Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator
Gespeichert in:
| Datum: | 2002 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/80252 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator / G.G. Lesnyakov, V.G. Kotenko, E.D. Volkov // Вопросы атомной науки и техники. — 2002. — № 4. — С. 67-69. — Бібліогр.: 10 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-80252 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-802522015-04-15T03:01:57Z Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator Lesnyakov, G.G. Kotenko, V.G. Volkov, E.D. Magnetic confinement 2002 Article Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator / G.G. Lesnyakov, V.G. Kotenko, E.D. Volkov // Вопросы атомной науки и техники. — 2002. — № 4. — С. 67-69. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 52.55.-s http://dspace.nbuv.gov.ua/handle/123456789/80252 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Magnetic confinement Magnetic confinement |
| spellingShingle |
Magnetic confinement Magnetic confinement Lesnyakov, G.G. Kotenko, V.G. Volkov, E.D. Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator Вопросы атомной науки и техники |
| format |
Article |
| author |
Lesnyakov, G.G. Kotenko, V.G. Volkov, E.D. |
| author_facet |
Lesnyakov, G.G. Kotenko, V.G. Volkov, E.D. |
| author_sort |
Lesnyakov, G.G. |
| title |
Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator |
| title_short |
Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator |
| title_full |
Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator |
| title_fullStr |
Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator |
| title_full_unstemmed |
Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator |
| title_sort |
mapping of magnetic field lines at the edge of the l=1 helical-axis yamator |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2002 |
| topic_facet |
Magnetic confinement |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80252 |
| citation_txt |
Mapping of magnetic field lines at the edge of the l=1 helical-axis Yamator / G.G. Lesnyakov, V.G. Kotenko, E.D. Volkov // Вопросы атомной науки и техники. — 2002. — № 4. — С. 67-69. — Бібліогр.: 10 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT lesnyakovgg mappingofmagneticfieldlinesattheedgeofthel1helicalaxisyamator AT kotenkovg mappingofmagneticfieldlinesattheedgeofthel1helicalaxisyamator AT volkoved mappingofmagneticfieldlinesattheedgeofthel1helicalaxisyamator |
| first_indexed |
2025-07-06T04:13:13Z |
| last_indexed |
2025-07-06T04:13:13Z |
| _version_ |
1836869426171871232 |
| fulltext |
MAPPING OF MAGNETIC FIELD LINES AT THE EDGE
OF THE L=1 HELICAL-AXIS YAMATOR
G. G. Lesnyakov, V. G. Kotenko, E. D. Volkov
Institute of Plasma Physics, National Science Center
"Kharkov Institute of Physics and Technology", Kharkov 61108, Ukraine
PACS: 52.55.-s
1. INTRODUCTION
Compared to magnetic traps of the same group, the
l=1 Yamator magnetic field coils are the simplest in
structure. In principle, the l=1 Yamator should be put in a
row with Heliac-type spatial-axis stellarators. By
definition given in papers [1,2], the term ”Heliac” means
the magnetic configuration, where the vertically elongated
magnetic-surface cross-section shape rotates
simultaneously with the principal normal with respect to
the magnetic axis. The YAMATOR is a new magnetic
system [3, 4], where two-wire lines wound round the
torus are plunged into an axisymmetric toroidal magnetic
field. The wires of the two-wire line with equal and
opposite currents have the same pitch of winding L and
are placed on the nested tori of one and the same major
radius Ro and of different minor radii a1 and a2 =a1+h, h
being the line wire spacing. The number of two-wire lines
determines Yamator’s multipolarity l.
In the l=1, m=3 toroidal Yamator (m is the number of
magnetic field periods) with a low aspect ratio
Ah1=Ro/a1≈3.333 and rax/a1≤0.4 (rax is the radius of the
helical magnetic axis), a significant magnetic well,
(-U)≥8%, and a rotational transform, i≤0.6 (i is given in
units of 2π), greater than the one in other-multipolarity
Yamators [3, 4] can be formed. At the same time, the l=1
Yamator has the unique peculiarity [4]: there are two
separatrix ribs (two X-points are the intersection points of
two separatrices) that lie on both sides of the two-wire
helical winding (line), symmetrically about its the
azimuth.
It is known that the magnetic field toroidicity leads to
the destruction of the separatrix region, which is replaced
by a stochastic layer. Therefore, it is interesting to know
how it occurs in the low aspect ratio magnetic system.
The aims of this paper have been: (i) to investigate
numerically the special features in the behavior of l=1
Yamator magnetic field lines, the trajectories of which
start immediately after the last closed magnetic surface,
(ii) to obtain footprints of magnetic field lines near the
conventional separatrix and X – points, and (iii) to
conceive even if in part the possibilities of this magnetic
divertor.
2. SPECIAL FEATURES OF LOW ASPECT
RATIO YAMATOR
Numerical calculations have been carried out for the
filamentary-wire model and are the continuation of
investigations of the l=1, m=3 Yamator [3]. The basic
parameters of the configurations were as follows: Θ=mϕ
is the law of the two-wire helical winding, Θ is the
poloidal angle, ϕ is the toroidal angle; h/Ro=0.15;
Ah1=Ro/a1=3.333 and Ah2= Ro/a2=2.222 are the aspect
ratios of the nested tori. The Bo/bo ratio (Bϕ=BoRo/R, bo is
the helical current-generated field of radius a1) determines
the structure and properties of magnetic surfaces in the
configuration. A controlling uniform transverse magnetic
field Bz was not applied.
2.1. Behavior of magnetic field lines in the
periphery region
The structure of magnetic field lines in the l=1, m=3
Yamator in the region of closed magnetic surfaces (CMS)
and in their periphery is shown in Fig.1. This structure
exhibits the following special features. It the edge region
of CMS there is a chain of large-size “natural” magnetic
islands, i=3/6. Outside the CMS region a wide stochastic
region lies. The stochastic region originates as a layer,
where large-size island chains overlap and break down
with simultaneous formation of the whisker field lines
that extend up to the conventional separatrix. The
whiskers are the hypertrophied remains resulting from the
break of the large-size island chains, where the islands
with i=5/10 predominate. The general pattern of the
behavior of magnetic field lines that form the whiskers in
the Yamator does not differ, in principle, from the field
line pattern in the l=2 heliotron/torsatron [5]. To that
description it should be added only that “folding” of the
whiskers takes place towards the rotational transform and
is stipulated by magnetic line shear. The conventional
separatrix is assumed to be the toroidal surface of minor
radius rs=0.3674 that is the same as for the separatrix
surface of a straight magnetic configuration. The
magnetic system toroidicity destroys the separatrices and
the CMS neighbouring with them, i. e. the ones that must
embrace the separatrices as a whole, and thus the
stochastic region is extended. As a result of this process, a
crescent region is formed on the inside of the torus.
When obtaining the structure of the magnetic field
lines in the stochastic region, 1086 field lines started
along 24 radial directions equally spaced in the angle Θ in
the ϕ=0° poloidal cross-section. The idea about the
starting point positions can be gained from the points of
the first intersection of the field lines with the cross-
sections of the torus ϕ=30°, Fig.1b, and ϕ=60°, Fig.1c.
In order to gain more the idea about the magnetic
divertor, all open field lines were traced up to their
intersection with the toroidal surface of conventional
separatrix rs. The map of toroidal coordinates Θ, ϕ
footprint intersection with the rs surface is shown in Fig.2.
Of the total number of the starting field lines only 67%
(731) of lines intersect the rs surface without being
Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 67-69 67
noticeably affected due to the hardness phenomenon at
numerical integration of magnetic field line differential
equations [6], this being the case in the Yamator. The
footprints of these field lines lie to the left from the
helical winding and present the divertor possibilities. The
magnetic field line trajectories on the outside of the CMS
region must be calculated with a substantially improved
integration accuracy of differential equations (a small step
of integration is also needed). In our calculations the
fourth-order Runge-Kutta integration method of
differential equations with a double accuracy was used
(step of integration λ, λ/2πRo≤5×10-4). Nevertheless, the
fulfillment of these requirements does not eliminate the
hardness phenomenon for differential equations [6]. As it
is seen from Fig.2, the location of magnetic field line
footprints in the first magnetic field period extends in rays
along the ϕ coordinate, and in the other two magnetic
field periods similar rays are transformed into the nested
spiral loops.
In integration of magnetic field line equations 355
(33%) field lines out of 1086 field lines undergo jumps. In
Fig.2, the footprints of the onset of jumps lie to the
0.5 1.0 1.5
R / Ro
-0.5
0.0
0.5
Z
/ R
o
a
0.5 1.0 1.5
R / Ro
-0.5
0.0
0.5
Z
/
R
o
b
0.5 1.0 1.5
R / Ro
-0.5
0.0
0.5
Z
/ R
o
c
Fig.1. Magnetic field line structure in the l=1, m=3
Yamator, Bo/bo=1.0, Bz/Bo=0, rcl/Ro=0.16, i(0)=0.44 и
icl=0.52 (in units of 2π), (-U)=10% (rcl and icl are the
minor radius and the rotational transform of the last
closed magnetic surface, corresponding): a) the φ=0˚
poloidal cross-section (the dashed circumference in the
central part shows the excursion of the helical magnetic
axis); b) φ=30˚ (one quarter of the field period); c)
φ=60˚ (middle of the field period).
0.0 1.0 2.0 3.0 4.0 5.0 6.0
ϕ
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
Θ
Fig.2. Footprints of magnetic field lines on the toroidal
surface with the minor radius of the conventional
separatrix rs=(a1a2)1/2=0.3674. The solid lines show the
position of helical winding.
right and closer to the helical winding. The coordinates of
the onset of field line jumps (footprints) indicate where
the hardness phenomenon takes place. In Yamators, the
hardness is caused by magnetic field inhomogeneities. It
appears at sites, where there is a close neighbourhood of
two regions with different magnetic fields, in which the
derivatives, great and small in magnitude, occur during
the integration process, this being especially in the
direction, where the high-value fields quickly decrease.
68
2.2. Magnetic field line trajectories in the vicinity
of the conventional separatrix
A general idea about the behavior of short magnetic
field lines, the lengths of which are shorter than the length
of one transit along the torus, can be gained from Fig.3.
The chosen trajectories of short magnetic field lines for
the starting points of four radial directions are
Fig.3. Magnetic field lines that reach the conventional
separatrix for less than one transit along the torus in the
l=1, m=3 toroidal Yamator straightened along the ϕ
coordinate (solid thin lines show the two-wire helical
winding). The magnetic field line starting points lie along
four radial directions: Θ=0˚ (lines 1); Θ=90˚ (2);
Θ=180˚ (3); Θ=270˚ (4).
typical, in principle, for the end segments of all field lines
that approach both the X-points and the conventional
separatrix rs. In this series of magnetic field lines the thick
line numbered as 3 is an example of the line, the
trajectory of which goes round on the inside of the two-
wire helical winding and ends with a jump because of the
hardness phenomenon.
Fig.4. Magnetic field line trajectories that belong to the
spiral structure of the footprints indicated in Fig.2 by
bold dots.
The field line trajectories of five footprints (they were
chosen along the spiral, bold dots in Fig.2) allow an
understanding of the mechanism of spiral structure
formation in the Yamator. The starting points of these
trajectories are located only along the radial direction Θ
=195° in the range 0.25853≤r≤0.30053. The three-
dimensional tracing of these lines in the straightened torus
(Fig.4) shows the region being a peculiar kind of a field-
line scattering generator as these field lines approach to
the X-points. In this region, the hardness of the set of
magnetic field line equations and, as a consequence, the
field line jumps take place. Both these phenomena are due
to the magnetic field gradient being opposite to the
direction of field line motion (see above). The impact
parameter of the scattering process is determined by the
magnetic field line trajectory in the stochastic region, and,
therefore, each spiral is spread due to a stochastic
instability of motion.
3. SUMMARY
The present investigations give the general idea about
the magnetic field line structure in the edge region of the
l=1 Yamator with a low aspect ratio and a helical
magnetic axis. The locations of the magnetic footprints on
the surface of the conventional separatrix give evidence
for both their broadening along the torus, and formation
of nested spiral structures. The main mechanism of spiral
structure formation of the footprints is the chaotic
scattering of magnetic field lines on the magnetic field
inhomogeneities, typical of Yamator, when the field lines
approach the X-points and the conventional separatrix.
The degree of scattering is stipulated by the degree of
hardness of the set of magnetic field line equations. The
spiral structure of the magnetic footprints in the Yamator
resembles partly the spiral structure of the magnetic
footprints in the two-wire model of the divertor field
configuration with a separatrix (magnetic geometry is
similar to the DIII-D tokamak) [7-9]. In the l=1 Yamator,
as also in the l=2 Yamator [10], the magnetic field lines
forming the divertor arrive for the most part at the surface
of the conventional separatrix only on the outside half of
the torus, that points to the possibility of creating a
discrete divertor.
REFERENCES
1. V. D. Shafranov. J. Plasma Fusion Res. SERIES, Vol.
1, 70-74 (1998).
2. M. Isaev et al. J. Plasma Fusion Res. SERIES, Vol. 1,
66-69 (1998).
3. V. G. Kotenko et al. Problems of Atomic Science and
Technology. Series: Plasma Physics, Issues 1(1), 2(2),
49-51 (NSC "KIPT", Kharkov, 1999).
4. V. G. Kotenko et al. J. Plasma Fusion Res. SERIES,
Vol. 3, 154-157 (2000).
5. H. Akao. J. Phys. Soc. of Japan, Vol. 59, No. 5, 1633-
1646 (1990).
6. A. B. Arushanyan, S. F. Zaletkin. Numerical solution
of ordinary differential equations using FORTRAN.
M.: Moscow University, 1990 (in Russian).
7. S. S. Abdullaev, G. M. Zaslavsky. Phys. Plasmas 2
(12), 4533-4541 (1995).
8. N. Pomphrey, A. Reiman. Phys. Fluids B4, 938-948
(1992).
9. S. S. Abdullaev, G. M. Zaslavsky. Phys. Plasmas 3 (2),
516-528 (1996).
10. G. G. Lesnyakov et al. Problems of Atomic Science
and Technology. Series: Plasma Physics (6), N 6, 68-
70 (NSC "KIPT", Kharkov, 2000).
69
References
|