  Vortices in protoplanetary disks can capture solid particles and form
planetary cores within shorter timescales than those involved in the standard
core-accretion model. We investigate vortex generation in thin unmagnetized
protoplanetary disks with an embedded giant planet with planet to star mass
ratio $10^{-4}$ and $10^{-3}$. Two-dimensional hydrodynamical simulations of a
protoplanetary disk with a planet are performed using two different numerical
methods. The results of the non-linear simulations are compared with a
time-resolved modal analysis of the azimuthally averaged surface density
profiles using linear perturbation theory. Finite-difference methods
implemented in polar coordinates generate vortices moving along the gap created
by Neptune-mass to Jupiter-mass planets. The modal analysis shows that unstable
modes are generated with growth rate of order $0.3 \Omega_K$ for azimuthal
numbers m=4,5,6, where $\Omega_K$ is the local Keplerian frequency.
Shock-capturing Cartesian-grid codes do not generate very much vorticity around
a giant planet in a standard protoplanetary disk. Modal calculations confirm
that the obtained radial profiles of density are less susceptible to the growth
of linear modes on timescales of several hundreds of orbital periods.
Navier-Stokes viscosity of the order $\nu=10^{-5}$ (in units of $a^2 \Omega_p$)
is found to have a stabilizing effect and prevents the formation of vortices.
This result holds at high resolution runs and using different types of boundary
conditions. Giant protoplanets of Neptune-mass to Jupiter-mass can excite the
Rossby wave instability and generate vortices in thin disks. The presence of
vortices in protoplanetary disks has implications for planet formation, orbital
migration, and angular momentum transport in disks.
