Landau Damping

Collisionless damping of a plasma wave by resonant particles moving near the wave's phase velocity.

The Effect

Landau damping is the transfer of wave energy to particles even in the complete absence of collisions. It arises from the resonant interaction between a wave of phase velocity omega/k and particles whose velocity is close to that value. Slightly slower particles are accelerated by the wave and gain energy; slightly faster particles are decelerated and lose energy. Because a Maxwellian has more slow than fast particles near any point on its falling tail, the net effect drains the wave.

Mathematics

The damping rate is proportional to the slope of the velocity distribution f evaluated at the phase velocity: gamma is proportional to df/dv at v = omega/k. For a Maxwellian this slope is negative, giving damping. The rate emerges from correctly handling the pole in the Vlasov dispersion integral using the Landau contour, which Landau introduced by treating the initial-value problem rather than assuming a steady oscillation.

Inverse Landau Damping

If the distribution has a positive slope at the resonant velocity, for example a beam or a bump-on-tail, energy flows from particles to the wave and the mode grows. This inverse Landau damping is the free-energy source for many velocity-space instabilities, and it is the kinetic origin of the bump-on-tail and two-stream instabilities.

Relevance

Landau damping sets the absorption of radio-frequency and lower-hybrid waves used for heating and current drive, and it damps or destabilizes Alfven eigenmodes through resonance with fast ions. In a burning plasma the fusion-born fast-ion population can drive modes through inverse Landau damping, a stability concern for any D-T device including the Hyperion breeder concept. All such analysis is simulation-stage.