the noise ${\boldsymbol\xi}$ and use has been made, in converting the advection by molecular noise into a diffusion term, of Itô’s lemma [@gardiner]. We see (it is assumed that the limit $\eta\to 0$ is already taken) that there is a renormalization of the damping kernel $\gamma$: $$\gamma_k\to\gamma_k+\sigma k^2
\eqno(3.29)$$ which leads to a cut-off for the velocity at the inverse diffusive scale $$\eta^{-1}_\sigma=(\rho\Ckol^\frac{1}{2})^\frac{3}{4}\flux^\frac{1}{4}\sigma^{-\frac{3}{4}}
\eqno(3.30)$$ We have then: $$\langle \v_\k(t)\v_{-\k}(0)\rangle
=\frac{\exp(-\sigma k^2|t|)}{1+(k\eta_\sigma)^\frac{4}{3}}\langle \u_\k(t)\u_{-\k}(0)\rangle,
\eqno(3.31)$$ and for small space separations $r/\eta_\sigma\to 0$, we have a quadratic behavior for the velocity structure function: $$\langle[v_r(\x+\r,t)-v_r(\x,t)]^2\rangle
=2\Ckol\flux^\frac{2}{3}r^2\eta_\sigma^{-\frac{4}{3}}
\int_0^\infty\frac{[1-\J_0(x)-\J_2(x)]\d x}{x^\frac{5}{3}(x^\frac{4}{3}+(r/\eta_\sigma)^\frac{4}{3})}$$ $$\simeq \frac{1}{4}\Ckol\flux^\frac{2}{3}\eta_\sigma^{-\frac{4}{3}}r^2|\log r/\eta_\sigma |
\eqno(3.32)$$ The transport of a passive scalar $\theta(\x,t)$ will be described by the equation $$(\partial_t+\v(\x,t)\cdot\nabla)\theta(\x,t)=\sigma\nabla^2\theta(\x,t)+f(\x,t)
\eqno(3.33)$$ with $f(\x,t)$ a source term. An interesting quantity to calculate is the fluctuation spectrum for $\theta$ in the case $f$ is random in time and concentrated at large scale: $$\langle f(\x+\r,t)f(\x,0)\rangle=F(r)\delta(t),\qquad 
F(r)=
\begin{cases}
2\flux_\theta, & k_0r<1
\\
0, & k_0r>0
\end{cases}
\eqno(3.34)$$ We can thus consider $\langle\theta\rangle=0$. The equation for the steady state passive scalar correlation $\Theta(r)=\langle\theta(\x+\r,t)\theta(\x,t)\rangle$ will then be, for $k_0r\ll 1$: $$\langle\theta(\x,t)[\v(\x+\r,t)-\v(\x,t)]\cdot\nabla\theta(\x+r,t)\rangle
=2\sigma\nabla^2\Theta(r)+4\flux_\theta
\eqno(3.35)$$ For $r\to 0$, the left hand side of this equation is zero; we thus obtain $\flux_\theta=\frac{\sigma}{2}\langle|\nabla\theta|^2\rangle$, i.e. $\flux_\theta$ is the 