1. 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
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. ..

2. Vortices appear in many fields of nature!
Vortices in Japanese sea
Galaxy
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.
Vortex tangle in superfluid helium
Vortex lattice in a rotating Bose condensate

3. 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
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.

4. 1. Introduction A quantized vortex is a vortex of superflow in a BEC.
(i) The circulation is quantized.
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.
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.
The vortex is stable. 〜Å ρ
(r) s
(iii) The core size is very small.
rot v
The order of the coherence length. s r

5. 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. r s
The Gross-Pitaevskii equation for the macroscopic wave function
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.

6. 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.
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.

7. 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
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.
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 1940’s 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 Fyenman’s famous article, showing large vortices are broken up to smaller vortices through vortex reconnections like a cascade process. In the 50’s, Hall and Vinen observed experimentally superfluid turbulence in which the mutual friction between vortices and normal fluid causes that dissipation.

8. Lots of experimental studies were done chiefly for thermal counterflow of superfluid 4He.
Vortex tangle
Heater
Normal flow
Super flow
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 80’s, Klaus Schwarz made the direct numerical simulation of the three-dimensional dynamics of quantized vortices and succeeded in understanding quantitatively the observed temperatures difference.
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 .

9. Three-dimensional dynamics of quantized vortex filaments (1)
Superfluid flow made by a vortex　　 Biot-Savart law r s
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.
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.
When two vortices approach, they are made to reconnect.

10. Three-dimensional dynamics of quantized vortex filaments (2)
Vortex reconnection
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.
This was confirmed by the numerical analysis of the GP equation J. Koplik and H. Levine, PRL71, 1375(1993).

11. Three-dimensional dynamics of quantized vortex filaments (1)
Superfluid flow made by a vortex　　 Biot-Savart law r r s s
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.
By considering all those effect, we make the simulation of the vortex dynamics.
When two vortices approach, they are made to reconnect.

12. 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(vns, T) agreed quantitatively with experimental data.
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 Schwarz’s old works.
vs vn

13. 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.
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.
What is the relation between superfluid turbulence and classical turbulence？

14. 2. Classical turbulence
When we raise the flow velocity around a sphere,
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?
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.

15. Classical turbulence and vortices
Gird turbulence
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.
Numerical analysis of the Navier-Stokes equation made by Shigeo Kida
Vortex cores are visualized by tracing pressure minimum in the fluid.

16. 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.
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.
Numerical analysis of the Navier-Stokes equation made by Shigeo Kida
Vortex cores are visualized by tracing pressure minimum in the fluid.

17. Classical turbulence Energy spectrum of the velocity field
Energy-containing range
The energy is injected into the system at
Richardson cascade process
Inertial range
Kolmogorov law
Dissipation does not work. The nonlinear interaction transfers the energy from low k region to high k region.
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.
E(k)=Cε2/3 k- -5/3
Kolmogorov law :
Energy-containing range
Inertial range
Energy-dissipative range
Energy-dissipative range
Energy spectrum of turbulence
The energy is dissipated with the rate ε at the Kolmogorov wave number kc = (ε/ν3 )1/4.

18. Kolmogorov spectrum in classical turbulence
Experiment
Numerical analysis
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.

19. 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).
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.
How is the intermittency?
The tangle may give an interesting model with the Kolmogorov law.

20. Summary of the motivation
Classical turbulence ？ ？
Vortices
It is possible to consider quantized vortices as elements in the fluid and derive the essence of turbulence.
Superfluid turbulence
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.
Quantized vortices

21. 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.
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 Oregon’s group studied the decay of grid turbulence, whose data was consistent with a classical model with the Kolmogorov spectrum.

22. Our work attacks this problem directly!
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.
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.
What happens at very low temperatures? Is there still the similarity or not?
Our work attacks this problem directly!

23. 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, (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.
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 equation

24. 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＝２ ４
They 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. ６ ８ １０ １２

25. C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)
△: 2 < k < ○: 2 < k < □: 2 < k < 16
E(k)=A(t) k -n(t)
E(k)
n(t)
5/3 t k
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.
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.
t=5.5でのエネルギースペクトル　（非圧縮性運動エネルギー）

26. Energy spectrum of a vortex tangle in a late stage
Energy spectrum of superfluid turbulence with no normal-fluid component T.Araki, M.Tsubota, S.K.Nemirovskii, PRL89, (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.
l : intervortex spacing
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 aren’t 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.
C= The dissipation ε arises from eliminating smallest vortices whose size becomes comparable to the numerical space resolution at k〜300cm-1.

27. The spectrum depends on the length scale.
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.
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, (2003)). ST does not mimic CT.
l : intervortex spacing
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 doesn’t 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 doesn’t mimic classical turbulence.

28. Kelvin waves
They are Kelvin waves, helical waves excited along each vortex.

29. The spectrum depends on the length scale.
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.
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, (2003)). ST does not mimic CT.
l : intervortex spacing
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 doesn’t 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 doesn’t mimic classical turbulence.

30. Energy spectrum of the Gross-Pitaevskii turbulence
M . Kobayashi and M. Tsubota
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..
The dissipation is introduced so that it may work only in the scale smaller than the healing length.

31. Starting from the uniform density and the random phase, we obtain a turbulent state.
Vorticity
Phase
0 < t < 5.76
γ=1
2563 grids
5123 grids
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.

32. Energy spectrum of the incompressible kinetic energy
Time development of the exponent
Energy spectrum at t = 5.76
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 month’s result has just wider inertial range. Physics is improving month by month.
July 2004 ( Trento and Prague)
August 2004 ( Lammi)

33. 3-3. Rotating superfluid turbulence Tsubota, Araki, Barenghi,PRL90, (‘03); Tsubota, Barenghi, et al., PRB69, (‘04)
Vortex array under rotation
Vortex tangle in turbulence Ω v v n s
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.
order
disorder
What happens if we combine both effects?

34. The only experimental work?
DG instability ?
C.E. Swanson, C.F. Barenghi, RJ. Donnelly, PRL 50, 190(1983).
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.
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 Feyman’s 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 didn’t know what happened to vortices in these regions.
This work has lacked theoretical interpretation!

35. 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.
Initial vortices：vortex array with small random noise
Boundary condition：　　Periodic（z-axis） Solid boundary（x,y-axis）
Counterflow is applied along z-axis.
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.

36. 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：
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. ,

37. Numerical confirmation of Glaberson instability
Ω=9.97×10-3 rad/sec
Glaberson’s theory:Vc=0.010[cm/s]
v =0.008[cm/s]
v =0.015[cm/s]
v =0.03[cm/s] ns ns ns
This figure shows the Glaberson’s instability. For this value of the angular velocity Omega, the Glaberson’s 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.
v =0.05[cm/s]
v =0.06[cm/s]
v =0.08[cm/s] ns ns ns

38. What happens beyond the Glaberson instability?
Vortex line density
Polarization
A polarized tangle！
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.

39. 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.
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.
Competition between order ( rotation) and disorder ( flow)

40. 4. Dynamics of quantized vortices in rotating BECs 4-1 Vortex lattice formation Tsubota, Kasamatsu, Ueda, Phys.Rev.A64, (2002) Kasamatsu, Tsubota, Ueda, Phys.Rev.A67, (2003)
Now we are moving to the topics of Bose-Einstein condensates.The first is vortex lattice formation.
cf. A. A. Penckwitt, R. J. Ballagh, C. W. Gardiner, PRL89, (2002)
E. Lundh, J. P. Martikainen, K-A. Suominen, PRA67, (2003)
C. Lobo, A. Sinatora, Y. Castin, PRL92, (2004)

41. Rotating superfluid and the vortex lattice
W < Wc
W > Wc
Minimizing the free energy in a rotating frame.
The triangular vortex lattice sustains the solid body rotation.
Yarmchuck and Packard
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 doesn’t 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.
Vortex lattice observed in rotating superfluid helium

42. Observation of quantized vortices in atomic BECs
ENS
K.W.Madison, et.al PRL 84, 806 (2000)
JILA
P. Engels, et.al
PRL 87, (2001)
MIT
J.R. Abo-Shaeer, et.al Science 292, 476 (2001)
Oxford
E. Hodby, et.al
PRL 88, (2002)
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.

43. How can we rotate the trapped BEC？
K.W.Madison et.al Phys.Rev Lett 84, 806 (2000) z
5mm
Axisymmetric potential y x
100mm
“cigar-shape”
Total potential
Non-axisymmetric potential
Rotation frequency W
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.
Optical spoon
20mm
16mm

44. 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
1. The BEC becomes elliptic, then oscillating.
2. The surface becomes unstable.
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. Rx
3. Vortices enter the BEC from the surface. Ry
4. The BEC recovers the axisymmetry, the vortices forming a lattice.

45. The Gross-Pitaevskii(GP) equation in a rotating frame
Interaction
Wave function
s-wave scattering length
in a rotating frame
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.
Two-dimensional
simplified Ω Ω

46. The GP equation with a dissipative term
S.Choi, et.al. PRA 57, 4057 (1998)
I.Aranson, et.al. PRB 54, (1996)
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 There are some discussions about this dissipation which comes from the collision between condensate and noncondensate.
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)

47. Profile of a single quantized vortex
A vortex
A quantized vortex
Velocity field
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.
Vortex core= healing length

48. Dynamics of the vortex lattice formation (1)
Time development of the condensate density
Tsubota et al., Phys. Rev. A 65, (2002)
Experiment
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.
Grid 256×256
Time step 10-3

49. Dynamics of the vortex lattice formation (2)
Time-development of the condensate density
t=0
67ms
340ms
Are these holes actually quantized vortices?
390ms
410ms
700ms
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?

50. Dynamics of the vortex lattice formation (3)
Time-development of the phase
This question is answered by investigating the phase profile as I said. This movie shows the development of the phase for the same dynamics.

51. Dynamics of the vortex lattice formation (4)
Time-development of the phase
t=0
67ms
340ms
Ghost vortices
Becoming real vortices
390ms
410ms
700ms
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.

52. This dynamics is quantitatively consistent with the observations.Rx Ry
K.W.Madison et.al. Phys. Rev. Lett.
86 , 4443 (2001)
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.

53. Simultaneous display of the density and the phase
As a summary of this topic, I will show you the simultaneous display of the density and the phase.

54. Three-dimensional calculation of the vortex lattice formation
M. Machida (JAERI), N. Sasa, M.Tsubota, K.Kasamatsu
Grid : 128×128×128
γ=0.03
Condensate density
γ=0.1
Riminding us of superfluidity of a neutron star!
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 The dynamics is qualitatively similar to the two-dimensional one. However this movie reminds us of superfluidity of a neutron star, doesn’t it?

55. 4-2. Giant vortex in a fast rotating BEC Kasamatsu,Tsubota,Ueda, Phys
4-2. Giant vortex in a fast rotating BEC Kasamatsu,Tsubota,Ueda, Phys.Rev.A67, (2002)
The next topic is Giant vortex….
cf. A. L. Fetter, PRA64, (2001)
U. R. Fischer and G. Baym, PRL90, (2003)
E. Lundh, PRA65, (2002)
T. L. Ho, PRL87, (2001) many references

56. A fast rotating BEC in a quadratic-plus-quartic potential
A quadratic potential W
A centrifugal potential
When , the trapping potential is no longer effective.
A quadratic-plus-quartic potential can trap the BEC even if
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? W
What happens to vortices when ？

57. A giant vortex in a fast rotating BEC
(1) Ω=2.5ω⊥
(2) Ω=3.2ω⊥
Vortices gather in the central hole.
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.
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.
Mexican hat
・This giant vortex is not a quantized vortex with multi-quanta but a lattice of ghost vortices.

58. 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, (2003), cond-mat ）