  The Rossby wave instability (RWI) is the fundamental non-axisymmetric radial
shear instability in disks. The RWI can facilitate disk accretion, set the
shape of planetary gaps and produce large vortices. It arises from density
and/or temperature features, such as radial gaps, bumps or steps. A general,
sufficient condition to trigger the RWI is lacking, which we address by
studying the linear RWI in a suite of simplified models, including
incompressible and compressible shearing sheets and global, cylindrical disks.
We focus on enthalpy amplitude and width as the fundamental properties of disk
features with various shapes. We find analytic results for the RWI boundary and
growth rates across a wide parameter space, in some cases with exact
derivations and in others as a description of numerical results. Features wider
than a scale-height generally become unstable about halfway to Rayleigh
instability, i.e.\ when the squared epicyclic frequency is about half the
Keplerian value, reinforcing our previous finding. RWI growth rates
approximately scale as enthalpy amplitude to the 1/3 power, with a weak
dependence on width, across much of parameter space. Global disk curvature
affects wide planetary gaps, making the outer gap edge more susceptible to the
RWI. Our simplified models are barotropic and height-integrated, but the main
results should carry over to more complex and realistic scenarios.
