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

Context. The Hall drift, namely, the 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 theoretical efforts, is still not fully understood. Aims. Through mostly analytic arguments and solutions, we intend to help understand this highly nonlinear process. Methods. 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. Results. We show that the evolution of any toroidal magnetic field can be described by Burgers' equation, as previously found by Vainshtein and collaborators in a plane-parallel geometry. This evolution leads to sharp current sheets, which 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 the instabilities found numerically by Rheinhardt and Geppert. 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. There may be additional, stable configurations, corresponding to restricted local minima or maxima of the magnetic energy. Conclusions. Hall equilibria can be described by a simple variational principle. Long-lived, toroidal fields are not expected in neutron star crusts or other regions where Hall drift is the dominant evolution mechanism. However, other stable configurations do exist, such as current-free poloidal fields and possibly others.