Clebsch Confinement and Instantons in Turbulence

TL;DR

This paper introduces Clebsch instantons as singular vorticity sheets with nontrivial helicity, explaining their role in turbulence, intermittency, and critical phenomena, supported by theoretical analysis and numerical simulations.

Contribution

It proposes a novel framework linking Clebsch instantons to turbulence intermittency and provides a detailed analysis of their effects on vorticity and velocity circulation.

Findings

Instantons dominate enstrophy and velocity circulation PDF.

Large loops exhibit exponential circulation distribution matching simulations.

Small loops relate to multifractal scaling laws and Liouville theory.

Abstract

We introduce a concept of Clebsch confinement related to unbroken gauge invariance and study Clebsch instantons: singular vorticity sheets with nontrivial helicity. This is realization of the "Instantons and intermittency" program we started back in the 90ties\cite{FKLM}. These singular solutions are involved in enhancing infinitesimal random forces at remote boundary leading to critical phenomena. In the Euler equation vorticity is concentrated along the random self-avoiding surface, with tangent components proportional to the delta function of normal distance. Viscosity in Navier-Stokes equation smears this delta function to the Gaussian with width $h \propto \nu^{\nicefrac{3}{5}}$ at $\nu \ra 0$ with fixed energy flow. These instantons dominate the enstrophy in dissipation as well as the PDF for velocity circulation $\Gamma_C$ around fixed loop $C$ in space. At large loops, the resulting symmetric exponential distribution perfectly fits the numerical simulations\cite{IBS20} including pre-exponential factor $1/\sqrt{|\Gamma|}$. At small loops, we advocate relation of resulting random self-avoiding surface theory with multi-fractal scaling laws observed in numerical simulations. These laws are explained as a result of fluctuating internal metric (Liouville field). The curve of anomalous dimensions $\zeta(n)$ can be fitted at small $n$ to the parabola, coming from the Liouville theory with two parameters $\alpha, Q$. At large $n$ the ratios of the subsequent moments in our theory grow linearly with the size of the loop, which corresponds to finite value of $\zeta(\infty)$ in agreement with DNS.