Nonlinear stability of the ideal magnetohydrodynamic interchange mode at marginal conditions in a transverse magnetic field

TL;DR

This paper analyzes the nonlinear stability of the ideal MHD interchange mode near marginal conditions, revealing that the system can be nonlinearly unstable even when linearly stable, with simulation results confirming the analytical predictions.

Contribution

It provides a third-order perturbation analysis of nonlinear effects on MHD interchange stability at marginal conditions, including the derivation of critical amplitude scaling.

Findings

Nonlinear instability occurs in the short wavelength limit.

Critical amplitude scales as |b_2/B_c|^{1/2}.

Simulation results agree with analytical predictions.

Abstract

The stability of the ideal magnetohydrodynamic (MHD) interchange mode at marginal conditions is studied. A sufficiently strong constant magnetic field component transverse to the direction of mode symmetry provides the marginality conditions. A systematic perturbation analysis in the smallness parameter, $|b_2/B_c|^{1/2}$, is carried out, where $B_c$ is the critical transverse magnetic field for the zero-frequency ideal mode, and $b_2$ is the deviation from $B_c$. The calculation is carried out to third order including nonlinear terms. It is shown that the system is nonlinearly unstable in the short wavelength limit, i.e., a large enough perturbation results in instability even if $b_2/B_c>0$ (linearly stable). The normalized amplitude for instability is shown to scale as $|b_2/B_c|^{1/2}$. A nonlinear, compressible, MHD simulation is done to check the analytic result. Good agreement is found, including the critical amplitude scaling.