Vortices and the saturation of the vertical shear instability in protoplanetary discs

Abstract

If sufficiently irradiated by its central star, a protoplanetary disks falls into an equilibrium state exhibiting vertical shear. This state may be subject to a hydrodynamical instability, the `vertical shear instability' (VSI), whose breakdown into turbulence transports a moderate amount of angular momentum while also facilitating planet formation, possibly via the production of small-scale vortices. In this paper, we show that VSI modes (a) exhibit arbitrary spatial profiles and (b) remain nonlinear solutions to the incompressible local equations, no matter their amplitude. The modes are themselves subject to parasitic Kelvin-Helmholtz instability, though the disk rotation significantly impedes the parasites and permits the VSI to attain large amplitudes (fluid velocities < 10% the sound speed). This `delay' in saturation probably explains the prominence of the VSI linear modes in global simulations. More generally, the parasites may set the amplitude of VSI turbulence in strongly irradiated disks. They are also important in breaking the axisymmetry of the flow, via the unavoidable formation of vortices. The vortices, however, are not aligned with the orbital plane and thus express a pronounced $z$-dependence. We also briefly demonstrate that the vertical shear has little effect on the magnetorotational instability, whereas magnetic fields easily quench the VSI, a potential issue in the ionised surface regions of the disk and also at larger radii.