Perturbation solution of the bounce-averaged Fokker-Planck equation in magnetic mirrors

Sloshing-ion distributions are a crucial feature in the end cells of Tandem Mirror Reactor designs. They provide the ambipolar potential that confines central-cell ions and often have the function of creating the electron thermal barrier with a potential shape that traps enough cold ions at the midplane for the stabilization of loss-cone modes. A perturbation method is developed to find solutions of sloshing ion distributions. This method uses an expansion in the ratio of electrostatic potential to average ion energy to simplify the bounce averaged Fokker-Planck equation. The zero-th order equation obtained is separated into equations for the angular- and velocity-dependent parts of the distribution function. An analytical solution of the angular equation is derived for small charge-exchange to ionization ratios. For any value of this ratio finite-element techniques, which provide rapid numerical solutions for parametric studies of sloshing ions, are used to derive the zero-th order angular and velocity equations. The first-order two-dimensional equation was also expanded into finite-element hat functions. Application of Galerkin's method gives a linear system of equations where all matrix and source elements are calculated analytically