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; two-dimensional 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-Vaisala 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 $\mu$-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.