Formation, survival, and destruction of vortices in accretion disks

Two-dimensional hydrodynamical disks are nonlinearly unstable to the formation of vortices. Once formed, these vortices essentially survive forever. What happens in three dimensions? We show with incompressible shearing box simulations that in three dimensions, a vortex in a short box forms and survives just as in two dimensions. But a vortex in a tall box is unstable and is destroyed. In our simulation, the unstable vortex decays into a transient turbulent-like state that transports angular momentum outward at a nearly constant rate for hundreds of orbital times. The three-dimensional instability that destroys vortices is a generalization of the two-dimensional instability that forms them. We derive the conditions for these nonlinear instabilities to act by calculating the coupling between linear modes, and thereby derive the criterion for a vortex to survive in three dimensions as it does in two dimensions: the azimuthal extent of the vortex must be larger than the scale height of the accretion disk. When this criterion is violated, the vortex is unstable and decays. Because vortices are longer in azimuthal than in radial extent by a factor that is inversely proportional to their excess vorticity, a vortex with given radial extent will only survive in a three-dimensional disk if it is sufficiently weak. This counterintuitive result explains why previous threedimensional simulations always yielded decaying vortices: their vortices were too strong. Weak vortices behave two-dimensionally even if their width is much less than their height because they are stabilized by rotation, and behave as Taylor-Proudman columns. We conclude that in protoplanetary disks, weak vortices can trap dust and serve as the nurseries of planet formation. Decaying strong vortices might be responsible for the outward transport of angular momentum that is required to make accretion disks accrete.