ON THE DISPERSION-RELATIONS FOR PARAMETRIC-INSTABILITIES OF PARALLEL-PROPAGATING ALFVEN WAVES

We consider the dispersion relation for the parametric instabilities of large-amplitude, circularly polarized Alfven waves, propagating parallel to the ambient magnetic field. A linear perturbation analysis is employed, and the perturbations are taken to propagate along the ambient field. The standard analysis which has been used previously assumes that density perturbations vary as exp [i(kz - omegat)]; this defines the meaning of omega and k. However, the differential equations have periodic coefficients, implying that Floquet analysis should be used. We here present an analysis based on Floquet's theorem. The result is a hierarchy of dispersion relations. However, all the dispersion relations are found to be equivalent to the one obtained via the standard analysis; the differences between them are due only to how omega and k are defined. Thus we conclude that physically there is really only one dispersion relation, namely the ''electrostatic dispersion relation,'' which is in agreement with earlier works. However, we disagree with Vinas and Goldstein (1991b), who obtained additional dispersion relations which they have called the ''electromagnetic dispersion relations.'' Their additional dispersion relations are a consequence of first truncating the dispersion relation for obliquely propagating perturbations and then taking the limit of parallel-propagating perturbations.