Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint

TL;DR

This paper introduces an augmented Lagrangian approach to approximate the ideal Ohm's Law constraint in relaxed magnetohydrodynamics, bridging the gap with ideal MHD and enabling a continuum of models.

Contribution

It develops a novel augmented Lagrangian method to enforce the ideal Ohm's Law constraint in relaxed MHD, allowing for iterative enforcement or a continuum of models between RxMHD and IMHD.

Findings

Derived dispersion relations for linear waves on MHD equilibrium.

Demonstrated the method's ability to approximate the ideal Ohm's Law.

Established a framework for transitioning between relaxed and ideal MHD models.

Abstract

The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium.