On the vertical shear instability in magnetized protoplanetary discs

The vertical shear instability (VSI) is a robust phenomenon in irradiated protoplanetary discs (PPDs). While there is extensive literature on the VSI in the hydrodynamic limit, PPDs are expected to be magnetized and their extremely low ionization fractions imply that non-ideal magnetohydrodynamic effects should be properly considered. To this end, we present linear analyses of the VSI in magnetized discs with Ohmic resistivity. We primarily consider toroidal magnetic fields, which are likely to dominate the field geometry in PPDs. We perform vertically global and radially local analyses to capture characteristic VSI modes with extended vertical structures. To focus on the effect of magnetism, a locally isothermal equation of state is employed. We find that magnetism provides a stabilizing effect to dampen the VSI, with surface modes, rather than body modes, being the first to vanish with increasing magnetization. Subdued VSI modes can be revived by Ohmic resistivity, where sufficient magnetic diffusion overcomes magnetic stabilization, and hydrodynamic results are recovered. We also briefly consider poloidal magnetic fields to account for the magnetorotational instability (MRI), which may develop towards surface layers in the outer regions of PPDs. The MRI grows efficiently at small radial wavenumbers, in contrast to the VSI. When resistivity is considered, the VSI dominates over the MRI for Ohmic Elsässer numbers ≲0.09 at plasma beta parameter β_Z ~ 10⁴.