FTLEs

Finite-Time Lyapunov Exponents (FTLEs) quantify the average exponential rate at which nearby trajectories diverge over a finite time interval in a time-dependent dynamical system. In fluid dynamics, FTLEs are commonly computed from a velocity field v(x,t) to reveal coherent structures that organize transport, such as barriers to mixing.

Choose t0 and a finite horizon T. For many initial positions x0 on a grid, integrate the trajectory x(t) with dx/dt = v(x,t) from t0 to t0+T. The flow map F maps x0 to x(t0+T). Approximate the deformation gradient ∂F/∂x0 via finite differences, assemble the Cauchy–Green strain tensor C = (∂F/∂x0)^T (∂F/∂x0). Let λ_max be the largest eigenvalue of C. The FTLE is σ(x0,t0;T) = (1/|T|) log sqrt(λ_max).

Computing FTLE with forward time horizon highlights repelling material lines; backward time horizon highlights attracting lines.

Ridges of the FTLE field are often interpreted as Lagrangian Coherent Structures (LCS), indicating transport barriers or regions of strong stretching. FTLE fields have become a standard diagnostic in ocean and atmospheric flows.

Practical considerations include the choice of T, grid resolution, and noise sensitivity. FTLE is a local, finite-time measure and may produce spurious or ambiguous ridges; it does not provide a complete description of LCS. Complementary approaches include geodesic or variational methods for identifying coherent structures.