Quantum Solution of Classical Turbulence Decaying Energy Spectrum

Alexander Migdal

doi:10.20944/preprints202312.1012.v10

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preprints.org > physical sciences > fluids and plasmas physics > doi: 10.20944/preprints202312.1012.v10

Preprint Article Version 10 Preserved in Portico This version is not peer-reviewed

Quantum Solution of Classical Turbulence Decaying Energy Spectrum

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 summarizes and elaborates on recently found \cite{migdal2023exact} exact reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a Number Theory problem of 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 \textbf{quantum} statistical theory of $N$ fermions on a ring, with an external field related to the random fractions of $\pi$. We find the solution of this system in the turbulent limit $N\to \infty, \nu \to 0$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the 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 approximations or fitted parameters. We compute resulting integrals in \Mathematica{} and present effective index $n(t) = -t E'(t)/E(t) $ for the energy decay as a function of time Fig.\ref{fig::NPlot} and $s(k)= -k\partial_k \log E(k,t) $ for the energy spectrum Fig.\ref{fig::SPIndex}. The asymptotic values are $n(\infty) = \frac{5}{4}, s(\infty) = \frac{7}{2}$, but the universal functions $n(t), s(t)$ are neither constant nor linear due to quantum effects (complex poles of the Mellin transform at zeros of Riemann zeta function). The theoretical value $n(\infty)=\sfrac{5}{4}$ matches the grid turbulence experiments \cite{GridTurbulence_1966} within experimental errors $\sim 2\%$.

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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