dissipation of passive scalar fluctuations. Following the same approach of the previous section, the velocity difference $\v(\x+\r,t)-\v(\x,t)$ is approximated by a white noise. From Itô’s Lemma, its contribution in Eqn. (3.35) will be an eddy diffusivity $D_{\alpha\beta}^v(\r)$, whose expression will coincide, for $r\gg \eta_\sigma r$, with the one for $D_{\alpha\beta}$ provided by Eqns. (3.16-17). Drift terms coming from the Stratonovich prescriptions are ruled out with the same arguments used in the previous section. The resulting diffusion equation will then read: $$\partial_\alpha\partial_\beta(\frac{1}{2}D_{\alpha\beta}^v(\r)+2\sigma\delta_{\alpha\beta})
\Theta(r)+4\flux_\theta=0
\eqno(3.36)$$ For $r\gg\eta_\sigma$ $D^v_{ij}$ is essentially a correction to the molecular diffusivity, and will read, from Eqn. (3.31): $$D_{\alpha}^v(\r)=
\int\d t\langle[v_\alpha(\r,t)-v_\alpha(0,t))][v_\beta(\r,0)-v_\beta(0,0))]\rangle$$ $$\simeq
\Big[\frac{r_\alpha r_\beta}{r^2}+\frac{13}{3}\Big(\delta_{\alpha\beta}-\frac{r_\alpha 
r_\beta}{r^2}\Big)\Big]
\frac{3\pi\sigma}{8\rho^2}(r/\eta_\sigma)^\frac{10}{3}
\eqno(3.37)$$ For $\eta_\sigma\ll r$, $D^v_{ij}$ is approximated by Eqn. (3.17), the molecular diffusivity $\sigma$ can be neglected and Eqn. (3.35) takes the form: $$\Theta''+\frac{7}{3r}\Theta'=-\frac{8\rho\flux_\theta}
{4\alpha_\smalsev\Ckol^\frac{1}{2}\flux^\frac{1}{3}r^\frac{4}{3}}
\eqno(3.38)$$ Solution of this equation gives automatically the passive scalar structure function $\langle[\theta(\x+\r,t)-\theta(\x,t)]^2\rangle=2[\Theta(0)-\Theta(r)]$ in the inertial range for $\theta$: $\eta_\sigma\ll r\ll k_0^{-1}$. This structure function scales like $r^\frac{2}{3}$ and can be written in the form: $$\langle[\theta(\x+\r,t)-\theta(\x,t)]^2\rangle
=\frac{B\flux_\theta r^\frac{2}{3}}{\Ckol^\frac{1}{2}\flux^\frac{1}{3}}
\eqno(3.39)$$ with the parameter $B=\frac{3\rho}{\alpha_\smalsev}$ the so called Batchelor constant of the flow. As with relative diffusion, the case of a velocity field with smooth time correlation described by 