Heliotron/torsatron configurations with minimal field ripples
Numerical studies were undertaken to elucidate the magnetic field ripple behavior as a function of the uniform transverse magnetic field strength in the l=2,3 polarity helical magnetic system models, that are similar to some actual heliotron\torsatron without additional longitudinal magnetic field c...
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irk-123456789-1101392016-12-31T03:01:42Z Heliotron/torsatron configurations with minimal field ripples Kotenko, V. Volkov, E. Yamazaki, K. Magnetic confinement Numerical studies were undertaken to elucidate the magnetic field ripple behavior as a function of the uniform transverse magnetic field strength in the l=2,3 polarity helical magnetic system models, that are similar to some actual heliotron\torsatron without additional longitudinal magnetic field coils, such as LHD and U-3M. The existence of vacuum magnetic surface configurations with a minimal field ripple is demonstrated. Проведені чисельні розрахунки залежності значень пробкового відношення магнітного поля від величини однорідного поперечного магнітного поля в моделях гвинтових магнітних систем з заходністю l=2,3, що подібні до деяких актуальних геліотрон-торсатронних гвинтових магнітних систем, таких як LHD та У-3М. Показана можливість існування конфігурацій магнітних поверхонь з мінімальним значенням пробкового відношення. Численным методом в зависимости от величины однородного поперечного магнитного поля изучено поведение величины пробочного отношения магнитного поля в моделях винтовых магнитных систем с заходностью l=2,3, подобных некоторым актуальным гелиотрон-торсатронным винтовым магнитным системам без катушек продольного магнитного поля, таким как LHD и У-3М. Показана возможность существования конфигураций магнитных поверхностей с минимальным пробочным отношением. 2003 Article Heliotron/torsatron configurations with minimal field ripples / V. Kotenko, E. Volkov, K. Yamazaki // Вопросы атомной науки и техники. — 2003. — № 1. — С. 19-22 — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 52.55.Hc http://dspace.nbuv.gov.ua/handle/123456789/110139 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic confinement Magnetic confinement Kotenko, V. Volkov, E. Yamazaki, K. Heliotron/torsatron configurations with minimal field ripples Вопросы атомной науки и техники |
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Numerical studies were undertaken to elucidate the magnetic field ripple behavior as a function of the uniform transverse magnetic field strength in the l=2,3 polarity helical magnetic system models, that are similar to some actual heliotron\torsatron without additional longitudinal magnetic field coils, such as LHD and U-3M. The existence of vacuum magnetic surface configurations with a minimal field ripple is demonstrated. |
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Kotenko, V. Volkov, E. Yamazaki, K. |
| author_facet |
Kotenko, V. Volkov, E. Yamazaki, K. |
| author_sort |
Kotenko, V. |
| title |
Heliotron/torsatron configurations with minimal field ripples |
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Heliotron/torsatron configurations with minimal field ripples |
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Heliotron/torsatron configurations with minimal field ripples |
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Heliotron/torsatron configurations with minimal field ripples |
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Heliotron/torsatron configurations with minimal field ripples |
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heliotron/torsatron configurations with minimal field ripples |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2003 |
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Magnetic confinement |
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http://dspace.nbuv.gov.ua/handle/123456789/110139 |
| citation_txt |
Heliotron/torsatron configurations with minimal field ripples / V. Kotenko, E. Volkov, K. Yamazaki // Вопросы атомной науки и техники. — 2003. — № 1. — С. 19-22 — Бібліогр.: 16 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT kotenkov heliotrontorsatronconfigurationswithminimalfieldripples AT volkove heliotrontorsatronconfigurationswithminimalfieldripples AT yamazakik heliotrontorsatronconfigurationswithminimalfieldripples |
| first_indexed |
2025-07-08T00:10:11Z |
| last_indexed |
2025-07-08T00:10:11Z |
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1837035330414313472 |
| fulltext |
HELIOTRON/TORSATRON CONFIGURATIONS WITH MINIMAL FIELD
RIPPLES
V. Kotenko, E.Volkov, K. Yamazaki*
Institute of Plasma Physics, NSC KIPT, 61108 Kharkov, Ukraine;
*National Institute for Fusion Science, Oroshi-cho 322-6, Toki, Gifu-ken, 509-5292, Japan
Numerical studies were undertaken to elucidate the magnetic field ripple behavior as a function of the uniform
transverse magnetic field strength in the l=2,3 polarity helical magnetic system models, that are similar to some actual
heliotron\torsatron without additional longitudinal magnetic field coils, such as LHD and U-3M. The existence of
vacuum magnetic surface configurations with a minimal field ripple is demonstrated.
PACS: 52.55.Hc
INTRODUCTION
The field ripple γ (γ=Bmax/Bmin, Bmax, Bmin are the
maximum and minimum magnetic field strength,
respectively) on the magnetic surfaces in closed magnetic
plasma traps characterizes the degree of magnetic field
nonuniformity. According to the neoclassical transport
theory [1], the plasma particle diffusion is supposed to be
enhanced on these nonuniformities at low particle
collision frequencies that are typical of the fusion reactor
conditions. A few methods of decreasing the γ value have
been suggested for both the tokamak magnetic systems
[2-4] and the stellarator-type magnetic systems [5,6].
From the standpoint of finding magnetic configurations
with a minimal γ versus the uniform transverse magnetic
field, numerical calculations were carried out here to
investigate helical magnetic system models with l=2,3
polarity, that are similar to the magnetic systems of some
conventional (i.e., without additional longitudinal field
coils) heliotrons/torsatrons now in operation, such as
LHD [7] and U-3M [5].
GENERAL FEATURES OF
CALCULATION MODELS
A poloidal cross-section ϕ =const. (ϕ is a toroidal
angle) for the torus of the calculation model is
schematically presented in Fig.1, where Ro is the major
radius of the torus, a is the minor radius. The electrical
currents in filament-like helical conductors (not shown in
Fig.1) located on the torus surface generate a longitudinal
magnetic-field component bo on the circular axis of the
system. To form closed magnetic surfaces inside the
torus, it is necessary to apply a transverse magnetic field
Bz, which was assumed to be uniform throughout the
calculations, z being the principal (straight) torus axis.
Fig.1 also shows the magnetic axis geometry. In the cases
under consideration, the undistorted magnetic axis has a
form of a helical line, which closes on itself after one go-
round over the length of the torus and lies on the surface
of an imaginary torus, the major radius of which is
denoted by Roax (magnetic-axis major radius), and the
minor radius – by rax(magnetic-axis minor radius), Rax(ϕ )
is the radial position of magnetic axis trace in the ϕ
poloidal cross-section. In calculations, the magnetic field
line was considered closed on itself if its trace position
after one go-round over the length of the torus coincided
with the starting point of its calculation to an accuracy no
worse than 1.0⋅10-4 (here and all over the text the lengths
are in Ro units).
Rax
Roax
R
rax
a
Bz
bo
o
( )
Fig.1. Torus cross-section and the magnetic axis
geometry
The magnetic axis radii were defined by
rax=│Rax(φ1)-Rax(φ2)│/2, Roax=(Rax(φ1)+Rax(φ2))/2, (1)
where ϕ1, ϕ2 are the toroidal angles of the poloidal cross-
sections, which are spaced apart by the ½ magnetic field
period, and where the magnetic axis traces occur in the central
plane of the torus. If ϕ1=0, then ϕ2=180°/ml, m is the number
of helical conductor pitches over the length of the torus. The
magnetic field ripple values were calculated for both the
magnetic axis, γax, and the last closed magnetic surface
(LCMS), γlc. The scatter of γax values did not exceed 1.0⋅10-4.
The scatter of γlc values was about ~1-2⋅10-2, as the starting
point of magnetic field line calculation for the LCMS
identification was set to an accuracy of no better than 1.0⋅10-3.
The LCMS was considered existent if it showed no breakage
over the length of the magnetic field line corresponding to no
less than 100 go-rounds over the length of the torus. The
calculation results depend on the winding law of helical
conductors. To carry out the calculations, the winding law was
written in a convenient explicit form:
θ = θ(m,ϕ,α) = θ1-kθ(θ2-2arctg(tg(θ1/2))), (2)
where 0≤ ϕ ≤2π,θ - poloidal angle, α =a/Ro, θ1= mϕ -
cylindrical law of winding, θ2= 2arctg (((1+α)/(1-α))
0.5tg(mϕ/2))- equi-inclined law of winding, kθ- numerical
coefficient. For the principal values 0≤ ϕ ≤π/m the
winding law permits a simple geometric interpretation: θ
= θ1-kθ(θ2-θ1). In this paper the special case kθ=-1, i.е., θ =
θ2, is discussed.
Problems of Atomic Science and Technology. 2003. № 1. Series: Plasma Physics (9). P. 19-22 19
THE L=2 SYSTEM
The calculation model of the l=2 helical magnetic
system had the following parameters: α=a/Ro=0.25, the
number of helical pitches m=5, the equi-inclined winding
law θ=2arctg(1.29099tg(2.5ϕ)).
B /b
0.12 0.16 0.20 0.24
1.00
1.10
1.20
1.30
ax lc
z o
1.00
1.50
2.00
2.50
3.00
Fig. 2. Field ripples on the magnetic axis (lower solid
curve), γax, and on the LCMS, γlc (empty circles) versus
the uniform transverse magnetic field Bz in the
l=2 system
Fig.2 presents the calculated magnetic field ripple
versus the transverse magnetic field Bz (in bo units
throughout) for the magnetic axis γax (lower solid curve)
and for the LCMS γlc (empty circles). It is seen from the
figure that the γax curve is monotone and has a minimum
in the vicinity of Bz= (Bz)b≈0.202. The γlc values show a
wide scatter exceeding 5 to15 times the calculation
accuracy. In Fig.2 a boundary curve of the highest γlc
values is drawn. Similarly to the γax curve, it has its
minimum in the vicinity of (Bz)b but, in contrast to the γax
curve, shows a systematic falloff near the boundaries of
the range of uniform transverse magnetic field variations,
that most likely continues beyond the range boundaries.
In the l=2 helical system the magnetic surface shape is
stable. Therefore, the fall off of γlc values as well as their
scatter are due, first of all, to the associated changes in the
LCMS average radius, as the uniform transverse magnetic
field strength (or magnetic axis position, see below Fig.3)
changes.
Fig.3 presents the geometrical characteristics of
magnetic axes versus the transverse magnetic field Bz.
From the comparison between Fig.2 and Fig.3 it follows
that γax, γlc are minimal if the magnetic-axis minor radius
of the magnetic surface configuration is equal to zero,
rax=0 (magnetic axis is plane in principle), and the
magnetic-axis major radius (Roax)b≈0.9594. The existence
of the plane magnetic axis follows from the fact that the
difference Rax(0o)-Rax(18o) in Eq.1 reverses its sign in the
vicinity of the magnetic-axis major radius (Roax)b≈0.9594.
Obviously, the region of closed magnetic surface
existence is displaced inward the torus if (Roax)b<1. The
region of closed magnetic surface existence will be well
centered, (Roax)b=1 for (Bz)b=0.34,.if kθ=0.45 in Eq.(2):
θ=5ϕ-0.9(arctg(1.29099tg(2.5ϕ))-arctg(tg(2.5ϕ/))).
0.12 0.16 0.20 0.24
0.000
0.005
0.010
0.015
0.020
0.025 r
B /b
ax
z o
0.85
0.90
0.95
1.00
1.05
R oax
Fig. 3. Magnetic-axis minor radius rax, magnetic axis
major radius Roax versus the uniform transverse magnetic
field Bz in the l=2 system
Magnetic surface parameters versus magnetic- surface
average radius r for the rax =0 regime in the l=2 system
show the following characteristic features: a great
magnetic hill value U≈0÷0.318, variations in the
rotational transform angle i within i≈0.43÷1.33 (in 2π
units, positive shear), and in the field ripple γ ≈1.06÷2.07.
THE L=3 SYSTEM
The calculation model of the l=3 helical magnetic
system had the following parameters: α= a/Ro=0.27, the
number of helical pitches over the length of the torus
m=3, the equi-inclined winding law θ
=2arctg(1.318987tg(1.5ϕ)).
It is known that in toroidal l=3 helical magnetic
systems, as the transverse magnetic field varies, a
conventional magnetic surface configuration with a single
undistorted (primary) magnetic axis can go over into the
magnetic surface configuration with the inner island
structure [8-12]. The island structure contains two
magnetic axes (secondary) and an 8-shaped inner
separatrix with a bifurcation line (line of hyperbolic-type
singular points of the magnetic surface function).
Calculations have shown that similar to the primary
magnetic axis (line of elliptic-type singular points of the
magnetic surface function), the bifurcation line, takes the
form of the helical line which closes on itself after one
go-round over the length of the torus and lies on the
surface of an imaginary torus, the major radius of which
is denoted as Rolb (bifurcation-line major radius), and the
minor radius as rlb(bifurcation-line minor radius); Rlb(ϕ )
is the radial position of bifurcation-line trace in the ϕ
poloidal cross-section. The island structure is surrounded
on the outside by external magnetic surfaces, which are
topologically equivalent to the magnetic surfaces in the
configuration with one magnetic axis. Presumably, the
bifurcation line plays the role of the magnetic axis with
respect to the external magnetic surfaces.
Fig.4 shows the calculated magnetic-axis minor radius
rax, magnetic-axis major radius Roax (solid curves),
bifurcation-line minor radius rlb, bifurcation-line major
radius Rolb (dashed curves) versus the transverse magnetic
field Bz in the l=3 system. It is seen, that at the boundaries
20
of the range of uniform magnetic field variations there
exists the configuration with one magnetic axis. Fig 4 also
shows the marked segments (Bz)left=0.289±1⋅10-3,
(Bz)right=0.367±1⋅10-3 of the horizontal axis, within which
one can observe the evidence of appearance
(disappearance) of the island structure, relying on the
change in the paraxial magnetic surface shape [13].
0.28 0.32 0.36 0.40
0.000
0.004
0.008
0.012
0.016
0.8
0.9
1.0
1.1rax
rlb
Roax
Rolb
B /bz o
Fig.4. Magnetic-axis minor radius rax, magnetic axis
major radius Roax (solid curves), bifurcation-line minor
radius rlb, bifurcation-line major radius Rolb (dashed
curves) versus the uniform transverse magnetic field Bz in
the l=3 system
The appearance (disappearance) of the island structure is
evidently connected with the transitions rax↔rlb, Roax↔Rolb.
Examination of Fig.4 indicates that these transitions are
monotone within the accuracy of calculations. So, the central
part of the range of Bz values forms the island structure
configuration. For Bz= (Bz)b=0.325 the island structure is
maximum developed, the bifurcation-line minor radius is
minimal, rlb=0.00025 (not equal to 0 in principle, because the
difference Rlb(0o)-Rlb(20o) in Eq.(1) does not reverse its sign),
the bifurcation-line major radius Rolb=0.9582<1, i.e. the
magnetic surface existence region is displaced inward the torus.
Fig.5 presents the calculated magnetic field ripple
versus the transverse magnetic field Bz for the magnetic
axis γax (lower solid curve), for the bifurcation line γlb
(dashed curve) and for the LCMS, γlc (empty circles). It is
seen that within the calculation accuracy, a smooth
(forward and reverse) conjugation of the γax and γlb
functions occurs within the marked segments of the Bz
range. Close to Bz= (Bz)b the γlb curve has its minimum. In
contrast to the l=2 system, the field ripple values γlc on the
LCMS in the l=3 system exhibit a scatter not exceeding
the calculation accuracy. This points to a monotone
change in the LCMS size and shape in the l=3 system
when the uniform magnetic field (the magnetic axis-
bifurcation line position) varies. In Fig.5, the boundary of
the highest γlc values is drawn (upper solid curve). This
curve, similarly to the γax curve, has a minimum at about
(Bz)b value, but in contrast to the γax curve, it exhibits a
tendency to a systematic falloff near the boundaries of the
range of uniform transverse magnetic field changes, this
tendency obviously becoming stronger beyond the range
boundaries.
The parameters of magnetic surfaces, both inner
(r<0.05Ro) and outer (r>0.05Ro) with respect to the inner
separatrix, were calculated as functions of the magnetic
surface average radius r in the minimum rlb regime. Inside
the domain of the inner separatrix the magnetic well value
is U=0÷-0.001, the rotational transform angle i is varying
within i≈0.5÷0.3 (in 2π units, negative shear), and the
field ripple value is γ≈1.07÷1.15. Outside the domain of
the inner separatrix we have U≈0.003÷0.024 (magnetic
hill), i≈0.125÷0.44 (positive shear), and γ≈1.16÷1.275.
Limited by the inner separatrix, the volume of the island
structure takes 0.4 of the LCMS volume.
0.24 0.28 0.32 0.36 0.40
1.00
1.04
1.08
1.12
1.00
1.10
1.20
1.30
1.40
1.50
ax
lb
lc
B /bz o
Fig. 5. Field ripples on the magnetic axis, γax, (solid
curves), on the bifurcation line, γlb, (dashed curve) and on
the LCMS, γlc, (empty circles) versus the uniform
transverse magnetic field Bz in the l=3 system
MAGNETIC AXIS POSITION
MEASUREMENT
The localization of the magnetic axis in any poloidal
cross-section may be useful for a prompt identification of
a magnetic surface configuration during the running
experiment. A simple scheme of measurements [14] can
be of the form presented in Fig.6.
A
CP
m. axis
Fig.6. The scheme of magnetic axis position
measurements
The basis for it is a miniature electron gun-probe (one
or several). It contains a hot cathode C and a probe P on
the back side of the anode box A. This probe can be made
as a hot cathode and can perform its function, too, if
necessary.
Owing to the possibility of decreasing essentially the
electron injection potential V during probe detection of
21
electrons after their go-round (full or partial) along the
system, one can minimize the electron trajectory
distortions δr ∼V 0.5/i [15,16] in order to provide high
accuracy of determining the magnetic axis trace position
in the chosen cross-section. Trace position of other
singular, closed-on themselves field lines can also be
determined with high accuracy. These may be, for
example, magnetic axes and ribs of various resonance
structures, as well as X-point of divertor configurations.
CONCLUSION
The numerical calculations have demonstrated the
existence of the regime specified by a certain uniform
transverse magnetic field value, Bz=(Bz)b in the l=2,3
helical systems. The field ripple on the LCMS and in the
central part of the region of magnetic surface existence
(magnetic axis in the l=2 system, bifurcation line in the
l=3 system) is close to the minimum value in this regime.
The indications of this regime are the zero minor radius of
the magnetic axis (l=2 system), the developed inner island
structure (l=3 system) and the minimum minor radius of
the inner-separatrix bifurcation line, the displacement of
region of magnetic surface existence inward the torus (in
particular, for the equi-inclined law of helical conductor
winding). For Bz=(Bz)b, the behavior of the magnetic-
surface parameters versus the magnetic-surface average
radius does not differ from the standard.
Taking into account that in the straight, with
undistorted helical symmetry, l=2,3 helical magnetic
systems the magnetic axis is coincident with the
geometrical axis (rax≡0) and the field ripple value includes
only the helical component (toroidal component is
absent), one can assume that for fixed toroidicity the
regime considered is the regime with minimum
distortions of the helical-symmetry.
The calculations have also shown that the magnetic
axis geometry depends on the winding law of filament-
like helical conductors. Consideration must be given to
the winding law of each turn in the multiturn real-size
helical coil, where several parameters can vary
simultaneously and insignificantly in the transition from
turn to turn (e.g., kθ and α in Eq.(2)). So, for comparison
between the calculated and real parameters of the
magnetic surface configuration, accurate measurements of
the plane magnetic axis position become of particular
importance.
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ГЕЛІОТРОН-ТОРСАТРОННІ КОНФІГУРАЦІЇ З МІНІМАЛЬНИМ ПРОБКОВИМ ВІДНОШЕННЯМ
В.Г. Котенко, Є.Д. Волков, К. Ямазакі
Проведені чисельні розрахунки залежності значень пробкового відношення магнітного поля від величини
однорідного поперечного магнітного поля в моделях гвинтових магнітних систем з заходністю l=2,3, що подібні
до деяких актуальних геліотрон-торсатронних гвинтових магнітних систем, таких як LHD та У-3М. Показана
можливість існування конфігурацій магнітних поверхонь з мінімальним значенням пробкового відношення.
ГЕЛИОТРОН-ТОРСАТРОННЫЕ КОНФИГУРАЦИИ С МИНИМАЛЬНЫМ ПРОБОЧНЫМ
ОТНОШЕНИЕМ
В.Г. Котенко, Е.Д. Волков, К. Ямазаки
Численным методом в зависимости от величины однородного поперечного магнитного поля изучено
поведение величины пробочного отношения магнитного поля в моделях винтовых магнитных систем с
заходностью l=2,3, подобных некоторым актуальным гелиотрон-торсатронным винтовым магнитным системам
без катушек продольного магнитного поля, таким как LHD и У-3М. Показана возможность существования
конфигураций магнитных поверхностей с минимальным пробочным отношением.
22
V. Kotenko, E.Volkov, K. Yamazaki*
Numerical studies were undertaken to elucidate the magnetic field ripple behavior as a function of the uniform transverse magnetic field strength in the l=2,3 polarity helical magnetic system models, that are similar to some actual heliotron\torsatron without additional longitudinal magnetic field coils, such as LHD and U-3M. The existence of vacuum magnetic surface configurations with a minimal field ripple is demonstrated.
PACS: 52.55.Hc
Introduction
General features of
calculation models
A poloidal cross-section =const. ( is a toroidal angle) for the torus of the calculation model is schematically presented in Fig.1, where Ro is the major radius of the torus, a is the minor radius. The electrical currents in filament-like helical conductors (not shown in Fig.1) located on the torus surface generate a longitudinal magnetic-field component bo on the circular axis of the system. To form closed magnetic surfaces inside the torus, it is necessary to apply a transverse magnetic field Bz, which was assumed to be uniform throughout the calculations, z being the principal (straight) torus axis. Fig.1 also shows the magnetic axis geometry. In the cases under consideration, the undistorted magnetic axis has a form of a helical line, which closes on itself after one go-round over the length of the torus and lies on the surface of an imaginary torus, the major radius of which is denoted by Roax (magnetic-axis major radius), and the minor radius – by rax(magnetic-axis minor radius), Rax( ) is the radial position of magnetic axis trace in the poloidal cross-section. In calculations, the magnetic field line was considered closed on itself if its trace position after one go-round over the length of the torus coincided with the starting point of its calculation to an accuracy no worse than 1.010-4 (here and all over the text the lengths are in Ro units).
Fig.2 presents the calculated magnetic field ripple versus the transverse magnetic field Bz (in bo units throughout) for the magnetic axis ax (lower solid curve) and for the LCMS lc (empty circles). It is seen from the figure that the ax curve is monotone and has a minimum in the vicinity of Bz= (Bz)b0.202. The lc values show a wide scatter exceeding 5 to15 times the calculation accuracy. In Fig.2 a boundary curve of the highest lc values is drawn. Similarly to the ax curve, it has its minimum in the vicinity of (Bz)b but, in contrast to the ax curve, shows a systematic falloff near the boundaries of the range of uniform transverse magnetic field variations, that most likely continues beyond the range boundaries. In the l=2 helical system the magnetic surface shape is stable. Therefore, the fall off of lc values as well as their scatter are due, first of all, to the associated changes in the LCMS average radius, as the uniform transverse magnetic field strength (or magnetic axis position, see below Fig.3) changes.
Fig.3 presents the geometrical characteristics of magnetic axes versus the transverse magnetic field Bz. From the comparison between Fig.2 and Fig.3 it follows that ax, lc are minimal if the magnetic-axis minor radius of the magnetic surface configuration is equal to zero, rax=0 (magnetic axis is plane in principle), and the magnetic-axis major radius (Roax)b0.9594. The existence of the plane magnetic axis follows from the fact that the difference Rax(0o)-Rax(18o) in Eq.1 reverses its sign in the vicinity of the magnetic-axis major radius (Roax)b0.9594. Obviously, the region of closed magnetic surface existence is displaced inward the torus if (Roax)b<1. The region of closed magnetic surface existence will be well centered, (Roax)b=1 for (Bz)b=0.34,.if k=0.45 in Eq.(2):
Fig.5 presents the calculated magnetic field ripple versus the transverse magnetic field Bz for the magnetic axis ax (lower solid curve), for the bifurcation line lb (dashed curve) and for the LCMS, lc (empty circles). It is seen that within the calculation accuracy, a smooth (forward and reverse) conjugation of the ax and lb functions occurs within the marked segments of the Bz range. Close to Bz= (Bz)b the lb curve has its minimum. In contrast to the l=2 system, the field ripple values lc on the LCMS in the l=3 system exhibit a scatter not exceeding the calculation accuracy. This points to a monotone change in the LCMS size and shape in the l=3 system when the uniform magnetic field (the magnetic axis-bifurcation line position) varies. In Fig.5, the boundary of the highest lc values is drawn (upper solid curve). This curve, similarly to the ax curve, has a minimum at about (Bz)b value, but in contrast to the ax curve, it exhibits a tendency to a systematic falloff near the boundaries of the range of uniform transverse magnetic field changes, this tendency obviously becoming stronger beyond the range boundaries.
The numerical calculations have demonstrated the existence of the regime specified by a certain uniform transverse magnetic field value, Bz=(Bz)b in the l=2,3 helical systems. The field ripple on the LCMS and in the central part of the region of magnetic surface existence (magnetic axis in the l=2 system, bifurcation line in the l=3 system) is close to the minimum value in this regime. The indications of this regime are the zero minor radius of the magnetic axis (l=2 system), the developed inner island structure (l=3 system) and the minimum minor radius of the inner-separatrix bifurcation line, the displacement of region of magnetic surface existence inward the torus (in particular, for the equi-inclined law of helical conductor winding). For Bz=(Bz)b, the behavior of the magnetic-surface parameters versus the magnetic-surface average radius does not differ from the standard.
Taking into account that in the straight, with undistorted helical symmetry, l=2,3 helical magnetic systems the magnetic axis is coincident with the geometrical axis (rax0) and the field ripple value includes only the helical component (toroidal component is absent), one can assume that for fixed toroidicity the regime considered is the regime with minimum distortions of the helical-symmetry.
The calculations have also shown that the magnetic axis geometry depends on the winding law of filament-like helical conductors. Consideration must be given to the winding law of each turn in the multiturn real-size helical coil, where several parameters can vary simultaneously and insignificantly in the transition from turn to turn (e.g., k and in Eq.(2)). So, for comparison between the calculated and real parameters of the magnetic surface configuration, accurate measurements of the plane magnetic axis position become of particular importance.
References
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