Numerical simulation of the interaction of a fast electron beam moving through a bounded plasma

A two-dimensional Particle-In-Cell simulation program which assumes the potential to be equal to zero on the boundary and which allows particle injection and absorption has been developed and applied to the study of the Coulomb interaction between an electron plasma and an injected electron beam. The object is to obtain a more realistic simulation of experiments in which electron beams are injected into plasmas. Previous analyses and simulations considered the beam to exist throughout the plasma and use completely periodic boundary conditions which constitute a powerful but unphysical form of feedback, as well as a neglect of possibly dominant initial injection transients. The initial small amplitude coherent longitudinal oscillation seen for a weak injected beam is observed to have both a slightly smaller frequency and a much smaller temporal growth rate than that predicted using the standard one-dimensional linear theory for an infinite system containing a uniform beam and a real wave vector initial perturbation. Including the boundary conditions and the effect of finite size of the simulation particles, solutions are found with four complex wave vectors for each frequency, each wave satisfying the linear dispersion relation and the combination of four the boundary conditions. The simulation results agree very well with the most unstable four wave solution's frequency, temporal grovrth rate, wave number and spatial growth. A small curvature is observed on the wave front and this is explained by a simplified two-dimensional model which otherwise has small effect on the results. A nonlinear oscillation pileup phenomenon (the strong oscillation region moves slowly back towards the injection pla ) is also observed in agreement with the predictions of the semiquantitative theory of Fainberg and Shapiro, using products of linear terms. Probably the most significant result of the nonlinear simulation is the verification of the standard formulae for the temporal growth of oscillations and related phenomena as a function of the beam velocity distribution function with both hydrodynamic and quasilinear conditions with a fairly large degree of nonlinearity. (The beam temperature however often shows signs of a limiting or power-law time behaviour—somewhat surprisingly like t[raised 2] ). The transition from hydrodynamic to quasilinear behaviour can be seen but usually only the most deeply penetrating parts of the beam at times late in a run have reached a velocity plateau stage of something like quasilinear equilibrium. In the transition period no particle trapping is seen and the particles usually manage to spill backwards from a wave crest to the next trough behind. This is probably due to a number of factors. The strong fields near the injection plane where some coherence is enforced by the boundary give the incoming particles a fairly large spread in velocity but the particles drift on into weaker random phase field regions so trapping does not occur. Also the two-dimensional dynamics allow sideways scattering from wave crests, and in fact the infinite magnetic field results (no y motion) show more siscns of trapping. Generally after the quasilinear regime is reached in the beam phase space, strips are observed moving and turning ballistically into the plasma, but occasionally there are smaller kinks in beam phase space plots which might be related to the clumps recently discussed by theorists.Physics, Department o