\eqno(3.24)$$ where $$\alpha_\smalfi=\int_0^\infty\d x\ x^{-\frac{5}{3}}[1-\J_0(x)-\J_2(x)]
\simeq 2.149
\eqno(3.25)$$ and, using Eqn. (3.22), for $\gamma_{x^{-1}}t\gg 1$, we find the normal diffusion behavior: $$\langle [u^\smalL_r(\r_0,0)-u^\smalL_r(0,0)]^2\rangle=\tilde c\flux t,
\qquad
\tilde c=\frac{16\alpha_\smalfi\alpha_\smalsev\Ckol^\frac{3}{2}}{3\rho}
\eqno(3.26)$$ Passing to the smoothed out in time version of the velocity field provided by Eqn. (3.7), is accomplished, as in the case of $\tau_L$, by exchanging $\rho\to\frac{\rho\hat\rho}{\rho+\hat\rho}$. In [@boffetta00], both a sub-exponential behavior for the function $f(R)$ and Richardson diffusion were observed in a direct DNS (direct numerical simulation) of two-dimensional turbulence in the inverse cascade regime. Based on the results of that paper, extrapolating applicability of our leading order expressions in $\rho$ would give then (taking also $\hat\rho\to\infty$): $\rho\simeq 2$.

[**3. The role of diffusivity and the Batchelor constant**]{} The dynamics of passive tracers, contrary to that of fluid elements, feels the effect of molecular diffusivity. Due to finiteness of the turbulent correlation times, this effect does not consist purely of an additive noise contribution to the tracer velocity. Indicating by $\sigma$ the molecular diffusivity, the passive tracer velocity will have the form: $$\v(\x,t)+(2\sigma)^\frac{1}{2}{\boldsymbol\xi}(\x,t)
\eqno(3.27)$$ with $\langle\xi_\alpha(\x,t)\xi_\beta(0,0)\rangle=\delta_{\alpha\beta}\delta(\x)\delta(t)$ and $\v$ obeying an equation in the form: $$(\partial_t+\v(\x,t)\cdot\nabla)q_v(\x,t)+\int\d^2y\gamma(\x-\y)q_v(\y,t)
-\int\d^2yh(\x-\y)\xi(\y,t)$$ $$=-(2\sigma)^\frac{1}{2}\langle{\boldsymbol\xi}(\x,t)\cdot\nabla q_v(\x,t)\rangle_\xi
\simeq\sigma\nabla^2q_v(\x,t)
\eqno(3.28)$$ where $q_v=\nabla_\perp\cdot\v$, $\langle .\rangle_\xi$ is an average limited to 