Viscous-convective scalar range ($S_C(r)\sim {ϵ}_c\sqrt{\nu /ϵ}\log (r/{\eta }_B)$)

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.