Conformal invariance of the Lungren–Monin–Novikov equations for vorticity fields in 2D turbulence

Abstract

We study the statistical properties of the vorticity field in two-dimensional turbulence. The field is described in terms of the infinite Lundgren–Monin–Novikov (LMN) chain of equations for multi-point probability density functions (pdf’s) of vorticity. We perform a Lie group analysis of the first equation in this chain using the direct method based on the canonical Lie-Bäcklund transformations devised for integro-differential equations. We analytically show that the conformal group is broken for the first LMN equation i.e. for the 1-point pdf at least for the inviscid case but the equation is still conformally invariant on the associated characteristic with zero-vorticity. Then, we demonstrate that this characteristic is conformally transformed. We find this outcome coincides with the numerical results about the conformal invariance of the statistics of zero-vorticity isolines, see e.g. Falkovich (2007 Russian Math. Surv. 63 497–510). The conformal symmetry can be understood as a ‘local scaling’ and its traces in two-dimensional turbulence were already discussed in the literature, i.e. it was conjectured more than twenty years ago in Polyakov (1993 Nucl. Phys. B 396 367–85) and clearly validated experimentally in Bernard et al (2006 Nat. Phys. 2 124–8).

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

“…where f is some arbitrary function in the spatial coordinates x = (x 1 , x 2 ) -sure, when including also the two nonlocal equations (Eqs. [5][6] in Frewer & Khujadze (2021a)) into the symmetry analysis, then f reduces to the harmonic function f (x) = 6c 11 (x), (1.3) as correctly described in Grebenev et al (2017Grebenev et al ( , 2021a. We therefore agree with what is said in the first part of point (a) for Eqs.…”

“…We therefore agree with what is said in the first part of point (a) for Eqs. [10][11] in Grebenev et al (2021a), but not for what is said in the second part, particularly that "Frewer and Khujadze did not consider the above classification problem", or later in point (d) that "neither differential nor integral consequences of this [LMN] equation with respect to ω can be considered".…”

“…Secondly, the symmetry analysis itself involves a differential operator X that contains a derivative in ω (see Eq. [27] in Grebenev et al (2017))…”

“…And it is exactly this approach we follow in Frewer & Khujadze (2021a). For example, if we want to construct an overall consistent symmetry solution through the differential generator X (1.5) for the special case ω = 0, as intended in Grebenev et al (2017), then we should consider and evaluate X| ω=0 as…”

“…

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.

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