Exact Coherent Structures in Fully Developed Two-Dimensional Turbulence

TL;DR

This paper discovers new classes of weakly unstable recurrent solutions of the 2+1-dimensional Euler equation, revealing their relevance to high-Reynolds-number turbulence and their connection to coherent structures observed in simulations.

Contribution

It introduces high-dimensional families of solutions and shows their connection to turbulent flow structures, extending the concept of exact coherent structures to fully developed turbulence.

Findings

Turbulence exhibits large-scale structures similar to time-periodic solutions.

Solutions form continuous families and connect different types.

Many solutions are dynamically relevant at high Reynolds numbers.

Abstract

This paper reports several new classes of weakly unstable recurrent solutions of the 2+1-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions have a number of remarkable properties which distinguish them from analogous solutions of the Navier-Stokes equation describing transitional flows. First of all, they come in high-dimensional continuous families. Second, solutions of different types are connected, e.g., an equilibrium can be smoothly continued to a traveling wave or a time-periodic state. Third, and most important, many of these solutions are dynamically relevant for turbulent flow at high Reynolds numbers. Specifically, we find that turbulence in numerical simulations exhibits large-scale coherent structures resembling some of our time-periodic solutions both frequently and over long temporal intervals. Such solutions are analogous to exact coherent structures originally introduced in the context of transitional flows.