J. Plasma Fusion Res. SERIES, Vol.5 (2002) 360-365

On the Gharacteristic Difference of Neoclassical Bootstrap
Current and lts Effects on MHD Equilibria between
CHS Heliotron/Torsatron and GHS-qa
Ouasi-Axisymmetric Stellarator
ISOBE Mitsutaka, OKAMURA Shoichi, NAKAJIMA Noriyoshi, SHIMIZU Akihiro"
NISHIMURA Shin, SUZUKI Chihiro, MURAKAMI Sadayoshi, FUJISAWA Akihideand MATSUOKA Keisuke

National Institute for Fusion Science, Toki, Gifu 509-5292, Japan' Nagoya University, Nagoya, Aichi 464-8603, Japan(Received: 11 December 2001 / Accepted: 17 May 2002)

Abstract
The characteristic difference of neoclassical bootstrap current and its effects on MHD equilibria aredescribed for the CHS heliotron/torsatron and the CHS-qa quasi-axisymmetric stellarator. The directionof bootstrap current strongly depends on collisionality in CHS, whereas it does not in CHS-qa because ofquasi-axisymmetry. In the CHS configuration, it appears that enhanced bumpy (Bs1) and sideband
components of helical ripple (By1) play an important role in reducing the magnetic geometrical factor,
which is a key factor in evaluating the value of bootstrap cuffent, and determining its polarity. The
bootstrap current in CHS-qa is theoretically predicted to be larger than that in CHS and produces

significant effects on the resulting rotational transform and magnetic shear. In the finite B plasmas, themagnetic well becomes deeper in both CHS and CHS-qa and its region is expanded in CHS. The
existence of co-flowing bootstrap current makes the magnetic well shallow in comparison with that incurrentless equilibrium.

Keywords:
quasi-axisymmetric stellarator, heliotron/torsatron, geometrical factor, bootstrap current

symmetry, and quasi-axisymmetry have been proposed.
A basic idea in developing new types of stellarators is toCompact Helical System (CHS) experiment has been
undertaken to investigate the confinement properties of
low aspect ratio (Ao - 5) heliotron/torsatron plasmas. In
order to explore further good confinement regime of acompact stellarator, we propose a quasi-axisymmetricstellarator CHS-qa as a post-CHS device [6-9]. The
primary concept of CHS-qa is to be MHD stable and torealize configurations having magnetohydrodynamicallystable characteristics as well as good neoclassicalimprove trapped particle orbits with realizing atokamak-like, quasi-axisymmetric magnetic field

confinement.

configuration on the Boozer coordinates and1. lntroduction
In recent years, many theoretical and computational
efforts have been made to explore advanced stellarator
configurations [-5] and as a result, new stellarator
concepts such as quasi-isodynamicity, quasi-helicalIn the National Institute for Fusion Science, the@2002 by The Japan Society of Plasma
Science and Nuclear Fusion Research

C o r r e s p o ndin g autho r' s e - mail : i s ob e @ nifs. a c.j p

360

Isobe M. et al., On the Characteristic Difference of Neoclassical Bootstrao Current and Its Effects on MHD Eouilibria

2. Bootstrap Current Property in CHS andmaintaining the compactness, which is the policy of
CHS. In CHS-qa, the magnetic well is realized in the
improved cornpared with that in CHS. Moreover, in

CHS-qa2.1 Basic Features of CHS and CHS-qa
CHS is a low aspect rutio (Rla - 5) heliotron/

actual experiments, the quasi-axisymmetry may lead to a

torsatron device with a toroidal period number of N = 8

drastically improved confinement regime such as thatobserved in tokamaks.
One of the remarkable differences between CHS
and CHS-qa concerns the bootstrap current property. In

and polarity I = 2 ll0l. The device's major radius R is I

m. The magnetic field is produced by two helicalwinding coils and three sets of poloidal coils. CHS is
flexible in its wide variability of R"* and ellipticity oftokamaks, the bootstrap current is recognized to be very

the plasma cross section. The toroidal magnetic field B,

important in regard to the steady state operation.
Although this current is not essentially required for
confining helical plasmas, it is important in helicalsystems becatrse it certainly affects not only MHD

can be increased up to 2 T. T\e CHS experiment began

entire plasma region and neoclassical transport is largely

in 1988 and we have studied confinement properties of
low aspect ratio (Rla - 5) heliotron/torsatron plasmas.
CHS-qa is a quasi-axisymmetric stellarator with N = 2.
R is 1.5 m and its aspect ratio is fairly low, Rla = 3.2 inthe present design. The magnetic field is produced bytwenty modular coils, which are capable of a maximum
B, of 1.5 T I l]. Three sets of poloidal coils and eight

equilibrium :Lnd stability but also transport andtrajectories of energetic ions through the change of therotational transform t/2n and magnetic field spectra.
Neoclassical theory predicts that the bootstrap current

behavior is complicated in CHS. The direction of

additional toroidal coils will be equipped to control theproperties of a magnetic field configuration such as Ru*,bootstrap current can be changed from co- to counter-,and vice versa, depending on collisionality and magnetic
field structure, which can be changed through magnetic
axis position Ru* and/or the ellipticity of the plasma
cross section. On the other hand, in CHS-qa, bootstrap
current always flows co-directionally and its magnitude

tl2n and, the magnetic well depth. Figure I shows themagnetic field spectra of the vacuum magnetic field of
both CHS and CHS-qa. In an inwardly shifted
configuration of CHS (Ru* = O.921 m), the magnetic
field structure is dominated by B1s(toroidicity) andis predicted to be larger than that in CHS because of821(helical ripple) (see Fig. 1(a)). The amplitude of By

quasi-axisymmetry, resulting in much lower neoclassicalsteeply increases toward the plasma boundary. It should

viscosity. In this article, we compare neoclassicalbootstrap current properties of CHS and CHS-qa interms of the magnetic geometrical factor. In addition,
the effects of bootstrap current on MHD equilibria arealso discussed tbr both CHS and CHS-qa configurations.

be noted that there is a .B16-dominated structure in thecore domain. An outwardly shifted configuration (R- =

0.3

0.3
a) CHS 0.921m

o.2

o
o

.a

-0.3

o
o

.a''

-Eo
E
-0.2

structure than that in Ru* of 0.921 m. In addition to 8,6
and 821,811(the sideband component of a helical ripple)

0.1

.'

b) CHS 0.995mc) GHS-qa 1.5m

0.2

0.1

t -0.10.995 m) has a more complicated magnetic field

0.1

-eo

a-.+__

tE -0.1

1,0)-----( l'-1)-'-" ( 2,-1)
-(

-o.2

- - -( o, 1)

o.2 0.4 0.6

-0.3
0.8

0.05

-->e';i-li=--":==__.________(.t,0)----' ( I,-l )-'-" ( 2,-l )'( 3,-l )1)---(0,
0.2

rla$ -o.os
-0.1

-0.15

o.4

0.6
r/a

0.8

-.-.---'-:'-

8o

tll

---.-(0,1)-(1,0)(2,-2)---(1,-1)--- ( I,-2)------(1,1)

0.2

o.4

0.6

0.8

r/a

Fig. 1 Magnetic field spectra on the Boozer coordinates in vacuum configurations. (a) CHS (R*=0.921 m), (b) CHS (B", =0.995 m), and (c) CHS-qa.

36r

Isobe M. et al., Qn the Characteristic Difference of Neoclassical Bootstrap Current and Its Effects on MHD Equilibria

teau regimes. The magnitude and polarity of G5. is sen-and 861(bumpiness) are largely enhanced especially inthe core domain (see Fig. 1(b)). The magnetic field

sitive to R* and the collisionality regime in CHS. Figure2(a) suggests that in CHS, the neoclassical bootstrap
current flows in the co-direction if the plasma is in thebanana regime. The B1s-dominated structure in the core
domain of Ru* = 0.921 m gives large Gb.h"n", which iscomparable to that in CHS-qa. On the other hand, in Rof 0.995 m, the magnitude of Gb"bunun" in the core domain is much smaller than that in Ro of 0.921 m. Judging from Figure 1(b), it is supposed that augmented amplitude of 861 and 811 plays an important role in reducing the value of Gb,oun*u. If the CHS plasma is in theplateau regime, the magnitude of Gbsplateau becomes
smaller than that of Gbrb"n"nu and the direction of thebootstrap current can be switched from the co- to thecounter-direction. Especially, at Ru* of 0.995 m, the polarity of Go,otut"uu is opposite in the whole region to thatof Gbrbununu. This means that in the plateau regime, thebootstrap curent flows in the counter-direction in thewhole region of a plasma having Ru* = 0.995 m. The
magnitude of G6, in CHS-qa is larger than that in CHS
and its polarity does not depend on the collisionality regime. The significant difference from CHS is that there
is no large difference in the magnitude of G6, between

structure of CHS-qa is quite different from that of CHS.
In CHS-qa, a tokamak like, B16-dominated magnetic
field structure is achieved in the whole region (see Fig.
1(c)), although residual non-axisymmetric magnetic
field components still exist in the peripheral region. The
profiles of tl2n and magnetic well depth of CHS and
CHS-qa are shown in Sec. 3 in detail.

2.2 Magnetic Geometrical Factor
The neoclassical bootstrap current density j5, of a

non-axisymmetric helical system in the banana (l/v)
regime is scaled as [12,13],{ti. B>. oc(,[/i) Gurb-*u P (n'ln + ar Ti'lTi * a2T"'/7")
The bootstrap current is calculated in terms of the mag-netic geometrical factor Gu". I is the magnetic fieldstrength and < >, denotes the flux surface &verage. d1
and &2 are constants, andfr andf, are the fractions oftrapped particles and circulating particles, respectively.
P, n, Tu and T; are the total pressure ; P = n" T" + ni Ti,
density, electroR temperature, and ion temperature, respectively. The primes stand for dldry, ry = Qyl2n where
p1 is the toroidal flux. Figure 2 shows the ratio of G6, of
CHS and CHS-qa to that of an equivalent tokamak, G6,lGsrtok for vacuum magnetic fields in the banana and pla-a)the banana and plateau regimes. The neoclassical theory

predicts that in CHS-qa, the bootstrap current is larger
than that in CHS and always flows in the co-direction. Itshould be noted that the ratio G6r/G6rtok is not very close

to I in CHS-qa because of the residual non-b)?il,?l;CHS-qaCHS-qa
T

I

$
d

cF:.
0.921m

0.5

o

0.5
cHs

3
(J

-

0.921m\-.
0:--:-:_- - -\r----\0.995m

0.995m

Co0

\

cHs

cHs
o

*1tr:
reglme0.2 0.4 0.6

-0.5
0.8

ila

0 a.2 0.4 0.6 0.8
rla

Fig. 2 Magnetic geometrical factors in vacuum configurations. (a) CHS (&, = 0.921 m). (b) CHS (F", = 0.995 m), and (c)CHS-qa.

362

Isobe M. et al., On the Characteristic Difference of Neoclassical Bootstrap Current and Its Effects on MHD Equilibria

axisymmetric rnagnetic field components.

current self-consistently in the whole collisionality range
from the banana to the Pfirsch-Schliiter regime [14] hasbeen employed. It has been so far confirmed that this

3. Effects of Bootstrap Current on MHD
Equilibrium
In order to investigate the effects of bootstrap

code reproduces well the experimental bootstrap

current on finite B equilibria of both CHS and CHS-qa,
the SPBSC code which calculates the three-dimensional

currents in ECRH plasmas of CHS at moderate density(n"- 1.0 x l0re m-3) [5,161. Figure 3 shows rhe profilesof bootstrap current density d16,/ds, tlzn and magnetic

MHD equilibrium including neoclassical bootstrap

well depth in the vacuum and finite p equilibria of CHS

CHS 0.921m

40

CHS 0.995m
L

30

a-l)

20

-bsc.highno
bsc : low n'

80

10

;

10

6

0

s

0

-10

960
o

-bsc. high n

-20t
0

...i
- - -.7.J--\

-20

'

0.2

0

.F
(U

TE

o.4

0

(6

-/--\'\
0.2

0.4

0.6

0.8

rla

o

E 03

0.6

td

o.4

.E 0.2

tf o.2

tr 0.1IE(dbsc : lofr n.

0 0.2 0.4 0.6

0o 0.2 0.4 0.6

0.8

rla
o.2

a-3)

0.1

rla

0.12

b-3)

0.1

c-3)
VaCUUm=0a
9 -o.r=0a
9 -o.r

=0.08

O -o.zQ -o.z

€o.o+

-0.3

-0.3

a.o2

-0.4

\

E 0'4

C

.9

-----curenlless-bsc. hioh rc-VaCUUm

0.2
0.1

s

--41
..";4
----

1

o
o 0.8

,tt

o.2

\

c-21
E

o
a 0.8
c
0,6

...

rla

1.2

1

E

i

I

20

a-21
E

./"'\-.-...-...

..t

440

0.2 0.4 0.6

ila

1.2

c-1) =-.:m;i]flI#

6

.-: /

-10

----.- bsc : low n€

CHS-qa 1.5m

t\

J

o
rct

---.-

20

-v

o

lJ'at
'\

30

100

^90.06?

0 0.2 0.4 0.6 0.8
rla

-o.4
1

?

b
0 0.2 0.4 0.6 0.8
ila0---.-curreille8g- -bsc'hioh n----- bsc : bW ne

j

i
t
it

i,

o o.2 0.4 0.6
ilaFig. 3 Bootstrap current density, rotational transform, and magnetic well depth in the vacuum, currentless finite p equilibrium, and finite B equilibrium including neoclassical bootstrap current self-consistently for high n" (n"(0) = 1.0 x1020 m-3) and low n" {n"(0) = 2.0 x 101s m'3) cases. (a) CHS {F", = 0.921 m), (b) CHS (8"* 0.995 m}, and (c) CHS-qa.
=
The central beta value P(0) is Z.g o/o.The calculations were carried out under the fixed boundary condition. The radial electric field E was not taken into account.

Isobe M. et al., On the Characteristic Difference of Neoclassical Bootstrap Current and lts Effects on MHD Equilibria

plasma center and the Brt component is also slightly
increased in the finite p equilibrium in comparison withthose in the vacuum equilibrium. In Ru* of 1.106 m,(R"* = 0.921 m, 0.995 m) and CHS-qa. The calculationsof MHD equilibria were performed for the following
four cases ; 1) vacuum, 2) currentless finite p, 3) finite P
with self-consistently including bootstrap current in highne case, and 4) in low n" case. The plasma parameters
used here are ; Z"(0) = 0.4 keV, 4(0) = 0.3 keV, andn"(0) = ni(0) =1.0 x 1020 m-3 for high ne case, r"(0) =2.0 keV, 4(0) =1.5 keV, and n"(0) - ni(0) = 2.0 x l01e
m-3 for low n" case with profiles of I = Z(0)(l - s), n =n(OXl - 0.8s + 1.3s2 - 1.5s3). and B, = I T. Here, s is
the label representing the normalized toroidal flux. Then" gradually declines toward s = 0.9 and then rapidly
drops to zero toward the plasma boundary (s = l). These
parameters give a central B value of 0(0) = 2.8 Vo. The
normalized collision frequency v* in low and high n"

which is the most outwardly shifted configuration of
CHS, the amplitude of 861 and 811 in the vacuum islarger than that in the finite B equilibrium of Rax = 0.995
m and the polarity of Gb,bununu is negative in the domainof rla < 0.45. Based on these facts, the counter-flowing
bootstrap current seen in Fig. 3(b-l) is supposed to be
due to augmented 861 and Btr components. A similar
discussion is available in ref. [13] for the Large Helical
Device (LHD), which has a magnetic field configuration
analogous to that of CHS. The effect of counter-directedbootstrap current on Llzft is clearly seen in the core
domain (see Fig. 3(b-2)) and plays a desirable role inexpanding magnetic well region (see Fig. 3(b-3)) fromcases for both inwardly and outwardly shifted
configurations of CHS are (l - 21 x lO-2 and 1.5 - 3 atrla (= sttzl of 0.7, respectively. For CHS-qa, v* is - 0.4
for low n" and - 50 for high n". Here, the condition of
y* << I and v* >> 1 correspond to the limit of the llv

the viewpoint of MHD stability. In CHS-qa, thebootstrap current always flows co-directionally (see Fig.

3(c-1)), increasing tl2tt.ln the low necase,16, is +56 kA
and exerts significant effects on tl2tt profrle.16, of +10.6
kA in the high n" case effects the value of il2n less but

regime and of the plateau regime, respectively. The
plasma potential is not considered in this analysis.
In Ru* of 0.921 m, the neoclassical bootstrap current
flows co-directionally in both low and high n. cases (seeFig. 3(a-1)) and increases t/2tt as a result (see Fig. 3(a2)). The total bootstrap currents 16, for low and high n"

it produces almost no magnetic shear. It should be
mentioned that in CHS-qa, L/27t tends to decrease as B
increases in the currentless equilibrium, therefore it isvery important to take into account of bootstrap current
in the finite B equilibrium. /5, steeply increases as B
increases, reaching 100 kA when <B> exceeds 2Vo in
the low collisionality regime [17]. The effect of finite B

cases are +20 kA and +8.7 kA, respectively. Here, "+"indicates the co-direction, while "-" represents thecounter-direction. There is no magnetic well region in
the vacuum configuration of Ru* = 0.921 m, but in thefinite p case, the magnetic well is formed in the regionof rla < 0.6 due to a Shafranov shift. It is seen that theon magnetic well depth is very clear. It enhances themagnetic well from 57o (vacuum) to more than lOVo
(finite p) at the plasma boundary. The co-flowing
bootstrap current and resulting higher tl2n make themagnetic well shallower than that in the currentlessbootstrap current suppresses the formation of thefinite B equilibrium through suppression of themagnetic well compared with the currentless
equilibrium because it pushes tp tl2n and then reduces
the Shafranov shift. In finite B plasmas of an outwardly
shifted configuration (R* = 0.995 m), 16. for low andhigh n" plasmas are +4.1 kA and -3.9 kA, respectively.
The bootstrap current in the high collisionality plasma
flows in the counter-direction over the whole region as
expected from Fig. 2. It is interesting to note that in the
low collisionality plasma, the bootstrap current flows in

Shafranov shift.

4. Summary
We have investigated the collisionality and Ru*
dependence of the bootstrap cunent properties and theireffects on MHD equilibria for CHS heliotron/torsatronand CHS-qa quasi-axisymmetric stellarator. It was
shown that the behavior of bootstrap current iscomplicated in CHS. Especially in the outwardly shifted
configuration, the bootstrap current can easily change its
direction from co- to counter-, and vice versa,
depending on collisionality. The analyses of magnetic
field spectra suggest that the enhanced amplitude of8e1
and .B11 components plays an important role in reducing
the magnitude of G6.. Judging from the magnitude andpolarity of G6", a larger bootstrap current, which always

opposite directions at the center and edge. As discussed
in Sec. 2.2, in the vacuum equilibrium, the direction of
the bootstrap current in the core domain is expected tobe co- in the low collisionality regime, whereas the selfconsistent bootstrap calculation for finite B equilibriumin the low collisionality case shows that the direction of
the current in the core domain is in counter-direction.
The .861 component is augmented, by about double at the364

Isobe M. et al., On the Characteristic Difference of Neoclassical Bootstrap Current and Its Effects on MHD Eouilibria

flows in the co-direction, is predicted in CHS-qa

gna,1994).

because of qulsi-axisymmetry and the tl2n proflle is

t4l A.Reiman et al.,Plasma Phys. Control. Fusion 41,

significantly affected by the existence of bootstrap
current especially in the low collisionality high p
regime. In regard to the effect of finite B on themagnetic well. it results in forming and expanding themagnetic well region in CHS. In CHS-qa, the magnetic
well is realizt>d in the whole plasma domain in the8273 (1999\.

l5l D.A. Spong et al.,Phys. Plasmas 5,1752 (1998).
t6l K. Matsuoka et al.,Plasma Physics Reports 23,542(1997).

I7l S. Okamura et al., J. Plasma Fusion Res. SERIES
1, 164 (1998).
S.
[8] Okamura et al., in Plasma Physics and Controlled Nuclear Fusion Research 2000 (Proc. 18th
Int. Conf. Sorrent, 2000) IAEA-CN-77/ICP/16.
t9l K. Matsuoka et al., J. Plasma Fusion Res. SERIES
4, r11 (2001).
t10l K. Matsuoka et al., in Plasma Physics and Controlled Nuclear Fusion Research 1988 (Proc. l2th
Int. Conf. Nice,1988), Vol.2, IAEA, Vienna, 411(l989).

vacuum equilitrrium and becomes deeper as B increases.

It is clearly shown that the existence of co-flowing
bootstrap cun'ent reduces the magnetic well depth
compared wittr that of currentless finite B equilibrium
for both configurations. Therefore, we see that it isimportant to consider MHD equilibrium including selfconsistent bootstrap current for physics design studies ofa new stellarator CHS-qa.

Acknowlednnents
We are grateful to Profs M. Fujiwara and O.
Motojima for t.heir continuous encouragement. One of
the authors (MI) thanks Dr. K.Y. Watanabe for the use

tl1l A. Shimizu et al., submitted to Fusion Eng. Design.

ll2l K.C. Shaing and J.D. Callen, Phys. Fluids 26,3315
fl983).

[3] K.Y. Watanabe and N. Nakajima, Nucl. Fusion 41,
63 (2001).

of the SPBSC r:ode.

[14] K.Y. Watanabe et al.,Nlcl. Fusion 35,335 (1995).
ll5l H. Yamada et al.,Nucl. Fusion 34,641 (1994).
[6] M. Isobe et al., Plasma Phys. Contol. Fusion.44

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