A critical examination of the conformal invariance in the statistical equations of 2D turbulent scalar fields

TL;DR

This paper critically analyzes previous claims about conformal invariance in 2D turbulent scalar fields, clarifying misconceptions and demonstrating that the proposed invariance does not hold as a symmetry or relate directly to observed phenomena.

Contribution

The paper corrects three key misconceptions about conformal invariance in 2D turbulence and clarifies its mathematical and physical limitations.

Findings

The proposed conformal invariance is not related to SLE of zero-isolines.

It violates the smoothness axiom of Lie-group actions, leading to non-physical PDFs.

The invariance is a weaker equivalence transformation, not a symmetry or solution-mapping.

Abstract

The recent study by Waclawczyk et al. [Phys. Rev. Fluids 6, 084610 (2021)] on conformal invariance in 2D turbulence is misleading as it makes three incorrect claims that form the core of their work. We will correct these claims and put them into the right perspective: First, the conformal invariance as proposed by Waclawczyk et al. is not related to the result that zero-isolines of the scalar field in the inverse energy cascade display a Schramm-Loewner evolution (SLE). Second, the conformal invariance is not a Lie-group for all values of the scalar field since it inherently violates the smoothness axiom of a Lie-group action, with the effect that a physical PDF gets mapped to a non-physical one. Third, although Waclawczyk et al. recognize that their conformal invariance does not constitute a symmetry but only a weaker equivalence transformation, it is still not classified correctly. The claim that their equivalence can map between solutions is not true. This fact will be demonstrated by using an illustrative example.