Non-linear mixing of waves in a plasma with velocity-dependent collisions

The frequency difference, sum and second harmonic currents, due to two interacting waves in a plasma with no external magnetic field, are calculated. The oscillation frequencies are taken to be sufficiently high that ion motion can be neglected. We consider the regime where Landau damping is negligible compared to collisional damping. The non-linearity is due to gradients in the a.c. electric fields. The form of the zero-order isotopic and non-oscillatory part of the velocity distribution function is left arbitrary. The velocity dependences of the collision frequency of electrons with neutrals, ions and other electrons are properly accounted for. The Fokker-Planck equation in its integro-differential form and including non-linear terms is used for electron-electron collisions. The self-consistent electric field is included with non-linear terms. Results are given (a) for any degree of ionization provided the lowest frequency in the mixing process is greater than the total collision frequency and (b) for a slightly ionized gas for any ratio of oscillation to collision frequency. In the former case, it is shown that electron-electron effects do not contribute to the non-linear current arising from wave mixing but contribute to the non-linear self-consistent electric field.