A Renormalisation Group Study of Three Dimensional Turbulence

TL;DR

This paper applies renormalisation group analysis to three-dimensional turbulence, showing convergence to a fixed point with a Kolmogorov energy spectrum, inspired by Polyakov's work on two-dimensional turbulence.

Contribution

It introduces a renormalisation group framework for 3D turbulence, identifying fixed points and their properties, including the energy spectrum and scaling dimensions.

Findings

Velocity potential probability law converges to a fixed point.

At the fixed point, the energy spectrum matches Kolmogorov's spectrum.

Scaling dimension of velocity potential is -4/3.

Abstract

We study the three dimensional Navier-Stokes equation with a random Gaussian force acting on large wavelengths. Our work has been inspired by Polyakov's analysis of steady states of two dimensional turbulence. We investigate the time evolution of the probability law of the velocity potential. Assuming that this probability law is initially defined by a statistical field theory in the basin of attraction of a renormalisation fixed point, we show that its time evolution is obtained by averaging over small scale features of the velocity potential. The probability law of the velocity potential converges to the fixed point in the long time regime. At the fixed point, the scaling dimension of the velocity potential is determined to be ${-{4\over 3}}$. We give conditions for the existence of such a fixed point of the renormalisation group describing the long time behaviour of the velocity potential. At this fixed point, the energy spectrum of three dimensional turbulence coincides with a Kolmogorov spectrum.