Dynamical relevance of periodic orbits under increasing Reynolds number and connections to inviscid dynamics

Abstract

Large numbers of relative periodic orbits (RPOs) have been found recently in doubly-periodic, two-dimensional Kolmogorov flow at moderate Reynolds numbers $Re \in \{40, 100\}$. While these solutions lead to robust statistical reconstructions at the $Re$-values where they were obtained, it is unclear how their dynamical importance evolves with increasing $Re$. We perform arclength continuation on this library of solutions to show that large numbers of RPOs quickly become dynamically irrelevant, reaching dissipation values either well above or below those associated with the turbulent attractor at high $Re$. The scaling of the high dissipation RPOs is shown to be consistent with a direct connection to solutions of the unforced Euler equation, and is observed for a wide variety of states beyond the 'unimodal' solutions considered in previous work (Kim & Okamoto, Nonlinearity 28, 2015). On the other hand, the weakly dissipative states have properties indicating a connection to exact solutions of a forced Euler equation. The apparent dynamical irrelevance is associated with poor statistical reconstructions away from the $Re$ values where the RPOs were originally converged. Motivated by the connection to solutions of the Euler equation, we show that many of these states can be well described by exact relative periodic solutions in a system of point vortices. The point vortex RPOs are converged via gradient-based optimisation of a scalar loss function which (1) matches the dynamics of the point vortices to the turbulent vortex cores and (2) insists the point vortex evolution is itself time-periodic.