Alfvén wave boundary condition for responsive magnetosphere-ionosphere coupling

The solution of electric fields and currents in a height-resolved ionosphere is traditionally solved as an elliptic equation with Dirichlet or Neumann boundary condition in which the magnetosphere is represented as an unresponsive (prescribed) voltage generator or current source. In this paper we derive an alternative boundary condition based upon Alfvén waves in which only the Alfvén wave from the magnetosphere that is incident upon the ionosphere (E_xi) is prescribed. For a uniform magnetosphere the new boundary condition reduces to

and is evaluated at the magnetosphere-ionosphere interface. The resulting
solution is interpreted as a responsive magnetosphere and establishes a  key stage in the full self-consistent and nonlinear coupling of the  magnetosphere and ionosphere.