Stellar evolution with rotation. II. A new approach for shear mixing.

Recent observations show He- and N-enhancements in fast rotating O-type stars, while studies on shear mixing in rotating stars have shown that the μ-gradients generally inhibit mixing processes. However, these studies (including ours) ignored the other sources of turbulence, such as semiconvection and horizontal turbulence, which may be present in the medium and may modify the onset of shear mixing. Indeed, in massive stars most of the zone where μ-gradients would inhibit mixing according to the Richardson criterion is semiconvective, i.e. such a zone would experience some turbulence anyway. This leads us to introduce the following working hypothesis: in a semiconvection region (or in any zone with other sources of turbulence) some fraction of the local energy excess in the shear is degraded by turbulence to change the local entropy gradient. Consistently with this hypothesis we derive the effects of the shear on the entropy gradient, then on the T- and μ-gradients, and express the fraction of the μ-gradient which can be diffused. From the basic equations describing the evolution of temperature perturbations we obtain the velocities of fluid elements under some specific conditions, and then we get the associated diffusion coefficient D. Interestingly enough, the values of D tend towards the diffusion coefficient for semiconvection, when the shears become negligible, and towards the coefficient by Zahn (1992A&A...265..115Z; cf. also Maeder and Meynet 1996A&A...313..140M) when shears dominate. We also examine the coupling of shear mixing and thermal transport; a third order equation expressing the combined effects is obtained. The solutions for shears in semiconvective and radiative zones are examined in detail. The main result of the above developments is the slight progressive erosion of the μ-barriers in stars, for cases where the usual form of the Richardson criterion would have imposed an insuperable threshold. In Appendices A and B we rediscuss the diffusion coefficient for semiconvection and find a more general expression, not limited to the adiabatic case.