  Pancake-like vortices are often generated by turbulence in geophysical flows.
Here, we study the inertia-gravity oscillations that can exist within such
geophysical vortices, due to the combined action of rotation and gravity. We
consider a fluid enclosed within a triaxial ellipsoid, which is stratified in
density with a constant Brunt-V\"ais\"al\"a frequency (using the Boussinesq
approximation) and uniformly rotating along a (possibly) tilted axis with
respect to gravity. The wave problem is then governed by a mixed
hyperbolic-elliptic equation for the velocity. As in the rotating
non-stratified case considered by Vantieghem (2014, Proc. R. Soc. A, 470,
20140093, doi:10.1098/rspa.2014.0093), we find that the spectrum is pure point
in ellipsoids (i.e. only consists of eigenvalues) with smooth polynomial
eigenvectors. Then, we characterise the spectrum using numerical computations
(obtained with a bespoke Galerkin method) and asymptotic spectral theory.
Finally, the results are discussed in light of natural applications (e.g. for
Mediterranean eddies or Jupiter's vortices).
