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Direct Energy Transformation in a System of Nonlinear Second Sound Waves
V.B.Efimov1,2 and P.V.E.McClintock2 1) Institute of Solid State Physics RAS, Chernogolovka, Moscow distr., Russia2) Physics department, Lancaster University, Lancaster, UK
The VI-th international conference: Solitons, Collapses and TurbulenceNovosibirsk Akademgorodok, 7 June 2012
Solitons, Collapses and Turbulence
OutlineIntroduction
◦Why superfluid helium?◦Burgers waves
Experiments◦Resonator of second sound wave;◦Low resonance number – a direct energy
cascade of nonlinear waves;◦Far resonances – an inverse energy
cascade;◦Very low resonances – fractional frequency
transformation.Discussion Akademgorodok,
Novosibirsk, 07 June 2012
Solitons, Collapses and Turbulence
Akademgorodok, Novosibirsk, 07 June
2012
Sounds in Superfluid Helium
0 1 2 3 40
20
40
60
80
100
120
140
160
180
200
220
240
First and Second sound velosity in 4He at SVP
v1 / 3
v2
v1
v, m
/s
T, K
TStSP
t n
s
2
2
2
2
2
,
SPu )/(1
n
s
CTSu
2
2
The differentiation of two equations with respect to time and substituting into another relationships we obtain for small disturbance (linear waves)
Solutions are wave of density (co-moving motion normal and superfluid
component)
waves of temperature or entropy (counter-movement motion normal and superfluid components, =n+s=const
Laboratory of Quantum Crystals
4
Second sound shock waves in superfluid helium
Temperature dependence of second sound nonlinear coefficient. Right hump is connected with roton waves, left one – with phonon’s temperature excitation
Nonlinear coefficient is positive for first sound
Nonlinear coefficient may be positive as well as negative for second sound. It means, the breakdown appears on front (a2>0) or on backside (for a2<0) of wave depending from T.
0 20 40 60 80 100
-4
-2
0
2
4
6
8
10
12
3
4
5
6
7
8
21
P=3atm., U =7.32V, =10s.1 T=2.103K2 T=2.071K3 T=2.044K4 T=1.999K5 T=1.968K6 T=1.81K7 T=1.783K8 T=1.774K
Arb
. uni
ts.
Time, s.
Why second sound in superfluid helium?
Low velocity of wave of second soundWeak wave dissipation Very strong dependence of velocity from
second sound wave amplitude (δT~mK) Linear dispersion law ω~k
Burgers equation
Dependence of second sound velocity from wave amplitude:
Resonator of second sound waves
We applied harmonic signal to heater U~sin(w*t), P~sin2(w*t)=[1+cos(2w*t)]/2
Resonant frequency fG=c0/4L
Laboratory of Quantum Crystals 7
Resonator In experiments was used high quality
cylindrical resonator (cover and bottom were parallel, wall was with high accurateness cylindrical – in some experiments we used cut off syringe), D~1.5 cm, L=2-8cm.
Heating meander occupied all lid area and we generated one dimensional wave
The eigenfrequencies are multiple harmonics
Quality of resonator 70 mm
The penetration depth is of the order of 2.5 μm for resonance frequency ~ 1000 Hz. Lets us suppose that the part of helium in length of Λ balks a motion of wave. The absolute value of resonator quality is of the order of
Bulk dissipation stays appreciable at frequencies higher 10 kHz
3 March 2010
The mechanical quality of the resonator is determined by parallel of reflecting walls Δl/l ~ 5* 10-4, which corresponds to highest resonance frequency ~400 kHz
Model. The counterflow of the superfluid helium in resonator yields a some viscous friction at penetration depth, which depends from a frequency of a resonance vibrations
The energy loss at motion along walls in layer Λ at distance L, which is defined by interference conditions.. The wave disappears in resonator as only the sum of reflection N multiplies on Δl will be order of wavelength λ=c/f=c/2 πω~Δl*N, where c - is sound velocity. Wave at this case runs a distance
and quality of resonator will be defined as verses value of losses
100 1000 40000 80000
1000
10000
Qua
lity,
f/
f
Frequency, Hz
Q~f 3/2
Phys.Rev. E
Solitons, Collapses and Turbulence
Measurements of standing waves at different resonances.
Low frequency f~1000 Hz, N~10
Akademgorodok, Novosibirsk, 07 June
2012
100100 1 ∞
Bad quality Good quality Balk dissipation
10
Energy Transformation in Superfluid Helium-4
11
Experimental studies of one dimensional nonlinear second sound waves in the cylindrical cell
Spectrum of temperature oscillations of the nonlinear second sound waves in a
resonator
Applied signal A=A0*sin(ωt)
100 1000 10000
1E-5
1E-4
1E-3
0.01
Frequency (Hz)
Am
plitu
de
A~f -1.62
0,001 0,002
-0,005
0,000
0,005
0,010
A, a
rb. u
nits
t, ms
-12
-6
0
6
12
Fast Fourier Transformation
Temperature in second sound wave
12
Formation of Kolmogorov’s spectrum
FFT spectrum
Laboratory of Quantum Crystals 13
Decay of nonlinear energy spectrum
100 1000 10000
1E-5
1E-4
1E-3
0,01
Frequency (Hz)
Am
plitu
de
100 1000 10000
1E-5
1E-4
1E-3
0,01
Frequency (Hz)
Am
plitu
de
We applied harmonic (~sin(ωt)) signal from generator to heater in cylinder resonator. After formation the nonlinear wave spectrum (a) we switched off the pumping signal and have observed transformation of the harmonics with time (b).
a)
b)
14
Laboratory of Quantum Crystals 15
Chaos in harmonics after switch off the pump
-8 0 81E-5
1E-4
1E-3
0,01
Change of regime
A=a0*exp(t/5.2)
Am
plitu
de, a
.u.
Time, s
B1 C2 D3 E4 F5 G6 H7 I8 J9 K10 L11 M12 N13 O14
0 1
1E-4
1E-3
Am
plitu
de, a
.u.
Time, s
Stop of energy pumping in the resonator drastically reduces multiple harmonics. The amplitudes of harmonics chaotically derange. Typically time of chaotic energy random walk is order of hundreds wave periods. After 2 seconds the amplitude of the main (first) harmonic reduces with slower rate and can be described by an exponent dependence which correspond to the quality of resonator about 5000.
Chaos in energy spectrum
We found in Fourier Transformation of recording signal change in behavior of harmonic amplitude:
Linear and nonlinear times After ceasing of pumping the main harmonic losses the energy into two
channels the linear process, corresponding to quality of resonator at low excitation and nonlinear energy flux into higher harmonics: . the last term appreciably accelerates the process of vibrations decay of the main harmonic. The prime time of the free decay nonlinear interaction transmits energy into multiple harmonics, amplitudes of which reduce, too.
3 March 2010 Phys.Rev. E
After 2 s the energy flux into multiple harmonics stays vanishing and main harmonic vibration decays with linear time
The difference between should be ascribed to a nonlinear flux into direct energy cascade.
Solitons, Collapses and Turbulence
Measurements of standing waves at different resonances.
Low frequency f~1000 Hz, N~10;Higher frequency f~5 kHz, N~100
Akademgorodok, Novosibirsk, 07 June
2012
100100 1 ∞
Bad quality Good quality Balk dissipation
Laboratory of Quantum Crystals 18
Formation of Inverse Cascade
A
f
12
34
5
ω1+ω2→ω3ω3 → ω2+ωf
Dissipation
Laboratory of Quantum Crystals 19
Formation of Inverse Cascade
A
f
2040
80100
120
104 5
ω f → ω2+ωf
Laboratory of Quantum Crystals
20
Inverse Energy CascadeA
f
2040
80100
120
104 5
ω f → ω2+ωf ω1+ω2→ω3
1000 10000 1000001E-5
1E-4
1E-3
0.01
46 R43 R
35 R31 R
26 R16 R
8 R5 R
51 R
3
2
fd=9594.8 Hz
Am
plitu
de, a
.u.
Frequency, Hz3 March 2010 Chernogolovka, LT seminar
Chernogolovka, June 2008
5.140 5.141 5.142 5.143-0.04
-0.02
0.00
0.02
0.04
0.06
7.590 7.591 7.592 7.593-0.04
-0.02
0.00
0.02
0.04
0.06
8.312 8.313 8.314 8.315-0.04
-0.02
0.00
0.02
0.04
0.06
3.280 3.281 3.282 3.283-0.04
-0.02
0.00
0.02
0.04
0.06
Y A
xis
Title
X Axis Title
B
A B
CD
100 1000 10000
1E-5
1E-4
1E-3
0.01
Frequency (Hz)
Am
plitu
de
A~f -1.62
Direct and inverse cascades
1000 10000 1000001E-5
1E-4
1E-3
0.01
46 R43 R
35 R31 R
26 R16 R
8 R5 R
51 R
3
2
fd=9594.8 Hz
Am
plitu
de, a
.u.
Frequency, Hz
ω1→ω2+ω3
ω1 + ω2 → ω3
Solitons, Collapses and Turbulence
Akademgorodok, Novosibirsk, 07 June
2012
Energy transformation at acoustic turbulence
0 20000 40000 60000 80000 100000 1200000.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14 Total Left Right
Ene
rgy,
a.v
.
Time, ms
Solitons, Collapses and Turbulence
Measurements of standing waves at different resonances.
Low frequency f~1000 Hz, N~10;Higher frequency f~5 kHz, N~100;Very low frequency f~100 Hz, N~1
Akademgorodok, Novosibirsk, 07 June
2012
100100 1 ∞
Bad quality Good quality Bulk dissipation
Solitons, Collapses and Turbulence
δT~100 -200 μK
Direct formation of higher harmonics
4 Resonance fG =185.9 HzfG =194.2 Hz (fW =8* fG =4* fD)
Akademgorodok, Novosibirsk, 07 June
2012
Solitons, Collapses and Turbulence
Akademgorodok, Novosibirsk, 07 June
2012
Solitons, Collapses and Turbulence
Conclusion One-dimensional system of strong
nonlinear wave of second sound allowed to modelling a behaviour of Burgers waves.
The direct cascade of energy flux was observed in discrete system.
At some condition the inverse cascade of energy formats in the system of nonlinear second sound waves
Was observed a new effect of direct formation of multiple harmonics
Akademgorodok, Novosibirsk, 07 June
2012
Solitons, Collapses and Turbulence
Direct formation of higher harmonics
fG =198 Hz (fW =6* fG =3* fD)fG =194.2 Hz (fW =8* fG =4* fD)
Akademgorodok, Novosibirsk, 07 June
2012
fG =206.5 Hz (fW =4* fG =2* fD)
Resonance position
50 100 150 200 250 300 3500
2
4
6
8
10
5 643
Upp
=3.6 V f=54.9 Hz U
pp=4.22 V
N o
f pea
ks
f, Hz
Resonance 2
Fractional frequency transformation Standing waves
0 1 2 3 4 5Heater Bolometer
We have 5 waves at distance 6 length of resonatorThe resonance circumstance:
2*N*L=n*λ
Solitons, Collapses and Turbulence
THANK YOU FOR YOUR ATTENTION!
Akademgorodok, Novosibirsk, 07 June
2012