振動物体が生成する超流動ヘリウムの量子乱流 -...
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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?)