Short wavelength magnetic buoyancy instability

TL;DR

This paper analyzes the short wavelength magnetic buoyancy instability in a compressible isothermal fluid with a decreasing horizontal magnetic field, revealing how perturbations localize vertically and identifying the most unstable modes.

Contribution

It provides a detailed Rayleigh-Schr"odinger perturbation analysis of the instability at large horizontal wavenumbers, extending previous work to three-dimensional perturbations.

Findings

Fastest growing modes become vertically localized at high wavenumbers

Two-dimensional perturbations are most unstable with strong magnetic field gradients

Analysis is supported by comparison with initial value problem simulations

Abstract

Magnetic buoyancy instability plays an important role in the evolution of astrophysical magnetic fields. Here we revisit the problem introduced by \citet{Gilman_1970} of the short wavelength linear stability of a plane layer of compressible isothermal fluid permeated by a horizontal magnetic field of strength decreasing with height. Dissipation of momentum and magnetic field is neglected. By the use of a Rayleigh-Schr\"odinger perturbation analysis, we explain in detail the limit in which the transverse horizontal wavenumber of the perturbation, denoted by $k$, is large (i.e.\ short horizontal wavelength) and show that the fastest growing perturbations become localized in the vertical direction as $k$ is increased. The growth rates are determined by a function of the vertical coordinate $z$ since, in the large $k$ limit, the eigenmodes are strongly localized in the vertical direction. We consider in detail the case of two-dimensional perturbations varying in the directions perpendicular to the magnetic field, which, for sufficiently strong field gradients, are the most unstable. The results of our analysis are backed up by comparison with a series of initial value problems. Finally we extend the analysis to three-dimensional perturbations.