Surface explosive instabilities in a beam plasma system

Abstract

Explosive instability due to non?linear wave–wave interaction is one of the most interesting phenomena in non?linear plasma theory. Since it is widely recognized that plasma, either in nature or in laboratory, is bounded in one way or the other, in the present work the author examines the explosive instabilities of a bounded plasma and their saturation through the mechanism of a non?linear frequency shift. The thesis starts with the description of the motivation and summary of the work, in the introductory chapter. In the second chapter, the high?frequency three?wave non?linear “explosive” interaction of the surface modes of a semi?infinite beam–plasma system under no external field is investigated. The conditions that favour non?linear instability keep the plasma linearly stable. The beam runs parallel to the surface and the density of beam (n?) is less than the density of plasma (n). The explosion time of bulk and surface waves is determined. A comparative study of the explosion time of the bulk and surface waves for different cases of wave propagation reveals some interesting results. If at least one of the three wave vectors of the surface modes is parallel to the beam, explosive interaction at the surface takes place after it has happened in the plasma bulk, provided the bulk waves propagate almost perpendicular to the surface and are of short wavelength. On the other hand, if bulk modes have long wavelength and are parallel to the surface, the surface modes can explode first. In the third chapter, the study of the saturation of explosive instability due to non?linear frequency shift is taken up. This chapter is devoted to the derivation of surface and bulk coupling coefficients for four?wave interaction. For four?wave interaction, only those three waves creating explosive instabilities (as discussed in the previous chapter) are taken into consideration along with the fourth one which is the complex conjugate of any of those three waves. The non?linear frequency shift for both the bulk and surface waves is calculated. The frequency shift is found to be different for different cases of wave propagation of the bulk as well as surface waves. Since it is well known that a frequency mismatch in resonance conditions can stabilize an explosive instability, this non?linear frequency shift can saturate the explosively growing amplitudes of the bulk and surface waves. Secondly, a change in beam density within the limit n? < n can saturate explosive instabilities for different types of wave propagation along the plane of the boundary surface. Increasing the beam density, the parallel waves saturate earlier than the antiparallel wave (k?? > k?? — where k?? and k?? are the wave vectors of the antiparallel wave lying along the boundary surface). Within a small variation of beam density this is followed by the saturation of the antiparallel wave (k?? < k??). In the concluding chapter, some of the main results of this thesis are briefly reviewed.