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Shear-induced turbulence could play a significant role in mixing momentum andchemical species in stellar radiation zones, as discussed by Zahn (1974). Inthis paper we analyze the results of direct numerical simulations of stratifiedplane Couette flows, in the limit of rapid thermal diffusion, to measure theturbulent diffusivity and turbulent viscosity as a function of the local shearand the local stratification. We find that the stability criterion proposed byZahn (1974), namely that the product of the gradient Richardson number and thePrandtl number must be smaller than a critical values $(J\Pr)_c$ forinstability, adequately accounts for the transition to turbulence in the flow,with $(J\Pr)_c \simeq 0.007$. This result recovers and confirms the priorfindings of Prat et al. (2016). Zahn's model for the turbulent diffusivity andviscosity (Zahn 1992), namely that the mixing coefficient should beproportional to the ratio of the thermal diffusivity to the gradient Richardsonnumber, does not satisfactorily match our numerical data when applied as is. Itfails (as expected) in the limit of large stratification where the Richardsonnumber exceeds the aforementioned threshold for instability, but it also failsin the limit of low stratification where the turbulent eddy scale becomeslimited by the computational domain size. We propose a revised model forturbulent mixing by diffusive stratified shear instabilities, that now properlyaccounts for both limits, fits our data satisfactorily, and recovers Zahn's1992 model in the limit of large Reynolds numbers.

astrophysical journal/the astrophysical journal

Turbulent transport by diffusive stratified shear flows: from local to  global models. Part I: Numerical simulations of a stratified plane Couette  flow