Self-consistent chaos in the beam-plasma instability

Abstract

The effect of self-consistency on Hamiltonian systems with a large number of degrees of freedom is investigated for the beam-plasma instability using the single-wave model of O'Neil, Winfrey, and Malmberg. The single-wave model is reviewed and then rederived within the Hamiltonian context, which leads naturally to canonical action-angle variables. Simulations are performed with a large (10 4) number of beam particles interacting with the single wave. It is observed that the system relaxes into a time asymptotic periodic state where only a few collective degrees are active; namely, a clump of trapped particles oscillating in a modulated wave, within a uniform chaotic sea with oscillating phase space boundaries. Thus self-consistency is seen to effectively reduce the number of degrees of freedom. A simple low degree-of-freedom model is derived that treats the clump as a single macroparticle, interacting with the wave and chaotic sea. The uniform chaotic sea is modeled by a fluid waterbag, where the waterbag boundaries correspond approximately to invariant tori. This low degree-of-freedom model is seen to compare well with the simulation.