finite-time braiding exponents

Department of Mathematics

SIAM Conference on Applications of Dynamical Systems Snowbird, UT, May 2015

Marko Budii Jean-Luc Thiffeault

FINITE-TIME BRAIDING EXPONENTS (ARXIV: 1502.02162)

Supported by NSF grants DMS-0806821 (JLT) and CMMI-1233935 (MB, JLT)

Budii, Thiffeault: Finite-Time Braiding ExponentsMay 20, 2015

Complexity of the material transport

2

Complexity of the flow rate of growth of material interfaces. Chaotic advection increases the rate of diffusion and/or reactions.

Newhouse, Pignataro (1993): Fastest growth of material line is given by the topological entropy of the flow.

Growth of material lines is computed by front tracking requires velocity fields and delicate numerical algorithms.

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(a) (b)

FIG. 2 (a) Numerical simulation of dye advection for a wallrotating at velocity T = 0.2, with the rod moving in afigure-eight pattern. (b) Poincare section, which shows a largechaotic region and closed orbits near the wall, with a separa-trix in between (From Thieault et al. (2011).) (** Removex) [please explain what needs removing here.]

Compare this to the exponential case, Eq. (8): the decayof concentration variance is now algebraic (1/t2), with alogarithmic correction. The form (12) has been verifiedin experiments and using a simple map model (Gouillartet al., 2008, 2007).

E. Dynamics near a no-slip moving wall

Now consider the case of a rotating wall, where weadd a constant speed > 0 to the velocity u in (10).Again we look for fixed points: all the parabolic fixedpoints on the wall have disappeared, as well as the twoseparatrices. Since u1() is continuous, has two zeros,and u01(0) > 0, u1() must have a minimum at someangle , where u01() = 0 and hence v(, y) = 0 forall y. Enforcing that the along-wall velocity also van-ish, there will a fixed point at y = /u1(). Nowwe look at the linearised dynamics near the fixed point.Let (, y) = ( +,/u1() + Y ); then

= u1()Y +O2, Y 2,Y

, (13a)

Y = 12u001()y2 +O2, Y 2,Y

. (13b)

The linearized motion thus has eigenvalues = =pu001()/2u1()U where the argument in the squareroot is non-negative since u1() < 0 and u001() 0.For u001() > 0 and > 0, this is a hyperbolic fixedpoint, and the approach along its stable manifold is givenby Y (t) Y0 exp( t) for (0, Y0) initially on the stablemanifold. Compare this to the 1/t approach for a fixedwall: the approach to the fixed point is now exponential,at a rate proportional to the speed of rotation of the wall.One expects that this exponential decay will dominate ifit is slower than the mixing rate in the bulk. Otherwise,if is large enough, then the rate of mixing in the bulkdominates.

(a) (b)

FIG. 3 The epitrochoid stirring protocol. (a) Experiment;(b) Poincare section, also showing the rods trajectory. Closedorbits are present near the wall, even though the wall is fixed.(From Thieault et al. (2011).)

Figure 2(a) shows a numerical simulation of the flowpattern for a wall rotating at a rate T = 0.2. Thehyperbolic fixed point is indicated by a dot, as is thedistance d(t) between the mixing pattern and the hy-perbolic point. Note the unmixed region between therotating wall and the mixing pattern. Figure 2(b) is aPoincare section that shows the presence of the unmixedregion, which consists of closed orbits. Numerical simu-lations (Thieault et al., 2011) have confirmed that thedecay rate of a passive scalar in the central region is in-deed exponential, so the rotating wall can help recoverexponential mixing. However, the price to pay is thatthere is now an unmixed region surrounding the regionof good mixing. Whether this is a price worth payingdepends on the specific application.Another strategy for mimicking a moving wall, and

thus recover exponential mixing, is to move the rod in alooping motion, as shown in Fig. 3. This motion cre-ates closed trajectories near the wall, as is evident in thePoincare section Fig. 3(b). Thieault et al. (2011) haveverified experimentally the the decay of the passive scalarin this case is indeed exponential, for the same reason asthe moving wall. However, the analysis of the near-wallmap for this system is more complicated than for a mov-ing wall and has not been carried out. Three-dimensionaleects also remain to be investigated: these could holdsome surprises, since the nature of separatrices at thewall is potentially much richer.

III. THE NEW FRONTIER: 3D UNSTEADY FLOW

A. Introduction

1. Motivation and background

Lagrangian methods have proven their worth in trans-port studies on deterministic flows. This flow class en-compasses a vast field, with length scales ranging frommicrons to hundreds of kilometers, and includes sys-


Sparse Lagrangian Sampling

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(WHOI)

(Rocky DEM) (Wilderness Films India Ltd.)

What if we have no access to velocity fields? Lagrangian trajectories the only

data available: ocean drifters, granular flows, crowds Fast characterization of mixing: no need to re-seed tracer computation is cheaper


Trajectories are represented by braids.

4

60 40 20

40

50

60

70

Latitude

Long

itude

Spaghetti plot of planar trajectories

050

100

4050

607060

40

20

TimeLongitudeLa

titud

e

Trajectories in space-time, physical braid

0 50 10040

50

60

70

Time

Latit

ude

Project onto a plane and monitor exchanges

Braid encodes trajectory crossings as symbols (generators).

k 1k

k

k + 1

k

k + 1

2 211 2

11

Crossing

Topology retained, geometry discarded.


Braids of punctured disks in dynamical systems

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More generally: braids are labels for flow maps (homeomorphisms) We use Artin generators: easy to deduce from data (other generator

sets are also useful) braids form a non-commutative group with additional relations

1

2

1 2

1

23

1

1

3

Partial commutativityBraid relation

Takeaway: Number of generators is not a good measure of complexity.

111 1

11 = e


Advected material is represented by loops rubber bands.

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2 211 2

11

11

2

Reduced modelFull modelBraids of N trajectoriesNon-autonomous ODEsDynamicsLoops pulled tightDetailed curvesMaterial

Rate of mixing Material line growth Braid entropy?

P/w linear maps on Front/interface trackingAdvection[Dynnikov, 2002] [Hall, Yurtas, 2009]

R2N4


Topological entropy of a braid (braid entropy)

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Topological entropy: exponential rate of loop growth under iterated braid.

Loop length: # of intersections with the horizontal

b = 11|`|, |b`|, . . . , |bn`| eh(b)n

Braid t. entropy Flow t. entropy independent of the choice of trajectories

Specifying stirrer trajectories according to a high-entropy braid, forces the increase in flow entropy. (Aref, Boyland, Finn, Stremler, Thiffeault,)

h(b) = limn!1

1

nlog |bn`|


Finite-Time Braiding Exponent

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Why not stick with braid entropy?

iteration of a braid justified only for periodic trajectories note: avoiding iteration removes the

need for braid closure coarse: non-exponential regions all

assigned zero braid entropy

Duration of recorded data instead of iterate no.

FTBE =1

Tlog(|bT `|/|`|)

Whats in a name? FTBE to braid entropy is parallel to FTLE relative to Lyapunov exponents.


Dependence of FTBE on parameters

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2 211 2

110

50100

4050

607060

40

20

TimeLongitude

Latit

ude

FTBE

number N of strands sampled duration T of trajectories loop used to compute FTBE locations of initial conditions

trajectory integration step angle of projection of trajectories

(rotational frame of trajectories)

Significant parameters: Nuisance parameters:

FTLyapE is a scalar field over the flow domain. FTBraidE depends on N>1 initial conditions.

Thiffeault (2005), (2010) has preliminary statistics


Numerical study: mixing Aref Blinking Vortex

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Circular domain, periodic alternation between two integrable Rankine vortices

Circulation varied within mixing regime. Mixing zone is the entire domain.

Classical expectations from ergodic theory: locations of initial conditions forgotten with time flow entropy depends on mag. of circulationBraids (and FTBEs) are not classical observables they live on a configuration space of particles, instead of the state space.


In mixing flows, initial conditions quickly stop mattering.

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at each n we seed 100 n-tuples of trajectories compute mean and relative standard deviation

Trajectory length T0 50 100 150 200 250 300

MeanFTBE

0

1

2

3

4

5

n = 5

n = 20

n = 100

Trajectory length T10

010

110

210

3

RSD

ofFTBE

10-3

10-2

10-1

100

n = 5

n = 20

n = 100

1/2 slope

Mean settles quickly

Limits are different

Variance decay consistent with Central Limit Theorem (mixing dynamics)

Mixing CLT carries through braid & FTBE calculations. What is the significance of the value of mean FTBE.


Mean FTBE flow entropy trajectories added.

12

Initial conditions drawn uniformly at random (100 ensembles for each point). All trajectories long enough for mean FTBE to stabilize.

n strands

0 100 200 300 400 500

MeanFTBE

0

1

2

3

= 3

= 6

= 9

Topological entropy h1 1.5 2 2.5 3

MeanFTBE

0.5

1

1.5

2

2.5

3

n=10

n=75

n=200

n=500

} Mean FTBE approaches flow entropy as strands added

Spatial mean FTBE colors are different circulation strengths

Topological entropy estimated by material advection.

Limit mean FTBEs and flow entropy correlate across different circulations.

FTBE2.5 2.505 2.51 2.515 2.52

Frac.

ofsamples

0

0.2

0.4

0.6

Indep. braids

Subbraids


Rebraiding saves simulation time.

13n strands

0 100 200 300 400 500

MeanFTBE

0

1

2

3

Indep. braids

Subbraids

Top. Entropy

n strands10

010

110

210

3

RSD

ofFTBE

10-4

10-3

10-2

10-1

100

Indep. braids

Subbraids

1 slope

Indepedent sampling: Each braid gets its fresh set of simulated trajectories.

Resampling: From a pool of strands (gray) we choose a subset (red) to form into braid.

N trajectories: 1 N-strand braid

2N

N

4N

2N trajectories:

N-strand braids

Mean FTBE reproduced. Variance decay reproduced. FTBE dist. at worst pointRebraiding of 550 strands vs. naive sampling using 150k strands (200x more!):

Initial conditions


Finite-Time Braiding Exponents

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arXiv: 1502.02162 Try it out: braidlab MATLAB toolbox

https://github.com/jeanluct/braidlab

FTBEs robust to braid construction parameters (sampling, rotated frame)

FTBEs topological entropy as # strands increased

rebraiding: Lagrangian data can be reused to compute statistics

Goal: characterize planar flows using sparse Lagrangian trajectories

Objective: establish consistency between discrete braids and continuous flow

Future: couple FTBE calculation with detection of coherent subbraids (Allshouse, Thiffeault)

Experiments (w/Peacock Lab MIT), theory

finite-time braiding exponents

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