Relaxation to magnetohydrodynamics equilibria via collision brackets

TL;DR

This paper introduces a relaxation method combining Hamiltonian and dissipative dynamics, inspired by collision operators, to compute equilibria in fluid systems like MHD and Euler equations.

Contribution

It develops a metriplectic framework for equilibria computation, integrating symplectic and metric structures, with applications to various fluid dynamical models.

Findings

Successfully computes MHD and Euler equilibria using the proposed method.

Demonstrates the method's effectiveness through case studies.

Provides a new approach linking collision operators with fluid equilibrium calculations.

Abstract

Metriplectic dynamics is applied to compute equilibria of fluid dynamical systems. The result is a relaxation method in which Hamiltonian dynamics (symplectic structure) is combined with dissipative mechanisms (metric structure) that relaxes the system to the desired equilibrium point. The specific metric operator, which is considered in this work, is formally analogous to the Landau collision operator. These ideas are illustrated by means of case studies. The considered physical models are the Euler equations in vorticity form, the Grad-Shafranov equation, and force-free MHD equilibria.