PARAMETRIC-INSTABILITIES OF PARALLEL-PROPAGATING ALFVEN WAVES - SOME ANALYTICAL RESULTS

We consider the stability of a circularly polarized Alfven wave (the pump wave) which propagates parallel to the ambient magnetic field. Only parallel-propagating perturbations are considered, and we ignore dispersive effects due to the ion cyclotron frequency. The dissipationless MHD equations are used throughout; thus possibly important effects arising from Landau and transit time damping are omitted. We derive a series of analytical approximations to the dispersion relation using A = DELTAB/B0)2 as a small expansion parameter; DELTAB is the pump amplitude, and B0 is the ambient magnetic field strength. We find that the plasma beta (the square of the ratio of the sound speed to the Alfven speed) plays a crucial role in determining the behavior of the parametric instabilities of the pump. If 0 < beta < 1 we find the familiar result that the pump decays into a forward propagating sound wave and a backward propagating Alfven wave with maximum growth rate gamma(max) is-proportional-to A1/2 , but beta cannot be too close to 0 or to 1. If beta almost-equal-to 1, we find gamma(max), is-proportional-to A3/4; if beta > 1, we find gamma(max) is-proportional-to A3/2, while if beta almost-equal-to 0, we obtain gamma(max) is-proportional-to A1/3; moreover, if beta almost-equal-to 0 there is a nearly purely growing instability. In contrast to the familiar decay instability, for which the backward propagating Alfven wave has lower frequency and wavenumber than the pump, we find that if beta greater than or similar to 1 the instability is really a beat instability which is dominated by a transverse wave which is forward propagating and has frequency and wavenumber which are nearly twice the pump values. Only the decay instability for 0 < beta < 1 can be regarded as producing two recognizable normal modes, namely, a sound wave and an Alfven wave. We discuss how the different characteristics of the instabilities may affect the evolution of Alfven waves in the solar wind. However, for a solar wind in which beta almost-equal-to 1 the growth times of the instabilities are probably too long for these instabilities to have an appreciable effect inside 1 AU.