and the integral scales of the flow $k_0$ and $\tau_L$: $$u_T^2
=3\Ckol\flux^\frac{2}{3}k_0^{-\frac{2}{3}}=6\rho\Ckol^\frac{3}{2}\flux\tau_L
\eqno(3.3)$$ The correlation time $\tau_L$ is determined by the particular form of $\U^\smalL_\k$ we have chosen at small $k$, which is non-universal. It is more interesting, and relevant from the point of view of Lagrangian dispersion modeling [@thomson87; @sawford91], to calculate the Lagrangian time structure function: $$\langle[u_\alpha^\smalL(\x,t)-u_\alpha^\smalL(\x,0)]
[u_\beta^\smalL(\x,t)-u_\beta^\smalL(\x,0)]\rangle
=\frac{1}{2}\langle|\u^\smalL(\x,t)-\u^\smalL(\x,0)|^2\rangle\delta_{\alpha\beta}
\eqno(3.4)$$ We discover immediately that, in order to have a self similar spectrum for the inertial range, the time correlations should have continuous time derivative at $t=0$, a property not satisfied by Eqn. (3.1).

This self-similarity violation can be illustrated in a simple way imagining the turbulence field in the neighborhood of the fluid parcel as a superposition of nested eddies with scale $l_n$, velocity $u_n$ and eddy turn-over time $\tau_n$: $$l_n=l_02^{-n},\qquad u_n=u_02^{-\frac{n}{3}},\qquad\tau_n=\tau_02^{-\frac{2n}{3}}
\eqno(3.5)$$ If the time correlation decayed linearly for $t\to 0$, we would have: $$\langle |\u^\smalL(\x,t)-\u^\smalL(\x,0)|^2\rangle
\sim\sum_{\tau_n<t}u_n^2\frac{t}{\tau_n}+\sum_{\tau_n>t}u_n^2
\sim u_0^2\log(\tau_0/t)\frac{t}{\tau_0};
\eqno(3.6)$$ Thus, identical scaling of $u_n^2$ and $\tau_n$, and linear decay of correlations cause the largest space scale to contribute to the structure function at arbitrary short time separation $t$, in the same way as a vortex with eddy turn-over time $\tau_n\sim t$, whence the logarithmic correction involving 