振動物体が生成する超流動ヘリウムの量子乱流振動物体が生成する超流動ヘリウムの量子乱流Steady Quantum Turbulence Generated by Oscillating Structures Steady Quantum Turbulence Generated by Oscillating Structures

in Superfluid in Superfluid 44HeHe

CollaboratorsCollaboratorsExperiment:Experiment: Y. Nago, T. Ogawa, A. Y. Nago, T. Ogawa, A. NishijimaNishijima, ,

K. Andachi, Y. Miura, M. Chiba, S. Mio, M. Inui, R. GotoK. Andachi, Y. Miura, M. Chiba, S. Mio, M. Inui, R. GotoK. Obara, O. Ishikawa, T. HataK. Obara, O. Ishikawa, T. Hata

Theory:Theory: S. Fujiyama, M. TsubotaS. Fujiyama, M. Tsubota

Osaka City UniversityOsaka City University Hideo YanoHideo Yano

Quantum turbulence generated by a thin vibrating wireQuantum turbulence generated by a thin vibrating wire1.1. Turbulent transitionTurbulent transition

• Vortex-free vibrating wire and seed vortices

2.2. Steady quantum turbulenceSteady quantum turbulence3.3. Critical behaviors of turbulenceCritical behaviors of turbulence

Riken2011, 5 Jan. 2011

Superfluid and Quantized VortexSuperfluid and Quantized VortexSimple 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

Vortex nucleation during cooling through the superfluid transition• Remnant vortex lines are still present, attached between boundaries.

quantumn circulatio : 4mhds

=⋅= ∫ rvκ

00.20.40.60.8

11.2

0 0.5 1 1.5 2 2.5

ρ s or

ρn /

ρ

T (K)

ρs ρn

4He

0=×∇ sv

Helium under rotation

Response of vibrating wire in superfluid Response of vibrating wire in superfluid 44HeHe

Generation of turbulence by vibrating wireGeneration of turbulence by vibrating wire

F : Lorentz forceB : magnetic field I : electric current

B

F vibrating wire(thickness ~ μm)

I ~mm

0

50

100

150

200

0 0.05 0.1 0.15 0.2 0.25

peak

vel

ocity

(m

m/s

)

driving force (nN)

HY, N. Hashimoto, A. Handa, et al, Phys. Rev. B 75, 012502 (2007).

VortexVortex--free vibrating wirefree vibrating wire

0

500

1000

0 0.5

peak

vel

ocity

(m

m/s

)

driving force (nN)

N. Hashimoto, R. Goto, HY, et al, Phys. Rev. B 76, 020504(R) (2007)HY, T. Ogawa, Y. Nago, Y. Miura, et al, J. Low Temp. Phys. 156, 132–144 (2009)

0100200300

0 0.1 0.2 0.3v (

mm

/s)

driving force (nN)

Vortex-free status requires1. to prevent vortex nucleation2. to cease flow3. smooth boundary

0

500 updown

velo

city

(m

m/s

)

1558

1560

1562

0 0.1 0.2 0.3

freq

uenc

y (H

z)

driving force (nN)

vortex-free vibrating wire

Response of vibrating wire Response of vibrating wire with attached vorticeswith attached vortices

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

f : resonance frequency

k : spring constant

m : effective mass

mkf /∝

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

Oscillation of bridge vortex lines generates turbulence.

Bridge vortex lines

Resonance frequency increases by 0.3 Hz

N. Hashimoto, R. Goto, HY, et al, Phys. Rev. B 76, 020504(R) (2007).

Kelvin wave oscillation of vortex lines

dispersionrelation

unstable if amplitude R ≳ wave-length λ

generation of turbulencegeneration of turbulence

λπ

πκω 2 , 1ln4 0

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛≈ k

kak

superfluidsuperfluidflowflow

R

vortexcore

λ

Kelvin wave

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

radius core-vortex:quantumn circulatio:length wave:

frequency:number wave:

0a

kκλω

(R. H(R. Häänninen, M. Tsubota, W.F. Vinen, PRB 75, 064502 (2007))nninen, M. Tsubota, W.F. Vinen, PRB 75, 064502 (2007))

Kelvin wave instabilityKelvin wave instability

sphere

vortex lines

oscillatingsuperfluid

flow

InitialInitialconditionconditiona. Kelvin waves arise on the bridged vortex line.

b.Vortex rings nucleate by reconnection.

c. Turbulence develops.

R. HR. Häänninen, M. Tsubota, W.F. Vinen, nninen, M. Tsubota, W.F. Vinen, Phys. Rev. B Phys. Rev. B 7575, 064502 (2007), 064502 (2007)

Time evolution of turbulenceTime evolution of turbulence simulated by Hänninen et al.

Turbulence due to oscillation of bridge vorticesTurbulence due to oscillation of bridge vortices

150 ms

232 ms

326 ms

obstacle: sphere 200 μmOscillating superfluid

velocity: 150 mm/sfrequency: 200 Hz

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

0

500

1000

1500

0 0.5 1 1.5

up

velo

city

(m

m/s

)

driving force (nN)

Vortex rings triggerthe turbulent transition.Vortex rings triggerVortex rings triggerthe turbulent transition.the turbulent transition.

Detector @30mK

generator ofvortex rings

detector(vortex free)

generator ON

generator

OFF

Transition to turbulence triggered by vortex ringsTransition to turbulence triggered by vortex rings1

0

500

1000

1500

0 0.5 1 1.5

down

velo

city

(m

m/s

)

driving force (nN)

generator OFF

2

Simulation of turbulence triggered by vortex ringsSimulation of turbulence triggered by vortex rings

Numerical simulation by Numerical simulation by Fujiyama and Tsubota Fujiyama and Tsubota

oscillating obstacle: sphere 6 μmvelocity: 137 mm/sfrequency: 1.59 kHz

A turbulence forms in the path of the sphere.

R. Goto, S. Fujiyama, M. Tsubota, HY, et al, PRL 100, 045301 (2008)

0

500

1000

1500

0 0.5 1 1.5

down

velo

city

(m

m/s

)

driving force (nN)

generator OFF

2

Energy flux in steady quantum turbulenceEnergy flux in steady quantum turbulence

Injection Power

⎪⎩

⎪⎨

⎧

=

factor lgeometrica : velocity wire:

force drag :

g

FFgP

turb

turb

v

v

Energy Dissipation• vortex rings• high energy phonons

Energy Cascade• Richardson cascade• Kelvin wave cascade

• steady quantum turbulenceVortex line density is constantduring turbulence generation.

• turbulence restrictedin the path of oscillation

Observation of vortex fluctuation in turbulenceObservation of vortex fluctuation in turbulenceusing the turbulent using the turbulent laminar transitionlaminar transition

0

500

1000

1500

0 0.5 1 1.5

down

velo

city

(m

m/s

)

driving force (nN)

generator OFF

2

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

generator ofvortex rings

detector(vortex free)

0

100

200

300

400

0 0.05 0.1 0.15 0.2

velo

city

(mm

/s)

driving force (pN)

Detector

Turbulent Turbulent Laminar transitionLaminar transition

0

50

velo

city

(mm

/s)

0

100

200

300

50 60 70 80 90 100

velo

city

(mm

/s)

time (s)

Generator

Detector

t

FG=90→0 pNat t=60

FD=100 pN(Fturb=76 pN) Fturb

Lifetime of turbulence generation

t =15 sec

on off

s 5.13=τ

Lifetime t of the turbulence

Exponential distribution( )

e) turbulencof lifetimemean :(

/exp

τ

τt−=events ofnumber normalized

Lifetime of turbulenceLifetime of turbulence

0

5

10

15

20

25

0 20 40 60

num

ber

of e

vent

s

lifetime (s)

0.01

0.1

1

0 10 20 30 40 50 60 70

prob

abili

ty

lifetime (s)

Fturb=76 pN

Number of eventsNumber of events

HY, Y. Nago, et al, Phys. Rev. B 81, 220507(R) (2010)

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

2

20

2

0 exp ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

LL

ττ

)density linevortex :(L

Critical behaviors of lifetimeCritical behaviors of lifetimeAbove 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 power below which the critical behaviors arise.

⎩⎨⎧

==

⎟⎟⎠

⎞⎜⎜⎝

⎛= pW88.0

s5.1exp0

02

0

2

0 PPP τττ

0.1

1

101

102

103

104

0 0.5 1 1.5 2 2.5

mea

n lif

etim

e (

s)

injection power (pW)

Mean lifetime of turbulenceMean lifetime of turbulenceMean lifetime vs. driving forceMean lifetime vs. driving force

HY, Y. Nago, et al, Phys. Rev. B 81, 220507(R) (2010)

Energy flux of Energy flux of classicalclassical quantumquantum cascadecascade

1−l 1−ξ

Vortex line density L between cascades

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

• critical energy: P0 = 0.88 pWl0 = (L0)-1/2 ≈ 7 μm

≈ oscillating amplitudep-p 9 μm(assuming a ≈ 1)

3/ κMPaL =

( )

( ) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=Λ

==

0

-1/2

/lnquantumn circulatio : number wave: fluid turbulentof mass :

spacing line vortex : power dissipated :

alkM

LlFgP turb

κ

v

Turbulence ceases when vortex lines are absent in the wire path.

bottleneck

Dissipation in turbulenceDissipation in turbulence (assuming a half-circular vortex)

Vortex expands if Magnus force Fv > vortex tension 2Tv.

Loop size R is limited to vibrating wire amplitude vw / ω.

Critical velocity at turbulent Critical velocity at turbulent laminar transitionlaminar transition

)12( 4

velocitycritical

≈Λ≈Λ

=

•

κωκωπcv

Λ=>= πκρ

κρ 422 22

vwvs

s TRF v

( )yinstabilitDonnelly -Glaberson 4w ≈Λ> Rπκv

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≡Λ

radius corevortex : radius loopvortex :

/2lnncirculatio of quantum:

density superfluid: frequency wire: velocitywire:

0

0

w

aR

aR

s

κρωv

ww 4 v

vπκω Λ

>> Rπ

κω4

2w

Λ>v

R

vortex loop

wirewire0

100

200

300

400

0 0.05 0.1 0.15 0.2

velo

city

(mm

/s)

driving force (pN)

10

100

400

4 10 100

v c

(mm

/s)

κω (mm/s)

3 μm VW in 4He at 50 mK

Critical velocity for thin wires (~3 Critical velocity for thin wires (~3 μμm)m)

Critical velocity Critical velocity vvcc vs. frequency vs. frequency ωω

The critical velocity of turbulence generation is well fitted to

present case a = 1.65

κωac =v

⎩⎨⎧

frequencyangular :quantumn circulatio :

ωκ

cv

Y. Nago, M. Inui, HY, et al, Phys. Rev. B 82, 224511 (2010)

critical velocity of turbulencevibrating wire in superfluid 3He-B at 0.2 Tc.

Coupled oscillation of a wire and a vortex loop Coupled oscillation of a wire and a vortex loop (in a linear response)

Resonance frequency ω

vortex frequency for R < vc / ωw

Resonance frequency in a turbulent stateResonance frequency in a turbulent state

( )⎪⎩

⎪⎨

⎧

≡Λradius corevortex :

/2lnncirculatio of quantum:

radius loopvortex :

0

0a

aR

Rκ 1558

1560

1562

0 0.1 0.2 0.3

Resonance frequency of a vibrating wire

freq

uenc

y (H

z)

driving force (nN)

w2v 343 ωπ

κω >Λ

=R

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>−−

≈±

<−

=−

−−

−

)()1(

)(

)()1(

1

wvw

v12

wvw

v

wvw

v12

2w

2

ωωα

ωω

ωωα

ωω

mm

mm

kk

⎟⎟⎠

⎞⎜⎜⎝

⎛≡≡≡

w

v

v

v2v

w

w2w , , ω

ωαωω mk

mk

mw kv mvkw

wire vortex

potential flow

turbulent flow

Summary & Future worksSummary & Future worksQuantum turbulence generated by a vibrating wire

1. Turbulent transition• seed vortices: bridge vortex line, vortex ring

2. Energy flux of steady quantum turbulence

3. Critical behaviors of turbulence• lifetime of turbulence • fluctuation of vortex lines in turbulence• critical velocity of turbulence generation

4. Future works• motion of vortex line (vortex mass, Kelvin wave) • energy dissipation (vortex ring, phonon?)