Hidden spatiotemporal symmetries and intermittency in turbulence

TL;DR

This paper investigates hidden symmetries in turbulence, showing how normalized flows with invariant measures reveal power law scaling in structure functions, providing new insights into intermittency phenomena in fluid dynamics.

Contribution

It introduces a framework for identifying hidden symmetries in infinite-dimensional dynamical systems, especially turbulence, using normalized flows and invariant measures.

Findings

Hidden symmetries lead to power law scaling in structure functions.

Formulas for structure function exponents are derived in terms of normalized measures.

The approach applies to Euler and Navier-Stokes systems through Quasi--Lagrangian description.

Abstract

We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing the equivalence relation with respect to temporal scalings and Galilean transformations, we define a representative set containing a single element within each equivalence class. Temporal scalings and Galilean transformations do not commute with the evolution operator (flow) and, hence, the equivalence relation is not invariant. Despite of that, we prove that a normalized flow with an invariant probability measure can be introduced on the representative set, such that symmetries are preserved in the statistical sense. We focus on hidden symmetries, which are broken in the original system but restored in the normalized system. The central motivation and application of this construction is the intermittency phenomenon in turbulence. We show that hidden symmetries yield power law scaling for structure functions, and derive formulas for their exponents in terms of normalized measures. The use of Galilean transformation in the equivalence relation leads to the Quasi--Lagrangian description, making the developed theory applicable to the Euler and Navier-Stokes systems.