# Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models

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## Abstract

**:**

## 1. Introduction

## 2. Basic 2D Model Equations

## 3. CG Invariance of Zero-Scalar Characteristic Line of the First LMN-Type Equation with Zero Viscosity and Friction

## 4. CG Invariance of Zero-Scalar Characteristic Line of the First LMN-Type Equation with Viscosity and Friction

## 5. Example of CG Transformation

## 6. Approximate Shape of the Scalar Isolines for the Conformal Transformations of Both x and ${x}^{\prime}$

## 7. Discussion

## 8. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CG | conformal group |

LMN | Lundgren–Monin–Novikov |

probability density function | |

SQG | surface quasi-geostrophic |

## Appendix A

## References

- Polyakov, A.M. The theory of turbulence in two dimensions. Nucl. Phys. B
**1993**, 396, 367–385. [Google Scholar] [CrossRef] [Green Version] - Belavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.A. Conformal field theory. Nucl. Phys. B
**1984**, 241, 333–380. [Google Scholar] [CrossRef] [Green Version] - Vorobieff, P.; Rivera, M.; Ecke, R.E. Soap film flows: Statistics of two-dimensional turbulence. Phys. Fluids
**1999**, 11, 2167. [Google Scholar] [CrossRef] - Bouchet, F.; Venaille, A. Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep.
**2009**, 515, 227–295. [Google Scholar] [CrossRef] [Green Version] - Bernar, D.; Boffetta, G.; Celani, A.; Falkovich, G. Conformal invariance in two-dimensional turbulence. Nat. Phys.
**2006**, 2, 124–128. [Google Scholar] [CrossRef] - Bernard, D.; Boffetta, G.; Celani, A.; Falkovich, G. Inverse Turbulent Cascades and Conformally Invariant Curves. Phys. Rev. Lett.
**2007**, 98, 024501–024504. [Google Scholar] [CrossRef] [Green Version] - Falkovich, G. Conformal invariance in hydrodynamic turbulence. Russ. Math. Surv.
**2007**, 63, 497–510. [Google Scholar] [CrossRef] [Green Version] - Schramm, O. Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math.
**2000**, 118, 221288. [Google Scholar] [CrossRef] [Green Version] - Grebenev, V.N.; Wacławczyk, M.; Oberlack, M. Conformal invariance of the Lungren-Monin-Novikov equations for vorticity fields in 2D turbulence. J. Phys. A Math. Theor.
**2017**, 50, 435502. [Google Scholar] [CrossRef] - Wacławczyk, M.; Grebenev, V.N.; Oberlack, M. Lie symmetry analysis of the Lundgren-Monin-Novikov equations for multipoint probability density functions of turbulent flow. J. Phys. A Math. Theor.
**2017**, 50, 175502. [Google Scholar] [CrossRef] - Lundgren, T.S. Distribution functions in the statistical theory of turbulence. Phys. Fluids
**1967**, 10, 969–975. [Google Scholar] [CrossRef] - Monin, A.S. Equations of turbulent motion. Prikl. Mat. Mekh.
**1967**, 31, 1057–1068. [Google Scholar] [CrossRef] - Novikov, E.A. Kinetic equations for a vortex field. Sov. Phys.—Dokl.
**1968**, 12, 1006–1008. [Google Scholar] - Oberlack, M.; Rosteck, A. New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discret. Contin. Dyn. Syst.-Ser. S
**2010**, 3, 451–471. [Google Scholar] [CrossRef] - Oberlack, M.; Wacławczyk, M.; Rosteck, A.; Avsarkisov, V. Symmetries and their importance for statistical turbulence theory. Mech. Eng. Rev.
**2015**, 2, 15–00157. [Google Scholar] [CrossRef] - Wacławczyk, M.; Staffolani, N.; Oberlack, M.; Rosteck, A.; Wilczek, M.; Friedrich, R. Statistical symmetries of the Lundgren-Monin-Novikov hierarchy. Phys. Rev. E
**2014**, 90, 013022. [Google Scholar] [CrossRef] [Green Version] - Grebenev, V.N.; Wacławczyk, M.; Oberlack, M. Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence. J. Phys. A Math. Theor.
**2019**, 50, 335501. [Google Scholar] [CrossRef] - Falkovich, G.; Musacchio, S. Conformal invariance in inverse turbulent cascades. arXiv
**2010**, arXiv:1012.3868. [Google Scholar] - Hasegawa, A.; Mima, K. Pseudo-three-dimensional turbulence in magnetized nonuniform plasma. Phys. Fluids
**1978**, 21, 87–92. [Google Scholar] [CrossRef] - Horton, W.; Hasegawa, A. Quasi-two-dimensional dynamics of plasmas and fluids. Chaos: Interdiscip. J. Nonlinear Sci.
**1994**, 4, 227. [Google Scholar] [CrossRef] [Green Version] - Friedrich, R.; Daitche, A.; Kamps, O.; Lülff, J.; Voßkuhle, M.; Wilczek, M. The Lundgren-Monin-Novikov hierarchy: Kinetic equations for turbulence. C. R. Phys.
**2012**, 13, 929–953. [Google Scholar] [CrossRef] [Green Version] - Ovsyannikov, L.V. Group Analysis of Differential Equations; Nauka: Moscow, Russia, 1978. [Google Scholar]
- Zawistowski, Z.J. Symmetries of Integro-Differential Equations. Rep. Math. Phys.
**2001**, 48, 269–275. [Google Scholar] [CrossRef] - Boffetta, G.; Ecke, R.E. Two-Dimensional Turbulence. Annu. Rev. Fluid Mech.
**2012**, 44, 427–451. [Google Scholar] [CrossRef] - Kraichnan, R.H. Inertial ranges in two-dimensional turbulence. Phys. Fluids
**1967**, 10, 1417–1423. [Google Scholar] [CrossRef] [Green Version] - Boffetta, G.; Celani, A.; Musacchio, S.; Vergassola, M. Intermittency in two-dimensional Ekman-Navier-Stokes turbulence. Phys. Rev. E
**2002**, 66, 026304. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Boffetta, G.; Cenedese, A.; Espa, S.; Musacchio, S. Effects of friction on 2D turbulence: An experimental study of the direct cascade. Europhys. Lett.
**2005**, 71, 590–596. [Google Scholar] [CrossRef] [Green Version] - Thalabard, S.; Rosenberg, D.; Pouquet, A.; Mininni, P.D. Conformal Invariance in Three-Dimensional Rotating Turbulence. Phys. Rev. Lett.
**2011**, 106, 204503. [Google Scholar] [CrossRef]

**Figure 1.**Illustrative zero-scalar Lagrangian particle path (

**left panel**) and its conformal transformation (

**right panel**). The growing tip of the curve is denoted by the symbol *.

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**MDPI and ACS Style**

Wacławczyk, M.; Grebenev, V.N.; Oberlack, M.
Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models. *Symmetry* **2020**, *12*, 1482.
https://doi.org/10.3390/sym12091482

**AMA Style**

Wacławczyk M, Grebenev VN, Oberlack M.
Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models. *Symmetry*. 2020; 12(9):1482.
https://doi.org/10.3390/sym12091482

**Chicago/Turabian Style**

Wacławczyk, Marta, Vladimir N. Grebenev, and Martin Oberlack.
2020. "Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models" *Symmetry* 12, no. 9: 1482.
https://doi.org/10.3390/sym12091482