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Journal of Vibration Testing and System Dynamics 8(1) (2024) 33--45 | DOI:10.5890/JVTSD.2024.03.003

Vladimir N. Grebenev, Alexandre N. Grishkov

Federal Research Center for Information and Computational Technologies, Novosibirsk, 630090, Russia

Institute of Mathematics and Statistics, University of Sao Paulo, Sao Paulo, 66281, Brazil

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Abstract

The conformal invariance of certain statistics in the inviscid two-dimensional turbulence is derived. For this we investigate the transport equation for the two-point probability density functions of vorticity from the infinite Lundgren-Monin-Novikov hierarchy and give the conditions under which the probability measure is conformally invariant. This is an extension of our previous analyses of the one-point statistics, which also paves the way for further generalisation to arbitrary $n$-point statistics.

References

1. [1]	Polyakov, A.M. (1993), The theory of turbulence in two dimensions, Nuclear Physics B, 396, 367-385.

2. [2]	Belavin, A.A., Polyakov, A.M., and Zamolodchikov, A.A. (1984), Conformal field theory, Nuclear Physics B, 383, 333-380.

3. [3]	Bernard, D., Boffetta, G., Celani, A., and Falkovich, G. (2006), Conformal invariance in two-dimensional turbulence, Nature Physics, 2, 124-128.

4. [4]	Falkovich, G. (2007), Conformal invariance in hydrodynamic turbulence, Russian Mathematical Surveys, 63, 497-510.

5. [5]	Hasegawa, A. and Mima, K. (1978), Pseudo-three-dimensional turbulence in magnetized nonuniform, Physics Fluids, 21, 87-92.

6. [6]	Horton, W. and Hasegawa, A. (1994), Quasi--two--dimensional dynamics of plasmas and fluids, Chaos, 4, 227.

7. [7]	Grebenev, V.N., Wac\l{}awczyk, M., and Oberlack, M. (2017), Conformal invariance of the Lungren-Monin-Novikov equations for vorticity fields in 2D turbulence, Journal of Physics A: Mathematical and Theoretical, 50, 435502.

8. [8]	Wac\l{}awczyk, M., Grebenev, V.N., and Oberlack, M. (2017), Lie symmetry analysis of the Lundgren-Monin-Novikov equations for multi-point probability density functions of turbulent flow, Journal of Physics A: Mathematical and Theoretical, 50, 175501.

9. [9]	Thalabard, S. and Bec, J. (2020), Turbulence of generalised flows in two dimensions, Journal of Fluid Mechanics, 883, A49.

10. [10]	Grebenev, V.N., Wac\l{}awczyk, M., and Oberlack, M. (2019), Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence, Journal of Physics A: Mathematical and Theoretical, 52, 335501.

11. [11]	Wac\l{}awczyk, M., Grebenev, V.N., and Oberlack, M. (2020), Conformal invariance of characteristic lines in a class of hydrodynamic models, Symmetry, 12, 1482.

12. [12]	Wac\l{}awczyk, M., Grebenev, V.N., and Oberlack, M. (2021), Conformal invariance of the $1$-point statistics of the zero isolines of $2d$ scalar fields in inverse turbulent, Physical Review Fluids, 6, 084610.

13. [13]	Friedrich, R., Daitche, A., Kamps, O., L{\"u}lff, J., Michel Vo{\ss} kuhle, M., and Wilczek, M. (2012), The Lundgren-Monin-Novikov hierarchy: Kinetic equations for turbulence, Comptes Rendus Physique, 13, 929-953.

14. [14]	Ovsyannikov, L.V. (1978), Group Analysis of Differential Equations, Moscow: Nauka.