  The gravito-inertial waves propagating over a shellular baroclinic flow
inside a rotating spherical shell are analysed using the Boussinesq
approximation. The wave properties are examined by computing paths of
characteristics in the non-dissipative limit, and by solving the full
dissipative eigenvalue problem using a high-resolution spectral method.
Gravito-inertial waves are found to obey a mixed-type second-order operator and
to be often focused around short-period attractors of characteristics or
trapped in a wedge formed by turning surfaces and boundaries. We also find
eigenmodes that show a weak dependence with respect to viscosity and heat
diffusion just like truly regular modes. Some axisymmetric modes are found
unstable and likely destabilized by baroclinic instabilities. Similarly, some
non-axisymmetric modes that meet a critical layer (or corotation resonance) can
turn unstable at sufficiently low diffusivities. In all cases, the instability
is driven by the differential rotation. For many modes of the spectrum, neat
power laws are found for the dependence of the damping rates with diffusion
coefficients, but the theoretical explanation for the exponent values remains
elusive in general. The eigenvalue spectrum turns out to be very rich and
complex, which lets us suppose an even richer and more complex spectrum for
rotating stars or planets that own a differential rotation driven by
baroclinicity.
