A stationary fluid model of field‐aligned electric fields and closure of Birkeland currents

We introduce and analyze a two-dimensional stationary self-consistent model of the coupling of the magnetosphere and the ionosphere within the framework of fluid theory. This model contains localized field-aligned electric fields as well as a steady state current circuit closing in the ionosphere. For the acceleration region above the ionosphere the magnetic field is determined by the solution of a Grad-Shafranov equation, whereas the flow field and the electric field are determined by a set of integrals along the magnetic field lines. At ionospheric heights the dissipative interaction of the plasma with the neutral gas is included. We show that the problem may again be reduced, in this case to a coupled system of two nonlinear differential equations plus the continuity equation. Within this theory, simple examples of solutions are presented to demonstrate the capabilities of the model.