Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a spherical domain

TL;DR

This paper explores how the geometry of a spherical domain influences the statistical mechanics of 2D Euler flows, revealing new invariants and analyzing their impact on equilibrium states and energy condensation.

Contribution

It introduces additional invariants specific to spherical domains and develops a mean-field theory incorporating these invariants for the 2D Euler equations.

Findings

Existence of new invariants related to angular momentum, energy, and enstrophy.

Analytical solution of the variational problem showing partial energy condensation.

Discussion of thermodynamic properties of the system.

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.