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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 β = ∈V=∈D, with ∈V the kinetic energy dissipation and ∈D = u3rms/Lint its dimensional expression, with w; urms the vertical and root mean square velocities, and Lint 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 10243 points, with Ro/Fr ≥ 2.5, and with 1600 ≤ Re ≈ 5.4 × 104; the Prandtl number is one, initial conditions are either isotropic and at large scale for the velocity and zero for the temperature θ, 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 τtr = Nτ2NL, with τNL = Lint/urms the turnover time and N the Brunt-Väisälä frequency, leads to β growing linearly with Fr in the intermediate regime, with a saturation at β ≈ 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 Rf = Bf/[Bf + ∈V], with Bf = N〈wθ〉 the buoyancy flux, transitions for approximately the same parameter values as for β. These regimes for the present study are delimited by B = ReFr2 ≈ 2 and RB ≈ 200. With Γf = Rf/[1 - Rf] the mixing efficiency, putting together the three relationships of the model allows for the pbkp_rediction of the scaling Γf ∼ Fr-2 ∼ B-1 in the low and intermediate regimes for high Re, whereas for higher Froude numbers, Γf ∼ -1/2B, a scaling already found in observations: as turbulence strengthens, β ∼ 1, w ≈ urms, and smaller buoyancy fluxes together correspond to a decoupling of velocity and temperature fluctuations, the latter becoming passive.