Large-scale Eigenfunctions and Mixing

Transcription

1 Large-scale Eigenfunctions and Mixing Jean-Luc Thiffeault Department of Mathematics Imperial College London with Steve Childress Courant Institute of Mathematical Sciences New York University jeanluc Large-scale Eigenfunctions and Mixing p.1/14

2 Persistent Pattern Disordered array (i = 2, 20, 50, 50.5) [Rothstein, Henry, and Gollub, Nature 401, 770 (1999)] Large-scale Eigenfunctions and Mixing p.2/14

3 Evolution of Pattern Striations Smoothed by diffusion Eventually settles into pattern (eigenfunction) Large-scale Eigenfunctions and Mixing p.3/14

4 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Large-scale Eigenfunctions and Mixing p.4/14

5 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky and Fouxon, PRE (1999)] [Son, PRE (1999)] Average over angles Statistical model Statistical model Large-scale Eigenfunctions and Mixing p.4/14

6 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky and Fouxon, PRE (1999)] [Son, PRE (1999)] Global theory: Eigenfunction of advection diffusion operator. Average over angles Statistical model Statistical model Large-scale Eigenfunctions and Mixing p.4/14

7 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky and Fouxon, PRE (1999)] [Son, PRE (1999)] Global theory: Eigenfunction of advection diffusion operator. [Pierrehumbert, Chaos Sol. Frac. (1994)] [Fereday et al., Wonhas and Vassilicos, PRE (2002)] [Sukhatme and Pierrehumbert, PRE (2002)] [Fereday and Haynes (2003)] [Thiffeault and Childress (2003)] Average over angles Statistical model Statistical model Strange eigenmode Baker s map Unified description Modified Cat map Large-scale Eigenfunctions and Mixing p.4/14

8 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky and Fouxon, PRE (1999)] [Son, PRE (1999)] Global theory: Eigenfunction of advection diffusion operator. Map allows analytical results. Average over angles Statistical model Statistical model Large-scale Eigenfunctions and Mixing p.4/14

9 The Map We consider a diffeomorphism of the 2-torus T 2 = [0, 1] 2, M(x) = M x + φ(x), where M = ( ) ; φ(x) = K 2π ( ) sin 2πx 1 sin 2πx 1 ; M x is the Arnold cat map. The map M is area-preserving and chaotic. For K = 0 the stretching of fluid elements is homogeneous in space. Large-scale Eigenfunctions and Mixing p.5/14

10 Advection and Diffusion Iterate the map and apply the heat operator to a scalar field (which we call temperature for concreteness) distribution θ (i 1) (x), θ (i) (x) = H ɛ θ (i 1) (M 1 (x)) where ɛ is the diffusivity, with the heat operator H ɛ. In other words: advect instantaneously and then diffuse for one unit of time. After Fourier expanding θ (i) (x), the effect of advection and diffusion becomes ˆθ (i) (x) = q T kq ˆθ(i 1) q, with the transfer matrix, T kq = e ɛ q2 δ 0,Q2 i Q 1 J Q1 ((k 1 + k 2 ) K), Q := k M q. Large-scale Eigenfunctions and Mixing p.6/14

11 Variance: A measure of mixing In the absence of diffusion (ɛ = 0), the variance σ (i) σ (i) := θ (i) (x) 2 dx = σ (i) k, σ(i) k := ˆθ(i) k T 2 k is preserved. (We assume the spatial mean of θ is zero.) For ɛ > 0 the variance decays. We consider the case ɛ 1, of greatest practical interest. 2 Large-scale Eigenfunctions and Mixing p.7/14

12 Variance: A measure of mixing In the absence of diffusion (ɛ = 0), the variance σ (i) σ (i) := θ (i) (x) 2 dx = σ (i) k, σ(i) k := ˆθ(i) k T 2 k is preserved. (We assume the spatial mean of θ is zero.) For ɛ > 0 the variance decays. We consider the case ɛ 1, of greatest practical interest. Three phases: The variance is initially constant; 2 Large-scale Eigenfunctions and Mixing p.7/14

13 Variance: A measure of mixing In the absence of diffusion (ɛ = 0), the variance σ (i) σ (i) := θ (i) (x) 2 dx = σ (i) k, σ(i) k := ˆθ(i) k T 2 k is preserved. (We assume the spatial mean of θ is zero.) For ɛ > 0 the variance decays. We consider the case ɛ 1, of greatest practical interest. Three phases: The variance is initially constant; It then undergoes a rapid superexponential decay; 2 Large-scale Eigenfunctions and Mixing p.7/14

14 Variance: A measure of mixing In the absence of diffusion (ɛ = 0), the variance σ (i) σ (i) := θ (i) (x) 2 dx = σ (i) k, σ(i) k := ˆθ(i) k T 2 k is preserved. (We assume the spatial mean of θ is zero.) For ɛ > 0 the variance decays. We consider the case ɛ 1, of greatest practical interest. Three phases: The variance is initially constant; It then undergoes a rapid superexponential decay; θ (i) settles into an eigenfunction of the A D operator that sets the exponential decay rate. Large-scale Eigenfunctions and Mixing p.7/14 2

15 Decay of Variance 10 0 K = 10 3 variance e 15.2i 10 5 lacements ɛ = iteration 0.5 Large-scale Eigenfunctions and Mixing p.8/14

16 Variance: 5 iterations for K = 0.3 and ɛ = Large-scale Eigenfunctions and Mixing p.9/14

17 Eigenfunction for K = 0.3 and ɛ = 10 3 (Renormalised by decay rate) i = i = 30 Large-scale Eigenfunctions and Mixing p.10/14

18 Decay Rate as ɛ log µ lacements K Large-scale Eigenfunctions and Mixing p.11/14

19 Eigenfunction: One Iteration The wavenumbers are mapped back to themselves, with their variance decreased by a uniform factor µ 2 < 1 (vertical arrows). But at the same time the modes are mapped to next one down the lacements cascade following the diagonal arrows. k 0 k 1 k 2 k 3 iteration i 1 µ 2 ν 0 ν 1 ν 2 iteration i Large-scale Eigenfunctions and Mixing p.12/14

20 Spectrum of Variance relative variance K = 10 3 ɛ = 10 4 i = 12 eplacements k Large-scale Eigenfunctions and Mixing p.13/14

21 Conclusions Three phases of chaotic mixing: constant variance, superexponential decay, exponential decay. Large-scale Eigenfunctions and Mixing p.14/14

22 Conclusions Three phases of chaotic mixing: constant variance, superexponential decay, exponential decay. Large-scale eigenmode dominates exponential phase, as for baker s map [Fereday et al., Wonhas and Vassilicos, PRE (2002)] and others [Sukhatme and Pierrehumbert (2002), Fereday and Haynes (preprint)]. Large-scale Eigenfunctions and Mixing p.14/14

23 Conclusions Three phases of chaotic mixing: constant variance, superexponential decay, exponential decay. Large-scale eigenmode dominates exponential phase, as for baker s map [Fereday et al., Wonhas and Vassilicos, PRE (2002)] and others [Sukhatme and Pierrehumbert (2002), Fereday and Haynes (preprint)]. The spectrum of this eigenmode is determined by a balance between the eigenfunction property and a cascade to large wavenumbers. Large-scale Eigenfunctions and Mixing p.14/14

24 Conclusions Three phases of chaotic mixing: constant variance, superexponential decay, exponential decay. Large-scale eigenmode dominates exponential phase, as for baker s map [Fereday et al., Wonhas and Vassilicos, PRE (2002)] and others [Sukhatme and Pierrehumbert (2002), Fereday and Haynes (preprint)]. The spectrum of this eigenmode is determined by a balance between the eigenfunction property and a cascade to large wavenumbers. For our case of a map with nearly uniform stretching, most of the variance is concentrated at large wavenumbers. Large-scale Eigenfunctions and Mixing p.14/14

25 Conclusions Three phases of chaotic mixing: constant variance, superexponential decay, exponential decay. Large-scale eigenmode dominates exponential phase, as for baker s map [Fereday et al., Wonhas and Vassilicos, PRE (2002)] and others [Sukhatme and Pierrehumbert (2002), Fereday and Haynes (preprint)]. The spectrum of this eigenmode is determined by a balance between the eigenfunction property and a cascade to large wavenumbers. For our case of a map with nearly uniform stretching, most of the variance is concentrated at large wavenumbers. The decay rate is unrelated to the Lyapunov exponents and their distribution. Large-scale Eigenfunctions and Mixing p.14/14

26 Conclusions Three phases of chaotic mixing: constant variance, superexponential decay, exponential decay. Large-scale eigenmode dominates exponential phase, as for baker s map [Fereday et al., Wonhas and Vassilicos, PRE (2002)] and others [Sukhatme and Pierrehumbert (2002), Fereday and Haynes (preprint)]. The spectrum of this eigenmode is determined by a balance between the eigenfunction property and a cascade to large wavenumbers. For our case of a map with nearly uniform stretching, most of the variance is concentrated at large wavenumbers. The decay rate is unrelated to the Lyapunov exponents and their distribution. Global structure matters! Large-scale Eigenfunctions and Mixing p.14/14

27 Conclusions Three phases of chaotic mixing: constant variance, superexponential decay, exponential decay. Large-scale eigenmode dominates exponential phase, as for baker s map [Fereday et al., Wonhas and Vassilicos, PRE (2002)] and others [Sukhatme and Pierrehumbert (2002), Fereday and Haynes (preprint)]. The spectrum of this eigenmode is determined by a balance between the eigenfunction property and a cascade to large wavenumbers. For our case of a map with nearly uniform stretching, most of the variance is concentrated at large wavenumbers. The decay rate is unrelated to the Lyapunov exponents and their distribution. Global structure matters! Large-scale Eigenfunctions and Mixing p.14/14