Solvable Phase Diagrams and Ensemble Inequivalence for Two-Dimensional and Geophysical Turbulent Flows

Abstract

Using explicit analytical computations, generic occurrence of inequivalence between two or more statistical ensembles is obtained for a large class of equilibrium states of two-dimensional and geophysical turbulent flows. The occurrence of statistical ensemble inequivalence is shown to be related to previously observed phase transitions in the equilibrium flow topology. We find in these turbulent flow equilibria, two mechanisms for the appearance of ensemble equivalences, that were not observed in any physical systems before. These mechanisms are associated respectively with second-order azeotropy (simultaneous appearance of two second-order phase transitions), and with bicritical points (bifurcation from a first-order to two second-order phase transition lines). The important roles of domain geometry, of topography, and of a screening length scale (the Rossby radius of deformation) are discussed. It is found that decreasing the screening length scale (making interactions more local) surprisingly widens the range of parameters associated with ensemble inequivalence. These results are then generalized to a larger class of models, and applied to a complete description of an academic model for inertial oceanic circulation, the Fofonoff flow.

Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a Spherical Domain

“…Besides, this sub-class of solutions already possess very interesting properties. From a practical point of view, they correspond to flow topologies which are indeed observed in real flows: monopoles, dipoles [25] and Fofonoff flows [92,70] in a rectangular basin, solid-body rotations and dipoles on a sphere [48,49], bottom-trapped currents in the ocean [90]... From a theoretical point of view, they exhibit peculiar thermodynamic properties, like bicritical points [91], second-order azeotropy [92], marginal ensemble equivalence [48]... Finally, the energy-enstrophy variational problem has been connected to the full MRS variational problem in several limiting physical cases: the strong-mixing limit [25] and the Gaussian small-scale vorticity prior [36,37,22] for instance (see also the discussions in [48,49,14,31]).…”

“…From the pioneering theory of point vortices initiated by Onsager ([74], see also [39]) and developed by many others [67,75,59,43,6,17,38,30,52,53] to the mean-field theory of Robert, Sommeria and Miller (RSM) [80,78,65] (see also [66,84,20,21,60,14]) through the spectral approach by Kraichnan [54,55], several theories are available, with their strengths and weaknesses. The models for turbulent flows investigated range from the Euler equations to quasi-geostrophic [82,41,63,34,13,92,49] or shallow-water equations [27,24]. In all these cases, the major feature that statistical mechanics enabled us to better understand is the large-scale organization of the flow.…”

Ribbon turbulence

“…Such a low energy limit allows us to compute analytically phase diagrams for the flow structure and to describe how this flow structure changes when the energy or the enstrophy of the flow are varied. For instance statistical equilibria associated with a linear q − ψ relation have been classified for various flow model in an arbitrary close domain 29,30 and on a channel 31 . In particular, it was shown in these studies that when the flow domain is sufficiently stretched in the x direction, then the equilibrium state is a dipolar flow.…”

The catalytic role of the beta effect in barotropization processes

“…This apparent paradox is solved when one realizes that microcanonical and canonical ensembles are, in general, not equivalent for long-ranged interacting systems, see e.g. Ellis et al (2000) for general considerations and Venaille & Bouchet (2011b) for application to two-dimensional and geophysical flows.…”