We present the results of high-resolution, three-dimensional (3D) hydrodynamic simulations of the dynamics and formation of coherent, long-lived vortices in stably-stratified protoplanetary disks. Tall, columnar vortices that extend vertically through many scale heights in the disk are unstable to small perturbations; such vortices cannot maintain vertical alignment over more than a couple scale heights and are ripped apart by the Keplerian shear. Short, finite-height vortices that extend only one scale height above and below the midplane are also unstable, but for a different reason: we have isolated an antisymmetric (with respect to the midplane) eigenmode that grows with an e-folding time of only a few orbital periods; the nonlinear evolution of this instability leads to the destruction of the vortex. Serendipitously, we observe the formation of 3D vortices that are centered not in the midplane, but at one to three scale heights above and below. Breaking internal gravity waves create vorticity; anticyclonic regions of vorticity roll-up and coalesce into new vortices, whereas cyclonic regions shear into thin azimuthal bands. Unlike the midplane-centered vortices that were placed ad hoc in the disk and turned out to be linearly unstable, the off-midplane vortices form naturally out of perturbations in the disk, and are stable and robust for many hundreds of orbits.
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