Internal symmetry in the multifractal spectrum of fully developed turbulence

Abstract

In the context of multifractal theory and She-Lévêque's model describing the statistics of intermittency in fully developed turbulence, we show that the multifractal dimensions can be simply written F(alpha)=1+alpha*-alpha*ln(alpha*/2) with alpha*=(2 beta-1-alpha)/ln beta=2 beta(p), where p is the order associated to the moment <epsilon(p)(r)> (with p> or =0) based on the rate of energy dissipation epsilon(r) and beta=[(1+3/the square root of 8)(1/3)+(1-3/the square root of 8)(1/3](3) approximately equal to 0.68). Introducing the fractal dimensions Delta(p)=F(alpha)+alpha*ln(alpha*/2), this leads to the recursive relation beta=(Delta(p+1)-Delta(infinity))/(Delta(p)-Delta(infinity)) with Delta(infinity)=1. This suggests the existence of an internal symmetry in the multifractal spectrum of fully developed turbulence, which reduces considerably the number of parameters necessary to characterize intermittency statistics.