  The effect of magnetic field diffusion on the stability of accretion disks is
a problem that has attracted considerable interest of late. In particular, the
Hall effect has the potential to bring about remarkable changes in the
dynamical behavior of disks that are without parallel. In this paper, we
conduct a systematic examination of the linear eigenmodes in a weakly
magnetized differentially rotating gas with special focus on Hall diffusion. We
first develop a geometrical representation of the eigenmodes and provide a
detailed quantitative description of the polarization properties of the
oscillatory modes under the combined influence of the Coriolis and Hall
effects. We also analyze the effects of magnetic diffusion on the structure of
the unstable modes and derive analytical expressions for the kinetic and
magnetic stresses and energy densities associated with the non-ideal MRI. Our
analysis explicitly demonstrates that, if the dissipative effects are
relatively weak, the kinetic stresses and energies make up the dominant
contribution to the total stress and energy density when the equilibrium
angular momentum and magnetic field vectors are anti-parallel. This is in sharp
contrast to what is observed in the case of the ideal or dissipative MRI. We
conduct shearing box simulations and find very good agreement with the results
derived from linear analysis. As the modes in consideration are also exact
solutions of the non-linear equations, the unconventional nature of the kinetic
and magnetic stresses may have significant implications for the non-linear
evolution in some regions of protoplanetary disks.
