Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature
Dispersion equation is analyzed for field aligned ion-cyclotron waves in a two-dimensional (2D) magnetospheric plasma with circular magnetic field lines. The steady-state bi-Maxwellian distribution function is used to model the energetic protons in a hydrogen plasma at the geostationary orbit. As in...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Цитувати: | Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature / N.I. Grishanov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 78-80. — Бібліогр.: 7 назв. — англ. |
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Grishanov, N.I. Azarenkov, N.A. 2017-01-04T12:11:03Z 2017-01-04T12:11:03Z 2007 Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature / N.I. Grishanov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 78-80. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 94.30.cq, 94.30.Hn https://nasplib.isofts.kiev.ua/handle/123456789/110410 Dispersion equation is analyzed for field aligned ion-cyclotron waves in a two-dimensional (2D) magnetospheric plasma with circular magnetic field lines. The steady-state bi-Maxwellian distribution function is used to model the energetic protons in a hydrogen plasma at the geostationary orbit. As in the uniform magnetic field, the growth rate of the proton-cyclotron instability (PCI) in a 2D magnetospheric plasma is defined by the contribution of the energetic ions/protons to the imaginary part of the transverse permittivity elements. It is shown that the PCI growth rate in 2D axisymmetric magnetosphere can be significantly smaller than that is for the straight magnetic field case with the same macroscopic bulk parameters. Проаналізовано дисперсійне рівняння для іонно-циклотронних хвиль, які поширюються вздовж геомагнітного поля у двовимірній (2D) магнітосферній плазмі з силовими лініями магнітного поля у вигляді кола. Бімаксвелівська функція розподілу використана при моделюванні розподілу енергійних протонів у водневій плазмі поблизу геостаціонарних орбіт. Як і у випадку однорідного магнітного поля, інкремент зростання протонно-циклотронної нестійкості (ПЦН) у 2D магнітосфері визначається внеском енергійних протонів в уявну частину поперечної діелектричної проникливості. Доведено, що інкремент ПЦН у 2D аксіально-симетричній магнітосфері може бути значно нижчим, ніж для плазми у прямому магнітному полі з тими ж самими макроскопічними параметрами. Проанализировано дисперсионное уравнение ионно-циклотронных волн, распространяющихся параллельно геомагнитному полю в двумерной (2D) магнитосферной плазме с круговыми силовыми линиями удерживающего магнитного поля. Бимаксвелловская функция распределения использована при моделировании распределения энергичных протонов в водородной плазме вблизи геостационарных орбит. Как и в однородном магнитном поле, инкремент нарастания протонно-циклотронной неустойчивости (ПЦН) в 2D магнитосфере определяется вкладом энергичных протонов в мнимую часть поперечной диэлектрической проницаемости. Показано, что инкремент ПЦН в 2D аксиально-симметричной магнитосфере может быть значительно ниже, чем для плазмы в прямом магнитном поле с теми же макроскопическими параметрами. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature Нестійкість іонно-циклотронних хвиль у 2D-магнітосферній плазмі з анізотропною температурою Неустойчивость ионно-циклотронных волн в 2D-магнитосферной плазме с aнизотропной температурой Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature |
| spellingShingle |
Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature Grishanov, N.I. Azarenkov, N.A. Basic plasma physics |
| title_short |
Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature |
| title_full |
Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature |
| title_fullStr |
Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature |
| title_full_unstemmed |
Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature |
| title_sort |
instability of ion-cyclotron waves in a 2d magnetospheric plasma with anisotropic temperature |
| author |
Grishanov, N.I. Azarenkov, N.A. |
| author_facet |
Grishanov, N.I. Azarenkov, N.A. |
| topic |
Basic plasma physics |
| topic_facet |
Basic plasma physics |
| publishDate |
2007 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Нестійкість іонно-циклотронних хвиль у 2D-магнітосферній плазмі з анізотропною температурою Неустойчивость ионно-циклотронных волн в 2D-магнитосферной плазме с aнизотропной температурой |
| description |
Dispersion equation is analyzed for field aligned ion-cyclotron waves in a two-dimensional (2D) magnetospheric plasma with circular magnetic field lines. The steady-state bi-Maxwellian distribution function is used to model the energetic protons in a hydrogen plasma at the geostationary orbit. As in the uniform magnetic field, the growth rate of the proton-cyclotron instability (PCI) in a 2D magnetospheric plasma is defined by the contribution of the energetic ions/protons to the imaginary part of the transverse permittivity elements. It is shown that the PCI growth rate in 2D axisymmetric magnetosphere can be significantly smaller than that is for the straight magnetic field case with the same macroscopic bulk parameters.
Проаналізовано дисперсійне рівняння для іонно-циклотронних хвиль, які поширюються вздовж геомагнітного поля у двовимірній (2D) магнітосферній плазмі з силовими лініями магнітного поля у вигляді кола. Бімаксвелівська функція розподілу використана при моделюванні розподілу енергійних протонів у водневій плазмі поблизу геостаціонарних орбіт. Як і у випадку однорідного магнітного поля, інкремент зростання протонно-циклотронної нестійкості (ПЦН) у 2D магнітосфері визначається внеском енергійних протонів в уявну частину поперечної діелектричної проникливості. Доведено, що інкремент ПЦН у 2D аксіально-симетричній магнітосфері може бути значно нижчим, ніж для плазми у прямому магнітному полі з тими ж самими макроскопічними параметрами.
Проанализировано дисперсионное уравнение ионно-циклотронных волн, распространяющихся параллельно геомагнитному полю в двумерной (2D) магнитосферной плазме с круговыми силовыми линиями удерживающего магнитного поля. Бимаксвелловская функция распределения использована при моделировании распределения энергичных протонов в водородной плазме вблизи геостационарных орбит. Как и в однородном магнитном поле, инкремент нарастания протонно-циклотронной неустойчивости (ПЦН) в 2D магнитосфере определяется вкладом энергичных протонов в мнимую часть поперечной диэлектрической проницаемости. Показано, что инкремент ПЦН в 2D аксиально-симметричной магнитосфере может быть значительно ниже, чем для плазмы в прямом магнитном поле с теми же макроскопическими параметрами.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/110410 |
| citation_txt |
Instability of ion-cyclotron waves in a 2D magnetospheric plasma with anisotropic temperature / N.I. Grishanov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 78-80. — Бібліогр.: 7 назв. — англ. |
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AT grishanovni instabilityofioncyclotronwavesina2dmagnetosphericplasmawithanisotropictemperature AT azarenkovna instabilityofioncyclotronwavesina2dmagnetosphericplasmawithanisotropictemperature AT grishanovni nestíikístʹíonnociklotronnihhvilʹu2dmagnítosferníiplazmízanízotropnoûtemperaturoû AT azarenkovna nestíikístʹíonnociklotronnihhvilʹu2dmagnítosferníiplazmízanízotropnoûtemperaturoû AT grishanovni neustoičivostʹionnociklotronnyhvolnv2dmagnitosfernoiplazmesanizotropnoitemperaturoi AT azarenkovna neustoičivostʹionnociklotronnyhvolnv2dmagnitosfernoiplazmesanizotropnoitemperaturoi |
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2025-11-25T15:37:13Z |
| last_indexed |
2025-11-25T15:37:13Z |
| _version_ |
1850516973671153664 |
| fulltext |
78 Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 78-80
INSTABILITY OF ION-CYCLOTRON WAVES IN A 2D
MAGNETOSPHERIC PLASMA WITH ANISOTROPIC TEMPERATURE
N.I. Grishanov, N.A. Azarenkov
V.N. Karazin Kharkov National University, Department of Physics and Technology, Ukraine
Dispersion equation is analyzed for field aligned ion-cyclotron waves in a two-dimensional (2D) magnetospheric plasma with
circular magnetic field lines. The steady-state bi-Maxwellian distribution function is used to model the energetic protons in a
hydrogen plasma at the geostationary orbit. As in the uniform magnetic field, the growth rate of the proton-cyclotron instability
(PCI) in a 2D magnetospheric plasma is defined by the contribution of the energetic ions/protons to the imaginary part of the
transverse permittivity elements. It is shown that the PCI growth rate in 2D axisymmetric magnetosphere can be significantly
smaller than that is for the straight magnetic field case with the same macroscopic bulk parameters.
PACS: 94.30.cq, 94.30.Hn
1. INTRODUCTION
Cyclotron waves are an important constituent of
plasmas in solar corona, solar wind and planetary
magnetospheres. As it is known, energetic particles with
anisotropic temperature can excite a wide class of
cyclotron wave instabilities. Kinetic theory of such waves
in the straight magnetic field plasmas is developed very
well, see e.g. Refs. [1-6]. However, plasma models in the
straight magnetic field are quite rough for planetary
magnetospheres which are three-dimensional in the
general case. As more suitable, the Earth’s magnetosphere
can be considered as a two-dimensional (2D) dipole
magnetic field configuration. Another interesting 2D
magnetospheric plasma model is a configuration with
circular magnetic field lines, which is artificial but
simpler and helpful to describe the principal wave
processes in the Earth’s magnetosphere. The dispersion
equations for cyclotron waves in magnetospheric plasmas
with dipole and circular magnetic field lines were derived
in Ref. [7]. In this paper we analyze the dispersion
characteristics of the electromagnetic ion-cyclotron
(EMIC) waves in the hydrogen plasma confined in the
last plasma model including the energetic protons with the
bi-Maxwellian distribution function.
2. DISPERSION EQUATION FOR FIELD-
ALIGNED CYCLOTRON WAVES
According to Ref. [7], the dispersion equations for
field aligned cyclotron waves in magnetospheric plasmas
with circular magnetic field lines can be rewritten by
analogy with the straight magnetic field case in the form:
∑+=
σ
σε
θω
π )(21 ,
)(,
2
0
L
LR
cn nn
l
o
, (1)
where σ denotes the particle species (electron, proton,
heavy ions); n is the mode number along the geomagnetic
field; )cos/( 0 θRRL = is the non-dimensional L-shell
parameter; θ is the geographical latitude; and the
transverse permittivity elements are
×
−
−= ∑∫ ∫
∞
−∞=
∞
⊥ p l
n
lp
n
lp
To
ponn
l
o
Zpu
YAu
d
Tv
TLR ν
ν
νν
θωπ
ω
ε
0 0
'
,,
4
3
||
5.1
||0
2
',
)(
)21(
2
.1)21(1exp ||22 du
T
T
u
−−−−×
⊥
ν (2)
Here we have used the following definitions:
×
−+
−−
+−
= ∫
− ⊥
⊥
t
t
T
T
LR
nuv
b
TTb
uY
o
Tn
lp
θ
θ θω
π
θν
θ
ν ||
0
||
2
||
, 1
)()21(1
/1)(
),(
,)()(2cos)1(
)()(2cos
||
2
0
||
2
0
θθ
ω
θτ
τ
π
θ
θ
π
θ
ω
θτ
τ
π
θ
θ
π
dC
uvL
lRpn
C
uvL
lRpn
T
co
bo
p
T
co
bo
++−+
+
−−×
)()21(1
)()()(2cos)1(
)()(2cos),(
2
||
2
0
||
2
0
,
θν
θθ
θ
ω
θτ
τ
π
θ
θ
π
θ
ω
θτ
τ
π
θ
θ
π
ν
θ
θ
b
dbC
uvL
lRpn
C
uvL
lRpnuA
T
co
bo
p
T
co
bo
n
lp
t
t
−−
++−
+
−−= ∫
−
( ) ( )
5.15.1 )21(2
),(,),(
)21(
)1(2)(
ν
νθα
ννθα
ν
ν
θ
−
−Π
−
−
=
FC ,
−−
−
= θ
θν
ν
θα sin
sin)1(2
21arcsin)( 2 ,
( )νθα
ν
ν
ν
ν
θα
ν
ν
νθτ ),(
2
21,
21
),(
21
)1(2
),(
2
F−
−
−
−
Π
−
−
=
Π−= νν
π
νντ ,2,
2
)21(8)(b ,
∫
−−
=Π
α
ηνηδ
η
νδα
0
22 sin1)sin1(
),,( d ,
∫
−
=
α
ην
η
να
0
2sin1
),( dF ,
θ
θ
4cos
1)( =b ,
θ
θν
θ
τ
θ
d
b
bb
t
b
∫
−−
=
0
2 )()21(1
)(4 ,
||Tv
vu = ,
ννθ 2arcsin)( =t ,
b
T
b LR
v
τ
π
ω
0
||2
= ,
M
Ne
po
2
2 4π
ω = ,
+= b
L
lZ co
b
l 3
1)( ω
ω
ω
ν ,
2
2
2
1
L
L
o
−
=ν ,
=
Lo
1arccosθ .
Note, that Eq. (2) describes the contribution of any
kind of the trapped particles to nn
l
,
)(, σε . The corresponding
expressions for electrons and ions can be obtained from
(2) by replacing the temperatures ||T and ⊥T , density N,
mass M, charge e by the electron Te, T⊥e, Ne, me, ee and
ion Ti, T⊥i, Ni, Mi, ei parameters, respectively.
79
Our dispersion equation is suitable to analyze the
instabilities of both the right-hand (if l=1) and left-hand
(if l=-1) polarized waves. Further, Eq. (1) should be
resolved numerically for the real and imaginary parts of
the wave frequency, ω =Reω +i Imω, to define the
instability conditions. As it is well known, in the straight
magnetic field case, the squared refractive index of the
EMIC waves (l = -1) in the hydrogen plasma is defined by
the expression
)Re(Re 00
22
||
ωω −ΩΩ
Ω
≈
cc
ppck , (3)
where pppp MeN /4 22 π=Ω is the squared proton plasma
frequency, Np= Nc+Nh; Nc and Nh are the densities of the
cold and hot protons, respectively;
30
)0,(
LcM
LeB co
p
c
ω
==Ω
is the cyclotron frequency of the L-shell protons at the
equatorial plane. Since the parallel wavenumber ||k is
connected with the mode numbers n as ]/[ 0|| oLRnk θπ= ,
for plasmasphere with circular magnetic field lines, so
that the mode numbers can be estimated as
)Re(
Re
00
0
ωπ
ωθ
−ΩΩ
Ω
≈
cc
ppo
c
LR
n . (4)
Where as the increment (decrement) of the EMIC waves
in a hydrogen plasma γ, if ωωγ ReIm <<= , is defined
by the expression
( )
( ) h
cpp
c
c
s
,1
0
2
2
0
0
Im
Re2
ReRe
2 −−ΩΩ
−Ω
−≈
Ω
ε
ω
ωωγ , (5)
where
Ω−
−×
×
−
Ω
−−
Ω
ΩΩ
= ⊥
−
2
||||
0
||00||||
2
0
2
,1
Reexp
1Re1Re
)(Re2
Im
hT
c
h
h
cchT
cph
h
vk
T
T
vk
ω
ωω
ω
π
ε
By index ’h’ we denote the plasma parameters for the
resonant hot protons. As follows from Eqs. (3), the proton
cyclotron instability (PCI) of EMIC waves, 0>γ , is
possible if 0Im ,1 <− hε , i.e., if hh TT ||>⊥ .
3. NUMERICAL RESULTS
Now, let us compare the PCI growth rates in the
plasmas confined in the straight magnetic field γs, and in
the 2D magnetosphere with circular magnetic field lines
γc. For simplicity, there are considered the hydrogen
plasmas at the geostationary orbit, L=6.6, including the
cold electrons with Ne=11 cm-3, cold protons with
Nc=10 cm-3, and energetic protons with Nh=1 cm-3. The
parallel and transverse temperatures of the energetic
protons are equal to 10|| =hT keV and 30=⊥hT keV,
whereas the temperature of the cold particles is small and
isotropic. In this case, the mode numbers n of the field
aligned EMIC waves can be defined by Eqs. (3), (4); the
corresponding dependence n(ω) is plotted in Fig. 1.
Fig. 1. Dependence of the mode numbers on the wave
frequency for EMIC waves in a hydrogen plasma
The PCI growth rate γs for EMIC waves in the straight
magnetic field plasma we estimate, as usually, by Eq. (5).
As for γc, for EMIC waves in the magnetospheric-like
plasma with circular magnetic field lines, we use the
similar expression :
( )
( )
nn
h
cpp
c
c
c ,
,1
0
2
2
0
0
Im
Re2
ReRe
2 −−ΩΩ
−Ω
−≈
Ω
ε
ω
ωωγ , (6)
where
∑ ∫
∞
=
−
−
⊥
−
−
Ω
=
1
0
,1
1,
3
5
||
||0
2
,
,1 ,)21(
2
Im
p
hn
p
To
phnn
h
o
p
Z
A
pTv
TLR ν
νν
θπω
ε
ννν d
T
T
p
Z
Z
p
Z
Y h
h
hn
p
−−−−
×
⊥
−
−
−
−
||2
2
2
,14
,1
,1
1, 1)21(1exp,
Fig. 2. The PCI growth rates versus ω for EMIC waves
in the hydrogen plasmas confined in the straight uniform
magnetic field (a) and in the 2D magnetosphere (b)
The PCI growth rates versus ω are presented in
Fig. 2a for EMIC waves in the straight magnetic field
plasma by Eq. (5), and in Fig. 2b for EMIC waves in the
2D magnetosphere-like plasma with circular magnetic
field lines. The computations of γc are carried out in the
interval Hz7Hz2 ≤≤ ω , whereas the minimal
gyrofrequency of the protons at L=6.6 is closed to
Hz110 ≈Ωc . As shown in Fig. 2a and Fig. 2b, the
instability of EMIC waves is possible for both plasma
models in the frequency range 0cΩ<ω . It should be noted
that the proton-cyclotron instability is impossible for EMIC
waves in the frequency range )(00 occ b θω Ω<<Ω , where
)(0 oc b θΩ is the maximal gyrofrequency of the protons at
the given L-shell magnetic field line.
80
As one can see, the dependence γs(ω) and γc(ω) on the
wave frequency ω are similar; however, γc(ω)<<γs(ω)
under the same bulk parameters. The ratio 104/s ÷∝cγγ
versus ω for considered magnetospheric-like plasmas is
presented in Fig. 3. This dependence is not linear; the
difference is very large (by factor 10) for EMIC waves in
the range of Hz2~ω and is smaller (by factor 4) in the
range of high frequencies Hz7~ω .
Fig. 3. The ratio γ s/γ c versus ω for EMIC waves in the
hydrogen plasmas
The large difference between γs and γc is connected
with the fact that the wave-particle interaction in the
straight magnetic field plasma is more effective since the
resonant particles move along the uniform magnetic field
line with the constant parallel velocity and interact
permanently (in time) with the wave according to the well
known resonance condition ||||0 vkc =Ω−ω . As for 2D
magnetospheric plasmas, since constv ≠|| for the trapped
particles, there is another wave-particle resonance
condition involving the particle energy, pitch angle,
cyclotron and bounce frequencies. As a result, the trapped
particle bouncing between the reflection points only part
of the bounce-time can interact effectively with the wave.
CONCLUSIONS
Dispersion equation is analyzed for EMIC waves in a
hydrogen magnetospheric plasma with circular B-field lines.
As in the straight B-field plasmas, the growth rate of PCI is
defined by the contribution of the resonant particles to the
imaginary part of the transverse permittivity elements. The
comparison of the growth rates is carried out for EMIC
waves in the hydrogen plasmas with the straight and circular
B-field lines under the same macroscopic bulk parameters at
the geostationary orbit L=6.6. It is shown that the PCI growth
rate in the 2D magnetosphere is much less than it is in the
straight uniform B-field case. Of course, the similar approach
can be used to analyze the dispersion characteristics of the
EMIC waves in 2D magnetospheric multi-ions plasmas with
dipole and circular magnetic field lines including the protons
and heavy ions (He+, O+) with the temperature anisotropy.
REFERENCES
1. J.M. Cornwall. Cyclotron instabilities and
electromagnetic emission in the ultra low frequency and very
low frequency ranges // J. Geophys. Res. 1965, v. 70, p. 61.
2. C.F. Kennel. H.E. Petschek. Limit on stably trapped
particle fluxes // J. Geophys. Res. 1966, v. 71, p. 1.
3. S. Cuperman. Electromagnetic kinetic instabilities in
multicomponent space plasmas. Theoretical predictions and
computer simulation experiments // Rev. Geophys. 1981,
v. 19, p. 307.
4. L. Gomberoff, R. Neira. Convective growth rate of ion
cyclotron waves in a H+-He+ and H+-He+-O+ plasma// J.
Geophys. Res. 1983, v. 88, p. 2170.
5. S.P. Gary. Theory of space plasma microinstabilities.
Cambridge Univ. Press, 1993.
6. S. Xue, R.M. Thorne, D. Summers. Parametric study of
electromagnetic ion cyclotron instability in the Earth
magnetosphere // J. Geophys. Res. 1996, v. 101, N A7, p. 15,
467.
7. N.I. Grishanov, M.A. Raupp, A.F.D. Loula, J. Pereira
Neto. Dispersion equations for field-aligned cyclotron waves
in axisymmetric magnetospheric plasmas // Annales
Geaphysicae. 2006, v. 24, p. 589.
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