Comment on ‘Conformal invariance of the Lundgren–Monin–Novikov equations for vorticity fields in 2D turbulence’

Abstract

The recent claim by Grebenev et al (2017 J. Phys. A: Math. Theor. 50 435502) that the inviscid 2D Lundgren–Monin–Novikov (LMN) equations on a zero-vorticity characteristic naturally would reveal local conformal invariance when only analyzing these by means of a classical Lie-group symmetry approach, is invalid. To note is that within this comment the (possible) existence of conformal invariance in 2D turbulence is not questioned, only the conclusion as given by Grebenev et al and their approach how this invariance was derived is criticized and refuted herein. In fact, the algebraic derivation of conformal invariance of the 2D LMN vorticity equations in Grebenev et al (2017 J. Phys. A: Math. Theor. 50 435502) is flawed. Its correction will reveal a globally constant scaling in the fields, instead of a local one as claimed.

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

“…is therefore an invalid statement as we have clearly shown in [3,4]. Still it is declared as a valid statement and essentially summarizes section 3 in [1]: the results shown therein are exactly the same from [2] we already refuted.…”

“…However, statement (b) we have already clearly refuted in our previous comments [3,4]. Hence, statement (b) cannot be used as a valid assumption to prove (c).…”

“…Instead, as said in (a) and (b), only an invariance analysis on the LMN-truncated and thus dimensionally nonreduced equation for f 1 was carried out, where in the end the results were just restricted to a zero-vorticity isoline f 1 | ω 1 =0 . And it is exactly this system we have proven in [3,4] that it does not admit conformal invariance.…”

“…Their statement in section 4.1 on p 12 that 'the invariance of the normalisation conditions (4) under the action of the group G (25)-( 31), ( 41) and (42) is evident and was derived before in [1]' is not true, and its falsity can be easily proven, even in the rationale of [1]. To prove this, we will proceed in the following only with the information and reasoning as it is provided and presented in [1], to demonstrate that if the authors would have done a thorough transformation of the normalization condition (4) along the lines of their own reasoning, they would have immediately realized that the conformal invariance need to get broken in order to have compatibility with (4). But such a thorough transformation on (4) has not been done by them, and therefore they miss this crucial fact.…”

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.…”