Non-linear effects in electron plasmas

Abstract

In this investigation of non-linear effects in electron plasmas, the complications of finite boundaries, external fields, non-zero temperature, multi-stream flow and collisions are ignored. Non-linear effects are studied by a perturbation process, the method of variation of parameters, which yields the dominant effects in weakly excited plasmas. The effects fall into three groups: the excitation of harmonics; coherent interaction; and incoherent interaction. 'Coherent’ interaction represents all effects which do not result in a redistribution of energy in wave-number space. Such effects, which arise also in the familiar theory of electron plasmas at non-zero temperature, may be summarized by a dispersion relation. The principal aim of this work has been to study the incoherent inter­action which is responsible for spectral decay, that is for the redistribution of energy in wave-number space—the breakdown of organized large-scale motion into disorganized small-scale motion. It is found that, to a certain stage of approximation, purely one-dimensional spectra are unique in exhibiting no spectral decay. The decay of a certain spectrum which is almost one-dimensional may be followed analytically and leads to the following result: plasma oscillations which initially have depth of modulation D and are collimated within an angle ε tend to isotropy in (4 ε D 2 ) -1 plasma periods. This effect may be attributed to a ‘plasma viscosity’ with coefficient ½ ω -1 P < v 2 >. The incoherent interaction between plasma oscillations and transverse (non-divergent) motion may be interpreted as viscosity or as a scattering process, depending upon the relative scales of the components. The effects studied in this paper may be interpreted as the result of collisions between plasmons; this interpretation illuminates the problem of spectral decay and suggests that the rate of decay of a one-dimensional spectrum is proportional to D 4 . Estimates of the damping of plasma oscillations by the mechanism of spectral decay are compared with the results of certain experiments.