Mixing with Ghost Rods

Jean-Luc Thiffeault

Matthew Finn

Emmanuelle Gouillart

http://www.ma.imperial.ac.uk/˜jeanluc

Department of Mathematics

Imperial College London

Mixing with Ghost Rods – p.1/19

Experiment of Boyland, Aref, & Stremler

[P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)]

[P. L. Boyland, M. A. Stremler, and H. Aref, Physica D 175, 69 (2003)]

Mixing with Ghost Rods – p.2/19

The Connection with Braiding

6

t

PSfrag replacements

σ1

PSfrag replacements

σ−2

Mixing with Ghost Rods – p.3/19

Generators of the n-Braid GroupPSfrag replacementsσ−1

i

σi

i − 1 i + 1 i + 2iPSfrag replacements

σ−1

i

σi

i − 1 i + 1 i + 2i

A generator

σi , i = 1, . . . , n − 1

is the clockwise interchange ofthe i th and (i + 1)th rod.The generators obey the presen-tation

σi+1 σi σi+1 = σi σi+1 σi

σiσj = σjσi, |i − j| > 1

These generators are used to characterise the motion of the rods.

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The Two BAS Stirring Protocols

σ1σ2 protocol

σ−1

1σ2 protocol

[P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)]Mixing with Ghost Rods – p.5/19

Topological Entropy of a Braid

Practically speaking, the topological entropy of a braid is a lowerbound on the line-stretching exponent of the flow!This is reasonable:

I II

σ2=⇒ II’

I’

In practice, we compute the topological entropy of a braid using atrain-track algorithm due to Bestvina & Handel. The end result isa transition matrix showing the how each edge is mapped underthe action of the braid.

[M. Bestvina and M. Handel, Topology 34, 109 (1995)]

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The Difference between BAS’s Two Protocols

• The matrices associated the generators have eigenvalues onthe unit-circle (but their product doesn’t necessarily).

• The first (bad) stirring protocol has eigenvalues on the unitcircle

• The second (good) protocol has largest eigenvalue(3 +

√5)/2 = 2.6180.

• So for the second protocol the length of a line joining therods grows exponentially!

• That is, material lines have to stretch by at least a factorof 2.6180 each time we execute the protocol σ−1

1σ2.

• This is guaranteed to hold in some neighbourhood of the rods(Thurston–Nielsen theorem).

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One Rod Mixer: The Kenwood Chef

Mixing with Ghost Rods – p.8/19

Poincaré Section

Mixing with Ghost Rods – p.9/19

Stretching of Lines

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Particle Orbits are Topological Obstacles

Choose any fluid particle orbit (green dot).

Material lines must bend around the orbit: it acts just like a “rod”![J-LT, Phys. Rev. Lett. 94, 084502 (2005)]

Today: focus on periodic orbits.

How do they braid around each other?Mixing with Ghost Rods – p.11/19

Motion of Islands

Make a braid from the motion ofthe rod and the periodic islands.Most (74%) of the topologicalentropy is accounted for by thisbraid.

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Motion of Islands and Unstable Periodic Orbits

Now we also include unsta-ble periods orbits as well asthe stable ones (islands).Almost all (99%) of the topo-logical entropy is accountedfor by this braid.

But are the periodic orbits re-ally “ghost rods”?That is, do material lines re-ally get out of their way?

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Periodic Orbits as Ghost Rods

[movie 3: sf_periodic.avi]

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Blowup of the Tip

0.06 0.07 0.08 0.09 0.10.47

0.48

0.49

0.5

0.51

0.52

Mixing with Ghost Rods – p.15/19

So Which Orbits Make Good Rods?

A good ghost rod should “look” like a real rod: material lineswrap around it.

Look at linearisation of the period-1 map:Two eigenvalues Λ and Λ−1 (Floquet multipliers),with |Λ| ≥ |Λ|−1.

Λ complex Elliptic Good rod (obvious).|Λ| > 1 Hyperbolic Crap rod.Λ = 1 Parabolic Good rod?

The parabolic points are the most interesting: they are associatedwith the crucial property of folding.

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Normal Form near a Parabolic Point

X ′ = X + quadratic termsY ′ = Y + quadratic terms

or

X ′ = X + Y + X2

Y ′ = Y + X2

The first of these does not lead to folding.The coefficient of X2 determines the “size” of the rod, which is afunction of time.[movie 4: folditer.avi]

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Curvature of the Tip at the Periodic Orbit

0 5 10 150

2000

4000

6000

8000

10000

12000

PSfrag replacements

curv

atur

e

periods

slope = (3/2)√

3π5

Mixing with Ghost Rods – p.18/19

Conclusions

• Topological chaos involves moving obstacles in a 2D flow,which create nontrivial braids.

• Periodic orbits make perfect obstacles (in periodic flows).• This is a good way to “explain” the chaos in a flow —

accounts for stretching of material lines.• Islands (elliptic orbits) look just like rods, and parabolic

orbits can look a lot like rods.• No need for infinitesimal separation of trajectories or

derivatives of the velocity field — this is an inherently globaldescription.

• The size of the rods is important — for islands this is obviousbut for parabolic points the apparent size is a function of time.

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