# Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509

LMFA - UMR 5509
Laboratoire de Mécanique des Fluides et d’Acoustique
Lyon
France

## Nos partenaires

Article dans J. Fluid Mech. (2018)

## Scaling laws for mixing and dissipation in unforced rotating stratified turbulence

Annick Pouquet, Duane Rosenberg, Raffaele Marino and Corentin Herbert

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 $\beta=\epsilon_\nu/\epsilon_D$, with $\epsilon_\nu$ the kinetic energy dissipation and $\epsilon_D=u_{rms}^3/L_{int}$ 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 $Ro/Fr\ge2.5$ , and with $1600\le Re\approx5.4 \times10^4$ ; the Prandtl number is one, initial conditions are either isotropic and at large scale for the velocity and zero for the temperature $\theta$, 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 $\tau_{tr}=N\tau_{NL}^2$, with $\tau_{NL}=u_{rms}/L_{int}$ the turnover time and $N$ the Brunt–Väisälä frequency, leads $\beta$ to growing linearly with $Fr$ in the intermediate regime, with a saturation at $\beta\approx0.3$ 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 $R_f=B_f/[B_f+\epsilon_\nu]$, with $B_f=N\langle w\theta\rangle$ the buoyancy flux, transitions for approximately the same parameter values as for $\beta$. These regimes for the present study are delimited by ${\cal R}_B=ReFr^2\approx2$ and ${\cal R}_B\approx200$. With $\Gamma_f=R_f/[1-R_f]$ the mixing efficiency, putting together the three relationships of the model allows for the prediction of the scaling $\Gamma_f\sim Fr^{-2}\sim {\cal R}_B^{-1}$ in the low and intermediate regimes for high , whereas for higher Froude numbers, $\Gamma_f\sim {\cal R}_B^{-1/2}$, a scaling already found in observations : as turbulence strengthens, $\beta\sim1$, $w\approx u_{rms}$, and smaller buoyancy fluxes together correspond to a decoupling of velocity and temperature fluctuations, the latter becoming passive.