Universality in Turbulence: an Exactly Soluble Model

TL;DR

This paper discusses a simple exactly solvable model of turbulence that exhibits restricted universality, highlighting the importance of infrared renormalization for understanding universal behavior in turbulent systems.

Contribution

It introduces a solvable turbulence model demonstrating restricted universality and emphasizes the role of infrared renormalization in achieving true universality.

Findings

Model shows universal but anomalous scaling in correlation functions

Universal amplitudes diverge with system size before renormalization

Renormalized correlators exhibit normal scaling

Abstract

The present note contains the text of lectures discussing the problem of universality in fully developed turbulence. After a brief description of Kolmogorov's 1941 scaling theory of turbulence and a comparison between the statistical approach to turbulence and field theory, we discuss a simple model of turbulent advection which is exactly soluble but whose exact solution is still difficult to analyze. The model exhibits a restricted universality. Its correlation functions contain terms with universal but anomalous scaling but with non-universal amplitudes typically diverging with the growing size of the system. Strict universality applies only after such terms have been removed leaving renormalized correlators with normal scaling. We expect that the necessity of such an infrared renormalization is a characteristic feature of universality in turbulence.