Exact Coherent Structures in Fully Developed Two-Dimensional Turbulence

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.