We present the results of high-resolution, three-dimensional 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 few scale heights and are ripped apart by the Keplerian shear. Short, finite-height vortices that extend only 1 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 three-dimensional vortices that are centered not in the mid-plane, but at 1-3 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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