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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

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SLIDE 1

Vortex Dynamics in Quantum Turbulence of Superfluid Vortex Dynamics in Quantum Turbulence of Superfluid 4

4He

He at the Turbulent Transition at the Turbulent Transition

Collaborators Collaborators Experiment: Experiment: Y. Nago, K. Andachi, Y. Miura, T. Ogawa, S. Mio, M. Chiba

  • 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

Osaka City University Osaka City University Hideo Yano Hideo Yano

Quantum Turbulence generated by a thin vibrating wire Quantum Turbulence generated by a thin vibrating wire 1.

  • 1. Vortex dynamics at the laminar

Vortex dynamics at the laminar-

  • to

to-

  • turbulent transition

turbulent transition

  • Seed vortices of turbulence
  • Kelvin wave instability

bridge vortices

2.

  • 2. Critical behaviors at the turbulent

Critical behaviors at the turbulent-

  • to

to-

  • laminar transition

laminar transition

PSM2010 (Superclean Materials), Yokohama, 9 Mar. 2010 O4

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SLIDE 2

Superfluid and Quantized Vortex Superfluid and Quantized Vortex

Simple superfluids (4He; 3He-B; cold atoms) exhibit

  • Two fluid behaviour: a viscous normal component

+ an “inviscid” superfluid component.

  • Normal component disappears at the 0 K limit.

Quantization of rotational motion: ,

  • except on quantized vortex lines,

each with one quantum of circulation round a core of radius (ξ ~0.05 nm for 4He).

  • Helium under rotation Array of vortex lines

Nucleation of vortices, during cooling through the superfluid transition

  • Remnant vortex lines are still present, attached between boundaries.

quantum n circulatio :

4

m h d

s

= ⋅ = ∫ r v κ

0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 2 2.5

ρs or ρn / ρ T (K) ρs ρn

4He

= × ∇

s

v

Helium under rotation

slide-3
SLIDE 3

Response of a vibrating wire in superfluid Response of a vibrating wire in superfluid 4

4He

He

Generation of turbulence by a vibrating wire Generation of turbulence by a vibrating wire

20 40 60 80 100 120 140 1 2 3 4 5 6 7 up down

peak velocity (mm/s) driving force (nN)

T=1.2 K

F : Lorentz force B : magnetic field I : electric current

B F vibrating wire

(thickness ~ μm)

I

~mm

slide-4
SLIDE 4

Oscillating obstacles in superfluid Oscillating obstacles in superfluid 4

4He.

He.

  • M. Blažková, D. Schmoranzer,

and L. Skrbek,

  • Phys. Rev. E 75, 025302(R)

(2007).

Fork Fork

H.A. Nichol, L. Skrbek, P.C. Hendry, P.V.E. McClintock,

  • Phys. Rev. Lett. 92, 244501

(2004).

Grid Grid

60 mm/s

200 μm

Microsphere Microsphere

  • J. Jager, B. Shuderer, W.

Schoepe,

  • Phys. Rev. Lett. 74, 566

(1995).

42 mm/s 115 mm/s

The velocity of generating turbulence ( (~ 50 mm/s ~ 50 mm/s) is much lower than an intrinsic velocity of vortex nucleation ( ~30 ~30 m/s m/s ). Remanent vortices should cause the generation of turbulence !!

slide-5
SLIDE 5

Study on the vortex dynamics Study on the vortex dynamics at the laminar at the laminar-

  • to

to-

  • turbulent transition

turbulent transition

Vortex free wire in superfluid Vortex free wire in superfluid 4

4He

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 Turbulence will be generated ? + seed vortices

slide-6
SLIDE 6

Vibrating wires (NbTi φ 3 μm) in superfluid 4He at 30 mK

500 1000 1500 0.5 1 1.5 up

velocity (mm/s) driving force (nN)

Vortex rings trigger the turbulent transition. Vortex rings trigger Vortex rings trigger the turbulent transition. the turbulent transition.

Detector @30mK generator of vortex rings detector (vortex free) generator ON g e n e r a t

  • r

O F F

Transition to turbulence triggered by vortex rings Transition to turbulence triggered by vortex rings

1

500 1000 1500 0.5 1 1.5 down

velocity (mm/s) driving force (nN)

generator OFF

2

slide-7
SLIDE 7

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

See a joint paper: R. Goto, S. Fujiyama, M. Tsubota, HY, et al,

  • Phys. Rev. Lett. 100, 045301 (2008)
  • scillating obstacle: sphere 6 μm

velocity: 137 mm/s frequency: 1.59 kHz

A turbulence forms in the path of the sphere.

slide-8
SLIDE 8

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 Tλ
  • 2. Cooling to 30 mK

How vortex lines attached to a wire cause turbulence?

  • Responses of the vibrating wire
slide-9
SLIDE 9

500 up down

velocity (mm/s)

1558 1560 1562 0.1 0.2 0.3

frequency (Hz) drive force (nN)

vortex-free vibrating wire

Response of a vibrating wire Response of a vibrating wire with attached vortices with attached vortices

Transition to turbulence due to attached vortices Transition to turbulence due to attached vortices

f : resonance frequency k : spring constant m : effective mass

m k f / ∝

Vortex lines are nucleated through the superfluid transition, attaching to a vibrating wire.

Oscillation of bridge vortex lines generates turbulence. Kelvin wave instability causes turbulence.

Bridge vortex lines

Resonance frequency increases by 0.3 Hz

  • N. Hashimoto, R. Goto, HY, et al, Phys. Rev. B 76, 020504 (2007).
slide-10
SLIDE 10

Study on the vortex dynamics Study on the vortex dynamics at the turbulent at the turbulent-

  • to

to-

  • laminar transition

laminar transition

500 1000 1500 0.5 1 1.5 down

velocity (mm/s) driving force (nN)

generator OFF

2

slide-11
SLIDE 11

100 200 300 400 0.05 0.1 0.15 0.2 velocity (mm/s) driving force (pN)

Detector

Turbulent Turbulent-

  • to

to-

  • Laminar transition

Laminar transition

50 velocity (mm/s) 100 200 300 50 60 70 80 90 100 velocity (mm/s) time (s)

Generator Detector

t

FG= 90→0 pN at t= 60

FD=100 pN (Fturb=76 pN)

Fturb

Lifetime of turbulence generation

  • exponential distribution

mean lifetime τ of turbulence

t =15 sec

  • n
  • ff

( )

) / exp( τ t − ∝

slide-12
SLIDE 12

Critical behaviors of lifetime Critical behaviors of lifetime

Above 0.9 pW,

the mean lifetime τ

Below 0.9 pW,

the lifetime τ decreases greatly from the equation.

The fitting parameter P0 reflects the critical injection energy below which the critical behaviors arise.

⎩ ⎨ ⎧ = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = pW 88 . s 5 . 1 exp

2 2

P P P τ τ τ

The lifetime is attributable to the statistical fluctuations of the vorticity [Schoepe, PRL2004].

2 2 2 0 exp

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = L L τ τ

) density line vortex : (L 0.1 1 101 102 103 104 0.5 1 1.5 2 2.5

mean lifetime (s) injection power (pW)

Mean lifetime of turbulence Mean lifetime of turbulence

Mean lifetime vs. driving force Mean lifetime vs. driving force

slide-13
SLIDE 13

500 1000 1500 0.5 1 1.5 down

velocity (mm/s) driving force (nN)

generator OFF

2

Energy flux in quantum turbulence Energy flux in quantum turbulence

Injected Power

⎪ ⎩ ⎪ ⎨ ⎧ = factor l geometrica : velocity wire : force drag : g F F g P

turb turb

v v

Energy Dissipation

  • vortex rings
  • high energy phonons
  • steady quantum turbulence
  • restricted volume

bottleneck Energy Cascade

  • Richardson cascade
  • Kelvin wave cascade
slide-14
SLIDE 14

Bottleneck of energy flux Bottleneck of energy flux

1 −

l

1 −

ξ

Prediction

bottleneck of energy cascade

Vortex line density L

due to the bottleneck

  • a ≈ 1

: unpolarized vortex tangle (bottleneck at kl ~ 1)

  • a ≈Λ5 : polarized vortex tangle

(bottleneck at kl ~ Λ-5/4, Λ ≈ 12)

[V.S. L’vov, et al., Phys. Rev. B 76, 024520 (2007)]

3

/ κ M P a L =

( ) ( )

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = Λ = =

  • 1/2

/ ln quantum n circulatio : number wave : fluid turbulent

  • f

mass : spacing line vortex : power dissipated : a l k M L l F g P

turb

κ v

slide-15
SLIDE 15

Vortex line density L due to the bottleneck

( ) ( )

⎪ ⎩ ⎪ ⎨ ⎧ = Λ = = =

  • 1/2

3

/ ln quantum n circulatio : number wave : fluid turbulent

  • f

mass : spacing line vortex : power injection : / a l k M L l F g P M P a L

turb

κ κ v

Vortex line spacing at the critical energy Vortex line spacing at the critical energy

Vortex line spacing at the critical energy (P0 = 0.88 pW) (assuming unpolarized vortex tangle (a ≈ 1) at low driving forces) l0 = (L0)-1/2 ≈ 7 μm ≈ oscillating amplitude 9 μm (=ampp-p) Turbulence ceases when vortex lines are absent in the wire path.

slide-16
SLIDE 16

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)