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 nth order correlation function of velocity differences Fₙ(R₁,R₂,) exhibits its own crossover length ηₙ to dissipative behavior as a function of R₁. This depends on n and on the remaining separations R₂,R₃,. One result is that when separations are of the same order R, this scales as ηₙ(R)~η(R/L)^xₙ with ${x}ₙ{{0ex}{0ex}}={{0ex}{0ex}}({{ζ}}ₙ{-}{{ζ}}ₙ₊₁+{{ζ}}₃{-}{{ζ}}₂)/(2{-}{{ζ}}₂)$, ${{ζ}}ₙ$ the scaling exponent of the $nth$ order structure function. We derive an infinite set of scaling relations that bridge the exponents of correlations of gradient fields to the exponents ${{ζ}}ₙ$, including the ``bridge relation'' for the scaling exponent of dissipation fluctuations ${μ}{{0ex}{0ex}}={{0ex}{0ex}}2{-}{{ζ}}₆$.