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Zonal flows as statistical equilibria

Zonal jets are striking and beautiful examples of the propensity for geophysical turbulent flows to spontaneously self-organize into robust, large scale coherent structures. There exist many dynamical mechanisms for the formation of zonal jets: statistical theories (kinetic approaches, second order or larger oder closures), deterministic approaches (modulational instability, $β$-plumes, radiating instability, zonostrophic turbulence, and so on). A striking remark is that all these different dynamical approaches, each of them possibly relevant in some specific regimes, lead to the same kind of final jet structures. Is it then possible to have a more general explanation of why all these different dynamical regimes, from fully turbulent flows to gentle quasilinear regime, consistently lead to the same jet attractors ? Equilibrium statistical mechanics provides an answer to this general question. Here we we present the salient features of this theory and review applications of this approach to the description of zonal jets. We show that equilibrium states on a beta plane or a sphere are usually zonal or quasi-zonal, and that increasing the energy leads to bifurcations breaking the zonal symmetry.