How Fast Does a Passive Scalar Decay? (Decay of Chaotically Advected - PowerPoint PPT Presentation

How Fast Does a Passive Scalar Decay? (Decay of Chaotically Advected Passive Scalars in the Zero Diffusivity Limit) Yue-Kin Tsang Courant Institute of Mathematical Sciences New York University Thomas M. Antonsen, Jr. and Edward Ott University of Maryland, College Park

Decay of Variance ∂φ u · ∇ φ = κ ∇ 2 φ ∂t + � ∇ · � u = 0 (incompressible) � 2 π � φ ( � x, 0) ∼ sin ( x + y ) L D � u ( � x, t ) : doubly periodic with period L f Mean is conserved: d � φ � = 0 dt � φ 2 � ( take � φ � = 0 ) Vairance =

Decay of Variance � φ 2 � d |∇ φ | 2 � � = − 2 κ dt variance decay due to diffusion ( κ � = 0 ) decay rate increases with |∇ φ |

Decay of Variance � φ 2 � d |∇ φ | 2 � � = − 2 κ dt variance decay due to diffusion ( κ � = 0 ) decay rate increases with |∇ φ | stirring/stretching of fluid ⇒ filaments ⇒ large |∇ φ | ⇒ enhanced diffusion ⇒ faster mixing/variance decay

Exponential Decay Rate γ 0 We are interested in long time behavior of φ as κ → 0 . numerical simulations and experiments show: φ 2 � ∼ e − γ ( κ ) t � some numerical evidence support the prediction: κ → 0 + γ ( κ ) ≡ γ 0 lim Question: Given a certain flow � u ( � x, t ) , can we predict the decay rate γ 0 ? 1. R.T. Pierrehumbert, Chaos, Solitons and Fractals 4 , 1091 (1994) 2. Voth el at. , Phys. Fluids 15 , 2560 (2003)

Wave Packet Model x j ( t )) = A j ( t ) sin[ � ϕ j ( � k j ( t ) · � x j ( t ) + ϑ j ( t )] ϕ 2 � � ω j ( t ) = j � k j ( t ) � x j ( t ) � k j (0) x j (0) �

Wave Packet Model x j ( t )) = A j ( t ) sin[ � ϕ j ( � k j ( t ) · � x j ( t ) + ϑ j ( t )] ϕ 2 � � ω j ( t ) = j � k j ( t ) � x j ( t ) � φ ( � x, t ) = ϕ j � k j (0) x j (0) � � � φ 2 � C ( t ) ≡ = ω j ( t )

Wave Packet Model x j ( t )) = A j ( t ) sin[ � ϕ j ( � k j ( t ) · � x j ( t ) + ϑ j ( t )] ϕ 2 � � ω j ( t ) = j � k j ( t ) � x j ( t ) � φ ( � x, t ) = ϕ j � k j (0) x j (0) � � � φ 2 � C ( t ) ≡ = ω j ( t ) dω j 2 ω j dt = − 2 κk j � k j ( t ) is determined by the stretching of fluid elements induced by the smooth velocity field � u

Characterizing Stretching x(t) Along a fluid trajectory, δ d� x ( t ) = � u ( � x ( t ) , t ) dt δ x(0) Finite-time Lyapunov Exponent, h x (0) | e ht | δ� x ( t ) | = | δ� Probability Distribution Function for h , P ( h t ) P ( h t ) ∼ exp[ − tG ( h )] (Reference: R.S. Ellis, “Entropy, Large Deviations and Statistical Mechanics”, 1985)

P ( h t ) and G ( h ) 2 t=1 t=5 t=5 t=10 3 t=10 t=15 t=15 t=20 t=20 P(h | t) 2 G(h) 1 1 0 0 −1.0 0.0 1.0 2.0 −1.0 −0.5 0.0 0.5 1.0 1.5 h h

Wave Packet Model x j ( t )) = A j ( t ) sin[ � ϕ j ( � k j ( t ) · � x j ( t ) + ϑ j ( t )] ϕ 2 � � ω j ( t ) = j � k j ( t ) � φ ( � x, t ) = ϕ j � x j ( t ) � � φ 2 � � k j (0) C ( t ) ≡ = ω j ( t ) x j (0) � dω j 2 ω j dt = − 2 κk j | � k j ( t ) |≈ | � k j (0) | cos θ e h j t

Wave Packet Model x j ( t )) = A j ( t ) sin[ � ϕ j ( � k j ( t ) · � x j ( t ) + ϑ j ( t )] ϕ 2 � � ω j ( t ) = j � k j ( t ) � φ ( � x, t ) = ϕ j � x j ( t ) � � φ 2 � � k j (0) C ( t ) ≡ = ω j ( t ) x j (0) � dω j 2 ω j dt = − 2 κk j | � k j ( t ) | ≈ | � k j (0) | cos θ e h j t γ 0 = min h [ h + G ( h )] Antonsen el at. , Phys. Fluids 8 , 3094 (1996)

Comparison with Numerics Flow Model: T = 1 , U = π ( L f = L D = 2 π )  U cos(2 π y + α n )ˆ nT ≤ t < ( n + 1 i , 2 ) T   L f  u ( � � x, t )= U cos(2 π x + β n ) ˆ j , ( n + 1 2 ) T ≤ t < ( n +1) T    L f 0.20 0.18 0.16 0.185 0.14 0.180 0.12 0.175 γ(κ) 0.10 0.170 0.08 0.06 0.165 0.04 0.160 0.02 0 -8 -8 -8 -8 -8 1 × 10 2 × 10 3 × 10 4 × 10 5 × 10 0.00 0 -8 -8 -8 -8 -8 1 × 10 2 × 10 3 × 10 4 × 10 5 × 10 κ T/L 2 D

Laboratory Experiment G. A. Voth, T.C. Saint, Greg Dobler, and J.P . Gollub, Phys. Fluids 15 , 2560 (2003)

Laboratory Experiment G. A. Voth, T.C. Saint, Greg Dobler, and J.P . Gollub, Phys. Fluids 15 , 2560 (2003) measured decay rate is 10 times smaller than predicted γ 0 !! Reason: the ratio L D /L f is an important factor

Variance Damping Mechanisms L D ≈ L f decay rate controlled by processes at small length scales (large k ) γ 0 predicted by Lagrangian stretching theory (short wavelength mechanism) L D ≫ L f variance being “leaked” out of the longest wavelength mode (smallest k ) decay rate limited by spatial diffusion on the large scales (long wavelength mechanism) (D.R. Fereday, P .H. Haynes, A. Wonhas and J.C. Vassilicos, Phys. Rev. E 65 035301(R), (2002))

Wavenumber Spectrum (2 π ) 2 δ ( k − | k ′ | ) | ˜ φ ( k ′ , t ) | 2 � d k ′ ∼ S ( k ) e − γ ( κ ) t S ( k, t ) = L 2 D “strange eigenmode” S ( k ) = � S ( k, t ) /C ( t ) � t S(k) Local stretching theory operating at small scales ln k 1/L D k d

L D = L f for each time period t > 20 T , remove all Fourier modes of φ with | k x | and | k y | less than k filter = a (2 π/L D ) S(k) Local stretching theory operating at small scales ln k 1/L D k d k filter

L D = L f decay rate (controlled by large k processes) is not affected by this filtering (fixed k d /k filter ) 0.25 0 0.24 0.23 -5 0.22 0.21 γ(κ) ln C(t) 0.20 -10 κ T/L D 2 0.19 -7 ( a=1 ) 2.78 x 10 a=0 0.18 a=1 -8 ( a=2 ) 6.95 x 10 a=2 -15 0.17 -8 ( a=4 ) a=4 1.74 x 10 a=8 -9 ( a=8 ) 4.35 x 10 0.16 a=16 -9 ( a=16 ) a=32 1.09 x 10 0.15 -15 -14 -13 -12 -11 -10 -20 1/2 0 25 50 75 100 log 2 ( κ T/L ) 2 t/T D

L D = ML f ( M > 1) at each time step n , remove all but the lowest k mode ( i.e. remove everything that leaks out of the lowest k mode) decay rate = rate of "leaking" from the lowest k mode S(k) Local stretching theory operating at small scales ln k 1/L D k d

L D = ML f ( M > 1 ) φ n +1 = [ J 0 ( η )] 2 φ n where η = πUT/ ( ML f ) leaking rate = − ln[ J 0 ( η )] 4 /T (dashed line) 0.20 0.15 γ(κ) 0.10 0.05 0.00 1 2 3 4 5 6 7 8 M

Upper Bound on γ 0 � ∞ S ( k ) e − γ 0 t = dk ′ S ( k ′ ) δ ( k − k ′ | cos θ | e ht ) � � h,θ 0 Assuming S ( k ) ∼ k − ψ (can generalize to anisotropic case), one can show γ 0 = min h [ h + G ( h ) − | ψ | h ] Two consequences: γ 0 < min h [ h + G ( h )] � G ( h ) − γ 0 � ψ = 1 + min h h

Wavenumber Spectra Exponent ψ short wavelength mechanism ⇒ flat spectra long wavelength mechanism ⇒ power-law spectra M = 1 M > 1 -3 0 -9 2 M=2 κ T/L =1.09 x 10 D -1 -4 -2 -9 κ T/L =4.35 x 10 2 D M=4 -5 -3 (k) ] (k) ] -4 -8 κ T/L =1.74 x 10 2 -6 D avg avg -5 M=6 ln [ S ln [ S -7 -6 -8 2 κ T/L =6.95 x 10 D -7 -8 -8 M=8 -9 -9 -10 -10 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ln k ln k

Summary For L D ≈ L f , short wavelength mechanism applies and γ 0 can be predicted using local stretching theory For L D ≫ L f , long wavelength mechanism applies, γ 0 limited by the decay of the longest wavelength mode Decay rate predicted by the local stretching theory provides an upper bound on γ 0 Long wavelength mechanism gives a power-law power spectrum, k − ψ with ψ > 0 , short wavelength mechanism gives a flat power spectrum ( ψ = 0) Tsang, Antonsen and Ott, Phys. Rev. E 71 , 066301 (2005)