behaviors. In fact, the divergences occurring for large $k$ in a loop diagram will not change if we exchange $\delta\z(t|\x,\y)\to\z(t|\x,0)+\z(t|\y,0)-(\x+\y)$; this because each small eddy contribute to the separation $\x-\y$ an amount which is of the same order of the one to sweep. Now, the logarithmic divergence predicted at small $k$ in a loop diagram comes indeed, from equating the scaling of the sweep $\z(t|\x,0)+\z(t|\y,0)-(\x+\y)$ with that of the trajectory separation $\delta\z(t|\x,\y)$ also at small $k$, which is incorrect. For small $k$, this scaling should be corrected by a factor $k$ per field $\delta\z$ involved in the lines of the loop, and this is enough to eliminate divergence.

**III. Passive tracer transport**

[**1. Self-diffusion of a fluid parcel**]{} Lagrangian correlation functions in the form $\langle\u^\smalL(\x,t)\u^\smalL(\x,0)\rangle=
\langle\u(\z(t|\x,0),t)\u(\x,0)\rangle$ are the simplest objects one may try to calculate from the random velocity field introduced in section II. The starting point, to lowest order in $\rho^{-1}$, and after sending the Kolmogorov scale $\eta$ to zero, is the following modification of Eqn. (2.6): $$\U^\smalL_\k(t)=
4\pi\Ckol\flux^\frac{2}{3}\frac{\k_\perp\k_\perp}{k^2}
\frac{\exp(-\gamma_k|t|)}{(k^2+k_0^2)^\frac{4}{3}}
\eqno(3.1)$$ The Lagrangian correlation time $\tau_L$ is then readily calculated: $$\tau_L^{-1}=
\langle|\u^\smalL|^2\rangle
\Big[\int\d t\langle\u^\smalL(\x,t)\cdot\u^\smalL(\x,0)\rangle\Big]^{-1}=
2\rho\Ckol^\frac{1}{2}\flux^\frac{1}{3}k_0^\frac{2}{3}
\eqno(3.2)$$ and we have the following relation between the turbulence level $u_T^2=\langle u^2\rangle=\langle |\u^\smalL|^2\rangle$ 