Chaotic Mixing in a Diffeomorphism of the Torus

Transcription

1 Chaotic Mixing in a Diffeomorphism of the Torus Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University with Steve Childress Courant Institute of Mathematical Sciences jeanluc Chaotic Mixing in a Diffeomorphism of the Torus p.1/28

2 Experiment of Rothstein et al. (1999) Regular array of magnets [Rothstein, Henry, and Gollub, Nature 401, 770 (1999)] Chaotic Mixing in a Diffeomorphism of the Torus p.2/28

3 Persistent Pattern Disordered array (i = 2, 20, 50, 50.5) Chaotic Mixing in a Diffeomorphism of the Torus p.3/28

4 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Chaotic Mixing in a Diffeomorphism of the Torus p.4/28

5 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Applies if initial scale small. Chaotic Mixing in a Diffeomorphism of the Torus p.4/28

6 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Applies if initial scale small. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky & Fouxon, PRE (1999)] [Son, PRE (1999)] Average over angles Statistical model Statistical model Chaotic Mixing in a Diffeomorphism of the Torus p.4/28

7 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Applies if initial scale small. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky & Fouxon, PRE (1999)] [Son, PRE (1999)] Average over angles Statistical model Statistical model Global theory: Eigenfunction of advection diffusion operator. Chaotic Mixing in a Diffeomorphism of the Torus p.4/28

8 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Applies if initial scale small. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky & Fouxon, PRE (1999)] [Son, PRE (1999)] Average over angles Statistical model Statistical model Global theory: Eigenfunction of advection diffusion operator. Applies if initial scale large. Chaotic Mixing in a Diffeomorphism of the Torus p.4/28

9 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Applies if initial scale small. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky & Fouxon, PRE (1999)] [Son, PRE (1999)] Average over angles Statistical model Statistical model Global theory: Eigenfunction of advection diffusion operator. Applies if initial scale large. [Pierrehumbert, Chaos Sol. Frac. (1994)] [Fereday et al., PRE (2002)] [Sukhatme and Pierrehumbert, PRE (2002)] Strange eigenmode Baker s map Unified description Chaotic Mixing in a Diffeomorphism of the Torus p.4/28

10 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Applies if initial scale small. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky & Fouxon, PRE (1999)] [Son, PRE (1999)] Average over angles Statistical model Statistical model Global theory: Eigenfunction of advection diffusion operator. Applies if initial scale large. Today: Focus on Global theory. Chaotic Mixing in a Diffeomorphism of the Torus p.4/28

11 Local vs Global Regimes of Mixing Local theory: Based on distribution of Lyapunov exponents. Applies if initial scale small. [Antonsen et al., Phys. Fluids (1996)] [Balkovsky & Fouxon, PRE (1999)] [Son, PRE (1999)] Average over angles Statistical model Statistical model Global theory: Eigenfunction of advection diffusion operator. Applies if initial scale large. Today: Focus on Global theory. Map allows analytical results. Chaotic Mixing in a Diffeomorphism of the Torus p.4/28

12 A Diffeomorphism of the Torus 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 ; so that M x is the Arnold cat map. The map M is area-preserving and chaotic. For K = 0 the stretching of phase-space elements is uniform in space (homogeneous). Chaotic Mixing in a Diffeomorphism of the Torus p.5/28

13 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, and the heat operator H ɛ and kernel h ɛ are H ɛ θ(x) := h ɛ (x y)θ(y) dy; T 2 h ɛ (x) = k exp(2πik x k 2 ɛ). Chaotic Mixing in a Diffeomorphism of the Torus p.6/28

14 Transfer Matrix Fourier expand θ (i) (x), θ (i) (x) = k ˆθ (i) k e2πik x. The effect of advection and diffusion becomes ˆθ (i) (x) = q T kq ˆθ(i 1) q, with the transfer matrix, T kq := exp ( 2πi (q x k M(x)) ɛ q 2) dx, T 2 = e ɛ q2 δ 0,Q2 i Q 1 J Q1 ((k 1 + k 2 ) K), Q := k M q, where the J Q are the Bessel functions of the first kind. Chaotic Mixing in a Diffeomorphism of the Torus p.7/28

15 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 Chaotic Mixing in a Diffeomorphism of the Torus p.8/28

16 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 Chaotic Mixing in a Diffeomorphism of the Torus p.8/28

17 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 Chaotic Mixing in a Diffeomorphism of the Torus p.8/28

18 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. Chaotic Mixing in a Diffeomorphism of the Torus p.8/28 2

19 Decay of Variance 10 0 K = 10 3 variance e 15.2i 10 5 lacements ɛ = iteration 0.5 Chaotic Mixing in a Diffeomorphism of the Torus p.9/28

20 Constant (Stirring) Phase Initially, the variance is essentially constant because of the tiny diffusivity. Chaotic Mixing in a Diffeomorphism of the Torus p.10/28

21 Constant (Stirring) Phase Initially, the variance is essentially constant because of the tiny diffusivity. However, there is a cascade of the variance to larger wavenumbers under the action of M 1 in the map. (Neglect K term.) Chaotic Mixing in a Diffeomorphism of the Torus p.10/28

22 Constant (Stirring) Phase Initially, the variance is essentially constant because of the tiny diffusivity. However, there is a cascade of the variance to larger wavenumbers under the action of M 1 in the map. (Neglect K term.) This is the well-known filamentation effect in chaotic flows: the stretching and folding action of the flow causes rapid variation of the temperature across the folds. Chaotic Mixing in a Diffeomorphism of the Torus p.10/28

23 Constant (Stirring) Phase Initially, the variance is essentially constant because of the tiny diffusivity. However, there is a cascade of the variance to larger wavenumbers under the action of M 1 in the map. (Neglect K term.) This is the well-known filamentation effect in chaotic flows: the stretching and folding action of the flow causes rapid variation of the temperature across the folds. Can no longer neglect diffusion after a number of iterations i (log ɛ 1 / log Λ 2 ) 6 for ɛ = 10 5, where Λ = (3 + 5)/2 is the largest eigenvalue of M 1. Chaotic Mixing in a Diffeomorphism of the Torus p.10/28

24 Variance: 5 iterations for K = 0.3 and ɛ = Chaotic Mixing in a Diffeomorphism of the Torus p.11/28

25 Superexponential Phase For small K and k, we have J 0 ((k 1 + k 2 )K) J 1 ((k 1 + k 2 )K), so we set K = 0 and retain only the Q 1 = 0 term in the transfer matrix, T kq = e ɛ q2 δ 0,Q + O ( (k 1 + k 2 ) 2 K 2) ; The nonvanishing matrix elements of T have k = q M 1. Chaotic Mixing in a Diffeomorphism of the Torus p.12/28

26 Superexponential Phase For small K and k, we have J 0 ((k 1 + k 2 )K) J 1 ((k 1 + k 2 )K), so we set K = 0 and retain only the Q 1 = 0 term in the transfer matrix, T kq = e ɛ q2 δ 0,Q + O ( (k 1 + k 2 ) 2 K 2) ; The nonvanishing matrix elements of T have k = q M 1. If initially the variance is concentrated in a single wavenumber q 0, then after one iteration it will all be in q 0 M 1, after two in q 0 M 2, etc. The length of q is multiplied by Λ at each iteration. Chaotic Mixing in a Diffeomorphism of the Torus p.12/28

27 Superexponential Phase For small K and k, we have J 0 ((k 1 + k 2 )K) J 1 ((k 1 + k 2 )K), so we set K = 0 and retain only the Q 1 = 0 term in the transfer matrix, T kq = e ɛ q2 δ 0,Q + O ( (k 1 + k 2 ) 2 K 2) ; The nonvanishing matrix elements of T have k = q M 1. If initially the variance is concentrated in a single wavenumber q 0, then after one iteration it will all be in q 0 M 1, after two in q 0 M 2, etc. The length of q is multiplied by Λ at each iteration. But each time the variance is multiplied by the diffusive decay factor exp( ɛ q 2 ), with q getting exponentially larger; the net decay is thus superexponential. Chaotic Mixing in a Diffeomorphism of the Torus p.12/28

28 Exponential Phase In the superexponential phase we completely neglected the effect of the wave term in the map. Chaotic Mixing in a Diffeomorphism of the Torus p.13/28

29 Exponential Phase In the superexponential phase we completely neglected the effect of the wave term in the map. We described the action as a perfect cascade to large wavenumbers, so that the variance was irrevocably moved to small scales and dissipated extremely rapidly. Chaotic Mixing in a Diffeomorphism of the Torus p.13/28

30 Exponential Phase In the superexponential phase we completely neglected the effect of the wave term in the map. We described the action as a perfect cascade to large wavenumbers, so that the variance was irrevocably moved to small scales and dissipated extremely rapidly. There can be no eigenfunction in such a situation, since the mode structure changes completely at each iteration. Chaotic Mixing in a Diffeomorphism of the Torus p.13/28

31 Exponential Phase In the superexponential phase we completely neglected the effect of the wave term in the map. We described the action as a perfect cascade to large wavenumbers, so that the variance was irrevocably moved to small scales and dissipated extremely rapidly. There can be no eigenfunction in such a situation, since the mode structure changes completely at each iteration. This direct cascade process dominates at first, but it is so efficient that eventually we must examine the effect of the the wave term, which is felt through the Bessel functions in the transfer matrix. Chaotic Mixing in a Diffeomorphism of the Torus p.13/28

32 Exponential Phase In the superexponential phase we completely neglected the effect of the wave term in the map. We described the action as a perfect cascade to large wavenumbers, so that the variance was irrevocably moved to small scales and dissipated extremely rapidly. There can be no eigenfunction in such a situation, since the mode structure changes completely at each iteration. This direct cascade process dominates at first, but it is so efficient that eventually we must examine the effect of the the wave term, which is felt through the Bessel functions in the transfer matrix. Can the wave term lead to the formation of an eigenfunction of the advection diffusion operator, which would imply exponential decay? Chaotic Mixing in a Diffeomorphism of the Torus p.13/28

33 An Eigenfunction? Recall: T kq = e ɛ q2 δ 0,Q2 i Q 1 J Q1 ((k 1 + k 2 ) K), Q := k M q, Consider a matrix element for which Q 1 0. This means that the initial (q) and final (k) wavenumbers connected by that matrix element can differ from k M = q by Q 1 in their first component. Chaotic Mixing in a Diffeomorphism of the Torus p.14/28

34 An Eigenfunction? Recall: T kq = e ɛ q2 δ 0,Q2 i Q 1 J Q1 ((k 1 + k 2 ) K), Q := k M q, Consider a matrix element for which Q 1 0. This means that the initial (q) and final (k) wavenumbers connected by that matrix element can differ from k M = q by Q 1 in their first component. Is it possible for a wavenumber to be mapped back onto itself by such a coupling? Seek solutions to (q 1 q 2 ) M = (q 1 + Q 1 q 2 ) = (q 1 q 2 ) = (0 Q 1 ). The matrix element connecting the (0 Q 1 ) mode to itself is T (0 Q1 ),(0 Q 1 ) = e ɛ Q2 1 i Q 1 J Q1 (Q 1 K). Chaotic Mixing in a Diffeomorphism of the Torus p.14/28

35 Eigenfunction for K = 0.3 and ɛ = 10 3 (Renormalized by decay rate) i = i = 30 Chaotic Mixing in a Diffeomorphism of the Torus p.15/28

36 Decay Rate For small K, the dominant Bessel function is J 1, so the decay factor µ 2 for the variance is µ = T(0 1),(0 1) = e ɛ J 1 (K) = 1 2 K + O( ɛ K, K 2). Hence, for small K the decay rate is limited by the (0 1) mode. The decay rate is independent of ɛ for ɛ 0. Chaotic Mixing in a Diffeomorphism of the Torus p.16/28

37 Decay Rate For small K, the dominant Bessel function is J 1, so the decay factor µ 2 for the variance is µ = T(0 1),(0 1) = e ɛ J 1 (K) = 1 2 K + O( ɛ K, K 2). Hence, for small K the decay rate is limited by the (0 1) mode. The decay rate is independent of ɛ for ɛ 0. This is an analogous result to the baker s map [Fereday et al., PRE (2002)], except that here the agreement with numerical results is good for K quite close to unity. Chaotic Mixing in a Diffeomorphism of the Torus p.16/28

38 Decay Rate For small K, the dominant Bessel function is J 1, so the decay factor µ 2 for the variance is µ = T(0 1),(0 1) = e ɛ J 1 (K) = 1 2 K + O( ɛ K, K 2). Hence, for small K the decay rate is limited by the (0 1) mode. The decay rate is independent of ɛ for ɛ 0. This is an analogous result to the baker s map [Fereday et al., PRE (2002)], except that here the agreement with numerical results is good for K quite close to unity. This is because in the baker s map the discontinuity generates many slowly-decaying harmonics at each step. Chaotic Mixing in a Diffeomorphism of the Torus p.16/28

39 Decay Rate as ɛ log µ lacements K Chaotic Mixing in a Diffeomorphism of the Torus p.17/28

40 Transition from Superexponential to Exponential Now that the mechanism of exponential decay is understood, we can go back and describe the condition for breakdown of the superexponential solution. Chaotic Mixing in a Diffeomorphism of the Torus p.18/28

41 Transition from Superexponential to Exponential Now that the mechanism of exponential decay is understood, we can go back and describe the condition for breakdown of the superexponential solution. The superexponential decay depletes the variance very rapidly until all that is left is variance in the exponentially decaying mode (0 1). Chaotic Mixing in a Diffeomorphism of the Torus p.18/28

42 Transition from Superexponential to Exponential Now that the mechanism of exponential decay is understood, we can go back and describe the condition for breakdown of the superexponential solution. The superexponential decay depletes the variance very rapidly until all that is left is variance in the exponentially decaying mode (0 1). The superexponential phase thus ends when the variance at large wavenumbers equals that in mode (0 1). Chaotic Mixing in a Diffeomorphism of the Torus p.18/28

43 Transition from Superexponential to Exponential Now that the mechanism of exponential decay is understood, we can go back and describe the condition for breakdown of the superexponential solution. The superexponential decay depletes the variance very rapidly until all that is left is variance in the exponentially decaying mode (0 1). The superexponential phase thus ends when the variance at large wavenumbers equals that in mode (0 1). Assuming that the variance resides entirely in the k 0 = (0 1) mode initially, the condition for breakdown is ( µ i 2 = exp ɛ k0 M (i 2 1) 2), where µ 2 is the decay factor of the variance in k 0. Chaotic Mixing in a Diffeomorphism of the Torus p.18/28

44 Transition (continued) After substituting k0 M (i 2 1) Λ i 2 1, solve numerically for i 2. For K = 10 3 and ɛ = 10 5, we have i 2 9.2, numerical results. Chaotic Mixing in a Diffeomorphism of the Torus p.19/28

45 Transition (continued) After substituting k0 M (i 2 1) Λ i 2 1, solve numerically for i 2. For K = 10 3 and ɛ = 10 5, we have i 2 9.2, numerical results. For ɛ 1, approximate solution given by i log ( ɛ 1 log µ 1) / log Λ 2, which gives i 2 8 for K = 10 3, ɛ = Chaotic Mixing in a Diffeomorphism of the Torus p.19/28

46 Transition (continued) After substituting k0 M (i 2 1) Λ i 2 1, solve numerically for i 2. For K = 10 3 and ɛ = 10 5, we have i 2 9.2, numerical results. For ɛ 1, approximate solution given by i log ( ɛ 1 log µ 1) / log Λ 2, which gives i 2 8 for K = 10 3, ɛ = Subtracting i 1 = 1 + log ɛ 1 / log Λ 2, the onset of the superexponential phase, we find the duration of the phase is i 2 i 1 log log µ 1 log Λ 2. Chaotic Mixing in a Diffeomorphism of the Torus p.19/28

47 Duration of Superexponential Phase i 2 i 1 log log µ 1 log Λ 2 Independent of ɛ ; Chaotic Mixing in a Diffeomorphism of the Torus p.20/28

48 Duration of Superexponential Phase i 2 i 1 log log µ 1 log Λ 2 Independent of ɛ ; Very weak dependence on the decay rate log µ ; Chaotic Mixing in a Diffeomorphism of the Torus p.20/28

49 Duration of Superexponential Phase i 2 i 1 log log µ 1 log Λ 2 Independent of ɛ ; Very weak dependence on the decay rate log µ ; Unless µ is small (recall that 0 < µ < 1), the superexponential phase is short and may not be noticeable at all, resembling instead a smooth transition; Chaotic Mixing in a Diffeomorphism of the Torus p.20/28

50 Duration of Superexponential Phase i 2 i 1 log log µ 1 log Λ 2 Independent of ɛ ; Very weak dependence on the decay rate log µ ; Unless µ is small (recall that 0 < µ < 1), the superexponential phase is short and may not be noticeable at all, resembling instead a smooth transition; For log µ 1 > 1 there is no superexponential phase at all; Chaotic Mixing in a Diffeomorphism of the Torus p.20/28

51 Duration of Superexponential Phase i 2 i 1 log log µ 1 log Λ 2 Independent of ɛ ; Very weak dependence on the decay rate log µ ; Unless µ is small (recall that 0 < µ < 1), the superexponential phase is short and may not be noticeable at all, resembling instead a smooth transition; For log µ 1 > 1 there is no superexponential phase at all; Observed in experiments? There µ tends to be closer to unity, so unlikely. But... Chaotic Mixing in a Diffeomorphism of the Torus p.20/28

52 Variance Spectrum of the Eigenfunction The long-wavelength mode (0 1) is the bottleneck that determines the decay rate, for small K. Chaotic Mixing in a Diffeomorphism of the Torus p.21/28

53 Variance Spectrum of the Eigenfunction The long-wavelength mode (0 1) is the bottleneck that determines the decay rate, for small K. But this dominant mode does not determine the structure of the eigenfunction. Chaotic Mixing in a Diffeomorphism of the Torus p.21/28

54 Variance Spectrum of the Eigenfunction The long-wavelength mode (0 1) is the bottleneck that determines the decay rate, for small K. But this dominant mode does not determine the structure of the eigenfunction. In fact, a very small amount of the total variance actually resides in that bottleneck mode: the variance is concentrated at small scales. Chaotic Mixing in a Diffeomorphism of the Torus p.21/28

55 Cascade The variance is taken out of the (0 1) mode by the map: there is a cascade to larger wavenumber through the action of M 1 : (0 1) ( 1 2) ( 3 5) ( 8 13).... These become more and more aligned with the stable (contracting) direction of the map. Chaotic Mixing in a Diffeomorphism of the Torus p.22/28

56 Cascade The variance is taken out of the (0 1) mode by the map: there is a cascade to larger wavenumber through the action of M 1 : (0 1) ( 1 2) ( 3 5) ( 8 13).... These become more and more aligned with the stable (contracting) direction of the map. The amplitude of the wavenumbers is multiplied at each step by a factor Λ = (3 + 5)/ , the largest eigenvalue of M 1. Chaotic Mixing in a Diffeomorphism of the Torus p.22/28

57 Cascade The variance is taken out of the (0 1) mode by the map: there is a cascade to larger wavenumber through the action of M 1 : (0 1) ( 1 2) ( 3 5) ( 8 13).... These become more and more aligned with the stable (contracting) direction of the map. The amplitude of the wavenumbers is multiplied at each step by a factor Λ = (3 + 5)/ , the largest eigenvalue of M 1. But we have seen that the exponential decay rate suggests that the scalar concentration is in an eigenfunction of the advection diffusion operator. What is going on? Chaotic Mixing in a Diffeomorphism of the Torus p.22/28

58 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 Chaotic Mixing in a Diffeomorphism of the Torus p.23/28

59 Eigenfunction and Cascade The decrease in variance for each of the diagonal arrows is diffusive and is given by the factor ν n = exp( 2ɛ k 2 n). Chaotic Mixing in a Diffeomorphism of the Torus p.24/28

60 Eigenfunction and Cascade The decrease in variance for each of the diagonal arrows is diffusive and is given by the factor ν n = exp( 2ɛ kn). 2 If we denote by σ n (i) := ˆθ kn 2 the variance in mode k n at the ith iteration, we have σ (i) n σ (i) n = µ 2 σ (i 1) n, n = 0, 1,..., = ν n 1 σ (i 1) n 1, n = 1, 2,.... Chaotic Mixing in a Diffeomorphism of the Torus p.24/28

61 Eigenfunction and Cascade The decrease in variance for each of the diagonal arrows is diffusive and is given by the factor ν n = exp( 2ɛ kn). 2 If we denote by σ n (i) := ˆθ kn 2 the variance in mode k n at the ith iteration, we have σ (i) n σ (i) n = µ 2 σ (i 1) n, n = 0, 1,..., = ν n 1 σ (i 1) n 1, n = 1, 2,.... These two recurrences can be combined to give Σ (i) n := σ(i) n σ (i) = ν ( n 1 ν n 2 ν 0 µ 2n = µ 2n exp 2ɛ 0 where Σ (i) n is the relative variance in the nth mode. n 1 m=0 k 2 m ), Chaotic Mixing in a Diffeomorphism of the Torus p.24/28

62 Eigenfunction and Cascade (cont d) The wavenumber is given by the exponential recursion, k n Λ k n 1 = k n Λ n k 0 = Λ n. Chaotic Mixing in a Diffeomorphism of the Torus p.25/28

63 Eigenfunction and Cascade (cont d) The wavenumber is given by the exponential recursion, k n Λ k n 1 = k n Λ n k 0 = Λ n. Solve for n = log k n / log Λ and rewrite the relative variance as Σ (i) n k n 2 log µ/ log Λ exp ( 2ɛk 2 n/λ 2), where we retained only the k 2 n 1 (last) term of the sum. Chaotic Mixing in a Diffeomorphism of the Torus p.25/28

64 Eigenfunction and Cascade (cont d) The wavenumber is given by the exponential recursion, k n Λ k n 1 = k n Λ n k 0 = Λ n. Solve for n = log k n / log Λ and rewrite the relative variance as Σ (i) n k n 2 log µ/ log Λ exp ( 2ɛk 2 n/λ 2), where we retained only the kn 1 2 (last) term of the sum. Does not (and should not) depend on the iteration number, i, and depends only on n through k n. Find Σ(k) = k 2ζ exp ( 2ɛk 2 /Λ 2), ζ := log µ/ log Λ, the spectrum of relative variance. Chaotic Mixing in a Diffeomorphism of the Torus p.25/28

65 Spectrum of Variance relative variance K = 10 3 ɛ = 10 4 i = 12 eplacements k Chaotic Mixing in a Diffeomorphism of the Torus p.26/28

66 Spectrum of Variance (cont d) Σ(k) = k 2ζ exp ( 2ɛk 2 /Λ 2), ζ := log µ/ log Λ Since µ < 1 and Λ > 1, we have ζ > 0. Chaotic Mixing in a Diffeomorphism of the Torus p.27/28

67 Spectrum of Variance (cont d) Σ(k) = k 2ζ exp ( 2ɛk 2 /Λ 2), ζ := log µ/ log Λ Since µ < 1 and Λ > 1, we have ζ > 0. This implies that there is more variance at the large wavenumbers than at the slowest-decaying mode k 0. Chaotic Mixing in a Diffeomorphism of the Torus p.27/28

68 Spectrum of Variance (cont d) Σ(k) = k 2ζ exp ( 2ɛk 2 /Λ 2), ζ := log µ/ log Λ Since µ < 1 and Λ > 1, we have ζ > 0. This implies that there is more variance at the large wavenumbers than at the slowest-decaying mode k 0. Find the maximum of Σ(k), k m = Λ (ζ/2ɛ) 1/2, Σ(k m ) = k 2ζ m e ζ = k 2ζ m µ log Λ. The peak wavenumber thus scales as ɛ 1/2, the same scaling as the dissipation scale. The relative variance in that peak wavenumber scales as ɛ ζ. k m largest wavenumber that must be included in a numerical calculation to capture the decay of variance correctly (fewer?). Chaotic Mixing in a Diffeomorphism of the Torus p.27/28

69 Conclusions Three phases of chaotic mixing: constant variance, superexponential decay, exponential decay. Chaotic Mixing in a Diffeomorphism of the Torus p.28/28

70 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., PRE (2002)] Chaotic Mixing in a Diffeomorphism of the Torus p.28/28

71 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., PRE (2002)] The spectrum of this eigenmode is determined by a balance between the eigenfunction property and a cascade to large wavenumbers. Chaotic Mixing in a Diffeomorphism of the Torus p.28/28

72 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., PRE (2002)] 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. Chaotic Mixing in a Diffeomorphism of the Torus p.28/28

73 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., PRE (2002)] 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 exponent or its distribution. Chaotic Mixing in a Diffeomorphism of the Torus p.28/28

74 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., PRE (2002)] 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 exponent or its distribution. Global structure matters! Chaotic Mixing in a Diffeomorphism of the Torus p.28/28

75 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., PRE (2002)] 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 exponent or its distribution. Global structure matters! Large K? Chaotic Mixing in a Diffeomorphism of the Torus p.28/28