Asymptotic shallow models arising in magnetohydrodynamics

TL;DR

This paper introduces a new asymptotic shallow model for the plasma-vacuum problem in magnetohydrodynamics, simplifying complex large-scale astrophysical plasma flows into a manageable set of equations.

Contribution

It derives the magnetic analogue of the Green-Naghdi equations, extending hydrodynamic reduction techniques to magnetohydrodynamics with weak vorticity and magnetic currents.

Findings

Derived a new shallow asymptotic model for plasma-vacuum interactions.

Extended Green-Naghdi equations to magnetic contexts.

Provided a closed set of equations for large-scale plasma flows.

Abstract

In this paper, we derive a new shallow asymptotic model for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equation, vital in describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green-Naghdi equations for water waves in the presence of weakly shared vorticity and magnetic currents. The method is inspired by developed ideas for hydrodynamics flows in by Castro and Lannes (2014) to reduce the $(d+1)$-dimensional dynamics of the problem to a finite cascade of equations which can be closed at the precision of the model.