Bohm criterion and ion-acoustic sound barrier

In the limit of a small Debye length λD the plasma boundary layer is split up into a thin space charge sheath and a quasineutral presheath. A necessary condition for the formation of a stationary sheath is the validity of Bohm’s criterion at the sheath edge. In its most simple form, it requires that the ions enter the sheath region at least with ion sound velocity vB. Usually, the quasineutral plasma approximation is terminated by a singularity at v=vB and Bohm’s criterion is fulfilled in the equality form. In widening geometries, however, the plasma flow may become supersonic within the presheath, and the sheath edge exhibits no singularity. This hydrodynamic result is in apparent contradiction to general kinetic considerations. The contradiction is exhibited (and finally resolved) with the example of a collisionless cylindrical column, where ionization is restricted to a small region near the axis.