1. Visualization Research Center University of Stuttgart On the Finite-Time Scope for Computing Lagrangian Coherent Structures from Lyapunov Exponents TopoInVis 2011 Filip Sadlo, Markus Üffinger, Thomas Ertl, Daniel Weiskopf VISUS - University of Stuttgart

2. Different Finite-Time Scopes Finite-Time Scope for LCS from Lyapunov Exponents2 Aletsch Glacier Image region:  5 km Flow speed:  100 m/y  Time scope:  10 9 s But: “same river”! Rhone in Lake Geneva Image region:  1 km Flow speed:  10 km/h  Time scope:  10 2 s Lagrangian coherent structures

3. LCS by Ridges in FTLE Lagrangian coherent structures (LCS) can be obtained as Ridges in finite-time Lyapunov exponent (FTLE) field FTLE = 1/|T| ln (  /  ) Lyapunov exponent (LE) LE = lim T  1/|T| ln (  /  ) LCS behave like material lines (advect with flow) Finite-Time Scope for LCS from Lyapunov Exponents3 Shadden et al. 2005  T T>0  repelling LCS T<0  attracting LCS

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 Finite-Time Scope for LCS from Lyapunov Exponents4 T = 0.5 s T = 3 s CFD example

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 Finite-Time Scope for LCS from Lyapunov Exponents5 T = 2 s T = 8 s Double gyre example

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) Finite-Time Scope for LCS from Lyapunov Exponents6 u u ? FTLE ridge

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 u i = (i 1 - i 0 ) /  t Require lim  t  0 u i = u( (i 0 + i 1 )/2, t +  t / 2 ) Finite-Time Scope for LCS from Lyapunov Exponents7 forw. FTLE ridge rev. FTLE ridge t t +  t  path line tt i0i0 i1i1 Find corresponding intersection: Advect i 0 (by path line) and get nearest intersection (i 1 ) Allow prescription of threshold on discrepancy  Problem: Accuracy of ridge extraction in order of FTLE sampling cell size  Ridge extraction error dominates for small  t

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  Finite-Time Scope for LCS from Lyapunov Exponents8 forw. FTLE ridge rev. FTLE ridge t t +  t  path line tt i0i0 i1i1 Find corresponding intersection: Advect i 0 (by path line) and get nearest intersection (i 1 ) Allow prescription of threshold on discrepancy 

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, T min and T max, 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 Exponents9

10. Example: Buoyant Flow with Obstacles Finite-Time Scope for LCS from Lyapunov Exponents10 T = 0.2 sT = 0.4 sT = 1.0 s 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

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 Exponents11

12. Thank you for your attention! Finite-Time Scope for LCS from Lyapunov Exponents12