Abstract:
We use the concept of gauge transformations in the proof of the invariance of the statistics of zero-vorticity lines in the case of the inverse energy cascade in wave optical turbulence; we study it in the framework of the hydrodynamic approximation of the two-dimensional nonlinear Schrödinger equation for the weight velocity field $\mathbf u$. The multipoint probability distribution density functions $f_n$ of the vortex field $\Omega=\nabla\times\mathbf u$ satisfy an infinite chain of Lundgren–Monin–Novikov equations {(}statistical form of the Euler equations{\rm)}. The equations are considered in the case of the external action in the form of white Gaussian noise and large-scale friction, which makes the probability distribution density function statistically stationary. The main result is that the transformations are local and conformally transform the $n$-point statistics of zero-vorticity lines or the probability that a random curve $\mathbf x(l)$ passes through points $\mathbf x_i\in\mathbb R^2$ for $l=l_i$, $i=1,\dots,n$, where $\Omega=0$, is invariant under conformal transformations.
Keywords:optical turbulence, $n$-point statistics of a vortex field, Lundgren–Monin–Novikov equations, gauge transformations, conformal invariance.
Fulltext:
PDF file (424 kB)