Trapping: The "Waterfall" of Vlasov-Poisson Dynamics and the Post-Transient Emergence of Drift-Independent Hole Structures in Collisionless Plasmas
A key aspect of structure formation in collisionless plasmas is particle trapping or, more precisely, the gap this process leaves in the theoretical description. While the dynamics prior to the onset of trapping are governed by Landau physics and phase mixing, the full effects of nonlinearity only emerge during the trapping phase, for which an explicit description is lacking. Consequently, this gap divides the temporal evolution into a linear Vlasov-Poisson phase preceding the transition and a post-transient Vlasov-Poisson phase characterized by a wide spectrum of particle trapping scenarios and associated Schamel equilibria. Metaphorically, this trapping process can thus be compared to a waterfall in fluid dynamics, where, similarly, no explicit dynamic link can be established between the regions above and below the cascade. To resolve this problem of the gap, an analysis using the method of matched asymptotic expansions is proposed. Applied to the singular boundary of the separatrix zone, this approach employs local smoothing techniques, such as those mediated by the two-particle correlation function. It offers a suitable mechanism for selecting hole equilibria in the post-transition phase. Furthermore, we report the existence of a spatially periodic Langmuir hole that vanishes if regularity requirements are too stringent, and we demonstrate that self-acceleration, accompanying the transition from a slow electron acoustic hole to a fast Langmuir hole, is driven by the release of deeply trapped electrons.