Stabilization of beam instability as a result of development of local instability in wave-wave interaction
The new mechanism of stabilization of beam instability is proposed. The considered mechanism plays a special role at stabilization of beam instability in plasma systems with small size of interaction area of a beam of particles with field of exited waves. The basis of this mechanism is the process o...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2003 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2003
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| Цитувати: | Stabilization of beam instability as a result of development of local instability in wave-wave interaction / V.А .Buts, I.K. Kovalchuk, E.A. Kornilov, D.V. Tarasov // Вопросы атомной науки и техники. — 2003. — № 4. — С. 109-113. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860170657220788224 |
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| author | Buts, V.А. Kovalchuk, I.K. Kornilov, E.A. Tarasov, D.V. |
| author_facet | Buts, V.А. Kovalchuk, I.K. Kornilov, E.A. Tarasov, D.V. |
| citation_txt | Stabilization of beam instability as a result of development of local instability in wave-wave interaction / V.А .Buts, I.K. Kovalchuk, E.A. Kornilov, D.V. Tarasov // Вопросы атомной науки и техники. — 2003. — № 4. — С. 109-113. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The new mechanism of stabilization of beam instability is proposed. The considered mechanism plays a special role at stabilization of beam instability in plasma systems with small size of interaction area of a beam of particles with field of exited waves. The basis of this mechanism is the process of three-wave decay with participation of a wave which easily abandons the field of interaction and also the process of chaotization of the fields at nonlinear interaction of waves.
|
| first_indexed | 2025-12-07T17:58:50Z |
| format | Article |
| fulltext |
STABILIZATION OF BEAM INSTABILITY AS A RESULT OF DEVEL-
OPMENT OF LOCAL INSTABILITY IN WAVE-WAVE INTERACTION
V.А .Buts, I.K. Kovalchuk, E.A. Kornilov, D.V. Tarasov
NSC “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
vbuts@kipt.kharkov.ua
The new mechanism of stabilization of beam instability is proposed. The considered mechanism plays a special
role at stabilization of beam instability in plasma systems with small size of interaction area of a beam of particles
with field of exited waves. The basis of this mechanism is the process of three-wave decay with participation of a
wave which easily abandons the field of interaction and also the process of chaotization of the fields at nonlinear in-
teraction of waves.
PACS: 52.35.Mw
1. INTRODUCTION
The essence of the mechanism can be explained by
the fact that in short electrodynamic systems the known
mechanisms of instability stabilization such as capture
of beam particles by the field of exited waves and
stochastic instability of movement of beam particles oc-
cur at mush bigger intensities of exited fields than it is
observed in long systems. It is caused by short time of
flight of beam particles through area of interaction with
a field. Local instability of wave - wave process can be
essential in such conditions. At the same time dynamics
of fields becomes chaotic. Efficiency of interaction of a
charged beam with fluctuating field is much lower than
with fields of regular waves. In addition fluctuating
fields rapidly convey their energy to heating of plasma
particles. Thus a new channel of rapid dissipation of en-
ergy of excited waves appears.
As a result stabilization of instability or even failure
of process of excitation of waves takes place. Chaotiza-
tion of wave fields exited by beam may result in fast
heating both particles of a beam, and particles of plas-
ma. The results of some experiments at which, apparent-
ly, the described mechanism of failure of plasma-beam
instability is realized, are described.
It is necessary to note, that the stabilization of insta-
bilities caused by the regular mechanism of decay has
been discussed in the literature for a long time (for ex-
ample [1]).
2. STOCHASTIC INSTABILITY OF DY-
NAMICS OF WEAK NONLINEAR IN-
TERACTION OF WAVES
At rather high amplitudes of the waves exited in
plasma it is possible that effective nonlinear interactions
of these waves with other proper waves of plasma elec-
trodynamic structure take place. Dynamics of this inter-
action can be both regular and chaotic. We are interest-
ed in chaotic regimes. Such regimes appear in different
schemes of nonlinear wave-wave interaction.
The more simple are modified decay and also the
case of three-wave interaction, when during interaction
the fourth wave can participate in this interaction. The
characteristics of this wave are close, for example, to a
low-frequency wave participating in the interaction. The
last case we shall term quasi-four-wave. The stochastic
instability develops only when amplitude of a decaying
wave (pump wave) exceeds some threshold value. Let's
consider these two cases in details.
2.1. QUASI-FOUR-WAVE INTERACTION
Let the wave with amplitude a1 wave number k1
and frequency ω 1 decay into two waves a k2 2 2, ,ω
and a k3 3 3, ,ω . Besides that let us assume, that there is
one more wave with the following parameters
a k4 4 4, ,ω ; k k4 3= , ω ω ω3 4 1− < < .Let us consid-
er that the fourth wave does not influence the process of
decay. The equation, which describes dynamics of com-
plex amplitudes at interaction of the first three waves
can be presented as [3]:
*a iV a a1 1 2 3= ,
*a iV a a2 1 1 3= , (1)
*a iV a a3 1 1 2= ,
where V V i o1 1= | |exp( )Φ is the matrix element of in-
teraction, a a ij j j= | |exp( )Φ . On the linear stage (
| | ,a const const1 1= =Φ ) of decay the amplitudes
| |a1 and | |a2 growth exponentially with increment
G a V= | || |1 1 . The phase change
Φ Φ Φ Φ Φ= − − +2 1 2 3 0( ) obeys equation of
mathematical pendulum:
( | || |) sinΦ Φ+ =2 01 1
2a V . (2)
It is seen from Eq.(2) that the half width of nonlinear
resonance equals 4G . If we replace the third wave by
forth wave we obtain the following set of equations:
exp( )*a iV a a i1 2 2 4= − δ τ ,
exp( )*a iV a a i2 2 1 4= δ τ , (3)
exp( )*a iV a a i3 2 1 2= δ τ .
On the linear stage phase
Ψ Φ Φ Φ Φ= − − + +2 1 2 4 0( )δ τ satisfies Eq.(2)
too, where G a V2 1 2= | || | , δ ω ω ω= − −1 2 4 . This
means that the distance between nonlinear resonances is
equal to 2δ . Assuming the width of nonlinear reso-
nance for the forth wave is small ( G G> > 2 ) we obtain
the condition of the nonlinear resonance overlapping
and, correspondingly, the criterion of stochastic instabil-
ity:
2 1G / δ > . (4)
2.2. MODIFIED DECAY
The important case of three-wave interaction is the
case of modified decay. At such decay the increment of
linear stage is larger then the frequency of a low-fre-
quency wave, which participates in three-wave interac-
tion. As we showed before [4] the modified decay is al-
ways chaotic.
Let us consider the decay of the HF electromagnetic
wave (i) with frequency ω i , wave vector
k i and ampli-
tude
Ei , which propagates in uniform, unbounded
plasma in the HF (s) electromagnetic wave ( ω s s sk E, ,
) and LF Langmuir wave (ω φpe pk, ,
). In order to de-
scribe this process we started from Maxwell's equations
for electromagnetic fields and hydrodynamic equations
for plasma electrons. We neglected the movement of
ions, assuming that background ions serve for compen-
sation of electron charge. Time averaging to (
t t tslw o fst> > > > , t slw pe∝ 1 / ω , t fst i∝ 1 / ω -
periods of slow and fast variables, respectively) leads to
the following system of coupled equations:
∂
∂
~ ~v
t
e
m
Ee = ,
∇ − − =2
2
2
2
2
2
2
2
1
~
~
~ ~
E
c
E
t c
E
c
n
n
Epe pe e
eo
∂
∂
ω ω δ
, (5)
( ) ~∂
∂
ω
δ2
2
2 2 2 2 21
2t
V
n
n
vpe Te
e
eo
e+ − ∇ = ∇ < >
,
where
~ve - HF electron velocity (varies on the fast
time-scale - t fst i∝ 1 / ω ),
~
E - HF component of the
electric field (varies on the fast time-scale -
t fst i∝ 1 / ω ), neo - equilibrium electron density,
δ ne - electron density perturbation (varies on the slow
time-scale - t slw pe∝ 1 / ω ), e m, - charge and mass of
the electron, respectively, c - speed of light in vacuum,
ω
π
pe
eoe n
m
2
24
= - plasma frequency, V
T
mTe
e2 = -
electrons thermal velocity. < >... - time averaging:
< > =
+
∫x t
t
x d
o t
t to
( ) ( )
1
τ τ .
Note, that we used the similar approach described in
[3] to receive the set of equations (5). Assuming follow-
ing form for HF electromagnetic field and LF plasma
density:
..)exp()(
2
1
)exp()(
2
1),(
~
ccrkititE
rkititEtrE
sss
iii
++−+
++−=
ω
ω
,
δ δn r t n t ik r c ce p p( , ) ( ) exp( ) . . = +
1
2
where
E t and E ti s( ) ( ) - slowly varying in time am-
plitudes of the HF pumping wave and LF scattered
wave, respectively, one can obtain from (5):
i
d
d
ii
s⋅ =
ε
τ
ε ρ τexp( )∆ ,
i
d
d
is
i⋅ = −
ε
τ
ε ρ τ* exp( )∆ , (6)
d
d
ii s
2
2
2ρ
τ
ρ ε ε τ+ = − −Ω ∆*exp( ) ,
where we neglected thermal term ∝ VTe
2 ,
ε i
i
io
E
E
= , ε
ω
ωs
s
io
s
i
E
E
=
ρ
δ ω α π
ω ω
=
n
n
mn
k E
p
eo
pe eo
p i s io
[
cos( )
/
] /
2
2 2
1 3
2
,
τ α
ω
ω ω
= t E
e k
mio
p pe
i s
[ cos ( ) ] /2 2
2 2 2
2 2
1 3
8
,
Ω 2
2 2
2 2 2 2
2 3
8
= [
cos ( )
] /
m
e E k
pe i s
io p
ω ω ω
α
,
∆ = −( )[
cos ( )
] /ω ω
ω ω
α ωi s
i s
io p pe
m
e E k
8 2 2
2 2 2 2 2
1 3 ,
where cos( )α - angle between
Ei and
E s ,
E E tio i= =( )0 . In order to obtain the set of equa-
tions (6) we assume that there is a spatial synchronism
between coupling waves:
k k ki s p− = .
On the linear stage of the decay when | |ε i const=
we can obtain from (6) the dispersion relation:
( )( )ω ω2 2 1− + =Ω ∆ , (7)
and following expressions for maximum values of the
growth rates:
G = =Im /ω 1 2Ω , Ω 2 1> > ;
G = =Im /ω 3 2 , Ω 2 1< < .
In the first case, when the amplitude of the pumping
wave is small, parameter K < < ∝1, ( )∆ Ω and dy-
namics of the decay according to (4) must be regular. At
large amplitudes of the incident wave
K > > ∝1 0, ( )∆ and the decay must be chaotic.
Note that the region of parameters where K > > 1 is re-
lated to the modified decay.
3. EFFICIENCY OF TRANSMISSION OF
ELECTRONIC BEAM ENERGY TO
FLUCTUATING FIELD
It was noted earlier that as a result of wave - wave
interaction dynamics of field becomes chaotic. It could
cause diminution of effectiveness of interaction of parti-
cles with field in electrodynamic system and respective-
ly stabilization of beam instability. For the study of pro-
cess of energy interchange with fluctuating field in re-
stricted area of space let’s use the set of equations given
in [5]:
,11
,
2
coscos
ph
ph
Vd
d
Vd
d
−=Φ
+Φ+Φ=
εξ
ξ
µ
ξ
ε
(8)
Where ε = V V2
0
2/ –dimensionless energy of par-
ticle, 0, VV –current and initial velocity of particle,
V v Vph ph= / 0 , v ph – phase velocity of wave,
1)/()2( 0 < <= VmE ωµ – dimensionless velocity of
wave, Φ = −ω t kz –phase of wave, L –length of a
system, ξ ω= z V/ 0 – normalized coordinate of a par-
ticle.
The set of equations (8) describes moving of a
charged particles in a field of a standing wave and can
be solved by the method of successive approximations
using the small parameter µ . Suspecting that the phase
of a wave has fluctuation parts ( ∆ Φ ), the system (8)
can be reduced to:
2cos cos cos
2sin sin sin ,
1 1 .
ph
ph
ph
d
d V
V
d
d V
ε ξµ
ξ
ξµ
ξ ε
= Φ + Φ + ∆ Φ −
− Φ + Φ + ∆ Φ
Φ = −
(9)
Initial conditions for dimensionless energy e m, .
We guess that ...1 )3(3)2(2)1( ++++= εµεµµ εε Then
in zero-order approximation from the second equation
of the system (9) for a phase of a wave in which there is
a particle we shall receive the following expression:
0
1
Φ+
−
=Φ ξ
ph
ph
V
V
, (10)
Where 0Φ - phase of field at the moment of en-
trance of a particle in a cavity. Substituting expression
for Φ as (10) in the first equation of the system (9) and
integrating it we shall receive expression for )1(ε :
(1)
0
0
0
0
2 cos( ) cos cos ( )
2 sin( ) cos sin ( )
ph
ph
d
V
d
V
ξ
ξ
ξε ξ ξ ξ
ξξ ξ ξ
′′ ′ ′= + Φ ∆ Φ −
′′ ′ ′− + Φ ∆ Φ
∫
∫
(11)
By analogy with ε the phase can be submitted as:
...
1 )2(2)1(
0 +Φ+Φ+Φ+
−
=Φ µµξ
ph
ph
V
V
.
From the second equation of the system (9) it is pos-
sible to receive the expression for correction of the first
degree to the phase:
∫ ′′−=Φ
ξ
ξξε
0
)1()1( )(
2
1 d .
Prolonging iterative procedure we shall receive the
correction of the second degree to dimensionless ener-
gy:
[ ]{ [ ]
[ ][ ]
[ ][ ]
[ ][ ]})(sin)(sin)sin()2sin(
)(cos)(sin)cos()2cos(
)(sin)(cos)cos()2cos(
)(cos)(cos)sin()2sin(
coscos
2202
2202
2202
2202
0 0 0
2
21
)2(
1
ξξξξξξ
ξξξξξξ
ξξξξξξ
ξξξξξξ
ξξξξξε
ξ ξ ξ
∆ Φ′∆ Φ−′−Φ++′
+∆ Φ′∆ Φ−′+Φ++′
+∆ Φ′∆ Φ−′−Φ++′
+∆ Φ′∆ Φ−′+Φ++′
×
′′= ∫ ∫ ∫
′
phph VV
ddd
(12)
First of all, we are interested in the corrections of the
first and the second degree in the expression for dimen-
sionless energy, which are defined in relations (11) and
(12). From (11), in particular, it follows that at injection
of continuous homogeneous monoenergetic beam, an
addend )1(µ ε (i.e. linear with respect to the amplitude of
the field) does not give any contribution in the expres-
sion for interchange of energy of beam with a field. It
will influence only on modulated beam. The influence
of fluctuations of a phase ∆ Φ on interchange of energy
is of interest. For this purpose it is necessary to average
the expressions (11) and (12) at realization of random
function )(ξ∆ Φ . Analytically it can be made in the ele-
mentary case by guessing that the density of probabili-
ties of distribution )(ξ∆ Φ is uniform. Let us guess that
the peak value of fluctuations of phase is equal m∆ Φ
and average value is 0)( > =∆ Φ< ξ .
It is possible to show, that 0)(sin > =∆ Φ< ξ and
m
m
∆ Φ
∆ Φ
> =∆ Φ<
sin
)(cos ξ . Using these relations for
(11), we shall receive for average value >< )1(ε :
m
m
∆ Φ
∆ Φ
> =<
sin)1()1( εε . (13)
In the case when δ - correlated fluctuations, i.e. at
realization of requirement
)()()( ξξδξξ ′−> =′∆ Φ∆ Φ< N
for average value of the correction of the second degree
to dimensionless energy we shall receive:
2
2
)2()2( sin
m
m
∆ Φ
∆ Φ
> =< εε . (14)
Thus, from expressions (13) and (14) it follows that
at presence of fluctuations of the phase of electromag-
netic field in restricted area, contribution of addends
(linear and square-law in amplitude of a field) in the ex-
pression ε decreases. Therefore, the efficiency of inter-
action of the electron beam with fluctuating electromag-
netic fields reduces.
4. HEATING OF PLASMA PARTICLES BY A
FIELD OF NOISE WAVES
As it was mentioned above stochastic instability re-
sults in chaotization of excited fields. The field energy
with randomly varying parameters is transmitted rather
effectively into thermal energy of charged particles,
which move in this field. The transmission of energy
from the field to particles is powerful mechanism of
wave attenuation. The presence of such mechanism of
energy sink, together with mechanism reducing efficien-
cy of energy transmission from beam to exited waves
(at development of stochastic instability) can break the
process of excitation of waves by electronic beam. Let
us estimate the efficiency of energy transmission from
random field to the particles. For this purpose we shall
choose the most prime model. Let us consider that the
charged particles move in random field where there are
no correlations, i.e. 2
1 2 1 2( ) ( ) ( )E t E t A t tδ⋅ = ⋅ − . From
general equations of motion of charged particles in field
of electromagnetic waves it is possible to receive the
following equation for definition of variation of particle
energy ( )γ τ in time:
v Fγ = ⋅
(15)
In equation (15) the following designations are used:
, , / ,d d t v v cγ γ τ τ ω= = ⋅ →
/F q E m c ω= ⋅ ⋅ ⋅
- parameter of wave force, ω - some
average frequency of spectral distribution of electro-
magnetic field.
Taking into account that there is no field correlation
from the equation (15) it is easy to determine the fol-
lowing estimation for energy which can be gained by
the particles in such field:
( )2 2 2( ) ( ) (0) v Aγ γ τ γ τ∆ = − = ⋅ ⋅ (16)
For nonrelativistic moving it is convenient to rewrite
equation (13) in dimensional unities:
0 0
0
2 cW W W A t
v
ω− = ⋅ ⋅ ⋅ ⋅ , (17)
where W - kinetic energy of particles, 0v - initial ve-
locity of particles.
As it is seen from (17) the energy of plasma elec-
trons can vary from several eV (electronvolt) up to keV
(kiloelectronvolt) in a time about hundreds of periods of
high-frequency field, electric field amplitude of which is
equal ~100 V/cm.
5. CONCLUSIONS
Above we have considered the processes which de-
velop in time. In beam amplifiers the processes pass in
space. Many of described above peculiarities of interac-
tion of beam particles and plasma with electromagnetic
waves will take place in this case too. Thus, the equa-
tions of three-wave interaction will differ, for example,
from the equations (1) only by the fact that time deriva-
tive will be replaced by coordinate derivative and the
coupling coefficients will gain a multiplier gv equal to
group velocity of the appropriate wave in a denomina-
tor.
In general case the processes take place both in time
and in space in plasma-beam experiments. Therefore it
is difficult to expect good quantitative agreement of the
results of the analysis of the simplified theoretical mod-
els with the results of experiments. Only in the specially
posed experiments it is possible to rely on such coinci-
dence. Below we shall shortly describe the results of the
experiment, in which above described mechanism of
stabilization of level of waves exited by beam was prob-
ably observed.
In the experiment the electronic beam with current
1-10 A and energy 10-40 keV was injected in the inter-
acting region. As a result beam-plasma discharge devel-
opment the plasma was created. Density of plasma var-
ied from 5⋅1011 up to 1⋅1013 e/sm-3. All system was locat-
ed in a constant external homogeneous magnetic field.
The strength of this field was made ~0.2Т. During cre-
ation of plasma it got the form of the tubular cylinder.
As a result of development beam-plasma instability
the proper wave of plasma electrodynamic structure
were exited. The pulse radiation of electromagnetic
waves, which were directed practically as perpendicular
to the axis of plasma electrodynamic structure, were ob-
served in experiment. The duration of pulses varied
from several µs (3-10) down to 0.5 µs.
The higher level of electromagnetic wave intensity,
the shorter duration radiation pulses were transforming
to intense single pulses. The radiation pulses from plas-
ma follow through approximately equal time intervals.
At the large levels of capacity of an electron beam the
duration of a radiation pulse is reduced and the time in-
terval between them becomes larger transforming to in-
dividual pulses. During failure of electromagnetic wave
radiation the heating both of plasma electrons and elec-
trons of beam was observed.
The details of experimental researches are in detail
represented in the report [6].
At theoretical description, apparently, it is possible
to consider that electrodynamic structure is tubular plas-
ma. As it is known there can be two eigen slow surface
waves in such plasma. The frequencies of these waves
are of order / 2pω . These waves are exited by beam.
When amplitude of exited waves reaches some thresh-
old level, the mechanism of three-wave decay of these
waves on low-hybrid wave and on electromagnetic
transverse wave takes part. Let us note, that electromag-
netic transverse wave is an improper wave of plasma
electrodynamic structure. The appropriate dispersion
curves and scheme of three-wave decay are shown in
figure 1.
We do not give the expressions for coupling coefficients
of three-wave decay as they are enough unwieldy and
what is more important, we have no opportunity to carry
out quantitative comparison of the experimental results
with the results of calculations. The values of many pa-
rameters defining the dynamics of model are unknown
for us. In particular we don’t know field intensities. Fur-
thermore the plasma cylindrical stratum is nonuniform
in the experiment. And, typical size of inhomogeneity is
of order of wave length. In experiment the process takes
place both in time and in space. All this suggests that
the quantitative comparison won’t be correct. It is possi-
ble to speak only about qualitative description of the
process.
kz
Ω h
ω p/ 2
ω
Fig.1. Dispersion curves of a tubular plasma waveguide
The qualitative situation is prime enough. Let us
give its brief description. At the beginning the beam ex-
cites one of eigen surface waves. These waves do not
abandon the plasma cylinder. When the level of excited
waves is high enough, eigen wave decays into trans-
verse electromagnetic wave and on low-hybrid one. As
the transverse wave is not eigen, it easily abandons plas-
ma. At this moment the effective radiation from the
plasma cylinder is observed. And the radiation is practi-
cally perpendicular to the axe of the plasma cylinder.
This part of the process is in good qualitative agreement
with the experiment. Really in experiment the excitation
of low-hybrid waves was always observed simultane-
ously with radiation of a transverse wave (perpendicular
to the axe of a waveguide). Heating of both plasma par-
ticles and particles of a beam was also observed. It
shows that the process of decay becomes chaotic. Let's
note that heating of the particles of a beam could be ob-
served as a result of local instability of dynamics of the
particles of a beam in fields of excited waves. However
for plasma particles this mechanism of heating is imped-
ed. Thus, at the moment of radiation of transverse
waves from plasma two channels of sink of energy ap-
pear. The first channel is outlet of energy together with
improper electromagnetic waves. The second one is ef-
fective heating of particles of a beam and plasma by
fields of fluctuating waves. These two channels cause
depression of level of fields in plasma and as a result
failure of three-wave decay. The radiation from plasma
stops. Let us note one feature of the considered mecha-
nism which can appear in such rather long system. In
ordinary dynamics of plasma-beam instability the stabi-
lization of level of exited field occurs as a result of in-
verse influence of the field on dynamics of particles of a
beam. As a result of such influence the particles of a
beam move from braking phase into accelerating phase.
The electromagnetic energy of the field is returned to
particles of a beam. As a result of three-wave interac-
tions the level of the field is reduced. The inverse action
of the field on particles weakens. Efficiency of energy
transmission from particles to the field can be enlarged.
Besides if the process of decay becomes chaotic, the
phases of the field vary at random fashion and the in-
verse action of the field (which causes extraction of en-
ergy of the wave by the particles of a beam) is broken.
The particles of the beam will prolong transmission of
the energy to the wave though with smaller efficiency.
This pattern of the process of excitation of waves in
plasma-beam system qualitatively coincides well with a
pattern, which is observed in experiment. We can not
speak about quantitative coincidence yet.
References
1. B. Atamanyuk, A.S Volokitin. The current instabil-
ity stabilization by decay processes // Fizika Plas-
my. 2001, v.27, № 7, p.637-646.
2. J.Weiland, H.Wilhelmsson. Coherent non-linear
interaction of waves in plasmas. Pergamon Press.
1977, 223 p.
3. V.A. Buts, O.V. Manuilenko, K.N.Stepanov, A.P.
Tolstoluzhskii // Fizika Plasmy. 1994, v.20, p.794.
4. V.A.Buts, O.V.Manuilenko, A.P.Tolstoluzhskii,
Yu.A.Turkin. Stochastic instability of the modified
decay // International Congress on Plasma Physics
combines with the 25-th EPS conference on
Controlled Fusion and Plasma Physics. Prague,
Czech Rep., June 29-July 3, 1998, Conf. Proc.,
p.252-255; ECA. 1998, v. 22C, p. 252-255.
5. V.A.Buts, І.К.Коvalchuk. The dynamics of
conjugate linier oscillators systems under acting
multiplicate noise // Ukranian Phys. Journal. 2000,
v.45, № 12, p. 1426-1430.
6. O.F. Kovpic, E.A. Kornilov, V.A. Buts. The influ-
ence of nonleniar interaction of waves on beam-
plasma interaction under plasma-beam discharge. //
VIII Interstate Workshop “Plasma Electronics And
New Acceleration Methods”, 2003, Kharkov,
Ukraine.
V.А .Buts, I.K. Kovalchuk, E.A. Kornilov, D.V. Tarasov
|
| id | nasplib_isofts_kiev_ua-123456789-110982 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:58:50Z |
| publishDate | 2003 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, V.А. Kovalchuk, I.K. Kornilov, E.A. Tarasov, D.V. 2017-01-07T16:18:24Z 2017-01-07T16:18:24Z 2003 Stabilization of beam instability as a result of development of local instability in wave-wave interaction / V.А .Buts, I.K. Kovalchuk, E.A. Kornilov, D.V. Tarasov // Вопросы атомной науки и техники. — 2003. — № 4. — С. 109-113. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.35.Mw https://nasplib.isofts.kiev.ua/handle/123456789/110982 The new mechanism of stabilization of beam instability is proposed. The considered mechanism plays a special role at stabilization of beam instability in plasma systems with small size of interaction area of a beam of particles with field of exited waves. The basis of this mechanism is the process of three-wave decay with participation of a wave which easily abandons the field of interaction and also the process of chaotization of the fields at nonlinear interaction of waves. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы Stabilization of beam instability as a result of development of local instability in wave-wave interaction Article published earlier |
| spellingShingle | Stabilization of beam instability as a result of development of local instability in wave-wave interaction Buts, V.А. Kovalchuk, I.K. Kornilov, E.A. Tarasov, D.V. Нелинейные процессы |
| title | Stabilization of beam instability as a result of development of local instability in wave-wave interaction |
| title_full | Stabilization of beam instability as a result of development of local instability in wave-wave interaction |
| title_fullStr | Stabilization of beam instability as a result of development of local instability in wave-wave interaction |
| title_full_unstemmed | Stabilization of beam instability as a result of development of local instability in wave-wave interaction |
| title_short | Stabilization of beam instability as a result of development of local instability in wave-wave interaction |
| title_sort | stabilization of beam instability as a result of development of local instability in wave-wave interaction |
| topic | Нелинейные процессы |
| topic_facet | Нелинейные процессы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110982 |
| work_keys_str_mv | AT butsva stabilizationofbeaminstabilityasaresultofdevelopmentoflocalinstabilityinwavewaveinteraction AT kovalchukik stabilizationofbeaminstabilityasaresultofdevelopmentoflocalinstabilityinwavewaveinteraction AT kornilovea stabilizationofbeaminstabilityasaresultofdevelopmentoflocalinstabilityinwavewaveinteraction AT tarasovdv stabilizationofbeaminstabilityasaresultofdevelopmentoflocalinstabilityinwavewaveinteraction |