Conformal invariance and the Lundgren-Monin-Novikov equations for vorticity fields in 2D turbulence: Refuting a recent claim

TL;DR

This paper refutes a recent claim that the inviscid 2D Lundgren-Monin-Novikov equations exhibit local conformal invariance, demonstrating that the original derivation was flawed and that the correct transformations do not support such invariance.

Contribution

The paper critically analyzes and corrects the algebraic derivation of conformal invariance in the 2D LMN vorticity equations, refuting the previous claim and clarifying the true symmetry properties.

Findings

The original derivation of conformal invariance was flawed.

Correct transformations lead to only global scaling, not local conformal invariance.

Different Lie-group infinitesimals result from the corrected analysis.

Abstract

The recent claim by Grebenev et al. [J. Phys. A: Math. Theor. 50, 435502 (2017)] that the inviscid 2D Lundgren-Monin-Novikov (LMN) equations on a zero vorticity characteristic naturally would reveal local conformal invariance when only analyzing these by means of a classical Lie-group symmetry approach, is invalid and will be refuted in the present comment. To note is that within this comment the (possible) existence of conformal invariance in 2D turbulence is not questioned, only the conclusion as is given in Grebenev et al. (2017) and their approach how this invariance was derived is what is being criticized and refuted herein. In fact, the algebraic derivation for conformal invariance of the 2D LMN vorticity equations in Grebenev et al. (2017) is flawed. A key constraint of the LMN equations has been wrongly transformed. Providing the correct transformation instead will lead to a breaking of the proclaimed conformal group. The corrected version of Grebenev et al. (2017) just leads to a globally constant scaling in the fields and not to a local one as claimed. In consequence, since in Grebenev et al. (2017) only the first equation within the infinite and unclosed LMN chain is considered, also different Lie-group infinitesimals for the one- and two-point probability density functions (PDFs) will result from this correction, replacing thus the misleading ones proposed.