  (abridged) We study the nonlinear evolution of the Rossby wave instability in
thin disks using global 2D hydrodynamic simulations. The key questions we are
addressing in this paper are: (1) What happens when the instability becomes
nonlinear? Specifically, does it lead to vortex formation? (2) What is the
detailed behavior of a vortex? (3) Can the instability sustain itself and can
the vortex last a long time? Among various initial equilibria that we have
examined, we generally find that there are three stages of the disk evolution:
(1) The exponential growth of the initial small amplitude perturbations. This
is in excellent agreement with the linear theory; (2) The production of large
scale vortices and their interactions with the background flow, including
shocks. Significant accretion is observed due to these vortices. (3) The
coupling of Rossby waves/vortices with global spiral waves, which facilitates
further accretion throughout the whole disk. Even after more than 20
revolutions at the radius of vortices, we find that the disk maintains a state
that is populated with vortices, shocks, spiral waves/shocks, all of which
transport angular momentum outwards. We elucidate the physics at each stage and
show that there is an efficient outward angular momentum transport in stages
(2) and (3) over most parts of the disk, with an $\alpha$ in the range from
$10^{-4}$ to $10^{-2}$. We show why such vortices prove to be almost ideal
``units'' in transporting angular momentum outwards, namely by positively
correlating the radial and azimuthal velocity components. We find some special
cases in which entropy can remain the same while angular momentum is
transported. We conclude that Rossby wave/vortex instability is an efficient,
purely hydrodynamic mechanism for angular momentum transport in thin disks.
