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 $\rho (t)$ decouples from the phase due to the spatial homogeneity of the equilibrium, and the resulting one-dimensional dynamics is analyzed using an expansion in $\rho$. As the linear growth rate $\gamma$ vanishes, the expansion coefficients diverge; a rescaling $\rho (t)\equiv {\gamma }^{2}r(\gamma t)$ of the mode amplitude absorbs these singularities and reveals that the mode electric field exhibits trapping scaling $|E_1|\sim {\gamma }^2$ as $\gamma \to 0$. The dynamics for $r(\tau )$ depends only on the phase $e^{i\xi }$ where $d{\epsilon }_k/dz=|{\epsilon }_k|e^{-i\xi /2}$ is the derivative of the dielectric as $\gamma \to 0$.

11 pages (Latex/RevTex), 2 figures available in hard copy from the Author ([email protected]); paper accepted by Physical Review Letters

References in corpus (1)

• Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators

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