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If sufficiently irradiated by its central star, a protoplanetary disks fallsinto an equilibrium state exhibiting vertical shear. This state may be subjectto a hydrodynamical instability, the `vertical shear instability' (VSI), whosebreakdown into turbulence transports a moderate amount of angular momentumwhile also facilitating planet formation, possibly via the production ofsmall-scale vortices. In this paper, we show that VSI modes (a) exhibitarbitrary spatial profiles and (b) remain nonlinear solutions to theincompressible local equations, no matter their amplitude. The modes arethemselves subject to parasitic Kelvin-Helmholtz instability, though the diskrotation significantly impedes the parasites and permits the VSI to attainlarge amplitudes (fluid velocities < 10% the sound speed). This `delay' insaturation probably explains the prominence of the VSI linear modes in globalsimulations. More generally, the parasites may set the amplitude of VSIturbulence in strongly irradiated disks. They are also important in breakingthe axisymmetry of the flow, via the unavoidable formation of vortices. Thevortices, however, are not aligned with the orbital plane and thus express apronounced $z$-dependence. We also briefly demonstrate that the vertical shearhas little effect on the magnetorotational instability, whereas magnetic fieldseasily quench the VSI, a potential issue in the ionised surface regions of thedisk and also at larger radii.

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