Convective growth of parametrically unstable modes in inhomogeneous media

By a suitable definition of an amplification coefficient, a convenient analytic criterion for convective instability of damped parametrically coupled unstable modes in an inhomogeneous medium is obtained. The net growth factor F3 on propagation through the resonant interaction region is found to be given by F3 = exp[(πγ20/κ′v1v2)F(√ν1ν2/γ0)], where the reduction factor F(x) = (2/π)[acos(x) − x√1−x2]. This can readily be seen to reduce to the usual inhomogeneous gain formula when damping is negligible and F=1, and to manifest the homogeneous convective threshold, γ20 ≳ ν1ν2. This result should find useful application in laser fusion plasmas, where damping and spatial inhomogeneity are frequently suppression mechanisms of comparable importance for laser-driven parametric instabilities.