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

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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

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5/Dec/2005, Warwick Turbulence Symposium

Direct Numerical Simulation of Gross-Pitaevskii Turbulence

Kyo Yoshida, Toshihico Arimitsu (Univ. Tsukuba)

START:⊲

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Abstract

Gross-Pitaevskii (GP) equation describes the dynamics of low-temperature superfluids and Bose-Einstein Condensates (BEC). We performed a numerical simulation of turbulence

  • beying 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 ⊲

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1 Background (Statistical theory of turbulence)

⊳ 3 ⊲

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1.1 Governing equations of Turbulence (Classical) Navier-Stokes equations ( in real space )

∂u ∂t + (u · ∇)u = −∇p + ν∇2u + f, ∇ · u = 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 + νk2

  • ui

k =

  • dpdqδ(k − p − q)M iab

k ua pub q + f i k

M iab

k

= − i 2

  • kaDib

k + kbDia k

  • ,

Dab

k = δij − kikj

k2 .

Symbolically,

∂ ∂t + νL

  • u = Muu + f

⊳ 4 ⊲

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1.2 Turbulence as a dynamical System 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 ⊲

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1.3 Violation of equipartition (1) Energy spectrum

E(k) = 1 2

  • dk′δ(|k′| − k)|uk′|2

Inviscid truncated system (ITS)

  • ν = 0, f = 0 (energy conserved system) and cutoff wavenumber kc is

introduced.

  • The law of equipartition holds. E(k) ∝ k2.

Navier-Stokes turbulence (NS)

  • Energy cascades from large scales to small scales.
  • Kolmogorov spectrum E(k) = Ckǫ2/3k−5/3. (ǫ, energy dissipation rate).

⊳ 6 ⊲

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1.4 Violation of equipartition (2) ITS ( ∼ 1283 modes )

1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1 10 100 E(k) k k2 t=0 t=T

Forced NS ( ∼ 1283 modes )

1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1 10 100 E(k) k k-5/3 t=0 t=T

⊳ 7 ⊲

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1.5 Non-Gaussianity

Longitudinal velocity increment δu(r) = ui(x + rei) − ui(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 14, 1065 (2002)

⊳ 8 ⊲

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1.6 Statistical Theory of Turbulence

  • cf. (for equilibrium states)

Thermodynamics

The macroscopic state is completely characterized by the free energy, F(T, V, N).

Statistical mechanics

Macroscopic variables are related to microscopic characteristics (Hamiltonian). 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 turbulence?

  • ǫ? (Kolmogorov Theory ?)
  • Fluctuation of ǫ? (Multifractal

models?) How to relate statistical variables to Navier-Stokes equations?

  • Lagrangian Closures?

⊳ 9 ⊲

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1.7 Classical Turbulence to Quantum Turbulence 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

  • f the vortex line.

⊳ 10 ⊲

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2 Quantum turbulence

⊳ 11 ⊲

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2.1 Dynamics of order parameter Hamiltonian of locally interacting boson field ˆ ψ(x, t)

ˆ H =

  • dx
  • − ˆ

ψ† ¯ h2 2m∇2 ˆ ψ − µ ˆ ψ† ˆ ψ + g 2 ˆ ψ† ˆ ψ† ˆ ψ ˆ ψ

  • µ : chemical potential,

g: coupling constant

Heisenberg equation

i¯ h∂ ˆ ψ ∂t = −

  • ¯

h2 2m∇2 + µ

  • ˆ

ψ + g ˆ ψ† ˆ ψ ˆ ψ ˆ ψ = ψ + ˆ ψ′, ψ = ˆ ψ ψ(x, t): Order parameter ψ(x, t) ∼ O(N) (N: number density of all particles) for T < Tc. Dynamics equations of ψ is obtained by neglecting ˆ ψ′. ⊳ 12 ⊲

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2.2 Governing equations of Quantum Turbulence Gross-Pitaevskii (GP) equation

i¯ h∂ψ ∂t = − ¯ h 2m∇2 + µ

  • ψ + g|ψ|2ψ,

µ = g¯ n, n = |ψ|2 ¯ · : volume average. Normalization

˜ x = x L, ˜ t = g¯ n ¯ h t, ˜ ψ = ψ √¯ n

Normalized GP equation

i∂ ˜ ψ ∂˜ t = −˜ ξ2 ˜ ∇2 ˜ ψ − ˜ ψ + | ˜ ψ|2 ˜ ψ,

  • ξ =

¯ h √2mg¯ n, ˜ ξ = ξ L

  • ξ: Healing length ( ∼ 0.5 ˚

A in Liquid 4He ) Hereafter, ˜ · is omitted. ⊳ 13 ⊲

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2.3 Superfluid velocity and quantized vortex line

ψ(x, t) =

  • ρ(x, t)eiϕ(x,t),

v(x, t) = 2ξ2∇ϕ(x, t) ∂ ∂tρ + ∇ · (ρv) = ∂ ∂tv + (v · ∇)v = −∇pq

  • pq = 2ξ4ρ − 2ξ4 ∇2√ρ

√ρ

  • ρ : Superfluid (condensate) density

v: Superfluid (condensate) velocity Quantized vortex line (ρ = 0) ω = ∇ × v = 0 (for ρ = 0)

  • C

dl · v = (2πn)2ξ2 (n = 0, ±1, ±2 · · · )

ρ = 0 C

⊳ 14 ⊲

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2.4 Experiments

Maurer and Tabeling, Europhysics Lett. 43, 29 (1998)

  • k−5/3 spectrum is observed in superfluid turbulence (well below Tc).
  • PDF of velocity increment

δu(r) = δu(x + r) − u(x) deviates from Gaussian for small r (Intermittency). ⊳ 15 ⊲

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2.5 Preceding Numerical Simulations

  • Nore, Abid, and Brachet (1997), Abid et al (2003)
  • Kobayashi and Tsubota (2005)

– With dissipation and random forcing. [i − γ(x, t)] ∂ ∂tψ(x, t) = [−∇2 − µ(t) + g|ψ(x, t)|2]ψ(x, t) +V (x, t)ψ(x, t) (in non-normalized form) – Ewi(k) ∼ k−5/3 is observed. w = √ρv Ewi(k) is the energy spectrum related to the incompressible part of w. ⊳ 16 ⊲

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3 Numerical simulation

⊳ 17 ⊲

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3.1 Dissipation and Forcing

GP equation (in wave vector space) i ∂ ∂tψk = ξ2k2ψk − ψk +

  • dpdqdrδ(k + p − q − r)ψ∗

pψqψr

−iνk2ψk+iαkψk

  • Dissipation

– The normal viscosity type model. ν = ξ2 is chosen. – The dissipation term acts mainly in the high wavenumber range ( k ∼> 1/ξ ).

  • Forcing (Pumping of condensates)

αk =    α (k < kf) (k ≥ kf) – α is determined at every time step so as to keep ¯ ρ almost constant. – The forcing acts in the low wavenumber range k < kf. ⊳ 18 ⊲

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3.2 Simulation conditions

  • (2π)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 kmaxξ = 3.
  • ν = ξ2.

N kmax ξ ν(×10−3) kf ∆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 ⊲

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3.3 Energy

Energy density per unit volume E = Ekin + Eint Ekin = 1 V

  • dxξ2|∇ψ|2 =
  • dkξ2k2|ψk|2 =
  • dkEkin(k)

Eint = 1 2V

  • dx(ρ′)2 = 1

2

  • dx|ρ′

k|2 =

  • dkEint(k)

(ρ′ = ρ − ¯ ρ) Ekin = Ewi + Ewc + Eq Ewi = 1 2V

  • dx|wi|2 = 1

2

  • dk|wi

k|2 =

  • dkEwi(k)
  • w =

1 √ 2ξ √ρv

  • Ewc

= 1 2V

  • dx|wc|2 = 1

2

  • dk|wc

k|2 =

  • dkEwc(k)

Eq = 1 V

  • dxξ2|∇√ρ|2 =
  • dkξ2k2|(√ρ)k|2 =
  • dkEq(k)

⊳ 20 ⊲

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3.4 Energy in the simulation

1e-04 0.001 0.01 0.1 1 10 100 2 4 6 8 10 12 14 t E Ekin Eint Ewi Ewc Eq

  • Ewc > Ewi. Different from Kobayashi and Tubota (2005).
  • Dissipation and forcing are different from those of KT.

⊳ 21 ⊲

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3.5 Energy spectrum

1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1 10 100 1000 k k-5/3 k4/3 Ekin(k) Eint(k) 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1 10 100 1000 k k-5/3 Ewi(k) Ewc(k) Eq(k)

  • Eint ∼ k−5/3, Ekin ∼ k4/3.
  • Ewi ∼ k−5/3?

⊳ 22 ⊲

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3.6 PDF of the density field

ρ(x) = |ψ(x)|2,

  • ρ(x) = |ψ(x)|

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 Pn n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.5 1 1.5 2 2.5 3 P/sqrt n /sqrt n

  • The system is not nearly incompressible (ρ ∼= const).

– due to the non-separation of the scales (L ∼ 100ξ) ? ⊳ 23 ⊲

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3.7 PDF of order parameter increment

δψ(r) = ψ(x + r) − ψ(x) PDF of Re[δψ(r)]

1e-06 1e-05 1e-04 0.001 0.01 0.1 1 10

  • 15
  • 10
  • 5

5 10 15 PRe[δ ψ(r)] Re[δ ψ(r)] r=1.96ξ r=7.85ξ r=31.4ξ r=126ξ Gaussian

⊳ 24 ⊲

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3.8 PDF of density increment

δρ(r) = ρ(x + r) − ρ(x) PDF of δρ(r).

1e-06 1e-05 1e-04 0.001 0.01 0.1 1 10

  • 15
  • 10
  • 5

5 10 15 Pδ n(r) δ n(r) r=1.96ξ r=7.85ξ r=31.4ξ r=126ξ Gaussian

⊳ 25 ⊲

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3.9 Low density region

N = 128 ξ = 0.05 ρ < 0.01 ⊳ 26 ⊲

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N = 256 ξ = 0.025 ρ < 0.005 ⊳ 27 ⊲

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4 Summary

Numerical simulations of Gross-Pitaevskii equation with forcing and dissipation are performed up to 5123 grid points.

  • Eint(k) ∼ k−5/3, Ekin(k) ∼ k4/3.
  • Ewi < Ewc.
  • Ewi(k) ∼ k−5/3 is not so clearly observed as in Kobayashi and

Tsubota (2005).

  • PDF of δψ(r) deviates from Gaussian as r decrease.
  • Deviation from Gaussian of PDF of δρ(r) is larger than that
  • f δψ(r).

⊳ 28 ⊲

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5 Future Studies

  • Closure analysis based on ψ, ρ = |ψ|2.

– Can Eint(k) ∼ k−5/3 and Ekin(k) ∼ k4/3 be derived?

  • Investigation of singularities in the physical space.

– Relation between the spatial and temporal structures of quantized vortex lines (reconnection etc.) and intermittency. – Singularity spectrum f(α).

⊳ 29 ⊲