The vertical shear instability in poorly ionized, magnetized protoplanetary discs

Protoplanetary discs should exhibit a weak vertical variation in their rotation profiles. Typically, this 'vertical shear' issues from a baroclinic effect driven by the central star's radiation field, but it might also arise during the launching of a magnetocentrifugal wind. As a consequence, protoplanetary discs are subject to a hydrodynamical instability, the 'vertical shear instability' (VSI), whose breakdown into turbulence could transport a moderate amount of angular momentum and facilitate, or interfere with, the process of planet formation. Magnetic fields may suppress the VSI, however, either directly via magnetic tension or indirectly through magnetorotational turbulence. On the other hand, protoplanetary discs exhibit notoriously low ionization fractions, and non-ideal effects, if sufficiently dominant, may come to the VSI's rescue. In this paper, we develop a local linear theory that explores how non-ideal magnetohydrodynamics influences the VSI, while exciting additional diffusive shear instabilities. We derive a set of analytical criteria that establish when the VSI prevails, and then show how it can be applied to a representative global model of a protoplanetary disc. Our calculations suggest that within ∼10 au the VSI should have little trouble emerging in the main body of the disc, but beyond that, and in the upper regions of the disc, its onset depends sensitively on the size of the preponderant dust grains.