Parametric instabilities of circularly polarized small-amplitude Alfvén waves in Hall plasmas

Abstract

AbstractWe study the stability of circularly polarized Alfvén waves (pump waves) in Hall plasmas. First we re-derive the dispersion equation governing the pump wave stability without making anad hocassumption about the dependences of perturbations on time and the spatial variable. Then we study the stability of pump waves with small non-dimensional amplitudea(a≪ 1) analytically, restricting our analysis tob< 1, wherebis the ratio of the sound and Alfvén speed. Our main results are the following. The stability properties of right-hand polarized waves are qualitatively the same as in ideal MHD. For any values ofband the dispersion parameter τ they are subject to decay instability that occurs for wave numbers from a band with width of ordera. The instability increment is also of ordera. The left-hand polarized waves can be subject, in general, to three different types of instabilities. The first type is the modulational instability. It only occurs whenbis smaller than a limiting value that depends on τ. Only perturbations with wave numbers smaller than a limiting value of orderaare unstable. The instability increment is proportional toa2. The second type is the decay instability. It has the same properties as in the case of right-hand polarized waves; however, it occurs only whenb< 1 τ. The third type is the beat instability. It occurs for any values ofband τ, and only perturbations with the wave numbers from a narrow band with the width of ordera2are unstable. The increment of this instability is proportional toa2, except for τ close to τcwhen it is proportional toa, where τcis a function ofb.