  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. For 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 macrostates for the two ensembles.
