Study of the formation of a charge separation at a plasma edge. Part I. The numerical integration of a Vlasov equation possessing an invariant Original Research Article

A computer-time saving method is presented and applied to study the problem of the formation of a charge separation and an electric field at a plasma edge. In this method, electrons are treated with an adiabatic law and a one-dimensional in space (1D) fully kinetic Vlasov code (three velocity dimensions) for the main ion species and the impurity ions. We make use of the fact that the characteristics of the Vlasov equation possess an exact invariant to apply a method of solution which expresses the distribution function in terms of the invariant itself. The dimensionality of the phase space is reduced, since the invariant only appears as a label of the Vlasov equation and can be coarsely discretized. There is a factor close to 10 gain in computation speed with respect to the case where the Vlasov equation is directly integrated using a fractional step method, and the results are obtained with the same accuracy. Also, the parallelization of these equations is more straightforward since the different equations for the different invariant labels are independent, which will further increase the computation speed. The results show the importance of the finite ions gyro-radius in establishing a charge separation at a plasma edge, and also the important role played by small fractions of impurity ions. The extension and application of the invariant method for a two-dimensional (2D) problem, in which electrons are treated using a drift-kinetic equation, and the ions are treated using a 2D fully kinetic code, is presented. The results of the 2D code do confirm the existence at the plasmas edge of a stable 1D equilibrium for the problem of the formation of a charge separation in the presence of a steep gradient.