  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 pseudospectral simulations that
in 3D a vortex in a short box forms and survives just as in 2D. 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 3D
instability that destroys vortices is a generalization of the 2D 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 3D as it does in 2D: 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 3D disk if it is sufficiently weak. This counterintuitive
result explains why previous 3D simulations always yielded decaying vortices:
their vortices were too strong. 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 outwards transport of angular
momentum that is required to make accretion disks accrete.
