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  • Open Access

Exact Coherent Structures as Building Blocks of Turbulence on Spatially Extended Domains

Dmitriy Zhigunov* and Jacob Page†

  • *Contact author: dmitriy.zhigunov@ed.ac.uk
  • †Contact author: jacob.page@ed.ac.uk
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Phys. Rev. Lett. 137, 164001 – Published 14 October, 2026

DOI: https://doi.org/10.1103/lh7v-r983

Abstract

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 localized regions of the flow. We first identify 25 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 to be a pair of weakly coupled small-box systems. The new large-scale states are the first examples of solutions featuring multiple interacting substructures and support the long-standing hypothesis that turbulence can be built up as a spatiotemporal tapestry of smaller domain solutions.

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Phys. Rev. Lett. 137, 164001– Published 14 October, 2026
  • Received 29 January 2026
  • Accepted 14 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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