Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

TL;DR

This paper introduces new relaxation equations for 2D turbulence based on a maximum entropy principle with a prior vorticity distribution, offering a tool for modeling and computing equilibrium states.

Contribution

It derives a novel class of relaxation equations using MEPP with a prescribed prior vorticity distribution, extending previous models and applicable for numerical simulations.

Findings

Numerical simulations show geometry-induced phase transitions in geophysical flows.

The equations can serve as small-scale parametrizations of 2D turbulence.

Comparison with existing models highlights differences and potential advantages.

Abstract

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a small-scale parametrization of 2D turbulence or serve as numerical algorithms to compute maximum entropy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows.