Can We Connect a Dynamical Description and a Statistical Description of Turbulence?

For a class of chaotic systems, periodic orbit theory (POT) aims to connect a dynamical description with a statistical one by means of unstable solutions of the governing equations. It has been conjectured that a suitable extension of POT to systems with discrete and continuous symmetries can be used to describe the statistics of turbulent fluid flows. To test POT predictions, we investigate a weakly turbulent, small aspect-ratio Taylor–Couette flow driven by counter-rotating cylinders. This flow is realizable in experiment and has both discrete and continuous symmetries. Consequently, most of the unstable solutions found for this flow are relative periodic orbits (RPOs). We find that weighted sums over a collection of twelve RPOs accurately predict the turbulent average of kinetic energy, but not energy dissipation. Neither observables’ higher order moments are well predicted by the sums over RPOs. This discrepancy can be traced to two separate issues. First of all, theoretical predictions for the statistical weights are found to be poorly correlated with the statistics of visits by turbulent flow to the neighborhoods of individual RPOs. Second, on accessible time scales, turbulent flow shadows multiple RPOs without ever coming particularly close to any of them.