Detecting coherent structures with automatic differentiation

Transport in complex flows is difficult to quantify using traditional tools from dynamical systems theory. Lagrangian coherent structures (LCS) seek to partition the flow into long-lived manifolds, such as persistent vortices, that dominate global transport and serve as an organizing skeleton for the flow. Many popular methods for detecting LCS implicitly require the spatial gradient of a flow to be computed along trajectories, in order to measure the degree to which virtual tracer particles spread apart along different directions over time. However, in a data-limited regime, the ability to precisely compute local spatial gradients rapidly degrades---a problem worsened by the tendency of chaotic flows to rapidly disperse nearby tracers. Here, we demonstrate a new approach to estimating LCS that takes advantage of recent computational advances in automatic differentiation. These approaches allow spatial gradient information to be accumulated during forward propagation of a complex flow, improving gradient estimates without substantially increasing computational cost. We demonstrate several example applications of our approach to diverse flows, including sparse experimental datasets and three-dimensional turbulence.