Statistical mechanics of 2D turbulence with a prior vorticity distribution

Abstract

We adapt the formalism of the statistical theory of 2D turbulence in the case where the Casimir constraints are replaced by the specification of a prior vorticity distribution. A phenomenological relaxation equation is obtained for the evolution of the coarse-grained vorticity. This equation monotonically increases a generalized entropic functional (determined by the prior) while conserving circulation and energy. It can be used as a thermodynamical parametrization of forced 2D turbulence, or as a numerical algorithm to construct (i) arbitrary statistical equilibrium states in the sense of Ellis-Haven-Turkington (ii) particular statistical equilibrium states in the sense of Miller-Robert-Sommeria (iii) arbitrary stationary solutions of the 2D Euler equation that are formally nonlinearly dynamically stable according to the Ellis-Haven-Turkington stability criterion refining the Arnold theorems.

arXiv commentProceedings of the international conference "Euler Equations: 250 Years" (Aussois, France 18-23 June 2007). Edited by G. Eyink, U. Frisch, R. Moreau and A. Sobolevski

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• Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts