Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions
Formation of low frequency harmonics on turbulent distribution in the system of waves on the surface of liquid hydrogen has been studied in the frequency range 1–100 Hz (capillary-gravity waves). It is shown that the geometry of the experimental cell has a significant influence on the direct casca...
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irk-123456789-1279382018-01-01T03:02:54Z Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions Brazhnikov, M.Yu. Levchenko, A.A. Mezhov-Deglin, L.P. Remizov, I.A. 10th International Conference on Cryocrystals and Quantum Crystals Formation of low frequency harmonics on turbulent distribution in the system of waves on the surface of liquid hydrogen has been studied in the frequency range 1–100 Hz (capillary-gravity waves). It is shown that the geometry of the experimental cell has a significant influence on the direct cascade of capillary waves generated by monochromatic force as well as on the direction of the wave energy transfer from the range of pumping towards that of dissipation. Besides a direct turbulent cascade, single half-frequency harmonic generation was observed in a cylindrical cell under high pump power. In a square cell we observed not only a half-frequency harmonic but a number of low frequency harmonics below the driving frequency generated by the nonlinear threewave interaction. In the case of a rectangular cell we observed formation of incommensurate low frequency harmonics caused by the three-wave interaction of capillary waves and generation of a wave mode of ~1 Hz in the frequency range of gravity waves which could be attributed to the four-wave interaction. 2015 Article Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions / M.Yu. Brazhnikov, A.A. Levchenko, L.P. Mezhov-Deglin, I.A. Remizov // Физика низких температур. — 2015. — Т. 41, № 6. — С. 615-619. — Бібліогр.: 12 назв. — англ. 0132-6414 PACS: 61.30.–v, 47.27.Gs http://dspace.nbuv.gov.ua/handle/123456789/127938 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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English |
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10th International Conference on Cryocrystals and Quantum Crystals 10th International Conference on Cryocrystals and Quantum Crystals |
| spellingShingle |
10th International Conference on Cryocrystals and Quantum Crystals 10th International Conference on Cryocrystals and Quantum Crystals Brazhnikov, M.Yu. Levchenko, A.A. Mezhov-Deglin, L.P. Remizov, I.A. Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions Физика низких температур |
| description |
Formation of low frequency harmonics on turbulent distribution in the system of waves on the surface of liquid
hydrogen has been studied in the frequency range 1–100 Hz (capillary-gravity waves). It is shown that the
geometry of the experimental cell has a significant influence on the direct cascade of capillary waves generated
by monochromatic force as well as on the direction of the wave energy transfer from the range of pumping towards
that of dissipation. Besides a direct turbulent cascade, single half-frequency harmonic generation was observed
in a cylindrical cell under high pump power. In a square cell we observed not only a half-frequency harmonic
but a number of low frequency harmonics below the driving frequency generated by the nonlinear threewave
interaction. In the case of a rectangular cell we observed formation of incommensurate low frequency harmonics
caused by the three-wave interaction of capillary waves and generation of a wave mode of ~1 Hz in the
frequency range of gravity waves which could be attributed to the four-wave interaction. |
| format |
Article |
| author |
Brazhnikov, M.Yu. Levchenko, A.A. Mezhov-Deglin, L.P. Remizov, I.A. |
| author_facet |
Brazhnikov, M.Yu. Levchenko, A.A. Mezhov-Deglin, L.P. Remizov, I.A. |
| author_sort |
Brazhnikov, M.Yu. |
| title |
Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions |
| title_short |
Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions |
| title_full |
Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions |
| title_fullStr |
Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions |
| title_full_unstemmed |
Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions |
| title_sort |
wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2015 |
| topic_facet |
10th International Conference on Cryocrystals and Quantum Crystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/127938 |
| citation_txt |
Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions / M.Yu. Brazhnikov, A.A. Levchenko, L.P. Mezhov-Deglin, I.A. Remizov // Физика низких температур. — 2015. — Т. 41, № 6. — С. 615-619. — Бібліогр.: 12 назв. — англ. |
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Физика низких температур |
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2025-07-09T08:02:50Z |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6, pp. 615–619
Wave turbulence on the surface of liquid hydrogen
in restricted geometry: the influence of the boundary
conditions
M.Yu. Brazhnikov, A.A. Levchenko, L.P. Mezhov-Deglin, and I.A. Remizov
Institute of Solid State Physics RAS, Chernogolovka, Moscow District 142432, Russia
E-mail: levch@issp.ac.ru
Received February 24, 2015, published online April 23, 2015
Formation of low frequency harmonics on turbulent distribution in the system of waves on the surface of liq-
uid hydrogen has been studied in the frequency range 1–100 Hz (capillary-gravity waves). It is shown that the
geometry of the experimental cell has a significant influence on the direct cascade of capillary waves generated
by monochromatic force as well as on the direction of the wave energy transfer from the range of pumping to-
wards that of dissipation. Besides a direct turbulent cascade, single half-frequency harmonic generation was ob-
served in a cylindrical cell under high pump power. In a square cell we observed not only a half-frequency har-
monic but a number of low frequency harmonics below the driving frequency generated by the nonlinear three-
wave interaction. In the case of a rectangular cell we observed formation of incommensurate low frequency har-
monics caused by the three-wave interaction of capillary waves and generation of a wave mode of ~1 Hz in the
frequency range of gravity waves which could be attributed to the four-wave interaction.
PACS: 61.30.–v Liquid crystals;
47.27.Gs Isotropic turbulence; homogeneous turbulence.
Keywords: liquid hydrogen, wave turbulence, low frequency harmonics.
Introduction
The dispersion law for waves on a free surface of liquid
in unrestricted geometry is given by the expression
2 3( ) /k gk kω = + σ ρ (1)
where ω and k are the frequency and the wave vector
length of the surface wave, σ and ρ are the surface tension
and density of liquid, g is the free-fall acceleration. The
first term in expression (1) describes pure gravity waves,
the second term corresponds to capillary waves. For capil-
lary waves the main processes of their nonlinear interac-
tion are the three-wave processes of decay and merging,
and for gravity waves the main process is the four-wave
interaction [1]. All the processes should satisfy the energy
(frequency) and the wave vector conservation laws.
In an experimental cell of finite size the spectrum
of surface waves becomes discrete. The distance between
the two nearest resonances in k-space is determined by
the boundary conditions and in the general case it is in-
versely proportional to cell size D. In ω space the distance
between two resonances ∆Ω for capillary waves increases
as ∆Ω ~ ω1/3 and decreases for gravity waves, ∆Ω ~ ω−1.
Due to the discreteness of the eigenmodes of the surface
oscillations of liquid in a basin of finite size the frequency
and wave vector conservation laws cannot always be satis-
fied, for instance for capillary waves in a rectangular basin
three-waves processes are prohibited [2]. In real systems
strict restrictions are softened due to viscous ωv and non-
linear ωn broadening of the resonances [3,4], and the con-
servation laws for the three-wave interaction could be re-
written in the form
1 2 3 ,ω + ω = ω + δ
1 2 3+ =κ κ κ
where ,n∆Ω > ∆ω = ∆ω + ∆ωv and .δ < ∆ω
In the case of radial waves on the surface of cylindrical
cells we deal with one-dimensional space, the wave vector
κ might be considered as an integer and the spectrum of
capillary waves can be described by the expression
1/2 3/2 3/2( ) (/ 2 4) )/ 3( /D nω = σ ρ π − (2)
where n is the resonance number.
© M.Yu. Brazhnikov, A.A. Levchenko, L.P. Mezhov-Deglin, and I.A. Remizov, 2015
mailto:levch@issp.ac.ru
M.Yu. Brazhnikov, A.A. Levchenko, L.P. Mezhov-Deglin, and I.A. Remizov
When using square or rectangular cells the κ space be-
comes two-dimensional. The resonant frequencies for the
capillary waves in the rectangular cell are described by the
expression:
1/2 2 2 3/4[( / ( ) () / ]/ ) n a m bω = σ ρ π + π (3)
and for the standing gravity waves:
1/2 2 2 1/4/ ]/[( ) ( )g n a m bω = π + π (4)
where a and b are the length and width of the working cell,
n and m are the integer numbers.
In our experiments the excitation of surface waves was
performed in the frequency range from 10 to 100 Hz.
It should be noted that waves with frequencies from 1 to 100
Hz on the surface of liquid hydrogen should be considered
as capillary-gravity ones and the three-wave processes along
with the four-wave ones are not prohibited for them at fre-
quencies lower or higher some critical frequency cf
1/43
0 1/22
.1 gf
ρ
= σπ
(5)
It is clear that the capillary waves correspond to fre-
quencies ,cf f>> and at cf f<< we deal with the gra-
vity waves on the liquid surface. For liquid hydrogen
16 Hzcf = at Т = 15.6 К.
In the turbulent state the energy pumped into the system
by an external source is transferred from the pumping re-
gion to the domain of dissipation owing to the nonlinear
wave interaction. The theory [1] predicted that in a system
of capillary waves in unrestricted geometry only direct
turbulent cascade should be generated and the wave energy
should flow towards the high frequency domain. In a sys-
tem of gravity waves formation of the both direct and
inverse cascades is possible and the wave energy could
be directed into both high and low frequency domains.
From our experiment and subsequent numerical simu-
lations [5–7] it follows that in a finite size cell, where the
spectrum of the wave excitations is discrete, one might
observe an energy flow from the pumping region towards
both high and low frequency domains even for capillary
waves. It should be mentioned that in all the numerical
simulations the authors preferred to use asymptotic approx-
imations, 1/2~ ,ω κ for capillary waves and 3/2~ω κ for
gravity waves. For the case of our experimental conditions
when the capillary-gravity waves prevail in the cells, espe-
cially in the range of 120 Hz, the dispersion law of wave
excitations on the charged hydrogen surface is close to
linear ~ω κ [8], numerical simulations being absent.
In this paper we present the recent results of our inves-
tigations of the wave turbulence on the charged surface of
liquid hydrogen in the cells of three different geometries,
namely, cylindrical, square and rectangular cells of the
same depth and comparable dimensions. The main goal
of this study was to clarify the influence of the boundary
conditions on the turbulent spectra and mechanisms of
wave energy transfer in the system of surface waves in
restricted geometry.
Experimental technique
The experiments were carried out at constant tempera-
ture T = 15 K. Hydrogen gas was condensed into a copper
cup of inner diameter 60 mm and depth 4 mm. Square and
rectangular cells were formed by flat metal inserts into the
original cylindrical cell. The inner dimensions of the
square cell were 40×40 mm, and those of the rectangular
one 40×20 mm. The metal collector (copper plate) sus-
pended at a distance of 4 mm above the experimental cell
and surface of liquid hydrogen formed a flat capacitor. A
source of charges was placed at the bottom of the cup. The
liquid hydrogen surface was charged with positive ions
when dc voltage of positive polarity Udc ≈ 1 kV was ap-
plied between the copper cup and collector. Low-
frequency ac voltage Uac = Up sin(ωt) applied to the cup in
addition to the dc voltage excited waves on the charged
surface of liquid hydrogen. It should be noted that waves
on the charged surface are generated owing to a strong
spatial heterogeneity of the electric field near the edges of
the experimental cell.
The surface waves were detected by means of a laser
beam reflected from the liquid surface and then focused
onto a photodetector. Detailed method of registration of
surface elevations is given in [5]. Any variations of the
angle between the gliding laser beam and the oscillating
surface led to modulation of the power of the reflected
beam. The ac signal from the photodetector was amplified
and digitized by a 24-bit analog-to-digital converter with
sampling frequency 102.4 kHz. The amplitude of digitized
signal P(t) is proportional to power variation of the reflect-
ed laser beam. As it was shown earlier [9], power spectrum
2Pω is proportional to the spectrum of the pair correlation
function of the surface elevation 2 .ω< η > The angular am-
plitudes of the waves in our experiments were close to the
maximum allowed value in the geometry of our experi-
ment which is equal 0.05 rad.
It is necessary to note that the applied perpendicular
electric field was less the critical value at which the flat
charged surface of liquid loses its stability. Moreover, the
perpendicular electric field modifies the dispersion law of
the surface waves: in our experiments the fundamental fre-
quency is shifted to low frequencies compared to Eq. (1).
Results and discussion
Cylindrical cell
The power spectra of oscillations 2Pω on the surface of
liquid hydrogen in the cylindrical cup presented in Figs. 1
and 2 show changes in the turbulent distribution 0.3 s and
616 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions
32 s after switching on monochromatic driving force (pump)
at the frequency of the fifteenth resonance: /2 p pf = ω π =
= 58.6 Hz. The power spectrum 2Pω was obtained by pro-
cessing the P(t) dependence recorded over a time window
of duration ∆t = 0.3 s. In Fig. 1 the main peak at driving
frequency pω dominates in distribution 2 ,Pω and the di-
rect turbulent cascade extends up to 10 kHz. At frequen-
cies above 2 kHz distribution 2Pω displays power-law de-
pendence 2 2Pω ω with exponent n close to – 2.5. This
value of exponent n is closer to the value n predicted by
the theory [1] for the case of broadband pumping. At fre-
quencies close to and above 10 kHz we observed a high
frequency edge of the inertial interval.
Within 32 s the turbulent cascade shown changed sig-
nificantly. In Fig. 3 one can see the appearance of a low
frequency subharmonic at half driving frequency ω0/2 and
additional combination harmonics like ωp + ωp/2, etc. in
the direct cascade. The three peaks at frequencies ωp/2, ωp
and ωp + ωp/2, which dominate in the low frequency do-
main play the role of a wide pumping range, although the
surface is excited by an electric force at single frequency
ωp. The turbulent distribution over the frequency range
from 100 Hz to 10 kHz can be described by a power law
function with an exponent close to – 3. It is seen that the
dissipative domain of the turbulent distribution can be
well described by an exponential dependence of the fre-
quency with the characteristic frequency of about 10 kHz,
close to the high frequency edge of the inertial range
[10,11]. The time of appearance of the subharmonic at
half-frequency harmonic fs = f0/2 in distribution 2Pω de-
pends on the amplitude of the driving force and varies
from several seconds to several minutes after switching on
the excitation.
Square cell
Figure 3 shows turbulent distribution 2Pω in the system
of waves on the surface of liquid hydrogen excited by har-
monic force at frequency 42 Hz marked by the arrow. At
low frequencies there is a half-frequency peak at frequency
ω0/2 = 21 Hz with an amplitude much higher than that at the
frequency of excitation ω0. The frequency dependence of the
turbulent spectrum is very close to ω–3 within the frequency
range from 100 Hz to 2 kHz. The dissipation interval is ob-
served at frequencies higher than 2 kHz.
When the frequency of excitation was decreased to
41.8 Hz (Fig. 4) at the same excitation amplitude dramatic
events occurred in the turbulent distribution. In addition
to the half-frequency harmonic we observed two other peaks
at high amplitude frequencies of 30.6, 20.9, 11.4, 9.7. In
the low frequency domain there are two high amplitude
peaks at frequencies 1.7 Hz and 3.5 Hz. Near the peaks we
can also see numerous satellite harmonics. We suggest that
the wave at frequency fp = ωp/2π = 41.8 Hz decays into two
peaks with frequencies 11.4 Hz and 30.6 Hz. And the wave
at frequency 11.4 Hz decays into two waves with frequen-
cies 1.7 Hz and 9.7 Hz. The remaining low frequency har-
monics are the result of the nonlinear wave interaction:
41.8 Hz ⇒11.4 Hz + 30.6 Hz
11.4 Hz ⇒ 1.7 Hz + 9.7 Hz
Fig. 1. Turbulent cascade on the surface of liquid hydrogen in
the cylindrical cup 0.45 s after switching on pumping at fre-
quency 58.6 Hz.
Fig. 2. Turbulent cascade on the surface of liquid hydrogen in
the cylindrical cup 32 s after switching on pumping at frequen-
cy f0 = 58.6 Hz.
Fig. 3. Turbulent cascade on the surface of liquid hydrogen in the
square cell. Excitation frequency fp = ω0/2π = 42 Hz.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 617
M.Yu. Brazhnikov, A.A. Levchenko, L.P. Mezhov-Deglin, and I.A. Remizov
Rectangular cell
In the rectangular cell a developed turbulent cascade is
easily excited well under pumping in the frequency range
10–40 Hz. In Fig. 5 we present distribution 2Pω when the
surface of the liquid in the rectangular cell was excited by
a harmonic force at frequency fp = ωp/2π = 28.0 Hz, as
indicated by the arrow. The half-frequency peak at fre-
quency fp/2 = 14 Hz is well seen. Note that its amplitude is
much higher than that of the peak at the pumping fre-
quency. The origin of this harmonic is the decay of the
main peak into two,
28 Hz ⇒14 Hz + 14 Hz.
As the frequency of pumping is increased by 0.5 Hz,
the turbulent cascade is significantly modified (Fig. 6). We
should point out several observations. First, the main peak
decays into two incommensurate harmonics:
28.5 Hz ⇒ 9.9 Hz + 18.7 Hz. (6)
Second, in the low frequency range there is a peak with a
frequency equal to 1.1 Hz. The wave length of this mode is
exactly equal to the first resonant mode of the surface os-
cillations along the long side of the experimental cell. This
peak should be attributed to the four-wave interaction like
9.9 Hz + 9.9 Hz ⇒ 18.7 Hz +1.1 Hz. (7)
Third, the lowest subharmonic itself generates a series of
other harmonics with frequency multiples of 1.1 Hz. The
peaks at frequencies from 9.9 Hz and higher are surround-
ed by satellites which result from the interaction of the
harmonics with lowest frequency mode ω = 1.1 Ηz. In
principle, we can state that we have observed transport of
wave energy from the range of excitation to the lowest
mode of the experimental cell (resonator) as a result of the
four-wave interaction.
In Fig. 7 we show the time dependence of the amplitude
of Fourier harmonics in the turbulent cascade observed
after stepwise lowering of the frequency of the exciting
force from 29.3 to 28.5 Hz. During the first 75 s the sur-
face was excited by a monochromatic force at frequency
f = 29.3 Hz. And at the 75th s the frequency of excitation
was lowered stepwise to 28.5 Hz without changing the
amplitude of the excitation.
After changing the frequency we observed the decay of
the amplitudes of the harmonics at frequencies 29.3 Hz and
1.1 Hz with characteristic time about 1 s. This time is very
close to that of viscous relaxation of capillary waves τν at
frequency 29.3 Hz [12]. Note that the low frequency mode
Fig. 4. Turbulent cascade on the surface of liquid hydrogen in the
square cell. Excitation at frequency fp = ω0/2π = 41.8 Hz.
Fig. 5. Turbulent cascade on the surface of liquid hydrogen in the
rectangular cell. Excitation at frequency fp = 28 Hz.
Fig. 6. Turbulent cascade on the surface of liquid hydrogen in the
rectangular cell. Excitation at frequency fp = 28.5 Hz.
Fig. 7. Time dependence of amplitude of Fourier harmonics gen-
erated at the surface of liquid hydrogen in the rectangular cell
after changing the frequency of the exciting force from 29.3 Hz
to fp = 28.5 Hz: 28.5 Hz (1), 18.7 Hz (2), 9.9 Hz (3), 1.1 Hz (4).
618 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
Wave turbulence on the surface of liquid hydrogen in restricted geometry: the influence of the boundary conditions
was excited exactly at the moment of changing frequency.
These peaks disappeared completely by 100 s. However,
the low frequency mode appears again at 140th s 65 s after
changing the frequency of the pumping force. Formation of
oscillations at frequencies 18.7 Hz and 9.9 Hz starts imme-
diately after changing the frequency of the exciting force. It
is seen that characteristic time τ of the three harmonic
growth is of the order of several seconds and larger than the
time of decay of the main peak. The low frequency peak
starts growing when the amplitude of the harmonics at fre-
quencies 9.9 Hz and 18.7 Hz reach appreciable values. This
experimental fact confirms our assumption that the lowest
peak is the result of the four-wave interaction (7).
Conclusion
From the comparison of results of our measurements
obtained at the cells of different geometry it follows that
the boundary conditions (shape of the cell) affect signifi-
cantly the spectra of the wave turbulence on the surface of
liquid hydrogen and the direction of the wave energy flow
from the range of excitation towards the dissipative do-
mains. At a high excitation level we could observe appear-
ance of a half-frequency subharmonic in the cylindrical
cell, while in the case of the square or rectangular cells
under similar conditions we observed generation of a num-
ber of incommensurate subharmonics down to ~1 Hz in the
low frequency region. The wave energy is transferred from
the region of excitation towards low frequencies only by
the three-wave interaction in the case of cylindrical and
square cells or by the three-wave and four-wave processes
in the case of a rectangular cell. Since the waves in the
frequency range of 1–20 Hz correspond to a capillary-
gravity region, further numerical simulations in this fre-
quency range taking into account the real dispersion law
for the surface waves in our cells are required.
This work was supported by the Russian Science Foun-
dation, Grant No.14-22-00259.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 619
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http://jetpletters.ac.ru/ps/1968/article_29771.shtml
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http://jetpletters.ac.ru/ps/1968/article_29771.shtml
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http://jetpletters.ac.ru/ps/1968/article_29771.shtml
http://jetpletters.ac.ru/ps/1968/article_29771.shtml
http://jetpletters.ac.ru/ps/1968/article_29771.shtml
http://jetpletters.ac.ru/ps/1968/article_29771.shtml
Introduction
Experimental technique
Results and discussion
Cylindrical cell
Square cell
Rectangular cell
Conclusion
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