Topological Vortexes, Asymptotic Freedom and Multifractals

Alexander Migdal

doi:10.20944/preprints202303.0383.v5

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preprints.org > physical sciences > theoretical physics > doi: 10.20944/preprints202303.0383.v5

Preprint Article Version 5 Preserved in Portico This version is not peer-reviewed

Topological Vortexes, Asymptotic Freedom and Multifractals

Alexander Migdal

Version 1 : Received: 21 March 2023 / Approved: 22 March 2023 / Online: 22 March 2023 (02:03:40 CET)
Version 2 : Received: 5 April 2023 / Approved: 5 April 2023 / Online: 5 April 2023 (13:48:34 CEST)
Version 3 : Received: 5 April 2023 / Approved: 6 April 2023 / Online: 6 April 2023 (05:28:57 CEST)
Version 4 : Received: 6 April 2023 / Approved: 7 April 2023 / Online: 7 April 2023 (03:01:11 CEST)
Version 5 : Received: 18 April 2023 / Approved: 19 April 2023 / Online: 19 April 2023 (03:52:40 CEST)

A peer-reviewed article of this Preprint also exists.

Migdal, A. Topological Vortexes, Asymptotic Freedom, and Multifractals. Fractal Fract. 2023, 7, 351. Migdal, A. Topological Vortexes, Asymptotic Freedom, and Multifractals. Fractal Fract. 2023, 7, 351.

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.

Keywords

Turbulence; Multifractals; Anomalous dissipation; Fixed point; Velocity circulation; Burgers vortex; asymptotic freedom

Subject

Physical Sciences, Theoretical Physics

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Comments (1)

Comment 1

Received: 19 April 2023
Commenter: Alexander Migdal
Commenter's Conflict of Interests: Author

Comment: Replied to the comments of referees in MDPI, added discussion of the phase transition in Turbulence and comparison with old RG.

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