Relaxation to magnetohydrodynamics equilibria via collision-like metric brackets: toward three-dimensional equilibria

In our previous work (Bressan et al., J. Phys.: Conf. Series, 2018) we have shown how the idea of metriplectic dynamics (Morrison, Phys. Letters A, 1984) can be employed to design relaxation methods for computing equilibria of Euler's and magnetohydrodynamics (MHD) equations in two-dimensions. Specifically, a metriplectic dynamical system is obtained from a Hamiltonian system by adding a dissipation mechanism in the form of a metric bracket which preserves the Hamiltonian function while dissipating a specific entropy. We observed that the Landau operator for Coulomb collisions can be used as a template for the construction of a specific class of metric brackets that in turn lead to favorable relaxation methods. Here we develop those ideas toward the application to three-dimensional MHD equilibria. We discuss in details a relaxation method for Beltrami fields in three-dimensions. For numerical computations, we show how the key geometric properties of the metric brackets must be preserved at the discrete level and propose a structure-preserving scheme based on finite element exterior calculus. At last a generalization of the method from Beltrami fields to full three-dimensional MHD equilibria is presented.