Four wave mixing of finite amplitude Alfven waves

Abstract

Summary form only given. Past treatments of coherent wave mixing of Alfven waves have generally been restricted to three wave interactions. This paper presents the first results that show that finite amplitude Alfven waves can also undergo four wave mixing. This observation opens the possibility of a variety of new low frequency nonlinear phenomena in plasmas. The wave mixing process is also unique in the sense that all the interacting waves are exact solutions of the nonlinear equation. We use the resistive derivative nonlinear Schrodinger (DNLS) equation to study the four wave interaction of Alfven waves. This equation describes the finite amplitude Alfven waves and solitons. It is shown that the frequencies and wave numbers of time harmonic solution of the DNLS equation satisfy the criterion for four wave mixing. Four such waves can, therefore, interact coherently with each other in the plasma. Assuming that the complex wave amplitudes evolve slowly in time and space, the evolution equations governing the wave amplitudes are derived. These equations are then used to identify quantities that remain invariant in time in a dissipationless plasma. One such quantity is the total wave magnetic energy. We also use the evolution equations and conservation relations to explore two applications. First, we investigate the evolution of two small amplitude Alfven waves in the presense of two large amplitude pump waves. The second application considers the interaction of four waves in a dissipative plasma. We also describe extensions of the basic plasma model and geometry that can be useful in studying other applications of four wave mixing.