Exact coherent structures in fully developed two-dimensional turbulence

Abstract

This paper reports several new classes of unstable recurrent solutions of the two-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions are in many ways analogous to recurrent solutions of the Navier–Stokes equation which are often referred to as exact coherent structures. In particular, we find that recurrent solutions of the Euler equation are dynamically relevant: they faithfully reproduce large-scale flows in simulations of turbulence at very high Reynolds numbers. On the other hand, these solutions have a number of properties which distinguish them from their Navier–Stokes counterparts. First of all, recurrent solutions of the Euler equation come in infinite-dimensional continuous families. Second, solutions of different types are connected, e.g. an equilibrium can be smoothly continued to a travelling wave or a time-periodic state. Third, and most important, they are only weakly unstable and, as a result, fully developed turbulence mimics some of these solutions remarkably frequently and over unexpectedly long temporal intervals.

Recurrent flow patterns as a basis for two-dimensional turbulence: Predicting statistics from structures

“…Even if complete statistical coverage is challenging, the ability to search for exact recurrent flows with particular physical properties is exceptionally useful if the goal is to understand the importance of individual dynamical events within well-known statistical phenomena like the energy cascade. It is also worth noting that the three-dimensional case may differ substantially from the two-dimensional flows studied here: in 2D we expect that some of the solutions we obtain at higher may connect to solutions of the Euler equations as ( 41 , 43 ), which are expected to take the form of domain-filling vortices, while wall-bounded turbulence is expected to require a more complex multiscale description with eddies of varying sizes attached to the boundaries ( 44 ). Early studies of weak, transient turbulence in 3D minimal flow units have shown that the statistics of individual UPOs can sometimes closely resemble the turbulence ( 8 ) itself which provides cautious optimism for the 3D case.…”

A 3-D minimum-enstrophy vortex in stratified quasi-geostrophic flows

“…In the absence of forcing, dissipation, bottom topography and beta effect, the final state of such a turbulent flow is a pair of contra-rotating vortices. The search for the spatial structure of final vortex states in 2-D turbulence has motivated the original work on minimum-enstrophy vortices (hereafter MEVs) (Leith 1984 a , b ), and more recently, a search for exact coherent structures in 2-D turbulent flows (Zhigunov & Grigoriev 2023). In his seminal paper, Leith used variational principles to derive the velocity profile of an axisymmetric vortex in 2-D incompressible flows, with minimal enstrophy for a given energy, angular momentum or circulation.…”