Singular limits of anisotropic weak solutions to compressible magnetohydrodynamics

TL;DR

This paper rigorously justifies the reduced magnetohydrodynamic (RMHD) model for strongly anisotropic plasmas by analyzing the asymptotic behavior of weak solutions to the compressible MHD equations, highlighting anisotropic incompressibility and convergence in both periodic and unbounded domains.

Contribution

It provides a rigorous mathematical proof of the convergence of anisotropic weak solutions to the RMHD equations in the context of strongly magnetized plasmas, including both periodic and unbounded domains.

Findings

Weak solutions converge to RMHD equations in the anisotropic limit.

Incompressibility is only in the perpendicular direction to the magnetic field.

The analysis employs tools from low Mach number limits and oscillatory group techniques.

Abstract

The aim is to justify rigorously the so-called reduced magnetohydrodynamic model (abbreviated as RMHD), which is widely used in fusion, space and astrophysical plasmas. Motivated by physics, the focus is on plasmas that are simultaneously strongly magnetized and anisotropic. We consider conducting fluids that can be described by viscous and resistive barotropic compressible magnetohydrodynamic equations. The purpose is to study the asymptotic behaviour of global weak solutions, which do exist, for strongly anisotropic plasmas such as the large aspect ratio framework. We prove that such anisotropic weak solutions converge to the weak solutions of the RMHD equations. Rigorous justification of this limit is performed both in a periodic domain and in the whole space. It turns out that the resulting system is incompressible only in the perpendicular direction to the external strong magnetic field, whereas it involves compressible features in the parallel direction. In order to pass to the singular limit in the perpendicular direction we exploit, among others, tools elaborated for proving the low Mach number limit of compressible fluid flows such as the introduction of a fast oscillatory unitary group associated to the dynamics of transverse fast magnetosonic waves. In the parallel direction, we bring out compactness arguments and particular cancellations coming from the structure of our equations.