  Equilibrium statistical mechanics tools have been developed to obtain
indications about the natural tendencies of nonlinear energy transfers in
two-dimensional and quasi two-dimensional flows like rotating and stratified
flows in geostrophic balance. In this article, we consider a simple model of
such flows with a non-trivial vertical structure, namely two-layer
quasi-geostrophic flows, which remain amenable to analytical study. We obtain
the statistical equilibria of the system in the case of a linear
vorticity-stream function relation, build the corresponding phase diagram, and
discuss the most probable outcome of nonlinear energy transfers, both on the
horizontal and on the vertical, in the presence of stratification and rotation.
