A Gauge-Invariant Lagrangian Determined by the n-Point Probability Density Function of a Vorticity Field of Wave Optical Turbulence

The geometric methods for Yang–Mills fields of gauge transformations are applied to find an invariant Lagrangian in the fiber bundle of the configuration of 2d space X of a turbulent flow determined by the n-point probability density function (PDF)  f_n. The two-dimensional wave optical turbulence is considered in the case of an inverse cascade of turbulence energy transfer under external impacts in the form of white Gaussian noise and large-scale friction. The n-point PDF of the vorticity field satisfies the  f_n-equation from the Lundgren–Monin–Novikov hierarchy, and the conditions of equation invariance under external action are found. A Lagrangian, which is invariant relative to the \(H \subset G\) subgroup (a group of the gauge transformations in the fiber bundle of the space X), and the conserved currents are constructed.