A nonlinear theory of the parallel firehose and gyrothermal instabilities

Weakly collisional magnetized plasmas tend to develop pressure anisotropies which trigger fast ($\sim$ ion cyclotron period) plasma instabilities at scales between the ion Larmor radius $\rho_i$ and the mean free path $\lambda_{mfp}$. These can dramatically affect the global ($\gg\lambda_{mfp}$) dynamics and their nonlinear evolution should drive pressure anisotropies towards marginal stability values, controlled by the plasma beta $\beta_i$. This nonlinear evolution is worked out in an {\em ab initio} kinetic calculation for the parallel ($k_\perp=0$) firehose instability in a high-beta plasma. We use a particular physical asymptotic ordering to derive a closed nonlinear equation for the firehose turbulence, which we solve. We find secular ($\propto t$) growth of magnetic fluctuations and a $k_\parallel^{-3}$ spectrum, starting at scales $\ga \rho_i$. When a parallel ion heat flux is present, the parallel firehose instability mutates into the new {\em gyrothermal instability}. Its nonlinear evolution also involves secular magnetic energy growth, but its spectrum is eventually dominated by modes with a maximal scale $\sim\rho_il_T/\lambda_{mfp}$, ($l_T$ is the parallel temperature gradient scale).