  Vortices are readily produced by hydrodynamical instabilities, such as the
Rossby wave instability, in protoplanetary disks. However, large-scale
asymmetries indicative of dust-trapping vortices are uncommon in sub-millimeter
continuum observations. One possible explanation is that vortices have short
lifetimes. In this paper, we explore how radiative cooling can lead to vortex
decay. Elliptical vortices in Keplerian disks go through adiabatic heating and
cooling cycles. Radiative cooling modifies these cycles and generates
baroclinicity that changes the potential vorticity of the vortex. We show that
the net effect is typically a spin down, or decay, of the vortex for a
sub-adiabatic radial stratification. We perform a series of two-dimensional
shearing box simulations, varying the gas cooling (or relaxation) time, $t_{\rm
cool}$, and initial vortex strength. We measure the vortex decay half-life,
$t_{\rm half}$, and find that it can be roughly predicted by the timescale
ratio $t_{\rm cool}/t_{\rm turn}$, where $t_{\rm turn}$ is the vortex
turnaround time. Decay is slow in both the isothermal ($t_{\rm cool}\ll t_{\rm
turn}$) and adiabatic ($t_{\rm cool}\gg t_{\rm turn}$) limits; it is fastest
when $t_{\rm cool}\sim0.1\,t_{\rm turn}$, where $t_{\rm half}$ is as short as
$\sim300$ orbits. At tens of au where disk rings are typically found, $t_{\rm
turn}$ is likely much longer than $t_{\rm cool}$, potentially placing vortices
in the fast decay regime.
