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Evolution of plumes and turbulent dynamics in deep-ocean convection
- Anikesh Pal, Vamsi K. Chalamalla
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- Journal:
- Journal of Fluid Mechanics / Volume 889 / 25 April 2020
- Published online by Cambridge University Press:
- 28 February 2020, A35
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Three-dimensional numerical simulations are performed to investigate the dynamics of deep-ocean convection. Organized structures of denser fluid moving downwards, known as plumes, are formed during the initial evolution. We propose a scaling for the diameter and velocity of these plumes based on surface flux magnitude $B_{0}$ and the thermal/eddy diffusivity. Rotation effects are found to be negligible during this initial evolution. At a later time $t>2\unicode[STIX]{x03C0}/f$, where $f$ is the Coriolis parameter, the flow comes under the influence of rotation, which stabilizes the flow by inhibiting the conversion of potential energy to turbulent kinetic energy. At moderate to low rotation rates, the dense fluid plummets and spreads laterally as a gravity current along the bottom boundary. However, at high rotation rates, the flow reaches a quasi-geostrophic state (before the dense fluid reaches the bottom boundary) with an approximate balance between the pressure gradient and the Coriolis forces. We also see the formation of baroclinic vortices and a rim current at the interface of the mixed and surrounding fluids at high rotation rates. A quantitative analysis of the root-mean-square velocities reveals that higher rotation rates result in lower turbulence intensities. Closure of the turbulent kinetic energy budget is also achieved with an approximate balance between the buoyancy flux and the dissipation rate.
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.
PSI in the case of internal wave beam reflection at a uniform slope
- Vamsi K. Chalamalla, Sutanu Sarkar
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- Journal:
- Journal of Fluid Mechanics / Volume 789 / 25 February 2016
- Published online by Cambridge University Press:
- 21 January 2016, pp. 347-367
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Two-dimensional numerical simulations are performed to examine internal wave reflection at a sloping boundary. Owing to reflection, the reflected wave amplitude and wavenumber increase. At low values of the incoming wave amplitude, the reflected wave beam is linear and its properties agree well with linear inviscid theory. Linear theory overestimates the reflected wave Froude number, $Fr_{r}$, for higher values of incoming wave amplitude. Nonlinearity sets in with increasing value of incoming wave Froude number, $Fr_{i}$, leading to parametric subharmonic instability (PSI) of the reflected wave beam: two subharmonics emerge from the reflection region with frequencies $0.33{\it\Omega}$ and $0.67{\it\Omega}$ and wavenumbers that add up to those of the reflected wave. The amplification of Froude number due to reflection must be sufficiently large for PSI to occur implying that the off-criticality in wave angle cannot be too large. The simulations also show that, all other parameters being fixed, a threshold in beam amplitude is required for the onset of PSI in the reflected beam, consistent with results from a previous weakly-nonlinear asymptotic theory for a freely propagating finite-width beam. Growth rates of subharmonic modes at moderate reflected wave amplitude are in reasonable agreement with that theory. However, for $Fr_{r}>0.5$, small scale fluctuations becomes prominent and the subharmonic energy growth rates saturate in the simulations in contrast to the theoretical prediction. Increasing the incoming beam thickness (number of carrier wavelengths) increases the strength of PSI. Keeping the incoming Froude number constant and increasing the incoming Reynolds number by a factor of 50 does not have an effect on the unequal division of frequencies among the subharmonic modes that is found in the simulations.
Turbulence during the reflection of internal gravity waves at critical and near-critical slopes
- Vamsi K. Chalamalla, Bishakhdatta Gayen, Alberto Scotti, Sutanu Sarkar
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- Journal:
- Journal of Fluid Mechanics / Volume 729 / 25 August 2013
- Published online by Cambridge University Press:
- 19 July 2013, pp. 47-68
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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.