Conformal invariance of the 1-point statistics of the zero-isolines of 2d scalar fields in inverse turbulent cascades

Wacławczyk M., Grebenev V.N. and Oberlack M.

Physical Review Fluids

6(8), 2021, art. 084610, 10.1103/PhysRevFluids.6.084610

This study concerns conformal invariance of certain statistics in 2 d turbulence. Namely, there exists numerical evidence by Bernard et al. [Nature Phys. 2, 124 (2006)], that the zero-vorticity isolines x ( l , t ) for the 2 d Euler equation with an external force and a uniform friction belong to the class of conformally invariant random curves. Based on this evidence, the CG invariance was formally proven by Grebenev et al. [J. Phys. A: Math. Theor. 50, 435502 (2017)] by a Lie group analysis for the 1-point probability density function (PDF) governed by the inviscid Lundgren-Monin-Novikov (LMN) equations for 2 d vorticity fields under the zero external force field. In this work we consider the first equation from the LMN chain for 2 d scalar fields under Gaussian white-in-time forcing and large-scale friction. With this, the flow can be kept in a statistically steady state and the analysis is performed for the stationary LMN. Specifically, for the inviscid case we prove the CG invariance of the 1-point statistics of the zero-isolines x ( l ) of a scalar field, i.e., the CG invariance of the probability f 1 ( x ( l ) , ϕ ) d ϕ that a random curve x ( l ) passes through the point x with ϕ = 0 for l = l 1 . We show an example, where the proposed transformations represent a change between PDF's describing homogeneous and nonhomogeneous fields. Possible implications of this result are discussed.