Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows

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Abstract
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 energy and circulation invariants are treated microcanonically give richer families of equilibria than models in which they are treated canonically. For each model, a variational principle that characterizes its equilibrium states is derived by large deviation techniques. An analysis of the two different variational principles resulting from the canonical and microcanonical ensembles reveals that their equilibrium states coincide if and only if the microcanonical entropy function is concave. Numerical computations implemented for geostrophic turbulence over topography in a zonal channel demonstrate that nonequivalence of ensembles occurs over a wide range of the model parameters and that physically interesting equilibria are often omitted by the canonical model. The nonlinear stability of the steady mean flows corresponding to microcanonical equilibria is established by a new Lyapunov argument. These stability theorems refine the well-known Arnold stability theorems, which do not apply when the microcanonical and canonical ensembles are not equivalent.