The kinetic Bohm condition in collisionless and collisional plasmas*

The kinetic Bohm condition [1] is an inequality that must be satisfied by integrals of the electron and the ion distribution functions at the entrance of a Debye sheath. As well as requiring that the average velocity of ions be larger than a certain threshold, this condition ensures that the number of slow ions is sufficiently small that they do not dominate Poisson’s equation near the sheath entrance. Without the kinetic Bohm condition, the sheath equations cannot be solved, and as a result, one needs it to be satisfied at the sheath entrance in quasineutral models of the plasma. Unfortunately, the kinetic Bohm condition cannot be imposed as a boundary condition because it depends only on the ions that enter in the Debye sheath and hence are completely determined by the quasineutral bulk plasma. We show that, in collisionless systems, the condition is not satisfied in general by time-independent solutions, but it can be imposed by kinetic rarefaction waves [2]. In contrast, we will show that when Fokker-Planck collisions are included, all steady state solutions must satisfy the Bohm condition.

 

[1] E.R. Harrison and W.B. Thompson, Proc. Roy. Soc. 74, 145 (1959).

[2] A.V. Gurevich, L.V. Pariiskaya and L.P. Pitaevskii, Sov. Phys. JETP 22, 449 (1966).