Obukhov-Corrsin theory ($S_C(r)\sim {ϵ}_c{ϵ}^{-1/3}{r}^{2/3}$)

Obukhov-Corrsin theory concerns transfer of scalar variance across scales in turbulent mixing. In the inertial-convective range, a constant scalar-variance flux and an eddy turnover time ${ϵ}^{-1/3}{r}^{2/3}$ predict the displayed second-order scalar structure function. It assumes a passive scalar, local transfer, approximate statistical equilibrium and separation from the molecular diffusion and forcing scales.

For scalar diffusivity much larger than kinematic viscosity, scalar diffusion dominates at lengths between the Kolmogorov and scalar cutoff scales. The velocity may still have inertial-range variations, but the scalar is smooth to leading order. The leading isotropic scalar structure function is quadratic, with coefficient fixed by the mean-square gradient and the scalar dissipation rate, rather than by a continuing inertial-convective two-thirds law.

In the large-diffusivity regime, equate diffusion rate $\alpha /r^2$ with inertial eddy rate ${ϵ}^{1/3}{r}^{-2/3}$. This gives the Corrsin scalar microscale, which exceeds the Kolmogorov length scale when $\alpha \gg \nu$.

For scalar diffusivity much smaller than kinematic viscosity, a scalar has structure below the Kolmogorov length scale while the velocity is smooth. The strain rate $s\sim (ϵ/\nu {)}^{1/2}$ transfers scalar variance, giving $d{S}_C/dr\sim {ϵ}_c/(sr)$. Integration produces a logarithm plus an additive matching constant.

At the Batchelor scalar microscale, molecular diffusion rate $\alpha /r^2$ balances the smooth Kolmogorov strain rate $(ϵ/\nu {)}^{1/2}$. For $\alpha \ll \nu$ this scale is much smaller than the Kolmogorov length scale.

This range lies above both the Kolmogorov length scale and the scalar diffusion cutoff, but below the forcing scale. Molecular viscosity and scalar diffusion have negligible direct influence on transfer at those scales. The Obukhov-Corrsin theory predicts a two-thirds scalar structure function there.