Variational method for the nonlinear dynamics of an elliptic magnetic stagnation line

The nonlinear evolution of the kink instability of a plasma with an elliptic magnetic stagnation line is studied by means of an amplitude expansion of the ideal magnetohydrodynamic equations. Wahlberg et al. [12] have shown that, near marginal stability, the nonlinear evolution of the stability can be described in terms of a two-dimensional potential U(X,Y), where X and Y represent the amplitudes of the perturbations with positive and negative helical polarization. The potential U(X,Y) is found to be nonlinearly stabilizing for all values of the polarization. In our paper a Lagrangian and an invariant variational principle for two coupled nonlinear ordinal differential equations describing the nonlinear evolution of the stagnation line instability with arbitrary polarization are given. Using a trial function in a rectangular box we find the functional integral. The general case for the two box potential can be obtained on the basis of a different ansatz where we approximate the Jost function by polynomials of order n instead of a piecewise linear function. An example for the second order is given to illustrate the general case. Some considerations concerning solar filaments and filament bands (circular or straight) are indicated as possible applications besides laboratory experiments with cusp geometry corresponding to quadripolar cusp geometries for some clouds and thunderstorms.