Beat, modulational, and decay instabilities of a circularly polarized Alfvén wave

A circularly polarized low‐frequency electromagnetic pump wave propagating along an ambient magnetic field is known to be unstable to the growth of several parallel‐propagating parametric instabilities. If ion‐cyclotron effects are retained in a two‐fluid description, the dispersion relation is a sixth‐order polynomial. We present a series of new analytical approximations to this dispersion relation. We emphasize new results for the beat instability that occurs as an interaction of the forward propagating upper sideband with the backward propagating lower sideband. The nature of the beat instability depends on β = (υ sound /υ A )² and on the sense of polarization of the pump wave. The beat and decay instabilities can occur together if the pump is left‐handed (i.e., ion resonant) and if β ≲ 1, but they cannot occur together if the pump is right‐handed. For a left‐handed pump the beat mode is the only instability if β ≳ 1. If the pump is right‐handed and β ≳ 1, then the beat instability exists only when the pump amplitude exceeds a threshold value, and the beat will be the only instability if the pump amplitude is large enough to stabilize the modulational instability. If the pump is left‐handed and β ≲ 1, then the beat mode is stabilized when the pump amplitude becomes sufficiently large. The beat instability primarily produces a forward propagating transverse wave in the upper sideband. Thus if β ≳ 1, the instabilities considered here do not produce the backward propagating waves which are thought to affect turbulence and the evolution of cross helicity in the solar wind. New analytical results are presented also for the decay and modulational instabilities when β ≈ 1.