A GAUGE-INVARIANT LAGRANGIAN DETERMINED BY THE <i>n</i>-POINT PROBABILITY DENSITY FUNCTION OF VORTICITY FIELD OF THE WAVE OPTICAL TURBULENCE

Abstract

The geometry methods for Yang–Mills fields of the gauge transformations are applied to finding an invariant Lagrangian in fiber bundle of the configuration \(2d\) space \(X\) of the turbulent flow defined by the \(n\)-point probability density function \({{f}_{n}}\) (PDF). The two-dimensional wave optical turbulence is considered in the case of the inverse cascade of energy. The n-point PDF of the vorticity field satisfies the \({{f}_{n}}\)-equation from the Landgren–Monin–Novikov (LMN) hierarchy. The basic result reads: we construct the Lagrangian which is invariant under a subgroup \(H \subset G\) – the group of the gauge transformations in fiber bundles of the space X and the conserved currents.