Labelling forced two-dimensional turbulence with vortex crystals

Recent methods for finding unstable periodic orbits (UPOs) in turbulence have dramatically expanded the number of known solutions in two-dimensional, turbulent Kolmogorov flow (Page et al, Proc. Nat. Acad. Sci., 121 (23), 2024). Despite this, it is still unknown which of these remain dynamically relevant as Re → ∞, while the self-sustaining processes encapsulated in this set of solutions have not been identified. In this talk, we perform an arclength continuation of a library of O(150) UPOs upwards from Re = 100 to beyond Re ≈ 1000, with several apparently connecting to solutions of the Euler equation. Motivated by this connection, we compute exact coherent solutions for a system of point vortices that mimic the Navier-Stokes UPOs. This is achieved via gradient-based optimisation of a scalar loss function which seeks to both (1) match the positions of the point vortices to the turbulent vortex cores and (2) insist that the point vortex evolution is itself time-periodic. We categorise the Kolmogorov UPOs into classes based on the corresponding point vortex solution. We also modify our approach to find stationary vortex crystals which can describe the large-scale vortices in decaying two-dimensional turbulence.