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Astron. Astrophys. 323, 831-838 (1997)

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1. Introduction

Rotation and shear mixing are becoming increasingly important physical ingredients that are not to be omitted in the new run of evolutionary calculations. They are both believed to be responsible for the onset of macroscopic motions in the stellar plasma resulting in enhanced mixing of chemical elements and the maintenance of angular momentum transport phenomena. In recent years, in particular, shear mixing has been frequently invoked to explain the unexpected degree of enrichment in heavy elements in the atmospheres of massive MS stars (Herrero et al. 1992; Venn 1995). Also, mixing processes can be inferred to operate within the shear layer at the base of the convective envelope of the Sun, thus possibly explaining the shallower helium profile deduced by helioseismological data as compared to solar model solutions with helium settling (Gough et al. 1996). Unfortunately, the thresholds for shear induced instabilities and the magnitude of the ensuing transport phenomena are badly known, and the few works devoted to the study of their effects on stellar evolution calculations remain in an exploratory phase.

The problem of the observed enrichment in metals at the surface of fast rotating massive stars has been studied in a first paper by Meynet & Maeder (1996), who remark that the common understanding of the Richardson criterion proved of formidable efficacy in preventing any significant mixing in regions where large µ-gradients are generated in consequence of nuclear evolution. In that respect let us recall that the Richardson criterion, in its original form, imposes a threshold, namely [FORMULA] for shear mixing in a plane-parallel, radiative zone (U being the horizontal velocity; the other symbols are defined below). According to Maeder (1996), it is possible to solve the present theoretical discrepancies if one supposes that a fraction of the local energy excess available in radiative shear flows is degraded by turbulence. This working hypothesis implies that even in regions stable according to the Ledoux criterion, partial, turbulent mixing occurs within a fraction of the hydrogen burning timescale, determining the progressive erosion of the µ-barriers and the consequent He- and N- enrichments in fast rotating O-stars. Of course, the existence of semiconvective shear zones, where the Richardson number [FORMULA], can only be assessed by confronting all their logical consequences with carefully selected observations.

As a matter of fact, turbulent transport in stellar radiation zones has a rather long history, which goes back to the first proposition of Schatzman (1969) to explain the chemical composition at the surface of the Sun and solar-like stars by a mild but efficient transport of matter. Since then, much theoretical work has been devoted to the study of its possible origin, soon recognizing that only shear instabilities may reach a turbulent state strong enough to actually mix the stellar material (Zahn 1983, 1990). In general, shear instabilities depend on the exact profile of U, the horizontal components are likely more vigorous and of larger extent, so that one should account for the three dimensional character of the motion field in deriving turbulent diffusivities that are necessarily anisotropic (Zahn 1991). Here, however, and more modestly in view of the aforementioned considerations on massive stars evolution, our purpose is just to demonstrate that shear induced semiconvection is in fact a natural prediction of a linear stability analysis of the basic hydrodynamic equations. Our approach is an extension of the classical work of Kato (1966) on chemically stratified Boussinesq mediums, when the inertial corrections are included in the force law equation. Additionally, we obtain a revised criterion for the onset of convection, and provide a way to compute the diffusion coefficient of scalar fields in presence of rotation, shear and thermal conductivity. These results, of modest computational effort and easy implementation in existing codes, could help in gaining some first order, deeper insight on massive stellar structure and evolution.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998