Scaling laws for mixing and dissipation in unforced rotating stratified turbulence

Abstract

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.