LCS and FTLE#
This section discusses Lagrangian Coherent Structures (LCS) and Finite-Time Lyapunov Exponents (FTLE)—formally, Lagrangian Coherent Structures and Finite-Time Lyapunov Exponents—and their role in describing tracer transport in fluid flows.
The transport of a tracer in a fluid is closely linked to the emergence of patterns commonly referred to as coherent structures, or Lagrangian Coherent Structures (LCS) when defined using fluid trajectories. In time-independent dynamical systems, these structures correspond to the stable and unstable manifolds of hyperbolic trajectories [1]. Coherent structures delineate regions of swirling, stretching, or contracting tracer distributions [2]: contraction occurs along stable manifolds, whereas unstable manifolds mark divergent directions where tracers are stretched. Stable and unstable manifolds are material curves that act as transport barriers, exhibiting the strongest local attraction, repulsion, or shear in the flow over a finite time interval [3]. In practice, LCSs are often identified as ridges in the field of Finite-Time Lyapunov Exponents (FTLE) [4], [5], [6], [7], [8].
FTLE is defined as the largest eigenvalue of the Cauchy–Green strain tensor of the flow map (see below for more details). The corresponding eigenvector is called the Finite-Time Lyapunov Vector. FTLE and FSLE are widely used to characterize transport and mixing processes in oceanographic flows, especially where the velocity field is only available as a finite dataset [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Recent studies have also demonstrated the potential of FTLE for tracer image assimilation in geophysical models [22], [23].
See also
[24] for a review of FTLE applications in detecting LCSs in other contexts.
FTLE and FSLE: Definitions and Properties#
FTLE is a local scalar measure of the rate at which initially neighbouring particles separate over a finite-time window \([t, t+T]\). The FTLE at a point \(x\) is defined as the growth rate of the norm of a small perturbation \(\delta\) initiated around \(x\) at time \(t\) and advected by the flow for a duration \(T\). Maximal stretching occurs when \(\delta\) aligns with the eigenvector associated with the largest eigenvalue \(\lambda_{\max}\) of the Cauchy–Green strain tensor \(\Delta = M^T M\), where \(M : x(t) \mapsto x(t+T)\) is the flow map of the advection system. This eigenvector is referred to as the forward Finite-Time Lyapunov Vector. The forward FTLE at \(x\) is then defined as:
\(FTLE(x) = \frac{\log(\lambda_{\max})}{2T}\).
FSLE is similar to FTLE, but the integration time \(T\) is not fixed a priori; instead, it is determined so that the particle separation reaches a prescribed final distance. The FSLE at point \(x\) for a separation distance \(d\) is defined as:
\(FSLE(x, d) = \frac{\log(\lambda_{\max}(d))}{2T(d)}\).
Seeding a domain with particles initially located on a grid enables the computation of discretized FSLE and FSLV fields.
Backward FSLE-Vs are defined in the same way, but with the time direction reversed in the advection equation. Ridges in backward FTLE (BFTLE) fields approximate attracting LCSs [25].
Backward FTLE fields reveal contours that correspond closely to major structures—such as filaments, fronts, and spirals—found in geophysical and biogeochemical tracer fields [10], [14], [15], [26]. The orientation of the gradient of a passive tracer has been shown to converge towards that of backward FTLVs in freely decaying two-dimensional turbulence [27]. This behaviour has also been observed in realistic oceanic flows and tracers [20].
Furthermore, [11] and [28] demonstrated, using real data, that the properties of FSLE and FTLE remain valid at the mesoscale, i.e., when the resolution of the velocity field—used to compute FTLE-V—is much lower than the resolution of the observed tracer field [20], [21].