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Rates of mixing in models of fluid flow - …rsturman/talks/leeds_new_rates.pdfChaotic mixing Decay...
Transcript of Rates of mixing in models of fluid flow - …rsturman/talks/leeds_new_rates.pdfChaotic mixing Decay...
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Rates of mixing in models of fluid flow
Rob Sturman
Department of MathematicsUniversity of Leeds
Leeds Fluids Seminar, 29 November 2012Leeds
Joint work with James Springham, now in Perth
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Introduction
Chaotic motion leads to exponential behaviour...
Topological entropy is computable (Thurston-Nielsenclassification theorem)Lower bound on the complexity.
[P. L. Boyland, H. Aref, & M. A. Stremler, JFM. 403, 277 (2000)]
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Introduction
...but walls seem to slow things down
“regions of low stretching which slow downmixing and contaminate the whole mixingpattern...”[Gouillart et al., PRE, 78, 026211 (2008)]Also:[Chernykh & Lebedev, JETP, 87(12), 682 (2008)][Lebedev & Turitsyn, PRE, 69, 036301, (2004)]
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Dynamical systems approach
Dynamical systems modelling fluids
Fluidsincompressible fluid someextra pointless stuffspatial/temporal periodicityfregion of unmixed(stationary) fluid‘chaotic advection’
Dynamical systemsinvertible, area-preservingdynamical systemDiscrete time map,f : M → Minvariant (periodic) setf (A) = Astretching & folding
horseshoe (topological)(non)uniformhyperbolicity (smoothergodic theory)
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Dynamical systems approach
Mixing
f is (strong) mixing if
limn→∞
µ(f n(A) ∩ B) = µ(A)µ(B)
Equivalently in functional form:
Cn(ϕ,ψ) =
∣∣∣∣∫ (ϕ ◦ f n)ψdµ−∫ϕdµ
∫ψdµ
∣∣∣∣→ 0
as n→∞ for scalar observables ϕ and ψ.At what rate does Cn decay to zero?
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Dynamical systems approach
Mixing
f is (strong) mixing if
limn→∞
µ(f n(A) ∩ B) = µ(A)µ(B)
Equivalently in functional form:
Cn(ϕ,ψ) =
∣∣∣∣∫ (ϕ ◦ f n)ψdµ−∫ϕdµ
∫ψdµ
∣∣∣∣→ 0
as n→∞ for scalar observables ϕ and ψ.At what rate does Cn decay to zero?
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
Stretching in alternating directions
Egg-beater flows
Blinking vortex
Pulsed source-sink mixers
Partitioned pipe mixer
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
Arnold Cat Map
+
A simple model of alternating shears givingchaotic dynamics on the 2-torus.
F (x , y) = (x + y , y),G(x , y) = (x , y + x)
H(x , y) = G ◦ F = (x + y , x + 2y)
DH = A =
(1 11 2
)is a hyperbolic matrix
and the Cat Map is uniformly hyperbolic.The Cat Map is strong mixing (not difficult toshow).
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
Arnold Cat Map
+
A simple model of alternating shears givingchaotic dynamics on the 2-torus.
F (x , y) = (x + y , y),G(x , y) = (x , y + x)
H(x , y) = G ◦ F = (x + y , x + 2y)
DH = A =
(1 11 2
)is a hyperbolic matrix
and the Cat Map is uniformly hyperbolic.The Cat Map is strong mixing (not difficult toshow).
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
Linked Twist Maps on the torus
Define annuli P and Q onthe torus T2 which inter-sect in region S.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
Linked Twist Maps on the torus
The horizontal annulusP has a horizontal twistmap....
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
Linked Twist Maps on the torus
F (x , y) = (x + f (y), y)
for points in P
F is the identity outsideP.
f (y) could be linear...
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
Linked Twist Maps on the torus
...or not, but must bemonotonic
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
Linked Twist Maps on the torus
After F , apply a verticaltwist
G(x , y) = (x , y + g(x))
to points in Q.
The combined mapH(x , y) = G ◦ F is thelinked twist map.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
LTMs on the plane
R\P R\Q
S = P ∩ Q
R\PR\Q
S = P ∩ Q
Or we can define on the plane (3-punctured disk), giving amodel of the Aref blinking vortex.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Simple models
A simple LTM
Linear twists on annuli half the width of the torus.F (x , y) = (x + 2y , y), G(x , y) = (x , y + 2x)
H(x , y) = G ◦ F (x , y)
LTMs as defined here are strong mixing (Pesin theory).
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Direct computation
Decay of correlations for the Cat Map
Assume w.l.o.g. that∫ψdµ = 0 and compute
Cn(ϕ,ψ) =
∫(ϕ ◦ Hn)ψdµ
Expanding (analytic) ϕ and ψ as Fourier series,
ϕ(x) =∑k∈Z2
akeik.x, ψ(x) =∑j∈Z2
bjei j.x,
Linearity and orthogonality means that
Cn(ϕ,ψ) =
∫ ∑k∈Z2
akeik.Anx∑j∈Z2
bjei j.xdx =∑k∈Z2
akb−kAn .
Since A is a hyperbolic matrix, the exponential growth of |kAn|together with the exponential decay of Fourier coefficientsyields (super)exponential decay of Cn.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Young Towers
Young Towers (Lai-Sang Young, 1999)
Young Towers give a procedure for computing decay ofcorrelations for more general systems (with some hyperbolicity).
Locate a set Λ with ‘good hyperbolic properties’ (actuallyhyperbolic product structure)
Run the system forwards and check when iterates of Λintersect Λ (in the right way)
Measure and record the return times for Λ
Obtain the statistical behaviour for the return time system
Pass the findings back to the original system
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Young Towers
Young Towers (Lai-Sang Young, 1999)
Construct return time function R : Λ→ Z+ and measureµ(x ∈ Λ|R(x) > n):
If∫
Rdm <∞, then H is ergodic
If∫
Rdm <∞ and gcd of R is 1, then H is strong mixing.
If
µ(R > n) =
O(n−α), α > 1 polynomial decay O(n−α+1)
Cθn, θ < 1 exponential decay C′θ′n
O(n−α), α > 2 central limit theorem holds
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Young Towers
Cat Map
v+v−
ΛChoose Λ to be anyrectangle with sidesaligned along stable andunstable eigenvectors.Iterate Λ under the CatMap until its imageintersects Λ.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Young Towers
Cat Map
v+v−
Λ
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Young Towers
Cat Map
v+v−
Λ
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Young Towers
Cat Map
R(x) = n1
R(x) = n1
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Young Towers
Linked Twist Maps
0 12 1
0
12
1
S = P ∩Q R\Q
R\P
Appears a perfect system toapply Young Towers.
But...local stable and unstablemanifolds only existµ-almost everywhere...... and Λ is defined as theintersection of suchmanifolds...... so must contain holes.In general fornon-uniformly hyperbolicsystems, constructing Λis hard.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Young Towers
Linked Twist Maps
0 12 1
0
12
1
S = P ∩Q R\Q
R\P
Appears a perfect system toapply Young Towers.But...
local stable and unstablemanifolds only existµ-almost everywhere...... and Λ is defined as theintersection of suchmanifolds...... so must contain holes.In general fornon-uniformly hyperbolicsystems, constructing Λis hard.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A related scheme
Chernov, Zhang, Markarian conditions
Study the return map to S and check:
SmoothnessHyperbolicityReturn map is mixingBounded distortionBounded curvatureAbsolute continuityAdmissible curves in the singularity setOne-step growth of unstable manifolds
This gives exponential decay for the return map. To getpolynomial decay for the original map, we need
infrequently returning points (Nightmare)
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A related scheme
Chernov, Zhang, Markarian conditions
Study the return map to S and check:
SmoothnessHyperbolicityReturn map is mixingBounded distortionBounded curvatureAbsolute continuityAdmissible curves in the singularity setOne-step growth of unstable manifolds
This gives exponential decay for the return map. To getpolynomial decay for the original map, we need
infrequently returning points (Nightmare)
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A related scheme
Chernov, Zhang, Markarian conditions
Study the return map to S and check:
Smoothness (Structure of the singularity set)Hyperbolicity (Easy)Return map is mixing (Surprisingly difficult)Bounded distortion (Trivial)Bounded curvature (Trivial)Absolute continuity (Easy)Admissible curves in the singularity set (Not too bad)One-step growth of unstable manifolds (Hard)
This gives exponential decay for the return map. To getpolynomial decay for the original map, we need to study
infrequently returning points (Nightmare)
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A related scheme
Chernov, Zhang, Markarian conditions
Study the return map to S and check:
Smoothness (Structure of the singularity set)Hyperbolicity (Easy)Return map is mixing (Surprisingly difficult)Bounded distortion (Trivial)Bounded curvature (Trivial)Absolute continuity (Easy)Admissible curves in the singularity set (Not too bad)One-step growth of unstable manifolds (Hard)
This gives exponential decay for the return map. To getpolynomial decay for the original map, we need
infrequently returning points (Nightmare)
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A related scheme
Chernov, Zhang, Markarian conditions
Study the return map to S and check:
Smoothness (Structure of the singularity set)Hyperbolicity (Easy)Return map is mixing (Surprisingly difficult)Bounded distortion (Trivial)Bounded curvature (Trivial)Absolute continuity (Easy)Admissible curves in the singularity set (Not too bad)One-step growth of unstable manifolds (Hard)
This gives exponential decay for the return map. To getpolynomial decay for the original map, we need
infrequently returning points (Nightmare)
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A related scheme
Decay of correlations for linear LTMs
Theorem (Springham & Sturman, ETDS, (2012))
For α-Hölder observables and for a linear LTM H,
Cn = O(1/n)
This is an upper bound on the worst behaviourCompare with the topological result of a lower bound onthe best behaviourAs it’s an upper bound we haven’t yet shown polynomialdecay rates actually happen. Cn could still decayexponentially fast.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A related scheme
Decay of correlations for linear LTMs
Theorem (Springham & Sturman, ETDS, (2012))
For α-Hölder observables and for a linear LTM H,
Cn = O(1/n)
This is an upper bound on the worst behaviourCompare with the topological result of a lower bound onthe best behaviourAs it’s an upper bound we haven’t yet shown polynomialdecay rates actually happen. Cn could still decayexponentially fast.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A simple computation
Contribution of the boundary
For LTMs we can compute explicitly the contribution to thedecay integral of a particular region near the boundary.
0 12 1
0
12
1
WnConsider all points whichtake n iterates to enter theoverlap S.These form wedge-shapedregions Bn.We will concentrate on Wn.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A simple computation
In =
∫Wn
(ϕ ◦ Hn)ψdµ
=43
∫ 1
12
∫ (1−y)/2n
0ϕ(x , y + 2nx)ψ(x , y)dxdy
Consider the related double integral
Jn =43
∫ 1
12
∫ (1−y)/2n
0ϕ(0, y + 2nx)ψ(0, y)dxdy .
Substitute t = y + 2nx :
Jn =2
3n
∫ 1
12
ψ(0, y)
∫ 1
yϕ(0, t)dtdy ∼ K/n
Finally show limn→∞ n|In − Jn| = 0, that is, the contributionmade to the correlation function by points near the boundary isasympotically the same as the contribution made by points atthe boundary.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A simple computation
In =
∫Wn
(ϕ ◦ Hn)ψdµ
=43
∫ 1
12
∫ (1−y)/2n
0ϕ(x , y + 2nx)ψ(x , y)dxdy
Consider the related double integral
Jn =43
∫ 1
12
∫ (1−y)/2n
0ϕ(0, y + 2nx)ψ(0, y)dxdy .
Substitute t = y + 2nx :
Jn =2
3n
∫ 1
12
ψ(0, y)
∫ 1
yϕ(0, t)dtdy ∼ K/n
Finally show limn→∞ n|In − Jn| = 0, that is, the contributionmade to the correlation function by points near the boundary isasympotically the same as the contribution made by points atthe boundary.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A simple computation
In =
∫Wn
(ϕ ◦ Hn)ψdµ
=43
∫ 1
12
∫ (1−y)/2n
0ϕ(x , y + 2nx)ψ(x , y)dxdy
Consider the related double integral
Jn =43
∫ 1
12
∫ (1−y)/2n
0ϕ(0, y + 2nx)ψ(0, y)dxdy .
Substitute t = y + 2nx :
Jn =2
3n
∫ 1
12
ψ(0, y)
∫ 1
yϕ(0, t)dtdy ∼ K/n
Finally show limn→∞ n|In − Jn| = 0, that is, the contributionmade to the correlation function by points near the boundary isasympotically the same as the contribution made by points atthe boundary.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A simple computation
In =
∫Wn
(ϕ ◦ Hn)ψdµ
=43
∫ 1
12
∫ (1−y)/2n
0ϕ(x , y + 2nx)ψ(x , y)dxdy
Consider the related double integral
Jn =43
∫ 1
12
∫ (1−y)/2n
0ϕ(0, y + 2nx)ψ(0, y)dxdy .
Substitute t = y + 2nx :
Jn =2
3n
∫ 1
12
ψ(0, y)
∫ 1
yϕ(0, t)dtdy ∼ K/n
Finally show limn→∞ n|In − Jn| = 0, that is, the contributionmade to the correlation function by points near the boundary isasympotically the same as the contribution made by points atthe boundary.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A cautionary note
A cautionary note
Now we have
Cn ∼ Kn
+
∫R\Bn
(ϕ ◦ Hn)ψdµ.
Certainly the contribution from the remaining integral is noslower than O(1/n) (from earlier).[Sturman & Springham, PRE, (2012)]So we see polynomial decay at rate 1/n
providing:1 K 6= 0, i.e., the contributions from the wedges do not
cancel each other out (do not choose non-genericsymmetric observables)
2 The decay rate of the remaining integral is not exactly−K/n to leading order.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A cautionary note
A cautionary note
Now we have
Cn ∼ Kn
+
∫R\Bn
(ϕ ◦ Hn)ψdµ.
Certainly the contribution from the remaining integral is noslower than O(1/n) (from earlier).[Sturman & Springham, PRE, (2012)]So we see polynomial decay at rate 1/n providing:
1 K 6= 0, i.e., the contributions from the wedges do notcancel each other out (do not choose non-genericsymmetric observables)
2 The decay rate of the remaining integral is not exactly−K/n to leading order.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A cautionary note
A cautionary note
Now we have
Cn ∼ Kn
+
∫R\Bn
(ϕ ◦ Hn)ψdµ.
Certainly the contribution from the remaining integral is noslower than O(1/n) (from earlier).[Sturman & Springham, PRE, (2012)]So we see polynomial decay at rate 1/n providing:
1 K 6= 0, i.e., the contributions from the wedges do notcancel each other out (do not choose non-genericsymmetric observables)
2 The decay rate of the remaining integral is not exactly−K/n to leading order.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
A cautionary note
A cautionary note
Now we have
Cn ∼ Kn
+
∫R\Bn
(ϕ ◦ Hn)ψdµ.
Certainly the contribution from the remaining integral is noslower than O(1/n) (from earlier).[Sturman & Springham, PRE, (2012)]So we see polynomial decay at rate 1/n providing:
1 K 6= 0, i.e., the contributions from the wedges do notcancel each other out (do not choose non-genericsymmetric observables)
2 The decay rate of the remaining integral is not exactly−K/n to leading order.
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Different boundary conditions
More general boundary behaviour
In a neighbourhood of the boundaries, replace linear twists with
F̃ (x , y) = (x + 2yp, y) and G̃(x , y) = (x , y + 2xp).
Then
Lemma ∫Bn
(ϕ ◦ H̃n)ψdµ ∼ Kn1/p
and so
Conjecture ∫LTM
(ϕ ◦ H̃n)ψdµ ∼ K ′
n1/p
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Different boundary conditions
Numerics — linear case
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Different boundary conditions
Numerics — linear case
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Different boundary conditions
Numerics — quadratic case
f̃ (y) = 1− cos−1(4y − 1)
π
g̃(x) = 1− cos−1(4x − 1)
π
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Different boundary conditions
Numerics — quadratic case
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Different boundary conditions
Numerics — maximum width of striations
Red: linear, O(1/n)
Blue: quadratic, O(1/n2)
Green: exponential de-cay of Cat Map
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Rigorous extensions
. Planar linked twist maps
R\P R\Q
S = P ∩ Q
R\PR\Q
S = P ∩ Q
. Introduce curvature to the Young Tower argument
. Combine with topological ideas to get a more detailed pictureof mixing
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Non-rigorous ideas
Young Tower method is practically tractableDon’t work in Fourier spaceDon’t need to think about observablesCan measure return timesCan ignore (maybe) fine details of the mathematics
Residence-time distributionsCompute return times for more realistic modelsConsequences of presence of islandsConnection with lobe dynamics
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Do it with data
Can the non-rigorous ideas be turned into a practical schemefor understanding data?Fundamental ingredients:
Periodicity (but what is the consequence of randomforcing?)‘Good’ hyperbolic region (product structure?)Trajectories which repeatedly return to the hyperbolicregion ‘in the right way’ (dynamic renewal?)Method of recording return timesAtmospheric or oceanographic?
Chaotic mixing Decay of correlations Achieving the upper bound Future directions
Singularity set for HS