# 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

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**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