Abstract: A central subject is the symmetries of turbulence statistics. The extension of symmetry of the Navier-Stokes equations to conformal invariance of statistics is the program proposed by Polyakov [Nucl. Phys. B396, 1993] for 2d statistical turbulence. There exists numerical evidence by Bernard et al. [Nature Phys. 2, 124 (2006)], that the zero-vorticity isolines for the 2d Euler equation with an external force and a uniform friction belong to the class of conformally invariant random curves: Schramm-Lowner evolution curves. This enables one to predict and find quantitative relations going far beyond what was known previously about the turbulence. It suggests relations between phenomena like the Euler equation and critical percolation.
This talk aims at reviewing the exact analytic results based on Lie group analysis about the conformal invariance of certain statistics in the case of the inverse energy cascade. For the hydrodynamic turbulence, we refer to our works published in [Phys. Rev. Fluids., (2021)], [J. Phys. A: Math. Theor. (2017, 2019)], [Symmetry, (2020)]. For the optical (photonic) turbulence, we review [Quantum electronics, (2022)], Theor. Math. Phys. (2023)]. We also demonstrate how the geometric methods for Yang–Mills fields of gauge transformations can be applied for the statistical turbulence. These works are collaborated with Marta Waclawczyk and Martin Oberlack.
