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    (a) Schematic view of a channelized bottom gravity current with the arrow denoting the dominant down-channel flow direction. (b) Coordinate system and channel geometry looking up-channel with the direction of the down-channel flow as indicated. Thin lines denote isopycnals of the tilted, wedge-shaped stratification in the interface.

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    (left) Cross-sectional view of the buoyancy at 0.01 m s−2 intervals, and (center) profiles of the down-channel, and (right) cross-channel velocity in the channel center. The rows correspond to different points of time as indicated (Tf is the local inertial period).

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    Ratio of bottom stress and down-channel pressure-gradient (full line), and ratio of bulk and geostrophic down-channel velocities (dashed), both shown for the channel center.

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    (a) Down-channel and (b) cross-channel velocity for t = 2Tf . Buoyancy is plotted as black contour lines at 0.02 m s−2 intervals. The dashed blue line in (a) denotes the bulk interface position, zH = zb + H. The black arrows in (b) illustrate the transverse circulation, and profile 1 and profile 2 refer to locations discussed below. All quantities are shown for t = 2Tf .

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    Cross-channel variation of (a) zm, zH, and zt (from bottom to top); (b) bulk buoyancy B and bottom buoyancy bb (dashed); (c) down-channel bottom stress τbx; (d) cross-channel bottom stress τby; and (e) entrainment velocity wb computed from (14). All quantities are shown for t = 2Tf .

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    Cross-channel variation of (a) friction velocity squared, down-channel pressure gradient, and their sum; and (b) bulk and geostrophic down-channel velocities.

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    Cross-channel variation of (a) Froude number, (b) Ekman number, and (c) drag coefficient as defined in (15) and (17) for t = 2Tf . The gray lines in panels (a) and (b) indicate the theoretical estimates for the Froude number Fr = (Sx/Cd)1/2, and that for the Ekman number given in (18), respectively. Both are derived using a constant Cd = 2 × 10−3.

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    (a) Initial tracer distribution and (b) tracer distribution for t = 2Tf . Black lines denote isopycnals as in Fig. 4.

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    (a) Isopycnals (black lines) for t = 2Tf as in Fig. 2. Gray-shaded areas indicate the subvolumes VR and VL, separated by the position y = yυ. (b) Lateral transport across y = yυ, and rate of change of the subvolumes; (c) averaged entrainment velocity for subvolumes VR (circles) and VL (squares).

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    Initial distribution of (a) ambient fluid and (b) interior fluid. The arrow in (a) schematically indicates the sign convention for positive transport Qi of interfacial fluid across the position y = yυ. (c) Time series of the relative contributions of ambient, interior, and interface fluid to the transport Qi.

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    Profiles of cross-channel velocity (full line), geostrophic velocity defined in (3) (dashed line), and frictional velocity defined in (21) (dot–dashed line). The thin dashed lines denote the upper and lower edges, z = zm and z = zt, of the interface, and the thin full line corresponds to the bulk interface position, zH = zb + H. All quantities are shown for t = 2Tf in the channel center.

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    (a) The cross-channel variation of the total and geostrophically balanced (gray) interfacial transport for t = 2Tf . (b)–(d) Vertical averages across the interface of (b) inverse Richardson number with the values corresponding to Ri = 0.25 marked by dashed line, (c) dissipation rate, and (d) mixing efficiency.

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    (a) Mixing efficiency, (b) dissipation rate, and (c) ratio of down-channel to total shear production of turbulence. All quantities are plotted for t = 2Tf for profile 1 (see Fig. 4a). (d)–(e) As in (a)–(c), but for profile 2 on the left slope of the channel. Thick dashed lines in (b) and (e) correspond to the law-of-the-wall relations, and the thin horizontal lines in all plots denote the interface geometry as explained in Fig. 11.

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    (a)–(c) Buoyancy and down- and cross-channel velocities for t = 2Tf for profile 1 (see Fig. 4a). The thick dashed line in (b) indicates the geostrophic down-channel velocity. (d)–(g) The vertical buoyancy flux, and the fluxes of down- and cross-channel momentum, respectively. Thin horizontal lines in all plots denote the interface geometry as explained in Fig. 11.

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    As in Fig. 14 but for profile 2 on the left slope of the channel (see Fig. 4a).

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Entrainment in Shallow Rotating Gravity Currents: A Modeling Study

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  • 1 Leibniz-Institute for Baltic Sea Research, Warnemünde, Germany
  • | 2 Department of Earth Sciences, University of Gothenburg, Göteborg, Sweden
  • | 3 Leibniz-Institute for Baltic Sea Research, Warnemünde, Germany
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Abstract

The physics of shallow gravity currents passing through a rotating channel at subcritical Froude number is investigated here with a series of idealized numerical experiments. It is found that the combined effects of friction and rotation set up a complex transverse circulation that has some crucial implications for the entrainment process. A key component of this secondary circulation is a geostrophically balanced transverse jet in the interface that laterally drains fluid from the interface. This effect is shown to result in a strong cross-channel asymmetry and a spatial separation of the entrainment process: drained interfacial fluid is partly replaced by entrained ambient fluid on the deep side of the gravity current, whereas the downward mixing of buoyant fluid occurs on the shallow side. These results, closely corresponding to recent measurements in a shallow, channelized gravity current in the western Baltic Sea, illustrate that the description of entrainment as a strictly vertical mixing process with the help of local bulk parameters like the Froude number is not generally applicable in rotating gravity currents.

Corresponding author address: Lars Umlauf, Leibniz-Institute for Baltic Sea Research, Seestrasse 15, 18119 Warnemünde, Germany. Email: lars.umlauf@io-warnemuende.de

Abstract

The physics of shallow gravity currents passing through a rotating channel at subcritical Froude number is investigated here with a series of idealized numerical experiments. It is found that the combined effects of friction and rotation set up a complex transverse circulation that has some crucial implications for the entrainment process. A key component of this secondary circulation is a geostrophically balanced transverse jet in the interface that laterally drains fluid from the interface. This effect is shown to result in a strong cross-channel asymmetry and a spatial separation of the entrainment process: drained interfacial fluid is partly replaced by entrained ambient fluid on the deep side of the gravity current, whereas the downward mixing of buoyant fluid occurs on the shallow side. These results, closely corresponding to recent measurements in a shallow, channelized gravity current in the western Baltic Sea, illustrate that the description of entrainment as a strictly vertical mixing process with the help of local bulk parameters like the Froude number is not generally applicable in rotating gravity currents.

Corresponding author address: Lars Umlauf, Leibniz-Institute for Baltic Sea Research, Seestrasse 15, 18119 Warnemünde, Germany. Email: lars.umlauf@io-warnemuende.de

1. Introduction

Submarine channels, canyons, and topographic corrugations have long been recognized for their potential importance in modifying the pathway, the dynamics, and the entrainment of dense bottom currents. Submarine channels, for example, have been identified as hot spots of gravity current entrainment at various locations (Peters et al. 2005; Mauritzen et al. 2005; Baringer and Price 1997), and canyons and other small-scale corrugations are believed to play a key role for the effective downward transport of dense shelf waters toward the deep ocean (Wåhlin 2004; Chapman and Gawarkiewicz 1995).

An interesting feature often observed in such rotating, channelized gravity currents is a strong asymmetry of the cross-channel density structure with a tilted, wedge-shaped interface, and a “downward-bending” of near-bottom isopycnals on the shallow, downwelling-favorable side (Umlauf and Arneborg 2009a; Mauritzen et al. 2005; Petrén and Walin 1976). While there is general agreement that the cross-channel interface tilt is a manifestation of a geostrophic balance, the physical explanation for the lateral distortion of the density field has proven to be less evident. Velocity measurements in oceanic gravity currents of rather different scales (Fer et al. 2010; Sherwin 2010; Arneborg et al. 2007; Johnson and Sanford 1992), numerical simulations (Burchard et al. 2009; Arneborg et al. 2007; Ezer 2006), and laboratory experiments on rotating tables (Wåhlin et al. 2008; Davies et al. 2006; Ohlson 1994) have consistently demonstrated that the observed transverse (secondary) circulation plays a key role here, but the mechanisms controlling it are presently not well understood. Potential explanations include rotating hydraulic theory (e.g., Hogg 1983), and a number of suggestions emphasizing the importance of the bottom and interfacial Ekman transports (Sherwin 2010; Ezer 2006; Jungclaus and Vanicek 1999; Johnson and Sanford 1992).

The effects of friction and rotation on the dynamics of bottom gravity currents have also been the focus of an extensive field program conducted in the western Baltic Sea (Umlauf and Arneborg 2009a; Arneborg et al. 2007; Umlauf et al. 2007) that has resulted in a new theoretical model for shallow rotating gravity currents (Umlauf and Arneborg 2009b). Here, our major goals are 1) to investigate the performance of a second-moment turbulence model for the description of gravity current entrainment by comparing model results with the detailed turbulence dataset obtained by Umlauf and Arneborg (2009a), and 2) to evaluate the theoretical framework suggested by Umlauf and Arneborg (2009b) with a series of idealized numerical experiments, providing additional parameters that cannot be measured and avoiding the complications of three-dimensionality, nonstationary, exterior flow, etc., that may have affected the field observations.

2. Geometry and model formulation

We investigate a bottom gravity current passing through an infinitely long and deep channel with a constant down-channel tilt, Sx ≪ 1, rotating at a constant rate, f/2, around the vertical axis (Fig. 1a). The down- and cross-channel coordinates are denoted by x and y, respectively, such that the z axis exhibits a small tilt, Sx, with respect to the vertical. The gravity current is driven by its negative buoyancy relative to the ambient fluid,
i1520-0485-40-8-1819-e1
where g denotes the acceleration of gravity, ρ the density, ρ0 a constant reference density, and ρ the density of the ambient fluid at z → ∞. Changes in the down-channel direction are ignored (∂/∂x ≡ 0) such that the problem becomes two-dimensional in the yz plane (Fig. 1b).

a. Two-dimensional description

Assuming a flow with small aspect ratio, the evolution of the down- and cross-channel velocities, u and υ, and the buoyancy, b, is described by the hydrostatic Boussinesq equations,
i1520-0485-40-8-1819-e2
where
i1520-0485-40-8-1819-e3
are the geostrophic velocities, τx and τy the turbulent fluxes of momentum, G the turbulent buoyancy flux, and w the z-component of the velocity vector computed from the continuity equation,
i1520-0485-40-8-1819-e4
For this idealized study, we assume (u, υ, b) → 0 for z → ∞ above the interface of the gravity current.
The vertical turbulent fluxes of momentum, τx and τy, and the buoyancy flux G appearing in (2) are computed from
i1520-0485-40-8-1819-e5
where the turbulent diffusivities, νt and νtb, follow from the “quasi-equilibrium” version (Galperin et al. 1988) of the second-moment turbulence closure model suggested by Canuto et al. (2001). In contrast to the latter authors, however, we compute the turbulent kinetic energy, k, and the dissipation rate, ε, from two prognostic transport equations; that is, we do not assume that the budget for k is in equilibrium. A complete description of the turbulence model and parameters is provided in Umlauf and Burchard (2005). Note that the most critical parameter for the correct prediction of entrainment rates with this class of second-moment models is the so-called stationary Richardson number Rist, which largely determines the model performance in stratified shear layers (Umlauf 2009; Ilicak et al. 2008; Umlauf and Burchard 2003; Burchard and Baumert 1995). Based on earlier work, and on available data for such flows, we chose Rist = 0.25 here (see Shih et al. 2000, 2005, and the references therein). For simplicity, all subgrid-scale fluxes in the cross-channel direction are ignored.

b. Vertically integrated description

As a useful tool for the following analysis we define the bulk buoyancy B and the bulk plume thickness H according to
i1520-0485-40-8-1819-e6
where z = zb at the bottom. From (6), it can be shown that z = zb + H/2 corresponds to the position of the center of mass, which relates any change in H to a corresponding change in local potential energy. Arneborg et al. (2007) have shown that the definitions given in (6), and the evolution equation for the buoyancy in (2), can be combined into an equation describing the rate of change of H due to irreversible mixing:
i1520-0485-40-8-1819-e7
which is a positive quantity since both, B, and the buoyancy flux G are negative in stably stratified flows. Other, reversible, processes like internal waves or advective effects are superimposed to this irreversible change of H.
Using the definition for H implied by (6), we define the bulk plume velocities, U and V, as
i1520-0485-40-8-1819-e8
such that the vertical integration of (2) yields
i1520-0485-40-8-1819-e9
where
i1520-0485-40-8-1819-e10
are the geostrophically balanced parts of the down-channel and cross-channel transports. For convenience, we have also introduced the bottom friction velocities,
i1520-0485-40-8-1819-e11
where τbx and τby denote the values of τx and τy at the bottom. Note that the entrainment stress does not explicitly appear in the vertically integrated momentum budget in (9) as discussed in more detail by Arneborg et al. (2007).
Vertical integration of (4) yields
i1520-0485-40-8-1819-e12
indicating that if there is any lateral transport divergence inside the gravity current; for example, because of geostrophic adjustment or asymmetric entrainment, a nonvanishing vertical velocity, w, will be observed above the interface.

c. Definition of entrainment

In gravity current studies, the concept of entrainment is frequently used to describe the turbulent transport of ambient fluid across the upper edge of the flow, separating the typically strongly turbulent gravity current from the weakly turbulent interior. The relative speed of this edge with respect to the ambient fluid is the entrainment velocity, which we define here as
i1520-0485-40-8-1819-e13
where zt denotes the position of the upper edge of the gravity current, and w the speed of the ambient fluid as defined in (12). Horizontal integration of (13) thus yields the net flux of ambient fluid entering the gravity current because of the entrainment process.

Turner (1986) carefully reviewed different entrainment concepts, and pointed out that the key assumption in the classical description of entrainment states that we scales with the characteristic speed of the largest eddies in the flow, and thus with a typical flow speed. Based on earlier work by Ellison and Turner (1959), Turner (1986) also concluded that in stratified flows this concept can be extended to include the damping effect of stratification on turbulence and entrainment. However, an additional complication arises from the fact that that entrained ambient fluid may not be distributed immediately across the entraining flow (as implicitly assumed in the classical view of entrainment) if turbulence is not energetic enough to overcome the stratification: entrainment remains “incomplete” with a substantial fraction of entrained fluid remaining in the interface region for long times as observed, for example, by Peters et al. (2005) and Peters and Johns (2005).

Complementary to (13), Arneborg et al. (2007) have therefore introduced a velocity associated with the change of plume thickness due to mixing,
i1520-0485-40-8-1819-e14
which, according to (7), is proportional to the irreversible rate of change of potential energy due to mixing. Only for gravity currents with perfect two-layer structure (13) and (14) can be shown to coincide.

Different aspects of the entrainment process are highlighted by the entrainment velocities defined in (13) and (14), respectively. According to the first definition, (13), fluid is considered as “entrained” immediately after it has crossed to upper edge of the gravity current. This definition ignores the fact that, from an energetic point of view, there is a large difference between entrainment where fluid is fully mixed down inside the gravity current, and “incomplete” entrainment where entrained fluid remains in the interface region. In contrast to (13), the second definition of entrainment in (14) takes these energetic differences into account; however, the entrainment velocity wb is not necessarily directly associated with the entrainment of ambient fluid.

Thus, in short, we describes where buoyant fluid from the ambient enters the gravity current, whereas wb indicates where it is mixed down. Recent applications of these concepts are described in Fer et al. (2010) and Umlauf and Arneborg (2009a,b).

d. Model setup and numerical implementation

Our goal in this study is to construct a model setup that is idealized enough to serve as a simple abstraction of the problem but, on the other hand, is comparable to the situation and parameter space encountered by Umlauf and Arneborg (2009a,b) in order to have a sound dataset available for comparison and validation.

With this idea in mind, we numerically solve (2) and (4) for a parabolic cross-channel topography with a width of 20 km at z = 25 m (z = 0 m corresponds to the deepest point in the center of the channel). For this purpose, we use the General Estuarine Transport Model (GETM), described in detail in Burchard and Bolding (2002), with a few adaptations. Specifically, the implementation of an “infinitely deep” ocean in a numerical model with a finite computational domain requires some comments here. Practically, our domain ends at z = 30 m, which is well above the gravity current during all our simulations. The standard kinematic boundary condition associated with the presence of the free surface has been replaced by 1) ignoring the barotropic pressure gradient, and 2) imposing a fixed but permeable boundary at z = z. The permeability condition at z = z is implemented as zero-gradient boundary conditions for the prognostic variables (u, υ, b), which, in the absence of a barotropic pressure gradient, insures that these variables remain at their initial values above the gravity current, u = υ = b = 0, for all times. Note, however, that in general w ≠ 0 above the gravity current, which, according to (12), is required in order to balance any lateral flow divergence inside the gravity current by a volume flux across the permeable upper boundary. It is easy to show that these boundary conditions are consistent with the equations of motion for a hydrostatic Boussinesq fluid.

The initial stratification consists of a layer with constant buoyancy (b = −0.1 m s−2), ranging from the bottom up to z = 9 m, topped by a density interface of 2 m thickness with linearly varying buoyancy, and zero buoyancy above. The width of the gravity current at the height of the interface is thus 12.5 km. These parameters approximately correspond to those encountered during the field study in the western Baltic Sea described in Umlauf and Arneborg (2009a) and Umlauf et al. (2007). The value chosen for the bottom roughness is z0 = 5 × 10−4 m, close to the value estimated by Umlauf and Arneborg (2009a) from their direct bottom stress measurements. The down-channel interface slope analyzed by Umlauf and Arneborg (2009b) showed some variability across the channel, but was, on the average, close to Sx = 5 × 10−4, which is the value we use in this study. Consistent with the latitude of the site investigated by Umlauf and Arneborg (2009a), we set the Coriolis parameter to f = 1.19 × 10−4 s−1, corresponding to an inertial period of Tf = 14.7 h. All model simulations were initialized with zero velocities.

As an extension of the numerical implementation described in Burchard and Bolding (2002), here we employ a vertically adaptive numerical grid recently introduced by Hofmeister et al. (2010). With this new technique, the vertical grid spacing dynamically adapts to regions with strong shear and stratification, resulting in a vertical resolution of approximately 0.05 m in the interface region. Given the complex velocity structure in the interface, and its small vertical extent, we found this approach necessary to reduce numerical errors to insignificant levels. Our numerical grid consists of 150 adaptive layers in the vertical with a horizontal resolution of 50 m. The down-channel slope, Sx, linearly increases from zero to the value mentioned above during the first hour of each run, allowing the vertical grid to adapt to the local stratification and shear parameters. After this initial adjustment, Sx remains constant.

3. Temporal evolution

a. Evolution of density and velocity structure

The temporal evolution of the cross-channel density structure is shown in Fig. 2. Already after half an inertial period, the interface exhibits the characteristic cross-channel tilt associated with the geostrophic adjustment. The down-channel speed in the center reaches 0.6 m s−1 at the lower edge of the interface; that is, at the vertical location where nearly the full down-channel pressure gradient has built up but frictional effects are still weak because of the suppression of mixing by stable stratification. The most striking feature observed in the cross-channel velocity is a transverse jet inside the interface with a maximum speed of approximately 0.2 m s−1, which is only partly compensated by a small return current in the interior of the gravity current.

At subsequent times (Fig. 2) the density structure inside the gravity current undergoes some important transformations. Most remarkable is the appearance of a region with a lateral buoyancy gradient in the interior of the gravity current on the left slope, evolving behind a front that slowly propagates toward the center of the channel. The propagation of this front is associated with a jumplike feature at the upper edge of the interface that leads to a sudden change of the interfacial stratification. Outside this region, on the right-hand side of the domain, no lateral stratification in the interior is observed. While the interface tilt does not significantly change for t > Tf/2, the interface geometry evolves into a wedge-shaped structure with a minimum thickness of less than 0.5 m on the deep side of the gravity current at t = 2Tf . Figure 2 illustrates that the remarkable stationarity of this pattern on the right-hand side of the domain is contrasted by the steadily evolving density field on the left slope.

It is worth pointing out that, in spite of the idealized nature of this experiment, all features reported above closely correspond, both qualitatively and quantitatively, to the observations reported by Umlauf and Arneborg (2009a, see e.g., their Fig. 8).

b. Bulk dynamics

First insight into the dynamics of the gravity current comes from an investigation of the dominant terms in the vertically integrated down- and cross-channel momentum budget in (9), evaluated here in the center of the channel. As discussed in more detail below, this location is representative for the bulk dynamics at other locations of the channel as well.

Figure 3 illustrates that after a short adjustment time of approximately Tf/2 the down-channel momentum budget is dominated by a balance between bottom friction, , and pressure gradient, SxBH, indicating that the effects of advection and entrainment drag are dynamically negligible. Using 0.4u*bx/f suggested by Umlauf and Arneborg (2009a) as a rough estimate for the Ekman layer thickness, we find that for the typical value u*bx ≈ 0.02 m s−1 for the friction velocity in the channel center (see Fig. 6a below), the Ekman layer exceeds the total plume thickness by almost one order of magnitude. This “shallowness” of the gravity current with respect to the Ekman layer thickness was also pointed out by Umlauf and Arneborg (2009a,b) based on their analysis of field data.

Figure 3 also shows that the ratio of U and Ug quickly approaches unity after an initial, strongly damped inertial response. Concluding, the cross-channel momentum budget is governed by a geostrophic balance, whereas in the down-channel direction the internal pressure gradient is balanced by bottom friction. This simple form of the integrated momentum budget agrees with the analysis of recent observations discussed in Umlauf and Arneborg (2009b), and is consistent with the key assumptions in the vertically integrated model of Wåhlin (2002, 2004) for frictionally controlled gravity currents.

4. Cross-channel structure

Based on the observation that at t = 2Tf the bulk dynamical balance has reached a steady state, and the fact that the density and velocity structure at that time strikingly resembles the patterns observed by Umlauf and Arneborg (2009a), we will now focus more closely on the situation during this stage of the evolution.

a. Velocity structure

The cross-channel structure of the velocity field at t = 2Tf is displayed in Fig. 4. The down-channel velocity confirms the vertical structure observed in Fig. 2 also for other locations across the gravity current, with a general tendency for a reduction of the speed near the edges where frictional effects become more dominant. Except in the region with the lateral interior buoyancy gradient, the vertical structure of the cross-channel velocity displayed in Fig. 4b is everywhere similar to that shown in Fig. 2 for the center of the channel: A strong transverse jet with speeds between 0.1 and 0.2 m s−1 is confined to the wedge-shaped interface, and a weak return current is observed in the bulk. On the left slope, in the region with lateral interior stratification, the situation is rather different. The two-layer transverse circulation is replaced here by a three-layer pattern with a broader but weaker jet in the interface, a strong return current just below the interface, and a third layer with weak upslope currents in the near-bottom region. This structure, in particular the flow reversal in the near-bottom layer, is in close agreement with the field data presented in Fig. 8 of Umlauf and Arneborg (2009a), and indicates an interesting dynamical behavior that will be explored in greater detail below.

b. Bulk description

For the following discussion it is helpful to define a few geometrical parameters describing the vertical structure of the gravity current. Figure 4a illustrates that the location of the interface is well described by zH = zb + H, where the bulk plume thickness, H, is computed from (6). The top of the plume, defined here as the position where the local buoyancy corresponds to 1% of the bottom buoyancy, is denoted by zt. This definition is somewhat arbitrary but, given the strong stratification at the top of the interface, not overly sensitive to variations of this threshold. The lower edge of the interface, zm, is defined here as the position of the velocity maximum of the down-channel velocity, or, likewise, as the position of vanishing down-channel stress [see (5)]. This definition corresponds to a suggestion by Peters et al. (2005), and has advantages for the dynamical separation of the plume’s interior from the interface as discussed in greater detail below. The thickness of the interior region is thus given by Hb = zmzb, and the thickness of the interface is Hi = ztzm. Note that in general HHb + Hi.

The cross-channel variation of zm, zH, and zt for t = 2Tf is displayed in Fig. 5a, illustrating the wedge-shaped interface and the increased interface tilt in the region with the interior lateral buoyancy gradient (Fig. 5b), as theoretically predicted by Umlauf and Arneborg (2009b). Figure 5c illustrates the variation of the dominant down-channel stress with maximum values around 0.5 Pa in the center of the channel that are in good agreement with the microstructure-derived stress estimates by Umlauf and Arneborg (2009a). The cross-channel stress (Fig. 5d) is more than one order of magnitude smaller, and changes its sign consistent with the reversal in the near-bottom cross-channel flow (Fig. 4b). The reversal occurs exactly at the front separating the regions with and without lateral buoyancy gradient (Fig. 5b), and is thus analogous to a similar reversal of the near-bottom velocity observed and explained theoretically by Umlauf and Arneborg (2009a,b).

Support for the model performance in stratified regions of the gravity current comes from a comparison of the energy-based entrainment velocity wb, defined in (14) with the corresponding values estimated from microstructure measurements (see Fig. 5e and Figs. 9 and 10 in Umlauf and Arneborg 2009a). The model reproduces the strong asymmetry of mixing with increased mixing rates in the region affected by the lateral interior buoyancy gradient, and, moreover, it predicts values for wb that are within a factor of 2 of the measurements. Given the complexity of the mixing phenomena occurring in this flow this is rather remarkable.

The dominant forces acting on the gravity current are displayed in Fig. 6, illustrating an almost perfect balance between bottom friction and the down-channel pressure gradient in the down-channel direction, and a (geostrophic) balance between the cross-channel pressure gradient and the Coriolis acceleration in the cross-channel direction. This extends our conclusions from Fig. 3 to the whole width of the gravity current.

c. Nondimensional parameters

Based on the data shown in Figs. 5 and 6, a number of nondimensional key parameters can be computed and compared to the parameters derived by Umlauf and Arneborg (2009b). These authors suggested that the relative effects of interfacial shear, stratification, bottom friction, and rotation are described by the Froude and the Ekman numbers,
i1520-0485-40-8-1819-e15
where
i1520-0485-40-8-1819-e16
denotes the total down-channel friction velocity composed of parts related to bottom friction, defined in (11), and the “entrainment drag” associated with the acceleration of entrained fluid as discussed in more detail below. Note that gravity currents in which the characteristic wave speed, in (15) approximated by (−BH)1/2, exceeds the flow speed U are typically referred to as subcritical (Fr < 1).
For bottom friction, Umlauf and Arneborg (2009b) have shown good agreement with a quadratic drag law based on the bulk speed,
i1520-0485-40-8-1819-e17
where the measured bulk drag coefficient was in the range Cd ≈ (1 − 2) × 10−3.

The cross-channel variation of these nondimensional parameters is shown in Fig. 7. Apart from the region above the left slope, the Froude number (Fig. 7a) is close to Fr = 0.5; that is, clearly subcritical and well inside the range of values observed by Umlauf and Arneborg (2009b). With Cd = 2 × 10−3 (Fig. 7c), this value coincides with the theoretical estimate, Fr = (Sx/Cd)1/2, derived from inserting (17) into the first of (9), and assuming a balance between the pressure gradient and bottom friction as suggested by Fig. 6a. A reduction of the drag coefficient by a factor of 2 is observed on the left slope, which is still inside the range of values measured by Umlauf and Arneborg (2009b) but may indicate a possible influence of the lateral buoyancy gradient on the turbulent momentum transport (see below).

The Ekman number (Fig. 7b) increases from Ek ≈ 0.8 in the center of the channel to Ek > 2 on the right edge of the gravity current, clearly indicating the importance of frictional effects. With the same assumptions used above to derive a theoretical estimate for the Froude number it is easy to show that
i1520-0485-40-8-1819-e18
Even with the assumption of a constant Cd = 2 × 10−3, (18) reproduces the observed cross-channel variability, and explains the increase of Ek on the right slope from the decrease of H alone, since all other parameters in (18) are exactly or approximately constant. Consistent with the drag reduction on the left slope mentioned above, Fr and Ek show a small increase and decrease, respectively, with respect to the theoretical estimates also shown in Figs. 7a,b.

5. The effect of rotation on entrainment

The effect of the secondary circulation on the entrainment process can conveniently be studied with an asymmetric experiment, in which ambient and interfacial fluid on the right-hand side of the domain are marked with a passive tracer of concentration 1 (see Fig. 8a). Since lateral diffusion is ignored in this idealized study, we expect that in a nonrotating gravity current, interfacial and ambient fluid will be vertically mixed into the gravity current’s interior with no associated lateral transport. This picture of strictly vertical entrainment is strongly contrasted by the tracer distribution for the rotating case as illustrated in Fig. 8b for t = 2Tf . Instead of being locally entrained into the interior, most of the interfacial and entrained ambient fluid is laterally advected with the interfacial jet (Fig. 4b), and ultimately mixed into the interior on the opposite side of the channel.

We further investigate this remarkable asymmetry of the entrainment process by laterally dividing the gravity current into two subvolumes, VR and VL, located to the right and left of the position y = yυ (see Fig. 9a). Choosing yυ = 7.5 km allows us to distinguish between entrainment processes in regions with lateral buoyancy gradient in the interior (VL) and those without (VR). Integrating the continuity Eq. (4) across VR and VL yields
i1520-0485-40-8-1819-e19
where QeR and QeL denote the fluxes of entrained ambient fluid across z = zt.

Figure 9b illustrates that during the initial geostrophic adjustment process the changes of both VR and VL are dominated by the lateral transport, VH, across the inner boundary of the subvolumes at y = yυ. For times larger than t = Tf , however, VR is seen to become stationary, implying, according to (19), that the lateral transport −VH is balanced by the flux QeR of entrained ambient fluid. In contrast to the stationarity of VR, the volume VL shows a continuous increase that is to a large extent balanced by lateral transport, indicating that entrainment forms only a small contribution (Fig. 9b). Thus, there is a substantially larger entrainment of ambient fluid in VR.

Denoting the lateral widths of VR and VL as ΔyR and ΔyL, respectively, averaged entrainment velocities, weR = QeRyR and weL = QeLyL, can be defined for both regions. These quantities, corresponding to the lateral averages of the entrainment velocity we defined in (13), reach stable values for t > Tf , and confirm the dominance of entrainment in VR (Fig. 9c). Comparing these values for t = 2Tf with Fig. 5e, showing the entrainment velocity wb defined in (14), provides an interesting energetic interpretation of the two-stage mixing process suggested in Fig. 8. On the right-hand side of the domain, we find weRwb, which is a consequence of the fact that partial entrainment of ambient fluid into the interface region requires substantially less potential energy than complete mixing across the whole thickness of the gravity current. Note, however, that weR is comparable to the values of wb observed on the left-hand side of the domain (Fig. 5e), thus illustrating where the fluid entrained in VR is ultimately mixed down into the bulk of the gravity current.

Finally, in order to study the composition of fluid advected with the interfacial jet from VR toward VL, two further experiments were conducted in which ambient and interior fluid, but not the fluid in the interface, were marked at the beginning of each experiment (Figs. 10a,b). With this setup, it is straightforward to trace the origin of interfacial fluid advected with the jet across y = yυ. The result shown in Fig. 10c illustrates that for t < Tf mainly fluid that was initially located inside the interface is transported with the jet. This is consistent with the idea of lateral “draining” of interfacial fluid, leading to the wedge-shaped interface illustrated in Fig. 2. For t > Tf , the shape and hence the volume of the interface on the right hand side of the channel becomes stationary (Fig. 2), implying that the transport of interfacial fluid with the jet is balanced by entrainment. Figure 10c corroborates this conclusion, and illustrates that the amounts of fluid entrained from the ambient and from the interior are comparable.

The results from this section clearly illustrate that the wide-spread idea of “entrainment” as a vertical mixing process described by local bulk parameters like the Froude number defined in (15) is not generally applicable for rotating gravity currents. This forms one of the major conclusions of this paper.

6. Transverse dynamics

a. Bulk and interface dynamics

A convenient way to analyze the dynamical reasons for the observed vertical structure of the cross-channel velocity is to rewrite the x component of (2) in the form
i1520-0485-40-8-1819-e20
where υ is split into a geostrophic part υg defined in (3), a frictional part,
i1520-0485-40-8-1819-e21
and a part related to the material derivative, Du/Dt, that is, to the acceleration of a material particle inside the gravity current. Note that the vertical integral of (21) corresponds to the bottom Ekman transport, implying that the net interfacial Ekman transport resulting from the entrainment stress is associated with the last term in (20). We will come back to this rather general argument below.

Vertical profiles of υ, υg, and υf evaluated at t = 2Tf in the center of the channel are displayed in Fig. 11. Below the interface, υg and υf are seen to balance almost perfectly, which is consistent with the theoretical model of Wåhlin (2002, 2004), who pointed out that compensating cross-channel geostrophic and Ekman transports are the necessary condition for downward channeling. In the lower part of the interface, however, turbulent friction is increasingly suppressed because of stable stratification, and υf rapidly decays (Fig. 11). Near the center of the transverse jet, we find υf ≈ 0 and thus υυg, indicating that the dominant balance in this region is geostrophic. The last term in (11) becomes comparable to the others only in the thin entrainment layer near the upper edge of the interface, where it represents the down-channel acceleration, Du/Dt > 0, of a material particle that is entrained from the ambient fluid into the gravity current.

To investigate this dynamical aspect of entrainment in more detail, we integrate (20) across the interface to obtain an expression for the lateral transport inside the interface:
i1520-0485-40-8-1819-e22
where the contribution of υf vanishes since, by definition, τx = 0 at the lower edge of the interface. This indicates that any net Ekman transport in the interface must be associated with the acceleration term in (22). Since Du/Dt > 0 in the entrainment layer, this net Ekman transport may partly compensate the negative geostrophic transport in the interface, depending on the relative importance of entrainment. For the gravity currents studied here, this effect only results in relatively small reduction of the geostrophic transport that dominates the interfacial jet (Fig. 12a), which is consistent with similar conclusions reached by Arneborg et al. (2007) and Umlauf and Arneborg (2009b). This may, however, be different for gravity currents with strong entrainment.

b. Interface mixing

From the previous discussion, we expect that the interfacial jet impacts in at least two ways on mixing and entrainment in the interface region: 1) by sharpening the interface through lateral draining of interfacial fluid, and 2) by providing additional shear in the upper and lower flanks of the jet. Both effects have a tendency to lower the gradient Richardson number, thus promoting shear instabilities and turbulence in the interface.

Figure 12b illustrates that on the right-hand side of the domain, the average Richardson number in the interface remains close to Ri = 0.25, the value predicted by the model for stationary stratified shear layers (Umlauf 2009; Umlauf and Burchard 2005). The interface remains fully turbulent with modeled dissipation rates of the order of ε ≈ 10−6 W kg−1 (Fig. 12c), comparable to the values measured by Umlauf and Arneborg (2009a). The shear production of turbulence associated with the cross-channel jet (Fig. 13c,f) provides a substantial contribution to the total interfacial shear production, in particular in the entrainment layer and at the lower edge of the interface, where the shear from the down-channel velocity is zero by definition. This effect, that could not be detected in the field with the available instrumentation (Umlauf and Arneborg 2009a), is in accordance with the theoretical prediction of Umlauf and Arneborg (2009b) for gravity currents with Ek = O(1).

We therefore conclude that turbulence responds to the lateral interface draining (that sharpens the interface and lowers the Richardson number) by additional entrainment of both ambient and interior fluid into the interface (see Fig. 10c) in order to adjust the interface thickness exactly such that Ri remains near the stationary threshold. This is the key mechanism for entrainment on the deep (right hand) side of the gravity current.

Interfacial mixing on the left slope is more variable but large values of ε are generally correlated with small values of Ri (Figs. 12b,c). The interfacial mixing efficiency, γ = −G/ε, shown in Fig. 12d is close to the canonical value of γ = 0.2 (Osborn 1980; Shih et al. 2005), and in accordance with the theoretical value for stationary turbulence in stratified shear layers for this turbulence model (Burchard and Hetland 2010; Umlauf 2009). Similar to the observations by Arneborg et al. (2007) and Umlauf and Arneborg (2009a), we find a well-defined local maximum of ε in the interface in the center of the channel (Fig. 13b), whereas interfacial mixing on the left slope exhibits a strong vertical variability (Fig. 13e).

c. Interior mixing

It is evident from Figs. 4b and 15c that the cross-channel vertical shear on the left slope, acting on nearly vertical isopycnals in the interior of the gravity current, has a tendency to create stable stratification below the interface. This process is compensated by vertical mixing, which results in a striking difference between the small interior buoyancy flux in the center of the channel (Fig. 14d), and the large values observed on the slope (Fig. 15d). This downward mixing of buoyant fluid into the near-bottom layer also explains the puzzling propagation of the interior buoyancy front (Fig. 2) against the upslope near-bottom current on the left side of the channel (Fig. 4b). Crucial for this process is the fact that the creation of weak interior stratification strongly enhances the mixing efficiency throughout the interior region except for a thin, energetic bottom boundary layer (Fig. 13a), which is contrasted by the situation in the center of the channel (Fig. 13d).

Note that the generation of internal stratification on the slope described above also has a tendency to suppress the turbulent fluxes. However, since the bottom momentum flux (i.e., the bottom stress) is imposed by the down-channel pressure gradient (see Fig. 6a), a larger shear is required in order to generate this imposed momentum flux. This implies a relative increase of U and thus, via (17), a decrease of Cd. We believe that this, rather than the potential generation of a “slippery” Ekman layers discussed in the following section, results in the drag reduction on the left slope observed in Fig. 7c.

d. The role of thermal wind shear

The presence of the lateral buoyancy gradient observed in the interior of the gravity current on the left slope implies, according to (3), a reduction of the geostrophic down-channel velocity ug with depth. This effect is commonly referred to as “thermal wind” shear. In the channel center (Fig. 14b), appreciable thermal wind shear is only observed in the interface region, where it is caused by the cross-channel tilt of the interface. On the slope, however, the cross-channel buoyancy gradient results in thermal wind shear also below the interface, comparable to the total observed shear, and sufficient to reduce the velocity to zero near the bottom (Fig. 15b). In such a situation, the appearance of “slippery” boundary layers (CD = 0) is expected according to the theory of MacCready and Rhines (1993), who argued that no turbulent stress would then be required to satisfy the no-slip boundary condition at the bottom. This, however, is neither observed (Umlauf nor Arneborg 2009b) nor modeled (Fig. 7c) because the crucial down-channel pressure-gradient setting the bottom stress was ignored by MacCready and Rhines (1993) such that their theory does not directly apply here.

This is further corroborated by an investigation of the turbulent momentum fluxes in the center (Fig. 14e) and on the slope (Fig. 15e) of the channel. At both locations, almost perfectly linear profiles are observed (as expected for a simple pressure-driven flow) with no discernible impact of the thermal wind shear. Slightly smaller values in the center (Fig. 14e) are exclusively caused by the fact that there a larger pressure gradient has to be balanced by the bottom stress (Fig. 6a). It is worth noting that the small values of the turbulent momentum flux in the interface region (see inset in Fig. 14e) confirm that the entrainment drag is negligible compared to the bottom drag, consistent with the discussion of Fig. 11 above.

Although irrelevant for the down-channel dynamics, the thermal wind shear has some crucial implications for the secondary circulation, which, as shown above, largely controls the entrainment process. This can be understood from the observation that differences between the total down-channel velocity u and the geostrophically balanced part ug, visible in both Figs. 14b and 15b, imply that a cross-channel frictional force is required to close the cross-channel momentum budget (the rate terms are negligible below the interface). In the near-bottom region, this friction force is due to bottom friction, and thus closely related to the near-bottom flow in the cross-channel direction. According to this argument, the sign of the near-bottom velocity is given by the sign of uug. This is confirmed by the reversal of both the cross-channel bottom velocity (Fig. 15c) and the stress (Fig. 15f) on the slope, where uug > 0 (in contrast to the situation in the center, where uug < 0).

Umlauf and Arneborg (2009b) have derived a similar result from a mathematically more rigorous approach that is not repeated here for brevity. We want to point out, however, that the vertical variability of the cross-channel stress shown in Figs. 14f and 15f can be fully explained with this approach (which was not possible for the field data discussed by Umlauf and Arneborg (2009b) since the stress could not be measured). The overall conclusion is that the three-layer transverse circulation with a near-bottom flow reversal observed on the slope (see Fig. 4b) is a purely two-dimensional response of the system to the interior cross-channel buoyancy gradient. The implications of the shear-dispersion caused by this effect for the overall evolution of the gravity current have been discussed above.

7. Conclusions

In close agreement with recent turbulence measurements in channelized gravity currents in the western Baltic Sea (Umlauf and Arneborg 2009a; Arneborg et al. 2007), our numerical results indicate substantial interfacial mixing and entrainment even far below the critical Froude number. Entrainment in this low Froude number range is in contradiction with the classical experiments (Ellison and Turner 1959; Turner 1986) but consistent with more recent laboratory studies (e.g., Cenedese and Adduce 2008), and has considerable implications for predicting the final water properties of dense gravity currents, and thus the basin-scale stratification (Wåhlin and Cenedese 2006; Hughes and Griffiths 2006).

However, in contrast to the traditional view of entrainment as a strictly vertical mixing process that is governed by the local bulk Froude number, our simulations indicate that for the class of gravity currents investigated here entrainment is a complex, strongly nonlocal mechanism. Entrainment was demonstrated to occur in a two-stage, spatially separated process with “incomplete” entrainment of ambient fluid into the interface layer on the deep side of the gravity current, and downward mixing of buoyant fluid into the gravity current’s interior occurring on the opposite side of the channel. This nonlocal mechanism is largely different from entrainment in nonrotating flows, and clearly cannot be described by any combination of local nondimensional bulk parameters like those defined in (15). In spite of the complexity of the entrainment processes encountered here, a second-moment turbulence closure model with prognostic equations for k and ε showed a remarkable agreement with measured mixing parameters, corroborating the conclusion of Ilicak et al. (2008) and Arneborg et al. (2007) that this class of turbulence models provides a comparatively reliable tool for the prediction of entrainment rates in overflows, at least if sufficient vertical resolution can be afforded.

The gravity currents investigated in this study are characterized by 1) small entrainment rates, 2) small thickness compared to the Ekman layer thickness (i.e., large Ekman number), and 3) width comparable to the internal Rossby radius. Future work will focus on the question about how the conclusions derived here can be extended for gravity currents with a broader range of geometrical and physical parameters.

Acknowledgments

The authors are grateful for the support by the German Research Foundation (DFG) and the German Federal Ministry for the Environment, Nature Conservation and Nuclear Safety through the projects QuantAS-Nat and QuantAS-Off, respectively. Lars Arneborg was supported by the Swedish Research Council. Constructive and helpful comments from two anonymous reviewers are greatly appreciated. Karsten Bolding provided valuable support during the initial model setup.

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Fig. 1.
Fig. 1.

(a) Schematic view of a channelized bottom gravity current with the arrow denoting the dominant down-channel flow direction. (b) Coordinate system and channel geometry looking up-channel with the direction of the down-channel flow as indicated. Thin lines denote isopycnals of the tilted, wedge-shaped stratification in the interface.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 2.
Fig. 2.

(left) Cross-sectional view of the buoyancy at 0.01 m s−2 intervals, and (center) profiles of the down-channel, and (right) cross-channel velocity in the channel center. The rows correspond to different points of time as indicated (Tf is the local inertial period).

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 3.
Fig. 3.

Ratio of bottom stress and down-channel pressure-gradient (full line), and ratio of bulk and geostrophic down-channel velocities (dashed), both shown for the channel center.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 4.
Fig. 4.

(a) Down-channel and (b) cross-channel velocity for t = 2Tf . Buoyancy is plotted as black contour lines at 0.02 m s−2 intervals. The dashed blue line in (a) denotes the bulk interface position, zH = zb + H. The black arrows in (b) illustrate the transverse circulation, and profile 1 and profile 2 refer to locations discussed below. All quantities are shown for t = 2Tf .

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 5.
Fig. 5.

Cross-channel variation of (a) zm, zH, and zt (from bottom to top); (b) bulk buoyancy B and bottom buoyancy bb (dashed); (c) down-channel bottom stress τbx; (d) cross-channel bottom stress τby; and (e) entrainment velocity wb computed from (14). All quantities are shown for t = 2Tf .

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 6.
Fig. 6.

Cross-channel variation of (a) friction velocity squared, down-channel pressure gradient, and their sum; and (b) bulk and geostrophic down-channel velocities.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 7.
Fig. 7.

Cross-channel variation of (a) Froude number, (b) Ekman number, and (c) drag coefficient as defined in (15) and (17) for t = 2Tf . The gray lines in panels (a) and (b) indicate the theoretical estimates for the Froude number Fr = (Sx/Cd)1/2, and that for the Ekman number given in (18), respectively. Both are derived using a constant Cd = 2 × 10−3.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 8.
Fig. 8.

(a) Initial tracer distribution and (b) tracer distribution for t = 2Tf . Black lines denote isopycnals as in Fig. 4.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 9.
Fig. 9.

(a) Isopycnals (black lines) for t = 2Tf as in Fig. 2. Gray-shaded areas indicate the subvolumes VR and VL, separated by the position y = yυ. (b) Lateral transport across y = yυ, and rate of change of the subvolumes; (c) averaged entrainment velocity for subvolumes VR (circles) and VL (squares).

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 10.
Fig. 10.

Initial distribution of (a) ambient fluid and (b) interior fluid. The arrow in (a) schematically indicates the sign convention for positive transport Qi of interfacial fluid across the position y = yυ. (c) Time series of the relative contributions of ambient, interior, and interface fluid to the transport Qi.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 11.
Fig. 11.

Profiles of cross-channel velocity (full line), geostrophic velocity defined in (3) (dashed line), and frictional velocity defined in (21) (dot–dashed line). The thin dashed lines denote the upper and lower edges, z = zm and z = zt, of the interface, and the thin full line corresponds to the bulk interface position, zH = zb + H. All quantities are shown for t = 2Tf in the channel center.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 12.
Fig. 12.

(a) The cross-channel variation of the total and geostrophically balanced (gray) interfacial transport for t = 2Tf . (b)–(d) Vertical averages across the interface of (b) inverse Richardson number with the values corresponding to Ri = 0.25 marked by dashed line, (c) dissipation rate, and (d) mixing efficiency.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 13.
Fig. 13.

(a) Mixing efficiency, (b) dissipation rate, and (c) ratio of down-channel to total shear production of turbulence. All quantities are plotted for t = 2Tf for profile 1 (see Fig. 4a). (d)–(e) As in (a)–(c), but for profile 2 on the left slope of the channel. Thick dashed lines in (b) and (e) correspond to the law-of-the-wall relations, and the thin horizontal lines in all plots denote the interface geometry as explained in Fig. 11.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 14.
Fig. 14.

(a)–(c) Buoyancy and down- and cross-channel velocities for t = 2Tf for profile 1 (see Fig. 4a). The thick dashed line in (b) indicates the geostrophic down-channel velocity. (d)–(g) The vertical buoyancy flux, and the fluxes of down- and cross-channel momentum, respectively. Thin horizontal lines in all plots denote the interface geometry as explained in Fig. 11.

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

Fig. 15.
Fig. 15.

As in Fig. 14 but for profile 2 on the left slope of the channel (see Fig. 4a).

Citation: Journal of Physical Oceanography 40, 8; 10.1175/2010JPO4367.1

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