Abstract
This study concerns conformal invariance of certain statistics in turbulence. Namely, there exists numerical evidence by Bernard et al. [Nature Phys. 2, 124 (2006)], that the zero-vorticity isolines for the 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 vorticity fields under the zero external force field. In this work we consider the first equation from the LMN chain for 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 of a scalar field, i.e., the CG invariance of the probability that a random curve passes through the point with for . 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.
- Received 27 October 2020
- Accepted 26 July 2021
DOI:https://doi.org/10.1103/PhysRevFluids.6.084610
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