Theory of Strong Trapped-Particle Turbulence

Amodelis presented for the final state of the streaming instability driven by a DC-electric field in a cold collisionless plasma. This state consists of large-amplitude travelling electric field fluctuations, which limit electron runaway by trapping most electrons. Poisson' s equation dictates the existence of a few untrapped electrons, which may be artibrarily presumed to be in a strip just outside the separatrix. without significantly influencing the velocity or energy of the turbulent wave. The ions are considered in the adiabatic approximation. The time dependence of this state is examined through an ansatz of multiple time scaling. One finds a transition between adjacent metastable states, which proceeds slowly compared to the time scale of wave motion, but rapidly compared to the time scale of wave decay. The trapping of the bulk electrons appears to be dynamically maintained, with the ions remaining marginally untrapped. The thermodynamic properties of these quasi-stationary states are computed in detail after first obtaining the distribution of the trapped electrons as a function of the electric potential according to the Bernstein-Greene-Kruskal formalism. Although a sinusoidal potential is used as a first example, the quasi-linear effects of short-wavelength instabilities on the BGK background state allow the calculation of the actual wave shape. The wave frame distributions of electrons and ions, together with the condition for momentum conservation, are sufficient to determine the ratio of the ion sound speed to the wave phase velocity, and the ratio of the potential energy to the kinetic energy of electrons or ions. Significantly, the ion sound speed is found to exceed the phase velocity. The transition between metastable states further entails continued heating of the trapped particles and growth of the wave, while at the same time the phase velocity of the wave slowly declines. The coherent sloshing energy of the ions approaches 40% of the electron thermal energy. If this asymptotic state can be reached experimentally, the consequences will be: (1) much higher temperatures than previously anticipated can be reached; (2) rapid penetration of electromagnetic fields at speeds above even that of the usual turbulent diffusive velocity. (author)