Parametric instabilities of parallel-propagating Alfven waves: Some analytical results

Details Parametric instabilities of parallel-propagating Alfven waves: Some analytical results Parametric instabilities of parallel-propagating Alfven waves: Some analytical results

Nov 1993

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1993jgr....9819049j&link_type=abstract

Journal of Geophysical Research (ISSN 0148-0227), vol. 98, no. A11, p. 19,049-19,063

Computer Science

Sound

38

Beat Frequencies, Beta Factor, Circular Polarization, Magnetohydrodynamic Stability, Magnetohydrodynamic Waves, Wave Attenuation, Wave Dispersion, Wave Propagation, Interplanetary Medium, Magnetohydrodynamic Turbulence, Solar Activity, Solar Wind

Scientific paper

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 possibibly important effects arising from Landau and transit time damping are omitted. We derive a series of analytical approximations to the dispersion relation using A = (Delta B/BO)2 as a small expansion parameter; Delta B is the pump amplitude, and BO 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 less than beta less than 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 gammamax varies A1/2, but beta cannot be too close to 0 or to 1. If beta approx. 1, we find gammamax varies A3/4, if beta greater than 1, we find gammamax varies A3/2, while if beta approx. 0, we obtain gammamax varies A1/3; moreover, if beta approx. 0 there is a nearly purely growing instability. In constrast 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 approx. equal 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 less than beta less than 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 approx. 1 the growth times of the instabilities are probably too long for these instabilities to have an appreciable effect inside 1 AU.

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