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Shear mixing in stellar radiative zones I. Effect of thermal diffusion and chemical stratification

Turbulent transport of chemical elements in radiative zones of stars is considered in current stellar evolution codes thanks to phenomenologically derived diffusion coefficients. Recent local numerical simulations (Prat & Lignières 2013, A&A, 551, L3) suggest that the coefficient for radial turbulent diffusion due to radial differential rotation satisfies $D_{\rm t}\simeq0.058κ/Ri$, in qualitative agreement with Zahn's model. However, this model does not apply when differential rotation is strong with respect to stable thermal stratification or when chemical stratification has a significant dynamical effect, a situation encountered at the outer boundary of nuclear-burning convective cores. We extend our numerical study to consider the effects of chemical stratification and of strong shear, and compare the results with prescriptions used in stellar evolution codes. We performed local, direct numerical simulations of stably stratified, homogeneous, sheared turbulence in the Boussinesq approximation. The regime of high thermal diffusivities, typical of stellar radiative zones, is reached thanks to the so-called small-Péclet-number approximation, which is an asymptotic development of the Boussinesq equations in this regime. The dependence of the diffusion coefficient on chemical stratification was explored in this approximation. Maeder's extension of Zahn's model in the strong-shear regime is not supported by our results, which are better described by a model found in the geophysical literature. As regards the effect of chemical stratification, our quantitative estimate of the diffusion coefficient as a function of the mean gradient of mean molecular weight leads to the formula $D_{\rm t}\simeq 0.45κ(0.12-Ri_μ)/Ri$, which is compatible in the weak-shear regime with the model of Maeder & Meynet (1996, A&A, 313, 140).