Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode

Abstract

An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the two-dimensional unstable manifold of the equilibrium. The mode amplitude $ρ(t)$ decouples from the phase due to the spatial homogeneity of the equilibrium, and the resulting one-dimensional dynamics is analyzed using an expansion in $ρ$. As the linear growth rate $γ$ vanishes, the expansion coefficients diverge; a rescaling $ρ(t)\equivγ^2\,r(γt)$ of the mode amplitude absorbs these singularities and reveals that the mode electric field exhibits trapping scaling $|E_1|\simγ^2$ as $γ\rightarrow0$. The dynamics for $r(τ)$ depends only on the phase $e^{iξ}$ where $dε_{k} /dz=|{ε_{k}}|e^{-iξ/2}$ is the derivative of the dielectric as $γ\rightarrow0$.