16 - Damping and decay

Summary

Physical description

If you look for a particular type of response mode in analyzing a mechanical system you might find it. But, if you do not look for it you will never find it. Lack of attention to this truism has often held back progress by promoting the feeling that a system is understood when its Fourier modes have been examined. Thus, it came as a surprise in plasma physics when Landau (1946) discovered that waves in a collisionless plasma, described by the collisionless Boltzmann equation, can damp and decay. At the time it was especially unexpected because damping had been associated with viscosity, and there is no viscosity in such plasmas. Similar behavior occurs in gravitating systems.

The physical reason for collisionless damping arises from the detailed interaction of a wave with the orbits of background stars which are not part of the wave. Thus, this process would not show up in a continuum approximation where the waves could merrily propagate without growth or decay.

To see how this works, consider the idealized triangular (rather than sinusoidal) wave shown in Figure 16. We look at it in a co-moving frame, traveling along with the wave's phase velocity v = ω/k. This ripple has been set up at t = 0 by suddenly imposing a periodic triangular perturbation in the density which gives a similar perturbation to the gravitational potential φ throughout the entire volume.