All opinions expressed herein are those of the authors and should not be reproduced, quoted in publications, or
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Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the U.S. Department of Energy.
Published by Fusion Energy Division, Oak Ridge National Laboratory
Building 5700 P.O. Box 2008 Oak Ridge, TN 37831-6169, USA
Editor: James A. Rome Issue 126 June 2010
E-Mail: jar@ornl.gov Phone (865) 482-5643
On the Web at http://www.ornl.gov/sci/fed/stelnews
LHD-type heliotron reactor
design
Based on the physics and engineering achievements of the
Large Helical Device (LHD) project [1], conceptual
designs of an LHD-type heliotron reactor designated the
Force Free Helical Reactor (FFHR) have been advanced.
Starting in 1991 [2,3], these studies have been aimed at
clarifying the critical issues of core plasma physics and
power plant engineering.
One of the critical issues in the design of LHD-type heliotron
reactors is to secure sufficient space for the blanket.
The blanket space is directly coupled to the helical coil
configuration, which affects the core plasma performance
and engineering design constraints. To examine these
interrelations, a conceptual FFHR-1 system with l = 3
(where l is the number of helical coils) was studied. As a
more modern successor, we have performed an optimization
study mainly focused on blanket space, neutronics
performance, a large superconducting magnet system, and
plasma operation, based on an LHD-type concept with l =
2, m = 10, where m is toroidal pitch number. Several ideas
have been proposed to create adequate blanket space. In
the current design study, a similar extension to reactor size
has been considered because recent studies have clarified
that the total plant capital cost does not increase so much
with an increase in the reactor size when the required confinement
improvement is kept constant. Since this enlargement
enables increased fusion power with constant
neutron wall loading, the cost of electricity (COE) can be
reduced. It also enables flexibility in the selection of the
magnetic configuration. Therefore, in the current design
study, a magnetic configuration that is consistent with
LHD high-beta operation has been adopted, resulting in a
further reliable extrapolation of LHD achievements.
Candidates to expand blanket space
In helical systems, the space between the plasma and
external coils is limited. In an LHD-type heliotron system,
this space has its minimum on the poloidal cross-section at
which the plasma exhibits a vertically elongated shape, as
shown in Fig. 1.
Fig. 1. Cross-sectional view of magnetic surface, blanket,
shield, and superconducting coils of an LHD-type heliotron
reactor at the cross-section with minimum blanket space at
the inboard side.
In this issue . . .
LHD-type heliotron reactor design
Conceptual designs for Large Helical Device (LHD)-
type heliotron reactor FFHR have been advanced. A
similar extension of the reactor size (Rc ~17 m)
enables simultaneous achievement of sufficient blanket
space (~1 m) and a magnetic configuration consistent
with LHD high-beta operation. At this size a
design with commercial-scale fusion output (~3 GW)
becomes possible using a foreseeable extrapolation of
the current physics and engineering achievements. 1
Remembering Paul Garabedian
Paul Garabedian passed away on May 13, 2010, after
a long and productive life. ....................................... 5
Stellarator News -2- June 2010
On the other hand, neutronics calculations indicate that a
blanket with a thickness of ~1 m is necessary to achieve a
sufficient tritium breeding ratio (TBR) plus effective
shielding of superconducting coils from fast (>0.1 MeV)
neutrons. Given these constraints, several methods have
been considered to expand the blanket space.
The first one is to control the magnetic surface structure
using external coils. Two key parameters determine the
magnetic surface structure of a LHD-type heliotron system
(LHDHS): the helical pitch parameter γ, and the magnetic
axis position Rax. The helical pitch parameter is defined as
γ = mac/lRc, where ac and Rc are the minor and major radii
of the helical coils, respectively. In the case of a LHDHS (l
= 2, m = 10), the helical pitch parameter corresponds to the
inverse aspect ratio of the helical coil. Thus, the minor
radius of the helical coils decreases with decreasing γ
when the major radius is kept constant. But plasma volume
also decreases and the space between the coil and
plasma increases with decreasing γ. The reduction of γ
also leads to a decrease in the magnetic hoop force.
The second parameter, Rax, can be varied by adjusting the
vertical field strength by controlling the current in the
poloidal coils. Thus, in the design study of FFHR-2, γ =
1.15, which is smaller than in normal LHD operation (γ
=1.25), and an outward-shifted configuration (Rax = Rc)
was selected to expand the blanket space [3]. However, the
blanket space at the inboard side is still insufficient
because of the interference between the first walls and the
ergodic layers of magnetic field lines surrounding the
nested flux surfaces. To overcome this problem, a helical
X-point divertor (HXD) concept [4] has been proposed.
However, influence of ergodic layers on the confinement
of high-energy ions, especially 3.5 MeV alpha particles,
has been found to be important in collisionless orbit simulation
[5]. Therefore, three alternatives without an HXD
have been considered. One is to reduce the shielding thickness
only at the inboard side. Neutronics calculations
show that the use of advanced materials (e.g., WC)
enables the reduction of shield thickness by ~ 0.2 m compared
with the standard (B4C and low-activation ferritic
steel JLF-1) case. The second is to improve the symmetry
of the magnetic surfaces around the magnetic axis, without
shifting the magnetic axis inward, by increasing the current
density at the inboard side of the helical coils while
decreasing it at the outboard side. Modulation of the current
density can be obtained in practice by splitting the
helical coils [6]. The third is a photographic enlargement
of the reactor size. Since the International Stellarator Scaling
(ISS) [7] predicts that the energy confinement time of
a helical system is almost proportional to the plasma volume,
the magnetic field strength can be reduced for a
larger reactor when the required confinement improvement
factor is maintained. Thus the stored magnetic
energy, which is an index of the total plant capital cost,
does not increase so much with reactor size. A larger reactor
enables an increase of fusion power at a constant neutron
wall loading, resulting in a reduction in COE [8]. This
enlargement also enables a flexible selection of magnetic
configuration. Consequently, a design concept based on
LHD high-beta operation has been proposed.
Consideration of the design point
It is quite important to consider the engineering design
feasibility of a large-scale helical coil. The base design for
the superconducting magnet system has been proposed
using the engineering base of the ITER TF coils as a conventional
option. The use of cable-in-conduit conductor
(CICC) with Nb3Al strands has been considered. A stored
magnetic energy of 120–140 GJ can be achieved with a
small extension of the ITER technology [9], and the
achievable maximum value is expected to be 160 GJ. Neutronics
calculations indicate that a blanket system with a
thickness of ~1 m enables a net TBR above 1.05 for the
standard design of Flibe (LiF + BeF2) + Be/JLF-1, as well
as a long lifetime for the spectral-shifter and tritium breeding
(STB) blanket [3]. In this concept, the average neutron
wall loading needs to be reduced to < 1.5 MW/m2. This
1 m thickness also keeps nuclear heating in the superconducting
coils below the cryogenic power limit (less than
1% of fusion power). The fast neutron flux to the superconducting
coil is also suppressed to an acceptable level:
neutron fluence < 1022 n/m2 for 30 years of operation [8].
In order to find a feasible design window in a quantitative
way, parametric scans were carried out over a wide design
space by a newly developed system design code for heliotron
reactors. This system code can deal with the actual
geometry of helical and poloidal coils in the calculation of
stored magnetic energy and blanket space. Plasma performance
is estimated by a simple volume-averaged (0-D)
power balance model. In this case, an inward-shifted configuration
(Rax/Rc = 3.6/3.9) and γ = 1.2 were adopted.
This configuration is the same one as for LHD high-beta
operation (volume-averaged beta ~5% via diamagnetic
measurement) [10]. A coil system that basically has a similar
shape to that of LHD was considered. However, the
minor radius of helical coils was changed to satisfy γ =
1.2. The width-to-height ratio of helical coils was also set
to 2 (larger than that of LHD) to expand the blanket space.
The number of pairs of poloidal coils is also reduced from
3 (for LHD) to 2 in order to secure large spaces for maintenance.
To calculate plasma confinement properties, physics
parameters related to the magnetic configuration are
needed. It was found that the magnetic surface structure
including the ergodized layers depends strongly on the
geometry and the current of not only the helical coils, but
also the poloidal coils. Therefore, the system code utilizes
Stellarator News -3- June 2010
the parameters obtained from detailed field line trace calculations
of the vacuum equilibrium.
Figure 2 shows the relation between the stored magnetic
energy Wmag and HLHD (the required confinement
improvement factor H relative to LHD, which is 0.93
times ISS04v3 scaling [7]) for design points with fusion
power ~3 GW. The design point that satisfies both Wmag <
160 GJ and average neutron wall load Γn < 1.5 MW/m2 is
found with a volume-average beta value > 5.5% and HLHD
> 1.3. Figure 3 shows that the design point with a blanket
space ~ 1 m can be found under the same conditions.
Therefore, this point (shown as red star in both Figs. 2 and
3) was selected as the candidate. The main design parameters
are listed in Table 1.
Table 1. Main design parameters of the selected
design point
As mentioned above, the calculation of the system design
code is based on parameters related to the magnetic surface
structure of vacuum equilibrium. On the other hand, a
large Shafranov shift has been observed in LHD high-beta
discharges. Shrinking of the nested surface volume due to
ergodization of peripheral region is also predicted by
numerical simulations with the HINT code [11]. To study
this, finite-beta equilibrium calculations were carried out
using the VMEC code to examine the self-consistency of
the design. These calculations showed that almost the
same plasma volume and plasma stored energy as estimated
by the system design code can be obtained by adding
adequate vertical field which can be done by changing
the current of poloidal coils [12].
Consequently, it is concluded that the design of an LHDtype
heliotron reactor with sufficient fusion output for
commercial operation (~3 GW) is possible with a foreseeable
extrapolation of the current physics and engineering
achievements.
Conclusion and future work
Based on the physics and engineering achievements of the
LHD project, conceptual designs for a LHD-type heliotron
reactor FFHR have been proposed. In the current design
study, a similar extension of reactor size has been considered
as a method to secure sufficient blanket space. This
enlargement enables commercial-scale (~ 3 GW) fusion
output with a low average neutron wall load (< 1.5 MW/
m2), which enables a long-life blanket concept. It also
leads to flexibility in selection of the magnetic configuration,
resulting in the adoption of a magnetic configuration
consistent with LHD high-beta operation. Consequently,
the design of LHD-type helical fusion reactors becomes
Fig. 2. The stored magnetic energy vs the required energy
confinement improvement factor relative to present LHD
experiments for design points with fusion power of ~3 GW.
The candidate design point is shown as a red star.
Coil major/minor radius Rc/ac [m] 17.0 / 4.08
Plasma major/minor radius Rp/ap [m] 15.7 / 2.50
Plasma volume Vp [m3] 1927
Average toroidal field on axis Bax [T] 5.0
Volume-averaged beta value <β> [%] 5.5
Confinement improvement factor HISS04v3 1.2
Averaged neutron wall load Γn [MW/m2] 1.5
Blanket space Δ [m] 0.985
Stored magnetic energy Wmag [GJ] 160
Fig. 3. The blanket space vs the required energy confinement
improvement factor relative to present LHD experiments
for design points with fusion power ~3 GW and
volume-averaged beta of ~5.5%. The candidate design
point is shown as a red star.
Stellarator News -4- June 2010
possible with a foreseeable extrapolation of the current
physics and engineering achievements.
Several physics and engineering issues remain to be
solved. One of the most critical is the development of the
technology for winding a large-scale continuous helical
coil. Optimization of the coil system, including the supporting
structure and a cooling method, is required. The
detailed design of a three-dimensional divertor is also
important. Effective fueling and heating of the core
plasma to realize steady-state sustainment of a desirable
pressure profile is another big issue. Continuous efforts on
both experimental and numerical research are enthusiastically
expected.
Acknowledgment
This work has been performed with the support from the
members of the FFHR design group in National Institute
for Fusion Science.
Akio Sagara and Takuya Goto
Fusion System Research Division
National Institute for Fusion Science
322-6 Oroshi-cho, Toki, Gifu 509-5292 Japan
E-mail: sagara.akio@LHD.nifs.ac.jp
References
[1] A. Komori et al., “Development of net-current free heliotron
plasmas in the Large Helical Device,” Nucl. Fusion
49 (2009) 104015.
[2] A. Sagara et al., “Improved structure and long-life blanket
concepts for heliotron reactors,” Nucl. Fusion 45
(2005) 258–263.
[3] A. Sagara et al., “Conceptual design activities and key
issues on LHD-type reactor FFHR,” Fusion Eng. Des.
81 (2006) 2703–2721.
[4] T. Morisaki et al., “Numerical study of magnetic field
configuration for FFHR from a viewpoint of divertor
and edge field structure,” Fusion Eng. Des. 81 (2006)
2749–2754.
[5] T. Watanabe and A. Sagara, “Alpha Particle Loss Fraction
in the FFHR,” NIFS Annual Report, 2006–2007.
[6] N. Yanagi et al., “Split and Segmented-Type Helical
Coils for the Heliotron Fusion Energy Reactor,” Plasma
Fusion Res. 5 (2010) S1026.
[7] H. Yamada et al., “Confinement Study of Net-Current
Free Toroidal Plasmas Based on Extended International
Stellarator Database,” Nucl. Fusion 45 (2005) 1684–
1693.
[8] A. Sagara et al., “Optimization activities on design
studies of LHD-type reactor FFHR,” Fusion Eng. Des.
83 (2008) 1690–1695.
[9] S. Imagawa et al., “Concept of magnet systems for
LHD-type reactor,” Nucl. Fusion 49 (2009) 075017.
[10] K. Y. Watanabe et al., “Recent study of the high performance
confinement and the high beta plasmas on the
Large Helical Devices,” Proc. 18th International Toki
Conference (ITC18), Toki, Japan, December 9–12,
2008, I-04.
[11] K. Y. Watanabe et al., “Change of plasma boundaries
due to beta in heliotron plasma with helical divertor
configuration,” Plasma Phys. Control. Fusion 49 (2007)
605–618.
[12] T. Goto et al., “Core Plasma Design of a Heliotron Reactor,”
to be published in Contributions to Plasma Physics.
Stellarator News -5- June 2010
Remembering
Paul Garabedian
Paul R. Garabedian, a leader in the field of computational
science, passed away on May 13, 2010, at his home in
Manhattan at the age of 82 after a long battle with cancer.
During the course of 60 years of research on the faculties
at Stanford and New York University (NYU) he maintained
an active research program in magnetohydrodynamics
(MHD) and computational fluid dynamics, and
was one of the premier applied mathematicians of his
time.
Paul (shown in Fig. 1) was born in Cincinnati, Ohio, in
1927. He grew up in an academic family, and did not
attend school before setting off for college; his formal education
first began as a precocious undergraduate at Brown
University, from which he graduated in 1946 [1]. His graduate
training was in pure mathematics at Harvard University,
where he received his PhD in 1948 in the field of
complex analysis [2]. He spent a year teaching at the University
of California, Berkeley, followed by nine years on
the mathematics faculty at Stanford. During this period
Paul made outstanding contributions to the theory of partial
differential equations and to problems in the theory of
functions of a complex variable. A highlight of his work at
this time was his celebrated proof of a special case of an
outstanding problem in complex analysis known as the
“Bieberbach Conjecture,” which he published with Max
Schiffer in 1955 [3]. He also served as a Scientific Liaison
Officer for the Office of Naval Research in London and
led Stanford’s research program in hydrodynamic free
boundary problems. A noteworthy example is his analysis
of the bow shock generated by a blunt body traveling at
supersonic velocities [4]. The fluid passing through the
shock undergoes substantial heating, and it is necessary to
characterize the degree of heating to ensure that the body
can sustain these extreme conditions without mechanical
failure. Computing the geometry of the shock wave and
fluid flow past a given body is an extremely challenging
nonlinear problem in partial differential equations. Paul’s
ingenious solution was to prescribe the shape of the bow
wave in advance, and then determine the shape of the body
that generated the shock wave and flow field using
advanced techniques from the theory of complex variables.
He was aided in this work by his first wife, Gladys,
who programmed the numerical procedure.
Paul joined the Courant Institute of Mathematical Sciences
at NYU in 1959. A few years after his arrival at NYU,
Paul’s book, Partial Differential Equations, was first published
[5]. This book is a unique blend of pure and applied
mathematics and includes discussions of free boundary
problems arising in plasma physics. Paul advised 27 doctoral
students in all [6], in addition to a number of master’s
students and postdoctoral fellows, and was the Director of
the Division of Computational Fluid Dynamics at the Courant
Institute for 32 years (1978–2010).
During the 1960s and 1970s, Paul and his students and coworkers
developed computer codes to study supercritical
wing technology as part of the development of aircraft
designed to fly near the speed of sound. The basic principle
involved concerns the suppression of boundary layer
separation by shifting shock waves that occur on the wing
toward the trailing edge and making the shock waves as
weak as possible. The resulting wing increases lift, fuel
efficiency, and the speed of aircraft, and the ideas that
flowed from this work influence much of commercial aircraft
design today. A NASA Award and a NASA Certificate
of Recognition acknowledged this research, which
resulted in three books [7,8,9] with longtime collaborator
Frances Bauer (whom he met during his undergraduate
studies at Brown, while she was a graduate student), former
Ph.D. student David Korn, and colleague Antony
Jameson.
Paul started working on problems related to fusion energy
in the 1970s, using the techniques he had developed for
problems in classical fluid dynamics and extending them
to study the magnetic confinement problem. He first
worked on free boundary models to understand plasma
physics experiments carried out at Los Alamos National
Laboratory and the Max-Planck Institute for Plasma Physics
in Germany. Over the years the sophistication of the
plasma modeling tools in Paul’s research group grew
steadily [10,11,12], resulting in a suite of computer codes
used to study plasma equilibrium, transport, and stability.
The NSTAB equilibrium code [13] employs a divergence-
Fig. 1. Paul R. Garabedian, 8/2/1927–5/13/2010.
Stellarator News -6- June 2010
free representation of the magnetic field that minimizes
the magnetic energy using a combination of spectral techniques
and finite differences. The stability of the nonlinear
equilibrium states is assessed by applying appropriate perturbations
to the system and determining whether they
lead to multiple steady states that indicate the occurrence
of bifurcated solutions. Transport properties of the ions
and electrons are estimated by following the orbits of test
particles that track the magnetic field lines between particle
collisions while maintaining a quasineutrality condition
[14]. The codes have been used to design
quasiaxisymmetric stellarators [15] as possible candidates
for DEMO experiments that may eventually be performed
after the completion of the planned ITER experiments (see
Fig. 2). Paul’s computations suggest that his quasisymmetric
designs have desirable transport properties, comparable
to those in axisymmetric tokamaks, while enjoying the
increased stability of stellarators by avoiding the need for
a large toroidal current [16,17]. Paul’s many contributions
in MHD were recognized by the American Physical Society's
Division of Plasma Physics with his election as an
APS Fellow in 2004.
Paul actively pursued his research until his death [19].
Among his many honors, Paul received the Birkhoff Prize
[awarded by the American Mathematical Society (AMS)
and the Society for Industrial and Applied Mathematics
(SIAM)] for an outstanding contribution to “applied mathematics
in the highest and broadest sense,” the Theodore
von Karman Prize awarded by SIAM, the National Academy
of Sciences Award in Applied Mathematics and
Numerical Analysis, and the Boris Pregal Award from the
New York Academy of Sciences. He was a fellow of
SIAM, and a member of the American Mathematical Society,
the American Physical Society, the National Academy
of Sciences, and the American Academy of Arts & Sciences.
Although only one-quarter Armenian, he was proud
of his heritage, and a hero to the Armenian mathematical
community. A scientific conference in his honor will be
held at the Courant Institute during the 2010–2011 academic
year.
Paul’s marriage to Gladys Rappaport ended in divorce.
Paul is survived by his wife, Lynnel; daughters, Emily and
Cathy; two grandchildren, and a sister.
Acknowledgments
Contributions by April Bacon, Frances Bauer, Leslie
Greengard, Peter Lax, and Harold Weitzner are gratefully
recognized.
Geoffrey B. McFadden
NIST
100 Bureau Drive, Stop 8910
Gaithersburg, MD 20899-8910
E-mail: geoffrey.mcfadden@nist.gov
References:
[1] For an informative interview with Paul Garabedian, see
the archive on the History of Numerical Analysis and
Scientific Computing at the Society for Industrial and
Applied Mathematics web site at
http://history.siam.org/oralhistories/garabedian.html
[2] P. R. Garabedian, “Schwarz’s lemma and the Szego
kernel function,” Trans. Am. Math. Soc. 67, 1–35
(1949).
[3] P. Garabedian and M. Schiffer, “A proof of the Bieberbach
conjecture for the fourth coefficient,” J. Rat.
Mech. Anal. 4, 427–465, (1955).
[4] P. R. Garabedian, “Numerical construction of detached
shock waves,” J. Math. Phys. 36, 192–205 (1957).
[5] P. R. Garabedian, Partial Differential Equations, (Wiley,
New York, 1964).
[6] For a listing of students, see the Mathematics Genealogy
Project at http://genealogy.math.ndsu.nodak.edu/
id.php?id=8298&fChrono=1.
[7] F. Bauer, P. Garabedian, and D. Korn, “A Theory of Supercritical
Wing Sections, with Computer Programs
and Examples,” Lecture Notes in Economics and Mathematical
Systems 66 (Springer-Verlag, New York,
1972).
[8] F. Bauer, P. Garabedian, D. Korn, and A. Jameson. “Supercritical
Wing Sections II,” Lecture Notes in Economics
and Mathematical Systems 108 (Springer-
Verlag, New York, 1975).
[9] F. Bauer, P. Garabedian, and D. Korn, “Supercritical
Wing Sections III,” Lecture Notes in Economics and
Mathematical Systems 150 (Springer-Verlag, New
York, 1977).
[10] F. Bauer, O. Betancourt, and P. Garabedian, “A Computational
Method in Plasma Physics,” Springer Series
Fig. 2. A quasiaxisymmetric stellarator design with two field
periods [18]. The plasma surface is shaded to indicate the
magnetic field strength, and the modular coils are separated
enough so that they can be constructed at reactor
sizes.
Stellarator News -7- June 2010
in Computational Physics (Springer-Verlag, New York,
1978).
[11] F. Bauer, O. Betancourt, and P. Garabedian, “Magnetohydrodynamic
Equilibrium and Stability of Stellarators”
(Springer-Verlag, New York, 1984).
[12] F. Bauer, O. Betancourt, P. Garabedian, and M. Wakatani,
“The BETA Equilibrium, Stability and Transport
Codes: Applications to the Design of Stellarators” Academic
Press, Boston, 1987.
[13] M. Taylor, “A high performance spectral code for nonlinear
MHD stability,” J. Comput. Phys. 110, 407–418
(1994).
[14] N. Kuhl, “Monte Carlo simulation of transport,” J.
Comput. Phys. 129, 170–180 (1996).
[15] P. R. Garabedian, “Stellarators with the magnetic symmetry
of a tokamak,” Phys. Plasmas 3, 2483–2485
(1996).
[16] P. Garabedian and L. Ku, “Reactors with stellarator stability
and tokamak transport,” Fusion Sci. Technol. 47,
400–405 (2005).
[17] P. Garabedian, “Theory of stellarators and tokamaks in
three dimensions,” Stellarator News Issue 111, ORNL,
Oct. 2007.
[18] P. R. Garabedian and G. B. McFadden, “Design of the
DEMO fusion reactor following ITER,” J. Res. Natl.
Inst. Standards Technol. 114, 229–236 (2009).
[19] P. R. Garabedian and G. B. McFadden, “The DEMO
quasisymmetric stellarator,” Energies 3, 277–284
(2010).