Correlation function and response function in shell model of - PowerPoint PPT Presentation

Nonequilibrium Dynamics in Astrophysics and Material Science, 2011/11/2, Kyoto Univ. Correlation function and response function in shell model of turbulence T. Ooshida, M. Otsuki 1 , S. Goto 2 , A. Nakahara 3 , T. Matsumoto 4 Tottori U., Aoyama Gakuin U. 1 , Okayama U. 2 , Nihon U. 3 , Kyoto U. 4 1

Fluctuation response relation • X ( t ) : quantity of some statistically steady-state system (many degrees of freedom) – Auto correlation function ( � � : ensemble average ). C ( t − s ) = � X ( t ) X ( s ) � – Response function to fluctuation f ( t ) � δX ( t ) � G ( t − s ) = δf ( s ) ∫ t In an integral form, X ( t ) + δX ( t ) = X ( t ) + 0 G ( t − s ) f ( s ) ds . • Fluctuation response relation (fluctuation dissipation relation) G ( t − s ) = β C ( t − s ) ( β : inverse temperature in equilibrium systems). • Formal expression of the response function � � � X ( t ) ∂ ln ρ ( X, t ) � G ( t − s ) = − � ∂X � t = s ρ ( X, t ) : probability distribution function of X . If the distribution ρ is not Gaussian, G �∝ C in general! 2

Typical examples • Correlation function C ( t − s ) = � X ( t ) X ( s ) � � � δX ( t ) G ( t − s ) = Response function δf ( s ) 1.2 1.2 G G β C C 1 1 0.8 0.8 C ( t − s ) , G ( t − s ) C ( t − s ) ,G ( t − s ) 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 0 0.002 0.004 0.006 0.008 0.01 0 0.5 1 1.5 2 2.5 t − s t − s Gaussian system G ∝ C non-Gaussian system G �∝ C 3

Implication to statistical theory of turbulence • Incompressible Navier-Stokes eq. with forcing ∂ t u + ( u · ∇ ) u = −∇ p + ν ∇ 2 u + f , ∇ · u = 0 . (1) u ( k , t )e i k · x ] • Expression in the Fourier space [ u ( x , t ) = ∑ k ˆ 3 u j ( k , t ) = − i u j ( k , t ) + ˆ ∑ ∑ u m ( − q , t ) − ν | k | 2 ˆ ∂ t ˆ P jlm ( k ) u l ( − p , t )ˆ ˆ f j ( k , t ) 2 p , q l,m =1 k + p + q = o (2) Holy grail: closure eq. of the correlation func. C jl ( k , t, s ) = � ˆ u j ( k , t )ˆ u l ( − k , s ) � • Direct interaction approximation (DIA), (R.H. Kraichnan 1959) – Decompose ˆ u ⇒ ˆ u + δ ˆ u and get the linearized eq. of δ ˆ u from (2). � δ ˆ � u j ( k , t ) – Response func. of the linearized eq.: G jl ( k , t, s ) = δ ˆ u l ( − k , s ) – Closure eqs. of C and G (under statistical homogeneity and isotropy): [ ∂ t + νk 2 + F 1 ( C, k, t, t ′ )] C ( k, t, t ′ ) = 0 , [ ∂ t + νk 2 + F 1 ( C, k, t, t ′ )] G ( k, t, t ′ ) = 0 , ( ∂ t + 2 νk 2 ) C ( k, t, t ′ ) = ∫ t ∫ dk ′ ∫ dk ′′ F 2 ( k, k ′ , k ′′ ) −∞ dsC ( k ′ , t, t ′ )[ G ( k, t, t ′ ) C ( k ′′ , t, t ′ ) − G ( k ′ , t, t ′ ) C ( k, t, t ′ )] – These closure eqs. are solved by (naturally) assuming G ∝ C . 4

• Our final goal is G and C of turbulence but.... � δ ˆ � u j ( k , t ) • Problem : calculation of the response function G jl ( k , t, s ) = δ ˆ u l ( − k , s ) is numerically costly ! • Less-costly models of turbulence – Turbulence in two dimensions – Burgers equation (Burgers turbulence) ∂ t u + ( u · ∇ ) u = ν ∇ 2 u . – “shell models” of turbulence – ... ∗ shell models (reduced dynamical-system models) · homogeneous, isotropic turbulence (Obukhov 1971; Gledzer 1973; Yamada & Ohkitani 1987). · thermal convection turbulence (Suzuki & Toh 1995). · magnetohydrodynamic (MHD) turbulence (Hattori & Ishizawa 2001). · quantum turbulence (Wacks & Barenghi 2011). · ... 5

Dynamical-system model of turbulence: shell model • Gledzer-Ohkitani-Yamada shell model (Yamada & Ohkitani 1987) ( d ) dt + νk 2 u j ( t ) = i[ k j u j +2 u ∗ j +1 − 1 2 k j − 1 u j +1 u ∗ j − 1 − 1 2 k j − 2 u j − 1 u j − 2 ] + f j j · ∗ complex conjugate. k j = k 0 2 j ; u j ( t ) ∈ C ; j = 1 , . . . , N ; – “shell” : annulus in the wavenumber space 2 j ≤ | k | ≤ 2 j +1 – This is a minimalistic model of the Navier-Stokes eq. in the Fourier space ∑ u ( k , t )e i k · x , ˆ u ( x , t ) = k 3 u n ( k , t ) = − i u m ( − q , t ) + ˆ ∑ ∑ ( ∂ t + ν | k | 2 )ˆ P nlm ( k ) u l ( − p , t )ˆ ˆ f n ( k , t ) . 2 l,m =1 p , q k + p + q = o u j ( t ) of k j is a representative of ˆ u ( k , t ) in the j -th shell k j ≤ | k | ≤ k j +1 . • This Gledzer-Ohkitani-Yamada shell model has a lot of success. 6

• Success of the Gledzer-Ohkitani-Yamada shell model ( d ) dt + νk 2 u j ( t ) = i[ k j u j +2 u ∗ j +1 − 1 2 k j − 1 u j +1 u ∗ j − 1 − 1 2 k j − 2 u j − 1 u j − 2 ] + f j j k j = k 0 2 j ; ( u j ( t ) ∈ C ; j = 1 , . . . , 24 ; f j is zero except f 1 = 0 . 5 + 0 . 5i ). The shell model can reproduce some characteristics of the Navier-Stokes turbulence. 10 2 10 0 10 0 k −5 / 3 − ζ p k j 10 −5 10 −2 〈 |u j | p 〉 |u j | 2 10 −4 k j 10 −10 10 −6 p= 2 , ζ 2 = 0 . 696 10 −15 p= 3 , ζ 3 = 1 . 00 10 −8 p= 4 , ζ 4 = 1 . 28 p= 5 , ζ 5 = 1 . 53 10 −20 10 −10 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 k j k j Energy spectrum | u j | 2 p -th order moments �| u j | p � ∝ k − ζ p j k j • Scaling exponents ζ p of the moments coincides with those of the NS turbulence �{ } p � [ u ( x + r ) − u ( r )] · r ∝ | r | ζ p . | r | 7

Correlation and response functions of the shell model ( d ) dt + νk 2 u j ( t ) = i[ k j u j +2 u ∗ j +1 − 1 2 k j − 1 u j +1 u ∗ j − 1 − 1 2 k j − 2 u j − 1 u j − 2 ] + f j j k j = k 0 2 j ; ( u j ( t ) ∈ C ; j = 1 , . . . , 24 ; f j is zero except f 1 = 0 . 5 + 0 . 5i ) • Correlation and response functions in the inertial range 1.2 G C 1 0.8 C ( t − s ) , G ( t − s ) 0.6 10 0 0.4 k −5 / 3 10 −5 0.2 |u j | 2 k j 10 −10 0 10 −15 -0.2 0 0.5 1 1.5 2 2.5 10 −20 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 t − s k j � δu 11 ( t ) � G ( t − s ) = Re ( δu 11 ( s ) is purely real.) Response function δu 11 ( s ) Auto correlation function C ( t − s ) = � Re[ u 11 ( t )] Re[ u 11 ( s )] � (normalized) �{ Re[ u 11 ( t )] } 2 � 8

• The fluctuation-dissipation relation G ∝ C breaks down for the shell model (Biferale et al . 2002). 10 0 1.2 G Gauss C 1 10 −1 0.8 10 −2 C ( t − s ) , G ( t − s ) 0.6 p.d.f. 10 −3 0.4 10 −4 0.2 10 −5 0 10 −6 -0.2 0 0.5 1 1.5 2 2.5 -10 -5 0 5 10 t − s Re[ u 11 ] / σ • An expression for this discrepancy between G and C for the shell model? ✓ ✏ After trial and error, we find : a relation between G and C can be obtained if we add noises to the shell model dt u j = i[ k j u j +2 u ∗ j +1 − 1 2 k j − 1 u j +1 u ∗ j − 1 − 1 2 k j − 2 u j − 1 u j − 2 ]+ f j − νk 2 d j u j + ξ j . ξ j : Gaussian white noise: � ξ j ( t ) � = 0 , � ξ j ( t ) ξ ∗ ℓ ( s ) � = 2 νk 2 j T δ j,ℓ δ ( t − s ) . ✒ ✑ 9

Property of the shell model with noise dt u j = i[ k j u j +2 u ∗ d j +1 − 1 2 k j − 1 u j +1 u ∗ j − 1 − 1 2 k j − 2 u j − 1 u j − 2 ] − νk 2 j u j + f j + ξ j , � ξ j ( t ) ξ ∗ ℓ ( s ) � = 2 νk 2 j T δ j,ℓ δ ( t − s ) . 10 2 10 0 k -1 10 -2 10 -4 |u j | 2 / k j k -5/3 10 -6 T = 1.0 10 -1 10 -2 10 -8 10 -3 10 -4 10 -10 T = 0 10 -12 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 k j If “the heat bath temperature” T is small enough, the noisy model is close to the Navier-Stokes turbulence. 10

C �∝ G The noisy shell model: correlation and response functions 2 k j − 2 u j − 1 u j − 2 ] − νk 2 d dt u j = i[ k j u j +2 u ∗ j +1 − 1 2 k j − 1 u j +1 u ∗ j − 1 − 1 j u j + f j + ξ j , ℓ ( s ) � = 2 νk 2 � ξ j ( t ) ξ ∗ j T δ j,ℓ δ ( t − s ) . 1 10 2 G 0.8 10 0 <u j (t) u * j (0)>/ <|u j (0)| 2 > k -5/3 10 -2 0.6 C(t), G(t) 10 -4 |u j | 2 / k j 0.4 10 -6 0.2 10 -8 T = 10 -4 0 10 -10 T = 0 10 -12 -0.2 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 0 0.2 0.4 0.6 0.8 1 k j t 10 0 Gauss 10 -1 10 -2 10 -3 P.D.F. 10 -4 10 -5 10 -6 10 -7 10 -8 -15 -10 -5 0 5 10 15 Re[ u 14 ]/ σ 11

Noisy shell model: Correlation function C and response function G 2 k j − 2 u j − 1 u j − 2 ] − νk 2 dt u j = i[ k j u j +2 u ∗ d j +1 − 1 2 k j − 1 u j +1 u ∗ j − 1 − 1 j u j + f j + ξ j , ℓ ( s ) � = 2 νk 2 � ξ j ( t ) ξ ∗ j T δ j,ℓ δ ( t − s ) . 1 G 0.8 H 10 2 <u j (t) u * j (0)>/ <|u j (0)| 2 > 0.6 10 0 C(t), G(t) k -5/3 10 -2 0.4 10 -4 |u j | 2 / k j 0.2 10 -6 10 -8 0 T = 10 -4 10 -10 T = 0 -0.2 10 -12 0 0.2 0.4 0.6 0.8 1 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 t k j δu j ( t ) C jj ( t ) = � u j ( t ) u ∗ G jj ( t ) = j (0) � , δu j (0) , 1 1 [ ] � u j ( t )Λ ∗ j (0) � + � u j (0)Λ ∗ H jj ( t ) = T C jj ( t ) − j ( t ) � . 2 νk 2 j T (Λ j ( t ) = i[ k j u j +2 u ∗ 2 k j − 1 u j +1 u ∗ j +1 − 1 j − 1 − 1 2 k j − 2 u j − 1 u j − 2 ] + f j ) . 12