On the Theory of Trapped Particle Instability. II. Stability of a Small-Amplitude B-G-K Wave

In a preceding report, (J. Phys. Soc. Japan 30 (1971) 1722) a general theory is developed to investigate the stability of a collisionless, one-dimensional plasma which sustains a large-amplitude, monochromatic electron plasma wave. In this paper, this theory is used to investigate further details of the instability by assuming a specific from of the unperturbed distribution, such that it is consistent with the small-amplitude limit of a Bernstein-Greene-Kruskal distribution. The properties of the dispersion relation are characterised by three parameters, the wavenumber of the carrier, its amplitude and the excess trapped-electron density Δ N t (compared with the value expected from the linear theory). In the weak-coupling limit, stable and unstable regions are shown in the plane spanned by Δ N t and bounce frequency. A numerical calculation for a moderately strong-coupling case shows that even in this case growing waves are separated into several distinct wavenumber regions.