Exact results on scaling exponents in the 2D enstrophy cascade.

Abstract

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 $〈({\ensuremath{\Delta}}_{\mathit{l}}\ensuremath{\omega}{)}^{p}〉\ensuremath{\sim}{\ensuremath{\ell}}^{{\ensuremath{\zeta}}_{p}}$ must have ${\ensuremath{\zeta}}_{2}\ensuremath{\le}2/3$ (Sulem-Frisch bound) and ${\ensuremath{\zeta}}_{p}\ensuremath{\le}0$, for $p\ensuremath{\ge}3$. If the minimum H\"older singularity of the vorticity is negative, ${h}_{\mathrm{min}}<0$, then the bounds can be improved to ${\ensuremath{\zeta}}_{p}\ensuremath{\le}{\ensuremath{\tau}}_{p/3}$, where ${\ensuremath{\tau}}_{p}$ is the scaling exponent of a local enstrophy flux: $〈|{Z}_{\ensuremath{\ell}}{|}^{p}〉\ensuremath{\sim}{\ensuremath{\ell}}^{{\ensuremath{\tau}}_{p}}$. However, if ${h}_{\mathrm{min}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$, then ${\ensuremath{\zeta}}_{p}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ for $p\ensuremath{\ge}2$ and Kraichnan theory is exact.