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

Abstract

The role of the domain geometry for the statistical mechanics of 2D Euler flows is investigated. It is shown that for a spherical domain, there exists invariant subspaces in phase space which yield additional angular momentum, energy and enstrophy invariants. The microcanonical measure taking into account these invariants is built and a mean-field, Robert-Sommeria-Miller theory is developed in the simple case of the energy-enstrophy measure. The variational problem is solved analytically and a partial energy condensation is obtained. The thermodynamic properties of the system are also discussed.Keywords Incompressible Euler fluid flow on S 2 · Turbulence · Robert-Sommeria-Miller theory · Dynamical Invariants IntroductionThe tools of equilibrium statistical mechanics have been applied to the study of turbulent flows in a variety of ways. 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. While Onsager's theory gave birth to the early prediction of the existence of negative temperature, and Kraichnan's work yielded the notion of energy condensation for fluid flows, it is arguably the RSM theory which provides the most convenient framework to effectively compute the statistical equilibria of the system. Very often, several statistical equilibria can coexist for a given value of the external parameters, which leads to interesting phase transitions. The long-range nature of the interactions at work in turbulent flows is responsible for new types of phase transitions [10]. Like with many other systems with long-range interactions [33,16], the energy is not additive in turbulent flows, which has the crucial consequence that the different statistical ensembles may not give equivalent results, even in the thermodynamic limit [36]. Ensemble inequivalence has many manifestations at the thermodynamic level, like the existence of negative specific heats, non concave entropies,... [92] For all the above mentioned properties, the domain geometry plays an important role. In the first place, it bears connections with the dynamical invariants of the system, which are the cornerstones

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