Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields
The time evolution of the formation of a plasma density cavity caused by inhomogeneous stochastic electric fields is investigated. The Fokker-Planck equation, which governs the temporal evolution of the plasma electron density due to localized stochastic inhomogeneous electric fields in the frequenc...
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| Цитувати: | Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields / N.A. Azarenkov, D.V. Chibisov, O.D. Chibisov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 21-26. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1961672023-12-11T13:48:26Z Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields Azarenkov, N.A. Chibisov, D.V. Chibisov, O.D. Relativistic and nonrelativistic plasma electronics The time evolution of the formation of a plasma density cavity caused by inhomogeneous stochastic electric fields is investigated. The Fokker-Planck equation, which governs the temporal evolution of the plasma electron density due to localized stochastic inhomogeneous electric fields in the frequency range of lower hybrid oscillations, is solved numerically. The spatial dependence of the plasma electron density for various times is obtained. Досліджено часову еволюцію утворення плазмової порожнини, спричиненої неоднорідними стохастичними електричними полями. Рівняння Фоккера-Планка, яке визначає часову еволюцію електронної густини плазми, що викликана локалізованими стохастичними неоднорідними електричними полями в діапазоні частот нижньогібридних коливань, розв’язано чисельно. Отримано просторову залежність густини електронів плазми для різних значень часу. 2023 Article Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields / N.A. Azarenkov, D.V. Chibisov, O.D. Chibisov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 21-26. — Бібліогр.: 18 назв. — англ. 1562-6016 PACS: 52.35.Ra, 52.35.Mw DOI: https://doi.org/10.46813/2023-146-021 http://dspace.nbuv.gov.ua/handle/123456789/196167 en Problems of Atomic Science and Technology Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Relativistic and nonrelativistic plasma electronics Relativistic and nonrelativistic plasma electronics |
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Relativistic and nonrelativistic plasma electronics Relativistic and nonrelativistic plasma electronics Azarenkov, N.A. Chibisov, D.V. Chibisov, O.D. Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields Problems of Atomic Science and Technology |
| description |
The time evolution of the formation of a plasma density cavity caused by inhomogeneous stochastic electric fields is investigated. The Fokker-Planck equation, which governs the temporal evolution of the plasma electron density due to localized stochastic inhomogeneous electric fields in the frequency range of lower hybrid oscillations, is solved numerically. The spatial dependence of the plasma electron density for various times is obtained. |
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Article |
| author |
Azarenkov, N.A. Chibisov, D.V. Chibisov, O.D. |
| author_facet |
Azarenkov, N.A. Chibisov, D.V. Chibisov, O.D. |
| author_sort |
Azarenkov, N.A. |
| title |
Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields |
| title_short |
Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields |
| title_full |
Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields |
| title_fullStr |
Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields |
| title_full_unstemmed |
Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields |
| title_sort |
temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2023 |
| topic_facet |
Relativistic and nonrelativistic plasma electronics |
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http://dspace.nbuv.gov.ua/handle/123456789/196167 |
| citation_txt |
Temporal evolution of the plasma density cavity caused by inhomogeneous stochastic electric fields / N.A. Azarenkov, D.V. Chibisov, O.D. Chibisov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 21-26. — Бібліогр.: 18 назв. — англ. |
| series |
Problems of Atomic Science and Technology |
| work_keys_str_mv |
AT azarenkovna temporalevolutionoftheplasmadensitycavitycausedbyinhomogeneousstochasticelectricfields AT chibisovdv temporalevolutionoftheplasmadensitycavitycausedbyinhomogeneousstochasticelectricfields AT chibisovod temporalevolutionoftheplasmadensitycavitycausedbyinhomogeneousstochasticelectricfields |
| first_indexed |
2025-07-17T00:39:57Z |
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2025-07-17T00:39:57Z |
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1837852574468800512 |
| fulltext |
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 21
https://doi.org/10.46813/2023-146-021
TEMPORAL EVOLUTION OF THE PLASMA DENSITY CAVITY
CAUSED BY INHOMOGENEOUS STOCHASTIC ELECTRIC FIELDS
N.A. Azarenkov
1,2
, D.V. Chibisov
2
, O.D. Chibisov
3
1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine;
2
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
3
H.S. Skovoroda Kharkiv National Pedagogical University, Kharkiv, Ukraine
E-mail: dmitriychibisov@karazin.ua
The time evolution of the formation of a plasma density cavity caused by inhomogeneous stochastic electric
fields is investigated. The Fokker-Planck equation, which governs the temporal evolution of the plasma electron
density due to localized stochastic inhomogeneous electric fields in the frequency range of lower hybrid oscillations,
is solved numerically. The spatial dependence of the plasma electron density for various times is obtained.
PACS: 52.35.Ra, 52.35.Mw
INTRODUCTION
In the plasma of the Earth's ionosphere and magne-
tosphere near the auroral zone, satellites and sounding
rockets have detected regions with a depleted density of
plasma and an increased level of oscillations in the low-
er hybrid frequency range which are called lower hybrid
cavities (LHC) [1 - 8]. The measurements showed that
the LHC are cylindrical regions elongated along the
geomagnetic field. Their perpendicular size is
10…100 m which is equal to a few ion gyroradii, while
along the geomagnetic field LHC extends for several
kilometers, and possibly for tens or hundreds of kilome-
ters. As an explanation for this phenomenon, it was
suggested in [9 - 12] that LHCs appear as a result of
modulation instability and lower hybrid collapse. How-
ever, it was shown in [13] that this model does not cor-
respond to the properties of most of the observed cavi-
ties. In [14], another mechanism for the occurrence of
LHC was proposed, which assumes the appearance of
an ion density cavity due to their expulsion from a cer-
tain volume by inhomogeneous stochastic oscillations
with a frequency of the order of the lower hybrid fre-
quency. A similar effect of electron expulsion due to an
inhomogeneous harmonic electric field was considered
in [15].
In [16], the diffusion and drift motion of both ions
and electrons across the magnetic field due to the action
of inhomogeneous stochastic electric fields was studied.
It was shown that the movement of electrons is much
faster than that of ions, so that the formation of an elec-
tron cavity occurs much earlier than an ionic one. In
[17, 18], the influence of the thermal motion of plasma
particles on the conditions for the formation of a cavity
was studied. In particular, it was shown that an electron
density cavity can form if, during the formation of the
cavity, the electrons do not leave the region with an
increased level of oscillations along the magnetic field.
Based on the Fokker-Planck equation, the steady state
plasma density distribution was obtained [16 - 18], as-
suming that the evolution of the distribution function
has ended.
In this paper, we study the temporal evolution of the
formation of an electron cavity. Solving the Fokker-
Planck equation, we obtain the electron density distribu-
tion at various points in time.
1. DIFFUSION AND DRIFT OF ELECTRONS
We consider collisionless and initially homogeneous
plasma in a constant magnetic field H directed along the
z axis, in which at some time a region with a stochastic
electric field appears, which is inhomogeneous along
the x axis and homogeneous in other directions. It is
assumed that the frequency range of stochastic oscilla-
tions is near the lower hybrid frequency ωlh, which is
much lower than the electron cyclotron frequency ωce.
To study changes in time of the spatial distribution
of the electron density n(x,t) due to inhomogeneous sto-
chastic electrostatic oscillations, we use the one-
dimensional Fokker-Planck equation
2
2
,
, ,
2
n x t B x
A x n x t n x t
t x x
, (1)
where A(x) is the drift velocity and B(x) / 2 is the diffu-
sion coefficient of electrons. These values are defined
from the particle motion equation as the quasi-linear
drift velocity averaged over a long time and the dis-
placement velocity of the squared rms electron dis-
placement.
The equation of motion of an electron in constant
magnetic and stochastic electric fields is
, ,
e e
dv e e
F x E r t v H
dt m m
, (2)
where ,E r t is the electric field strength of stochastic
oscillations far from the region with a high level of tur-
bulence, F(x) is the envelope of stochastic oscillations,
which has a maximum at x = 0 and F(∞) = 1.
Neglecting in the zero approximation the influence
of the stochastic electric field, we obtain the solution of
equation (2) in the form of integrals of motion
, ,
y x
z e z
ce ce
v v
X x Y y p m v
, (3)
where X and Y are the coordinates of the guiding center
of electron and pz is the electron momentum along the
magnetic field. The first two integrals mean the invaria-
bility in time of the coordinates of the guiding center of
an electron rotating in a magnetic field.
The next approximation in (2) takes into account the
effect of stochastic electric fields, which leads to small
changes in the integrals of motion (3). In this case, the
22 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146)
solution of equation (2) is represented as
0 1e ev v v ,
where
0ev is the electron velocity vector at the initial
moment of time the components of which are random
variables distributed according to the normal law, that
is, 0e Tev v is the thermal velocity of electrons,
1ev is
the fluctuation of the velocity, caused by stochastic
electric fields and which is defined by the equation
1
1, ,
e e
dv e e
F x E r t v B
dt m m
. (4)
The solutions of (4) for the components of velocity
are
1 2
,
,
x
x y
e cee ce
eF x dE r t eF x
v E r t
dt mm
, (5)
1 2
,
,
y
y x
e cee ce
dE r teF x eF x
v E r t
dt mm
, (6)
0
1 ,
t
z z
e t
e
v F x E r t dt
m
. (7)
Since it is assumed that the stochastic electric field is
inhomogeneous along the x axis, the electron density
changes only along this axis, while in other directions
the density gradient is zero. Therefore, across the mag-
netic field, the influence of the stochastic field is con-
sidered only on the change in the X-component of the
coordinate of the guiding center. To define the change
in X due to stochastic electric fields, the X coordinate is
written as
0 1X X X , (8)
where X1 is the random displacement of the coordinate
of the guiding center due to stochastic electric fields.
Using (3) we write
1 1 1
1
y
ce
X x v
. (9)
In equation (9), the value of x1 is defined by integrat-
ing v1x (5) over time
1 1
0
t
xx v t dt
2
0
, ,
t
x ce y
e ce
eF x
E r t E r t
m
. (10)
Substituting (6) and (10) into (9) yields
1 3
0
,
,
t
y
y
e ce e ce
dE r teF x eF x
X E r t dt
m dtm
. (11)
Differentiating (9) with respect to time and using (5)
and (6), we find the rate of change of X1
11
1
1 1
,
y
x y
ce e ce
dvdX e
v F x E r t
dt dt m
. (12)
Then multiplying (11) by (12) and averaging over a
large time interval, we obtain the rate of change of the
root-mean-square displacement
2 22
11
1 2 2
1
2 e ce
d X F xdX e
X
dt dt m
2
0
,1 1
, , .
2
t
y
y y
ce
d E r t
E r t E r t dt
dt
(13)
Neglecting the second term in (13) which is much
smaller than the first one, we obtain
2 22
1
2 2
0
1
, , .
2
t
y y
e ce
d X F xe
E r t E r t dt
dt m
(14)
We assume that the strength of stochastic electric
fields satisfies the conditions
, 0,E r t
2 2, , ,E r t E r t E r t E r , (15)
where 2E r is the square of the amplitude of stochas-
tic oscillations which does not depend on time. Taking
into account the condition (15) we obtain from (14)
2 2
1 2 2
2 2
1
2 2
y
e ce
d X B x e
F x E r t
dt m
. (16)
Equation (16) can also be written as
2 2 ,
2
dx
B x
F x v r t t , (17)
where
y
dx
cE
v
H
, (18)
is the velocity of the drift motion of electrons in crossed
electric and magnetic fields along the x-axis. Thus, the
rate of the root-mean-square displacement of the coor-
dinate of the guiding center along the x-axis is defined
by the mean value of the square of y-component of the
stochastic electric field, or, otherwise, by the mean val-
ue of the square of the electron drift stochastic velocity
in crossed fields.
The formation of the electron density cavity is influ-
enced by their motion along the magnetic field. In order
for the cavity to form, it is necessary that during the
time of the formation of the cavity, the electrons along
the magnetic field would not leave the region with an
increased level of oscillations. For this estimate, we
need to define the electron diffusion coefficient along
the magnetic field. From (3) and (7) we get
2
2
2 2 2
2 2
1
2 2
z Te
e
d z t e
F x E r t v t
dt m
,
where Δω is the width of the spectrum of stochastic
oscillations which is the order of ωlh [16, 17]. However,
the effect of the stochastic electric field on diffusion
along the magnetic field can be neglected in comparison
with thermal motion, so
2
2
0
1
2
e
d z t
v t
dt
. (19)
Comparison of (17) and (19) shows that the diffu-
sion coefficient along the magnetic field is much greater
than across the magnetic field.
In order to find the speed of the drift motion of elec-
trons along the x axis A(x) due to the ponderomotive
force, which caused by the inhomogeneity of stochastic
oscillations, we represent the random displacement of
the coordinate of the guiding center as the sum of the
oscillating and quasilinear components
1X X X , (20)
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 23
where
1X X and 0X . Substituting (20) into
(12), yields
1 1
,y
e ce
dX dX dX e
F x E r t
dt dt dt m
. (21)
Expand envelop function F(x) in a Taylor series
about the initial value of the position of the coordinate
of the guiding center
0 0 1F x F X F X x (22)
and substitute (22) into (21)
1
0 1 0
1
,y
e ce
dX e
F X x F X E r t
dt m
. (23)
Then averaging over a large time interval we obtain
the rate of quasi-linear change in the coordinate of the
guiding center
1
0 1
1
,y
e ce
d X dX e
F X x E r t
dt dt m
. (24)
Find in (24) 1 ,yx E r t substituting the value x1
(10):
1 ,yx E r t
0
2
0
, , ,
t
x ce y y
e ce
eF X
E r t E r t dt E r t
m
.
Since , , 0x yE r t E r t we obtain
2
0 02 2
0
, ,
t
e
y y
e ce
edX
F X F X E r t E r t dt
dt m
.(25)
Integration (25) using condition (15) yields
2
2
2
02 22
y
e ce
dX e
F X E r t
dt m
and then applying the notation (18) we write this equa-
tion as
2 2
0
1
,
2
dx
dX
A x F X v r t t
dt
. (26)
Equation (26) defines the drift velocity A(x) of the
guiding center of electron along the x-axis.
2. TIME EVOLUTION OF THE PLASMA
ELECTRON DENCITY DISTRIBUTION
Let us study the evolution of the development of
electron density cavity in time, assuming that the size of
the region with an increased level of oscillations along
the magnetic field is large enough, so that the electrons
do not have time to leave this region before the cavity is
formed. Substituting the diffusion coefficient (17) and
drift velocity (26) into equation (1) yields
2
,
,dx
n x t
v r t t
t
2
2
,1
,
2
F x n x t
F x n x t
x x
. (27)
It is assumed that at the initial time t = 0 the plasma
density has a uniform distribution n(x) = n0 = const. It
was shown [17, 18] that due to the action of localized
stochastic electric fields with an envelope F(x) at the
end of evolution in a stationary state, i.e., at t = ∞ the
dependence of the electron density on the x-axis is de-
fined by the relation
0n
n x
F x
. (28)
Now, using the equation (27), we study the evolu-
tion of the spatial distribution of the electron density,
i.e. how the density distribution changes over time from
uniform to (28). To solve the equation (27), we intro-
duce a new unknown function N(x,t), which is related to
n(x) by
, ,N x t n x t F x . (29)
The initial condition for this function is
0,0 ,0N x n x F x n F x . (30)
Substituting (29) into (27) yields
2
,1
,dx
N x t
v r t t
F x t
2 2
0
1
2
N x N x
F X F x
x F x x F x
and after simplifications this equation becomes
2
,
,dx
N x t
v r t tF x
t
0F X N x F x N x
x x
.
Writing here the derivative of the product and sim-
plifying, we get
2
,
,dx
N x t N x
v r t tF x F x
t x x
. (31)
Denote
2 ,dxv r t D , (32)
then (31) is written as
2
, 1
2
N x N x
DF x F x
x xt
. (33)
Equation (33) is the diffusion equation, which, un-
like (27), no longer has a drift term. Introducing the
renormalized coordinate ξ defined as
dx
d
F x
, (34)
or otherwise
dx
f x
F x
, (35)
equation (33) reduced to the classical diffusion equation
2
2 2
, 1
2
N t N
D
t
. (36)
To solve (36), we write the Fourier transform for
N(ξ,t)
1
, ,
2
iss t N t e d
, (37)
and then perform the Fourier transform of (36) by mul-
tiplying it by 1/ 2 exp is and integrating over ξ
2
,1
exp
2
N
is d
t
24 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146)
2
2
1 1
exp
22
N
D is d
. (38)
Integrating the right side of (38) twice by parts and
using (37) we get
2
2
1
,
2
s D t s
t
. (39)
Equation (39) has a solution
2 21
2,
Ds t
s t Ce
.
The constant C is found from the initial condition
(30) and relation (37)
0
1
,0
2
isC s n F x e d
.
And thus
2 21
0 2,
2
Ds t
isn
s t e F x e d
. (40)
After the inverse Fourier transform in (40), we ob-
tain
2 21
0 2,
2
is is Ds tn
N t F x e dsd
. (41)
Integrating in (41) over s yields
2
20 2
2
2
,
2
Dt
n
N t F x e d
Dt
. (42)
To obtain the dependency N(x,t), it is necessary to
perform in (42) the inverse substitutions of variables
(34) and (35)
2
0
22
, exp
22
f x f xn
N x t dx
DtDt
. (43)
Finally, using (29), we obtain from (43) an expres-
sion for the distribution of plasma electrons density at
an arbitrary moment of time
0
2
,
2
n
n x t
F x Dt
2
2
exp
2
f x f x
dx
Dt
. (44)
This completes the formal part of the solution, and
further, to obtain a specific dependence of the density
on the coordinate, it is necessary to choose the form of
the envelope function F(x). At that the form of F(x) is
limited by the necessary requirements to have a maxi-
mum at x = 0 and decrease to F=1 at x=±∞. We choose
F(x) as
2 2
0
1 1
1
1
a a
F x
rx
x
, (45)
which satisfies these requirements. Here x0 is the size of
the inhomogeneity of the region of stochastic oscilla-
tions, parameter a defines the height of the envelope
above the background value of stochastic oscillations in
the environment, and r is the normalized coordinate
along the x-axis. At x = 0, the value (1+a) defines how
many times the oscillation amplitude exceeds the noise
level in the environment. The function f(x) (35) in this
case which is in the exponent in (44) is equal to
0
0
arctan
1 1
x a x
f x x
a x a
,
or using denoting r = x/x0 is
0 0arctan
1 1
a r
f x x r x f r
a a
.
However, the analytical calculation of the integral
(44) is associated with great difficulties, and further
solution is possible only numerically.
The distributions of the plasma electron density at
different times at a = 3, obtained from (44), by numeri-
cal integration, are shown in Figure.
The dependence of the electron density
on the radius for different moments of time
Here τ is the normalized time defined as
0
t
t
, (46)
where 0 0/x D t is the average time for which an
electron drifts over a distance of the size of the envelope
inhomogeneity x0. Estimate t0 which corresponds τ = 1.
Observational data show that the electric field of hiss in
the lower hybrid frequency range in the environment is
of the order 10…40 mV/m depending on the height. The
Earth's geomagnetic field is about 0.5 Oe. Then (18)
2 4~ 5 10 cm/sdrD v .
The characteristic value x0 is usually on the order of
10 m. With these parameters, we get t0 ~ 0.02 s and thus
a normalized time corresponding to τ = 1 is t = 0.02 s.
At the initial moment of time, when a burst of inho-
mogeneous stochastic oscillations occurs the envelope
of which is defined by (45), the plasma density is uni-
form. Almost immediately, in the region with the max-
imum amplitude of oscillations, the plasma density be-
comes depleted, at that the depth and width of the re-
sulting cavity increase up to a time τ ~ 5, after which
they practically do not change. In the area adjacent to
the cavity the plasma electrons density increases due to
electrons that are pushed out of the cavity by inhomo-
geneous stochastic electric fields. Further these elec-
trons, moving away from the cavity, and their density
decreases due to spreading in space.
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 25
Estimate the dimensions of the region along the
magnetic field with an increased level of oscillations
necessary for the formation of electron cavity. As al-
ready mentioned, the size of such region must exceed
the distance that an electron travels along the magnetic
field with a thermal velocity in the time required to form
a cavity, in this case, t=0.02 s. For 0.25 eV of the ther-
mal energy of electrons at a height of 600 km, the ther-
mal velocity of electrons is vTe ~ 3·10
7
cm/s. Then the
distance that an electron travels in t = 0.02 s is 6 km.
Thus, the size of the region with an increased level of
oscillations along the magnetic field should at least ex-
ceed 6 km. According to the data of the work [7], the
size of LHC along the geomagnetic field is certainly a
few kilometers and probably a few hundred kilometers.
And it can be argued that cavities with such dimensions
can be formed due to the transfer of electrons across the
magnetic field due to inhomogeneous stochastic fields.
CONCLUSIONS
Inhomogeneous stochastic oscillations of the electric
field with frequencies on the order of the lower hybrid
frequency lead to the formation of an electron density
cavity in a magnetized plasma. It has been established
that the cavity formation time for the ionospheric plasma
parameters is about 0.02 s. The development of the cavity
proceeds as follows. Initially, the cavity depth as well as
its width across to the magnetic field are small. Over
time, both the depth and width of the cavity increase.
The electrons displaced from the cavity form an in-
creased density at the edge of the cavity and move away
from it. The formation of an electron density cavity ends
in a time of about τ ~ 5 where τ is defined by (46).
A possible hindrance to the formation of a cavity is
the escape of electrons along the geomagnetic field from
a region with an increased level of oscillations due to
thermal motion. Estimates have shown that the size of
such a region along the magnetic field should be at least
6 km. At the same time, spacecraft observations have
shown that the dimensions of the LHC along the geo-
magnetic field are tens and hundreds of kilometers.
Thus, in order for the electrons to remain inside the re-
gion and form a cavity, its dimensions along the mag-
netic field of the order of 6 km turn out to be quite suf-
ficient.
During the formation of the electron cavity, ions due
to drift in inhomogeneous stochastic oscillations are
displaced by a distance much less than electrons [16-
18]. Therefore, in the region of the electron cavity a
stationary electric field is formed, which, in turn, accel-
erates ions from this cavity and, as a result, a neutral
plasma density cavity is formed.
This work was supported by the Ministry of Educa-
tion and Science of Ukraine, under the grant
0121U109922.
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Article received 10.06.2023
26 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146)
ЧАСОВА ЕВОЛЮЦІЯ ПЛАЗМОВОЇ ПОРОЖНИНИ, ЯКА ВИКЛИКАНА НЕОДНОРІДНИМИ
СТОХАСТИЧНИМИ ЕЛЕКТРИЧНИМИ ПОЛЯМИ
М.О. Азарєнков, Д.В. Чібісов, О.Д. Чібісов
Досліджено часову еволюцію утворення плазмової порожнини, спричиненої неоднорідними стохастич-
ними електричними полями. Рівняння Фоккера-Планка, яке визначає часову еволюцію електронної густини
плазми, що викликана локалізованими стохастичними неоднорідними електричними полями в діапазоні час-
тот нижньогібридних коливань, розв’язано чисельно. Отримано просторову залежність густини електронів
плазми для різних значень часу.
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