$\xi_\k(t)$ is the Fourier transform of a zero mean fully uncorrelated noise term of unitary amplitude: $$\langle\xi(\x,t)\xi(0,0)\rangle=\delta(\x)\delta(t)
\eqno(2.3)$$ The damping and forcing Kernels $\gamma_k$ and $h_k$ are chosen, for $k\ll\eta^{-1}$, with $\eta$ the Kolmogorov length of the flow, as: $$\gamma_k=\rho\Ckol^\frac{1}{2}\flux^\frac{1}{3}(k^2+k_0^2)^\frac{1}{3}
\eqno(2.4)$$ $$H_k=|h_k|^2=\frac{8\pi\rho\Ckol^\frac{3}{2}\flux k^2}{k^2+k_0^2}
\eqno(2.5)$$ while, for $k\eta>1$, some cut-off is imposed on the forcing amplitude $H_k$. In this way, the velocity spectrum $\U_\k(t)$, defined by $\langle\u_\k(t)\u_\p(0)\rangle=\U_\k(t)(2\pi)^2$ $\delta(\k+\p)$, will read, for $k\eta\ll 1$: $$\U_\k(t) =4\pi\Ckol\flux^\frac{2}{3}\frac{\k_\perp\k_\perp}{k^2}
\frac{\exp(-\gamma_k|t|)}{(k^2+k_0^2)^\frac{4}{3}}
\eqno(2.6)$$ where $\Ckol$ and $\flux$ play the role, respectively, of the Kolmogorov constant and the inertial range energy flux in a real turbulent field having this correlation spectrum. For $k_0\ll k\ll\eta^{-1}$, we thus have the energy spectrum: $E_k=\Ckol\flux^\frac{2}{3}k^{-\frac{5}{3}}$. Identifying $\gamma_k^{-1}$ with the decay time and $k^{-2}U_k^{-\frac{1}{2}}(0)$ with the turn-over time of an eddy at scale $k^{-1}$, we see that $\rho$ gives the ratio of the eddy turn-over and eddy decay time in the inertial range. The effect of sweep by the large scales, however, is not accounted for in this way.

The most natural way to impose Lagrangian correlations in the random velocity field is to include an advection term in Eqn. (2.1), which will take the form in real space: $$(\partial_t+\u(\x,t)\cdot\nabla)q(\x,t)+\int\d^2y\gamma(\x-\y)q(\y,t)=\int\d^2yh(\x-\y)\xi(\y,t)
\eqno(2.7)$$ This has 