Plasma confinement by means of poloidal-toroidal magnetic surfaces
The paper proposes to perform plasma magnetic confinement in the constructions where plasma streams flow by knotted trajectories. It is shown that in this case due to more complicated topology of plasma circulation, a quasi-stationary equilibrium may be carried out without an external magnetic field...
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| Cite this: | Plasma confinement by means of poloidal-toroidal magnetic surfaces / M.V. Maksyuta, G.P. Golovach, Ye.V. Martysh // Вопросы атомной науки и техники. — 2011. — № 1. — С. 38-40. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-906162016-01-04T15:48:01Z Plasma confinement by means of poloidal-toroidal magnetic surfaces Maksyuta, M.V. Golovach, G.P. Martysh, Ye.V. Магнитное удержание The paper proposes to perform plasma magnetic confinement in the constructions where plasma streams flow by knotted trajectories. It is shown that in this case due to more complicated topology of plasma circulation, a quasi-stationary equilibrium may be carried out without an external magnetic field, and i.e. plasma is self-confined by the self-generation of poloidal-toroidal magnetic surfaces. This phenomenon may serve as a mechanism of a ball-lightning origination and existence. Пропонується магнітне утримання плазми здійснювати в таких конструкціях, в яких плазмові потоки рухаються завузленими траєкторіями. Показується, що в цьому випадку за рахунок більш складної топології циркуляції плазми квазістійка рівновага може існувати і без зовнішнього магнітного поля, тобто плазма утримує сама себе за допомогою породжуваних нею полоїдально-тороїдальних поверхонь. Це явище може слугувати механізмом виникнення та існування кульової блискавки. Предлагается магнитное удержание плазмы осуществлять в конструкциях, в которых плазменные потоки движутся заузленными траекториями. Показывается, что в этом случае за счет более сложной топологии циркуляции плазмы квазиустойчивое равновесие может существовать и без внешнего магнитного поля, т.е. плазма удерживает сама себя с помощью порождаемых ею полоидально-тороидальных магнитных поверхностей. Это явление может служить механизмом возникновения и существования шаровой молнии. 2011 Article Plasma confinement by means of poloidal-toroidal magnetic surfaces / M.V. Maksyuta, G.P. Golovach, Ye.V. Martysh // Вопросы атомной науки и техники. — 2011. — № 1. — С. 38-40. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 52.55. – s http://dspace.nbuv.gov.ua/handle/123456789/90616 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Магнитное удержание Магнитное удержание Maksyuta, M.V. Golovach, G.P. Martysh, Ye.V. Plasma confinement by means of poloidal-toroidal magnetic surfaces Вопросы атомной науки и техники |
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The paper proposes to perform plasma magnetic confinement in the constructions where plasma streams flow by knotted trajectories. It is shown that in this case due to more complicated topology of plasma circulation, a quasi-stationary equilibrium may be carried out without an external magnetic field, and i.e. plasma is self-confined by the self-generation of poloidal-toroidal magnetic surfaces. This phenomenon may serve as a mechanism of a ball-lightning origination and existence. |
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Maksyuta, M.V. Golovach, G.P. Martysh, Ye.V. |
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Maksyuta, M.V. Golovach, G.P. Martysh, Ye.V. |
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Maksyuta, M.V. |
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Plasma confinement by means of poloidal-toroidal magnetic surfaces |
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Plasma confinement by means of poloidal-toroidal magnetic surfaces |
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Plasma confinement by means of poloidal-toroidal magnetic surfaces |
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Plasma confinement by means of poloidal-toroidal magnetic surfaces |
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Plasma confinement by means of poloidal-toroidal magnetic surfaces |
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plasma confinement by means of poloidal-toroidal magnetic surfaces |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma confinement by means of poloidal-toroidal magnetic surfaces / M.V. Maksyuta, G.P. Golovach, Ye.V. Martysh // Вопросы атомной науки и техники. — 2011. — № 1. — С. 38-40. — Бібліогр.: 13 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-06T18:48:33Z |
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1836924496789897216 |
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PLASMA CONFINEMENT BY MEANS OF POLOIDAL-TOROIDAL
MAGNETIC SURFACES
M.V. Maksyuta, G.P. Golovach, Ye.V. Martysh
Taras Shevchenko Kiev National University, Radio Physics Department, Kiev, Ukraine
E-mail: maksyuta@univ.kiev.ua
The paper proposes to perform plasma magnetic confinement in the constructions where plasma streams flow by
knotted trajectories. It is shown that in this case due to more complicated topology of plasma circulation, a quasi-
stationary equilibrium may be carried out without an external magnetic field, and i.e. plasma is self-confined by the
self-generation of poloidal-toroidal magnetic surfaces. This phenomenon may serve as a mechanism of a ball-lightning
origination and existence.
PACS: 52.55. – s
1. INTRODUCTION
It is shown that plasma magnetic confinement may be
carried out not only in tokamaks and stellarators (see, for
example, [1, 2]), but in quite new constructions where
plasma streams flow by knotted trajectories. If it aimed at
the creation of thermonuclear synthesis reaction, such
constructions may be called knotted thermonuclear
synthesis reactors (KTSR). Note that first the idea
concerning such constructions on heuristic level was
expressed in [3, 4].
38 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2011. № 1.
Series: Plasma Physics (17), p. 38-40.
Thus, let in KTSR the magnetic axis, i.e. the line
corresponding to maximum pressure, is written down in
the following vector parametric form:
( ) ( ) ( ){ }cos cos , cos sin , sinn n nR r R r rρ φ = + φ φ + φ φ ± φ
r ,
(1)
where ( )1 2n nφ = + φ , , – tore
parameters, on the surface of which (Seifert surfaces [5])
one may trace this space curve (here the signs “
2 2− π ≤ φ ≤ π ,R r
± ”
correspond to right- and left-hand n-foils). In Fig. 1
possible magnetic surfaces with magnetic axis (1) in the
case of the meanings of 3,5,7,9n = are represented.
Besides, in this figure as an example of 5-foils, a current
toroidal line is expressed schematically by a dotted line
from foil 3 to foil 4 and originated by the poloidal
magnetic field (heavy line). This field in concern to foil 1
is also simultaneously toroidal. Thus, an arising magnetic
surface may be called poloidal-toroidal. It is generated by
a toloidal-poloidal plasma current and vice versa. One
may say that in the case of non-dissipative plasma the
external field pressure is not in need of its confinement.
At the expense of a complicated topology of KTSR,
plasma in it, in a stationary regime will be considerably
confined by itself. In such constructions some internally
distant regions in an external region are neighboring
which leads to their magnetodynamic interaction. Maybe
just such constructions will be the most optimal for
magnetic (correctly, self-confinement) of plasma and at
last, the confinement solution of one of the main problems
is the realization of the controllable thermonuclear
synthesis. Further we will try to substantiate the above
mentioned just from the mathematician point of view.
n = 3 n = 5
1
2
3
4
5
n = 7 n = 9
Fig. 1. Schematic representation of knotted poloidal-
toroidal-magnetic surfaces at various meanings of the
parameter n
2. THE ORIGINATION OF POLOIDAL-
TOROIDAL MAGNETIC SURFACES IN
KNOTTED PLASMA VOLUMES
As in [6], we will start the investigation of the
equilibrium condition of knotted plasma stream from the
notes of the magnetic hydrodynamics equations
1 4, 0, rot
4 , div 0,
p j H H
c c
G H
π⎡ ⎤−∇ + +ρ∇Φ = =⎣ ⎦
ΔΦ = − π ρ =
,j
r rr r
r
(2)
where , pρ – density and pressure of a conducting
plasma, j
r
, H
r
– current density and a magnetic field
strength, Φ , – gravitation potential and gravitation
constant. However, now besides a local
magnetohydrodynamic interaction leading to the
generation of poloidal magnetic field plasma by a toroidal
current there is a non-local interaction leading to the
G
generation of poloidal currents by toroidal magnetic
fields.
Analogous to [7] at the calculation of electrostatic
potential in non-local polarizable medium, in our case at
the calculation of magnetic values instead of conduction
current ( )j r
r r in the second equation of magnetostatic we
substitute the current
( ) ( ) ( )rot
4
r
cj r H r F r r dr
′′
⎡ ⎤κ ′ ′⎢ ⎥+ −
π ⎢ ⎥⎣
∫
r
rr r r r
39
′
⎦
r r , (3)
which takes into account the reverse effect on the current
in point of of magnetic fields of rr ( )H r′
r r in the points
. Here – a dimensionless coefficient depending on
the parameters of medium and the system geometry,
r′r κ
( )F r r′−
r r – an averaging nucleus taken as a function of
( ) ( )2 2 3/ 2exp 3F r r r r a a⎡ ⎤′ ′− = − − π⎢ ⎥⎣ ⎦
r r r r , where a – an
averaging length, depending on medium parameters.
Further, substituting (3) into the second equation of the
system (2), we get integro-differential equation
( ) ( ) ( ) ( )4rot
r
H r H r F r r dr j
c
′′
⎧ π⎪ ′ ′ ′− κ − =⎨
⎪⎩
∫
r
r r rr r r r r r
⎫⎪
⎬
⎪⎭
r , (4)
which should be used in the system (2) instead of the
second equation. From here it is evident that since
magnetic hydrostatics equation remains invariable,
magnetic field lines and lines of flux are as usual arranged
at the surfaces of the constant pressure of (see,
for example, [8, 9]), in spite of the fact that now these
lines must not be self-orthogonal. It allows to take
advantage of the results of [6, 9], where by means of
Grad-Shafranov equation for axial-symmetric systems it
is shown that at some parameters we get limited toroidal
configurations nearby magnetic axis with circular cross-
sections. One may admit that just in the case of magnetic
axis like a space curve (1) there arise nearby limited
configurations such as, for example, in Fig. 1. It could be
shown strictly mathematical using orthogonal system of
Mercier coordinates (see for example, [10]), connecting
with the curve (1). Giving a spiral-like lines of flux
winding on a knotted limited surface one may by means
of equation (4) find magnetic field lines. Using Fourier
transformation to the equation (4), we find
constp =
( ) ( ) ( )4 ,
r
H r j r G r r
c
′
π ⎡ dr⎤′ ′ ′ , (5) = −⎣ ⎦∫
r
rr rr r r r r
where
( ) ( ) ( )
( )2 2 2
sin3
1 exp 4
2
k k r r dk
k k a
k
G r r
⎡ ⎤′−− ⎣ ⎦
⎡ +κ −⎣ ⎦
′− = π ∫
r r r
3. THE OBTAINING OF THE CONDITION
OF A STEADY BALANCE IN KNOTTED
CONFIGURATIONS
It is known [8, 9], that in the case of plasma
confinement in tokamaks and stellarators there is so-
called virial theorem, leading to a condition of
2
3 0
8
r
Hp dr
⎛ ⎞
+ =⎜ ⎟⎜ ⎟π⎝ ⎠
∫
r
r , (6)
which fulfills only at the meanings of and 0p = 0H =
r
,
when plasma is absent.
Substituting current density from the equation (4) into
the first equation of the system (2) (we neglect gravitation
potential Φ in this case), after the transformations given
in [8], we get an essential condition of a steady
equilibrium
( ) ( ) ( )
2
3
8
, ,
4
r
r
r r
Hp dr
r j r F r r H r dr dr
′
⎛ ⎞
+ =⎜ ⎟⎜ ⎟π⎝ ⎠
⎛ ⎞κ ⎡ ⎤⎜ ⎟⎡ ⎤′ ′ ′= ⋅ ∇ −⎣ ⎦⎣ ⎦⎜ ⎟π ⎝ ⎠
∫
∫ ∫
r
r
r r
r
rr
,r r r r r r r
(7)
which fulfills just at some zero meanings of andp H
r
,
that admits plasma presence. Thus, plasma confinement is
possible just at the expense of own magnetic field.
4. ON THE MECHANISM OF BALL-
LIGHTNINGS GENERATION FROM
THE POINT OF VIEW OF KNOTTED
OR BUNCHING PLASMA STREAMS
There are some statements (see, for example, [6, 11])
that ball-lightnings are closed currents. On the basis of
above mentioned it is more natural to suppose that
currents in ball-lightnings are knotted (or bunching). May
be the mechanism of ball-lightnings generation is in that
magnetic field lines of the current curved lines at the
expense of bunching from the side of positive curvature
curve plasma streams more strongly and at last knot them.
And maybe only in this case, as it follows from the
condition (7), they may be self-confined at the expense of
own magnetic fields (in many papers it is also stated that
gravitation forces including into the first equation of the
system (2) also play an essential role). In this connection
the most frequently met ball-lightnings of 20...30 cm [12]
in dimension are evidently simple knots (right- or left-
hand trifoils). In nature we find some other kinds of ball-
lightnings in the form of ellipsoids, strips, cylinders,
spindles having tails, etc. [12]. Evidently, they may be as
well compared with various knotted structures [5]. Just
there may origin ball-lightnings in the form of bunching
plasma vortices. For example, such a scenario may be
represented by one of the frames of the cinerama in [12,
13], shown in Fig. 2 on the left. It represents the
formation of three nearby placed plasmoids associated
with Borromean rings (picture in Fig. 2 on the right).
⎤
r r
r
r r r – vector Green’s
function, transforming at into the function of 0κ =
( ) ( ) 34G r r r r r r′ ′− = − π −
r r r r r r r′ , and the formula (5) into
the law of the Bio-Savara-Laplasa.
3. M.V. Maksyuta, E.V. Martysh, G.P. Golovach. Knotted
fusion reactors // Proceedings of the IV International
conferece “Electronics and applied physics”. October,
23-25, 2008, Kyiv, Ukraine, p. 108-109 (In Ukrainian).
4. M.V. Maksyuta, E.V. Martysh. Physical vacuum as
crystal-like Plank plasma // Problems of Atomic Science
and Technology. Series “Plasma Physics”. 2009, N 1,
p. 89-91.
5. V.V. Prasolov, A.B. Sossinsky. Knots, bunches, plaits
and three-dimensional varieties. Moscow:
“МTcNMO”, 1997 (In Russian).
Fig. 2. The comparison of three forming plasmoids with
Borromean bunching
6. V.D. Shafranov. On equlibrium magneto
hydrodynamical configurations // ZhETF. 1957, v. 33,
N 3(9), p. 710-722 (In Russian).
At the end of this paragraph it should be mentioned, as
it is stated in [13], that sometimes in ball-lightnings we
observe an internal structure like, for example, shining
layers or moving sparks. It may serve one more argument
in favor of the proposed mechanism of ball-lightnings
origination.
7. Ye. Koskin, G.S. Dragan. The electrostatic potential in
nonlocal polarizable media // Ukr. J. Phys. 2010, v. 55,
N 7, p. 763-768 (In Ukrainian).
8. A.G. Sitechko, V.N. Mal’nev. The principles of plasma
theory. Кyiv: “Naukova dumka”, 1994 (In Ukrainian).
5. CONCLUSIONS 9. V.D. Shafranov. Plasma equilibrium in magnetic field
// Edited by Leontovich M.A. The questions of plasma
theory. Issue 2. M.: “Atomizdat”, 1963, p. 92-131 (In
Russian).
Thus on the basis of above mentioned we may
suppose that in the proposed knotted constructions plasma
confinement at the expense of poloidal-toroidal magnetic
surfaces will be longer than in tokamaks and stellarators
which may lead to the achievement of Lousson criterion
in the conditions of self-compensated plasma instabilities.
Maybe exotic natural phenomena in the form of long-
lived ball-lightnings are just empiric confirmations of the
correct geometry choice while constructing thermonuclear
synthesis generators in the form of knotted constructions.
10. S.L. Solovyev, V.D. Shafranov. Closed magnetic
configurations for plasma confinement // Edited by
Leontovich M.A. The questions of plasma theory. Issue
5. Moscow: “Atomizdat”, 1967, p. 3-208 (In Russian).
11. V.V. Balyberdin, G.A. Bryzgalov, V.G. Kasyan. The
methods of obtaining of a steady system of plasma
vortices // Pis’ma v ZhETF. 1968, v. 7, N 8, p. 262-264
(In Russian).
REFERENCES 12. B.M. Smirnov. The physics of ball-lifgtnings // UFN.
1990, v. 160, N 4, p. 1-45 (In Russian).
1. B.B. Kadomtsev, V.D. Shafranov. Plasma magnetic
restriction // UFN. 1983, v. 139, N 3, p. 399-434 (In
Russian).
13. G.S. Paiva, J.V. Ferreira, C.C. Bastos, M.V.P. dos
Santos, A.C. . Energy density calculations for
ball-lightning-like luminous silicon balls // UFN. 2010,
v. 180, N 2, p. 218-222 (In Russian).
Pavao%
2. L.M. Kovrizhnykh. Modern status of stellarator
programme // UFN. 2009, v. 179, N 7, p. 772-779 (In
Russian).
Article received 12.10.10
УДЕРЖАНИЕ ПЛАЗМЫ С ПОМОЩЬЮ ПОЛОИДАЛЬНО-ТОРОИДАЛЬНЫХ МАГНИТНЫХ
ПОВЕРХНОСТЕЙ
Н.В. Максюта, Г.П. Головач, Е.В. Мартыш
Предлагается магнитное удержание плазмы осуществлять в конструкциях, в которых плазменные потоки
движутся заузленными траекториями. Показывается, что в этом случае за счет более сложной топологии
циркуляции плазмы квазиустойчивое равновесие может существовать и без внешнего магнитного поля, т.е.
плазма удерживает сама себя с помощью порождаемых ею полоидально-тороидальных магнитных
поверхностей. Это явление может служить механизмом возникновения и существования шаровой молнии.
УТРИМАННЯ ПЛАЗМИ ЗА ДОПОМОГОЮ ПОЛОЇДАЛЬНО-ТОРОЇДАЛЬНИХ МАГНІТНИХ
ПОВЕРХОНЬ
М.В. Максюта, Г.П. Головач, Є.В. Мартиш
Пропонується магнітне утримання плазми здійснювати в таких конструкціях, в яких плазмові потоки
рухаються завузленими траєкторіями. Показується, що в цьому випадку за рахунок більш складної топології
циркуляції плазми квазістійка рівновага може існувати і без зовнішнього магнітного поля, тобто плазма
утримує сама себе за допомогою породжуваних нею полоїдально-тороїдальних поверхонь. Це явище може
слугувати механізмом виникнення та існування кульової блискавки.
40
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