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 to study the dynamics of baroclinic stellar envelopes is presented. Methods. This model treats the stellar fluid with the Boussinesq approximation and assumes that it is contained in a rigid spherical domain. The temperature field and the rotation of the system generate the baroclinic flow. Results. We 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 to investigate the fluid dynamics of rotating stars in particular for the study of the various instabilities affecting baroclinic flows or a dynamo effect.