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Astron. Astrophys. 334, 1000-1006 (1998)

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1. Introduction

The accumulation of observational evidences in favour of some additional mixing processes occurring during stellar evolution, particularly in massive stars (cf. Maeder 1995a), shows the necessity to deepen the study of stellar evolution with rotational mixing. Several models have been elaborated in the past (e.g. Endal and Sofia 1975; Schatzman and Maeder 1981) which often contained a large number of free parameters (cf. Pinsonneault et al. 1989). Progress has been achieved by taking into account the turbulence which is likely to arise from differential rotation (Zahn 1992); it is then possible to build a self-consistent picture of the transport of angular momentum and chemical elements, including the effects of meridional circulation and of turbulent diffusion.

In Paper I (Meynet and Maeder 1997) a number of evolutionary models with rotation were calculated, according to Zahn's prescriptions (1992). It was found that the µ-gradients developing during nuclear evolution are always too large to allow any significant mixing, despite the fact that the threshold for mixing imposed by Richardson's criterion was lowered by the inclusion of thermal effects (Maeder and Meynet 1996). In paper II (Maeder 1997) the modification of the Richardson criterion was considered when other sources of turbulence exist, such as semiconvection or horizontal turbulence. A new diffusion coefficient was derived with the assumption that the energy excess present in the shears is degraded by turbulence, which changes the local entropy gradient und consequently the local T - and µ-gradients. On their side Talon and Zahn (1997) treated the effect of horizontal mixing much as that of radiative damping, and they established an alternate form of the turbulent diffusivity. This latter prescription was applied to describe the rotational mixing of a [FORMULA] star (Talon et al. 1997).

In the present work we extend our theoretical approach by removing a few restrictive assumptions, in particular regarding the effects of µ-gradients on meridional circulation, as well as regarding the stationarity of the solutions. In Sect. 2 we re-examine the eulerian equation for the transport of angular momentum in non-stationary models and we recall in Sect. 3 the effect of a strong horizontal diffusion on the transport of chemical species. In Sect. 4 we express the thermal imbalance and the meridional velocity, allowing for non-stationarity and for a general equation of state. Special care will be taken in treating the large effects due to µ-gradients. Finally in Sect. 5 we discuss the results.

In an Appendix, we give the expression of the entropy change for a simple mixture of perfect gas and radiation.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998