Nonlinear Interchange Modes Near Marginal Stability

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The nonlinear stability of the ideal magnetohydrodynamic interchange mode, at marginal conditions, is studied. The interchange mode is made to be at marginal conditions by providing a constant magnetic field that is just sufficiently strong enough to balance the mode growth. How nonlinearity affects the stability of the interchange mode is analyzed for three different systems. We first consider introducing small amplitude perturbations on a two-dimensional system. We show that if the fractional deviation from marginality is given by a small parameter b, then perturbation amplitudes of order b1/2 can cause the system to become nonlinearly unstable. The analysis is corroborated by a nonlinear, compressible, magnetohydrodynamic simulation that shows excellent agreement with the result, including the amplitude scaling. We then extend the analysis to a three-dimensional system where, we show that, the perturbations separate into two different modes. The first mode is shown to be isomorphic to the two-dimensional case and, thus, has the same dynamics, i.e. nonlinearly unstable; however, we show that the second mode is nonlinearly stable. The latter modes are shown to satisfy line-tied boundary conditions. The third system we consider is a two-dimensional system with perturbations introduced as deformations of the boundaries. We show that these small distortions can penetrate deep in the magnetized plasma and become globally amplified. The amplification is shown to be inversely proportional to b. Additionally, we show that nonlinearities can cause the system to become unstable for distortions of order b3/2.