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 Soap film Paret and Tabeling (1998) Burgess et al. (1999) http://www.fluid.tue.nl driving belt f ( x , t ) ∼ I ( t ) × B ( x ) induced motion drag from bottom wall friction surrounding gas

Forced-dissipative 2D Systems Conducting fluid layer Soap film The Ocean wind ✲ Paret and Tabeling (1998) Burgess et al. (1999) J. M. Toole (1996) http://www.fluid.tue.nl driving belt wind forcing, f ( x , t ) ∼ I ( t ) × B ( x ) induced motion lunisolar tide drag from sea floor drag, bottom wall friction surrounding gas instabilities

Forced-dissipative 2D Systems Conducting fluid layer Soap film The Ocean wind ✲ Paret and Tabeling (1998) Forcing (energy input) Nonlinear interaction (energy redistributed) Dissipation (energy removed) J. M. Toole (1996) 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 − µ v = mdv dt Steady state velocity, v = F µ Energy injection rate (Power input), � � F ∼ µ − 1 ε = Fv = F µ

Dependence of ε on µ Forced harmonic oscillator x + ω 2 ¨ 0 x = − µ ˙ x + A cos ω t instantaneous : ε int ( t ) = ˙ x ( t ) A cos ω t � T ε = 1 averaged : ε int ( t ′ ) dt ′ T 0

Dependence of ε on µ Forced harmonic oscillator x + ω 2 ¨ 0 x = − µ ˙ x + A cos ω t instantaneous : ε int ( t ) = ˙ x ( t ) A cos ω t � T ε = 1 averaged : ε int ( t ′ ) dt ′ T 0 ε ∼ µ − 1 ∼ µ 0 µ ∗ ( ω ) µ

Two-dimensional turbulence ζ t + u · ∇ ζ = f ( x , t ) − µζ + ν ∇ 2 ζ u = ( u , v ) ζ ≡ v x − u y = ∇ 2 ψ Energy injection rate ε = −� ψ f � Power Integral (conservation of energy) ε = µ � u 2 + v 2 � + ν � ζ 2 � ���� � ������ �� ������ � ε ν ε µ small-scale forcing: f = τ − 2 f cos( k f x ), k − 1 ≪ box size f drag is the main dissipative mechanism: ε µ ≫ ε ν

Numerical Model ζ t + u · ∇ ζ = cos x − µζ − ν ∇ 8 ζ L = 32(2 π ) N = 1024 2 µ = 0 . 007 ν = 10 − 5 ε = � ζ cos x � x , y ε int ( t ) = ζ cos x

Instantaneous energy injection rate 0.3 0.3 instantaneous energy injection rate, ε int ( t ) mean = 0.154 energy dissipation due to drag standard deviation = 0.019 hyperviscous energy dissipation 0.2 0.2 0.1 0.1 0.0 0 0 1000 2000 3000 4000 0 5 10 15 20 t/ τ f Probability density function of ε int ( t ) �� ε int ( t ) = 1 ζ ( x , y , t ) cos x dxdy L 2 � t 1 1 ε int ( t ′ ) dt ′ ε = t 1 − t 0 t 0

Energy Injection Rate vs. Drag ζ L = cos x 1.0 µ + ν 1 ε ≈ ✚ 2 µ ✚ ❂ 0.8 ❏ ❪ ❆ ❑ ❏ ❆ 0.6 ε 0.4 ✛ 0.2 0.0 0.0 0.2 0.4 0.8 0.6 µ

Scaling Law for ε ( µ ) 0.16 0.5 0.45 0.14 µ =0.004 0.12 4 8 16 32 64 L /2 π ε / 1 3 −5 2 L =32(2 π ), ν=10 , N =1024 -3 2 0.1 L =128(2 π ), ν=2.5 x 10 , N =2048 0.001 0.010 0.100 µ Results are insensitive to ν and (large enough) L

Theory: The Model + ˜ ζ ( x , y , t ) = A ( t ) cos( k f x ) + B ( t ) sin( k f x ) ζ ( x , y , t ) � ����������������������������� �� ����������������������������� � forced mode, ˆ ζ ( x , t ) ε = ( k f /τ f ) 2 � ˆ � ζ cos( k f x ) Random Sweeping Model ζ t + U ˆ ˆ ζ x + V ˆ f cos( k f x ) − µ ˆ ζ − η ˆ ζ y = τ − 2 ζ (1) ε ≈ µ � U 2 + V 2 � ≈ 2 µ � U 2 � (2)

Theory: The Model + ˜ ζ ( x , y , t ) = A ( t ) cos( k f x ) + B ( t ) sin( k f x ) ζ ( x , y , t ) � ����������������������������� �� ����������������������������� � forced mode, ˆ ζ ( x , t ) ε = ( k f /τ f ) 2 � ˆ � ζ cos( k f x ) Random Sweeping Model ζ t + U ˆ ˆ ζ x + V ˆ f cos( k f x ) − µ ˆ ζ − η ˆ ζ y = τ − 2 ζ (1) ε ≈ µ � U 2 + V 2 � ≈ 2 µ � U 2 � (2) advection by large-scale eddies ( U , V ) isotropic: � U 2 � = � V 2 � = U 2 rms vary on scales ≫ k − 1 f large-eddy turnover time ∼ µ − 1 ≫ U rms k f

Theory: The Model + ˜ ζ ( x , y , t ) = A ( t ) cos( k f x ) + B ( t ) sin( k f x ) ζ ( x , y , t ) � ����������������������������� �� ����������������������������� � forced mode, ˆ ζ ( x , t ) ε = ( k f /τ f ) 2 � ˆ � ζ cos( k f x ) Random Sweeping Model ζ t + U ˆ ˆ ζ x + V ˆ f cos( k f x ) − µ ˆ ζ − η ˆ ζ y = τ − 2 ζ (1) ε ≈ µ � U 2 + V 2 � ≈ 2 µ � U 2 � (2) advection by large-scale eddies ( U , V ) isotropic: � U 2 � = � V 2 � = U 2 rms vary on scales ≫ k − 1 f large-eddy turnover time ∼ µ − 1 ≫ U rms k f nonlinear energy transfer out of the forced mode η ≫ µ ≫ ν

Theory: The Solution ζ t + U ˆ ˆ ζ x + V ˆ ζ y = τ − 2 f cos( k f x ) − µ ˆ ζ − η ˆ ζ Neglect µ ( η ≫ µ ) and seek steady-state solution, U ˆ ζ x = τ − 2 f cos( k f x ) − η ˆ ζ Since U ( x , y ) varies on the large scales, cos( k f x − φ ) Uk f ˆ ζ ≈ , tan φ = � η τ 2 η 2 + ( Uk f ) 2 f

Theory: The Solution ζ t + U ˆ ˆ ζ x + V ˆ ζ y = τ − 2 f cos( k f x ) − µ ˆ ζ − η ˆ ζ Neglect µ ( η ≫ µ ) and seek steady-state solution, U ˆ ζ x = τ − 2 f cos( k f x ) − η ˆ ζ Since U ( x , y ) varies on the large scales, cos( k f x − φ ) Uk f ˆ ζ ≈ , tan φ = � η τ 2 η 2 + ( Uk f ) 2 f large U small U φ → π/ 2 φ → 0 ˆ ζ Forcing

Theory: The Scaling Law η � η � transfer rate typical ∼ sweeping rate ≡ α = Uk f U rms k f  1 1 2 ) � η � ∼ shear at k f ∼ ε ( shear ∼ k [ kE ( k )]  3  4 1  3 ) (anticipate ε ∼ µ α ∼ µ 9  2 µ − 1 1 ( ε ≈ 2 µ U 2  U rms k f ∼ ε rms )  2

Theory: The Scaling Law η � η � transfer rate typical ∼ sweeping rate ≡ α = Uk f U rms k f  1 1 2 ) � η � ∼ shear at k f ∼ ε ( shear ∼ k [ kE ( k )]  3  4 1  3 ) (anticipate ε ∼ µ α ∼ µ 9  1 2 µ − 1 ( ε ≈ 2 µ U 2  U rms k f ∼ ε rms )  2 From definition of ε , � � � ˆ η � ζ cos( k f x ) ε ∼ ∼ η 2 + ( Uk f ) 2 Let U ′ = U / U rms and η ′ = η/ � η � , � αη ′ 1 ( αη ′ ) 2 + U ′ 2 P ( U ′ , η ′ ) dU ′ d η ′ ε ∼ U rms

Theory: The Scaling Law η � η � transfer rate typical ∼ sweeping rate ≡ α = Uk f U rms k f  1 1 2 ) � η � ∼ shear at k f ∼ ε ( shear ∼ k [ kE ( k )]  3  4 1  3 ) (anticipate ε ∼ µ α ∼ µ 9  2 µ − 1 1 ( ε ≈ 2 µ U 2  U rms k f ∼ ε rms )  2 From definition of ε , small φ � � � ˆ η � ζ cos( k f x ) ε ∼ ∼ η 2 + ( Uk f ) 2 Let U ′ = U / U rms and η ′ = η/ � η � , � αη ′ 1 P ( U ′ , η ′ ) dU ′ d η ′ ε ∼ ( αη ′ ) 2 + U ′ 2 U rms � ��������� �� ��������� � δ ( U ′ ) as α → 0

Theory: The Scaling Law η � η � transfer rate typical ∼ sweeping rate ≡ α = Uk f U rms k f  1 1 2 ) � η � ∼ shear at k f ∼ ε ( shear ∼ k [ kE ( k )]  3  4 1  3 ) (anticipate ε ∼ µ α ∼ µ 9  2 µ − 1 1 ( ε ≈ 2 µ U 2  U rms k f ∼ ε rms )  2 From definition of ε , small φ � � � ˆ η � ζ cos( k f x ) ε ∼ ∼ η 2 + ( Uk f ) 2 Let U ′ = U / U rms and η ′ = η/ � η � , � αη ′ 1 P ( U ′ , η ′ ) dU ′ d η ′ ε ∼ ( αη ′ ) 2 + U ′ 2 U rms � ��������� �� ��������� � δ ( U ′ ) as α → 0 So, ε ∼ U − 1 rms and ε ≈ 2 µ U 2 rms imply: 1 ε ∼ µ 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: 1 ε ∼ µ as µ → 0 3 random sweeping model suggests energy input is mainly due to regions with small velocity 0.16 0.5 0.45 0.14 µ =0.004 0.12 4 8 16 32 64 L /2 π ε / 1 3 2 −5 L =32(2 π ), ν=10 , N =1024 -3 2 0.1 L =128(2 π ), ν=2.5 x 10 , N =2048 0.001 0.010 0.100 µ