Superﬂuid helium and atomic condensates: classical and …profs.scienze.univr.it/~caliari/mmqf09/talks/Barenghi.pdf · 2009. 9. 21. · Superﬂuid helium and atomic condensates:

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

Outline

Energy spectrum

The limit of absolute zero

Velocity statistics

Improving experimental techniques


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

Superfluid turbulence

Vortex line densityL = vortex length/volume

Average intervortex spacingℓ ≈ L−1/2

ξ

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


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) ?


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

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


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.


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

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)


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)


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


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


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 ?


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 ?


Reconnection of superfluid vortex bundles

The vortex bundles retain their identity in a reconnection(Alamri, Youd & Barenghi, PRL 101, 215302, 2008):


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).


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).


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


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.


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


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


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)


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


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


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



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

)



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


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/


Documents

Superﬂuid helium and atomic condensates: classical and …profs.scienze.univr.it/~caliari/mmqf09/talks/Barenghi.pdf · 2009. 9. 21. · Superﬂuid helium and atomic condensates: