Visualization Research Center University of Stuttgart

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

TopoInVis 2011Filip Sadlo, Markus Üffinger, Thomas Ertl, Daniel WeiskopfVISUS - University of Stuttgart

Finite-Time Scope for LCS from Lyapunov Exponents 2

Different Finite-Time Scopes

Aletsch GlacierImage region: 5 kmFlow speed: 100 m/y Time scope: 109 s

But: “same river”!

Rhone in Lake GenevaImage region: 1 kmFlow speed: 10 km/h Time scope: 102 s

Lagrangian coherent structures

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 = lim T 1/|T| ln ( / )

LCS behave like material lines (advect with flow)

Shadden et al. 2005

T

T>0 repelling LCST<0 attracting LCS

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

T = 0.5 s T = 3 s

CFD

exam

ple

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

T = 2 s T = 8 s

Doub

le g

yre

exam

ple

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)

u

u

?FTLE ridge

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 limt0 ui = u( (i0 + i1)/2, t + t / 2 )

forw. FTLE ridge

rev. FTLE ridget t + t

path lineti0 i1

Find corresponding intersection:• Advect i0 (by path line) and get

nearest intersection (i1)• 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

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

rev. FTLE ridget t + t

path lineti0 i1

Find corresponding intersection:• Advect i0 (by path line) and get

nearest intersection (i1)• Allow prescription of threshold on

discrepancy

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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

discre

pancy

in FT

LE cel

l size

FTLE advection time T

average discrepancyminimum discrepancy

intersections count

T = 0.2 s T = 0.4 s T = 1.0 s

disc

repa

ncy

in F

TLE

sam

plin

g ce

ll siz

e

• 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

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!