John David Crawford

The amplitude equation for an unstable electrostatic wave is analyzed using an expansion in the mode amplitude $A(t)$. In the limit of weak instability, i.e. $γ\to 0^+$ where $γ$ is the linear growth rate, the nonlinear coefficients are singular and their singularities predict the dependence of $A(t)$ on $γ$. Generically the scaling $|A(t)|=γ^{5/2}r(γt)$ as $γ\to 0^+$ is required to cancel the coefficient singularities to all orders. This result predicts the electric field scaling $|E_k|\simγ^{5/2}$ will hold universally for these instabilities (including beam-plasma and two-stream configurations) throughout the dynamical evolution and in the time-asymptotic state. In exceptional cases, such as infinitely massive ions, the coefficients are less singular and the more familiar trapping scaling $|E_k|\simγ^2$ is recovered.

The amplitude equation for an unstable electrostatic wave in a multi-species Vlasov plasma has been derived. The dynamics of the mode amplitude $ρ(t)$ is studied using an expansion in $ρ$; in particular, in the limit $γ\rightarrow0^+$, the singularities in the expansion coefficients are analyzed to predict the asymptotic dependence of the electric field on the linear growth rate $γ$. Generically $|E_k|\sim γ^{5/2}$, as $γ\rightarrow0^+$, but in the limit of infinite ion mass or for instabilities in reflection-symmetric systems due to real eigenvalues the more familiar trapping scaling $|E_k|\sim γ^{2}$ is predicted.