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

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, 1 the journal nevertheless decided to publish the Replies.

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

“…The transformation (1.1) is a new proposal by Wac lawczyk et al for a conformal equivalence. It is not the full global form of the original Lie-group infinitesimals determined in their earlier studies [3,4,12] and refuted as a conformal Lie-group in [13][14][15][16]. The decisive difference is that (1.1) is missing the infinitesimal Lie-group constraint for the scalar field ξ φ x = ξ φ y = 0, which translates to the global constraint ∂ x γ(x) = ∂ y γ(x) = 0.…”

“…The decisive difference is that (1.1) is missing the infinitesimal Lie-group constraint for the scalar field ξ φ x = ξ φ y = 0, which translates to the global constraint ∂ x γ(x) = ∂ y γ(x) = 0. A consistent Lie-group invariance analysis shows that this constraint has to hold for all isolines of φ, including the one for φ = 0, thus breaking the conformal group [13][14][15][16]. On the other hand, this constraint can surely be switched off for φ = 0, but it then comes at the price of leading to internal inconsistencies in the Lie-group invariance analysis, as clearly demonstrated in [13][14][15][16].…”

“…A consistent Lie-group invariance analysis shows that this constraint has to hold for all isolines of φ, including the one for φ = 0, thus breaking the conformal group [13][14][15][16]. On the other hand, this constraint can surely be switched off for φ = 0, but it then comes at the price of leading to internal inconsistencies in the Lie-group invariance analysis, as clearly demonstrated in [13][14][15][16]. Only when breaking the conformal group for all scalar isolines an overall consistent analysis is restored.…”

An example from turbulence how not to use the invariant function method of Lie-group symmetries

“…These are the so-called non-negativity, normalization, coincidence, separation and conditional constraints[14,16]. In particular, the reviews[19][20][21][22] explicitly show what happens if one of these constraints are violated 5. Obviously, if an invariant transformation already violates from the outset one of the dynamical constraints as causality, as the two invariant transformations Eqs.…”