Consistent matching of plasma and sheath

Due to the sheath edge singularity, the asymptotic ($\epsilon\sim\lambda_D/L \to 0$) plasma- and sheath solutions cannot be matched smoothly. The sheath edge singularity, however, can be bridged by an intermediate scale analysis accounting in lowest order both for plasma processes (e.g. collisions, ionization, and non-planar geometry) and space charge. The possibility to construct a uniformly valid matched asymptotic expression from the plasma -, intermediate -, and sheath solutions was questioned in the literature. Problems arise from the asymptotic singularities and from the ionization eigenvalue problem of bounded plasmas (plasma balance). To clarify the topic we analyze the plasma-sheath problem both analytically and numerically. We show that the validity of the intermediate scale analysis is limited to a very narrow vicinity ($|\Delta \varphi| \leq 0.1$) of the sheath edge and formulate a matched asymptotic expression uniformly valid from the plasma core to the wall. The approximations obtained by matching are compared with exact solutions and discontinuities in the derivatives are investigated analytically.