Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He
We discuss the results of recent studies of acoustic turbulence in a system of nonlinear second sound waves in a high-quality resonator filled with superfluid ⁴He. It was found that, when the driving amplitude was sufficiently increased, a steady-state direct wave cascade is formed involving a flu...
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| Цитувати: | Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He / A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, P.V.E. McClintock, L.P. Mezhov-Deglin // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 367–372. — Бібліогр.: 24 назв. — англ. |
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irk-123456789-1169172017-05-19T03:02:57Z Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He Ganshin, A.N. Efimov, V.B. Kolmakov, G.V. McClintock, P.V.E. Mezhov-Deglin, L.P. Жидкий гелий We discuss the results of recent studies of acoustic turbulence in a system of nonlinear second sound waves in a high-quality resonator filled with superfluid ⁴He. It was found that, when the driving amplitude was sufficiently increased, a steady-state direct wave cascade is formed involving a flux of energy towards high frequencies. The wave amplitude distribution follows a power law over a wide range of frequencies. Development of a decay instability at high driving amplitudes results in the formation of subharmonics of the driving frequency, and to a backflow of energy towards the low-frequency spectral domain, in addition to the direct cascade. 2008 Article Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He / A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, P.V.E. McClintock, L.P. Mezhov-Deglin // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 367–372. — Бібліогр.: 24 назв. — англ. 0132-6414 PACS: 67.25.dg;67.25.dt;47.27.Cn;05.20.Dd http://dspace.nbuv.gov.ua/handle/123456789/116917 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
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Жидкий гелий Жидкий гелий |
| spellingShingle |
Жидкий гелий Жидкий гелий Ganshin, A.N. Efimov, V.B. Kolmakov, G.V. McClintock, P.V.E. Mezhov-Deglin, L.P. Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He Физика низких температур |
| description |
We discuss the results of recent studies of acoustic turbulence in a system of nonlinear second sound
waves in a high-quality resonator filled with superfluid ⁴He. It was found that, when the driving amplitude
was sufficiently increased, a steady-state direct wave cascade is formed involving a flux of energy towards
high frequencies. The wave amplitude distribution follows a power law over a wide range of frequencies.
Development of a decay instability at high driving amplitudes results in the formation of subharmonics of
the driving frequency, and to a backflow of energy towards the low-frequency spectral domain, in addition to
the direct cascade. |
| format |
Article |
| author |
Ganshin, A.N. Efimov, V.B. Kolmakov, G.V. McClintock, P.V.E. Mezhov-Deglin, L.P. |
| author_facet |
Ganshin, A.N. Efimov, V.B. Kolmakov, G.V. McClintock, P.V.E. Mezhov-Deglin, L.P. |
| author_sort |
Ganshin, A.N. |
| title |
Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He |
| title_short |
Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He |
| title_full |
Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He |
| title_fullStr |
Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He |
| title_full_unstemmed |
Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴He |
| title_sort |
observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid ⁴he |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Жидкий гелий |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/116917 |
| citation_txt |
Observation of acoustic turbulence in a system of
nonlinear second sound waves in superfluid ⁴He
/ A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, P.V.E. McClintock, L.P. Mezhov-Deglin
// Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 367–372. — Бібліогр.: 24 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
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| first_indexed |
2025-07-08T11:18:54Z |
| last_indexed |
2025-07-08T11:18:54Z |
| _version_ |
1837077402060062720 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5, p. 367–372
Observation of acoustic turbulence in a system of
nonlinear second sound waves in superfluid 4He
A.N. Ganshin1, V.B. Efimov1,2, G.V. Kolmakov1,2, P.V.E. McClintock1, and
L.P. Mezhov-Deglin2
1
Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
2
Institute of Solid State Physics RAS, Chernogolovka, Moscow region, 142432, Russia
E-mail: mezhov@issp.ac.ru
Received October 30, 2007
We discuss the results of recent studies of acoustic turbulence in a system of nonlinear second sound
waves in a high-quality resonator filled with superfluid 4He. It was found that, when the driving amplitude
was sufficiently increased, a steady-state direct wave cascade is formed involving a flux of energy towards
high frequencies. The wave amplitude distribution follows a power law over a wide range of frequencies.
Development of a decay instability at high driving amplitudes results in the formation of subharmonics of
the driving frequency, and to a backflow of energy towards the low-frequency spectral domain, in addition to
the direct cascade.
PACS: 67.25.dg Transport, hydrodynamics, and superflow;
67.25.dt Sound and excitations;
47.27.Cn Transition to turbulence;
05.20.Dd Kinetic theory.
Keywords: superfluid helium, second sound, acoustic turbulence, Kolmogorov spectrum.
1. Introduction
First, we are much indebted to the Editorial Board of
the Low Temperature Physics journal for their invitation
to report the results of our recent studies in this special is-
sue to mark the 100th anniversary of the liquefaction of
helium.
In what follows, we review our experimental investi-
gations of the acoustic turbulence created in a system of
nonlinear second sound standing waves in a high-quality
resonator filled with He II, the superfluid phase of 4He.
It is well-known that He II supports two quite separate
sound propagation modes: first sound (conventional pres-
sure/density waves) and second sound (temperature/en-
tropy waves) [1,2]. Second sound waves of infinitely
small amplitude running in bulk superfluid He II are char-
acterized by a linear dispersion relation between fre-
quency � and wave vector k [2]
�k u k� 20 , (1)
which is also typical of acoustic waves in condensed mat-
ter and in gases. Here u 20 is the second sound velocity,
which depends on the temperature of the helium sample. At
temperatures close to that of the superfluid-to-normal transi-
tion at T� � 2.177 K, the velocity u 20 tends to zero. At tem-
peratures down to T �1 K, the second sound velocity is
u 20 � 20 m/s, which is much smaller than the velocity of
conventional sound in condensed media, u10 � 3�102 m/s.
We emphasize that in contrast to ordinary (classical) me-
dia, where a temperature wave is damped over a distance
of the order of its wavelength [1], the second sound wave
mode has very small dissipation within the experimen-
tally convenient temperature range T � 1.5–2.1 K: typi-
cally, a second sound pulse of duration � �1 s (i.e., of
wavelength � � 2�10–3 cm) can propagate through the
superfluid over a distance of metres before being damped
by viscous losses.
Second sound is characterized by rather strong nonlin-
ear properties [2–4]. For example, a traveling second
sound pulse of amplitude
T � 1 mK (i.e., with relative
amplitude
T/T � 10–3) transforms into a shock wave
over a distance L � 1 cm from the source [5–7]. To a first
approximation, the velocity of a second sound wave of fi-
nite amplitude depends on the wave amplitude
T as
© A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, P.V.E. McClintock, and L.P. Mezhov-Deglin, 2008
u u T2 20 1� �( )
.
Here
�
�
�
�
�
�
�
�
�
T
u
C
T
ln 20
3
is the nonlinearity coefficient of second sound and C is
the specific heat per unit mass at constant pressure. Under
the saturated vapor pressure, in the region of roton second
sound, T � 0.9 K, the nonlinearity coefficient is positive
( )
� 0 for temperatures T T� �
1.88 K (like the non-
linearity coefficient of conventional sound waves in ordi-
nary media); but it is negative in the range T T T
�� �
(and many times larger than the nonlinearity coefficient
of first sound) [4]. At T T�
the nonlinearity coefficient
passes through zero.
In the studies reported below, we exploit these special
properties of second sound in He II for an investigation
of turbulence in a system of nonlinear acoustic waves
(acoustic turbulence). This is a state in which a large
number of acoustic wave modes are excited and interact-
ing strongly with each other. It is characterized by a di-
rected energy flux through frequency scales [8–11].
Acoustic turbulence has been at the focus of numerous
investigations during the last few decades because of its
importance for basic nonlinear physics and in view of nu-
merous applications in engineering and fundamental sci-
ence [8]. Well-known examples of acoustic turbulence
include the turbulence of sound waves in oceanic wave-
guides [12], magnetic turbulence in interstellar gases
[13], and shock waves in the solar wind and their coupling
with the Earth’s magnetosphere [14].
Second sound is ideal for modelling the dynamics of
nonlinear waves because of the way in which its nonli-
nearity coefficient, which determines the strength of the
wave interactions, can be tuned over a wide range simply
by changing the bath temperature. It allows one to study
the dynamics of both nearly linear and strongly nonlinear
waves, with both positive (like conventional sound) and
negative nonlinearity, while using exactly the same ex-
perimental techniques. Such possibilities are unavailable
in conventional experiments. The fact that the velocity of
second sound u 20 is relatively small permits one to in-
crease time resolution of the measurements. Note that the
acoustic turbulent state is radically different from quan-
tum turbulence (QT) [15], which is also formed in He II,
because the density of quantized vortices is close to zero.
Furthermore, the motions of both the normal and super-
fluid components can be considered as being to a first
approximation potential.
Based on measurements of nonlinear second sound
waves in a high-quality resonator, we observed formation
of a steady-state wave-energy cascade in He II involving
a flux of energy through the spectral range towards high
frequencies. Initial results of the studies were published
in Ref. 11. Since then, we have found that, under some
circumstances, wave energy in the acoustic system can
also flow in the reverse direction. Below we discuss these
observations in more detail.
2. Experimental techniques
The experimental arrangements were similar to those
used in our earlier studies of nonlinear second sound
waves [16,17]. The cryoacoustical resonator was made of
a cylindrical quartz tube of nominal length L � 7 cm and
internal diameter D � 1.5 cm, filled with superfluid he-
lium. The low-inertial film heater and bolometer were de-
posited on the surfaces of flat glass plates capping the
ends of the tube. The heater was driven by a harmonic
voltage generator in the frequency range 0.1–100 kHz.
The frequency of the second sound (twice the frequency
of the voltage generator) was set close to the frequency of
a longitudinal resonance in the resonator. The amplitude
of the standing wave
T could be changed from 0.05 mK
up to a few mK by adjustment of the power to the heater.
The measured heat flux density W into the liquid is sub-
ject to systematic uncertainties of up to � 10% associated
with estimation of the resonator cross-section and the
resistances of the leads, and possible small inho-
mogeneities in the heater film thickness. The Q factor of
the resonator determined from the widths of longitudinal
resonances at small heat fluxes W � 4 mW/cm2 (nearly
linear regime) was Q � 3000 for resonance numbers
20 100� �p and decreased to about 500 at frequencies be-
low 500 Hz. A typical resonance curve is shown in Fig. 1
(the 32nd resonance).
Use of a high-Q resonator enables us to create nonlinear
second sound standing waves of high amplitude (
T � 1 mK)
accompanied by only small heat input at the source
W � 55 mW/cm2, thus avoiding possible complications
368 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, P.V.E. McClintock, and L.P. Mezhov-Deglin
0.02
0.01
0
3224 3228 3232 3236
T
,
ar
b
.
u
n
it
s
� �d/2 , Hz
Fig. 1. Second sound resonance curve measured in the He II
filled resonator at a temperature close to 2.08 K. The system
was driven at a frequency close to its 32nd resonance with an
ac heat flux density of W � 4 mW/cm
2
.
[18] due to vortex creation in the bulk He II and nonlinear
phenomena at the heater/superfluid interface. The second
sound waveform registered by the bolometer was Fou-
rier-analyzed and its power spectrum was computed.
3. Results and discussion
Figure 2 shows the evolution of the second sound
wave spectrum with increasing ac heat flux density W
from the heater, measured at temperature close to 2.08 K
when driving at the frequency of the 31st resonance. For
small W � 4 mW/cm2 we observed a nearly linear regime
of wave generation, where a small number of harmonics
of the driving frequency were excited due to nonlinearity
(see Fig. 2,a), and the shape of the recorded signal was
close to sinusoidal. An increase of the excitation above
12 mW/cm2 led to visible deformation of the signal shape
and to the generation of a large number of harmonics in
the second sound wave spectrum as shown in Fig. 2,b.
It is evident from Fig. 2 that the main spectral peak
(marked by the arrow) lies at the driving frequency �d ,
and that high-frequency peaks appear at its harmonics
� �n dn� with n = 2, 3, ... It can be seen in Fig. 2,b that a
cascade of waves is formed over the frequency range up
to 80 kHz, i.e., up to a frequency 25 times higher than the
driving frequency. As also shown in Fig. 2,b, the depen-
dence of peak height on frequency may be described by a
power-law-like function
�T � const��–s for frequencies
lower than some cut-off frequency �b that increases with in-
creasing W . Note there are systematic uncertainties of
about � 10% in the values of s and �b extracted from plots
of this kind, depending on the range of � through which the
straight line is drawn. For sufficiently high ac heat flux den-
sities W � 12 mW/cm2 (i.e., for the developed cascade), the
scaling index tends towards s � 1.5.
Formation of the spectra observed in the experiments
is evidently attributable to the cascade transfer of wave
energy through the frequency scales due to nonlinearity,
thus establishing an energy flux in K space directed from
the driving frequency towards the high-frequency do-
main. In accordance with basic ideas formulated in
Refs. 8–10 we may infer that, at relatively high driving
amplitudes, we observe acoustic turbulence formed in the
system of second sound waves within the inertial
(nondissipative) range of frequencies. Formation of the
observed direct cascade is similar to creation of the
Kolmogorov distribution of fluid velocity over frequency
in the bulk of a classical fluid [19].
We observed also that, when the ac heat flux density
was raised above some critical value at even resonance
numbers p � 30, a spectral peak appeared at the frequency
equal to half the driving frequency (i.e., formed on the left
of the fundamental peak) and at its harmonics. Figure 3
shows the evolution of the wave spectrum with increasing
ac heat flux density when driving on the 32nd resonance.
It is evident that, at the relatively small heat flux density
W � 4 mW/cm2, the wave spectrum shown in Fig. 3,a is
quite similar to that observed under similar conditions
when driving at the 31st resonance (i.e., at the nearest odd
numbered resonance), see Fig. 2,a. Formation of the
low-frequency harmonic (subharmonic) at � �� d /2 at
W � 16 mW/cm2 is clearly seen in Fig. 3,b. It was found that
the threshold value for generation of subharmonics obtained
in measurements with 30 95� �p was about 12 mW/cm2.
The formation of subharmonics may be attributed to
development of a decay instability of the periodic wave.
In accordance with general theory [8,20,21], the instabil-
ity is controlled mainly by nonlinear decay of the wave
into two waves of lower frequency, and by the opposite
process of the confluence of two waves to form one wave.
Acoustic turbulence in superfluid 4He
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 369
10
–2
10
–4
10
–6
�d
�b
a
10
3
10
4
10
3
10
4
10
5
�d
�b
b
10
–2
10
–4
10
–6
T
,
ar
b
.
u
n
it
s
�
T
,
ar
b
.
u
n
it
s
�
� �/2 , Hz
� �/2 , Hz
Fig. 2. Evolution in the power spectrum of second sound stan-
ding waves as the ac heat flux density is increased from W =
= 4 (a) to 25 mW/cm
2
(b). The dashed line in (a) is a guide to
the eye, whereas that in (b) corresponds to
�T � �1 7. . The ar-
rows indicate the positions of the fundamental spectral peak
formed at the driving frequency �d and of the high-frequency
edge �b of the inertial frequency range. The system was driven
at its 31st resonance, at a temperature close to 2.08 K.
The energy (or frequency) conservation law for this
3-wave process is
� � �1 2 3� � , (2)
where �i iu k� 20 is the frequency of a linear wave of wave
vector k i . The case shown in Fig. 3,b evidently corresponds
to generation of subharmonics with � � �2 3 2� � d / .
We also observed the high-frequency cutoff of the
wave spectrum due to viscosity. As shown in Fig. 2,a, it
manifests itself as an abrupt decrease in the amplitudes of
the harmonics at W � 12 mW/cm2, and as a change in
slope of the spectrum at higher W (Fig. 2,b) when plotted
on double-log scales, which occurs at some characteristic
frequency �b . At � �� b the nonlinear mechanism for
nearly nondissipative transfer of the wave energy changes
to viscous damping of the waves (cf. observations of the
high-frequency edge of the inertial range of frequencies
of capillary turbulence on the surface of liquid hydrogen
[22]). It causes a faster reduction of sound amplitudes at
frequencies � �� b , as observed.
The dependence of the boundary frequency � ��b / on
the standing wave amplitude
T is shown in Fig. 4. It is
seen that the inertial range is extended towards higher fre-
quencies when the driving force is increased. When driv-
ing at resonant frequencies with odd resonance numbers
p, with sufficiently large driving amplitudes, the bound-
ary frequency depends linearly on wave amplitude
� �� �
b d/ T T� const( , )
(the filled symbols in Fig. 4) in agreement with our nu-
merical calculations [11].
When driving at even p (open symbols in Fig. 4) the
boundary frequency is noticeably lower than that mea-
sured for the nearest odd resonance number, with
W � 12 mW/cm2. This reduction may be connected with a
change in the mechanism of energy relaxation in the wave
system caused by the generation of subharmonics with
frequencies lower than �d .
One can see from Fig. 4 that the energy balance in the
wave system is highly nonlocal in K space: energy is
pumped into the system in the low-frequency (long-wave)
domain and it flows to the high frequency (short-wave)
domain where it is absorbed by dissipative mechanisms.
It was found that, when driving at sufficiently high
ac heat flux densities W � 23 mW/cm2 and at resonance
numbers p � 50, multiple subharmonics were generated in
the low-frequency spectral domain � �� d , see Fig. 5.
Here we present the initial results of our study of the de-
cay instability in the acoustic system, a phenomenon that
promises to be of huge interest for nonlinear sound wave
dynamics. The results of the more detailed investigations
now in progress will be published and discussed else-
where in due course.
The formation of subharmonics in the wave spectrum
is evidently attributable to further development of the de-
370 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, P.V.E. McClintock, and L.P. Mezhov-Deglin
10
–2
10
–4
10
–6
�d a
10
3
10
4
10
3
10
4
�d b
10
–2
10
–4
10
–6
T
,
ar
b
.
u
n
it
s
�
T
,
ar
b
.
u
n
it
s
�
� �/2 , Hz
� �/2 , Hz
�d/2
Fig. 3. Second sound wave spectra measured when driving on
the frequency of the 32nd resonance with W � 4 mW/cm
2
(a)
and 16 mW/cm
2
(b). The arrows indicate the fundamental
peaks at the driving frequency � �� d and a subharmonic
formed at � �� d /2. The temperature was close to 2.08 K.
p = 31, 2.08 K
p = 32, 2.08 K
p = 31, 1.77 K
p = 32, 1.77 K
0
25
50
75
100
0.05 0.10 0.15 0.20 0.25 0.30
T, arb. units
�
�
b
/2
,
k
H
z
Fig. 4. Dependence of the viscous cutoff frequency � ��b/ on
the amplitude
T of the standing wave for different tempera-
tures T and resonance numbers p. Dashed lines indicate the re-
sults of our numerical computations [11]; data points represent
experimental measurements.
cay instability, when not only a wave with frequency
� ��b / is created due to nonlinearity, but a number of
waves with frequencies obeying the conservation law (2)
are generated. This regime is quite similar to the kinetic
instability known for weak turbulent systems [21]. It is
seen from Fig. 5 that, when the instability develops, the
wave spectrum becomes almost continuous: all possible
modes seem to be excited.
We may interpret the generation of waves of frequency
lower than the driving frequency as the establishment of
an energy backflow towards low frequencies. Inverse en-
ergy cascades are known in two-dimensional incompress-
ible liquids [23] and Bose gases [24] but, to our knowl-
edge, this phenomena has not been observed earlier for
nonlinear acoustic waves. Absorption of the wave energy
at low frequencies is probably attributable to viscous drag
of the normal fluid component on the resonator walls,
given that bulk second sound damping is negligibly small
in this frequency range: this would be consistent with the
observed strong decrease of the resonator Q factor below
2 kHz. We have also observed that formation of the in-
verse energy cascade is accompanied by a reduction of
wave amplitude in the high-frequency spectral domain,
and by a contraction of the inertial range. This observa-
tion is consistent with our inference that, after onset of the
instability, the energy flux is shared between the direct
and inverse energy cascades.
4. Conclusions
We have demonstrated that the system of second sound
waves in a high-Q resonator filled with He II can be used
as an effective tool for the detailed modelling and investi-
gation of acoustic turbulence. We observed a smooth
crossover in the system of second sound waves from a
nearly linear regime at low driving amplitudes to a non-
linear regime at moderate driving amplitudes, and,
further, to developed turbulence at high driving ampli-
tudes (a Kolmogorov-like cascade). In the high-frequen-
cy domain a cutoff of the cascade is observed, caused by a
change in the mechanism of energy transfer, from nonlin-
ear wave transformation to viscous damping. When driv-
ing at moderate amplitudes at resonances of even reso-
nance number, a decay instability develops in the system
due to 3-wave interactions. It results in the generation of a
subharmonic of frequency equal to half the driving fre-
quency. At relatively high driving amplitudes multiple
subharmonics are generated in the wave spectrum, corre-
sponding to the formation of an inverse energy cascade
directed towards the low-frequency spectral domain.
The investigations were supported by the Russian
Foundation for Basic Research, project Nos. 06-02-17253
and 07-02-00728, by the Presidium of the Russian Acad-
emy of Sciences in frames of the programs «Quantum
Macrophysics» and «Mathematical Methods in Nonlinear
Dynamics», and by the Engineering and Physical Sci-
ences Research Council (UK).
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Acoustic turbulence in superfluid 4He
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10
3
10
4
10
5
�d10
–1
10
–3
10
–5
T
,
ar
b
.
u
n
it
s
�
� �/2 , Hz
Fig. 5. A second sound wave spectrum measured when driving
at the frequency of the 96th resonance with an ac heat flux
density W � 25 mW/cm
2
. The temperature is close to 2.08 K.
The arrow marks the fundamental peak at the driving fre-
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