Equilibrium and low-frequency stability of a uniform density, collisionless, spherical Vlasov system
Equilibrium and stability of a collisionless, spherical Vlasov system with uniform density are considered. Such an electron system is useful for the Periodically Oscillating Plasma Sphere (POPS) fusion system. In POPS the space charge of a uniform-density spherical electron cloud provides a harmonic well for an under-dense thermal ion population. Previous special solutions [D. C. Barnes, Phys. Plasmas 6, 4472 (1999)] are extended to arbitrary energy dependence. These equilibrium distribution functions and their first derivatives may be made nonsingular, in contrast to the previous solutions. Linear stability of general spherical equilibria is considered, and reduced to a one-dimensional calculation by the introduction of a spherical harmonic decomposition. All azimuthal mode numbers are degenerate. Using this formalism, the low-frequency stability of a collisionless, spherical Vlasov electron system coupled to a minority ion cloud is studied for the class of uniform-density electron equilibria found. In the low-frequency (adiabatic) limit, the general kinetic stability formalism can be integrated to find a closed form for the response of electron number density. The adiabatic response operator is shown to be self-adjoint. Computation of its eigenvalues proves the constant-density electrons/thermal ions system in POPS to be mostly stable to ion-electron electrostatic modes. Unstable modes are avoided unless central electrons have an extremely small energy spread. These results may also be useful for the consideration of gravitational and beam systems.