Mixing with Ghost Rods
Jean-Luc Thiffeault
Matthew Finn
Emmanuelle Gouillart
http://www.ma.imperial.ac.uk/˜jeanluc
Department of Mathematics
Imperial College London
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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)]
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The Connection with Braiding
6
t
PSfrag replacements
σ1
PSfrag replacements
σ−2
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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
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Poincaré Section
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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
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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
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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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