Relaxation to magnetohydrodynamics equilibria via collision brackets

C Bressan,P J Morrison,M Kraus,O Maj

doi:10.1088/1742-6596/1125/1/012002

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.

Highlights

The computation of general 3D magnetohydrodynamic (MHD) equilibria plays a fundamental role in simulations of stellarators and it is important for tokamaks as well, due to deviations from axisymmetry
Such a structure has been proposed by Morrison [6, 7] and it is referred to as metriplectic dynamics since it combines the symplectic structure of Hamilton’s equations with the metric structure of gradient flows
The geometric properties of metriplectic flows can be exploited to design artificial dynamical systems that relax to an equilibrium of the considered physical system

Summary

Introduction

The computation of general 3D magnetohydrodynamic (MHD) equilibria plays a fundamental role in simulations of stellarators and it is important for tokamaks as well, due to deviations from axisymmetry (magnetic islands, ripples, and resonant magnetic perturbations). The Boltzmann equation has the additional property that a solution of the initial value problem relaxes, as time goes to infinity, to an equilibrium because of the celebrated H theorem [19]; equilibria can be identified by time-evolution of properly chosen initial conditions In general this is not the case: ideal systems with no dissipation mechanisms will not relax to an equilibrium. The idea proposed by Morrison [6] shows the possibility to define a dissipative dynamics that relaxes to a solution of the variational problem for the equilibrium These concepts will be explained in more detail here with the help of specific physical models.

We restrict the entropy functional to be of the form

The Hamiltonian and entropy functionals must satisfy the conditions

Li δu