Analysis of statistical equilibrium models of geostrophicturbulence

[摘要] Statistical equilibrium lattice models of coherent structures in geostrophic turbulence,formulated by discretizing the governing Hamiltonian continuumdynamics, are analyzed. The first set of results concern large deviation principles(LDP's) for a spatially coarse-grained process with respect to either the canonicaland/or the microcanonical formulation of the model. These principles are derivedfrom 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 theHamiltonian equations. The second set of results addresses the equivalence ornonequivalence of the microcanonical and canonical ensembles. In particular, necessaryand sufficient conditions for a correspondence between microcanonicalequilibria and canonical equilibria are established in terms of the concavity of themicrocanonical entropy. A complete characterization of equivalence ofensembles is deduced by elementary methods of convex analysis. Themathematical results proved in this paper complement the physical reasoning andnumerical computations given in a companion paper, where it is argued that thestatistical equilibrium model defined by a prior distribution on potential vorticityfluctuations and microcanonical conditions on total energy and circulation isnatural from the perspective of geophysical applications.

[学科分类] 应用数学