Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits

In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating that turbulent flows in both experiment and numerics transiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). We show that UPOs we computed are also statistically significant, with turbulent flows spending a sizable fraction of the total time near these solutions. As a result, the average rates of energy input and dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.

Figure 1

Experimental setup to generate quasi-2D Kolmogorov-like flow. (a) Top view indicating magnet array (dashed lines) and directions of magnetic field $\mathbf{B}$, current density $\mathbf{J}=J\widehat{\mathbf{y}}$, and electromagnetic forcing $\mathbf{F}$. (b) Side view showing stably stratified immiscible two-layer configuration.

Figure 2

(a) Low-dimensional projection of state space showing turbulent trajectory from experiment (black curve) shadowing ${\mathrm{UPO}}_{3A}$ (red loop). Each point on these curves represents a flow field. The segment in black (gray) lies in (outside) the neighborhood of ${\mathrm{UPO}}_{3A}$. The sphere and square indicate instantaneously closest points on the turbulent trajectory and ${\mathrm{UPO}}_{3A}$. The corresponding flow snapshots are shown in (b) and (c), where color represents vorticity $\omega =(\nabla \times \mathbf{u}{)}_z$. The projection method is detailed in the SM [31].

Figure 3

(a) Instantaneous normalized separation $D_1$ between a turbulent trajectory in experiment and periodic orbit ${\mathrm{UPO}}_{3A}$. The dashed black line ($D_1=0.45$) indicates the limit for closeness in state space. (b) $t^{\prime }$ and $t$ parametrize time along ${\mathrm{UPO}}_{3A}$ and the turbulent trajectory, respectively. $T=113.2$ s is the period of ${\mathrm{UPO}}_{3A}$.

Figure 4

Statistical significance of UPOs. (a) Probability (in %) to find a turbulent trajectory at a (normalized) distance $D_1\le \epsilon$ from the UPOs we computed. Dashed line indicates the upper limit $\epsilon =0.45$ for closeness in state space. (b) Conditional probabilities for turbulent trajectories visiting neighborhoods ($\epsilon =0.45$) of UPO clusters. Error bars indicate changes to probabilities when $\epsilon$ is varied between [0.4, 0.5].

Figure 5

Energy input rate $\mathcal{I}$ versus the difference between input and dissipation rates ($\mathcal{I}-\mathcal{D}$) for turbulent time series in experiment (scatter plot) and UPOs (closed loops). Inset: Probability density function of $\mathcal{I}(t)$ for turbulent flow in experiment (solid gray curve) and DNS (dashed gray curve). Colored symbols show the mean values of $\mathcal{I}$ for each of the seven UPOs and the dashed black lines represent the range of $\mathcal{I}$ for ${\mathrm{UPO}}_{2A–2C}$ and ${\mathrm{UPO}}_{3A,3B}$.