  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 laws (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 study,
we demonstrate analytically that the Miller-Robert-Sommeria theory leads to
non-trivial statistical equilibria for quasi-geostrophic flows on a rotating
sphere, with or without bottom topography. We first consider flows without
bottom topography and with an infinite Rossby deformation radius, with and
without conservation of angular momentum. When the conservation of angular
momentum is taken into account, we report a case of second order phase
transition associated with spontaneous symmetry breaking. In a second step, we
treat the general case of a flow with an arbitrary bottom topography and a
finite Rossby deformation radius. Previous studies were restricted to flows in
a planar 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-scale flow through their sole macroscopic features. We build the
phase diagrams of the system and discuss the relations of the various
statistical ensembles.
