Refuting the claim of conformal invariance for a zero-vorticity characteristic equation in 2D turbulence

TL;DR

This paper refutes the claim that conformal invariance holds for a zero-vorticity characteristic equation in 2D turbulence, clarifying misunderstandings and emphasizing the symmetry-breaking nature of the normalization constraint.

Contribution

It provides a critical correction to previous claims of conformal invariance in 2D turbulence, demonstrating the symmetry-breaking property of the normalization constraint.

Findings

The claim of conformal invariance is incorrect.

The symmetry-breaking property of the normalization constraint is confirmed.

Previous proofs were misrepresented and misconstrued.

Abstract

Although the current Reply by Grebenev et al. (2021a) makes their original analysis in Grebenev et al. (2017) more transparent, the actual problem remains. Their claim to have analytically proven conformal invariance in 2D turbulence for a zero-vorticity characteristic equation is not true. We refuted this claim in Frewer & Khujadze (2021a,b), which we will briefly summarize here again with respect to the presented Reply. In particular our proof on the symmetry-breaking property of the integral normalization constraint is misrepresented and misconstrued, especially in their second Reply (Grebenev et al., 2021b). Although the journal's only selected expert reviewer clearly agreed with our proof in his final conclusion, the journal nevertheless decided to publish the Replies.