Parametric instabilities of circularly polarized smallamplitude Alfvén waves in Hall plasmas
Abstract
We study the stability of circularly polarized Alfvén waves (pump waves) in Hall plasmas. First we rederive 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 nondimensional 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 righthand 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 lefthand 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 a^{2}. The second type is the decay instability. It has the same properties as in the case of righthand 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 a^{2} are unstable. The increment of this instability is proportional to a^{2}, except for τ close to τ_{c} when it is proportional to a, where τ_{c} is a function of b.
 Publication:

Journal of Plasma Physics
 Pub Date:
 February 2008
 DOI:
 10.1017/S0022377807006691
 Bibcode:
 2008JPlPh..74..119R