The transformation of long scale Alfven waves in space dusty plasma
A nonlinear mechanism of the generation of kinetic Alfv´en waves (KAW) on dust plasma with small plasma parameter β is proposed. As the generation mechanism, the parametric instability where a pumping wave is the MHD Alfv´en wave is considered. On the basis of the three-fluid MHD, the nonlinear disp...
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| Zitieren: | The transformation of long scale Alfven waves in space dusty plasma / A.K. Yukhimuk, V.M. Fedun, A.D. Voitsekhovska, O.K. Cheremnykh // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 192-195. — Бібліогр.: 10 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860170352995336192 |
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| author | Yukhimuk, A.K. Fedun, V.M. Voitsekhovska, A.D. Cheremnykh, O.K. |
| author_facet | Yukhimuk, A.K. Fedun, V.M. Voitsekhovska, A.D. Cheremnykh, O.K. |
| citation_txt | The transformation of long scale Alfven waves in space dusty plasma / A.K. Yukhimuk, V.M. Fedun, A.D. Voitsekhovska, O.K. Cheremnykh // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 192-195. — Бібліогр.: 10 назв. — англ. |
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| container_title | Кинематика и физика небесных тел |
| description | A nonlinear mechanism of the generation of kinetic Alfv´en waves (KAW) on dust plasma with small plasma parameter β is proposed. As the generation mechanism, the parametric instability where a pumping wave is the MHD Alfv´en wave is considered. On the basis of the three-fluid MHD, the nonlinear dispersion equation describing the three-wave interaction is deduced and its solution is derived. Obtained instability growth rate is determined by parameters of dust plasma particles. The nonlinear process under consideration can take place both in laboratory and in space plasma with small plasma parameter β. As an application of theoretical results, we consider Saturn’s F-ring.
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| first_indexed | 2025-12-07T17:58:03Z |
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THE TRANSFORMATION OF LONG SCALE ALFVÉN WAVES
IN SPACE DUSTY PLASMA
A. K. Yukhimuk1, V. M. Fedun1,2, A. D. Voitsekhovska1, O. K. Cheremnykh3
1Department of Space Plasma Physics, Main Astronomical Observatory, NAS of Ukraine
27 Akademika Zabolotnoho Str., 03680 Kyiv, Ukraine
e-mail: fedun@mao.kiev.ua
2Department of Astronomy and Space Physics, National Taras Shevchenko University of Kyiv, Ukraine
3Department of Space Plasma, Space Research Institute of NASU–NSAU, Kyiv, Ukraine
A nonlinear mechanism of the generation of kinetic Alfvén waves (KAW) on dust plasma with
small plasma parameter β is proposed. As the generation mechanism, the parametric instability
where a pumping wave is the MHD Alfvén wave is considered. On the basis of the three-fluid
MHD, the nonlinear dispersion equation describing the three-wave interaction is deduced and its
solution is derived. Obtained instability growth rate is determined by parameters of dust plasma
particles. The nonlinear process under consideration can take place both in laboratory and in
space plasma with small plasma parameter β. As an application of theoretical results, we consider
Saturn’s F-ring.
INTRODUCTION
Now dust plasma is the object of the intensive researches from a set of application, both in laboratory experi-
ments, and in space. The extensive scientific literature, including reviews [3, 4] and monographies [1, 6], shows
large interest to dust plasma. Practically, any plasma (including space) contains some number of dust grains.
The dust grains are ices, graphite, silicates, etc. The interplanetary space, rings of giant planets, tail of comets
and, certainly, the Earth’s magnetosphere and ionosphere are typical objects of Solar System which contain
a lot of dust. Typical parameters of dust plasma in this regions are reduced in Table 1.
Table 1. Dust plasma parameters
Characteristics Interplanetary space The giant planets rings Comet tail Earth’s ionosphere
ne (cm−3) ≈ 5 0.1 – 102 103 − 104 ≈ 103
Te (K) ≈ 105 105 – 106 ≤ 0.1 (eV) ≈ 150
nd (cm−3) ≈ 10−12 10−7 – 10 10−10 − 10−3 ≈ 10 − 103
The most important source of dust in the Earth’s atmosphere is simulated contamination (aerosols). The pre-
sence of dust grains in the rings of the giant planets was established during satellite missions of Voyagers 1
and 2 [7, 8].
In the present work we consider nonlinear parametric interaction MHD Alfvén pump wave (ω0, �k0) with
kinetic Alfvén waves (ω1, �k1, ω2, �k2) in dust plasma. The frequency ω0 and wave vector �k0 are related by linear
dispersion relation:
ω2
0 = k2
0zV
2
Ad,
where VAd = B0/
√
4πndmd is the dust Alfvén velocity, nd and md is the density and mass of dust grains.
We consider equation in Cartezian coordinates (x, y, z), supposing that all wave vectors are situated in xz plane
�k = (kx, 0, kz), �B0 = B0�ez. It is assumed that the wave synchronism conditions are satisfied:
ω0 = ω1 + ω2,
�k0 = �k1 + �k2.
c© A. K. Yukhimuk, V. M. Fedun, A. D. Voitsekhovska, O. K. Cheremnykh, 2004
192
DISPERSION EQUATION FOR DUST KAW
Nonlinear three-wave interaction in the dust plasma is considered on the basis of magnetic hydrodynamics.
In this case the main system of equations describes electrons, ions and charged dust particles as conductive
liquids connected with each other by electromagnetic fields. Thus, the main system of equations looks like:
∂
−→
Vα
∂t
=
1
mα
(
eα
�E + �Fα
)
+
(−→
V α × ωBα
)
− Tα
mαnα
−→∇nα; (1)
∂nα
∂t
= −−→∇
(
nα
−→
Vα
)
; (2)
�∇× �B =
4π
c
�j; (3)
�∇× �E = −1
c
∂ �B
∂t
; (4)
�∇ · �E = 4πρ; (5)
where �j = e
(
ni
�Vi − ne
�Ve − Znd
�Vd
)
, ρ = e (ni − ne − Znd), �Fα =
eα
c
(
�Vα × �B
)
− mα
(
�Vα∇
)
�Vα. Index
α = i, e, d correspond to the ion, electron and dust (negative charge) plasma components. As the frequence
range ω � ωBd, ωBi is considered, the influence of displacement current is not essential.
Nonlinear dispersion equation for KAWs in electron-ion plasma was considered in [9, 10]. It is obvious that
in case of three-component plasma this equation will be a little modified, as the plasma approximation in this
case is of the form:
ñi = ñe + Zñd, (6)
where ñe, ñi, ñd are perturbations of the ion, electron and dust grains number densities, respectively. This can
be used, since the wave modes are considered at the frequencies less than ion cyclotron frequency. Using
the motion (1) and continuity (2) equations we find:
ñe =
en0e
kzTe
(
kzϕ − ωAz
c
+ i
Fz
ekz
)
; (7)
ñi = − en0i
kzTi
(
kzϕ − ωAz
c
)
; (8)
ñd = −Zen0d
mdω2
[(
k2
z − ω2k2
x
ω2 − ω2
Bd
)
ϕ − kzω
c
Az
]
, (9)
where ϕ and Az are scalar potential and z-component of the vector potential KAW, ωBd = eZB0/mdc is
the cyclotron frequency of grains with charge Ze.
Substituting (7)–(9) in (6) we obtain nonlinear dispersion equation:[(
ω2 − k2
zC2
d
) (
ω2 − k2
zV 2
Ad
) − k2
zV 2
Ad μd ω2
]
ϕ = k2
zV 2
Ad ω2QNL, (10)
where
QNL = −i
mdnoeC
2
dFz
en0dZ2kzTe
, Cd =
[
Z2TeTin0d
(noeTi + noiTe)md
]1/2
, VAd =
B0√
4πnodmd
,
μd = k2
⊥ρ2
d, ρd =
Cd
ωBd
is dust particle Larmor radius.
In the absence of pump wave (QNL = 0) and μd �= 0 we have:
ω2 = k2
zV 2
Ad (1 + μd) , (11)
ω2 =
k2
zC2
d
1 + μd
. (12)
193
The expression (11) describes the dispersion law for KAW, and (12) describes the dispersion law for ion-
acoustic waves in dust plasma.
Taking into account that VAd � Cd we can note the equation (10) as:
εA (ω, k)ϕ = PNL, (13)
where εA (ω, k) = ω2 − k2
zV 2
Ad (1 + μd), PNl = k2
zV 2
AdQNl.
From (13) we obtain KAW dispersion equation:
ε1ϕ1 = η1 (E0xϕ∗
2) , (14)
where coupling coefficient is:
η1 = i
emdn0eC
2
dV 2
Adω2
n0dZ2T 2
e ω0
k0zk1z
k2x
μe.
The dispersion equation for second KAW follows from (14), where the index “1” and “2” are exchanged:
ε2ϕ2 = η2 (E0xϕ∗
1) , (15)
where
η2 = i
emdn0eC
2
dV 2
Adω1
n0dZ2T 2
e ω0
k0zk2z
k1x
μe.
NONLINEAR DISPERSION EQUATION
Using dispersion equation for two KAW (14) and (15), we can find a nonlinear dispersion equation describing
three-wave interaction:
ε1ε
∗
2 = η1η
∗
2 |E0x|2. (16)
Assuming in (16) ω1 = ω1r + iγ, ω2 = ω2r + iγ (|γ| � ω1r, ω2r) and expanding ε1 and ε2 in Taylor series in
the small parameter γ, we find:
γ =
W
2
(πn0iTi)
1
2 emdn0eC
2
dV 2
Adk0z
n0dZ2
dT 2
e ω0
(
k1zk2z
k1xk2x
) 1
2
μe, (17)
where W =
|E0x|√
4πn0Ti
.
CONCLUSION
The nonlinear parametric interaction of the pump MHD Alfvén wave with kinetic Alfvén waves in dust plasma
with small plasma parameter β is considered. To describe the nonlinear interaction, the three-fluid magneto-
hydrodynamics is used. From (17) it follows that the instability growth rate depends on parameters of plasma
charged grains.
We consider Saturn’s F-ring and laboratory plasma as an application of our theoretical results. Typical
parameters of Saturn’s F-ring are: Zd ∼ 104, n0d ∼ 1 cm−3, n0i ∼ 104 cm−3, Te ∼Ti ∼ 1 eV, B0 ∼ 0.05G, md
∼=
10−12– 10−6 g. The instability growth rate and dust cyclotron frequency dependences on a mass of dust particles
are shown in Fig. 1 and Fig. 2, respectively.
It is obvious that the maximum value of the instability growth rate γ ∼= 10−7 s−1 is reached when a mass of
dust particles md
∼= 10−12 g. Cyclotron frequency at the same parameters of dust particles is ωBd
∼= 10−5 s−1.
For an estimation of a decrement of the dust KAW we use the expression (12) from [5]:
γL
∼= −
√
π
8
(
ck⊥
ωpi
)2
kIIVTi
[
1 +
Ti
δTe
]−2
[
1 +
me
mi
1
δ
(
Ti
Te
)3/2
]
.
194
Figure 1. Dependence of the instability growth rate on a mass of dust particles
Figure 2. Dependence of the cyclotron dust frequency on a mass of dust particles
Using the same parameters, we obtain decrements which meet the condition γL � γ. Therefore, Landau
damping is enough small, and thus, the considered instability is practically thresholdless.
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1996.–53, N 4.–P. 4051–4055.
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[5] Salimullah M., Rosenberg M. On kinetic Alfvén waves in a dusty plasma // Phys. Lett. A.–1999.–A254, N 4.–
P. 347–350.
[6] Shukla P. K., Mamun A. A. Introduction to dusty plasma physics.–Bristol: Institute of Physics Publishing, 2002.–
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[7] Smith B. A., Soderblom L., Beebe R., et al. Encounter with Saturn –Voyager 1 imaging results // Science.–1981.–
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195
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| id | nasplib_isofts_kiev_ua-123456789-79642 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0233-7665 |
| language | English |
| last_indexed | 2025-12-07T17:58:03Z |
| publishDate | 2005 |
| publisher | Головна астрономічна обсерваторія НАН України |
| record_format | dspace |
| spelling | Yukhimuk, A.K. Fedun, V.M. Voitsekhovska, A.D. Cheremnykh, O.K. 2015-04-03T16:59:48Z 2015-04-03T16:59:48Z 2005 The transformation of long scale Alfven waves in space dusty plasma / A.K. Yukhimuk, V.M. Fedun, A.D. Voitsekhovska, O.K. Cheremnykh // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 192-195. — Бібліогр.: 10 назв. — англ. 0233-7665 https://nasplib.isofts.kiev.ua/handle/123456789/79642 A nonlinear mechanism of the generation of kinetic Alfv´en waves (KAW) on dust plasma with small plasma parameter β is proposed. As the generation mechanism, the parametric instability where a pumping wave is the MHD Alfv´en wave is considered. On the basis of the three-fluid MHD, the nonlinear dispersion equation describing the three-wave interaction is deduced and its solution is derived. Obtained instability growth rate is determined by parameters of dust plasma particles. The nonlinear process under consideration can take place both in laboratory and in space plasma with small plasma parameter β. As an application of theoretical results, we consider Saturn’s F-ring. en Головна астрономічна обсерваторія НАН України Кинематика и физика небесных тел MS2: Physics of Solar Atmosphere The transformation of long scale Alfven waves in space dusty plasma Article published earlier |
| spellingShingle | The transformation of long scale Alfven waves in space dusty plasma Yukhimuk, A.K. Fedun, V.M. Voitsekhovska, A.D. Cheremnykh, O.K. MS2: Physics of Solar Atmosphere |
| title | The transformation of long scale Alfven waves in space dusty plasma |
| title_full | The transformation of long scale Alfven waves in space dusty plasma |
| title_fullStr | The transformation of long scale Alfven waves in space dusty plasma |
| title_full_unstemmed | The transformation of long scale Alfven waves in space dusty plasma |
| title_short | The transformation of long scale Alfven waves in space dusty plasma |
| title_sort | transformation of long scale alfven waves in space dusty plasma |
| topic | MS2: Physics of Solar Atmosphere |
| topic_facet | MS2: Physics of Solar Atmosphere |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79642 |
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