Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 73 / 2 / v / Solution

Let the two-dimensional enstrophy cascade spectrum be $E(k)=C{k}^{-3}$ between $k_0$ and $k_d$, and choose a separation well inside the range: $k_0\ll k_c=\pi /r\ll k_d$. The high-wavenumber energy isThe low-wavenumber correction instead givesIts ratio to the preceding energy contribution is $({\pi }^2/4)\log (k_c/k_0)$, which is large when the separation is sufficiently far below the integral scale of turbulence. Consequentlyup to the stated piecewise-filter approximation and subleading contributions. Equivalently, if $Z_{<k_c}={\int }_0^{k_c}{k}^{2}E(k)dk$ is the cumulative enstrophy in motions larger than $r$, then $S_2\simeq r^2{Z}_{<k_c}/4$.

The leading increment variance is controlled by the accumulated enstrophy at scales larger than the separation, rather than the kinetic energy at smaller scales. For a $k^{-3}$ spectrum, each logarithmic wavenumber interval contributes the same amount to that enstrophy integral, so the conclusion does not mean that only the largest eddy supplies it. The logarithm is a marginal failure of a pure local $r^2$ estimate. The stipulated wide power-law range is essential, and the dominance need not hold at its upper spatial edge where $k_c/k_0$ is only of order one.