  Astrophysical disks with localized radial structure, such as protoplanetary
disks containing dead zones or gaps due to disk-planet interaction, may be
subject to the non-axisymmetric Rossby wave instability (RWI) that lead to
vortex-formation. The linear instability has recently been demonstrated in
three-dimensional (3D) barotropic disks. It is the purpose of this study to
generalize the 3D linear problem to include an energy equation, thereby
accounting for baroclinity in three-dimensions. Linear stability calculations
are presented for radially structured, vertically stratified,
geometrically-thin disks with non-uniform entropy distribution in both
directions. Polytropic equilibria are considered but adiabatic perturbations
assumed. The unperturbed disk has a localized radial density bump making it
susceptible to the RWI. The linearized fluid equations are solved numerically
as a partial differential equation eigenvalue problem. Emphasis on the ease of
method implementation is given. It is found that when the polytropic index is
fixed and adiabatic index increased, non-uniform entropy has negligible effect
on the RWI growth rate, but pressure and density perturbation magnitudes near a
pressure enhancement increases away from the midplane. The associated
meridional flow is also qualitatively changed from homentropic calculations.
Meridional vortical motion is identified in the nonhomentropic linear solution,
as well as in a nonlinear global hydrodynamic simulation of the RWI in an
initially isothermal disk evolved adiabatically. Numerical results suggest
buoyancy forces play an important role in the internal flow of Rossby vortices.
