{
  "id": "2309.12754",
  "version": "v1",
  "published": "2023-09-22T09:53:54.000Z",
  "updated": "2023-09-22T09:53:54.000Z",
  "title": "Exact coherent structures in two-dimensional turbulence identified with convolutional autoencoders",
  "authors": [
    "Jacob Page",
    "Joe Holey",
    "Michael P. Brenner",
    "Rich R. Kerswell"
  ],
  "categories": [
    "physics.flu-dyn"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-22T09:53:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact coherent structures",
      "convolutional autoencoders",
      "two-dimensional turbulence",
      "individual latent fourier modes decode",
      "periodic latent fourier modes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}