Author\´s reply to "Good news: you can patch active plasma and collisionless sheath"

Godyak and Sternberg (2003) reassert their contention that one can obtain a satisfactory physical solution to the active plasma-collisionless sheath by patching together plasma and sheath. They choose to do it at an arbitrary point where the sheath electric field is kT_e /eλ_D. If one tacks their sheath solution to the full plasma solution, then the field is infinity on the plasma side and finite on the sheath side. Alternatively, if one terminates the plasma solution where the plasma field is kT_e/eλ_D, then one has continuity of electric field, but not of its gradient, since on the sheath side it is zero and on the plasma side of order L/λ_D, where L is the size of the plasma. Furthermore, in achieving continuity of the field, one has introduced discontinuities in the ion speed and in the particle densities. Thus, in no sense is a joining which denies the existence of a transition layer, smooth. J. Ockendon and H. Ockendon, my colleagues in the production of our paper describing the transition layer (Franklin et al., 1970), privately expressed disappointment in not finding a proof of the existence and uniqueness of our solution. Such a formal mathematical proof has been given recently by Slemrod (2002). Smooth joining of active plasma and collisionless sheath within the context of a fluid model or free fall model of the ion motion, does require a transition layer and of length scale intermediate between L and λ_D.