In the Boussinesq framework, velocity couples to density fluctuations whereas in magnetohydrodynamic turbulence, the velocity field is coupled to the magnetic field. Both systems support waves (inertia-gravity in the presence of rotation, or Alfvén), with anisotropic dispersion relations. What kind of turbulence regimes result from the interactions between waves and nonlinear eddies in such flows? And what is delimiting these regimes?

I shall sketch the phenomenological framework for rotating stratified turbulence within which one is led to scaling laws in terms of the Froude number, Fr=U/[LN], which measures the relative celerity of gravity waves and nonlinear eddies, with U and L characteristic velocity and length scale, and N the Brunt-V\"ais\"al\"a frequency. These laws apply to the mixing efficiency of such flows, indicating the relative roles of the buoyancy flux due to the waves, and of the measured kinetic and potential energy dissipation rates. Various measures of mixing are found to follow power laws in terms of the Froude number, and may differ for the three regimes that can be identified, namely the wave-dominated, wave-eddy balance and eddy-dominated domains [1]. In particular, in the intermediate regime, the effective dissipation varies linearly with Fr, in agreement with simple wave-turbulence arguments. This analysis is inspired by and corroborates results from a large parametric study using direct numerical simulations (DNS) on grids of 1024^3 points, as well as from atmospheric and oceanic observations.

Such scaling laws can be related to previous DNS results concerning the existence for the energy of bi-directional constant-flux cascades to both the small scales and to the large scales, due to the presence of rotation in such flows, as measured for example in the ocean. These dual energy cascades lead to an alteration, and a decrease, of the mixing and available energy to be dissipated in the small scales [2]. Some perspectives might also be given at the end of the talk.

[1] A. Pouquet, D. Rosenberg, R. Marino & C. Herbert, Scaling laws for mixing and dissipation in unforced rotating stratified turbulence. J. Fluid Mechanics 844, 519, 2018.
[2] R. Marino, A. Pouquet & D. Rosenberg, Resolving the paradox of oceanic large-scale balance and small-scale mixing. Physical Review Letters 114, 114504, 2015.