Logarithmic Separable Solutions of Force-Free Magnetic Fields in Plane-Parallel and Axial Symmetry

TL;DR

This paper develops a systematic method using separation of variables to find new logarithmic solutions for force-free magnetic fields with plane-parallel and axial symmetry, revealing properties distinct from known solutions.

Contribution

It introduces a novel logarithmic family of solutions for force-free magnetic fields, expanding the analytical solution space with potential astrophysical and plasma applications.

Findings

Logarithmic solutions allow bounded, periodic, and infinite magnetic field configurations.

The method transforms the PDE into decoupled ODEs, facilitating solution analysis.

Distinct properties from linear and nonlinear solutions are demonstrated.

Abstract

This work introduces a systematic method for identifying analytical and semi-analytical solutions of force-free magnetic fields with plane-parallel and axial symmetry. The method of separation of variables is used, allowing the transformation of the non-linear partial differential equation, corresponding to force-free magnetic fields, to a system of decoupled ordinary differential equations, which nevertheless, are in general non-linear. It is then shown that such solutions are feasible for configurations where the electric current has a logarithmic dependence to the magnetic field flux. The properties of the magnetic fields are studied for a variety of physical parameters, through solution of the systems of the ordinary differential equations for various values of the parameters. It is demonstrated that this new logarithmic family of solutions has properties that are highly distinct from the known linear and non-linear equations, as it allows for bounded solutions of magnetic fields, for periodic solutions and for solutions that extend to infinity. Possible applications to astrophysical fields and plasmas are discussed as well as their use in numerical studies, and the overall enrichment of our understanding of force-free configurations.