Numerical Simulation of Quantum Fluid Turbulence Kyo Yoshida and - PowerPoint PPT Presentation

01234 11/Sep/2006, IUTAM Symposium NAGOYA 2006 “Computational Physics and New Perspectives in Turbulence” 56789 Numerical Simulation of Quantum Fluid Turbulence Kyo Yoshida and Toshihico Arimitsu START: ⊲

01234 0 Abstract 56789 Gross-Pitaevskii (GP) equation describes the dynamics of low-temperature superfluids and Bose-Einstein Condensates (BEC). We performed a numerical simulation of turbulence obeying GP equation (Quantum fluid turbulence). Some results of the simulation are reported. Outline 1 Background (Statistical theory of turbulence) 2 Quantum Fluid turbulence 3 Numerical Simulation ⊲ ⊳ 2

01234 1 Background (Statistical Theory of Turbulence) 56789 Characteristics of (classical fluid) turbulence as a dynamical system are • Large number of degrees of freedom • Nonlinear ( modes are strongly interacting ) • Non-equilibrium ( forced and dissipative ) Why quantum fluid turbulence ? • Another example of such a dynamical system. Another test ground for developing the statistical theory of turbulence. – What are in common and what are different between classical and quantum fluid turbulence? ⊲ ⊳ 3

01234 2 Quantum fluid turbulence 56789 2.1 Dynamics of order parameter Hamiltonian of locally interacting boson field ˆ ψ ( x , t ) � � h 2 ψ † ¯ ψ + g � 2 m ∇ 2 ˆ ψ † ˆ ψ † ˆ ψ † ˆ ˆ − ˆ ψ − µ ˆ ˆ ψ ˆ H = d x ψ 2 µ : chemical potential, g : coupling constant Heisenberg equation h∂ ˆ � � h 2 ψ ¯ 2 m ∇ 2 + µ ψ † ˆ ψ + g ˆ ˆ ψ ˆ i ¯ ∂t = − ψ ψ = ψ + ˆ ˆ ψ = � ˆ ψ ′ , ψ � ψ ( x , t ) : Order parameter ⊲ ⊳ 4

01234 2.2 Governing equations of Quantum Turbulence 56789 Gross-Pitaevskii (GP) equation � ¯ h∂ψ h � 2 m ∇ 2 + µ ψ + g | ψ | 2 ψ, i ¯ = − ∂t n = | ψ | 2 µ = g ¯ n, ¯ · : volume average. Normalization t = g ¯ n ψ x = x ˜ ˜ ˜ √ ¯ L, h t, ψ = ¯ n Normalized GP equation i∂ ˜ ψ � h ¯ ξ = ξ � ξ 2 ˜ ∇ 2 ˜ ψ | 2 ˜ t = − ˜ ψ − ˜ ψ + | ˜ ˜ ψ, ξ = √ 2 mg ¯ n, ∂ ˜ L ξ : Healing length ( ∼ 0 . 5 ˚ A in Liquid 4 He ) Hereafter, ˜ · is omitted. ⊲ ⊳ 5

01234 2.3 Quantum fluid velocity and quantized vortex line 56789 � ρ ( x , t ) e iϕ ( x ,t ) , v ( x , t ) = 2 ξ 2 ∇ ϕ ( x , t ) ψ ( x , t ) = ∂ ∂tρ + ∇ · ( ρ v ) = 0 p q = 2 ξ 4 ρ − 2 ξ 4 ∇ 2 √ ρ � � ∂ ∂t v + ( v · ∇ ) v = −∇ p q √ ρ ρ : Quantum fluid density v : Quantum fluid velocity Quantized vortex line ( ρ = 0 ) ρ = 0 ω = ∇ × v = 0 ( for ρ � = 0) C � (2 πn )2 ξ 2 d l · v = ( n = 0 , ± 1 , ± 2 · · · ) C ⊲ ⊳ 6

01234 3 Numerical simulation 56789 3.1 Dissipation and Forcing GP equation (in wave vector space) i ∂ � ξ 2 k 2 ψ k − ψ k + d p d q d r δ ( k + p − q − r ) ψ ∗ ∂tψ k = p ψ q ψ r − iνk 2 ψ k + iα k ψ k • Dissipation – The dissipation term acts mainly in the high wavenumber range ( k ∼ > 1 /ξ ). • Forcing (Pumping of condensates)  α ( k < k f )  α k = 0 ( k ≥ k f )  – α is determined at every time step so as to keep ¯ ρ almost constant. ⊲ ⊳ 7

01234 3.2 Simulation conditions 56789 • (2 π ) 3 box with periodic boundary conditions. • An alias-free spectral method with a Fast Fourier Transform. • A 4th order Runge-Kutta method for time marching. • Resolution k max ξ = 3 . • ν = ξ 2 . ν ( × 10 − 3 ) N k max ξ k f ∆ t ρ ¯ 128 60 0.05 2.5 2.5 0.01 0.998 256 120 0.025 0.625 2.5 0.01 0.999 512 241 0.0125 0.15625 2.5 0.01 0.998 ⊲ ⊳ 8

01234 3.3 Energy 56789 Energy density per unit volume E kin + E int E = 1 � � � d x ξ 2 |∇ ψ | 2 = d k ξ 2 k 2 | ψ k | 2 = E kin dkE kin ( k ) = V 1 d x ( ρ ′ ) 2 = 1 � � � k | 2 = ( ρ ′ = ρ − ¯ E int dkE int ( k ) d x | ρ ′ = ρ ) 2 V 2 E wi + E wc + E q E kin = 1 d x | w i | 2 = 1 � 1 √ ρ v � � � � k | 2 = E wi d k | w i dkE wi ( k ) √ = w = 2 V 2 2 ξ 1 d x | w c | 2 = 1 � � � k | 2 = E wc d k | w c dkE wc ( k ) = 2 V 2 1 � d x ξ 2 |∇√ ρ | 2 = � d k ξ 2 k 2 | ( √ ρ ) k | 2 = � E q dkE q ( k ) = V ⊲ ⊳ 9

01234 3.4 Energy in the simulation 56789 10 E E kin E int 1 E wi E wc E q 0.1 E wi 0.01 E wc 0.001 Kobayashi and Tsubota 1e-04 0 5 10 15 20 25 30 35 t (J. Phys. Soc. Jpn. 57 , 3248(2005)) • E wc > E wi . Different from KT. • Dissipation and forcing are different from those of KT. ⊲ ⊳ 10

01234 3.5 Energy spectrum 56789 1 1 E kin (k) E wi (k) k -5/3 E int (k) E wc (k) E q (k) 0.1 0.1 0.01 0.01 k -3/2 k 4/3 0.001 0.001 k -5/3 1e-04 1e-04 1e-05 1e-05 1e-06 1e-06 1e-07 1e-07 1 10 100 1000 1 10 100 1000 k k • E int ∼ k − 3 / 2 . – Consistent with the weak turbulence theory. (Dyachenko et. al. Physica D 57 96 (1992)) • E kin ∼ k 4 / 3 . • E wi ∼ k − 5 / 3 is not observed. – E wi ∼ k − 5 / 3 is observed in KT. Difference in the forcings? ⊲ ⊳ 11

01234 3.6 PDF of the density field 56789 ρ ( x , t ) = | ψ ( x , t ) | 2 , � ρ ( x , t ) = | ψ ( x , t ) | In the weak turbulence theory, ρ ( x , t ) = ρ + δρ ( x , t ) , | δρ | ≪ ρ. 1.2 0.8 0.7 1 0.6 0.8 0.5 P / sqrt n P n 0.6 0.4 0.3 0.4 0.2 0.2 0.1 0 0 0 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 n /sqrt n The turbulence is not weak? ⊲ ⊳ 12

01234 3.7 Low density region 56789 N = 512 ξ = 0 . 0125 ρ < 0 . 0025 ⊲ ⊳ 13

01234 4 Frequency spectrum 56789  | ψ k ,ω | 2 + | ψ k , − ω | 2 � t 0 + T ( ω � = 0) ψ k ,ω := 1  dt ψ k ( t ) e − iω ( t − t 0 ) . Ψ k ( ω ) := , 2 π | ψ k ,ω | 2 ( ω = 0) t 0  In the weak turbulence theory, it is assumed that � 2 + ξ 2 k 2 . Ψ k ( ω ) ∼ δ ( ω − Ω k ) , Ω k := ξk 2500 k=6 k=8 2000 k=10 1500 Ψ k ( ω ) 1000 500 0 0 5 10 15 20 ω / Ω k The assumption is not satisfied, i.e. , the turbulence is not weak. ⊲ ⊳ 14

01234 5 Summary 56789 Numerical simulations of Gross-Pitaevskii equation with forcing and dissipation are performed up to 512 3 grid points. • E int ( k ) ∼ k − 3 / 2 . – The scaling coincides with that in the weak turbulence theory. However, it is found that the turbulence is not weak, i.e. , | δρ | ∼ ρ and Ψ k ( ω ) � = δ ( ω − Ω k ) . – A possible scenario for the explanation of the scaling is to introduce the time scale of decorrelation τ ( k ) ∼ Ω − 1 k . Closure analysis (DIA, LRA)? • E wi ( k ) ∼ k − 5 / 3 is not so clearly observed. – The present result is different from that in Kobayashi and Tsubota (2005). Presumably, the forcing in the present simulation injects little to E wi of the system. ⊲ ⊳ 15