Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma
It is shown that in a plasma located in an external magnetic field (magnetically active plasma) it is possible to convert the energy of low-frequency oscillations into the energy of high-frequency oscillations. Such a transformation is possible due to the fact that in such a plasma it is possible to...
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Buts, V.A. 2023-12-08T14:45:19Z 2023-12-08T14:45:19Z 2023 Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma / V.A. Buts // Problems of Atomic Science and Technology. — 2023. — № 1. — С. 21-24. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 04.30.Nk; 52.35.Mw; 78.70.Gq DOI: https://doi.org/10.46813/2023-143-021 https://nasplib.isofts.kiev.ua/handle/123456789/195961 It is shown that in a plasma located in an external magnetic field (magnetically active plasma) it is possible to convert the energy of low-frequency oscillations into the energy of high-frequency oscillations. Such a transformation is possible due to the fact that in such a plasma it is possible to create conditions for nonreciprocal coupling between high-frequency waves. Показано, що в плазмі, яка знаходиться у зовнішньому магнітному полі (магнітоактивна плазма), є можливість перетворювати енергію низькочастотних коливань в енергію високочастотних коливань. Таке перетворення можливе завдяки тому, що в такій плазмі можна створити умови для невзаємного зв’язку між високочастотними хвилями. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Basic plasma physics Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma Перетворення енергії низькочастотних коливань на енергію високочастотних коливань у магнітоактивній плазмі Article published earlier |
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Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma |
| spellingShingle |
Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma Buts, V.A. Basic plasma physics |
| title_short |
Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma |
| title_full |
Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma |
| title_fullStr |
Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma |
| title_full_unstemmed |
Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma |
| title_sort |
conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma |
| author |
Buts, V.A. |
| author_facet |
Buts, V.A. |
| topic |
Basic plasma physics |
| topic_facet |
Basic plasma physics |
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2023 |
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English |
| container_title |
Problems of Atomic Science and Technology |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Перетворення енергії низькочастотних коливань на енергію високочастотних коливань у магнітоактивній плазмі |
| description |
It is shown that in a plasma located in an external magnetic field (magnetically active plasma) it is possible to convert the energy of low-frequency oscillations into the energy of high-frequency oscillations. Such a transformation is possible due to the fact that in such a plasma it is possible to create conditions for nonreciprocal coupling between high-frequency waves.
Показано, що в плазмі, яка знаходиться у зовнішньому магнітному полі (магнітоактивна плазма), є можливість перетворювати енергію низькочастотних коливань в енергію високочастотних коливань. Таке перетворення можливе завдяки тому, що в такій плазмі можна створити умови для невзаємного зв’язку між високочастотними хвилями.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/195961 |
| citation_txt |
Conversion of the energy of low-frequency oscillations into the energy of high-frequency oscillationsin magnetoactive plasma / V.A. Buts // Problems of Atomic Science and Technology. — 2023. — № 1. — С. 21-24. — Бібліогр.: 10 назв. — англ. |
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AT butsva conversionoftheenergyoflowfrequencyoscillationsintotheenergyofhighfrequencyoscillationsinmagnetoactiveplasma AT butsva peretvorennâenergíínizʹkočastotnihkolivanʹnaenergíûvisokočastotnihkolivanʹumagnítoaktivníiplazmí |
| first_indexed |
2025-11-24T11:50:13Z |
| last_indexed |
2025-11-24T11:50:13Z |
| _version_ |
1850846241533984768 |
| fulltext |
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143).
Series: Plasma Physics (29), p. 21-24. 21
https://doi.org/10.46813/2023-143-021
CONVERSION OF THE ENERGY OF LOW-FREQUENCY
OSCILLATIONS INTO THE ENERGY OF HIGH-FREQUENCY
OSCILLATIONS IN MAGNETOACTIVE PLASMA
V.A. Buts
1,2
1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine;
2
Institute of Radio Astronomy of NAS of Ukraine, Kharkiv, Ukraine
E-mail: vbuts1225@gmail.com
It is shown that in a plasma located in an external magnetic field (magnetically active plasma) it is possible to
convert the energy of low-frequency oscillations into the energy of high-frequency oscillations. Such a
transformation is possible due to the fact that in such a plasma it is possible to create conditions for nonreciprocal
coupling between high-frequency waves.
PACS: 04.30.Nk; 52.35.Mw; 78.70.Gq
INTRODUCTION
Non-reciprocal mediums in many of their properties,
differ significantly from the usual reciprocal mediums.
Such properties of media and such systems, in which
nonreciprocity arises, arise practically in all areas of
physics. This special property of systems and media is
well studied and widely used in electrodynamics and
optics (see, for example, [1-5]). Below we will be
interested in that special property of nonreciprocal
systems, the use of which makes it possible to convert
the energy of low-frequency oscillations into the energy
of high-frequency oscillations. In our previous works
[6-9], it was shown that the non-reciprocal coupling of
high-frequency oscillatory systems allows them to draw
energy from a low-frequency source. In particular, the
work [8] experimentally shows the possibility of
excitation of high-frequency oscillations of non-
mutually coupled circuits by a low-frequency source,
the frequency of which is forty times lower than the
frequency of high-frequency exited circuits. Note that
the non-reciprocal coupling of high-frequency
oscillations can be created artificially (as in [8]). In
addition, it can exist naturally in media that have the
property of nonreciprocity. There are many such media.
Well-known examples are ferrites and plasmas in an
external magnetic field. In the present work, it is shown
that conditions can indeed be created in a magnetoactive
plasma under which electromagnetic waves do not
mutually interact with each other. Such connection
between these high-frequency waves makes it possible
to excite them using low-frequency sources (for
example, using low-frequency longitudinal plasma
oscillations).
Below, in the second section, the problem is
formulated, and the system of equations is written out.
In the third section, a system of truncated equations is
obtained, which allows analytical methods to find a
solution to the original equations. Conditions for the
excitation of high-frequency waves using the energy of
low-frequency longitudinal plasma oscillations are
obtained. In particular, there found increment of
parametric instability, which is proportional to the
amplitude of longitudinal plasma oscillations. The
fourth section describes the nonlinear dynamics of the
three-wave interaction for the case when the interaction
matrix elements for high-frequency waves are
proportional to the first power of their wave vectors. In
conclusion, the main results of the work are formulated.
1. STATEMENT OF THE PROBLEM
AND BASIC EQUATIONS
The system of initial equations are the Maxwell
equations for fields and the equations of hydrodynamics
for plasma:
1 H
rotE
c t
; 4divE en ;
1 4E
rotH j
c t c
; j env ;
v e e
v v E v H
t m mc
0
n
div nv
t
. (1)
Fig. 1. Dispersion diagram of interacting waves. The
interaction occurs with the participation of longitudinal
plasma waves. Case
1 20; 0k k
mailto:vbuts1225@gmail.com
22 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143)
To simplify the form of general formulas below, we will
proceed from the diagrams of interacting waves, which
are presented in Figs. 1, 2. It follows from them that we
will consider the interaction of three waves. Two of
them are transverse (
1 2, ), and the third is
longitudinal, i.e. type interaction is t t l considered.
We will also consider a spatially one-dimensional case,
i.e., the entire process of interaction will depend on only
one spatial coordinate z . The whole system is placed in
an external magnetic field
0H , which is directed along
the z axis. In this case, the system of equations (1) for
transverse waves will take the form:
1 yx
HE
z c t
;
1y x
E H
z c t
;
1 1
4
yx
y
EH
nev
z c t c
;
1 1
4
y x
x
H E
nev
z c t c
; (2)
x x
x H y z y
v ve e
E v v H
t m mc z
;
y y
y H x z x
v ve e
E v v H
t m mc z
.
Let us write out the equations for low-frequency
longitudinal waves:
z z
z z x y y x
v ve e
E v v H v H
t m z mc
; 4zE
en
z
;
0
n
nv
t z
. (3)
Fig. 2. Dispersion diagram of the same trio of waves as
in Fig. 1. Distinction: high-frequency waves are
directed in one direction:
1 20; 0k k
Let a low-frequency longitudinal wave has been
excited in the plasma. The electric field of this wave can
be written as:
exp . .zE E i k c (4)
Here ( )t z .
The dimensionless velocity of plasma particles
under the action of such a field is
0 exp( ) . .zv V i k c ,
0 ( / )V i eE mc . The main result we are interested in
can be obtained by assuming that this speed is given. It
is not perturbation. Then we substitute this speed into
the system of equations that describe the high-frequency
dynamics of fields and particles, i.e. into the system of
equations (2). In addition, we will assume that the
interaction process occurs in a spatially limited volume
(in a resonator). Therefore, the dynamics of the process
under study depends only on time
( ; / zz ik ). As a
result, after simple but cumbersome calculations, the
following system of nonlinear equations with varying
coefficients can be obtained to determine the dynamics
of high-frequency fields and high-frequency velocities:
2 2
, 1 ,( / ) 1j j jE k c E v n nv ;
, , , , , ,j j H j j z z j j jv E i v k v E i v . (5)
Here
x yE E iE ; x yH H iH ; , , /j jE dE d ;
1t ; x yv v iv ; 1,2j ;
1/E eE mc ;
0/n n n ;
2
2
1/p ; p ‒
plasma frequency. System (5) is closed since the
longitudinal velocity of particles (low-frequency
velocity) is given (see (3) and (4)).
2. ANALYTICAL SOLUTION
The system of equations (5) is a system of four
second-order ordinary differential equations for fields
(two equations for the first and second high-frequency
waves, as well as two equations for different
polarizations), as well as four first-order ordinary
differential equations for determining high-frequency
particle velocities plasma. Thus, the system of equations
(5) is equivalent to the system of twelve ordinary
differential equations of the first order. Note that each
polarization can be considered independently. The
system (5) does not contain terms that describe the
interaction of these polarizations. Note that such terms
appear when it is necessary to take into account the
dynamics of low-frequency waves. The system of
equations (5) does not contain equations describing the
low-frequency dynamics of fields and velocities.
Despite this simplification, it is still quite complex.
Solutions of system (5) will be sought in the form:
1 1 2 2exp exp . .E A i A i k c ;
1 1 2 2exp exp . .v a i a i k c ;
3 exp( ) . .zv a i k c , (6)
where 1,2 1,2 1,2t k z ,
3 0a V const ‒ is given.
The factors in front of the high-frequency exponents
in the linear approximation are constants and slow
functions of time when nonlinear terms are taken into
account:
1,2 1,2 1,2 1,2 1,2( ); ;A A t A A
1,2 1,2 1,2 1,2 1,2( );a a t a a .
The first terms in formulas (6) describe (in a
complex form) a wave that propagates along the axis,
and the second terms describe a wave that propagates
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143) 23
towards it (
2 0k , see Fig. 1) or in the same direction
(
2 0k , see Fig. 2). We will also assume that the
frequencies and wave vectors of the interacting waves
satisfy the synchronism conditions:
1 2 ;
1 2k k . (7)
In the linear approximation, the connection between
the field amplitudes and velocities is described by the
formula
j j j HA a i . (8)
Nonlinear terms determine the slow dynamics of the
amplitudes:
1 3 2 2HA i a A i a ; 2 3 1 1 2/HA i a A i a ;
1 2 3 2 2a k a A ia ; 2 1 1 1 1a k a A ia . (9)
When obtaining (9), we took into account relations
(7).
Let us substitute expressions (8) into the second
equation of system (9):
1 1 3 2 Ha k a a ; 2 2 3 1 Ha k a a . (10)
Equations (10) are equivalent to the pendulum equation:
2 2
1 3 1 2 1 0Ha a k k a . (11)
It follows from this equation that if high-frequency
waves are unidirectional (see Fig. 2), then only
oscillatory dynamics of high-frequency waves occurs.
The amplitude of these oscillations is determined by the
initial conditions. If the interacting high-frequency
waves propagate in opposite directions (see Fig. 1), then
equation (11) describes the dynamics of an unstable
pendulum. The amplitude of such a pendulum grows
exponentially. The instability increment is
1 2H
e E
k k
m c
. (12)
Let us substitute (8) into the first two equations of
system (9). We obtain the following equations for
determining the amplitudes of slowly changing high-
frequency fields:
1 3 2 2 2/ HA i a A ; 2 3 1 / 1 HA i a A . (13)
System (13) corresponds to the linear oscillator
equation:
2 22
1 3 1/ 1 0HA a A
. (14)
It follows from Eq. (14) that the slow dynamics of
high-frequency fields due to non-linear terms leads only
to slow oscillatory dynamics of these fields. It is only
due to the nonlinear dynamics of particles (see (12)) the
amplitude of the fields can also increase.
3. GENERAL MODEL OF THREE-WAVE
INTERACTION IN THE PRESENCE OF NON-
RECIPROCAL COUPLING BETWEEN
HIGH-FREQUENCY WAVES
Let's see what the linear dependence of the matrix
coupling elements on the wave vectors of interacting
high-frequency waves leads to. In the general case, the
system of equations that describes such a three-wave
interaction will look like:
1 1 1 2 3a k V a a ; 2 2 2 1 3a k V a a ; 3 3 1 2a V a a . (15)
Multiply the left and right parts of each equation in
system (15) by. We multiply each equation of the
complex conjugate system by. We add the resulting
equations. The left side of the resulting equation will be
the derivative of the total energy of the interacting
waves. Introducing the notation adopted in [10], the
resulting equation will take the form
3
2 1 1 2 2
1 3 1 1 3
0j j
j
k V k Vd
N
d k V V
. (16)
Here
2
j jN a – number of quanta in j-wave.
Multiplying each equation of system (15) by j jk a ,
and the complex conjugate system by j jk a , we
obtain a relation expressing the momentum conservation
law:
2 1 1 2 2
3 1 1 3
0j j
k k V k Vd
k N
d k k V V
. (17)
In the one-dimensional case, from equations (16) and
(17) we obtain the following relations between the
matrix elements of system (15):
3 1 1 1 1 2 2; .V k V k V k V (18)
Let us substitute (18) into the system of equations
(15).
From the resulting system of equations (taking into
account the complex conjugate system), we can obtain
the following integrals:
1 2N N const ;
1 3N N const . (19)
Integrals (19) are obtained for the case when interacting
waves propagate in opposite directions (see Fig. 1).
Integrals (19) indicate that high-frequency waves can
simultaneously increase their amplitude (the first
integral of system (19)). The second integral shows that
the energy of high-frequency waves is drawn from the
energy of low-frequency waves. If high-frequency
waves propagate in one direction (see Fig. 2), then their
energy does not change (
1 2N N const ).
24 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143)
CONCLUSIONS
Thus, in a magnetoactive plasma there is a range of
parameters in which conditions can be created for
converting the energy of low-frequency oscillations into
the energy of high-frequency oscillations. Let us note
that after works [6-9] it might seem that in all
nonreciprocal media such conditions can be created.
The results of the work show that finding such
conditions is a rather difficult task, and it is not always
obvious that such conditions can be found. It should
also be noted that the reason for the non-reciprocal
coupling between high-frequency waves is the particle
dynamics. More specifically, these are nonlinear terms
due to the magnetic Lorentz force, as well as nonlinear
terms associated with hydrodynamics. It is important to
note that in the absence of an external magnetic field,
these two nonlinear terms annihilate each other.
REFERENCES
1. B. Lax, K.J. Button. Microwave ferrites and
ferrimagnetics. New York: “McGraw-Hill”, 1962.
2. M. Born, E. Wolf. Principles of Optics:
Electromagnetic Theory of Propagation, Interference
and Diffraction of Light. Cambridge: “University
Press”, 1999.
3. B.E.A. Saleh, M.C. Teich. Fundamentals of
Photonics / 2nd Ed. New York: “Wiley-Interscience”,
2007.
4. D.M. Pozar. Microwave Engineering / 4nd Ed. New
York: “Wiley-Interscience”, 2011, chap 9.
5. A. Ishimaru. Electromagnetic Wave Propagation,
Radiation and Scattering: From Fundamentals to
Applications / 2nd Ed. New York: “Wiley-Interscience”,
2017.
6. V.A. Buts, D.M. Vavriv. The role of nonreciprocity
in the theory of oscillations // Radiophysics and radio
astronomy. 2018, № 1, v. 23, p. 60-71.
7. V.A. Buts. Review. Mechanisms for increasing the
frequency and degree of radiation coherence //
Problems of Theoretical Physics. Series “Problems of
Theoretical and Mathematical Physics. Scientific
works”. KSU, 2014, № 1, p. 82-247.
8. V.A. Buts, D.M. Vavriv, O.G. Nechayev, D.V. Tara-
sov. A Simple Method for Generating Electromagnetic
Oscillations // IEEE Transactions on circuits and
systems – II. Express Briefs, January 1, 2015, v. 62,
p. 36-40.
9. V.A. Buts, D.M. Vavriv. The effect of nonreciprocity
on the dynamics of coupled oscillators and coupled
waves // Problems of Atomic Science and Technology.
Series “Physics of Radiation Effect and Radiation
Materials Science” (116). 2018, № 4, p. 213-216.
10. B.B. Kadomtsev. Collective phenomena in plasma.
M.: “Nauka”, 1988.
Article received 02.11.2022
ПЕРЕТВОРЕННЯ ЕНЕРГІЇ НИЗЬКОЧАСТОТНИХ КОЛИВАНЬ НА ЕНЕРГІЮ
ВИСОКОЧАСТОТНИХ КОЛИВАНЬ У МАГНІТОАКТИВНІЙ ПЛАЗМІ
В.О. Буц
Показано, що в плазмі, яка знаходиться у зовнішньому магнітному полі (магнітоактивна плазма), є
можливість перетворювати енергію низькочастотних коливань в енергію високочастотних коливань. Таке
перетворення можливе завдяки тому, що в такій плазмі можна створити умови для невзаємного зв'язку між
високочастотними хвилями.
https://vant.kipt.kharkov.ua/CONTENTS/CONTENTS_2016_4.html
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