= Solution
Let the <two-dimensional enstrophy cascade> spectrum be $E(k)=Ck^{-3}$ 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 is
$$
\int_{k_c}^{k_d}E(k)\,dk=\frac C2(k_c^{-2}-k_d^{-2})\simeq\frac{Cr^2}{2\pi^2}.
$$
The low-wavenumber correction instead gives
$$
\frac{r^2}{8}\int_{k_0}^{k_c}k^2E(k)\,dk
=\frac{Cr^2}{8}\log\frac{k_c}{k_0}.
$$
Its 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>. Consequently
$$
\boxed{S_2(r)\simeq\frac{Cr^2}{4}\log\frac{\pi}{k_0r}},
$$
up to the stated piecewise-filter approximation and subleading contributions. Equivalently, if $Z_{<k_c}=\int_0^{k_c}k^2E(k)\,dk$ is the cumulative <enstrophy> in motions larger than $r$, then $S_2\simeq r^2Z_{<k_c}/4$.
\b[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.
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