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Rates of Mixing in Models of Fluid Flow

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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('; ) = (' f ) dµ 'dµ dµ 0 ◦ − ! as n for scalar observables ' and . ! 1 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!1 \ 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!1 \ Equivalently in functional form: Z Z Z n Cn('; ) = (' f ) dµ 'dµ dµ 0 ◦ − ! as n for scalar observables ' and . ! 1 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 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 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 dµ = 0 and compute Z n Cn('; ) = (' H ) dµ ◦ Expanding (analytic) ' and as Fourier series, X ik:x X ij:x '(x) = ake ; (x) = bje ; 2 2 k2Z j2Z Linearity and orthogonality means that Z X ik:Anx X ij:x X Cn('; ) = ake bje dx = akb−kAn : 2 2 2 k2Z j2Z k2Z Since A is a hyperbolic matrix, the exponential growth of kAn together with the exponential decay of Fourier coefficientsj j 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): ! 2 j If R Rdm < , then H is ergodic 1 If R Rdm < and gcd of R is 1, then H is strong mixing. 1 If 8 (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 Λ v+ v− Chaotic mixing Decay of correlations Achieving the upper bound Future directions Young Towers Cat Map Λ 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. 1 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.
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