  In the solid crusts of neutron stars, the advection of the magnetic field by
the current-carrying electrons, an effect known as Hall drift, should play a
very important role as the ions remain essentially fixed (as long as the solid
does not break). Although Hall drift preserves the magnetic field energy, it
has been argued that it may drive a turbulent cascade to scales at which Ohmic
dissipation becomes effective, allowing a much faster decay in objects with
very strong fields. On the other hand, it has been found that there are "Hall
equilibria", i.e., field configurations that are unaffected by Hall drift.
Here, we address the crucial question of the stability of these equilibria
through axially symmetric (2D) numerical simulations of Hall drift and Ohmic
diffusion, with the simplifying assumption of uniform electron density and
conductivity. We demonstrate the 2D-stability of a purely poloidal equilibrium,
for which Ohmic dissipation makes the field evolve towards an attractor state
through adjacent stable configurations, around which damped oscillations occur.
For this field, the decay scales with the Ohmic timescale. We also study the
case of an unstable equilibrium consisting of both poloidal and toroidal field
components that are confined within the crust. This field evolves into a stable
configuration, which undergoes damped oscillations superimposed on a slow
evolution towards an attractor, just as the purely poloidal one.
