Improving global stability analysis of Kolmogorov fl ows using - PowerPoint PPT Presentation

Improving global stability analysis of Kolmogorov fl ows using enstrophy Yue-Kin Tsang School of Mathematics and Statistics University of St Andrews William R. Young ( SIO, UCSD) Richard R. Kerswell (Mathematics, Bristol)

Kolmogorov Flows 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 single-scaled sinusoidal body force (at k f = 1) initially used by Kolmogorov (with β = µ = 0) to study bifurcations as Reynolds number increases subsequent work by others: viscous linear stability ( Meshalkin & Sinai ), weakly nonlinear theory ( Sivashinsky ), energy stability ( Fukuta & Murakami ) quasi-2D laboratory experiments with approximate sinusoidal forcing (e.g. Rivera & Wu, Burgess et al. )

Kolmogorov Flows 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 Two-dimensional fl ows: dual conservation of energy E and enstrophy Z , E = 1 Z = 1 � |∇ ψ | 2 � � ( ∇ 2 ψ ) 2 � , 2 2 nonlinear interactions transfer E simultaneously up ( k < 1) and down ( k > 1) scale large-scale dissipation (e.g. sidewall drag, Ekman friction) needed to achieve statistically steady state

Kolmogorov Flows 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 Geophysical fl ows: variation of the Coriolis parameter with latitude modeled using the β -plane approximation Kolmogorov fl ow on a β -plane as a model of zonal jet formation

Kolmogorov Flows 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 Geophysical fl ows: variation of the Coriolis parameter with latitude modeled using the β -plane approximation Kolmogorov fl ow on a β -plane as a model of zonal jet formation We shall fi rst consider the inviscid case: ν = 0

Stability of the Laminar Solution ζ L ( x ) = a cos( x − x β ) x β = tan − 1 β 1 , a = µ � β 2 + µ 2 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 = µ � β 2 + µ 2 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 = µ � β 2 + µ 2 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 ζ t + u ζ x + v ζ y + βψ x = − µζ + cos x STABLE � UNSTABLE 0 µ

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

Energy Method ∇ 2 ψ t + J ( ψ, ∇ 2 ψ ) + βψ x = − µ ∇ 2 ψ + cos x ψ L ( x ) = − a cos( x − x β ) Time evolution equation for ϕ ( ψ = ψ L + ϕ ) : ∇ 2 ϕ t + J ( ψ L , ∇ 2 ϕ ) + J ( ϕ, ∇ 2 ψ L ) + J ( ϕ, ∇ 2 ϕ ) + βϕ x = − µ ∇ 2 ϕ dE ϕ � � � |∇ ϕ | 2 � ϕ J ( ψ L , ∇ 2 ϕ ) − µ dt = ∴ � � = a ϕ x ϕ y cos x − 2 µ E ϕ

Energy Method dE ϕ � � dt = 2 a R [ ϕ ] − µ E ϕ Φ � � ϕ x ϕ y cos x where R [ ϕ ] ≡ ϕ ( t ) � |∇ ϕ | 2 � Now de fi ne R ∗ ≡ max ϕ ∈ Φ R [ ϕ ] Φ : set of all functions satisfying periodic boundary conditions Then, dE ϕ dt < 2 ( a R ∗ − µ ) E ϕ

Energy Method By Gronwall’s inequality, dE ϕ dt < 2 ( a R ∗ − µ ) E ϕ E ϕ ( t ) < E ϕ (0) e 2( a R ∗ − µ ) t ⇒ E ϕ ( t →∞ ) → 0 if a R ∗ − µ < 0 ∴ Neutral condition � R 2 a = 1 ∗ µ 2 − µ 2 ⇒ µ β = R ∗

Variational Results � � ϕ x ϕ y cos x Maximize: R [ ϕ ] ≡ 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 ( a = 2 µ ) β = 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 ( a = 2 µ ) β = 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 � 0 t the most ef fi cient 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 dZ ϕ � � dt = a ϕ x ϕ y cos x − 2 µ Z ϕ Recall, dE ϕ � � dt = a ϕ x ϕ y cos x − 2 µ E ϕ Then, d dt ( E ϕ − Z ϕ ) = − 2 µ ( E ϕ − Z ϕ ) E ϕ ( t ) − Z ϕ ( t ) = e − 2 µ t [ E ϕ (0) − Z ϕ (0)]

Energy-Enstrophy Balance E ϕ ( t ) − Z ϕ ( t ) → 0 as t → ∞ Φ EZ = { ϕ ∈ Φ such that E ϕ = Z ϕ } ⇒ Φ EZ attracts all initial conditions 1.0 Φ ϕ ( t ) 0.5 Z � Φ EZ : E ϕ = Z ϕ µ � 0.40 , 0.00 0.61 , 0.00 0.0 0.0 0.1 0.2 0.3 0.4 0.5 E �

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

Energy-Enstrophy (EZ) Stability ( ν = 0) � 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 ( ν = 0) � 2 0 . 13 µ 2 − µ 2 ( a = 2 . 8 µ ) β = E ϕ 1 3 EZ stability Energy stability Linear stability 0 0 5 10 time 2 STABLE 1.7 β C 1 1 E ϕ ? 1.6 0.0 0.1 0.2 0.3 UNSTABLE B D 0 0.0 0.2 0.4 0.6 0.8 0 0 2 4 time µ

The viscous case: ν > 0 ∇ 2 ψ t + J ( ψ, ∇ 2 ψ ) + βψ x = − µ ∇ 2 ψ + cos x + ν ∇ 2 ζ ψ L ( x ) = − a cos( x − x β ) 1 a = β 2 + ( µ + ν ) 2 � dE ϕ � � dt = a ϕ x ϕ y cos x − 2 µ E ϕ + 2 ν Z ϕ dZ ϕ � � dt = a ϕ x ϕ y cos x − 2 µ Z ϕ + 2 ν P ϕ P ϕ = 1 � |∇ ( ∇ 2 ϕ ) | 2 � 2

Extended Energy-Enstrophy (EEZ) Stability Consider a family of norm with the parameter α : Q ( α ) = (1 − α ) E ϕ + α Z ϕ , 0 � α � 1 d Q � � d t = 2 R Q [ ϕ ; α, ν, a ] − µ Q Global stability : Q ( α ) → 0 as t → ∞ for some α .

Extended Energy-Enstrophy (EEZ) Stability Consider a family of norm with the parameter α : Q ( α ) = (1 − α ) E ϕ + α Z ϕ , 0 � α � 1 d Q � � d t = 2 R Q [ ϕ ; α, ν, a ] − µ Q Global stability : Q ( α ) → 0 as t → ∞ for some α . R ∗ Q ( α, ν, a ) = max ϕ ∈ Φ R Q [ ϕ ; α, ν, a ] For each α , neutral condition : R ∗ Q ( α, ν, a ) = µ 3 STABLE 2 R ∗ R ∗ Q ( α 2 ) < R ∗ Q ( α 1 ) Q ( α 1 ) β 1 ? R ∗ Q ( α 2 ) B 0 0.0 0.2 0.4 0.6 0.8 µ

Extended Energy-Enstrophy (EEZ) Stability Consider a family of norm with the parameter α : Q ( α ) = (1 − α ) E ϕ + α Z ϕ , 0 � α � 1 d Q � � d t = 2 R Q [ ϕ ; α, ν, a ] − µ Q Global stability : Q ( α ) → 0 as t → ∞ for some α . R ∗ Q ( α, ν, a ) = max ϕ ∈ Φ R Q [ ϕ ; α, ν, a ] For each α , neutral condition : R ∗ Q ( α, ν, a ) = µ R ∗ "Optimal" neutral condition : min Q ( α, ν, a ) = µ α 3 STABLE 2 R ∗ R ∗ Q ( α 2 ) < R ∗ Q ( α 1 ) Q ( α 1 ) β 1 ? R ∗ Q ( α 2 ) B 0 0.0 0.2 0.4 0.6 0.8 µ

Extended EZ Stability ( ν = 10 − 3 ) 0.018 Q ( α ) = (1 − α ) E ϕ + α Z ϕ 0.016 0.014 0.012 3 α ∗ 0.010 Linear stability 0.008 EEZ stability 0.006 Energy stability 0.004 2 0.002 0.00 0.01 0.02 ν/µ STABLE β 0.416 1 ? 0.414 l ∗ 0.412 UNSTABLE 0.410 B 0 0.0 0.2 0.4 0.6 0.408 µ 0.00 0.01 0.02 ν/µ

Extended EZ Stability for different ν 3 EZ ( � =0) � 5 Energy � =10 � 5 ��� � =10 � 3 Energy � =10 � 3 EEZ � =10 � 1 Energy 2 � =10 � 1 EEZ � =10 � STABLE 1 ? B 0 0.0 0.2 0.4 0.6 µ

Summary By incorporating information based on the enstrophy, we develop the EZ and EEZ stability method which allows transient growth in E ϕ ( t ) ( ϕ ( t = 0) � Φ EZ ) identi fi es a physically realistic most-unstable disturbance lies closer to the linear stability neutral curve 3 Linear stability EEZ stability Energy stability 2 STABLE β 1 ? UNSTABLE B 0 0.0 0.2 0.4 0.6 µ EZ stability: Tsang & Young, Phys. Fluids 20 , 084102 (2008)