Yue-Kin Tsang Scripps Institution of Oceanography University of - PowerPoint PPT Presentation

Enstrophy-constrained stability analysis of β -plane Kolmogorov flow with drag Yue-Kin Tsang Scripps Institution of Oceanography University of California, San Diego William R. Young

Kolmogorov Flow ζ t + u ζ x + v ζ y + βψ x = − µζ + cos x + ν ∇ 2 ζ velocity: ( u , v ) = ( − ψ y , ψ x ) (2-D periodic domain) vorticity: ζ ( x , y ) = v x − u y = ∇ 2 ψ Kolmogorov flow: sinusoidal forcing (single scale)

Kolmogorov Flow ζ t + u ζ x + v ζ y + βψ x = − µζ + cos x + ν ∇ 2 ζ velocity: ( u , v ) = ( − ψ y , ψ x ) (2-D periodic domain) vorticity: ζ ( x , y ) = v x − u y = ∇ 2 ψ Kolmogorov flow: sinusoidal forcing (single scale) µ = bottom drag quasi-2D experiments: friction from the container walls or surrounding air geophysical flows: Ekman friction

Kolmogorov Flow ζ t + u ζ x + v ζ y + βψ x = − µζ + cos x + ν ∇ 2 ζ velocity: ( u , v ) = ( − ψ y , ψ x ) (2-D periodic domain) vorticity: ζ ( x , y ) = v x − u y = ∇ 2 ψ Kolmogorov flow: sinusoidal forcing (single scale) µ = bottom drag quasi-2D experiments: friction from the container walls or surrounding air geophysical flows: Ekman friction β = gradient of Coriolis parameter along y important in differentially rotating systems

Stability of the Laminar Solution ζ L ( x ) = a cos( x − x β ) x β = tan − 1 β 1 a = , β 2 + µ 2 µ � 10 1 5 0.5 y y 0 0 −0.5 −5 −1 −10 x x µ = 0 . 5 β = 1 . 0 µ = 0 . 1 β = 0 . 0

Stability of the Laminar Solution ζ L ( x ) = a cos( x − x β ) x β = tan − 1 β 1 a = , β 2 + µ 2 µ � 10 1 5 0.5 y y 0 0 −0.5 −5 −1 −10 x x µ = 0 . 5 β = 1 . 0 µ = 0 . 1 β = 0 . 0

Stability of the Laminar Solution ζ L ( x ) = a cos( x − x β ) x β = tan − 1 β 1 a = , β 2 + µ 2 µ � 10 1 5 0.5 y y 0 0 −0.5 −5 −1 −10 x x µ = 0 . 5 β = 1 . 0 µ = 0 . 1 β = 0 . 0 stable unstable

Goal: Neutral Curve ∇ 2 ψ t + J ( ψ, ∇ 2 ψ ) + βψ x = − µ ∇ 2 ψ + cos x STABLE β UNSTABLE 0 µ

Stability Analysis ψ ( x , y , t ) = ψ L ( x ) + ϕ ( x , y , t ) Linear Instability assume infinitesimal disturbance ϕ ∼ e − i ω t ℑ{ ω } > 0 ⇒ ψ L is unstable gives sufficient condition for instability Global Stability (Asymptotic Stability) ϕ is not assumed to be small disturbance energy E ϕ ( t ) = 1 � |∇ ϕ | 2 � → 0 t → ∞ as 2 gives sufficient condition for stability

Energy Method dE ϕ � � a R [ ϕ ] − µ dt = 2 E ϕ Φ � � ϕ x ϕ y cos x R [ ϕ ] ≡ where � |∇ ϕ | 2 � ϕ ( t ) R ∗ ≡ max Now define ϕ ∈ Φ R [ ϕ ] Φ : set of all functions satisfying periodic boundary conditions E ϕ ( t ) < E ϕ (0) e 2( a R ∗ − µ ) t → 0 a R ∗ − µ < 0 Then, if Neutral condition � R 2 a = 1 ∗ µ 2 − µ 2 ⇒ µ β = R ∗

An Optimization Problem � � ϕ x ϕ y cos x R [ ϕ ] ≡ Maximize: over the set Φ . � |∇ ϕ | 2 � Optimal solution R ∗ = R [ ϕ ∗ ] = 1 2 � � l →∞ cos � l ( y + sin x ) � exp l ϕ ∗ ( x , y ) ≈ lim 2 cos 2 x y x

Energy Stability Curve � 1 4 µ 2 − µ 2 β = ( a = 2 µ ) 3 Energy stability 2 STABLE ( a < 2 µ ) β C 1 ? ( a > 2 µ ) B D 0 0.0 0.2 0.4 0.6 0.8 µ

Energy Stability and Linear Stability Curve � 1 4 µ 2 − µ 2 β = ( a = 2 µ ) 3 Energy stability Linear stability 2 STABLE β 1 ? UNSTABLE B 0 0.0 0.2 0.4 0.6 0.8 µ

Limitations of the Energy Method requires E ϕ ( t ) to decrease monotonically for all ϕ , thus excludes transient growth of E ϕ ( t ) E ϕ 0 t 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

Energy-Enstrophy Balance � ( ∇ 2 ϕ ) 2 � Disturbance enstrophy: Z ϕ = 1 2 d dt ( E ϕ − Z ϕ ) = − 2 µ ( E ϕ − Z ϕ ) t → ∞ E ϕ = Z ϕ as 1.0 Φ ϕ ( t ) 0.5 Z ϕ Φ EZ : E ϕ = Z ϕ µ β 0.40 , 0.00 0.61 , 0.00 0.0 0.0 0.1 0.2 0.3 0.4 0.5 E ϕ

Optimization with Constraints � � ϕ x ϕ y cos x R [ ϕ ] ≡ Maximize: � |∇ ϕ | 2 � � |∇ ϕ | 2 � = � ( ∇ 2 ϕ ) 2 � with constraint Optimal solution R ∗ = R [ ϕ ∗ ] = 0 . 3571 e i l y ˜ � � ϕ ∗ ( x , y ) = ℜ l ≈ 0 . 4166 ϕ ( x ) with x y

Energy-Enstrophy (EZ) Stability � 0 . 13 µ 2 − µ 2 ( a = 2 . 8 µ ) β = 3 EZ stability Energy stability Linear stability 2 STABLE β C 1 ? UNSTABLE B D 0 0.0 0.2 0.4 0.6 0.8 µ

Energy-Enstrophy (EZ) Stability 2 � 0 . 13 µ 2 − µ 2 ( a = 2 . 8 µ ) β = E ϕ 3 1 EZ stability Energy stability Linear stability 0 0 5 10 time 2 STABLE β C 1 1 E ϕ ? UNSTABLE B D 0 0.0 0.2 0.4 0.6 0.8 0 0 2 4 time µ

Summary E ϕ ( t ) = Z ϕ ( t ) as t → ∞ , Based on the observation: we develop the Energy -Enstrophy (EZ) stability method which allows transient growth in E ϕ ( t ) ( ϕ ( t = 0) � Φ EZ ) identifies a physically realistic most-unstable disturbance lies closer to the linear stability neutral curve 3 EZ stability Energy stability Linear stability 2 STABLE β C 1 ? UNSTABLE B D 0 0.0 0.2 0.4 0.6 0.8 µ