Nonlinear theory of parametric instabilities in plasmas

The weak plasma turbulence theory recently formulated by one of us is applied to parametrically unstable plasmas. A dispersion relation for normal modes of the turbulent plasma is derived. The condition of marginal stability of these modes provides a nonlinear equation describing the quasistationary turbulent spectrum. Two classes of nonlinearities arise: (i) Diagonal corrections, similar to the renormalized frequencies and damping rates of usual weak turbulence theory but which must be computed using the frequencies of the coupled parametric modes, and (ii) coupling saturation corrections which are renormalized mode coupling coefficients related to spectral cross coupling coefficients such as $〈 E(\overrightarrow{\mathrm{k}},\omega )E(\overrightarrow{\mathrm{k}}-{\overrightarrow{\mathrm{k}}}_0,\omega -{\omega }_0)〉$. The coupling saturation theory is applied to the electron-electron decay instability $\omega \sim 2{\omega }_{\mathrm{pe}}$ and strongly inhibits the conversion of pump radiation into plasma waves.