makoto tsubota- quantum turbulence: from superfluid helium to atomic bose-einstein condensates
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8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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4XDQWXP7XUEXOHQFH4XDQWXP7XUEXOHQFH--From Superfluid Helium to AtomicFrom Superfluid Helium to Atomic
BoseBose--Einstein CondensatesEinstein Condensates--
Makoto TSUBOTADepartment of Physics,
Osaka City University, JapanThanks to: M. Kobayashi, S. Kida and many friends
Review article: M. Tsubota, arXive: 0806.2737 (J. Phys. Soc. Jpn. (in press))
Progress in Low Temperature Physics, vol.16 (Elsevier), eds. W. P. Halperin and M. Tsubota
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8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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My message
1. Quantum turbulence (QT) has recently become one of the most important fields in
low-temperature physics.
2. QT consists of quantized vortices, which are stabletopological defects. Since each ³element´ is clear and
well-defined, QT is able to give a much simpler prototype
than classical turbulence.
3. QT has been studied in superfluid helium for more than
50 years. Since the mid 90s, QT research has tended
toward a new direction, now involving atomic Bose-
Einstein condensates (BECs) too.
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8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Another Da Vinci code Another Da Vinci code
Leonardo Da Vinci
(1452-1519) Da Vinci observed turbulent flow
in water and found that turbulence
consisted of many vortices.
Turbulence is not a simple disordered state but
rather it consists of some structures with vortices.
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Turbulence appears to have many vortices«
Turbulence in the wake of a dragonfly:
«however, these vortices are unstable
- they repeatedly appear, diffuse and disappear.
http://www.nagare.or.jp/mm/2004/gallery/iida/dragonfly.html
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8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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A quantized vortex is a vortex of superflow in a BEC.
Any rotational motion in superfluid is sustained by
quantized vortices.(i) The circulation is quantized.
(iii) The core size is very small.
v s d s ! On´ n = 0,1,2,
A vortex with n2 is unstable.
(ii) Free from the decay mechanism of the viscous diffusion of the vorticity.
Every vortex has the same circulation.
The vortex is stable.
r
s
(r )
rot v
sThe order of the coherence
length.
O ! h / m
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C lassical Turbulence ( C T) vs. Quantum Turbulence (QT)C lassical Turbulence ( C T) vs. Quantum Turbulence (QT)
Classical turbulence Quantum turbulence
- The vortices are unstable. Not easy
to identify each vortex.
-The circulation varies with time.
- it is not conserved.
- Quantized vortices are stable
topological defects.- Each vortex has the same
circulation.
- Circulation is conserved.
Motion of
vortex
cores
QT is much simpler
than CT, because eachelement of turbulence
is definite and clear.
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8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Contents of this talk:Contents of this talk:
1.1. History of QT researchHistory of QT research
2.2. Recent QT studies in superfluid heliumRecent QT studies in superfluid helium
3.3. QT in atomic BoseQT in atomic Bose--Einstein condensatesEinstein condensates
Research into QT can make a breakthrough into one of the
great mysteries of nature.
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1. History of QT research
Liquid 4He enters the superfluid state below 2.17 K (P point)
with Bose-Einstein condensation (BEC).
Its hydrodynamics are well described by the two-fluid model:
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Temperature
point
The two-fluid model
The system is a mixture of inviscid
superfluid and viscous normal fluid.
V ! V s
Vn
j ! V sv
s Vnvn
Density Velocity Viscosity Entropy
Superfluid 0 0
Normal fluid
V sT
Vn T
v sr
vnr L
nT sn T
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The two-fluid model can explain various experimentally
observed phenomena of superfluidity (e.g., thethermomechanical effect, film flow, etc.)
However, «
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(ii) v > vs c
vs
A tangle of quantized vortices develops. The two fluids
interact through mutual friction generated by tangling, and
the superflow decays.
v = 0s
Superfluidity breaks down in fast flow
(i) v < vs c
vs
The two fluids do not interact so that the superfluid canflow forever without decaying.
vst p g
(some critical velocity)
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1955: Feynman proposed that ³superfluid turbulence´ consists
of a tangle of quantized vortices.
1955 ± 1957: Vinen observed ³superfluid turbulence´.
Mutual friction between the vortex tangle and the normalfluid causes dissipation of the flow.
Progress in Low Temperature Physics
Vol. I (1955), p.17
Such a large vortex should
break up into smaller vortices
similar to the cascade processin classical turbulence.
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Many experimental studies were conducted chiefly on thermal
counterflow of superfluid 4He.
1980s K . W. Schwarz Phys. Rev. B38, 2398 (1988)Performed a direct numerical simulation of the three-dimensional
dynamics of quantized vortices and succeeded in quantitatively
explaining the observed temperature difference (T .
Vortex tangle
Heater
Normal flow Superflow
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Development of a vortex tangle in a thermal counterflow
Schwarz, Phys. Rev. B38, 2398 (1988).
Schwarz obtained numerically the
statistically steady state of a vortex
tangle, which is sustained by the
competition between the appliedflow and the mutual friction. The
calculated vortex density L(vns, T )
agreed quantitatively with
experimental data.
vs vn
Counterflow turbulence has been
successfully explained.
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What is the relation between superfluid
turbulence and classical turbulence
Most studies of superfluid turbulence have focused onthermal counterflow.
No analogy with classical turbulence
When Feynman drew the above figure, he was thinking of a
cascade process in classical turbulence.
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How can we study
the similarities and differences between QT and CT?
One of the best ways is to focus on the most importantstatistical law in CT, namely, Kolmogorov¶s law:
E (k ) ! C I2 / 3k 5 / 3
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E nergy spectrum of fully developed turbulence
Energy-
containing
range
Inertial
range
Energy-dissipative
range
E nergy spectrum of turbulence
Kolmogorov¶s law
Energy spectrum of the velocity field
Energy-containing range
Energy is in jected into the system at .k } k 0 !1/ "0
Inertial range
Dissipation does not work. The nonlinear
interaction transfers energy from low k region
to high k region.
K olmogorov¶s law : E ( k )=C 2/3 k -5/3
Energy-dissipation range
Energy is dissipated at a rate I at the
K olmogorov wavenumber, k c = (I/R3 )1/4.
Richardson cascade process
E !1
2v´
2
d r ! E (k )d k ´
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Everybody believes
that turbulence issustained by
Richardson cascade.
However, this is only
based on an image ;nobody has ever
clearly confirmed it.
One reason is that it is
too difficult toidentify each eddy in
a classical fluid.
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For QT to be a simple model of turbulence,
it should show express the essence of turbulence.
Based on this motivation, QT studies have Based on this motivation, QT studies have
entered a new stage since the mid 1990s!entered a new stage since the mid 1990s!
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Recent studies on energy spectra of superfluid heliumExperimental confirmation of the Kolmogorov spectra:
J. Maurer and P. Tabeling, Europhy. Lett. 43, 29 (1998) 1.4 K < T < T
S.R. Stalp, L. Skrbek and R.J. Donnely, PRL 82, 4831 (1999) 1.4 < T < 2.15 K
D.I. Bradley, D.O. Clubb, S.N. Fisher, A. M. Guenault, R.P. Haley, C.J.
Matthews, G.R. Pickett, V. Tsepelin, and K. Zaki, PRL 96, 035301 (2006) 3He
Theoretical consideration of QT at finite temperatures:
W.F. Vinen, Phys. Rev. B 61, 1410 (2000)D. Kivotides, J. C. Vassilicos, D. C. Samuels, C. F. Barenghi, Europhy. Lett. 57, 845 (2002)
What happens to QT and energy spectra at very low
temperatures?
2. Recent QT studies of superfluid helium
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Decaying K olmogorov turbulence in a model of superflow
C. Nore, M. Abid and M.E. Brachet, Phys. Fluids 9, 2644 (1997)
The Gross-Pitaevskii (GP) model
Energy spectrum of superfluid turbulence with no normal-fluidcomponent
T. Araki, M.Tsubota and S.K.Nemirovskii, Phys. Rev. Lett. 89, 145301(2002)
The vortex-filament model
K olmogorov spectrum of superfluid turbulence: Numerical analysis of the Gross-Pitaevskii equation with a small-scale dissipation
M. Kobayashi and M. Tsubota,Phys. Rev. Lett. 94, 065302 (2005), J. Phys. Soc. Jpn.74, 3248 (2005).
Three studies have directly investigated numerically QT
energy spectrum at zero temperature:
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C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)
By using the GP model, they
obtained a vortex tangle with
starting from the Taylor-Green
vortices.t
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In order to study the K olmogorov spectrum, it is necessary to
decompose the total energy into some components. (Nore et al., 1997)
Total energy
The kinetic energy is divided into
the compressible part with
and
the incompressible part with .
E !1
d x V´d x** 2
g
2*
2«-¬
»½¼´ *
E ! E int E q E k in
* ! V exp i U
E k in !1
d x V´d x * U ´
2
E k inc
!1
d x V´d x * U c? A´
2
E k ini
!1
d x V´d x * U i? A´
2
div * U i
! 0
rot * U c! 0
This incompressible kinetic energy E kini should
obey the Kolmogorov spectrum.
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C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)
: 2 < k < 12
: 2 < k < 14: 2 < k < 16
The right figure shows the energy spectrum at a moment. The left figure
shows the development of the exponent n(t). The exponent n(t) goesthrough 5/3 on the way of the dynamics.
n ( t )
t k
E ( k )
5/3
E (k ) k -n(t )
In the late stage, however, the exponent deviates from 5/3, because the
sound waves resulting from vortex reconnections disturb the cascade
process of the inertial range.
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K olmogorov spectrum of quantum turbulenceM. K obayashi and M. Tsubota, Phys. Rev. Lett. 94, 065302 (2005), J.
Phys. Soc. Jpn. 74, 3248 (2005)
1. We solved the GP equation in wavenumber spacein order to use fast Fourier transform.
2. We achieved steady-state turbulence by:2-1 Introducing a dissipative term that dissipateshigh-wavenumber Fourier components (i.e., short-wavelength phonons).
2-2 Exciting the system on a large scale by movingthe random potential.
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To solve the GP equation numerically with high accuracy, we use the
Fourier spectral method in space with cubic periodic boundary
conditions.
GP equation in Fourier space:
(healing length giving the vortex core size)
i
x
xt * k , t ! k 2
Q * k ,t
g
V 2* k 1, t ** k 2, t * k k 1 k 2, t
k 1 ,k 2
§
\2!
1 g *
2
GP equation in real space:
i
x
x t * r, t !
2
Q g * r,t 2
? A* r,t
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GP equation with small-scale dissipation
(healing length giving the vortex core size)
We introduce the dissipation that operates only in the scale smaller than \.
\ 2! 1 g *
2
{i K (k )}x
xt * k , t ! k
2 Q * k , t
g
V 2* k 1,t ** k 2, t * k k 1 k 2, t
k 1 ,k 2
§
K (k ) ! K 0 U k 2T / \
H ow to dissipate the energy at small scales?
This type of dissipation is justified by coupled analysis of the GP
and Bogoliubov- de Gennes equations.
cf., M. Kobayashi and M. Tsubota, PRL 97, 145301 (2006)
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This is done by moving the random potential satisfying the
space-time correlation:
V (x, t )V ( dx , dt ) ! V 02exp
x dx 2
2 X 02
t dt 2
2T 02
«
-¬¬
»
½¼¼
The variable X 0 determines thescale of the energy-containing
range.
V 0=50, X 0=4 and T 0=6.410-2
H ow to inject the energy at large scales?
X0
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Thus, steady-state turbulence is obtained (1)
Time development of each energy component
Vortices Phase in a central plane Moving random potential
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Thus, steady-state turbulence is obtained (2)
Time development of each energy component
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Picture of the cascade process
Quantized vortices
Phonons
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In order to confirm this picture, we calculate
(1) the energy dissipation rate of E kini
(2) the energy flux of the Richardson cascade.
Quantized vortices
Phonons
Is comparable to ?
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(1) the energy dissipation rate of E kini
In the steady state, we turn off
the large-scale excitation
suddenly and monitor the time
development of E kini .
Thus we obtain
from the decay.
I ! d E kin
i
dt } 12.5 s 2.3
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(2) the energy flux of the Richardson cascade
Dissipation rate 12.5
1. is about constant in the
inertial range.
2. is comparable to .
They confirm the picture of
the inertial range as written
in textbooks.
Ensemble averaged over 50 states.
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Energy spectrum of steady-state turbulence
The energy spectrum obeys the Kolmogorov law.
Quantum turbulence is
found to express the
essence of classicalturbulence!
2 / X0 2 /
The inertial range is
sustained by a genuine
Richardson cascade of quantized vortices.
M. K obayashi and M. Tsubota, Phys. Rev. Lett. 94,
065302 (2005), J. Phys. Soc. Jpn. 74, 3248 (2005)
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Overall picture of energy spectrum of QT at 0 K
Kelvin-wave cascadeD. Kivotides, J. C. Vassllicos, D. C. Samuels,
C. F. Barenghi, PRL 86, 3080 (2001)
W. F. Vinen, M. Tsubota, A. Mitani, PRL91,
135301 (2003)
E. Kozik, B. V. Svistunov, PRL 92, 035301
(2004); PRL 94, 025301(2005); PRB 72,
172505 (2005)
Classical-quantum crossover
V. S. L¶vov, S. V. Nazarenko, O. Rudenko,
PRB 76, 024520 (2007)E. Kozik, B. V. Svistunov, PRB 77, 060502
(2008)
: mean distance between vortices
Classical Quantum
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3. QT in atomic Bose-Einstein condensates(BECs)M. Kobayashi and M. Tsubota, Phys. Rev. A76, 045603 (2007)
Vortex array Vortex tangle
Superfluid He
Atomic BEC
Two principal cooperative phenomena of quantized
vortices are vortex arrays and vortex tangles.
None
Packard (Berkeley)
Ketterle (MIT)
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Usual setup of atomic BECs
K.W.Madison et.al Phys.Rev Lett 84, 806 (2000)
Optical spoon
(oscillating
laser beam)
Rotation frequency
;
z
x
y 100Qm
5Qm
20Qm 16Qm
³cigar-shape´
Laser cooling
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Observation of quantized vortices in atomic BECs
K.W. Madison, et. al PRL 84, 806 (2000)
J.R. Abo-Shaeer, et. alScience 292, 476 (2001)
P. Engels, et. al
PRL 87, 210403
(2001)
ENS
MIT
JILA
E. Hodby, et. al
PRL 88, 010405
(2002)
Oxford
8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Dynamics of vortex lattice formation by the GP model
Time development of condensate density n0
; ! 0.7[B Experiment
M. Tsubota, K. Kasamatsu
and M. Ueda, Phys. Rev. A
65, 023603 (2002)
V trap(r ) !1
2m[
B
2r
2
=(r )! n0 (r )ei U(r )
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K.W. Madison et.al., PRL 86,
4443 (2001)
8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Is it possible to produce turbulence in a trapped BEC?
(1) We cannot apply any flow to this system.
(2) This is a finite-size system. Can we make
turbulence with a sufficiently wide inertial range?
The coherence length is not much smaller than the system size.
However, we could confirm Kolmogorov¶s law.
M. Kobayashi and M. Tsubota, Phys. Rev. A76, 045603 (2007)
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8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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How to produce turbulence in a trapped BEC
x
y
z 1. Trap the BEC in a weak
elliptical potential.
U x !m [
2
2
1 I1 1 I2 x 2 1 I1 1 I2 y 2 1 I2 z2? A
2. Rotate the system first
around the x-axis, then
around the z-axis.
; t ! ; x, ; zsin; xt ,; z cos; xt
8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Actually, this idea has been already used in CT.
S. Goto, N. Ishii, S. Kida, and M. Nishioka, Phys. Fluids 19, 061705 (2007)
Rotation
around
one axis
Rotationaround
two axes
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Thanks to S. Goto
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8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Condensate density Quantized vortices
Two precessions (xz) Three precessions (y x z)
Condensate density Quantized vortices
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8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Energy spectra for two cases
Three precessions produce more isotropic QT, whose is closer to 5/3.
M. Kobayashi and M. Tsubota, Phys. Rev. A76, 045603 (2007)
Two precessions Three precessions
8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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What can we learn from QT of atomic BEC?
Controlling the transition to turbulence by changingthe rotational frequency or interaction parameters, etc.
We can visualize quantized vortices. We can consider
the relation of real space cascade of vortices and the
wavenumber space cascade (Kolmogorov¶s law).
Changing the trapping potential or the rotational
frequency leads to dimensional crossover (2D3D)
in turbulence.
8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Controlling the transition to turbulence
x
z
Vortex lattice
V o r t e x l a
t t i c e
Vortex tangle
?
0
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xz
8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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What can we learn from quantum turbulence of atomic BEC?
Controlling the transition to turbulence by changing therotational frequency or interaction parameters, etc.
We can visualize quantized vortices. We can consider
the relation of real space cascade of vortices and the
wavenumber space cascade (Kolmogorov¶s law).
Changing the trapping potential or the rotational
frequency leads to dimensional crossover (2D3D) in
turbulence.
We can obtain the most controllable turbulence!
y z
x
y
z
x
y z
x
8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates
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Summary 16th century
21st century
We discussed the recent interests
in quantum turbulence.
Quantum turbulence consists of
quantized vortices, which are
stable topological defects.
Quantum turbulence may give a
prototype of turbulence that is
much simpler than classical
turbulence.
Review article: M. Tsubota, arXive:0806.2737
(J. Phys. Soc. Jpn. (in press))
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