Multi-scale theory of rotating turbulence

Aims. To understand the dynamics of stellar interiors, we study the effect of rotation on turbulence. Methods. We consider turbulence induced by an arbitrary forcing and derive turbulence amplitude and turbulent transport coefficients (turbulent viscosity and diffusivity), first by using a quasi-linear theory and then by using a multi-scale renormalisation analysis. Results. With an isotropic forcing, the quasi-linear theory gives that the turbulent transport coefficients, both parallel and perpendicular to the rotation vector, have the asymptotic scaling Ω−1 for rapid rotation (i.e. when the rotation rate Ω is larger than the inverse of the correlation time of the forcing and the diffusion time), while the renormalisation analysis suggests a weaker dependence on Ω, with Ω−1/2 scaling. The turbulence amplitude is found to scale as Ω − Ω−1 in the rapid rotation limit depending on the property of the forcing. In the case of an anisotropic forcing with inhibited motion in the vertical direction, as should be relevant in a strongly stratified medium, we find that non-diffusive fluxes of angular momentum scale as Ω−2 − Ω−1 for rapid rotation, depending on the temporal correlation of the forcing. We discuss the implications of our result for the dynamics of stellar interiors.