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

TL;DR

This paper refines the theoretical understanding of the turbulent area law by deriving the missing terms in the minimal surface solution, confirming the scaling index α=1/2 through comparison with experimental data.

Contribution

It identifies and calculates the missing delta function terms in the minimal surface solution, confirming the scaling index α=1/2 in turbulence.

Findings

The scaling index α is approximately 0.49, consistent with theoretical predictions.

The asymptotic formula for λ(p) fits experimental data for p=3 to 10.

The derived terms improve the theoretical model of turbulence surface integrals.

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.