Plasma expansion in vacuum: Modeling the breakdown of quasi neutrality

We consider a system consisting of a vacuum gap delimited by two electrodes. A quasi-neutral plasma (ions and electrons) is injected from the cathode and expands. Electrons are emitted from the plasma-vacuum interface to the anode forming a beam. In this paper, from a two fluid isentropic system coupled with the Poisson equation, we perform a formal asymptotic analysis. This leads to a quasi-neutral model for the plasma region and a Child-Langmuir model for the beam region. The main point of the analysis is the connection between these two models. This is done by studying a transmission layer problem. Finally, we numerically show the accuracy of the asymptotic model, comparing it to the original two-fluid model.