Asymptotic matching of plasma bulk and sheath: convergence properties and approximation failure

Summary form only given. The numerical simulation of low pressure high density plasmas becomes inherently complicated due to the presence of vastly different length scales in the dynamics. Particularly, the sheath thickness H (which typically amounts to a few Debye lengths) is often much smaller than the characteristic reactor scale L. This fact is often referred to as the "numerical stiffness" of the plasma equations, and has motivated many authors to apply the technique of matched asymptotic expansions to arrive at plasma descriptions which combine a geometrically extended bulk model with an infinitesimally thin sheath model. In these formulations, the bulk dynamics is represented by a set of partial differential equations and the sheath by their boundary conditions. There are two principal problems connected with the approach. One is the fact that the underlying mathematical technique of matched asymptotic expansions seem to pose serious difficulties of its own. In fact, a lively discussion is currently underway in which some contributors seem to doubt the validity of the approach altogether. The method of matched asymptotic expansions yield approximations which contain a modeling error and fail when the underlying scaling assumptions break down. It is this problem that is addressed by our contribution. We study a simplified one-dimensional toy model of the dynamics of a low pressure plasma. This description contains the equation of continuity and the equation of motion for the ion component, assumes Boltzmann equilibrium for the electrons, and couples both fluids under the condition of quasi-neutrality. The model is compared with an even simpler description where the equation of motion is replaced by a drift-diffusion ansatz. This, replacement is equivalent to neglecting of ion inertia in the limit of a small collisional mean free path /spl lambda/ and it requires the choice of a boundary condition (to substitute Bohms criterion which is not any longer availabl- ).