LCS Tool : A Computational Platform for Lagrangian Coherent Structures

1 Introduction

• LCSs organize trajectory patterns in non-autonomous dynamical systems and are applied across fluid, biological, aeronautical, celestial, crowd, and mechanical settings.Source: §1 · p. 1
• FTLE ridges are heuristic indicators of hyperbolic LCSs, but treating them as LCSs can produce theoretical inconsistencies and false positives or negatives.The role of FTLE in detecting elliptic and parabolic LCSs remains an open question.Source: §1 · p. 1
• Recent variational formulations characterize hyperbolic and parabolic LCSs through averaged material shear and elliptic LCSs through averaged strain, yielding null-geodesics of Lorentzian metrics.Source: §1 · pp. 1–2
• LCS Tool provides an algorithmic introduction to geodesic LCS detection and reviews a MATLAB function library for extracting LCSs from two-dimensional unsteady flows.The paper's examples are programmed as demonstration scripts.Source: §1 · p. 2
• Existing packages primarily generate FTLE plots for visual assessment of hyperbolic LCSs, whereas LCS Tool targets geodesic extraction of parametrized material curves and extends extraction to elliptic LCSs.Source: §1 · p. 2

2 Theory

• The framework considers finite-time, two-dimensional unsteady velocity fields and uses the flow map and its right Cauchy–Green strain tensor to describe Lagrangian strain.Source: §2 · pp. 2–3

2.1 Elliptic LCSs

• Elliptic LCSs are closed stationary curves of averaged tangential strain, computed as closed orbits whose subsets stretch uniformly by a factor λ.The outermost member of a closed family of λ-lines is a coherent Lagrangian vortex boundary.Source: §2.1 · p. 3

2.2 Hyperbolic LCSs

• Hyperbolic LCSs are stationary curves of averaged shear; repelling shrinklines and attracting stretchlines are eigenvector-field orbits and null-geodesics of a Lorentzian metric.Source: §2.2 · p. 4
• Normal perturbations to shrinklines grow by λ2, while those to stretchlines decay by λ1 under the flow map.Source: §2.2 · p. 4

3.1 Computing the invariants of the Cauchy-Green strain tensor

• The common first step for calculating elliptic and hyperbolic LCSs is computing the Cauchy-Green strain tensor field; four symmetrically placed auxiliary points around each Cartesian-grid point increase finite-difference accuracy.Experience suggests auxiliary-grid spacing of 1–10% of main-grid spacing. Vectorized trajectory integration typically takes five to ten minutes with default error tolerances, though memory requirements may become excessive at high resolutions.Source: §3.1 · pp. 5–6

3.1.1 Special case: incompressible velocity fields

• For incompressible flows, λ1λ2 = 1, so λ1 and its eigenvector can be derived after computing λ2 and its eigenvector.The passage reports that this calculation order gives higher accuracy than the reverse order.Source: §3.1.1 · p. 7

3.1.2 Special case: data-defined velocity fields

• Dataset-defined velocity fields require spatial and temporal interpolation so velocity can be evaluated at arbitrary positions and times.The article points to its ocean-dataset example for possible interpolation functions.Source: §3.1.2 · pp. 7–8

3.2 Computing elliptic LCSs

• Elliptic boundaries are found by integrating λ-lines from Poincaré sections, locating closed returns, and selecting the outermost closed orbit.Section placement is manual; the λ-line field is monitored and reoriented to correct orientation discontinuities before interpolation.Source: §3.2 · pp. 8, 11

3.3 Computing hyperbolic LCSs

• Repelling shrinklines start at local maxima of λ2, while attracting stretchlines start at local minima of λ1; integrations stop at the maximum length or domain boundary.The algorithm suppresses nearby remaining λ2 maxima as later seeds and removes shrinkline segments inside elliptic LCSs by default.Source: §3.3 · pp. 11–12
• Repelling hyperbolic LCSs continue to affect tracer patterns inside the elliptic LCS.The illustration encodes tracer x1(t)-positions in red and x2(t)-positions in green.Source: §3.3 · p. 13

4.1 Double gyre

• The double-gyre demo uses FTLE-guided Poincaré sections and a λ sweep, retaining the outermost closed orbit as the Lagrangian vortex boundary.It varies λ from 0.93 to 1.07 in increments of 0.01.Source: §4.1 · pp. 14–15
• For the double gyre, larger stretchline than shrinkline maximization distances separate LCSs and help balance their counts because shrinklines track λ2 ridges while stretchlines are normal to them.Equal distances would produce more stretchlines than shrinklines overall.Source: §4.1 · p. 16

4.2 Bickley jet

• The Bickley-jet demo uses chaotic Duffing-attractor forcing, periodic x-boundaries, and five Poincaré sections to identify coherent eddies and display elliptic and hyperbolic LCSs.The closed-orbit λ range is [0.80, 1.20] with step 0.01.Source: §4.2 · p. 20, Figure 7 · p. 23

4.3 Ocean velocity data from satellite altimetry

• For satellite-derived South Atlantic velocities, discrete in space and time, LCS Tool uses spline interpolation to evaluate the velocity field for particle integration.The velocity data have 1/4° spatial and 7-day temporal resolution.Source: §4.3 · p. 24
• The ocean-flow demo integrates for 30 days on a 400 × 400 initial-condition grid, computes elliptic LCSs over λ = [0.90, 1.10] in 0.02 steps, and also computes hyperbolic LCSs.The tracer grid resolution is 15 times higher than that of the velocity field; incompressibility is enforced and eigenvalues are computed from an auxiliary grid.Source: §4.3 · p. 26
• Five coherent eddy boundaries are found, including a large eddy at (3, −32) with a non-stretching boundary (λ = 1.00) and four smaller cores.The large eddy corresponds to eddy #2 in Figure 3 of; the smaller cores do not remain coherent for 90 days.Source: §4.3 · p. 26, Figure 8 · p. 27
• Hyperbolic LCSs determine fluid deformation between coherent eddy cores.Repelling LCSs are shown in red and attracting LCSs in blue.Source: §4.3 · p. 26, Figure 8 · p. 27
• Figure 8 shows elliptic LCSs with λ values 1.00, 1.08, 0.94, 0.90, and 1.06, with repelling and attracting LCSs overlaid on FTLE values.The analysis uses T = 30 days rather than 90 days in to avoid length tangling of hyperbolic LCSs; all closed orbits are stable under increased spatial resolution.Source: §4.3 · p. 26, Figure 8 · p. 27

• LCS Tool implements recent variational results for LCSs in two-dimensional unsteady flows and is demonstrated on two analytic models and a geophysical velocity dataset.Source: §5 · p. 28
• The authors hope future computational advances will make LCS Tool more suitable for large-scale real-time forecasting applications, including environmental-contaminant tracking.Source: §5 · p. 28
• The authors hope LCS Tool will support numerical implementation of geodesic theory for parabolic LCSs and variational theories for hyperbolic and elliptic LCSs in three-dimensional flows.Source: §5 · p. 28