Logarithmic structure function in an enstrophy cascade
= Logarithmic structure function in an enstrophy cascade
{title2=$S_2\simeq Cr^2\log[\pi/(k_0r)]/4$}
For a wide $Ck^{-3}$ range and separation well below its outer scale, the second-order <longitudinal structure function> is controlled by $\int_{k_0}^{\pi/r}k^2E(k)\,dk$. The low-wavenumber filter term gives $S_2(r)\simeq Cr^2\log[\pi/(k_0r)]/4$, larger than the order-$r^2$ energy above the cutoff. This is a marginal dependence on larger-scale <enstrophy>.