Reactive and resistive nonlinear instabilities

Abstract

It is argued that familiar nonlinear instabilities, such as three-wave decay instabilities and scattering off quasi-modes, have reactive and resistive versions. In appropriate limits these versions correspond to fixed-phase parametric instabilities and to random phase growth in weak turbulence theory, respectively. Using a simplified model it is shown that the reactive version passes over into the resistive version when the band-width of the pump is comparable with the growth rate.

Unified theory of monochromatic and broadband modulational and decay instabilities of Langmuir waves

“…1 Equation ͑20͒ differs from Melrose's result 1 in that ⌬ has the opposite sign to ␥ S within the square root, whereas Melrose argued that the replacement ͑5͒ should be made throughout to obtain the broadband case from the monochromatic one. The difference occurs because the replacement ͑5͒ is actually only required in the square brackets in ͑4͒, not in the Green function.…”

“…In the latter case, the interaction is often called nonlinear Landau damping, or induced scattering by ions, because it takes energy from Langmuir waves without producing any other propagating product waves. 1,3,4 For ӷ1 the form ͑13͒ of the response applies and we find…”

“…The threshold has the form ⌬ϭ⌫ 0 for W small, where ⌫ 0 is the monochromatic growth rate at the same W. At large W, one finds ⌬ ϭ⌫ 0 2 /␥ S at threshold, which is equivalent to results in the literature. 1,13 If we assume Ϸ1, weakly damped ion-sound waves are replaced by a nonresonant virtual mode described by a Green function of the form ͑16͒ or ͑17͒. In this case, ͑4͒ can be approximated ͑in dimensionless units͒ as…”

“…This was demonstrated in detail for three-wave decay instabilities in which a parent Langmuir wave decays into another Langmuir wave and a low-frequency disturbance ͑e.g., an ion-sound wave͒. [1][2][3][4] However, Melrose 2 also noted that there is no recognized resistive counterpart to four-wave modulational instabilities in which two Langmuir waves interact via a nonresonant density response to produce two product Langmuir waves.…”

“…7,8 We then introduce a finite bandwidth for the Langmuir waves using a procedure developed by Melrose. 1 The resulting nonlinear dispersion equation involves a Green function for the plasma density response. In a Maxwellian plasma, this function involves the Fried-Conte plasma dispersion function, 9 which is related to the complex error function.…”