Modulational instability of finite amplitude dispersive Alfvén waves

Abstract

The stability of a finite amplitude plane monochromatic circularly polarized Alfvén wave is studied by using an approximate two‐fluid model obtained by performing an amplitude and dual time scale expansion in which temporal changes, observed when moving with the wave, are assumed slow. The lowest order dispersive effects and coupling terms to sound waves are of the same order. A large‐amplitude Alfvén wave obeying an approximate dispersion relation is an exact solution of the resulting model equations. Small perturbations of this solution consisting of two sideband Alfvén waves and a sound wave are then introduced. The resulting dispersion relation, which is of fourth order, is examined analytically for small wave amplitudes of the parent wave. It reveals different regions of stability and instability in a plot of wave number k0 of the unperturbed wave versus β0, the ratio of sound speed to Alfvén speed. For long wave lengths, the left‐hand polarized mode is found to be stable for β0 > 1 and the right‐hand polarized mode for β0 < 1. Modulational instability of the former mode occurs for β0 < 1 and of the latter for β0 > 1. Decay instability is predicted near β0 = 0 and β0 = 1. Formulas for growth rates and unstable wave number bands are given.