Explosive instability in the plasma-beam systems
The results of investigation of explosive instability in the plasma-beam systems are presented. It has been shown that in the general case, the dynamics of this instability can be essentially changed, right up to its full suppression, if one takes into account a fast beam wave (wave with positive en...
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irk-123456789-1092222016-11-22T03:03:33Z Explosive instability in the plasma-beam systems Buts, V.A. Koval’chuk, I.K. Плазменная электроника The results of investigation of explosive instability in the plasma-beam systems are presented. It has been shown that in the general case, the dynamics of this instability can be essentially changed, right up to its full suppression, if one takes into account a fast beam wave (wave with positive energy). The parameters when explosive instability is realized and when it is suppressed are defined. It was shown that in the case of such four wave interaction there are regimes with attributes of irregularity. Представлены результаты исследования динамики взрывной неустойчивости в плазменно-пучковых системах. Показано, что в общем случае учет быстрой пучковой волны (волны с положительной энергией) может существенно менять динамику этой неустойчивости, вплоть до ее срыва. Определены параметры, при которых реализуется взрывная неустойчивость, и параметры, при которых происходит ее подавление. Показано, что при таком четырехволновом взаимодействии существуют режимы с признаками нерегулярности. Представлені результати дослідження динаміки вибухової нестійкості в плазмово-пучкових системах. Показано, що в загальному випадку врахування швидкої пучкової хвилі (хвилі з позитивною енергією) може істотно змінити динаміку цієї нестійкості, навіть до зриву. Визначені параметри, при яких реалізується вибухова нестійкість, та параметри, при яких виникає її зрив. Показано, що при такій чотирьоххвилевій взаємодії існують режими з ознаками нерегулярності. 2012 Article Explosive instability in the plasma-beam systems / V.A. Buts, I.K. Koval’chuk // Вопросы атомной науки и техники. — 2012. — № 6. — С. 152-154. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/109222 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. Explosive instability in the plasma-beam systems Вопросы атомной науки и техники |
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The results of investigation of explosive instability in the plasma-beam systems are presented. It has been shown that in the general case, the dynamics of this instability can be essentially changed, right up to its full suppression, if one takes into account a fast beam wave (wave with positive energy). The parameters when explosive instability is realized and when it is suppressed are defined. It was shown that in the case of such four wave interaction there are regimes with attributes of irregularity. |
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Article |
| author |
Buts, V.A. Koval’chuk, I.K. |
| author_facet |
Buts, V.A. Koval’chuk, I.K. |
| author_sort |
Buts, V.A. |
| title |
Explosive instability in the plasma-beam systems |
| title_short |
Explosive instability in the plasma-beam systems |
| title_full |
Explosive instability in the plasma-beam systems |
| title_fullStr |
Explosive instability in the plasma-beam systems |
| title_full_unstemmed |
Explosive instability in the plasma-beam systems |
| title_sort |
explosive instability in the plasma-beam systems |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2012 |
| topic_facet |
Плазменная электроника |
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http://dspace.nbuv.gov.ua/handle/123456789/109222 |
| citation_txt |
Explosive instability in the plasma-beam systems / V.A. Buts, I.K. Koval’chuk // Вопросы атомной науки и техники. — 2012. — № 6. — С. 152-154. — Бібліогр.: 5 назв. — англ. |
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Вопросы атомной науки и техники |
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AT butsva explosiveinstabilityintheplasmabeamsystems AT kovalchukik explosiveinstabilityintheplasmabeamsystems |
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2025-07-07T22:43:34Z |
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2025-07-07T22:43:34Z |
| _version_ |
1837029880227692544 |
| fulltext |
152 ISSN 1562-6016. ВАНТ. 2012. №6(82)
EXPLOSIVE INSTABILITY IN THE PLASMA-BEAM SYSTEMS
V.A. Buts, I.K. Koval’chuk
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
The results of investigation of explosive instability in the plasma-beam systems are presented. It has been shown
that in the general case, the dynamics of this instability can be essentially changed, right up to its full suppression, if
one takes into account a fast beam wave (wave with positive energy). The parameters when explosive instability is
realized and when it is suppressed are defined. It was shown that in the case of such four wave interaction there are
regimes with attributes of irregularity.
PACS: 52.35.Mw
INTRODUCTION
Processes of nonlinear wave interaction belong to
the key processes of the plasma theory and are well
investigated. It is possible to mark out two subsections
of the theory of wave interaction: (i) coherent nonlinear
wave interaction, and (ii) interaction of waves in the
random phase approximation. Here we will consider the
coherent interaction only. The process of three wave
interaction is the main element of such interactions.
Similar interaction can be useful, for example, at the
wave excitation in the scheme of acceleration in beat-
wave. This interaction may essentially restrict the level
of exciting oscillations in the plasma-beam generators
(e.g., [1–5]).
Among three wave interaction the decay processes
are essentially interest. Especially, when decaying wave
has negative energy. The explosive instability arises in
this case. Usually stabilization of this instability occurs
due to cubic nonlinearities, and therefore the level of
exciting oscillations can be high enough. Often, when
considering such processes, the authors do restrict
themselves by a three wave interaction.
However in many cases besides decaying wave with
negative energy there are waves with positive energy. If
the characteristic of such waves is near the
characteristics of wave with negative energy, such
waves can take part in the nonlinear interaction and
significantly change the character of such interaction.
Especially this concerns the beam systems. In reality,
both the fast and slow waves exist in such systems. The
slow wave has a negative energy and fast one has a
positive energy. The wave characteristics of these waves
are always close. So, in the general case it is naturally
and necessary to take into account both these waves in
the nonlinear dynamics.
This article is devoted just to this problem. We show
that decay processes of the beam waves, when taking
into account the fast and slow waves, can significantly
differ from the processes when only one of these waves
takes part in the process.
1. BASIC EQUATIONS
Let us consider electrodynamic system through
which electron beam moves. There are two beam waves
in this system, the dispersion of which may be presented
as follows:
11,12 1k Vω δω= ± , (1)
where 11,12ω – are frequencies of beam waves; k1 –
projection of wave vector on the direction of beam
propagation; V – its velocity, δω ~ ωb, ωb – is the
plasma frequency of beam. The second of these two
waves is the wave with negative energy. Besides, it is
supposed that in this electrodynamics system there are
two natural waves with frequencies less than 11,12ω . We
will consider interaction of two beam modes and natural
waves of system which is effectively realized when
synchronism conditions are satisfied (conservation
laws), Fig 1:
12 2 3 1 2 3, ,k k kω ω ω= + = +
r r r
(2)
where 2,3ω – frequencies of third and fourth waves
(system of eigen waves) taking part in interaction, k2,3 –
are their wave numbers.
Fig. 1. The possible scheme of wave interaction
Such nonlinear interaction, in which slow beam
wave (wave with negative energy) takes part, ordinarily
results in rising an explosive instability. If in
expressions (2) to change index 12 to 11, this scheme
will correspond to a decay process. Thus in the beam
system there are two close waves participating in the
fully different processes with two other modes of
electrodynamics system.
Usually in theoretical investigations only explosive
instability is considered in which slow beam wave takes
part but the influence of the fast one is not taken in
account. It is interesting to define how accounting of
fast wave influences the explosive instability.
The set of shortened equations for dimensionless
slowly varying amplitudes of all four interacting modes
ISSN 1562-6016. ВАНТ. 2012. №6(82) 153
was obtained in the standard way from Maxwell
equations for electromagnetic fields and the
hydrodynamical equation for particles. This set is as
following:
( )
( )
11
2 3
12
2 3
exp ( ) ;
exp ( ) ;
dE
E E i
d
dE
E E i
d
μ δω τ
τ
μ δω τ
τ
= − Δ +
= Δ −
(3)
( ) ( )
( ) ( )
*2
11 12 3
*3
11 12 2
exp ( ) exp ( ) ;
exp ( ) exp ( ) ,
dE
E i E i E
d
dE
E i E i E
d
δω τ δω τ
τ
δω τ δω τ
τ
= − Δ + + − Δ −⎡ ⎤⎣ ⎦
= − Δ + + − Δ −⎡ ⎤⎣ ⎦
where E11 and E12 – (E→eE/(mcω), m – is electron
mass; с – the velocity of light) – are dimensionless
slowly varying complex amplitudes of HF and LF beam
waves; E2 and E3 – dimensionless slowly varying
complex amplitudes of any other natural waves of the
electrodynamic system, which can take part in the
investigated process; µ – dimensionless coefficient
which is proportional to cube of beam density.
Dimensionless time is measured in the periods of k1V.
Dimensionless frequencies are normalized to k1V.
∆ – characterizes synchronism conditions 2 and 3 waves
with beam modes and is defined by means correlation:
2 3 1ω ω+ = −Δ . (4)
When there is synchronism with slow beam wave the
condition ∆ = δω is satisfied. If there is synchronism
with fast beam wave the condition ∆ = – δω is satisfied.
2. RESULTS OF INVESTIGATION
The set of equations (3) has integral:
2 2
2 3E E const− = . (5)
The equations describing three wave explosive
processes have similar integral. It follows from (5) that
amplitudes of second and third waves can grow
infinitely.
The set of equations (3) was investigated
numerically for two cases: ∆ = δω when second and
third waves are in synchronism with slow beam mode
(wave with negative energy), and ∆ = – δω when these
waves are in synchronism with fast beam mode.
When condition 0δωΔ = → is satisfied the set of
equations has the integral:
11 12 0E E C+ = , (6)
and for non beam modes the next expressions can be
found:
( ) ( )
( ) ( )
2 21 0 22 0
3 31 0 32 0
exp exp ,
exp exp ,
E C C C C
E C C C C
τ τ
τ τ
= + −
= + −
(7)
what corresponds to infinite exponential growth. For
amplitudes of beam modes similar expressions can be
obtained but in an exponent the constant C0 has to be
changed to 2|C0|. This exponential growth is not
connected with linear stage of beam instability, but is
the result of nonlinear four wave interaction.
When there is synchronism of second and third
waves with fast and slow beam waves, the following
values of parameter δω were selected: δω = 1.0×10-6,
0.001, 0.01, 0.1, 0.2. Besides, two regimes
corresponding to different initial conditions were
investigated. In the first case the initial dimensionless
values of beam mode amplitudes were taken equal to
0.1 in dimensionless units, and initial values of two
other modes of the investigated electrodynamic system
were much less. In the second regime the initial value of
slow beam mode having negative energy was chosen
equal to zero. The characteristic temporal dependence of
slow wave amplitude in the logarithm scale for case
∆ = δω ≠ 0 is shown in Fig. 2.
The amplitudes of other waves have qualitatively
similar temporal dependence. As follows from Fig. 2,
the exponential growth is changed by explosive
instability. The time of its rising decreases when
parameter δω increases. The threshold value of
amplitudes corresponding to the beginning of the
explosive instability decreases also. When there is
synchronism of second and third waves with fast beam
mode (the condition ∆= – δω is satisfied) for the values
δω = 0…0.1 the dynamics of the process is practically
identical with the case ∆ = δω. But for δω = 0.2 it
essentially changes.
Fig.2. Temporal dependence of slow beam wave
amplitude in the logarithmic scale
The explosive instability arises noticeably later than
in the considered cases when characteristic time of
instability growth was approximately 40…60 time units.
Now this instability appears at more than 400 units. Up
to this moment the well expressed exponential growth is
absent and the dynamics has an oscillating character
with an attribute of nonregularity. This is confirmed by
investigations of frequency spectra and autocorrelation
functions. The spectrum is wide and autocorrelation
function decreases. The time of the explosive instability
start is very sensitive to initial conditions in this case.
In the case when initial amplitude of slow beam
wave equals zero for values δω = 1.0×10-6, 0.001, 0.01
the explosive instability arises. But when δω = 0.1 it is
absent. If to approach to this value from left, the time of
explosive instability start tends to infinity.
CONCLUSIONS
Thus, in the general way, for the beam and plasma-
beam systems as well as for the systems where there are
waves with positive and negative energy and closed
wave characteristics, it is necessary to consider the
dynamics of four waves instead of three. Such four
154 ISSN 1562-6016. ВАНТ. 2012. №6(82)
wave dynamics can essentially differ from the three
wave one. Most impotent distinctive features of four
wave dynamics are:
1. At three wave interaction the decay of fast beam
wave takes place. This decay is characterized by a
periodical dynamics. If we take into consideration close
waves with negative energy, the periodical dynamics
does not occur, but the result will be an exponential
growth of amplitudes of interacting waves.
2. If the beam wave with negative energy decays
then the existence of fast beam wave can break the
process of explosive instability.
3. Four wave processes have the interval of
parameters with non regular dynamics.
REFERENCES
1. B.B. Kadomtsev. Collective phenomena in plasma.
Moscow: “Nauka”, Gl. Red. Phis.-mat. Lit., 1988 (in
Russian).
2. H. Wilhelmsson, J. Weiland. Coherent non-linear
interaction of waves in plasmas. Moscow:
“Energoatomizdat”, 1981 (in Russian).
3. M.V. Kuzelev, A.A. Rukhadze. Electrodynamics of
dense electron beam in plasma. Moscow: “Nauka”, Gl.
Red. Phis.-mat. Lit., 1990 (in Russian).
4. V.A. Buts, O.V. Manujlenko, K.N. Stepanov,
A.P. Tolstoluzhsky. Chaotic dynamics of charged
particles at wave-particle type interaction and chaotic
dynamics at weak nonlinear interaction of wave-wave
type // Fizika Plazmy. 1995, v. 20, №9, p. 794-801 (in
Russian).
5. 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.
Article received 06.09.12
ВЗРЫВНАЯ НЕУСТОЙЧИВОСТЬ В ПЛАЗМЕННО-ПУЧКОВЫХ СИСТЕМАХ
В.А. Буц, И.К. Ковальчук
Представлены результаты исследования динамики взрывной неустойчивости в плазменно-пучковых
системах. Показано, что в общем случае учет быстрой пучковой волны (волны с положительной энергией)
может существенно менять динамику этой неустойчивости, вплоть до ее срыва. Определены параметры, при
которых реализуется взрывная неустойчивость, и параметры, при которых происходит ее подавление.
Показано, что при таком четырехволновом взаимодействии существуют режимы с признаками
нерегулярности.
ВИБУХОВА НЕСТІЙКІСТЬ У ПЛАЗМОВО-ПУЧКОВИХ СИСТЕМАХ
В.А. Буц, І.К. Ковальчук
Представлені результати дослідження динаміки вибухової нестійкості в плазмово-пучкових системах.
Показано, що в загальному випадку врахування швидкої пучкової хвилі (хвилі з позитивною енергією) може
істотно змінити динаміку цієї нестійкості, навіть до зриву. Визначені параметри, при яких реалізується
вибухова нестійкість, та параметри, при яких виникає її зрив. Показано, що при такій чотирьоххвилевій
взаємодії існують режими з ознаками нерегулярності.
|