An exact, time-dependent analytical solution for the magnetic field in the inner heliosheath

We derive an exact, time-dependent analytical magnetic field solution for the inner heliosheath, which satisfies both the induction equation of ideal magnetohydrodynamics in the limit of infinite electric conductivity and the magnetic divergence constraint. To this end, we assume that the magnetic field is frozen into a plasma flow 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 first-order system of ordinary differential equations for the magnetic vector potential. Then, using the general solution of this system, we compute the magnetic field via the magnetic curl relation. Finally, we analyze the well-posedness of the corresponding Dirichlet-type initial-boundary value problem, specify compatibility conditions for the initial-boundary values, and outline the implementation of initial-boundary conditions.

Automated summary

We present an exact and time-dependent analytical solution for the magnetic field in the inner heliosheath. The solution satisfies both the induction equation of ideal magnetohydrodynamics with infinite electric conductivity and the constraint of magnetic divergence. We assume that the magnetic field is frozen into a plasma flow similar to the interaction between the solar wind and the interstellar medium. By utilizing the ideal Ohm's law and using suitable gauge conditions, we derive a system of ordinary differential equations for the magnetic vector potential. We obtain the magnetic field by solving this system and analyzing the well-posedness of the corresponding initial-boundary value problem.