Nonlinear evolution of unstable electrostatic waves in a multiple species Vlasov plasma.

摘要

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 → 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)| = γ5/2r(γt) as γ → 0+ is required to cancel the coefficient singularities to all orders. This result predicts that the electric field scaling will hold universally for these instabilities 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 is recovered.; From the associated amplitude expansions, the asymptotic features of the electric field and distribution functions are studied in the limit of weak instability. The asymptotic electric field is monochromatic at the wavelength of the linear mode with a nonlinear time dependence. The structure of the distributions outside the resonant region is given by the linear eigenfunction but in the resonant region the distribution is nonlinear. The details depend on whether the ions are fixed or mobile; in either case this generally derived physical picture corresponds to the “single wave model” originally proposed by O'Neil, Winfrey, and Malmberg [Phys. Fluids 14 1204 (1971)] for the special case of a cold weak beam instability in a plasma of fixed ions.; The above results are used to propose a generalized “single wave model” (SWM) for electrostatic instabilities for multiple species plasma in the weak growth rate limit. This reduced model is vastly simpler to solve numerically than the full Vlasov system. The generalized SWM is shown to preserve the important physical properties of the full Vlasov system, such as the Hamiltonian structure and the saturation scaling of the unstable wave.