Statistical mechanics of 2D turbulence with a prior vorticity distribution

TL;DR

This paper develops a statistical mechanics framework for 2D turbulence incorporating a prior vorticity distribution, leading to a relaxation equation that models the evolution towards equilibrium states while conserving key quantities.

Contribution

It introduces a novel formalism replacing Casimir constraints with a prior distribution, and provides a relaxation equation for modeling 2D turbulence evolution.

Findings

Derived a relaxation equation increasing a generalized entropy functional.

Can generate various equilibrium states including those stable under specific criteria.

Provides a thermodynamical parametrization and numerical method for 2D turbulence.

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.