Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen
We report on the experimental observation of energy accumulation near the high frequency boundary of the inertial range in the spectrum of turbulence in a system of capillary waves on the surface of liquid hydrogen driven by a harmonic force. The effect is manifested as a local maximum in the spectr...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2016
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irk-123456789-1293312018-01-19T03:04:19Z Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen Remizov, I.A. Brazhnikov, M.Yu. Levchenko, A.A. Квантовые жидкости и квантовые кpисталлы We report on the experimental observation of energy accumulation near the high frequency boundary of the inertial range in the spectrum of turbulence in a system of capillary waves on the surface of liquid hydrogen driven by a harmonic force. The effect is manifested as a local maximum in the spectrum of pair correlation function of the surface elevation. This phenomenon is dynamical and can be seen only during reconfiguration of the turbulent cascade caused by waves generation of below the driving frequency. 2016 Article Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen / I.A. Remizov, M.Yu. Brazhnikov, A.A. Levchenko // Физика низких температур. — 2003. — Т. 42, № 12. — С. 1363-1367. — Бібліогр.: 11 назв. — агл. 0132-6414 PACS: 47.27.Gs http://dspace.nbuv.gov.ua/handle/123456789/129331 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Remizov, I.A. Brazhnikov, M.Yu. Levchenko, A.A. Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen Физика низких температур |
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We report on the experimental observation of energy accumulation near the high frequency boundary of the inertial range in the spectrum of turbulence in a system of capillary waves on the surface of liquid hydrogen driven by a harmonic force. The effect is manifested as a local maximum in the spectrum of pair correlation function of the surface elevation. This phenomenon is dynamical and can be seen only during reconfiguration of the turbulent cascade caused by waves generation of below the driving frequency. |
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Article |
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Remizov, I.A. Brazhnikov, M.Yu. Levchenko, A.A. |
| author_facet |
Remizov, I.A. Brazhnikov, M.Yu. Levchenko, A.A. |
| author_sort |
Remizov, I.A. |
| title |
Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen |
| title_short |
Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen |
| title_full |
Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen |
| title_fullStr |
Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen |
| title_full_unstemmed |
Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen |
| title_sort |
observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2016 |
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Квантовые жидкости и квантовые кpисталлы |
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http://dspace.nbuv.gov.ua/handle/123456789/129331 |
| citation_txt |
Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen / I.A. Remizov, M.Yu. Brazhnikov, A.A. Levchenko // Физика низких температур. — 2003. — Т. 42, № 12. — С. 1363-1367. — Бібліогр.: 11 назв. — агл. |
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Физика низких температур |
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2025-07-09T11:09:03Z |
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2025-07-09T11:09:03Z |
| _version_ |
1837167384304025600 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12, pp. 1363–1367
Observation of dynamic maximum in a turbulent cascade
on the surface of liquid hydrogen
I.A. Remizov, M.Yu. Brazhnikov, and A.A. Levchenko
Institute of Solid State Physics RAS, Chernogolovka, Moscow district. 142432, Russia
E-mail: levch@issp.ac.ru
Received May 17, 2016, published online October 24, 2016
We report on the experimental observation of energy accumulation near the high frequency boundary of the
inertial range in the spectrum of turbulence in a system of capillary waves on the surface of liquid hydrogen
driven by a harmonic force. The effect is manifested as a local maximum in the spectrum of pair correlation
function of the surface elevation. This phenomenon is dynamical and can be seen only during reconfiguration of
the turbulent cascade caused by waves generation of below the driving frequency.
PACS: 47.27.Gs Isotropic turbulence; homogeneous turbulence.
Keywords: low temperatures, liquid hydrogen, nonlinear waves, turbulence, surface waves.
1. Introduction
A wide set of nonlinear wave systems can be described
in the frame of the theory of weak wave turbulence [1].
Among them are capillary and gravity waves on the sur-
face of water, Rossby waves in the atmosphere and oceans
of planets, Langmuir waves in plasma, and spin waves in
magnetics. A relatively low viscosity of liquid hydrogen
and the possibility to excite waves on the charged surface
of cryogenic liquids by electrical force offer a unique op-
portunity for experimental studies of wave turbulence. The
use of liquid hydrogen for experiments on wave turbulence
has already allowed us to study phenomena predicted by
the theory, e.g., Kolmogorov–Zakharov steady state spec-
tra of capillary turbulence in a wide range of frequencies,
as well to observe new ones which have been explained
successfully in the framework of the weak turbulence ap-
proximation: quasi-adiabatic decay of capillary turbulence
and suppression of high-frequency turbulent oscillations by
additional low-frequency driving force [2].
The dispersion of waves on the surface of liquid is giv-
en by:
2 3( ) ( / )k gk kω = + σ ρ , (1)
where ω and k are the frequency and the wave vector of a
surface wave, correspondingly, σ and ρ are the surface
tension and the density of liquid, and g is the free-fall ac-
celeration. The first term in (1) prevails in the long wave
region and describes gravity waves, the second one corre-
sponds to capillary waves (ripples).
The main processes of the nonlinear interaction for ca-
pillary waves are three-wave processes of decay and merg-
ing that satisfy the conservation laws of frequency and
wave vector:
1 2 3 1 2 3, ω = ω + ω = +k k k . (2)
Due to discreteness of the eigenmodes of the surface
oscillation of liquid in a finite basin these relations cannot
be always satisfied, e.g. for capillary waves in a square
basin three-wave processes are prohibited: the set of equa-
tions has no mathematical solution [3]. In real systems
these resonant conditions are softened due to the viscous
and nonlinear broadening of the resonances.
Neglecting viscous and nonlinear effects a radially
symmetric surface wave in a cylindrical basin is described
by the Bessel function. The boundary conditions on the
resonator wall define the eigenvalues of wavenumber k:
J1(kD/2) = 0, where D is the resonator diameter, J1(x) is
the Bessel function of the first order. Using the asymptotic
form of the Bessel function for large k we can find that the
resonance wavenumbers are approximately equidistant
with spacing ∆k ∼ 2π/D.
The pair correlation function of the surface deviation from
an equilibrium state in the Fourier representation Iω in the
inertial range is described by the power function of frequency
~ .I −β
ω ω For the case of broadband high-frequency excita-
tion of the surface with spectrum (1) the value of the index β
equals 17/6 [1], and narrowband excitation β equals 23/6 [4].
© I.A. Remizov, M.Yu. Brazhnikov, and A.A. Levchenko, 2016
I.A. Remizov, M.Yu. Brazhnikov, and A.A. Levchenko
A nonlinear wave interaction leads to energy redistribu-
tion from a low frequency domain, where the energy is
injected by an external force, to higher frequencies, where
the energy is dissipated due to viscous losses. The position
of high-frequency boundary of the inertial range bω can be
found assuming that a nonlinear energy flux is comparable
with the amount of energy leaving the cascade because of
viscous damping. In the case of continuous spectrum [5]:
( )6/52 17/6
0 0 / ,bω ∼ η ω ν
where 0η is the wave amplitude at the pumping frequency
0.ω The estimation shows that the high-frequency bounda-
ry ωb for liquid hydrogen at 15 K and water at normal con-
ditions are related as 5:1 under the same condition of exci-
tation. From this point of view liquid hydrogen is a
preferable fluid to study the capillary turbulence.
Turbulent distribution may deviate from the power de-
pendence. One of the possible reasons for the deviation can
be associated with a discrete spectrum of surface excita-
tions instead of continuous spectrum (1) [6], which leads to
the energy accumulation on several resonance modes near
the edge of the inertial interval [7].
The frequency distance between the nearest resonant
modes for capillary waves in a cylindrical basin at high
frequencies is given by:
( ) ( )1/3 1/33 / /D∆Ω = π ⋅ σ ρ ω . (3)
Strictly speaking, in the ideal system of capillary waves the
laws of energy and momentum conservation cannot be
satisfied simultaneously. This restriction is removed taking
into account the broadening of the resonance peaks due to
viscous losses and non-linear interaction of waves. At high
frequencies the broadening due to viscosity becomes com-
parable with the distance between the resonance modes
∆Ω at the frequency
(3 /4 )( / ).c Dω ∼ π σ νρ
For example, in our experiments D = 60 mm, the value
/2cω π for liquid hydrogen at the temperature of 15 K, and
for water under normal conditions equals 4 kHz, and
3 kHz, respectively.
Another reason for deviation of the turbulent distribu-
tion Iω from the power function may be connected with an
insufficiently effective absorption of energy within the
dissipation domain due to viscosity. This leads to an in-
crease of the harmonic amplitudes (accumulation of ener-
gy) near the high-frequency edge of the inertial interval to
keep an energy flux propagated on turbulent cascade con-
stant [8].
In the experiments presented here we were able to ob-
serve the accumulation of energy near the high-frequency
edge of the inertial interval in the time-dependent (dynam-
ic) conditions.
2. Experimental technique
The experimental method of wave registration on the
surface of a cryogenic liquid was described in [9]. The
investigations were carried out at the liquid hydrogen tem-
perature T = 15 K. Hydrogen gas was condensed into cop-
per cup of the inner diameter 60 mm and 4 mm deep. The
cup and a copper plate which was fixed 4 mm above the
cup form a flat capacitor. The liquid was ionized with a
source of charges placed on the bottom of the cup. The
liquid surface is charged with positive ions extracted from
the bulk of liquid. dc voltage of about 1 kV is applied be-
tween the capacitor plates.
A low-frequency ac voltage applied in addition to the
dc voltage excites waves on the charged liquid surface.
Waves are detected by means of a laser beam reflected
from the liquid surface and then focused into a
photodetector. Variation of the angle between the laser
beam and the oscillating surface leads to the modulation of
the reflected light power. The ac signal from the
photodetector is amplified and digitized by a 24-bit analog-
to-digital converter with a sampling frequency of
102.4 kHz. The digitized signal ( )P t is proportional to the
variation of power of the reflected laser beam. As we
showed earlier [9] the power spectrum 2Pω is proportional
to the spectrum of the pair correlation function of the sur-
face elevation 2 .ω< η >
3. Results
Oscillations on the surface of liquid hydrogen were ex-
cited at the frequency of the fifteenth resonance in the ex-
perimental cylindrical cell, 0 ω = 58.6 Hz. Figure 1 exhibits
the power spectrum 2Pω , obtained by processing ( )P t rec-
orded in a time window of ∆t = 0.3 s. duration in two se-
conds after switching on the driving force. A subharmonic
at half frequency of the driving force can be seen apart
from the main peak at the driving frequency and a turbu-
lent cascade, which extents to 10 kHz. At frequencies
Fig. 1. Spectrum in two seconds after switching on the driving
voltage at frequency of 58.6 Hz. Straight line corresponds to the
power-law dependence 2.5.−ω
1364 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 121
Observation of dynamic maximum in a turbulent cascade on the surface of liquid hydrogen
above 200 Hz the distribution 2Pω displays the power-law
dependence with the index β ≈ 2.5, which is close to the
theoretical prediction for the case of broadband pumping.
The time of the subharmonic appearance in the distribu-
tion 2Pω depends on the driving force amplitude and var-
ies from a few seconds to several minutes after switching
on the excitation.
A peak at the frequency of 0 /2ω on the turbulent cas-
cade after 5 seconds of evolution from the moment of
switching on the pumping is clearly visible in Fig. 2. The
wave amplitude at the frequency 0 /2ω has not reached
its maximum value yet, but it has already started affecting
the turbulent distribution: peaks which are multiples of
the half-harmonic frequency 0 /2ω start to emerge in the
frequency range from 100 to 1000 Hz. The height of the
peaks in this frequency range can be described by the
power function mω with an exponent close to – 4.1.
Above 1 kHz the height of the peaks in the spectrum 2Pω
decreases more slowly, in the frequency range from 1 to
10 kHz it is described well by the power function with an
exponent of – 2.5.
A local maximum on the distribution 2Pω is clearly seen
in the frequency domain from 10 to 20 kHz. Its amplitude
is several times larger than the height of neighboring har-
monics. At frequencies above 14 kHz the turbulent cascade
decays due to viscous losses. Note that during the evolu-
tion of the turbulent cascade the local maximum remains at
the same position on the frequency scale.
Figure 3 presents the distribution 2Pω formed on the sur-
face in 30 seconds after switching on the excitation. Three
peaks at frequencies 0 /2ω , ω0 and 0 0 /2,ω + ω which dom-
inate in the low-frequency domain play the role of pump-
ing range, although the surface excitation by the electric
force still goes on at a single frequency 0ω . The turbulent
distribution over the frequency range from 100 Hz to
20 kHz can be described by the power law function with an
exponent close to – 2.8. It is seen that at high frequencies
the local extremum disappears, and the inertial interval
spreads to 20 kHz. The dissipative domain on the turbulent
distribution is not seen.
The time evolution of the peaks at the driving force fre-
quency ω0, at the frequency 0 /2ω and at frequency
0 0 /2ω + ω is presented in Fig. 4. The wave amplitude at
the pumping frequency decreases by three times within the
first 9 seconds after switching on the excitation. However,
after 15 seconds of evolution its amplitude exceeds initial
magnitude almost by 1.5 times.
The same time intervals can be distinguished for the
subharmonic at 0 /2ω . While its amplitude increases in
both intervals, it grows much faster during the second time
interval. The wave at frequency 0 0 /2,ω + ω appears simul-
taneously with the subharmonic and its amplitude gradual-
ly increases all 15 s from the beginning, and then remains
almost constant. The vertical lines in Fig. 4 denote the time
domain during which the local maximum on the turbulent
distribution at high frequencies is observed. It can be em-
phasised that the local maximum exists during the first
time interval when the harmonic amplitude at the driving
frequency decreases, while the amplitude of the
subharmonic increases.
Fig. 2. Spectrum 2Pω in five seconds after switching on the driv-
ing voltage. The solid lines corresponds to power-law functions
4.1−ω and 2.5.−ω
Fig. 3. Spectrum 2Pω in thirty seconds after switching on the
excitation. The solid line corresponds to power-law function
2.8.−ω
Fig. 4. Time dependence of amplitudes of the first three harmon-
ics in the distribution of 2Pω : 0 /2ω (curve 1), 0ω (curve 2),
0 0 /2,ω + ω (curve 3), 0 /2ω π = 58.6 Hz. Vertical lines mark the
time interval when the local maximum on the turbulent cascade
can be identified.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1365
I.A. Remizov, M.Yu. Brazhnikov, and A.A. Levchenko
Figure 5 shows two cases of time dependence of the peak
amplitude at frequencies 15 and 23 kHz after switching on
pumping. The peak at 15 kHz is in the middle of the dissipa-
tion domain and the peak at the frequency of 23 kHz — be-
yond the dissipation area. It is seen that the peak amplitude
at the frequency of 15 kHz after switching on the excita-
tion increases almost by three times for the first 5 seconds,
then decreases and reaches a steady state level at the 10th
second. The amplitude of the peak at the frequency of 23 kHz
is practically independent of time, that is, the amplitude is
not sensitive to switching on the excitation force.
4. Discussion
All spectra of capillary turbulence presented in this pa-
per were obtained under excitation of the surface oscilla-
tions by an external harmonic force. Although the spec-
trum of the eigenfrequencies of the surface oscillations in a
finite basin is of discrete nature, it can be considered quasi-
continuous above several kilohertz due to viscous broaden-
ing of the eigenmodes. When the surface oscillation is ex-
cited by a monochromatic pumping, a turbulent cascade is
formed which consists of a multiple harmonic of a pump-
ing frequency and extends for more than two decades. The
appearance of the first subharmonic mode at half the driv-
ing frequency leads to a reconstruction of the cascade and
is accompanied by the formation of a local maximum in
the power spectrum near the high frequency boundary of
the inertial range. However, the maximum manifests itself
for a period of several seconds, and disappears completely
before the turbulent cascade reaches its new steady state
distribution. The time of local maximum existence coin-
cides with the time of the decrease of the main harmonic
amplitude at the driving frequency and growth of the
subharmonic and combination harmonics (Fig. 4).
The observed phenomenon cannot be explained as re-
lated to the formation of a bottleneck preventing the energy
transfer towards high frequencies due to the detuning ef-
fect in discrete systems as in the case of liquid helium ex-
periment [7], because the eigenmode spectrum of the sur-
face oscillations is quasi-continuous above 4 kHz. The
influence of discreetness on the turbulent distribution in
the system of waves on the liquid hydrogen surface was
studied before in [10,11].
The formation of the local maximum on the turbulent
cascade near the high frequency boundary may be attribut-
ed to the bottleneck effect caused by the viscous damping
in a high frequency domain. As was shown in [8] the fi-
niteness of the dissipation scale leads to an increase of the
wave amplitudes in the inertial range. The energy accumu-
lation near the high frequency edge of the inertial range is
caused by a reduction of the energy flux due to low occu-
pation numbers (waves amplitudes) in the dissipation do-
main. Though the original bottleneck effect [9] was de-
scribed for steady state spectra of turbulence, the same
rationale can be extended to our dynamic case. Note that
the local maximum is observed during an intensive energy
loss of the main harmonic (Fig. 5), the main harmonic re-
leases about 90% of its initial energy during the first 5 se-
conds. Under a permanent pumping by the external force
this energy loss cannot be related to the viscous dissipa-
tion, but to the nonlinear energy transfer to the growing
subharmonic and multiple harmonics. The viscous damp-
ing (Eq. 4) of waves within the inertial range is not suffi-
cient to dissipate the released energy, and the energy is
transferred to higher frequencies by the nonlinear wave
interaction towards the dissipation domain, where the
mechanism of the bottleneck begins to act. The delay be-
tween the growth of the subharmonic at half of the driving
frequency and the appearance of the local maximum at
high frequencies enables the estimation of the velocity of
local disturbance propagation over the turbulent cascade as
/ ~ / ~ d dt tω ω 104 Hz/s.
5. Conclusion
We experimentally observed the accumulation of ener-
gy near the high frequency boundary of the inertial range
on the spectrum of turbulence because of reduction of en-
ergy flux in dissipation domain — the bottleneck effect.
Authors are grateful to G.V. Kolmakov and L.P.
Mezhov-Deglin for useful discussions. This work was sup-
ported in part by the program project of the Presidium of
the Russian Academy of Sciences «Modern problems of
low temperature physics».
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1366 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 121
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http://dx.doi.org/10.1016/S0167-2789(99)00069-X
http://dx.doi.org/10.1134/S0021364010060032
http://dx.doi.org/10.1023/A:1021418819539
http://dx.doi.org/10.1023/A:1021418819539
http://dx.doi.org/10.1063/1.4922104
http://dx.doi.org/10.1063/1.4922104
http://dx.doi.org/10.1063/1.4915913
http://dx.doi.org/10.1063/1.4915913
1. Introduction
2. Experimental technique
3. Results
4. Discussion
5. Conclusion
|