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

Grebenev V.N., Wacławczyk M. and Oberlack M. 
Publication type:

Journal of Physics A: Mathematical and Theoretical,50, 2017, art. 435502, 10.1088/17518121/aa8c69 
Organization unit:

We study the statistical properties of the vorticity field in twodimensional turbulence. The field is described in terms of the infinite Lundgren–Monin–Novikov (LMN) chain of equations for multipoint 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 LieBäcklund transformations devised for integrodifferential equations. We analytically show that the conformal group is broken for the first LMN equation i.e. for the 1point pdf at least for the inviscid case but the equation is still conformally invariant on the associated characteristic with zerovorticity. 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 zerovorticity 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 twodimensional 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).