The wide-channel model (Fig. 4) di!ers from the narrow-channel model by having a wide #at base to the channel, so, while the total width of the channel can be the same, the depth pro"le is very di!erent. This channel topography closely mimics the topography of the Northeast Channel, which has very steep sides and a relatively #at bottom. Over the #at bottom of the channel there is no vortex stretching due to topography, so the scalings described above are less obviously applicable. Again, the shelf width and channel length are as large as possible so as to closely simulate the adjustment of the #ow at the shelfbreak to an in"nitely long channel.
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Fig. 2. The computational model domain for the corner model. The upper panel is a plan view showing the isobaths (thin lines) every 100 m from 100 to 1600 m, the free-slip boundaries (thick lines), and the in#ow jet. The value of the in#ow is variable. The lower panel is a cross-section of the bathymetry used.
3. Numerical model The numerical model used to solve for the #ow "elds shown is the Semi-spectral Primitive Equation Model version 5.1 (SPEM 5.1), an earlier version of which is described in Haidvogel et al. (1991). This model is a rigid-lid, p-coordinate, 3-D primitive-equation, "nite-di!erence model and, while it is more complex than the present study requires, it allows for the study of the shallow water equations and for subsequent modeling of the e!ects of strati"cation, wind-stress and bottom friction. The equations solved by SPEM 5.1 are
*u *u 1 *P *u *u #(u ) )u!fv"! #A # #A , & *x *y 4 *z o *x *t
(6)
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Fig. 3 . The computational model domain for the narrow-channel model. The upper panel is a plan view showing the isobaths (thin lines) every 10 m from 100 to 600 m, the free-slip boundaries (thick lines), and the in#ow jet. The values of the in#ow and the depth and width of the channel are variable. The lower panel is a cross section of the channel bathymetry used.
*v *v *v 1 *P *v #(u ) )v#fu"! #A # #A , & *x *y 4 *z *t o *y *P "!og, *z *u *v *w # # "0, *x *y *z
(7) (8) (9)
subject to the top and bottom boundary conditions of no #ow through the rigid lid and the bottom, and no stress on the rigid lid and the bottom, w"0 at z"0,
(10)
uh #vh "0 at z"!h, V W
(11)
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Fig. 4 . The computational model domain for the wide-channel model. The upper panel is a plan view showing the isobaths (thin lines) every 10 m from 100 to 600 m, the free-slip boundaries (thick lines), and the in#ow jet. The values of the in#ow and the depth and width of the channel are variable. The lower panel is a cross section of the channel bathymetry used.
u "0, v "0 at z"0, (12) X X A u "0, A v "0 at z"!h. (13) 4 X 4 X The horizontal viscosity A is set to a small value (50 m s\) to provide numerical stability by & damping numerical noise in the model. It makes a very small quantitative di!erence in the #ow "elds generated and is dynamically insigni"cant compared with other terms in the momentum equation. The vertical viscosity A is set to 0.002 m s\ to dampen numerical noise in the vertical 4 and ensure that the #ow is depth-independent. The numerical solution is then essentially that of Eqs. (1) and (2). The model domains used are shown in Figs. 2}4. The closed boundaries of the model domains are free-slip walls, so on the wall u "0, ,
*u "0, *x ,
(14)
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where u is the horizontal velocity perpendicular to the boundary, u is the horizontal velocity , parallel to the boundary, and x is the horizontal coordinate perpendicular to the boundary. The , open out#ow boundary in each model domain is a no-gradient out#ow condition on all variables *
"0, (15) *x , where is any variable. This boundary condition is dynamically consistent with the geostrophic out#ow that is expected for steady states obtained in this model. The in#ow boundary in each model domain has a speci"ed in#ow perpendicular to that boundary that is independent of depth. The in#ow velocity is typically a Gaussian jet centered on the shelfbreak
u ";exp ! ,
x !x , u "0, ¸
(16)
where x is the x -position of the shelfbreak, ¸ is the width scale of the in#ow jet, and ; is the maximum speed of the in#ow jet. For this study, the model was con"gured with the minimum of three grid points in the vertical and a horizontal resolution of 2.083 km in each direction. The time step was chosen so that advection, wave propagation, and di!usion were properly resolved. Typically, there were 100 time steps in 24 h. Each experiment was run until the initial conditions had evolved to a steady, or statistically steady, state. A few experiments were done at twice the horizontal resolution to test the accuracy of the #ow "elds at the lower resolution.
4. Corner model The corner model is used to explore the adjustment of the shelfbreak jet to strong channel topography (large S) where the #ow is expected to stay attached to the topography. Fig. 5 shows four di!erent #ow "elds after the initial transients have decayed away. Identical model parameters are used in these solutions except the maximum in#ow speed ; was increased from 0.1 m s\ in Fig. 5(a), to 0.5 m s\ in Fig. 5(b), to 1.0 m s\ in Fig. 5(c), and to 2.0 m s\ in Fig. 5(d). All these model "elds are qualitatively similar: the #ow is steady, smoothly turns the corner, and is nearly symmetrical about the corner. There is no eddy formation or overshoot at the corner in these "nal #ow "elds even though the initial transients for these model runs contain these features. The symmetry of the streamlines about the corner implies that a #uid column nearly returns to its initial depth after #owing past the corner. Fig. 6 shows the depth following a #uid column along three streamlines from Fig. 5(c). The #ow shoals as it approaches the corner and then deepens on leaving the corner. The dynamical balance associated with this #ow is best described by considering the conservation of potential vorticity along a streamline
d f#u "0. h dt
(17)
Conservation of potential vorticity implies that accompanying the shoaling of the #ow at the corner is a gain in anticyclonic relative vorticity. In streamwise coordinates, the relative vorticity
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Fig. 5. Contour plots of the transport streamfunction for the corner model. The in#ow in each case is a Gaussian jet centered on the shelfbreak with ¸"20 km. The maximum in#ow speeds are 0.1, 0.5, 1.0, and 2.0 m s\ in (a), (b), (c) and (d), respectively. The dashed contours are the streamlines and the solid contours mark the shelfbreak and the bottom of the slope.
may be written *u u u"! ! , *n r
(18)
where u is the speed of the #ow, r is the local radius of curvature of the #ow, and n is the cross-stream coordinate. The "rst term of this equation is due to the local radius of curvature of the #ow and the second term is due to the shear in the #ow. To negotiate the corner, the #ow turns anticyclonically and so the relative vorticity due to the turning of the #ow is also anticyclonic. There is then a possible balance between the shoaling of the #ow and the turning of the #ow where
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Fig. 6. The depth of three streamlines for ;"1.0 m s\ (Fig. 5(c)) plotted versus the distance along the streamline around the corner, which is located at +105 km. The dash}dot line is a streamline from entirely over the shelf, the dashed line from entirely over the slope, and the solid line is from a streamline between the other two.
the anticyclonic relative vorticity of the turning of the #ow around the corner is provided by the gain in anticyclonic relative vorticity due to shoaling of the #ow. Such a #ow could occur without requiring a dramatic change in the shear of the #ow, and thus the structure of the jet may remain largely unchanged. Figs. 7 and 8 show pro"les across the jet at the corner and at the in#ow from the #ow "eld shown in Fig. 5(c). The velocity pro"le of the jet is both wider and slower at the corner than at the in#ow. The maximum speed is reduced from 1 to 0.79 m s\, and the jet width increases by +50%. The size of the shear in the jet is reduced in the wide, slow jet at the corner. On the right-hand side of the jet, over the shelf, the shear becomes more cyclonic, and on the left-hand side of the jet, over the slope, the shear becomes more anticyclonic. The vorticity balance at the corner, therefore, is di!erent on either side of the jet since the change in the vorticity due to the change in the shear of the #ow is of opposite sign. On the right-hand side of the jet, the total relative vorticity in the jet becomes slightly more anticyclonic at the corner, indicating the small relative change of depth of the #ow over the shelf of only +0.1 (Fig. 6). The increase in cyclonic vorticity due to the shear of the jet is opposite to the overall anticyclonic gain in vorticity. The anticyclonic vorticity due to the turning of the jet compensates for this. On the left-hand side of the jet, the total relative vorticity in the jet goes from being strongly cyclonic to being strongly anticyclonic, indicating the large relative change of depth of the #ow over the slope which reaches a maximum of 0.52 near the top of the slope (Fig. 6). The increase in
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Fig. 7. Comparison of the velocity pro"le of the jet at the in#ow (dashed line) and at the corner (solid line) for ;"1.0 m s\ (Fig. 5(c)).
Fig. 8 . Comparison between the vorticity pro"le of the jet at the in#ow (dashed line) and at the corner (solid line) for ;"1.0 m s\ (Fig. 5(c)). The dash-dot line is the vorticity due to the shear in the #ow at the corner.
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anticyclonic vorticity is largely due to the anticyclonic turning of the #ow around the corner, with a smaller contribution from the change in shear of the jet. At the corner of the shelfbreak, the maximum Rossby number of the #ow is about 0.4. If the jet were #owing around a corner in the coastline, the Rossby number would be expected to be larger (Klinger, 1993). It is the broad adjustment to the corner in the shelfbreak, which is possible when the coastline is distant, that leads to the lower Rossby number. Robinson and Niiler (1967) described the path of thin jets over weak topography. They obtained a path equation that is appropriate here,
*h * ; #f ;M "0, *s *s r
(19)
where ; is the speed of the jet, s is the along-stream coordinate, h is the water depth, r is the local radius of curvature of the jet, and M is the integral of over the cross-stream coordinate. It is necessary to make the approximation that the velocity pro"le of the jet does not change as the jet #ows over the topography for this path equation. The approximation holds if the radius of curvature of the jet r is always larger than the width of the jet ¸ (a thin jet). A constant velocity pro"le means that changes in relative vorticity due to the changing depth of the jet are entirely expressed in the turning of the jet. The path of the jet then can be calculated. Without an approximation that allows calculation of the turning of the jet as a function of its depth, a path equation cannot be formed. The path equation (19) is used here to predict the path of individual streamlines around the corner. Assuming the speed of the #ow is constant along a streamline and that the cross-stream shear in the #ow is also constant, the above path equation may be rewritten as
f *h * 1 # "0, (20) ;h *s *s r where ; is the speed of the jet on the streamline at the in#ow and h is the initial depth of the streamline at the in#ow. Numerical integration of Eq. (20) was performed to give streamlines that cross perpendicularly to the apex of the corner, as do the streamlines in the steady numerical results from SPEM 5.1. A comparison of streamlines from Fig. 5(c) with the predictions of the path equation (20) is shown in Fig. 9. There is good agreement between the streamline on the left-side of the jet (over the slope) and the path equation prediction. The agreement is not as good between the streamline on the right-side of the jet (over the shelf) and the path equation prediction. Here, the results from the path equation give a larger radius of curvature at the corner than the model solution. The larger radius of curvature is expected because the path equation takes no account of the reduction of shear in the jet at the corner that is found in the numerical model solution. From the scaling arguments of Section 2, it is expected that the radius of curvature of the #ow at the corner r scales as ;h , (21) r JL, L" f h
which is the same scaling given by the path equation (20). Using the streamline from the center of the jet for all model solutions, which vary in#ow speed and jet width, this scaling is found to hold
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Fig. 9 . Comparison between two streamlines for ;"1.0 m s\ (Fig. 5(c)) (dashed contours) and the results of the path equation (thin solid contours). The two thick solid contours mark the shelfbreak and the bottom of the slope.
(Fig. 10). The least-squares line through the data in Fig. 10 gives r "1.3e , B where B"h / h and e";/fB is a Rossby number. The scaling can be rewritten as r "1.3L, to show the expected scaling holds.
(22)
(23)
5. Narrow-channel model The corner model of the previous section addresses the case of large S topography where the #ow is expected to stay attached to the topography. We now wish to explore the case of arbitrarily weak channel topography (small S) where the #ow is not expected to stay attached to the topography. The model domain used is that of the narrow channel described in Section 2. Figs. 11(a)}(c) shows three di!erent #ow "elds from the narrow-channel model after the initial
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Fig. 10 . Shown is r /B plotted against e where e";/fB and B"h / h. The data are taken from the center of the jet A for model runs with in#ow velocities of 0.1, 0.5, 0.75, 1.0, 1.5, and 2.0 m s\ for the base case jet (#), and in#ow velocities of 0.1, 0.2, and 0.4 m s\ for an in#ow jet half as wide (o). The solid line is the &best "t' line to the data in the least-squares sense and the dash}dot line is the scaling that the simple path model would give.
transients have decayed. Identical model parameters are used in the three model runs except for the maximum in#ow speed ;, which equals 0.6 m s\ in Fig. 11(a), 0.65 m s\ in Fig. 11(b), and 0.85 m s\ in Fig. 11(c). The topography of the slope in all these cases is strong (S+4), and the in#ow that originates over the slope #ows straight along the slope. The in#ow that originates over the shelf has to negotiate the channel topography embedded in the shelf. In Fig. 11(a), the in#ow is slow enough that the #ow over the shelf splits from the #ow over the slope and #ows around the edge of the channel. Note that we do not know if the #ow would cross the channel, rather than #owing around the edge, if the length of the channel were increased or made in"nite. For the slightly faster in#ow shown in Fig. 11(b), very little of the #ow goes around the edge of the channel. The majority of the #ow over the shelf crosses the channel part way along the channel from the shelfbreak. In Fig. 11(c), nearly all the #ow over the shelf crosses the channel with only a small de#ection along the channel. In this case, the #ow over the shelf does not fully separate from the #ow over the slope. The dynamical balance of the #ow over the channel can be described by considering the conservation of potential vorticity following a streamline close to the center of the jet in streamwise coordinates. In the following discussion, left and right refer to an observer facing the direction of #ow of the jet. On approaching the channel, the jet begins to separate from the #ow over the slope
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Fig. 11 . Contours of the transport stream function (dashed lines) for various narrow-channel model runs. The thick solid lines mark the shelfbreak, the top and bottom of the channel and the bottom of the slope. The solid Gaussian on the left-hand border of each plot marks the in#ow jet.
and turns to the right. The jet is over the #at shelf, so the gain of anticyclonic relative vorticity due to the turning is compensated by a gain of cyclonic relative vorticity due to the shear to keep the potential vorticity constant. (This is achieved in the center of the jet by the maximum speed of the jet moving to the right relative to the streamlines.) As the #ow continues and crosses over the upstream slope of the channel, the depth increases and there must be a gain in cyclonic relative vorticity. The gain is achieved both in the turning of the jet and the shear of the jet. The anticyclonic vorticity due to the turning of the jet over the top of the slope gradually becomes cyclonic relative vorticity as the jet straightens over the slope and then turns to the left in the center of the channel. The width of the jet ¸ is comparable to the width of the slope into the channel ¸ . Cyclonic vorticity is then gained in the shear of the jet because the left hand side of the jet is deeper than the right-hand side, and so slower. In the center of the channel, the average vorticity of the jet is cyclonic and entirely due to the turning of the jet. The vorticity at the center of the jet can be estimated simply by integrating the steady shallow water vorticity equation
) [( f#u)u]"0,
(24)
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over a patch. The patch is bounded on either side by the streamlines that bound the jet when it is in the center of the channel and extends from the in#ow to the center of the channel. The integration gives
h !h 2;h ! (25) u"(2 f h p¸h and assumes that the jet in the center of the channel has a constant curvature across the jet, is Gaussian, and is entirely separated from the #ow over the slope. The "rst term on the right-hand side represents the vortex stretching due to the change in depth from the shelf into the channel. The second term is a correction due to the anticyclonic relative vorticity of the #ow impinging on the channel and its separation from the #ow over the slope as it #ows into the channel. In the above cases the #ow is essentially steady, but a steady state did not form in all model runs. A typical unsteady #ow is shown in Fig. 11(f), which has a deeper channel (200-m deep) and faster in#ow (1.2 m s\) than Figs. 11(a)}(c). The bulk of the #ow that crosses the channel does so as a quasi-steady jet. The edge of this jet varies with time as a series of growing eddies propagate downstream along the edge of the jet and out of the domain. The eddies are of interest but are not discussed here because they do not appear to interfere with the bulk of the #ow crossing the channel. Model runs were conducted to attempt to "nd an empirical scaling for the distance P from the shelf break that the #ow crosses the channel. P is de"ned as the di!erence in > of the streamline with the maximum speed in the middle of the channel from its > value at the in#ow. Figs. 11(d) and (e) show two such model runs, one with a wider jet (¸"40 km) and the other with a wider channel (¸ "80 km). Fig. 12 shows a plot of (;/f¸)(h /(h !h )) against (P/¸ )(;/f¸ )(h /(h !h )) for 21 model runs. The line drawn through this graph gives the scaling ¸ P "1.3S! , (26) ¸ ¸ which applies when the jet in the channel is fully separated from the jet over the slope. The "rst term on the right-hand side of this scaling S is proportional to c /;, where c is the group velocity of long-wavelength, low-mode topographic Rossby waves along the channel bathymetry. S can then represent the competition between the advection of #ow across the channel and the propagation down the channel due to topographic Rossby waves. The second term on the right-hand side of (26), ¸ /¸, is a correction due to the anticyclonic vorticity of the #ow that impinges on the channel. The data points that do not "t this scaling, at large x and small y in the plot, are those in which the jet crosses the channel but only partially separates from the #ow over the slope.
6. Wide-channel model The wide-channel model also was used to explore the adjustment of the shelfbreak jet to arbitrarily weak channel topography (small S) where the #ow is not expected to stay attached to the topography. Fig. 13 shows two di!erent #ow "elds from the wide-channel model, after the initial transients have decayed. The #ow "elds have di!erent maximum in#ow speed ;, but all other model parameters are identical.
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Fig. 12 . A plot showing the scaling found for the distance along the channel P that the jet crosses the channel. (P/¸ )(;/f¸ )(h /(h !h )) is plotted against (;/f¸)(h /(h !h )) where ; is the maximum in#ow speed, h is the shelf depth, h is the channel depth, f is the Coriolis parameter, ¸ is the half-width of the jet, and ¸ is the half-width of the channel. The asterisks are data from various model runs and the straight line is the line x#y"1.3.
The adjustment of the jet is similar to that in the narrow-channel model. For the slower in#ow speed of 0.5 m s\, (Fig. 13(a)), the jet over the shelf #ows around the edge of the channel, and for the faster in#ow speed of 1.0 m s\, (Fig. 13(b)), the #ow over the shelf crosses the channel part way down the channel. The potential vorticity dynamics of these #ows are also similar to the narrow channel model except where #ow crosses the #at bottom of the wide channel where there is no stretching of the water column. No scaling has been developed here for the distance down the channel that the jet crosses, but it will now depend on both length scales of the topography, i.e., the width of the slope into the channel ¸ and the width of the #at base of the channel ¸ . The wide-channel model most closely mimics the topography of the Northeast Channel. We found for a model run in which h "100 m, h "200 m, ¸ "10 km, ¸ "30 km, ¸"20 km, and ;"1.0 m s\, that the #ow does not cross the channel but #ows around the edge of the channel. This channel topography is less extreme than that of the Northeast Channel, and the in#ow speed is the largest that can reasonably be expected at the shelfbreak of the Scotian Shelf. Hence, steady, inviscid, barotropic vorticity dynamics do not allow for #ow across the Northeast Channel and some additional physical process must be introduced to explain the observed #ows.
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Fig. 13 . Contours of the transport stream function (dashed lines) for two wide-channel model runs. The thick solid lines mark the shelfbreak, the top and bottom of the channel and the bottom of the slope. The solid Gaussian on the left-hand border of each plot marks the in#ow jet. The width of the slope into the channel ¸ is 10 km, the width of the #at base to the channel ¸ is 50 km, the depth of the channel h is 150 m, the depth of the shelf h is 100 m the width of the in#ow jet ¸ is 20 km and speed of the in#ow jet ; is 0.5 and 1.0 m s\ in (a) and (b), respectively.
7. Discussion 7.1. Irreversible yow All of the models explored here have the coast to the right of the direction of #ow. The mean #ow is in the same direction as the propagation of long topographic Rossby waves. Topographic wave energy, such as the initial transients or the eddying edge to the jet, propagates out of the domain through the downstream open boundary. If the direction of the #ow is reversed, so the coast is to the left of the direction of #ow, the propagation of long topographic Rossby waves is against the mean #ow and towards the prescribed in#ow boundary. The energy in these waves will only move downstream and leave the domain if the speed of the mean #ow ; is greater than the group speed of the waves c or if, on re#ection at the in#ow boundary, they are converted into short waves that then propagate with the mean #ow. The propagation of topographic Rossby waves against the mean #ow suggests the possibility of wave energy being trapped at its generation point if the waves are propagating upstream at the same speed as their downstream advection by the mean #ow. The corners in the topography in the models used here are the most plausible generation points for these waves. If wave energy were to build up at these points, time-dependent eddying motions may result, producing a qualitatively di!erent #ow to that found for the #ow "elds in this model. A di!erent kind of adjustment to the corner is also required by steady, potential-vorticity dynamics when the coast is to the left of the direction of #ow. The vorticity dynamics are fundamentally asymmetrical with respect to the direction of the in#ow relative to the corner.
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A reversal of the #ow changes the sign of relative vorticity in the #ow and so potential vorticity is not conserved. In the simplest case of #ow around a corner, if the corner is a left-hand one rather than a right-hand one, then the relative vorticity associated with the turning of the #ow around that corner is cyclonic. If the #ow smoothly turns the corner, as in the corner model, then it must shoal and gain anticyclonic relative vorticity. There must be a large gain in anticyclonic relative vorticity in the shear of the #ow at the corner to counter the cyclonic relative vorticity in the curvature of the #ow and provide a total gain in anticyclonic relative vorticity. A simple broadening and slowing of the jet as it #ows around the corner does not provide the anticyclonic relative vorticity required and is not allowed. The narrow-channel model has a similar problem as the #ow crosses the middle of the channel if the direction of the jet is reversed. Cyclonic relative vorticity of the jet in the middle of the channel cannot be expressed in the turning of the jet there if the direction of the mean #ow is reversed. 7.2. Stability of the yow The in#ow has been restricted to rather wide Gaussian jets centered on the shelfbreak. This restriction has the advantage that the jet is stable, when running over straight bathymetry, for the parameters chosen in these model runs (see the appendix). The e!ect of instability in the jet in the adjustment to the topography need not be considered for these wide jets. However, the features of a narrow, unstable jet may be needed to explain the #ow of Scotian Shelf water across the Northeast Channel. While the #ow over the shelf in these model runs may not be too unrealistic, the #ow over the slope produced by the Gaussian jet is rather unusual. Typically, the #ow over the shelfbreak does not extend very far over the slope and, if it does, it is surface-intensi"ed rather than depthindependent. In the narrow-channel and the wide-channel models, the #ow over the slope #ows directly across the mouth of the channel and so there is always #ow of `Scotian Shelf watera across the `Northeast Channela in these model runs. Having the #ow centered on the shelfbreak means that the integrated relative vorticity of the #ow over the shelf that impinges on the channel is cyclonic, hence the correction term due to this in Eqs. (25) and (26). If the jet were entirely over the shelf, then the integrated relative vorticity of the in#ow over the shelf would be zero, and we expect that P/¸JS for the steady #ow in the narrow-channel model. However, in this case, the #ow over the shelf is also not necessarily stable, and so the steady vorticity dynamics may not apply. 7.3. Further modeling Results from the wide-channel model, presented in Section 6, indicate that steady barotropic potential vorticity dynamics do not allow #ow over the Scotian Shelf to cross the Northeast Channel to Georges Bank. The #ows over the Scotian Shelf are too slow and the Northeast Channel too deep and wide to allow the #ow to cross. Other factors need to be considered, such as the time variability of the #ow over the Scotian Shelf (Smith and Petrie, 1983). A study of time varying barotropic #ows impinging on channel topography is a possible way forward. The variation of the #ow could be achieved by specifying an unstable jet or time varying in#ow that mimicked the time variability observed at the shelfbreak of the Scotian Shelf.
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The depth-independent, barotropic #ow considered here is unlike the real #ows observed near the Northeast Channel. In particular, the #ow directly across the Northeast Channel observed in 1992 was measured to be only about 130-m deep and so did not "ll the Northeast Channel. Typically, the water "lling the bottom of the Northeast Channel below 150 m is warm salty Slope Water #owing into the Gulf of Maine. This water is denser than the surface water and will provide some decoupling of any #ow across the channel from the bottom of the channel. Hence, the topography of the Northeast Channel may be weaker than it "rst appears and could be better modeled as a 1-layer model with the lower quiescent layer partially "lling the channel. Wind forcing is also likely to play a role in the episodic #ows across the Northeast Channel. The occurrence of Scotian Shelf water over Georges Bank during late winter coincides with the winter storms that pass over this region. The wind stress vector rotates during the passage of a storm and so at some point the wind-driven #ow is likely to be from the Scotian Shelf to Georges Bank. If Scotian Shelf water can only be found over Georges Bank in late-winter/early-spring, then a wind-driven mechanism is attractive. Strati"cation in the Northeast Channel is largest during the summer, and so, if strati"cation is the dominant factor, then Scotian Shelf water might be expected over Georges Bank more frequently in summer than in winter. This is not the case, although it is harder to distinguish Scotian Shelf water over Georges Bank in the summer due to a decrease in temperature contrast.
8. Conclusions For the range of parameters used in this paper, the adjustment of a steady, inviscid, barotropic shelfbreak jet to channel topography embedded in the shelf is generally smooth and steady: there is no eddying or overshoot at the edge of the channel. The adjustment of the jet to the channel topography is governed by the strength of the topography S and the relative vorticity of the #ow impinging on the channel. For large S, the #ow is con"ned to the topography, but as S is reduced, #ow can cross the channel. The distance down the channel that the #ow crosses decreases with decreasing S. Channel topography similar to the Northeast Channel does not allow #ow to cross the channel for parameters within the oceanographic range. Hence, other factors need to be introduced to explain the episodic #ow of Scotian Shelf water across the Northeast Channel to Georges Bank. In particular, the in#uence of strati"cation, time varying in#ow, and wind forcing need to be examined.
Acknowledgements Thanks to Peter Smith for input on the structure of the #ow over the Scotian Shelf and to Joseph Pedlosky for helpful comments on the choice of modeling domain. Financial support was provided by the National Science Foundation under grants OCE93-13671, OCE96-32357 and OCE9632348. Computer facilities at the National Center for Atmospheric Research in Boulder Colorado were used for the numerical calculations. The National Center for Atmospheric Research is funded by the National Science Foundation. This is contribution number 142 of the US GLOBEC program, funded jointly by NSF and NOAA.
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Appendix. Stability of a barotropic Gaussian jet over sloping bathymetry For the inviscid, shallow water equations with a rigid lid, the relevant criterion to ensure that a jet #owing parallel to straight bathymetry is stable to small perturbations is that the initial potential vorticity gradient across the jet does not change sign. For a Gaussian jet of the form u"u e\W\* 5 #owing along straight bathymetry with a constant bottom slope b, and the
coast on the right, this criterion implies the jet is stable for a '2, e
(A.1)
where a"=/¸, e"u /f¸, and ¸ is the distance to the coast as in the simple model. Note that
a/eJ1/¸, so that as ¸ decreases, the jet becomes more stable for a given u and =. The jet
broadens as a increases and comes into contact with the coast for a'0.5, so we cannot consider very broad jets in the numerical model independently of the boundary conditions at the coast, and the range of Rossby numbers is restricted to 0(e(0.1. The shelf/slope topography used in the numerical model stabilizes the in#ow jet in the base case. If the slope is continued upward it reaches the surface very close to the center of the jet so that for the left-hand side of the jet which is over the slope, ¸ is small and a/e'2 for the range of e and a used in this study. The right-hand side of the jet is over the shelf where ¸ is large and a/e(2. The jet as a whole is found to be stable.
References Bisagni, J.J., Beardsley, R.C., Ruhsam, C.M., Manning, J.P., Williams, W.J., 1996. Historical and recent evidence concerning the presence of Scotian Shelf Water on southern Georges Bank. Deep-Sea Research II 43 (7}8), 1439}1471. Butman, B., Beardsley, R.C., 1987. Physical oceanography: introduction.. In: Backus, R. (Ed.), Georges Bank. MIT Press, Cambridge, MA, pp. 88}98. Chapman, D.C., Barth, J.A., Beardsley, R.C., Fairbanks, R.G., 1986. On the continuity of the mean #ow between the Scotian Shelf and the Middle Atlantic Bight. Journal of Physical Oceanography 16 (4), 758}772. Haidvogel, D.B., Wilkin, J.L., Young, R., 1991. A semi-spectral primitive equation ocean model using vertical sigma and orthogonal curvilinear horizontal coordinates. Journal of Computational Physics 94 (1), 151}185. Klinger, B.A., 1993. Gyre formation at a corner by rotating barotropic coastal #ows along a slope. Dynamics of Atmospheres and Oceans 19, 27}63. Robinson, A.R., Niiler, P.P., 1967. The theory of free inertial currents. Tellus XIX (2), 269}291. Smith, P.C., 1983. The mean and seasonal circulation o! southwest Nova Scotia. Journal of Physical Oceanography 13 (6), 1034}1054. Smith, P.C., Petrie, B.D., 1983. Low-frequency circulation at the edge of the Scotian Shelf. Journal of Physical Oceanography 12, 28}46.
The adjustment of a shelfbreak jet to cross-shelf topography夽 William J. Williams*, Glen G. Gawarkiewicz, Robert C. Beardsley Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Received 7 October 1998; received in revised form 14 July 1999; accepted 10 December 1999
Abstract A rigid-lid, "nite-di!erence numerical model is used to study the adjustment of inviscid, along-shelf, barotropic shelfbreak jets to cross-shelf, channel topography. The channel is embedded in the shelf topography perpendicularly to the shelfbreak, the shelfbreak jet #ows with the direction of propagation of long-wavelength, topographic Rossby waves, and the coast is su$ciently distant so as not to a!ect the #ow. Three models are used that vary the strength of the channel topography S"(f*hD/;h, where f is the Coriolis parameter, *h is the di!erence between the shelf depth and the channel depth, D is the width of the slope into the channel, ; is the maximum speed of the jet at the in#ow, and h is the depth of the shelf. Estimation of the path of the jet from the in#ow parameters and the geometry of the channel is possible in some cases. Generally, for large S the #ow follows the topography of the channel and for small S the #ow crosses the channel. The motivation for this study is the episodic #ow of Scotian Shelf water from the Scotian Shelf across the Northeast Channel to Georges Bank. The steady, inviscid, non-linear, barotropic dynamics studied here do not allow such a #ow for a channel of similar dimensions to the Northeast Channel and for #ow speeds within the oceanographic range. Other factors such as strati"cation, wind stress and time variability need to be introduced to account for this phenomenon. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Scotian Shelf water episodically #ows from the Scotian Shelf across the Northeast Channel to Georges Bank (Bisagni et al., 1996). Georges Bank is on the western edge of the North Atlantic Ocean at approximately 663W 423N (Fig. 1). To the northeast of Georges Bank lies the Northeast Channel which separates Georges Bank from the Scotian Shelf. The channel is 45-km wide and has 夽
Paper published in December 2000. * Corresponding author; Present address: Institute of Marine Sciences, University of Alaska, P.O. Box 757220, Fairbanks, AK 99775-7220, USA. Fax: #1-907-474-7204. E-mail address: [email protected] (W.J. Williams). 0967-0645/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 4 5 ( 0 0 ) 0 0 0 8 5 - 0
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Fig. 1. A map of the Northeast Channel area showing the mean circulation (solid arrows), and the episodic #ow across the Northeast Channel (dashed arrow). The Northeast Channel is marked by NEC and the x- and y-axis are marked in degrees longitude and degrees latitude, respectively.
a sill depth of 240 m. Typical shelf depths in this region are +100 m, and so the channel forms a relatively deep, wide gap in the otherwise continuous shelf topography. The channel is also the deepest route for water to #ow between the Atlantic Ocean and the Gulf of Maine. Surface circulation over the shelf in this region is generally from the northeast to the southwest (Chapman et al., 1986; Butman and Beardsley, 1987) (Fig. 1), with a component of the #ow being close to the shelfbreak (Smith and Petrie, 1983; Smith, 1983). This southwestward #ow is complicated by the presence of the Northeast Channel and the Gulf of Maine. The typical route for the #ow over the Scotian Shelf is for it to turn to the right, before reaching the Northeast Channel, and #ow past Cape Sable into the Gulf of Maine (Smith and Petrie, 1983; Smith, 1983). Only after #owing around the Gulf of Maine does the water reach Georges Bank and continue southwestward (Butman and Beardsley, 1987). This circuitous route is due to the presence of the Northeast Channel. A barotropic, geostrophic #ow is constrained to follow isobaths (conserving potential vorticity) so such a #ow over the Scotian Shelf #owing towards the Northeast Channel would be required to turn and #ow into the Gulf of Maine rather than crossing the channel. Episodic #ows of Scotian Shelf water across the Northeast Channel have been observed many times (Bisagni et al., 1996). Observations of these are clearest in late-winter/early-spring when the
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di!erence in temperature and salinity between Scotian Shelf water and the water over Georges Bank is the greatest. Perhaps the best set of observations was collected in 1992 and is described in Bisagni et al. (1996). During February of that year, a plume of cold, fresh Scotian Shelf water crossed the Northeast Channel to the southern #ank of Georges Bank. The plume was clearly visible from AVHRR imagery and persisted into June. Flows that cross the isobaths of the Northeast Channel cannot be considered simply barotropic, geostrophically balanced currents. Factors that could allow episodic #ow from the Scotian Shelf to Georges Bank are strati"cation, wind stress, variable in#ow, and inertia. To begin to study these factors, the possible role of inertia is discussed here by considering homogeneous #ows in the presence of topography similar to the Northeast Channel. Simple scalings of the shallow-water potential vorticity equations are developed in the next section. These scalings give an estimate of the dynamical size of the channel-like topography in comparison to the shelfbreak jet. Three model domains are then described which allow exploration of the adjustment of a shelfbreak jet to strong, weak, and wide channel topography. A numerical model, used to solve for the #ow "elds in each domain, is then introduced and described in Section 3. Next, we present the results from using the numerical model in each model domain, and then consider the irreversibility of the #ow "elds generated by the numerical model, the stability of the #ow and directions for future research.
2. Models Isobaths near the shelfbreak of the southwestern Scotian Shelf turn sharply 903 to the right at the Northeast Channel with a radius of curvature r of about 5 km. The Rossby number R of a #ow M following these isobaths is ;/fr, where ; is the #ow speed scale and f"10\ s\ is the Coriolis parameter. For a #ow with ;'0.2 m s\, R '0.4. R is not small in this case and the dynamical M M importance of #ow inertia needs to be considered. The simplest equations of motion that can be used to consider inertial e!ects are the inviscid, shallow water equations on an f-plane with a rigid lid 1 *u #(u ) )u#f;u"! P, o *t
) (hu)"0,
(1) (2)
where u is the velocity vector, t is time, o "1000 kg m\ is the density, P is the pressure on the rigid lid, and h is the depth. Density is assumed constant throughout the domain. The rigid-lid approximation is expected to be very good since the barotropic Rossby radius is much larger than the width of the Northeast Channel and the depth variations are large. In the models presented here, a geostrophic jet, centered close to the shelfbreak and running parallel to the shelfbreak, impinges on channel-like topography running perpendicular to the shelfbreak (for example, see Fig. 3). The orientation of the jet and channel is such that it mimics the #ow over the shelfbreak of the Scotian Shelf towards the Northeast Channel. Ageostrophic adjustments of the model jet to the topography are of interest here and will contain some departure of the #ow from its initial depth. An estimate of the scale of this departure can be obtained from the
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potential-vorticity equation
f#u *u #u ) u! u ) h"0, h *t
(3)
where u is the relative vorticity de"ned by *v/*x!*u/*y. If the #ow is a balance between the non-linear advection of relative vorticity u ) u and the stretching of vortex columns due to the topography (( f#u)/h)u ) h, then the length scale for the #ow used to estimate u must be
; . (4) b 2 ; is again the #ow speed scale and b "f*h/Dh is the topographic b-e!ect, where *h is a scale for 2 the change in height of the topography, D a scale for the width of the topography, and h a scale for the typical depth. If L is compared with the scale for the topography D, a measure of the strength of the topography is obtained
L"
L f*hD S" " . ;h D
(5)
Three models are used in this paper: the corner model (Fig. 2), the narrow-channel model (Fig. 3), and the wide-channel model (Fig. 4). They consider the e!ect on the #ow of varying the strength of the topographic parameter S. The corner model (Fig. 2) contains only a shelf with a sharp right hand turn in the shelfbreak, which is meant to mimic the southwestern Scotian Shelf and its border with the Northeast Channel. For jet speeds within the oceanographic range (0.1';'2.0 m s\), the topography of the slope is strong (20'S'4) and the jet is expected to follow the topography. The manner of adjustment of the jet to the corner is of interest. Klinger (1993) studied the adjustment of a barotropic coastal jet to a sharp bend in the coastline and topography. Due to its inertia, this coastal jet overshoots the corner in the coastline and a stationary eddy forms between the jet and the coast immediately downstream from the corner, the Rossby number of the #ow there being O(1). In the corner model used here, the coast is su$ciently distant so as to play no role in the adjustment of the jet to the sharp corner in the shelf topography, and so it is not required that the jet overshoot at the corner. The narrow-channel model (Fig. 3) features a long channel embedded in the shelf with arbitrarily weak topography (small S) so that it is possible to have #ow crossing the topography. This provides a route for #ow to cross the channel or #ow around the edge of the channel. The shelf-width and channel-length are as large as possible so as to closely simulate the adjustment of the #ow at the shelfbreak to an in"nitely long channel. The wide-channel model (Fig. 4) di!ers from the narrow-channel model by having a wide #at base to the channel, so, while the total width of the channel can be the same, the depth pro"le is very di!erent. This channel topography closely mimics the topography of the Northeast Channel, which has very steep sides and a relatively #at bottom. Over the #at bottom of the channel there is no vortex stretching due to topography, so the scalings described above are less obviously applicable. Again, the shelf width and channel length are as large as possible so as to closely simulate the adjustment of the #ow at the shelfbreak to an in"nitely long channel.
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Fig. 2. The computational model domain for the corner model. The upper panel is a plan view showing the isobaths (thin lines) every 100 m from 100 to 1600 m, the free-slip boundaries (thick lines), and the in#ow jet. The value of the in#ow is variable. The lower panel is a cross-section of the bathymetry used.
3. Numerical model The numerical model used to solve for the #ow "elds shown is the Semi-spectral Primitive Equation Model version 5.1 (SPEM 5.1), an earlier version of which is described in Haidvogel et al. (1991). This model is a rigid-lid, p-coordinate, 3-D primitive-equation, "nite-di!erence model and, while it is more complex than the present study requires, it allows for the study of the shallow water equations and for subsequent modeling of the e!ects of strati"cation, wind-stress and bottom friction. The equations solved by SPEM 5.1 are
*u *u 1 *P *u *u #(u ) )u!fv"! #A # #A , & *x *y 4 *z o *x *t
(6)
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Fig. 3 . The computational model domain for the narrow-channel model. The upper panel is a plan view showing the isobaths (thin lines) every 10 m from 100 to 600 m, the free-slip boundaries (thick lines), and the in#ow jet. The values of the in#ow and the depth and width of the channel are variable. The lower panel is a cross section of the channel bathymetry used.
*v *v *v 1 *P *v #(u ) )v#fu"! #A # #A , & *x *y 4 *z *t o *y *P "!og, *z *u *v *w # # "0, *x *y *z
(7) (8) (9)
subject to the top and bottom boundary conditions of no #ow through the rigid lid and the bottom, and no stress on the rigid lid and the bottom, w"0 at z"0,
(10)
uh #vh "0 at z"!h, V W
(11)
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Fig. 4 . The computational model domain for the wide-channel model. The upper panel is a plan view showing the isobaths (thin lines) every 10 m from 100 to 600 m, the free-slip boundaries (thick lines), and the in#ow jet. The values of the in#ow and the depth and width of the channel are variable. The lower panel is a cross section of the channel bathymetry used.
u "0, v "0 at z"0, (12) X X A u "0, A v "0 at z"!h. (13) 4 X 4 X The horizontal viscosity A is set to a small value (50 m s\) to provide numerical stability by & damping numerical noise in the model. It makes a very small quantitative di!erence in the #ow "elds generated and is dynamically insigni"cant compared with other terms in the momentum equation. The vertical viscosity A is set to 0.002 m s\ to dampen numerical noise in the vertical 4 and ensure that the #ow is depth-independent. The numerical solution is then essentially that of Eqs. (1) and (2). The model domains used are shown in Figs. 2}4. The closed boundaries of the model domains are free-slip walls, so on the wall u "0, ,
*u "0, *x ,
(14)
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where u is the horizontal velocity perpendicular to the boundary, u is the horizontal velocity , parallel to the boundary, and x is the horizontal coordinate perpendicular to the boundary. The , open out#ow boundary in each model domain is a no-gradient out#ow condition on all variables *
"0, (15) *x , where is any variable. This boundary condition is dynamically consistent with the geostrophic out#ow that is expected for steady states obtained in this model. The in#ow boundary in each model domain has a speci"ed in#ow perpendicular to that boundary that is independent of depth. The in#ow velocity is typically a Gaussian jet centered on the shelfbreak
u ";exp ! ,
x !x , u "0, ¸
(16)
where x is the x -position of the shelfbreak, ¸ is the width scale of the in#ow jet, and ; is the maximum speed of the in#ow jet. For this study, the model was con"gured with the minimum of three grid points in the vertical and a horizontal resolution of 2.083 km in each direction. The time step was chosen so that advection, wave propagation, and di!usion were properly resolved. Typically, there were 100 time steps in 24 h. Each experiment was run until the initial conditions had evolved to a steady, or statistically steady, state. A few experiments were done at twice the horizontal resolution to test the accuracy of the #ow "elds at the lower resolution.
4. Corner model The corner model is used to explore the adjustment of the shelfbreak jet to strong channel topography (large S) where the #ow is expected to stay attached to the topography. Fig. 5 shows four di!erent #ow "elds after the initial transients have decayed away. Identical model parameters are used in these solutions except the maximum in#ow speed ; was increased from 0.1 m s\ in Fig. 5(a), to 0.5 m s\ in Fig. 5(b), to 1.0 m s\ in Fig. 5(c), and to 2.0 m s\ in Fig. 5(d). All these model "elds are qualitatively similar: the #ow is steady, smoothly turns the corner, and is nearly symmetrical about the corner. There is no eddy formation or overshoot at the corner in these "nal #ow "elds even though the initial transients for these model runs contain these features. The symmetry of the streamlines about the corner implies that a #uid column nearly returns to its initial depth after #owing past the corner. Fig. 6 shows the depth following a #uid column along three streamlines from Fig. 5(c). The #ow shoals as it approaches the corner and then deepens on leaving the corner. The dynamical balance associated with this #ow is best described by considering the conservation of potential vorticity along a streamline
d f#u "0. h dt
(17)
Conservation of potential vorticity implies that accompanying the shoaling of the #ow at the corner is a gain in anticyclonic relative vorticity. In streamwise coordinates, the relative vorticity
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Fig. 5. Contour plots of the transport streamfunction for the corner model. The in#ow in each case is a Gaussian jet centered on the shelfbreak with ¸"20 km. The maximum in#ow speeds are 0.1, 0.5, 1.0, and 2.0 m s\ in (a), (b), (c) and (d), respectively. The dashed contours are the streamlines and the solid contours mark the shelfbreak and the bottom of the slope.
may be written *u u u"! ! , *n r
(18)
where u is the speed of the #ow, r is the local radius of curvature of the #ow, and n is the cross-stream coordinate. The "rst term of this equation is due to the local radius of curvature of the #ow and the second term is due to the shear in the #ow. To negotiate the corner, the #ow turns anticyclonically and so the relative vorticity due to the turning of the #ow is also anticyclonic. There is then a possible balance between the shoaling of the #ow and the turning of the #ow where
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Fig. 6. The depth of three streamlines for ;"1.0 m s\ (Fig. 5(c)) plotted versus the distance along the streamline around the corner, which is located at +105 km. The dash}dot line is a streamline from entirely over the shelf, the dashed line from entirely over the slope, and the solid line is from a streamline between the other two.
the anticyclonic relative vorticity of the turning of the #ow around the corner is provided by the gain in anticyclonic relative vorticity due to shoaling of the #ow. Such a #ow could occur without requiring a dramatic change in the shear of the #ow, and thus the structure of the jet may remain largely unchanged. Figs. 7 and 8 show pro"les across the jet at the corner and at the in#ow from the #ow "eld shown in Fig. 5(c). The velocity pro"le of the jet is both wider and slower at the corner than at the in#ow. The maximum speed is reduced from 1 to 0.79 m s\, and the jet width increases by +50%. The size of the shear in the jet is reduced in the wide, slow jet at the corner. On the right-hand side of the jet, over the shelf, the shear becomes more cyclonic, and on the left-hand side of the jet, over the slope, the shear becomes more anticyclonic. The vorticity balance at the corner, therefore, is di!erent on either side of the jet since the change in the vorticity due to the change in the shear of the #ow is of opposite sign. On the right-hand side of the jet, the total relative vorticity in the jet becomes slightly more anticyclonic at the corner, indicating the small relative change of depth of the #ow over the shelf of only +0.1 (Fig. 6). The increase in cyclonic vorticity due to the shear of the jet is opposite to the overall anticyclonic gain in vorticity. The anticyclonic vorticity due to the turning of the jet compensates for this. On the left-hand side of the jet, the total relative vorticity in the jet goes from being strongly cyclonic to being strongly anticyclonic, indicating the large relative change of depth of the #ow over the slope which reaches a maximum of 0.52 near the top of the slope (Fig. 6). The increase in
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Fig. 7. Comparison of the velocity pro"le of the jet at the in#ow (dashed line) and at the corner (solid line) for ;"1.0 m s\ (Fig. 5(c)).
Fig. 8 . Comparison between the vorticity pro"le of the jet at the in#ow (dashed line) and at the corner (solid line) for ;"1.0 m s\ (Fig. 5(c)). The dash-dot line is the vorticity due to the shear in the #ow at the corner.
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anticyclonic vorticity is largely due to the anticyclonic turning of the #ow around the corner, with a smaller contribution from the change in shear of the jet. At the corner of the shelfbreak, the maximum Rossby number of the #ow is about 0.4. If the jet were #owing around a corner in the coastline, the Rossby number would be expected to be larger (Klinger, 1993). It is the broad adjustment to the corner in the shelfbreak, which is possible when the coastline is distant, that leads to the lower Rossby number. Robinson and Niiler (1967) described the path of thin jets over weak topography. They obtained a path equation that is appropriate here,
*h * ; #f ;M "0, *s *s r
(19)
where ; is the speed of the jet, s is the along-stream coordinate, h is the water depth, r is the local radius of curvature of the jet, and M is the integral of over the cross-stream coordinate. It is necessary to make the approximation that the velocity pro"le of the jet does not change as the jet #ows over the topography for this path equation. The approximation holds if the radius of curvature of the jet r is always larger than the width of the jet ¸ (a thin jet). A constant velocity pro"le means that changes in relative vorticity due to the changing depth of the jet are entirely expressed in the turning of the jet. The path of the jet then can be calculated. Without an approximation that allows calculation of the turning of the jet as a function of its depth, a path equation cannot be formed. The path equation (19) is used here to predict the path of individual streamlines around the corner. Assuming the speed of the #ow is constant along a streamline and that the cross-stream shear in the #ow is also constant, the above path equation may be rewritten as
f *h * 1 # "0, (20) ;h *s *s r where ; is the speed of the jet on the streamline at the in#ow and h is the initial depth of the streamline at the in#ow. Numerical integration of Eq. (20) was performed to give streamlines that cross perpendicularly to the apex of the corner, as do the streamlines in the steady numerical results from SPEM 5.1. A comparison of streamlines from Fig. 5(c) with the predictions of the path equation (20) is shown in Fig. 9. There is good agreement between the streamline on the left-side of the jet (over the slope) and the path equation prediction. The agreement is not as good between the streamline on the right-side of the jet (over the shelf) and the path equation prediction. Here, the results from the path equation give a larger radius of curvature at the corner than the model solution. The larger radius of curvature is expected because the path equation takes no account of the reduction of shear in the jet at the corner that is found in the numerical model solution. From the scaling arguments of Section 2, it is expected that the radius of curvature of the #ow at the corner r scales as ;h , (21) r JL, L" f h
which is the same scaling given by the path equation (20). Using the streamline from the center of the jet for all model solutions, which vary in#ow speed and jet width, this scaling is found to hold
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Fig. 9 . Comparison between two streamlines for ;"1.0 m s\ (Fig. 5(c)) (dashed contours) and the results of the path equation (thin solid contours). The two thick solid contours mark the shelfbreak and the bottom of the slope.
(Fig. 10). The least-squares line through the data in Fig. 10 gives r "1.3e , B where B"h / h and e";/fB is a Rossby number. The scaling can be rewritten as r "1.3L, to show the expected scaling holds.
(22)
(23)
5. Narrow-channel model The corner model of the previous section addresses the case of large S topography where the #ow is expected to stay attached to the topography. We now wish to explore the case of arbitrarily weak channel topography (small S) where the #ow is not expected to stay attached to the topography. The model domain used is that of the narrow channel described in Section 2. Figs. 11(a)}(c) shows three di!erent #ow "elds from the narrow-channel model after the initial
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Fig. 10 . Shown is r /B plotted against e where e";/fB and B"h / h. The data are taken from the center of the jet A for model runs with in#ow velocities of 0.1, 0.5, 0.75, 1.0, 1.5, and 2.0 m s\ for the base case jet (#), and in#ow velocities of 0.1, 0.2, and 0.4 m s\ for an in#ow jet half as wide (o). The solid line is the &best "t' line to the data in the least-squares sense and the dash}dot line is the scaling that the simple path model would give.
transients have decayed. Identical model parameters are used in the three model runs except for the maximum in#ow speed ;, which equals 0.6 m s\ in Fig. 11(a), 0.65 m s\ in Fig. 11(b), and 0.85 m s\ in Fig. 11(c). The topography of the slope in all these cases is strong (S+4), and the in#ow that originates over the slope #ows straight along the slope. The in#ow that originates over the shelf has to negotiate the channel topography embedded in the shelf. In Fig. 11(a), the in#ow is slow enough that the #ow over the shelf splits from the #ow over the slope and #ows around the edge of the channel. Note that we do not know if the #ow would cross the channel, rather than #owing around the edge, if the length of the channel were increased or made in"nite. For the slightly faster in#ow shown in Fig. 11(b), very little of the #ow goes around the edge of the channel. The majority of the #ow over the shelf crosses the channel part way along the channel from the shelfbreak. In Fig. 11(c), nearly all the #ow over the shelf crosses the channel with only a small de#ection along the channel. In this case, the #ow over the shelf does not fully separate from the #ow over the slope. The dynamical balance of the #ow over the channel can be described by considering the conservation of potential vorticity following a streamline close to the center of the jet in streamwise coordinates. In the following discussion, left and right refer to an observer facing the direction of #ow of the jet. On approaching the channel, the jet begins to separate from the #ow over the slope
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Fig. 11 . Contours of the transport stream function (dashed lines) for various narrow-channel model runs. The thick solid lines mark the shelfbreak, the top and bottom of the channel and the bottom of the slope. The solid Gaussian on the left-hand border of each plot marks the in#ow jet.
and turns to the right. The jet is over the #at shelf, so the gain of anticyclonic relative vorticity due to the turning is compensated by a gain of cyclonic relative vorticity due to the shear to keep the potential vorticity constant. (This is achieved in the center of the jet by the maximum speed of the jet moving to the right relative to the streamlines.) As the #ow continues and crosses over the upstream slope of the channel, the depth increases and there must be a gain in cyclonic relative vorticity. The gain is achieved both in the turning of the jet and the shear of the jet. The anticyclonic vorticity due to the turning of the jet over the top of the slope gradually becomes cyclonic relative vorticity as the jet straightens over the slope and then turns to the left in the center of the channel. The width of the jet ¸ is comparable to the width of the slope into the channel ¸ . Cyclonic vorticity is then gained in the shear of the jet because the left hand side of the jet is deeper than the right-hand side, and so slower. In the center of the channel, the average vorticity of the jet is cyclonic and entirely due to the turning of the jet. The vorticity at the center of the jet can be estimated simply by integrating the steady shallow water vorticity equation
) [( f#u)u]"0,
(24)
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over a patch. The patch is bounded on either side by the streamlines that bound the jet when it is in the center of the channel and extends from the in#ow to the center of the channel. The integration gives
h !h 2;h ! (25) u"(2 f h p¸h and assumes that the jet in the center of the channel has a constant curvature across the jet, is Gaussian, and is entirely separated from the #ow over the slope. The "rst term on the right-hand side represents the vortex stretching due to the change in depth from the shelf into the channel. The second term is a correction due to the anticyclonic relative vorticity of the #ow impinging on the channel and its separation from the #ow over the slope as it #ows into the channel. In the above cases the #ow is essentially steady, but a steady state did not form in all model runs. A typical unsteady #ow is shown in Fig. 11(f), which has a deeper channel (200-m deep) and faster in#ow (1.2 m s\) than Figs. 11(a)}(c). The bulk of the #ow that crosses the channel does so as a quasi-steady jet. The edge of this jet varies with time as a series of growing eddies propagate downstream along the edge of the jet and out of the domain. The eddies are of interest but are not discussed here because they do not appear to interfere with the bulk of the #ow crossing the channel. Model runs were conducted to attempt to "nd an empirical scaling for the distance P from the shelf break that the #ow crosses the channel. P is de"ned as the di!erence in > of the streamline with the maximum speed in the middle of the channel from its > value at the in#ow. Figs. 11(d) and (e) show two such model runs, one with a wider jet (¸"40 km) and the other with a wider channel (¸ "80 km). Fig. 12 shows a plot of (;/f¸)(h /(h !h )) against (P/¸ )(;/f¸ )(h /(h !h )) for 21 model runs. The line drawn through this graph gives the scaling ¸ P "1.3S! , (26) ¸ ¸ which applies when the jet in the channel is fully separated from the jet over the slope. The "rst term on the right-hand side of this scaling S is proportional to c /;, where c is the group velocity of long-wavelength, low-mode topographic Rossby waves along the channel bathymetry. S can then represent the competition between the advection of #ow across the channel and the propagation down the channel due to topographic Rossby waves. The second term on the right-hand side of (26), ¸ /¸, is a correction due to the anticyclonic vorticity of the #ow that impinges on the channel. The data points that do not "t this scaling, at large x and small y in the plot, are those in which the jet crosses the channel but only partially separates from the #ow over the slope.
6. Wide-channel model The wide-channel model also was used to explore the adjustment of the shelfbreak jet to arbitrarily weak channel topography (small S) where the #ow is not expected to stay attached to the topography. Fig. 13 shows two di!erent #ow "elds from the wide-channel model, after the initial transients have decayed. The #ow "elds have di!erent maximum in#ow speed ;, but all other model parameters are identical.
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Fig. 12 . A plot showing the scaling found for the distance along the channel P that the jet crosses the channel. (P/¸ )(;/f¸ )(h /(h !h )) is plotted against (;/f¸)(h /(h !h )) where ; is the maximum in#ow speed, h is the shelf depth, h is the channel depth, f is the Coriolis parameter, ¸ is the half-width of the jet, and ¸ is the half-width of the channel. The asterisks are data from various model runs and the straight line is the line x#y"1.3.
The adjustment of the jet is similar to that in the narrow-channel model. For the slower in#ow speed of 0.5 m s\, (Fig. 13(a)), the jet over the shelf #ows around the edge of the channel, and for the faster in#ow speed of 1.0 m s\, (Fig. 13(b)), the #ow over the shelf crosses the channel part way down the channel. The potential vorticity dynamics of these #ows are also similar to the narrow channel model except where #ow crosses the #at bottom of the wide channel where there is no stretching of the water column. No scaling has been developed here for the distance down the channel that the jet crosses, but it will now depend on both length scales of the topography, i.e., the width of the slope into the channel ¸ and the width of the #at base of the channel ¸ . The wide-channel model most closely mimics the topography of the Northeast Channel. We found for a model run in which h "100 m, h "200 m, ¸ "10 km, ¸ "30 km, ¸"20 km, and ;"1.0 m s\, that the #ow does not cross the channel but #ows around the edge of the channel. This channel topography is less extreme than that of the Northeast Channel, and the in#ow speed is the largest that can reasonably be expected at the shelfbreak of the Scotian Shelf. Hence, steady, inviscid, barotropic vorticity dynamics do not allow for #ow across the Northeast Channel and some additional physical process must be introduced to explain the observed #ows.
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Fig. 13 . Contours of the transport stream function (dashed lines) for two wide-channel model runs. The thick solid lines mark the shelfbreak, the top and bottom of the channel and the bottom of the slope. The solid Gaussian on the left-hand border of each plot marks the in#ow jet. The width of the slope into the channel ¸ is 10 km, the width of the #at base to the channel ¸ is 50 km, the depth of the channel h is 150 m, the depth of the shelf h is 100 m the width of the in#ow jet ¸ is 20 km and speed of the in#ow jet ; is 0.5 and 1.0 m s\ in (a) and (b), respectively.
7. Discussion 7.1. Irreversible yow All of the models explored here have the coast to the right of the direction of #ow. The mean #ow is in the same direction as the propagation of long topographic Rossby waves. Topographic wave energy, such as the initial transients or the eddying edge to the jet, propagates out of the domain through the downstream open boundary. If the direction of the #ow is reversed, so the coast is to the left of the direction of #ow, the propagation of long topographic Rossby waves is against the mean #ow and towards the prescribed in#ow boundary. The energy in these waves will only move downstream and leave the domain if the speed of the mean #ow ; is greater than the group speed of the waves c or if, on re#ection at the in#ow boundary, they are converted into short waves that then propagate with the mean #ow. The propagation of topographic Rossby waves against the mean #ow suggests the possibility of wave energy being trapped at its generation point if the waves are propagating upstream at the same speed as their downstream advection by the mean #ow. The corners in the topography in the models used here are the most plausible generation points for these waves. If wave energy were to build up at these points, time-dependent eddying motions may result, producing a qualitatively di!erent #ow to that found for the #ow "elds in this model. A di!erent kind of adjustment to the corner is also required by steady, potential-vorticity dynamics when the coast is to the left of the direction of #ow. The vorticity dynamics are fundamentally asymmetrical with respect to the direction of the in#ow relative to the corner.
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A reversal of the #ow changes the sign of relative vorticity in the #ow and so potential vorticity is not conserved. In the simplest case of #ow around a corner, if the corner is a left-hand one rather than a right-hand one, then the relative vorticity associated with the turning of the #ow around that corner is cyclonic. If the #ow smoothly turns the corner, as in the corner model, then it must shoal and gain anticyclonic relative vorticity. There must be a large gain in anticyclonic relative vorticity in the shear of the #ow at the corner to counter the cyclonic relative vorticity in the curvature of the #ow and provide a total gain in anticyclonic relative vorticity. A simple broadening and slowing of the jet as it #ows around the corner does not provide the anticyclonic relative vorticity required and is not allowed. The narrow-channel model has a similar problem as the #ow crosses the middle of the channel if the direction of the jet is reversed. Cyclonic relative vorticity of the jet in the middle of the channel cannot be expressed in the turning of the jet there if the direction of the mean #ow is reversed. 7.2. Stability of the yow The in#ow has been restricted to rather wide Gaussian jets centered on the shelfbreak. This restriction has the advantage that the jet is stable, when running over straight bathymetry, for the parameters chosen in these model runs (see the appendix). The e!ect of instability in the jet in the adjustment to the topography need not be considered for these wide jets. However, the features of a narrow, unstable jet may be needed to explain the #ow of Scotian Shelf water across the Northeast Channel. While the #ow over the shelf in these model runs may not be too unrealistic, the #ow over the slope produced by the Gaussian jet is rather unusual. Typically, the #ow over the shelfbreak does not extend very far over the slope and, if it does, it is surface-intensi"ed rather than depthindependent. In the narrow-channel and the wide-channel models, the #ow over the slope #ows directly across the mouth of the channel and so there is always #ow of `Scotian Shelf watera across the `Northeast Channela in these model runs. Having the #ow centered on the shelfbreak means that the integrated relative vorticity of the #ow over the shelf that impinges on the channel is cyclonic, hence the correction term due to this in Eqs. (25) and (26). If the jet were entirely over the shelf, then the integrated relative vorticity of the in#ow over the shelf would be zero, and we expect that P/¸JS for the steady #ow in the narrow-channel model. However, in this case, the #ow over the shelf is also not necessarily stable, and so the steady vorticity dynamics may not apply. 7.3. Further modeling Results from the wide-channel model, presented in Section 6, indicate that steady barotropic potential vorticity dynamics do not allow #ow over the Scotian Shelf to cross the Northeast Channel to Georges Bank. The #ows over the Scotian Shelf are too slow and the Northeast Channel too deep and wide to allow the #ow to cross. Other factors need to be considered, such as the time variability of the #ow over the Scotian Shelf (Smith and Petrie, 1983). A study of time varying barotropic #ows impinging on channel topography is a possible way forward. The variation of the #ow could be achieved by specifying an unstable jet or time varying in#ow that mimicked the time variability observed at the shelfbreak of the Scotian Shelf.
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The depth-independent, barotropic #ow considered here is unlike the real #ows observed near the Northeast Channel. In particular, the #ow directly across the Northeast Channel observed in 1992 was measured to be only about 130-m deep and so did not "ll the Northeast Channel. Typically, the water "lling the bottom of the Northeast Channel below 150 m is warm salty Slope Water #owing into the Gulf of Maine. This water is denser than the surface water and will provide some decoupling of any #ow across the channel from the bottom of the channel. Hence, the topography of the Northeast Channel may be weaker than it "rst appears and could be better modeled as a 1-layer model with the lower quiescent layer partially "lling the channel. Wind forcing is also likely to play a role in the episodic #ows across the Northeast Channel. The occurrence of Scotian Shelf water over Georges Bank during late winter coincides with the winter storms that pass over this region. The wind stress vector rotates during the passage of a storm and so at some point the wind-driven #ow is likely to be from the Scotian Shelf to Georges Bank. If Scotian Shelf water can only be found over Georges Bank in late-winter/early-spring, then a wind-driven mechanism is attractive. Strati"cation in the Northeast Channel is largest during the summer, and so, if strati"cation is the dominant factor, then Scotian Shelf water might be expected over Georges Bank more frequently in summer than in winter. This is not the case, although it is harder to distinguish Scotian Shelf water over Georges Bank in the summer due to a decrease in temperature contrast.
8. Conclusions For the range of parameters used in this paper, the adjustment of a steady, inviscid, barotropic shelfbreak jet to channel topography embedded in the shelf is generally smooth and steady: there is no eddying or overshoot at the edge of the channel. The adjustment of the jet to the channel topography is governed by the strength of the topography S and the relative vorticity of the #ow impinging on the channel. For large S, the #ow is con"ned to the topography, but as S is reduced, #ow can cross the channel. The distance down the channel that the #ow crosses decreases with decreasing S. Channel topography similar to the Northeast Channel does not allow #ow to cross the channel for parameters within the oceanographic range. Hence, other factors need to be introduced to explain the episodic #ow of Scotian Shelf water across the Northeast Channel to Georges Bank. In particular, the in#uence of strati"cation, time varying in#ow, and wind forcing need to be examined.
Acknowledgements Thanks to Peter Smith for input on the structure of the #ow over the Scotian Shelf and to Joseph Pedlosky for helpful comments on the choice of modeling domain. Financial support was provided by the National Science Foundation under grants OCE93-13671, OCE96-32357 and OCE9632348. Computer facilities at the National Center for Atmospheric Research in Boulder Colorado were used for the numerical calculations. The National Center for Atmospheric Research is funded by the National Science Foundation. This is contribution number 142 of the US GLOBEC program, funded jointly by NSF and NOAA.
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Appendix. Stability of a barotropic Gaussian jet over sloping bathymetry For the inviscid, shallow water equations with a rigid lid, the relevant criterion to ensure that a jet #owing parallel to straight bathymetry is stable to small perturbations is that the initial potential vorticity gradient across the jet does not change sign. For a Gaussian jet of the form u"u e\W\* 5 #owing along straight bathymetry with a constant bottom slope b, and the
coast on the right, this criterion implies the jet is stable for a '2, e
(A.1)
where a"=/¸, e"u /f¸, and ¸ is the distance to the coast as in the simple model. Note that
a/eJ1/¸, so that as ¸ decreases, the jet becomes more stable for a given u and =. The jet
broadens as a increases and comes into contact with the coast for a'0.5, so we cannot consider very broad jets in the numerical model independently of the boundary conditions at the coast, and the range of Rossby numbers is restricted to 0(e(0.1. The shelf/slope topography used in the numerical model stabilizes the in#ow jet in the base case. If the slope is continued upward it reaches the surface very close to the center of the jet so that for the left-hand side of the jet which is over the slope, ¸ is small and a/e'2 for the range of e and a used in this study. The right-hand side of the jet is over the shelf where ¸ is large and a/e(2. The jet as a whole is found to be stable.
References Bisagni, J.J., Beardsley, R.C., Ruhsam, C.M., Manning, J.P., Williams, W.J., 1996. Historical and recent evidence concerning the presence of Scotian Shelf Water on southern Georges Bank. Deep-Sea Research II 43 (7}8), 1439}1471. Butman, B., Beardsley, R.C., 1987. Physical oceanography: introduction.. In: Backus, R. (Ed.), Georges Bank. MIT Press, Cambridge, MA, pp. 88}98. Chapman, D.C., Barth, J.A., Beardsley, R.C., Fairbanks, R.G., 1986. On the continuity of the mean #ow between the Scotian Shelf and the Middle Atlantic Bight. Journal of Physical Oceanography 16 (4), 758}772. Haidvogel, D.B., Wilkin, J.L., Young, R., 1991. A semi-spectral primitive equation ocean model using vertical sigma and orthogonal curvilinear horizontal coordinates. Journal of Computational Physics 94 (1), 151}185. Klinger, B.A., 1993. Gyre formation at a corner by rotating barotropic coastal #ows along a slope. Dynamics of Atmospheres and Oceans 19, 27}63. Robinson, A.R., Niiler, P.P., 1967. The theory of free inertial currents. Tellus XIX (2), 269}291. Smith, P.C., 1983. The mean and seasonal circulation o! southwest Nova Scotia. Journal of Physical Oceanography 13 (6), 1034}1054. Smith, P.C., Petrie, B.D., 1983. Low-frequency circulation at the edge of the Scotian Shelf. Journal of Physical Oceanography 12, 28}46.