Nonlinear Asymptotic Analysis of the Positive Column

The positive column of a slightly ionized gas discharge confined by cold, insulating walls is described by a set of nonlinear fluid equations. The inertia, space charge, and collision terms are retained. A zeroth-order solution uniformly convergent to the exact solution in both plasma and sheath regions is derived using asymptotic boundary-layer analysis. The value of potential at the wall is calculated by means of a kinetic model. It is found that the density at the wall can be a significant fraction of the value at the center and that it vanishes only in the low electron temperature limit. The original Bohm criterion is recovered as a necessary condition for sheath stability and is interpreted to mean that the ambipolar sound speed (1) asymptotically separates the plasma from the sheath, and (2) is the maximum ambipolar diffusion velocity.