Weakly nonlinear dynamics of electrostatic perturbations in marginally stable plasmas

A single-wave model equation describing the weakly nonlinear evolution and saturation of localized electrostatic perturbations in marginally stable plasmas, with or without collisions, is derived using matched asymptotic expansions. The equation is universal in the sense that it is independent of the equilibrium, and it contains as special cases the beam-plasma and the bump-on-tail instability problems among others. In particular, the present work offers a systematic justification of the single-wave, beam-plasma model originally proposed by O'Neil, Winfrey, and Malmberg. The linear theory of the single-wave model is studied using the Nyquist method, and solutions of the linear initial value problem of stable perturbations which exhibit transient growth and do not Landau damp are presented. Families of exact nonlinear solutions are constructed, and numerical results showing the growth and saturation of instabilities, transient growth of stable perturbations, and marginal stability relaxation are presented. The single-wave model equation is analogous to the equation describing vorticity dynamics in marginally stable shear flows and thus, all the results presented are directly applicable to fluid dynamics.