Nonlinear Saturation of an Electrostatic Wave: Mobile Ions Modify Trapping Scaling

The amplitude equation for an unstable electrostatic wave in a multispecies Vlasov plasma has been derived. The dynamics of the mode amplitude $\rho \left(t\right)$ is studied using an expansion in $\rho$; in particular, in the limit $\gamma \to 0^{+}$, the singularities in the expansion coefficients are analyzed to predict the asymptotic dependence of the electric field on the linear growth rate $\gamma$. Generically $|E_k|\sim {\gamma }^{5/2}$, as $\gamma \to 0^{+}$, but in the limit of infinite ion mass or for instabilities in reflection-symmetric systems due to real eigenvalues the more familiar trapping scaling $|E_k|\sim {\gamma }^2$ is predicted.