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# Exact Solution of Decaying Turbulence

Version 1
: Received: 29 August 2023 / Approved: 29 August 2023 / Online: 30 August 2023 (09:01:51 CEST)

Version 2 : Received: 31 August 2023 / Approved: 4 September 2023 / Online: 4 September 2023 (12:00:54 CEST)

Version 3 : Received: 6 September 2023 / Approved: 7 September 2023 / Online: 7 September 2023 (09:28:55 CEST)

Version 4 : Received: 15 September 2023 / Approved: 18 September 2023 / Online: 18 September 2023 (13:56:35 CEST)

Version 2 : Received: 31 August 2023 / Approved: 4 September 2023 / Online: 4 September 2023 (12:00:54 CEST)

Version 3 : Received: 6 September 2023 / Approved: 7 September 2023 / Online: 7 September 2023 (09:28:55 CEST)

Version 4 : Received: 15 September 2023 / Approved: 18 September 2023 / Online: 18 September 2023 (13:56:35 CEST)

A peer-reviewed article of this Preprint also exists.

Migdal, A. To the Theory of Decaying Turbulence. *Fractal Fract.* **2023**, *7*, 754.
Migdal, A. To the Theory of Decaying Turbulence. Fractal Fract. 2023, 7, 754.

## Abstract

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N$ Ising variables $\sigma_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum \sigma_i = q r$. This equivalence provides a \textbf{dual} theory describing a strong turbulent phase of the Navier-Stokes flow in $\mathbb R_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. This is a \textbf{quantum} statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N \rightarrow \infty$ or its chemical potential $\mu \rightarrow 0$. The system with fixed $N$ has different asymptotics at odd and even $N\rightarrow \infty$, but the limit $\mu \rightarrow 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu/\mu^2$ in the continuum limit $\mu \rightarrow 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu} \to 0$. The same method is used to compute all the moments of the enstrophy distribution. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda}$, where the anomalous dimensions $\lambda$ satisfy the spectral equation \eqref{spectralEq}. The spectrum becomes a continuum in the statistical limit $N \to \infty$, leading to multifractal phenomena.

## Keywords

Turbulence; Fractal; Anomalous dissipation; Fixed point; Velocity circulation; Loop Equations; Euler Phi; Prime numbers

## 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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Commenter: Alexander Migdal

Commenter's Conflict of Interests: Author

Fixed a few typos and added a few comments in the abstract and in the Conclusions, elucidating the meaning of the discovered quantum analogy.