Statistical Mechanics of Quasi-geostrophic flows on a rotating sphere

Statistical Mechanics provides an elegant explanation to the appearance of coherent structures in two-dimensional inviscid turbulence: while the fine-grained vorticity field, described by the Euler equation, becomes more and more filamented through time, its dynamical evolution is constrained by some global conservation constraints (Energy, Casimir invariants). As a consequence the coarse-grained vorticity field can be predicted through standard statistical mechanics arguments (relying on the Hamiltonian structure of the two-dimensional Euler flow), for any given set of the integral constraints. It has been suggested that the theory applies equally well to geophysical turbulence; specifically in the case of the quasi-geostrophic equations, with potential vorticity playing the role of the advected quantity. In this contribution, we demonstrate analytically that the Miller-Robert-Sommeria theory leads to non-trivial statistical equilibria for quasi-geostrophic flow on a rotating sphere, with or without topography. In the context of geophysics, this approach allows to study directly, in a thermodynamic framework, the statistical equilibria of the large-scale general circulation, given a set of macroscopic conserved quantities. In practice, it provides a way to avoid the difficult computation of the effects of the small-scales on the dynamics, at the expanse of a loss of information, inherent to the thermodynamical approach, since the fine-grained potential vorticity field cannot be recovered from the coarse-grained field alone. From a theoretical viewpoint, we are able to classify the statistical equilibria for the large-scale flow through their sole macroscopic features.