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Relativistic density profiles and current flow in a cross-field relativistic electron vacuum device

We use the cold-fluid plasma equations to consider the nonlinear effects of a strong, relativistic RF electric field (with a frequency, w, and a wavevector, k) which is propagating on the background electron density profile in a relativistic crossed-field, electron vacuum device. Earlier, we had shown that in the nonrelativistic case, when k and w are such that a wave-particle resonance, w equals vdk, can occur at the edge of a Brillouin sheath, then the Brillouin sheath becomes strongly unstable to a Rayleigh instability, with the instability being driven by the strong negative density gradient at the edge of the Brillouin sheath. As a consequence of this instability, the average DC density profile becomes strongly modified and is driven away from the classical Brillouin flow by the RF field, and is driven toward stationary solutions of a nonlinear diffusion equation. From this nonlinear diffusion equation, one can predict the DC current flow through a device and also can predict the shape of the stationary DC electron density profile. Also we have demonstrated that such stationary solutions do exist and can be calculated. Further, we showed that when one combined these stationary solutions with the RF field solutions, then the total solution would generate the standard spoke structure, long seen in numerical simulations. Here, we shall extend these calculations into the relativistic regime and discuss their form.