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  • View in gallery

    Example of model output at t = 7.5 tidal cycles. (top left) Across-slope component υ, (top right) along slope component u, (bottom left) vertical velocity w, and (bottom right) dissipation rate ε. Black lines depict isotherms at equal intervals; uforcing = 0.1 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1.

  • View in gallery

    Vertical profiles of (left) cross-slope velocity υ, (middle) temperature fluctuation ΔT, and (right) dissipation rate ε at a fixed location y = −37 900 m, spanning one tidal cycle starting at t = 6.5; uforcing = 0.2 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1.

  • View in gallery

    Evolution of cross-slope velocity υ during one tidal cycle starting at t = 6.5 tidal cycles. Red lines depict isotherms at equal intervals; uforcing = 0.15 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1.

  • View in gallery

    As in Fig. 3, but for along-slope velocity u.

  • View in gallery

    As in Fig. 3, but for dissipation rate field ε.

  • View in gallery

    (a) Average vertical profile of across-slope velocity magnitude savg in the midslope region, where d is the height above the sloping bottom. Dashed lines indicate the determination of the boundary layer height Hu (uforcing = 0.15 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1). (b),(c) Boundary layer height Hu for υbt = 0.1 m s−1 (blue circles) and υbt = 0.2 m s−1 (red triangles) as function of stratification N and rotation f, respectively. (d) Boundary layer height Hu as function of barotropic velocity υbt for given Coriolis parameters f. Dashed line in (b) depicts power-law exponent −1.

  • View in gallery

    (a) Average normalized profile of dissipation in the midslope region, where d is the height above the sloping bottom. Dashed line indicates the determination of the boundary layer height Hε (uforcing = 0.15 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1). (b),(c) Height of dissipation boundary layer Hε for υbt = 0.1 m s−1 (blue circles) and υbt = 0.2 m s−1 (red triangles) as function of stratification N and rotation f, respectively. (d) Height of dissipation boundary layer Hε as function of barotropic velocity υbt, for given Coriolis parameters f. Dashed lines depict power-law exponent −1 (b), 0.5 (c), and 1 (d).

  • View in gallery

    Normalized across-slope boundary current υbbl/co as function of (a) normalized barotropic flow speed υbt/co and (b) normalized coarse-resolution across-slope current υcr/co. For different rotation values f (open symbols) and different stratification N (crosses). The red dashed line depicts power-law dependence with exponent 0.87; the black dashed line depicts a 1:1 fit.

  • View in gallery

    Time evolution of (a) across-slope velocity and (b) average energy dissipation within control box E(t) normalized by its maximum value Emax for different forcing υbt = [0.05, 0.1, 0.2, 0.3, 0.4] m s−1 (black to light gray).

  • View in gallery

    Energy dissipation D in midslope region as a function of (a) rotation, (b) stratification, and (c) bottom boundary velocity. Forcing velocity for (a) and (b) is uforcing = 0.1 (blue circles) and 0.2 m s−1 (red triangles). In (c), N = 5.2 × 10−3 s−1 and f = [0.5, 1.0, 1.2] × 10−4 s−1 (cyan, red, gray). The dashed lines depict power-law dependences, with exponents 3/2, −1, 3.

  • View in gallery

    Dissipation in the midslope region obtained from the model Dmodel vs dissipation Dbt, parameterized based on the barotropic flow. For (top) different Coriolis parameters f, (middle) different stratification N, and (bottom) relative error as function of barotropic flow υbt.

  • View in gallery

    As in Fig. 11, but for the coarse-resolution boundary layer flow.

  • View in gallery

    Model results for isolated ridge with flat top (black cross) and with supercritical peak (gray circle). (a) Height of dissipation boundary layer Hε, (b) velocity boundary layer Hu, (c) boundary velocity υbbl as function of barotropic speed υbt, and (d) dissipation in the midslope region obtained from the model D vs dissipation Dbbl, parameterized based on the full-resolution boundary layer flow; f = 1 × 10−4 s−1 and N = 5.2 × 10−3 s−1.

  • View in gallery

    Average energy dissipation as function of slope criticality. For different forcing, uforcing = 0.05 (light gray) to 0.4 m s−1 (black); f = 1 × 10−4 s−1 and N = 5.2 × 10−3 s−1.

  • View in gallery

    Model results for no-slip (black cross) and free-slip (gray circle) boundary condition. (a) Height of dissipation boundary layer Hε, (b) velocity boundary layer Hu, (c) boundary velocity υbbl as function of barotropic speed υbt and (d) dissipation in the midslope region obtained from the model D vs dissipation Dbbl, parameterized based on the full-resolution boundary layer flow; f = 1 × 10−4 s−1 and N = 5.2 × 10−3 s−1.

  • View in gallery

    As in Fig. 2, but for no-slip boundary condition.

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Dissipation of Internal Wave Energy Generated on a Critical Slope

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  • 1 University of Victoria, Victoria, British Columbia, Canada
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Abstract

Two-dimensional simulations of stratified flow over an isolated ridge are used to evaluate energy dissipation associated with barotropic tidal flow over topography with critical or near-critical slope. In the midslope region, a shallow borelike flow forms along the bottom in a layer where dissipation rates are increased by several orders of magnitude, and the flow speed is about twice the barotropic background velocity. The height and turbulence in this layer depend on predictable functions of stratification, rotation, and the characteristic forcing speed. A physically sound power-law parameterization of the total energy dissipation associated with this turbulent layer is presented. This simple parameterization is also applicable to coarser-resolution models, where it may be included to compute energy dissipation above continental slopes, even for cases where the slope angle differs somewhat from criticality.

Denotes Open Access content.

Corresponding author address: Johannes Gemmrich, Department of Physics and Astronomy, University of Victoria, P.O. Box 1700, Victoria BC V8W 2Y2, Canada. E-mail: gemmrich@uvic.ca

Abstract

Two-dimensional simulations of stratified flow over an isolated ridge are used to evaluate energy dissipation associated with barotropic tidal flow over topography with critical or near-critical slope. In the midslope region, a shallow borelike flow forms along the bottom in a layer where dissipation rates are increased by several orders of magnitude, and the flow speed is about twice the barotropic background velocity. The height and turbulence in this layer depend on predictable functions of stratification, rotation, and the characteristic forcing speed. A physically sound power-law parameterization of the total energy dissipation associated with this turbulent layer is presented. This simple parameterization is also applicable to coarser-resolution models, where it may be included to compute energy dissipation above continental slopes, even for cases where the slope angle differs somewhat from criticality.

Denotes Open Access content.

Corresponding author address: Johannes Gemmrich, Department of Physics and Astronomy, University of Victoria, P.O. Box 1700, Victoria BC V8W 2Y2, Canada. E-mail: gemmrich@uvic.ca

1. Introduction

Deep-ocean mixing takes place via a number of processes that are hard to predict a priori from large-scale forcing. Sources of large-scale ocean energy include balanced currents, eddies, tides, and the winds. The energy in these motions is largely believed to be converted to smaller scales before it is dissipated, usually due to interactions with the rough seafloor. The effect of the wind on deep mixing is unproven and is a challenge to observe (Alford 2003). Mean flows and eddies have been the focus of recent work in the Southern Ocean, where these effects are strong (e.g., Watson et al. 2013), and this source of turbulence appears amenable to parameterization (e.g., Nikurashin and Ferrari 2010).

Much more work has been done on turbulence due to tidal flow over topography. Efforts over the Mid-Atlantic Ridge (Polzin et al. 1997) and Hawaii (Klymak et al. 2006) have demonstrated increased dissipation over deep, rough topography. Similar observations have been made elsewhere in a growing body of literature (e.g., Gemmrich and van Haren 2002; Nash et al. 2007; Alford et al. 2011, 2014; van Haren and Gostiaux 2012). Progress has also been made in predicting the dissipation observed near topography. For topography with characteristic slopes that are less steep than internal tide beams, a priori estimates of the expected dissipation are possible from linear models of the generation (St. Laurent et al. 2002; Polzin 2009). A fraction of the upward-radiating energy is assumed to dissipate because of wave–wave interactions. Similarly, if the topography is steeper than the tidal beams, such as at sharp ocean ridges like Hawaii, the lateral radiation of the internal tide is very nonlinear for high vertical mode waves, which then break locally in a systematic manner that lends itself to a priori parameterization (Klymak et al. 2010a, 2013).

These efforts to understand and predict near-topography turbulence due to the tides break down when the slope of the topography is near the tidal beam propagation angle, that is, when the topography is “near critical.” This has long been recognized to be a singularity in the reflection of linear internal waves (Eriksen 1982), with nonlinearity and turbulence as the result (McPhee-Shaw and Kunze 2002). Observations have revealed elevated near-bottom mixing at midslope regions that are near critical to the tidal forcing. This enhanced mixing is associated with internal wave propagation onto a critical slope by the setup of borelike structures due to wave reflection (White 1994) or through the collapse of unstable stratification associated with oblique internal waves (Gemmrich and van Haren 2001).

There have been recent efforts to understand near-critical internal waves in numerical models. Gayen and Sarkar (2011a,b) demonstrate in direct numerical simulations and large-eddy simulations that barotropic forcing of a stratified flow over a near-critical slope generates turbulence on a tidal time scale, and that longer slopes generate more turbulent kinetic energy. They also find that normalizing turbulent profiles through the “beam width” and maximum slope velocity somewhat collapses the results. Finally, they find a substantial fraction of energy dissipated near the slope (12% of the energy that fluxes away). Legg (2014) considers low-mode internal tides impacting critical and subcritical slopes over a range of topographic heights h relative to the water depth H. The fraction of dissipation on the slope is found to roughly scale with h/H. The fraction of incoming energy dissipated had almost no dependence on the forcing strength for the two forcings considered. Encouragingly, the effect of variable stratification was found to be amenable to a Wentzel–Kramers–Brillouin (WKB) scaling, an effect also argued by Klymak et al. (2010b) and Hall et al. (2013), and three-dimensional seamounts had similar response to two-dimensional ones.

Here, we extend the effort of Legg (2014) to the somewhat simpler case of oscillating barotropic flow over an isolated obstacle and attempt to quantify and understand the turbulence along the slope. The obstacle is made to be large relative to the excursion of the tide and the height of the turbulent region in order to develop a region of homogenous turbulence on the slope. The goal is to understand the turbulence in the homogenous region and to exclude the regions where the slope changes at the seafloor and at the top of the topography. This setup is explored under different forcing, stratification, and values of the Coriolis parameter. Unlike others (Klymak et al. 2011; Hall et al. 2013; Legg 2014), we are not concerned with the full energy budget of this system.

2. Model setup

Simulations were undertaken with a two-dimensional hydrostatic setup of the MITgcm with a free-slip bottom boundary condition. Energy dissipation rates were calculated based on the scheme proposed by Klymak and Legg (2010) and are based on finding overturns in the simulation and relating the size of the overturns to the dissipation via the Ozmidov scale.

As a reference case, we chose a domain of −739.3 km ≤ y ≤ 739.3 km, −Zz < 0, Z = 2000 m, with an isolated single ridge of height H = 1200 m centered at y = 0 (Fig. 1). The water column is uniformly thermally stratified giving a constant buoyancy frequency N = 5.2 × 10−3 s−1, and a barotropic flow uforcing oscillating at semidiurnal frequency ω = 1.41 × 10−4 s−1 is prescribed at the boundaries with a sponge. Thus, internal waves propagate at an angle
e1
where f = 2Ω sinφ is the Coriolis parameter, Ω is the earth rotation rate, and φ is the latitude. The Coriolis parameter is set to be constant for the entire model domain. For this stratification and a Coriolis parameter of f = 10−4 s−1, the slope is α = 1.08°.
Fig. 1.
Fig. 1.

Example of model output at t = 7.5 tidal cycles. (top left) Across-slope component υ, (top right) along slope component u, (bottom left) vertical velocity w, and (bottom right) dissipation rate ε. Black lines depict isotherms at equal intervals; uforcing = 0.1 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

The ridge has a slope equal to the critical slope s = α up to a height h ≤ 0.67H (i.e., −1875 m ≤ z ≤ −1165 m) and supercritical slope s = 2α at the top h > 0.67H (i.e., −1150 m ≤ z ≤ −906 m), with smooth transitions in between. The supercritical slope near the top of the ridge ensures that locally generated internal waves are not affecting the midslope region because the beams of high shear radiating from the transition point are situated above the region of interest.

Here, we are interested in mixing and energy dissipation in the midslope region, and therefore the model has high resolution in this region: Δz = 5 m for −1815 m ≤ z ≤ −1365 m, increasing to Δz = 30 m toward z = −2000 m and z = 0 (independent of y). The horizontal resolution is highest in the slope region −257 km < y < 0, where Δy = 400 m and increases to Δy = 4000 m at the edges of the domain, with a smooth transition in between. The size of the model domain grid is 178 × 1200.

Model runs were performed for seven different values of the barotropic forcing speed uforcing = [0.05, 0.07, 0.10, 0.15, 0.20, 0.30, 0.40] m s−1. These seven model runs build the reference set, upon which the effects of stratification N, slope criticality γs = α/αcrit, and rotation (Coriolis parameter f) are evaluated. Five different stratifications were evaluated for N = rsN0, with reference N0 = 5.2 × 10−3 s−1 and scaling factor rs = [0.25, 0.4, 0.5, 1.0, 1.1]. To maintain the same slope criticality and model resolution, all vertical grid parameters (including the topography) are stretched by a factor , so the total water depth and the topography height increase inversely to the scaling factor, but the total number of grid points in the slope region remains constant.

Slope criticality γs and latitudinal dependence f were tested for the nominal stratification N0 and uforcing = [0.05, 0.10, 0.20, 0.30, 0.40] m s−1 only, with γs = [0.5, 0.75, 1.0, 1.5, 2.0], f = [0, 0.3, 0.5, 0.7, 1.0, 1.2, 1.3]f0, and f0 = 1 × 10−4 s−1.

All model runs were started from a fluid at rest and run for t = 96 hours, corresponding to t = 7.8 tidal cycles, with a time step Δt = 12.4 s. An example of the fields of the three velocity components u, υ, w in along-slope, cross-slope, and vertical direction, respectively, as well as temperature T and energy dissipation rate ε is given in Fig. 1.

Sensitivity of the results on grid resolution were tested with model runs with doubled grid resolution (in y and z) for the reference case with uforcing = [0.05, 0.20, 0.30] m s−1.

Additional model runs were performed for different boundary conditions and for a modified ridge topography. Overall, this yields a total of 108 different parameter sets, which are summarized in Tables 14.

Table 1.

Model parameters for reference set (35 cases).

Table 1.
Table 2.

Model parameters for assessing rotation (35 cases).

Table 2.
Table 3.

Model parameters for assessing slope criticality (25 cases).

Table 3.
Table 4.

Model parameters for assessing ridge shape (five cases), no-slip boundary condition (five cases), and grid resolution (three cases).

Table 4.

3. Model results

Our main interest is in the processes at the midslope region as revealed by the temporal and spatial evolution of the velocity, temperature, and dissipation fields. The temporal evolution at a fixed location, analogous to observations with a bottom-mounted mooring, is given in Fig. 2. The magnitude of the across-slope velocity in the lowest O(100) m above the bottom is about twice that higher up in the water column. The flow near the bottom leads the interior flow with a phase shift of slightly less than 90°. The temperature record also shows the strongest variability near the bottom. In the interior, the temperature varies approximately sinusoidally. However, near the bottom the negative temperature anomalies are about twice as big as the positive fluctuations. Temperature and across-slope velocity are approximately 90° out of phase. The dissipation rate in the lowest O(50) m increases by two orders of magnitude to almost 10−6 m2 s−3 for a period of about a third of the tidal cycle. Away from the boundary, dissipation rates are several orders of magnitude smaller and nearly constant. In terms of turbulent diffusivities against the background stratification of N = 5.2 × 10−3 s−1, the turbulent peak is equivalent to κυ = 7 × 10−3 m2 s−1 or 700 times the background 10−5 m2 s−1.

Fig. 2.
Fig. 2.

Vertical profiles of (left) cross-slope velocity υ, (middle) temperature fluctuation ΔT, and (right) dissipation rate ε at a fixed location y = −37 900 m, spanning one tidal cycle starting at t = 6.5; uforcing = 0.2 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

The vertical structure found at the fixed location is representative of the entire midslope region. The two-dimensional fields show two vertical scales associated with processes near the boundary layers; the velocity fields (Figs. 3, 4) are dominated by the signature of the internal wave set up by the large-scale obstacle, whereas the temperature and dissipation fields (Fig. 5) show the existence of a bottom boundary layer, with a height generally less than the vertical scale of the velocity field.

Fig. 3.
Fig. 3.

Evolution of cross-slope velocity υ during one tidal cycle starting at t = 6.5 tidal cycles. Red lines depict isotherms at equal intervals; uforcing = 0.15 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for along-slope velocity u.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

Fig. 5.
Fig. 5.

As in Fig. 3, but for dissipation rate field ε.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

a. Vertical scale

In the presence of stratification, internal waves propagating throughout the entire domain are generated by the flow constriction of the topography (Figs. 1a–c). In the critical slope region, the internal wave generates a current up and down the slope close to the boundary, which is also associated with high energy dissipation rates (Figs. 35). In this example, the current is restricted to a narrow vertical range of about 100 m, with its maximum speed occurring at midheight. A convenient measure to characterize the velocity field is the barotropic velocity in cross-topography direction:
e2
evaluated at y = −400 km. In the slope region, the maximum speed near the seafloor is up to 4 times the barotropic flow speed (Fig. 3). The strength of the boundary currents in the up- and downslope phases are comparable (see also Gayen and Sarkar 2011a,b), so it is possible to characterize the structure of the cross-shelf velocity by considering just its magnitude. Defining the average speed profile over our control volume −48 km = y1 < y < y2 = −28 km and over a tidal cycle gives
e3
the profiles of which show a local maximum approximately 10 grid points above the bottom (Fig. 6a). Note that the model does not include bottom friction, and the local current maximum is entirely due to the flow structure of the internal wave field. Here, we define a vertical scale Hu associated with this enhanced current as the lowest level where savg(Hu) ≤ savg(0). This velocity scale height Hu is inversely correlated with stratification, HuN−1 (Fig. 6b), and does not depend on rotation (Fig. 6c). For moderate forcing, υbt < 0.2 m s−1, Hu increases with forcing but saturates for larger forcing speeds (Fig. 6d).
Fig. 6.
Fig. 6.

(a) Average vertical profile of across-slope velocity magnitude savg in the midslope region, where d is the height above the sloping bottom. Dashed lines indicate the determination of the boundary layer height Hu (uforcing = 0.15 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1). (b),(c) Boundary layer height Hu for υbt = 0.1 m s−1 (blue circles) and υbt = 0.2 m s−1 (red triangles) as function of stratification N and rotation f, respectively. (d) Boundary layer height Hu as function of barotropic velocity υbt for given Coriolis parameters f. Dashed line in (b) depicts power-law exponent −1.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

The shape of the vertical profile of the cross-slope velocity has a profound impact on the temperature/density field above the slope region. In the upslope current phase, the strongest advection of dense water occurs at the position of the maximum current, that is, at some distance above the bottom (Fig. 3). This results in unstable stratification, overturns, and increased dissipation rates near the bottom (Fig. 5). The transition from high but strongly depth-dependent dissipation rates to much lower, nearly constant dissipation rates above characterizes the height Hε of this bottom boundary layer. We take Hε as the location of the local maximum of the second derivative of the dissipation rate profile in this transition region (Fig. 7a). The height Hε depends on stratification, rotation, and the forcing by the barotropic velocity (Figs. 7b–d). The turbulent layer height is limited by the vertical velocity scale HεHu, where this limit is reached for weak rotation and for strong forcing. For all three parameters the functional dependence can be described by a power law at moderate barotropic forcing, yielding
e4
Fig. 7.
Fig. 7.

(a) Average normalized profile of dissipation in the midslope region, where d is the height above the sloping bottom. Dashed line indicates the determination of the boundary layer height Hε (uforcing = 0.15 m s−1, N = 5.2 × 10−3 s−1, and f = 1 × 10−4 s−1). (b),(c) Height of dissipation boundary layer Hε for υbt = 0.1 m s−1 (blue circles) and υbt = 0.2 m s−1 (red triangles) as function of stratification N and rotation f, respectively. (d) Height of dissipation boundary layer Hε as function of barotropic velocity υbt, for given Coriolis parameters f. Dashed lines depict power-law exponent −1 (b), 0.5 (c), and 1 (d).

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

b. Velocity

The barotropic velocity υbt describes the large-scale forcing of the flow field, and although there is a strong correlation between υbt and the height of the turbulent bottom boundary layer Hε, the relation between the boundary layer velocity and the barotropic velocity is not given a priori. However, the flow in the boundary layer is the most direct driver of overturns and thus energy dissipation, and a characterization of the boundary layer velocity is warranted.

The velocity in the bottom layer is the superposition of the potential flow on the slope υ0(y, z) due to the vertical constriction and the flow pattern associated with the internal waves. We define a characteristic speed of the boundary current as the maximum speed in the layer of height Hu:
e5
In the following, velocities are nondimensionalized by a scaling velocity co, which we take as coHN = πc1, where c1 is the phase speed of a mode-1 internal wave. Obviously, the boundary flow is forced by the barotropic flow, and we find it scales as a power-law dependence
e6
with a = 2 and n = 0.87, based on best fits throughout the entire f- and N-space (Fig. 8a).
Fig. 8.
Fig. 8.

Normalized across-slope boundary current υbbl/co as function of (a) normalized barotropic flow speed υbt/co and (b) normalized coarse-resolution across-slope current υcr/co. For different rotation values f (open symbols) and different stratification N (crosses). The red dashed line depicts power-law dependence with exponent 0.87; the black dashed line depicts a 1:1 fit.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

The grid resolution in this simple 2D model by far exceeds any resolution that can currently be achieved by large-scale circulation models. Therefore, we also define a coarse-resolution velocity υcr as the amplitude of the average velocity within the lowest 50 m, which approximately corresponds to the lowest grid box of a high-resolution general circulation model. It turns out that this coarse grid velocity is a close representation of the boundary layer velocity υcr/υbbl = 1.02 ± 0.04 (Fig. 8b).

c. Energy dissipation and parameterization

The main focus of this paper is the energy dissipation due to current–topography interaction in the midslope region. The vertical shear of the boundary current causes the advection of colder, denser water above warmer water during the upslope phase of the boundary current. This generates unstable stratification near the bottom, resulting in overturns and high dissipation rates (Figs. 3, 5; t = 7.0). Thus, at the beginning of the downslope phase of the current, the near-bottom layer is mixed and the advection of warmer water beneath colder water yields only sporadic unstable conditions, mainly at some distance above the bottom (Figs. 3, 5; t = 7.33). The strongest dissipation occurs roughly at the time of the strongest upslope flow (Fig. 2), Note, the model does not include bottom friction, and the obtained dissipation rates are entirely due to prescribed high vertical diffusivities linked to these density overturns (Klymak and Legg 2010; see the appendix).

The total, depth-integrated dissipation D in the midslope region is
e7
where z = h is the lowest grid above the sloping seafloor, and Δh = 500 m is the height of the integration, chosen to include all bottom enhanced mixing, even for the most intense cases. Above the shallow bottom boundary layer, dissipation drops off rapidly by several orders of magnitude (Fig. 5), therefore, the depth integration of (7) is insensitive to the choice Δh of the upper integration bound. The angle brackets 〈 〉 indicate an average in time over one tidal cycle from 6.5 < t ≤ 7.5 tidal cycles. The spatial average is taken along the midslope section −45.5 km < y < −29.9 km, that is, 40 grid points, in order to smooth out small fluctuations. The choice of the box width does not affect the resulting values of E. Furthermore, results are not affected by the given grid resolution, as test runs with doubled grid resolution (in y and z) confirmed.

Dissipation in the control section E(t) fluctuates significantly during a tidal cycle and with forcing (Fig. 9). At weak forcing, enhanced dissipation is restricted to the upslope phase of the current cycle, and the magnitudes fluctuate by more than two orders of magnitude. In the case of the strongest forcing, enhanced turbulence is also generated during the second half of the downslope current phase, albeit at turbulence levels that are a factor of 20 to 50 less than the maximum levels during the upslope current phase. Independent of forcing, the maximum dissipation occurs just before the maximum in the upslope current component is reached. Toward the end of the upslope current phase, the unstable stratification has already been eroded and dissipation rates drop off significantly. During the downslope phase, only strong flows (υbt > 0.1 m s−1) generate unstable stratification with overturns and significant dissipation. This is in qualitative agreement with LES model results that found maximum dissipation rates just after the flow reversal from downslope to upslope motion (Gayen and Sarkar 2011b). However, observations above a supercritical slope at Kaena Ridge showed the dissipation maximum right during flow reversal and a second, somewhat smaller maximum coinciding with the strongest upslope current (Aucan et al. 2006). The dissipation maximum in these observations is likely associated with the sudden collapse of a statically strongly unstable stratification generated by oblique internal waves and is therefore not directly comparable to our model calculations. The dissipation scheme in our model does not allow the buildup of unstable stratification over several time steps and therefore can capture only the secondary maximum in these observations.

Fig. 9.
Fig. 9.

Time evolution of (a) across-slope velocity and (b) average energy dissipation within control box E(t) normalized by its maximum value Emax for different forcing υbt = [0.05, 0.1, 0.2, 0.3, 0.4] m s−1 (black to light gray).

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

The ensemble of the model results indicate that total, depth-integrated, time-averaged dissipation on a slope of critical angle depends on rotation, stratification, and forcing (Fig. 10). To nondimensionalize these parameters, we chose co = NH as the velocity scale and ω−1 as time scale. Note that these scaling parameters did not change throughout the entire range of model runs (N changed but NH did not). Based on all model runs covering the entire υbt, N, and f parameter space, we find the following empirical dependences:
e8
or, if we include the dependence of ,
e9
as per the power law found for (6).
Fig. 10.
Fig. 10.

Energy dissipation D in midslope region as a function of (a) rotation, (b) stratification, and (c) bottom boundary velocity. Forcing velocity for (a) and (b) is uforcing = 0.1 (blue circles) and 0.2 m s−1 (red triangles). In (c), N = 5.2 × 10−3 s−1 and f = [0.5, 1.0, 1.2] × 10−4 s−1 (cyan, red, gray). The dashed lines depict power-law dependences, with exponents 3/2, −1, 3.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

A comparison of the model dissipation Dmodel [(7)] versus parameterized dissipation based on the barotropic forcing Dbt [(9)], shows very good agreement (Fig. 11). As f approaches ω, dissipation values decrease, and it is likely that our model does not fully resolve the small overturns associated with these low dissipation rates, and the discrepancy between model results and parameterization increases, especially for weak forcing. Therefore, data for f = 1.3 × 10−4 s−1 (black triangles) were excluded from the fit, yielding a best-fit parameter A1 = 2.6 × 10−2 and a standard deviation σ(Dbt/Dmodel) = 0.23.

Fig. 11.
Fig. 11.

Dissipation in the midslope region obtained from the model Dmodel vs dissipation Dbt, parameterized based on the barotropic flow. For (top) different Coriolis parameters f, (middle) different stratification N, and (bottom) relative error as function of barotropic flow υbt.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

The best-fit parameterization based on the coarse-resolution velocity υcr is
e10
reflecting the fact that υcr scales linearly with υbbl (Fig. 8b), and A2 = 1.6 × 10−2. For f < 1.2 × 10−4 s−1, the agreement with the model result is excellent, with a standard deviation σ(Dcr/Dmodel) = 0.16 (Fig. 12). Furthermore, the exponent B2 = 3.0 provides a dimensionally consistent parameterization, even for dimensional speeds.
Fig. 12.
Fig. 12.

As in Fig. 11, but for the coarse-resolution boundary layer flow.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

Similarly, the parameterization based on the full-resolution boundary layer velocity υbbl is
e11
with A3 = 1.61 × 10−2, and standard deviation σ(Dbbl/Dmodel) = 0.16 (not shown).

d. Importance of geometry

For the simulations discussed so far, the critical slope was bounded downslope by the flat seafloor and upslope by a supercritical slope. We test if this has an effect on our findings by replacing the supercritical slope with a flat top. The total height of the ridge was kept the same in order to not modify the potential flow υ0.

We find the vertical scales Hu and Hε are slightly smaller in the case of a flat ridge top (Figs. 13a,b), whereas the velocity scale near the bottom υbbl increases for the flat top runs compared to the supercritical peak (Fig. 13c). As a consequence, for a given barotropic forcing, dissipation rates are higher for the flat top ridge (Fig. 13d). However, parameterizations based on velocity observations in the slope region Dcr or Dbbl [(10, 11)] are still valid.

Fig. 13.
Fig. 13.

Model results for isolated ridge with flat top (black cross) and with supercritical peak (gray circle). (a) Height of dissipation boundary layer Hε, (b) velocity boundary layer Hu, (c) boundary velocity υbbl as function of barotropic speed υbt, and (d) dissipation in the midslope region obtained from the model D vs dissipation Dbbl, parameterized based on the full-resolution boundary layer flow; f = 1 × 10−4 s−1 and N = 5.2 × 10−3 s−1.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

So far, all results were based on model runs with the topography of critical slope. In the case of a critical slope, the reflected internal wave propagates along the slope and thus will set up a boundary current along the entire slope section. For subcritical, as well as supercritical slopes, the region of interaction will be reduced, and thus the total dissipation can be expected to be reduced as well. We find the reduction of dissipation on subcritical slopes to be greater than for supercritical slopes (Fig. 14). Furthermore, the dependence on slope criticality increases with decreasing forcing. For example, for a slope of half the critical value (γs = 0.5) dissipation is reduced by a factor 800 in the case of the weakest forcing (uforcing = 0.05 m s−1) but only by a factor 20 for the strongest forcing case (uforcing = 0.4 m s−1). For small deviations from the critical slope, say γs = 1 ± 0.25, dissipation values are expected to be at least half the value of the dissipation above a critical slope.

Fig. 14.
Fig. 14.

Average energy dissipation as function of slope criticality. For different forcing, uforcing = 0.05 (light gray) to 0.4 m s−1 (black); f = 1 × 10−4 s−1 and N = 5.2 × 10−3 s−1.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

4. Discussion

For our simple forcing and geometry, we find a relatively robust parameterization for the dissipation on a critical slope under barotropic forcing [(9)]. It breaks down for weak forcing and low f, but that is because the model fails to resolve the turbulence properly. Here, we offer an explanation for the observed dependence on the flow parameters noted above and note some caveats and future directions this effort could take.

a. Oscillating wedge explanation for dissipation dependence

Conceptually, the tidal flow over a sloping ridge modeled here is equivalent to an oscillating wedge in a rotating stratified fluid at rest. Moving the wedge toward the fluid sets up a bore structure along the slope. Over time, the bore grows and reaches a steady-state thickness Hb. Let U, V be the velocity scale perpendicular and parallel to the wedge, respectively; N is the buoyancy frequency, ω is the oscillation frequency, and f is the rotation frequency. The potential energy (PE) density is proportional to the displacement versus stratification:
e12
If we know the near-bottom velocity, it is convenient to scale the PE as kinetic energy (KE) density:
e13
where the kinetic energy density within the bore is
e14
Thus, the height of the bore scales with the near-bottom cross-slope velocity as
e15
This scaling shows the same dependences on forcing, stratification, and rotation as the boundary layer thickness Hε [(4)].
To get the depth-integrated dissipation, it can be assumed that the potential energy in the bore region dissipates within one cycle of the oscillation:
e16
Combining (16) and (15), the scaling of the dissipation follows as
e17
and the dissipation associated with the wedge-generated bore displays the same dependencies on forcing, stratification, and rotation as the dissipation observed in the model runs [(9)].

b. Caveats

There are a large number of limitations to this study. First, we do not have any a priori estimate for the best-fit constant for the dissipation in the near-bottom bores. However, the scaling just provided indicates the underlying energetics.

A second caveat is that we have tried to avoid slope transitions in what we have done here, and no attempt has been made to narrow down how long a near-critical slope is necessary for the “infinite slope” approximation to be valid. Empirically, staying a vertical distance δzπU/N from the transitions provided us with results that were robust to slope changes and would agree with steady lee-wave theory (Klymak et al. 2010a). For the runs here, that would correspond to approximately 50 m in the vertical, a region we were well clear of.

The biggest limitation of this effort is that we do not have an a priori explanation for the maximum velocity along the slope υbbl as a function of the barotropic forcing υbt [(6)]. Neither this bottom velocity nor the observed thickness of the bottom boundary layer is directly linkable to any linear theory. Gayen and Sarkar (2011a) explored the near-bottom velocity and the bottom layer width and found that υbbl approximately scaled with υbt, but they varied the height of their topography and found both parameters changed. Here, we find an empirical relation between υbbl and the forcing, which holds approximately for the entire parameter space of forcing, stratification, and rotation. The bottom layer height scales linearly with υbt. However, we found a relatively large change of these dependences when moving from a supercritical peak to a flat top. This inability to predict υbbl a priori means that coarse models may have to be run to characterize this velocity scale under barotropic forcing for different geometries.

Energy dissipation at midslope does not require local generation of internal waves but applies to low-mode, far-propagating internal waves found throughout most ocean basins. Here, we mapped out dissipation above the slope of an isolated ridge. The parameter space explored spans typical deep-ocean conditions, and the underlying mechanism of boundary layer overturns is not restricted to isolated ridges but applies to processes above the continental slopes as well. Most of the continental slopes have steepness close to the critical slope associated with the most prevailing stratification (Cacchione et al. 2002), and (10) and (11) provide adequate parameterizations for the dissipation of internal wave energy above the slope. For small deviations from slope criticality, dissipation values can be scaled by a scaling factor β(γs, υ) = D/Dmax obtained from a best fit of the data presented in Fig. 14.

5. Summary

Simulations of a tidal flow over an isolated ridge were performed with the 2D MITgcm model. For critical slopes, that is, the slope of the topography matches the propagation angle α of the internal wave field [(1)], a borelike flow develops along the bottom boundary in the midslope region, which is associated with high dissipation rates. We derived a simple, physically sound, power-law parameterization, relating dissipation D to stratification N, rotation (Coriolis parameter f), tidal frequency ω, and the characteristic forcing speed U. To be applicable to coarser-resolution models, U can be represented by the coarse-resolution velocity υcr next to the bottom. In this case, we propose
e18
as an accurate, straightforward parameterization of the dissipation of internal wave energy per tidal cycle above the continental slope and midocean ridges. This parameterization is suitable for global general circulation models, where υcr represents the across-slope velocity in the lowest grid box above the bottom. For small deviations from slope criticality, dissipation values have to be reduced by an empirical scaling factor, depending on slope criticality and forcing. The velocity within the bore is roughly twice the barotropic velocity but so far can only be determined empirically or from in situ observations.

Acknowledgments

This work was supported by the U.S. Office of Naval Research (ONR 14-09-1-0274). Comments by the reviewers and by Chris Garrett helped to improve the clarity of the manuscript.

APPENDIX

Boundary Conditions

The bulk of the paper uses a free-slip bottom boundary condition, as implemented in the MITgcm. First, it should be noted that the topography is not “smooth,” as the vertical discretization introduces steps in the MITgcm. The MITgcm uses partial cells that better follow topography, so the steps did not exceed 0.5 m in the vertical.

A real fluid has a no-slip bottom boundary condition; however, implementing that implies that we believe we are resolving the bottom boundary layer, which we are not in these coarse simulations. Nonetheless, the results here are insensitive to the bottom boundary layer implementation. For a range of forcing there are almost identical no-slip and free-slip characteristics to the flow and energy terms (Fig. A1). Comparing the time series at the same point in each model (Figs. 2, A2) indicates that the flows are essentially identical.

Fig. A1.
Fig. A1.

Model results for no-slip (black cross) and free-slip (gray circle) boundary condition. (a) Height of dissipation boundary layer Hε, (b) velocity boundary layer Hu, (c) boundary velocity υbbl as function of barotropic speed υbt and (d) dissipation in the midslope region obtained from the model D vs dissipation Dbbl, parameterized based on the full-resolution boundary layer flow; f = 1 × 10−4 s−1 and N = 5.2 × 10−3 s−1.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

Fig. A2.
Fig. A2.

As in Fig. 2, but for no-slip boundary condition.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-14-0236.1

This emphasizes that the source of turbulence in these flows is breaking nonlinear internal waves, not shear stresses at the boundary destabilizing the fluid. The presence of the near-critical boundary is important for focusing and concentrating the internal waves (Eriksen 1982), but the exact boundary layer physics are of secondary importance at these Reynolds numbers.

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