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Energetics and mixing in buoyancy-driven near-bottom stratified flow
- Pranav Puthan, Masoud Jalali, Vamsi K. Chalamalla, Sutanu Sarkar
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- Journal:
- Journal of Fluid Mechanics / Volume 869 / 25 June 2019
- Published online by Cambridge University Press:
- 23 April 2019, pp. 214-237
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Turbulence and mixing in a near-bottom convectively driven flow are examined by numerical simulations of a model problem: a statically unstable disturbance at a slope with inclination $\unicode[STIX]{x1D6FD}$ in a stable background with buoyancy frequency $N$. The influence of slope angle and initial disturbance amplitude are quantified in a parametric study. The flow evolution involves energy exchange between four energy reservoirs, namely the mean and turbulent components of kinetic energy (KE) and available potential energy (APE). In contrast to the zero-slope case where the mean flow is negligible, the presence of a slope leads to a current that oscillates with $\unicode[STIX]{x1D714}=N\sin \unicode[STIX]{x1D6FD}$ and qualitatively changes the subsequent evolution of the initial density disturbance. The frequency, $N\sin \unicode[STIX]{x1D6FD}$, and the initial speed of the current are predicted using linear theory. The energy transfer in the sloping cases is dominated by an oscillatory exchange between mean APE and mean KE with a transfer to turbulence at specific phases. In all simulated cases, the positive buoyancy flux during episodes of convective instability at the zero-velocity phase is the dominant contributor to turbulent kinetic energy (TKE) although the shear production becomes increasingly important with increasing $\unicode[STIX]{x1D6FD}$. Energy that initially resides wholly in mean available potential energy is lost through conversion to turbulence and the subsequent dissipation of TKE and turbulent available potential energy. A key result is that, in contrast to the explosive loss of energy during the initial convective instability in the non-sloping case, the sloping cases exhibit a more gradual energy loss that is sustained over a long time interval. The slope-parallel oscillation introduces a new flow time scale $T=2\unicode[STIX]{x03C0}/(N\sin \unicode[STIX]{x1D6FD})$ and, consequently, the fraction of initial APE that is converted to turbulence during convective instability progressively decreases with increasing $\unicode[STIX]{x1D6FD}$. For moderate slopes with $\unicode[STIX]{x1D6FD}<10^{\circ }$, most of the net energy loss takes place during an initial, short ($Nt\approx 20$) interval with periodic convective overturns. For steeper slopes, most of the energy loss takes place during a later, long ($Nt>100$) interval when both shear and convective instability occur, and the energy loss rate is approximately constant. The mixing efficiency during the initial period dominated by convectively driven turbulence is found to be substantially higher (exceeds 0.5) than the widely used value of 0.2. The mixing efficiency at long time in the present problem of a convective overturn at a boundary varies between 0.24 and 0.3.
Tidal flow over topography: effect of excursion number on wave energetics and turbulence
- Masoud Jalali, Narsimha R. Rapaka, Sutanu Sarkar
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- Journal:
- Journal of Fluid Mechanics / Volume 750 / 10 July 2014
- Published online by Cambridge University Press:
- 09 June 2014, pp. 259-283
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The excursion number, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}Ex = U_0/\varOmega l$, is a parameter that characterizes the ratio of streamwise fluid advection during a tidal oscillation of amplitude $U_0$ and frequency $\varOmega $ to the streamwise topographic length scale $l$. Direct numerical simulations are performed to study how internal gravity waves and turbulence change when $Ex$ is varied from a low value (typical of a ridge in the deep ocean) to a value of unity (corresponding to energetic tides over a small topographic feature). An isolated obstacle having a smoothed triangular shape and 20 % of the streamwise length at critical slope is considered. With increasing values of $Ex$, the near field of the internal waves loses its beam-like character, the wave response becomes asymmetric with respect to the ridge centre, and transient lee waves form. Analysis of the baroclinic energy balance shows significant reduction in the radiated wave flux in the cases with higher $Ex$ owing to a substantial rise in advection and baroclinic dissipation as well as a decrease in conversion. Turbulence changes qualitatively with increasing $Ex$. In the situation with $Ex \sim 0.1$, turbulence is intensified at the near-critical regions of the slope, and is also significant in the internal wave beams above the ridge where there is intensified shear. At $Ex = O(1)$, the transient lee waves overturn adjacent to the ridge flanks and, owing to convective instability, buoyancy acts as a source for turbulent kinetic energy. The size of the turbulent overturns has a non-monotonic dependence on excursion number: the largest overturns, as tall as twice the obstacle height, occur in the $Ex = 0.4$ case, but there is a substantial decrease of overturn size at larger values of $Ex$ simulated here.