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., Russia
2) Physics department, Lancaster University,
Lancaster, UK
The VI-th international conference:
Solitons, Collapses and Turbulence
Novosibirsk Akademgorodok, 7 June 2012
Outline

Introduction
◦ 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
Sounds in Superfluid Helium
The differentiation of two equations
with respect to time and substituting
into another relationships we obtain
for small disturbance
(linear waves)
240
220
200
180
v1 /  3
160
 S s S

T
2
t
n
2
2
140
v, m/s
 
 P,
2
t
2
120
u1  (P /  ) S

First and Second sound
4
velosity in He at SVP
100
80
v2
60
Solutions are
 wave of density (co-moving
 motion
normal
and
component)
v1
40
20
0
0
1
2
3
4
T, K
superfluid
TS 2  s
C n
waves of temperature or entropy
u2 
(counter-movement motion normal and
Akademgorodok, Novosibirsk,
superfluid components,
=n+07June
2012 Solitons, Collapses and Turbulence
s=const
Second sound shock waves in
 Nonlinear coefficient
superfluid helium
for first sound
Temperature dependence of second
sound nonlinear coefficient. Right
hump is connected with roton
waves, left one – with phonon’s
temperature excitation

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.
12
10
P=3atm., U =7.32V, =10s.
8
5
8
Arb. units.

is positive
7
4
6
6
4
3
2
2
1
2
3
4
5
6
7
8
T=2.103K
T=2.071K
T=2.044K
T=1.999K
T=1.968K
T=1.81K
T=1.783K
T=1.774K
1
0
-2
-4
0
20
40
60
80
100
Time, s.
Laboratory of
Quantum Crystals
4
Low velocity of wave of second sound
 Weak 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:
Why second sound in superfluid
helium?
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
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
Laboratory of Quantum
Crystals
7
Quality of resonator 70 mm

10000
Quality, f/ f

Q~f
3/2
1000
100
1000
40000
80000
Frequency, Hz


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
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
and quality of resonator will be defined as verses
length of Λ balks a motion of wave. The absolute value of resonator
value of losses
quality is of the order of
Bulk dissipation stays appreciable at frequencies higher 10 kHz
3 March 2010
Phys.Rev. E
Measurements of standing waves at
different resonances.
0
1
Bad quality

10
100
∞
Good quality Balk dissipation
Low frequency f~1000 Hz, N~10
Akademgorodok, Novosibirsk,
07 June 2012
Solitons, Collapses and Turbulence
Energy Transformation in
Superfluid Helium-4
10
Experimental studies of one dimensional
nonlinear second sound waves in the
cylindrical cell
12
0,010
Applied signal A=A0*sin(ωt)
6
Temperature in second
sound wave
A, arb. units
0,005
0
0,000
-6
-0,005
-12
0,001
0,002
t, ms
Fast Fourier
Transformation
Spectrum of temperature
oscillations of the nonlinear
second sound waves in a
resonator
Amplitude
0.01
A~f
-1.62
1E-3
1E-4
1E-5
100
1000
10000
Frequency (Hz)
11
Formation of
Kolmogorov’s
spectrum
FFT spectrum
12
Decay of nonlinear energy spectrum
Amplitude
0,01
1E-3
1E-4
1E-5
b)
1000
10000
Frequency (Hz)
a)
0,01
Amplitude
 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).
100
1E-3
1E-4
1E-5
100
1000
10000
Frequency (Hz)
Laboratory of Quantum
Crystals
13
14
We found in Fourier Transformation
of recording signal change in
behavior of harmonic amplitude:
B1
C2
D3
E4
F5
G6
H7
I8
J9
K10
L11
M12
N13
O14
Change of regime
Amplitude, a.u.
0,01
A=a0*exp(t/5.2)
1E-3
1E-4
 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
1E-3
Amplitude, a.u.
Chaos in harmonics after
switch off the pump
1E-4
1E-5
-8
0
8
Time, s
0
Laboratory of Quantum
Crystals
1
Time, s
15
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.

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.


3 March 2010
Phys.Rev. E
Measurements of standing waves at
different resonances.
0
1
Bad quality
10
100
∞
Good quality Balk dissipation
Low frequency f~1000 Hz, N~10;
 Higher frequency f~5 kHz, N~100

Akademgorodok, Novosibirsk,
07 June 2012
Solitons, Collapses and Turbulence
Formation of Inverse Cascade
ω3 → ω2+ωf
ω1+ω2→ω3
A
1
2
3
4
5
Dissipation
f
Laboratory of Quantum Crystals
18
Formation of Inverse Cascade
ω f → ω2+ωf
A
4 5
10
20
40
80
100
120
f
Laboratory of Quantum Crystals
19
Laboratory of
Quantum Crystals
20
ω f → ω2+ωf
A
10
4 5
ω1+ω2→ω3
Inverse
Energy
Cascade
20
40
80
100
120
46 R 51 R
43 R
fd=9594.8 Hz
35 R
31 R
26 R
2
16 R
0.01
Amplitude, a.u.
f
3
1E-3
8R
5R
1E-4
1E-5
1000
3 March 2010
Chernogolovka, LT seminar
10000
Frequency, Hz
100000
Direct and inverse cascades
A~f
ω1→ω2+ω3
ω 1 + ω2 → ω3
-1.62
46 R 51 R
43 R
fd=9594.8 Hz
35 R
31 R
26 R
2
16 R
1E-3
0.01
1E-4
Amplitude, a.u.
Amplitude
0.01
1E-5
100
1000
10000
Frequency (Hz)
3
1E-3
8R
5R
1E-4
B
Y Axis Title
0.06
0.06
A
0.04
0.04
0.02
0.02
0.00
0.00
-0.02
-0.02
-0.04
3.280
3.281
3.282
3.283
X Axis Title
0.06
1E-5
B
1000
-0.04
5.140
5.141
5.142
5.143
8.313
8.314
8.315
0.06
D
0.04
C
0.04
0.02
0.02
0.00
0.00
-0.02
-0.02
-0.04
7.590
7.591
7.592
7.593
-0.04
8.312
10000
Frequency, Hz
Chernogolovka, June 2008
100000
Energy transformation
at acoustic turbulence
Total
Left
Right
0.14
0.12
Energy, a.v.
0.10
0.08
0.06
0.04
0.02
0.00
0
20000
40000
60000
80000
100000
120000
Time, ms
Akademgorodok, Novosibirsk,
07 June 2012
Solitons, Collapses and Turbulence
Measurements of standing waves at
different resonances.
0
1
Bad quality
10
100
∞
Good quality Bulk dissipation
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
Solitons, Collapses and Turbulence
4 Resonance fG =185.9 Hz
fG =194.2 Hz (fW =8* fG =4* fD)
δT~100 -200 μK
Direct formation of higher
harmonics
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
fG =198 Hz (fW =6* fG =3* fD)
fG =194.2 Hz (fW =8* fG =4* fD)
fG =206.5 Hz (fW =4* fG =2* fD)
Direct formation of higher
harmonics
Akademgorodok, Novosibirsk,
07 June 2012
Solitons, Collapses and Turbulence
Resonance position
10
Upp=3.6 V f=54.9 Hz
Upp=4.22 V
N of peaks
8
6
4
2
Resonance
2
3
4
5
6
0
50
100
150
200
f, Hz
250
300
350
Fractional frequency transformation
Standing waves
Heater
0
Bolometer
1
2
3
4
5
 We have 5 waves at distance 6 length of resonator
 The resonance circumstance:
2*N*L=n*λ

THANK YOU FOR
YOUR ATTENTION!
Akademgorodok, Novosibirsk,
07 June 2012
Solitons, Collapses and Turbulence