Temporal evolution of reactive and resistive nonlinear instabilities

Abstract

A kinetic theory for nonlinear processes involving Langmuir waves, developed in an earlier paper, is extended through consideration of three aspects of the temporal evolution, (i) Following Falk & Tsytovich (1975). the dynamic equation for the rate of change of one amplitude at t is expressed as an integral over T of the product of two amplitudes at t – T and a kernel functionf(T); two generalizations of Falk & Tsytovich's form (f(T) ∝ T) that satisfy the requirement f(∞) = 0 are identified, (ii) It is shown that the low-frequency or beat disturbance may be described in terms of fluctuations in the electron number density, and that its time evolution involves an operator that is essentially the inverse of f(t). (iii) The transition from oscillatory evolution in the reactive or ‘coherent-wave’ version of the three-wave instability to the secular evolution of the resistive or ‘random-phase’ version is discussed qualitatively.

Mean field theory of the coherent to random-phase state transition in three-wave interactions

“…1,[3][4][5] Projectionoperator 6,7 and instability analyses 8 indicate that as ⌬ increases from small values, a smooth decrease in coherence occurs. Similarly the phase-coherent approximation applies for T⌬Ӷ1.…”