Finite-time Lyapunov exponents · hyperbolic manifolds · transport barriers
Click "Compute FTLE" to generate field
Forward FTLE (ridges = repelling LCS)
Instantaneous velocity field (t=0)
Lagrangian Coherent Structures (LCS) are material surfaces that act as transport
barriers in unsteady flows — they don't mix with surrounding fluid even over finite time.
The Finite-Time Lyapunov Exponent (FTLE) at point x₀ measures the maximum stretching
rate of fluid elements over time interval [t₀, t₀+T]: σ(x₀) = (1/|T|) log √λ_max(Δ)
where Δ = [∂ϕᵀ/∂x₀][∂ϕ/∂x₀] is the Cauchy-Green deformation tensor,
and ϕ is the flow map x(t₀+T) = ϕ(x₀).
Ridges of FTLE approximate repelling LCS (forward time) or attracting LCS
(backward time). These are the "hidden skeleton" of transport in geophysical and
engineering flows — separating ocean eddies, atmospheric lobes, and wake structures.
The double-gyre is the canonical benchmark from Shadden et al. (2005).