Hall drift of axisymmetric magnetic fields in solid neutron-star matter

TL;DR

This paper investigates the nonlinear Hall drift process in neutron star crusts, analyzing magnetic field evolution, stability, and energy conservation, revealing conditions for stable and unstable magnetic configurations.

Contribution

It provides a detailed theoretical analysis of Hall drift in neutron star crusts, including stability criteria and the derivation of a variational principle for Hall equilibria.

Findings

Toroidal fields evolve as Burgers' equation, forming current sheets.

Toroidal fields are unstable to poloidal perturbations.

Current-free poloidal fields are stable and long-lived.

Abstract

Hall drift, i. e., transport of magnetic flux by the moving electrons giving rise to the electrical current, may be the dominant effect causing the evolution of the magnetic field in the solid crust of neutron stars. It is a nonlinear process that, despite a number of efforts, is still not fully understood. We use the Hall induction equation in axial symmetry to obtain some general properties of nonevolving fields, as well as analyzing the evolution of purely toroidal fields, their poloidal perturbations, and current-free, purely poloidal fields. We also analyze energy conservation in Hall instabilities and write down a variational principle for Hall equilibria. We show that the evolution of any toroidal magnetic field can be described by Burgers' equation, as previously found in plane-parallel geometry. It leads to sharp current sheets that dissipate on the Hall time scale, yielding a stationary field configuration that depends on a single, suitably defined coordinate. This field, however, is unstable to poloidal perturbations, which grow as their field lines are stretched by the background electron flow, as in instabilities earlier found numerically. On the other hand, current-free poloidal configurations are stable and could represent a long-lived crustal field supported by currents in the fluid stellar core.