Hydrodynamic approximation for two-dimensional optical turbulence: symmetries of statistical distributions

Optical turbulence is described in terms of multipoint probability density functions (PDFs) f_n using the Lundgren–Monin–Novikov (LMN) equation (statistical form of the Euler equations) for the vortex field w = ∇ × u in a two-dimensional flow (u is the velocity weight field). Lagrangian particles are shown to evolve along the characteristics of the f_n-equation from the LMN hierarchy. The vorticity is preserved along the characteristics in the absence of an external random force. It is shown that the group G of conformal transformations invariantly transforms the characteristics of the equation with zero vorticity and the family of f_n-equations for PDFs along these lines, or the statistics of the line of zero vorticity. Along other lines of the level w = const ≠ 0, the statistics is not conformally invariant. Moreover, the action G preserves the PDF class.