On Linear Instability and Stability of the Rayleigh-Taylor Problem in Magnetohydrodynamics

TL;DR

This paper analyzes how magnetic fields influence the stability of the Rayleigh-Taylor problem in magnetohydrodynamics, revealing that strong horizontal magnetic fields can stabilize the instability similarly to vertical fields.

Contribution

It establishes a new instability/stability criterion for the linearized magnetic Rayleigh-Taylor problem and uncovers the stabilizing role of strong horizontal magnetic fields.

Findings

A criterion for instability/stability based on magnetic field strength.

Horizontal magnetic fields can stabilize the Rayleigh-Taylor instability.

Large magnetic fields stabilize the magnetic buoyancy (Parker) problem.

Abstract

We investigate the stabilizing effects of the magnetic fields in the linearized magnetic Rayleigh-Taylor (RT) problem of a nonhomogeneous incompressible viscous magnetohydrodynamic fluid of zero resistivity in the presence of a uniform gravitational field in a three-dimensional bounded domain, in which the velocity of the fluid is non-slip on the boundary. By adapting a modified variational method and careful deriving \emph{a priori} estimates, we establish a criterion for the instability/stability of the linearized problem around a magnetic RT equilibrium state. In the criterion, we find a new phenomenon that a sufficiently strong horizontal magnetic field has the same stabilizing effect as that of the vertical magnetic field on growth of the magnetic RT instability. In addition, we further study the corresponding compressible case, i.e., the Parker (or magnetic buoyancy) problem, for which the strength of a horizontal magnetic field decreases with height, and also show the stabilizing effect of a sufficiently large magnetic field.