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

We consider the stability of a circularly polarized Alfven wave (the p ump wave) which propagates parallel to the ambient magnetic field. Onl y parallel-propagating perturbations are considered, and we ignore dis persive effects due to the ion cyclotron frequency. The dissipationles s MHD equations are used throughout; thus possibly important effects a rising from Landau and transit time damping are omitted. We derive a s eries of analytical approximations to the dispersion relation using A = DELTAB/B0)2 as a small expansion parameter; DELTAB is the pump ampli tude, 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 paramet ric instabilities of the pump. If 0 < beta < 1 we find the familiar re sult that the pump decays into a forward propagating sound wave and a backward propagating Alfven wave with maximum growth rate gamma(max) i s-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; more over, if beta almost-equal-to 0 there is a nearly purely growing insta bility. In contrast to the familiar decay instability, for which the b ackward propagating Alfven wave has lower frequency and wavenumber tha n the pump, we find that if beta greater than or similar to 1 the inst ability is really a beat instability which is dominated by a transvers e wave which is forward propagating and has frequency and wavenumber w hich are nearly twice the pump values. Only the decay instability for 0 < beta < 1 can be regarded as producing two recognizable normal mode s, namely, a sound wave and an Alfven wave. We discuss how the differe nt characteristics of the instabilities may affect the evolution of Al fven 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 t oo long for these instabilities to have an appreciable effect inside 1 AU.