Exact coherent structures in fully developed two-dimensional turbulence

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.

The movies below provide material supplementing the figures and data contained in the manuscript. The flow is represented by the corresponding vorticity field with red (blue) indicating positive (negative) values.

Movie S1: Fully-resolved turbulent flow (bottom left), the corresponding large-scale flow (bottom right) and the recurrence diagram (top). The black line shows the current time instant.
Movie S2: Unstable periodic orbit with temporal period $T=10.02$ a snapshot of which is shown in Figure 2(d).	Movie S3: Unstable periodic orbit with the temporal period $T=1.05$ a snapshot of which is shown in Figure 11(c).
Movie S4: A family of solutions connecting unstable periodic orbits shown in Figures 12(a) and 12(c).	Movie S5: A family of solutions connecting the equilibrium shown in Figure 13(a) and the periodic orbit shown in Figure 13(b).