NONLINEAR GROWTH-RATE OF LANGMUIR-WAVES IN THE CASE OF A WIDE SPECTRUM

A nonlinear equation for the evolution of a Langmuir harmonic in a wide spectrum is derived in a consistent manner. This equation substitutes the usual quasilinear growth rate of the weak plasma turbulence theory. The derivation is based on the determination of the resonant particle distribution through the solution of the exact nonlinear equations of motion in a given field, composed of all effectively acting harmonics. The method requires a slow variation of the harmonic amplitudes over the nonlinear characteristic time of resonant interaction, and small enough amplitudes for the averaging method to be applicable. Two nonlinear terms appear in this equation: one is proportional to the particular harmonic considered, the other is a contribution from the whole effective spectrum around this harmonic. From this last term, it follows that a wave with negligible initial amplitude can grow spontaneously through the nonlinear wave-particle interaction. © 1995 American Institute of Physics.