Laboratoire de Mécanique des Fluides et d'Acoustique - UMR 5509

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Laboratoire de Mécanique des Fluides et d’Acoustique
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Accueil > Équipes > Turbulence & Instabilités > Publications T&I > Publications T&I 2018

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

Scaling laws for mixing and dissipation in unforced rotating stratified turbulence

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.
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