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Numerical simulations of superfluid vortex turbulence Vortex dynamics in superfluid helium and BEC

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

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Numerical simulations of superfluid vortex turbulence Vortex dynamics in superfluid helium and BEC

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  1. Numerical simulations of superfluid vortex turbulenceVortex 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 ..

  2. Vortices appear in many fields of nature! Vortices in Japanese sea Galaxy 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

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

  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.

  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, 1957Hall and Vinen Observing “superfluid turbulence” The mutual friction between the vortex tangle and the normal fluid causes the dissipation.

  8. Lots of experimental studies were done chiefly for thermal counterflow of superfluid 4He. Vortex tangle Heater Normal flow Super flow 1980’sK. W. SchwarzPhys.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 vortexBiot-Savart law r s In the absence of friction, the vortex moves with its local velocity. The mutual friction with the normal flowis considered at a finite temperature. When two vortices approach, they are made to reconnect.

  10. 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).

  11. Three-dimensional dynamics of quantized vortex filaments (1) Superfluid flow made by a vortexBiot-Savart law r r s s In the absence of friction, the vortex moves with its local velocity. The mutual friction with the normal flowis considered at a finite temperature. When two vortices approach, they are made to reconnect.

  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. vs vn

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

  14. 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.

  15. 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.

  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. 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. E(k)=Cε2/3k- -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

  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). 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 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, 2.3KMeasurements of local pressure in flows driven by two counterrotating disks finds theKolmogorov spectrum. • Stalp, Skrbek and Donnely, Phys.Rev.Lett. 82, 4831(1999)1.4 < T < 2.15KDecay of grid turbulence The data of the second sound attenuation was consistent with a classical model with the Kolmogorov spectrum.

  22. Vinen, Phys.Rev.B61, 1410(2000)Considering the relation between ST and CTThe 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!

  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, 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.

  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 ８ ６ １０ １２

  25. C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) △: 2 < k < 12 ○: 2 < k < 14 □: 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. 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 superfluid turbulence with no normal-fluid componentT.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. l : intervortex spacing C=1 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, 135301(2003)). ST does not mimic CT. l : intervortex spacing

  28. Kelvin waves

  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, 135301(2003)). ST does not mimic CT. l : intervortex spacing

  30. 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.

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

  32. Energy spectrum of the incompressible kinetic energy Time development of the exponent Energy spectrum at t = 5.76 July 2004 ( Trento and Prague) August 2004 ( Lammi)

  33. 3-3. Rotating superfluid turbulenceTsubota, Araki, Barenghi,PRL90, 205301(‘03); Tsubota, Barenghi, et al., PRB69, 134515(‘04) Vortex array under rotation Vortex tangle in turbulence Ω v v n s order disorder What happens if we combine both effects?

  34. ? DG instability ? C.E. Swanson, C.F. Barenghi, RJ. Donnelly, PRL 50, 190(1983). The only experimental work 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. 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.

  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： ,

  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 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！

  39. Polarization <s’z> as a function of vns and Ω The values of Ω are4.98×10-2 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)

  40. 4. Dynamics of quantized vortices in rotating BECs4-1 Vortex lattice formationTsubota, 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)

  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 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, 210403 (2001) MIT J.R. Abo-Shaeer, et.al Science 292, 476 (2001) Oxford E. Hodby, et.al PRL 88, 010405 (2002)

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

  47. Profile of a single quantized vortex A vortex A quantized vortex Velocity field 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, 023603 (2002) Experiment 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? 410ms 700ms 390ms

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

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