  Statistical equilibrium models of coherent structures in two-dimensional and
barotropic quasi-geostrophic turbulence are formulated using canonical and
microcanonical ensembles, and the equivalence or nonequivalence of ensembles is
investigated for these models. The main results show that models in which the
global invariants are treated microcanonically give richer families of
equilibria than models in which they are treated canonically. Such global
invariants are those conserved quantities for ideal dynamics which depend on
the large scales of the motion; they include the total energy and circulation.
For each model a variational principle that characterizes its equilibrium
states is derived by invoking large deviations techniques to evaluate the
continuum limit of the probabilistic lattice model. An analysis of the two
different variational principles resulting from the canonical and
microcanonical ensembles reveals that their equilibrium states coincide only
when the microcanonical entropy function is concave. These variational
principles also furnish Lyapunov functionals from which the nonlinear stability
of the mean flows can be deduced. While in the canonical model the well-known
Arnold stability theorems are reproduced, in the microcanonical model more
refined theorems are obtained which extend known stability criteria when the
microcanonical and canonical ensembles are not equivalent. A numerical example
pertaining to geostrophic turbulence over topography in a zonal channel is
included to illustrate the general results.
