Parametric Instabilities in Inhomogeneous Plasmas

The threshold ac electric field required for the excitation of the purely growing “oscillating two-stream” and decay instabilities is found to increase when the density-gradient scale length H becomes less than the mean free path. For a plasma with no magnetic field, the threshold field is given by E2/4πnTe≈2(k‖H)−1, where k‖≲(1/4)D−1 is the wavenumber of the instability in the direction orthogonal to the density gradient. (D is the electron Debye length.) Qualitative arguments suggest that this result should hold in the presence of a magnetic field. Recent experimental measurements of the threshold field agree well with these theoretical calculations. The reason for the threshold field increasing is that, in an inhomogeneous plasma, the unstable region has a finite extent spatially and energy (in the form of electron plasma waves) can propagate out of this region, thus creating an energy loss not found in uniform plasmas.