Features of three wave interaction in the magnetoactive plasma
The decay processes of waves propagating in the nonlinear gyrotropic media were investigated. It was shown that the dynamics of the process does essentially change if conditions for Faraday effect are realized. The most important result is the conclusion that the presence of gyrotropy can lead to su...
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irk-123456789-821172015-05-26T03:01:53Z Features of three wave interaction in the magnetoactive plasma Buts, V.A. Koval’chuk, I.K. Плазменная электроника The decay processes of waves propagating in the nonlinear gyrotropic media were investigated. It was shown that the dynamics of the process does essentially change if conditions for Faraday effect are realized. The most important result is the conclusion that the presence of gyrotropy can lead to suppression of the decay instability. This means that in such conditions the plasma wave will propagate without attenuation. The analytical conclusions are confirmed by numerical results. Исследованы процессы распадов волн, которые распространяются в гиротропных нелинейных средах. Показано, что в условиях реализации эффекта Фарадея динамика процессов существенно изменяется. Наиболее важным результатом является утверждение, что наличие гиротропии может привести к подавлению распадной неустойчивости. Это означает, что в таких условиях волны в плазме распространяются без затухания. Выводы аналитического рассмотрения подтверждаются результатами численного анализа. Досліджені процеси розпаду хвиль, що поширюються в гіротропних нелінійних середовищах. Показано, що в умовах реалізації ефекту Фарадея динаміка процесів істотно змінюється. Найбільш вагомим висновком є висновок, що наявність гіротропії може привести до зриву розпадної нестійкості. Це означає, що в таких умовах хвилі в плазмі розповсюджуються без згасання. Висновки аналітичного розгляду підтверджуються результатами чисельного аналізу. 2015 Article Features of three wave interaction in the magnetoactive plasma / V.A. Buts, I.K. Koval’chuk // Вопросы атомной науки и техники. — 2015. — № 1. — С. 152-155. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/82117 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Плазменная электроника Плазменная электроника |
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Плазменная электроника Плазменная электроника Buts, V.A. Koval’chuk, I.K. Features of three wave interaction in the magnetoactive plasma Вопросы атомной науки и техники |
| description |
The decay processes of waves propagating in the nonlinear gyrotropic media were investigated. It was shown that the dynamics of the process does essentially change if conditions for Faraday effect are realized. The most important result is the conclusion that the presence of gyrotropy can lead to suppression of the decay instability. This means that in such conditions the plasma wave will propagate without attenuation. The analytical conclusions are confirmed by numerical results. |
| format |
Article |
| author |
Buts, V.A. Koval’chuk, I.K. |
| author_facet |
Buts, V.A. Koval’chuk, I.K. |
| author_sort |
Buts, V.A. |
| title |
Features of three wave interaction in the magnetoactive plasma |
| title_short |
Features of three wave interaction in the magnetoactive plasma |
| title_full |
Features of three wave interaction in the magnetoactive plasma |
| title_fullStr |
Features of three wave interaction in the magnetoactive plasma |
| title_full_unstemmed |
Features of three wave interaction in the magnetoactive plasma |
| title_sort |
features of three wave interaction in the magnetoactive plasma |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2015 |
| topic_facet |
Плазменная электроника |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/82117 |
| citation_txt |
Features of three wave interaction in the magnetoactive plasma / V.A. Buts, I.K. Koval’chuk // Вопросы атомной науки и техники. — 2015. — № 1. — С. 152-155. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT butsva featuresofthreewaveinteractioninthemagnetoactiveplasma AT kovalchukik featuresofthreewaveinteractioninthemagnetoactiveplasma |
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2025-07-06T08:16:58Z |
| last_indexed |
2025-07-06T08:16:58Z |
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1836884769641594880 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №1(95)
152. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 152-155.
FEATURES OF THREE WAVE INTERACTION IN THE
MAGNETOACTIVE PLASMA
V.A. Buts, I.K. Koval’chuk
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukrane
E-mail: vbuts@kipt.kharkov.ua
The decay processes of waves propagating in the nonlinear gyrotropic media were investigated. It was shown that
the dynamics of the process does essentially change if conditions for Faraday effect are realized. The most important
result is the conclusion that the presence of gyrotropy can lead to suppression of the decay instability. This means
that in such conditions the plasma wave will propagate without attenuation. The analytical conclusions are
confirmed by numerical results.
PACS: 52.35.Mw
INTRODUCTION
By the present time, nonlinear wave interaction in
different media has been investigated quite minutely. As
examples, optics, hydrodynamics, plasma physics, etc.
can be exemplified. Especially thoroughly were studied
so called three wave processes, when three natural
waves of a physical system are taking part in nonlinear
interaction. The well-known example of such
interaction is the decay instability. On the other hand,
there is the Faraday effect in a gyrotropic plasma.
Separately both the Faraday effect and processes of
three wave interaction has been investigated quite fully
and in detail. However, the processes of nonlinear wave
interaction in magnetoactive plasma when Faraday
effect is taken into account were not investigated yet. In
this report the results of investigation of such processes
are presented.
First of all the equations describing such processes
were obtained. In the common case these are
complicated equations and their analysis may be carried
out by means of numerical methods only. But there are
some conditions when consideration of these processes
is essentially simplified. In particular, such simplify-
cation is possible at weak gyrotropy. In this case the
dispersion properties of the system are close to
properties of vacuum dispersion. Such conditions may
exist in rare plasma ( 2 2
p ) and frequencies ( )
are not close to electron cyclotron or to upper hybrid
frequencies.
The Faraday effect may be considered as energy
exchange between two linear oscillators which are E-
and H-waves accordingly. It may be expected that, if
the period of this exchange process is sufficiently short,
then all other processes having longer characteristic
time will be suppressed. The mechanism of stabilization
of such states is described in [1, 2].
Below we present: the increments of E- and H-wave
decays and parameter of linear energy exchange, the
ranges of parameters where linear transfer and decay
processes are possible, the assessments of threshold
amplitudes when decay process is possible, and the
results of numerical investigation of the equations
describing nonlinear processes in magnetoactive
plasma. It was shown that linear interaction between E-
and H-wave can result in suppression of decay
instability.
The obtained results can be useful to overcome the
limitation on amplitudes of waves excited at beam-
plasma interaction [3, 4].
1. PROBLEM DEFINITION AND BASIC
EQUATIONS
Let us consider an infinite magnetoactive plasma.
We suppose that external uniform magnetic field is
directed along z- axis. The permittivity of such plasma
is described by means of tensor of the form (see, for
example [5]):
2 2
2 2 2
2 2
22 2 2
0
0 , 1 ,
0 0
1 , ,
p
e
p p e
e
i
i
(1)
where p is plasma frequency,
e – electron cyclotron
frequency, – frequency of wave. If the waves
propagate at some angle to magnetic field the coordinate
axes can be directed in such a way that wave vector will
be in the yz plane. Only the case when electromagnetic
waves do not depend on x coordinate will be analyzed,
with additional supposition that wave vectors of
interacting waves are in the same plane.
To investigate dynamics of nonlinear wave
interaction, the Maxwell equations describing electro-
magnetic field components and hydrodynamic equations
for plasma electrons will be used. It is supposed that
spatial-temporal dependence of all electromagnetic field
components has the form:
, , ( )expi i i yiE H E H z i t ik y , (2)
where ωi is frequency of ith wave, kxi, is y component
of its wave vector. Expressing from Maxwell equations
all x and y components of electromagnetic field through
Ez and Hz, the following set of ordinary differential
equations can be obtained for these components:
2 2 2 2
22
2 2
2 4
,
i i i z
zi yi zi
i
i i i zi
z
i
d H
H k H
c dz
dE
rot j
c dz c
mailto:vbuts@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2015. №1(95) 153
2 2
2
2 2
2
2
4
.
ii zi
i zi yi zi
i i zi
z
i
d E
E k E
c dz
dH
i j
c dz c
(3)
The second terms in the right parts of these equations
describe nonlinear interaction. The index i means that
corresponding value belongs to characteristics of ith
wave. It is following from this set that in gyrotropic
plasma the E- and H-waves are not independent. Below
we will be interested in weak girotropy, when
2 1 .
This is the case of low density plasma when frequencies
of interacting wave are not near the electron cyclotron
frequency or upper hydride one. In such case dispersion
properties of media are close to vacuum dispersion.
Let’s consider the influence of linear coupling on the
nonlinear interaction of eigenmodes. To move further,
the dependence of z-components of the electric and
magnetic fields on z coordinate are presented as
follows:
, ( ), ( )expzi zi zi zi ziE H E z H z ik z , (4)
where Ei(z), Hi(z) are dimensionless and slowly varying
amplitudes of z components of electric and magnetic
fields for ith wave. We will limit ourselves by three
wave nonlinear interaction. To obtain shorted equations
describing dynamics of slowly varying amplitudes in
the three wave nonlinear interaction process, the method
presented, for example, in [6…8] can be used. The
conditions of synchronism may be taken according to:
1 2 3
1 2 3
1 2 3
,
,
,
y y y
z z z z
k k k
k k k k
(5)
where δkz – detuning for z component of wave vector
(
z zik k ).
After some mathematic transformation the set of
ordinary differential equations can be obtained, which
describe the dynamics of nonlinear interaction of natural
modes in magnetoactive plasma taking into con-
sideration the influence of energy exchange between E-
and H-waves:
2 2 2
11
12 2 2 2
1 1
2
1 1
23 2 32 2 2 2
1 2 3
23 2 3 23 2 3 23 2 3
( )
2
exp( )
2 ( )( )
,
p e pz
z
e p
y p z
h z z
z e e
h z z h z z h z z
dH
E
dz c
k k z
a H H
k
b E H c H E d E E
2 2
11
12 2 2 2
1 1
2 4 2 2 2
1 1
23 2 32 2 2 2
1 1 1
23 2 3 23 2 3
2
exp( )
2
,
p ez
z
e p
p e p z
e z z
z e p
e z z e z z
dE
H
dz c
k z
i b E H
k c
c H E d E E
2 2 2
22
22 2 2 2
2 2
2
2 1 *
13 1 32 2 2 2
2 1 3
* * *
13 1 3 13 1 3 13 1 3
2 2
22
22 2 2 2
2 2
2 3 2 2 2
1 2 1
( )
2
exp( )
2 ( )( )
,
2
exp( )
2
p e pz
z
e p
y p z
h z z
z e e
h z z h z z h z z
p ez
z
e p
p e p z
dH
E
dz c
k k z
a H H
k
b E H c H E d E E
dE
H
dz c
k z
k
*
13 1 32 2 2 2
2 2 2
* *
13 1 3 13 1 3 ,
e z z
z e p
e z z e z z
b E H
c
c H E d E E
(6)
2 2 2
33
32 2 2 2
3 3
2
3 1 *
12 1 22 2 2 2
3 1 2
* * *
12 1 2 12 1 2 12 1 2
2 2
33
32 2 2 2
3 3
2 3 2 2 2
1 3 3
( )
2
exp( )
2 ( )( )
,
2
exp( )
p e pz
z
e p
y p z
h z z
z e e
h z z h z z h z z
p ez
z
e p
p e p z
dH
E
dz c
k k z
a H H
k
b E H c H E d E E
dE
H
dz c
k z
i
*
12 1 22 2 2 2
1 3 3
* *
12 1 2 12 1 2
2
.
e z z
z e p
e z z e z z
b E H
k c
c H E d E E
The values a, b, c, d with different indexes are quite
complicated. They characterize the influence of
nonlinear terms. These coefficients, entering into the
equation for /zidE dz , contain denominators of the type
2 2
e . It is following from these equations that
process of three wave interaction in magnetoactive
plasma is substantially more complicated than it is
described, for example, in [6-8]. The linear interaction
between the E- and H- components plays significant
role in this case. The linear terms describing such
interaction can much exceed the terms describing the
decay instability. However, it may be expected that at
some electromagnetic field strength the contions can be
realized when the decay instability dominates over the
linear transfer.
2. RESULTS OF ANALYTICAL AND
NUMERICAL INVESTIGATIONS
By neglecting the nonlinear terms, the set (6) is
transformed into three set of equations, each describing
the linear energy transfer between E- and H-components
of one eigenmode. Every such set of first order
equations can be reduced to equation of the view:
2
2
2
0zi
Li zi
d H
H
dz
, (7)
where
4 2
2
2 2 2 2 2 2
.
4 ( )
p e
Li
i e i e pc
(8)
154 ISSN 1562-6016. ВАНТ. 2015. №1(95)
It follows from expression (8) that energy exchange
between E- and H-waves of one mode is possible in the
frequency region ω<ωe and 2 2
e p , the latter is
the upper hybrid frequency. Linear transfer becomes
faster when approaching the electron cyclotron
frequency or the upper hybrid one. However, it should
be taken into account that in the vicinity of these
frequencies the approximation of slow varying
amplitudes is not correct. From expression (8) the
characteristic length for energy exchange between E-
and H-waves can be estimated: L=1/λi..
As it was pointed out in the previous section, the subject
of our interest is the case when the terms of questions
describing the effect of magnetic field on the permittivity
tensor are small. With that the dispersive properties of
investigated system are close to those of vacuum. This is
possible if the conditions, mentioned in the previous
paragraph, are true. In the selected model when wave
vectors of interacting waves are in one plane, the
synchronism conditions (5) are satisfied if all interacting
waves are propagating in one direction (and the dispersion
properties are close to the vacuum dispersion properties).
To define the increment of decay instability it is
supposed that the decaying mode amplitude is a constant
value. In this case the set of shorted equations is linear. In
this case, every of natural modes has E- and
H-components which take place in the process of linear
energy exchange. The decay instability can be realized
differently depending on what components of natural
mode is dominant. The estimation of increment was
performed for two cases: (i) at initial moment the
decaying wave contains only H-component, and (ii) only
E- component. The expressions for these increments are
rather complicated. Both, E- and H-component
amplitudes, enter as multipliers in these expressions, and
this allows to point out those amplitude values when the
nonlinear decay dominates over the linear transfer.
If initially the H-wave dominates in the decaying
mode, then there are two increments of its decay into E-
and H-waves in second and third modes. On the
contrary, if E-wave is dominant initially, there is one
increment of decay into modes 2 and 3. The expressions
for these increments are complicated, especially if E-
wave decays. The expressions for these increments
allow to conclude that there are ranges of parameters
where decay is not possible although synchronism
conditions are hold. For every of indicated decays it is
possible to define a threshold value for the amplitude of
the decaying wave when the decay instability is
possible.
Thus the expressions for increments of decay H-
wave instability (for made above supposition) are:
24 2 2
1 12
2
4 2 2
2 3 1
22 2 4 6 2
2 3 1 12
2
4 4 2 2 2 2 2 2
1 2 3 1 2 3
,
4
,
4
p e z
he
z z e
y y p e z
hh
y z z e e e
H
k k c
k k H
k k k c
(9)
where he , hh – increments of decay instability H-wave
into E- and H-wave of the modes 2 and 3, respectively.
As can be seen from equations (9) there is frequency
range where transfer of H-wave into H-wave is not
possible. The increment of the decay instability of E-
wave is more complicated, so it was not shown here. It
should be noted that square of increment of this
instability can be not only a real value, but complex
value also. Thus, the decay of E-wave has a more more
complicated character, depending of frequencies of
natural modes that do take part in interaction.
Using expressions (8) and (9) together with condition
1L he it is possible to obtain threshold value for
dimensionless strength of H-wave of decaying mode
when the decay instability is possible. The criterion for
the decay instability H E to exist is:
2 3
1
1
z z
z
c k k
H
. (10)
As follows from these results the dynamics of the
decay instability in the magnetoactive plasma may be
more complicated comparing with the well-known cases
(e.g., see, [6-8]).
In addition to what was said above, we want to know
how the decay instability is influenced by the process of
linear transfer between E- and H-modes.
One might expect that if the period of this exchange
is below the inverse increment of the decay instability,
then the latter will be suppressed.
This expectation was confirmed by results of
numerical investigation of the equation (6). The plots
demonstrating influence of linear transfer on decay
instability are presented in Figs. 1-3.
.
Fig. 1. Nonlinear dynamics of decaying mode where
H-wave is dominant. The process, corresponding to
linear transfer is neglected. Red curve corresponds to
H-wave; blue curve – E-wave. The vertical axes
corresponds to amplitudes of the waves. The horizontal
axes corresponds z coordinate
Fig. 2. Linear transfer between E- and H-wave of mode
1. Red curve corresponds to H-wave; blue curve –
E-wave. The vertical axes corresponds to amplitudes of
the waves. The horizontal axes corresponds z
coordinate
0 200 400 600 800 1000
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0 200 400 600 800 1000
0.02
0.04
0.06
0.08
0.10
0.12
0.14
ISSN 1562-6016. ВАНТ. 2015. №1(95) 155
Fig. 3. The dynamics of the mode 1 when both linear
and nonlinear processes are taken into account. Red
curve corresponds to H-wave; blue curve – E-wave.
Nonlinear decay is suppressed. The vertical axes
corresponds to amplitudes of the waves. The horizontal
axes corresponds z coordinate
CONCLUSIONS
Nonlinear equations describing interaction of the
waves in magnetoactive plasma which propagate under
some angle to the magnetic field were obtained. The set
of equations describing slow varying amplitudes of
interacting modes was derived. As distinct from the
well-known three-wave approximation ([6-8]), in our
case the resulting system contains six equations. This is
because the every interacting mode contains two
connected components: E- and H-wave.
The expression for the length where linear transfer
takes place is presented. The increments of the decay
instability when the H-wave amplitude exceeds the E-
wave amplitude are presented. The estimation of
decaying mode amplitude was found when the decay
instability dominates on the process of linear energy
exchange.
The most important result of this study, on our point,
is the conclusion that the presence of gyrotropy can lead
to suppression of the decay instability, what means that
in such conditions the plasma wave will propagate
without attenuation.
REFERENCES
1. V.A. Buts. Stabilization of classic and quantum
systems // Problems of Atomic Acience and Technology.
2012, № 6(82), p. 146-148.
2. V.A. Buts. Stabilization of unstable states. 20th
International Conference on Microwaves, Radar and
Wireless Communications, June 16-18, 2014, Gdansk,
Poland, p. 681-685.
3. А.N. Аntonov, В.А. Buts, I.K. Koval`chuk,
О.F. Kovpik, E.А. Kornilov, В.G. Svichenskii,
D.V. Tarasov. Regular and stochastic decay waves in
the plasma resonator // Plasma Physics. 2012, № 38,
v. 8, p. 693-708.
4. V.A. Buts, I.K. Koval`chuk, E.A. Kornilov, and
D.V. Tarasov. Stabilization of Beam Instability by a
Local Instability Developing due to a Wave-Wave
Interaction // Plasma Physics Reports. 2006, v. 32, № 7,
p. 563-571.
5. Electrodynamics of plasma / Editor A.I. Akhiyezer.
Moscow: ”Nauka”. Gl. Red. Phys.-mat. Lit, 1974,
p. 720 (in Russian).
6. B.B. Kadomtsev. Collective phenomena in plasma.
Moscow: ”Nauka”. Gl. Red. Phys.-mat. Lit. 1988,
p. 394 (in Russian).
7. H. Wilhelmsson, J. Weiland. Coherent non-linear
interaction of waves in plasmas. M: “Energoizdat”.
1981, 223 p. (in Russian).
8. V.N. Tsitovich. Nonlinear collective effects in
plasma. Moscow: ”Nauka”, p. 268 (in Russian).
Article received 03.12.2014
ОСОБЕННОСТИ ТРЕХВОЛНОВОГО ВЗАИМОДЕЙСТВИЯ В МАГНИТОАКТИВНОЙ ПЛАЗМЕ
В.А. Буц, И.К. Ковальчук
Исследованы процессы распадов волн, которые распространяются в гиротропных нелинейных средах.
Показано, что в условиях реализации эффекта Фарадея динамика процессов существенно изменяется.
Наиболее важным результатом является утверждение, что наличие гиротропии может привести к
подавлению распадной неустойчивости. Это означает, что в таких условиях волны в плазме
распространяются без затухания. Выводы аналитического рассмотрения подтверждаются результатами
численного анализа.
ОСОБЛИВОСТІ ТРИХВИЛЕВОЇ ВЗАЄМОДІЇ В МАГНІТОАКТИВНІЙ ПЛАЗМІ
В.О. Буц, І.К. Ковальчук
Досліджені процеси розпаду хвиль, що поширюються в гіротропних нелінійних середовищах. Показано,
що в умовах реалізації ефекту Фарадея динаміка процесів істотно змінюється. Найбільш вагомим висновком
є висновок, що наявність гіротропії може привести до зриву розпадної нестійкості. Це означає, що в таких
умовах хвилі в плазмі розповсюджуються без згасання. Висновки аналітичного розгляду підтверджуються
результатами чисельного аналізу.
0 200 400 600 800 1000
0.02
0.04
0.06
0.08
0.10
0.12
0.14
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