Two-dimensional enstrophy cascade ($E(k)\propto k^{-3}$)

A two-dimensional forward two-dimensional enstrophy cascade has the idealized spectrum $E(k)\propto k^{-3}$. The energy integral above a wavenumber scales as its inverse square, whereas the cumulative enstrophy integral $\int k^{2}E(k)dk$ grows logarithmically. The spectrum used here omits possible further logarithmic corrections.

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.