Turbulent transport by diffusive stratified shear flows: from local to global models. Part I: Numerical simulations of a stratified plane Couette flow
Shear-induced turbulence could play a significant role in mixing momentum and chemical species in stellar radiation zones, as discussed by Zahn (1974). In this paper we analyze the results of direct numerical simulations of stratified plane Couette flows, in the limit of rapid thermal diffusion, to measure the turbulent diffusivity and turbulent viscosity as a function of the local shear and the local stratification. We find that the stability criterion proposed by Zahn (1974), namely that the product of the gradient Richardson number and the Prandtl number must be smaller than a critical values $(J\Pr)_c$ for instability, adequately accounts for the transition to turbulence in the flow, with $(J\Pr)_c \simeq 0.007$. This result recovers and confirms the prior findings of Prat et al. (2016). Zahn's model for the turbulent diffusivity and viscosity (Zahn 1992), namely that the mixing coefficient should be proportional to the ratio of the thermal diffusivity to the gradient Richardson number, does not satisfactorily match our numerical data when applied as is. It fails (as expected) in the limit of large stratification where the Richardson number exceeds the aforementioned threshold for instability, but it also fails in the limit of low stratification where the turbulent eddy scale becomes limited by the computational domain size. We propose a revised model for turbulent mixing by diffusive stratified shear instabilities, that now properly accounts for both limits, fits our data satisfactorily, and recovers Zahn's 1992 model in the limit of large Reynolds numbers.