Osaka City University Hideo Yano Osaka City University Hideo - PowerPoint PPT Presentation

PSM2010 (Superclean Materials), Yokohama, 9 Mar. 2010 O4 Vortex Dynamics in Quantum Turbulence of Superfluid 4 4 He He Vortex Dynamics in Quantum Turbulence of Superfluid at the Turbulent Transition at the Turbulent Transition Osaka City University Hideo Yano Osaka City University Hideo Yano Quantum Turbulence generated by a thin vibrating wire Quantum Turbulence generated by a thin vibrating wire 1. Vortex dynamics at the laminar Vortex dynamics at the laminar- -to to- -turbulent transition turbulent transition 1. • Seed vortices of turbulence � bridge vortices • Kelvin wave instability 2. Critical behaviors at the turbulent 2. Critical behaviors at the turbulent- -to to- -laminar transition laminar transition Collaborators Collaborators Experiment: Y. Nago, K. Andachi, Y. Miura, T. Ogawa, S. Mio, M. Chiba Experiment: Y. Nago, K. Andachi, Y. Miura, T. Ogawa, S. Mio, M. Chiba K. Obara, O. Ishikawa, T. Hata K. Obara, O. Ishikawa, T. Hata Theory: Theory: S. Fujiyama, M. Tsubota S. Fujiyama, M. Tsubota

Superfluid and Quantized Vortex Superfluid and Quantized Vortex 1.2 ρ s ρ n � Simple superfluids ( 4 He; 3 He-B; cold atoms) exhibit 1 ρ s or ρ n / ρ • Two fluid behaviour : a viscous normal component 0.8 0.6 + an “inviscid” superfluid component. 0.4 4 He 0.2 • Normal component disappears at the 0 K limit. 0 0 0.5 1 1.5 2 2.5 T (K) ∇ × = � Quantization of rotational motion: , v 0 s • except on quantized vortex lines, each with one quantum of circulation = ∫ κ ⋅ = v d r h m : circulatio n quantum 4 s round a core of radius ( ξ ~0.05 nm for 4 He). • Helium under rotation � Array of vortex lines Helium under rotation � Nucleation of vortices, during cooling through the superfluid transition • Remnant vortex lines are still present, attached between boundaries.

Generation of turbulence by a vibrating wire Generation of turbulence by a vibrating wire Response of a vibrating wire in superfluid 4 He Response of a vibrating wire in superfluid 4 He 140 peak velocity (mm/s) T=1.2 K 120 vibrating wire F (thickness ~ μ m) 100 80 ~mm B 60 F : Lorentz force I B : magnetic field 40 up I : electric current down 20 0 0 1 2 3 4 5 6 7 driving force (nN)

4 He. Oscillating obstacles in superfluid 4 He. Oscillating obstacles in superfluid Microsphere Microsphere Grid Grid Fork Fork 200 μ m 115 mm/s 60 mm/s 42 mm/s The velocity of generating turbulence ( (~ 50 mm/s ~ 50 mm/s) is much lower M. Bla ž ková, D. Schmoranzer, J. Jager, B. Shuderer, W. H.A. Nichol, L. Skrbek, P.C. than an intrinsic velocity of vortex nucleation ( ~30 ~30 m/s m/s ). Schoepe, Hendry, P.V.E. McClintock, and L. Skrbek, Phys. Rev. Lett. 74 , 566 Phys. Rev. Lett. 92 , 244501 Phys. Rev. E 75 , 025302(R) Remanent vortices should cause the generation of turbulence !! (1995). (2004). (2007).

Study on the vortex dynamics Study on the vortex dynamics at the laminar- -to to- -turbulent transition turbulent transition at the laminar Vortex free wire in superfluid 4 He Vortex free wire in superfluid 4 He to reduce remnant vortex lines 1. thin vibrating wire with smooth surface 2. liquefying superfluid below 100 mK A vortex-free wire does not generate turbulence, even at a velocity above 1 m/s. vibrating a vortex-free wire + seed vortices Turbulence will be generated ?

Transition to turbulence triggered by vortex rings Transition to turbulence triggered by vortex rings Detector @30mK 1 1500 F up F O Vortex rings trigger r Vortex rings trigger velocity (mm/s) o Vortex rings trigger generator ON t a 1000 r e n the turbulent transition. the turbulent transition. e the turbulent transition. g 500 Vibrating wires (NbTi φ 3 μ m) in superfluid 4 He 0 at 30 mK 0 0.5 1 1.5 generator of driving force (nN) vortex rings detector 2 (vortex free) 1500 down velocity (mm/s) 1000 generator OFF 500 0 0 0.5 1 1.5 driving force (nN)

Simulation of turbulence triggered by vortex rings Simulation of turbulence triggered by vortex rings Numerical simulation by Numerical simulation by Fujiyama and Tsubota Fujiyama and Tsubota oscillating obstacle: sphere 6 μ m velocity: 137 mm/s frequency: 1.59 kHz A turbulence forms in the path of the sphere. See a joint paper: R. Goto, S. Fujiyama, M. Tsubota, HY, et al , Phys. Rev. Lett. 100, 045301 (2008)

Study on the transition due to vortices Study on the transition due to vortices attached to a vibrating wire attached to a vibrating wire To attach vortex lines to a wire To attach vortex lines to a wire 1. Warming above T λ 2. Cooling to 30 mK How vortex lines attached to a wire cause turbulence? • Responses of the vibrating wire

Transition to turbulence due to attached vortices Transition to turbulence due to attached vortices Response of a vibrating wire Response of a vibrating wire with attached vortices with attached vortices Vortex lines are nucleated through Oscillation of bridge vortex up 500 velocity (mm/s) the superfluid transition, attaching down lines generates turbulence. to a vibrating wire. Kelvin wave instability causes turbulence. 0 Bridge vortex lines frequency (Hz) 1562 Resonance ∝ f k / m 1560 vortex-free vibrating wire frequency f : resonance frequency increases k : spring constant 1558 by 0.3 Hz 0 0.1 0.2 0.3 m : effective mass drive force (nN) N. Hashimoto, R. Goto, HY, et al , Phys. Rev. B 76, 020504 (2007).

Study on the vortex dynamics Study on the vortex dynamics at the turbulent- -to to- -laminar transition laminar transition at the turbulent 2 1500 down velocity (mm/s) 1000 generator OFF 500 0 0 0.5 1 1.5 driving force (nN)

Turbulent- -to to- -Laminar transition Laminar transition Turbulent velocity (mm/s) 50 F G = 90 → 0 pN Generator 400 on off at t= 60 Detector 0 300 velocity (mm/s) 300 velocity (mm/s) 200 t F D =100 pN 200 (F turb =76 pN) 100 F turb 100 Detector 0 0 0 0.05 0.1 0.15 0.2 t =15 sec 50 60 70 80 90 100 time (s) driving force (pN) Lifetime of turbulence generation ( ) • exponential distribution ∝ − τ exp( t / ) � mean lifetime τ of turbulence

Mean lifetime of turbulence Mean lifetime of turbulence Mean lifetime vs. driving force Mean lifetime vs. driving force 10 4 Critical behaviors of lifetime Critical behaviors of lifetime � Above 0.9 pW, 10 3 the mean lifetime τ mean lifetime (s) ⎛ ⎞ τ = ⎧ 2 P 1 . 5 s ⎜ ⎟ τ = τ ⎨ 0 10 2 exp ⎜ ⎟ = 0 2 ⎩ P 0 . 88 pW ⎝ ⎠ P 0 0 � Below 0.9 pW, 10 1 the lifetime τ decreases greatly from the equation. 1 0.1 The fitting parameter P 0 reflects 0 0.5 1 1.5 2 2.5 the critical injection energy below injection power (pW) which the critical behaviors arise. The lifetime is attributable to 2 ⎛ ⎞ 2 L ⎜ ⎟ τ = τ the statistical fluctuations of 0 exp ( L : vortex line density ) ⎜ ⎟ 2 L ⎝ ⎠ 0 the vorticity [Schoepe, PRL2004].

Energy flux in quantum turbulence Energy flux in quantum turbulence Injected Power • steady quantum turbulence • restricted volume = v P g F turb ⎧ F : drag force turb ⎪ v ⎨ : wire velocity ⎪ bottleneck ⎩ g : geometrica l factor 2 1500 Energy Dissipation down Energy Cascade • vortex rings velocity (mm/s) • Richardson cascade 1000 • high energy phonons • Kelvin wave cascade generator OFF 500 0 0 0.5 1 1.5 driving force (nN)

Bottleneck of energy flux Bottleneck of energy flux Vortex line density L due to the bottleneck L = κ 3 a P / M ( ) ⎛ ⎞ = = v -1/2 P g F : dissipated power l L : vortex line spacing ⎜ ⎟ turb ⎜ ⎟ M : mass of turbulent fluid k : wave number ( ) ⎜ ⎟ κ Λ = : circulatio n quantum ln l / a ⎝ ⎠ 0 a ≈ 1 • : unpolarized vortex tangle (bottleneck at kl ~ 1 ) a ≈ Λ 5 : polarized vortex tangle • − − ξ (bottleneck at kl ~ Λ -5/4 , Λ ≈ 12 ) 1 1 l [V.S. L’vov, et al ., Phys. Rev. B 76 , 024520 (2007)] Prediction bottleneck of energy cascade

Vortex line spacing at the critical energy Vortex line spacing at the critical energy Vortex line density L due to the bottleneck ( ) ⎧ = = = κ v -1/2 3 P g F : injection power l L : vortex line spacing L a P / M ⎪ turb ⎨ M : mass of turbulent fluid k : wave number ( ) κ Λ = ⎪ : circulatio n quantum ln / l a ⎩ 0 Vortex line spacing at the critical energy ( P 0 = 0.88 pW) (assuming unpolarized vortex tangle ( a ≈ 1) at low driving forces) l 0 = ( L 0 ) -1/2 ≈ 7 μ m ≈ oscillating amplitude 9 μ m (=amp p-p ) Turbulence ceases when vortex lines are absent in the wire path.

Summary & Future works Summary & Future works Quantum turbulence generated by thin vibrating wires 1. Vortex dynamics at the turbulent transition • seed vortices triggering the turbulent transition • turbulent transition due to Kelvin wave instability 2. Critical behaviors of turbulence • critical behaviors of mean lifetime • fluctuation of vortex lines • energy flux and its bottleneck 3. Future works • Detection of Kelvin waves (P77) • Vortex generation at high temperatures (P72) • Critical behaviors using high frequency oscillators (P73)