Stochastic averaging, large deviations and random transitions for the dynamics of 2D and geostrophic turbulent vortices

Geophysical turbulent flows are characterized by their self-organization into large scale coherent structures, in particular parallel jets. We will present a theory in order to describe the effective statistics and dynamics of these jets. We prove that this closure is exact in the limit of a timescale separation between the forcing and the inertial dynamics, which is rare in a turbulent flow. The equation obtained describes the attractors for the dynamics (alternating zonal jets) and the relaxation towards those attractors. At first order, these attractors are the same as the ones obtained from a quasi-Gaussian closure, already studied. Our work thus justifies this approximation and the corresponding asymptotic limit. We also present a new, very efficient algorithm to compute the terms appearing in this equation. The theory also goes beyond the quasi-Gaussian approximation, and indeed it can also describe the stationary distribution of the jets (fluctuations and large deviations).