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

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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, |b₂/Bc|¹/², is carried out, where Bc is the critical transverse magnetic field for the zero-frequency ideal mode, and b₂ is the deviation from Bc. 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 b₂/Bc>0 (linearly stable). The normalized amplitude for instability is shown to scale as |b₂/Bc|¹/². A nonlinear, compressible, MHD simulation is done to check the analytic result. Good agreement is found, including the critical amplitude scaling.