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

TL;DR

This paper presents an exact, time-dependent analytical solution for the magnetic field in the inner heliosheath, satisfying MHD equations and divergence constraints, based on plasma flow assumptions and gauge conditions.

Contribution

It introduces a novel analytical method to solve the magnetic field in the inner heliosheath using characteristic coordinates and gauge conditions, advancing theoretical understanding.

Findings

Derived an exact analytical magnetic field solution.

Analyzed well-posedness of the initial-boundary value problem.

Outlined implementation of initial-boundary conditions.

Abstract

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.