Quasilinear description of electron cloud dynamics in the solar corona plasma
In this study, we consider dynamics of maxwellian electron cloud with the temperature significantly exceeding that of surrounding plasma. We investigate the transition from initially free electron propagation to a quasilinear one. We show qualitatively and quantitatively that only a small part of th...
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| Zitieren: | Quasilinear description of electron cloud dynamics in the solar corona plasma / V.N. Mel’nik, E.P. Kontar, and V.I. Lapshin // Вопросы атомной науки и техники. — 2000. — № 1. — С. 265-269. — Бібліогр.: 18 назв. — англ. |
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Mel’nik, V.N. Kontar, E.P. Lapshin, V.I. 2015-05-19T08:51:44Z 2015-05-19T08:51:44Z 2000 Quasilinear description of electron cloud dynamics in the solar corona plasma / V.N. Mel’nik, E.P. Kontar, and V.I. Lapshin // Вопросы атомной науки и техники. — 2000. — № 1. — С. 265-269. — Бібліогр.: 18 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/81680 533.9 In this study, we consider dynamics of maxwellian electron cloud with the temperature significantly exceeding that of surrounding plasma. We investigate the transition from initially free electron propagation to a quasilinear one. We show qualitatively and quantitatively that only a small part of the cloud propagates into a plasma, whereas the main part of electrons is concentrated in the place of injection. These electrons turn out to be locked by the Langmuir turbulence generated by the electrons left that region. Authors are pleased to thank Supercomputing Systems AG (Switzerland) for the free access to the supercomputer Gigabooster. One of the authors (E.P.K.) would like to thank the Ukrainian Office of International Science Foundation for financial support in the framework of grant PSU082125. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы Quasilinear description of electron cloud dynamics in the solar corona plasma Article published earlier |
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Quasilinear description of electron cloud dynamics in the solar corona plasma |
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Quasilinear description of electron cloud dynamics in the solar corona plasma Mel’nik, V.N. Kontar, E.P. Lapshin, V.I. Нелинейные процессы |
| title_short |
Quasilinear description of electron cloud dynamics in the solar corona plasma |
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Quasilinear description of electron cloud dynamics in the solar corona plasma |
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Quasilinear description of electron cloud dynamics in the solar corona plasma |
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Quasilinear description of electron cloud dynamics in the solar corona plasma |
| title_sort |
quasilinear description of electron cloud dynamics in the solar corona plasma |
| author |
Mel’nik, V.N. Kontar, E.P. Lapshin, V.I. |
| author_facet |
Mel’nik, V.N. Kontar, E.P. Lapshin, V.I. |
| topic |
Нелинейные процессы |
| topic_facet |
Нелинейные процессы |
| publishDate |
2000 |
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English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| description |
In this study, we consider dynamics of maxwellian electron cloud with the temperature significantly exceeding that of surrounding plasma. We investigate the transition from initially free electron propagation to a quasilinear one. We show qualitatively and quantitatively that only a small part of the cloud propagates into a plasma, whereas the main part of electrons is concentrated in the place of injection. These electrons turn out to be locked by the Langmuir turbulence generated by the electrons left that region.
|
| issn |
1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/81680 |
| citation_txt |
Quasilinear description of electron cloud dynamics in the solar corona plasma / V.N. Mel’nik, E.P. Kontar, and V.I. Lapshin // Вопросы атомной науки и техники. — 2000. — № 1. — С. 265-269. — Бібліогр.: 18 назв. — англ. |
| work_keys_str_mv |
AT melnikvn quasilineardescriptionofelectronclouddynamicsinthesolarcoronaplasma AT kontarep quasilineardescriptionofelectronclouddynamicsinthesolarcoronaplasma AT lapshinvi quasilineardescriptionofelectronclouddynamicsinthesolarcoronaplasma |
| first_indexed |
2025-11-25T23:32:43Z |
| last_indexed |
2025-11-25T23:32:43Z |
| _version_ |
1850587134342201344 |
| fulltext |
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ 2000. №1.
Серия: Плазменная электроника и новые методы ускорения (2), с. 265-269.
265
UDK 533.9
QUASILINEAR DESCRIPTION OF ELECTRON CLOUD DYNAMICS IN
THE SOLAR CORONA PLASMA
V.N. Mel’nik*, E.P. Kontar**, and V.I. Lapshin**
*Institute of Radio Astronomy of National Academy of Sciences, Ukraine
**National Science Center "Kharkov Institute of Physics and Technology", Kharkov, Ukraine
In this study, we consider dynamics of maxwellian electron cloud with the temperature significantly exceeding
that of surrounding plasma. We investigate the transition from initially free electron propagation to a quasilinear
one. We show qualitatively and quantitatively that only a small part of the cloud propagates into a plasma, whereas
the main part of electrons is concentrated in the place of injection. These electrons turn out to be locked by the
Langmuir turbulence generated by the electrons left that region.
1. Introduction
The propagation of the electron stream through
plasma has been widely discussed [1-10]. The interest to
this problem is stimulated by different astrophysical
applications (activity induced by particle beams in the
Earth ionosphere, in the solar corona, pulsar
magnetospheres, etc.). The difficulty of the problem
originates from the fact that electrons generate
Langmuir waves, which influence on electron
propagation [9,10]. If the consideration is carried out in
the framework of weak turbulence theory, when the
level of Langmuir waves generated is quite low
2)(/ DknTW λ< ( n , T are the density and the
temperature of surrounding plasma, Dλ is Debye
radius, k is a wave number), then analytical results as
well as numerical solutions of the kinetic equations
point out that electrons can propagate at large distances
being a source of Langmuir turbulence. An analytical
solution of the problem when in the injection place
electron distribution function had 0/ >∂∂ vf was
found in [11]. Later in [14] analytical and numerical
solutions of the initial problem for 0/ >∂∂ vf was
derived. In both cases the solution is a beam-plasma
structure. It consists of electrons and plasmons and
moves at a constant velocity. The method used in
[11,14] for finding the solution based on the smallness
of quasilinear relaxation time in comparison with the
time of electron propagation. For the first time Ryutov
and Sagdeev [1] proposed this method at consideration
of the dynamics of maxwellian electron cloud
propagation. They supposed that electron propagation
would be self-similar at large distances and big times.
There were some doubts [9,10,12] because at the initial
moment 0/ <∂∂ vf and at first electrons propagate
freely. Only in some time when fast electrons overtake
slow ones a positive derivative vf ∂∂ / is appeared and
quasilinear relaxation begins. So free propagation of
electrons defines the next stage.
In this study numerically and analytically dynamics
of electron maxwellian cloud are considered beginning
from the free propagation. It was found that there were
some electrons groups that propagate in plasma in diffe-
rent regimes. Among them we recognize electrons with
low density that propagate in the form of beam-plasma
structure and the main part of electrons “locked” by
Langmuir turbulance generated with the left electrons.
2. Free and quasilinier electron propagation
Let the fast cloud electrons have the maxwellian
distribution function at the initial moment 0=t
)exp()exp()0,,( 2
0
2
2
2
0
0
v
v
d
x
v
ntxvf −−
π
== , (1)
where d is the cloud size, 0n , 0v are the density and
the thermal velocity of the fast electrons, and
Tevv >>0 is the thermal velocity of plasma electrons.
For estimations and numerical simulations we take the
following parameters: 38106 −×= cmn , KTe
610= ,
( scmvTe /104 8×= ), 3
0 2000 −= cmn , scmv /1010
0 = ;
cmd 9103 ×= . In the scope of the weak turbulence
theory, the one-dimensional electron distribution
function, ),,( txvf , and the one-dimensional spectral
energy density of Langmuir waves, ),,( txvW , are
described by the system of the kinetic equations [13]
v
f
v
W
vm
e
x
fv
t
f
∂
∂
∂
∂π
∂
∂
∂
∂ 224=+ , (2)
kv
v
fWv
nt
W
p
p == ω
∂
∂πω
∂
∂ ,2 . (3)
Since the initial electron distribution is stable as to
generation of plasma waves, then electrons initially
propagate freely and their distribution evolve following
)exp())(exp()0,,( 2
0
2
2
2
0
0
v
v
d
vtx
V
ntxvf −−−==
π
. (4)
For 0>x , there is a positive derivative, vf ∂∂ / , at
the electron distribution function (Fig. 1), because fast
266
p(x,t)
u(x,t)0 v
f(v,x,t)
Fig. 1 Unstable electron distribution function.
v3
v2
v1
x
t0 t0=d/v0
Fig.2 The curves in ),( tx plane for the three
different velocities 321 vvv >> , where 0/ =∂∂ vf
are below the corresponding curves.
electrons overtake slow ones. One derives the equation
of the curve separating regions with 0/ >∂∂ vf and
0/ <∂∂ vf [12]
)( 2
0
2
tv
dtvx += (5)
from 0/ =∂∂ vf and (4). Note, that positive derivative
can be only for 0>x , and therefore, propagation of
electrons is free in the region 0≤x . In Fig. 2 curves
are presented for three different velocities
321 vvv >> . It is obvious that the positive derivative
at a given point $x$ initially appears at small velocities.
The maximum velocity dxvv 2/* 0= , when
0/ >∂∂ vf $, is reached at the moment 00 / vdt = .A
positive derivative means that the condition for the
quasilinear relaxation exists.
Since only a small part of all the particles takes part in
the relaxation, then plateau is said to be formed up to
the velocity ),( txuv = ,
22
0
2
2
02),(
tvd
xtvtxu
+
≈ , (6)
which is derived from the approximate equality (see
Fig. 1)
),),,((),,0( txtxuftxf ≈ . (7)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
t= 1.5 s
M
ax
im
um
p
la
te
au
v
el
oc
ty
u
(x
)/
v
0
d is tance x/d
Fig. 3. Maximum velocity of the plateau electrons
),( txu Dashed and solid lines are analytical and
numerical results, correspondingly.
t0
3d x=
u 0
t
tmin
xmin
xН
0 t
x
Fig. 4 The curve (11) in ),( tx plane separating regions
where electrons propagates freely (not striped) and
under relaxation (striped)
Maximum of ),( txu (at the moment 00 / vdt = )
linearly grows with distance dxvxu /)( 0= (Fig. 3).
Taking into account (6), the number of relaxing
particles can be estimated as
∫ −=
),(
0
)),,0(),,(('
txu
dvtxftxvfn (8)
Consequently, the quasilinear relaxation time
1' −
ω=τ
n
n
pe
follows substituting the result of
integration from (8)
1
22
0
22
22
0
2
2
222
0
2
0 1
)(
exp)exp(
2
),(
−
−
+
+
τ=τ
tvdd
tvx
d
x
d
tvd
tx ,(9)
where penn ω=τ 00 / . It is natural to consider that the
quasilinear relaxation influences the dynamics of
electron propagation only if ),( txτ becomes equal to
the electron propagation time, t ,
ttx =τ ),( . (10)
The curve )(tx , where the condition (10) takes place,
is shown in Fig. 4. The long time lengths of the
quasilinear relaxation far and near the injection site are
connected with the fact that in the first case the number
of fast ( 0vv >> ) particles is vanishing, and in the
267
second case the number of slow ( 0vv < ) particles is
big. As a result, in both cases, vf ∂∂ / as well as 'n
turn out to be small.
For large x the curve )(tx has an
asymptotic tux 0= with
τ
=
0
0
00
2
ln
tvu , and for the
parameters chosen 00 3vu ≈ . Therefore, particles with
velocities 0uv > mainly do not participate in the
quasilinear relaxation, i.e. they propagate freely. At
small x
dvdxx low /00τ≈< (11)
electrons move freely also. At the same time, for
electrons with 0vv ≤ interaction with Langmuir waves
is displayed at the distances comparable with the size of
electron cloud d , at the moment
( ) svdt 04.0/ 3/12
0
2
0min ≈τ≅ , which corresponds to
dtxx ≈= )( minmin .
Let us consider the most interesting region
dxxlow 3<< , where electron propagation is ruled by
Langmuir waves. After the first stage of free spread,
plateau is formed at the electron distribution function
from 0=v to ),( mintxuv = due to quasilinear
relaxation, since mintt ≈ . Further, electrons arrived to a
given point x have a positive derivative vf ∂∂ / for
the greater velocities and as a result plateau becomes
wider. As it was mentioned above, its maximum width
dxvxu /)( 0= will be at 00 / vdt = .
In order to understand what occurs after the moment
00 / vdt = we need to consider the distribution of
Langmuir turbulence in this region. Since the plateau
height is
∫ −≅=
),(
2
2
0
0
0
0
0
0
)exp(),,(
),(
1),(
txu
o d
x
v
ndvtxvf
txu
txp
(12)
the spectral energy density follows
≈
−
ω
= ∫ ∫
v txu
pe
dvtxvfdvtxvf
txu
mvtxvW
0
),(
0
00
0
3
0
0
),,(),,(
),(
1),,(
−−
ω
≈ )/(
22
11)exp( 02
24
0 vverf
d
xvmn
pe
(13)
where ∫ −
π
≡
x
dxxxerf
0
2 )exp(2)( is the
probability integral. Taking into account (13) one
derives the spatial distribution of wave energy
≈ω=≡ ∫∫ dvvtxvWWdkE
txu
peW
2
),(
0
0 /),,(
0
)/exp(1.0 22
3
2
00 dx
d
xvmn −
≈ , (14)
and its maximum is close to d7.1 . In comparison to
the distribution of particle energy
≈≡ ∫ dvvtxvfmE
txu
e
2
),(
0
0
0
),,(
2
)/exp(2.0 22
3
2
00 dx
d
xvmn −
≈ , (15)
one concludes that Langmuir turbulence is placed
deeper in a plasma.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Electron energy density
Wave energy density
t=1.5 s
E w
,e
/m
n'
v 02
distance x/d
Fig. 5 The energy density of Langmuir waves (14) and
electrons (15). Dashes (solid) lines correspond to
theoretical (numerical) results.
The turbulence excited near d7.1 drastically
influences the electron spread. Two groups of electrons
are formed: the particles, whose propagation is ruled by
Langmuir turbulence, and those that pass through this
region without essential interaction with the turbulence.
Electrons with velocities greater than maximum plateau
velocity at this point 07.1)7.1( vdxuv ==> , are
related to the latter group. Generally, this value should
be considered to be the low limit, because the high level
of Langmuir turbulence exists for dx 7.1> , where
)(xu is greater, and therefore the real velocity must be
slightly more. This fact is also supported by the
numerical solution (Fig. 5).
The density of particles with velocity 07.1 vv >
can be estimated as
∫
∞
≈==≈
07.1
005.0)0,0,(*
v
ndvtxvfn (23)
and consequently only after the moment pennt ω= */
these particles begin to relax, but it happens at distances
dx 7.1> . Later electrons with greater velocities (up to
0uv = ) are added to these particles. Plateau from
0≈v to 0uv ≈ is formed at their distribution
function. The electrons are accompanied with Langmuir
waves and together they form beam-plasma structure
[11,14,15].
What happens with the electrons whose velocity is
268
less than 07.1 v ? If there were not Langmuir waves,
those electrons could propagate freely due to
0/ ≈∂∂ vf . Again, the fast electrons would overtake
the slow ones and the positive derivative vf ∂∂ /
would appear. But the derivative is small and therefore
free propagation continues till it will become great
enough to start quasilinear relaxation. However,
Langmuir turbulence presence in this region completely
changes the dynamics. As soon as 0/ >∂∂ vf , process
of relaxation starts ( 1−∝τ W ) and the electron
distribution function becomes flatter, i.e. electrons slow
down immediately. This means that Langmuir
turbulence “locks” electrons within the region,
dx 7.1< .
If processes leading to a decrease in the level of
Langmuir turbulence are taken into consideration, the
conditions for separation of the next small part of
electrons appears. Since the maximum velocity of the
electrons locked is 0max 7.1 vv ≤ , then it is clear that
the velocity of the next group is less than that of the first
one. As a result beam-plasma structures propagate into a
plasma one by one with declining velocity.
3. Numerical solution of quasilinear
equations
The numerical solution [18] of the kinetic equations (2,
3) has been conducted for the case when at the initial
moment a cloud of maxwellian electrons is determined
by (1), and the spectral energy density is uniform and
equals
pe
Te
T
Tvn
txvW
ω
0510)0,,( −== (16)
We performed our calculations for the parameters
presented in the previous section. The results of
numerical simulations are presented in fig. 3, 5-8.
Generally, they support the conclusions of the
qualitative consideration in the previous section. Indeed,
at the moment svdt 1/ 00 ≈= a group of electrons
separates from the main part (Fig. 8). The full
separation is considered to be at the moment st 5.1=
(Fig. 5).For dx 3< (Fig. 6) plateau is formed at the
electron distribution function as a result of electron
interaction with Langmuir waves. The plateau height
rapidly declines with x , but the plateau maximum
velocity linearly grows with distance (Fig. 3), that was
predicted by the qualitative consideration. Congruent
agreement is found for the energy density distribution of
electrons and plasma waves (Fig. 5). During the
numerical simulations, a fact of “locking” takes place
after st 0.1= and the distribution of electrons near the
initial site seems to be steady till the end of the
simulations st 10= .
-3 0 3 6 9 12 15 18 21
0
1
2
3
4
t=1.02 s
ln
(1
+1
00
n(
x)
/n
')
distance x/d
Fig. 6 The electron density at st 02.1=
-3 0 3 6 9 12 15 18 21
0
1
2
3
4
t=1.5s
distance x/d
ln
(1
+1
00
n(
x)
/n
')
Fig.7 The electron density at st 5.1= .
4. Conclusion
So we see that four groups of electrons can be
recognized in the initially maxwelliam cloud. Electrons
with velocity
03vv > propagate in plasma without quasilinear
relaxation because of smallness of their density that
yields nonequality t>τ . The next group of electrons
with velocity 00 37.1 vvv << moves in plasma in the
form of beam-plasma structure. For them equality
t≤τ is fulfilled. A further group is concentrated in the
site dxxlow 3<< . These electrons are “locked” by
Langmuir turbulance generated by the previous group.
At every point plateau is formed at electron distribution
function because of interaction with Langmuir waves.
At last there are electrons (at lowxx < ) that has
negative derivative vf ∂∂ / and for this reason they
move freely. Thus our results show that self-similar
solution [1] is not realized.
According to observations of sporadic solar radio
emission fast electrons propagate out of flare site as
isolated beams. Problem of their formation is very
important [9] for understanding both beam radiation and
269
flare phenomenon. In the scope of our consideration we
see that for appearance of beams it is enough there was
cloud of hot electrons somewhere. In the time equals
0/3 vd a small group of electrons with velocity 02v≈
separates from the cloud and moves into plasma.
Because this group of electrons is accompanied by
Langmuir waves there is an opportunity to radiate with
plasma mechanism.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
-0.025
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275 t=1.02 s
x=1.33d
x=1.72d
x=2.11d
f(v
)v
0/n
'
velocity v/v0
Fig. 8 The electron distribution function at st 5.1= at
different x.
Authors are pleased to thank Supercomputing
Systems AG (Switzerland) for the free access to the
supercomputer Gigabooster. One of the authors (E.P.K.)
would like to thank the Ukrainian Office of
International Science Foundation for financial support
in the framework of grant PSU082125.
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