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Formation of a shelfbreak front by freshwater discharge
Dynamics of Atmospheres and Oceans 36 (2002) 103–124
Formation of a shelfbreak front by freshwater discharge Chandrasekher Narayanan a,∗ , Richard W. Garvine b,1 a
Center of Higher Learning, Room 109, Building 1103, Stennis Space Center, Mississippi, MS 39529, USA b College of Marine Studies, University of Delaware, Newark, DE 19716, USA Received 26 May 2000; received in revised form 20 August 2001; accepted 4 March 2002
Abstract The circulation and transport of freshwater generated by an idealized buoyant source is studied using a three-dimensional primitive equation model. Freshwater enters the continental shelf, turns anticyclonically and moves downstream in the direction of Kelvin wave propagation. In the region close to the source, the flow reaches an equilibrium in the bottom boundary layer so that freshwater does not spread offshore any further. This offshore equilibrium distance increases as we move downstream until the freshwater is able to feel the presence of the shelfbreak. A shelfbreak front forms and the shelfbreak prevents any further offshore spreading of freshwater in the bottom boundary layer. Two complimentary mechanisms are responsible for the slow cross-shelf migration of freshwater and subsequent trapping of shelfbreak fronts: bottom stress and topographic changes. The shelfbreak creates an active, dynamic process preventing leakage from the continental shelf region to the slope region. However, the dynamical process that traps the front to the shelfbreak is still unclear. The location of the shelfbreak front depends on four dimensionless parameters: scaled inlet volume transport, scaled breadth, scaled “diffusivity” and scaled shelf width. We develop empirical relations for predicting the location of the frontal bottom intersection, given these parameters. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Shelfbreak front; Freshwater discharge; Bottom boundary layer; Coastal currents; Fronts
1. Introduction Freshwater discharges from rivers and estuaries typically form plumes and fronts on many continental shelves. Often, strong gradients of water properties develop near the shelfbreak, ∗ Corresponding author. Tel.: +1-228-688-3763; fax: +1-228-688-4759. E-mail addresses: [email protected] (C. Narayanan), [email protected] (R.W. Garvine). 1 Tel.: +1-302-831-2169.
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where the relatively flat continental shelf gives way to the steeper continental slope, typically separating fresh and cool shelf waters from the warmer, saltier slope water, e.g. the Middle Atlantic Bight (Beardsley and Flagg, 1976), the eastern Bering sea (Coachman, 1986) and the Celtic sea (Pingree et al., 1982). These shelfbreak fronts are remarkably persistent features and are associated with high levels of biological activity (Marra et al., 1990). Understanding the processes and mechanisms related to the formation and maintenance of shelfbreak fronts is therefore of great importance. We will focus our study on the Middle Atlantic Bight (hereafter called MAB). Linder and Gawarkiewicz (1998) (hereafter called LG98) produced a climatology of the shelfbreak front in the MAB for different times of the year. Later, Gawarkiewicz et al. (2001) showed that the structure of the shelfbreak front, although very robust, was strongly variable. Small-scale hydrographic features appear within the frontal zone and then change or disappear within a few days. The salinity structure in the MAB remains fairly similar throughout the year and the density signal is mainly dependent on the changes in temperature which is mostly homogeneous in winter and strongly stratified during summer. In fall, the cross-shelf density gradients appear mostly uniform over a large part of the continental shelf region whereas during spring strong horizontal salinity gradients seem to be localized near the shelfbreak (Manning, 1991). What are the underlying dynamics and thermodynamics that maintain the shelfbreak front? There have been many theories enquiring into the formation and maintenance of shelfbreak fronts, but none of them have fully answered the basic question of how shelfbreak fronts form. Initial efforts by Wang (1984) and Ou (1984) were able to predict velocity fields that were consistent with the density front in the shelfbreak region. However, these two approaches were limited in their scope because they were two-dimensional models lacking alongshelf gradients in velocity and density. They also neglected bottom friction, which can produce a bottom boundary layer with an associated cross-shelf transport. Chapman (1986) showed that the net offshore flow due to bottom friction is significant. Later, Chapman (1986), noting that the shelfbreak front in the MAB has strong salinity and temperature gradients (but weak density gradients), investigated the possibility of shelfbreak fronts forming without density gradients. Using a barotropic model and examining a vertically uniform flow he showed that a strong tracer gradient can form near the shelfbreak. Gawarkiewicz and Chapman (1991) studied unstratified flow using a three-dimensional numerical model and found a shelfbreak front similar to that of Chapman (1986). In this case, the shelfbreak front did not act as a barrier between the shelf and open ocean and there was continuous flux of water from the shelf to the slope in the bottom boundary layer. Later, Gawarkiewicz and Chapman (1992) studied stratified flow, including nonlinear density advection. This allows for a feedback between density advection in the bottom boundary layer and the interior field. They found that the shelfbreak front indeed acted as a barrier to the exchange of water between the shelf and slope water. They also found that the offshore transport in the bottom boundary layer pushed lighter water under heavier water, eventually resulting in a well-mixed shelf. Chapman and Lentz (1994) (hereafter called CL94) also included nonlinear density advection and observed that the density front was “trapped” to an isobath. The front remained parallel to the isobath and did not move further offshore. Later, Yankovsky and Chapman (1997) derived a simple approximation for this trapping isobath in the absence of mixing processes. Chapman (2000) extended the study of CL94
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by investigating the establishment of shelfbreak fronts over relatively short alongshelf distances and showed that the frontal trapping mechanism is so robust that the presence of a shelfbreak has little effect on the location of the front. From the studies discussed above, it is clear that for the offshore spreading of freshwater and formation of a front, we need to include the effects of nonlinear density advection and a frictional boundary layer. All the studies discussed so far restrict their domain to a few hundred kilometers from the source. In contrast, Narayanan and Garvine (submitted for publication) (hereafter called NG00) studied the offshore spreading of buoyant coastal discharge over very long distances under the influence of diffusive processes. They showed that the maximum offshore penetration of density increases with alongshelf distance. Furthermore, the diffusive processes prevent the plume from being “trapped” to an isobath. However, they limited their work to mildly-sloping continental shelves which did not include a shelfbreak. In this paper we extend the work of NG00 to investigate the offshore spreading of freshwater in the presence of a shelfbreak. Can coastal freshwater discharge from rivers and estuaries spread offshore to form a shelfbreak front? If so, what is the structure of the front? How are the frontal characteristics dependent on the inflow conditions at the source? Does the shelfbreak act as a barrier to the transport of freshwater to the deep ocean? Is the frontal trapping mechanism dependent on the position of the shelfbreak? The paper is organized as follows. A three-dimensional primitive equation numerical model is used to shed light on the formation of a shelfbreak front and is described in Section 2. We describe the induced shelf circulation and the corresponding dynamics and thermodynamics (Section 3) and compare the results in a qualitative sense to observation (Section 4). A parameter study is conducted and described in Section 5. The paper concludes with a discussion in Section 6.
2. Numerical model We use the s-coordinate primitive equation model (SPEM 5.1) to solve the hydrostatic Boussinesq equations (Haidvogel et al., 1991). The model geometry appears in Fig. 1. The coordinates x, y form a Cartesian pair with x alongshelf and y cross-shelf. The model uses an s-coordinate transformation for the vertical. The dependent variables are u, v, w and T . Density ρ is calculated using a linear equation of state from T . The cross-shelf topography consists of a mildly-sloping shelf, a shelfbreak, a steep slope region and a flat abyssal plain. At y = 0, there is a coastal wall of depth h0 = 10 m and a solid wall offshore at y = 200 km. The isobaths parallel the x-axis. The domain length in the alongshelf direction is kept long enough to allow slow cross-shelf migration of density fronts as the flow evolves in the alongshelf direction. Inflow to the continental shelf occurs uniformly over the depth h0 and breadth b through a gap in the coastline. The interaction of the estuary and continental shelf is neglected in this work in order to keep the model simple. The number of grid points in x and y are 321 and 65, respectively. The grid size ranges from 2.5 to 4 km in different model runs. The grid size is made small enough so that the internal Rossby radius is everywhere resolved. The vertical structure is resolved using 30 s-coordinate levels. The time step, t, is held fixed at 960 s (90 time steps per day).
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Fig. 1. Model domain.
A Mellor–Yamada 2.0 turbulence closure scheme (Mellor and Yamada, 1982) with background vertical viscosity and diffusivity value of AV = KV = 2 × 10−5 m2 /s is chosen. An upper bound of AV = KV = 0.01 m2 /s is used so that excessive vertical mixing is not introduced into the model domain. The horizontal Laplacian diffusivity and viscosity values are KH = 10 m2 /s and AH = 50 m2 /s, respectively. The horizontal mixing is performed along constant s-coordinate surfaces. A central differencing advection scheme is used for both the momentum and density equations. The central differencing scheme introduces overshoots that need to be suppressed. Hence, whenever the density anomaly exceeds the allowed range, the horizontal diffusivity is raised locally to 150 m2 /s. This smooths the overshoots without numerical difficulties (Narayanan, 1999; Narayanan and Garvine, in preparation). A rigid lid is assumed at the surface (w = 0 at z = 0). No flow is permitted through the bottom or the side walls except for the buoyancy inflow introduced through a gap in the coast. At the upstream boundary a uniform inflow of u = uup is specified. This current is necessary to prevent buoyancy from being transported upstream (Yankovsky and Chapman, 1997). At the downstream boundary, an open boundary condition is used (Chapman and Lentz, 1994). The vorticity and depth-averaged velocity components are advected out with a radiation condition and a zero-gradient condition is applied to the depth-varying fields. At the surface, the stress is set to zero. At the bottom, the shear stress is parameterized using a linear bottom friction parameterization. There is no flux of buoyancy at the surface or the bottom.
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Table 1 Flow parameters for the standard case Parameter
Values
Buoyant inflow velocity (vi ) Depth at the source (h0 ) Width of the source (b) Internal Rossby Radius (Ri ) Inflow transport (Ti = vi h0 b) Geostrophic transport (g h20 /2f ) Bottom friction coefficient (r) Coriolis parameter (f ) Time step ( t) Alongshelf grid spacing ( x) Cross-shelf grid spacing ( y) Horizontal diffusivity (KH ) Horizontal viscosity (AH ) Background vertical diffusivity (KV ) Background vertical viscosity (AV ) Upper bound vertical diffusivity (KV ) Upper bound vertical viscosity (AV ) Shelf width (ysb ) Slope of continental shelf (α) Continental slope (β)
4 cm/s 10 m 15 km 6.26 km 6000 m3 /s 19,600 m3 /s 2.65 × 10−4 m/s 10−4 s− 960 s 3 km 2.5 km 10 m2 /s 50 m2 /s 2 × 10−5 m2 /s 2 × 10−5 m2 /s 0.01 m2 /s 0.01 m2 /s 37.5 km 0.001 0.03
3. Description of flow—the standard case NG00 studied the offshore spreading of freshwater over a constant slope. For their standard case, the maximum offshore penetration of density in the bottom boundary layer is 58.2 km. Here also (called the shelfbreak case), we use a standard case (Table 1) and study the variation from it. The inflow conditions are exactly the same as NG00. The topography now includes, in addition to the mildly-sloping shelf (α = 0.001), a shelfbreak, a slope region with bottom slope of 0.03, and a flat abyssal plain with a constant depth of 1500 km. For the same inflow parameters, the necessary condition for the formation of a shelfbreak front is for the shelf width (ysb ) to be less than 58.2 km. For the standard case, the shelfbreak lies 37.5 km offshore. Hence, the necessary condition is satisfied. The shelfbreak is defined as the point of maximum curvature in the topography. This topography is very similar to that of Gawarkiewicz and Chapman (1992). We will make frequent references and compare results with NG00. The channel length of the domain is 960 km while the offshore (solid) wall is at y = 200 km. The grid is uniform in x and non-uniform in y and z. Each calculation begins from rest. At the upstream boundary, a weak barotropic mean current uup = 4 cm/s is introduced. The transients generated by this flow are allowed to equilibrate for 10 days before the flux of freshwater is imposed. At t = 10 days, freshwater is introduced (vi = 4 cm/s and ρi = −4 kg/m3 ) through a gap in the coast centered at x = 43.5 km. The ambient density anomaly ρa is 0. We designate the plume boundary as the −0.2 density contour. This contour is representative of the seaward edge of the freshwater, as it is equal to 5% of the density anomaly introduced at the source.
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Fig. 2. Evolution of the plume boundary at times t = 80, 160, 240, 320 and 400 days for the case that includes the shelfbreak. The shelfbreak is 37.5 km offshore.
The freshwater plume turns anticyclonically upon entering the continental shelf. Fig. 2 shows the evolution of the plume boundary at the surface and the bottom from 80 to 400 days (intervals of 80 days) for the shelfbreak case. In contrast, Fig. 3 shows the same for the constant slope case. Comparing Figs. 3 and 4, one sees that the nearfields2 are reasonably similar. That is, the presence of the shelfbreak does not affect the flow in the nearfield. The flow there is described by NG00 (see Section 3, Figs. 3–8). The emphasis of this paper is on the farfield. In the farfield, the flow is strongly affected by the presence of the shelfbreak. While in the constant slope case, the plume continues to spread offshore as we progress downstream, it becomes arrested for the shelfbreak case. Fig. 4 plots the surface and bottom density contours at t = 400 days for flow in the presence of a shelfbreak. The −0.1 and −0.2 bottom density contours are parallel to the isobaths, whereas the surface contours spread further than the shelfbreak, indicating that the bottom boundary layer prevents flow from spreading offshore of the shelfbreak. In other words, the flow is trapped to an isobath close to the shelfbreak. Next, we show the evolution of the plume boundary at four different cross-shelf sections in the farfield: x = 300, 360, 450 and 600 km (Fig. 5). The top panel (x = 300 km) shows that as time progresses the vertical density gradient increases. There is offshore transport throughout the water column up to 320 days. However, the offshore transport decreases with time, which can be deduced by observing that successive plume boundary contours 2 We distinguish two horizontal regions in the flowfield. We define the region close to the source as the nearfield and the region far downstream as the farfield. More precisely, the nearfield is defined as the region within 15 internal Rossby radii from the source, while the farfield is at least 40 internal Rossby radii from the source. For the standard case, the nearfield is within 100 km of the source and the farfield is at least 250 km from the source.
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Fig. 3. Evolution of the plume boundary at times t = 80, 160, 240, 320 and 400 days for the case of constant bottom slope, α = 0.001.
become more tightly spaced as we march with time. The offshore transport in the bottom boundary layer shuts off at about 320 days. The second panel (x = 360 km) shows similar behavior in the bottom boundary layer, but within 240 days an equilibrium is reached. In contrast, the plume boundary at the surface varies in y due to the presence of instabilities
Fig. 4. The bottom and surface density contours are shown at t = 400 days for the case that includes the shelfbreak. The shelfbreak is 37.5 km offshore.
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Fig. 5. Evolution of the plume boundary at times t = 80, 160, 240, 320 and 400 days for four different cross-shelf sections: x = 300, 360, 450 and 600 km. The shelfbreak is 37.5 km offshore.
(see Fig. 4). While the plume boundary is nearly anchored to the bottom preventing leakage of freshwater from the continental shelf to the deep ocean, the surface is relatively free to move. Further downstream (x = 450 km), a surface-to-bottom, nearly vertical front forms at early times (t = 80 days). As freshwater moves offshore, though, the flow becomes
Fig. 6. The density, cross-shelf velocity and alongshelf velocity structure at a constant cross-shelf section x = 300 km is plotted at t = 400 days. Note the double alongshelf jet.
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Fig. 7. The density, cross-shelf velocity and alongshelf velocity structure at a cross-shelf section x = 600 km is shown at t = 400 days.
stratified. The plume boundary has not reached x = 600 km at 80 days. However, at 160 days, the plume boundary is nearly vertical. It is striking that for x > 320 km, the offshore trapping distance of the front remains nearly constant (about 30 km). Freshwater enters the continental shelf, turns anticyclonically and moves downstream in the direction of the Kelvin wave propagation. In the region close to the source, the flow
Fig. 8. The top figure shows the maximum alongshelf velocity plotted along the length of the channel. The lower figure plots the density anomaly between the surface and the bottom versus the alongshelf distance at the shelfbreak (at t = 400 days).
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reaches an equilibrium in the bottom boundary layer so that freshwater does not spread offshore any further. This offshore equilibrium distance increases as we move downstream until the freshwater is able to feel the presence of the shelfbreak. The shelfbreak prevents any further offshore spreading of freshwater in the bottom boundary layer. In order to understand the trapping close to the shelfbreak better, we plot the density and velocity structure for different cross-shelf sections at t = 400 days (Fig. 6). At x = 300 km the density field is strongly stratified especially near the surface. At the surface, there is an offshore flow seaward of the shelfbreak and an onshore flow landward. Also, there is a single strong alongshelf surface jet present (peak of 18 cm/s). As we move downstream to x = 360 km (not shown), the density field is less stratified. There is an offshore flow at the surface for y > 50 km and an onshore flow for y < 50 km. Also, there is a double jet with one strong alongshelf (peak of 14 cm/s) jet seaward of the shelfbreak and a weak alongshelf jet landward. Thus, there is a transition from a single alongshelf jet at x = 300 km to two alongshelf jets at x = 360 km. Further downstream at the cross-shelf section x = 600 km (Fig. 7), the stratification is still weaker and two alongshelf jets are still present. Fig. 8 (top) plots the maximum alongshelf velocity versus the alongshelf distance. The maximum velocities occur close to x = 300 km and reduce as we move further downstream. The degree of stratification along the shelfbreak is plotted in lower panel. Stratification is estimated using ρ (defined as ρsurface − ρbottom ) along the shelfbreak at t = 400 days. The stratification is maximum at about x = 300 km and reduces as we move downstream. Thus, as the stratification reduces so do the geostrophic velocities. We studied the nature of instabilities by collecting time-series data for the alongshelf and cross-shelf velocities at two different fixed locations at the surface: x = 600 km, y = 65 km and x = 600 km, y = 35 km. These points correspond to the two surface jets that form at the cross-shelf section x = 600 km. The data collected show the presence of instabilities landward of the shelfbreak. Further analysis showed that the necessary condition for the existence of baroclinic instabilities was satisfied. Two dominant periods were obtained: 11.4 and 0.7 days. While the 0.7-day period corresponds to the inertial frequency, the 11.4-day period is likely due to baroclinic instabilities. No dominant frequencies are obtained from the point measurement seaward of the shelfbreak. We use time-averaged heat and momentum balances to understand the processes and mechanisms related to the formation of the shelfbreak front. The dynamics and thermodynamics in the nearfield (not shown) are very similar to the case of constant slope (see NG00, Fig. 11). The dominant heat balance in the bottom boundary layer is between the time-averaged vertical diffusion, horizontal diffusion and cross-shelf density flux. At the surface, the alongshelf buoyancy flux also plays an important role. As we move downstream to x = 300 km, the dominant boundary layer heat balance is between cross-shelf and alongshelf buoyancy flux, vertical diffusion and horizontal diffusion (see top panel, Fig. 9). Further downstream (x = 600 km), the boundary layer heat balance is between vertical diffusion, horizontal diffusion and cross-shelf density flux. The ∂ρ/∂t is negligible everywhere implying that quasi steady state is reached. The cross-shelf momentum balance is geostrophic at the surface and interior and Ekman layer dynamics prevails at the bottom. The alongshelf momentum balance in the bottom boundary layer is between Coriolis force, pressure gradient force and vertical shear
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Fig. 9. The time-averaged heat balance at the bottom is shown for the cross-shelf sections x = 300 km (top) and 600 km (bottom). The time-averaging was done from 400 to 480 days.
divergence. At the surface, the primary balance is quasi geostrophic with other terms contributing to a lesser degree. In the interior, the balance is nearly geostrophic. This is very similar to the constant slope case (NG00). The time-averaged alongshelf buoyancy flux is calculated at four different cross-shelf sections, x = 300, 360, 450 and 600 km and is plotted over the entire cross-shelf distance (Fig. 10). The figure shows that at 300 km, there is one strong peak whereas at 600 km, the structure is more diffused and contains two peaks. At 600 km, the twin peaks correspond to the two alongshelf jets that form near the surface (Fig. 7). Advection is an important process in the nearfield (NG00), whereas both advection and diffusion play important roles in the farfield. Chapman (2000) studied a bottom-advected plume to investigate the establishment of a shelfbreak front. He finds very little influence by the shelfbreak on the location of the front. The frontal trapping isobath remains the same irrespective of whether there is a shelfbreak or not. Now, we investigate if the results of the present paper show the same. Freshwater over a constant slope continues to spread offshore due to the influence of diffusive processes as we move downstream limited only by the boundary layer control
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Fig. 10. The time-averaged alongshelf buoyancy flux is plotted at four different cross-shelf sections, x = 300, 360, 450 and 600 km. The time-averaging is done from 400 to 480 days.
due to bottom friction. In contrast, in the presence of a shelfbreak, the flow is trapped to an isobath. If the shelfbreak did not act as a dynamical barrier to prevent the offshore spreading of freshwater, then the sharp change in topography would not restrict the offshore transport of buoyancy flux in the bottom boundary layer. It would simply redistribute the buoyancy in the slope region. To illustrate this point, we plot the depth3 at which the flow reaches equilibrium in the bottom boundary layer as a function of alongshelf distance (measured from the source) for both the constant slope and the shelfbreak case (Fig. 11). While the equilibrium depth in the farfield continues to increase for the constant slope case, it remains nearly constant for the shelfbreak case. We also conducted five different runs with varying continental slope β for the same inflow conditions. The β values chosen are 0.015, 0.02, 0.03 (standard), 0.06 and 0.1. We plot the bottom density contour of the plume boundary at t = 400 days for the different runs (Fig. 12). The trapping always occurs inshore of the shelfbreak and the trapping distance decreases with increasing β values. This clearly shows that the presence of the shelfbreak is responsible for the formation of the shelfbreak front. The shelfbreak acts as the focus for a dynamic process preventing leakage from the continental shelf region to the slope region. However, the exact mechanism for the trapping is still unclear. To summarize, freshwater spreads offshore to form a shelfbreak front. The flow in the nearfield region is unaffected by the presence of the shelfbreak. As we move downstream, a single strong alongshelf surface jet is formed at the shelfbreak in strongly stratified waters. Further downstream, two alongshelf jets are formed in the presence of reduced stratification. The bottom cross-shelf penetration distance for density remains constant over the alongshelf distance (in the farfield). That is, the density contours remain trapped to a local isobath. The shelfbreak acts as the focus for a dynamic process for the formation of a shelf3 The maximum offshore spreading penetration distance for the −0.2 density contour is calculated for both the constant slope and shelfbreak cases. The depth that corresponds to this cross-shelf distance is then calculated.
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Fig. 11. The depth at which the flow reaches equilibrium in the bottom boundary layer is plotted as a function of alongshelf distance (measured from the source) for both the constant slope and shelfbreak case.
break front. The surface cross-shelf penetration distance varies in the alongshelf direction due to instabilities. The primary heat balance responsible for the equilibrium in the bottom boundary layer is between cross-shelf buoyancy flux, vertical diffusion and horizontal diffusion.
Fig. 12. The bottom density contour of the plume boundary is plotted at t = 400 days for different β. The β values are 0.015, 0.02, 0.03 (standard), 0.06 and 0.1.
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4. Qualitative comparison with observations The MAB shelfbreak front, although robust, is strongly variable (Gawarkiewicz et al., 2001). Small-scale hydrographic features appear within the frontal zone and then change or disappear within a few days. The numerical model results presented in this work are for highly idealized situations. We have neglected effects of solar insolation, winds, tides, alongshelf topographic changes and the influence of the deep ocean. Hence, we attempt only qualitative comparisons between the observed winter salinity field in the MAB and the numerical model results for the shape and slope of the isopycnals. The cross-shelf salinity gradient in the summer is strongest near the shelfbreak and weaker both offshore and onshore of the shelfbreak [Fig. 6b, LG98]. However, in winter (February/March), the cross-shelf density gradients appear to be nearly uniform (although slightly stronger near the shelfbreak). Manning (1991) shows the salinity structure in the New York Bight for different seasons [see Fig. 4a, Manning (1991)]. In fall, the cross-shelf density gradient appears mostly uniform over a large part of the continental shelf region. The numerical model results also show the cross-shelf density gradients to be nearly uniform. Fig. 13 plots the model cross-shelf density gradient along the surface and bottom as a function of cross-shelf distance at four different cross-shelf sections x = 150, 300, 450 and 600 km. At x = 300 km, there is one peak at the surface corresponding to the single alongshelf jet. Further downstream, two peaks are observed corresponding to the two surface jets found in the numerical model. Also, the cross-shelf surface density gradient at x = 300 km is stronger than at the other locations and is associated with a stronger alongshelf jet. In contrast, ∂ρ/∂y at the bottom has only one peak.
Fig. 13. The cross-shelf density gradient at the surface and bottom is plotted as a function of cross-shelf distance at four different cross-shelf sections x = 150, 300, 450 and 600 km. The units of ∂ρ/∂y are kg/m4 .
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The slope of the isohalines in the Middle Atlantic Bight front varies seasonally at different cross-shelf sections. LG98 notes that the slope of the 34.5 isohaline varies from 5 × 10−3 in October/November to 2 × 10−3 in November/December. The 34.5 isohaline does not intersect the surface within 40 km of the shelfbreak during any other time. The same feature was observed by Manning (1991). For the model standard shelfbreak case, the slope of the −0.2 density contour is about 2.1 × 10−3 at x = 300 km, 1.4 × 10−3 at x = 450 km and 3.7 × 10−3 at x = 600 km. These slopes are in the range observed by LG98. Gawarkiewicz et al. (2001) and Gawarkiewicz (personal communication) find double jets at different locations along the Middle Atlantic Bight and Scotian Shelf. They note that, while in the Scotian Shelf the double jets are separated by about 100 km, the double jets are much closer in the MAB. They found these jets both in summer and spring. The double jets in the model are found only in the farfield. However, the position of the double jets and their relative strengths depend on the parameters at the source, bottom slope, mixing characteristics and the shelf width. The distance between the two jets ranges between 20 and 30 km in the model. This is closer to what is observed in the MAB than in the Scotian shelf. The modeling results show the jet seaward of the shelfbreak to be stronger than the inshore jet, in contrast to the observations. However, note that Gawarkiewicz et al. (2001) mentions that the double jet may be due to the presence of an eddy that dominated the flow structure. Hence, the mechanism for the formation of the double jet that we find may not be the same as that observed by Gawarkiewicz et al. (2001). Also, other studies (e.g. Fratantoni et al., 2001) do not find evidence of a double jet.
5. Parameter studies NG00 studied the offshore spreading of freshwater over a sloping continental shelf. That flow turned anticyclonically and spread offshore while slowly evolving downstream. An equilibrium was reached in the bottom boundary layer at every alongshelf location. This cross-shelf equilibrium distance increased with alongshelf distance until advection and mixing reached a balance so that there was no further offshore spreading of freshwater. Because of horizontal diffusion, the flow was not trapped to an isobath. Nevertheless, a finite limit was reached through a balance between advection and diffusion. NG00 conducted numerous experiments to predict the maximum offshore distance to which the plume spread. The maximum offshore penetration distance depended on three dimensionless parameters: scaled inlet transport τi , scaled bottom friction r/(ci α) and inlet Kelvin number Ki where ci is the inlet internal phase speed ( g h0 ). The τi is the ratio of the inlet transport (Ti ) to the geostrophic transport (= g h20 /2f ) and measures the buoyant source strength. Inlet Kelvin number (Ki = bf/ci ) also characterizes the scale of the buoyant source. The r/(ci α) is the scaled effective “diffusivity” in the parabolic equation derived by Csanady (1978). Fig. 14 (from NG00) synthesizes the relationship between offshore spreading of freshwater and the inflow characteristics. Given the three parameters, one can use Fig. 14 to estimate the maximum offshore penetration distance (y∞ ) in the absence of a shelfbreak. If the shelfbreak lies offshore of this distance, then a shelfbreak front can never form. However, if the shelf width (xsb ) is less than y∞ for the constant slope case, then a shelfbreak front can form.
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Fig. 14. Y (τi , r/(ci α), Ki , y∞ /ri ) is plotted against the scaled volume transport, τi .
The primary objective of this section is to evaluate the influence of the shelfbreak. How does the presence of a shelfbreak change the cross-shelf penetration distance (yb )? How does yb depend on inflow parameters, bottom slope and shelf width? Under what conditions does the plume reach an equilibrium distance inshore of the shelfbreak? How long does the flow take to reach the shelfbreak? Over what alongshelf length scale does this happen (xb )? 5.1. Methodology The offshore spreading of freshwater depends on many dimensional parameters: the inlet breadth b, Coriolis parameter f , depth at the coast h0 , inlet density anomaly ρi , inlet velocity vi , slope of the continental shelf α, bottom friction coefficient r, shelf width ysb , slope of the continental slope β and upstream inflow uup . We ran many different experiments by changing some of these parameters. In all the model runs, h0 , r, ρi and β were held constant. In an approach similar to Garvine (1999) and NG00, we evaluate the dependence of xb and yb as a function of both dimensional and non-dimensional parameters. The flow is likely
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to depend on four dimensionless parameters: scaled volume transport τi , scaled “diffusivity” r/(ci α), dimensionless distance to the shelfbreak ysb /ri and inlet Kelvin number (Ki ). We calculate the maximum cross-shelf and alongshelf penetration distances (y∞ and x∞ ) for the shelfbreak case using a method similar to that used by NG00. First, we calculate the maximum cross-shelf penetration distance of the plume boundary (yb ) at the bottom at any particular time (say, t = 400 days). The data obtained were then fitted by an exponential curve of the form n
yb = y∞ (1 − e−kx )
(1)
The least squares fit calculates three unknown quantities: n, y∞ and k. The x∞ is evaluated from k and n as follows. At kxn = 3, the cross-shelf penetration distance is approximately 95% of the maximum cross-shelf penetration distance. We define the x-coordinate there as x∞ = (3/k)1/n . We obtain x∞ at four different times and calculate its average. Fig. 15 plots yb as a function of x at 400 days for the standard case. The solid line is the analytical equation fit and the dashed-dot line is the −0.2 density contour result obtained from the numerical model (at t = 400 days). For this case, x∞ = 205 km and y∞ = 30.35 km. That is, for the standard case, the alongshelf distance at which the freshwater reaches the shelfbreak is 205 km downstream of the source. 5.2. Experiments and results We ran many model experiments changing the total volume transport, shelf width and bottom slope. Note that we did not change the values of horizontal diffusion, upstream inflow and vertical diffusion because NG00 showed that these parameters changed the
Fig. 15. The −0.2 bottom density contour at t = 400 days is plotted along with the analytical fit obtained from Eq. (1). The solid line is the analytical fit and the dashed-dot line is the −0.2 density contour obtained from the numerical model.
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Table 2 Model runs τi
vi (cm/s)
α (cm/s)
ysb (km)
Symbol
0.31 0.31 0.78 0.78 1.57 1.57
4 4 10 10 20 20
0.001 0.002 0.001 0.002 0.001 0.002
37.5–52.5 27.5–35 37.5–52.5 32.5–45 37.5–57.5 37.5–55
∗
䉫
+
䊊
×
䊐
offshore penetration distance for density only minimally. Three different values of τi were chosen: 0.31 (standard), 0.78 and 1.57. The first two cases correspond to a surface-advected type plume, whereas the third case corresponds to an intermediate type (Yankovsky and Chapman, 1997). Also, these three values correspond to inlet velocities of 4, 10 and 20 cm/s, respectively. For fixed τi , we made several experiments changing the bottom slope and shelf width. Two different slopes were chosen: α = 0.001 and 0.002. These correspond to r/(ci α) values of 0.42 (standard) and 0.21, respectively. For fixed r/(ci α) and τi , the shelf width was varied. The ysb ranged from 27.5 to 57.5 km in different runs. For all the runs, we ensure that the shelf width (ysb ) is always less than the y∞ predicted for the constant slope case (for the same inflow conditions and shelf bottom slope α). Hence, there is always potential for the formation of a shelfbreak front. The β (0.03), f (10−4 s−1 ), uup (4 cm/s) and b (15 km) were held constant in all the model runs. The inlet Kelvin number (Ki ) was held constant (Table 2). Fig. 16 plots the maximum cross-shelf penetration distance as a function of the shelf width. For fixed τi and r/(ci α), increasing the shelf width increases the offshore trapping
Fig. 16. The maximum cross-shelf penetration distance (y∞ ) is plotted against shelf width (ysb ) for all the model runs. The different symbols correspond to the different runs and can be found in Table 2.
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distance. This is because y∞ for the constant slope case is much greater than ysb and there is no dynamical mechanism to trap the buoyant inflow much inshore of the shelfbreak. Also, for a fixed bottom slope and shelf width, increasing the inlet volume transport results in fronts being trapped at a greater offshore distance. However, the freshwater plume is always trapped inshore of the shelfbreak. For a fixed inlet volume transport and shelf width, greater offshore spreading of density and subsequent trapping occurs on milder slopes. Note that for all cases considered, the maximum cross-shelf penetration distance (y∞ ) never exceeds the shelf width (ysb ). This shows that the freshwater is always trapped to the shelfbreak. We did not run any bottom-advected flow cases. Bottom-advected type plumes typically have much larger transport than the surface-advected plume. The rate of offshore migration of freshwater is relatively fast and occurs over short alongshelf distances. Also, advective processes are likely to dominate over diffusion. In this case, the presence of a shelfbreak may not have as profound an affect on the final location of the shelfbreak front. There may be leakage of freshwater from the continental shelf to the slope region. Chapman (2000) finds this behavior in his investigation of bottom-advected plumes. Similarly, we evaluate the behavior of the alongshelf penetration distance. The results obtained can easily be understood without illustrations. For a fixed volume transport and bottom slope, the alongshelf distance at which the fronts are trapped increases with increasing shelf width. For a fixed bottom slope and shelf width, the alongshelf distance at which trapping occurs decreases with increasing volume transport. Also, for fixed τi and ysb , x∞ is greater for lower r/(ci α) values. For the flow over a topography that includes a shelfbreak, we evaluated the results with respect to three dimensionless parameters: τi , r/(ci α) and ysb /ri . Now, we collapse all the cross-shelf penetration distances from the different model runs onto a single curve when
Fig. 17. Y (τi , r/(ci α), y∞ /ri ) is plotted against the scaled shelf width, ysb /ri .
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Fig. 18. X(τi , r/(ci α), x∞ /ri ) is plotted against the scaled shelf width, ysb /ri .
plotted against the scaled shelf width, ysb /ri (Fig. 17). Thus, for a given τi , r/(ci α) and ysb /ri , this curve predicts the offshore trapping distance of the plume. An estimate for y∞ /ri can be obtained using the following equation: y∞ r ysb log10 = 0.04 log10 τi + 0.26 log10 + 0.342 (2) + 0.076 ri ci α ri The companion curve, Fig. 18, predicts the alongshelf distance at which the freshwater reaches the shelfbreak. For this case, x∞ /ri can be expressed as x∞ r ysb log10 = −0.5 log10 τi − log10 − 0.599 (3) + 0.284 ri ci α ri Therefore, from Figs. 14, 17 and 18, it is possible first to predict whether the freshwater reaches the shelfbreak, and if so, to predict the location of the bottom of the shelfbreak front.
6. Discussion Freshwater discharges from rivers and estuaries produce plumes and coastal currents locally. Can the freshwater spread offshore under the influence of diffusive processes and form a shelfbreak front? Chapman and Lentz (1994) concluded from their investigation of bottom-advected fronts that a steady state is reached in the bottom boundary layer between vertical mixing and onshore advection of density such that the density front remains parallel to an isobath and does not move further offshore. However, they omitted nonlinearity in their
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evaluation of momentum terms and omitted horizontal diffusion in the density equation in their study of the spreading of density fronts. Later, Chapman (2000) extended this study further by investigating the frontal mechanism in the presence of a shelfbreak. He concluded that the frontal mechanism is remarkably robust and that the presence of a shelfbreak does not affect the final location of the front. In other words, the shelfbreak topography has no dynamical influence on the formation of the shelfbreak front. The focus of both these papers were on processes that occur over relatively short spatial and temporal scales. The scale over which diffusion is effective in influencing the flow field is much larger. The focus of the present study is on the formation of large-scale buoyancy driven currents. NG00 show that there is a slow cross-shelf migration of a surface-advected plume as the flow evolves in the alongshelf direction. In the farfield, due to the influence of diffusive processes, the plume transforms to an intermediate or bottom-advected type. Their study was restricted to flow over continental shelves in the absence of a shelfbreak. The present paper extends that study to include a shelfbreak. We show that freshwater can spread offshore and form a shelfbreak front. There are two notable differences between the present study and Chapman (2000). First, as mentioned before, there is a difference in spatial and temporal scales. Second, Chapman (2000) studied a bottom-advected plume, whereas we have studied a surface-advected type. Bottom-advected flows typically have large inflow transport and spread offshore to larger distances over short time scales. Hence, they are mainly driven by advective processes. In contrast, surface-advected plumes are strongly affected by both advective and diffusive processes. The cross-shelf migration of freshwater is then much slower for bottom-advected plumes. The shelfbreak topography has a strong influence on the offshore spreading of surface-advected plumes. Indeed, it acts as the focus for a dynamical mechanism for the formation of shelfbreak fronts. Shelfbreak fronts are formed and maintained when slow cross-shelf migration of freshwater occurs. The freshwater feels the presence of the shelfbreak and gets trapped to an isobath close to the shelfbreak. An equilibrium is reached such that the offshore buoyancy flux is balanced by the horizontal diffusion and vertical mixing in the bottom boundary layer. This mechanism prevents leakage of freshwater from the continental shelf to the slope. Within the bottom boundary layer on a sloping bottom, the Ekman transport that is driven by bottom stress acts to either upwell or downwell stratified water. The bottom boundary layer itself exerts a powerful control over buoyant coastal currents, providing a mechanism for an equilibrium to be reached in the bottom boundary layer. This mechanism is responsible for the boundary layer equilibrium (or trapping) observed by Chapman and Lentz (1994), Yankovsky and Chapman (1997) and NG00. In the present paper, another mechanism also acts along with the bottom stress control. This mechanism is due to topographic changes and can result in the formation of shelfbreak fronts. Furthermore, topographic trapping is felt only when there is slow cross-shelf migration of buoyancy. Diffusive processes are as important as advective processes to drive the shelf circulation in this case. However, note that how the presence of topography helps to trap the front to the shelfbreak is not yet understood and is the subject of ongoing research. NG00 established the relationship between the maximum offshore penetration of freshwater, in the absence shelfbreak, and the inflow characteristics (Fig. 14). If the shelfbreak lies offshore of this distance, then a shelfbreak front can never form. However, if the shelf
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width is less than this maximum cross-shelf penetration distance for the constant slope case, then a shelfbreak front can form. If so, then the location of the shelfbreak front along the bottom depends on four dimensionless parameters: scaled inlet volume transport, scaled breadth, scaled “diffusivity” and scaled shelf width. References Beardsley, R.C., Flagg, C.N., 1976. The water structure, mean currents, and shelf/slope water front on the New England continental shelf. Mem. Soc. R. Sci. Liege. 6 (X), 209–225. Coachman, L.K., 1986. Circulation, water masses, and fluxes on the southeastern Bering Sea Shelf. Cont. Shelf Res. 5, 23–108. Csanady, G.T., 1978. The arrested topographic wave. J. Phys. Oceanogr. 8, 47–62. Chapman, D.C., 1986. A simple model of the formation and maintenance of the shelf/slope front in the Middle Atlantic Bight. J. Phys. Oceanogr. 7, 1273–1279. Chapman, D.C., Lentz, S.J., 1994. Trapping of a coastal density front by the bottom boundary layer. J. Phys. Oceanogr. 24, 1464–1479. Chapman, D.C., 2000. Boundary layer control of buoyant coastal currents and the establishment of a shelfbreak front. J. Phys. Oceanogr. 30 (11), 2941–2955. Fratantoni, P.S., Pickart, R.S., Torres, D.J., Scotti, A., 2001. Mean structure and dynamics of the shelfbreak jet in the Middle Atlantic Bight during fall and winter. J. Phys. Oceanogr. 31 (8), 2135–2156. Garvine, R.W., 1999. Penetration of buoyant coastal discharge onto the continental shelf: a numerical model experiment. J. Phys. Oceanogr. 29, 1892–1909. Gawarkiewicz, G., Chapman, D.C., 1991. Formation and maintenance of shelf-break fronts in an unstratified flow. J. Phys. Oceanogr. 22, 1225–1239. Gawarkiewicz, G., Chapman, D.C., 1992. The role of stratification in the formation and maintenance of shelf-break fronts. J. Phys. Oceanogr. 17, 1877–1896. Gawarkiewicz, G., Bahr, F., Beardsley, R.C., Brink, K.H., 2001. Interaction of a slope eddy with the shelfbreak front in the Middle Atlantic Bight. J. Phys. Oceanogr. 31 (9), 2783–2796. Haidvogel, D.B., Wilkin, J.L., Young, R., 1991. A semi-spectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear horizontal coordinates. J. Comp. Phys. 94 (1), 151–184. Linder, C.A., Gawarkiewic, G.G., 1998. A climatology of the shelfbreak front in the Middle Atlantic Bight. J. Geophys. Res. 103, 18405–18423. Manning, J., 1991. Middle Atlantic Bight salinity: inter annual variability. Cont. Shelf Res. 11, 123–137. Marra, J., Houghton, R.W., Garside, C., 1990. Phytoplankton growth at the shelfbreak front in the Middle Atlantic Bight. J. Mar. Res. 48, 851–868. Mellor, G.L., Yamada, T., 1982. Development of a turbulent closure model for geophysical fluid problems. Rev. Geo. Space Phys. 20, 851–875. Narayanan, C., 1999. Offshore spreading of a buoyant coastal discharge. Ph.D. thesis, University of Delaware. 267 pp. Narayanan, C., Garvine, R.W., submitted for publication. Large scale buoyancy driven circulation on the continental shelf. Dyn. Atmos. Oceans. Narayanan, C., Garvine, R.W., in preparation. Effect of sub-grid scale parameterization and advection schemes on large scale coastal circulation. Ocean Modelling. Ou, H.W., 1984. Geostrophic adjustment: a mechanism for frontogenesis. J. Phys. Oceanogr. 14 (6), 994–1000. Pingree, R.D., Mardell, G.T., Holligan, P.M., Griffiths, D.K., Smithers, J., 1982. Celtic Sea and Armorican current structure and the vertical distributions of temperature and chlorophyll. Cont. Shelf Res. 1, 99–116. Wang, D.-P., 1984. Mutual intrusion of a gravity current and density front formation. J. Phys. Oceanogr. 14, 1191–1199. Yankovsky, A.E., Chapman, D.C., 1997. A simple theory for the fate of buoyant coastal discharges. J. Phys. Oceanogr. 27, 1386–1401.
Formation of a shelfbreak front by freshwater discharge Chandrasekher Narayanan a,∗ , Richard W. Garvine b,1 a
Center of Higher Learning, Room 109, Building 1103, Stennis Space Center, Mississippi, MS 39529, USA b College of Marine Studies, University of Delaware, Newark, DE 19716, USA Received 26 May 2000; received in revised form 20 August 2001; accepted 4 March 2002
Abstract The circulation and transport of freshwater generated by an idealized buoyant source is studied using a three-dimensional primitive equation model. Freshwater enters the continental shelf, turns anticyclonically and moves downstream in the direction of Kelvin wave propagation. In the region close to the source, the flow reaches an equilibrium in the bottom boundary layer so that freshwater does not spread offshore any further. This offshore equilibrium distance increases as we move downstream until the freshwater is able to feel the presence of the shelfbreak. A shelfbreak front forms and the shelfbreak prevents any further offshore spreading of freshwater in the bottom boundary layer. Two complimentary mechanisms are responsible for the slow cross-shelf migration of freshwater and subsequent trapping of shelfbreak fronts: bottom stress and topographic changes. The shelfbreak creates an active, dynamic process preventing leakage from the continental shelf region to the slope region. However, the dynamical process that traps the front to the shelfbreak is still unclear. The location of the shelfbreak front depends on four dimensionless parameters: scaled inlet volume transport, scaled breadth, scaled “diffusivity” and scaled shelf width. We develop empirical relations for predicting the location of the frontal bottom intersection, given these parameters. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Shelfbreak front; Freshwater discharge; Bottom boundary layer; Coastal currents; Fronts
1. Introduction Freshwater discharges from rivers and estuaries typically form plumes and fronts on many continental shelves. Often, strong gradients of water properties develop near the shelfbreak, ∗ Corresponding author. Tel.: +1-228-688-3763; fax: +1-228-688-4759. E-mail addresses: [email protected] (C. Narayanan), [email protected] (R.W. Garvine). 1 Tel.: +1-302-831-2169.
0377-0265/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 6 5 ( 0 2 ) 0 0 0 2 7 - 1
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where the relatively flat continental shelf gives way to the steeper continental slope, typically separating fresh and cool shelf waters from the warmer, saltier slope water, e.g. the Middle Atlantic Bight (Beardsley and Flagg, 1976), the eastern Bering sea (Coachman, 1986) and the Celtic sea (Pingree et al., 1982). These shelfbreak fronts are remarkably persistent features and are associated with high levels of biological activity (Marra et al., 1990). Understanding the processes and mechanisms related to the formation and maintenance of shelfbreak fronts is therefore of great importance. We will focus our study on the Middle Atlantic Bight (hereafter called MAB). Linder and Gawarkiewicz (1998) (hereafter called LG98) produced a climatology of the shelfbreak front in the MAB for different times of the year. Later, Gawarkiewicz et al. (2001) showed that the structure of the shelfbreak front, although very robust, was strongly variable. Small-scale hydrographic features appear within the frontal zone and then change or disappear within a few days. The salinity structure in the MAB remains fairly similar throughout the year and the density signal is mainly dependent on the changes in temperature which is mostly homogeneous in winter and strongly stratified during summer. In fall, the cross-shelf density gradients appear mostly uniform over a large part of the continental shelf region whereas during spring strong horizontal salinity gradients seem to be localized near the shelfbreak (Manning, 1991). What are the underlying dynamics and thermodynamics that maintain the shelfbreak front? There have been many theories enquiring into the formation and maintenance of shelfbreak fronts, but none of them have fully answered the basic question of how shelfbreak fronts form. Initial efforts by Wang (1984) and Ou (1984) were able to predict velocity fields that were consistent with the density front in the shelfbreak region. However, these two approaches were limited in their scope because they were two-dimensional models lacking alongshelf gradients in velocity and density. They also neglected bottom friction, which can produce a bottom boundary layer with an associated cross-shelf transport. Chapman (1986) showed that the net offshore flow due to bottom friction is significant. Later, Chapman (1986), noting that the shelfbreak front in the MAB has strong salinity and temperature gradients (but weak density gradients), investigated the possibility of shelfbreak fronts forming without density gradients. Using a barotropic model and examining a vertically uniform flow he showed that a strong tracer gradient can form near the shelfbreak. Gawarkiewicz and Chapman (1991) studied unstratified flow using a three-dimensional numerical model and found a shelfbreak front similar to that of Chapman (1986). In this case, the shelfbreak front did not act as a barrier between the shelf and open ocean and there was continuous flux of water from the shelf to the slope in the bottom boundary layer. Later, Gawarkiewicz and Chapman (1992) studied stratified flow, including nonlinear density advection. This allows for a feedback between density advection in the bottom boundary layer and the interior field. They found that the shelfbreak front indeed acted as a barrier to the exchange of water between the shelf and slope water. They also found that the offshore transport in the bottom boundary layer pushed lighter water under heavier water, eventually resulting in a well-mixed shelf. Chapman and Lentz (1994) (hereafter called CL94) also included nonlinear density advection and observed that the density front was “trapped” to an isobath. The front remained parallel to the isobath and did not move further offshore. Later, Yankovsky and Chapman (1997) derived a simple approximation for this trapping isobath in the absence of mixing processes. Chapman (2000) extended the study of CL94
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by investigating the establishment of shelfbreak fronts over relatively short alongshelf distances and showed that the frontal trapping mechanism is so robust that the presence of a shelfbreak has little effect on the location of the front. From the studies discussed above, it is clear that for the offshore spreading of freshwater and formation of a front, we need to include the effects of nonlinear density advection and a frictional boundary layer. All the studies discussed so far restrict their domain to a few hundred kilometers from the source. In contrast, Narayanan and Garvine (submitted for publication) (hereafter called NG00) studied the offshore spreading of buoyant coastal discharge over very long distances under the influence of diffusive processes. They showed that the maximum offshore penetration of density increases with alongshelf distance. Furthermore, the diffusive processes prevent the plume from being “trapped” to an isobath. However, they limited their work to mildly-sloping continental shelves which did not include a shelfbreak. In this paper we extend the work of NG00 to investigate the offshore spreading of freshwater in the presence of a shelfbreak. Can coastal freshwater discharge from rivers and estuaries spread offshore to form a shelfbreak front? If so, what is the structure of the front? How are the frontal characteristics dependent on the inflow conditions at the source? Does the shelfbreak act as a barrier to the transport of freshwater to the deep ocean? Is the frontal trapping mechanism dependent on the position of the shelfbreak? The paper is organized as follows. A three-dimensional primitive equation numerical model is used to shed light on the formation of a shelfbreak front and is described in Section 2. We describe the induced shelf circulation and the corresponding dynamics and thermodynamics (Section 3) and compare the results in a qualitative sense to observation (Section 4). A parameter study is conducted and described in Section 5. The paper concludes with a discussion in Section 6.
2. Numerical model We use the s-coordinate primitive equation model (SPEM 5.1) to solve the hydrostatic Boussinesq equations (Haidvogel et al., 1991). The model geometry appears in Fig. 1. The coordinates x, y form a Cartesian pair with x alongshelf and y cross-shelf. The model uses an s-coordinate transformation for the vertical. The dependent variables are u, v, w and T . Density ρ is calculated using a linear equation of state from T . The cross-shelf topography consists of a mildly-sloping shelf, a shelfbreak, a steep slope region and a flat abyssal plain. At y = 0, there is a coastal wall of depth h0 = 10 m and a solid wall offshore at y = 200 km. The isobaths parallel the x-axis. The domain length in the alongshelf direction is kept long enough to allow slow cross-shelf migration of density fronts as the flow evolves in the alongshelf direction. Inflow to the continental shelf occurs uniformly over the depth h0 and breadth b through a gap in the coastline. The interaction of the estuary and continental shelf is neglected in this work in order to keep the model simple. The number of grid points in x and y are 321 and 65, respectively. The grid size ranges from 2.5 to 4 km in different model runs. The grid size is made small enough so that the internal Rossby radius is everywhere resolved. The vertical structure is resolved using 30 s-coordinate levels. The time step, t, is held fixed at 960 s (90 time steps per day).
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Fig. 1. Model domain.
A Mellor–Yamada 2.0 turbulence closure scheme (Mellor and Yamada, 1982) with background vertical viscosity and diffusivity value of AV = KV = 2 × 10−5 m2 /s is chosen. An upper bound of AV = KV = 0.01 m2 /s is used so that excessive vertical mixing is not introduced into the model domain. The horizontal Laplacian diffusivity and viscosity values are KH = 10 m2 /s and AH = 50 m2 /s, respectively. The horizontal mixing is performed along constant s-coordinate surfaces. A central differencing advection scheme is used for both the momentum and density equations. The central differencing scheme introduces overshoots that need to be suppressed. Hence, whenever the density anomaly exceeds the allowed range, the horizontal diffusivity is raised locally to 150 m2 /s. This smooths the overshoots without numerical difficulties (Narayanan, 1999; Narayanan and Garvine, in preparation). A rigid lid is assumed at the surface (w = 0 at z = 0). No flow is permitted through the bottom or the side walls except for the buoyancy inflow introduced through a gap in the coast. At the upstream boundary a uniform inflow of u = uup is specified. This current is necessary to prevent buoyancy from being transported upstream (Yankovsky and Chapman, 1997). At the downstream boundary, an open boundary condition is used (Chapman and Lentz, 1994). The vorticity and depth-averaged velocity components are advected out with a radiation condition and a zero-gradient condition is applied to the depth-varying fields. At the surface, the stress is set to zero. At the bottom, the shear stress is parameterized using a linear bottom friction parameterization. There is no flux of buoyancy at the surface or the bottom.
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Table 1 Flow parameters for the standard case Parameter
Values
Buoyant inflow velocity (vi ) Depth at the source (h0 ) Width of the source (b) Internal Rossby Radius (Ri ) Inflow transport (Ti = vi h0 b) Geostrophic transport (g h20 /2f ) Bottom friction coefficient (r) Coriolis parameter (f ) Time step ( t) Alongshelf grid spacing ( x) Cross-shelf grid spacing ( y) Horizontal diffusivity (KH ) Horizontal viscosity (AH ) Background vertical diffusivity (KV ) Background vertical viscosity (AV ) Upper bound vertical diffusivity (KV ) Upper bound vertical viscosity (AV ) Shelf width (ysb ) Slope of continental shelf (α) Continental slope (β)
4 cm/s 10 m 15 km 6.26 km 6000 m3 /s 19,600 m3 /s 2.65 × 10−4 m/s 10−4 s− 960 s 3 km 2.5 km 10 m2 /s 50 m2 /s 2 × 10−5 m2 /s 2 × 10−5 m2 /s 0.01 m2 /s 0.01 m2 /s 37.5 km 0.001 0.03
3. Description of flow—the standard case NG00 studied the offshore spreading of freshwater over a constant slope. For their standard case, the maximum offshore penetration of density in the bottom boundary layer is 58.2 km. Here also (called the shelfbreak case), we use a standard case (Table 1) and study the variation from it. The inflow conditions are exactly the same as NG00. The topography now includes, in addition to the mildly-sloping shelf (α = 0.001), a shelfbreak, a slope region with bottom slope of 0.03, and a flat abyssal plain with a constant depth of 1500 km. For the same inflow parameters, the necessary condition for the formation of a shelfbreak front is for the shelf width (ysb ) to be less than 58.2 km. For the standard case, the shelfbreak lies 37.5 km offshore. Hence, the necessary condition is satisfied. The shelfbreak is defined as the point of maximum curvature in the topography. This topography is very similar to that of Gawarkiewicz and Chapman (1992). We will make frequent references and compare results with NG00. The channel length of the domain is 960 km while the offshore (solid) wall is at y = 200 km. The grid is uniform in x and non-uniform in y and z. Each calculation begins from rest. At the upstream boundary, a weak barotropic mean current uup = 4 cm/s is introduced. The transients generated by this flow are allowed to equilibrate for 10 days before the flux of freshwater is imposed. At t = 10 days, freshwater is introduced (vi = 4 cm/s and ρi = −4 kg/m3 ) through a gap in the coast centered at x = 43.5 km. The ambient density anomaly ρa is 0. We designate the plume boundary as the −0.2 density contour. This contour is representative of the seaward edge of the freshwater, as it is equal to 5% of the density anomaly introduced at the source.
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Fig. 2. Evolution of the plume boundary at times t = 80, 160, 240, 320 and 400 days for the case that includes the shelfbreak. The shelfbreak is 37.5 km offshore.
The freshwater plume turns anticyclonically upon entering the continental shelf. Fig. 2 shows the evolution of the plume boundary at the surface and the bottom from 80 to 400 days (intervals of 80 days) for the shelfbreak case. In contrast, Fig. 3 shows the same for the constant slope case. Comparing Figs. 3 and 4, one sees that the nearfields2 are reasonably similar. That is, the presence of the shelfbreak does not affect the flow in the nearfield. The flow there is described by NG00 (see Section 3, Figs. 3–8). The emphasis of this paper is on the farfield. In the farfield, the flow is strongly affected by the presence of the shelfbreak. While in the constant slope case, the plume continues to spread offshore as we progress downstream, it becomes arrested for the shelfbreak case. Fig. 4 plots the surface and bottom density contours at t = 400 days for flow in the presence of a shelfbreak. The −0.1 and −0.2 bottom density contours are parallel to the isobaths, whereas the surface contours spread further than the shelfbreak, indicating that the bottom boundary layer prevents flow from spreading offshore of the shelfbreak. In other words, the flow is trapped to an isobath close to the shelfbreak. Next, we show the evolution of the plume boundary at four different cross-shelf sections in the farfield: x = 300, 360, 450 and 600 km (Fig. 5). The top panel (x = 300 km) shows that as time progresses the vertical density gradient increases. There is offshore transport throughout the water column up to 320 days. However, the offshore transport decreases with time, which can be deduced by observing that successive plume boundary contours 2 We distinguish two horizontal regions in the flowfield. We define the region close to the source as the nearfield and the region far downstream as the farfield. More precisely, the nearfield is defined as the region within 15 internal Rossby radii from the source, while the farfield is at least 40 internal Rossby radii from the source. For the standard case, the nearfield is within 100 km of the source and the farfield is at least 250 km from the source.
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Fig. 3. Evolution of the plume boundary at times t = 80, 160, 240, 320 and 400 days for the case of constant bottom slope, α = 0.001.
become more tightly spaced as we march with time. The offshore transport in the bottom boundary layer shuts off at about 320 days. The second panel (x = 360 km) shows similar behavior in the bottom boundary layer, but within 240 days an equilibrium is reached. In contrast, the plume boundary at the surface varies in y due to the presence of instabilities
Fig. 4. The bottom and surface density contours are shown at t = 400 days for the case that includes the shelfbreak. The shelfbreak is 37.5 km offshore.
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Fig. 5. Evolution of the plume boundary at times t = 80, 160, 240, 320 and 400 days for four different cross-shelf sections: x = 300, 360, 450 and 600 km. The shelfbreak is 37.5 km offshore.
(see Fig. 4). While the plume boundary is nearly anchored to the bottom preventing leakage of freshwater from the continental shelf to the deep ocean, the surface is relatively free to move. Further downstream (x = 450 km), a surface-to-bottom, nearly vertical front forms at early times (t = 80 days). As freshwater moves offshore, though, the flow becomes
Fig. 6. The density, cross-shelf velocity and alongshelf velocity structure at a constant cross-shelf section x = 300 km is plotted at t = 400 days. Note the double alongshelf jet.
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Fig. 7. The density, cross-shelf velocity and alongshelf velocity structure at a cross-shelf section x = 600 km is shown at t = 400 days.
stratified. The plume boundary has not reached x = 600 km at 80 days. However, at 160 days, the plume boundary is nearly vertical. It is striking that for x > 320 km, the offshore trapping distance of the front remains nearly constant (about 30 km). Freshwater enters the continental shelf, turns anticyclonically and moves downstream in the direction of the Kelvin wave propagation. In the region close to the source, the flow
Fig. 8. The top figure shows the maximum alongshelf velocity plotted along the length of the channel. The lower figure plots the density anomaly between the surface and the bottom versus the alongshelf distance at the shelfbreak (at t = 400 days).
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reaches an equilibrium in the bottom boundary layer so that freshwater does not spread offshore any further. This offshore equilibrium distance increases as we move downstream until the freshwater is able to feel the presence of the shelfbreak. The shelfbreak prevents any further offshore spreading of freshwater in the bottom boundary layer. In order to understand the trapping close to the shelfbreak better, we plot the density and velocity structure for different cross-shelf sections at t = 400 days (Fig. 6). At x = 300 km the density field is strongly stratified especially near the surface. At the surface, there is an offshore flow seaward of the shelfbreak and an onshore flow landward. Also, there is a single strong alongshelf surface jet present (peak of 18 cm/s). As we move downstream to x = 360 km (not shown), the density field is less stratified. There is an offshore flow at the surface for y > 50 km and an onshore flow for y < 50 km. Also, there is a double jet with one strong alongshelf (peak of 14 cm/s) jet seaward of the shelfbreak and a weak alongshelf jet landward. Thus, there is a transition from a single alongshelf jet at x = 300 km to two alongshelf jets at x = 360 km. Further downstream at the cross-shelf section x = 600 km (Fig. 7), the stratification is still weaker and two alongshelf jets are still present. Fig. 8 (top) plots the maximum alongshelf velocity versus the alongshelf distance. The maximum velocities occur close to x = 300 km and reduce as we move further downstream. The degree of stratification along the shelfbreak is plotted in lower panel. Stratification is estimated using ρ (defined as ρsurface − ρbottom ) along the shelfbreak at t = 400 days. The stratification is maximum at about x = 300 km and reduces as we move downstream. Thus, as the stratification reduces so do the geostrophic velocities. We studied the nature of instabilities by collecting time-series data for the alongshelf and cross-shelf velocities at two different fixed locations at the surface: x = 600 km, y = 65 km and x = 600 km, y = 35 km. These points correspond to the two surface jets that form at the cross-shelf section x = 600 km. The data collected show the presence of instabilities landward of the shelfbreak. Further analysis showed that the necessary condition for the existence of baroclinic instabilities was satisfied. Two dominant periods were obtained: 11.4 and 0.7 days. While the 0.7-day period corresponds to the inertial frequency, the 11.4-day period is likely due to baroclinic instabilities. No dominant frequencies are obtained from the point measurement seaward of the shelfbreak. We use time-averaged heat and momentum balances to understand the processes and mechanisms related to the formation of the shelfbreak front. The dynamics and thermodynamics in the nearfield (not shown) are very similar to the case of constant slope (see NG00, Fig. 11). The dominant heat balance in the bottom boundary layer is between the time-averaged vertical diffusion, horizontal diffusion and cross-shelf density flux. At the surface, the alongshelf buoyancy flux also plays an important role. As we move downstream to x = 300 km, the dominant boundary layer heat balance is between cross-shelf and alongshelf buoyancy flux, vertical diffusion and horizontal diffusion (see top panel, Fig. 9). Further downstream (x = 600 km), the boundary layer heat balance is between vertical diffusion, horizontal diffusion and cross-shelf density flux. The ∂ρ/∂t is negligible everywhere implying that quasi steady state is reached. The cross-shelf momentum balance is geostrophic at the surface and interior and Ekman layer dynamics prevails at the bottom. The alongshelf momentum balance in the bottom boundary layer is between Coriolis force, pressure gradient force and vertical shear
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Fig. 9. The time-averaged heat balance at the bottom is shown for the cross-shelf sections x = 300 km (top) and 600 km (bottom). The time-averaging was done from 400 to 480 days.
divergence. At the surface, the primary balance is quasi geostrophic with other terms contributing to a lesser degree. In the interior, the balance is nearly geostrophic. This is very similar to the constant slope case (NG00). The time-averaged alongshelf buoyancy flux is calculated at four different cross-shelf sections, x = 300, 360, 450 and 600 km and is plotted over the entire cross-shelf distance (Fig. 10). The figure shows that at 300 km, there is one strong peak whereas at 600 km, the structure is more diffused and contains two peaks. At 600 km, the twin peaks correspond to the two alongshelf jets that form near the surface (Fig. 7). Advection is an important process in the nearfield (NG00), whereas both advection and diffusion play important roles in the farfield. Chapman (2000) studied a bottom-advected plume to investigate the establishment of a shelfbreak front. He finds very little influence by the shelfbreak on the location of the front. The frontal trapping isobath remains the same irrespective of whether there is a shelfbreak or not. Now, we investigate if the results of the present paper show the same. Freshwater over a constant slope continues to spread offshore due to the influence of diffusive processes as we move downstream limited only by the boundary layer control
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Fig. 10. The time-averaged alongshelf buoyancy flux is plotted at four different cross-shelf sections, x = 300, 360, 450 and 600 km. The time-averaging is done from 400 to 480 days.
due to bottom friction. In contrast, in the presence of a shelfbreak, the flow is trapped to an isobath. If the shelfbreak did not act as a dynamical barrier to prevent the offshore spreading of freshwater, then the sharp change in topography would not restrict the offshore transport of buoyancy flux in the bottom boundary layer. It would simply redistribute the buoyancy in the slope region. To illustrate this point, we plot the depth3 at which the flow reaches equilibrium in the bottom boundary layer as a function of alongshelf distance (measured from the source) for both the constant slope and the shelfbreak case (Fig. 11). While the equilibrium depth in the farfield continues to increase for the constant slope case, it remains nearly constant for the shelfbreak case. We also conducted five different runs with varying continental slope β for the same inflow conditions. The β values chosen are 0.015, 0.02, 0.03 (standard), 0.06 and 0.1. We plot the bottom density contour of the plume boundary at t = 400 days for the different runs (Fig. 12). The trapping always occurs inshore of the shelfbreak and the trapping distance decreases with increasing β values. This clearly shows that the presence of the shelfbreak is responsible for the formation of the shelfbreak front. The shelfbreak acts as the focus for a dynamic process preventing leakage from the continental shelf region to the slope region. However, the exact mechanism for the trapping is still unclear. To summarize, freshwater spreads offshore to form a shelfbreak front. The flow in the nearfield region is unaffected by the presence of the shelfbreak. As we move downstream, a single strong alongshelf surface jet is formed at the shelfbreak in strongly stratified waters. Further downstream, two alongshelf jets are formed in the presence of reduced stratification. The bottom cross-shelf penetration distance for density remains constant over the alongshelf distance (in the farfield). That is, the density contours remain trapped to a local isobath. The shelfbreak acts as the focus for a dynamic process for the formation of a shelf3 The maximum offshore spreading penetration distance for the −0.2 density contour is calculated for both the constant slope and shelfbreak cases. The depth that corresponds to this cross-shelf distance is then calculated.
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Fig. 11. The depth at which the flow reaches equilibrium in the bottom boundary layer is plotted as a function of alongshelf distance (measured from the source) for both the constant slope and shelfbreak case.
break front. The surface cross-shelf penetration distance varies in the alongshelf direction due to instabilities. The primary heat balance responsible for the equilibrium in the bottom boundary layer is between cross-shelf buoyancy flux, vertical diffusion and horizontal diffusion.
Fig. 12. The bottom density contour of the plume boundary is plotted at t = 400 days for different β. The β values are 0.015, 0.02, 0.03 (standard), 0.06 and 0.1.
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4. Qualitative comparison with observations The MAB shelfbreak front, although robust, is strongly variable (Gawarkiewicz et al., 2001). Small-scale hydrographic features appear within the frontal zone and then change or disappear within a few days. The numerical model results presented in this work are for highly idealized situations. We have neglected effects of solar insolation, winds, tides, alongshelf topographic changes and the influence of the deep ocean. Hence, we attempt only qualitative comparisons between the observed winter salinity field in the MAB and the numerical model results for the shape and slope of the isopycnals. The cross-shelf salinity gradient in the summer is strongest near the shelfbreak and weaker both offshore and onshore of the shelfbreak [Fig. 6b, LG98]. However, in winter (February/March), the cross-shelf density gradients appear to be nearly uniform (although slightly stronger near the shelfbreak). Manning (1991) shows the salinity structure in the New York Bight for different seasons [see Fig. 4a, Manning (1991)]. In fall, the cross-shelf density gradient appears mostly uniform over a large part of the continental shelf region. The numerical model results also show the cross-shelf density gradients to be nearly uniform. Fig. 13 plots the model cross-shelf density gradient along the surface and bottom as a function of cross-shelf distance at four different cross-shelf sections x = 150, 300, 450 and 600 km. At x = 300 km, there is one peak at the surface corresponding to the single alongshelf jet. Further downstream, two peaks are observed corresponding to the two surface jets found in the numerical model. Also, the cross-shelf surface density gradient at x = 300 km is stronger than at the other locations and is associated with a stronger alongshelf jet. In contrast, ∂ρ/∂y at the bottom has only one peak.
Fig. 13. The cross-shelf density gradient at the surface and bottom is plotted as a function of cross-shelf distance at four different cross-shelf sections x = 150, 300, 450 and 600 km. The units of ∂ρ/∂y are kg/m4 .
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The slope of the isohalines in the Middle Atlantic Bight front varies seasonally at different cross-shelf sections. LG98 notes that the slope of the 34.5 isohaline varies from 5 × 10−3 in October/November to 2 × 10−3 in November/December. The 34.5 isohaline does not intersect the surface within 40 km of the shelfbreak during any other time. The same feature was observed by Manning (1991). For the model standard shelfbreak case, the slope of the −0.2 density contour is about 2.1 × 10−3 at x = 300 km, 1.4 × 10−3 at x = 450 km and 3.7 × 10−3 at x = 600 km. These slopes are in the range observed by LG98. Gawarkiewicz et al. (2001) and Gawarkiewicz (personal communication) find double jets at different locations along the Middle Atlantic Bight and Scotian Shelf. They note that, while in the Scotian Shelf the double jets are separated by about 100 km, the double jets are much closer in the MAB. They found these jets both in summer and spring. The double jets in the model are found only in the farfield. However, the position of the double jets and their relative strengths depend on the parameters at the source, bottom slope, mixing characteristics and the shelf width. The distance between the two jets ranges between 20 and 30 km in the model. This is closer to what is observed in the MAB than in the Scotian shelf. The modeling results show the jet seaward of the shelfbreak to be stronger than the inshore jet, in contrast to the observations. However, note that Gawarkiewicz et al. (2001) mentions that the double jet may be due to the presence of an eddy that dominated the flow structure. Hence, the mechanism for the formation of the double jet that we find may not be the same as that observed by Gawarkiewicz et al. (2001). Also, other studies (e.g. Fratantoni et al., 2001) do not find evidence of a double jet.
5. Parameter studies NG00 studied the offshore spreading of freshwater over a sloping continental shelf. That flow turned anticyclonically and spread offshore while slowly evolving downstream. An equilibrium was reached in the bottom boundary layer at every alongshelf location. This cross-shelf equilibrium distance increased with alongshelf distance until advection and mixing reached a balance so that there was no further offshore spreading of freshwater. Because of horizontal diffusion, the flow was not trapped to an isobath. Nevertheless, a finite limit was reached through a balance between advection and diffusion. NG00 conducted numerous experiments to predict the maximum offshore distance to which the plume spread. The maximum offshore penetration distance depended on three dimensionless parameters: scaled inlet transport τi , scaled bottom friction r/(ci α) and inlet Kelvin number Ki where ci is the inlet internal phase speed ( g h0 ). The τi is the ratio of the inlet transport (Ti ) to the geostrophic transport (= g h20 /2f ) and measures the buoyant source strength. Inlet Kelvin number (Ki = bf/ci ) also characterizes the scale of the buoyant source. The r/(ci α) is the scaled effective “diffusivity” in the parabolic equation derived by Csanady (1978). Fig. 14 (from NG00) synthesizes the relationship between offshore spreading of freshwater and the inflow characteristics. Given the three parameters, one can use Fig. 14 to estimate the maximum offshore penetration distance (y∞ ) in the absence of a shelfbreak. If the shelfbreak lies offshore of this distance, then a shelfbreak front can never form. However, if the shelf width (xsb ) is less than y∞ for the constant slope case, then a shelfbreak front can form.
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Fig. 14. Y (τi , r/(ci α), Ki , y∞ /ri ) is plotted against the scaled volume transport, τi .
The primary objective of this section is to evaluate the influence of the shelfbreak. How does the presence of a shelfbreak change the cross-shelf penetration distance (yb )? How does yb depend on inflow parameters, bottom slope and shelf width? Under what conditions does the plume reach an equilibrium distance inshore of the shelfbreak? How long does the flow take to reach the shelfbreak? Over what alongshelf length scale does this happen (xb )? 5.1. Methodology The offshore spreading of freshwater depends on many dimensional parameters: the inlet breadth b, Coriolis parameter f , depth at the coast h0 , inlet density anomaly ρi , inlet velocity vi , slope of the continental shelf α, bottom friction coefficient r, shelf width ysb , slope of the continental slope β and upstream inflow uup . We ran many different experiments by changing some of these parameters. In all the model runs, h0 , r, ρi and β were held constant. In an approach similar to Garvine (1999) and NG00, we evaluate the dependence of xb and yb as a function of both dimensional and non-dimensional parameters. The flow is likely
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to depend on four dimensionless parameters: scaled volume transport τi , scaled “diffusivity” r/(ci α), dimensionless distance to the shelfbreak ysb /ri and inlet Kelvin number (Ki ). We calculate the maximum cross-shelf and alongshelf penetration distances (y∞ and x∞ ) for the shelfbreak case using a method similar to that used by NG00. First, we calculate the maximum cross-shelf penetration distance of the plume boundary (yb ) at the bottom at any particular time (say, t = 400 days). The data obtained were then fitted by an exponential curve of the form n
yb = y∞ (1 − e−kx )
(1)
The least squares fit calculates three unknown quantities: n, y∞ and k. The x∞ is evaluated from k and n as follows. At kxn = 3, the cross-shelf penetration distance is approximately 95% of the maximum cross-shelf penetration distance. We define the x-coordinate there as x∞ = (3/k)1/n . We obtain x∞ at four different times and calculate its average. Fig. 15 plots yb as a function of x at 400 days for the standard case. The solid line is the analytical equation fit and the dashed-dot line is the −0.2 density contour result obtained from the numerical model (at t = 400 days). For this case, x∞ = 205 km and y∞ = 30.35 km. That is, for the standard case, the alongshelf distance at which the freshwater reaches the shelfbreak is 205 km downstream of the source. 5.2. Experiments and results We ran many model experiments changing the total volume transport, shelf width and bottom slope. Note that we did not change the values of horizontal diffusion, upstream inflow and vertical diffusion because NG00 showed that these parameters changed the
Fig. 15. The −0.2 bottom density contour at t = 400 days is plotted along with the analytical fit obtained from Eq. (1). The solid line is the analytical fit and the dashed-dot line is the −0.2 density contour obtained from the numerical model.
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Table 2 Model runs τi
vi (cm/s)
α (cm/s)
ysb (km)
Symbol
0.31 0.31 0.78 0.78 1.57 1.57
4 4 10 10 20 20
0.001 0.002 0.001 0.002 0.001 0.002
37.5–52.5 27.5–35 37.5–52.5 32.5–45 37.5–57.5 37.5–55
∗
䉫
+
䊊
×
䊐
offshore penetration distance for density only minimally. Three different values of τi were chosen: 0.31 (standard), 0.78 and 1.57. The first two cases correspond to a surface-advected type plume, whereas the third case corresponds to an intermediate type (Yankovsky and Chapman, 1997). Also, these three values correspond to inlet velocities of 4, 10 and 20 cm/s, respectively. For fixed τi , we made several experiments changing the bottom slope and shelf width. Two different slopes were chosen: α = 0.001 and 0.002. These correspond to r/(ci α) values of 0.42 (standard) and 0.21, respectively. For fixed r/(ci α) and τi , the shelf width was varied. The ysb ranged from 27.5 to 57.5 km in different runs. For all the runs, we ensure that the shelf width (ysb ) is always less than the y∞ predicted for the constant slope case (for the same inflow conditions and shelf bottom slope α). Hence, there is always potential for the formation of a shelfbreak front. The β (0.03), f (10−4 s−1 ), uup (4 cm/s) and b (15 km) were held constant in all the model runs. The inlet Kelvin number (Ki ) was held constant (Table 2). Fig. 16 plots the maximum cross-shelf penetration distance as a function of the shelf width. For fixed τi and r/(ci α), increasing the shelf width increases the offshore trapping
Fig. 16. The maximum cross-shelf penetration distance (y∞ ) is plotted against shelf width (ysb ) for all the model runs. The different symbols correspond to the different runs and can be found in Table 2.
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distance. This is because y∞ for the constant slope case is much greater than ysb and there is no dynamical mechanism to trap the buoyant inflow much inshore of the shelfbreak. Also, for a fixed bottom slope and shelf width, increasing the inlet volume transport results in fronts being trapped at a greater offshore distance. However, the freshwater plume is always trapped inshore of the shelfbreak. For a fixed inlet volume transport and shelf width, greater offshore spreading of density and subsequent trapping occurs on milder slopes. Note that for all cases considered, the maximum cross-shelf penetration distance (y∞ ) never exceeds the shelf width (ysb ). This shows that the freshwater is always trapped to the shelfbreak. We did not run any bottom-advected flow cases. Bottom-advected type plumes typically have much larger transport than the surface-advected plume. The rate of offshore migration of freshwater is relatively fast and occurs over short alongshelf distances. Also, advective processes are likely to dominate over diffusion. In this case, the presence of a shelfbreak may not have as profound an affect on the final location of the shelfbreak front. There may be leakage of freshwater from the continental shelf to the slope region. Chapman (2000) finds this behavior in his investigation of bottom-advected plumes. Similarly, we evaluate the behavior of the alongshelf penetration distance. The results obtained can easily be understood without illustrations. For a fixed volume transport and bottom slope, the alongshelf distance at which the fronts are trapped increases with increasing shelf width. For a fixed bottom slope and shelf width, the alongshelf distance at which trapping occurs decreases with increasing volume transport. Also, for fixed τi and ysb , x∞ is greater for lower r/(ci α) values. For the flow over a topography that includes a shelfbreak, we evaluated the results with respect to three dimensionless parameters: τi , r/(ci α) and ysb /ri . Now, we collapse all the cross-shelf penetration distances from the different model runs onto a single curve when
Fig. 17. Y (τi , r/(ci α), y∞ /ri ) is plotted against the scaled shelf width, ysb /ri .
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Fig. 18. X(τi , r/(ci α), x∞ /ri ) is plotted against the scaled shelf width, ysb /ri .
plotted against the scaled shelf width, ysb /ri (Fig. 17). Thus, for a given τi , r/(ci α) and ysb /ri , this curve predicts the offshore trapping distance of the plume. An estimate for y∞ /ri can be obtained using the following equation: y∞ r ysb log10 = 0.04 log10 τi + 0.26 log10 + 0.342 (2) + 0.076 ri ci α ri The companion curve, Fig. 18, predicts the alongshelf distance at which the freshwater reaches the shelfbreak. For this case, x∞ /ri can be expressed as x∞ r ysb log10 = −0.5 log10 τi − log10 − 0.599 (3) + 0.284 ri ci α ri Therefore, from Figs. 14, 17 and 18, it is possible first to predict whether the freshwater reaches the shelfbreak, and if so, to predict the location of the bottom of the shelfbreak front.
6. Discussion Freshwater discharges from rivers and estuaries produce plumes and coastal currents locally. Can the freshwater spread offshore under the influence of diffusive processes and form a shelfbreak front? Chapman and Lentz (1994) concluded from their investigation of bottom-advected fronts that a steady state is reached in the bottom boundary layer between vertical mixing and onshore advection of density such that the density front remains parallel to an isobath and does not move further offshore. However, they omitted nonlinearity in their
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evaluation of momentum terms and omitted horizontal diffusion in the density equation in their study of the spreading of density fronts. Later, Chapman (2000) extended this study further by investigating the frontal mechanism in the presence of a shelfbreak. He concluded that the frontal mechanism is remarkably robust and that the presence of a shelfbreak does not affect the final location of the front. In other words, the shelfbreak topography has no dynamical influence on the formation of the shelfbreak front. The focus of both these papers were on processes that occur over relatively short spatial and temporal scales. The scale over which diffusion is effective in influencing the flow field is much larger. The focus of the present study is on the formation of large-scale buoyancy driven currents. NG00 show that there is a slow cross-shelf migration of a surface-advected plume as the flow evolves in the alongshelf direction. In the farfield, due to the influence of diffusive processes, the plume transforms to an intermediate or bottom-advected type. Their study was restricted to flow over continental shelves in the absence of a shelfbreak. The present paper extends that study to include a shelfbreak. We show that freshwater can spread offshore and form a shelfbreak front. There are two notable differences between the present study and Chapman (2000). First, as mentioned before, there is a difference in spatial and temporal scales. Second, Chapman (2000) studied a bottom-advected plume, whereas we have studied a surface-advected type. Bottom-advected flows typically have large inflow transport and spread offshore to larger distances over short time scales. Hence, they are mainly driven by advective processes. In contrast, surface-advected plumes are strongly affected by both advective and diffusive processes. The cross-shelf migration of freshwater is then much slower for bottom-advected plumes. The shelfbreak topography has a strong influence on the offshore spreading of surface-advected plumes. Indeed, it acts as the focus for a dynamical mechanism for the formation of shelfbreak fronts. Shelfbreak fronts are formed and maintained when slow cross-shelf migration of freshwater occurs. The freshwater feels the presence of the shelfbreak and gets trapped to an isobath close to the shelfbreak. An equilibrium is reached such that the offshore buoyancy flux is balanced by the horizontal diffusion and vertical mixing in the bottom boundary layer. This mechanism prevents leakage of freshwater from the continental shelf to the slope. Within the bottom boundary layer on a sloping bottom, the Ekman transport that is driven by bottom stress acts to either upwell or downwell stratified water. The bottom boundary layer itself exerts a powerful control over buoyant coastal currents, providing a mechanism for an equilibrium to be reached in the bottom boundary layer. This mechanism is responsible for the boundary layer equilibrium (or trapping) observed by Chapman and Lentz (1994), Yankovsky and Chapman (1997) and NG00. In the present paper, another mechanism also acts along with the bottom stress control. This mechanism is due to topographic changes and can result in the formation of shelfbreak fronts. Furthermore, topographic trapping is felt only when there is slow cross-shelf migration of buoyancy. Diffusive processes are as important as advective processes to drive the shelf circulation in this case. However, note that how the presence of topography helps to trap the front to the shelfbreak is not yet understood and is the subject of ongoing research. NG00 established the relationship between the maximum offshore penetration of freshwater, in the absence shelfbreak, and the inflow characteristics (Fig. 14). If the shelfbreak lies offshore of this distance, then a shelfbreak front can never form. However, if the shelf
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width is less than this maximum cross-shelf penetration distance for the constant slope case, then a shelfbreak front can form. If so, then the location of the shelfbreak front along the bottom depends on four dimensionless parameters: scaled inlet volume transport, scaled breadth, scaled “diffusivity” and scaled shelf width. References Beardsley, R.C., Flagg, C.N., 1976. The water structure, mean currents, and shelf/slope water front on the New England continental shelf. Mem. Soc. R. Sci. Liege. 6 (X), 209–225. Coachman, L.K., 1986. Circulation, water masses, and fluxes on the southeastern Bering Sea Shelf. Cont. Shelf Res. 5, 23–108. Csanady, G.T., 1978. The arrested topographic wave. J. Phys. Oceanogr. 8, 47–62. Chapman, D.C., 1986. A simple model of the formation and maintenance of the shelf/slope front in the Middle Atlantic Bight. J. Phys. Oceanogr. 7, 1273–1279. Chapman, D.C., Lentz, S.J., 1994. Trapping of a coastal density front by the bottom boundary layer. J. Phys. Oceanogr. 24, 1464–1479. Chapman, D.C., 2000. Boundary layer control of buoyant coastal currents and the establishment of a shelfbreak front. J. Phys. Oceanogr. 30 (11), 2941–2955. Fratantoni, P.S., Pickart, R.S., Torres, D.J., Scotti, A., 2001. Mean structure and dynamics of the shelfbreak jet in the Middle Atlantic Bight during fall and winter. J. Phys. Oceanogr. 31 (8), 2135–2156. Garvine, R.W., 1999. Penetration of buoyant coastal discharge onto the continental shelf: a numerical model experiment. J. Phys. Oceanogr. 29, 1892–1909. Gawarkiewicz, G., Chapman, D.C., 1991. Formation and maintenance of shelf-break fronts in an unstratified flow. J. Phys. Oceanogr. 22, 1225–1239. Gawarkiewicz, G., Chapman, D.C., 1992. The role of stratification in the formation and maintenance of shelf-break fronts. J. Phys. Oceanogr. 17, 1877–1896. Gawarkiewicz, G., Bahr, F., Beardsley, R.C., Brink, K.H., 2001. Interaction of a slope eddy with the shelfbreak front in the Middle Atlantic Bight. J. Phys. Oceanogr. 31 (9), 2783–2796. Haidvogel, D.B., Wilkin, J.L., Young, R., 1991. A semi-spectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear horizontal coordinates. J. Comp. Phys. 94 (1), 151–184. Linder, C.A., Gawarkiewic, G.G., 1998. A climatology of the shelfbreak front in the Middle Atlantic Bight. J. Geophys. Res. 103, 18405–18423. Manning, J., 1991. Middle Atlantic Bight salinity: inter annual variability. Cont. Shelf Res. 11, 123–137. Marra, J., Houghton, R.W., Garside, C., 1990. Phytoplankton growth at the shelfbreak front in the Middle Atlantic Bight. J. Mar. Res. 48, 851–868. Mellor, G.L., Yamada, T., 1982. Development of a turbulent closure model for geophysical fluid problems. Rev. Geo. Space Phys. 20, 851–875. Narayanan, C., 1999. Offshore spreading of a buoyant coastal discharge. Ph.D. thesis, University of Delaware. 267 pp. Narayanan, C., Garvine, R.W., submitted for publication. Large scale buoyancy driven circulation on the continental shelf. Dyn. Atmos. Oceans. Narayanan, C., Garvine, R.W., in preparation. Effect of sub-grid scale parameterization and advection schemes on large scale coastal circulation. Ocean Modelling. Ou, H.W., 1984. Geostrophic adjustment: a mechanism for frontogenesis. J. Phys. Oceanogr. 14 (6), 994–1000. Pingree, R.D., Mardell, G.T., Holligan, P.M., Griffiths, D.K., Smithers, J., 1982. Celtic Sea and Armorican current structure and the vertical distributions of temperature and chlorophyll. Cont. Shelf Res. 1, 99–116. Wang, D.-P., 1984. Mutual intrusion of a gravity current and density front formation. J. Phys. Oceanogr. 14, 1191–1199. Yankovsky, A.E., Chapman, D.C., 1997. A simple theory for the fate of buoyant coastal discharges. J. Phys. Oceanogr. 27, 1386–1401.