Direct Numerical Simulation of Gross-Pitaevskii Turbulence Kyo - PowerPoint PPT Presentation

01234 5/Dec/2005, Warwick Turbulence Symposium 56789 Direct Numerical Simulation of Gross-Pitaevskii Turbulence Kyo Yoshida, Toshihico Arimitsu (Univ. Tsukuba) 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 turbulence). We report some preliminary results of the simulation. Outline of the talk 1 Background (Statistical theory of turbulence) 2 Quantum turbulence 3 Numerical simulation ⊲ ⊳ 2

01234 56789 1 Background (Statistical theory of turbulence) ⊲ ⊳ 3

01234 1.1 Governing equations of Turbulence (Classical) 56789 Navier-Stokes equations ( in real space ) ∂ u −∇ p + ν ∇ 2 u + f , ∂t + ( u · ∇ ) u = ∇ · u = 0 u ( x , t ) :velocity field, p ( x , t ) : pressure field, ν : viscosity, f ( x , t ) : force field. Navier-Stokes equations ( in wave vector space ) � ∂ � � ∂t + νk 2 u i d p d q δ ( k − p − q ) M iab k u a p u b q + f i k = k = − i k = δ ij − k i k j M iab � k a D ib k + k b D ia � D ab , k 2 . k k 2 Symbolically, � ∂ � ∂t + νL u = Muu + f ⊲ ⊳ 4

01234 1.2 Turbulence as a dynamical System 56789 Characteristics of turbulence as a dynamical system • Large number of degrees of freedom • Nonlinear ( modes are strongly interacting ) • Non-equilibrium ( forced and dissipative ) Statistical mechanics of thermal equilibrium states can not be applied to turbulence. • The law of equipartition do not hold. • Probability distribution of physical variables strongly deviates from Gaussian (Gibbs distribution). ⊲ ⊳ 5

01234 1.3 Violation of equipartition (1) 56789 Energy spectrum E ( k ) = 1 � d k ′ δ ( | k ′ | − k ) | u k ′ | 2 2 Inviscid truncated system (ITS) • ν = 0 , f = 0 (energy conserved system) and cutoff wavenumber k c is introduced. • The law of equipartition holds. E ( k ) ∝ k 2 . Navier-Stokes turbulence (NS) • Energy cascades from large scales to small scales. • Kolmogorov spectrum E ( k ) = C k ǫ 2 / 3 k − 5 / 3 . ( ǫ , energy dissipation rate). ⊲ ⊳ 6

01234 1.4 Violation of equipartition (2) 56789 ITS ( ∼ 128 3 modes ) Forced NS ( ∼ 128 3 modes ) 1 1 t=0 t=0 k -5/3 t=T t=T 0.1 0.1 k 2 0.01 0.01 E(k) E(k) 0.001 0.001 1e-04 1e-04 1e-05 1e-05 1e-06 1e-06 1 10 100 1 10 100 k k ⊲ ⊳ 7

01234 1.5 Non-Gaussianity 56789 Longitudinal velocity increment δu ( r ) = u i ( x + r e i ) − u i ( x ) Probability density function (PDF) of δu ( r ) strongly deviates from Gaussian and has long tail for small r ( intermittency ). Gotoh, Fukayama, and Nakano, Phys. Fluids 1 4, 1065 (2002) ⊲ ⊳ 8

01234 1.6 Statistical Theory of Turbulence 56789 cf. (for equilibrium states) Statistical mechanics Thermodynamics Macroscopic variables are related to The macroscopic state is completely microscopic characteristics characterized by the free energy, (Hamiltonian). F ( T, V, N ) . F ( T, V, N ) = − kT log Z ( T, V, N ) Statistical theory of turbulence ? What are the set of variables that characterize the statistical state of How to relate statistical variables to turbulence? Navier-Stokes equations? • ǫ ? (Kolmogorov Theory ?) • Lagrangian Closures? • Fluctuation of ǫ ? (Multifractal models?) ⊲ ⊳ 9

01234 1.7 Classical Turbulence to Quantum Turbulence 56789 The statistical theory of (classical) turbulence is far from complete (to our knowledge). Why quantum turbulence ? • Quantum turbulence may provide a test ground for the existing empirical theories for classical turbulence. • Some new ideas may be obtained from the study of quantum turbulence. – Discrete structure of quantized vortex lines. Reconnection of the vortex line. ⊲ ⊳ 10

01234 56789 2 Quantum turbulence ⊲ ⊳ 11

01234 2.1 Dynamics of order parameter 56789 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 ψ ( x , t ) ∼ O ( N ) ( N : number density of all particles) for T < T c . Dynamics equations of ψ is obtained by neglecting ˆ ψ ′ . ⊲ ⊳ 12

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. ⊲ ⊳ 13

01234 2.3 Superfluid 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 : Superfluid (condensate) density v : Superfluid (condensate) 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 ⊲ ⊳ 14

01234 2.4 Experiments 56789 Maurer and Tabeling, Europhysics Lett. 4 3, 29 (1998) • k − 5 / 3 spectrum is observed in superfluid turbulence (well below T c ). • PDF of velocity increment δu ( r ) = δu ( x + r ) − u ( x ) deviates from Gaussian for small r (Intermittency). ⊲ ⊳ 15

01234 2.5 Preceding Numerical Simulations 56789 • Nore, Abid, and Brachet (1997), Abid et al (2003) • Kobayashi and Tsubota (2005) – With dissipation and random forcing. [ i − γ ( x , t )] ∂ [ −∇ 2 − µ ( t ) + g | ψ ( x , t ) | 2 ] ψ ( x , t ) ∂tψ ( x , t ) = + V ( x , t ) ψ ( x , t ) (in non-normalized form) – E wi ( k ) ∼ k − 5 / 3 is observed. w = √ ρ v E wi ( k ) is the energy spectrum related to the incompressible part of w . ⊲ ⊳ 16

01234 56789 3 Numerical simulation ⊲ ⊳ 17

01234 3.1 Dissipation and Forcing 56789 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 normal viscosity type model. ν = ξ 2 is chosen. – 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. – The forcing acts in the low wavenumber range k < k f . ⊲ ⊳ 18

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 ⊲ ⊳ 19

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 ⊲ ⊳ 20

01234 3.4 Energy in the simulation 56789 100 E E kin E int 10 E wi E wc E q 1 0.1 0.01 0.001 1e-04 0 2 4 6 8 10 12 14 t • E wc > E wi . Different from Kobayashi and Tubota (2005). • Dissipation and forcing are different from those of KT. ⊲ ⊳ 21

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 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 − 5 / 3 , E kin ∼ k 4 / 3 . • E wi ∼ k − 5 / 3 ? ⊲ ⊳ 22

01234 3.6 PDF of the density field 56789 ρ ( x ) = | ψ ( x ) | 2 , � ρ ( x ) = | ψ ( x ) | 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 system is not nearly incompressible ( ρ ∼� = const ). – due to the non-separation of the scales ( L ∼ 100 ξ ) ? ⊲ ⊳ 23

01234 3.7 PDF of order parameter increment 56789 δψ ( r ) = ψ ( x + r ) − ψ ( x ) PDF of Re[ δψ ( r )] 10 r=1.96 ξ r=7.85 ξ r=31.4 ξ 1 r=126 ξ Gaussian 0.1 P Re[ δ ψ (r)] 0.01 0.001 1e-04 1e-05 1e-06 -15 -10 -5 0 5 10 15 Re[ δ ψ (r)] ⊲ ⊳ 24