  We study ambipolar diffusion in strongly magnetised neutron stars, with
special focus on the effects of neutrino reaction rates and the impact of a
superfluid/superconducting transition in the neutron star core. For
axisymmetric magnetic field configurations, we determine the deviation from
$\beta-$equilibrium induced by the magnetic force and calculate the velocity of
the slow, quasi-stationary, ambipolar drift. We study the temperature
dependence of the velocity pattern and clearly identify the transition to a
predominantly solenoidal flow. For stars without superconducting/superfluid
constituents and with a mixed poloidal-toroidal magnetic field of typical
magnetar strength, we find that ambipolar diffusion proceeds fast enough to
have a significant impact on the magnetic field evolution only at low core
temperatures, $T \lesssim 1-2\times10^8$ K. The ambipolar diffusion timescale
becomes appreciably shorter when fast neutrino reactions are present, because
the possibility to balance part of the magnetic force with pressure gradients
is reduced. We also find short ambipolar diffusion timescales in the case of
superconducting cores for $T \lesssim 10^9$ K, due to the reduced interaction
between protons and neutrons. In the most favourable scenario, with fast
neutrino reactions and superconducting cores, ambipolar diffusion results in
advection velocities of several km/kyr. This velocity can substantially
reorganize magnetic fields in magnetar cores, in a way that can only be
confirmed by dynamical simulations.
