Statistical mechanics of quasi-geostrophic flows on a rotating sphere

Abstract

Statistical mechanics provides an elegant explanation for the appearance of coherentstructures in two-dimensional inviscid turbulence: while the fine-grained vorticity field,described by the Euler equation, becomes more and more filamentary through time, itsdynamical evolution is constrained by some global conservation laws (energy, Casimirinvariants). As a consequence, the coarse-grained vorticity field can be predicted throughstandard statistical mechanics arguments (relying on the Hamiltonian structure ofthe 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 playingthe role of the advected quantity. In this study, we demonstrate analytically thatthe Miller–Robert–Sommeria theory leads to non-trivial statistical equilibria forquasi-geostrophic flows on a rotating sphere, with or without bottom topography.We first consider flows without bottom topography and with an infinite Rossbydeformation radius, with and without conservation of angular momentum. When theconservation of angular momentum is taken into account, we report a second-order phasetransition associated with spontaneous symmetry breaking. In a second step,we treat the general case of a flow with an arbitrary bottom topography and afinite Rossby deformation radius. Previous studies were restricted to flows in aplanar domain with fixed or periodic boundary conditions with a beta-effect.In these different cases, we are able to classify the statistical equilibria for the large-scaleflow through their macroscopic features. We build the phase diagrams of the system anddiscuss the relations of the various statistical ensembles.