Topological vortexes, asymptotic freedom, and multifractals

TL;DR

This paper develops a theoretical framework linking turbulence multifractality to asymptotic freedom in QCD, using Kelvinon solutions and RG equations to predict multifractal dimensions consistent with turbulence data.

Contribution

It introduces a novel analogy between turbulence and QCD, deriving running multifractal dimensions through RG analysis of Kelvinon solutions.

Findings

Derived multifractal dimensions match turbulence data.

Established a logarithmic dependence of fractal dimensions on loop size.

Predicted slow variation of fractal dimensions at high Reynolds numbers.

Abstract

We study the Kelvinons: monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid limit. However, in the anomalous Hamiltonian, some terms are growing as logarithms of Reynolds number; these terms come from the core of the Burgers vortex. In our theory, the turbulent multifractal phenomenon is similar to asymptotic freedom in QCD, with these logarithmic terms summed up by an RG equation. The small effective coupling does not imply small velocity; on the contrary, velocity is large compared to its fluctuations, which opens the way for a quantitative theory. In the leading order in the perturbation theory in this effective coupling constant, we compute running multifractal dimensions for high moments of velocity circulation, in good agreement with the data for quantum Turbulence and available data for classical Turbulence. The logarithmic dependence of fractal dimensions on the loop size comes from the running coupling in anomalous dimensions. This slow logarithmic drift of fractal dimensions would be barely observable at Reynolds numbers achievable at modern DNS.