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 ∂t + u · ∇φ = κ∇2φ ∇ · u = 0

(incompressible)

φ( x, 0) ∼ sin 2π LD (x + y)

• u(

x, t) : doubly periodic with period Lf

Mean is conserved: d φ

dt = 0

Vairance =

• φ2

(take φ = 0)

Decay of Variance

d

• φ2

dt = −2κ

• |∇φ|2

variance decay due to diffusion (κ = 0) decay rate increases with |∇φ|

Decay of Variance

d

• φ2

dt = −2κ

• |∇φ|2

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 γ0

We are interested in long time behavior of φ as κ → 0.

numerical simulations and experiments show:

• φ2

∼ e−γ(κ)t some numerical evidence support the prediction: lim

κ→0+ γ(κ) ≡ γ0

Question: Given a certain flow u( x, t), can we predict the decay rate γ0?

• 1. R.T. Pierrehumbert, Chaos, Solitons and Fractals 4, 1091 (1994)
• 2. Voth el at., Phys. Fluids 15, 2560 (2003)

Wave Packet Model

• xj(0)
• xj(t)
• kj(0)
• kj(t)

ϕj( xj(t)) = Aj(t) sin[ kj(t) · xj(t) + ϑj(t)] ωj(t) =

• ϕ2

j

Wave Packet Model

• xj(0)
• xj(t)
• kj(0)
• kj(t)

ϕj( xj(t)) = Aj(t) sin[ kj(t) · xj(t) + ϑj(t)] ωj(t) =

• ϕ2

j

• φ(

x, t) =

• ϕj

C(t) ≡

• φ2

=

• ωj(t)

Wave Packet Model

• xj(0)
• xj(t)
• kj(0)
• kj(t)

ϕj( xj(t)) = Aj(t) sin[ kj(t) · xj(t) + ϑj(t)] ωj(t) =

• ϕ2

j

• φ(

x, t) =

• ϕj

C(t) ≡

• φ2

=

• ωj(t)

dωj dt = −2κkj

2ωj

• kj(t) is determined by the stretching of fluid

elements induced by the smooth velocity field

u

Characterizing Stretching

x(0) δ x(t) δ

Along a fluid trajectory,

d x(t) dt = u( x(t), t)

Finite-time Lyapunov Exponent, h

|δ x(t)| = |δ x(0)|eht

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)

−1.0 0.0 1.0 2.0

h

1 2 3

P(h | t)

t=1 t=5 t=10 t=15 t=20 −1.0 −0.5 0.0 0.5 1.0 1.5

h

1 2

G(h)

t=5 t=10 t=15 t=20

Wave Packet Model

• xj(0)
• xj(t)
• kj(0)
• kj(t)

ϕj( xj(t)) = Aj(t) sin[ kj(t) · xj(t) + ϑj(t)] ωj(t) =

• ϕ2

j

• φ(

x, t) =

• ϕj

C(t) ≡

• φ2

=

• ωj(t)

dωj dt = −2κkj

2ωj

| kj(t)|≈ | kj(0)| cos θ ehjt

Wave Packet Model

• xj(0)
• xj(t)
• kj(0)
• kj(t)

ϕj( xj(t)) = Aj(t) sin[ kj(t) · xj(t) + ϑj(t)] ωj(t) =

• ϕ2

j

• φ(

x, t) =

• ϕj

C(t) ≡

• φ2

=

• ωj(t)

dωj dt = −2κkj

2ωj

| kj(t)| ≈ | kj(0)| cos θ ehjt γ0 = min

h [h + G(h)]

Antonsen el at., Phys. Fluids 8, 3094 (1996)

Comparison with Numerics

Flow Model: T = 1, U = π (Lf = LD = 2π)

• u(

x, t)=        U cos(2π Lf y + αn)ˆ i , nT ≤ t < (n+ 1

2)T

U cos(2π Lf x + βn) ˆ j , (n+ 1

2)T ≤ t < (n+1)T

1×10

• 8

2×10

• 8

3×10

• 8

4×10

• 8

5×10

• 8

κT/L

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

γ(κ)

1×10

• 8

2×10

• 8

3×10

• 8

4×10

• 8

5×10

• 8

0.160 0.165 0.170 0.175 0.180 0.185

D 2

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 γ0!! Reason: the ratio LD/Lf is an important factor

Variance Damping Mechanisms

LD ≈ Lf

decay rate controlled by processes at small length scales (large k)

γ0 predicted by Lagrangian stretching theory (short

wavelength mechanism)

LD ≫ Lf

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

S(k, t) =

• d k′

(2π)2 δ(k − |k′|) |˜ φ(k′, t)|2 L2

D

∼ S(k)e−γ(κ)t

“strange eigenmode”

S(k) = S(k, t)/C(t)t

S(k) ln k kd 1/LD

Local stretching theory

• perating at small scales

LD = Lf

for each time period t > 20T, remove all Fourier modes of φ with |kx| and |ky| less than kfilter = a(2π/LD)

S(k) ln k kd 1/LD

Local stretching theory

• perating at small scales

kfilter

LD = Lf

decay rate (controlled by large k processes) is not affected by this filtering (fixed kd/kfilter)

25 50 75 100

t/T

• 20
• 15
• 10
• 5

ln C(t)

2.78 x 10

• 7 (a=1)

6.95 x 10

• 8 (a=2)

1.74 x 10

• 8 (a=4)

4.35 x 10

• 9 (a=8)

1.09 x 10

• 9 (a=16)

κT/LD

2

• 15
• 14
• 13
• 12
• 11
• 10

log2(κT/L )

1/2

0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

γ(κ)

a=0 a=1 a=2 a=4 a=8 a=16 a=32 D 2

LD = MLf (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) ln k kd 1/LD

Local stretching theory

• perating at small scales

LD = MLf (M > 1)

φn+1 = [J0(η)]2φn where η = πUT/(MLf) leaking rate = − ln[J0(η)]4/T (dashed line)

1 2 3 4 5 6 7 8

M

0.00 0.05 0.10 0.15 0.20

γ(κ)

Upper Bound on γ0

S(k)e−γ0t = ∞ dk′ S(k′)

• δ(k − k′| cos θ|eht)
• h,θ

Assuming S(k) ∼ k−ψ (can generalize to anisotropic case), one can show γ0 = min

h [h + G(h) − |ψ|h]

Two consequences: γ0 < min

h [h + G(h)]

ψ = 1 + min

h

G(h) − γ0 h

Wavenumber Spectra Exponent ψ

short wavelength mechanism ⇒ flat spectra long wavelength mechanism ⇒ power-law spectra

M = 1 M > 1

1 2 3 4 5 6 7

ln k

• 10
• 9
• 8
• 7
• 6
• 5
• 4
• 3

ln [S (k)]

κT/L =1.74 x 10

• 8

κT/L =4.35 x 10

• 9

κT/L =1.09 x 10

• 9

κT/L =6.95 x 10

• 8

D D D D 2 2 2 2

avg

1 2 3 4 5 6 7

ln k

• 10
• 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

ln [S (k)]

avg

M=2 M=4 M=6 M=8

Summary

For LD ≈ Lf, short wavelength mechanism applies and γ0 can be predicted using local stretching theory For LD ≫ Lf, long wavelength mechanism applies,

γ0 limited by the decay of the longest wavelength

mode Decay rate predicted by the local stretching theory provides an upper bound on γ0 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)