Fluid and kinetic stability of virtual cathodes for the periodically oscillating plasma sphere

Recent theoretical work [R. A. Nebel and D. C. Barnes, Fusion Technol. 38, 28 (1998); D. C. Barnes and R. A. Nebel, Phys. Plasmas 5, 2498 (1998)] has suggested that a tiny oscillating ion cloud (referred to as the periodically oscillating plasma sphere or POPS) may undergo a self-similar collapse that can result in the periodic and simultaneous attainment of ultrahigh densities and temperatures. However, a major uncertainty in this plasma system is the behavior of the electron cloud that forms a virtual cathode. Here it is demonstrated that the required electron cloud (which forms a harmonic oscillator potential) is susceptible to an instability related to buoyancy-driven modes present in compressible fluids. Although it is demonstrated that no absolutely stable profiles with uniform electron density exist, stable profiles that are close to the required harmonic oscillator potential are found. A simple two-stream analysis indicates that kinetic effects lead to a critical limit in λD/a above which the virtual cathodes are stable. This result is consistent with previous experimental observations.