On Inhibition of Rayleigh--Taylor Instability by Horizontal Magnetic Field in an Inviscid MHD Fluid with Velocity Damping

TL;DR

This paper rigorously proves that a sufficiently strong horizontal magnetic field can inhibit Rayleigh--Taylor instability in an inviscid, incompressible MHD fluid with velocity damping, marking a first for nonlinear equations in this context.

Contribution

It provides the first nonlinear mathematical proof that a horizontal magnetic field can suppress RT instability in a 2D inviscid MHD fluid with damping, identifying a critical magnetic strength.

Findings

Existence of a critical magnetic field strength for stability

Exponential stability when magnetic field exceeds critical strength

Nonlinear instability when magnetic field is below critical strength

Abstract

It is still an open problem whether the inhibition phenomenon of Rayleigh--Taylor (RT) instability by horizontal magnetic field can be mathematically proved in a non-resistive magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it had been roughly verified by a 2D linearized motion equations in 2012 \cite{WYC}. In this paper, we find that this inhibition phenomenon can be rigorously verified in the inhomogeneous, incompressible, inviscid case with velocity damping. More precisely, there exists a critical number $m_{\rm{C}}$ such that if the strength $|m|$ of horizontal magnetic field is bigger than $m_{\rm{C}}$, then the small perturbation solution around the magnetic RT equilibrium state is exponentially stable in time. Our result is also the first mathematical one based on the nonlinear motion equations for the proof of inhibition of flow instabilities by a horizontal magnetic field in a horizontal slab domain. In addition, we also provide a nonlinear instability result for the case $|m|\in [0,m_{\rm{C}})$. Our instability result presents that horizontal magnetic field can not inhibit the RT instability, if it's strength is to small.