AC space charge self-fields in gyroiron beams are studied via a set of particle equations. The study addresses the linear stability of beams in tapered magnetic fields as well as the nonlinear-multi frequency saturation of space charge instabilities.
H. Li, H. Li,
"Space charge instabilities in gyrotron beams", Proc. SPIE 1576, 16th International Conference on Infrared and Millimeter Waves, 15761T (1 October 1991); doi: 10.1117/12.2297778; https://doi.org/10.1117/12.2297778