  Seismic waves sensitive to the outermost part of the Earth's liquid core seem
to be affected by a stably stratified layer at the core-mantle boundary. Such a
layer could have an observable signature in both long-term and short-term
variations of the magnetic field of the Earth, which are used to probe the flow
at the top of the core. Indeed, with the recent SWARM mission, it seems
reasonable to be able to identify waves propagating in the core with period of
several months, which may play an important role in the large-scale dynamics.
In this paper, we characterize the influence of a stratified layer at the top
of the core on deep quasi-geostrophic (Rossby) waves. We compute numerically
the quasi-geostrophic eigenmodes of a rapidly rotating spherical shell, with a
stably stratified layer near the outer boundary. Two simple models of
stratification are taken into account, which are scaled with commonly adopted
values of the Brunt-V{\"a}is{\"a}l{\"a} frequency in the Earth's core. In the
absence of magnetic field, we find that both azimuthal wavelength and frequency
of the eigenmodes control their penetration into the stratified layer: the
higher the phase speed, the higher the permeability of the stratified layer to
the wave motion. We also show that the theory developed by Takehiro and Lister
[2001] for thermal convection extends to the whole family of Rossby waves in
the core. Adding a magnetic field, the penetrative behaviour of the
quasi-geostrophic modes (the so-called fast branch) is insensitive to the
imposed magnetic field and only weakly sensitive to the precise shape of the
stratification. Based on these results, the large-scale and high frequency
modes (1 to 2 month periods) may be detectable in the geomagnetic data measured
at the Earth's surface, especially in the equatorial area where the modes can
be trapped.
