Structure of the edge magnetic field of the L=2 Yamator
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Цитувати: | Structure of the edge magnetic field of the L=2 Yamator / G.G. Lesnyakov, V.G. Kotenko, E.D. Volkov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 68-70. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-785052015-03-19T03:02:26Z Structure of the edge magnetic field of the L=2 Yamator Lesnyakov, G.G. Kotenko, V.G. Volkov, E.D. Magnetic confinement 2000 Article Structure of the edge magnetic field of the L=2 Yamator / G.G. Lesnyakov, V.G. Kotenko, E.D. Volkov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 68-70. — Бібліогр.: 7 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/78505 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Magnetic confinement Magnetic confinement |
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Magnetic confinement Magnetic confinement Lesnyakov, G.G. Kotenko, V.G. Volkov, E.D. Structure of the edge magnetic field of the L=2 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 |
Structure of the edge magnetic field of the L=2 Yamator |
| title_short |
Structure of the edge magnetic field of the L=2 Yamator |
| title_full |
Structure of the edge magnetic field of the L=2 Yamator |
| title_fullStr |
Structure of the edge magnetic field of the L=2 Yamator |
| title_full_unstemmed |
Structure of the edge magnetic field of the L=2 Yamator |
| title_sort |
structure of the edge magnetic field of the l=2 yamator |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2000 |
| topic_facet |
Magnetic confinement |
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http://dspace.nbuv.gov.ua/handle/123456789/78505 |
| citation_txt |
Structure of the edge magnetic field of the L=2 Yamator / G.G. Lesnyakov, V.G. Kotenko, E.D. Volkov // Вопросы атомной науки и техники. — 2000. — № 6. — С. 68-70. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT lesnyakovgg structureoftheedgemagneticfieldofthel2yamator AT kotenkovg structureoftheedgemagneticfieldofthel2yamator AT volkoved structureoftheedgemagneticfieldofthel2yamator |
| first_indexed |
2025-07-06T02:34:54Z |
| last_indexed |
2025-07-06T02:34:54Z |
| _version_ |
1836863240478392320 |
| fulltext |
UDC533.9
68 Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 68-70
Structure of the edge magnetic field of the l=2 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
1. Introduction
It is probable, that in future reactor-size toroidal-
confinement devices a divertor would become
indispensable, and this requirement would determine the
choice of the magnetic system also. It is known [1] that
a divertor is an integral part of helical stellarator-type
magnetic systems. However, its properties depend on
the magnetic field configuration (e. g., see, for example,
ref. [2]) and thus govern the practical design of the
divertor.
A new helical magnetic system called Yamator [3-5]
also belongs to the class of stellarator-type systems. At
moderate values of basic magnetic-surface
characteristics and a large relative volume of closed
magnetic surfaces (CMS), a significant magnetic well
can be created in Yamators. The Yamator magnetic
system is analogous to the one of a classical stellarator
with the only difference that the poloidal magnetic field
components are here formed with the help of 2-wire
lines wound round the torus. The wires of each of 2-
wire lines, wound with equal and opposite currents,
have the same pitch of winding L and are placed on the
nested tori of the same major radius Ro and of different
minor radii a1 and a2=a1+h, h being the line wire
spacing. The number of 2-wire lines determines the
Yamator polarity l.
The aim of this paper is to elucidate special features
of field lines behaviour in the magnetic field edge
structure of the l=2 Yamator and to gain an idea about
field-lines divertor functions.
2. Linear configuration of the l=2 Yamator
In accordance with [6], if a magnetic field has a
helical symmetry and (2πa2/L)2<<1, the magnetic
surface function in the l=2 straight Yamator can be
defined analytically as:
{
},]2cos)/(2
)/(1ln[]2cos)/(2
)/(1ln[/)4/(),(
2
2
4
2
2
1
4
1
2
1
2
0
Θ−
++Θ−
+−=Ψ
ar
arar
ararIr ηπµθ
where I is the helical wire current, η=2πεa1Bo/µoI, ε=2
πa1/L, Bo is the longitudinal magnetic field and Θ=θ-2π
ζ in terms of cylindrical coordinates r, θ, ζ. At
coincident the signs of Bo and bo (bo being the field on
the geometrical axis of the system, caused by the helical
current I at radius a1), and in accordance with the η
parameter there exist closed contours of magnetic-
surface cross-sections that correspond to the solution of
the Ψ=const equation, Fig.1.
Compared to a straight classical stellarator or a
torsatron, the edge magnetic field structures in these
new systems have a double number of separatrix ribs
(so-called "X-points"). The X-points are the intersection
points of two separatrices (thick curves in the figure),
one of which is small and it embraces the 2-wire line. In
the Yamator, at a given current I, its shape and size
depend only on the longitudinal magnetic field value.
The other, the larger separatrix is determined from the
condition ∂Ψ/∂Θ=0 and it has the minor radius
rs=(a1a2)1/2. The X-points lie symmetrically relative to
the azimuth of this helical coil, and their angles are
determined from the condition ∂Ψ/∂r rs=0,
.
)(
1
2
arccos
2
1
2
2
2
1
2
2
1
2
2
2
2
2
2
1
+
−−+±=Θ
aar
aa
r
aa
ss η
For the magnetic surface structure shown in Fig.1, the
angle positions of X-points, Θ≈±23.9°, are determined
by the given expression exactly.
Bo
Fig.1 The magnetic surface structure in the l=2 straight
Yamator with the dimensionless parameter η=1.35 that
determines the ratio of the longitudinal magnetic field
Bo to the helical current-generated field bo of the
radius a1. The coils of the longitudinal magnetic field
are not shown
3. Toroidal Yamator
In the previous paper [3], study was made into the
structure and properties of the magnetic surfaces in the
l=2, m=3 toroidal configurations of the Yamator system:
the average minor radius of the last CMS was
rlc/Ro=0.23÷0.28, the rotational transform ilc=0.25÷
0.45 (in units of 2π), the magnetic well was (-U)=10÷
25%.
The present paper is a continuation of those
investigations of the l=2, m=3 Yamator with the
emphasis on the study of the magnetic field edge
structure. Numerical calculations have been carried out
for the models with filamentary wires. The basic
parameters of the configurations were as follows: a 2-
wire line wound a round the torus along the helical line
Θ=mϕ, Θ - poloidal angle, ϕ - toroidal angle, m - the
number of helical pitches over the length of the torus;
h/Ro=0.15; the aspect ratio of the nested tori
69
Ah1=Ro/a1=3.333 (a1/Ro=0.3) and Ah2=Ro/a2=2.222
(a2/Ro=0.45).
A set of 2-wire lines is plunged into the
axisymmetric toroidal magnetic field Bϕ=BoRo/R,
where R is the radial position of the observation point,
reckoned from the straight axis of the system, z. The
ratio Bo/bo determines the structure and properties of
magnetic surfaces in the configuration. The
superposition of a controlling uniform transverse
magnetic field Bz is possible.
3.1. Characteristics of edge magnetic fields in
the Yamator
Numerical tracing of magnetic field lines was used
to obtain the magnetic field structure of different
configurations in the l=2, m=3 low aspect ratio
Yamator, Fig.2, was obtained. The variations of the
edge field structure (beyond the last CMS) were caused
by both the transverse magnetic field (Fig.2a,b) and the
toroidal magnetic field (Fig.2c). In the toroidal
geometry, there occur a shift to the outside and a
disturbance of the magnetic surfaces that embrace all
helical wires as a whole (see Fig.1) and the separatrices,
too. As a result, crescent magnetic surfaces can be
observed on the left side of the edge region (Fig.2a). It
also turns out that with variations in the transverse
magnetic field there exists its optimum value,
Bz/Bo=-0.0166, at which it appears possible to hold a
comparatively thin layer of the magnetic surfaces
immediately around the conventional separatrix. In this
case, the magnetic surface structure in the toroidal
geometry, near the separatrix, become similar to the
straight configuration, see Fig.1. It can be also seen, that
at the left and at the right from the CMS the shape and
size of the small separatrix are inversely proportional to
the magnetic field value and they obey its radial
dependence. Fig.2c shows the magnetic field structure
in the configuration, when by decreasing the toroidal
magnetic field one can attain i=0.41 the CMS region,
still keeping a rather high magnetic well. The choice of
equal inclination law of winding for the 2-wire helical
lines in the configuration provides the largest volume of
CMS (rlc≥a1) with the magnetic axis near the
geometrical axis of the torus and the region, around the
conventional separatrix, gets symmetrized, Fig.2d.
However, such a configuration looses the magnetic
well.
The structure of mod-B contours, Fig.3, appears
helpful to reveal the peculiarities of magnetic field
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)
0.5 1.0 1.5
R / Ro
-0.5
0.0
0.5
Z
/ R
o
(d)
Fig.2 Magnetic field line structure in the ϕ=0 poloidal
cross-section for four different configurations: a)
Bo/bo=2.5, Bz/Bo=0, rlc/Ro=0.27, ilc=0.26 (in units of
2π), (-U)=25%; b) Bo/bo=2.5, Bz/Bo=-0.0166,
rlc/Ro=0.225, ilc=0.34, (-U)=15%; c) Bo/bo=1.833,
Bz/Bo=0, rlc/Ro=0.255, ilc=0.41, (-U)=11%, no shear;
d) equi-inclined 2-wire helical winding, Bo/bo=2.5,
Bz/Bo=0, rlc/Ro=0.331, ilc=0.55, the magnetic hill is
(+U)=15%, shear
formation and magnetic field line behaviour in the l=2
Yamator configurations. In contrast to the classical
stellarator [7], the Bmin regions in the Yamator (closed
contours of B/Bo=0.633,0.972 in the ϕ=0° cross-section
(Fig.3a) and of B/Bo=0.66,1.067 at the half of the
magnetic field period, in the ϕ=30° cross-section
(Fig.3b)) are removed from the operating volume and
are located on each side of the 2-wire helical winding,
beyond the minor radius a1. The large separatrix passes
through these regions, and the X-points coincide with
the Bmin points. It can be stated that the ergodic
70
magnetic field line intersects the local region Bmin in
the vicinity of the X-point.
0.5 1.0 1.5
R / Ro
-0.5
0.0
0.5
Z
/ R
o
0.633
0.633
0.
76
7
0.
76
7
0.
9
0.
90.
94
5
0.
95
6
0.956
1.
01
67
0.
97
2
1.1
1.1
1.056
(a)
0.5 1.0 1.5
R / Ro
-0.5
0.0
0.5
Z
/ R
o
1.
18
7
1.
1
1.0
67
0.
7
0.
9
0.66
0.
76
7
(b)
Fig.3. Constant B contours and different magnetic
surfaces of the toroidal example (l=2, m=3 Yamator, the
configuration of Fig.2a): a) ϕ=0 cross-section; b) ϕ
=30° cross-section
The connection length of the ergodic lines reaching
the conventional separatrix of radius rs=0.367 falls off
rather rapidly outside the CMS in the configurations
understudy. The toroidal coordinates ϕ, Θ of the
footprints of intersection of the escaping field lines with
the toroidal surface of minor radius rs are shown in
Fig.4. The locations of these footprints are of vital
importance for the choice of the divertor concept. Fig.5
furnishes insight into the behaviour of the edge
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.4. Footprints of edge field lines on the toroidal
surface with the minor radius of the conventional
separatrix rs=(a1a2)1/2. The solid lines show the
positions of helical coils
Fig.5. Magnetic field lines in the l=2, m=3 toroidal
Yamator straightened along the ϕ coordinate: 1 - field
lines start in succession order along R on the inner side
of the torus beyond the closed magnetic surfaces (CMS);
2 - the short line begins in the outside region of the
torus, beyond CMS. Solid thin lines are the 2-wire
helical lines
magnetic field lines that intersect the rs toroidal surface.
Here, the magnetic field lines are shown in the three-
dimensional presentation in the toroidal configuration
straightened along the ϕ coordinate.
4. Summary
The present investigations give a general idea about
the edge magnetic field structure in both the l=2 straight
and with the low aspect ratio Ah=Ro/a1≈3.33 toroidal
Yamators and enable us to discuss a feasible concept of
divertor. The positions of footprints of more that 370
magnetic field lines show that the edge field lines,
forming a divertor region in the toroidal Yamator, come
to the outside half of the torus, that points to the
possibility of creating a discrete divertor. The three-
dimensional representation of edge field lines
apparently permits a more detailed discussion of the
divertor plate geometry.
References
[1] C. Gourdon et al, Nucl. Fusion 11 (1971) 161.
[2] T. Mizuuchi et al,. J. Plasma Fusion Res. SERIES,
Vol. 1 (2000) 209.
[3] V. G. Kotenko et al,. Voprosy Atomnoj Nauki i
Tekhniki (NNTs "KhFTI", Kharkov, 1999).
Problems of Atomic Science and Technology,
Series: Plasma Physics. NSC "KhIPT", Kharkov,
1999. Issues 1(1), 2(2), 49 (in English).
[4] V. G. Kotenko et al, 7th Ukraine Conf. on Control.
Nucl. Fus. and Plasma Phys., Kiev, September 20-
21, 1999. Book of Abstracts, p. 48 (in Ukraine).
[5] V. G. Kotenko et al,. 10th Intern. Toki Conf. on
Plasma. Phys. and Control. Nucl. Fus. (ITC)
"Physics and Technology for Steady State
Plasmas". Abstracts, January 18-21, 2000, Toki-
city, Japan. PI-8 (the paper will be published in J.
Plasma Fusion Res. SERIES, Vol. 3 (2000)).
[6] V. F. Aleksin, Zhurn. Tekhn. Fiziki 31 (1961) 1284
(in Russian).
[7] K. Miyamoto, Phys. Fluids 14 (1971) 722.
"Kharkov Institute of Physics and Technology", Kharkov 61108, Ukraine
References
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