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

We 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 an ad hoc assumption about the dependences of perturbations on time and the spatial variable. Then we study the stability of pump waves with small non-dimensional amplitude a (a 1) analytically, restricting our analysis to b < 1, where b is 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 of b and the dispersion parameter τ they are subject to decay instability that occurs for wave numbers from a band with width of order a. The instability increment is also of order a. 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 when b is smaller than a limiting value that depends on τ. Only perturbations with wave numbers smaller than a limiting value of order a are unstable. The instability increment is proportional to a2. 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 when b < 1 τ. The third type is the beat instability. It occurs for any values of b and τ, and only perturbations with the wave numbers from a narrow band with the width of order a2 are unstable. The increment of this instability is proportional to a2, except for τ close to τc when it is proportional to a, where τc is a function of b