Classical capillary turbulence on the surface of quantum liquid He-II
Superfluid helium-4 is a unique liquid for experimental study of capillary wave turbulence due to its very low viscosity. We have studied the influence of amplitude and spectral characteristics of the excitation force on the behavior of the turbulent cascade of capillary waves in the cylindrical cel...
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irk-123456789-1185742017-05-31T03:03:51Z Classical capillary turbulence on the surface of quantum liquid He-II Abdurakhimov, L.V. Brazhnikov, M.Yu. Remizov, I.A. Levchenko, A.A. 8th International Conference on Cryocrystals and Quantum Crystals Superfluid helium-4 is a unique liquid for experimental study of capillary wave turbulence due to its very low viscosity. We have studied the influence of amplitude and spectral characteristics of the excitation force on the behavior of the turbulent cascade of capillary waves in the cylindrical cell of diameter 30 mm. We have found that the experimental results can be explained within the framework of the wave turbulence theory (WTT) when the amplitude of pumping is relatively high. However, a very interesting phenomenon was observed at moderate amplitudes of harmonic surface excitation. It was detected the deviation of the turbulent spectrum from the power law function predicted by WTT at high frequencies — local maximum — that can be interpreted as wave energy accumulation. Our estimations show that the special case of wave turbulence is realized in our measurements, namely, discrete turbulence which is characterized by the strong influence of discreteness of cell resonant frequencies on the mechanism of the nonlinear interaction. 2011 Article Classical capillary turbulence on the surface of quantum liquid He-II / L.V. Abdurakhimov, M.Yu. Brazhnikov, I.A. Remizov, A.A. Levchenko // Физика низких температур. — 2011. — Т. 37, № 5. — С. 512–516. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 68.03.Kn, 47.27.Gs http://dspace.nbuv.gov.ua/handle/123456789/118574 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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8th International Conference on Cryocrystals and Quantum Crystals 8th International Conference on Cryocrystals and Quantum Crystals Abdurakhimov, L.V. Brazhnikov, M.Yu. Remizov, I.A. Levchenko, A.A. Classical capillary turbulence on the surface of quantum liquid He-II Физика низких температур |
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Superfluid helium-4 is a unique liquid for experimental study of capillary wave turbulence due to its very low viscosity. We have studied the influence of amplitude and spectral characteristics of the excitation force on the behavior of the turbulent cascade of capillary waves in the cylindrical cell of diameter 30 mm. We have found that the experimental results can be explained within the framework of the wave turbulence theory (WTT) when the amplitude of pumping is relatively high. However, a very interesting phenomenon was observed at moderate amplitudes of harmonic surface excitation. It was detected the deviation of the turbulent spectrum from the power law function predicted by WTT at high frequencies — local maximum — that can be interpreted as wave energy accumulation. Our estimations show that the special case of wave turbulence is realized in our measurements, namely, discrete turbulence which is characterized by the strong influence of discreteness of cell resonant frequencies on the mechanism of the nonlinear interaction. |
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Abdurakhimov, L.V. Brazhnikov, M.Yu. Remizov, I.A. Levchenko, A.A. |
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Abdurakhimov, L.V. Brazhnikov, M.Yu. Remizov, I.A. Levchenko, A.A. |
| author_sort |
Abdurakhimov, L.V. |
| title |
Classical capillary turbulence on the surface of quantum liquid He-II |
| title_short |
Classical capillary turbulence on the surface of quantum liquid He-II |
| title_full |
Classical capillary turbulence on the surface of quantum liquid He-II |
| title_fullStr |
Classical capillary turbulence on the surface of quantum liquid He-II |
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Classical capillary turbulence on the surface of quantum liquid He-II |
| title_sort |
classical capillary turbulence on the surface of quantum liquid he-ii |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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8th International Conference on Cryocrystals and Quantum Crystals |
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Classical capillary turbulence on the surface of quantum liquid He-II / L.V. Abdurakhimov, M.Yu. Brazhnikov, I.A. Remizov, A.A. Levchenko // Физика низких температур. — 2011. — Т. 37, № 5. — С. 512–516. — Бібліогр.: 21 назв. — англ. |
| series |
Физика низких температур |
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AT abdurakhimovlv classicalcapillaryturbulenceonthesurfaceofquantumliquidheii AT brazhnikovmyu classicalcapillaryturbulenceonthesurfaceofquantumliquidheii AT remizovia classicalcapillaryturbulenceonthesurfaceofquantumliquidheii AT levchenkoaa classicalcapillaryturbulenceonthesurfaceofquantumliquidheii |
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2025-07-08T14:15:45Z |
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1837088529613586432 |
| fulltext |
© L.V. Abdurakhimov, M.Yu. Brazhnikov, I.A. Remizov, and A.A. Levchenko, 2011
Fizika Nizkikh Temperatur, 2011, v. 37, No. 5, p. 512–516
Classical capillary turbulence on the surface of quantum
liquid He-II
L.V. Abdurakhimov, M.Yu. Brazhnikov, I.A. Remizov, and A.A. Levchenko
Institute of Solid State Physics RAS, Chernogolovka, Moscow district. 142432, Russia
E-mail: abdurl@issp.ac.ru
Received June 14, 2010
Superfluid helium-4 is a unique liquid for experimental study of capillary wave turbulence due to its very low
viscosity. We have studied the influence of amplitude and spectral characteristics of the excitation force on the
behavior of the turbulent cascade of capillary waves in the cylindrical cell of diameter 30 mm. We have found
that the experimental results can be explained within the framework of the wave turbulence theory (WTT) when
the amplitude of pumping is relatively high. However, a very interesting phenomenon was observed at moderate
amplitudes of harmonic surface excitation. It was detected the deviation of the turbulent spectrum from the pow-
er law function predicted by WTT at high frequencies — local maximum — that can be interpreted as wave
energy accumulation. Our estimations show that the special case of wave turbulence is realized in our measure-
ments, namely, discrete turbulence which is characterized by the strong influence of discreteness of cell resonant
frequencies on the mechanism of the nonlinear interaction.
PACS: 68.03.Kn Dynamics (capillary waves);
47.27.Gs Isotropic turbulence; homogeneous turbulence.
Keywords: capillary turbulence, discrete turbulence, superfluid helium-4.
Introduction
Wave turbulence (WT) is a non-equilibrium state of the
weakly nonlinear wave system wherein scales of energy
pumping and energy dissipation are well-separated [1].
Examples of this nonlinear phenomenon can be found in
diverse fields such as plasma physics, nonlinear optics,
ferromagnetism, and hydrodynamics. In particular, an en-
semble of nonlinear capillary waves on the liquid surface is
a well-known system in which wave turbulence can be
realized. For capillary waves the main processes of the
nonlinear interaction are three-wave processes of decay
and merging that satisfy the conservation laws of frequen-
cy ω and wave-vector :k 1 2 3= ,ω +ω ω 1 2 3= .+k k k
The WT theory predicts that a direct turbulent cascade is
formed in the inertial range between the energy pumping
scale and energy dissipation scale with the wave spectrum
described by the power law function of frequency (Kolmo-
gorov–Zakharov spectrum). In recent years, due to the de-
velopment of technical equipment and computational tools
for data processing, capillary turbulence has been in focus
of experimental studies of capillary waves on the surface
of water [2– 4], ethanol [4], silicon oil [5] and mercury [6].
Our previous experiments [7,8] showed that the use of
quantum fluids (liquid 4He and liquid hydrogen) for inves-
tigations of capillary turbulence can provide additional
advantages with respect to traditional liquids due to their
low viscosity. In the present paper we focus on the pecu-
liarities of the shape of the turbulent wave energy distri-
bution at high frequencies observed in experiments with
He II, a superfluid phase of liquid 4He (preliminary results
were published in [9]). For the first time we observed that
under certain conditions a deviation from the power law
spectrum is formed at frequencies of the order of the visc-
ous cut-off scale. We also suggest a qualitative model of
this phenomenon that takes into account the influence of
discreteness arising due to the finite sizes of the experi-
mental cell.
Experimental set-up
In the reported experiments with superfluid 4He we
used the technique similar to that of our previous studies of
capillary turbulence on the surface of liquid hydrogen [10].
The measurements were made at temperature T ≈ 1.7 K
(for comparison, the temperature of the normal-to-
superfluid state transition for liquid 4He at saturated vapor
pressure is =Tλ 2.17 K). At this temperature superfluid
helium has kinematic viscosity 5= 8.9 10−ν ⋅ cm2/s, densi-
ty = 0.145ρ g/cm3, surface tension = 0.32σ din/cm [11].
Liquid helium was condensed into a copper cup of inner
diameter = 30D mm and depth 4 mm. At the bottom of
Classical capillary turbulence on the surface of quantum liquid He-II
Fizika Nizkikh Temperatur, 2011, v. 37, No. 5 513
Fig. 1. Turbulent spectrum of capillary waves on He II surface
(T = 1.7 K) obtained when the surface was pumped by sinusoidal
force with frequency 80 Hz, amplitude of pumping voltage 11 V.
Straight line corresponds to power law 2 3.7P −
ω ω∼ predicted by
the theory for narrow-band pumping. Dashed curve corresponds
to exp ( /150)f−∼ .
100 1000
Frequency, Hz
10
12
10
10
10
8
10
6
10
4
10
2
P
,
ar
b
.
u
n
it
s
f2
Fig. 2. Turbulent spectrum of capillary waves in the case of noise
pumping at 90–200 Hz. Straight line corresponds to power law
2 2.8P −
ω ω∼ predicted by the theory for broad-band pumping.
Dashed curve corresponds to exp ( / 800)f−∼ .
100 1000
Frequency, Hz
10
11
10
9
10
7
10
5
10
3
P
f2
,
ar
b
.
u
n
it
s
10
10
4
the cup a radioactive thin plate was placed. Emitted from
the plate β-electrons ionized 4He atoms which led to for-
mation of positively charged ions and electrons. Between
the cup and the upper electrode (a copper plate placed in
vapor at a distance of 3 mm above the liquid surface) con-
stant dc voltage 600U ≈ V of appropriate polarity was
applied, so that the liquid surface was charged by positive
ions. By applying low-frequency ac voltage with ampli-
tudes from 1 up to 20 V to the cup, in addition to dc vol-
tage, the capillary waves were excited on the charged sur-
face. The advantage of this method of wave pumping is
that the surface is excited directly by the driving force,
rather than instability as in the experiments with parametr-
ically excited waves [2,4,5] and not by blade-wavemakers
as in [6] which probably resulted in any motion in the bulk.
Oscillations of the surface were detected by measurements
of the power of the laser beam reflected from the surface.
The laser beam fell on the surface at sliding angle
0.15α ≈ producing an elliptic spot of size 1 mm×7 mm.
Typical wavelengths of capillary waves in the experiments
were smaller than the spot dimensions. The laser beam
reflected from the oscillating surface was focused by the
lens on a Si photodiode (Hamamatsu s3590-08). Time var-
iations of the signal from the photodiode were amplified by
a SR570 amplifier and then digitized by a 24-bit analogue-
to-digital converter Leonardo-II [12] with discretization
frequency 102.4 kHz. Amplified signal from the diode was
proportional to the time variations of the reflected laser
beam power ( )P t . Signal ( )P t was recorded to the hard
disk into a series of 16 files of approximately 5 seconds
long each. From Fast Fourier Transform (FFT) of every
data file, power spectrum 2| |Pω was computed, and aver-
aging was performed over all the 16 files. As it was shown
in our previous works (see, for example, [8,10]) the re-
flected laser beam power spectrum 2| |Pω was proportional
to the spectrum of the correlation function of surface eleva-
tion 2= | |Iω ωη , hence 2 2| | | | .Pω ωη∼
In the reported measurements two types of low-
frequency driving force were used. In the first type expe-
riments the surface was driven by a sinusoidal force with
frequency pumpω that was equal to that of resonant fre-
quencies of the cell. In the second type measurements driv-
ing by a noise force was used. In this case the electrical
noise was produced by a programmable noise generator.
Results
Figure 1 shows the turbulent spectrum of capillary
waves 2Pω obtained in our experiments when the surface
was pumped by ac voltage at frequency 80 Hz and ampli-
tude 11 V. Turbulent distribution 2Pω looks similar to that
observed earlier in our experiments on the surface of liquid
hydrogen [7] and in our first experiments on the surface of
liquid 4He [8]. The spectrum consists of equidistant har-
monics: the first harmonic frequency corresponds to the
pumping frequency, while other harmonics are generated
due to the nonlinear interactions at frequencies multiple to
the pumping frequency. In the inertial range the amplitudes
of the harmonics depend on frequency as power law func-
tion 2 mP −
ω ω∼ with index 3.7m ≈ which is in accordance
with the theoretical prediction for narrow-band pumping
[13]. At frequencies of about 4 kHz the wave amplitudes
were reduced significantly as a result of viscous dissipation
and tended to the level of instrumental noise.
Similarly, when the surface was forced by an intense
noise pumping in the frequency range 90–200 Hz, power
law spectrum 2 2.8P −
ω ω∼ was obtained (Fig. 2). Index ∼2.8
is predicted by the theory for broad-band pumping [1], and
the difference of the indices in the experiments with har-
L.V. Abdurakhimov, M.Yu. Brazhnikov, I.A. Remizov, and A.A. Levchenko
514 Fizika Nizkikh Temperatur, 2011, v. 37, No. 5
Fig. 3. Turbulent spectrum after reduction of pumping amplitude.
Frequency of pumping 80 Hz, amplitude of pumping voltage
10 V. At frequencies of about 2.5 kHz, a local maximum is ob-
served (emphasized by dashed curve). Dashed straight line cor-
responds to 2 3.7P −
ω ω∼ .
100 1000
Frequency, Hz
10
12
10
10
10
8
10
6
10
4
10
2
�b
P
f2
,
ar
b
.
u
n
it
s
Fig. 4. Turbulent spectrum in the case of pumping at
pump / 2 = 34ω π Hz. Formation of a maximum at the spectrum is
clearly seen on the main plot (circled) and in the insert on an
enlarged scale.
1000
100
10
0.1
1
2000
10
0.1
100 1000
Frequency, Hz
10
11
10
9
10
7
10
5
10
3P
f2
,
ar
b
.
u
n
it
s
10
4
monic pumping and broad-band noise pumping is in ac-
cordance with our previous experiments on liquid hydro-
gen [14]. It is interesting to note, that, within a dissipative
frequency range, the wave amplitudes in the turbulent
spectrum generated by a harmonic force are decreased
more rapidly than those in the spectrum generated by a
noise force (exponential dashed curves in Figs 1, 2).
When the amplitude of harmonic pumping was de-
creased to 10 V, the shape of the turbulent spectrum is
changed (Fig. 3). High-frequency edge of the turbulent cas-
cade is shifted towards low frequencies which is in accor-
dance with our previous experiments [8,15]. However a
new phenomenon was observed, namely, formation of a
well-pronounced local maximum in the wave amplitudes at
frequencies of the order of high-frequency edge of turbulent
cascade bω (shown by the dashed curve in Fig. 3). With
further reduction of the pumping amplitude, the local max-
imum was shifted towards low frequencies. Eventually, at
very low pumping amplitudes the spectrum was consisted
only of a few harmonics, and no local maximum was found.
Formation of a local maximum was seen in a number of
experiments with harmonic pumping. The common feature
observed in the experiments was that the local maximum
was formed near the high-frequency edge of the turbulent
cascade although the shape of the local maximum and its
exact position depended on the pumping frequency as well
as on the pumping amplitude. For example, when the sur-
face was pumped by the sinusoidal force at 34 Hz, the
wave spectrum with a distinct local maximum was ob-
tained (Fig. 4). However, the local maximum was located
in the dissipation region of turbulent distribution rather
than in the inertial range as is shown in Fig. 3.
In the case of noise pumping, we never observed a local
maximum. To illustrate this, in Fig. 5 we present a turbulent
distribution obtained when the surface was pumped by a
noise signal in the frequency range 60–130 Hz. The ampli-
tude of pumping force was chosen so that the high-
frequency edge of inertial interval was located approximate-
ly at those frequencies at which it was seen in Figs. 3, 4. The
spectrum differs drastically from the spectra observed in the
experiments with harmonic pumping: it is continuous and
the spectrum shape at high frequencies is more smooth.
Discussion
Formation of the local maximum can be interpreted as
the wave energy accumulation at the high-frequency edge
of the turbulent cascade where transition from the nonli-
near wave energy transfer to viscous damping occurs. In
the theoretical work [16] it is shown that viscous dissipa-
tion (which leads to finiteness of the inertial range) can in
principle result in the energy accumulation due to the bot-
tleneck effect. However, it is not clear in which manner the
dependence of formation of the local maximum on the
Fig. 5. Turbulent spectrum of capillary waves in the case of noise
pumping at 60–130 Hz.
100 1000
Frequency, Hz
10
4
10
11
10
9
10
7
10
5
10
3
10
P
f2
,
ar
b
.
u
n
it
s
Classical capillary turbulence on the surface of quantum liquid He-II
Fizika Nizkikh Temperatur, 2011, v. 37, No. 5 515
Fig. 6. When broadening of resonance δω is smaller than distance
between two neighbour resonances ,Δω some harmonics are out of
the resonances of surface oscillations in the experimental cell.
�N –1 �N �N + 1
��
��
n��pump
pumping amplitude observed in our measurements can be
explained by the influence of viscous dissipation. On the
other hand, it was pointed out by E. A. Kartashova [17,18]
that there were peculiarities in weakly nonlinear wave inte-
ractions in resonators due to discreteness of wavenumber
space. Theory of discrete wave turbulence was developed
later in a number of works, but of particular interest for us
is the work [19] where the model of “frozen” turbulence
was proposed. It follows from this model that, at small
amplitudes of pumping, discreteness of wavenumber space
can result in an oscillatory wave spectrum. We suppose
that the local maximum is also caused by the discreteness
of k -space. However, the above-mentioned work [19] con-
siders the case of waves on the surface with square boun-
dary, when the wavenumber k-space is two-dimensional.
Otherwise, in our experiments the boundary geometry is
cylindrical, surface oscillations should be described by the
Bessel functions, and the wavenumber space is one-
dimensional. It is known that the form of the basin boun-
dary strongly affects the density of resonant wavenumbers
[18]. Therefore, we performed the following alternative
estimations to understand the influence of discreteness on
spectrum in our experiments. We suppose that the pheno-
menon is caused by the detuning of frequencies of the
harmonics (shown in Figs. 1, 3, 4) and the resonant fre-
quencies of the cell. For surface waves in a cylindrical re-
sonator with diameter D the eigenvalues of wavenumber
k are the roots of boundary condition equation
1( / 2) = 0J kD , where 1( )J x is a Bessel function of the
first order. When k is large, the asymptotic form of the
Bessel function can be used, and the resonance wave-
numbers are equidistant with step 2 / .k DΔ ≈ π Hence, the
distance between two neighbour resonance frequencies
should grow with increasing frequency as
1/3
1/32 3= = .k
k k D D
⎛ ⎞∂ω ∂ω π π σ
Δω Δ ≈ ω⎜ ⎟∂ ∂ ρ⎝ ⎠
(1)
Thus, for capillary waves the resonant frequencies are
not equidistant which is in contrast to the frequencies of
harmonics that are multiple to the pumping frequency
pumpω (it should be mentioned, that in our experiments
pump< ).Δω ω
Obviously, such detuning is important at frequencies
where broadening of resonance δω is smaller than the dis-
tance between two neighbour resonances Δω (see Fig. 6)
< 1.δω
Δω
(2)
Resonance broadening δω can be presented as a sum
of viscous broadening νδω and nonlinear broadening
nlδω ,
= .nlνδω δω + δω (3)
The viscous resonance broadening corresponds to the
viscous damping [20]
2/3
2 4/3= 4 = 4 ,kν ω
ρ⎛ ⎞δω ν ν ω⎜ ⎟σ⎝ ⎠
(4)
and is related to characteristic time of viscous dissipation
ντ as 1= .−
ν νδω τ Nonlinear resonance broadening corres-
ponds to nonlinear transfer of wave energy and is deter-
mined from characteristic time of nonlinear interaction nlτ
as 1= .nl nl
−δω τ We suppose that for the case of harmonic
pumping nonlinear time 1/6
nlτ ω∼ [21] and consequently
1/6.nl
−δω ω∼ With increase of the amplitude of pumping
A the nonlinearity should increase too, therefore it can be
assumed that
1/6( ) ,nl A −δω ≈ ε ω (5)
where ( )Aε is the increasing function of driving ampli-
tude A.
Thus, by substituting (1),(3),(4),(5) in (2), we obtain
that detuning is important at frequencies where
2/3 4/3 1/6
1/3 1/3
4 ( / ) ( ) < 1.
(3 / )( / )
A
D
−ν ρ σ ω + ε ω
π σ ρ ω
(6)
At the high-frequency edge of inertial range bω , where
damping of the turbulent cascade is observed, the nonlinear
wave energy transfer is changed by viscous dissipation.
Therefore, it can be assumed that the characteristic viscous
time is close to the characteristic nonlinear time at bω ,
( ) ( )b nl bντ ω ≈ τ ω . Thus, although function ( )Aε is un-
known, it can be estimated from the condition that at the
edge frequency bω the nonlinear broadening 1=nl nl
−δω τ
and the viscous broadening 1= −
ν νδω τ are equal,
( ) ( ).b nl bνδω ω ≈ δω ω (7)
L.V. Abdurakhimov, M.Yu. Brazhnikov, I.A. Remizov, and A.A. Levchenko
516 Fizika Nizkikh Temperatur, 2011, v. 37, No. 5
It follows from our estimations that, in our experiments,
condition (6) is satisfied in the frequency domain near cut-
off viscous frequency bω (both below and above bω , see
Fig. 7). In this frequency domain the detuning of the har-
monic frequencies and of the resonant frequencies becomes
essential, and it determines the specific shape of turbulent
distribution 2.Pω However, based upon a simple qualitative
reasoning it is impossible to predict the exact shape and po-
sition of the local maximum. Thus, detailed theoretical stu-
dies and numerical simulations are required to understand
the mechanism of formation of the local maximum.
The model proposed implies that the following condi-
tions should be satisfied for observation of energy accumu-
lation in the case of capillary turbulence: low viscous
broadening (low viscosity of liquid), low nonlinear broa-
dening (harmonic pumping with a moderate amplitude),
and large distance between the resonant frequencies (a
relatively small size of the cell). In our experiments these
conditions were fulfilled. It is interesting to note that the
use of superfluid 4He allowed us to observe the energy
accumulation in turbulent cascade due to its very low vis-
cosity in comparison with classical liquids.
We are grateful to L.P. Mezhov-Deglin, G.V. Kolma-
kov, E.A. Kuznetsov for fruitful discussions and A.V.
Lokhov for experimental assistance. The work was sup-
ported by RFBR grants 09-02-01146, 07-02-00728 and the
programs of the Presidium RAS “Quantum Physics of
Condensed Matter” and “Fundamental Problems of Nonli-
near Dynamics”.
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Fig. 7. Results of estimation of relative broadening /δω Δω .
Ascending thin curve represents viscous relative broadening
/νδω Δω , descending thin curve —nonlinear relative broadening
/nlδω Δω , thick curve with minimum — total relative broaden-
ing /δω Δω .
1.5
1
0.5
10 100 1000 10
4
R
el
at
iv
e
b
ro
ad
en
in
g
Frequency, Hz
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