On Inhibition of Rayleigh--Taylor Instability by a Horizontal Magnetic Field in Non-resistive MHD Fluids: the Viscous Case

TL;DR

This paper proves that a sufficiently strong horizontal magnetic field can stabilize Rayleigh--Taylor instability in viscous MHD fluids, extending previous linear results to the nonlinear case with boundary conditions.

Contribution

It establishes the nonlinear stability of RT instability inhibition by a horizontal magnetic field in viscous MHD fluids, identifying a critical field strength threshold.

Findings

Existence of a critical magnetic field strength m_C for stability.

Stability occurs when |m| > m_C.

Instability persists when |m| < m_C.

Abstract

It is still open whether the phenomenon of inhibition of Rayleigh--Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive \emph{viscous} magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it was roughly proved in the linearized case by Wang in \cite{WYC}. In this paper, we prove such inhibition phenomenon by the (nonlinear) inhomogeneous, incompressible, \emph{viscous case} with \emph{Navier (slip) boundary condition}. More precisely, we show that there is a critical number of field strength $m_{\mm{C}}$, such that if the strength $|m|$ of a horizontal magnetic field is bigger than $m_{\mm{C}}$, then the small perturbation solution around the magnetic RT equilibrium state is {algebraically} stable in time. In addition, we also provide a nonlinear instability result for the case $|m|\in[0, m_{\mm{C}})$. The instability result presents that a horizontal magnetic field can not inhibit the RT instability, if it's strength is too small.