Proceedings of ITC/ISHW2007

Applications of HINT2 code to stellarator/heliotron
Yasuhiro Suzuki, N. Nakajima, J. Geigera, S. Yamamotob, K. Y. Watanabe, S. Sakakibara,
A. Wellera, A. Wernera, M. Drevlaka, Y. Nakamurac, W. Kernbichlerd
V. V. Nemovd and J. NührenbergaaNational Institute for Fusion Science,322-6 Oroshi-cho, Toki 509-5292, Japan
Max-Planck-Institut für Plasmaphysik, IPP-Euratom Assoc. Greifswald D-17491, GermanybKyoto University Pioneering Research Unit for Next Generation, Kyoto University,Gokasho Uji 611-0011, JapancGraduate School of Energy Science, Kyoto University, Gokasho, Uji 611-0011, JapandAssociation EURATOM-ÖAW, TU Graz, Graz 8010, Austria
To investigate the nature of three-dimensional (3D) MHD equilibrium and interpret the experiments, anequilibrium calculation code without the assumption of nested flux surfaces, HINT2 is developing. This code
applied many stellarator/ heliotron devices and properties of MHD equilibrium were clarified. Some applications ofHINT2 are shown and future plans are discussed.
Keywords: MHD, equilibrium, stochasticity, HINT2

1. Inroduction
The MHD equilibrium is the basis of both mosttheoretical considerations and physics interpretation of
the experimental results. As a standard technique tocalculate the 3D MHD equilibrium, inverse equilibrium
solver VMEC [1], assuming the existence of perfectnested flux surfaces, is widely used. In such a technique,a flux coordinate system is directly constructed so as tosatisfy the force balance or MHD equilibrium equation:
𝐽 × 𝐵 = 𝛻𝑝. For the low- equilibrium, if the vacuum
magnetic field sustains clear flux surfaces, the standard
technique is acceptable. However, by nature, the 3D
MHD equilibrium may exhibit magnetic islands andstochastic regions in the plasma because of the absence of
toroidal symmetry. For the high- equilibrium, thedegradation of flux surfaces by the finite  effect has tobe determined, so that the standard technique based on
the nested flux surfaces could not be directly applicableto them and other techniques are required such as
HINT/HINT2 [2,3] and PIES [4] codes.
The HINT2 code is one of such solvers, where arelaxation method based on the dissipative MHDequations of the magnetic field. A special feature ofHINT2 is the coordinates system, which uses a‘non-orthogonal rotating helical coordinate’. Since thehelical coordinate system is Eulerian, HINT2 can treat
magnetic islands and stochastic regions in plasmas.
The HINT2 code has been applied to the study ofMHD equilibrium in many helical configurations, inauthor’s e-mail: ysuzukil@LHD.nifs.ac.jp

order to clarify the properties inherent to 3D MHD
equilibrium in various types of helical systems, which areLHD, W-7AS, W-7X, Heliotron-J. In this study, we show
applications of HINT2 for high- stellarator/heliotron
researches.

2. Application of LHD
In order to understand how about HINT2 shows thefinite- MHD equilibrium, several quantities of a finite-
equilibrium with ~3.5% for an inward shifted (Rax=3.6,
=1.254, BQ=100%) are shown in fig.1; (a) Puncture mapof field lines, (b) rotational transform  and distance along
the magnetic field LC, (c) a profile of the normalized
plasma pressure p/p0 (p0 is the pressure at the magneticaxis), (d) contour lines of p/p0, and (e) contour lines ofsquare of the local residual force. Vertical lines in fig.1(a)and arrows in fig.1(b) and (c) indicate positions of thevacuum LCFS. In this finite  equilibrium, the initialpressure profile is set to p = p0 1 − s (1 − s 4 ), where sis the normalized toroidal flux defined as s = /edge. Infig.1(a), a region colored with blue corresponds to theregion with clear flux surfaces and a region with red(green) corresponds to Puncture map of stochastic
magnetic field lines started from 4.419 < R < 4.55 (R >4.55). For ~3.5%, a considerable part in the plasma
periphery is ergodized. The position of the LCFS is much
inside the vacuum LCFS. However, from the comparison
among fig.1(a), (c) and (d), it is understood that theplasma pressure spreads over the stochastic region andthe pressure gradient ∇p exists in the stochastic region.

Proceedings of ITC/ISHW2007
This fact comes from the relaxation method of thepressure used in HINT2. The condition for a fixed B iswritten asi+1p=p=

F=L indlℱpi−L inB
L in dl−L in B(1)

1:for 𝐿𝐶 ≥ 𝐿𝑖𝑛
0:for 𝐿𝐶 < 𝐿𝑖𝑛

where i means a step number of iterations, LC is theconnection length of a magnetic field line starting eachgrid point (LC is finite for open magnetic field lines), andLin is the length along a magnetic field line followed fromeach grid point and prescribed as an input parameter. If
the magnetic field line cannot be traced up to theprescribed value or the connection length LC is shorter
than Lin (LC < Lin), the averaged plasma pressure is set tozero. That is, the distribution of relaxed plasma pressure
depends on the relative magnitude between theconnection length LC and the parameter Lin prescribing
the length of the field line trace. In other words, although
HINT or HINT2 does not need the explicit boundary
condition between plasma and the vacuum, the ratio of
the connection length LC to the length for pressure
average Lin plays a role in controlling the distribution of
the plasma pressure near the periphery. In fig.1, Lin is set
to 30m, which corresponds to the length connecting theoutside to the inside of the torus near the magnetic axis. Itis seen from the comparison between fig.1(a) and (b) thatthe structure of magnetic field lines becomes stochastic
outside the region with ~1, because magnetic islands
corresponding to some rational surfaces (i.e. n/m = 10/10,10/9, 10/8, 10/7, 10/6, ...) overlap. Although these fieldlines are ergodized some magnetic field lines have longconnection lengths LC exceeding Lin as shown in fig.1(b),leading to the existence of the pressure and pressure
gradient in the stochastic region. As mentioned above, inthe case of HINT or HINT2 not assuming the existence ofthe nested flux surfaces, the relaxation method of thepressure profile mainly plays the role of determining theperipheral pressure profile instead of the plasma-vacuum
boundary condition.

3. Application of Wendelstein 7-AS
The standard divertor configuration (SDC) for
HDH-discharges has a boundary -value of the vacuum
field of 5/9, where 9 natural islands form a clear
separatrix. Additionally, correction coils located inside
the vacuum vessel are used to increase the island size andkeep a larger distance to the divertor plates. Thecorrugated separatrix boundary of such configurations
can not be treated by free-boundary VMEC and thus

Fig.1 Various quantities of a finite- equilibrium with
~3.5% on the horizontally elongated poloidal cross sectionin the inward-shifted LHD configuration: (a) Puncture map ofmagnetic field lines, (b) the rotational transform  and thedistance along the magnetic field LC, (c) profile of thenormalized plasma pressure, p/p0, along the equatorial plane,(d) contour lines of p/p0, and (e) contour lines of F 2 =2
∇p − J × B .

equilibrium calculations were not available up to now. We
compare the influence of different pressure profile forms
on the configuration. The initial profiles used in thisstudy are shown in Fig.2. They were choosen forvariability and to have a zero pressure gradient at theboundary. Figure 3 shows how the initially flat -profile
builds up shear by a central increase due to theShafranov-shift. Thus, the 5/9-resonance appears in the-profile and islands develop whose position depends onthe  and the underlying profile form. The broader profile(HDH-2) leads to larger islands closer to the boundary.

Proceedings of ITC/ISHW2007

4. Application of Wendelstein 7-X

Fig2. Pressure profiles used in this study as the function ofnormalized toroidal flux s.

Fig.3 -profiles of the vacuum and finite- calculations withdifferent pressure profiles.

Fig.4 Axis position and separatrix depending on  (left) andpressure profiles (right).

The identations of the separatrix formed by the 5/9
boundary islands increase with  as seen in Fig.3. We
also note that the x-points move poloidally which wasalready observed in the study performed with the old
HINT-code [2]. Both effects are due to the poloidal
expansion of the flux tubes due to the Shafranov-shift.
Additionally, the island fans are moving radially on thetarget plates of the divertor which is also seen in theexperiment. We note, that the separatrix structure seems
not to depend strongly on the profile changes if theenergy content is kept constant which is approximately
fulfilled for the two profiles compared here.

Figure 4 shows the Large-Volume configuration forvacuum and for ~4.36%. Puncture plots of magnetic
field lines are plotted at =0 plane. The profile of therotational transform  of this configuration has a low
shear, clear flux surfaces in the core and clear 5/5 island
structures in the edge region of the vacuum field. Thus,
this configuration is sensitive to finite- effects. The
configuration had already been investigated with PIES [5].
Our study finds the following results; (i) the Shafranov
shift and the change of  is very small, (ii) the edge region
is ergodized and the last closed flux surface (LCFS)
retreats, that is, the plasma volume decreases.
In figs, green dots indicate the LCFS. The initialpressure distribution has been set to in this study, where sis the normalized toroidal flux /edge.conforming to theprofile used in Ref [5]. For b~4.36%, the edge region isergodized and 5/5 islands evolve. Thus, the shape of theLCFS is changing according to the change of the edge
region. However, clear flux surfaces are maintained
inside 5/5 islands and other large resonances dot notappear.
We compare these results to those of PIES. In Ref[5], The LCFS of HINT2 is larger than the one of PIES.
In both cases, 10/11 rational surfaces appear inside theLCFS. For PIES, the LCFS locates just outside 10/11
rational surface, whereas for HINT2, some more closed
flux surfaces exist outside 10/11 rational surface. The
LCFS of HINT2 locates outside 15/16 rational surface. Inorder to do a detailed comparison, the plasma volume
inside the LCFS, VLCFS, is studied. The volume VLCFS forthe vacuum is about 33.6m3 (see Ref [5]). In the HINT2
calculation, VLCFS is decreasing to ~ 31.6m3, a volume
reduction of ~ 6%. However, PIES sees a reduction ofVLCFS of ~ 21%, being much larger than the one ofHINT2. This is a significant difference in the HINT2 andPIES results. There are some possibility explaining thisdifference. One is the numerical scheme of both codes.
HINT2 is based on the relaxation method andcalculations are done on a rectangular grid without theassumption of nested flux surfaces. On the other hand,
PIES is an iterative solver on a quasi-flux coordinates
system. This suggests differences in the effect of thefinite pressure in the ergodic region. Another effectconcerns the evolution of the pressure profile. Since
HINT2 is based on the relaxation method, the pressure
profile evolves during the relaxation process. The relaxed
profile is almost the same as the initial profile but
nevertheless slightly different. In order to confirm such
effects more clearly, further studies and benchmarking arenecessary.

Proceedings of ITC/ISHW2007

Fig.4 Puncture map of magnetic file lines for the vacuum field (left) and finite- equilibrium (right).

5. Application of Heliotron J
Heliotron J device is an L=1/M=4 helical-axis
heliotron with flat -profile and deep magnetic well for thevacuum field. Since the magnetic shear is low, theconfiguration is very sensitive to the pressure-induced
perturbation and net toroidal currents. In experiments, thechange of magnetic configuration due to the evolution ofbeam driven currents was observed and it is suggested thechange of the magnetic configuration might be the trigger
of a spontaneous transition [6]. In order to study effects ofnet toroidal currents to the configuration, the equilibrium
including the net toroidal current is studied for the standard
configuration. For the direction of toroidal currents ispositive, the edge field structure is changed by appearing
large magnetic islands. This affects the confinement
through the plasma volume. However, for the negative
direction, clear flux surfaces are kept by disappearing
dangerous resonances.

6. Summary
HINT2 is useful and powerful tool to study 3D MHD
equilibrium and it applies to many configurations.
Future plans of HINT2 are twofold. First direction isthe further improvement of the code. One example isshown. A critical issue of stellarator/heliotron researches is
the stochastic property of edge field lines due to thefinite-. Since HINT2 does not assume nested flux

surfaces, the code can calculate the equilibrium withstochastic field. However, in the stochastic region, theplasma pressure is the flux surface quantity and the radial
force is not balanced. In order to treat the plasma pressurein the stochastic region consistently, the anisotropicpressure is necessary. Second direction is the application tointerpret the experiments.

References
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