 Logarithmic structure function in an enstrophy cascade

For a wide $C{k}^{-3}$ range and separation well below its outer scale, the second-order longitudinal structure function is controlled by ${\int }_{k_0}^{\pi /r}{k}^{2}E(k)dk$. The low-wavenumber filter term gives $S_2(r)\simeq C{r}^2\log [\pi /(k_{0}r)]/4$, larger than the order-$r^2$ energy above the cutoff. This is a marginal dependence on larger-scale enstrophy.