Saturation of kinetic plasma instabilities by particle trapping

The amplitude at which a single unstable Langmuir or ion-acoustic wave in a collisionless plasma saturates due to electron trapping is estimated using both the adiabatic and sudden approximations, and is shown to obey the approximate relation ωb∼3γlin. This is found from momentum conservation. It is found that energy conservation implies a nonlinear dispersion relation that is linearly independent of that derived from Poisson's equation, but that a reasonable distribution of resonant particles can be found that satisfies both relations.