Magnetic field of heliotron and mirror-type magnetic system combination
In numerical calculations for the model of combined magnetic system a possibility of existence of closed magnetic surfaces is shown. The model comprises the magnetic system of l=2 torsatron without additional toroidal magnetic field coils (heliotron) with a single current-carrying turn as an ele...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-811692015-05-12T03:02:04Z Magnetic field of heliotron and mirror-type magnetic system combination Kotenko, V.G. Магнитное удержание In numerical calculations for the model of combined magnetic system a possibility of existence of closed magnetic surfaces is shown. The model comprises the magnetic system of l=2 torsatron without additional toroidal magnetic field coils (heliotron) with a single current-carrying turn as an element of the mirror-type magnetic system. The turn encircles the heliotron closed magnetic surface region and produces a magnetic field of opposite direction to the heliotron magnetic field. Численными расчётами показана возможность существования замкнутых магнитных поверхностей в модели комбинированной магнитной системы. В состав модели входят магнитная система 2-заходного торсатрона без катушек дополнительного тороидального магнитного поля (гелиотрон) и магнитная система пробкотрона в виде одиночного витка с током. Виток охватывает область существования замкнутых магнитных поверхностей гелиотрона и по отношению к магнитному полю гелиотрона создает магнитное поле встречного направления. Чисельними розрахунками показана можливість існування замкнутих магнітних поверхонь у моделі комбінованої магнітної системи. До складу моделі належить магнітна система 2-заходного торсатрона без котушок додаткового тороїдального магнітного поля (геліотрон) та магнітна система пробкотрона у вигляді одного кільця зі струмом. Кільце охоплює область існування магнітних поверхонь геліотрона і створює по відношенню до магнітного поля геліотрона магнітне поле зустрічного напрямку. 2014 Article Magnetic field of heliotron and mirror-type magnetic system combination / V.G. Kotenko // Вопросы атомной науки и техники. — 2014. — № 6. — С. 22-25. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.55.Hc http://dspace.nbuv.gov.ua/handle/123456789/81169 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Магнитное удержание Магнитное удержание |
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Магнитное удержание Магнитное удержание Kotenko, V.G. Magnetic field of heliotron and mirror-type magnetic system combination Вопросы атомной науки и техники |
| description |
In numerical calculations for the model of combined magnetic system a possibility of existence of closed
magnetic surfaces is shown. The model comprises the magnetic system of l=2 torsatron without additional toroidal
magnetic field coils (heliotron) with a single current-carrying turn as an element of the mirror-type magnetic system.
The turn encircles the heliotron closed magnetic surface region and produces a magnetic field of opposite direction
to the heliotron magnetic field. |
| format |
Article |
| author |
Kotenko, V.G. |
| author_facet |
Kotenko, V.G. |
| author_sort |
Kotenko, V.G. |
| title |
Magnetic field of heliotron and mirror-type magnetic system combination |
| title_short |
Magnetic field of heliotron and mirror-type magnetic system combination |
| title_full |
Magnetic field of heliotron and mirror-type magnetic system combination |
| title_fullStr |
Magnetic field of heliotron and mirror-type magnetic system combination |
| title_full_unstemmed |
Magnetic field of heliotron and mirror-type magnetic system combination |
| title_sort |
magnetic field of heliotron and mirror-type magnetic system combination |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2014 |
| topic_facet |
Магнитное удержание |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/81169 |
| citation_txt |
Magnetic field of heliotron and mirror-type magnetic system combination / V.G. Kotenko // Вопросы атомной науки и техники. — 2014. — № 6. — С. 22-25. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT kotenkovg magneticfieldofheliotronandmirrortypemagneticsystemcombination |
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2025-07-06T05:32:49Z |
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2025-07-06T05:32:49Z |
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1836874479098134528 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2014. №6(94)
22 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2014, № 6. Series: Plasma Physics (20), p. 22-25.
MAGNETIC FIELD OF HELIOTRON AND MIRROR-TYPE MAGNETIC
SYSTEM COMBINATION
V.G. Kotenko
Institute of Plasma Physics NSC KIPT, Kharkov, Ukraine
In numerical calculations for the model of combined magnetic system a possibility of existence of closed
magnetic surfaces is shown. The model comprises the magnetic system of l=2 torsatron without additional toroidal
magnetic field coils (heliotron) with a single current-carrying turn as an element of the mirror-type magnetic system.
The turn encircles the heliotron closed magnetic surface region and produces a magnetic field of opposite direction
to the heliotron magnetic field.
PACS: 52.55.Hc
INTRODUCTION
As a fusion neutron source for the sub-critical fast
hybrid reactor the magnetic plasma trap has been
proposed on the basis of the combination of magnetic
systems including a stellarator-type magnetic system and a
conventional mirror-type system [1]. In numerical
calculations for the combined magnetic system (CMS)
model the study of a magnetic field has been carried out
[2]. The model comprises a stellarator-type magnetic
system of the l=2 torsatron with additional toroidal
magnetic field coils with a single current-carrying turn as
an element of the mirror magnetic system. The turn
encircled the torsatron closed magnetic surface region
and produced a magnetic field of unidirectional or
opposite direction relative to the torsatron magnetic
field. In the CMS model, taking into account the width
of helical coils and that of additional toroidal magnetic
field coils, the mirror-type magnetic system was realized
by switching off one of the additional toroidal magnetic
field coils [3].
In this paper the magnetic field of the CMS where
the stellarator-type magnetic system is an l=2 torsatron
without additional toroidal magnetic field coils, i.e. the
model of a heliotron-type magnetic system is investigated.
As is known [4], a similar magnetic system is in operation
in the LHD heliotron magnetic plasma trap of steady-
state action.
1. CALCULATION MODEL OF A
HELIOTRON MAGNETIC SYSTEM
The calculation model of a heliotron magnetic system
is schematically represented in Fig. 1. The main
geometrical characteristics of the initial calculation
model of the heliotron magnetic system are as follows:
– toroidicity α=a/R0=0.25, a is the minor radius of the
torus, R0 is the major radius of the torus;
– l=2 is the polarity;
– m=6 is the number of helical coil pitches along the
length of the torus, i.e., helical coil pitch parameter is
p=mα =1.5;
– each of the helical coils of the calculation model
comprises one filament-like conductor;
– the helical coils are wound on the torus according to
the cylindrical helix law, 1=m , and denotes the
poloidal angle and the toroidal angle consequently.
Fig. 1 also presents the projection of the last closed
magnetic surface (LCMS), onto the equatorial torus
plane. It is seen that the LCMS minor radius has a
comparable value with the torus minor radius.
a b
Fig. 1. Top view of the heliotron helical coils and the LCMS (a) and the CMS with the current-carrying turn
( =180º) and the LCMS (b). The toroidal azimuths of poloidal cross-sections are indicated (see Fig. 2, 4)
ISSN 1562-6016. ВАНТ. 2014. №6(94) 23
φ=0º 7.5º 15º 180º
Fig. 2. Magnetic surface cross-sections in the initial heliotron
Fig. 2 shows the poloidal cross-sections of the
magnetic surface configurations calculated for heliotron
initial calculation model. The outer circle represents the
torus cross-sections with traces of helical coils (large
black points). The cross-sections are spaced apart by the
toroidal angle φ (see Fig. 1) within the limits of the
magnetic field half-period, φ=0º, 7.5º, 15º. The cross-
section φ=180º at the assumed position of the turn is
also presented. From the figure one can see that the size
of the region of magnetic surface existence does not
depend on the cross-section toroidal azimuth. In all the
cross-sections the average radius of the last closed
magnetic surface is the same, rlc/R0=0,18
(rlc/a=0,72). The magnetic surface configuration shapes
in the cross-sections =0º and =180º are identical.
It is also seen from Fig.2, that in all the cross sections
the magnetic axis traces are disposed in the equatorial
torus plane, its major radius R0ax/R0=0,978<1, i.e. the
initial configuration of magnetic surfaces is in the mode
with a planar magnetic axis and is shifted inward the torus.
The mode can be realized with a uniform transverse
compensating magnetic field Bz/B0=0.25, where B0 is
the amplitude of the toroidal component of the magnetic
field generated by helical coils on the circular axis of the
torus. From Fig. 2 one can see, that magnetic surfaces
are elongated along the vertical z-axis of the torus in the
central part of the magnetic surface configuration. The
magnetic surface ellipticity parameter is not lesser than
2 in the vicinity of the magnetic axis. So, on the analogy
with present-day tokamaks, which have a noncircular D-
shaped poloidal cross-section, in the central part of the
plasma core of the heliotron under consideration one can
expect the double increase in the limiting β-value.
The magnetic surface parameters as a function of
their average radius r/R0 are shown in Fig. 5 by dotted
lines. From the figure it follows that the rotational
transform angle ( is in 2 units) increases with radius
increasing =0.1 0.91, there is a magnetic hill
U=0 0.23, and the mirror ratio
γ=Bmax/Bmin=1.003 2.66. The parameter values for the
last closed magnetic surface together with its average
radius (in brackets) are indicated by lettering to the
curves.
2. CALCULATION MODEL OF THE CMS
In the CMS under consideration the current-
carrying turn takes place in the torus poloidal cross-
section at the toroidal azimuth =180º (see Fig.1,b).
The turn radius at/R0=0.35, the turn center is on the
circular axis of the torus The ratio of the turn current to
the helical coil current is It/Ih=-0.36. Thus in the turn
center the current produces a magnetic field B0t in the
opposite direction relative to the torsatron guiding
magnetic field B0t=-0.27B0. Fig. 1,b presents the LCMS
projection onto the equatorial torus plane. It is seen that
superposition of the turn magnetic field diminishes the
LCMS radius by a factor of ~2.
Fig. 3 represents the calculated poloidal cross
sections of magnetic surfaces in the CMS. The average
value of the last closed magnetic surface radius
gradually increases from rlc/R0=0.09 in the cross-section
φ=0º to rlc/R0=0.11 in the cross-section φ=180º.The
island structure arises at the edge of the magnetic
surface configuration. The magnetic axis looses the
circle shape in the CMS. It is evidenced by the observed
dependence of the magnetic axis trace position at the
toroidal -azimuth of the poloidal cross-section. So,
Rax/Ro=0.983 in the φ=0º cross-section and Rax/Ro=1.003
in φ=180º cross-section.
φ=0º 7.5º 15º 180º
Fig. 3. Poloidal cross-sections of magnetic surfaces in the CMS
24 ISSN 1562-6016. ВАНТ. 2014. №6(94)
Fig. 4. Parameters of magnetic surfaces as a function of their average radius in the heliotron (dotted lines) and
CMS (solid lines)
The radial dependences of the CMS magnetic surface
parameters are shown in Fig. 4 by solid lines. It is seen
from the figure that the value of the rotational transform
angle changes (ι is in 2 units) within the limits
ιaxis ιlcms=0.12 0.31. The mirror ratio on the last
closed magnetic surface is slightly decreased and that on
the magnetic surfaces near by the magnetic axis is
significantly increased as compared with the initial
configuration =1,4 2.28. The high magnetic hill in the
initial configuration is almost vanished, U=0.002.
3. MIRROR-TYPE REGION
The main aim of CMS is to create a mirror region
with a decreased value of magnetic field strength in the
heliotron field lines. The mirror region is formed in the
vicinity of the turn. Below we shall estimate the
longitudinal and transverse sizes of the mirror region
and the “effective” value of the mirror ratio.
3.1. THE LONGITUDINAL SIZE OF THE
MIRROR REGION
To estimate the longitudinal size of the mirror region
the behaviour of the CMS magnetic axis characteristics
has been investigated (see Fig.5). For comparison the
magnetic axis characteristics of the initial heliotron are
also presented in Fig. 5 by dotted lines.
The top part of Fig. 5 represents the magnetic field
strength along the full length of the CMS magnetic axis.
One can see from the figure that the mirror region of the
magnetic field appears in the vicinity of the turn. Its
longitudinal size can be limited by a short interval
marked on the abscissa axis around the toroidal angle
∆φ=150...210º. Within these interval the value of the
mirror ratio [γax] on the CMS magnetic axis is
[γax]=0.95γax=1.33. According to the designations in the
figure [γax]=[Bax]/[Bax]min, and γax=Baxmax/[Bax]min. The
interval comprises two heliotron magnetic field periods,
i.e., the mirror region half-length is L/R0≈0.5. In the
initial heliotron γax=1.003 on the full magnetic axis
length.
The middle part of Fig. 5 shows that the major radius
of the CMS magnetic axis exceeds the magnetic axis
major radius in the initial heliotron. The major radius of
the CMS magnetic axis reaches the maximum value
(Rax/R0=1.003) inside the interval at the turn toroidal
azimuth.
0 60 120 180 240 φ(deg.)
Fig. 5. Characteristics of the magnetic axis along its full
length in the initial heliotron (dotted lines) and in the
CMS (solid lines): Bax(a.u.) − the magnetic field
strength, Rax/R0 − the projection of the magnetic axis
major radius onto the equatorial torus plane, Zax/R0-
magnetic axis deviation from equatorial torus plane
The lower part of Fig. 5 shows the CMS magnetic
axis declination from the torus equatorial plane, Zax/R0.
The declination value gradually increases as oncoming
to the turn azimuth, takes on the extreme values
Zax/R0=±0.003 inside the interval, and gets across zero at
the turn azimuth. The initial heliotron magnetic axis
declination is estimated by the accuracy of the
calculation carrying out, |Zax/R0|≈0.001.
3.2. TRANSVERSE SIZE OF THE MIRROR
REGION
It is evident, that the mirror region comprises not
only the sections of field lines forming regular magnetic
surfaces, but the sections of field lines of destroyed
magnetic surfaces at the periphery. Therefore the radial
or cross size of the mirror region exceeds the LCMS
cross size in all the poloidal cross sections of the torus.
In the calculations the transverse mirror region size was
determined under condition that the field line does not
escape the torus surface in the interval ∆φ=150…210°.
ISSN 1562-6016. ВАНТ. 2014. №6(94) 25
φ=210º 195º 180º 165º 150º
Fig. 6. Mirror region cross sections
Fig. 6 shows calculated cross sections of the mirror
region (tinted). In the cross-sections with φ=150°, 210°
at the interval ends the average radius of the mirror
region boundary is rend/R0=0.13, inside the interval −
rin/R0=0.17. Hence, according to the magnetic flow
conservation law an “effective” value of the mirror ratio
γeff within the mirror region boundary is equal to
γeff=(rin)
2
/(rend)
2
=1.7. The “effective” value of the mirror
ratio within the LCMS boundary has the intermediate
value, γlceff=1.5.
CONCLUSIONS
In this study, the numerical calculations of the
magnetic field produced by the model of a combined
magnetic system were carried out. The model comprises
a heliotron-type magnetic system with a single current-
carrying turn as an element of the mirror-type magnetic
system.The calculations show that the magnetic field
produced by the current-carrying turn, being opposite to
the heliotron magnetic field diminishes the region of
existence of closed magnetic surfaces in the heliotron. In
the place of the current-carrying turn a curvilinear
mirror region appears. The region comprises the
sections of magnetic surface field lines, as well as,
the sections of field lines in the edge layer of
destroyed magnetic surfaces. The curvilinear mirror
region half-length exceeds its transverse maximum size
(2rin/R0) by a factor of 1.5 (nonparaxial mirror trap), the
average “effective” mirror ratio value is γeff=1.5.
Numerical calculations have been obtained using the
idealized model regarded as heliotron and mirror-type
magnetic systems. The next stage of investigations will
evidently meet the necessity to determine the influence
of finite size of CMS current-carrying coils.
REFERENCES
1. V.E. Moiseenko, K. Noack, O. Ågren. Stellarator-
mirror based fusion driven fission reactor // J. Fusion
Energy. 2010, v. 29, p. 65-69.
2. V.G. Kotenko, V.E. Moiseenko. Influence of the
magnetic field single ripple on the torsatron magnetic
surfaces // Voprosy atomnoj nauki i tekhniki.
Series”Termoyadrnyj sintex”. 2011, №3, p. 74-80 (in
Russian).
3. V.G. Kotenko, V.E. Moiseenko, Yu.F. Sergeev,
E.L. Sorokovoy, O. Ågren. Magnetic Surfaces of a
Combined Magnetic System // Problems of Atomic
Science and Technology. Ser. “Plasma Physics”. 2012,
№ 6, p. 22-24.
4. O. Motojima. Status of LHD project and construction
// A Collection of Papers Presented at the IAEA
Technical Committee Meeting on Stellarators and Other
Helical Confinement Systems at Garching, Germany 10-
14 May 1993, IAEA,Vienna, Austria, 1993, p. 41.
Article received 03.09.2014
МАГНИТНОЕ ПОЛЕ КОМБИНАЦИИ МАГНИТНЫХ СИСТЕМ ГЕЛИОТРОНА И ПРОБКОТРОНА
В.Г. Котенко
Численными расчётами показана возможность существования замкнутых магнитных поверхностей в
модели комбинированной магнитной системы. В состав модели входят магнитная система 2-заходного
торсатрона без катушек дополнительного тороидального магнитного поля (гелиотрон) и магнитная система
пробкотрона в виде одиночного витка с током. Виток охватывает область существования замкнутых
магнитных поверхностей гелиотрона и по отношению к магнитному полю гелиотрона создает магнитное
поле встречного направления.
МАГНІТНЕ ПОЛЕ КОМБІНАЦІЇ МАГНІТНИХ СИСТЕМ ГЕЛІОТРОНА ТА ПРОБКОТРОНА
В.Г. Котенко
Чисельними розрахунками показана можливість існування замкнутих магнітних поверхонь у моделі
комбінованої магнітної системи. До складу моделі належить магнітна система 2-заходного торсатрона без
котушок додаткового тороїдального магнітного поля (геліотрон) та магнітна система пробкотрона у вигляді
одного кільця зі струмом. Кільце охоплює область існування магнітних поверхонь геліотрона і створює по
відношенню до магнітного поля геліотрона магнітне поле зустрічного напрямку.
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