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

Abstract

Statistical equilibrium models of coherent structures intwo-dimensional and barotropic quasi-geostrophic turbulence areformulated using canonical and microcanonical ensembles, and theequivalence or nonequivalence of ensembles is investigated for thesemodels. The main results show that models in which the energy andcirculation invariants are treated microcanonically give richerfamilies of equilibria than models in which they are treatedcanonically. For each model, a variational principle thatcharacterizes its equilibrium states is derived by large deviationtechniques. An analysis of the two different variational principlesresulting from the canonical and microcanonical ensembles reveals thattheir equilibrium states coincide if and only if the microcanonicalentropy function is concave. Numerical computations implemented forgeostrophic turbulence over topography in a zonal channel demonstratethat nonequivalence of ensembles occurs over a wide range of the modelparameters and that physically interesting equilibria are oftenomitted by the canonical model. The nonlinear stability of the steadymean flows corresponding to microcanonical equilibria is establishedby a new Lyapunov argument. These stability theorems refine thewell-known Arnold stability theorems, which do not apply when themicrocanonical and canonical ensembles are not equivalent.