Parametric instabilities in the presence of space-time random fluctuations

The effect of a finite bandwidth driver or of turbulence on the parametric instability is considered. First, exact equations are obtained for special values of the group velocity of the fluctuations V0, with bandwidth modeled by a Kubo–Anderson process. Next, a method which enables one to deal with the space-time problem for arbitrary V0 is given. For the averaged amplitudes it reduces to the Bourret approximation and for the correlation function it gives the kinetic equation for random phase waves. Convective and absolute instability thresholds are given. It is found that for absolute instabilities the threshold obtained from the averaged intensities is lower than the one obtained from the averaged amplitudes.