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The Traditional Approximation of Rotation (TAR) is a treatment of thehydrodynamic equations of rotating and stably stratified fluids in which theaction of the Coriolis acceleration along the direction of the entropy andchemical stratifications is neglected because it is weak in comparison with thebuoyancy force. The dependent variables in the equations for the dynamics ofgravito-inertial waves (GIWs) then become separable into radial and horizontalparts as in the non-rotating case. The TAR is built on the assumptions that thestar is spherical (i.e. its centrifugal deformation is neglected) and uniformlyrotating. We study the feasibility of carrying out a generalisation of the TARto account for the centrifugal acceleration in the case of strongly deformeduniformly and rapidly rotating stars (and planets), and to identify thevalidity domain of this approximation. We built analytically a completeformalism that allows the study of the dynamics of GIWs in spheroidalcoordinates which take into account the flattening of rapidly rotating stars byassuming the hierarchies of frequencies adopted within the TAR in the sphericalcase and by deriving a generalised Laplace tidal equation for the horizontaleigenfunctions of the GIWs and their asymptotic wave periods, which can be usedto probe the structure and dynamics of rotating deformed stars withasteroseismology. Using 2D ESTER stellar models, we determine the validitydomain of the generalised TAR as a function of the rotation rate of the starnormalised by its critical angular velocity and its pseudo-radius. Thisgeneralisation allows us to study the signature of the centrifugal effects onGIWs in rapidly rotating deformed stars. We found that the effects of thecentrifugal acceleration in rapidly rotating early-type stars on GIWs aretheoretically detectable in modern space photometry using observations fromKepler.

The traditional approximation of rotation for rapidly rotating stars and planets. I. The impact of strong deformation