Effect of spatial turbulence on parametric instabilities

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Abstract
Three-wave backscattering instabilities are investigated in a plasma slab with stationary density fluctuations. The noise is of sufficient amplitude to induce random variation in the phase of the coupled waves. The analysis is formulated in terms of the Nyquist theorem. The ensemble average number and distribution of the zeroes of the exact dispersion relation are obtained from the solution of a Fokker–Planck/Boltzmann equation for the probability density of the phase angle of the dispersion relation. Threshold lengths and growth rates obtained are less optimistic for stability than those found in previous random phase investigations. The stabilizing influence of inhomogeneity is weaker in turbulent plasmas than in nonturbulent ones. The analytic results are confirmed in detail by direct numerical simulation of the fundamental equations.