Dual Theory of Decaying Turbulence

Alexander Migdal

doi:10.20944/preprints202312.1012.v5

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preprints.org > physical sciences > fluids and plasmas physics > doi: 10.20944/preprints202312.1012.v5

Preprint Article Version 5 Preserved in Portico This version is not peer-reviewed

Dual Theory of Decaying Turbulence

Alexander Migdal

Version 1 : Received: 13 December 2023 / Approved: 13 December 2023 / Online: 14 December 2023 (08:24:06 CET)
Version 2 : Received: 24 December 2023 / Approved: 26 December 2023 / Online: 26 December 2023 (09:54:09 CET)
Version 3 : Received: 7 March 2024 / Approved: 7 March 2024 / Online: 7 March 2024 (14:17:24 CET)
Version 4 : Received: 24 March 2024 / Approved: 25 March 2024 / Online: 26 March 2024 (08:24:14 CET)
Version 5 : Received: 26 March 2024 / Approved: 27 March 2024 / Online: 27 March 2024 (14:15:31 CET)
Version 6 : Received: 27 March 2024 / Approved: 28 March 2024 / Online: 28 March 2024 (13:31:01 CET)
Version 7 : Received: 2 April 2024 / Approved: 3 April 2024 / Online: 3 April 2024 (14:11:11 CEST)
Version 8 : Received: 7 April 2024 / Approved: 8 April 2024 / Online: 8 April 2024 (11:36:43 CEST)
Version 9 : Received: 11 April 2024 / Approved: 12 April 2024 / Online: 12 April 2024 (14:39:31 CEST)
Version 10 : Received: 13 April 2024 / Approved: 15 April 2024 / Online: 17 April 2024 (07:19:09 CEST)
Version 11 : Received: 3 May 2024 / Approved: 3 May 2024 / Online: 6 May 2024 (08:40:49 CEST)
Version 12 : Received: 31 May 2024 / Approved: 31 May 2024 / Online: 3 June 2024 (08:33:04 CEST)
Version 13 : Received: 17 June 2024 / Approved: 18 June 2024 / Online: 18 June 2024 (08:23:11 CEST)
Version 14 : Received: 8 July 2024 / Approved: 8 July 2024 / Online: 9 July 2024 (08:23:24 CEST)

A peer-reviewed article of this Preprint also exists.

Migdal, A. (2024). Quantum solution of classical turbulence: Decaying energy spectrum. Physics of Fluids, 36(9). Migdal, A. (2024). Quantum solution of classical turbulence: Decaying energy spectrum. Physics of Fluids, 36(9).

Abstract

This paper investigates the recently found \cite{migdal2023exact} reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a Number Theory problem of finding the statistical limit of the Euler ensemble. We reformulate the Euler ensemble as a Markov chain and show the equivalence of this formulation to the quantum statistical theory of free fermions on a ring, with an external field related to the random fractions of $\pi$. We find the solution of this system in the statistical limit $N\to \infty$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the charge density of these fermions. This results in an analytic formula for the observable correlation function of vorticity in wavevector space. This is a full solution of decaying turbulence from the first principle without assumptions, approximations, or fitted parameters. We compute resulting integrals in \Mathematica{} and present effective indexes for the energy decay as a function of time Fig.\ref{fig::NPlot} and the energy spectrum as a function of the wavevector at fixed time Fig.\ref{fig::SPIndex}. In particular, the asymptotic value of the effective index in energy decay $n(\infty) = \frac{7}{4}$, but the universal function $n(t)$ is neither constant nor linear.

Keywords

Turbulence; fractal; anomalous dissipation; fixed point; velocity circulation; loop equations; Euler Phi; prime numbers; Fermi particles; instanton

Subject

Physical Sciences, Fluids and Plasmas 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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