Rates of Mixing in Models of Fluid Flow

Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Rates of mixing in models of ﬂuid ﬂow

Rob Sturman

Department of Mathematics University of Leeds

Leeds Fluids Seminar, 29 November 2012 Leeds 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-Nielsen classiﬁcation 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 down mixing and contaminate the whole mixing pattern...” [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

Fluids Dynamical systems incompressible ﬂuid some invertible, area-preserving extra pointless stuff dynamical system spatial/temporal periodicity Discrete time map, f f : M M → region of unmixed invariant (periodic) set (stationary) ﬂuid f (A) = A ‘chaotic advection’ stretching & folding horseshoe (topological) (non)uniform hyperbolicity (smooth ergodic theory) Equivalently in functional form: Z Z Z n Cn(ϕ, ψ) = (ϕ f )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

lim µ(f n(A) B) = µ(A)µ(B) n→∞ ∩ Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Dynamical systems approach Mixing

f is (strong) mixing if

lim µ(f n(A) B) = µ(A)µ(B) n→∞ ∩

Equivalently in functional form: Z Z Z n Cn(ϕ, ψ) = (ϕ f )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 Pulsed source-sink mixers

Blinking vortex Partitioned pipe mixer 1 1 DH = A = is a hyperbolic matrix 1 2 and the Cat Map is uniformly hyperbolic. The Cat Map is strong mixing (not difﬁcult to show).

Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Simple models Arnold Cat Map

A simple model of alternating shears giving chaotic 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) + ◦ Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Simple models Arnold Cat Map

A simple model of alternating shears giving chaotic 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) + ◦ 1 1 DH = A = is a hyperbolic matrix 1 2 and the Cat Map is uniformly hyperbolic. The Cat Map is strong mixing (not difﬁcult to show). Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Simple models Linked Twist Maps on the torus

Deﬁne annuli P and Q on the torus T2 which intersect in region S. Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Simple models Linked Twist Maps on the torus

The horizontal annulus P has a horizontal twist map.... 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 outside P.

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 be monotonic Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Simple models Linked Twist Maps on the torus

After F, apply a vertical twist

G(x, y) = (x, y + g(x))

to points in Q.

The combined map H(x, y) = G F is the linked twist map.◦ Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Simple models LTMs on the plane

S = P Q ∩

R P R Q R P R Q \ \ \ \

S = P Q ∩

Or we can deﬁne on the plane (3-punctured disk), giving a model 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 R ψdµ = 0 and compute Z n Cn(ϕ, ψ) = (ϕ H )ψdµ ◦ Expanding (analytic) ϕ and ψ as Fourier series,

X ik.x X ij.x ϕ(x) = ake , ψ(x) = bje , 2 2 k∈Z j∈Z Linearity and orthogonality means that Z X ik.Anx X ij.x X Cn(ϕ, ψ) = ake bje dx = akb−kAn . 2 2 2 k∈Z j∈Z k∈Z Since A is a hyperbolic matrix, the exponential growth of kAn together with the exponential decay of Fourier coefﬁcients| | yields (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 of correlations for more general systems (with some hyperbolicity).

Locate a set Λ with ‘good hyperbolic properties’ (actually hyperbolic product structure)

Run the system forwards and check when iterates of Λ intersect 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 measure µ(x Λ R(x) > n): → ∈ | If R Rdm < , then H is ergodic ∞ If R Rdm < and gcd of R is 1, then H is strong mixing. ∞ If

 (n−α), α > 1 polynomial decay (n−α+1)   O O  µ(R > n) = Cθn, θ < 1 exponential decay C0θ0n    (n−α), α > 2 central limit theorem holds O Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Young Towers Cat Map

Choose Λ to be any Λ rectangle with sides aligned along stable and unstable eigenvectors. Iterate Λ under the Cat Map until its image v+ intersects Λ. v− Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Young Towers Cat Map

v+ v− Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Young Towers Cat Map

v+ v− Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Young Towers Cat Map

R(x) = n1

R(x) = n1 But... local stable and unstable manifolds only exist µ-almost everywhere...... and Λ is deﬁned as the intersection of such manifolds...... so must contain holes. In general for non-uniformly hyperbolic systems, constructing Λ is hard.

Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Young Towers Linked Twist Maps

Appears a perfect system to apply Young Towers.

R\P

1 2

S = P ∩ Q R\Q

0 1 0 2 1 Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Young Towers Linked Twist Maps

Appears a perfect system to apply Young Towers. But... 1 local stable and unstable R\P manifolds only exist µ-almost everywhere... 1 2 ... and Λ is deﬁned as the intersection of such S = P ∩ Q R\Q manifolds...... so must contain holes. 0 1 In general for 0 2 1 non-uniformly hyperbolic systems, constructing Λ is hard. Smoothness Hyperbolicity Return map is mixing Bounded distortion Bounded curvature Absolute continuity Admissible curves in the singularity set One-step growth of unstable manifolds This gives exponential decay for the return map. To get polynomial 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: 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 Hyperbolicity Return map is mixing Bounded distortion Bounded curvature Absolute continuity Admissible curves in the singularity set One-step growth of unstable manifolds This gives exponential decay for the return map. To get polynomial 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 get polynomial decay for the original map, we need to study infrequently returning points (Nightmare) 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 get polynomial decay for the original map, we need 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 get polynomial decay for the original map, we need infrequently returning points (Nightmare) This is an upper bound on the worst behaviour Compare with the topological result of a lower bound on the best behaviour As it’s an upper bound we haven’t yet shown polynomial decay rates actually happen. Cn could still decay exponentially 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 = (1/n) O 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 = (1/n) O This is an upper bound on the worst behaviour Compare with the topological result of a lower bound on the best behaviour As it’s an upper bound we haven’t yet shown polynomial decay rates actually happen. Cn could still decay exponentially 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 the decay integral of a particular region near the boundary. 1

Consider all points which Wn take n iterates to enter the 1 2 overlap S. These form wedge-shaped regions Bn. We will concentrate on Wn.

0 1 0 2 1 Consider the related double integral 4 Z 1 Z (1−y)/2n Jn = ϕ(0, y + 2nx)ψ(0, y)dxdy. 3 1 2 0 Substitute t = y + 2nx: 2 Z 1 Z 1 Jn = ψ(0, y) ϕ(0, t)dtdy K /n 3n 1 ∼ 2 y

Chaotic mixing Decay of correlations Achieving the upper bound Future directions

A simple computation

Z n In = (ϕ H )ψdµ Wn ◦ 4 Z 1 Z (1−y)/2n = ϕ(x, y + 2nx)ψ(x, y)dxdy 3 1 2 0 Substitute t = y + 2nx: 2 Z 1 Z 1 Jn = ψ(0, y) ϕ(0, t)dtdy K /n 3n 1 ∼ 2 y

Chaotic mixing Decay of correlations Achieving the upper bound Future directions

A simple computation

Z n In = (ϕ H )ψdµ Wn ◦ 4 Z 1 Z (1−y)/2n = ϕ(x, y + 2nx)ψ(x, y)dxdy 3 1 2 0 Consider the related double integral 4 Z 1 Z (1−y)/2n Jn = ϕ(0, y + 2nx)ψ(0, y)dxdy. 3 1 2 0 Finally show limn→∞ n In Jn = 0, that is, the contribution made to the correlation| function− | by points near the boundary is asympotically the same as the contribution made by points at the boundary.

Chaotic mixing Decay of correlations Achieving the upper bound Future directions

A simple computation

Z n In = (ϕ H )ψdµ Wn ◦ 4 Z 1 Z (1−y)/2n = ϕ(x, y + 2nx)ψ(x, y)dxdy 3 1 2 0 Consider the related double integral 4 Z 1 Z (1−y)/2n Jn = ϕ(0, y + 2nx)ψ(0, y)dxdy. 3 1 2 0 Substitute t = y + 2nx: 2 Z 1 Z 1 Jn = ψ(0, y) ϕ(0, t)dtdy K /n 3n 1 ∼ 2 y Chaotic mixing Decay of correlations Achieving the upper bound Future directions

A simple computation

Finally show limn→∞ n In Jn = 0, that is, the contribution made to the correlation| function− | by points near the boundary is asympotically the same as the contribution made by points at the boundary. providing: 1 K = 0, i.e., the contributions from the wedges do not cancel6 each other out (do not choose non-generic symmetric 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 Z K n Cn + (ϕ H )ψdµ. ∼ n R\Bn ◦

Certainly the contribution from the remaining integral is no slower than (1/n) (from earlier). [Sturman & Springham,O PRE, (2012)] So we see polynomial decay at rate 1/n 1 K = 0, i.e., the contributions from the wedges do not cancel6 each other out (do not choose non-generic symmetric 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 Z K n Cn + (ϕ H )ψdµ. ∼ n R\Bn ◦

Chaotic mixing Decay of correlations Achieving the upper bound Future directions

A cautionary note A cautionary note

Now we have Z K n Cn + (ϕ H )ψdµ. ∼ n R\Bn ◦

Certainly the contribution from the remaining integral is no slower than (1/n) (from earlier). [Sturman & Springham,O PRE, (2012)] So we see polynomial decay at rate 1/n providing: 1 K = 0, i.e., the contributions from the wedges do not cancel6 each other out (do not choose non-generic symmetric observables) Chaotic mixing Decay of correlations Achieving the upper bound Future directions

A cautionary note A cautionary note

Now we have Z K n Cn + (ϕ H )ψdµ. ∼ n R\Bn ◦

Certainly the contribution from the remaining integral is no slower than (1/n) (from earlier). [Sturman & Springham,O PRE, (2012)] So we see polynomial decay at rate 1/n providing: 1 K = 0, i.e., the contributions from the wedges do not cancel6 each other out (do not choose non-generic symmetric 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

F˜(x, y) = (x + 2y p, y) and G˜ (x, y) = (x, y + 2xp).

Then Lemma Z K (ϕ H˜ n)ψdµ 1/p Bn ◦ ∼ n

and so Conjecture Z K 0 (ϕ H˜ n)ψdµ 1/p LTM ◦ ∼ n 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

cos−1(4y 1) ˜f (y) = 1 − − π cos−1(4x 1) g˜(x) = 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, (1/n) O Blue: quadratic, (1/n2) O Green: exponential decay of Cat Map Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Rigorous extensions

. Planar linked twist maps

S = P Q ∩

R P R Q R P R Q \ \ \ \

S = P Q ∩

. Introduce curvature to the Young Tower argument

. Combine with topological ideas to get a more detailed picture of mixing Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Non-rigorous ideas

Young Tower method is practically tractable Don’t work in Fourier space Don’t need to think about observables Can measure return times Can ignore (maybe) ﬁne details of the mathematics Residence-time distributions Compute return times for more realistic models Consequences of presence of islands Connection 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 scheme for understanding data? Fundamental ingredients: Periodicity (but what is the consequence of random forcing?) ‘Good’ hyperbolic region (product structure?) Trajectories which repeatedly return to the hyperbolic region ‘in the right way’ (dynamic renewal?) Method of recording return times Atmospheric or oceanographic? Chaotic mixing Decay of correlations Achieving the upper bound Future directions

Singularity set for HS