Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion
The problem of evolution and disappearance of the lower hybrid cavities that are observed in the plasma of the Earth’s ionosphere is solved. It is assumed that the destruction of the cavity is caused by turbulent diffusion of plasma, which arises due to the drift instability of radially inhomogeneou...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1951612023-12-03T16:35:12Z Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion Azarenkov, N.A. Chibisov, D.V. Kovalenko, N.I. Maslennikov, D.I. Non-relativistic and relativistic electronics The problem of evolution and disappearance of the lower hybrid cavities that are observed in the plasma of the Earth’s ionosphere is solved. It is assumed that the destruction of the cavity is caused by turbulent diffusion of plasma, which arises due to the drift instability of radially inhomogeneous plasma. The initial plasma density distribution on the radius in the cavity is considered to be the inverse Gaussian distribution. A solution of the diffusion equation is obtained, which at any time determines the radial dependence of the plasma density in the cavity. In the asymptotic limit t →∞ the plasma density in the cavity becomes equal to the density of the surrounding plasma. Вирішується проблема еволюції і зникнення нижньогібридних порожнин, які спостерігаються в плазмі іоносфери Землі. Передбачається, що руйнування порожнини викликане турбулентною дифузією плазми, яка виникає через дрейфову нестійкість радіально-неоднорідної плазми. В якості початкового радіального розподілу щільності плазми в порожнині розглядається зворотній розподіл Гаусcа. Отримано рішення рівняння дифузії, яке в будь-який момент часу визначає залежність від радіуса щільності плазми в порожнині. Показано, що в асимптотичній границі t → ∞ щільність плазми в порожнині стає рівною щільності навколишньої плазми. Решается проблема эволюции и исчезновения нижнегибридных полостей, наблюдаемых в плазме ионосферы Земли. Предполагается, что разрушение полости вызвано турбулентной диффузией плазмы, которая возникает из-за дрейфовой неустойчивости радиально-неоднородной плазмы. В качестве начального радиального распределения плотности плазмы в полости рассматривается обратное распределение Гаусса. Получено решение уравнения диффузии, которое в любой момент времени определяет зависимость от радиуса плотности плазмы в полости. Показано, что в асимптотическом пределе t → ∞ плотность плазмы в полости становится равной плотности окружающей плазмы. 2019 Article Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion / N.A. Azarenkov, D.V. Chibisov, N.I. Kovalenko, D.I. Maslennikov // Problems of atomic science and technology. — 2019. — № 4. — С. 27-30. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.35.Qz, 52.35.Ra, 94.20.wf http://dspace.nbuv.gov.ua/handle/123456789/195161 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| topic |
Non-relativistic and relativistic electronics Non-relativistic and relativistic electronics |
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Non-relativistic and relativistic electronics Non-relativistic and relativistic electronics Azarenkov, N.A. Chibisov, D.V. Kovalenko, N.I. Maslennikov, D.I. Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion Вопросы атомной науки и техники |
| description |
The problem of evolution and disappearance of the lower hybrid cavities that are observed in the plasma of the Earth’s ionosphere is solved. It is assumed that the destruction of the cavity is caused by turbulent diffusion of plasma, which arises due to the drift instability of radially inhomogeneous plasma. The initial plasma density distribution on the radius in the cavity is considered to be the inverse Gaussian distribution. A solution of the diffusion equation is obtained, which at any time determines the radial dependence of the plasma density in the cavity. In the asymptotic limit t →∞ the plasma density in the cavity becomes equal to the density of the surrounding plasma. |
| format |
Article |
| author |
Azarenkov, N.A. Chibisov, D.V. Kovalenko, N.I. Maslennikov, D.I. |
| author_facet |
Azarenkov, N.A. Chibisov, D.V. Kovalenko, N.I. Maslennikov, D.I. |
| author_sort |
Azarenkov, N.A. |
| title |
Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion |
| title_short |
Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion |
| title_full |
Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion |
| title_fullStr |
Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion |
| title_full_unstemmed |
Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion |
| title_sort |
temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2019 |
| topic_facet |
Non-relativistic and relativistic electronics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/195161 |
| citation_txt |
Temporal evolution of the lower hybrid cavities in the ionosphere plasma due to turbulent diffusion / N.A. Azarenkov, D.V. Chibisov, N.I. Kovalenko, D.I. Maslennikov // Problems of atomic science and technology. — 2019. — № 4. — С. 27-30. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT azarenkovna temporalevolutionofthelowerhybridcavitiesintheionosphereplasmaduetoturbulentdiffusion AT chibisovdv temporalevolutionofthelowerhybridcavitiesintheionosphereplasmaduetoturbulentdiffusion AT kovalenkoni temporalevolutionofthelowerhybridcavitiesintheionosphereplasmaduetoturbulentdiffusion AT maslennikovdi temporalevolutionofthelowerhybridcavitiesintheionosphereplasmaduetoturbulentdiffusion |
| first_indexed |
2025-07-16T23:00:05Z |
| last_indexed |
2025-07-16T23:00:05Z |
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1837846293158821888 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2019. №4(122) 27
TEMPORAL EVOLUTION OF THE LOWER HYBRID CAVITIES
IN THE IONOSPHERE PLASMA DUE TO TURBULENT DIFFUSION
N.A. Azarenkov1, D.V. Chibisov1, N.I. Kovalenko2, D.I. Maslennikov2
1V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
2Kharkiv National Agrarian University, «Dokuchaievske-2», Kharkiv region, Ukraine
E-mail: dmitriychibisov@karazin.ua
The problem of evolution and disappearance of the lower hybrid cavities that are observed in the plasma of the
Earth’s ionosphere is solved. It is assumed that the destruction of the cavity is caused by turbulent diffusion of plas-
ma, which arises due to the drift instability of radially inhomogeneous plasma. The initial plasma density distribu-
tion on the radius in the cavity is considered to be the inverse Gaussian distribution. A solution of the diffusion
equation is obtained, which at any time determines the radial dependence of the plasma density in the cavity. In the
asymptotic limit t →∞ the plasma density in the cavity becomes equal to the density of the surrounding plasma.
PACS: 52.35.Qz, 52.35.Ra, 94.20.wf
INTRODUCTION
Lower hybrid cavities (LHC) are axisymmetric soli-
tary structures in the Earth’s ionosphere, elongated
along the geomagnetic field, in which the plasma densi-
ty is depleted compared to the surrounding plasma [1].
A characteristic feature of cavities is also an enhanced
level of lower hybrid oscillations compared with ambi-
ent hiss. These two features are quite accurately reflect-
ed in the name of these structures. LHCs are detected by
satellites as well as by sounding rockets and are ob-
served only when they pass directly through cavities.
Since the time of cross of the cavity by the spacecraft is
short, the possibilities for detailed study of the LHC are
limited. In particular, the moments of appearance of
such structures, as well as their disappearance, have not
been detected. Despite the fact that quite a lot of re-
search data has been obtained, there is still no accepta-
ble theory of the origin of LHC. There is also no expla-
nation for their stability, but at the same time, the origin
of the oscillations is explained by the presence of an
azimuthal current caused by a non-uniform plasma den-
sity [1]. For a long time, no attention was paid to the
problem of the destruction of cavities. In our paper [2]
was suggested that the destruction of LHC is caused by
the anomalous diffusion of ions and electrons, which
occurs due to the development of drift turbulence of
radially inhomogeneous plasma. It was shown that radi-
al in homogeneity of plasma density leads to the devel-
opment of the drift instability and drift turbulence of
plasma in the cavity. For the conditions of LHC the es-
timates of values of the frequency, as well as the growth
rate of drift oscillations were made. In addition, the av-
erage time of plasma diffusion in the cavity was prelim-
inarily estimated. A comparison of these average times
showed that the plasma diffusion rate is less than the
oscillation growth rate. Apparently, this can explain the
absence of a low-frequency spectrum of oscillations in
the LHC, measured by a spacecraft.
In this paper, which is a development of work [2],
the problem of the plasma density temporal evolution in
a cavity due to the anomalous transport across the mag-
netic field caused by drift turbulence in radially inho-
mogeneous plasma is considered in detail. In particular,
more accurate estimates of the diffusion rate, which
correct the time of destruction of the cavity, are made.
EVOLUTION OF PLASMA DENSITY
DEPENDENCE ON RADIUS
In homogeneous magnetized plasma, the density of
which is 0n , we consider a cavity with axial symmetry,
whose axis coincides with the direction of the magnetic
field. It is assumed that the initial dependence of the
plasma density on the radius in the cavity is [2]
( )
−−= 2
0
2
0 2
exp1
r
ranrn , (1)
which is the inverse Gaussian distribution. In (1) 0r is
the length of the in homogeneity of plasma density, a is
the positive constant, which is the depth of the cavity.
The dependence (1) is confirmed by observations [1].
As a result of the development of drift turbulence due to
the radial in homogeneity of plasma, anomalous
transport of ions and electrons across the magnetic field
occurs, which is governed by the diffusion equation
, ,
,
1
⊥
∂ ∂ ∂
= ∂ ∂ ∂
e i e i
e i
n n
rD
t r r r
, (2)
where subscripts e and i denotes the electrons and
ions, , ⊥e iD is the diffusion coefficient across the mag-
netic field, which is the same for electrons and ions,
since diffusion is ambipolar. The value of , ⊥e iD is [3 - 5]
( ) ( ) ( )
( )
2 2
2
, 2 2 2
0
γ
ω⊥ ⊥= ∑ ∫ m
e i m e m
m m
kc mD dk J k R I k
B r k
,
where ( )γ m k and ( )ωm k are the growth rate and the
frequency of drift oscillations in LHC [2]. For drift tur-
bulence in the saturation state, the diffusion coefficient
is [3 - 5]
,
0 0
ρ
⊥ =
e s
e i
cT
D
eB r
. (3)
To solve the equation (2), we introduce the notation
,4 ⊥ =e iD D and rewrite (2) as
2
2 2
∂ ∂ ∂ = ∂ ∂ ∂
n nr D
t r r
. (4)
Take the Laplace transform of eq. (4) by multiplying
both sides of it by ( )2exp − pr
and integrate over 2r
ISSN 1562-6016. ВАНТ. 2019. №4(122) 28
( ) ( )2 22 2 2
2 2
0 0
,
,
∞ ∞
− − ∂ ∂ ∂
= ∂ ∂ ∂
∫ ∫pr pr r t
n r t e dr e r D dr
t r r
.
Denote
( ) ( )
2 2
0
, ,
∞
−= = ∫ prN N p t n r t e dr , (5)
which is the Laplace transform for plasma density by
the squared radial coordinate. Then we get the equation
2 2 2
2 2
0
∞
−∂ ∂ ∂ = ∂ ∂ ∂
∫ prN ne r D dr
t r r
. (6)
Now we integrate the right side of (6) by parts twice
and obtain:
( )2 22 2 2
2
0
∞
− −∂ ∂ = − − + ∂ ∂
∫ pr prN p n pe r D e r D dr
t r
.
Suppose that the diffusion coefficient does not de-
pend on the radius, and then this equationis simplified
22 2 2
0
∞
−∂
= −
∂ ∫ prN p D ne r dr pDN
t
. (7)
For the first term in (7), we use differentiation by pa-
rameter
22 2 2
0
∞
− =∫ prp D ne r dr
22 2 2
0
∞
− ∂ ∂
= − = − ∂ ∂
∫ pr Np D ne dr p D
p p
. (8)
As a result, the eq. (7) for ( ),N p t becomes
2∂ ∂
= − −
∂ ∂
N Np D pDN
t p
,
or
2∂ ∂
+ = −
∂ ∂
N Np D pDN
t p
. (9)
To solve the partial differential equation (9), we use
the method of characteristics. First integral of the eq. (9)
we find from the characteristic equation
21 Dp
dpdt
= . (10)
The solution of eq. (10), i.e. the first integral of
eq. (9) is
1
= +u t
Dp
. (11)
Now we get the second integral of eq. (9). For this
we rewrite eq. (9) as
2∂ ∂ ∂
+ = −
∂ ∂ ∂
N N Np D pDN
t p N
.
Then we obtain second characteristic equation for
(9)
= −
dp dN
p N
. (12)
So the second integral is
=v Np . (13)
First and second integrals give a general solution of
(9):
( )ugv = , (14)
where g is an arbitrary function. Substituting (11) and
(13) into (14) we obtain
1
= +
Np g t
Dp
,
or
( )
+=
Dp
tg
p
tpN
11
, . (15)
Now we take into account the initial dependence (i. e.
at 0=t ) of plasma density on the radius (1). Solving the
diffusion equation (9), we consider only second term in
(1), since the first does not depend on either time or coor-
dinate. Thus we study the evolution of the distribution
( )
2
2
02
0
−
=
r
rn r n ae . (16)
Find the Laplace transform of (16)
( )
2
2 2
02 2 0
0 0
0
2
0
1
2
−∞
−= =
+
∫
r
r pr n a
N p n a e e dr
p
r
, (17)
and substitute it into (15) where we assume 0=t .
Thereby we find the function ( )0N p for the initial
moment of time
( ) 0
0
2
0
1 1
1
2
= =
+
n a
N p g
p Dpp
r
. (18)
Thus we found the explicit form of the function g
(14):
( ) 0
2
0
1
2
=
+
n a
g x
D x
r
. (19)
Now we substitute into (19) instead of x the value
1
= +x t
Dp
,
that yields time dependence of ( ),N p t :
( ) 0
2
0
1, ,
11
2
=
+ +
n a
N p t
p D t
Dpr
or, otherwise
( )
+
+
⋅
+
=
t
r
Dr
pt
r
D
antpN
2
0
2
0
2
0
0
2
12
1
1
2
1
, . (20)
To obtain the dependency ( ),n r t , we take the in-
verse Laplace transform for ( ),N p t (20):
( ) ( ) == ∫
∞⋅+
∞⋅−
dpetpN
i
trn pr
is
is
2
,
2
1,
p
12
2 2
0 0
1
2 20
2
0
1
2
−
− +
=
+
r D t
r rn a
e
D t
r
. (21)
Now in (21) we back to the original diffusion coeffi-
cient ⊥= ieDD ,4 and obtain
ISSN 1562-6016. ВАНТ. 2019. №4(122) 29
( )
12 ,
2 2
0 0
2
1
20
,
2
0
,
2
1
−
⊥
− +
⊥
=
+
e iDr t
r r
e i
n a
n r t e
D
t
r
. (22)
Finally, we get the dependence of plasma density in
the cavity on the radius and on time as
( )
12 ,
2 2
0 0
2
1
2
0
,
2
0
, 1
2
1
−
⊥
− +
⊥
= − +
e iDr t
r r
e i
an r t n e
D
t
r
. (23)
In (23), the depth of the cavity is determined by
( )
,
2
0
2
1 ⊥
=
+
e i
aa t
D
t
r
. (24)
It follows from (24) that the depth of the cavity is
inversely proportional to time and in the asymptotic
limit ∞→t the plasma density in the cavity tends to
the density of the surrounding plasma. In addition, the
cavity expands with time; the root-mean-square radial
size of the cavity varies with time as
( ) ,
0 2
0
2
1σ ⊥= + e iD
t r t
r
. (25)
Let us estimate now the time of evolution of the
plasma density distribution in the cavity on the radius.
Characteristic values of plasma parameters in the cavity
are [1]: 3/0 sr ρ , r 5 00 m, 0 ~ 0.2B Gs , ~ 0.3eT eV .
For these parameter values the diffusion coefficient (3)
is 7 2 1
, 5 10 −
⊥ = ⋅ ⋅e iD cm s . Denote k the ratio of the depth
of the cavity at the initial and arbitrary points in time,
( )
,
2
0
2
1 ⊥
= = +
e iDak t
a t r
, (26)
and obtain the time kt , which corresponds to the change
in the depth of the cavity by k
( ) 2
0
,
1
2 ⊥
−
=k
e i
k
t r
D
. (27)
Substituting known values into (27), we get
( )0.25 1= −kt k . (28)
For example, a decrease in cavity depth by 2 times
will occur in 0.25 s, and a decrease in cavity depth by 5
times occurs in 1 s. The time evolution of the depend-
ence of the plasma density in the cavity on the radius is
shown in the Figure, where the plasma densities versus
radius are shown for different moments of time:
0 1 2 3 40 , 0.25 , 0.5 0.75 , 1= = = = =t s t s t s t s t s at initial
cavity depth 0.6=a .
Note that the time of development of the drift insta-
bility in the cavity is 0.5…1.5 s [2] and this time value
should be added to the diffusion time. Therefore the
complete disappearance of the cavity will occur in the
order of a few seconds.
The dependence of the plasma density
in the cavity on the radius for different moments of time
CONCLUSIONS
Radial in homogeneity of the plasma density in the
LHC causes the drift turbulence and anomalous diffu-
sion of ions and electrons to the center of the LHC
across the magnetic field. Plasma diffusion leads to the
temporal evolution of the dependence of the plasma
density on the radius in accordance with (23).
The depth of the cavity is inversely proportional to
time (24) and in the asymptotic limit →∞t the plasma
density in the cavity tends to the density of the sur-
rounding plasma. The time to decrease the depth of the
cavity in k times is proportional to )1( −k (27). Simul-
taneously with a decrease in the depth of the cavity, its
expansion occurs. Root-mean-square radial size of the
cavity is proportional to the square root of time (25).
Estimation of plasma diffusion time in the cavity,
taking into account the time of development of the drift
instability, shows that the destruction of the cavity oc-
curs within a few seconds. Thus, spacecraft are not able
to detect the process of destruction of the cavity, since
the observation time by them is much less than the time
of the existence of the cavity.
REFERENCES
1. P.W. Schuck, J.W. Bonnell, P.M. Kintner. A review
of lower hybrid solitary structures // IEEE Trans.
Plasma Sci. 2003, v. 31, № 6, p. 1125-1177.
2. N.A. Azarenkov, D.V. Chibisov. Anomalous diffu-
sion of plasma in the lower hybrid cavities observed
in the terrestrial ionosphere // Problems of Atomic
Science and Technology. Series “Plasma Physics”.
2018, № 6, p. 117-120.
3. D.V. Chibisov, V.S. Mikhailenko, K.N. Stepanov.
Drift wave turbulence of a radially inhomogeneous
plasma // Phys. Lett. A. 1991, v. 157, p. 141-145.
4. V.S. Mikhailenko, K.N. Stepanov, D.V. Chibisov.
Drift and drift-cyclotron turbulence of a radially
varying axisymmetric plasma // Soviet Journal of
Plasma Physics. 1991, v. 17, № 10, p. 710-716.
5. V.S. Mikhailenko, K.N. Stepanov, D.V. Chibisov.
Drift turbulence of an azimutally symmetric radially
nonuniform plasma // Plasma Phys. Rep. 1995,
v. 21, № 12, p. 141-150.
Article received 29.05.2019
ISSN 1562-6016. ВАНТ. 2019. №4(122) 30
ВРЕМЕННАЯ ЭВОЛЮЦИЯ НИЖНЕГИБРИДНЫХ ПОЛОСТЕЙ В ИОНОСФЕРНОЙ ПЛАЗМЕ
ВСЛЕДСТВИЕ ТУРБУЛЕНТНОЙ ДИФФУЗИИ
Н.А. Азаренков, Д.В.Чибисов, Н.И. Коваленко, Д.И. Масленников
Решается проблема эволюции и исчезновения нижнегибридных полостей, наблюдаемых в плазме ионо-
сферы Земли. Предполагается, что разрушение полости вызвано турбулентной диффузией плазмы, которая
возникает из-за дрейфовой неустойчивости радиально-неоднородной плазмы. В качестве начального ради-
ального распределения плотности плазмы в полости рассматривается обратное распределение Гаусса. Полу-
чено решение уравнения диффузии, которое в любой момент времени определяет зависимость от радиуса
плотности плазмы в полости. Показано, что в асимптотическом пределе t → ∞ плотность плазмы в полости
становится равной плотности окружающей плазмы.
ЕВОЛЮЦІЯ В ЧАСІ НИЖНЬОГІБРИДНИХ ПОРОЖНИН В ІОНОСФЕРНІЙ ПЛАЗМІ
ВНАСЛІДОК ТУРБУЛЕНТНОЇ ДИФУЗІЇ
М.О. Азарєнков, Д.В. Чібісов, М.І. Коваленко, Д.І. Масленніков
Вирішується проблема еволюції і зникнення нижньогібридних порожнин, які спостерігаються в плазмі
іоносфери Землі. Передбачається, що руйнування порожнини викликане турбулентною дифузією плазми,
яка виникає через дрейфову нестійкість радіально-неоднорідної плазми. В якості початкового радіального
розподілу щільності плазми в порожнині розглядається зворотній розподіл Гаусcа. Отримано рішення рів-
няння дифузії, яке в будь-який момент часу визначає залежність від радіуса щільності плазми в порожнині.
Показано, що в асимптотичній границі t → ∞ щільність плазми в порожнині стає рівною щільності навко-
лишньої плазми.
EVOLUTION OF PLASMA DENSITY DEPENDENCE ON RADIUS
In homogeneous magnetized plasma, the density of which is , we consider a cavity with axial symmetry, whose axis coincides with the direction of the magnetic field. It is assumed that the initial dependence of the plasma density on the radius in the c...
, (1)
which is the inverse Gaussian distribution. In (1) is the length of the in homogeneity of plasma density, is the positive constant, which is the depth of the cavity. The dependence (1) is confirmed by observations [1]. As a result of the development o...
, (2)
where subscripts and denotes the electrons and ions, is the diffusion coefficient across the magnetic field, which is the same for electrons and ions, since diffusion is ambipolar. The value of is [3 - 5]
,
where and are the growth rate and the frequency of drift oscillations in LHC [2]. For drift turbulence in the saturation state, the diffusion coefficient is [3 - 5]
. (3)
To solve the equation (2), we introduce the notation and rewrite (2) as
. (4)
Take the Laplace transform of eq. (4) by multiplying both sides of it by and integrate over
.
Denote
, (5)
which is the Laplace transform for plasma density by the squared radial coordinate. Then we get the equation
. (6)
Now we integrate the right side of (6) by parts twice and obtain:
.
Suppose that the diffusion coefficient does not depend on the radius, and then this equationis simplified
. (7)
For the first term in (7), we use differentiation by parameter
. (8)
As a result, the eq. (7) for becomes
,
or
. (9)
To solve the partial differential equation (9), we use the method of characteristics. First integral of the eq. (9) we find from the characteristic equation
. (10)
The solution of eq. (10), i.e. the first integral of eq. (9) is
. (11)
Now we get the second integral of eq. (9). For this we rewrite eq. (9) as
.
Then we obtain second characteristic equation for (9)
. (12)
So the second integral is
. (13)
First and second integrals give a general solution of (9):
, (14)
where is an arbitrary function. Substituting (11) and (13) into (14) we obtain
,
or
. (15)
Now we take into account the initial dependence (i. e. at ) of plasma density on the radius (1). Solving the diffusion equation (9), we consider only second term in (1), since the first does not depend on either time or coordinate. Thus we study the e...
. (16)
Find the Laplace transform of (16)
, (17)
. (18)
Thus we found the explicit form of the function (14):
. (19)
Now we substitute into (19) instead of the value
,
that yields time dependence of :
or, otherwise
. (20)
To obtain the dependency , we take the inverse Laplace transform for (20):
. (21)
Now in (21) we back to the original diffusion coefficient and obtain
. (22)
Finally, we get the dependence of plasma density in the cavity on the radius and on time as
. (23)
In (23), the depth of the cavity is determined by
. (24)
. (25)
Let us estimate now the time of evolution of the plasma density distribution in the cavity on the radius. Characteristic values of plasma parameters in the cavity are [1]: , m, , . For these parameter values the diffusion coefficient (3) is . Denote ...
, (26)
and obtain the time , which corresponds to the change in the depth of the cavity by
. (27)
Substituting known values into (27), we get
. (28)
For example, a decrease in cavity depth by 2 times will occur in 0.25 s, and a decrease in cavity depth by 5 times occurs in 1 s. The time evolution of the dependence of the plasma density in the cavity on the radius is shown in the Figure, where the ...
Note that the time of development of the drift instability in the cavity is 0.5…1.5 s [2] and this time value should be added to the diffusion time. Therefore the complete disappearance of the cavity will occur in the order of a few seconds.
The dependence of the plasma density in the cavity on the radius for different moments of time
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