A weak turbulence analysis of the two-stream instability

In this investigation, we study the effects of non-linear interaction of plasma waves in the weak turbulence approximation on a stream-plasma system which is subject to the two-stream instability. The momentum distribution of the stream is assumed to be a Gaussian function and the background plasma is assumed to be a Maxwellian with a temperature T. The non-linear wave-particle interaction of induced scattering on the polarization clouds of ions may lead to stabilization of the stream—plasma waves in the resonant cone are scattered into the nonresonant region. The non-linear wave-particle interaction of induced scattering on the polarization clouds of ions may lead to stabilization of the stream; plasma waves in the resonant cone are scattered into the non-resonant region and vice versa, leading to a well-defined average energy density in plasma waves which is much less than the energy density of the stream. Non-linear scattering also causes transformation of plasma waves into electromagnetic radiation near the fundamental of the electron plasma frequency of the background plasma, and combination of two plasma waves leads to electromagnetic radiation near the second harmonic of this frequency. Expressions are given for the latter process for emission and absorption for arbitrary values of the ratio of the wave number of plasma waves to the wave number of the electromagnetic wave. The major features of the whole phenomenon are illustrated with the help of a numerical example, with conditions typical for the solar corona in which case an ion stream would be stabilized, but an electron or neutral stream (with the chosen density) would not.