An Exact, Time-dependent Analytical Solution for the Magnetic Field in the Inner Heliosheath

Abstract

We derive an exact, time-dependent analytical magnetic eld solution for the inner heliosheath, which satis es both the induction equation of ideal magnetohydrodynamics in the limit of in nite electric conductivity and the magnetic divergence constraint. To this end, we assume that the magnetic eld is frozen into a plasma ow resembling the characteristic interaction of the solar wind with the local interstellar medium. Furthermore, we make use of the ideal Ohm's law for the magnetic vector potential and the electric scalar potential. By employing a suitable gauge condition that relates the potentials and working with a characteristic coordinate representation, we thus obtain an inhomogeneous rst-order system of ordinary di erential equations for the magnetic vector potential. Then, using the general solution of this system, we compute the magnetic eld via the magnetic curl relation. Finally, we analyze the well-posedness of the corresponding Dirichlettype initial-boundary value problem, specify compatibility conditions for the initial-boundary values, and outline the implementation of initial-boundary conditions.