Numerical simulations of superfluid
vortex turbulence
Vortex dynamics in superfluid helium and BEC
Makoto TSUBOTA
Osaka City University, Japan
Collaborators: UK W.F. Vinen, C.F. Barenghi
Finland M. Krusius, V.B. Eltsov, G.E. Volovik, A.P.Finne
Russia S.K. Nemirovskii
CR L. Skrbek
Tokyo M. Ueda
..
Osaka T. Araki, K. Kasamatsu, R.M. Hanninen,
M. Kobayashi, A. Mitani
Galaxy
Vortex tangle in superfluid helium
Vortices in Japanese sea
Vortex lattice in a rotating Bose
condensate
Outline
1. Introduction
2. Classical turbulence
3. Dynamics of quantized vortices in superfluid helium
3-1 Recent interests in superfluid turbulence
3-2 Energy spectrum of superfluid turbulence
3-3 Rotating superfluid turbulence
4. Dynamics of quantized vortices in rotating BECs
4-1 Vortex lattice formation
4-2 Giant vortex in a fast rotating BEC
1. Introduction
A quantized vortex is a vortex of superflow in a BEC.
(i) The circulation is quantized.
v
s
 ds   n
n = 0,1,2, 
 h/m
A vortex with n≧2 is unstable.
Every vortex has the same circulation.
(ii) Free from the decay mechanism of the viscous diffusion of the vorticity.
The vortex is stable.
〜Å
ρ s (r)
(iii) The core size is very small.
The order of the coherence
length.
rot v s
r
How to describe the vortex dynamics
Vortex filament formulation (Schwarz)
Biot-Savart law
 s  r  d s
vs r   
3
4
sr
r
s
A vortex makes the superflow of the Biot-Savart law, and moves with
this flow. At a finite temperature, the mutual friction should be
considered. Numerically, a vortex is represented by a string of points.
The Gross-Pitaevskii equation for the macroscopic wave function
(r )  n0 (r)e
(r,t)  2 2
2 
i
 
 Vext (r)  g (r,t) (r,t)
t
 2m

i (r )
What is the difference of vortex dynamics between
superfluid helium and an atomic BEC?
Superfluid helium (4He)
Core size (healing length) ξ〜Å ≪ System size L〜㎜
The vortex dynamics is local, compared with the scale of the whole
system.
Atomic Bose-Einstein condensate
Core size (healing length) ξ〜 subμm ≦ System size L〜 μm
The vortex dynamics is closely coupled with the collective motion of
the whole condensate.
Superfluid helium
Liquid 4He enters the superfluid state at
2.17 K with Bose condensation.
Its hydrodynamics is well described by
the two fluid model (Landau).
point
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Temperature (K)
Superfluid helium becomes dissipative when it flows above a critical
velocity.
1955 Feynman Proposing “superfluid turbulence” consisting of a tangle
of quantized vortices.
1955, 1957 Hall and Vinen
Observing “superfluid turbulence”
The mutual friction between the vortex
tangle and the normal fluid causes the
dissipation.
Lots of experimental studies were done chiefly for thermal
counterflow of superfluid 4He.
Vortex tangle
Heater
Normal flow
Super flow
1980’s K. W. Schwarz Phys.Rev.B38, 2398(1988)
Made the direct numerical simulation of the three-dimensional
dynamics of quantized vortices and succeeded in explaining
quantitatively the observed temperature difference △T .
Three-dimensional dynamics of quantized vortex filaments (1)
Superfluid flow made by a vortex
 s  r  d s
vs r   
3
4
sr
Biot-Savart law
s
In the absence of friction, the vortex moves with its local velocity.
d s0
dt
 vs s
The mutual friction with the normal flow vis
n considered at a finite
temperature.
d s d s0

dt
dt
ds0
  s ( vn 
)
dt
When two vortices approach, they are made to reconnect.
r
Three-dimensional dynamics of quantized vortex filaments (2)
Vortex reconnection

l
l  
This was confirmed by the numerical analysis of the GP equation.
J. Koplik and H. Levine, PRL71, 1375(1993).
Three-dimensional dynamics of quantized vortex filaments (1)
Superfluid flow made by a vortex
 s  r  d s
vs r   
3
4
sr
Biot-Savart law
s s
In the absence of friction, the vortex moves with its local velocity.
d s0
dt
 vs s
The mutual friction with the normal flow vis
n considered at a finite
temperature.
d s d s0

dt
dt
ds0
  s ( vn 
)
dt
When two vortices approach, they are made to reconnect.
rr
Development of a vortex tangle in a thermal counterflow
Schwarz, Phys.Rev.B38, 2398(1988).
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vs
vn
Schwarz obtained numerically the
statistically steady state of a vortex
tangle which is sustained by the
competition between the applied
flow and the mutual friction. The
obtained vortex density L(vns, T)
agreed quantitatively with
experimental data.
Most studies of superfluid turbulence were devoted to thermal
counterflow.
⇨ No analogy with classical turbulence
When Feynman showed the above figure, he thought of a
cascade process in classical turbulence.
What is the relation between superfluid
turbulence and classical turbulence？
2. Classical turbulence
When we raise the flow velocity around a sphere,
The understanding and control of turbulence have been one of the
most important problems in fluid dynamics since Leonardo Da Vinci,
but it is too difficult to do it.
Classical turbulence and vortices
Gird turbulence
Numerical analysis of the Navier-Stokes equation made by Shigeo Kida
Vortex cores are visualized by tracing pressure minimum in the fluid.
Classical turbulence and vortices
・The vortices have different
circulation and different core size.
・The vortices repeatedly appear,
diffuse and disappear.
It is difficult to identify
each vortex!
Compared with
quantized vortices.
Numerical analysis of the Navier-Stokes equation made by Shigeo Kida
Vortex cores are visualized by tracing pressure minimum in the fluid.
Classical turbulence
Energy spectrum of the velocity field
E   2 v2 dr   E(k)dk
Energy-containing range
The energyRichardson
is injected into
the system
at k  k0  1/
cascade
process
Kolmogorov law
0
..
Inertial range
Dissipation does not work. The nonlinear
interaction transfers the energy from low k
region to high k region.
Energycontaining
range
Kolmogorov law :
Inertial
range
Energy-dissipative
range
Energy spectrum of turbulence
E(k)=Cε2/3 k- -5/3
Energy-dissipative range
The energy is dissipated with the rate ε at the
Kolmogorov wave number kc = (ε/ν3 )1/4.
Kolmogorov spectrum in classical turbulence
Experiment
Numerical analysis
Vortices in superfluid turbulence
• ST consists of a tangle of quantized vortex
filaments
Characteristics of quantized vortices
•Quantization of the circulation
•Very thin core
•No viscous diffusion of the vorticity
•A quantized vortex is a stable and definite defect, compared
with vortices in a classical fluid. The only alive freedom is
the topological configuration of its thin cores.
•Because of superfluidity, some dissipation would work only
at large wave numbers (at very low temperatures).
The tangle may give an interesting model with the Kolmogorov law.
Summary of the motivation
Classical turbulence
？ Vortices ？
Superfluid turbulence
Quantized vortices
It is possible to consider
quantized vortices as elements
in the fluid and derive the
essence of turbulence.
3. Dynamics of quantized vortices in superfluid helium
3-1 Recent interests in superfluid turbulence
Does ST mimic CT or not?
• Maurer and Tabeling, Europhysics. Letters. 43, 29(1998) T =1.4, 2.08, 2.3K
Measurements of local pressure in flows driven by two counterrotating
disks finds the Kolmogorov spectrum.
• Stalp, Skrbek and Donnely, Phys.Rev.Lett. 82, 4831(1999) 1.4 < T < 2.15K
Decay of grid turbulence The data of the second sound attenuation was
consistent with a classical model with the Kolmogorov spectrum.
• Vinen, Phys.Rev.B61, 1410(2000) Considering the relation between
ST and CT The Oregon’s result is understood by the coupled
dynamics of the superfluid and the normal fluid due to the mutual
friction. Length scales are important, compared with the
characteristic vortex spacing in a tangle.
• Kivotides, Vassilicos, Samuels and Barenghi, Europhy. Lett. 57, 845(2002)
When superfluid is coupled with the normal-fluid turbulence that
obeys the Kolmogorov law, its spectrum follows the Kolmogorov law
too.
What happens at very low temperatures?
Is there still the similarity or not?
Our work attacks this problem directly!
3-2. Energy spectrum of superfluid turbulence
Decaying Kolmogorov turbulence in a model of superflow
C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)
By solving the Gross-Pitaevskii equation, they studied the energy spectrum of
a Tarlor-Green flow. The spectrum shows the -5/3 power on the way of the
decay, but the acoustic emission is concerned and the situation is complicated.
Energy Spectrum of Superfluid Turbulence with No NormalFluid Component
T. Araki, M.Tsubota and S.K.Nemirovskii, Phys.Rev.Lett.89, 145301(2002)
The energy spectrum of a Taylor-Green vortex was obtained under the vortex
filament formulation. The absolute value with the energy dissipation rate was
consistent with the Kolmogorov law, though the range of the wave number
was not so wide.
C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)
t＝２
Although the total energy is
conserved, its incompressible
component is changed to the
compressible one (sound
waves).
４
The total length of vortices
increases monotonically, with
the large scale motion
decaying.
８
６
１０
１２
C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)
△: 2 < k < 12
○: 2 < k < 14
□: 2 < k < 16
n(t)
E(k)
E(k)=A(t) k -n(t)
5/3
t
k
The right figure shows the energy spectrum att=5.5
a moment.
でのエネルギースペクトル
The left figure （非
shows the development of the exponent n(t) in圧縮性運動エネルギー）
the spectrum E(k)=A(t) k n(t) . The exponent n(t) goes through 5/3 on the way of the dynamics.
Energy spectrum of superfluid turbulence with no normal-fluid component
T.Araki, M.Tsubota, S.K.Nemirovskii, PRL89, 145301(2002)
Energy spectrum of a vortex
tangle in a late stage
E(k) [cm 3/sec 2]
We calculated the vortex filament
dynamics starting from the Taylor-Green
flow, and obtained the energy spectrum
directly from the vortex configuration.
10-6
C 2/3k -5/3
2/l
10-7
101
102
k [1/cm]
l : intervortex spacing
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C=1
The dissipation ε arises from
eliminating smallest vortices
whose size becomes comparable
to the numerical space
resolution at k〜300cm-1.
E(k) [cm 3/sec 2]
The spectrum depends on the length scale.
10-6
C 2/3k -5/3
2/l
10-7
101
• For k < 2π/ l, the spectrum is
consistent with the Kolmogorov law,
reflecting the velocity field made by
the tangle. The Richardson cascade
process transfers the energy from
small k region to large k region. ST
mimics CT even without normal fluid.
102
k [1/cm]
l : intervortex spacing
• For k > 2π/ l, the spectrum is k-1,
coming from the velocity field due to
each vortex. The energy is probably
transferred by the Kelvin wave
cascade process (Vinen, Tsubota, Mitani,
PRL91, 135301(2003)). ST does not
mimic CT.
Kelvin waves
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E(k) [cm 3/sec 2]
The spectrum depends on the length scale.
10-6
C 2/3k -5/3
2/l
10-7
101
• For k < 2π/ l, the spectrum is
consistent with the Kolmogorov law,
reflecting the velocity field made by
the tangle. The Richardson cascade
process transfers the energy from
small k region to large k region. ST
mimics CT even without normal fluid.
102
k [1/cm]
l : intervortex spacing
• For k > 2π/ l, the spectrum is k-1,
coming from the velocity field due to
each vortex. The energy is probably
transferred by the Kelvin wave
cascade process (Vinen, Tsubota,Mitani,
PRL91, 135301(2003)). ST does not
mimic CT.
Energy spectrum of the Gross-Pitaevskii turbulence
M . Kobayashi and M. Tsubota
The dissipation is introduced so that it may work only in the scale
smaller than the healing length.
Starting from the uniform density and the random phase, we obtain
a turbulent state. Vorticity
Density
Phase
0 < t < 5.76
γ=1
2563 grids
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5123 grids
Energy spectrum of the incompressible kinetic energy
Time development of the exponent
July 2004 ( Trento and Prague)
Energy spectrum at t = 5.76
August 2004 ( Lammi)
3-3. Rotating superfluid turbulence
Tsubota, Araki, Barenghi,PRL90, 205301(‘03); Tsubota, Barenghi, et al., PRB69, 134515(‘04)
Vortex array under rotation
2
Lrot 

Vortex tangle in turbulence
Lflow   v2ns
Ω
vn
order
vs
disorder
The only experimental work
C.E. Swanson, C.F. Barenghi, RJ. Donnelly, PRL 50, 190(1983).
DG instability
?
?
By using a rotating cryostat, they made
counterflow turbulence under rotation and
observed the vortex line density by the
measurement of the second sound attenuation.
1.There are two critical velocities vc1
and vc2; vc1 is consistent with the onset
of Donnely-Graberson(DG) instability
of Kelvin waves, but vc2 is a mystery.
L=2Ω/κ
Vortex array
2. Vortex tangle seems to be polarized, because
the increase due to the flow is less than
expected.
This work has lacked theoretical
interpretation!
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Time evolution of a vortex array
towards a polarized vortex tangle
  4.98 10 2 rad / sec, vns  0.08[cm / s], T  1.6K
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1. The array becomes
Initial vortices：vortex
array
unstable,exciting
Kelvin waves.
with small
random noise
Glaberson
instability
Boundary
condition：
2. When
the wave
amplitude
Periodic（z-axis）
becomes
comparable to theSolid
vortex
boundary（x,y-axis）
separation,
reconnections start to
make
lots of vortex
loops. along
Counterflow
is applied
z-axis.loops disturb the array,
3. These
leading to a tangle.
Glaberson instability
W.I. Glaberson et al., Phys. Rev. Lett 33, 1197 (1974).
Glaberson et al. discussed the stability of the vortex array
in the presence of axial normal-fluid flow.
If the normal-fluid is moving faster than the vortex wave with k, the
amplitude of the wave will grow (analogy to a vortex ring).
Dispersion relation of vortex wave in a rotating frame：

2
b 
  20  k ,

log
a 
4
b：vortex line spacing,
：angular
velocity of rotation

0
Landau critical velocity beyond

which the array becomes unstable：
 
vc     2 2 ,
k mi n
kc 

vc   
k mi n
2

2 0
k
Numerical confirmation of Glaberson instability
Ω=9.97×10-3 rad/sec
Glaberson’s theory:Vc=0.010[cm/s]
v ns=0.008[cm/s]
v ns=0.015[cm/s]
v ns=0.03[cm/s]
v ns=0.05[cm/s]
v ns=0.06[cm/s]
v ns=0.08[cm/s]
What happens beyond the Glaberson instability?
2
  4.98 10 rad / sec,vns  0.08 cm / s
Vortex line density
1
L   d
V
Polarization
1
s
dsz 
z 

L
A polarized tangle！
Polarization <s’z> as a function of vns and Ω
The values of Ω are
4.98×10-2 rad/sec(■),
2.99×10-2 rad/sec (▲),
9.97×10-3 rad/sec(●).
We obtained a new
vortex state, a polarized
vortex tangle.
Competition between
order ( rotation) and
disorder ( flow)
4. Dynamics of quantized vortices in
rotating BECs
4-1 Vortex lattice formation
Tsubota, Kasamatsu, Ueda, Phys.Rev.A64, 053605(2002)
Kasamatsu, Tsubota, Ueda, Phys.Rev.A67, 033610(2003)
cf. A. A. Penckwitt, R. J. Ballagh, C. W. Gardiner, PRL89, 260402 (2002)
E. Lundh, J. P. Martikainen, K-A. Suominen, PRA67, 063604 (2003)
C. Lobo, A. Sinatora, Y. Castin, PRL92, 020403 (2004)
Rotating superfluid and the vortex lattice
c
>c
F  E  Lz
Minimizing the free energy
in a rotating frame.
  n0 e
i

The triangular vortex
lattice sustains the solid
body rotation.
Yarmchuck and Packard
vs  
m

  vs  0
Vortex lattice
observed in rotating
superfluid helium
Observation of quantized vortices in atomic BECs
ENS K.W.Madison, et.al PRL 84, 806 (2000)
JILA
P. Engels, et.al
PRL 87, 210403
(2001)
MIT J.R. Abo-Shaeer, et.al
Science 292, 476 (2001)
Oxford
E. Hodby, et.al
PRL 88, 010405
(2002)
How can we rotate the trapped BEC？
K.W.Madison et.al Phys.Rev Lett 84, 806 (2000)
z
x
Axisymmetric potential
5mm
y
Vext (R)  Vtrap (R)  U stir (R)
100mm
Total potential
“cigar-shape”
Rotation frequency

Non-axisymmetric potential
m 2
Ustir (R)    (x X 2  yY 2 )
2

x  y
Optical spoon
20mm

16mm

Direct observation of the vortex lattice formation
K.W.Madison et.al. PRL 86 , 4443 (2001)
Snapshots of the BEC after turning on the rotation
R 2x  Ry2
 = 2
R x  Ry2
1. The BEC becomes elliptic,
then oscillating.
2. The surface becomes unstable.
Rx
Ry
3. Vortices enter the BEC
from the surface.
4. The BEC recovers the
axisymmetry, the vortices
forming a lattice.
The Gross-Pitaevskii(GP) equation in a rotating frame

2
2
i

   Vtrap   g  
t
2m
2
Wave function
(r,t)

4  2 as
as
g
m s-wave
Interaction
scattering length
in a rotating frame



2
2


i

   (Vtrap  U stir )  g    Lz 
t
2m
2
m 2
2
2
U
(R)


(

X


Y
)
Two-dimensional
stir

x
y
2
simplified

Ω
Ω
The GP equation with a dissipative term

  2 2
2
i
 
  (Vtrap  U stir )  g   m  Lz 
t  2m


(i   )
t
  0.03: dimensionless parameter
S.Choi, et.al. PRA 57, 4057 (1998)
I.Aranson, et.al. PRB 54, 13072 (1996)
This dissipation
 comes from the interaction between the condensate and
the noncondensate.
E.Zaremba, T. Nikuni, and A. Griffin, J. Low Temp. Phys. 116, 277 (1999)
C.W. Gardiner, J.R. Anglin, and T.I.A. Fudge, J. Phys. B 35, 1555 (2002)
Profile of a single quantized vortex

2
2m
2
   Vtrap  g    m
2
A quantized vortex
(r )  n0 (r)e
i (r )
A vortex
n0
0.02
0.015


0.01
0.005

Velocity field
0
0
2
4
r
6
8
Vortex core= healing length

2mgn0
10
Dynamics of the vortex lattice formation (1)
  0.7
Time development of the condensate density n0
Tsubota et al.,
Phys. Rev. A 65,
023603 (2002)
1
Vtrap (r)  m 2 r 2
2
(r )  n0 (r)e i(r )
Grid 256×256
Time step 10-3
Experiment
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ÅB
Dynamics of the vortex lattice formation (2)
Time-development of the condensate density n0
t=0
67ms
340ms
390ms
410ms
700ms
Dynamics of the vortex lattice formation (3)
Time-development of the phase
(r )  n0 (r)e

i (r )
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ÅB
Dynamics of the vortex lattice formation (4)
Time-development of the phase 
t=0
67ms
340ms
Ghost vortices
Becoming real vortices 410ms
390ms
700ms
This dynamics is quantitatively consistent with the observations.
0.3
Rx

0.2
Ry
R 2x  Ry2
 = 2
R x  Ry2
0.1
0
0
100
200
300
400
500
600
time (msec)
K.W.Madison et.al. Phys. Rev. Lett.
86 , 4443 (2001)
700
Simultaneous display of the density and the phase
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Three-dimensional calculation of the vortex lattice formation
M. Machida (JAERI), N. Sasa, M.Tsubota, K.Kasamatsu
γ=0.03
Condensate density
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Grid : 128×128×128
γ=0.1
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4-2. Giant vortex in a fast rotating BEC
Kasamatsu,Tsubota,Ueda, Phys.Rev.A67, 053606(2002)
cf. A. L. Fetter, PRA64, 063608 (2001)
U. R. Fischer and G. Baym, PRL90, 140402 (2003)
E. Lundh, PRA65, 043604 (2002)
T. L. Ho, PRL87, 060403 (2001)
many references
A fast rotating BEC in a quadratic-plus-quartic potential
1
Vtrap (r)  m 2 r 2
2
A quadratic potential

A centrifugal potential
1
  m 2 r 2
2
When  >  , the trapping potential is no longer effective.
A quadratic-plus-quartic potential can trap
the BEC even if  >  .
1
1 4
2 2
Vtrap (r)  m  r  Kr
2
4

What happens to vortices
when  >  ？
A giant vortex in a fast rotating BEC
(2) Ω=3.2ω⊥
(1) Ω=2.5ω⊥
Vortices gather in
the central hole.
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A giant
vortex
・ The effective potential takes the Mexican-hat
form, so the central region becomes dilute and
allows the vortices gather there, and the
superfluid rotates around the core.
1
1 2
2
2
2
Veff (r)  m(    )r  Kr
2
4
Mexican hat
・This giant vortex is not a quantized vortex with
multi-quanta but a lattice of ghost vortices.
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Summary
3. Dynamics of quantized vortices in superfluid helium
3-1 Recent interests in superfluid turbulence
3-2 Energy spectrum of superfluid turbulence
3-3 Rotating superfluid turbulence
4. Dynamics of quantized vortices in rotating BECs
4-1 Vortex lattice formation
4-2 Giant vortex in a fast rotating BEC
4-3 Vortex states in two-component BEC
（ Kasamatsu, Tsubota, Ueda,
Phys.Rev.Lett.91, 150406(2003), cond-mat 0406150）