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SYMMETRIES OF THE LUNDGREN–MONIN–NOVIKOV EQUATION FOR PROBABILITY OF THE VORTICITY FIELD DISTRIBUTION
- Authors: Grebenev V.N.1, Grishkov A.N.2, Oberlack M.3
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Affiliations:
- Federal Research Center for Information and Computational Technologies
- Institute of Mathematics and Statistics, The University of Sao Paulo
- Technical University of Darmstadt
- Issue: Vol 509, No 1 (2023)
- Pages: 50-55
- Section: МЕХАНИКА
- URL: https://journals.rcsi.science/2686-7400/article/view/135915
- DOI: https://doi.org/10.31857/S2686740023010054
- EDN: https://elibrary.ru/OXVIJA
- ID: 135915
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Abstract
A.M. Polyakov suggested the programme to expand the symmetries admitted by hydrodynamic models to the conformal invariance of statistics in the inverse cascade where the conformal group is infinite-dimensional. In the present work, the group of transformations G of the \(n\)-point probability density function fn (PDF) is presented for the infinite chain of Lundgren–Monin–Novikov equations (the statistical form of the Euler equations) for vorticity fields of the two-dimensional inviscid flow. The problem is written in the Lagrangian setting. The main result is that the group G transforms conformally the zero-vorticity characteristics and invariantly a family of the fn-equations for PDF along these lines. The equations are not invariant along other characteristics. Moreover, the action of G conserves the class of PDF. The results obtained can be used for studying the invariance of statistical properties of the optical turbulence.
About the authors
V. N. Grebenev
Federal Research Center for Information and Computational Technologies
Author for correspondence.
Email: vngrebenev@gmail.com
Russia, Novosibirsk
A. N. Grishkov
Institute of Mathematics and Statistics, The University of Sao Paulo
Author for correspondence.
Email: grishkov@ime.usp.br
Brazil, Sao Paulo
M. Oberlack
Technical University of Darmstadt
Author for correspondence.
Email: oberlack@fdy.tu-darmstadt.de
Germany, Darmstadt
References
- Polyakov A.M. The theory of turbulence in two dimensions // Nuclear Phys. B. 1993. V. 396. N. 23. P. 367–385.
- Belavin A.A.,Polyakov A.M., Zamolodchikov A. A. Conformal field theory // Nuclear Phys. B. 1984. V. 241. P. 333–380.
- Bernard D., Boffetta G., Celani A., Falkovich G. Conformal invariance in two-dimensional turbulence // Nature Physics. 2006. V. 2. P. 124–128.
- Bernard D., Boffetta G., Celani A., Falkovich G. Inverse Turbulent Cascades and Conformally Invariant Curves // Phys. Rev. Lett. 2007. V. 98. P. 024501–504.
- Falkovich G. Conformal invariance in hydrodynamic turbulence // Russian Math. Surveys. 2007. V. 63. P. 497–510.
- Lundgren T.S. Distribution functions in the statistical theory of turbulence // Phys. Fluids. 1967. V. 10. P. 969–975.
- Monin A.S. Equations of turbulent motion // Prikl. Mat. Mekh. 1967. V. 31. P. 1057–1068.
- Novikov E.A. Kinetic equations for a vortex field // Sov. Phys. Dokl. V. 12. P. 1006-8.
- Grebenev V.N., Wacławczyk M., Oberlack M. Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence // J. Phys. A: Math. Theor. 2019. V. 50. P. 335501.
- Wacławczyk M., Grebenev V.N., Oberlack M. Conformal invariance of characteristic lines in a class of hydrodynamic models // Symmetry. 2020. V. 12. P. 1482.
- Wacławczyk M., Grebenev V.N., Oberlack M. Conformal invariance of the -point statistics of the zero-isolines of scalar fields in inverse turbulent cascades // Physical Review Fluids. 2021. V. 6. P. 084610.
- Friedrich R., Daitche A., Kamps O., Lülff J., Michel Voßkuhle M., Wilczek M. The Lundgren-Monin-Novikov hierarchy: Kinetic equations for turbulence // C.R. Physique. 2012. V. 13. P. 929–953.
- Madelung E. Quantentheorie in hydrodynamischer form // Zeitschrift für Physik. 1927. V. 40. P. 322–326.
- Bustamante M.D., Nazarenko S.V. Derivation of the Biot–Savart equation from the nonlinear Schrödinger equation // Phys. Rev. E. 2015. V. 92. P. 053019.
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