Geodesic
Symmetry transformations of the vortex field statistics in
Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 2, pp. 438-451 Cet article a éte moissonné depuis la source Math-Net.Ru

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We use the concept of gauge transformations in the proof of the invariance of the statistics of zero-vorticity lines in the case of the inverse energy cascade in wave optical turbulence; we study it in the framework of the hydrodynamic approximation of the two-dimensional nonlinear Schrödinger equation for the weight velocity field $\mathbf u$. The multipoint probability distribution density functions $f_n$ of the vortex field $\Omega=\nabla\times\mathbf u$ satisfy an infinite chain of Lundgren–Monin–Novikov equations {(}statistical form of the Euler equations{\rm)}. The equations are considered in the case of the external action in the form of white Gaussian noise and large-scale friction, which makes the probability distribution density function statistically stationary. The main result is that the transformations are local and conformally transform the $n$-point statistics of zero-vorticity lines or the probability that a random curve $\mathbf x(l)$ passes through points $\mathbf x_i\in\mathbb R^2$ for $l=l_i$, $i=1,\dots,n$, where $\Omega=0$, is invariant under conformal transformations.
Keywords: optical turbulence, $n$-point statistics of a vortex field, Lundgren–Monin–Novikov equations, gauge transformations
Mots-clés : conformal invariance. 
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     title = {Symmetry transformations of the~vortex field statistics in},
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V. N. Grebenev; A. N. Grishkov; S. B. Medvedev. Symmetry transformations of the vortex field statistics in. Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 2, pp. 438-451. http://geodesic.mathdoc.fr/item/TMF_2023_217_2_a13/

[1] C. Wan, Q. Cao, J. Chen, A. Chong, Q. Zhan, “Toroidal vortices of light”, Nat. Photon., 16 (2022), 519–522 | DOI

[2] M. D. Bustamante, S. Nazarenko, “Derivation of the Biot–Savart equation from the nonlinear Schrödinger equation”, Phys. Rev. E, 92:5 (2015), 053019, 9 pp. | DOI | MR

[3] A. M. Polyakov, “The theory of turbulence in two dimensions”, Nucl. Phys. B, 396:2–3 (1993), 367–385 | DOI

[4] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory”, Nucl. Phys. B, 241:2 (1984), 333–380 | DOI

[5] V. N. Grebenev, A. N. Grishkov, S. B. Medvedev, M. P. Fedoruk, “Gidrodinamicheskoe priblizhenie dlya dvumernoi opticheskoi turbulentnosti: simmetrii statisticheskikh raspredelenii”, Kvantovaya elektronika, 52:11 (2022), 1023–1030

[6] V. N. Grebenev, A. N. Grishkov, M. Oberlak, “Simmetriii uravnenii Landgrena–Monina–Novikova dlya raspredelenii veroyatnosti polya vikhrya”, Dokl. RAN. Fiz. tekhn. nauki, 509:1 (2023), 50–55 | DOI

[7] V. N. Grebenev, M. Wacławczyk, M. Oberlack, “Conformal invariance of the Lundgren–Monin–Novikov equations for vorticity fields in 2D turbulence”, J. Phys. A: Math. Theor., 50:43 (2017), 435502, 22 pp. | DOI | Zbl

[8] V. N. Grebenev, M. Wacławczyk, M. Oberlack, “Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence”, J. Phys. A: Math. Theor., 52:33 (2019), 335501, 16 pp. | DOI

[9] M. Wacławczyk, V. N. Grebenev, M. Oberlack, “Conformal invariance of characteristic lines in a class of hydrodynamic models”, Symmetry, 12:9 (2020), 1482, 19 pp. | DOI

[10] M. Wacławczyk, V. N. Grebenev, M. Oberlack, “Conformal invariance of the $1$-point statistics of the zero-isolines of $2d$ scalar fields in inverse turbulent”, Phys. Rev. Fluids, 6:8 (2021), 084610, 15 pp. | DOI

[11] R. Panico, P. Comaron, M. Matuszewski, A. S. Lanotte, D. Trypogeorgos, G. Gigli, M. De Giorgi, V. Ardizzone, D. Sanvitto, D. Ballarini, “Onset of vortex clustering and inverse energy cascade in dissipative quantum fluids”, Nat. Photon., 17 (2023), 451–456, arXiv: 2205.02925 | DOI

[12] U. Bortolozzo, J. Laurie, S. Nazarenko, S. Residori, “Optical wave turbulence and the condensation of light”, J. Opt. Soc. Am. B, 26:12 (2009), 2280–2284 | DOI

[13] G. Falkovich, “Konformnaya invariantnost v gidrodinamicheskoi turbulentnosti”, UMN, 62:3(375) (2007), 193–206 | DOI | DOI | MR | Zbl

[14] E. Madelung, “Quantentheorie in hydrodynamischer Form”, Z. Phys., 40 (1927), 322–326 | DOI

[15] L. P. Pitaevskii, “Vikhrevye niti v neidealnom boze-gaze”, ZhETF, 40:2 (1961), 646–651

[16] R. Friedrich, A. Daitche, O. Kamps, J. Lülff, M. Voßkuhle, M. Wilczek, “The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence”, C. R. Physique, 13:9–10 (2012), 929–953 | DOI

[17] D. Bernard, G. Boffetta, A. Celani, G. Falkovich, “Conformal invariance in two-dimensional turbulence”, Nature Phys., 2 (2006), 124–128 | DOI

[18] D. Bernard, G. Boffetta, A. Celani, G. Falkovich, “Inverse turbulent cascades and conformally invariant curves”, Phys. Rev. Lett., 98:2 (2007), 024501, 4 pp. | DOI