The traditional approximation of rotation, including the centrifugal acceleration for slightly deformed stars

AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette Cedex, France
e-mail: stephane.mathis@cea.fr

Received: 8 April 2019
Accepted: 15 August 2019

Abstract

Context. The traditional approximation of rotation (TAR) is a treatment of the dynamical equations of rotating and stably stratified fluids in which 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 the neglect of the horizontal projection of the rotation vector in the equations for the dynamics of gravito-inertial waves (GIWs) that become separable, such as in the non-rotating case, while they are not separable in the case in which the full Coriolis acceleration is taken into account. This approximation, first introduced in geophysical fluid dynamics for thin atmospheres and oceans, has been broadly applied in stellar (and planetary) astrophysics to study low-frequency GIWs that have short vertical wavelengths. The appoximation is now being tested thanks to direct 2D oscillation codes, which constrain its domain of validity. The mathematical flexibility of this treatment allows us to explore broad parameter spaces and to perform detailed seismic modelling of stars.

Aims. The TAR treatment 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. In addition, their induced gravitational potential fluctuations is neglected. However, it has been recently generalised with including the effects of a differential rotation. We aim to carry out a new generalisation that takes into account the centrifugal acceleration in the case of deformed stars that are moderately and uniformly rotating.

Methods. We construct an analytical expansion of the equations for the dynamics of GIWs in a spheroidal coordinates system by assuming the hierarchies of frequencies and amplitudes of the velocity components adopted within TAR in the spherical case.

Results. We derive the complete set of equations that generalises TAR by taking the centrifugal acceleration into account. As in the case of a differentially rotating spherical star, the problem becomes 2D but can be treated analytically if we assume the anelastic and JWKB approximations, which are relevant for low-frequency GIWs. This allows us to derive a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods, which 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.