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Slide 2 Numerical simulations of superfluid vortex turbulence Vortex dynamics in superfluid helium and BEC Collaborators: UK W.F. Vinen, C.F. Barenghi Finland M. Krusius, V.B. Eltsov, G.E. Volovik, A.P.Finne Russia S.K. Nemirovskii CR L. Skrbek Tokyo M. Ueda Osaka T. Araki, K. Kasamatsu, R.M. Hanninen, M. Kobayashi, A. Mitani Makoto TSUBOTA Osaka City University, Japan.. Slide 3 Vortices in Japanese sea Galaxy Vortex lattice in a rotating Bose condensate Vortices appear in many fields of nature! Vortex tangle in superfluid helium Slide 4 3. Dynamics of quantized vortices in superfluid helium 3-1 Recent interests in superfluid turbulence 3-2 Energy spectrum of superfluid turbulence 3-3 Rotating superfluid turbulence 4. Dynamics of quantized vortices in rotating BECs 4-1 Vortex lattice formation 4-2 Giant vortex in a fast rotating BEC 1. Introduction 2. Classical turbulence Outline Slide 5 1. Introduction A quantized vortex is a vortex of superflow in a BEC. (i) The circulation is quantized. (iii) The core size is very small. A vortex with n 2 is unstable. (ii) Free from the decay mechanism of the viscous diffusion of the vorticity. Every vortex has the same circulation. The vortex is stable. r s (r)(r) rot v s The order of the coherence length. Slide 6 How to describe the vortex dynamics Vortex filament formulation (Schwarz) Biot-Savart law A vortex makes the superflow of the Biot-Savart law, and moves with this flow. At a finite temperature, the mutual friction should be considered. Numerically, a vortex is represented by a string of points. s r The Gross-Pitaevskii equation for the macroscopic wave function Slide 7 What is the difference of vortex dynamics between superfluid helium and an atomic BEC? Superfluid helium ( 4 He) Core size (healing length) System size L The vortex dynamics is local, compared with the scale of the whole system. Atomic Bose-Einstein condensate Core size (healing length) subm System size L m The vortex dynamics is closely coupled with the collective motion of the whole condensate. Slide 8 Superfluid helium 1955 Feynman Proposing superfluid turbulence consisting of a tangle of quantized vortices. 1955, 1957 Hall and Vinen Observing superfluid turbulence The mutual friction between the vortex tangle and the normal fluid causes the dissipation. Liquid 4 He enters the superfluid state at 2.17 K with Bose condensation. Its hydrodynamics is well described by the two fluid model (Landau). Temperature (K) Superfluid helium becomes dissipative when it flows above a critical velocity. point Slide 9 Lots of experimental studies were done chiefly for thermal counterflow of superfluid 4 He. 1980s K. W. Schwarz Phys.Rev.B38, 2398(1988) Made the direct numerical simulation of the three-dimensional dynamics of quantized vortices and succeeded in explaining quantitatively the observed temperature difference T. Vortex tangle Heater Normal flowSuper flow Slide 10 Three-dimensional dynamics of quantized vortex filaments (1) Superfluid flow made by a vortex Biot-Savart law s r In the absence of friction, the vortex moves with its local velocity. The mutual friction with the normal flow is considered at a finite temperature. When two vortices approach, they are made to reconnect. Slide 11 Three-dimensional dynamics of quantized vortex filaments (2) Vortex reconnection This was confirmed by the numerical analysis of the GP equation. J. Koplik and H. Levine, PRL71, 1375(1993). Slide 12 Three-dimensional dynamics of quantized vortex filaments (1) Superfluid flow made by a vortex Biot-Savart law s r In the absence of friction, the vortex moves with its local velocity. The mutual friction with the normal flow is considered at a finite temperature. When two vortices approach, they are made to reconnect. s r Slide 13 Development of a vortex tangle in a thermal counterflow Schwarz, Phys.Rev.B38, 2398(1988). Schwarz obtained numerically the statistically steady state of a vortex tangle which is sustained by the competition between the applied flow and the mutual friction. The obtained vortex density L(v ns, T) agreed quantitatively with experimental data. v s v n Slide 14 What is the relation between superfluid turbulence and classical turbulence Most studies of superfluid turbulence were devoted to thermal counterflow. No analogy with classical turbulence When Feynman showed the above figure, he thought of a cascade process in classical turbulence. Slide 15 2. Classical turbulence When we raise the flow velocity around a sphere, The understanding and control of turbulence have been one of the most important problems in fluid dynamics since Leonardo Da Vinci, but it is too difficult to do it. Slide 16 Classical turbulence and vortices Numerical analysis of the Navier-Stokes equation made by Shigeo Kida Vortex cores are visualized by tracing pressure minimum in the fluid. Gird turbulence Slide 17 Classical turbulence and vortices Numerical analysis of the Navier-Stokes equation made by Shigeo Kida Vortex cores are visualized by tracing pressure minimum in the fluid. The vortices have different circulation and different core size. The vortices repeatedly appear, diffuse and disappear. It is difficult to identify each vortex! Compared with quantized vortices. Slide 18 Classical turbulence Energy- containing range Inertial range Energy-dissipative range Energy spectrum of turbulence Kolmogorov law Energy spectrum of the velocity field Energy-containing range The energy is injected into the system at.. Inertial range Dissipation does not work. The nonlinear interaction transfers the energy from low k region to high k region. Kolmogorov law : E(k)=C 2/3 k - -5/3 Energy-dissipative range The energy is dissipated with the rate at the Kolmogorov wave number k c = (/ 3 ) 1/4. Richardson cascade process Slide 19 Kolmogorov spectrum in classical turbulence ExperimentNumerical analysis Slide 20 Vortices in superfluid turbulence ST consists of a tangle of quantized vortex filaments Characteristics of quantized vortices Quantization of the circulation Very thin core No viscous diffusion of the vorticity The tangle may give an interesting model with the Kolmogorov law. A quantized vortex is a stable and definite defect, compared with vortices in a classical fluid. The only alive freedom is the topological configuration of its thin cores. Because of superfluidity, some dissipation would work only at large wave numbers (at very low temperatures). How is the intermittency? Slide 21 Summary of the motivation Classical turbulence Vortices Superfluid turbulence Quantized vortices It is possible to consider quantized vortices as elements in the fluid and derive the essence of turbulence. Slide 22 Does ST mimic CT or not? Maurer and Tabeling, Europhysics. Letters. 43, 29(1998) T =1.4, 2.08, 2.3K Measurements of local pressure in flows driven by two counterrotating disks finds the Kolmogorov spectrum. Stalp, Skrbek and Donnely, Phys.Rev.Lett. 82, 4831(1999) 1.4 < T < 2.15K Decay of grid turbulence The data of the second sound attenuation was consistent with a classical model with the Kolmogorov spectrum. 3. Dynamics of quantized vortices in superfluid helium 3-1 Recent interests in superfluid turbulence Slide 23 Vinen, Phys.Rev.B61, 1410(2000) Considering the relation between ST and CT The Oregons result is understood by the coupled dynamics of the superfluid and the normal fluid due to the mutual friction. Length scales are important, compared with the characteristic vortex spacing in a tangle. Kivotides, Vassilicos, Samuels and Barenghi, Europhy. Lett. 57, 845(2002) When superfluid is coupled with the normal-fluid turbulence that obeys the Kolmogorov law, its spectrum follows the Kolmogorov law too. What happens at very low temperatures? Is there still the similarity or not? Our work attacks this problem directly! Slide 24 3-2. Energy spectrum of superfluid turbulence Decaying Kolmogorov turbulence in a model of superflow C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) By solving the Gross-Pitaevskii equation, they studied the energy spectrum of a Tarlor-Green flow. The spectrum shows the -5/3 power on the way of the decay, but the acoustic emission is concerned and the situation is complicated. Energy Spectrum of Superfluid Turbulence with No Normal- Fluid Component T. Araki, M.Tsubota and S.K.Nemirovskii, Phys.Rev.Lett.89, 145301(2002) The energy spectrum of a Taylor-Green vortex was obtained under the vortex filament formulation. The absolute value with the energy dissipation rate was consistent with the Kolmogorov law, though the range of the wave number was not so wide. Slide 25 C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) Although the total energy is conserved, its incompressible component is changed to the compressible one (sound waves). The total length of vortices increases monotonically, with the large scale motion decaying. t Slide 26 C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) t=5.5 : 2 < k < 12 : 2 < k < 14 : 2 < k < 16 The right figure shows the energy spectrum at a moment. The left figure shows the development of the exponent n(t) in the spectrum E(k)=A(t) k - n(t). The exponent n(t) goes through 5/3 on the way of the dynamics. n(t) tk E(k) 5/3 E(k)=A(t) k -n(t) Slide 27 l : intervortex spacing Energy spectrum of superfluid turbulence with no normal-fluid component T.Araki, M.Tsubota, S.K.Nemirovskii, PRL89, 145301(2002) Energy spectrum of a vortex tangle in a late stage We calculated the vortex filament dynamics starting from the Taylor-Green flow, and obtained the energy spectrum directly from the vortex configuration. C=1 The dissipation arises from eliminating smallest vortices whose size becomes comparable to the numerical space resolution at k 300cm -1. Slide 28 The spectrum depends on the length scale. For k < 2/ l, the spectrum is consistent with the Kolmogorov law, reflecting the velocity field made by the tangle. The Richardson cascade process transfers the energy from small k region to large k region. ST mimics CT even without normal fluid. For k > 2/ l, the spectrum is k -1, coming from the velocity field due to each vortex. The energy is probably transferred by the Kelvin wave cascade process (Vinen, Tsubota, Mitani, PRL91, 135301(2003)). ST does not mimic CT. l : intervortex spacing Slide 29 Kelvin waves Slide 30 The spectrum depends on the length scale. For k < 2/ l, the spectrum is consistent with the Kolmogorov law, reflecting the velocity field made by the tangle. The Richardson cascade process transfers the energy from small k region to large k region. ST mimics CT even without normal fluid. For k > 2/ l, the spectrum is k -1, coming from the velocity field due to each vortex. The energy is probably transferred by the Kelvin wave cascade process (Vinen, Tsubota,Mitani, PRL91, 135301(2003)). ST does not mimic CT. l : intervortex spacing Slide 31 Energy spectrum of the Gross-Pitaevskii turbulence M. Kobayashi and M. Tsubota The dissipation is introduced so that it may work only in the scale smaller than the healing length. Slide 32 Starting from the uniform density and the random phase, we obtain a turbulent state. Vorticity 0 < t < 5.76 =1 256 3 grids 512 3 grids Density Phase Slide 33 Energy spectrum of the incompressible kinetic energy Time development of the exponentEnergy spectrum at t = 5.76 July 2004 ( Trento and Prague)August 2004 ( Lammi) Slide 34 3-3. Rotating superfluid turbulence Tsubota, Araki, Barenghi,PRL90, 205301(03); Tsubota, Barenghi, et al., PRB69, 134515(04) Vortex array under rotation order Vortex tangle in turbulence disorder vv ns What happens if we combine both effects? Slide 35 The only experimental work C.E. Swanson, C.F. Barenghi, RJ. Donnelly, PRL 50, 190(1983). 2. Vortex tangle seems to be polarized, because the increase due to the flow is less than expected. 1.There are two critical velocities v c1 and v c2 ; v c1 is consistent with the onset of Donnely-Graberson(DG) instability of Kelvin waves, but v c2 is a mystery. This work has lacked theoretical interpretation! L=2/ Vortex array DG instability ? ? By using a rotating cryostat, they made counterflow turbulence under rotation and observed the vortex line density by the measurement of the second sound attenuation. Slide 36 Initial vortices vortex array with small random noise Boundary condition Periodic z-axis Solid boundary x,y-axis Counterflow is applied along z-axis. Time evolution of a vortex array towards a polarized vortex tangle 1. The array becomes unstable,exciting Kelvin waves. Glaberson instability 2. When the wave amplitude becomes comparable to the vortex separation, reconnections start to make lots of vortex loops. 3. These loops disturb the array, leading to a tangle. Slide 37 Glaberson instability W.I. Glaberson et al., Phys. Rev. Lett 33, 1197 (1974). Glaberson et al. discussed the stability of the vortex array in the presence of axial normal-fluid flow. If the normal-fluid is moving faster than the vortex wave with k, the amplitude of the wave will grow (analogy to a vortex ring). Dispersion relation of vortex wave in a rotating frame , b vortex line spacing, angular velocity of rotation Landau critical velocity beyond which the array becomes unstable , Slide 38 Numerical confirmation of Glaberson instability Glabersons theory:Vc=0.010[cm/s] v =0.008[cm/s] ns v =0.015[cm/s]v =0.03[cm/s] ns v =0.05[cm/s]v =0.06[cm/s]v =0.08[cm/s] ns =9.9710 -3 rad/sec Slide 39 What happens beyond the Glaberson instability? Vortex line densityPolarization A polarized tangle Slide 40 Polarization as a function of v ns and The values of are 4.9810 -2 rad/sec( ), 2.9910 -2 rad/sec ( ), 9.9710 -3 rad/sec( ). We obtained a new vortex state, a polarized vortex tangle. Competition between order ( rotation) and disorder ( flow) Slide 41 4. Dynamics of quantized vortices in rotating BECs 4-1 Vortex lattice formation Tsubota, Kasamatsu, Ueda, Phys.Rev.A64, 053605(2002) Kasamatsu, Tsubota, Ueda, Phys.Rev.A67, 033610(2003) cf. A. A. Penckwitt, R. J. Ballagh, C. W. Gardiner, PRL89, 260402 (2002) E. Lundh, J. P. Martikainen, K-A. Suominen, PRA67, 063604 (2003) C. Lobo, A. Sinatora, Y. Castin, PRL92, 020403 (2004) Slide 42 Rotating superfluid and the vortex lattice c c The triangular vortex lattice sustains the solid body rotation. Yarmchuck and Packard Minimizing the free energy in a rotating frame. Vortex lattice observed in rotating superfluid helium Slide 43 Observation of quantized vortices in atomic BECs K.W.Madison, et.al PRL 84, 806 (2000) J.R. Abo-Shaeer, et.al Science 292, 476 (2001) P. Engels, et.al PRL 87, 210403 (2001) ENS MIT JILA E. Hodby, et.al PRL 88, 010405 (2002) Oxford Slide 44 How can we rotate the trapped BEC K.W.Madison et.al Phys.Rev Lett 84, 806 (2000) Non-axisymmetric potential Optical spoon Total potential Rotation frequency z x y 100 m 5m5m 20 m 16 m Axisymmetric potential cigar-shape Slide 45 Direct observation of the vortex lattice formation Snapshots of the BEC after turning on the rotation 1. The BEC becomes elliptic, then oscillating. 2. The surface becomes unstable. 3. Vortices enter the BEC from the surface. 4. The BEC recovers the axisymmetry, the vortices forming a lattice. K.W.Madison et.al. PRL 86, 4443 (2001) RxRx RyRy Slide 46 The Gross-Pitaevskii(GP) equation in a rotating frame Wave function Interaction s-wave scattering length in a rotating frame Two-dimensional simplified Slide 47 The GP equation with a dissipative term S.Choi, et.al. PRA 57, 4057 (1998) I.Aranson, et.al. PRB 54, 13072 (1996) This dissipation comes from the interaction between the condensate and the noncondensate. E.Zaremba, T. Nikuni, and A. Griffin, J. Low Temp. Phys. 116, 277 (1999) C.W. Gardiner, J.R. Anglin, and T.I.A. Fudge, J. Phys. B 35, 1555 (2002) Slide 48 Profile of a single quantized vortex A quantized vortex Velocity field Vortex core= healing length A vortex Slide 49 Dynamics of the vortex lattice formation (1) Time development of the condensate density Experiment Tsubota et al., Phys. Rev. A 65, 023603 (2002) Grid 256256 Time step 10 -3 Slide 50 Dynamics of the vortex lattice formation (2) t=067ms340ms 390ms 410ms700ms Time-development of the condensate density Are these holes actually quantized vortices? Slide 51 Dynamics of the vortex lattice formation (3) Time-development of the phase Slide 52 Dynamics of the vortex lattice formation (4) t=067ms340ms 390ms 410ms700ms Ghost vortices Becoming real vortices Time-development of the phase Slide 53 This dynamics is quantitatively consistent with the observations. K.W.Madison et.al. Phys. Rev. Lett. 86, 4443 (2001) RxRx RyRy Slide 54 Simultaneous display of the density and the phase Slide 55 Three-dimensional calculation of the vortex lattice formation M. Machida (JAERI), N. Sasa, M.Tsubota, K.Kasamatsu Condensate density Grid : 128128128 =0.03=0.1 Riminding us of superfluidity of a neutron star! Slide 56 4-2. Giant vortex in a fast rotating BEC Kasamatsu,Tsubota,Ueda, Phys.Rev.A67, 053606(2002) cf. A. L. Fetter, PRA64, 063608 (2001) U. R. Fischer and G. Baym, PRL90, 140402 (2003) E. Lundh, PRA65, 043604 (2002) T. L. Ho, PRL87, 060403 (2001) many references Slide 57 A fast rotating BEC in a quadratic-plus-quartic potential A quadratic potential A centrifugal potential A quadratic-plus-quartic potential can trap the BEC even if. What happens to vortices when When, the trapping potential is no longer effective. Slide 58 A giant vortex in a fast rotating BEC (1) =2.5 (2) =3.2 A giant vortex The effective potential takes the Mexican-hat form, so the central region becomes dilute and allows the vortices gather there, and the superfluid rotates around the core. This giant vortex is not a quantized vortex with multi-quanta but a lattice of ghost vortices. Vortices gather in the central hole. Mexican hat Slide 59 3. Dynamics of quantized vortices in superfluid helium 3-1 Recent interests in superfluid turbulence 3-2 Energy spectrum of superfluid turbulence 3-3 Rotating superfluid turbulence 4. Dynamics of quantized vortices in rotating BECs 4-1 Vortex lattice formation 4-2 Giant vortex in a fast rotating BEC 4-3 Vortex states in two-component BEC Kasamatsu, Tsubota, Ueda, Phys.Rev.Lett.91, 150406(2003), cond-mat 0406150 Summary