  The magnetorotational instability (MRI) is the most promising mechanism by
which angular momentum is efficiently transported outwards in astrophysical
discs. However, its application to protoplanetary discs remains problematic.
These discs are so poorly ionised that they may not support magnetorotational
turbulence in regions referred to as `dead zones'. It has recently been
suggested that the Hall effect, a non-ideal magnetohydrodynamic (MHD) effect,
could revive these dead zones by enhancing the magnetically active column
density by an order of magnitude or more. We investigate this idea by
performing local, three-dimensional, resistive Hall-MHD simulations of the MRI
in situations where the Hall effect dominates over Ohmic dissipation. As
expected from linear stability analysis, we find an exponentially growing
instability in regimes otherwise linearly stable in resistive MHD. However,
instead of vigorous and sustained magnetorotational turbulence, we find that
the MRI saturates by producing large-scale, long-lived, axisymmetric structures
in the magnetic and velocity fields. We refer to these structures as zonal
fields and zonal flows, respectively. Their emergence causes a steep reduction
in turbulent transport by at least two orders of magnitude from extrapolations
based upon resistive MHD, a result that calls into question contemporary models
of layered accretion. We construct a rigorous mean-field theory to explain this
new behaviour and to predict when it should occur. Implications for
protoplanetary disc structure and evolution, as well as for theories of planet
formation, are briefly discussed.
