Resonant character of the hall instability in protoplanetary disks

We consider nonaxisymmetric magnetosonic oscillations of a radially stratified, weakly ionized protoplanetary disk with a vertical magnetic field. The combined effect of the Hall electric field and the density and magnetic field inhomogeneities present in the disk has been previously predicted to lead to an instability of its small azimuthal perturbations. We revise the previous results and take into account the effect of inhomogeneous ionization of the protoplanetary material related to the inhomogeneity of the disk medium. We show that the instability criterion is governed by three parameters: the magnetic field and ionization fraction gradients and the plasma β. We have found that at high values of β typical of protoplanetary disks, the instability does not manifest itself if the gradients are directed oppositely. In the case of codirectional gradients, the interaction of magnetosonic fluctuations with inhomogeneities of a fixed size is resonant in character, giving rise to an instability in a narrow range of wave numbers.