Sloshing ion distributions are a crucial feature in the end cells of recent tandem mirror reactor designs. They provide the ambipolar potentials that confine central ions and often have the function of making 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 angular and the velocity distribution functions. The density ratio and the ambipolar potential, as functions of axial distance, are computed from the angular distribution function. There is excellent agreement with results from the Lawrence Livermore National Laboratory bounce-averaged Fokker-Planck code with as much as 500 times less CRAY-1 computer time.
