Broken Mirror Symmetry of Tracer’s Trajectories in Turbulence

Topological properties of physical systems play a crucial role in our understanding of nature, yet their experimental determination remains elusive. We show that the mean helicity, a dynamical invariant in ideal flows, quantitatively affects trajectories of fluid elements: the linking number of Lagrangian trajectories depends on the mean helicity. Thus, a global topological invariant and a topological number of fluid trajectories become related, and we provide an empirical expression linking them. The relation shows the existence of long-term memory in the trajectories: the links can be made of the trajectory up to a given time, with particles positions in the past. This property also allows experimental measurements of mean helicity.

Figure 1

Definition of apparent crossing. A pair of experimental 3D trajectories (a) are projected onto a 2D plane [bottom of (a), and (b)]. Arrows and color gradients indicate time progression. A crossing is an intersection between the projections and is given by the particles’ history; it does not necessarily occur with the particles being near at the same instant. The sign of each crossing is defined by the right-hand rule [(b), bottom]: $+1$ when an anticlockwise rotation is needed to move from the tip of the arrow on top to the tip of the arrow below, and $-1$ in the clockwise case. Here, with two tracks projected in one plane, we see one negative and two positive crossings.

Figure 2

Linking number of tracers trajectories $\mathcal{K}$ as a function of normalized helicity $\mathcal{H}$ for DNSs of HIT and the LEM experiment. Error bars correspond to 95% confidence intervals. A least-squares linear fit taking into account the error bars is shown as a reference. Inset (a) shows the PDF of $\mathcal{K}$ in semilogarithmic scale for different subsets of tracers of the same DNS, indicated by the arrow. A Gaussian distribution with the same mean and standard deviation is shown for comparison. Inset (b) shows $\mathcal{K}$ as a function of $\mathcal{H}$ for the VK experiment, and cells in TG simulations; each of the cells corresponds to a VK-like swirling flow with nonzero mean helicity. A weighted least-squares linear fit for all TG data is shown, and it is compared with the linear fit obtained from HIT.

Figure 3

(a) Linking number $\mathcal{K}$ as a function of time in the eight TG cells, and in the VK experiment. The different points correspond to $\mathcal{K}$ estimated for ten subsets of 250 particles (in groups of connected points), at different consecutive time intervals $\mathrm{\Delta }T={\tau }_L$ up to time $6{\tau }_L$. Half the cells have $\mathcal{K}>0$ (with fluctuations in time), and the other half $\mathcal{K}<0$. The dashed line is the mean value of $\mathcal{K}$ in the VK experiment, with the colored band representing a 95% confidence interval considering time fluctuations. (b) Time evolution of $\mathcal{K}$ for all the cells with negative helicity, reconstructed by averaging the ten subsets from the data in (a). (c) PDF of $\mathcal{K}$ fluctuations in the VK experiment, with a normal distribution $N({\mu }_{\mathrm{VK}},{\sigma }_{\mathrm{VK}})$ with the same mean ${\mu }_{\mathrm{VK}}$ and standard deviation ${\sigma }_{\mathrm{VK}}$ for comparison.