  Neutron star magnetic field evolution is mediated through the Hall effect and
Ohmic dissipation in the crust while ambipolar diffusion is taking place in the
core. These effects have been studied in detail in either part of the star,
however, their combined, simultaneous evolution and interplay has not been
explored in detail yet. Here, we present simulation results of the simultaneous
evolution of the magnetic field in the core due to ambipolar diffusion and the
crust due to Hall effect and Ohmic decay, under the assumption of axial
symmetry. We find that a purely poloidal field generates a toroidal field in
the crust, due to the Hall effect, that sinks into the core. A purely toroidal
field remains toroidal and spreads into the core and the crust. Finally, for a
mixed poloidal-toroidal field, the north-south symmetry is broken due to the
Hall effect in the crust, however, ambipolar diffusion, tends to restore it. We
examine the role of ambipolar diffusion to the magnetic field decay and we
compare the rate of the conversion of magnetic field energy into heat, finding
that it enhances the magnetic field decay in neutron stars.
