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Superfluid helium and atomic condensates:classical and quantum aspects of turbulence
Carlo F. Barenghi
Verona, September 2009
AcknowlegdmentsAngela White, Anthony Youd, Nick Proukakis, Sultan Alamri, Yuri Sergeev
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Outline
Energy spectrum
The limit of absolute zero
Velocity statistics
Improving experimental techniques
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Quantum fluids
Quantum fluids available:
superfluid 3He-B
superfluid 4He
atomic Bose–Einstein condensates
Main properties:
superfluid component (zero viscosity)
normal fluid component, viscous, negligible for T → 0
complex order parameter ψ(r, t) = |ψ(r, t)|e iφ(r,t)
superfluid velocity vs = (h/m)∇φ
”vorticity” concentrated into thin filaments of fixed coreradius (ξ
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Superfluid turbulence
Vortex line densityL = vortex length/volume
Average intervortex spacingℓ ≈ L−1/2
ξ
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Classical energy spectrum
source + sink + nonlinear interaction → Kolmogorov spectrum
E (k) = Cǫ2/3k−5/3
k−5/3
1/D 1/η
E(k
)lo
g
log k
Energy injected at scale D and dissipated at scale η
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Superfluid energy spectrum
At high T : Kolmogorov spectrum bothobserved (Maurer & Tabeling, EPL 43,29, 1998) and computed (Kivotides,Vassilicos, Barenghi & Samuels, EPL57, 845, 2002)
Since the normal fluid is turbulent, this is expected (Barenghi,Hulton & Samuels, PRL 89, 275301, 2002), although at small T itis ”the tail wagging the dog”.
What happens in the T → 0 limit (no normal fluid) ?
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Superfluid energy spectrum
Computation of the Kolmogorov spectrum at T = 0: Nore & al,PRL 78, 3896, 1998; Araki & al, PRL 89, 145301, 2002; Kobayashi& Tsubota, PRL 94, 665302, 2005.
Partial polarization of vortex lines for k
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Energy sink at absolute zero
At T = 0 (no normal fluid, viscosity, friction) the sink of kineticenergy is sound.
Mechanisms of sound emission:
Classical mechanism: sound radiation by moving vortices
⇒ Kelvin wave cascade.
Quantum mechanism: sound emission at vortex reconnections
⇒ very dense tangles, atomic BEC
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Quantum mechanism of sound emission
Rarefaction pulse created at reconnection event (Leadbeater,Winiecki, Samuels, Barenghi & Adams, PRL 86, 1410, 2001 andPRA 67, 015601, 2002)
In atomic BECs, the pulses
are relevant, as vortices aretightly packed,
co–exist with radiation.
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Classical mechanism of sound emission
Leadbeater, Samuels, Barenghi &Adams, PRA 67, 015601, 2002.
Barenghi, Parker, Proukakis &Adams, JLTP 138, 629, 2005
Vinen (PRB 64, 13450, 2001): this mechanism is not sufficientunless there exist scales
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The Kelvin wave cascade
Kivotides, Vassilicos, Samuels &Barenghi, PRL 86, 3080, 2001
Reconnections → cusps → Kelvinwaves → shorter Kelvin waves
Kelvin wave cascade shifts energyto k large enough for soundemission
(Svistunov, PRB 52, 3647, 1995;Vinen & el, PRL, 91, 135301,2003; Kozik & Svistunov, PRL92, 035301, 2004)
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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From Kolmogorov to Kelvin
k > 1/ℓ: Kelvin wave cascadeIs there a bottleneck between thetwo cascades ?
yes (L’vov, Nazarenko & al, PRB 76, 024520, 2007)
no (Kozik & Svistunov PRL 100, 195302, 2008)
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Classical vortex stretching
In classical turbulence vortexstretching is held responsible forthe generation of small scales
Dω
Dt= (ω · ∇)v + ν∇2ω
where (ω · ∇)v‖ = |ω|dv‖
ds
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Superfluid vortex stretching ?
Can quantum vortices stretch ?No, not in the classical sense,because the core radius ξis fixed.However:
At T 6= 0 a vortex ring or aKelvin wave can extractenergy from the normal fluidand become longer
At T = 0 the vortex canbecome longer by changingits geometry (turning someinteraction energy intolength).
Vn
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Coherent structures
In classical turbulencevorticity is concentrated intubular regions (She & al,Nature 344, 266, 1990;Vincent & Meneguzzi, JFM225, 1, 1992 ; Farge & al,PRL 87, 054501, 2001;Goto, JFM 605, 355, 2008)
Are there coherentstructures in quantumturbulence ?
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Numerical evidence for coherent structures at T > 0
Kivotides, PRL 96, 175301, 2008
Morris and Koplik, PRL 101,015301, 2008
For T > 0 (classical) normal fluid coherent structures caninduce superfluid vortex bundles (Barenghi & al, PoF, 9,2631, 1997)
Do vortex bundles form also in the T = 0 case, without thenormal fluid ?
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Reconnection of superfluid vortex bundles
The vortex bundles retain their identity in a reconnection(Alamri, Youd & Barenghi, PRL 101, 215302, 2008):
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Reconnection of superfluid vortex bundles
Total length vs time, average curvature vs time, PDF of curvaturevs time (Alamri, Youd & Barenghi, PRL 101, 215302, 2008).
Note the large amount of streching, and the development of Kelvinwaves, as in Kerr’s Euler calculation (Nonlinearity 9, 271, 1996),confirming results on the generation of helicity in near singularevents of the Euler equation (Holm & Kerr, PRL 88, 244501,2002).
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Energy per unit length
The energy per unit length is a constant only for an isolatedstraight vortex
Toroidal, Um,1, and poloidal,U1,m vortex unknots, vortex knotT3,2 (Ricca, Samuels & Barenghi,JFM 391, 29, 1999).
Energy per unit length vsknot–type (Maggioni, Alamri,Barenghi & Ricca 2009).
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Ambiguity of the Kolmogorov spectrum
This flow is very smooth:
u(x, t) =
n=N∑
n=1
(Cn×k̂n cos (kn · x + ωnt)+Dn×k̂n sin (kn · x + ωnt))
and satisfies the Kolmogorov k−5/3 law:
Magnetic structures produced by the small-scale dynamo,Wilkin, Barenghi & Shukurov, PRL 99, 134501, 2007
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Ambiguity of the Kolmogorov spectrum
Very differently, in this flow thevorticity is concentrated in thinviscous tubes:
but the Kolmogorov law is stillsatisfied !
Interaction of solid particles with a tangle of vortex filaments in a
viscous fluid Kivotides, Barenghi, Sergeev & Mee, PRL 99, 074501,2007;Magnetic field generation by coherent turbulence structures
Kivotides, Mee & Barenghi, New J. Phys. 9, 291, 2007.
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Classical velocity statistics
PDF of velocity components are Gaussian
ExperimentNoullez, Frisch & al
J. Fluid Mech. 339, 287, 1997
CalculationVincent & Meneguzzi
J. Fluid Mech. 225, 1, 1991
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Quantum velocity statistics: experiment
Velocity components measured in superfluid helium using solidhydrogen tracers are non-Gaussian (Lathrop, Sreenivasan & al)
Vortex lattice in rotating heliumNature, 441, 588, 2006
PDF of vx and vy in turbulent heliumPRL 101, 154501, 2008
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Quantum velocity statistics: calculation
Turbulence in 3-dim atomic condensate:
i∂
∂tψ =
(
−1
2∇2 +
1
2r2 + C |ψ|2
)
ψ,
∫
dr|ψ|2 = 1.
PDF of velocity components are non-Gaussian(White, Proukakis & Barenghi 2009)
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Quantum velocity statistics
Simplify the problem: turbulence in 2-dim condensate. Again, wefind non-Gaussian PDF
No role played by vortex reconnections
Simplify further: 2-dim vortex points: again, PDF is non-Gaussian
• non-Gaussian PDF of velocity components is the signature ofquantum turbulence
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Pressure spectrum
Corresponding to the k−5/3 Kolmogorov spectrum, the spectrumof classical pressure fluctuations scales as Ep ∼ k
−7/3.
In superfluid helium it is Ep ∼ k−2 (Kivotides, Vassilicos, Barenghi,
Khan & Samuels PRL 87, 275301, 2001)
• Another signature of quantum turbulence
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Do tracers disturb vortices ?
Vortex ring of radius R containing N buoyant tracers of radius a
• Magnus force per unit lengthfM = ρsκ × (vL − vs − vsi )
• Drag force per unit lengthfD = ρsκ[Γ0(vn−vL)+Γ
′0κ̂×(vn−vL)]
• Stokes force induced by tracersFS = 6πaνnρnN(vn − vL)
Self-induced velocity:
vi =κ
4πR
[
ln (8Ri
ξ) −
1
2
]
Bare vortex ring:
2πR(fM + fD) = 0
Vortex ring with tracers:
drL
dt= vL
mNdvL
dt= 2πR(fM + fD) + FS
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Do tracers disturb vortices ?
unit of length: R0 = R(0)unit of speed: V0 = κ/R0unit if time: R20/κ
Dimensionless equations:
Ṙ = u
ż = v
v − vs − vi − Γ0u + Γ′0(vn − v) =
1
R(ǫu̇ + ζu)
−(1 − Γ ′0)u + Γ0(vn − v) =1
R(ǫv̇ + ζv)
where ǫ =2N
3
(
a
R0
)3 ( ρ
ρs
)
, ζ = 3N
(
R0
a
)(
ρnρs
)
(νnκ
)
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Do tracers disturb vortices ?
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2
R
t
a
bcd
e
R vs t
10
20
30
40
50
60
70
80
90
100
110
1.6 1.7 1.8 1.9 2 2.1 2.2
β
T
β vs T
Superfluid Stokes number S =FS
FD=
(
Na
2πR
)(
6π
Γ0
(νnκ
)
(
ρnρs
))
= δβ
Barenghi & Sergeev, PRB 80, 024514, 2009
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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Conclusions:
Is quantum turbulence ”classical” (the result of many quanta) ?
Energy spectrum is classical (but it may not be significant)
Velocity statistics and pressure spectrum are not classical, butrather the signature of quantum turbulence
References:
Alamri, Youd & Barenghi, PRL 101, 215302, 2008
Barenghi & Sergeev, PRB 80, 024514, 2009
Maggioni, Barenghi & Ricca, Nuovo Cimento C 32, 133, 2009
Roche, Barenghi & Leveque, to appear in EPL, 2009
http://www.mas.ncl.ac.uk/∼ncfb/
Carlo F. Barenghi Superfluid helium and atomic condensates: classical and quantum
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