Magnetic Field Evolution of Neutron Stars I: Basic formalism, numerical techniques, and first results

In this work we explore the evolution of magnetic fields inside strongly magnetized neutron stars in axisymmetry. We model numerically the coupled field evolution in the core and the crust. Our code models the Hall drift and Ohmic effects in the crust, the back-reaction on the field from magnetically-induced elastic deformation of the crust, the magnetic twist exchange between the crust and the core, and the drift of superconducting flux tubes inside the core. The correct hydromagnetic equilibrium is enforced in the core. We find that i) The Hall attractor found by Gourgouliatos and Cumming in the crust exists also for configurations when the B-field penetrates into the core. However, the evolution timescale for the core-penetrating fields is dramatically different than that of the fields confined to the crust. ii) The combination of Jones' flux tube drift and Ohmic dissipation in the crust can deplete the pulsar magnetic fields on the timescale of $150$ Myr if the crust is hot ($T\sim2\times10^8$ K), but acts on much slower timescales for cold neutron stars, such as recycled pulsars ($\sim 1.8 $ Gyr, depending on impurity levels). iii) The outward motion of superfluid vortices during the rapid spin-down of a young highly magnetized pulsar, can result in a partial expulsion of flux from the core when $B\lesssim 10^{13}$ G. However for $B\gtrsim2\times 10^{13}$ G, the combination of a stronger magnetic field and a longer spin period implies that the core field cannot be expelled.