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

摘要

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.