NONLINEAR EVOLUTION OF AN OBLIQUELY PROPAGATING LANGMUIR WAVE

The transition between regular and stochastic motion which occurs when resonances overlap plays an important role in present-day plasma physics research, and has recently been the subject of intensive study. However, investigations to date have been primarily concerned with single-particle motion in given wave fields, and there has been little attempt to determine how particle motion feeds back on the wave to affect its evolution. We explore this problem here in the simple context of an obliquely propagating Langmuir wave. Electrons are resonant with the wave whenever v(,'') = ((omega)-n(OMEGA))/k(,''), where n is any integer.

This wave has three distinct evolution regimes whose locations in parameter space are given roughly by the following relations: If (omega)(,b) (the electron bounce frequency) < (gamma) (the linear damping rate), then linear theory is valid, and the wave damps away. If (gamma) < (omega)(,b) < (OMEGA) (the cyclotron frequency), then electrons can be trapped in individual resonances. In this trapping regime, the wave evolution can be treated in a fashion analogous to O'Neil's treatment of a parallel-propagating wave. We find that the amplitude oscillations disappear at propagation angles greater than 14(DEGREES) due to a "super-phase-mixing" of the many bounce frequencies. Finally, if (gamma) < (OMEGA) < (omega)(,b), then resonances overlap and electron motion is stochastic. In this regime, particle motion is nearly diffusive in the region of velocity space where resonances overlap, and the wave evolution can be treated using quasilinear theory with resonance broadening. However, because most of the stochastic electrons are initially on the edge of the stochastic region in velocity space, where large regular regions exist, their motion is not entirely diffusive, and this theory's results, while qualitatively useful, are not quantitatively accurate, as comparison with numerical simulation shows.

In the transition regime between regular and stochastic motion, the electron orbits are complicated and an analytic solution does not appear possible. It is possible, however, to determine the asymptotic total damping of the wave using two different approaches. In the first, we assume that the distribution function is asymptotically flattened over the resonant regions. In the second, we use "mini-simulations," following the orbits of the resonant electrons numerically and treating the rest of the electrons as a background linear dielectric medium. Using this approach, the wave evolution is determined self-consistently by updating the field at each time step. The two methods lead to asymptotic amplitudes which agree to within a factor of two, and the wave's total damping is found to increase significantly when a transition is made between the regular and stochastic regimes, i.e. when resonances overlap.

In order to make contact with possible experiments, an idealized boundary-value problem in which (omega) and k(,(PERP)) are fixed and k(,'') is then determined by the plasma is considered. The principal qualitative results which should be observable in experiments and full-particle simulations are found also in the boundary-value problem.

We restate the principal qualitative results: (1) the amplitude oscillations in the trapping regime disappear due to super-phase-mixing of the many bounce frequencies when the angle of propagation is increased past 14(DEGREES); (2) the wave exhibits increased total damping when a transition is made between the regular and stochastic regimes.