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 th e beam-plasma and the bump-on-tail instability problems among others. In particular, the present work offers a systematic justification of t he single-wave, beam-plasma model originally proposed by O'Neil, Winfr ey, and Malmberg. The linear theory of the single- wave model is studi ed 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 saturati on of instabilities, transient growth of stable perturbations, and mar ginal stability relaxation are presented. The single- wave model equat ion is analogous to the equation describing vorticity dynamics in marg inally stable shear flows and thus, all the results presented are dire ctly applicable to fluid dynamics. (C) 1998 American Institute of Phys ics. [S1070-664X(98)03011-0].