### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Set Up
- 3. Results
- 4. Conclusion
- Acknowledgments
- References

[1] A numerical study based on large eddy simulation (LES) is performed to investigate near-bottom mixing processes in an internal wave beam over a critical slope. Transition to turbulence from an initial laminar state is followed by mixing events that occur at specific phases. Maximum turbulent kinetic energy and dissipation rate are found just after the zero velocity point when flow reverses from downslope to upslope motion. At this phase, convective instability leads to density overturns that originate in the upper flank of the beam and span the beam width of 60 m. Turbulence originating at the bottom and with smaller vertical extent is also present during the phases of peak upslope and downslope flow when the boundary layer shear is large. The present numerical simulations identify and characterize a process, internal wave beam at a critical slope during generation or after propagation from a nearby generation site, that may lead to the high turbulence levels, modulated at the tidal frequency, that is observed near the bottom in oceanic sites with near-critical topography.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Set Up
- 3. Results
- 4. Conclusion
- Acknowledgments
- References

[2] Localized regions of high boundary turbulence play an important role in setting average diapycnal mixing rates necessary to maintain the vertical stratification observed in the mid-ocean. Bottom boundary turbulence is enhanced at rough topography including sea-mounts [*Lueck and Mudge*, 1997; *Kunze and Toole*, 1997], submarine ridges [*Rudnick et al.*, 2003; *Aucan et al.*, 2006; *Levine and Boyd*, 2006; *Klymak et al.*, 2006], continental slopes [*Thorpe et al.*, 1990; *Moum et al.*, 2002; *Nash et al.*, 2007] and deep rough topography [*Polzin et al.*, 1997; *St. Laurent et al.*, 2001]. Measurements of currents, vertical displacements and turbulent dissipation rate at these sites often show a dominant modulation by internal tides that are generated by surface tides that oscillate over topographic features. Therefore, there is much current interest in understanding bottom turbulence processes associated with internal waves at sloping topography.

[3] Observations indicate that bottom turbulence, with order 100 m density overturns, is present at or near regions with critical slope where the slope angle, *β*, is approximately equal to the inclination, θ, of internal wave phase lines with the horizontal. *Moum et al.* [2002], using a towed body package that included turbulence sensors, found a large 5 km lateral extent of turbulent flow within a few hundred meters off a sloping region that is critical to the M2 internal tide. The dissipation profile averaged over the survey exhibited a value of 10^{−9} W/kg at 500 mab which is 10 times the open ocean value while the values closer to the bottom were about 100 times lager than the open ocean value. *Aucan et al.* [2006] examined tidal variations within 200 m of a deep (2425 m) flank of Kaena Ridge as part of the Hawaii Ocean Mixing Experiment (HOME). A series of temperature sensors detected tidally driven overturns with vertical scales of order 100 m and peak near-bottom dissipation of 2 × 10^{−6} W/kg inferred by Thorpe-scale analysis. The time-averaged dissipation was 1.2 × 10^{−8} W/kg, about 10–100 times the value in the ocean interior at similar depths and 50 km from the ridge. Furthermore, *Aucan et al.* [2006] found that enhanced dissipation occurred at specific phases of the internal tide. *Nash et al.* [2007] reported two deep ocean hotspots of turbulent mixing, both at near-critical regions of the Oregon continental slope, where they estimated time-averaged turbulent energy dissipation rates of 10^{−7} Wkg^{−1} and eddy diffusivities of *K*_{ρ} = 10^{−2} m^{2}/s.

[4] During the generation of internal tides at critical slopes, high vertical number modes are formed that combine to give internal tidal beams. Laboratory [*Lim et al.*, 2010] and numerical [*Gayen and Sarkar*, 2010, 2011] experiments have shown evidence of turbulence in tidal beams over slope topography with length of the order ��(1–30) m. However, the beam was at laboratory scale (width smaller than 0.2 m) and did not allow the large separation in scales between the viscous boundary layer and the oscillating core of the beam that occurs in the ocean. The objective of the present work is to numerically model a near-bottom beam with a larger, more realistic width and characterize its turbulence characteristics. We will also show that the phase dependence of turbulence in the numerical model agrees with that observed by *Aucan et al.* [2006] off Kaena Ridge and try to explain the underlying mechanism.

### 2. Problem Set Up

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Set Up
- 3. Results
- 4. Conclusion
- Acknowledgments
- References

[5] The boundary-conforming grid, curvilinear coordinates, and streamwise inhomogeneity in our earlier studies [*Gayen and Sarkar*, 2010, 2011] enabled accurate simulation of internal wave beam formation at a slope in response to the imposed barotropic forcing and subsequent formation of turbulence. In the present work, we assume that there is a beam with a given width, *l*_{b}, and peak near-bottom velocity, *U*_{b}, on a slope and simulate phase-dependent turbulence in a small patch of the beam. It would be impractical to perform a turbulence-resolving simulation in a streamwise non-periodic domain with the large slope length of several hundred meters that is required for beam generation with *l*_{b} = 60 m.

[6] Large-eddy simulation (LES) is used to obtain the filtered velocity and temperature fields by numerical solution of the Navier-Stokes equations under the Boussinesq approximation, written in rotated coordinates [*x*_{r}, *y*_{r}, *z*_{r}] in dimensional form as:

Here, *p** and *ρ** denote *deviation* from the background pressure and density, respectively. The quantity *τ* that denotes the subgrid scale stress tensor and *λ* that is the subgrid density flux are modeled as described by *Gayen et al.* [2010]. Briefly, *τ* is represented with a dynamic eddy viscosity model, and *λ* with a dynamic eddy diffusivity model. Each subgrid model has a single Smagorinsky coefficient that is evaluated through a dynamic procedure that involves the introduction of an additional test filter. The coefficient is averaged over the homogeneous directions (slope-parallel plane). The NS equations are numerically solved to obtain the velocity in rotated coordinates [*u*_{r}, *v*_{r}, *w*_{r}] and the deviations from background density and pressure using a mixed spectral/finite difference algorithm on a staggered grid. Spanwise and across slope derivatives are treated with a pseudo-spectral method, and the wall normal spatial derivatives are computed with second-order finite differences. A third-order Runge-Kutta-Wray method is used for time-stepping, and viscous terms are treated implicitly with the Crank-Nicolson method. The test domain, excluding the sponge region, consists of a rectangular box of 30 m length, 160 m height and 10 m width whose bottom boundary is coincident with the slope topography as shown in Figure 1a. The grid size in the test domain is 128 × 620 × 64 in the *x*_{r}, *z*_{r} and *y*_{r} directions, respectively, with stretching in *z*_{r} direction. The grid spacing (Δ*x*_{r} = 0.234 m, Δ*y*_{r} = 0.157 m, Δ*z*_{r,min} = 0.0037 m, Δ*z*_{r,max} = 3.5 m) is sufficient for LES as is confirmed by examining the spanwise and the streamwise spectra as shown in Figure 2. Periodicity is imposed in the spanwise, *y*_{r}, and streamwise, *x*_{r}, direction. Zero velocity and a zero value for wall-normal total density flux i.e. *dρ*/*dz*_{r} = 0 are imposed at the bottom. The sponge region, with damping to the background state, contains 15 points, extends from 160 m to 200 m, and has a maximum Δ*z*_{r} = 5.59 m.

[7] *Gayen and Sarkar* [2011] found that the beam width, *l*_{b}, and velocity amplitude, *U*_{b}, increased with increasing slope length, following a power-law dependence. The beam in the present simulation is initialized by taking advantage of the approximately self-similar velocity structure, when normalized with *l*_{b} and *U*_{b}, that was previously found. At *t* = 0, there are neither velocity nor density fluctuations. In the work by *Gayen and Sarkar* [2010], an oscillating barotropic tide with constant amplitude was prescribed on a near-critical slope, and the barotropic-to-baroclinic energy conversion compensated for the losses due to turbulent dissipation leading to a boundary jet with constant amplitude, *U*_{b}. In the current simulation, the amplitude, *U*_{b}, is maintained by adding a relaxation term, −*σ*_{f}(*z*)[*u*_{r}(**x**, *t*) − *u*_{r,f}(*z*)cos(Ω *t*)], to the right-hand side of the *x*_{r} -momentum equation, with the target shape function, *u*_{r,f} (*z*), plotted in Figure 1b. The shape of *σ*_{f}(*z*) is such that the forcing is limited to the beam core, *z*_{r} = 20 m − 60 m, allowing the bottom boundary layer to develop independently. The imposed forcing is not shear unstable since the minimum gradient Richardson number, *Ri*_{g}, based on background velocity and stratification exceeds 0.5, larger than the critical value of *Ri*_{g} = 0.25. Owing to periodicity in the *x*_{r} direction, mixed fluid recirculates in the computational domain so that turbulence at the flanks of the beam (caused by density overturns as will be shown) decreases after three tidal cycles. A simple way to overcome this limitation is to force *ρ* towards the linear background during a short period (0.5 hour) when the downslope boundary flow is maximum by adding the function −*σ*_{f}[*ρ**(**x**, *t*)] to the evolution equation for *ρ**.

[8] In the present simulation, *N*_{∞} = 1.6 × 10^{−3} rad s^{−1} and Ω = 1.4076 × 10^{−4} rad s^{−1}, which gives the wave angle θ = sin^{−1}(Ω/*N*_{∞}) ≈ 5°. Slope angle *β* = 5° sets the criticality parameter, ε = tan *β*/tan θ ≈ 1. The kinematic viscosity, *ν* = 10^{−6} m^{2}/s, is that of water. The Prandtl number is chosen to be *Pr* = 7. The beam velocity amplitude is chosen to be *U*_{b} = 0.125 m/s and the beam width is *l*_{b} = 60 m. Cycle-averaged and maximum values of turbulent Reynolds number are *Re*_{T} ∼ 4000 and ∼10^{5}, respectively. Here, *Re*_{T} is based on the velocity scale, *u*_{T} = (1/*l*_{b})*dz*_{r} where *K* = 1/2 〈*u*′_{i}u′_{i}〉 is the turbulent kinetic energy, the length scale, *l*_{b}, and the molecular viscosity, *ν*. Variable time stepping with a fixed CFL number 1.2 is used leading to Δ *t* ≃ ��(1) sec. One tidal cycle takes approximately 1000 CPU hours.

### 3. Results

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Set Up
- 3. Results
- 4. Conclusion
- Acknowledgments
- References

[9] The simulation starts with the phase of peak downslope velocity. There is transition to turbulence within approximately 3 hrs in the upper flank of the beam and in approximately 6 hrs at the bottom wall, followed by distinct turbulent mixing events that repeat periodically at different phases of the internal tide cycle as explained below.

[10] Figure 3 illustrates the phase variability of density and velocity by showing snapshots at four different times. Figure 3a at *t* = 12.8 hrs corresponds to a phase when the downslope flow is near its peak. The corresponding density deviation, *ρ**, is small and spans a short region, 0 < *z* < 20 m. Inspection of Figures 3a–3d shows that *ρ** lags *u* by 90°; it is maximum when *u* is minimum and vice-versa. At *t* = 12.8 hrs, the near-wall shear is large and, in the sheared zone of 0 < *z* < 20 m, the corresponding density field in Figure 3e is indicative of both shear instability and fine scale turbulence. From *t* = 12.8 hrs until *t* = 16 hr when the velocity becomes approximately zero, the downward flow continues to bring water from above. Consequently, the region between 5 m and 30 m which, at *t* = 12.8 hrs was occupied by water with density shown by green in Figure 3e, is replaced by lighter, warmer water at *t* = 16 hrs, shown by red and yellow in Figure 3f. Furthermore, this body of relatively lighter, warmer water passes underneath the colder water above it that, being at the edge of the beam, has low velocity. The density profile that results at *t* = 16 hrs indicates a density inversion as shown in Figure 3b. The corresponding density field in Figure 3f shows a mushroom shaped plume suggesting convective instability. The shear at this time is near zero. Note that the density inversion is even more prominent a little earlier at *t* = 14.5 hrs. Later in time, the large-scale overturns collapse and break into smaller structures. The shear starts increasing and at *t* = 18.7 hrs, the velocity profile in Figure 3c shows maximum upslope velocity and, correspondingly, the near wall flow is again susceptible to shear instability as shown in Figure 3g. The velocity in the next snapshot, Figure 3d, is almost zero and corresponds to *t* = 21.4 hrs when the flow reverses from up to down. Between *t* = 18.7 and 21.4 hrs, the flow decelerates but continues to move upslope and heavy, cold fluid associated with the central portion of the internal tide beam surges up beneath the lighter, low-velocity fluid above. The cumulative effect of this upsurge is to strengthen the stratification in the upper flank of the beam as shown by the region, 20 < *z* < 60 m, in the density profile of Figure 3d. A the same time, the upsurge overtakes low-velocity, lighter fluid near the wall leading to a density inversion with light fluid beneath dense fluid. Therefore, during flow reversal from up to down, a convective instability with the formation of a mushroom shaped structure occurs in the lower part of the flow as shown in Figure 3h. The height of the density overturn (<15 m) is significantly smaller than that observed earlier at *t* = 16 hr during flow reversal from down to up.

[11] Figures 4b–4d show the time evolution of vertical profiles of some turbulence statistics. 〈ϕ〉, denotes an average of the quantity ϕ calculated by averaging over an *x*_{r} − *y*_{r} plane parallel to the slope and we will show the dependence, 〈ϕ〉 (*z**, *t*), on height measured in meters above bottom (mab), *z**, and time, *t*. Figure 4a shows streamwise velocity at 15 mab, to locate the phase of the internal tidal cycle. In Figure 4a, temperature is shown as a function of time at different elevations. It can be seen in Figure 4a that, during downward motion, e.g. *t* = 10 hrs to *t* = 14 hrs, the temperature at each *z** increases owing to warmer water coming down the slope from above. The rate of temperature increase is largest at *z** = 20 mab which is near the center of the beam and progressively decreases for higher elevations. Therefore, the temperature (density) gradient in the region 20 < *z** < 60 progressively decreases in magnitude and, after *t* ≃ 14 hrs, the gradient reverses sign. The vertical span of the region with negative temperature gradient increases but such an unstable profile cannot be maintained for long and there is convective instability that develops into dissipative turbulence.

[12] Figure 4b shows turbulent kinetic energy (*TKE*) defined by *K* = 1/2 〈*u*′_{i}u′_{i}〉, where *u*′_{i} = *u*_{i} − 〈*u*_{i}〉. Figures 4c and 4d show analogous plots for the buoyancy flux, *B* = −(*g*/*ρ*_{0}) 〈*ρ*′*w*′〉 and the turbulent dissipation rate, ɛ, calculated as the sum of resolved and subgrid dissipation rates. The combined effect of boundary velocity and density fields results in significant vertical variability of turbulent kinetic energy over a tidal cycle. Maximum *TKE* occurs away from the boundary and during flow reversal mixing from down to up (we will refer to such mixing events *FRM* − *DU*) at various time spans, Δ *t* ∼ 3.1–4.8, 15.5–17.2 and 28–29.5 hrs during the tidal cycle as shown by vertical grey colored bins in Figure 4a. Large positive buoyancy flux indicative of energy transfer from potential to kinetic energy occurs only during *FRM* − *DU* time intervals as shown in Figure 4c. Turbulent production, *P*, as well as dissipation, ɛ, are also large at that time. It is noteworthy that during these time intervals, the regions of large *TKE* and ɛ are *detached* from the wall and extend up to a height of ∼60–70 m.

[13] Shear production results in significant levels of turbulence when the upslope and down slope velocity are large; these mixing events are termed as *SM*-*U* and *SM*-*D*, respectively. During late deceleration (e.g. 8 < *t* < 10), the turbulent dissipation peak also moves up and extends up a height of 20 m. Such an enhancement of dissipation away from the immediate vicinity of the bottom was not found by *Gayen et al.* [2010] and occurs here because of an additional physical mechanism, namely, convective instability during flow reversal mixing from up to down (*FRM* − *UD*) which was visualized earlier in Figure 3h.

[14] Figure 5a shows the temporal evolution of depth-averaged values of each term in the *TKE*-budget over a tidal period spanning *t* = 26–38 hrs. The depth-averaged buoyancy flux, 〈*B*〉, starts increasing at *t* = 27.5 at the onset of the *FRM* − *DU* event just prior to the zero-velocity phase, reaches its maximum value, and then plummets to zero at the end of the density overturn. Significant amount of negative production 〈*P*〉 occurs during the flow reversal event signaling the energy transfer from velocity fluctuations to the mean flow. The negative production, also seen in a companion DNS in a smaller domain, occurs because the turbulence is initiated by a positive buoyancy flux (not mean shear) and the mean shear changes sign after *u* passes through zero. Soon after peak 〈*B*〉, the dissipation, 〈ɛ〉, increases. Production, 〈*P*〉, attains positive values immediately after the *FRM* − *DU* event. During the *SM* − *U* event as shown in the inset in Figure 5a, 〈*P*〉 and 〈ɛ〉 are the only dominant terms in the *TKE*-budget and both peak at the same time during the decelerating phase of upslope flow similar to the oscillatory boundary layer on a non-sloping bottom. At the end of the *SM* − *U* event and during the beginning of the *FRM* − *UD* event, the buoyancy flux, 〈*B*〉 regains positive values which are smaller relative to those during the *FRM* − *DU* event.

[15] Figure 5b quantifies the phase dependence of turbulent dissipation rate, ɛ, through a depth-averaged value. The evolution of ɛ at *z** = 40 m, a location at the upper flank of the internal tide beam, is also given to isolate the contribution of detached mixing events. ɛ(*z** = 40, *t*) shows pronounced peaks, shown by the downward pointing arrows in red, that occur during *FRM* − *DU*. The elevated levels of dissipation occur over a time interval of approximately 1.5hrs. The depth-averaged ɛ also shows peaks during *FRM* − *DU*. It is worth noting that at the onset of *FRM* − *DU*, the depth-averaged buoyancy frequency is minimum. Figure 5b shows additional time periods of elevated values of depth-averaged dissipation: one that occurs over a period of 1.5 hrs during the deceleration of upslope flow and is associated with *SM* − *U* followed by *FRM* − *UD*, and another that occurs during shear mixing during down slope flow, *SM* − *D*.

[16] To place these simulation results in context, we compare with the bottom mooring observations by *Aucan et al.* [2006] taken during HOME. There is remarkable agreement between their observations and our simulations: (i) Density (potential temperature) records at different elevations converge at the *t* ≃ 11 point and then overturn as shown by *Aucan et al.* [2006, Figure 8a] similar to the shaded regions in Figure 4a, (ii) There is a large peak in turbulence and decrease in buoyancy frequency at the time corresponding to density overturns as shown by both *Aucan et al.* [2006, Figure 8b] and our Figure 4b, and (iii) Peak dissipation occurs when the across-flow velocity reverses from downslope to upslope, compare *Aucan et al.* [2006, Figure 8c] with blue curve in Figure 5a. *Gemmrich and van Haren* [2001] have observed thermal fronts associated with rapid fall of temperature above the continental slope in the Bay of Biscay. *Gemmrich and van Haren* [2001, Figure 4] show similarities with the sudden decrease of temperature after a plateau that occurs towards the end of FRM–DU events (shaded) in all the near-bottom temperature records shown in our Figure 4a.