  The Traditional Approximation of Rotation (TAR) is a treatment of the
dynamical equations of rotating stably stratified fluids where the action of
the Coriolis acceleration along the direction of the entropy (and chemicals)
stratification is neglected while assuming that the fluid motions are mostly
horizontal because of their inhibition in the vertical direction by the
buoyancy force. This leads to neglect the horizontal projection of the rotation
vector in the equations for the dynamics of gravito-inertial waves (GIWs) that
become separable as in the non-rotating case while they are not in the case
with the full Coriolis acceleration. This approximation has been broadly
applied in stellar (and planetary) astrophysics to study low-frequency GIWs.
TAR is built on the assumptions that the star is spherical (i.e. its
centrifugal deformation is neglected) and uniformly rotating while an adiabatic
treatment of the dynamics of the waves is adopted. However, it has been
recently generalised with including the effects of a differential rotation. We
aim to do a new generalisation that takes into account the centrifugal
acceleration in the case of moderately uniformly rotating deformed stars. As in
the case of a differentially rotating spherical star, the problem becomes 2D
but can be treated analytically if assuming the Cowling, anelastic and JWKB
approximations, which are relevant for low-frequency GIWs. It allows us to
derive a generalised Laplace tidal equation for the horizontal eigenfunctions
and asymptotic wave periods that can be used to probe the structure and
dynamics of rotating deformed stars thanks to asteroseismology. A first
numerical exploration of its eigenvalues and horizontal eigenfunctions shows
their variation as a function of the pseudo-radius for different rotation rates
and frequencies and the development of avoided crossings.

Solar and Stellar Astrophysics;Earth and Planetary Astrophysics;Atmospheric and Oceanic Physics;Fluid Dynamics