Exact Results on Scaling Exponents in the 2D Enstrophy Cascade

We establish rigorous inequalities for short-distance scaling exponents in 2D incompressible turbulence. Using only the condition of constant ultraviolet enstrophy flux, we show that $〈({\Delta }_{\mathit{l}}\omega {)}^p〉\sim {\ell }^{{\zeta }_p}$ must have ${\zeta }_2\le 2/3$ (Sulem-Frisch bound) and ${\zeta }_p\le 0$, for $p\ge 3$. If the minimum Hölder singularity of the vorticity is negative, $h_{\min }<0\mathrm{}$, then the bounds can be improved to ${\zeta }_p\le {\tau }_{p/3\mathrm{}}$, where ${\tau }_p$ is the scaling exponent of a local enstrophy flux: $〈|Z_{\ell }{|}^p〉\sim {\ell }^{{\tau }_p}$. However, if $h_{\min }=0\mathrm{}$, then ${\zeta }_p=0\mathrm{}$ for $p\ge 2$ and Kraichnan theory is exact.