Parametric Excitation and Mode Coupling

Abstract

We have described simple examples of nonlinear wave propagation in Sects.5.6 and 6.6. There we were mainly concemed with a single nonlinear wave mode. ln many cases of interest, nonlinear coupling of different linear wave modes can also be important Such couplings typically occur as a result of periodic modulation of a plasma parameter characterizing the dispersion relation of the wave. The modulation is due to the presence of another wave, therefore it can be considered as a mode coupling process. Such a mode coupling often causes an unstable beat wave, that is, a beat between the original wave and the modulation. This instability is called the parametrie instability [8.1.2]. In this chapter, we first discuss some general properties of parametrie instabilities by using model equations and then specifically discuss the case of the excitation of an electron plasma wave and an ion acoustic wave in an isotropie plasma. We further discuss the nonlinear wave-particle interaction for the case of the electron plasma wave and finally discuss the Langmuir wave turbulence with particular attention to the condensation and collapse of the wave. 8.1 Mathieu Equation Model The simplest example of parametrie excitation is the amplification of a pendulum oscillation by a periodic modulation of the string length. The equation describing the pendulum oscillation can be written as J2 X(t) + {p X(t) = 0 dt 2 (8.1.1) where X(t) is the displacement from the equilibrium position and [J is the frequency which depends on the string length. We denote its periodic modulation by [J2 = [J5(1 + 2e cos wot) , (8.1.2) where [Jo is the natural frequency of the pendulum, estands for the strength of the modulation and Wo the modulation frequency. Substitution of (8.1.2) into (8.1.1) gives the Mathieu equation whose periodic solutions are given by the Mathieu functions. For small !cl < 1), the solution of the Mathieu equation becomes unstable when [8.3] 136 K. Nishikawa et al., Plasma Physics