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56789

11/Sep/2006, IUTAM Symposium NAGOYA 2006“Computational Physics and New Perspectives in Turbulence”

Numerical Simulation ofQuantum Fluid Turbulence

Kyo Yoshida and Toshihico Arimitsu

START:.

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567890 Abstract

Gross-Pitaevskii (GP) equation describes the dynamics oflow-temperature superfluids and Bose-Einstein Condensates(BEC). We performed a numerical simulation of turbulenceobeying GP equation (Quantum fluid turbulence). Some resultsof the simulation are reported.

Outline

1 Background (Statistical theory of turbulence)

2 Quantum Fluid turbulence

3 Numerical Simulation

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567891 Background (Statistical Theory of Turbulence)

Characteristics of (classical fluid) turbulence as a dynamicalsystem 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 testground for developing the statistical theory of turbulence.

– What are in common and what are different betweenclassical and quantum fluid turbulence?

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567892 Quantum fluid turbulence

2.1 Dynamics of order parameter

Hamiltonian of locally interacting boson field ψ(x, t)

H =

∫dx

[−ψ† h2

2m∇2ψ − µψ†ψ +

g

2ψ†ψ†ψψ

]

µ : chemical potential, g: coupling constant

Heisenberg equation

ih∂ψ

∂t= −

(h2

2m∇2 + µ

)ψ + gψ†ψψ

ψ = ψ + ψ′, ψ = 〈ψ〉

ψ(x, t): Order parameter

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567892.2 Governing equations of Quantum Turbulence

Gross-Pitaevskii (GP) equation

ih∂ψ

∂t= −

(h

2m∇2 + µ

)ψ + g|ψ|2ψ,

µ = gn, n = |ψ|2

· : volume average.

Normalizationx =

x

L, t =

gn

ht, ψ =

ψ√n

Normalized GP equation

i∂ψ

∂t= −ξ2∇2ψ − ψ + |ψ|2ψ,

(ξ =

h√

2mgn, ξ =

ξ

L

)

ξ: Healing length ( ∼ 0.5A in Liquid 4He )

Hereafter, · is omitted.

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567892.3 Quantum fluid velocity and quantized vortex line

ψ(x, t) =√

ρ(x, t)eiϕ(x,t), v(x, t) = 2ξ2∇ϕ(x, t)

∂

∂tρ + ∇ · (ρv) = 0

∂

∂tv + (v · ∇)v = −∇pq

(pq = 2ξ4ρ − 2ξ4∇2√ρ

√ρ

)

ρ : Quantum fluid densityv: Quantum fluid velocity

Quantized vortex line (ρ = 0)

ω = ∇× v = 0 (for ρ 6= 0)∫

C

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

ρ = 0

C

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567893 Numerical simulation

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 dissipation term acts mainly in the high wavenumber range (k ∼> 1/ξ ).

• Forcing (Pumping of condensates)

αk =

α (k < kf )

0 (k ≥ kf )

– α is determined at every time step so as to keep ρ almost constant.

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567893.2 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

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567893.3 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)

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567893.4 Energy in the simulation

1e-04

0.001

0.01

0.1

1

10

0 5 10 15 20 25 30 35t

EEkin

Eint

Ewi

Ewc

Eq

Ewi

Ewc

Kobayashi and Tsubota

(J. Phys. Soc. Jpn. 57,

3248(2005))

• Ewc > Ewi. Different from KT.

• Dissipation and forcing are different from those of KT.

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567893.5 Energy spectrum

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

1 10 100 1000k

k-5/3

k-3/2

k4/3

Ekin(k)Eint(k)

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

1 10 100 1000k

k-5/3

Ewi(k)Ewc(k)Eq(k)

• Eint ∼ k−3/2.

– Consistent with the weak turbulence theory.(Dyachenko et. al. Physica D 57 96 (1992))

• Ekin ∼ k4/3.

• Ewi ∼ k−5/3 is not observed.

– Ewi ∼ k−5/3 is observed in KT. Difference in the forcings?

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567893.6 PDF of the density field

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.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

Pn

n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3P

/sqr

t n/sqrt n

The turbulence is not weak?

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567893.7 Low density region

N = 512

ξ = 0.0125

ρ < 0.0025

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567894 Frequency spectrum

Ψk(ω) :=

|ψk,ω|2 + |ψk,−ω|2 (ω 6= 0)

|ψk,ω|2 (ω = 0), ψk,ω :=

1

2π

∫ t0+T

t0

dt ψk(t)e−iω(t−t0).

In the weak turbulence theory, it is assumed that

Ψk(ω) ∼ δ(ω − Ωk), Ωk := ξk√

2 + ξ2k2.

0

500

1000

1500

2000

2500

0 5 10 15 20

Ψk(

ω)

ω / Ωk

k=6k=8

k=10

The assumption is not satisfied, i.e., the turbulence is not weak.

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567895 Summary

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

• Eint(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 Ψk(ω) 6= δ(ω − Ωk) .

– A possible scenario for the explanation of the scaling is to introduce thetime scale of decorrelation τ(k) ∼ Ω−1

k . Closure analysis (DIA, LRA)?

• Ewi(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 littleto Ewi of the system.

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