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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..

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Vortices in Japanese sea Galaxy Vortex lattice in a rotating Bose condensate Vortices 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 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

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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. ρ r s 〜Å (r)(r) rot v s The order of the coherence length.

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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

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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.

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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

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Lots of experimental studies were done chiefly for thermal counterflow of superfluid 4 He. 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. Vortex tangle Heater Normal flowSuper flow

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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.

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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).

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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

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Development of a vortex tangle in a thermal counterflow Schwarz, Phys.Rev.B38, 2398(1988). Schwarz obtained numerically the statistically steady state of a vortex tangle which is sustained by the competition between the applied flow and the mutual friction. The obtained vortex density L(v ns, T) agreed quantitatively with experimental data. v s v n

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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.

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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.

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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

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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.

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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)=Cε 2/3 k - -5/3 Energy-dissipative range The energy is dissipated with the rate ε at the Kolmogorov wave number k c = (ε/ν 3 ) 1/4. Richardson cascade process

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Kolmogorov spectrum in classical turbulence ExperimentNumerical analysis

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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?

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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. ？

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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 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!

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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.

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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 ＝２ ４ ６ ８ １０ １２

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C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) 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） △ : 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)

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l : intervortex spacing 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. C=1 The dissipation ε arises from eliminating smallest vortices whose size becomes comparable to the numerical space resolution at k 〜 300cm -1.

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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

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Kelvin waves

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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

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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.

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Starting from the uniform density and the random phase, we obtain a turbulent state. Vorticity 0 < t < 5.76 γ= grids grids Density 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, (‘03); Tsubota, Barenghi, et al., PRB69, (‘04) Vortex array under rotation order Vortex tangle in turbulence disorder Ω vv ns What 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 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=2Ω/κ 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.

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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.

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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 ：,

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Numerical confirmation of Glaberson instability Glaberson’s 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 Ω=9.97×10 -3 rad/sec

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

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Polarization as a function of v ns 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)

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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) 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)

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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

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

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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”

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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

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

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The GP equation with a dissipative term S.Choi, et.al. PRA 57, 4057 (1998) I.Aranson, et.al. PRB 54, (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 vortex A quantized vortex Velocity field Vortex core= healing length A vortex

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

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Dynamics of the vortex lattice formation (2) t=067ms340ms 390ms 410ms700ms Time-development of the condensate density Are 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=067ms340ms 390ms 410ms700ms Ghost vortices Becoming real vortices Time-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) RxRx RyRy

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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 density Grid : 128×128×128 γ=0.03γ=0.1 Riminding 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, (2002) 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

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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.

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A giant vortex in a fast rotating BEC (1) Ω=2.5ω ⊥ (2) Ω=3.2ω ⊥ 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

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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 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 ） Summary

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