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Numerical simulations of superfluid vortex turbulenceVortex
dynamics in superfluid helium and BECCollaborators: 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
Makoto TSUBOTAOsaka City University, Japan..
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Vortices in Japanese seaGalaxyVortex lattice in a rotating Bose
condensateVortices appear in many fields of nature!Vortex tangle in
superfluid helium
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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 turbulence4. Dynamics
of quantized vortices in rotating BECs 4-1 Vortex lattice formation
4-2 Giant vortex in a fast rotating BEC 1. Introduction2. Classical
turbulenceOutline
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1. IntroductionA quantized vortex is a vortex of superflow in a
BEC.(i) The circulation is quantized.(iii) The core size is very
small.A vortex with n2 is unstable.(ii) Free from the decay
mechanism of the viscous diffusion of the vorticity. rs(r)rot
vs
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How to describe the vortex dynamicsVortex filament formulation
(Schwarz)Biot-Savart law
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. srThe Gross-Pitaevskii equation for the macroscopic wave
function
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What is the difference of vortex dynamics between superfluid
helium and an atomic BEC?Superfluid helium (4He)Core size (healing
length) System size LThe vortex dynamics is local, compared with
the scale of the whole system.Atomic Bose-Einstein condensateCore
size (healing length) subm System size L mThe vortex dynamics is
closely coupled with the collective motion of the whole
condensate.
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Superfluid helium1955 Feynman Proposing superfluid turbulence
consisting of a tangle of quantized vortices.1955, 1957 Hall and
Vinen Observing superfluid turbulenceThe mutual friction between
the vortex tangle and the normal fluid causes the
dissipation.Liquid 4He enters the superfluid state at 2.17 K with
Bose condensation.Its hydrodynamics is well described by the two
fluid model (Landau).Temperature (K)Superfluid helium becomes
dissipative when it flows above a critical velocity.point
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Lots of experimental studies were done chiefly for thermal
counterflow of superfluid 4He.1980sK. 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 .
Vortex tangleHeater Normal flow Super flow
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Three-dimensional dynamics of quantized vortex filaments
(1)Superfluid flow made by a vortex Biot-Savart law
srIn the absence of friction, the vortex moves with its local
velocity.
The mutual friction with the normal flowis considered at a
finite temperature.
When two vortices approach, they are made to reconnect.
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Three-dimensional dynamics of quantized vortex filaments (2)
Vortex reconnectionThis was confirmed by the numerical analysis
of the GP equation. J. Koplik and H. Levine, PRL71, 1375(1993).
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Three-dimensional dynamics of quantized vortex filaments
(1)Superfluid flow made by a vortex Biot-Savart law
srIn the absence of friction, the vortex moves with its local
velocity.
The mutual friction with the normal flowis considered at a
finite temperature.
When two vortices approach, they are made to reconnect.sr
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Development of a vortex tangle in a thermal counterflowSchwarz,
Phys.Rev.B38, 2398(1988).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.vs vn
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What is the relation between superfluid turbulence and classical
turbulenceMost studies of superfluid turbulence were devoted to
thermal counterflow.No analogy with classical turbulenceWhen
Feynman showed the above figure, he thought of a cascade process in
classical turbulence.
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2. Classical turbulenceWhen 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.
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Classical turbulence and vorticesNumerical analysis of the
Navier-Stokes equation made by Shigeo KidaVortex cores are
visualized by tracing pressure minimum in the fluid.
Gird turbulence
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Classical turbulence and vorticesNumerical analysis of the
Navier-Stokes equation made by Shigeo KidaVortex cores are
visualized by tracing pressure minimum in the fluid.
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.
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Classical turbulenceEnergy-containing rangeInertial
rangeEnergy-dissipative rangeEnergy spectrum of turbulence
Kolmogorov lawEnergy spectrum of the velocity field
Energy-containing rangeThe energy is injected into the system at
.. Inertial rangeDissipation does not work. The nonlinear
interaction transfers the energy from low k region to high k
region.Kolmogorov law :E(k)=C2/3 k- -5/3Energy-dissipative rangeThe
energy is dissipated with the rate at the Kolmogorov wave number kc
= (/3 )1/4. Richardson cascade process
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Kolmogorov spectrum in classical turbulenceExperimentNumerical
analysis
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Vortices in superfluid turbulence
ST consists of a tangle of quantized vortex
filamentsCharacteristics of quantized vorticesQuantization of the
circulationVery thin coreNo viscous diffusion of the vorticityThe
tangle may give an interesting model with the Kolmogorov law. 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).How is the intermittency?
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Summary of the motivationClassical turbulenceVorticesSuperfluid
turbulenceQuantized vorticesIt is possible to consider quantized
vortices as elements in the fluid and derive the essence of
turbulence.
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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.3. Dynamics of quantized
vortices in superfluid helium 3-1 Recent interests in superfluid
turbulence
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Vinen, Phys.Rev.B61, 1410(2000) Considering the relation between
ST and CT The Oregons 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!
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3-2. Energy spectrum of superfluid turbulence Decaying
Kolmogorov turbulence in a model of superflowC. 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
Normal-Fluid ComponentT. 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.
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C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9,
2644(1997)Although the total energy is conserved, its
incompressible component is changed to the compressible one (sound
waves).The total length of vortices increases monotonically, with
the large scale motion decaying.t
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C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9,
2644(1997)t=5.5: 2 < k < 12 : 2 < k < 14 : 2 < k
< 16The right figure shows the energy spectrum at a moment. The
left figure shows the development of the exponent n(t) in the
spectrum E(k)=A(t) k -n(t) . The exponent n(t) goes through 5/3 on
the way of the dynamics. n(t)tkE(k)5/3E(k)=A(t) k -n(t)
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Energy spectrum of superfluid turbulence with no normal-fluid
componentT.Araki, M.Tsubota, S.K.Nemirovskii, PRL89,
145301(2002)
l : intervortex spacing Energy spectrum of a vortex tangle in a
late stage We calculated the vortex filament dynamics starting from
the Taylor-Green flow, and obtained the energy spectrum directly
from the vortex configuration. C=1 The dissipation arises from
eliminating smallest vortices whose size becomes comparable to the
numerical space resolution at k300cm-1.
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The spectrum depends on the length scale.For k < 2/ 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.For k > 2/ 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.l : intervortex spacing
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Kelvin waves
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The spectrum depends on the length scale.For k < 2/ 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.For k > 2/ 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.l : intervortex spacing
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Energy spectrum of the Gross-Pitaevskii turbulence M . Kobayashi
and M. TsubotaThe dissipation is introduced so that it may work
only in the scale smaller than the healing length.
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Starting from the uniform density and the random phase, we
obtain a turbulent state.Vorticity 0 < t < 5.76=1
2563 grids
5123 gridsDensity Phase
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Energy spectrum of the incompressible kinetic energy Time
development of the exponentEnergy spectrum at t = 5.76 July 2004 (
Trento and Prague)August 2004 ( Lammi)
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3-3. Rotating superfluid turbulence Tsubota, Araki,
Barenghi,PRL90, 205301(03); Tsubota, Barenghi, et al., PRB69,
134515(04)Vortex array under rotationorderVortex tangle in
turbulencedisordervvnsWhat happens if we combine both effects?
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The only experimental work C.E. Swanson, C.F. Barenghi, RJ.
Donnelly, PRL 50, 190(1983).2. Vortex tangle seems to be polarized,
because the increase due to the flow is less than expected.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. This work has lacked theoretical
interpretation!L=2/Vortex arrayDG 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.
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Time evolution of a vortex array towards a polarized vortex
tangle Initial vorticesvortex array with small random noise
Boundary conditionPeriodicz-axis Solid boundaryx,y-axisCounterflow
is applied along z-axis.
1. The array becomes unstable,exciting Kelvin waves. Glaberson
instability2. When the wave amplitude becomes comparable to the
vortex separation, reconnections start to make lots of vortex
loops.3. These loops disturb the array, leading to a tangle.
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Glaberson instabilityW.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,bvortex line spacing, angular velocity of rotation Landau
critical velocity beyond which the array becomes unstable,
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Numerical confirmation of Glaberson instabilityGlabersons
theory:Vc=0.010[cm/s]v =0.008[cm/s]nsv =0.015[cm/s]v
=0.03[cm/s]nsnsv =0.05[cm/s]v =0.06[cm/s]v
=0.08[cm/s]nsnsns=9.9710-3 rad/sec
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What happens beyond the Glaberson instability? Vortex line
densityPolarizationA polarized tangle
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Polarization as a function of vns and The values of are 4.9810-2
rad/sec(), 2.9910-2 rad/sec (), 9.9710-3 rad/sec().
We obtained a new vortex state, a polarized vortex
tangle.Competition between order ( rotation) and disorder (
flow)
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4. Dynamics of quantized vortices in rotating BECs4-1 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)
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Rotating superfluid and the vortex latticeW < WcW > WcThe
triangular vortex lattice sustains the solid body
rotation.Yarmchuck and PackardMinimizing the free energy in a
rotating frame.Vortex lattice observed in rotating superfluid
helium
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Observation of quantized vortices in atomic BECsK.W.Madison,
et.al PRL 84, 806 (2000)J.R. Abo-Shaeer, et.al Science 292, 476
(2001)P. Engels, et.al PRL 87, 210403 (2001)ENSMITJILAE. Hodby,
et.al PRL 88, 010405 (2002)Oxford
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How can we rotate the trapped BECK.W.Madison et.al Phys.Rev Lett
84, 806 (2000)Non-axisymmetric potentialOptical spoonTotal
potentialRotation frequency Wzxy100mm5mm20mm16mmAxisymmetric
potential cigar-shape
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Direct observation of the vortex lattice formationSnapshots of
the BEC after turning on the rotation1. The BEC becomes elliptic,
then oscillating.2. The surface becomes unstable.3. Vortices enter
the BEC from the surface.4. The BEC recovers the axisymmetry, the
vortices forming a lattice.K.W.Madison et.al. PRL 86 , 4443
(2001)RxRy
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The Gross-Pitaevskii(GP) equation in a rotating frameWave
functionInteractions-wave scattering lengthin a rotating frame
Two-dimensionalsimplified
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The GP equation with a dissipative termS.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)
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Profile of a single quantized vortexA quantized vortexVelocity
fieldVortex core= healing lengthA vortex
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Dynamics of the vortex lattice formation (1)Time development of
the condensate densityExperimentTsubota et al., Phys. Rev. A 65,
023603 (2002) Grid 256256Time step 10-3
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Dynamics of the vortex lattice formation
(2)t=067ms340ms390ms410ms700msTime-development of the condensate
densityAre these holes actually quantized vortices?
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Dynamics of the vortex lattice formation (3)Time-development of
the phase
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Dynamics of the vortex lattice formation
(4)t=067ms340ms390ms410ms700msGhost vorticesBecoming real
vorticesTime-development of the phase
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This dynamics is quantitatively consistent with the
observations.K.W.Madison et.al. Phys. Rev. Lett. 86 , 4443
(2001)RxRy
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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 Condensate
densityGrid : 128128128=0.03=0.1Riminding us of superfluidity of a
neutron star!
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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
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A fast rotating BEC in a quadratic-plus-quartic potentialW A
quadratic potentialA centrifugal potentialA quadratic-plus-quartic
potential can trap the BEC even if . WWhat happens to vortices when
When , the trapping potential is no longer effective.
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A giant vortex in a fast rotating BEC(1) =2.5(2) =3.2A 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. This giant vortex is
not a quantized vortex with multi-quanta but a lattice of ghost
vortices.Vortices gather in the central hole.Mexican hat
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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 turbulence4. 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 Summary
First of all, I would like to thank the organizing committee for
giving me a chance to give you this talk here. I am Makoto Tsubota,
coming from Osaka, Japan. I would like to talk about dynamics of
quantized vortices in superfluid helium and rotating Bose-Einstein
condensates, with emphasis on the recent activity of our group.
They are my collaborators. There a re many kinds of vortices in
nature. This is a tornado in United States. This is a galaxy in
space. This is a vortex in Japanese sea. And, quantized vortices
appear in superfluid helium and Bose-Einstein condensates
too.Physics of quantized vortices has been studied in the field of
superfluid helium, and recently it enters a new stage which is
rather different from the old works. One o the important
motivations is the relation between superfluid turbulence and
classical turbulence. After remembering briefly classical
turbulence in section 2, I will talk about the new research of
quantized vortices and superfluid turbulence in section 3. The
recent realization of Bose-Einstein condensation of alkali atomic
gases has opened the new research field of superfluidity and
vortices, which is the topic of Sec.4.A quantized vortex is a
vortex of superflow in a Bose-Einstein condensate. A quantized
vortex has some properties different from classical vortices.
First, the circulation is quantized by the quantum of circulation.
Since a vortex with the quantum number larger than two is unstable
, actually every vortex has the same circulation. Second, since
this is a vortex of inviscid superflow, it is free from the decay
mechanism of the viscous diffusion of the vorticity. For example,
this is a photo of a tornado. Usually a vortex consists of a core
and a flow circulating around the core. The core region has high
vorticity, but because of the viscous diffusion of vorticity, the
core becomes to diffuse, gradually disappearing. The point is a
quantized vortex is free from such decay mechanism. Hence a
quantized vortex is much more stable, compared with classical
vortices. Third, the core size is found to be very small, Angstrom
size in superfluid He4 and submicron even in atomic BECs. Generally
we have two methods of how to describe the dynamics of quantized
vortices. One is the vortex filament formulation, which was
pioneered by Klaus Schwarz. A vortex makes the superflow of the
Biot-Savart law, where r is an arbitrary point in the space and s
is a point on the vortex filament. The vortex moves with this
Biot-Savart flow. At a finite temperature, the mutual friction
should be considered. Numerically, a vortex is represented by a
string of points, and we follow the motion of these points. The
other is to solve the Gross-Pitaevskii equation for the macroscopic
wave function. Here I will show the results of both methods,
depending on the problems.Here I would like to note the difference
of vortex dynamics between superfluid helium. In the case of
superfluid helium 4He, the vortex core size is found to be the
order of angstrom, which is usually much smaller than the system
size. Hence the vortex dynamics is local compared with the scale of
the whole system. On the other hand, in the case of a dilute atomic
Bose-Einstein condensate, the vortex core size is submicron,
usually the same order as the system size of micron scale. So the
vortex dynamics is closely coupled with the collective motion of
the whole condensate, which will be shown later.As you know, liquid
He4 enters superfluid state at 2.17K with Bose-Einstein
condensation. Its hydrodynamics is well understood by the two fluid
model which states this liquid is a mixture of inviscid superfluid
and viscous normal fluid with a ration depending on temperature.
However, it was known in 1940s that superfluid helium becomes
dissipative when it flows above a critical velocity. About this
issue, Richard Feynman proposed this is superfluid turbulent state
consisting of a tangle of quantized vortices. This is a figure
appeared in the Fyenmans famous article, showing large vortices are
broken up to smaller vortices through vortex reconnections like a
cascade process. In the 50s, Hall and Vinen observed experimentally
superfluid turbulence in which the mutual friction between vortices
and normal fluid causes that dissipation.After that, lots of
experimental works have been done, chiefly for thermal counterflow.
This is a sketch. When heat is injected into superfluid helium,
normal fluid flows in this case from the left to the right, and
superfluid flows oppositely. Here appears a vortex tangle, and the
mutual friction causes dissipation. In the 80s, Klaus Schwarz made
the direct numerical simulation of the three-dimensional dynamics
of quantized vortices and succeeded in understanding quantitatively
the observed temperatures difference.The superfluid velocity flow
made by vortices is represented by the usual Biot-Savart law. In
the absence of friction, the vortex moves with the local superflow.
At a finite temperature, the mutual friction with the normal flow
is considered. When two vortices approach, they are made to
reconnect.As I told before, a vortex is represented by a string of
points, whose minimum distance delta gzi is the numerical space
resolution. When two vortices get close within delta gzi, they are
made to reconnect. Of course, this procedure is an assumption in
the vortex filament formulation, but later this picture was
confirmed by the numerical analysis of the Gross-Pitaevskii
equation by Koplik and Levine.By considering all those effect, we
make the simulation of the vortex dynamics.This animation, which
was made by our group, shows how initial six vortex rings grow up
to a vortex tangle in a thermal counterflow. Schwarz obtained
...This was a great success of the numerical research, but the
recent study is rather different from the Schwarzs old
works.However, most studies of superfluid turbulence have been
devoted to thermal counterflow, which has no analogy with classical
turbulence, so people of fluid dynamics have been hardly interested
in superfluid turbulence. When Feynman showed this figure, he
thought of a cascade process in classical turbulence. Then remains
an important question, what is the relation between superfluid
turbulence and classical turbulence.So here, we would like to
remember briefly classical turbulence. As you know, when we raise
the flow velocity around a shere, the flow changes from laminar via
emission of Karman vortices to turbulence. The understanding and
control of turbulence have been one of the most important problems
in fluid dynamics since Leonald Da Vinci, but it is too difficult
to do it. Why is it so difficult?This is a photo of grid
turbulence. When a flow passes through a grid, it becomes isotropic
and homogeneous turbulence far enough behind the grid. Certainly
the turbulence seems to have vortices. This figure shows a
numerical analysis of the Navier-Stokes equation made by Shigio
kida, Japanese fluid scientist. Here the vortex cores are
visualized by tracing pressure minimum in the fluid.However, these
vortices have different circulation. They are unstable, repeatedly
appear, diffuse by the viscous diffusion of the vorticity, and
disappear. Thus it is difficult to identify each vortex, these
properties should be compared with those of quantized
vortices.Before going into superfluid turbulence, we would like to
remember briefly classical turbulence. In a fully developed
turbulence, the energy spectrum is known to have the characteristic
behavior, as shown here. The region of the wave number is divided
into three ranges. The energy is injected into the fluid at some
low wave number kzero in the energy-containing range. In the
experiment of grid turbulence, for example, this wave number
corresponds to the inverse of the grid mesh size. In the
intermediate inertial range, the energy is transferred from small k
region to large k region without being dissipated. Here the
spectrum takes the famous Kolmogorov law which is the most
important statistical law in turbulence. The inertial range is
believed to be sustained by this Richardson cascade process, where
large eddies are broken up to smaller ones and epsilon is the
energy flux of this cascade process. The energy which is
transferred into the dissipation range is dissipated by the
viscosity with the dissipation rate epsilon at the Kolmogorov wave
number. They are Kolmogorov spectrum obtained in classical
turbulence. The left figure was obtained in a experiment of grid
turbulence. The right figure shows the spectrum numerically obtain
by solving the Navier-Stokes equation.We know superfluid turbulence
consists of a tangle of quantized vortex filaments. As I discussed
before, a quantized vortex is a stable and definite topological
defect, compared with usual vortices in a classical fluid. Because
of superfluidity, some dissipative mechanism would work only at
large wave numbers or small scales. Therefore our tangle may give a
definite inertial range with the Kolmogorov spectrum, or we may ask
whether such quantized vortices can still produce the essence of
turbulence or not.
This sketch summarizes my own motivation. Classical turbulence
is believed to consist of vortices, but their relation is not so
clear. This is not a trivial but a serious problem, for example,
being discussed in detail in the recent famous textbook by Frish,
but nobody knows the answer. On the other hand, we know superfluid
turbulence consists of quantized vortices, and one of the important
problems is to consider quantized vortices as elements in the fluid
and derive the essence of turbulence. Such kind of approaches is
impossible in classical turbulence. Recently appear some remarkable
studies about this topic. This Paris group observed local pressure
in a flow driven by two counterrotating disks, finding the
Kolmogorov spectra. The Oregons group studied the decay of grid
turbulence, whose data was consistent with a classical model with
the Kolmogorov spectrum. Being motivated by these studies, Vinen
investigated the relation between superfluid turbulence and
classical turbulence. According to him, the observed similarity is
caused by the coupling of the superfluid and the normal fluid due
to the mutual friction. He stressed the importance of the length
scale, compared with the characteristic vortex spacing ina tangle.
The Newcastle group studied the energy spectrum of superflow when
it is coupled with the normal flow having the Kolmogorov law,
finding that superflow is dragged to the normal flow to have the
Kolmogorov law.Then appears a very important question. What happens
in a vortex tangle at very low temperatures where there is no
normal fluid. Is there any similarity between superfluid and
classical turbulence or not? Our work attacks this problem
directly.Until now there are two works which study the energy
spectrum of a vortex tangle in the absence of mutual friction. One
is the contribution of the Paris group, and the other is ours. The
Paris group solved numerically the Gross-Pitaevskii equationThey
solved numerically the Gross-Pitaevskii equation to obtain a vortex
tangle starting from the Taylor-Green flow. Although the total
energy is conserved, the incompressive component is changed to the
compressive one. The total length of vortices increases
monotonically, with the large scale motion decaying. The situation
is just complicated. The right figure shows the energy spectrum at
a moment. By the fitting, they obtained the exponent of the
spectrum as a function of time. The left figure shows the
development of the exponent $n$. The three curves differ in the
wave-number range of the fitting. The exponent certainly goes
through -5/3 on the way of the decay.This is our work. We
calculated the vortex filament dynamics starting from the
Taylor-Green flow, and obtained the energy spectrum directly from
the vortex configuration. Initially the vortices arent a tangle,
but in the late stage become a tangle. The right figure shows the
energy spectrum in a late stage. The red line refers to the
spectrum obtained numerically, and the black solid line shows the
Kolmogorov spectrum. Here the Kolmogorov constant is unity and the
energy dissipation rate epsilon arises from eliminating smallest
vortices whose size becomes comparable to the numerical space
resolution. Please note the vertical axis is not arbitrary but the
absolute value.You can find the spectrum in low k region is
consistent with the Kolmogorov law, not only in the power but also
in the absolute value.The energy spectrum depends on the length
scale. The spectrum changes at 2pi over l. L is the intervortex
spacing. This quantity is proper to superfluid turbulence.
Classical turbulence doesnt have such quantity, because vortices
are obscure. When the wave number k is smaller than 2pi/l, the
spectrum is consistent with the Kolmogorov law, reflecting the
velocity field made by the tangle. The energy is transferred by the
Richardson cascade process from small k region to large k region,
which is confirmed by the study of the vortex size distribution.
Therefore superfluid turbulence mimics classical turbulence in this
region. On the other hand, when k is larger than 2pi/l, the
spectrum is k to the minus one, coming from the velocity field
around each vortex. The energy is probably transferred by the
Kelvin wave cascade process,which is studied in this paper of ours.
In this region, superfluid turbulence doesnt mimic classical
turbulence. They are Kelvin waves, helical waves excited along each
vortex.The energy spectrum depends on the length scale. The
spectrum changes at 2pi over l. L is the intervortex spacing. This
quantity is proper to superfluid turbulence. Classical turbulence
doesnt have such quantity, because vortices are obscure. When the
wave number k is smaller than 2pi/l, the spectrum is consistent
with the Kolmogorov law, reflecting the velocity field made by the
tangle. The energy is transferred by the Richardson cascade process
from small k region to large k region, which is confirmed by the
study of the vortex size distribution. Therefore superfluid
turbulence mimics classical turbulence in this region. On the other
hand, when k is larger than 2pi/l, the spectrum is k to the minus
one, coming from the velocity field around each vortex. The energy
is probably transferred by the Kelvin wave cascade process,which is
studied in this paper of ours. In this region, superfluid
turbulence doesnt mimic classical turbulence. In both works by the
Paris and our groups it was impossible to control the energy
dissipation. SO, very recently, we started another calculation to
solve the Gross-Pitaevskii equation in the wave number k-space.
Here the dissipation is introduced a a step function so that it may
work only in the scale smaller than the healing length..Starting
from the uniform density and the random phase of the wave function,
we obtain turbulence. They shows vorticity, phase and density at a
cross section. The upper figures show the result of 256cubed grids,
while the lower is 512cubed grids. Two results are equivalent.
The left figure shows the time development of the exponent of
the incompressible kinetic energy spectrum. You can find the
exponent converges to 5/3 in late stage. The right figure shows the
energy spectrum at a late stage. The dissipation works in these
scales. Certainly we can obtain the spectrum qualitatively
consistent with the Kolmogorov spectrum. Some of you saw this kind
of figure in my talk in Trento and Prage last month, but this is
different from what you saw . This is the result of the last month,
while this months result has just wider inertial range. Physics is
improving month by month.The next topic comes from this
consideration. We know two kinds of cooperative vortex states. One
is an ordered vortex array under rotation. The other is a
disordered vortex tangle in turbulent state. Our problem is what
happens if we combine these two states. We know only one
experimental work about the topic, which was made by Swanson,
Barenghi and Donnely 20 years ago. By using a rotating cryostat,
they made counterflow turbulence under rotation and observed the
vortex line density by the usual measurement of the second sound
attenuation. This figure shows the vortex line density as a
function of the rotation frequency and the counterflow velocity,
showing two critical velocities. In the left side of vc1, the
vortex density was consistent with the Feymans rule, which means
this state is a vortex array. And vc1 was consistent with the onset
of the Donnely-Glaberson instability of Kelvin wave, which will be
explained later. Second, the vortex tangle seemed to be polarized.
However the authors didnt know what happened to vortices in these
regions. Then I will show you some typical results. The initial
state consists of 33 parallel vortices under rotation along this
direction. The vortices are seeded with small random perturbations
to make the simulation realistic. When flow is applied along the
rotation axis, the array becomes unstable change to a vortex
tangle. What happens here? First, the array becomes unstable,
exciting Kelvin waves, which is called Glaberson instability . When
the amplitude of the Kelvin waves become comparable to the vortex
separation, reconnections start to make lots of vortex loops. The
vortex reconnections and vortex loops disturb the vortex array,
leading to a random vortex tangle. We will discuss what happens in
the dynamics. The key point in the early stage is Glaberson
instability. Glaberson et al. discussed the stability of a vortex
array in the presence of axial normal flow. The point is very
simple; if the normal flow is faster than the propagation of the
vortex wave with wave number k, its amplitude grows, whose
mechanism is similar to the expand or shrink of a vortex ring
subject to a flow. Glaberson obtained the dispersion relation of
the Kelvin wave, where b is the average distance between parallel
vortices and a is the small cut-off parameter corresponding to the
core size. This dispersion law has the Landau critical velocity,
and if the flow exceeds the critical velocity, the Kelvin wave with
that wave number becomes unstable, growing exponentially in time.
This figure shows the Glabersons instability. For this value of the
angular velocity Omega, the Glabersons theory gives the critical
velocity 0.01cm/sec. When the flow velocity is smaller than the
critical value, the vortex array is certainly stable. However if
the flow exceeds the critical value, the vortex array becomes
unstable to excite the Kelvin waves.
The next question is what happens to the vortices beyond the
Glaberson instability. The vortex tangle is characterized by these
quantities. The left figure shows the development of the vortex
line density. The Graberson instability causes the initial
exponential growth, followed by the saturation to a statistical
steady state.The right figure shows the polarization. The
polarization decreases from unity of the vortex array, but never
becomes to zero of a completely isotropic tangle. The saturation at
a finite value means this vortex tangle is polarized.This figure
shows the polarization of the saturated state as a function of vns
and Omega. The polarization decreases with the flow velocity and
increases with Omega, which shows the competition between the flow
and the rotation. Hence we obtained a new vortex state where it is
polarized with rotation. I do not mention the detail now, but these
results are qualitatively consistent with the old works of 20 years
ago.Now we are moving to the topics of Bose-Einstein
condensates.The first is vortex lattice formation.According to the
history of superfluid helium, one of the best ways for making
quantized vortices is to rotate the system. When the rotation
frequency is smaller than a critical value, the superfluid doesnt
rotate. However, the rotation above the critical value rotates the
system, the vortex lattice being formed to sustain the solid body
rotation of the fluid. This is the famous photograph taken for
superfluid helium by the Berkeley group.This idea was applied to
atomic BECs. Until now, as far as I know, four groups have
succeeded in making and observing vortex lattices. The first
experiment made by the Paris group revealed the exciting dynamics.
Being motivated by this experiment, we made theoretical and
numerical research. This viewgraph shows what was done by the Paris
group.The BEC was trapped by this ciger-shaped potential. They
rotated the BEC by using this laser beam, what they call, optical
spoon. The point is the optical spoon not only rotates the system
but also induces small anisotropy in the potential. The Paris group
observed for the first time the dynamical process of how the vortex
lattice is formed. They are the snapshots of the BEC after turning
on the rotation. At first, This figure shows the time development
of the anisotropic parameter corresponding to the deformation of
the shape of the condensate, something like the aspect ratio. The
point is that, at first this parameter oscillates, but it suddenly
vanishes and the condensate becomes circular when the vortices
enter the condensate from the surface. In order to understand this
phenomena, we made the numerical analysis of the GP equation. This
is the GP equation in a rotating frame, which includes the term of
the product of the rotation frequency and the angular momentum.
Since the system is long cigar shaped, we assume the system is
two-dimensional like this.Since a vortex lattice corresponds to the
energy minimum of the system, we need some dissipative mechanism in
order to obtain the state. So we introduce phenomenologically the
dissipative term into the GP equation. We tune this dissipative
parameter gamma to 0.03. There are some discussions about this
dissipation which comes from the collision between condensate and
noncondensate.
Before going to the dynamics of many vortices, we would like to
see the profile of a single quantized vortex. These figures show
the profile of density and phase of the wave function when there is
one quantized vortex at the center. The density has a hole in the
vortex core. And this figure shows the phase by color. You can find
a branch cut between 0 and 2pi, and its edge corresponds to the
vortex core around which the phase rotates by twice pi. Thus we can
identify a quantized vortex by both density and phase profiles. So
this is the highlight of this topic. We show the development of the
density after turning on the rotation whose frequency is 0.7 times
the trapping frequency. We can find the dynamics consistent with
the experimental results.They are the snapshots of the dynamics. At
first, the condensate makes the quadruple oscillation, then the
surface becomes unstable, then the surface ripples developing to
the density holes, eventually the lattice is formed. Then appears
an important question; are these density holes actually quantized
vortices?This question is answered by investigating the phase
profile as I said. This movie shows the development of the phase
for the same dynamics.These are the snapshots of the phase
dynamics. After the rotation starts, lots of branch cuts appear in
the low density region, coming to the surface of the condensate.
Their edges are quantized vortices. However, their further invasion
into the condensate costs the energy, so they cannot enter the BEC
easily. These vortices in the low density region are invisible in
the density profile, so we call them ghost vortices. Then ghost
vortices run along the surface of the condensate, causing surface
ripples in the density.Then some ghost vortices enter the BEC,
becoming real vortices with the density holes, eventually forming
the vortex lattice. On the other hand, the ghost vortices which
could not enter the condensate wander in the low density
region.This dynamics is quantitatively consistent with the
observations. The upper figure shows the development of the
anisotropic parameter of the simulation, which agrees well with the
experimental result of the lower figure. For example, the BEC
recovers its circular symmetry at about 400ms, which is consistent
with 300ms in the experiment.As a summary of this topic, I will
show you the simultaneous display of the density and the
phase.Recently we make the three-dimensional calculation of the
vortex lattice formation with Machida and Sasa. The grid is 126
cubed, and gamma is still 0.03. The dynamics is qualitatively
similar to the two-dimensional one. However this movie reminds us
of superfluidity of a neutron star, doesnt it?The next topic is
Giant vortex.This research comes from the motivation of what
happens to vortices when the BEC rotates very fast. The BEC is
usually trapped by a quadratic potential, while the rotation makes
this quadratic centrifugal potential. Thus, when the rotation
frequency increases and exceeds the trapping frequency, the
trapping potential is no longer effective. In order to trap the EBC
even under such fast rotation, we introduced the quartic potential.
Then, the question is what happens to vortices?The left movie shows
the density dynamics when the rotation frequency is 2.5times the
trapping frequency. The dynamics is similar to the previous case,
but vortices tend to gather in the center. When Omega is 3.2 omega,
all vortices are absorbed by the central hole. This behavior is
understood by the potential. In this case, 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. We call this structure a giant vortex. This phase
movie shows this giant vortex is not a quantized vortex with
multi-quanta but a lattice of ghost vortices. Such giant vortex was
observed by the JILA group, the method was just different from our
proposal though.