Amplitude Equations for Electrostatic Waves: universal singular behavior in the limit of weak instability

Abstract

An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the unstable manifold of the equilibrium $F_0(v)$.\\ The mode eigenvalue arises from a simple zero of the dielectric ${\epsilon }_k(z)$; as the linear growth rate $\gamma$ vanishes, the eigenvalue merges with the continuous spectrum on the imaginary axis and disappears. The evolution of the mode amplitude $\rho (t)$ is studied using an expansion in $\rho$. As $\glim$, the expansion coefficients diverge, but these singularities are absorbed by rescaling the amplitude: $\rho (t)\equiv {\gamma }^{2}r(\gamma t)$. This renders the theory finite and also indicates that the electric field exhibits trapping scaling $E\sim {\gamma }^2$. These singularities and scalings are independent of the specific $F_0(v)$ considered. The asymptotic dynamics of $r(\tau )$ depends on $F_0$ only through $\exp i\xi$ where $d{\epsilon }_k/dz=|{\epsilon }_k^{\prime }|\exp -i\xi /2$. Similar results also hold for the electric field and distribution function.

79 pages (Latex/RevTex), 4 figures available in hard copy from the Author ([email protected]). Paper has been accepted at Physics of Plasmas

References in corpus (2)

• Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators
• Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode

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