Nonlinear Landau Damping—The Spectrum

The spatial damping of large-amplitude electrostatic waves in a collisionless plasma is derived. The theory leads to differential equations which describe the nonlinear oscillation of the electric field amplitude, the broadening of the frequency spectrum, and the growth of sidebands. When driven at a single frequency the sidebands are displaced by a frequency ±0.8Ω = ±0.8(eκE0/m)1/2, where e and m are the electron charge and mass, κ is the wavenumber, and E0 is the amplitude of the electric field. The sidebands oscillate in space with a periodicity L = (2π/0.8Ω)(dω/dκ). The upper sideband is suppressed by linear Landau damping for large amplitudes.