An Iterative Method to Solve the Nonlinear Poisson's Equation in the Case of Plasma Tangential Discontinuities

In order to determine the electric potential in collisionless tangential discontinuities of a magnetized plasma, it is required to solve a non-linear Poisson's equation with sources of charge and current depending on the actual potential solution. This non-linear second-order differential equation is solved by an iterative method. This leads to an ordered sequence of non-linear algebraic equations for each successive approximation of the actual electric potential. It is shown that the method holds for transitions with characteristic thicknesses ( D) as thin as five Debye lengths (λ). For smaller thicknesses, when D shrinks to 3λ or less, the method fails because in that case the iteration procedure does no longer converge. Numerical results are shown for an ion-dominated layer ( D ∼ 10 ² - 10 ³λ), as well as for two electron-dominated layers characterized by D ≈ 5 λ and D ≈ 2.5 λ, respectively. In all cases considered in this paper, the relative error on the electric potential obtained as a solution of the quasi-neutrality approximation is of the order of the relative charge density. When the method holds, each successive approximation reduces the relative error on the potential by roughly a factor of 10. For space plasma boundary layers, the quasi-neutrality approximation can be used with much confidence since their thickness is always much larger than the local Debye length.