Direct numerical simulation (DNS) and large-eddy simulation (LES) are employed to study the mixing brought about by convective overturns in a stratified, oscillatory bottom layer underneath internal tides. The phasing of turbulence, the onset and breakdown of convective overturns, and the pathway to irreversible mixing are quantified. Mixing efficiency shows a systematic dependence on tidal phase, and during the breakdown of large convective overturns it is approximately 0.6, a value that is substantially larger than the commonly assumed value of 0.2 used for calculating scalar mixing from the turbulent dissipation rate. Diapycnal diffusivity is calculated using the irreversible diapycnal flux and, for tall overturns of O(50) m, the diffusivity is found to be almost 1000 times higher than the molecular diffusivity. The Thorpe (overturn) length scale is often used as a proxy for the Ozmidov length scale and thus infers the turbulent dissipation rate from overturns. The accuracy of overturn-based estimates of the dissipation rate is assessed for this flow. The Ozmidov length scale LO and Thorpe length scale LT are found to behave differently during a tidal cycle: LT decreases during the convective instability, while LO increases; there is a significant phase lag between the maxima of LT and LO; and finally LT is not linearly related to LO. Thus, the Thorpe-inferred dissipation rates are quite different from the actual values. Interestingly, the ratio of their cycle-averaged values is found to be O(1), a result explained on the basis of available potential energy.
Direct numerical simulation is performed with a focus on the characterization of nonlinear dynamics during reflection of a plane internal wave at a sloping bottom. The effect of incoming wave amplitude is assessed by varying the incoming Froude number, $Fr$, and the effect of off-criticality is assessed by varying the slope angle in a range of near-critical values. At low $\mathit{Fr}$, the numerical results agree well with linear inviscid theory of near-critical internal wave reflection. With increasing $\mathit{Fr}$, the reflection process becomes nonlinear with the formation of higher harmonics and the initiation of fine-scale turbulence during the evolution of the reflected wave. Later in time, the wave response becomes quasi-steady with a systematic dependence of turbulence on the temporal and spatial phase. Convective instabilities are found to play a crucial role in the formation of turbulence during each cycle. The cycle evolution of flow statistics is studied in detail and qualitative differences between off-critical and critical reflection are identified. The parametric dependence of turbulence levels on Froude number and slope angle is calculated. Interestingly, at a given value of $\mathit{Fr}$, the turbulent kinetic energy (TKE) can be higher for somewhat off-critical reflection compared to exactly critical reflection. For a fixed slope angle, as the Froude number increases in the simulated cases, the fraction of the input wave energy converted into the turbulent kinetic energy and the fraction of the input wave power dissipated by turbulence also increase.
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