Analytic Theory of Strong Nonlinear Wave Coupling in Plasma

Abstract

An analytic theory is given for the nonlinear, fixed-phase interaction between two high frequency waves and a low frequency field in a plasma, when the rate of interaction is much larger than the resonant low frequency, i.e. a quasidecay or modified decay process. Strong compensation occurs when the four wave interaction is included, which leads to an ion-dependent instability. Backscattering of electromagnetic waves due to quasidecay and induced scattering on particles is discussed.

Temporal evolution of reactive and resistive nonlinear instabilities

“…The two functions (lla, b) are plotted in figure 1 for T e /T t = 10. For sufficiently short times (t 4 OJJ 1 , dy^" 1 , (|k p -k D \ V i /-\/2)~1) the two forms (lla,b) coincide and reduce to the approximation assumed by Falk & Tsytovich (1975), i.e. to f(t) ~ (Opi tH(t) in the present notation.…”

“…The starting point for the discussion in paper I is kinetic theory (cf. also Falk & Tsytovich 1975), unlike most conventional treatments for nonlinear instabilities, which are based on simple models for the plasma and the nonlinear interaction (e.g. Sagdeev & Galeev 1969;Weiland & Wilhelmsson 1977).…”

“…Three related topics involving the temporal evolution of nonlinear instabilities are discussed in the present paper. Following Falk & Tsytovich (1975), dynamic equations that describe the evolution of the wave amplitudes may be derived from equations given in paper I by inverting Fourier transforms. The dynamic equation for the amplitude of one of the highfrequency waves, a D (t) say, gives its time derivative as an integral, over T say, of the product a D (t -T)a*(t -T) times a function /(T).…”

“…This function /(T) characterizes the nonlinear interaction. However, the form obtained for it by Falk & Tsytovich (1975), specifically /(T) OC TH{T), where H{T) is the step function, is unacceptable for large T. The effect of the nonlinear interaction must go to zero as the time delay goes to infinity. That is, the limit T -*• oo of/(r) must be zero.…”

Reactive and resistive nonlinear instabilities

“…Following an approach used by Falk & Tsytovich (1975), (1) may be reduced to a nonlinear dispersion equation by performing a statistical average (denoted by angular brackets ( » over the pump field. The statistical average is non-zero only for k t = -k 3 .…”