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

This study concerns conformal invariance of certain statistics in 2d 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 2d 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 2d vorticity fields under the zero external force field. In this work we consider the first equation from the LMN chain for 2d 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 f1(x(l),ϕ)dϕ that a random curve x(l) passes through the point x with ϕ=0 for l=l1. 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.