Landau damping

In physics, Landau damping, named after its discoverer, the eminent Soviet physicist Lev Davidovich Landau, is the effect of damping (exponential decrease as a function of time) of longitudinal space charge waves in plasma or a similar environment. This phenomenon prevents an instability from developing, and creates a region of stability in the parameter space.

Wave-particle interactions

Landau damping occurs due to the energy exchange between a wave with phase velocity $v_{ph}$ and particles in the plasma with velocity approximately equal to $v_{ph}$, who can interact strongly with the wave. Those particles having velocities slightly less than $v_{ph}$ will be accelerated by the wave electric field to move with the wave phase velocity, while those particles with velocities slightly greater than $v_{ph}$ will be decelerated by the wave electric field, losing energy to the wave.

In a collisionless plasma where the particle velocities have a Maxwellian distribution function. If the slope of the function is negative, the number of particles with velocities slightly less than the wave phase velocity is greater than the number of particles with velocities slightly greater. Hence, there are more particles gaining energy from the wave than losing to the wave, which leads to wave damping. If, however, the slope of the function is positive, the number of particles with velocities slightly less than the wave phase velocity is smaller than the number of particles with velocities slightly greater. Hence, there are more particles losing energy to the wave than gaining from the wave, which leads to a resultant increase in the waves energy.

Physical interpretation

Mathematical proof of Landau damping is somewhat involved, requiring the use of contour integration. But there is a simple physical interpretation (although not strictly correct) that helps to visualize this phenomenon.

It is possible to imagine Langmuir waves as waves in the sea, and the particles as surfers trying to catch the wave, all moving in the same direction. If the surfer is moving on the water surface at a velocity slightly less than the waves he will eventually be caught and pushed along the wave (gaining energy), while a surfer moving slightly faster than a wave will be pushing on the wave as he moves uphill (losing energy to the wave).

It is worth to note that only the surfers are playing an important role in this energy interactions with the waves; a beachball floating on the water (zero velocity) will go up and down as the wave goes by, not gaining energy at all. Also, a boat that moves very fast (faster than the waves) does not exchange much energy with the wave.

Bibliography

• Chen, Francis F. Introduction to Plasma Physics and Controlled Fusion. Second Ed., 1984 Plenum Press, New York.

• Tsurutani, B., and Lakhina, G. Some basic concepts of wave-particle interactions in collisionless plasmas. Reviews of Geophysics 35(4), p.491-502. Download