Analysis of statistical equilibrium models of geostrophic turbulence

Statistical equilibrium lattice models of coherent structures in geostrophic turbulence, formulated by discretizing the governing Hamiltonian continuum dynamics, are analyzed. The first set of results concern large deviation principles (LDP's) for a spatially coarse-grained process with respect to either the canonical and/or the microcanonical formulation of the model. These principles are derived from a basic LDP for the coarse-grained process with respect to product measure, which in turn depends on Cramér's Theorem. The rate functions for the LDP's give rise to variational principles that determine the equilibrium solutions of the Hamiltonian equations. The second set of results addresses the equivalence or nonequivalence of the microcanonical and canonical ensembles. In particular, necessary and sufficient conditions for a correspondence between microcanonical equilibria and canonical equilibria are established in terms of the concavity of the microcanonical entropy. A complete characterization of equivalence of ensembles is deduced by elementary methods of convex analysis. The mathematical results proved in this paper complement the physical reasoning and numerical computations given in a companion paper, where it is argued that the statistical equilibrium model defined by a prior distribution on potential vorticity fluctuations and microcanonical conditions on total energy and circulation is natural from the perspective of geophysical applications.