Scaling Index $\alpha = \frac{1}{2}$ In Turbulent Area Law

Abstract

We analyze the Minimal Area solution to the Loop Equations in turbulence \cite{M93}. As it follows from the new derivation in the recent paper \cite{M19}, the vorticity is represented as a normal vector to the minimal surface not just at the edge, like it was assumed before, but all over the surface. As it was pointed in that paper, the self-consistency relation for mean vorticity leads to $\alpha=\frac{1}{2}$, however the similar conditions for product of two and more vorticities cannot be satisfied without extra terms, which were left undetermined in that paper. In this paper we find these missing terms -- they are delta functions at coinciding points which must be taken into considerations in surface integrals. We compare this value of $\alpha$ with new measurements of the same team which confirmed the area law \cite{S19} and we find that asymptotic formula $\lambda(p) \approx 2 \alpha p + \beta \ln p$, with $\alpha =0.49 \pm 0.02, \beta= 0.92 \pm 0.01 $, fits all data at $p=3,...10$ within error bars.