Plasma and Fusion Research: Regular Articles

Volume 14, 3403100 (2019)

Vacuum Configurations of D Shape Heliotron∗)
Katsuji ICHIGUCHI1,2) and Yasuhiro SUZUKI1,2)
1)2)

National Institute for Fusion Science, Toki 509-5292, JapanSOKENDAI, The Graduate University for Advanced Studies, Toki 509-5292, Japan(Received 10 January 2019 / Accepted 16 April 2019)

The eﬀects of the deformation of the flux surfaces into a “D shape” on vacuum heliotron configurations are
examined numerically. In this study, one poloidal coil on the mid plane just inside the helical coils is assumed to
be installed so that the current in the coil can deform the surface shape. Such deformation gives the advantagethat the clearance between the helical coil and the outermost surface can be enlarged, which is crucial for thedivertor action in heliotrons. The eﬀects of the deformation on other vacuum properties are also discussed,
such as rotational transform, magnetic well, and |B|min contour. We also focus on such eﬀects on downsized
configurations.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: heliotron, vacuum magnetic configuration, numerical study, flux surface, divertor clearance, D shape
DOI: 10.1585/pfr.14.3403100

1. Introduction
The stellarators have the great advantage that no net
toroidal current is needed for the generation of the confinement field, while a large toroidal current is necessary intokamaks. This advantage allows us to operate continuous
discharges without any current-drive facilities and without
any disruptions. In addition, the heliotrons in the stellarator category have the further splendid feature that the helical divertor field lines are provided intrinsically. These
field lines are able to be generated only with the continuous
coils in the heliotrons. However, such field lines are notgenerated with the modular coils utilized in other types ofstellarators. In order to obtain good divertor performancein the heliotrons, we must maintain the clearance for thedivertor field lines between the helical coils and the outermost surface.
Considering the construction cost of the future reactor, downsizing the system is attractive. Furthermore, thesuperconducting coils are necessary for the long term operation. Since the current density of the superconducting
coils cannot be increased easily, the width of the cross section of the helical coils for obtaining a similar amplitude of
the magnetic field cannot be reduced even in the smallersized system. Therefore, the relative width of the coils
becomes large as the system size becomes small. In thiscase, the helical coils approach the outermost surface. The
clearance between the helical coils and the outermost surface, in particular in the inward side of the surface, shrinks,
which is necessary for the divertor action. Thus, we consider keeping or enlarging the clearance by deforming theshape of the outermost surface into a “D shape.” This deauthor’s e-mail: ichiguch@nifs.ac.jp∗) This article is based on the presentation at the 27th International Toki
Conference (ITC27) & the 13th Asia Pacific Plasma Theory Conference(APPTC2018).

formation reduces the inward convex structure of the flux
surfaces and therefore can enlarge the divertor clearance.
In the present study, we seek to achieve such a D shape byincorporating one additional poloidal field coil just inside
the helical coils. We also examine the eﬀects of the deformation on the properties in the vacuum configurations
such as rotational transform, magnetic well, and contours
of |B|min .
Concerning the D shape deformation of heliotron
plasmas, Matsumoto et al. [1] analyzed the eﬀects on thevacuum LHD (Large Helical Device) configuration [2] andthe finite beta equilibria. In their analysis, they applied ananalytic hexapole field on the original vacuum configuration to add the triangularity on the surface shape. However,they did not show the actual control method for the deformation, nor did they examine the deformation eﬀects on
the divertor clearance. Thus, we focus on the actual coilinstallation and the compatibility between the flux surface
properties and the divertor clearance enlargement in thispaper.

2. Calculation of Vacuum LHD
Configurations with D Shape Coil
The vacuum magnetic configurations are obtainedwith the KMAG code [3]. This code calculates the magnetic field based on the Biot-Savart’s law from the current
flowing in the finite size coils in the cylindrical coordinates(R, φ, Z). The current density is assumed constant in eachcoil. In order to examine the eﬀects of the D shape deformation, we consider the situation with a single poloidal
field coil installed on the mid plane just inside the helical coils, which we call “D shape coil,” as shown in Fig. 1
schematically.
Here, the eﬀects of the D shape coil current on the3403100-1

c 2019 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

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LHD configuration are examined. The LHD coil system consists of one pair of helical coils and three pairsof poloidal field coils or vertical field coils. Figure 2
shows the puncture plots of the field lines in the poloidal
cross sections without the D shape coil current. In thiscase, the major radius Rcnt and the pitch parameter γc ≡(M/)(ac /Rcnt ) of the helical coils are employed as Rcnt =3.9 m and γc = 1.254. Here M, , and ac are the numberof the toroidal field period, the pole number, and the minor radius of the helical coils, respectively, The numbersof M = 10 and  = 2 are fixed in the LHD configuration.
In the calculation for the case of Fig. 2, the current density in the helical coils jH = 37.5 A/mm2 is used and thetoroidal field at the coil center is obtained as BT = 2.77 T.
Note that the properties such as magnetic field line configuration, rotational transform, specific volume, and |B|min
contours do not depend on the magnetic field strength, andtherefore, do not depend on the current itself in each coil.
These properties depend only on the current ratio between
the coils.
Figure 3 shows an example of the puncture plots with
the D shape coil current. Here the D shape coil is assumed
to be located at RD = 2.6 m. The current flowing in thecoil is adjusted so that ID /IH = 0.370 in this case. HereID and IH are the D shape coil current and the current ineach helical coil, respectively. It is seen that the current

eﬀectively deforms the flux surfaces into the D shape. The
position of the inner edge of the outermost surface on themid plane is Rin = 3.27 m for ID /IH = 0 and Rin = 3.52 m
for ID /IH = 0.370. Therefore, the clearance between the
inward helical coil and the outermost surface that is neededfor drawing out the divertor field lines is enlarged due tothe deformation. Here, in the evaluation of Rin , we utilize
the starting position of the field line tracing. The starting position is shifted on the mid plane inwardly from themagnetic axis every 0.015 m. The field line is traced for
400 toroidal field periods from the starting position andwhether the field line is bounded around the magnetic axis
or not is examined. Then, we define Rin as the most inward
starting position from which the field line is bounded.
On the other hand, the horizontal position of the magnetic axis is shifted outward by the current as shown in
Fig. 4. This outward axis shift is similar to the shift obtained by the control of the current in the vertical fieldcoils. The outward axis shift enhances the magnetic well asshown in Fig. 5, where the profiles of the specific volume
are plotted as the functions of the average minor radiusr. The well enhancement is favorable for the stability
against the pressure driven modes. However, the axis shift
reduces the particle confinement. Figure 6 shows the contours of the |B|min which indicate a simple index of the orbit of the deeply trapped particles. The confinement regionof such particles shrinks in this D shape case. Figure 7
shows the change in the rotational transform profile. As
ID /IH increases, the rotational transform decreases up to
ID /IH = 0.074 slightly and increases in the range beyond
the value. This tendency is similar to the outward shift caseby vertical field control [4].

3. D Shape LHD Configurations
under Fixed Axis Position
Constraint

3.0

4.0

5.0

1.0
0.0
-1.0

0.0

1.0

Regarding good particle confinement, we consider the

-1.0

-1.0

0.0

1.0

Fig. 1 Schematic image of the D shape coil and the helical coils.

3.0

4.0

5.0

3.0

4.0

5.0

Fig. 2 Puncture plots of the field lines in the LHD configuration with Rax = 3.6m and γc = 1.254 with no D shape coil current at thepoloidal cross sections of φ = 0 (left), π/(2 M) (center), and π/M (right). Solid lines show the contours of the amplitude of themagnetic field. The helical coils are also plotted. The numbers along the horizontal and the vertical axes denote the measure of R
and Z coordinates in meters, respectively.

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Fig. 3 Puncture plots of the field lines in the same LHD configuration as in Fig. 1 with ID /IH = 0.370 at the poloidal cross sections of
φ = 0 (left), π/(2 M) (center), and π/M (right).

Fig. 4 Dependence of the magnetic axis position on the D shapecoil current.
Fig. 6 Contours of |B|min for ID /IH =0 (top) and 0.370 (bottom)
in the toroidally averaged flux surface of the LHD configuration. The thick solid line indicates the boundary of
the |B|min contour closed in the outermost surface. The
dashed line shows the position of the flux surface witha half average radius. The numbers along the horizontal and the vertical axes denote the measure of R and Z
coordinates in meters, respectively.

Fig. 5 Profiles of the specific volume for ID /IH =0, 0.185, and
0.370 in the LHD configuration.

D shape deformation under the constraint in which themagnetic axis position is fixed. For this purpose, the axis
position with no D shape coil current, Rax (ID = 0), should
be shifted inwardly under the control of the vertical fieldcoil currents. Here, the vertical field coil currents are adjusted so that the magnetic axis is located at Rax = 3.6 m
combined with the D shape coil current. Figure 8 shows
the puncture plots at the φ = 0 cross section in the caseFig. 7 Profiles of the rotational transform for ID /IH =0, 0.074,
and 0.370 in the LHD configuration.

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Fig. 8 Puncture plots of the field lines in the same LHD configuration as in Fig. 1 with Rax = 3.6m and ID /IH = 0.1622
at the poloidal cross section of φ = 0.

Fig. 10 Profiles of the specific volume for ID /IH =0, 0.0808, and
0.1622 in the LHD configuration with Rax = 3.6m.

Fig. 11 Contours of |Bmin | for ID /IH =0.1622 in the LHD configuration with Rax = 3.6m plotted in the same way as inFig.6.

Fig. 9 Dependence of the average minor radius and the inward
edge position on Rax (ID = 0) in the LHD configuration.
Blue solid line and red solid line show the average minor
radius and the inward edge position, Rin , respectively, in
the increase of the D shape coil current with Rax = 3.6m.
Blue dashed line shows the average minor radius with no
D shape coil current. Each number indicates the value of
ID /IH corresponding to Rax (ID = 0).

of Rax (ID = 0) = 3.42 m and ID /IH = 0.1622. Compared
to Fig. 2, the divertor clearance is also enlarged even under this constraint. The divertor clearance improves, that is
Rin increases, with the smaller Rax (ID = 0) and the larger
ID /IH as shown in Fig. 9. Figure 9 also shows that the minor radius is reduced in the increase of the divertor clearance. This is considered to be due to not only the enhancement of the D shape deformation but also the inward shiftby the poloidal field control, because the reduction of theminor radius in the case without the D shape coil current islarger than in the case with the current.
The specific volume itself is enhanced by the D shapedeformation as shown in Fig. 10. However, the magnetic
well corresponding to the derivative of the specific volume
is almost unchanged. This is attributed to the fact that themagnetic axis position is fixed in the change of the D shapecoil current. The confinement region of the trapped par-Fig. 12 Profiles of the rotational transform for ID /IH =0, 0.0808,
and 0.1622 in the LHD configuration with Rax = 3.6 m.

ticle is reduced in the vertical direction by the D shapedeformation as shown in Fig. 11, which is diﬀerent fromthe reduction in the horizontal direction in the case of theoutward shift of the magnetic axis. The D shape deformation slightly decreases the rotational transform in the region near the magnetic axis when the axis position is fixed,
as shown in Fig. 12.

4. Downsized D Shape Configurations
Now we consider downsizing the LHD configuration.
First, we examine the configuration downsized into the 2/3

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Volume 14, 3403100 (2019)size. The major radius of the helical coils is reduced to
Rcnt = 2.6 m, while the width of each coil is assumed tobe the same as that in the LHD. The same γc as in theoriginal size is employed and the magnetic axis is fixed as
Rax = 2.4 m. Other coil positions shrink proportionally.
Figure 13 shows the puncture plots of the magnetic fieldlines at the φ = 0 cross section. In the case of ID = 0,
the divertor clearance is smaller than that in the original
size. Figure 14 shows that Rin becomes larger and the divertor clearance is enlarged when the D shape coil current
is applied and increased with the magnetic axis position
fixed. However, the larger ID /IH is needed to obtain thesame amount of the enlargement increment compared tothe case of the original size as shown in Fig. 14. This figure also shows that the minor radius is reduced as the D
shape coil current increases. The specific volume is almost
unchanged as shown in Fig. 15. These tendencies are similar to those of the original size case. The rotational transform is also decreased near the magnetic axis as shown in
Fig. 16. The reduction is more remarkable than that in theoriginal size.
Next, we examine the configuration downsized into
the 1/2 size in the same manner. The major radius of thehelical coils is reduced to Rcnt = 1.95 m and the magnetic
axis is fixed as Rax = 1.8 m. The puncture plots at theφ = 0 cross section are shown in Fig. 17. In the caseFig. 14 Dependence of the average minor radius and the inward
edge position on Rax (ID = 0) in the 2/3 size of the LHD
configuration. Blue solid line and red solid line show theaverage minor radius and the inward edge position, Rin ,respectively, in the increase of the D shape coil current
with Rax = 2.4 m. Blue dashed line shows the averageminor radius with no D shape coil current. Each number
indicates the value of ID /IH corresponding to Rax (ID = 0).

Fig. 15 Profiles of the specific volume for ID /IH = 0, 0.1010,
and 0.1825 in the 2/3 size of the LHD configuration with
Rax = 2.4 m.

Fig. 13 Puncture plots of the field lines in the 2/3 size of the LHD
configuration as in Fig. 1 with Rax = 2.4 m and ID /IH = 0(top) and 0.1825 (bottom) at the poloidal cross section of
φ = 0.

Fig. 16 Profiles of the rotational transform for ID /IH = 0, 0.1010,
and 0.1825 in the 2/3 size of the LHD configuration with
Rax = 2.4 m.

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Fig. 17 Puncture plots of the field lines in the 1/2 size of the LHD
configuration as in Fig. 1 with Rax = 1.8 m and ID /IH = 0(top) and 0.2434 (bottom) at the poloidal cross section of
φ = 0.

of ID = 0, the outermost surface is attached to the helical coil and there is no divertor clearance. In the increaseof the D shape coil current, the magnetic axis is split at
ID /IH = 0.2434 before the appearance of the clearance.
Therefore, it is impossible to obtain the half size of the
LHD configuration with finite divertor clearance and a single magnetic axis by means of the D shape coil current.

5. Summary and Discussion
The eﬀects of the D shape deformation of the flux
surfaces on vacuum heliotron configurations are analyzed.
Here the existence of the coil current installed just inside
the helical coils is assumed. The current works well to enlarge the divertor clearance between the helical coil and theoutermost surface in the LHD configuration. On the other
hand, the current also contributes to the outward shift of theflux surfaces. To maintain the magnetic axis at a fixed position, an additional vertical field is necessary. Even in thiscase, the divertor clearance is enlarged eﬀectively. However, there is the tendency that the minor radius of the outermost surface and the closed |B|min region are decreased.
This tendency is not a favorable feature in the applicationof the D shape coil current. The rotational transform isslightly decreased. This decrease leads to the enhancementof the Shafranov shift at finite beta. Therefore, this property is favorable for the generation of the magnetic well.

However, this property is not favorable for the particle confinement. The negligible change in vacuum magnetic wellis also obtained.
On the other hand, it was found in the previous resultsby Matsumoto et al. [1] that the rotational transform at theaxis is unchanged and the magnetic well becomes deepas the triangularity increases. However, both the change
in the rotational transform in the present analysis and thechange in the magnetic well in the previous analysis are
not substantial. Therefore, we consider both results to beconsistent. The slight diﬀerences in the results would be
attributed to the diﬀerence in the deformation method andthe original circular configuration.
The D shape coil current can work similarly in the
LHD configuration downsized to 2/3 size under the assumption that the helical coil width is the same. The larger
D shape coil current is needed to acquire the same divertor
clearance increment as that in the original size. The decrease of the rotational transform is enhanced compared tothat in the original size. On the other hand, in the half size
case, it is impossible to obtain a configuration with finite
divertor clearance and a single magnetic axis.
In the analysis of the downsized configuration, we assumed the fixed width of the coils. Here we discuss thisassumption with respect to the energy confinement of theplasma. The scaling concerning the energy confinement
time τE for stellarators and heliotrons called ISS04 is given
S 04= 0.134a2.28 R0.64 P−0.61 n−0.54
B0.84ι0.41
by τISeE´2/3 [5]. Here,a, R, P, ne , B, and ι2/3 are minor and major radii, absorbed´
heating power, line-averaged density, magnetic field, andthe rotational transform at r/a = 2/3, respectively. According to this scaling, when both minor and major radii
are downsized to the 2/3 and the 1/2 sizes, the coil currents
must be 2.7 and 5.6 times larger than those in the original
size configuration to obtain the magnetic field corresponding to the same confinement time, respectively. Therefore,
more improvement of the coil current density is required
to maintain the ISS04 scaling if we employ the same widthof the coils as in the original size configuration. Or a wider
width of the coils is necessary if we employ the same current density.
One method for improving the D shape configuration
would be the addition of toroidal field coils. The additional
toroidal coil currents can increase the rotational transformand may contribute to the enlargement of the divertor clearance. On the other hand, this idea brings another challenging problem to the construction of the superconducting coilsystem, because there is diﬃculty in setting the toroidal
coils outside the helical coils.
The present study is limited to the vacuum configuration analysis. The eﬀects of the finite beta should beincorporated for the comprehensive understanding of theplasma performance.

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Acknowledgments
One of the authors (K.I.) thanks Prof. N. Yanagi
for the fruitful discussion. This work was supported by
the budget NIFS18KNST126 and NIFS18KNXN361 of
National Institute for Fusion Science (NIFS) and JSPS
KAKENHI Grant Number 15k06651. The super computer
Plasma Simulator in NIFS was utilized for the numericalcalculations.

[1] T. Matsumoto et al., J. Phys. Soc. Jpn. 62, 3904 (1993).
[2] A. Komori et al., Fusion Sci. Technol. 58, 1 (2010).
[3] J. Todoroki, private communication; Y. Nakamura, private
communication.
[4] K. Ichiguchi et al., Nucl. Fusion 33, 481 (1993).
[5] H. Yamada et al., Nucl. Fusion 45, 1684 (2005).

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