Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock
Model of the stripped monoenergetic electron beam with the sharp boundaries based on the results of CLUSTER measurements is proposed for the near-Earth foreshock vicinity. Dispersion equation is obtained and analyzed using numerical methods. Dependency of the equation roots corresponding to kinetic...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2005 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/79779 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock / I.O. Anisimov, V.V. Krasnosselskikh, K.S. Musatenko // Вопросы атомной науки и техники. — 2005. — № 2. — С. 152-154. — Бібліогр.: 2 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-79779 |
|---|---|
| record_format |
dspace |
| spelling |
Anisimov, I.O. Krasnosselskikh, V.V. Musatenko, K.S. 2015-04-04T18:35:13Z 2015-04-04T18:35:13Z 2005 Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock / I.O. Anisimov, V.V. Krasnosselskikh, K.S. Musatenko // Вопросы атомной науки и техники. — 2005. — № 2. — С. 152-154. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 94.30.Tz, 52.35.Fp https://nasplib.isofts.kiev.ua/handle/123456789/79779 Model of the stripped monoenergetic electron beam with the sharp boundaries based on the results of CLUSTER measurements is proposed for the near-Earth foreshock vicinity. Dispersion equation is obtained and analyzed using numerical methods. Dependency of the equation roots corresponding to kinetic mechanism of beam-plasma instability on the model parameters is studied. Запропоновано модель моноенергетичного стрічкоподібного пучка з різкими границями, яка базується на результатах вимірювань експерименту КЛАСТЕР в області форшоку ударної хвилі Землі. Було отримано дисперсійне співвідношення, яке проаналізовано числовими методами. Вивчені залежності положення коренів, що відповідають кінетичному механізму плазмово-пучкової нестійкості, від параметрів моделі. Предложена модель моноэнергетического ленточного пучка с резкими границами, которая базируется на результатах измерений эксперимента КЛАСТЕР в области форшока ударной волны Земли. Было получено дисперсионное уравнение и проанализировано числовыми методами. Изучены зависимости положения корней дисперсионного уравнения, соответствующих кинетическому механизму плазменно-пучковой неустойчивости, от параметров модели. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock Випромінювання ленгмюрівських хвиль електронним пучком обмеженого перерізу в області електронного форшоку Излучение ленгмюровских волн электронным пучком ограниченного сечения в области електронного форшока Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock |
| spellingShingle |
Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock Anisimov, I.O. Krasnosselskikh, V.V. Musatenko, K.S. Plasma electronics |
| title_short |
Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock |
| title_full |
Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock |
| title_fullStr |
Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock |
| title_full_unstemmed |
Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock |
| title_sort |
langmuir waves excitation by electron beam with the limited cross-section in the near-eearth electron foreshock |
| author |
Anisimov, I.O. Krasnosselskikh, V.V. Musatenko, K.S. |
| author_facet |
Anisimov, I.O. Krasnosselskikh, V.V. Musatenko, K.S. |
| topic |
Plasma electronics |
| topic_facet |
Plasma electronics |
| publishDate |
2005 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Випромінювання ленгмюрівських хвиль електронним пучком обмеженого перерізу в області електронного форшоку Излучение ленгмюровских волн электронным пучком ограниченного сечения в области електронного форшока |
| description |
Model of the stripped monoenergetic electron beam with the sharp boundaries based on the results of CLUSTER measurements is proposed for the near-Earth foreshock vicinity. Dispersion equation is obtained and analyzed using numerical methods. Dependency of the equation roots corresponding to kinetic mechanism of beam-plasma instability on the model parameters is studied.
Запропоновано модель моноенергетичного стрічкоподібного пучка з різкими границями, яка базується на результатах вимірювань експерименту КЛАСТЕР в області форшоку ударної хвилі Землі. Було отримано дисперсійне співвідношення, яке проаналізовано числовими методами. Вивчені залежності положення коренів, що відповідають кінетичному механізму плазмово-пучкової нестійкості, від параметрів моделі.
Предложена модель моноэнергетического ленточного пучка с резкими границами, которая базируется на результатах измерений эксперимента КЛАСТЕР в области форшока ударной волны Земли. Было получено дисперсионное уравнение и проанализировано числовыми методами. Изучены зависимости положения корней дисперсионного уравнения, соответствующих кинетическому механизму плазменно-пучковой неустойчивости, от параметров модели.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/79779 |
| citation_txt |
Langmuir waves excitation by electron beam with the limited cross-section in the near-Eearth electron foreshock / I.O. Anisimov, V.V. Krasnosselskikh, K.S. Musatenko // Вопросы атомной науки и техники. — 2005. — № 2. — С. 152-154. — Бібліогр.: 2 назв. — англ. |
| work_keys_str_mv |
AT anisimovio langmuirwavesexcitationbyelectronbeamwiththelimitedcrosssectionintheneareearthelectronforeshock AT krasnosselskikhvv langmuirwavesexcitationbyelectronbeamwiththelimitedcrosssectionintheneareearthelectronforeshock AT musatenkoks langmuirwavesexcitationbyelectronbeamwiththelimitedcrosssectionintheneareearthelectronforeshock AT anisimovio vipromínûvannâlengmûrívsʹkihhvilʹelektronnimpučkomobmeženogopererízuvoblastíelektronnogoforšoku AT krasnosselskikhvv vipromínûvannâlengmûrívsʹkihhvilʹelektronnimpučkomobmeženogopererízuvoblastíelektronnogoforšoku AT musatenkoks vipromínûvannâlengmûrívsʹkihhvilʹelektronnimpučkomobmeženogopererízuvoblastíelektronnogoforšoku AT anisimovio izlučenielengmûrovskihvolnélektronnympučkomograničennogosečeniâvoblastielektronnogoforšoka AT krasnosselskikhvv izlučenielengmûrovskihvolnélektronnympučkomograničennogosečeniâvoblastielektronnogoforšoka AT musatenkoks izlučenielengmûrovskihvolnélektronnympučkomograničennogosečeniâvoblastielektronnogoforšoka |
| first_indexed |
2025-11-27T02:57:06Z |
| last_indexed |
2025-11-27T02:57:06Z |
| _version_ |
1850795668219625472 |
| fulltext |
LANGMUIR WAVES EXCITATION BY ELECTRON BEAM WITH THE
LIMITED CROSS-SECTION IN THE NEAR-EARTH ELECTRON FORE
SHOCK
I.O. Anisimov1, V.V. Krasnosselskikh2, K.S. Musatenko1
1Taras Shevchenko National University of Kyiv, Radio Physics Faculty,
64 Volodymyrs’ka St., 01033, Kyiv, Ukraine, ksm@univ.kiev.ua;
2Laboratoire de Physique et Chimie de l’ Environnement, 3A Avenue de la Recherche Scienti
fique, 45071 Orleans CEDEX, Orleans, France
Model of the stripped monoenergetic electron beam with the sharp boundaries based on the results of CLUSTER
measurements is proposed for the near-Earth foreshock vicinity. Dispersion equation is obtained and analyzed using nu
merical methods. Dependency of the equation roots corresponding to kinetic mechanism of beam-plasma instability on
the model parameters is studied.
PACS: 94.30.Tz, 52.35.Fp
1. INTRODUCTION
Multisatellite measurements that were performed in
the international project CLUSTER fixed the electric field
oscillations on the border between Earth magnetosphere
and foreshock region [1]. Frequencies of these oscilla
tions correspond to the Langmuir and electron-acoustic
waves, respectively. Direct measurements of the electron
velocity distribution function indicated the presence of
electron beam that had been reflected from the shock
front. This beam hypothetically causes excitation of the
above-mentioned waves. Measurement results show that
specified beam is not solid and can be considered as a
system of separated radially restricted beams. Theoretical
investigation of the waves’ excitation by such a beam is a
purpose of this work.
2. MODEL DESCRIPTION AND DISPER
SION EQUATION
The simplest geometrical model is proposed where the
stripped monoenergetic electron beam with the sharp
boundaries pierces the warm plasma without magnetic
field (Fig.1).
Fig.1. Model of the stripped beam piercing plasma
Considering all the physical magnitudes having the
harmonic temporal dependence, f(t)~exp(-iωt), dielectric
permettivity for warm isotropic plasma without beam can
be written as:
( )
2
2
2 2
3 3 2 2
1
13 exp
2 2
pel
D
D D
k
ik
k k
ω
ε ω
ω
πλ
λ λ
, = − −
ж ц
− + − .з ч
и ш
r
(1)
Here ωpe is an electron plasma frequency, and λD is a De
bay radius. But taking into consideration the beam intro
duces the summand to expression (1):
( ) ( )
2 22
2 2 2
0 0
0 0 0
0 0 0
0 0 B xB
zz z
k
kk v k v
δε
ωω
ω ω
й щ
к ъ
к ъ
к ъ≈ .к ъ
к ъ− −к ъ− −к ъл ы
(2)
Here v0 is a beam velocity, kx and kz are the transversal
and longitudinal components of the wave vector, and ωB
is an electron plasma frequency of the beam.
Following dispersion equations can be obtained by
solving Poisson expression (div(ε∇ϕ)=0) for infinite plas
ma and infinite beam:
2
2 2 2
2
2 3 23
2
3 22 2 22 2
1
3
3 31 exp ,
2
x z
D
pe
pepe
k k k
i π
λ
ω ω ω
ω ω ωω ω
/ ж цж ц з чз ч и ши ш
= + = ґ
й щж ц
к ъз ч− + −ґ з чк ъ−− и шл ы (3)
( )
4
2 2 2
2
2 32 23
2
2 3 22 2 22 2
0
1
3
3 31 exp .
2
x z
D
pe B
pez pe
k k k
i
k v
π
λ
ω ωω ω
ω ω ωω ω ω
/ ж цж ц з чз ч и ши ш
= + = ґ
й щж ц
к ъз ч− − + −ґ
з чк ъ−− − и шл ы
(4)
Here kx2 is the Langmuir waves’ transversal wavenumber
for plasma without beam, kx4 is the transversal wavenum
ber for infinite beam in plasma, and 2d is a beam width.
152 Problems of Atomic Science and Technology. Series: Plasma Physics (11). 2005. № 2. P. 152-154
In order to find dispersion equation for the beam in plas
ma boundary conditions for the potential on the beam-plas
ma border should be written. Potentials are specified as:
( ) ( )( ) ( )1 1 2 2exp exp expm z m x zk x ik x ik zϕ ϕ ϕ= + − ;
( )
( ) ( )
3 7
8 2
( exp
exp ) exp
m z
m x z
k x
ik x ik z
ϕ ϕ
ϕ
= − +
+
; (5)
( ) ( )
( ) ( ) ( )
2 3 4
5 4 6 4
( exp exp
exp exp )exp .
m z m z
m x m x z
k x k x
ik x ik x ik z
ϕ ϕ ϕ
ϕ ϕ
= + − +
+ + −
Here ϕ1, ϕ2 and ϕ3 are the potentials in the areas 1, 2 and
3, respectively (see Fig.1). Boundary conditions that pro
vide continuity of the potentials, their derivatives and
Laplacians on the beam-plasma border have a form:
( )1 0
x dx
ϕ
=−
∂ =
∂
, ( )3 0
x dx
ϕ
=
∂ =
∂
,
( )2 0
x dx
ϕ
=±
∂ =
∂ , (6)
1 2x d x d
ϕ ϕ
=− =−
= , 2 3 ,
x d x d
ϕ ϕ
= =
=
1 2 ,x d x dϕ ϕ
=− =−
∆ = ∆ 2 3 .x d x dϕ ϕ
= =
∆ = ∆
As a result homogeneous equations’ set is obtained. It
has a non-trivial solution only in the case of its determi
nant is equal to zero. Desired dispersion equation found
from this condition has a form (see, e.g., [2]):
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ){
( ) ( ) ( ) }
2 2
4 2 4
2 2
2 4 4
2 2 2 2
2 4 4 2
2 2
4 4 2 4
, [ cos
sin ]
cos sin ( ) 0.
x z x z x
z x z x x z
x z x z x x z
x x z x x z
F k k k k sh k d k d
k ik k k k d ch k d
ik k k k k k sh k d
k d k k k k d ch k d
ω = + +
+ + + ґ
й щ+ + −ґ ґл ы
+ + =ґ
r
(7)
First co-factor in (7) corresponds to the antisymmetric
modes and the second one – to the symmetric modes.
Each co-factor was also obtained for symmetric and anti
symmetric modes separately.
3. NUMERICAL SOLUTION OF THE DISPER
SION EQUATION
Equation (7) was solved numerically. Dispersion func
tion F was studied as a function of real and imaginary
parts of frequency. Well defined maximums of the value
(–logF) on the plane of complex frequency correspond to
the roots of the dispersion equation. They become more
acute and unreservedly grow if they are built more accu
rately. Identification of the waves’ types was performed
by investigation of extreme cases of the model.
The roots obtained correspond to Langmuir waves
(Fig.2a) and beam-plasma modes (Fig.2b). Langmuir
waves have symmetrical and antisymmetrical branches.
Beam-plasma modes can be of stable and unstable type.
The dispersion curve and increment (decrement) depen
dence on the wave number are plotted on Fig.3a-b, re
spectively.
In the presence of the beam the root corresponding to
Langmuir wave in plasma is accompanied by a family of
roots with stronger damping. The discreteness of the roots
can be explained by the transversal restriction of the
beam. In the direction normal to the beam motion stand
ing waves occur.
a
b
Fig.2. Family of roots corresponding to Langmuir waves
(a) and beam-plasma modes (b): ω = ω1+iω2, ωb/ω
p=0.01, vb/vTe=4.58, d/λD=50, kzλD=0.24
a
b
Fig.3. Dispersion dependency of the real (a) and imagi
nary (b) parts of the frequency for one of the roots near
153
the point of Cherenkov resonance (ω = ω1+iω2,, ωb/ω
p=0.006, vb/vTe=4.58, d/λD=30)
Investigation shows that maximal increment does not
depend on the beam width, but only on its density and ve
locity. It results from the fact that maximal increment cor
responds to the purely longitudinal waves propagating
along the beam motion.
Dependence of the real part of the squared transversal
wavenumber in plasma is plotted at Fig.4.
Fig.4. Dependence of the squared real part of the
transversal wave number: ω =ω1+iω2, ωb/ωp=0.01,
vb/vTe=4.58, d/λD=50, kzλD=0.24
One can see that curve is lying in the area of the nega
tive and positive values and passes zero value in the syn
chronism point. Magnitude Re(kx2)2 defines en existence
of the waves’ excitation from the beam. So, in the posi
tive values’ area, on the right of the synchronism point,
the Langmuir waves’ emission from the beam takes place.
On the left of the synchronism point there is an exponen
tial reducing of the electric field out of the beam.
4. CONCLUSIONS
Dispersion equation for the stripped beam with sharp
boundaries moving in the warm isotropic plasma was ob
tained. This equation analysis indicates the presence of
two types of waves: Langmuir waves and beam-plasma
modes. Langmuir waves have symmetrical and antisym
metrical families of the roots and beam-plasma modes are
of dumping and growing types.
Large number of roots is a result of the presence of the
preferential direction in the system and transversal limita
tion of the beam. Due to that in the system standing wave
occurs.
Maximum increment corresponds to the purely longi
tudinal waves that is why it does not depend on the
transversal dimension of the beam but on its density and
velocity.
The real part of the squared transversal wave number
can be of negative and positive value. In positive values
area separate beams in periodical sequence of the beams
can interact via the Langmuir waves excitation even if the
distance between the beams is relatively large. Contrary,
in the negative values area interaction between the beams
is minimal and periodical sequence can be considered as a
set of the independent beams if the distance between
beams is more then specific length of the electric field re
ducing |kx2|-1.
REFERENCES
1. J.Sousek, T.Dudok de Wit, V.Krasnoselskikh, J.Pick
ett. Statistical properties high frequency electric field
oscillations in the foreshock. // Spatio-Temporal
Analysis and Multipoint Measurements in Space. Or
leans, France, 12-16 May 2003. P.33-34.
2. A.F.Aleksandrov, L.S.Bogdankevich, A.A.Rukhadze.
Basics of plasma electrodynamics. Moscow:
“Vys’shaja shkola”, 1978 (In Russian).
ИЗЛУЧЕНИЕ ЛЕНГМЮРОВСКИХ ВОЛН ЭЛЕКТРОННЫМ ПУЧКОМ
ОГРАНИЧЕННОГО СЕЧЕНИЯ В ОБЛАСТИ ЕЛЕКТРОННОГО ФОРШОКА
И.О. Анисимов, В.В. Красносельских, К.С. Мусатенко
Предложена модель моноэнергетического ленточного пучка с резкими границами, которая базируется на ре
зультатах измерений эксперимента КЛАСТЕР в области форшока ударной волны Земли. Было получено дис
персионное уравнение и проанализировано числовыми методами. Изучены зависимости положения корней дис
персионного уравнения, соответствующих кинетическому механизму плазменно-пучковой неустойчивости, от
параметров модели.
ВИПРОМІНЮВАННЯ ЛЕНГМЮРІВСЬКИХ ХВИЛЬ ЕЛЕКТРОННИМ ПУЧКОМ
ОБМЕЖЕНОГО ПЕРЕРІЗУ В ОБЛАСТІ ЕЛЕКТРОННОГО ФОРШОКУ
І.О. Анісімов, В.В. Красносельських, К.С. Мусатенко
Запропоновано модель моноенергетичного стрічкоподібного пучка з різкими границями, яка базується на ре
зультатах вимірювань експерименту КЛАСТЕР в області форшоку ударної хвилі Землі. Було отримано дис
персійне співвідношення, яке проаналізовано числовими методами. Вивчені залежності положення коренів, що
відповідають кінетичному механізму плазмово-пучкової нестійкості, від параметрів моделі.
154
|