Birkeland currents in an anisotropic, magnetostatic plasma

An expression for the parallel current density is derived for a plasma characterized by negligible bulk flow (magnetostatic) velocity and a two‐component (anisotropic) pressure tensor by expanding the equilibrium Vlasov equation for each species in the adiabatic parameter until such point as a nonvanishing moment j∥ = ∫ d³ υυ∥ ƒ is identified. The result is a nonlocal one: it relates j∥ at one point s along a field line to j∥ at another (reference) point s0 plus an integral function of the pressure and magnetic field between them. It is a generalization and elaboration of results obtained by Bostrom (1975), Heinemann (1990), and Heinemann and Pontius (1991). The expression could have been obtained by integrating the current continuity equation with −▽ · j⊥ as a source term and j⊥ given by perpendicular momentum balance. We explicitly show the equivalency. The widely used Vasyliunas (1970) equation follows when P⊥ is set equal to P∥ and s and s0 are taken to be at the ionosphere and the equator. An extended discussion of the relationship of results derived here to others in the literature is carried out in an effort to bring unity and perspective to this problem area.