Eqn. (3.6), diverges logarithmically for $\hat\rho/\rho\to\infty$.

[**2. Relative diffusion**]{} Analyzing the transport of a cluster of particles requires consideration of time intervals, during which the space separations involved cannot be approximated as constant. Over these timescales, the short correlation time limit leads to a perturbation scheme, which treats the velocity field to zero order as a white noise.

We focus on the case of a pair of particles. We have to study an equation in the form: $$\partial_t[z_\alpha(t|\r_0,0)-z_\alpha(t|0,0)]
=u_\alpha^\smalL(\r_0,t)-u_\alpha^\smalL(0,t)=U_{\alpha\beta}(\z(t|\r_0,0)-\z(t|0,0))
\xi_\beta(t)
\eqno(3.11)$$ with $\langle\xi_\alpha(t)\xi_\beta(0)\rangle=\delta_{\alpha\beta}\delta(t)$ and $U_{\alpha\beta}$ to be determined. Due to the multiplicative noise nature of this equation, attention must be paid to the possible presence of drift terms arising from the Stratonovich prescription implicit in its definition [@gardiner]. It is easy to show that this drift is identically zero, either by direct calculation of the increment $\delta\z(t|\x,0)$ for $t$ in the inertial range, or noticing that: $$\langle\delta\z(t|\r_0,0)\rangle=\int_0^t\d\tau[
\langle (\u^\smalL(\r_0,\tau)\rangle-\langle\u^\smalL(0,\tau)\rangle]=0;
\eqno(3.12)$$ this, because of homogeneity of turbulence. For this reason, the separation process is described simply by: $$\partial_t\langle[z_\alpha(t|\r_0,0)-z_\alpha(t|0,0)][z_\beta(t|\r_0,0)-z_\beta(t|0,0)]\rangle=
D_{\alpha\beta}(\z(t|\r_0,0)-\z(t|0,0))
\eqno(3.13)$$ with $$D_{\alpha\beta}(\r)=\int\d t\langle[u_\alpha^\smalL(\r,t)-u_\alpha^\smalL(0,t))]
[u_\beta^\smalL(\r,0)-u_\beta^\smalL(0,0))]\rangle
\eqno(3.14)$$ This tensor is easily calculated from $D_{11}(\r)$ for $\r=(r,0)$, exploiting incompressibility. Using $\int_0^{2\pi}\d\theta\sin^2\theta\sin^2(x\cos\theta)=
\frac{\pi}{2}[1-\J_0(2x)-\J_2(2x)]$, we find, in the limit $k_0\to 0$: $$D_{11}(\r)=\frac{4\alpha_\smalsev\Ckol^\frac{1}{2}\flux^\frac{1}{3}}{\rho}r^\frac{4}{3}
\eqno(3.15)$$ where $$\alpha_\smalsev=\int_0^\infty\d x\  