Exact coherent structures in the 2D Euler Equation

While the formalism of exact coherent structures (ECS) proved highly insightful for transitional flows, extension to fully developed turbulence proved challenging. As $Re$ increases ECSs become more unstable, numerous, and much harder to find. However, investigations of high-$Re$ 2D turbulence in a periodic box produced several surprises. Key among them is that, on large scales, turbulence shadows particular time-periodic solutions of the Euler equation over extremely long temporal intervals, which has serious implications for the lack of universality in the direct cascade. The Euler equation has substantially higher symmetry that Navier-Stokes and, as a result, has far more ECSs. Unexpectedly, these ECSs, at least those describing large scales, are both far easier to find than ECSs of the Navier-Stokes equation and very weakly unstable. Moreover, they come in continuous families spanned by what looks like an infinite number of parameters, which is unlike ECSs of Navier-Stokes where each parameter corresponds to a continuous spatial symmetry (e.g., rotation or translation about an axis). Finally, in Euler, different classes of ECS are all connected, i.e. an equilibrium can be continued to a traveling wave or a time-periodic state.