Journal Article

Scaling laws for mixing and dissipation in unforced rotating stratified turbulence

Rosenberg, D., Marino, R., Herbert, C., Pouquet, A.

2018

Overview

We present a model for the scaling of mixing in weakly rotating stratified flows characterized by their Rossby, Froude and Reynolds numbers $Ro,Fr$ , $Re$ . This model is based on quasi-equipartition between kinetic and potential modes, sub-dominant vertical velocity, $w$ , and lessening of the energy transfer to small scales as measured by a dissipation efficiency $\\unicode[STIX]{x1D6FD}=\\unicode[STIX]{x1D716}_{V}/\\unicode[STIX]{x1D716}_{D}$ , with $\\unicode[STIX]{x1D716}_{V}$ the kinetic energy dissipation and $\\unicode[STIX]{x1D716}_{D}=u_{rms}^{3}/L_{int}$ its dimensional expression, with $w,u_{rms}$ the vertical and root mean square velocities, and $L_{int}$ the integral scale. We determine the domains of validity of such laws for a large numerical study of the unforced Boussinesq equations mostly on grids of $1024^{3}$ points, with $Ro/Fr\\geqslant 2.5$ , and with $1600\\leqslant Re\\approx 5.4\\times 10^{4}$ ; the Prandtl number is one, initial conditions are either isotropic and at large scale for the velocity and zero for the temperature $\\unicode[STIX]{x1D703}$ , or in geostrophic balance. Three regimes in Froude number, as for stratified flows, are observed: dominant waves, eddy–wave interactions and strong turbulence. A wave–turbulence balance for the transfer time $\\unicode[STIX]{x1D70F}_{tr}=N\\unicode[STIX]{x1D70F}_{NL}^{2}$ , with $\\unicode[STIX]{x1D70F}_{NL}=L_{int}/u_{rms}$ the turnover time and $N$ the Brunt–Väisälä frequency, leads to $\\unicode[STIX]{x1D6FD}$ growing linearly with $Fr$ in the intermediate regime, with a saturation at $\\unicode[STIX]{x1D6FD}\\approx 0.3$ or more, depending on initial conditions for larger Froude numbers. The Ellison scale is also found to scale linearly with $Fr$ . The flux Richardson number $R_{f}=B_{f}/[B_{f}+\\unicode[STIX]{x1D716}_{V}]$ , with $B_{f}=N\\langle w\\unicode[STIX]{x1D703}\\rangle$ the buoyancy flux, transitions for approximately the same parameter values as for $\\unicode[STIX]{x1D6FD}$ . These regimes for the present study are delimited by ${\\mathcal{R}}_{B}=ReFr^{2}\\approx 2$ and $R_{B}\\approx 200$ . With $\\unicode[STIX]{x1D6E4}_{f}=R_{f}/[1-R_{f}]$ the mixing efficiency, putting together the three relationships of the model allows for the prediction of the scaling $\\unicode[STIX]{x1D6E4}_{f}\\sim Fr^{-2}\\sim {\\mathcal{R}}_{B}^{-1}$ in the low and intermediate regimes for high $Re$ , whereas for higher Froude numbers, $\\unicode[STIX]{x1D6E4}_{f}\\sim {\\mathcal{R}}_{B}^{-1/2}$ , a scaling already found in observations: as turbulence strengthens, $\\unicode[STIX]{x1D6FD}\\sim 1$ , $w\\approx u_{rms}$ , and smaller buoyancy fluxes together correspond to a decoupling of velocity and temperature fluctuations, the latter becoming passive.

• Publisher: Cambridge University Press,Cambridge University Press (CUP)
• Subject: Balances (scales) / Boussinesq approximation / Boussinesq equations / Brunt-Vaisala frequency / Buoyancy / Buoyancy flux / Computational fluid dynamics / Decoupling / Efficiency / Energy dissipation / Energy exchange / Energy transfer / Fluid mechanics / Fluxes / Froude number / Gravitational waves / Initial conditions / Interactions / JFM Papers / Kinetic energy / Mathematical models / Mechanics / Physics / Prandtl number / Reynolds number / Richardson number / Rotation / Saturation / Scaling / Scaling laws / Temperature / Turbulence / Turnover time / Velocity / Wave interaction / Wave interactions