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


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)]


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)]


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)



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?



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?


Simple models

Stretching in alternating directions

Egg-beater flows

Blinking vortex

Pulsed source-sink mixers

Partitioned pipe mixer


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).


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).


Simple models

Linked Twist Maps on the torus

Define annuli P and Q onthe torus T2 which inter-sect in region S.


Simple models


The horizontal annulusP has a horizontal twistmap....


Simple models


F (x , y) = (x + f (y), y)

for points in P

F is the identity outsideP.

f (y) could be linear...


Simple models


...or not, but must bemonotonic


Simple models


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.


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.


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).


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.


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


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


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.


Young Towers

Cat Map

v+v−

Λ


Young Towers

Cat Map

v+v−

Λ


Young Towers

Cat Map

R(x) = n1

R(x) = n1


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.


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.


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)


A related scheme



SmoothnessHyperbolicityReturn map is mixingBounded distortionBounded curvatureAbsolute continuityAdmissible curves in the singularity setOne-step growth of unstable manifolds




A related scheme



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



A related scheme







A related scheme







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.


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.


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.



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.



In =

∫Wn

(ϕ ◦ Hn)ψdµ

=43

∫ 1

12

∫ (1−y)/2n



Jn =43

∫ 1

12

∫ (1−y)/2n

0ϕ(0, y + 2nx)ψ(0, y)dxdy .


Jn =2

3n

∫ 1

12

ψ(0, y)

∫ 1





In =

∫Wn

(ϕ ◦ Hn)ψdµ

=43

∫ 1

12

∫ (1−y)/2n



Jn =43

∫ 1

12

∫ (1−y)/2n

0ϕ(0, y + 2nx)ψ(0, y)dxdy .


Jn =2

3n

∫ 1

12

ψ(0, y)

∫ 1





In =

∫Wn

(ϕ ◦ Hn)ψdµ

=43

∫ 1

12

∫ (1−y)/2n



Jn =43

∫ 1

12

∫ (1−y)/2n

0ϕ(0, y + 2nx)ψ(0, y)dxdy .


Jn =2

3n

∫ 1

12

ψ(0, y)

∫ 1




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.


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)



A cautionary note

A cautionary note

Now we have

Cn ∼ Kn

+

∫R\Bn

(ϕ ◦ Hn)ψdµ.





A cautionary note

A cautionary note

Now we have

Cn ∼ Kn

+

∫R\Bn

(ϕ ◦ Hn)ψdµ.





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



Numerics — linear case



Numerics — linear case



Numerics — quadratic case

f̃ (y) = 1− cos−1(4y − 1)

π

g̃(x) = 1− cos−1(4x − 1)

π



Numerics — quadratic case



Numerics — maximum width of striations

Red: linear, O(1/n)

Blue: quadratic, O(1/n2)

Green: exponential de-cay of Cat Map


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


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


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?


Singularity set for HS

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