  The bounded oscillations of rotating fluid-filled ellipsoids can provide
physical insight into the flow dynamics of deformed planetary interiors. The
inertial modes, sustained by the Coriolis force, are ubiquitous in rapidly
rotating fluids and Vantieghem (2014, Proc. R. Soc. A, 470, 20140093,
doi:10.1098/rspa.2014.0093) pioneered a method to compute them n incompressible
fluid ellipsoids. Yet, taking density (and pressure) variations into account is
required for accurate planetary applications, which has hitherto been largely
overlooked in ellipsoidal models. To go beyond the incompressible theory, we
present a Galerkin method in rigid coreless ellipsoids, based on a global
polynomial description. We apply the method to investigate the normal modes of
fully compressible, rotating and diffusionless fluids. We consider an idealized
model, which fairly reproduces the density variations in the Earth's liquid
core and Jupiter-like gaseous planets. We successfully benchmark the results
against standard finite-element computations. Notably, we find that the
quasi-geostrophic inertial modes can be significantly modified by
compressibility, even in moderately compressible interiors. Finally, we discuss
the use of the normal modes to build reduced dynamical models of planetary
flows.
