Comment on 'Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence'

TL;DR

This paper critically examines and refutes the claim that the inviscid 2D Lundgren-Monin-Novikov equations on a zero-vorticity Lagrangian path are conformally invariant, highlighting errors in previous analyses and clarifying the issue.

Contribution

It provides a detailed critique of prior claims of conformal invariance in 2D turbulence equations, correcting misconceptions and clarifying the mathematical analysis.

Findings

Previous claims of conformal invariance are invalid.

Errors in earlier analyses are identified and corrected.

The Lagrangian formulation does not admit conformal invariance.

Abstract

The current claim by Grebenev et al. [J. Phys. A: Math. Theor. 52, 335501 (2019)], namely that the inviscid and unclosed 2D Lundgren-Monin-Novikov (LMN) equations on a zero-vorticity Lagrangian path admit conformal invariance, is based on a flawed and misleading analysis published earlier by Grebenev et al. (2017). All false results and conclusions made before in the Eulerian picture were now extended by Grebenev et al. (2019) to the Lagrangian picture. Although we have already commented on these errors and consistently refuted their previous study (Frewer & Khujadze, 2018), we deem it necessary to address and discuss these errors again in the new formulation and notation of Grebenev et al. (2019) as it will offer new insights into this issue.