Critical Magnetic Number in the MHD Rayleigh-Taylor instability

TL;DR

This paper identifies a critical magnetic field strength in MHD Rayleigh-Taylor instability, showing how magnetic orientation influences stability and growth rates of perturbations in a viscous, incompressible fluid system.

Contribution

It introduces a variational characterization of the critical magnetic number and analyzes stability conditions for different magnetic field orientations in MHD Rayleigh-Taylor instability.

Findings

Vertical magnetic field stabilizes low frequency modes when above critical strength.

Horizontal magnetic field stabilizes high frequency modes when above critical strength.

Growth rates of unstable modes are bounded below the critical magnetic number.

Abstract

We reformulate in Lagrangian coordinates the two-phase free boundary problem for the equations of Magnetohydrodynamics in a infinite slab, which is incompressible, viscous and of zero resistivity, as one for the Navier-Stokes equations with a force term induced by the fluid flow map. We study the stabilized effect of the magnetic field for the linearized equations around the steady-state solution by assuming that the upper fluid is heavier than the lower fluid, $i. e.$, the linear Rayleigh-Taylor instability. We identity the critical magnetic number $|B|_c$ by a variational problem. For the cases $(i)$ the magnetic number $\bar{B}$ is vertical in 2D or 3D; $(ii)$ $\bar{B}$ is horizontal in 2D, we prove that the linear system is stable when $|\bar{B}|\ge |B|_c$ and is unstable when $|\bar{B}|<|B|_c$. Moreover, for $|\bar{B}|<|B|_c$ the vertical $\bar{B}$ stabilizes the low frequency interval while the horizontal $\bar{B}$ stabilizes the high frequency interval, and the growth rate of growing modes is bounded.