numerical simulation of quantum fluid turbulenceyoshida.kyo.fu/presentation/nagoya...numerical...
TRANSCRIPT
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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.
/ 5 .
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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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