$\tau_0$.

In order to have a quadratic behavior of the time correlation at $t=0$, it is necessary that the noise $\xi$ in Eqn. (2.7) be correlated in time, and the correlation must again be given along the trajectories. The appropriate modification to Eqn. (2.7) is therefore: $$\begin{cases}
(\partial_t+\u(\x,t)\cdot\nabla)q(\x,t)+\int\d^2y\gamma(\x-\y)q(\y,t)=r(\x,t)
\\
(\partial_t+\u(\x,t)\cdot\nabla)r(\x,t)+\int\d^2y\hat\gamma(\x-\y)r(\y,t)=\int\d^2yh(\x-\y)\xi(\y,t)
\end{cases}
\eqno(3.7)$$ where, for $k\ll\eta^{-1}$: $$\hat\gamma_k=\hat\rho\Ckol^\frac{1}{2}\flux^\frac{1}{3}k^\frac{2}{3}
\quad{\rm and}\quad
H_k=|h_k|^2=8\pi\rho\hat\rho(\rho+\hat\rho)\Ckol^\frac{5}{2}\flux^\frac{5}{3}
(k^2+k_0^2)^\frac{2}{3}
\eqno(3.8)$$ It is easy to show that also the field theory associated with Eqn. (3.7) is characterized by marginal interactions: $[\lambda_{\gamma_n}]=[\lambda_{\hat\gamma_n}]=[\lambda_{H_n}]=[\lambda_{G_n}]=0$ and the considerations in section II extend to the present case.

The zero order of the theory leads to the following correlation function: $$\U^\smalL_\k(t)
=\frac{\k_\perp\k_\perp}{k^2}
\frac{4\pi\Ckol\flux^\frac{2}{3}}{(k^2+k_0^2)^\frac{4}{3}}
\frac{\rho\ex^{-\hat\gamma_k|t|}-\hat\rho\ex^{-\gamma_k|t|}}{\rho-\hat\rho}
\eqno(3.9)$$ and the time correlation has a quadratic maximum at $t=0$. Calculation of the Lagrangian correlation time leads to the same result of Eqn. (3.2), with the substitution $\rho\to\frac{\rho\hat\rho}{\rho+\hat\rho}$, while smoothness of the time correlation eliminates the logarithmic correction to the scaling of the Lagrangian time structure function. This structure function obeys in fact, after sending $k_0\to 0$, the expected normal diffusion behavior: $$\langle|\u^\smalL(\x,t)-\u^\smalL(\x,0)|^2\rangle=2C_0\flux|t|
\qquad
{\rm with} 
\qquad
C_0
%\frac{\Ckol(\rho\hat\rho)^\frac{1}{2}}{2}
%\int_0^\infty x^{-\frac{5}{3}}\d x\Big[1-\frac{1}{\rho-\hat\rho}
%\Big(\rho\exp(-(\hat\rho/\rho)^\frac{1}{2}x^\frac{2}{3})
%-\hat\rho\exp(-(\rho/\hat\rho)^\frac{1}{2}x^\frac{2}{3})\Big)\Big]
%$$ =\^/; (3.10) $$ the constant $C_0$ is $\O(\rho)$ and, as expected from the discussion leading to 