Beam-Plasma Instability in a Nonuniform Plasma

The behavior of the beam-plasma instability in a nonuniform plasma depends only on the dimensionless quantity $\lambda \sim {k_i}^{-2\mathrm{}}\frac{\partial k}{\partial x}$, where $k_i$ is the spatial growth rate in the homogeneous plasma. If $\left|\lambda \right|>~1$ the instability is quenched.