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

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