  We study the effect of the non-linear process of ambipolar diffusion (joint
transport of magnetic flux and charged particles relative to neutral particles)
on the long-term behavior of a non-uniform magnetic field in a one-dimensional
geometry. Our main focus is the dissipation of magnetic energy inside neutron
stars(particularly magnetars), but our results have a wider application,
particularly to the interstellar medium and the loss of magnetic flux from
collapsing molecular cloud cores. Our system is a weakly ionized plasma in
which neutral and charged particles can be converted into each other through
nuclear beta decays (or ionization-recombination processes). In the
"weak-coupling" limit of infrequent inter-particle interactions, the evolution
of the magnetic field is controlled by the beta decay rate and can be described
by a non-linear partial integro-differential equation. In the opposite,
"strong-coupling" regime, the evolution is controlled by the inter-particle
collisions and can be modelled through a non-linear diffusion equation. We show
numerically that, in both regimes, ambipolar diffusion tends to spread out the
magnetic flux, but, contrary to the normal Ohmic diffusion, it produces sharp
magnetic field gradients with associated current sheets around those regions
where the magnetic field is weak.
