Numerical simulations of superfluid vortex turbulence Vortex dynamics

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

n i r a e p Galaxy p a s e c i t r o V Vortex tangle in superfluid helium s d l e i f y n a m ! e r u t of na Vortices in Japanese sea Vortex lattice in a rotating Bose condensate

Outline 1. Introduction 2. Classical turbulence 3. Dynamics of quantized vortices in superfluid helium 3 -1 Recent interests in superfluid turbulence 3 -2 Energy spectrum of superfluid turbulence 3 -3 Rotating superfluid turbulence 4. Dynamics of quantized vortices in rotating BECs 4 -1 Vortex lattice formation 4 -2 Giant vortex in a fast rotating BEC

1. Introduction A quantized vortex is a vortex of superflow in a BEC. (i) The circulation is quantized. A vortex with n≧ 2 is unstable. Every vortex has the same circulation. (ii) Free from the decay mechanism of the viscous diffusion of the vorticity. The vortex is stable. 〜Å ρ s (r) (iii) The core size is very small. The order of the coherence length. rot v s r

How to describe the vortex dynamics Vortex filament formulation (Schwarz)　　Biot-Savart law r s A vortex makes the superflow of the Biot-Savart law, and moves with this flow. At a finite temperature, the mutual friction should be considered. Numerically, a vortex is represented by a string of points. The Gross-Pitaevskii equation for the macroscopic wave function

What is the difference of vortex dynamics between superfluid helium and an atomic BEC? Superfluid helium (4 He) Core size (healing length) ξ〜Å ≪ System size L〜㎜ The vortex dynamics is local, compared with the scale of the whole system. Atomic Bose-Einstein condensate Core size (healing length) ξ〜 subμm ≦ System size L〜 μm The vortex dynamics is closely coupled with the collective motion of the whole condensate.

Superfluid helium Liquid 4 He 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.

Lots of experimental studies were done chiefly for thermal counterflow of superfluid 4 He. Vortex tangle Heater Normal flow Super flow 1980’s　K. W. Schwarz　 Phys. Rev. B 38, 2398(1988) Made the direct numerical simulation of the three-dimensional dynamics of quantized vortices and succeeded in explaining quantitatively the observed temperature difference △T.

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

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, PRL 71, 1375(1993).

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

Development of a vortex tangle in a thermal counterflow Schwarz, Phys. Rev. B 38, 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

Most studies of superfluid turbulence were devoted to thermal counterflow. 　　　　⇨　No analogy with classical turbulence When Feynman showed the above figure, he thought of a cascade process in classical turbulence. What is the relation between superfluid turbulence and classical turbulence？

2. Classical turbulence When we raise the flow velocity around a sphere, The understanding and control of turbulence have been one of the most important problems in fluid dynamics since Leonardo Da Vinci, but it is too difficult to do it.

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

Classical turbulence and vortices ・The vortices have different circulation and different core size. ・The vortices repeatedly appear, diffuse and disappear. It is difficult to identify each vortex! Compared with quantized vortices. Numerical analysis of the Navier-Stokes equation made by Shigeo Kida Vortex cores are visualized by tracing pressure minimum in the fluid.

Classical turbulence Energy spectrum of the velocity field Energy-containing range The energy. Richardson is injected into the system at cascade process Kolmogorov law . . Inertial range Dissipation does not work. The nonlinear interaction transfers the energy from low k region to high k region. Energycontaining range Kolmogorov law : Inertial range Energy-dissipative range Energy spectrum of turbulence E(k)=Cε 2/3 k- -5/3 Energy-dissipative range The energy is dissipated with the rate ε at the Kolmogorov wave number kc = (ε/ν 3 )1/4.

Kolmogorov spectrum in classical turbulence Experiment Numerical analysis

Vortices in superfluid turbulence • ST consists of a tangle of quantized vortex filaments Characteristics of quantized vortices • Quantization of the circulation • Very thin core • No viscous diffusion of the vorticity • A quantized vortex is a stable and definite defect, compared with vortices in a classical fluid. The only alive freedom is the topological configuration of its thin cores. y? c n e t t i erm t n i e h t • Because of superfluidity, How is some dissipation would work only at large wave numbers (at very low temperatures). The tangle may give an interesting model with the Kolmogorov law.

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

3. Dynamics of quantized vortices in superfluid helium 3 -1 Recent interests in superfluid turbulence Does ST mimic CT or not? • Maurer and Tabeling, Europhysics. Letters. 43, 29(1998) T =1. 4, 2. 08, 2. 3 K 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. 15 K Decay of grid turbulence The data of the second sound attenuation was consistent with a classical model with the Kolmogorov spectrum.

• Vinen, Phys. Rev. B 61, 1410(2000) Considering the relation between ST and CT The Oregon’s result is understood by the coupled dynamics of the superfluid and the normal fluid due to the mutual friction. Length scales are important, compared with the characteristic vortex spacing in a tangle. • Kivotides, Vassilicos, Samuels and Barenghi, Europhy. Lett. 57, 845(2002) When superfluid is coupled with the normal-fluid turbulence that obeys the Kolmogorov law, its spectrum follows the Kolmogorov law too. What happens at very low temperatures? Is there still the similarity or not? Our work attacks this problem directly!

3 -2. Energy spectrum of superfluid turbulence Decaying Kolmogorov turbulence in a model of superflow C. Nore, M. Abid and M. E. Brachet, Phys. Fluids 9, 2644(1997) By solving the Gross-Pitaevskii equation, they studied the energy spectrum of a Tarlor-Green flow. The spectrum shows the -5/3 power on the way of the decay, but the acoustic emission is concerned and the situation is complicated. Energy Spectrum of Superfluid Turbulence with No 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.

C. Nore, M. Abid and M. E. Brachet, Phys. Fluids 9, 2644(1997) t＝２ Although the total energy is conserved, its incompressible component is changed to the compressible one (sound waves). ４ The total length of vortices increases monotonically, with the large scale motion decaying. ８ ６ １０ １２

C. Nore, M. Abid and M. E. Brachet, Phys. Fluids 9, 2644(1997) △: 2 < k < 12 ○: 2 < k < 14 □: 2 < k < 16 n(t) E(k)=A(t) k -n(t) 5/3 t k The right figure shows the energy spectrum att=5. 5 a moment. でのエネルギースペクトル　（非 The left figure shows the development of the exponent n(t) in圧縮性運動エネルギー） the spectrum E(k)=A(t) k n(t). The exponent n(t) goes through 5/3 on the way of the dynamics.

Energy spectrum of superfluid turbulence with no normal-fluid component T. Araki, M. Tsubota, S. K. Nemirovskii, PRL 89, 145301(2002) We calculated the vortex filament dynamics starting from the Taylor-Green flow, and obtained the energy spectrum directly from the vortex configuration. Energy spectrum of a vortex tangle in a late stage l : intervortex spacing C=1 The dissipation ε arises from eliminating smallest vortices whose size becomes comparable to the numerical space resolution at k〜 300 cm-1.

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. l : intervortex spacing • For k > 2π/ l, the spectrum is k-1, coming from the velocity field due to each vortex. The energy is probably transferred by the Kelvin wave cascade process (Vinen, Tsubota, Mitani, PRL 91, 135301(2003)). ST does not mimic CT.

Kelvin waves

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. l : intervortex spacing • For k > 2π/ l, the spectrum is k-1, coming from the velocity field due to each vortex. The energy is probably transferred by the Kelvin wave cascade process (Vinen, Tsubota, Mitani, PRL 91, 135301(2003)). ST does not mimic CT.

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

Starting from the uniform density and the random phase, we obtain a turbulent state. Vorticity Density Phase 0 < t < 5. 76 γ=1 2563 grids 5123 grids

Energy spectrum of the incompressible kinetic energy Time development of the exponent July 2004 ( Trento and Prague) Energy spectrum at t = 5. 76 August 2004 ( Lammi)

3 -3. Rotating superfluid turbulence Tsubota, Araki, Barenghi, PRL 90, 205301(‘ 03); Tsubota, Barenghi, et al. , PRB 69, 134515(‘ 04) Vortex array under rotation Vortex tangle in turbulence Ω vn order vs disorder ? ts c e ff e th o b e in b m o c What happens if we

The only experimental work C. E. Swanson, C. F. Barenghi, RJ. Donnelly, PRL 50, 190(1983). DG instability By using a rotating cryostat, they made counterflow turbulence under rotation and observed the vortex line density by the measurement of the second sound attenuation. 1. There are two critical velocities vc 1 and vc 2; vc 1 is consistent with the onset of Donnely-Graberson(DG) instability of Kelvin waves, but vc 2 is a mystery. L=2Ω/κ Vortex array 2. Vortex tangle seems to be polarized, because the increase due to the flow is less than expected. This work has lacked theoretical interpretation! ? ?

Time evolution of a vortex array towards a polarized vortex tangle 1. The array becomes Initial vortices：vortex array unstable, exciting Kelvin waves. with small random noise Glaberson instability Boundary condition：　　 2. When the wave amplitude Periodic（z-axis） becomes comparable to the. Solid vortex boundary（x, y-axis） separation, reconnections start to make lots of vortex loops. along z Counterflow is applied -axis. loops disturb the array, 3. These leading to a tangle.

Glaberson instability W. I. Glaberson et al. , Phys. Rev. Lett 33, 1197 (1974). Glaberson et al. discussed the stability of the vortex array in the presence of axial normal-fluid flow. If the normal-fluid is moving faster than the vortex wave with k, the amplitude of the wave will grow (analogy to a vortex ring). Dispersion relation of vortex wave in a rotating frame： , b：vortex line spacing, ：angular velocity of rotation Landau critical velocity beyond which the array becomes unstable： ,

Numerical confirmation of Glaberson instability Ω=9. 97× 10 -3 rad/sec Glaberson’s theory: Vc=0. 010[cm/s] v ns =0. 008[cm/s] v ns =0. 015[cm/s] v ns =0. 03[cm/s] v ns =0. 05[cm/s] v ns =0. 06[cm/s] v ns =0. 08[cm/s]

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

Polarization <s’z> as a function of vns and Ω The values of Ω are 4. 98× 10 -2 rad/sec(■), 2. 99× 10 -2 rad/sec (▲), 9. 97× 10 -3 rad/sec(●). We obtained a new vortex state, a polarized vortex tangle. Competition between order ( rotation) and disorder ( flow)

4. Dynamics of quantized vortices in rotating BECs 4 -1 Vortex lattice formation Tsubota, Kasamatsu, Ueda, Phys. Rev. A 64, 053605(2002) Kasamatsu, Tsubota, Ueda, Phys. Rev. A 67, 033610(2003) cf. A. A. Penckwitt, R. J. Ballagh, C. W. Gardiner, PRL 89, 260402 (2002) E. Lundh, J. P. Martikainen, K-A. Suominen, PRA 67, 063604 (2003) C. Lobo, A. Sinatora, Y. Castin, PRL 92, 020403 (2004)

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 Vortex lattice observed in rotating superfluid helium

Observation of quantized vortices in atomic BECs ENS K. W. Madison, et. al PRL 84, 806 (2000) JILA P. Engels, et. al PRL 87, 210403 (2001) MIT J. R. Abo-Shaeer, et. al Science 292, 476 (2001) Oxford E. Hodby, et. al PRL 88, 010405 (2002)

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

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. Rx Ry 3. Vortices enter the BEC from the surface. 4. The BEC recovers the axisymmetry, the vortices forming a lattice.

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

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)

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

Dynamics of the vortex lattice formation (1) Time development of the condensate density Tsubota et al. , Phys. Rev. A 65, 023603 (2002) Grid 256× 256 Time step 10 -3 Experiment

Dynamics of the vortex lattice formation (2) Time-development of the condensate density t=0 390 ms 67 ms 340 ms s? e c i t r o v d e z i t an u q y l l a u 410 ms 700 ms t c a s e l o h e s e Are th

Dynamics of the vortex lattice formation (3) Time-development of the phase

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

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

Simultaneous display of the density and the phase

Three-dimensional calculation of the vortex lattice formation γ=0. 03 M. Machida (JAERI), N. Sasa, M. Tsubota, K. Kasamatsu Condensate density Grid : 128× 128 γ=0. 1 Rimind r e p u s f ing us o ! r a t s n o tr u e n a f fluidity o

4 -2. Giant vortex in a fast rotating BEC Kasamatsu, Tsubota, Ueda, Phys. Rev. A 67, 053606(2002) cf. A. L. Fetter, PRA 64, 063608 (2001) U. R. Fischer and G. Baym, PRL 90, 140402 (2003) E. Lundh, PRA 65, 043604 (2002) T. L. Ho, PRL 87, 060403 (2001) many references

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

A giant vortex in a fast rotating BEC (2) Ω=3. 2ω⊥ (1) Ω=2. 5ω⊥ Vortices gather in the central hole. A giant vortex ・ The effective potential takes the Mexican-hat form, so the central region becomes dilute and allows the vortices gathere, and the superfluid rotates around the core. Mexican hat ・This giant vortex is not a quantized vortex with multi-quanta but a lattice of ghost vortices.

Summary 3. Dynamics of quantized vortices in superfluid helium 3 -1 Recent interests in superfluid turbulence 3 -2 Energy spectrum of superfluid turbulence 3 -3 Rotating superfluid turbulence 4. Dynamics of quantized vortices in rotating BECs 4 -1 Vortex lattice formation 4 -2 Giant vortex in a fast rotating BEC 4 -3 Vortex states in two-component BEC （ Kasamatsu, Tsubota, Ueda, Phys. Rev. Lett. 91, 150406(2003), cond-mat 0406150）