Landau Waves

This article first reviews the steps which led from Thomson's and Vlasov's electron plasma waves to Landau's prediction that electrostatic waves should be damped, even in the absence of collisions. After considerable controversy, this damping mechanism can now be understood in terms of a balance between particle acceleration and wave amplification. Ionospheric radar experiments by Bowles, Evans and a laboratory experiment by Wong et al, verified, respectively, the validity of temporal and spatial Landau damping as applied to ion acoustic waves. Similar proof in the case of electron plasma waves came soon thereafter. Perkins et al, demonstrated the existence of temporal Landau damping in the ionosphere while Malmberg and Wharton measured spatial Landau damping in the laboratory. The results of the latter experiment and the dispersion measurements of Van Hoven and of Derfler and Simonen have verified the existence of Landau waves with and without an applied magnetic field. The application of Landau's theory to the above experiments is largely based on Ohm's Law in frequency- wave number space, i.e. iωϵ0 D(ω, k) E[ω, k] = I[ω, k]. Combined with Nyquist's fluctuation dissipation theorem it explains the density variations observed in the ionosphere. The signal II(t, x), spreading from a short localized current swing, I(t, x) = ϵ0 δ'(t)δ(x), predicts the ringing at the plasma frequency as observed by the Alouette ionospheric top side sounder. The Laplace transform, II[ω, x), and the Fourier transform, II(t, k], of this propagator basically solve the problem of spatial and temporal Landau damping. A perturbation of the velocity distribution established within a collision free plasma survives forever, while the moments of the distribution decay in time. An exciting new scheme due to Gould retrieves in part the original information long after Landau damping has destroyed the collective Coulomb field. (author)