23P066-P
Use of Twisted Cauchy-Condition Surface to Reconstruct
the Last Closed Magnetic Surface in the LHD
ひねりコーシー条件面を用いたLHDの最外殻磁気面逆解析
Gaku Okubo1), Masayuki Akazawa1), Yutaka Matsumoto1), Masafumi Itagaki1),
Ryosuke Seki2) , Yasuhiro Suzuki2) and Kiyomasa Watanabe2)
1)
大久保岳 ，赤澤眞之1)，松本裕1)，板垣正文1)，關良輔2)，鈴木康浩2，渡邊清政2)
1)

2)

Hokkaido University, Kita 13, Nishi 8, Kita-ku, Sapporo 060-8628, Japan
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki-city, GIFU 509-5292, Japan1)
北海道大学, 〒060-8628 札幌市北区北13条西8丁目2)
核融合科学研究所, 〒509-5292 岐阜県土岐市下石町322-6

The three-dimensional (3-D) Cauchy condition surface (CCS) method code is now under development toreconstruct the 3-D magnetic field profile in the Large Helical Device (LHD). A new “twisted CCS” hasbeen introduced, whose elliptic cross section rotates with the variation in plasma geometry in the toroidal
direction. Independent of the toroidal angle, this CCS can be placed at a certain distance from the lastclosed magnetic surface (LCMS). With this new CCS, the numerical accuracy in the reconstructed field has
been improved. Further, the magnetic field line tracing indicates the LCMS more precisely than with theuse of the axisymmetric CCS.

1. Introduction
The Cauchy condition surface (CCS), where boththe Dirichlet and Neumann conditions are unknown, isplaced in a domain that can be supposed to be inside
the plasma. In the analysis, no plasma current isassumed outside this CCS. Instead, the CCS plays thesame role as the plasma current in causing the fieldoutside the plasma. The choice of suitable shape andsize of the CCS is important to assure the accuracy of
the reconstructed solution. In the previous work for
the Large Helical Device (LHD) [1], the shape of the3-D CCS was simply assumed to be a torus that has acircular cross section. Instead of this axisymmetric
CCS, one here introduces a new idea named “twistedCCS” in order to reduce the numerical error.
2. Outline of the 3-D CCS Method [1]Assuming a vacuum field outside the CCS, three
types of boundary integral equations (BIEs), i.e., formagnetic field sensors, flux loops and points along
the CCS, are given in terms of 3-D vector potential.
They are solved simultaneously in such a way thatthe vector potential and its derivative on the CCS
will be consistent with the sensor signals.
In more detail, the BIEs are discretized, coupledand expressed in the matrix equation form as(1)
Dp  g .
Here the solution vector p contains the vector
potentials and their normal derivatives on the CCS.
In the present work, Eq. (1) is solved using thetruncated singular value decomposition technique,where the condition number is kept to be not larger
than 105 . Once all the components in p are known,

the magnetic fields for arbitrary points can becalculated.

3. Twisted CCS
A larger cross-section of CCS, i.e., a shorter distance
between the CCS and a sensor position is better forreceiving the sensor signal information. However, thefield solution inside the last closed magnetic surface(LCMS) given by the CCS method is, so to speak,turbulence or chaos that exerts a harmful influence on
the accuracy of the field profile outside the plasma.
From this standpoint, a small size of CCS is a good
choice to avoid the numerical instability. Because ofthis, the cross section of the axisymmetric CCS in theprevious work [1] had to be set very small, i.e. a circle
having 0.075m radius.

TwistedCCS

LCMS

Fig.1. Image of twisted CCS

One here introduces the “twisted CCS” as illustrated
in Fig.1, whose elliptic cross section rotates with thevariation in vacuum vessel geometry in the toroidal
direction. Independent of the toroidal angle, this CCS
can be placed at a certain distance from the LCMS. A
reduction in the numerical error can then be expected.
The ellipse given by(r  r0 )2 / a 2  ( z  z0 )2 / b2  1(2)rotates 180° clockwise in the poloidal direction when

it proceeds 36° counterclockwise in the toroidal
direction following the 10-fold rotational symmetry of
the LHD plasma. In this process the variation in thecoordinates (r , z ) on the CCS is described as r  r0   cos 5 sin 5  a cos(  5 ) .
 z  z0    sin 5 cos 5  b sin(  5 ) ( 0    180, 0    36 )(3)

4. Numerical Examples
One here considers the plasma with a volumeaveraged  being   2.7% in the LHD. The
reference field and the sensor signals (126 flux loops
and 440 field sensors) for this condition had been
calculated beforehand using the HINT2 code [2].
Test calculations were made for (i) the axisymmetric
CCS having a 0.075m radius circular cross section,and (ii) the twisted one where the values of a and bin Eq. (2) are 0.15m and 0.375m, respectively.

z (m)CCSabsolute error of Br respectively obtained using theaxisymmetric CCS and the twisted CCS, both ofwhich are on the r-z plane at   18 (the horizontalelongated cross section). The absolute errors in thestochastic region are summarized in Table I.
Based on the reconstructed field, magnetic field line
tracings were carried out. Figure 3 shows the Poincaré
plots on the r-z plane at   18 . The dashed closed
line shows the LCMS for the vacuum field, i.e.,  0% . This is shifted outward when  takes
the nonzero value   2.7% . The reference LCMS
in this case is the solid closed line. The round symbols
show the results of the trace originating at the samestarting point as the reference LCMS for   2.7% ,but based on the reconstructed field obtained using theaxisymmetric CCS. They do not form a sharp closed
surface, however, the round symbols are distributed
almost along the reference LCMS for   2.7% .
The results based on the twisted CCS are shown inFig. 4. The scatter of the plot points is narrower thanin the case of the axisymmetric CCS, i.e., the accuracy
in the reconstruction has been significantly improved.

LCMSr (m)Error (T)

z (m)

0.5

LCMS
(Vacuum)Starting
point

0(a) Axisymmetric CCS-0.5

Reconstructed

LCMS
3Preliminary

3.5r (m)

44.5

0.5

CCS

z (m)

z (m)

Fig.3. Reconstructed LCMS (Axisymmetric CCS)

LCMS
r (m)

Starting
point

0Error (T)(b) Twisted CCS
Fig.2. Distribution of absolute error of Br

LCMS(Vacuum)-0.5

Reconstructed

LCMS
33.5r (m)44.5

Fig.4. Reconstructed LCMS (Twisted CCS)
Table I. Tendency of the absolute errors of Br
Axisymmetric
CCS
43.9%

TwistedCCS
80.3%

24.3%

50.8%

5.9%

12.1%

Max. Error

0.042(T)0.025(T)Ave. Error

0.013(T)0.006(T)  0.01(T)  0.005(T)  0.001(T)Figures 2(a) and 2(b) show the distributions of

5. Conclusion
With the use of the twisted CCS rather than theaxisymmetric CCS, the numerical accuracy in thereconstructed field has been improved. Also, the fieldline tracing indicates the LCMS more precisely.
References[1] M. Itagaki et al.: Plasma Phys. Control. Fusion 53(2011) 105007.
[2] Y. Suzuki et al.: Nucl. Fusion 46 (2006) L19.