  We carry out three-dimensional, high resolution (up to $1024^2\times 256$)
hydrodynamic simulations of the evolution of vortices in vertically
unstratified Keplerian disks using the shearing sheet approximation. The
transient amplification of incompressible, linear amplitude leading waves
(which has been proposed as a possible route to nonlinear hydrodynamical
turbulence in disks) is used as one test of our algorithms; our methods
accurately capture the predicted amplification, converges at second-order, and
is free from aliasing. Waves expected to reach nonlinear amplitude at peak
amplification become unstable to Kelvin-Helmholtz modes when $\mid W_{\rm
max}\mid\gtrsim \Omega$ (where $W_{\rm max}$ is the local maximum of vorticity
and $\Omega$ the angular velocity). We study the evolution of a power-law
distribution of vorticity consistent with Kolmogorov turbulence; in
two-dimensions long-lived vortices emerge and decay slowly, similar to previous
studies. In three-dimensions, however, vortices are unstable to bending modes,
leading to rapid decay. Only vortices with a length to width ratio smaller than
one survive; in three-dimensions the residual kinetic energy and shear stress
is at least one order of magnitude smaller than in two-dimensions. No evidence
for sustained hydrodynamical turbulence and transport is observed in
three-dimensions. Instead, at late times the residual transport is determined
by the amplitude of slowly decaying, large-scale vortices (with horizontal
extent comparable to the scale height of the disk), with additional
contributions from nearly incompressible inertial waves possible. Evaluating
the role that large-scale vortices play in astrophysical accretion disks will
require understanding the mechanisms that generate and destroy them.
