Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons
The influence of resonant electrons in phase space on the nonlinear dynamics of electron-cyclotron waves is investigated. It is shown that in the case when the frequency of externally applied electromagnetic waves is close to that of the electron-cyclotron waves, the nonlinear coupling between the e...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2002 |
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| Мова: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
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| Цитувати: | Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons / H. Abbasi, H. Hakimi Pajouh // Вопросы атомной науки и техники. — 2002. — № 4. — С. 158-160. — Бібліогр.: 11 назв. — англ. |
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Abbasi, H. Hakimi Pajouh, H. 2015-04-14T17:52:31Z 2015-04-14T17:52:31Z 2002 Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons / H. Abbasi, H. Hakimi Pajouh // Вопросы атомной науки и техники. — 2002. — № 4. — С. 158-160. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 52.35.-g https://nasplib.isofts.kiev.ua/handle/123456789/80314 The influence of resonant electrons in phase space on the nonlinear dynamics of electron-cyclotron waves is investigated. It is shown that in the case when the frequency of externally applied electromagnetic waves is close to that of the electron-cyclotron waves, the nonlinear coupling between the electron-cyclotron waves and the resulting longitudinal perturbations is mainly caused by the pondermotive potential of the electrons. It is shown that in the competition of refraction caused by the non-resonant electrons and the relativistic nonlinearity, the influence of resonant electrons will increase strongly. The reaction of such longitudinal waves with electrons that are in resonance with high frequency electromagnetic field changes the character of the soliton propagation in an essential manner. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons |
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Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons Abbasi, H. Hakimi Pajouh, H. Plasma electronics |
| title_short |
Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons |
| title_full |
Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons |
| title_fullStr |
Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons |
| title_full_unstemmed |
Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons |
| title_sort |
stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons |
| author |
Abbasi, H. Hakimi Pajouh, H. |
| author_facet |
Abbasi, H. Hakimi Pajouh, H. |
| topic |
Plasma electronics |
| topic_facet |
Plasma electronics |
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2002 |
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English |
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Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| description |
The influence of resonant electrons in phase space on the nonlinear dynamics of electron-cyclotron waves is investigated. It is shown that in the case when the frequency of externally applied electromagnetic waves is close to that of the electron-cyclotron waves, the nonlinear coupling between the electron-cyclotron waves and the resulting longitudinal perturbations is mainly caused by the pondermotive potential of the electrons. It is shown that in the competition of refraction caused by the non-resonant electrons and the relativistic nonlinearity, the influence of resonant electrons will increase strongly. The reaction of such longitudinal waves with electrons that are in resonance with high frequency electromagnetic field changes the character of the soliton propagation in an essential manner.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/80314 |
| citation_txt |
Stationary dynamics of nonlinear electron-cyclotron waves at the presence of resonant electrons / H. Abbasi, H. Hakimi Pajouh // Вопросы атомной науки и техники. — 2002. — № 4. — С. 158-160. — Бібліогр.: 11 назв. — англ. |
| work_keys_str_mv |
AT abbasih stationarydynamicsofnonlinearelectroncyclotronwavesatthepresenceofresonantelectrons AT hakimipajouhh stationarydynamicsofnonlinearelectroncyclotronwavesatthepresenceofresonantelectrons |
| first_indexed |
2025-11-27T02:10:46Z |
| last_indexed |
2025-11-27T02:10:46Z |
| _version_ |
1850793061508972544 |
| fulltext |
STATIONARY DYNAMICS OF NONLINEAR ELECTRON-CYCLOTRON
WAVES AT THE PRESENCE OF RESONANT ELECTRONS
H. Abbasi 1,2 and H. Hakimi Pajouh 1
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
The influence of resonant electrons in phase space on the nonlinear dynamics of electron-cyclotron waves is
investigated. It is shown that in the case when the frequency of externally applied electromagnetic waves is close to that
of the electron-cyclotron waves, the nonlinear coupling between the electron-cyclotron waves and the resulting
longitudinal perturbations is mainly caused by the pondermotive potential of the electrons. It is shown that in the
competition of refraction caused by the non-resonant electrons and the relativistic nonlinearity, the influence of resonant
electrons will increase strongly. The reaction of such longitudinal waves with electrons that are in resonance with high
frequency electromagnetic field changes the character of the soliton propagation in an essential manner.
PACS: 52.35.-g
INTRODUCTION
Circularly polarized electromagnetic waves are useful
for heating of fusion plasma [1], modifications of the
lower part of the earth's ionosphere [2]. Accordingly, it is
essential to understand the linear as well as the nonlinear
propagation characteristics of cyclotron waves. The linear
propagation of electron cyclotron waves is fully
understood and the results have been summarized in a
monograph [3]. The nonlinear propagation involves the
study of three wave interaction, wave modulation, self-
filamentation, soliton formation, etc.
In the past, there have been substantial efforts [4] in
the investigating of electron cyclotron waves. However,
most of the studies were concerned with the study of
small amplitude case. Nevertheless, the development of
ultraintense short pulse lasers allows exploration of
fundamentally new parameter regimes for nonlinear laser
plasma interaction. In fact, a number of experiments have
been carried out in which plasmas are irradiated by laser
beams with intensities up to 219 /10 cmW . At such
intensities, the electron quiver velocity approaches the
speed of light and a host of phenomena have been
predicted such as parametric resonance in an electron
plasma [5], the relativistic self focusing [6] and the
generation of large amplitude plasma waves (wake field)
[7].
Among the nonlinear phenomena, trapping of plasma
particles in the trough of low frequency longitudinal
waves of plasma exert considerable influence on the
propagation characteristics of such waves.
Determination of collisionless distribution function
and its moments in nonstationary fields, which describes
both free and trapped particles, involves difficult
mathematics demanding simplifying assumptions.
Gurevich [8], assuming slowly varying longitudinal
fields and a collisionless trapping mechanism, calculate
the particle distribution function self-consistently.
The validity of his solution is demonstrated by simulation
and experimental results of Debye shielding and
antishielding [9]. When applied field changes slowly, the
distribution function is a continuous function of total
energy. Then the distribution of untrapped particle is
determined by that of particles coming from infinity with
a Maxwellian distribution. It then follows from the
continuity (continuity of particles current from the
trapped to untrapped region) conditions that trapped
particle disteribution reduces to a constant. In this way,
integrating over the free and trapped particle distributions
results in the following total density:
,21
0 αα
α
π
α
T
U
T
U
erfe
n
n effeffT
Ueff
−+
−−=
>< −
(1)
where effU describes the potential well and αT is
temperature of particles (α =species index; e,i). We will
later explain the meaning of brackets.
In this paper, we consider a class of problems
involving the interaction of relativistically intense
monochromatic radiation with a magnetized plasma when
the effect of trapped electrons is taken into account. To
our knowledge, the present study represents the first
investigation of the effects of the trapped electrons on the
soliton formation.
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:
158 Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 158-160
J
c
E
c
B t
π41 +∂=×∇ , (3)
B
c
1E t
∂−=×∇ , (4)
π ρ4=⋅∇ 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
characteristic time of the order of the high frequency
oscillation, ωπτ 2= , 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 τ .
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 [10]:
EEcEi
e
p
zzt
e
−
Ω−
−+ ∂∂ 2
2
22 ω
ω
ω ω
ω (9)
( )
,0
2 222
22
0
2
=
Ω−Ω−
−
Ω−
− E
cm
Ee
n
n
eee
e
e
pe
ωω
ωδ
ω
ω ω
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 τ to
find [11]:
( ) ,0
2
22
=〉〈
Ω−
+−
〉〈+〉〈
∂∂−
∂∂
ep
ee
z
ezzet
f
m
Ee
e
fvf
zωω
ϕ
(10)
where ef is the distribution of electrons. From here we
can define the effective potential energy as follow:
( ) .
2
22
ee
eff m
Ee
eU
Ω−
+−=
ωω
ϕ (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).
Substituting eq. (12) in eq. (9) and introducing the
dimensionless variables as follows
,
8
,,
0
2
2
E
Tn
Ezz
c
tt
e
pe →→→
πω
ωωω (13)
will leads to the following equation:
( )
,0
12
32
2
=−+
−
Ω−
−∂+∂
EEPEER
EEEi
e
p
zzt
e
ωω
ω
(14)
where,
( ) ( )
,
22
2
2
2
2
2
2
2
−
−
Ω−Ω−
=
s
sT
ee cu
c
c
v
R e
ω
ω
ω
ω
(15)
159
,
3
4 2
5
22
22
5
−
Ω−
=
s
s
ep cu
c
P
e
ω
ω
ω
ω
π
(16)
and eeT mTv
e
= .
The first term in eq. (15) describes the weak
relativistic effect and the second one is connected with the
density deviation of free electrons. It is essential that
these terms have different signs. The last term in eq. (14)
describes the influence of trapped electrons. It must be
mentioned that, as it is seen from eq. (12), in the case of
eΩ>ω , the trapping of electrons is possible only in the
supersonic regime when 22
scu > .
SOLUTION AND RESULTS
Now, we will consider the case when eΩ>ω . To
solve the eq. (14) together with expressions (15) and (16)
we use the standard method in which the amplitude of the
high frequency field is expressed in the following form:
)],,(exp[),(),( tzitzatzE ψ= (17)
where a and ψ are real functions. On substituting eq.
(17) in eq. (14), one can find for the real and imaginary
parts of eq. (14) the following equations:
( )( ) ,022 =∂+∂∂+∂ ψψ zzzzt aa (18)
( ) ( )
.0
12
43
2
2
=−+
−
Ω−
−∂−∂−∂
paRa
Eaa
e
p
ztzz
e
ωω
ω
ψψ
(19)
In the stationary state when the amplitude and phase
depend on the argument,
,t
c
uz −=ξ (20)
following the boundary conditions:
,0)(, →± ∞→ ξξ a (21)
eqs. (18) and (19) are integrable and the equation for the
amplitude can be presented in the following form:
( ) ( ) ,0
2
1 2 =+∂ aVaξ (22)
where,
( ) ( ) ,
5
1
4
11
2
1 542
2
22
PaRaa
c
uaV
e
pe −+
−−
Ω−
−=
ωω
ω
(23)
is the effective potential . Equation (22) is identical in
form to the energy equation of a single particle with
coordinate a and time ξ . The relation between the
maximum amplitude of soliton and its velocity is:
( ) .
5
2
2
11 3
max
2
max
2
2
2
PaRa
c
u
e
pe +−
−
Ω−
=
ωω
ω
(24)
A glance at the effective potential )(aV shows the
effect of trapped electron ( )551 Pa− . As it is seen from
eq. (23) to have a bound solution for the eq. (22), the
equation ,
,0)( =aV (25)
should at least have two roots. In the competition of
refraction caused by the non-resonant electrons and the
relativistic nonlinearity, R can becomes enough small
that the last term in )(aV (caused by the trapped
electrons) dominates R and consequently it can destroy
the localized structure of the amplitude.
REFERENCES
[1] K. Matsuda, Phys. Fluids 29, 3490 (1986).
[2] S. P. Kuo and M. C. Lee, Geophys. Res. Lett. 10, 979
(1983).
[3] D. G. Lominadze, Cyclotron waves in plasmas
(Pergamon, New York, 1981).
[4] V. I. Karpman and H. Washimi, J. Plasma Phys. 18,
173 (1977).
[5] N. L. Tsintsadze, Sov. Phys. JETP 32, 684 (1971).
[6] D. P. Garuchava, N. L. Tsintsadze, and D. D.
Tskhakaya, Sov. Phys. JETP 71, 873 (1990).
[7] T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267
(1979).
[8] A. V. Gurevich, Sov. Phys. JETP 53, 953 (1967).
[9] C. Hansen, A. B. Reimann, and J. Fajans, Phys.
Plasmas 3, 1820 (1996).
[10] N. N. Rao, P. K. Shukla and M. Y. Yu, Phys.
Fluids 27, 2664 (1984).
[11] L. M. Kerashvili and N. L. Tsintsadze, Fiz. Plaszmy
9, 570 (1983).
160
INTRODUCTION
BASIC EQUATIONS
SOLUTION AND RESULTS
REFERENCES
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