Energy Injection into Two-dimensional Turbulence: a Scaling Regime - - PowerPoint PPT Presentation

Energy Injection into Two-dimensional Turbulence: a Scaling Regime Controlled by Drag Yue-Kin Tsang Scripps Institution of Oceanography University of California, San Diego William R. Young

Forced-dissipative 2D Systems

Conducting fluid layer

Paret and Tabeling (1998) http://www.fluid.tue.nl

f(x, t) ∼ I(t) × B(x) bottom wall friction

Forced-dissipative 2D Systems

Conducting fluid layer

Paret and Tabeling (1998) http://www.fluid.tue.nl

f(x, t) ∼ I(t) × B(x) bottom wall friction Soap film

Burgess et al. (1999)

driving belt induced motion drag from surrounding gas

Forced-dissipative 2D Systems

Conducting fluid layer

Paret and Tabeling (1998) http://www.fluid.tue.nl

f(x, t) ∼ I(t) × B(x) bottom wall friction Soap film

Burgess et al. (1999)

driving belt induced motion drag from surrounding gas The Ocean

wind✲

• J. M. Toole (1996)

wind forcing, lunisolar tide sea floor drag, instabilities

Forced-dissipative 2D Systems

Conducting fluid layer

Paret and Tabeling (1998)

Soap film The Ocean

wind✲

• J. M. Toole (1996)

Forcing (energy input) Nonlinear interaction (energy redistributed) Dissipation (energy removed)

How much power is needed to drive these systems?? =

ε

Energy Injection Rate

Pulling a block on a rough surface by a constant force Newton’s second law, F − µv = mdv dt Steady state velocity, v = F µ Energy injection rate (Power input), ε = Fv = F

• F

µ

• ∼ µ−1

Dependence of ε on µ

Forced harmonic oscillator ¨ x + ω 2

0 x = −µ ˙

x + A cos ωt instantaneous : εint(t) = ˙ x(t) A cos ωt averaged : ε = 1 T T εint(t′) dt′

Dependence of ε on µ

Forced harmonic oscillator ¨ x + ω 2

0 x = −µ ˙

x + A cos ωt instantaneous : εint(t) = ˙ x(t) A cos ωt averaged : ε = 1 T T εint(t′) dt′

ε

µ µ∗(ω) ∼µ ∼µ−1

Two-dimensional turbulence

ζt + u · ∇ζ = f(x, t) − µζ + ν∇2ζ u = (u, v) ζ ≡ vx − uy = ∇2ψ Energy injection rate ε = −ψ f Power Integral (conservation of energy) ε = µ u2 + v2

• εµ

+ ν ζ2

• εν

small-scale forcing: f = τ−2

f cos(kfx), k−1 f

≪ box size drag is the main dissipative mechanism: εµ ≫ εν

Numerical Model

ζt + u · ∇ζ = cos x − µζ − ν∇8ζ L = 32(2π) N = 10242 µ = 0.007 ν = 10−5 ε = ζ cos x εint(t) = ζ cos x

x,y

Instantaneous energy injection rate

1000 2000 3000 4000

t/τf

0.0 0.1 0.2 0.3

instantaneous energy injection rate, energy dissipation due to drag hyperviscous energy dissipation

5 10 15 20 Probability density function of 0.1 0.2 0.3

mean = 0.154 standard deviation = 0.019

εint(t)

εint(t)

εint(t) = 1 L2

• ζ(x, y, t) cos x dxdy

ε = 1 t1 − t0 t1

t0

εint(t′) dt′

Energy Injection Rate vs. Drag

0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0

ε µ

ζL = cos x

µ+ν

ε ≈

1 2µ

✚ ✚ ❂ ❏ ❏ ❪ ❆ ❆ ❑ ✛

Scaling Law for ε(µ)

0.001 0.010 0.100 µ 0.1 0.5 ε

L=32(2π), ν=10

−5

, N=1024

2

L=128(2π), ν=2.5x10

• 3

, N=2048

2

4 8 16 32 64

L/2π

0.12 0.14 0.16

0.45

1 3

µ=0.004

/ Results are insensitive to ν and (large enough) L

Theory: The Model

ζ(x, y, t) = A(t) cos(k fx) + B(t) sin(k fx)

• forced mode, ˆ

ζ(x,t)

+ ˜ ζ(x, y, t) ε = (k f/τ f)2 ˆ ζ cos(k fx)

• Random Sweeping Model

ˆ ζt + U ˆ ζx + V ˆ ζy = τ−2

f cos(k fx) − µ ˆ

ζ − η ˆ ζ (1) ε ≈ µU2 + V2 ≈ 2µU2 (2)

Theory: The Model

ζ(x, y, t) = A(t) cos(k fx) + B(t) sin(k fx)

• forced mode, ˆ

ζ(x,t)

+ ˜ ζ(x, y, t) ε = (k f/τ f)2 ˆ ζ cos(k fx)

• Random Sweeping Model

ˆ ζt + U ˆ ζx + V ˆ ζy = τ−2

f cos(k fx) − µ ˆ

ζ − η ˆ ζ (1) ε ≈ µU2 + V2 ≈ 2µU2 (2) advection by large-scale eddies (U, V) isotropic: U2 = V2 = U2

rms

vary on scales ≫ k−1

f

large-eddy turnover time ∼ µ−1 ≫ Urmsk f

Theory: The Model

ζ(x, y, t) = A(t) cos(k fx) + B(t) sin(k fx)

• forced mode, ˆ

ζ(x,t)

+ ˜ ζ(x, y, t) ε = (k f/τ f)2 ˆ ζ cos(k fx)

• Random Sweeping Model

ˆ ζt + U ˆ ζx + V ˆ ζy = τ−2

f cos(k fx) − µ ˆ

ζ − η ˆ ζ (1) ε ≈ µU2 + V2 ≈ 2µU2 (2) advection by large-scale eddies (U, V) isotropic: U2 = V2 = U2

rms

vary on scales ≫ k−1

f

large-eddy turnover time ∼ µ−1 ≫ Urmsk f nonlinear energy transfer out of the forced mode η ≫ µ ≫ ν

Theory: The Solution

ˆ ζt + U ˆ ζx + V ˆ ζy = τ−2

f cos(k fx) − µ ˆ

ζ − η ˆ ζ Neglect µ (η ≫ µ) and seek steady-state solution, U ˆ ζx = τ−2

f cos(k fx) − η ˆ

ζ Since U(x, y) varies on the large scales, ˆ ζ ≈ cos(k fx − φ) τ2

f

• η2 + (Uk f)2

, tan φ = Uk f η

Theory: The Solution

ˆ ζt + U ˆ ζx + V ˆ ζy = τ−2

f cos(k fx) − µ ˆ

ζ − η ˆ ζ Neglect µ (η ≫ µ) and seek steady-state solution, U ˆ ζx = τ−2

f cos(k fx) − η ˆ

ζ Since U(x, y) varies on the large scales, ˆ ζ ≈ cos(k fx − φ) τ2

f

• η2 + (Uk f)2

, tan φ = Uk f η ˆ ζ Forcing

small U φ → 0 large U φ → π/2

Theory: The Scaling Law

typical η Ukf ∼ η Urmskf = transfer rate sweeping rate ≡ α η ∼ shear at kf ∼ ε

1 3

( shear ∼ k [kE(k)]

1 2 )

Urmskf ∼ ε

1 2 µ− 1 2

(ε ≈ 2µU2

rms)

       α ∼ µ

4 9

(anticipate ε ∼ µ

1 3 )

Theory: The Scaling Law

typical η Ukf ∼ η Urmskf = transfer rate sweeping rate ≡ α η ∼ shear at kf ∼ ε

1 3

( shear ∼ k [kE(k)]

1 2 )

Urmskf ∼ ε

1 2 µ− 1 2

(ε ≈ 2µU2

rms)

       α ∼ µ

4 9

(anticipate ε ∼ µ

1 3 )

From definition of ε, ε ∼ ˆ ζ cos(kfx)

• ∼
• η

η2 + (Ukf )2

• Let U′ = U/Urms and η′ = η/η ,

ε ∼ 1 Urms

• αη′

(αη′)2 + U′2 P(U′, η′) dU′dη′

Theory: The Scaling Law

small φ

typical η Ukf ∼ η Urmskf = transfer rate sweeping rate ≡ α η ∼ shear at kf ∼ ε

1 3

( shear ∼ k [kE(k)]

1 2 )

Urmskf ∼ ε

1 2 µ− 1 2

(ε ≈ 2µU2

rms)

       α ∼ µ

4 9

(anticipate ε ∼ µ

1 3 )

From definition of ε, ε ∼ ˆ ζ cos(kfx)

• ∼
• η

η2 + (Ukf )2

• Let U′ = U/Urms and η′ = η/η ,

ε ∼ 1 Urms

• αη′

(αη′)2 + U′2

• δ(U′) as α→0

P(U′, η′) dU′dη′

Theory: The Scaling Law

small φ

typical η Ukf ∼ η Urmskf = transfer rate sweeping rate ≡ α η ∼ shear at kf ∼ ε

1 3

( shear ∼ k [kE(k)]

1 2 )

Urmskf ∼ ε

1 2 µ− 1 2

(ε ≈ 2µU2

rms)

       α ∼ µ

4 9

(anticipate ε ∼ µ

1 3 )

From definition of ε, ε ∼ ˆ ζ cos(kfx)

• ∼
• η

η2 + (Ukf )2

• Let U′ = U/Urms and η′ = η/η ,

ε ∼ 1 Urms

• αη′

(αη′)2 + U′2

• δ(U′) as α→0

P(U′, η′) dU′dη′ So, ε ∼ U−1

rms and ε ≈ 2µU2 rms imply:

ε ∼ µ

1 3

Summary

study energy injection rate ε in two-dimensional turbulence with drag µ and a prescribed small-scale body force discover a new scaling regime: ε ∼ µ

1 3

as µ → 0 random sweeping model suggests energy input is mainly due to regions with small velocity

0.001 0.010 0.100 µ 0.1 0.5 ε

L=32(2π), ν=10

−5

, N=1024

2

L=128(2π), ν=2.5x10

• 3

, N=2048

2

4 8 16 32 64

L/2π

0.12 0.14 0.16

0.45

1 3

µ=0.004

/