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(v)/epsilon(D), with epsilon(v) 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 with Ro/Fr >= 2.5, and with 1600 <= Re approximate to 5.4 x 10(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 Fronde number, as for stratified flows, are observed: dominant waves, eddy-wave interactions and strong turbulence. A wave-turbulence balance for the transfer time tau(rr) = N tau(2)(NL), with tau(NL)= L-int/u(rms) the turnover time and N the Brunt-Vaisala frequency, leads to beta growing linearly with Fr in the intermediate regime, with a saturation at beta approximate to 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 + epsilon(v)], with B-f = N < w theta > the buoyancy flux, transitions for approximately the same parameter values as for beta. These regimes for the present study are delimited by R-B = ReFr2 approximate to 2 and R-B approximate to 200. 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) similar to Fr-2 similar to R-B(-1) in the low and intermediate regimes for high Re, whereas for higher Froude numbers, Gamma(f) similar to R-B(-1)/2, a scaling already found in observations: as turbulence strengthens, beta similar to 1, w approximate to u(rms), and smaller buoyancy fluxes together correspond to a decoupling of velocity and temperature fluctuations, the latter becoming passive.