Parametric instabilities of parallel‐propagating Alfvén waves: Some analytical results

<jats:p>We consider the stability of a circularly polarized Alfvén 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 possibly important effects arising from Landau and transit time damping are omitted. We derive a series of analytical approximations to the dispersion relation using <jats:italic>A</jats:italic> = (Δ<jats:italic>B</jats:italic>/<jats:italic>B</jats:italic><jats:sub>0</jats:sub>)² as a small expansion parameter; Δ<jats:italic>B</jats:italic> is the pump amplitude, and <jats:italic>B</jats:italic><jats:sub>0</jats:sub> is the ambient magnetic field strength. We find that the plasma β (the square of the ratio of the sound speed to the Alfvén speed) plays a crucial role in determining the behavior of the parametric instabilities of the pump. If 0 < β < 1 we find the familiar result that the pump decays into a forward propagating sound wave and a backward propagating Alfvén wave with maximum growth rate γ<jats:sub>max</jats:sub> ∝ <jats:italic>A</jats:italic><jats:sup>1/2</jats:sup>, but β cannot be too close to 0 or to 1. If β ≈ 1, we find γ<jats:sub>max</jats:sub>, ∝ <jats:italic>A</jats:italic><jats:sup>3/4</jats:sup>; if β > 1, we find γ<jats:sub>max</jats:sub> ∝ <jats:italic>A</jats:italic><jats:sup>3/2</jats:sup>, while if β ≈ 0, we obtain γ<jats:sub>max</jats:sub> ∝ <jats:italic>A</jats:italic><jats:sup>1/3</jats:sup>; moreover, if β ≈ 0 there is a nearly purely growing instability. In contrast to the familiar decay instability, for which the backward propagating Alfvén wave has lower frequency and wavenumber than the pump, we find that if β ≳ 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 < β < 1 can be regarded as producing two recognizable normal modes, namely, a sound wave and an Alfvén wave. We discuss how the different characteristics of the instabilities may affect the evolution of Alfvén waves in the solar wind. However, for a solar wind in which β ≈ 1 the growth times of the instabilities are probably too long for these instabilities to have an appreciable effect inside 1 AU.</jats:p>