Dynamics of three wave stochastic decays in nonlinear matter
In the present work the transition of decay instability into stochastic regime in a nonlinear media is analyzed for a magnetoactive plasma and for the ferrite magnetized to the saturation level. It was shown that parameter of nonlinear interaction can be anomalously strong near the resonance frequ...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1154382017-04-05T03:02:15Z Dynamics of three wave stochastic decays in nonlinear matter Buts, V.A. Kovalchuk, I.K. Plasma electronics In the present work the transition of decay instability into stochastic regime in a nonlinear media is analyzed for a magnetoactive plasma and for the ferrite magnetized to the saturation level. It was shown that parameter of nonlinear interaction can be anomalously strong near the resonance frequencies: electron and ion cyclotron frequencies in the magnetoactive plasma and ferromagnetic frequency in ferrite. It was shown that the threshold value of decaying wave amplitude, when transition into stochastic regime takes place, is reducing when frequency of the wave with the lowest frequency is decreasing. Анализируется переход распадной неустойчивости в стохастический режим в нелинейных средах: магнитоактивной плазме и намагниченном до насыщения феррите. Показано, что параметр нелинейного взаимодействия может быть аномально большим вблизи резонансных частот: электронной и ионной циклотронной в магнитоактивной плазме и частоты ферримагнитного резонанса в феррите. Отмечено, что с уменьшением частоты самой низкочастотной волны, участвующей в распаде, уменьшается пороговое значение амплитуды распадающейся волны, при котором происходит переход в стохастический режим. Аналізується перехід розпадної нестійкості в стохастичний режим у нелінійних середовищах: магнітоактивній плазмі і намагніченому до насичення фериті. Показано, що параметр нелінійної взаємодії може бути занадто великим поблизу резонансних частот: електронної та іонної циклотронної в магнітоактивній плазмі та частоти феромагнітного резонансу у фериті. Відзначено, що зі зменшенням частоти самої низькочастотної хвилі, що бере участь у розпаді, зменшується порогове значення амплітуди хвилі, що розпадається, при якому відбувається перехід у стохастичний режим. 2016 Article Dynamics of three wave stochastic decays in nonlinear matter / V.A. Buts, I.K. Kovalchuk // Вопросы атомной науки и техники. — 2016. — № 6. — С. 156-159. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/115438 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| language |
English |
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Plasma electronics Plasma electronics |
| spellingShingle |
Plasma electronics Plasma electronics Buts, V.A. Kovalchuk, I.K. Dynamics of three wave stochastic decays in nonlinear matter Вопросы атомной науки и техники |
| description |
In the present work the transition of decay instability into stochastic regime in a nonlinear media is analyzed for a
magnetoactive plasma and for the ferrite magnetized to the saturation level. It was shown that parameter of
nonlinear interaction can be anomalously strong near the resonance frequencies: electron and ion cyclotron
frequencies in the magnetoactive plasma and ferromagnetic frequency in ferrite. It was shown that the threshold
value of decaying wave amplitude, when transition into stochastic regime takes place, is reducing when frequency of
the wave with the lowest frequency is decreasing. |
| format |
Article |
| author |
Buts, V.A. Kovalchuk, I.K. |
| author_facet |
Buts, V.A. Kovalchuk, I.K. |
| author_sort |
Buts, V.A. |
| title |
Dynamics of three wave stochastic decays in nonlinear matter |
| title_short |
Dynamics of three wave stochastic decays in nonlinear matter |
| title_full |
Dynamics of three wave stochastic decays in nonlinear matter |
| title_fullStr |
Dynamics of three wave stochastic decays in nonlinear matter |
| title_full_unstemmed |
Dynamics of three wave stochastic decays in nonlinear matter |
| title_sort |
dynamics of three wave stochastic decays in nonlinear matter |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2016 |
| topic_facet |
Plasma electronics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/115438 |
| citation_txt |
Dynamics of three wave stochastic decays in nonlinear matter / V.A. Buts, I.K. Kovalchuk // Вопросы атомной науки и техники. — 2016. — № 6. — С. 156-159. — Бібліогр.: 11 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT butsva dynamicsofthreewavestochasticdecaysinnonlinearmatter AT kovalchukik dynamicsofthreewavestochasticdecaysinnonlinearmatter |
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2025-07-08T08:46:45Z |
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2025-07-08T08:46:45Z |
| _version_ |
1837067833157091328 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2016. №6(106)
156 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2016, № 6. Series: Plasma Physics (22), p. 156-159.
DYNAMICS OF THREE WAVE STOCHASTIC DECAYS IN NONLINEAR
MATTER
V.A. Buts
1,2
, I.K. Kovalchuk
1
1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2
V.N. Karazin Kharkiv National University, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua, kovalchuk-ik@rambler.ru
In the present work the transition of decay instability into stochastic regime in a nonlinear media is analyzed for a
magnetoactive plasma and for the ferrite magnetized to the saturation level. It was shown that parameter of
nonlinear interaction can be anomalously strong near the resonance frequencies: electron and ion cyclotron
frequencies in the magnetoactive plasma and ferromagnetic frequency in ferrite. It was shown that the threshold
value of decaying wave amplitude, when transition into stochastic regime takes place, is reducing when frequency of
the wave with the lowest frequency is decreasing.
PACS: 52.35.Mw
INTRODUCTION
A weakly nonlinear approach was proposed quite
long time ago for investigation of electromagnetic
phenomena in plasma. With the use of this approach
some important physical results were obtained (e.g.,
[1-3]). The simplest examples of weakly nonlinear
interaction are three-wave processes. Among these are
the decay and explosive instabilities. A well known
regime of decay instability is characterized by
periodical energy transition between three natural
modes of electrodynamic system with frequency and
wave number satisfying the synchronism conditions.
Such regime is regular. However, our investigations
showed that conditions can be realized when the decay
process is chaotic [4]. This is possible in the case of
modified decay. The presence of additional fourth wave
is needed in similar case, with its characteristics close to
the characteristics of one of the waves taking part in the
decay. In such a set of waves there is a possibility to
distinguish two triples of waves interacting between
themselves. Every such triple is described by equation
of mathematical pendulum which corresponds to the
nonlinear resonance. If the resonances corresponding to
different triples are overlapping, the transition to
stochastic regime takes place. The overlapping is
possible if the increment of decay instability exceeds the
frequency shift between third and forth waves. The
mathematical criterion of transition to chaos will be
presented in the second part.
Theses regimes allow to transform the regular
oscillations into chaotic oscillations. The theoretical
conclusions are confirmed by results of numerical and
experimental studies [5, 6]. In previous investigations
presented, in particular, in [4-7], decays of
electromagnetic wave into electromagnetic and plasma
ones were considered. Meanwhile, it is interesting to
investigate also the decay process of electromagnetic
wave into new electromagnetic wave and plasma wave
where plasma ions are taking part. Excitation of such
oscillations can be an effective way to heat plasma ions.
In this work the analysis of results of decay process
of high frequency waves into new high frequency waves
and the low frequency ones with the frequency close to
the ion cyclotron wave in magnetoactive plasma is
presented.
The possibility of decay processes in other nonlinear
matter, namely, in a ferrite magnetized to saturation was
also investigated. It was found out that decay processes
in ferrites are analogous to those inherent in plasma,
when the interaction of electromagnetic and
magnetostatic waves are considered.
1. IN MAGNETOACTIVE PLASMA
In the above mentioned works [4-7] mostly simple
case of stochastic decay was theoretically considered,
namely processes that are realized in isotropic plasma.
The analysis shows that their realization in a
nonisotropic (namely in giyrotropic) medium may be
more complicated. The examples of such media are
magnetoactive plasma and ferrites. The feature of
electromagnetic system filled with similar media is the
coupling of E- and H-waves. As a result, the linear
transfer between these components may cause the decay
process disrupted. In particular just such case was
considered in [8]. In the gyrotropic medium the
equations for slowly varying amplitudes are essentially
complicated. They include lot of nonlinear terms and
accounting of each of them requires an individual
separated analysis. In this work we will not cite
completely these equations but limit ourselves by their
characteristic features. Namely, we will take interest in
the region of low frequency range, of the order of an ion
cyclotron frequency.
To investigate decay processes in magnetoactive
plasma the Maxwell equations for electromagnetic field
components are used and hydrodynamics equations are
used for electrons and ions of plasma. It is supposed that
electrodynamic system is placed into an external
stationary uniform magnetic field directed along z axis.
The consideration will be limited by a simple case,
when magnetoactive plasma is spatially unlimited. All
components of the electromagnetic field are presenting
as propagating harmonic waves, which amplitudes are
slowly varying along z direction:
, , ( )expi i i iE H E H r i t ik r , (1)
ik – wave vector. We suppose that interacting waves
propagate under arbitrary angle to the external magnetic
field.
mailto:vbuts@kipt.kharkov.ua
mailto:kovalchuk-ik@rambler.ru
ISSN 1562-6016. ВАНТ. 2016. №6(106) 157
By the use of Maxwell equations the following
equations for longitudinal components of electric and
magnetic fields ,zj zjE H can be obtained:
2 2 2 2
22
2 2
2
2 2
2
2 2
2
2
4
,
4
.
i i i z
zi yi zi
i
i i i zi
z
i
ii zi
i zi yi zi
i i zi
z
i
d H
H k H
c dz
dE
rot j
c dz c
d E
E k E
c dz
dH
i j
c dz c
(2)
where 2, , are the components of permittivity
tensor of magnetoactive plasma depending on frequency
which has the structure:
2
2
0
0
0 0
i
i
, (3)
Here j is the perturbation of current density, yjk – y
component of wave vector, c – velocity of light.
The last terms in the right parts of equations (2)
describe nonlinear interaction of natural modes of
electrodynamics system. Performing the replacement
, ( ), ( )expzi zi zi zi ziE H E z H z ik z , (4)
and taking into account synchronism conditions
1 2 3 1 2 3 1 2 3, , ,y y y z z z zk k k k k k k (5)
(supposing that wave vectors of interacting waves lie in
one plane) one obtains the equations for slowly varying
amplitudes of interacting waves ( ), ( )zi ziE z H z . In view
of complexity of these equations they will not be
presented here in fully, instead we present below their
general structure and provide its analysis.
1
1 1 23 2 3 23 2 3
23 2 3 23 2 3
1
1 1 23 2 3 23 2 3
23 2 3 23 2 3
* *2
2 2 13 1 3 13 1 3
* *
13 1 3 13 1 3
*2
2 2 13 1 3
,
,
,
z
z h z z h z z
h z z h z z
z
z e z z e z z
e z z e z z
z
z h z z h z z
h z z h z z
z
z e z z
dH
E a H H b E H
dz
c H E d E E
dE
H a H H b E H
dz
c H E d E E
dH
E a H H b E H
dz
c H E d E E
dE
H a H H
dz
*
13 1 3
* *
13 1 3 13 1 3
* *3
3 3 12 1 2 12 1 2
* *
12 1 2 12 1 2
* *3
3 3 12 1 2 12 1 2
* *
12 1 2 12 1 2
,
,
.
e z z
e z z e z z
z
z h z z h z z
h z z h z z
z
z e z z e z z
e z z e z z
b E H
c H E d E E
dH
E a H H b E H
dz
c H E d E E
dE
H a H H b E H
dz
c H E d E E
(6)
Here the coefficients , , , , ,a b c d with different
indexes depend on parameters of the problem.
In the present work only qualitative analysis will be
done of these coefficients which describe process of
nonlinear interaction of different waves. For that the
expression for perturbation current density is used:
0 0( ) ( )l i i e ej e n n v e n n v , (7)
where e – charge of electron,
0n – equilibrium density
of plasma, ,i in v – perturbations of ion density and
velocity, correspondingly, ,e en v – perturbations of
electron density and velocity. For slowly varying
amplitudes of the j-th wave the nonlinear interaction is
described by terms ( )i i jn v and ( )e e jn v in the right parts
of equations. From the linear parts of the hydrodynamic
equations the expressions for perturbations of electron
and ion density and velocity can be obtained:
2 2
1
,ik ik
k i
n v
,
2 2
1
,ek ek
k e
n v
, (8)
where
k is the frequency of k-th wave taking part in
nonlinear interaction, ,i e are ion and electron
cyclotron frequencies correspondingly. As follows from
expressions (8) the dependence of coefficients in the
right parts of equations (5) on the characteristic
frequency has the form:
2 2 2 2
, ,
1 1
, , ,
k i e l i e
a b c d
, (9)
where
k ,
l are frequencies of interacting waves,
satisfying synchronism condition (5). It is following
from expression (9) that in the case when frequency of
any of interacting wave is close to the one of the
cyclotron waves, the coefficients of nonlinear
interaction are anomalously large. It should be noted
that condition of slowness of amplitudes change is not
right when frequency of one of the interacting waves is
near some of cyclotron frequencies.
The criterion for transition of decay instability into a
stochastic regime was obtained in [4]:
1 1
lf
VE
, (10)
where V is the matrix element of nonlinear interaction
(in this case it corresponds to nonlinear interaction
coefficients in the set of equations (5)),
1E – initial
amplitude of decaying wave, lf – lowest frequency of
one of waves taking part in nonlinear interaction. Thus,
in magnetoactive plasma there are favorable conditions
for stochastic decays to be realized with participation of
high frequency wave and low frequency ion wave which
is of the order of the ion cyclotron frequency. This
criterion is qualitative. The estimation of threshold
value of the decaying wave amplitude can be found
from this criterion for conditions when transition to
stochastic regime is possible.
2. DECAYS IN A FERROMAGNETIC
MATTER
The ferrites are an example of another matter where
nonlinear interaction of natural oscillations is possible.
The ferrites as well as magnetoactive plasma are
158 ISSN 1562-6016. ВАНТ. 2016. №6(106)
gyrotropic. The gyrotropic property of ferrites becomes
apparent when they are placed in a statical magnetic
field. The gyrotropic properties of magnetoactive
plasma are described by permittivity tensor while
gyrotropy of ferrites is described by permeability tensor.
However, the general regularities of nonlinear processes
in plasma and in ferrites are similar. The physical
processes in ferrites were minutely described, for
example, in [9-11].
To obtain components of permittivity tensor for the
magnetoactive plasma the hydrodynamic equations
describing perturbations of electron and ion velocity and
density are used. For ferromagnetic matter the Landau-
Lifshitz equation describing dynamics of magnetic
moment induced by external constant magnetic field is
used to obtain components of magnetic permeability
tensor. The high frequency tensor of magnetic
permeability for the ferrite magnetized to the saturation
level has the structure
0
0
0 0 1
a
a
i
i
(11)
The nonlinear plasma properties are conditioned by
the current terms in the equations (2). The nonlinear
wave interaction in ferrites is conditioned by the term
that takes into account ferrite magnetization arising
when ferrite is placed into external magnetic field.
To obtain nonlinear equations describing the
interaction of natural modes of the electrodynamics
system containing ferrite, the approach analogous to the
one used for magnetoactive plasma was employed. The
equations for slowly varying amplitudes ( ), ( )zi ziE z H z
of nonlinearly interacting natural waves of
electrodynamic system structure obtained from Maxwell
equations have the structure similar to the equations set
(5). The functional frequency dependence of
coefficients of the nonlinear terms is
2 2
1
, , ,
H j
a b c d
, (12)
where j is one of frequencies of the wave taking part
in nonlinear interaction, 0
H
e
eH
m c
– ferromagnetic
resonance frequency, , ee m – charge and mass of
electron,
0H is external magnetizing magnetic field. As
is seen, the ferromagnetic resonance frequency
coincides with electron cyclotron frequency, but it also
depends on the form of ferromagnetic pattern.
Thus in a ferrite placed into external constant
magnetic field there is the frequency range where
nonlinear interaction of parameters may be anomalously
strong. As follows from criterion for arising of
stochasticity (10), the most favorable conditions for
chaos rising in decay instability exist in the region of
small frequency values for lowest frequency wave that
takes part in nonlinear interaction. In ferrite it may be
magnetostatic waves.
CONCLUSIONS
In the present work the possibility of realization of
stochastic regimes for decay instability in nonlinear
matter is analyzed: in magnetoactive plasma and in the
ferrite magnetized to saturation. It was shown that in
both cases there is frequency region where nonlinear
interaction coefficients are anomalously large. In the
magnetoactive plasma these are two frequency regions
close to electron and ion cyclotron resonances. In the
magnetized ferrite this is frequency region near
ferromagnetic resonance.
As follows from the criterion for the rise of
stochastic regime. the less the frequency of the lowest
frequency wave taking part in nonlinear interaction the
lower the threshold of transition decay instability in
stochastic regime.
From the above presented results it follows that the
mode with chaotic dynamics is easily excited when two
conditions are met;
i) the frequency of one of interacting waves is near one
of the resonance frequencies, i.e., electron or ion
cyclotron frequencies in plasma, and ferromagnetic
resonance frequency in the ferrite magnetized to
saturation;
ii) the less frequency of this resonance the lower the
threshold for stochasticity rise.
In the noted above cases it may expect of effective
transition of decaying wave energy into other modes
with simultaneous chaotization. This fact can be used
for effective heating of plasma ions.
REFERENCES
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“Energoatomizdat”, 1981 (in Russian.)
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A.P. Tolstoluzhsky. Chaotic dynamics of charged
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D.V. Tarasov. Regular and Stochastic Decays of Waves
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O.F. Kovpik, E.A. Kornilov, V.G. Svichensky,
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stochastic heating of the plasma in plasma resonator //
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“Plasma Physics”. 2014, № 6(94), p. 87-90.
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Article received 05.10.2016
ДИНАМИКА ТРËХВОЛНОВЫХ СТОХАСТИЧЕСКИХ РАСПАДОВ В НЕЛИНЕЙНЫХ СРЕДАХ
В.А. Буц, И.К. Ковальчук
Анализируется переход распадной неустойчивости в стохастический режим в нелинейных средах:
магнитоактивной плазме и намагниченном до насыщения феррите. Показано, что параметр нелинейного
взаимодействия может быть аномально большим вблизи резонансных частот: электронной и ионной
циклотронной в магнитоактивной плазме и частоты ферримагнитного резонанса в феррите. Отмечено, что с
уменьшением частоты самой низкочастотной волны, участвующей в распаде, уменьшается пороговое
значение амплитуды распадающейся волны, при котором происходит переход в стохастический режим.
ДИНАМІКА ТРИХВИЛЕВИХ СТОХАСТИЧНИХ РОЗПАДІВ У НЕЛІНІЙНИХ СЕРЕДОВИЩАХ
В.О. Буц, І.К. Ковальчук
Аналізується перехід розпадної нестійкості в стохастичний режим у нелінійних середовищах:
магнітоактивній плазмі і намагніченому до насичення фериті. Показано, що параметр нелінійної взаємодії
може бути занадто великим поблизу резонансних частот: електронної та іонної циклотронної в
магнітоактивній плазмі та частоти феромагнітного резонансу у фериті. Відзначено, що зі зменшенням
частоти самої низькочастотної хвилі, що бере участь у розпаді, зменшується порогове значення амплітуди
хвилі, що розпадається, при якому відбувається перехід у стохастичний режим.
|