The parametric instability of Alfv\'en waves: effects of temperature anisotropy

We study the stability of large amplitude, circularly polarized Alfv\'en waves in an anisotropic plasma described by the double-adiabatic/CGL closure, and in particular the effect of a background thermal pressure anisotropy on the well-known properties of Alfv\'en wave parametric decay in Magnetohydrodynamics (MHD). Anisotropy allows instability over a much wider range of values of parallel plasma beta ($\beta_\parallel$) when $\xi=p_{0\bot}/p_{0\parallel} > 1$. When the pressure anisotropy exceeds a critical value, $\xi\geq\xi^*$ with $\xi^*\simeq2.7$, there is a new regime in which the parametric instability is no longer quenched at high $\beta_\parallel$ and in the limit $\beta_\parallel\gg1$ the growth rate becomes independent of $\beta_\parallel$. In the opposite case of $\xi< \xi^*$, the instability is strongly suppressed with increasing parallel plasma beta, similarly to the MHD case. We analyze marginal stability conditions for parametric decay in the $(\xi,\,\beta_\parallel)$ parameter space, and discuss possible implications for Alfv\'enic turbulence in the solar wind.