Nonlinear oscillations and nonstationary flow in a zero temperature plasma: Part II. General characteristics of the motion

Abstract

General nonlinear effects in plasma motion are investigated on the basis of a previous treatment. (1) A travelling-wave solution is studied and the adequate boundary and initial conditions for its excitation are investigated. The dispersion relation in approximation of any order is the same as in the linear theory. It can be shown, however, in virtue of the perturbation method applied, that in the case of a velocity distribution the result does not hold and nonlinear effects alter the dispersion relation. (2) An equipartition is derived between the particle energy and field energy. (3) The development of an original wave number and frequency spectrum is calculated. A procedure is outlined to calculate contribution of any order to the spectrum. The spectrum can be analyzed into higher harmonics of ω 0 and the only nonlinear effect in the development of the wave-number spectrum is the incidence of higher harmonics and no spectral decay or nonlinear dispersion relation exists. An analysis of the time variation of the energy content of the spectrum indicates a behavior analogous to the “constancy of big eddies” in hydrodynamical turbulence. The differences in the development of the frequency spectrum are in the lack of simple harmonic relation between wave numbers which transfer different parts of the spectrum, and in a relative stability of frequencies in the neighborhood of ω = ω 0 instead of κ = 0.