Equilibrium and Out of Equilibrium Phase Transitions for Two Dimensional and Geostrophic Turbulence

The equilibrium statistical mechanics of two dimensional and geostrophic flows (Robert-Sommeria-Miller theory), predicts the outcome for the large scales of the flow, resulting from the turbulent mixing. We describe applications of this theory to describe detailed properties of the Great Red Spot and other localized structures of Jupiter's troposphere. It explains the detailed ring structure and stability of the Great Red Spot velocity field.
Another aim of this talk is to discuss the range of applicability of this theory to ocean dynamics. This range is probably limited due the inertial assumption underlying this equilibrium approach. Still we will show that the theory is able to reproduce in much detail localized structures like the gulf stream rings. We also uncover the relations between strong eastward mid-basin inertial jets, like the Kuroshio extension and the Gulf Stream, and statistical equilibria. All these results cannot be obtained in the context of the older Salmon-Kraichnan theory, that mainly predicts only features related to the topography.
Forcing and dissipation play an essential role for ocean dynamics. We consider an out of equilibrium theory of the mixing of the potential vorticity that takes these into account. We describe results for ergodicity in this out of equilibrium configuration. We also show that we can predict out of equilibrium phase transitions, where the flow switches randomly between two different large scale patterns. The main interest of the theory is to predict the range of parameters of this bistability phenomena and to predict the mean streamfunction for each of these two states. We discuss possible future applications of these ideas to the bistability of the Kuroshio Current, and of other geophysical flows.
We will also describe briefly recent results of the asymptotic behavior of the 2D Euler equation and results on the 2D Euler equation with stochastic forces.