Viscous Lengths in Hydrodynamic Turbulence are Anomalous Scaling Functions

It is shown that the idea that scaling behavior in turbulence is limited by one outer length $L$ and one inner length η is untenable. Every $n\mathrm{th}$ order correlation function of velocity differences ${\mathcal{F}}_n({\mathit{R}}_1,{\mathit{R}}_2,\dots )$ exhibits its own crossover length ${\eta }_n$ to dissipative behavior as a function of $R_1$. This depends on $n$ and on the remaining separations $R_2{,R}_3,\dots$. One result is that when separations are of the same order $R$, this scales as ${\eta }_n\left(R\right)\sim \eta (R/L{)}^{x_n}$ with $x_n=({\zeta }_n-{\zeta }_{n+1\mathrm{}}+{\zeta }_3-{\zeta }_2)/(2-{\zeta }_2)$, ${\zeta }_n$ the scaling exponent of the $n\mathrm{th}$ order structure function. We derive an infinite set of scaling relations that bridge the exponents of correlations of gradient fields to the exponents ${\zeta }_n$, including the “bridge relation” for the scaling exponent of dissipation fluctuations $\mu =2-{\zeta }_6$.