  Protoplanetary disks with non-axisymmetric structures have been observed. The
Rossby wave instability (RWI) is considered as one of the origins of the
non-axisymmetric structures. We perform linear stability analyses of the RWI in
barotropic flow using four representative types of the background flow on a
wide parameter space. We find that the co-rotation radius is located at the
background vortensity minimum with large concavity if the system is marginally
stable to the RWI, and this allows us to check the stability against the RWI
easily. We newly derive the necessary and sufficient condition for the onset of
the RWI in semi-analytic form. We discuss the applicability of the new
condition in realistic systems and the physical nature of the RWI.
