Proceedings of ITC/ISHW2007

Results of the stellarator optimization with respect to theneoclassical 1/ν transport∗
Bernhard SEIWALD1) , Sergei V. KASILOV1,2) , Viktor V. NEMOV1,2) , Winfried KERNBICHLER1) ,Victor TRIBALDOS3) , Thomas SUNN PEDERSEN4) , Victor N. KALYUZHNYI2)
1)

Association EURATOM-ÖAW, Institut für Theoretische Physik - Computational Physics, TU Graz, Petersgasse 16,
A–8010 Graz, Austria
2)Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology”, AkademicheskayaStr. 1, 61108 Kharkov, Ukraine
3)
Asociación Euratom-CIEMAT, Madrid, Spain
4)
Columbia University, Department of Applied Physics and Applied Mathematics, New York, New York 10027, USA
Powerful tools have been developed in the last years for the design of new stellarator devices. This codes
are usually working in magnetic coordinates. Tasks addressed by these codes are minimization of neoclassical transport, maximizing equilibrium and stability properties, etc. However, optimizing existing stellarators istime consuming because costly coordinate transformations are involved. A procedure working in real space coordinates for maximizing the plasma energy content, based on reducing the most unfavorable 1/ν neoclassicaltransport, has been presented in [1]. This tool, named SORSSA, was developed especially for optimizing existing stellarator devices which are not fully optimized with respect to neoclassical transport. Some results for thestellarators Uragan 2M, U-2M, TJ-II and the Columbia Nonneutral Torus, CNT, are presented.
Keywords: Neoclassical Transport, Optimization, Stellarator

1

Introduction

One of the main disadvantages of non-axisymmetric confinement devices is their unfavorable, 1/ν, temperaturescaling in the long mean free path collisionality (lmfp)
regime, ∝ T 7/2 . The ∇B drift associated with their threedimensional asymmetries leads to rapid losses of trapped
particles. Therefore, improving stellarator transport starts,
precisely, by optimizing the configurations against this disadvantageous 1/ν lmfp regime [2].

2Energy confinement

The scheme behind SORSSA analyzes the neoclassical 1/νtransport properties of a magnetic field configuration. The
measure of the neoclassical transport is the effective ripple
eff [3]. For good confinement eff should be small. The
effective ripple can be used to compute the total stored energy in the plasma volume, which is the figure of merit.
The heat conductivity equation is solved under the premisethat the neoclassical transport is the dominant transportmechanism. Assuming that the temperature profile is de∗ This work, supported by the European Communities under the contract of Association between EURATOM and the Austrian Academy of
Sciences, was carried out within the framework of the European Fusion
Development Agreement. The views and opinions expressed herein do
not necessarily reflect those of the European Commission. Additional
funding is provided by the Austrian Science Foundation, FWF, under contract number P16797-N08.
author’s e-mail: seiwald@itp.tugraz.at

fined by the heat conductivity equation
dT
1 d
rκ⊥
+ Q(r) = 0 ,
r dr
dr

(1)

 
with the boundary conditions T (a) = 0 and lim r dT
dr = 0,r→0where a is the boundary of the plasma. The heat conduc3/2 7/2
3/2
tivity is proportional to effTand computation of eff
for sets of computed magnetic surfaces is an essential partof the optimization procedure. Integrating Eq. (1) leads tothe normalized stored energy [4, 5]
Ŵ =

Z a
0

Z a

dr r n̂(r) 
r

dr0
3/2 0
r0 eff
(r )

2/9

,


(2)

where n̂ = n/n0 is the normalized particle density profile
and a denotes the plasma radius. The integration variable
is the effective radius of the flux surfaces. It is convenient
to define the re-normalized stored energy Wn as
Wn = Ŵ/ŴS ,(3)with Ŵ the normalized stored energy defined in Eq. (2) andŴS the normalized stored energy of the standard configuration.
3/2
The effective ripple eff
, which is part of the 1/ν neoclassical transport coefficients and contains the characteristic features of the magnetic field geometry, is given by(see [3])
3/2
eff

=

 Ls   Ls
−2
Z
 Z

πR2
 ds   ds|∇ψ|√ lim 



 

L
→∞
B
B
s
8 2
0

0

Proceedings of ITC/ISHW2007
B(abs)
max
Z /B0

db0

·

jmax
X
Ĥ 2j
j=1

B(abs)
/B0
min

Iˆj

.

(4)

with
(max)

Ĥ j

=

sj
Z

1
b0

r

ds
B

b0 −

!
B
1
B0
,
4 − 0 |∇ψ|kG(5)B0Bbs(min)
j
(max)

Iˆj

=

sj
Z

ds
B

r
1−

B
,
B0 b0

(6)

s(min)
jand b0 = v2 /J⊥ B0 . The geodesic curvature of the magnetic
field line is given as kG = (h × (h · ∇)h) · ∇ψ/|∇ψ| withthe unit vector h = B/B. Equation (4) is computed by integration over the magnetic field line, s, over a sufficiently
large interval 0 − L s , and by integration over the perpendicular adiabatic invariant of trapped particles, J⊥ . Here,
(abs)
Bmin
and B(abs)max are the minimum and maximum values of
B within the interval [0, L s ]. The quantities s(min)and s(max)jjwithin the sum over j in (4) - (6) correspond to the turning
points of trapped particles. This integration takes into account all kinds of trapped particles, such as those trapped
within one magnetic field ripple as well as particles trapped
within several magnetic field ripples.
The total stored energy, defined in Eq. (2), depends
on eff and on the plasma volume. A decrease of eff at a
fixed plasma volume increases the stored energy as well as
an increase of the plasma volume at a fixed value of eff .
As the plasma volume and eff are not independent in astellarator, the optimum stored energy will not necessarily
coincide with the largest attainable volume or at the lowest
possible value of eff .

3

Application

The total stored energy, defined in Eq. (2), has been used as
figure of merit for optimizing the stellarators U-2M, TJ-II
and CNT.
The U-2M device [6] is an l = 2 torsatron. An additional toroidal magnetic field is produced by a system of
16 toroidal field coils (TF coils) which are uniformly distributed in angle along the major circumference (four coils
in each field period). The mean current in such a coil, IT FC ,is expressed in units of the helical winding current. For the
standard configuration this current is IT FC = 5/12, according Ref. [6]. In this case the parameter kϕ = Bth /(Bth + Btt )is kϕ = 0.375 where Bth and Btt are the toroidal componentsof the magnetic field produced by the helical winding and
TF coils, respectively.
The vertical field coil (VF coil) system plays an important role in achieving the magnetic configuration of thetorsatron. The total vertical magnetic field, B⊥ , is producedby the VF coils and the vertical magnetic field of the helical3/2
Fig. 1 Parameters eff
as functions of reff for kϕ = 0.31 andvarious fV FC ; 1: fV FC = 1, ∆I = 0; 2: fV FC = 1.166(B⊥ /B0 = 0), ∆I = 0; 3: fV FC = 1, ∆I = 0.06; 4 (standard configuration): kϕ = 0.375, fV FC = 1, ∆I = 0. (takenfrom [1])winding. The desired vertical field is obtained by adjusting the current in the VF coils. In the computations the VF
coil system variant [7] is used which allows to suppress
significantly the island structure of the magnetic surfaces.
Three control parameters are appropriate for the optimization. These parameters are related to the currents in
the helical winding and in TF as well as VF coils:(i) IT FC , the mean current of TF coils, in units of helical winding current. It is directly connected to the above
mentioned parameter kϕ (used in Refs. [6, 8]) by the ratio
kϕ = 1/(1 + 4IT FC ) as it follows from the kϕ definition.
(ii) ∆I, which is introduced in view of the results ofRef. [8]. The currents in the TF coils are presented further
in a form IT FC ± ∆I with a plus sign for the inner two coils
in each field period and with a minus sign for the outer twocoils (∆I is also expressed in units of the helical winding
current).
(iii) fV FC , a multiplying factor for the currents in the
VF coils. This factor is connected to an additional verticalmagnetic field. It enters linearly into the expression for themagnetic field of the VF coils in a way that for kϕ =0.375 it
results in B⊥ /B0 = 2.5% for fV FC = 1, B⊥ /B0 = 0 for fV FC= 1.166 and B⊥ /B0 =-2.5% for fV FC = 1.332. Here, B⊥
is the resulting vertical magnetic field and B0 is the mean
toroidal magnetic field.
Here we focus on results for the studies of the total
stored energy for a decreased parameter kϕ . Decreasingkϕ may reduce the helical ripple, may lead to an improvement of the stored energy in the 1/ν transport regime anda reduction of the magnetic island structure produced bycurrent-feeds and detachable joints of the helical winding.
3/2
In Fig. 1 results for eff
corresponding to kϕ ≈ 0.31
for fV FC = 1 and 1.166 are presented. The latter value corresponds to a full compensation of the mean vertical magnetic field of the helical winding. Besides, curve 3 shows
the results corresponding to ∆I , 0 ( fV FC = 1, kϕ ≈ 0.31).
For comparison, the results for the standard configuration( fV FC = 1, ∆I = 0) are shown.

Proceedings of ITC/ISHW2007
1.6

total stored energy
3.5

2.5
2

1.2
1
Wn

3

Wn

1.4

10
9
8
7
6
5
4
3
2
1

0.8
0.6
0.4

1.5
0.2

1

0
0

100

200

300

400

0.5
1.05

1.1

1.15

1.2

1.25

1.3

600

700

800

900 1000

1.35

fVFC

Fig. 2 Re-normalized stored energy (see Eqs. (2) and (3)) for
various fV FC (B⊥ /B0 ) and IT FC (kϕ ) with taking into account limitations by the chamber; 1: IT FC = 0.41667(kϕ = 0.375); 2: IT FC = 0.43928 (kϕ = 0.33627);
3: IT FC = 0.4619 (kϕ = 0.3512); 4: IT FC = 0.4845(kϕ = 0.3404); 5: IT FC = 0.5071 (kϕ = 0.3302); 6:
IT FC = 0.5298 (kϕ = 0.3206); 7: IT FC = 0.55555(kϕ = 0.3103); 8: IT FC = 0.5976 (kϕ = 0.2949); 9:
IT FC = 0.62024 (kϕ = 0.2873); 10: IT FC = 0.6429(kϕ = 0.28). (taken from [1])Fig. 3 The re-normalized stored energy (see Eqs. (2) and (3)) forthe computed TJ-II configurations (taken from [4]).
1

standard
best

0.1

εeff3/2

1

500
no.

0.01

0.001
0

It can be seen in Fig. 2 that with decreasing kϕ thestored energy increases and the maxima in the stored energy correspond to some optimum values of fV FC (B⊥ ). Forkϕ ≈ 0.31 (thick line in Fig. 2) it follows that the maximum
in the stored energy is approximately 2.5/1.5 ≈ 1.65 times
higher than the corresponding maximum for kϕ = 0.375.
The results corresponding to IT FC = 5/9 (kϕ ≈ 0.31) areof special interest because it follows from Ref. [11] thatfor kϕ ≈ 0.31 the magnetic configuration of U-2M is lesssensitive to the influence of current-feeds and detachable
joints of the helical winding. It has been shown that in thiscase for the major part of the configuration the rotationaltransform stays within the range 1/3 < ι < 1/2 and big
magnetic islands are absent.
The medium size heliac TJ-II (R = 1.5 m, a < 0.2 m)
has four field periods [9] and consists of 32 helicallydisplaced toroidal coils, one central helical coil wrappedaround the central circular coil and two vertical field coils.
The free parameters for the “common” TJ-II operation are(1) the toroidal coil current, (2) the current for the helical winding, (3) the current for the central circular coiland (4) the current for vertical field coils. For a run ofthe optimizer these parameters have been varied within therange of ±20% of the corresponding values for the standard
configuration, which is within the technical constraints.
Within the frame of optimization several configurations
with enhanced stored energy compared to the standard configuration could be found. The re-normalized stored energy, defined in Eq. (3) is shown in Fig. 3. It can be seen,that the energy of the best configuration is about 1.45 times
as high as the the energy of the standard configuration.
The radial dependency of eff on the effective radius ispresented in Fig. 4. It can be seen that the effective ripple

0.02

0.04

0.06

0.08
reff [m]

0.1

0.12

0.14

0.16

3/2
Fig. 4 Effective ripple eff
vs. effective radius reff for TJ-II standard configuration and the best configuration(taken from[4]).

of the best configuration is slightly smaller compared to thestandard configuration. Furthermore, the plasma radius of
the best configuration (a ≈ 15 cm) is markedly increased
compared to the standard configuration (a ≈ 11 cm).
The CNT was designed as a simple and compact stellarator with only two pairs of circular, planar coils [10].
These are one pair of interlocking (IL) coils inside the vacuum vessel and another pair of coils, the poloidal field(PF) coils, outside the vacuum vessel. The achieved design
goals have been the error field resilience, a large flux surface volume relative to the experimental footprint and thatit should be easy to build the experiment. However, minimizing the neoclassical transport was not a design goal.
The two free parameters for the optimization are the current of the poloidal field coils (IPF ) and the angle between
the interlocking coils, called tilt angle. The current for theinterlocking coils is fixed at 170 kA which is the design
current (see [10]). The two dimensional grid spanned by
the PF coil current and the tilt angle has been scanned toget a good knowledge how the re-normalized stored energy
Wn depends on the two free parameters. Two different values of the coil separation, which is the vertical distance
between the centers of the two IL coils, have been used.
One scan was done for the nominal value with 63 cm. The
other scan was done for a IL coil separation of of 62.6 cm,
because for 64 deg tilt angle the edges of the two IL coils
would touch each other if they were at the 63 cm coil sep-Proceedings of ITC/ISHW2007
std
best 630
best 626

εeff

3/2

10

1

0.1
0

(a)

0.02 0.04 0.06 0.08

0.1 0.12 0.14 0.16 0.18
reff [m]0.2

3/2
vs. effective radius reff for the stanFig. 6 Effective ripple eff
dard configuration and the best configurations for the two
considered coil separations (taken from [5]).

4Conclusion

A scheme for optimizing stellarators in real space with respect to the total stored energy based on neoclassical transport, has been developed and implemented numerically inthe code SORSSA. Applying SORSSA to the fusion devices U-2M, TJ-II and CNT, it could be shown that, witha proper choice of the free parameters, configurations withenhanced total stored energy compared to well known standard configurations could be achieved.
(b)Fig. 5 The re-normalized stored energy for CNT (see Eqs. (2)and (3)) for 63 cm coil separation (a) and 62.6 cm coilseparation (b) (taken from [5]). The standard configuration is marked with “+” and the best configuration of eachscan is marked with “×”.

aration. The results of these scans are presented in Fig. 5.
From these plots can be seen, that Wn is increasing whenapproaching the area where the PF coil current is -30 kA
and the tilt angle is 58 deg. Configurations where the tilt
angle is about 90 deg and the PF coil current is in the rangeof -60 to -30 kA consist mainly of islands and stochastic
zones.
The best configuration with a coil separation of 63 cm(referred as “best 630”) exhibits an energy which is enhanced by 17% compared to the standard configuration.
The energy of the best configuration with 62.6 cm coil separation (referred as “best 626”) is about 1.1 times the energy of the standard configuration. The standard configuration exhibits the lowest values of eff of the three considered configurations. The plasma radius for the standard
configuration is slightly smaller compared to the plasma
radii of the two other configurations. The positive volume
effect is stronger than the negative effect of an increasedeff so that the stored energy is higher for the best configurations compared to the stored energy of the standard
configuration.

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