Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction
Modulational instability of a circularly polarized electromagnetic wave, propagation along an external constant magnetic filed in plasma is investigated. The method is based on the derivation of a nonlinear Schrodinger equation, which contains nonlinear terms associated with trapped electrons and al...
Збережено в:
| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2002 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/80319 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction / H. Hakimi Pajouh, H. Abbasi // Вопросы атомной науки и техники. — 2002. — № 4. — С. 112-114. — Бібліогр.: 6 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859829098111565824 |
|---|---|
| author | Hakimi Pajouh, H. Abbasi, H. |
| author_facet | Hakimi Pajouh, H. Abbasi, H. |
| citation_txt | Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction / H. Hakimi Pajouh, H. Abbasi // Вопросы атомной науки и техники. — 2002. — № 4. — С. 112-114. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Modulational instability of a circularly polarized electromagnetic wave, propagation along an external constant magnetic filed in plasma is investigated. The method is based on the derivation of a nonlinear Schrodinger equation, which contains nonlinear terms associated with trapped electrons and also relativistic electron quiver velocity. Among different mechanisms of trapping the collision-less process is considered. Since we are interested in characteristic spatial scales larger than the Debye length, the condition of quasi-neutrality is assumed. The maximum growth rate is calculated and the result shows that the trapped particles prevent the spatial localization of the electron cyclotron waves.
|
| first_indexed | 2025-12-07T15:31:46Z |
| format | Article |
| fulltext |
MODULATIONAL INSTABILITY OF THE ELECTRON CYCLOTRON
WAVES IN AN ADIABATIC WAVE - PARTICLE INTERACTION
H. Hakimi Pajouh 1, H. Abbasi 1,2
1 Institute for Studies in Theoretical Physics and Mathematics,
P. O. Box 19395-5531, Tehran, Iran
2 Physics Department, Amir Kabir University,
P. O. Box 15875-4413, Tehran, Iran
Modulational instability of a circularly polarized electromagnetic wave, propagation along an external constant
magnetic filed in plasma is investigated. The method is based on the derivation of a nonlinear Schrodinger equation,
which contains nonlinear terms associated with trapped electrons and also relativistic electron quiver velocity. Among
different mechanisms of trapping the collision-less process is considered. Since we are interested in characteristic
spatial scales larger than the Debye length, the condition of quasi-neutrality is assumed. The maximum growth rate is
calculated and the result shows that the trapped particles prevent the spatial localization of the electron cyclotron waves.
PACS: 52.35.Hr
INTRODUCTION
The development of ultra intense short pulse lasers
allows exploration of fundamentally new parameter
regimes for nonlinear laser plasma interaction. Now we
have laser beams with intensities up to 1019 W/cm2 in
laboratories, at such a high intensities electrons quiver
velocity becomes relativistic. In such a situation,
consideration of the relativistic electron mass variation [1]
as well as the relativistic ponderomotive force [2] is very
essential in the study of nonlinear laser plasma
interactions.
On the other hands, propagation of high frequency
waves in plasma can generate the longitudinal (potential)
waves in which some of the plasma particles can be
trapped (electron here). Among different mechanisms of
trapping, the mechanism resulted from nonstationarity of
the field is considered [3, 4]. It means that τ the
characteristic time of variation of the field is small
compared with the electron mean free time, so that the
plasma is collisionless. Besides, we restrict ourseleves to
the adiabatic approximation. That means, τ should be
much larger than the period of electronic oscillations in
the field trough. By virtue of this approximation, the field
may be regarded as stationary during the passage of an
electron through it and to the same accuracy, the electron
distribution function in the field can be expressed by an
adiabatic invariant. Under this approximation the
electrons density can be analytically calculated as
follows:
,21
0 e
eff
e
effT
U
e
T
U
T
U
erfe
n
n
e
eff
−+
−−=
>< −
π
(1)
where effU describes the potential well and eT is the
temperature of electrons. We will later explain the
meaning of brackets.
In this paper, we consider the influence of trapped
electrons on the modulational instability of electron
cyclotron waves. The organization of the paper is as
follows. First, from basic equation, an envelope equation,
including trapped electrons, will be obtained. Then, it is
solved for the investigation of modulational instabilities.
Finally, a summary of results will be presented.
BASIC EQUATIONS
Consider the propagation of a right-handed circularly
polarized electromagnetic wave along the external
magnetic field zBB
00 = in the form of,
( ) ( ) ( ) .,.exp,
2
1 cctitzEyixE +−+= ω
(2)
where ω is the frequency of external electromagnetic
field and ..cc stands for the complex conjugate.
The complete set of equations is:
,41 J
c
E
c
B t
π+∂=×∇ (3)
,B
c
1E t
∂−=×∇ (4)
,4π ρ=⋅∇ E
(5)
( ),ei nne −=ρ (6)
( ),eeii VnVneJ
−= (7)
where ee = . In order to close the above systems of
equations, ρ and J should be properly defined. Since
we are going to study the effect of trapped electrons we
assume electron temperature, eT , is much larger that ions
one. Therefore, electrons are treated kinetically [eq. (1)]
while the fluid treatment is used for the ions dynamics.
In our formalism there are two distinguishable time
scales: one associated with the fast motion with the
112 Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 112-114
characteristic time of the order of the high frequency
oscillation, ωπτ 2=p , and another with the slow motion
resulting from the ions motion. Therefore, each quantity
can be decomposed into the slowly and rapidly varying
parts:
,~AAA +>= < (8)
where the brackets denote averaging over the time
interval pτ .
We consider the one-dimensional case in which
0== ∂∂ yx . Therefore, the nonlinear interaction of a
pump field with the background plasma that will generate
a slowly varying envelope for the amplitude of
electromagnetic wave propagating along the external
magnetic field involving the weak relativistic quiver
velocity is governed by [5]:
EEcEi
e
p
zzt
e
−
Ω−
−+ ∂∂ 2
2
22 ω
ω
ω ω
ω
( )
,0
2 222
22
0
2
=
Ω−Ω−
−
Ω−
− E
cm
Ee
n
n
eee
e
e
pe
ωω
ωδ
ω
ω ω
(9)
where c is light velocity, ep mne
e 0
22 4πω = the electron
plasma frequency, 0n the unperturbed electron density,
cmeB ee 0=Ω the electron cyclotron frequency, em the
electron rest mass and 0nnn ee −〉〈=δ .
In order to define the slowly varying deviation of the
density, enδ , taking into account the electron trapping we
have to proceed from eq. (1). In eq. (1) we need the
definition of effU . For this purpose one should calculate
the average of Vlasov equation over the time interval pτ
to find [6]:
( ) ,0
2
22
=〉〈
Ω−
+−
〉〈+〉〈
∂∂−
∂∂
ep
ee
z
ezzet
f
m
Ee
e
fvf
zωω
ϕ
(10)
where ef denotes the distribution of electrons. From here
we can define the effective potential energy as follow:
( )ee
eff m
Ee
eU
Ω−
+−=
ωω
ϕ
2
22
(11)
We assume the effective potential energy has the
form of a single potential well, vanishing at infinity
(soliton solution). Then, by substituting eq. (11) in eq. (1)
the density of electrons is clearly defined. As it is
mentioned, the ions will also be described by the
hydrodynamic equations assuming their temperature is
negligibly small. Then for the shallow well when
eeff TU < < ( 0nne < <δ ) and for the case when quasi-
neutrality condition is fulfilled ( ie nn δδ ≈ ) one can find
for the enδ the following expression:
( )
( ) ,
23
4
2
2
3
222
5
22
2
22
22
2
0
Ω−
−
+
Ω−−
=
eees
s
eees
se
Tm
Ee
cu
c
Tm
Ee
cu
c
n
n
ωωπ
ωω
δ
(12)
where u is the velocity of stationary envelope, and
ies mTc =2 ( im =mass of ion).
Equation (12) together with equation (9) construct a
complete set of equations for the investigation of
modulational instabilities of electron cyclotron waves.
MODULATIONAL INSTABILITY
Now we Consider the case of propagating wave
along the constant magnetic field. Let define the slowly
varying amplitude as ( ) ikzetzE ,ψ= . Substituting E in
eqs. (9) and (12), we have:
( )
( )
( )
( )
.0
2
||
2
||
3
4
2
||
22
222
22
2/3222/5
22
2
22
22
22
2
2
22
=
Ω−Ω−
−
Ω−
−
+
Ω−−Ω−
−
−
Ω−
−−∂+∂+∂
ψ
ω
ψ
ω
ω
ωω
ψ
π
ωω
ψ
ω
ωω
ψω
ω
ωω
ψψψψω
eee
eees
s
eees
s
e
p
e
p
zzzt
cm
e
Tm
e
cu
c
Tm
e
cu
c
kikci
(13)
To consider the effect of trapping on the modulational
instabilities, we perturbed eq. (13) (with a long time and
wavelength perturbation) around its plane wave solution,
as follows:
( ) ).(0 tiExp ωδ ψψψ ∆−+= (14)
Here 0ψ is real and ω∆ shows the nonlinear
frequency shift. On substituting eq. (14) in eq. (13) and
considering its zero order (respect to δ ψ ), one can find
the following expression for the nonlinear frequency shift
( )
( )
( )
( ) .
2
3
4
2
2
22
1
3
0
2/32
2/5
22
2
2
2
0222
2
2
22
2
2
222
2
ψ
ωω
πω
ω
ψ
ω
ω
ω
ωω
ω
ω
ω
ω
ω ω
ωω
Ω−
−
Ω−
+
Ω−
Ω−
−
Ω−−
Ω−
+
+−
Ω−
=∆
eees
s
e
p
ee
eeees
s
e
p
e
p
Tm
e
cu
c
cm
e
Tm
e
cu
c
ck
(15)
The first order of eq. (13) respect to δ ψ together with
eq. (15) results to the following equation for the complex
amplitude of the perturbation,
[ ]
( ) ( )
( )
.0
2
2
2
2
2/32
0
22/5
22
2
2/5
2
22
2
0
2
2
2
2
2
22
2
2
2
22
=
−Ω−
−
Ω−
−
−Ω−
−
∂+∂+∂
δ ψ
ψ
ωω
ω
π
δ ψ
ψ
ω
ω
ω
ω
δ ψδ ψω
ees
s
e
p
eeTs
s
e
p
zzzt
Tm
e
cu
c
cm
e
v
c
cu
c
ckci
(16)
By dividing the δ ψ into real and imaginary parts and
considering the plane wave solutions, as follows,
),(~
~
, tiziExp
N
M
N
M
iNM ωχδ ψ −
=
+=
(17)
we can find the nonlinear dispersion relation,
( )
( )
( )
.
2
4
1
4
2/32
0
22/5
222
22
2/52/3
222
22
2
0
2
2
2
222
22
2
22
2
2
22
2
0
2
22
22
2
4422
−ΩΩ−
+
−ΩΩ−
=
Ω−
−−
−Ω
ees
s
e
p
eTs
s
e
p
ee
p
Tm
e
c
cc
cm
e
v
c
c
cc
cm
e
c
ckc
ψ
χ
χ
ωωπ
ωχ
ψ
χ
χ
ω
χ
ω
ω
ψ
ωχ
ωω
ω
χχ
ω
(18)
Among different possible cases, we consider the case
when
.,
22
< <ΩΩ+=Ω χ
ω
δδχ
ω
kckc
(19)
The maximum value of the growth rate is achieved
at mχχ = , where,
( ) ( )
( )
,
41
11
2
2/3
22
2
2
2
22
2
2
2
2
2
2
22
−
Ω−
+
−
Ω−
−
Ω−
=
W
ckc
c
c
v
ckc
c
v
cW
c
s
s
T
e
s
s
Tee
pm
ω
ωω
ω
π
ω
ωωωω
ω
ω
χ
(20)
and it is equal to
.
2
1Im
2
22
ω
χ
δ m
Max
c
=Ω (21)
From analysis of growth rate we can see the term causes
by the trapped particles has a negative sign so we need a
harder restriction for the amplitude of the homogeneous
wave.
CONCLUSION
In this paper we considered the propagation of electron
cyclotron waves, propagating along an external constant
magnetic field in plasma. We included the adiabatic
trapping of the electrons in the longitudinal field. As it is
mentioned, according to the existence of two natural time
scales in the plasma, an envelope equation has been found
for the slowly varying part of the wave amplitude. It is
contained both the trapping of electrons and their
relativistic mass variations.
It was shown that this system is modulationally
unstable, but the trapped electrons have a stabilizing
influence. It means that at the presence of trapped
electrons for the creation of nonlinear structures, e.g.
solitons, we will encounter with harder conditions.
REFERENCES
[1] P. K. Kaw and J. M. Dawson, Phys. Fluids, 13, 472
(1970).
[2] T. Tajima and J. M. Dawson, Phys. Rep. 122, 173
(1985).
[3] H. Abbasi, M. R. Rouhani and D. D. Teskhakaya,
physica Scripta, 56, 619 (1997).
[4] A. V. Gurevich, JETP. 53, 953 (1967).
[5] N. N. Rao, P. K. Shukla and M. Y. Yu, Phys.
Fluids 27, 2664 (1984).
[6] L. M. Kerashvili and N. L. Tsintsadze, Fiz. Plaszmy 9,
570 (1983).
INTRODUCTION
BASIC EQUATIONS
MODULATIONAL INSTABILITY
CONCLUSION
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-80319 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:31:46Z |
| publishDate | 2002 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Hakimi Pajouh, H. Abbasi, H. 2015-04-14T17:58:04Z 2015-04-14T17:58:04Z 2002 Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction / H. Hakimi Pajouh, H. Abbasi // Вопросы атомной науки и техники. — 2002. — № 4. — С. 112-114. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.35.Hr https://nasplib.isofts.kiev.ua/handle/123456789/80319 Modulational instability of a circularly polarized electromagnetic wave, propagation along an external constant magnetic filed in plasma is investigated. The method is based on the derivation of a nonlinear Schrodinger equation, which contains nonlinear terms associated with trapped electrons and also relativistic electron quiver velocity. Among different mechanisms of trapping the collision-less process is considered. Since we are interested in characteristic spatial scales larger than the Debye length, the condition of quasi-neutrality is assumed. The maximum growth rate is calculated and the result shows that the trapped particles prevent the spatial localization of the electron cyclotron waves. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction Article published earlier |
| spellingShingle | Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction Hakimi Pajouh, H. Abbasi, H. Basic plasma physics |
| title | Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction |
| title_full | Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction |
| title_fullStr | Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction |
| title_full_unstemmed | Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction |
| title_short | Modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction |
| title_sort | modulational instability of the electron cyclotron waves in an adiabatic wave - particle interaction |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80319 |
| work_keys_str_mv | AT hakimipajouhh modulationalinstabilityoftheelectroncyclotronwavesinanadiabaticwaveparticleinteraction AT abbasih modulationalinstabilityoftheelectroncyclotronwavesinanadiabaticwaveparticleinteraction |