The dynamics of the radiative envelope of rapidly rotating stars I. A spherical Boussinesq model

Context. The observations of rapidly rotating stars are increasingly detailed and precise thanks to interferometry and asteroseismology; twodimensional models taking into account the hydrodynamics of these stars are very much needed. Aims. A model for studying the dynamics of baroclinic stellar envelope is presented. Methods. This models treats the stellar fluid at the Boussinesq approximation and assumes that it is contained in a rigid spherical domain. The temperature field along with the rotation of the system generate the baroclinic flow. Results. We manage to give an analytical solution to the asymptotic problem at small Ekman and Prandtl numbers. We show that, provided the Brunt-Väisälä frequency profile is smooth enough, differential rotation of a stably stratified envelope takes the form a fast rotating pole and a slow equator while it is the opposite in a convective envelope. We also show that at low Prandtl numbers and without μ-barriers, the jump in viscosity at the core-envelope boundary generates a shear layer staying along the tangential cylinder of the core. Its role in mixing processes is discussed. Conclusions. Such a model provides an interesting tool for investigating the fluid dynamics of rotating stars in particular for the study of the various instabilities affecting baroclinic flows or, even more, of a dynamo effect.