Sharp Nonlinear Stability Criterion of Viscous Non-resistive MHD Internal Waves in 3D

We consider the dynamics of two layers of incompressible electrically conducting fluid interacting with a magnetic field, which are confined within a 3D horizontally infinite slab and separated by a free internal interface. We assume that the upper fluid is heavier than the lower fluid so that the fluids are susceptible to Rayleigh–Taylor instability, yet we show that the viscous and non-resistive problem around the equilibrium is nonlinearly stable provided that the strength of the vertical component of the steady magnetic field, |B¯3|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|{\bar{B}_3}|}$$\end{document}, is greater than the critical value, Mc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}_c}$$\end{document}, which we identify explicitly. We also prove that the problem is nonlinearly unstable if |B¯3|<Mc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|{\bar{B}_3}| < \mathcal{M}_c}$$\end{document}. Our results indicate that the non-horizontal magnetic field has a strong stabilizing effect on the Rayleigh–Taylor instability but the horizontal one does not have this in 3D.