Logarithmic correction in the two-dimensional direct enstrophy cascade
In plain words
Flat turbulence also sends enstrophy (the mean squared spin of the fluid) to small scales, and theory predicts the energy spectrum there with a small logarithmic correction. That correction has not been confirmed.
Precise statement
In the direct enstrophy cascade of forced 2D turbulence, Kraichnan (1971) predicted $E(k) = C' \eta^{2/3} k^{-3} [\ln(k/k_f)]^{-1/3}$, with $\eta$ the enstrophy flux and $k_f$ the forcing wavenumber. In the limit of vanishing large-scale friction and $k/k_f \to \infty$, verify or refute the exponent $-1/3$ of the logarithm and determine C'.
What would settle it
DNS with negligible large-scale friction over about three decades of $k/k_{f}$, fitting the logarithmic factor with error bars.
Status in the literature
Unverified note
Linear friction steepens the spectrum and masks the correction (arXiv:2511.08220, 2025).