x^{-\frac{7}{3}}[1-\J_0(x)-\J_2(x)]
\simeq 0.265,
\eqno(3.16)$$ with $\J_n$ the Bessel function of the first kind, is evaluated in terms of Gamma functions [@gradshteyn] using the formula $\int_0^\infty\d x\ x^\mu J_\nu(x)=2^\mu
\frac{\Gamma(\frac{1}{2}(1+\nu+\mu))}{\Gamma(\frac{1}{2}(1+\nu-\mu))}$. From incompressibility we find therefore: $$D_{\alpha\beta}(\r)=\Big[\frac{r_\alpha r_\beta}{r^2}+\frac{7}{3}\Big(\delta_{\alpha\beta}
-\frac{r_\alpha r_\beta}{r^2}\Big)\Big]
\frac{4\alpha_\smalsev\Ckol^\frac{1}{2}\flux^\frac{1}{3}}{\rho}r^\frac{4}{3}
\eqno(3.17)$$ We want to study the asymptotics of the separation process of two particles in the inertial range. The procedure is standard (see e.g [@borgas94]); we introduce the distribution $P$ for the separation $\r$ at time $t$, which will obey the diffusion equation (the summation over repeated indices convention is adopted throughout the paper): $$\partial_tP=\frac{1}{2}\partial_\alpha\partial_\beta D_{\alpha\beta}P
\eqno(3.18)$$ and look for an isotropic similarity solution in the form $$P(\r,t)=t^{-3}f(t^{-\frac{3}{2}}r)=t^{-3}f(R)
\eqno(3.19)$$ Equation (3.18) takes then the form $$\frac{3}{2}\partial_\alpha(R_\alpha f)
+\frac{4\alpha_\smalsev\Ckol^\frac{1}{2}\flux^\frac{1}{3}}{2\rho}\partial_\alpha R^\frac{1}{3}
R_\alpha\partial_Rf=0
\eqno(3.20)$$ This equation has an unphysical solution, which is divergent in $R=0$, and a finite one: $$f(R)=\exp\Big(-\frac{9\rho R^\frac{2}{3}}{8\alpha_\smalsev\Ckol^\frac{1}{2}\flux^\frac{1}{3}}\Big)
\eqno(3.21)$$ whose moments are: $$\langle R^n\rangle=\int_0^\infty R^{1+n}\d Rf(R)=\frac{3}{2}
\Big(\frac{8\alpha_\smalsev\Ckol^\frac{1}{2}\flux^\frac{1}{3}}{9\rho}\Big)^{3+\frac{3n}{2}}
\Gamma(3+\frac{3n}{2})
\eqno(3.22)$$ with $\Gamma$ the standard gamma function.

From here, the expression for the particle space separation is obtained in a straightforward manner; for $\gamma_{x^{-1}}t\gg 1$, indicating $\r(t)=\z(t|\r_0,0)-\z(t|0,0)$: $$\langle r^2(t)\rangle=c\flux t^3,
\qquad
c=\frac{10240\alpha_\smalsev^3\Ckol^\frac{3}{2}}{243\rho^3}
\eqno(3.23)$$ i.e. the space separation obeys Richardson diffusion. For the relative velocity, we have, from Eqn. (2.6): $$\langle [u^\smalL_r(\r_0,0)-u^\smalL_r(0,0)]^2\rangle=2\alpha_\smalfi\Ckol\flux^\frac{2}{3}
\langle r^\frac{2}{3}(t)\rangle
