Parametric instabilities of Alfven waves in the solar wind

We consider the stability of a circularly-polarized Alfven wave (the pump wave) propagating along a uniform ambient magnetic field B$\sb{\rm O}$. The system is linearly perturbed to study the stability of the Alfven wave. The perturbations are also assumed to propagate along the ambient field. Four different problems are addressed relating to the stability of the Alfven wave. The first involves using Floquet\u27s theorem to obtain a dispersion relation for studying the stability. The result is a hierarchy of dispersion relations. However, all the dispersion relations are found to be equivalent. This technique showed that some results of other workers are incorrect. This method is very useful to obtain a dispersion relation for obliquely propagating perturbations. The second problem is to obtain analytical approximations to the dispersion relation using A = ($\rm \Delta B/B\sb{O})\sp2$ as a small expansion parameter; $\Delta$B is the pump amplitude. The analysis shows the crucial role played by plasma beta ($\beta$) in determining the behavior of the parametric instabilities of the pump. Expressions for the growth rates are presented for four ranges of $\beta$. The polarizations are also computed to give some physical insight into the properties of the daughter waves (the modes generated as a result of the instability are called daughter waves). The third problem is to study the effects of streaming He$\sp{++}$. The growth rates for new instabilities due to streaming He$\sp{++}$ are presented as a function of plasma beta, pump wave frequency, and $\Delta$B. The studies show that these new instabilities could compete with the well known decay instability. The final problem is to develop a methodology to study kinetic effects on the instabilities. This was done by breaking the plasma into beams, and treating each beam as a fluid. The nonlinear fluid equations are solved iteratively to obtain the perturbed densities and velocities. These are then used to derive the kinetic dispersion relation for the decay instability. The dispersion relation obtained by this procedure agrees with a previously known result which was obtained by a different method