On the Finite-Time Scope for Computing Lagrangian Coherent Structures from Lyapunov Exponents

1. On the Finite-Time Scope for Computing Lagrangian Coherent Structures fromLyapunov Exponents TopoInVis 2011 FilipSadlo, Markus Üffinger, Thomas Ertl, Daniel Weiskopf VISUS - University of Stuttgart

2. Different Finite-Time Scopes Lagrangian coherent structures Aletsch Glacier Image region:  5 km Flow speed:  100 m/y  Time scope:  109 s Rhone in Lake Geneva Image region:  1 km Flow speed:  10 km/h  Time scope:  102 s But: “same river”! Finite-Time Scope for LCS from Lyapunov Exponents

3. LCS by Ridges in FTLE • Lagrangian coherent structures (LCS)can be obtained asRidges in finite-time Lyapunov exponent (FTLE) field FTLE = 1/|T| ln ( / ) Lyapunov exponent (LE) LE = limT1/|T| ln ( / ) LCS behave like material lines (advect with flow) T>0  repelling LCS T<0  attracting LCS   T Shadden et al. 2005 Finite-Time Scope for LCS from Lyapunov Exponents

4. Finite-Time Scope: Upper Bound • “Time scope T can’t be too large” • For T : FTLE = LE • Well interpretable • But LCS tend to grow as T grows • Sampling problems & visual clutter • Upper bound is application dependent CFD example T = 3 s T = 0.5 s Finite-Time Scope for LCS from Lyapunov Exponents

5. Finite-Time Scope: Lower Bound • “Time scope T must not be too small” (for topological relevance) • For T 0: FTLE  major eigenvalue of (u + (u)T)/2 • Ridges of “instantaneous FTLE” cannot satisfy advection property • No transport barriers for too small T • Lower bound can be motivated by advection property Double gyre example T = 8 s T = 2 s Finite-Time Scope for LCS from Lyapunov Exponents

6. Testing Advection Property: State of the Art • Shadden et al. 2005 • Measure cross-flow of instantaneous velocity through FTLE ridges • Theorem 4.4: • Larger time scopes T better advection property • Sharper ridges  better advection property • But: zero cross-flow is necessary but not sufficient for advection property • Reason: tangential flow discrepancy not tested: • Problem: tangential speed of ridge not available (Ridges are purely geometric, not by identifiable particles that advect) FTLE ridge u ? u Finite-Time Scope for LCS from Lyapunov Exponents

7. Testing Advection Property • Our approach (only for 2D fields) • If both ridges in forward and reverse FTLE satisfy advection property,then also their intersections • Intersections represent identifiable points that have to advect • Approach 1: • Velocity of intersection ui = (i1- i0) / t • Require limt0ui = u( (i0+ i1)/2, t + t / 2 ) forw. FTLE ridge Find corresponding intersection: • Advecti0 (by path line) and get nearest intersection (i1) • Allow prescription of threshold on discrepancy   t i1 i0 path line rev. FTLE ridge t + t t Problem: • Accuracy of ridge extraction in order of FTLE sampling cell size • Ridge extraction error dominates for small t Finite-Time Scope for LCS from Lyapunov Exponents

8. Testing Advection Property • Our approach (only for 2D fields) • If both ridges in forward and reverse FTLE satisfy advection property,then also their intersections • Intersections represent identifiable points that have to advect • Approach 2: • Use comparably large t (several cells) and measure  • Analyze  for all intersections • We used average  forw. FTLE ridge Find corresponding intersection: • Advecti0 (by path line) and get nearest intersection (i1) • Allow prescription of threshold on discrepancy   t i1 i0 path line rev. FTLE ridge t + t t Finite-Time Scope for LCS from Lyapunov Exponents

9. Overall Method • A fully automatic selection of T is not feasible • Parameterization of FTLE visualization depends on goal, typically by trial-and-error • User selects sampling grid, filtering thresholds, Tmin and Tmax, etc. • Our technique takes over these parameters and provides • Plot • Local and global minima • Smallest T that satisfies prescribed discrepancy • … Finite-Time Scope for LCS from Lyapunov Exponents

10. Example: Buoyant Flow with Obstacles discrepancy in FTLE sampling cell size • Accuracy of ridge extraction in order of FTLE sampling cell size • Discrepancy can even grow with increasing T because ridges get sharper, introducing aliasing • LCS by means of FTLE ridges is highly sampling dependent,in space and time FTLE vs. advected repelling ridges (black) after t’ = 0.05 s T = 0.2 s T = 0.4 s T = 1.0 s Finite-Time Scope for LCS from Lyapunov Exponents

11. Conclusion • We presented a technique for • analyzing the advection quality w.r.t. to T • selecting T w.r.t. to a prescribed discrepancy • We confirmed findings of Shadden et al. 2005 • Advection property increases with increasing T and ridge sharpness • However, ridge extraction accuracy seems to be a major limiting factor • Needs future work on accuracy of height ridges • We only test intersections • Could be combined with Shadden et al. 2005 • Comparison of accuracy of both approaches • Extend to 3D fields Finite-Time Scope for LCS from Lyapunov Exponents

12. Thank you for your attention! Finite-Time Scope for LCS from Lyapunov Exponents