23P067-P
Numerical Determination of the Last Closed Magnetic Surface
for the Reconstructed Magnetic Field Profile in the LHD
LHDの逆解析された磁場分布に対する最外殻磁気面の数値的決定法
Masayuki Akazawa1), Gaku Okubo1), Yutaka Matsumoto1), Masafumi Itagaki1),
Ryosuke Seki2) , Yasuhiro Suzuki2) and Kiyomasa Watanabe2)
1)
赤澤眞之 ，大久保岳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 method has been developed to reconstruct themagnetic field profile in the Large Helical Device. The reconstructed field shows acceptable accuracy,
however, the Poincaré plot does not form the last closed magnetic surface (LCMS) clearly. One here
proposes a technique to determine the LCMS numerically. The Poincaré plot is converted to contours of a„quasi‟ magnetic surface function using the expansion of radial basis functions. Introducing the„inside/outside‟ ratio related to the scatters in the Poincaré plot, the most probable contour is extracted as
the LCMS, which agrees well with the reference LCMS.

1. Introduction
The three-dimensional (3-D) Cauchy condition
surface method [1,2] was recently developed toreconstruct the 3-D magnetic field profile outside thenon-axisymmetric plasma from the sensor signals inthe Large Helical Device (LHD). A test calculation
was made for the plasma with a volume-averaged beta
being   2.7% in the LHD. The reference fieldfor this condition had been calculated beforehand
using the HINT2 code [3].

Magnetic field line tracing was carried out using thereconstructed field. Figure 1 shows the Poincaré plotson the r-z plane at   18 . The outer surface of thestochastic region was identified precisely. However, thePoincaré plot does not form the last closed magnetic
surface (LCMS) clearly. The plot points in Fig. 2 aredistributed along the reference LCMS. One here
proposes a method to identify the LCMS numericallyfor the distributed plot points.

2. Numerical Scheme to Identify the LCMS2.1 Quasi magnetic surface function
In the field line tracing, the starting points of thetraces were set askkrstart 4.30  0.01k (k  0, 1, , 40) m, zstart 0.0m ,kand start18 . For convenience, one recognizes ther-coordinate of the starting points as „quasi‟ magneticksurface functions, say,  k  rstart. One here introduces
the radial basis function (RBF) expansion
N

Fig.1. Poincaré plot: Peripheral regionz (m)0.5-0.5

fi  r , z; ri , zi   exp   r  ri    z  zi Startingpoint
ReconstructedLCMS
3i 1with the Gaussian type RBF

LCMS(Vacuum)0k k (r , z )   wi fi (r , z; ri , zi )  rstart(k  1, 2, , K ) (1)

3.5

r (m)

4Fig.2. Poincaré plot: LCMS

4.522

   , (2)2where (ri , zi ) means the center of each RBF. The
weights wi are determined in a least-square manner.
With the use of the RBFs, the Poincaré plot shown inFig. 2 can be converted into a contour map of „quasi‟
surface function as shown in Fig. 3. Contours arefound even outside the LCMS in Fig. 3, however, they
are not the true magnetic surface. One needs toexclude them to determine the true LCMS.

defines the „inside/outside‟ ratio as
2
2R  sinsidesoutside2
inside(4)2outsidewith sand sbeing the scatters calculatedusing Eq. (3) respectively for the points inside andoutside the contour under consideration.

2.2 Definitions of four regions
In addition to the vacuum region and the stochasticregion, one here defines the following two regions asshown in Fig. 4 and Table I. The „dirty‟ region is thedomain sandwiched between the LCMS and the CCS.
The „black‟ region is the region inside the CCS, which
is out of the analysis under consideration

Inside/Outside Ratio

Fig.3. Contours of quasi magnetic surface.

3. Results
Figure 5 shows the variation in „inside/outside‟ ratio
as a function of rstart . The ratio jumps where rstart isreduced to a value smaller than 4.47m. Because of this,one can judge rstart  4.47m to be the most probablevalue that corresponds to the LCMS.
150150100

1004.47m
50

050

4.4
4.5
4.6
r-coordinate of starting point (m)0Fig.5. Variation in „inside/outside‟ ratio

Fig.4. Definitions of the four regions

In the CCS analysis, the reconstructed field in thedirty region has a large error which causes the large
scatter in the Poincaré plot. Note here that in thevicinity of the LCMS the scatter inside the LCMS ismuch larger than that outside the LCMS (see Fig. 2).
This is because the plots inside the LCMS passthrough the dirty region.

Among the contours drawn by the RBF expansionscheme, the contour corresponding to rstart  4.47m
was extracted. This estimated LCMS agrees well withthe reference LCMS as shown in Fig. 6.

Table I. Features of the four regions
RegionVacuumStochastic“Dirty”“Black”

Current
densityNo
WeakStrongStrongAccuracy of thereconstructed field
Acceptable
Fair
Large error
Out of the analysis

Fig.6. Reconstructed LCMS

4. Conclusion
The Poincaré plot based on the reconstructed magnetic
field was converted into contours of a smooth magnetic
surface function using the expansion of the radial basis
functions. Introduction of the „inside/outside‟ ratio
enables one to extract the LCMS from the contour map.
The LCMS thus determined agrees well with thereference LCMS.

2.3 Method of ‘inside/outside’ ratio
One here introduces the „scatter‟ given by
s2 

1 m
 ( j  0 )2 .
m j 1(3)

kHere  0 denotes rstartin Eq. (1), while  j means

the value at the point (rj , z j ) of quasi magnetic
surface function that is taken by the RBF approxik
mation for m points originating at rstart. Next, one

References[1] M. Itagaki et al.: Plasma Phys. Control. Fusion 53(2011) 105007.
[2] G. Okubo et al.: Proc. PLASMA2011, Kanazawa,2011, 23P067-P[3] Y. Suzuki et al.: Nucl. Fusion 46 (2006) L19.