Exact coherent structures in two-dimensional turbulence identified with convolutional autoencoders

Abstract

Convolutional autoencoders are used to deconstruct the changing dynamics of two-dimensional Kolmogorov flow as $Re$ is increased from weakly chaotic flow at $Re=40$ to a chaotic state dominated by a domain-filling vortex pair at $Re=400$. The highly accurate embeddings allow us to visualise the evolving structure of state space and are interpretable using `latent Fourier analysis' (Page {\em et. al.}, \emph{Phys. Rev. Fluids} \textbf{6}, 2021). Individual latent Fourier modes decode into vortical structures with a streamwise lengthscale controlled by the latent wavenumber, $l$, with only a small number $l \lesssim 8$ required to accurately represent the flow. Latent Fourier projections reveal a detached class of bursting events at $Re=40$ which merge with the low-dissipation dynamics as $Re$ is increased to $100$. We use doubly- ($l=2$) or triply- ($l=3$) periodic latent Fourier modes to generate guesses for UPOs (unstable periodic orbits) associated with high-dissipation events. While the doubly-periodic UPOs are representative of the high-dissipation dynamics at $Re=40$, the same class of UPOs move away from the attractor at $Re=100$ -- where the associated bursting events typically involve larger-scale ($l=1$) structure too. At $Re=400$ an entirely different embedding structure is formed within the network in which no distinct representations of small-scale vortices are observed; instead the network embeds all snapshots based around a large-scale template for the condensate. We use latent Fourier projections to find an associated `large-scale' UPO which we believe to be a finite-$Re$ continuation of a solution to the Euler equations.