Logarithmic Correlation Functions in Two Dimensional Turbulence

TL;DR

This paper uses conformal field theory to analyze two-dimensional turbulence, revealing logarithmic factors in correlation functions and energy spectra, consistent with recent simulations and experiments.

Contribution

It demonstrates that logarithmic conformal field theory describes 2D turbulence with and without perturbations, providing new analytical insights into turbulence spectra.

Findings

Logarithmic factors appear in correlation functions under perturbation.

Energy spectrum follows a modified power law with a logarithmic correction.

Predictions align with recent numerical and experimental results.

Abstract

We consider the correlation functions of two-dimensional turbulence in the presence and absence of a three-dimensional perturbation, by means of conformal field theory. In the persence of three dimensional perturbation, we show that in the strong coupling limit of a small scale random force, there is some logarithmic factor in the correlation functions of velocity stream functions. We show that the logarithmic conformal field theory $c_{8,1}$ describes the 2D- turbulence both in the absence and the presence of the perturbation. We obtain the following energy spectrum $E(k) \sim k^{-5.125 } \ln(k )$ for perturbed 2D - turbulence and $E(k) \sim k^{-5 } \ln(k )$ for unperturbed turbulence. Recent numerical simulation and experimental results confirm our prediction.