Braid Entropy of Two-Dimensional Turbulence

The evolving shape of material fluid lines in a flow underlies the quantitative prediction of the dissipation and material transport in many industrial and natural processes. However, collecting quantitative data on this dynamics remains an experimental challenge in particular in turbulent flows. Indeed the deformation of a fluid line, induced by its successive stretching and folding, can be difficult to determine because such description ultimately relies on often inaccessible multi-particle information. Here we report laboratory measurements in two-dimensional turbulence that offer an alternative topological viewpoint on this issue. This approach characterizes the dynamics of a braid of Lagrangian trajectories through a global measure of their entanglement. The topological length NE of material fluid lines can be derived from these braids. This length is found to grow exponentially with time, giving access to the braid topological entropy S-Braid. The entropy increases as the square root of the turbulent kinetic energy and is directly related to the single-particle dispersion coefficient. At long times, the probability distribution of N-E is positively skewed and shows strong exponential tails. Our results suggest that S-Braid may serve as a measure of the irreversibility of turbulence based on minimal principles and sparse Lagrangian data.