Hans Schamel

This review presents an upgraded wave theory adapted to the high fluctuation level of driven realistic i.e. non-idealized plasmas. Above all, this means giving up the well-known concept of a linear wave theory in favor of a thoroughly nonlinear theory. Based on the author's early publication (H. Schamel, Plasma Phys. 14 (1972) 905) and supported by recent Vlasov-Poisson simulations of realistic noisy plasmas, an extended framework is presented which not only covers the essential features of coherent hole structures, but which also enables to make the necessary corrections to the current wave theory. These corrections are long overdue and can be briefly summarized under the heading: loss of linear Vlasov dynamics when adequately addressing equilibrium states (i.e. failure of linear Landau theory and of continuous van Kampen spectra, respectively). A linear stability analysis for single harmonic waves that successfully incorporates trapped particle effects (in contrast to previous analyses) shows an unconditional marginal stability independent of the drift between electrons and ions, which irrevocably contradicts Landau's theory. Moreover, holes of negative energy are of particular interest because they act as attractors in the dynamical system. They are the source for the release of further modes and thus increase the level of intermittent turbulence. In summary, pattern formation in collisionless plasmas is inherently nonlinear and kinetic. However, to have a satisfactory, if not yet complete, understanding of its processes a twofold paradigm shift is imperative: one from the conventional linear, discrete wave models to the nonlinear wave models dealing with continuous spectra due to trapping and a second from the BGK to the present method for the correct handling of equilibria.

Electron holes (EH) are highly stable non-linear structures met omnipresently in driven collision-less hot plasmas. A mechanism destabilizing small perturbations into holes is essential for an often witnessed but less understood sub-critically driven intermittent plasma turbulence. In this paper we show how a tiny, eddy-like, non-topological seed fluctuation can trigger an unstable evolution deep in the linearly damped region, a process being controlled by the trapping non-linearity and hence being beyond the realm of the Landau scenario. After a (transient) transition phase modes of the privileged spectrum of cnoidal EH are excited which in the present case consist of a solitary electron hole (SEH), two counter-propagating "Langmuir" modes (plasma oscillation), and an ion acoustic mode. A quantitative explanation involves employing non-linear eigen-modes, yielding a non-linear dispersion relation with a forbidden regime and the negative energy character of the SEH, properties being inherent in Schamel's model of undamped Vlasov-Poisson structures identified here as lowest order trapped particle equilibria. An important role in the final adaption of nonlinear plasma eigenmodes is played by a deterministic response of trapped electrons which facilitates transfer of energy from electron thermal energy to an ion acoustic non-uniformity, accelerating the SEH and positioning it into the right place assigned by the theory.