Journal of Statistical Physics, Vol. 101, Nos. 5/6, 2000. p. 999-1064.
We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. Foi each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macro-states for the two ensembles. Microcanonical equilibrium macrostates an characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions ol an unconstrained minimization problem in which the constraint in the firsi problem is replaced by a Lagrange multiplier...