Exact coherent structures as building blocks of turbulence on large domains

Exact unstable solutions of the Navier-Stokes equations are thought to underpin the dynamics of turbulence, but are usually computed in minimal computational domains. Here, we extend this dynamical systems approach to spatially extended turbulent flows featuring multiple interacting 'substructures', and show how new simple invariant solutions can be constructed by spatial tiling of exact solutions from small-box calculations. Candidate solutions are found via gradient-based optimization of a scalar loss function which targets autorecurrence in spatially-masked regions of the flow. We apply these ideas to a vertically-extended Kolmogorov flow, where we first identify large numbers of relative periodic orbits (RPOs) which are combinations of high-dissipation, small-box solutions with laminar patches. We then show that vertically-stacked combinations of pairs of distinct small-box RPOs can form robust guesses for dynamically-relevant two-tori in the larger domain. Finally, we show how our optimization procedure can identify 'turbulent' trajectories which locally shadow a small-box RPO for multiple periods in a subdomain. These small-box combinations are possible as the flow spends prolonged periods in a regime where it can be effectively considered as a pair of weakly-coupled small-box systems, due to shielding effects associated with higher-dissipation flow structures.