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Yue-Kin Tsang Scripps Institution of Oceanography University of - - PowerPoint PPT Presentation
Yue-Kin Tsang Scripps Institution of Oceanography University of - - PowerPoint PPT Presentation
Enstrophy-constrained stability analysis of -plane Kolmogorov flow with drag Yue-Kin Tsang Scripps Institution of Oceanography University of California, San Diego William R. Young Kolmogorov Flow t + u x + v y + x = +
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Kolmogorov Flow
ζt + uζx + vζy + βψx = −µζ + cos x + ν∇2ζ velocity: (u, v) = (−ψy, ψx) (2-D periodic domain) vorticity: ζ(x, y) = vx − uy = ∇2ψ Kolmogorov flow: sinusoidal forcing (single scale) µ = bottom drag
quasi-2D experiments: friction from the container walls
- r surrounding air
geophysical flows: Ekman friction
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Kolmogorov Flow
ζt + uζx + vζy + βψx = −µζ + cos x + ν∇2ζ velocity: (u, v) = (−ψy, ψx) (2-D periodic domain) vorticity: ζ(x, y) = vx − uy = ∇2ψ Kolmogorov flow: sinusoidal forcing (single scale) µ = bottom drag
quasi-2D experiments: friction from the container walls
- r surrounding air
geophysical flows: Ekman friction
β = gradient of Coriolis parameter along y
important in differentially rotating systems
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Stability of the Laminar Solution
ζL(x) = a cos(x − xβ)
a = 1
- β2 + µ2
, xβ = tan−1 β µ
−1 −0.5 0.5 1
x y µ = 0.5 β = 1.0
−10 −5 5 10
x y µ = 0.1 β = 0.0
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Stability of the Laminar Solution
ζL(x) = a cos(x − xβ)
a = 1
- β2 + µ2
, xβ = tan−1 β µ
−1 −0.5 0.5 1
x y µ = 0.5 β = 1.0
−10 −5 5 10
x y µ = 0.1 β = 0.0
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Stability of the Laminar Solution
ζL(x) = a cos(x − xβ)
a = 1
- β2 + µ2
, xβ = tan−1 β µ
−1 −0.5 0.5 1
x y stable µ = 0.5 β = 1.0
−10 −5 5 10
x y unstable µ = 0.1 β = 0.0
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Goal: Neutral Curve
∇2ψt + J(ψ, ∇2ψ) + βψx = −µ∇2ψ + cos x
µ β
STABLE UNSTABLE
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Stability Analysis
ψ(x, y, t) = ψL(x) + ϕ(x, y, t) Linear Instability assume infinitesimal disturbance ϕ ∼ e−iωt ℑ{ω} > 0 ⇒ ψL is unstable gives sufficient condition for instability Global Stability (Asymptotic Stability) ϕ is not assumed to be small disturbance energy Eϕ(t) = 1 2
- |∇ϕ|2
→ 0 as t → ∞ gives sufficient condition for stability
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Energy Method
Φ ϕ(t) dEϕ dt = 2
- aR[ϕ] − µ
- Eϕ
where R[ϕ] ≡
- ϕx ϕy cos x
- |∇ϕ|2
Now define R∗ ≡ max
ϕ∈Φ R[ϕ]
Φ : set of all functions satisfying periodic boundary conditions Then, Eϕ(t) < Eϕ(0) e2(aR∗−µ)t → 0 if aR∗ − µ < 0 Neutral condition a = 1 R∗ µ ⇒ β =
- R2
∗
µ2 − µ2
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An Optimization Problem
x y
Maximize: R[ϕ] ≡
- ϕx ϕy cos x
- |∇ϕ|2
- ver the set Φ.
Optimal solution R∗ = R[ϕ∗] = 1 2 ϕ∗(x, y) ≈ lim
l→∞ cos l(y + sin x) exp
- l
2 cos 2x
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Energy Stability Curve
β =
- 1
4µ2 − µ2 (a = 2µ)
0.0 0.2 0.4 0.6 0.8 1 2 3
Energy stability
B
STABLE
C D
?
β µ
(a > 2µ) (a < 2µ)
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Energy Stability and Linear Stability Curve
β =
- 1
4µ2 − µ2 (a = 2µ)
0.0 0.2 0.4 0.6 0.8 1 2 3
Energy stability Linear stability
B
STABLE
?
UNSTABLE
β µ
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Limitations of the Energy Method
requires Eϕ(t) to decrease monotonically for all ϕ, thus excludes transient growth of Eϕ(t)
t Eϕ
the most efficient energy-releasing disturbance ϕ∗(x, y) is unphysical: l → ∞ a gap between the energy stability curve and the neutral curve from linear stability analysis
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Energy-Enstrophy Balance
0.0 0.1 0.2 0.3 0.4 0.5
Eϕ
0.0 0.5 1.0
0.40 , 0.00 0.61 , 0.00
µ
Zϕ
β
Φ ϕ(t)
ΦEZ : Eϕ=Zϕ
Disturbance enstrophy: Zϕ = 1
2
- (∇2ϕ)2
d dt(Eϕ − Zϕ) = −2µ(Eϕ − Zϕ) Eϕ = Zϕ as t → ∞
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Optimization with Constraints
x y Maximize: R[ϕ] ≡
- ϕx ϕy cos x
- |∇ϕ|2
with constraint |∇ϕ|2 = (∇2ϕ)2 Optimal solution R∗ = R[ϕ∗] = 0.3571 ϕ∗(x, y) = ℜ
- e i l y ˜
ϕ(x)
- with
l ≈ 0.4166
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Energy-Enstrophy (EZ) Stability
β =
- 0.13
µ2 − µ2 (a = 2.8µ)
0.0 0.2 0.4 0.6 0.8 1 2 3
EZ stability Energy stability Linear stability
B
STABLE
C D
?
UNSTABLE
β µ
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Energy-Enstrophy (EZ) Stability
β =
- 0.13
µ2 − µ2 (a = 2.8µ)
0.0 0.2 0.4 0.6 0.8 1 2 3
EZ stability Energy stability Linear stability
B
STABLE
C D
?
UNSTABLE
β µ
5 10
time
1 2
Eϕ
2 4
time
1
Eϕ
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Summary
0.0 0.2 0.4 0.6 0.8 1 2 3
EZ stability Energy stability Linear stability
B STABLE C D
?
UNSTABLE