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arXiv:2201.12547v2 [physics.plasm-ph] 3 Feb 2022
Existence of an optimized stellarator with simple coils
Guodong Yu, Zhichen Feng, Peiyou Jiang, and GuoYong Fu∗
Institute for Fusion Theory and Simulation and Department of physics, Zhejiang University, Hangzhou 310027, China
An optimized compact stellarator with four simple coils is obtained from direct optimization
via coil shape. The new stellarator consists of two interlocking coils and two vertical field coils
similar to those of the Columbia Non-neutral Torus (CNT)[Pedersen et al. Phys. Rev. Lett. 88,
205002 (2002)]. The optimized configuration has global magnetic well and a low helical ripple level
comparable to that of Wendelstein 7-X (W7-X)[Wolf et al. Nucl. Fusion 57, 102020 (2017)]. The
two interlocking coils have a smooth three-dimensional shape much simpler than those of advanced
stellarators such as W7-X. This result opens up possibilities of future stellarator reactors with
simplified coils.
The two main approaches of magnetic fusion energy
(MCF) are tokamak’s and stellarator’s. Tokamak is cur-
rently the dominant approach with advantages of axisym-
metric geometry and achieved plasma parameters signif-
icantly better than those of other MCF devices. How-
ever, stellarators have recently enjoyed a renaissance as
recent results of the advanced stellarator Wendelstein 7-
X (W7-X)[1] demonstrated the reduced neoclassical en-
ergy transport[2][3]. It is expected that plasma perfor-
mance of W7-X can reach a level comparable to that of an
equivalent tokamak in next few years. Stellarators have
advantages of naturally steady state operations because
their magnetic fields are mainly generated by external
coils and plasma current is not needed for confinement
and is usually quite small. The harmful current-driven
instabilities such as disruptions in tokamaks are absent
in stellarators. The confinement properties of stellarator
plasmas are largely determined by external coils. The
recent success of W7-X demonstrates that it is possible
to build the optimized three dimensional (3D) coils to
such high precisions that the designed plasma confine-
ment performance can be achieved as predicted in actual
experiments. It has been argued recently that the stel-
larator approach provides the fastest track to the realiza-
tion of fusion energy because favorable plasma confine-
ment properties can be designed, realized and controlled
almost fully by external 3D coils[4].
The advanced stellarators such as W7-X are usually de-
signed for improved neoclassical confinement and MHD
stability by two stage approaches. First, the shape of
plasma boundary is varied to obtain optimized proper-
ties. Second, the optimized boundary shape is realized
by designing the shapes of 3D coils. Unfortunately, the
resulting 3D coils are usually quite complex and are diffi-
cult and costly to build. This is evidenced by the long de-
lay of W7-X project [5] and the cancellation of the NCSX
project [6] due to difficulties in engineering and building
of 3D coils. Therefore it is necessary to simplify the 3D
coils for stellarator to become an economical platform for
fusion reactors. For this reason, recently, coil simplifica-
tion of stellarator has been a subject of intensive studies
including reducing coil complexity in the optimization[7]
[8][9] as well as using permanent magnets to replace 3D
coils[10][11][12].
In this work, we explore the possibilities of optimized
stellarators with simplified coils. We use a direct opti-
mization method of varying coil shape to optimize both
plasma confinement and MHD stability. In this way we
can effectively control the complexity of coils while opti-
mizing the plasma properties. The details of optimization
method will be given later. Following our recent work[13],
we choose the coil topology of the Columbia Non-neutral
Torus (CNT)[14] for our optimization space. CNT con-
sists of two circular interlocking(IL) coils and two circular
vertical field (VF) coils. It is arguably the simplest com-
pact stellarator ever built and successfully operated[14].
Here we carry out a global optimization of both neo-
classical confinement and MHD stability by varying the
3D shape of the two IL coils. An initial phase of this
work has been done recently by targeting only the neo-
classical transport. The results showed that the effective
helical ripple level was reduced by an order of magni-
tude as compared to that of CNT indicating significant
improvement in neoclassical confinement[13]. This was
realized by two planar IL coils with elliptical shape. In
present work, a much broader parameter space of coil
shape is explored allowing 3D shape of the two IL coils.
As a result, an optimized Compact Stellarator with Sim-
ple Coils(CSSC), to be called CSSC, has been discovered.
The CSSC has both global magnetic well and a low level
of helical ripple comparable to that of W7-X. This break-
through opens up possibilities of stellarator reactors with
simplified coils. The design of CSSC can be used as basis
for a low-cost experiment to study the physics of compact
stellarators. The design could also be used as a candidate
stellarator for electron-positron plasma experiments[14].
We now describe the configuration specification and
physics properties of CSSC in details. In our optimiza-
tion, the radius of the two circular VF coils is varied
while the distance between them is fixed. The shapes of
the two IL coils are varied and are described by following
Fourier representation[15]
2
x=xc,0+
nf
X
n=1
xc,n cos(nt),(1)
y=
nf
X
n=1
ys,n sin(nt),(2)
z=
nf
X
n=1
zs,n sin(nt),(3)
where (x, y, z ) are Cartesian coordinates of line coils, t
is an angle variable ranges [0,2π] and nis the harmonic
number. The total number of Fourier coefficients for each
IL coil is 3×nf+1 with nfbeing the maximum harmonic
number. In order to maintain two periods of flux sur-
faces, the two IL coils are constrained to have same shape.
Furthermore, nf= 3 is chosen in this work. This choice
of nf= 3 is a compromise between allowing enough de-
gree of freedom while constraining the coil complexity.
Finally, taking into account the current ratio between IL
coils and VF coils, the optimization parameter space has
a total of 12 degree of freedom.
Table I shows the Fourier coefficients of the optimized
configuration CSSC (4th column) along with the opti-
mization ranges for each Fourier coefficient (3rd column).
Fig. 1 shows the 3D schematic of coils and the last
closed flux surface (LCFS) of CSSC. We observe that the
two IL coils have a smooth shape with modest 3D varia-
tion. The averaged radius of the IL coils is 1.11m. The
radius of vertical field coils have a radius of 1.85m. The
distance between the two vertical field coils is 1.54m. The
current ratio of IL coils and VF coils is 1.323. The minor
and major radius of LCFS is 0.15mand 0.58mrespec-
tively, corresponding to a low aspect ratio of R/a = 3.87.
The volume of LCFS is 0.27m3. The Poincare plots of
vacuum magnetic surfaces are shown in Fig. 2 at toroidal
angels 0◦and 90◦respectively. The configuration has
Parameter Range Results
IL coil
xc,00.3∼0.6 0.509
xc,10.6∼1.2 1.051
xc,2-0.3∼0.3 0.227
xc,3-0.1∼0.1 -2.622e-2
ys,10.6∼1.2 0.806
ys,2-0.3∼0.3 0.162
ys,3-0.1∼0.1 -6.257e-2
zs,1-0.4∼0.8 0.667
zs,2-0.2∼0.2 -3.87e-2
zs,3-0.1∼0.1 -6.052e-2
PF coil ys,1=xc,11.0∼2.0 1.850
zc,00.77 0.770
Current ratio IIL /IP F 1∼3 1.323
TABLE I. The optimization ranges of coil Fourier coefficients
and coil current ratio are in the third column. The optimized
results are in the fourth column.
FIG. 1. The optimized stellarator with four simple coils: the
gold color denotes the two inner interlocked 3D coils located
between two circular vertical field coils; the purple color de-
notes the last closed flux surface.
good magnetic flux surfaces except for a small magnetic
island at the ι= 0.25 surface. The rotational transform
profile is shown in Fig. 3 and is seen to vary from 0.2 at
the magnetic axis to 0.28 at LCFS with modest magnetic
shear.
0.4 0.5 0.6 0.7 0.8 0.9
R(m)
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Z(m)
(a) φ= 0◦
0.3 0.4 0.5 0.6 0.7 0.8
-0.2
-0.1
0
0.1
0.2
Z(m)
R(m)
(b) φ= 90◦
FIG. 2. Poincare plot of vacuum magnetic surfaces at toroidal
angels 0◦(left) and 90◦(right).
One of our main optimization targets is neoclassical
confinement, particularly the so-called 1/ν neoclassical
transport with νbeing the collision frequency of plas-
mas. This 1/ν scaling is very unfavorable for stellarator
confinement for high temperature regime of fusion plas-
mas because the transport increases strongly with plasma
temperature[16]. Therefore, the minimization of neoclas-
sical transport in the 1/ν regime is necessary for stel-
larator reactors. It has been shown that the 1/ν trans-
port is proportional the effective helical ripple coefficient
ǫ3/2
eff [17]. We use this coefficient as a target. Fig. 4 shows
the radial profile of ǫ3/2
eff of the CSSC as well as W7-X
and the Large Helical Device (LHD)[18]. We observe that
ǫ3/2
eff of CSSC is comparable to that of W7-X and is much
3
0 0.2 0.4 0.6 0.8 1
/edge
0.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
Iota
FIG. 3. Rotational transform iota versus normalized toroidal
flux.
0 0.2 0.4 0.6 0.8 1
10-4
10-3
10-2
3/2
CSSC
W7-X
LHD shifted in
FIG. 4. Effective helical ripple ǫ3/2
eff versus square root of the
normalized toroidal flux pψ/ψedge for the optimized configu-
ration (red line), W7-X (blue line)[3] and LHD (green line)[3].
smaller than that of LHD.
The other important target of optimization is magnetic
well which is a key parameter for MHD stability. The
magnetic well is defined via the surface-averaged mag-
netic pressure[19] as following:
W= 2 V
hB2i
d
dV B2
2(4)
where Bis magnetic field strength, V(ψ) is the volume
within flux surface ψ. A positive gradient of the surface-
averaged magnetic field corresponds to a magnetic well.
The magnetic well profile of CSSC is calculated with free
boundary condition using the VMEC code for several
plasma beta values. The results are plotted in Fig. 5.
We see that CSSC possess a global magnetic well profile
0 0.2 0.4 0.6 0.8 1
/edge
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Well depth
=0
=0.005
=0.01
FIG. 5. Magnetic well depth Wversus the normalized toroidal
flux ψ/ψedge for three values of plasma β.
with well depth increasing with plasma beta. In addition
to magnetic well, we have also carried out initial MHD
stability calculation using the 3D MHD stability code
TERPSICHORE [20] in order to confirm a more com-
plete MHD stability of the CSSC. The results show that
low-n global MHD modes are stable up to the volume-
averaged beta of β= 1%. It should be noted that our
optimization is done for vacuum magnetic field. Future
work will consider optimization at finite plasma beta.
So far, we have shown that the optimized configura-
tion CSSC has both magnetic well and low level of heli-
cal ripple. Now we show that these favorable properties
are achieved with relatively simple coils, i.e., the shape
of the two IL coils is relatively smooth with modest 3D
variation as compared to that of W7-X. Fig. 6 compares
the curvature of the optimized configuration with that
of one of W7-X’s module coils at the same coil length of
1m. We observe that the coil curvature of CSSC is sig-
nificantly smaller than that of the W7-X coil. Also the
variation of coil curvature along the coil loop is much less
complex indicating that the coil complexity of CSSC is
much less in comparison.
We now describe some details of the optimization
method used to obtain above results. In addition to
neoclassical transport and magnetic well, we also tar-
get plasma volume, rotational transform and flux sur-
face quality. The magnetic field generated by current-
carrying-coils is calculated from the Biot-Savart law. The
flux surfaces and rotational transform profile are calcu-
lated by following the magnetic field lines. The effective
ripple of each surface is calculated by integrating along
a magnetic field line [17] as done in our recent work[13].
We vary the shape of the two IL coils via the ten Fourier
coefficients as described above to search for desirable con-
figurations that meet our targets. Since the total degree
of freedom is not a small number, a multi-stage random
4
0 0.2 0.4 0.6 0.8 1
Coil length(m)
0
5
10
15
20
(m-1)
CSSC
W7-X
FIG. 6. Coil curvature κvariation along the loop of coil for
the optimized configuration (black line) and W7-X (red).
search algorithm is adopted in the optimization process.
Specifically, at the first stage, the initial ranges of each
degree of freedom are selected as given in the TABLE I. It
should be noted that the range of each Fourier coefficient
is constrained to be smaller for higher nin order to keep
the coil complexity low. Given the ranges of parameters,
a large collection of stellarator configurations are gener-
ated in the 12-dimension parameter space with value of
each parameter randomly selected within its range. For
the first stage, a total of 80 million cases are evaluated.
A few good configurations are found that meet the crite-
ria of ǫ3/2
eff <0.01, ι > 0.2, plasma volume V > 0.25m3,
and having a global magnetic well profile. This concludes
the first stage of search. At the second stage, this ran-
dom search process is repeated starting from each of the
good configurations found in the first stage. The ranges
of parameters are updated based on each new configura-
tion and are chosen to be narrower than those of the first
stage. A total of four such stages of search have been
carried out in arriving the final optimized configuration
CSSC as defined in TABLE I. It should be noted that
there are other optimized configurations with compara-
ble quality. The details of other configurations will be
given in another paper.
In conclusion, a direct optimization from coils demon-
strates the existence of a new optimized stellarator with
four simple coils. The optimized configuration has favor-
able properties of magnetic well and a low effective heli-
cal ripple level comparable to that of W7-X. The two 3D
interlocking coils have a smooth shape much simpler as
compared to those of advanced stellarators such as W7-
X. This work opens up possibilities of future stellarator
reactors with simplified coils. In near term, the opti-
mized configuration can be used as a basis for a low-cost
experiment to study the MHD stability and plasma con-
finement of compact stellarators. The design could also
be used as a candidate stellarator for electron-positron
plasma experiments.
ACKNOWLEDGEMENT
We thank Dr. Steve Hirshman for the use of the 3D
equilibrium code VMEC code. We also thank Dr. W. A.
Cooper for use of the 3D MHD stability code TERPSI-
CHORE, and Dr. David Gates and Dr. Caoxiang Zhu
for use of the stellarator optimization code STELLOPT.
This work was funded by the start-up funding of Zhe-
jiang University for one of the authors (Prof. Guoyong
Fu).
∗corresponding author’s Email: gyfu@zju.edu.cn
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