Slide 2 Numerical simulations of superfluid vortex turbulence
Vortex dynamics in superfluid helium and BEC 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
Makoto TSUBOTA Osaka City University, Japan.. Slide 3 Vortices in
Japanese sea Galaxy Vortex lattice in a rotating Bose condensate
Vortices appear in many fields of nature! Vortex tangle in
superfluid helium Slide 4 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 2. Classical turbulence Outline Slide 5 1.
Introduction A 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 n 2 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)(r) rot v s
The order of the coherence length. Slide 6 How to describe the
vortex dynamics Vortex 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. s r The Gross-Pitaevskii equation for the macroscopic wave
function Slide 7 What is the difference of vortex dynamics between
superfluid helium and an atomic BEC? Superfluid helium ( 4 He) Core
size (healing length) System size L The vortex dynamics is local,
compared with the scale of the whole system. Atomic Bose-Einstein
condensate Core size (healing length) subm System size L m The
vortex dynamics is closely coupled with the collective motion of
the whole condensate. Slide 8 Superfluid helium 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. Liquid 4 He 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
Slide 9 Lots of experimental studies were done chiefly for thermal
counterflow of superfluid 4 He. 1980s 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.
Vortex tangle Heater Normal flowSuper flow Slide 10
Three-dimensional dynamics of quantized vortex filaments (1)
Superfluid flow made by a vortex Biot-Savart law s r In the absence
of friction, the vortex moves with its local velocity. The mutual
friction with the normal flow is considered at a finite
temperature. When two vortices approach, they are made to
reconnect. Slide 11 Three-dimensional dynamics of quantized vortex
filaments (2) Vortex reconnection This was confirmed by the
numerical analysis of the GP equation. J. Koplik and H. Levine,
PRL71, 1375(1993). Slide 12 Three-dimensional dynamics of quantized
vortex filaments (1) Superfluid flow made by a vortex Biot-Savart
law s r In the absence of friction, the vortex moves with its local
velocity. The mutual friction with the normal flow is considered at
a finite temperature. When two vortices approach, they are made to
reconnect. s r Slide 13 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 applied flow and the
mutual friction. The obtained vortex density L(v ns, T) agreed
quantitatively with experimental data. v s v n Slide 14 What is the
relation between superfluid turbulence and classical turbulence
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. Slide 15 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. Slide 16 Classical turbulence and vortices Numerical
analysis of the Navier-Stokes equation made by Shigeo Kida Vortex
cores are visualized by tracing pressure minimum in the fluid. Gird
turbulence Slide 17 Classical turbulence and vortices Numerical
analysis of the Navier-Stokes equation made by Shigeo Kida Vortex
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. Slide 18
Classical turbulence Energy- containing range Inertial range
Energy-dissipative range Energy spectrum of turbulence Kolmogorov
law Energy spectrum of the velocity field Energy-containing range
The energy is injected into the system at.. Inertial range
Dissipation does not work. The nonlinear interaction transfers the
energy from low k region to high k region. Kolmogorov law : E(k)=C
2/3 k - -5/3 Energy-dissipative range The energy is dissipated with
the rate at the Kolmogorov wave number k c = (/ 3 ) 1/4. Richardson
cascade process Slide 19 Kolmogorov spectrum in classical
turbulence ExperimentNumerical analysis Slide 20 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
The 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? Slide
21 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. Slide 22 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 Slide 23 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! Slide 24 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
Normal- Fluid 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. Slide 25 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
Slide 26 C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9,
2644(1997) t=5.5 : 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) 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) tk E(k) 5/3 E(k)=A(t) k -n(t) Slide
27 l : intervortex spacing 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 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 k 300cm -1. Slide 28 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 Slide 29 Kelvin waves Slide 30 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 Slide
31 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. Slide 32
Starting from the uniform density and the random phase, we obtain a
turbulent state. Vorticity 0 < t < 5.76 =1 256 3 grids 512 3
grids Density Phase Slide 33 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) Slide 34
3-3. Rotating superfluid turbulence Tsubota, Araki, Barenghi,PRL90,
205301(03); Tsubota, Barenghi, et al., PRB69, 134515(04) Vortex
array under rotation order Vortex tangle in turbulence disorder vv
ns What happens if we combine both effects? Slide 35 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 v c1 and v c2 ; v c1 is consistent with the
onset of Donnely-Graberson(DG) instability of Kelvin waves, but v
c2 is a mystery. This work has lacked theoretical interpretation!
L=2/ Vortex array 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. Slide 36 Initial vortices vortex array with small
random noise Boundary condition Periodic z-axis Solid boundary
x,y-axis Counterflow is applied along z-axis. Time evolution of a
vortex array towards a polarized vortex tangle 1. The array becomes
unstable,exciting Kelvin waves. Glaberson instability 2. 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. Slide 37 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 , b vortex line spacing, angular velocity of
rotation Landau critical velocity beyond which the array becomes
unstable , Slide 38 Numerical confirmation of Glaberson instability
Glabersons theory:Vc=0.010[cm/s] v =0.008[cm/s] ns v =0.015[cm/s]v
=0.03[cm/s] ns v =0.05[cm/s]v =0.06[cm/s]v =0.08[cm/s] ns =9.9710
-3 rad/sec Slide 39 What happens beyond the Glaberson instability?
Vortex line densityPolarization A polarized tangle Slide 40
Polarization as a function of v ns 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) Slide 41 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) Slide 42 Rotating
superfluid and the vortex lattice c c The triangular vortex lattice
sustains the solid body rotation. Yarmchuck and Packard Minimizing
the free energy in a rotating frame. Vortex lattice observed in
rotating superfluid helium Slide 43 Observation of quantized
vortices in atomic BECs K.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) ENS MIT JILA E. Hodby, et.al PRL 88, 010405 (2002)
Oxford Slide 44 How can we rotate the trapped BEC K.W.Madison et.al
Phys.Rev Lett 84, 806 (2000) Non-axisymmetric potential Optical
spoon Total potential Rotation frequency z x y 100 m 5m5m 20 m 16 m
Axisymmetric potential cigar-shape Slide 45 Direct observation of
the vortex lattice formation Snapshots of the BEC after turning on
the rotation 1. 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) RxRx RyRy Slide
46 The Gross-Pitaevskii(GP) equation in a rotating frame Wave
function Interaction s-wave scattering length in a rotating frame
Two-dimensional simplified Slide 47 The GP equation with a
dissipative term 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) Slide 48 Profile of a single quantized vortex A
quantized vortex Velocity field Vortex core= healing length A
vortex Slide 49 Dynamics of the vortex lattice formation (1) Time
development of the condensate density Experiment Tsubota et al.,
Phys. Rev. A 65, 023603 (2002) Grid 256256 Time step 10 -3 Slide 50
Dynamics of the vortex lattice formation (2) t=067ms340ms 390ms
410ms700ms Time-development of the condensate density Are these
holes actually quantized vortices? Slide 51 Dynamics of the vortex
lattice formation (3) Time-development of the phase Slide 52
Dynamics of the vortex lattice formation (4) t=067ms340ms 390ms
410ms700ms Ghost vortices Becoming real vortices Time-development
of the phase Slide 53 This dynamics is quantitatively consistent
with the observations. K.W.Madison et.al. Phys. Rev. Lett. 86, 4443
(2001) RxRx RyRy Slide 54 Simultaneous display of the density and
the phase Slide 55 Three-dimensional calculation of the vortex
lattice formation M. Machida (JAERI), N. Sasa, M.Tsubota,
K.Kasamatsu Condensate density Grid : 128128128 =0.03=0.1 Riminding
us of superfluidity of a neutron star! Slide 56 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 Slide 57 A fast
rotating BEC in a quadratic-plus-quartic potential A quadratic
potential A centrifugal potential A quadratic-plus-quartic
potential can trap the BEC even if. What happens to vortices when
When, the trapping potential is no longer effective. Slide 58 A
giant vortex in a fast rotating BEC (1) =2.5 (2) =3.2 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. 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 Slide 59
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 Summary