J. Plasma Fusion Res. SERIES, Vol.5 (2002) 124-l3O

Study of Time Evolution of Toroidal current in LHD
WATANABE Kiyomasa, SAKAKIBARA Satoru, SASAO Hajime', YAKOVLEV Michael,
NISHIMURA Kiyohiko, MURAKAMI Sadayoshi, NAKAJIMA Noriyoshi, YAMADA Hiroshi,
NARIHARA Kazushige, YAMADA Ichihiro, TANAKA Kenji, KAWAHATA Kazuo,
TOKUZAWA Tokihiko, IDA Katsumi, MORITA Shigeru, KANEKO Osamu, TSUMORI Katsuyoshi,
IKEDA Katsunori, NARUSHIMA Yoshirou, MORISAKI Tomohiro, KOMORI Akio,

LHD experimental group and LHD device engineering group
National Institute for Fusion Science, Toki, Gifu 509-5292, Japant Japan Atomic Energy Research Institute, Ibaragi, 3tt-0193, Japan(Received: 2l December 2001 / Accepted: 18 June 2002)

Abstract
We study the properties of non-inductive current in LHD experiments. The experimentally observed
dependence on the plasma stored energy and the magnetic axis position agrees with a theoreticalprediction based on bootstrap current under NB balanced injection. We apply a calculation model of time

evolution in toroidal current profile based on bootstrap current and Ohkawa current to analyzing the
dynamic change of the rotational transform in a discharge with fairly high beta and high positive toroidal
current. The numerical analysis suggests that the disappearance of L=ll2 rational surface leads to thesudden disappearance of the m/n=211 magnetic fluctuation experimentally observed.

Keywords:
bootstrap current, Ohkawa current, neutral beam, heliotron, LHD, MHD stability

1. lntroduction
In helical devices, net toroidal current is notnecessary to produce the magnetic field for plasma
confinement. Over l00kA of net toroidal current has
been observed in NBI plasmas of LHD (a device of
heliotron) []. In typical LHD experiments, we do not
drive ohmic current actively. Bootstrap current andthe suppression of Shafranov shift which restrains theformation of magnetic well. The above effect is moresensitive on the current profile than the net current. In
helical systems, the direct measurement of toroidal
current profile is difficult because the additional changeof the poloidal field to the vacuum field should be
detected. Then it is important to identify the driving
mechanism of the toroidal current, which leads to theconstruction of the reasonable model for thedetermination of current profile in present experiments
as well as in reactor designs.
In the typical LHD-NBI discharge, the duration
time of heating plasma is less than 5s. On the contrary,
the relaxation time of the toroidal current is over 9s for

Ohkawa current are theoretically considered as thecandidates of the non-inductive current source. The
observed toroidal current is not large enough to activate

current driven instabilities. However, there is apossibility that it affects the MHD stability and transport
through changing of the magnetic configurations [2,3].
The toroidal current with the direction to increase therotational transform, which we call co-direction or
positive, leads to the decrease in the magnetic shear andthe plasma with 3 keV as the central electron@20O2 by The Japan Society of Plasma

Corresponding author's e-mail: kiyowata@ LHD.nifs.ac.jp

r24

Science and Nuclear Fusion ResearchWatanabe K.'t . et al., Study of Time Evolution of Toroidal current in LHD

temperature, which we have obtained in LHD

and supporting structures are measured with Rogowski

experiments [4]. Then it is necessary to analyze the time
evolution of net toroidal current in order to estimate
non-inductive current. In LHD device, the coil power

coils. The current increment of the helical coils andpoloidal coils are also measured. The magnetic
fluctuation is measured with 64 three-axial magnetic

supply is controlled to keep coil currents constant evenwhile the plasrna exists. However, the response time of
the control is over 10s. As the results, any feedback

probes along the helical coils, which are distributed at 6

toroidal locations. The electron density is obtained by
the measurements with 13-channel far infrared laserinterferometer. The electron temperature profile ismeasured with the multi-channel Thomson scatterins

control of the coil current is not effective during atypical plasma. discharge, and the helical coil currents
and poloidal r:oil currents change during the plasma
discharge, whrch might affect to the estimation of the
non-inductive currents. In this paper, we analyze thetime evolution of toroidal current taking the mutual
coupling due to coil currents into account. The
properties of the non-inductive current are studied insystem.

The non-inductive toroidal current, /yy, is estimatedby the following equation,(1)

Roln = Yl*e + Re/Nr .

Here 1o is the observed toroidal current and Vlooo is theone turn voltage. In LHD, there is no loop coil to measure the one turn voltage inside the vacuum vessel. The
one turn voltage is estimated by the following relation-balanced NB injected plasma.

In LHD erperiments with fairly high beta and highpositive toroidal current, the mln=2ll magnetic
fluctuation mode, which is observed from the initial
phase, disappears suddenly as the beta and the net
toroidal current increase [5]. The most probableexplanation is the disappearance of t=l12 rationalsurface in the plasma region due to the fairly high beta
and high positive toroidal current. In order to identify
the current driven mechanism in LHD experiments, weapply a calculltion model of time evolution in toroidal

ship in this paper,. ilo

Vroop=-Lptaf * S.,

d1i

4Mipd,

Q)

where the lst term and the 2nd term in right hand side
are the self-induced part and the mutual coupling parts,
respectively. dlildt is the current increment in components surrounding the plasma. i denotes a component
surrounding the plasma such as helical coils, poloidal

current profile based on bootstrap current and Ohkawa
current to analyzing the dynamic change of rotationaltransform in ttLe above-mentioned discharge.
This papr:r is organized as follows: in Sec. 2, weshow the experimental setup and the way to estimate thenon-inductive current. The properties of the non-coils and so on. lo and Mio are the self-inductance ofplasma and the mutual inductance between plasma and acomponent surrounding the plasma, respectively. In thispaper, the following expressions are used as Lp, and Rp.

Lo = L"*r+ li, L"*t=pon(fog$ - Z),

inductive current under NB balanced injection aret,= ! UoR, Ro = 2nR I I odS ,shown, and tbe comparison between the experimental
results and the theoretical prediction is shown in Sec. 3.

d = o,pir. ( I - /-s) (1 + o.o:ere)l( | + 0.47 r,z),

In Sec. 4, we show a calculation model of time
evolution of toroidal current profile and result of its

where Z"*, is the external self-inductance, /; is the inter-

application. In Sec. 5, the results are summarized.

nal self-inductance, o is the neoclassical electric con-

2. Experimental Setup

ductivity, o,pit, is so called Spitzer conductivity and e isthe toroidal ripple of magnetic field strength. Here we

LHD is a heliotron device with L = 2 and M = l0

adopt the equivalent tokamak's model as the neoclassical conductivity. The validity of the above assumption
should be made clear in future. A calculated database
based on multi-filament model is used as M;0, which
agrees with experimental results [7].
Figure I shows the time evolution of the observed
toroidal current, the non-inductive current, the one turnvoltage, the self-induced part and the mutual couplingparts in a NBI discharge. One turn voltage increases in
the initial phase of discharge (/=0.5-ls), and thensuper-conducting helical coils and three pairs of poloidal

coil [6]. }{erc L is a pole number and M is the field
period of helical coil system. The device major radius is

3.9 m, and the plasma minor radius is 0.64 m in a
typical operation. The net toroidal current and plasmastored energy are measured with Rogowski coils anddiamagnetic loops installed inside vacuum vessel,
respectively.'lhe eddy current in the helical coil can,and the summation of the eddv current in vacuum vesselt25

Watanabe K.Y. et al., Study of Time Evolution of Toroidal current in LHD

3. Non-lnductive Current Under Balanced

decreases. That reason is because the density rapidly
increases in the initial phase, which leads to the increase

NB Injection

ofthe non-inductive current (driven by NB here) and theresistivity. After t=ls, the density and the temperature

In LHD balanced NBI discharges, we analyze thenon-inductive current experimentally obtained from
aspect of the dependence on the plasma stored energy

are almost constant, the non-inductive current and theresistivity are also constant. One turn voltage decreases
as the plasma current approaches to the non-inductive
current. Here so-called l,/R time is -3.8s because theelectron temperature keeps almost constant I"6-1.8 keV.
According to Fig. I, the self-induced part of the one turnvoltage is three times larger than the summation of themutual coupling parts. In the present power supply
control of the coil systems, the active control is noteffective within l0s. Then the increment of the current
in the components surrounding the plasma is induced by
the increment of the toroidal plasma current, whichleads to the mutual coupling part of one turn voltage inthe typical discharge. In the steady state, dlpld/=O, yroop

and the magnetic axis position.

Figure 2 shows the dependence of the toroidal noninductive current obtained experimentally on plasmastored energy in NB balanced injected case. Here theratio of deferential port-through power between co andcntr. (counter) injected NB to total power is less than
7%.Then we can ignore Ohkawa current here. Magnetic
axis is 3.75 m, operation magnetic field strength is 1.5 T
and hydrogen gas-puff is adopted. The closed triangles
are experimentally obtained data, which is estimatedbased on eq.(1) as mentioned in Sec.2. Here thecollisionalities of data in Fig. 2 belong to so-call 1/yregime at p-0.8. The experimental data are proportionalhelical coils contribution to one turn voltage is dominantamong the mutual coupling parts.

to plasma-stored energy. The open circles are thetheoretical prediction using SPBSC code, whichcalculates bootstrap current consistent with 3-

We adopt the time averaged value of the noninductive current, while the density and the electron

dimensional MHD equilibrium [8]. In the calculation,
the measurement data is used as the line averaged

temperature are almost constant, as the non-inductive

electron density, trs-6^. a;rrd the electron temperature, 2",current obtained experimentally. For example, the noninductive current is averased from r=1.5s to 3.0s in Fie.

but n" profile is assumed as -( 1-p8). Z,s;2 and T,=Ti atealso assumed. The magnitude and direction of theobserved plasma current fairly agree with the predictionof bootstrap current.

becomes almost 0. Here it should be noted that the inner

l.

-Y

o

t:

o

p(s
Y

I

50
40
30
20
10

40

I Experiment35

O Theo. Pred.

30'e'-4--

25

0
-'10

o

420
lc

0.3
0.2

o
(')

(E

=o

0.1

10

0

5

-0.1
-o.2

0

0 0.5 1 '1.5 2 2.5 3 3.5 4

o

o

0

50 100 150 200 250
we(kJ)

time (s)Fig. 2 The plasma stored energy dependence of the

non-inductive current obtained experimentally,

Fig. 1 Time evolution of the observed toroidal current,/", the estimated non-inductive current, /*,, and thecalculated one turn voltage, t(""".

and the comparison between experimental results
and the theoretical prediction.
126

Watanabe K.Y . et al., Study of Time Evolution of Toroidal current in LHD

the most probable candidate as the driving mechanismof non-inductive current in LHD experiments under

Figure 3 shows the dependence of non-inductive
current obtained experimentally on the magnetic
configurations with different magnetic axis. The betabalanced NB-injection.

values are almost same, <Pai">=0.33-0.41Vo. Here <Ba1S

4. MHD Activity and Central Rotational

is the volume averaged beta value. The data are obtained

Transform
In LHD experiments with "Ru"=J.f m (magnetic

under a same condition of NB injection; co NB has port
through power of 2 MW with beam energy of 150 keV,and two cntr. NBs have 0.6 MW with 120 keV and I

axis torus-inwardly shifted) configuration", the mln=2ll
mode of magnetic fluctuation is observed generally inthe rage of the beta,<Faiu>=0.3-2.3Vo with the low
toroidal current. The strong relationship between the

MW with 150 keV, respectively. Circles denote theexperimental data and a line denotes the theoreticalprediction of bootstrap current by SPBSC code. We findthat the experimentally obtained non-inductive current
decreases as the magnetic axis goes torus-outwardly.

observation of the m/n=2/1 magnetic fluctuation and
Mercier unstable condition is observed [5,10]. Here mand n are the poloidal and toroidal mode number,respectively. However, in experiments with fairly highbeta and high positive toroidal current, the rnln=2ll
mode observed by the magnetic probes from the initial
phase disappears suddenly as the beta and the net
toroidal current increase [5].
In Fig. 4, we show an example of the waveform of

The theoretical prediction also shows that the bootstrap

current decreases with the torus-outward shift of themagnetic axis. That reason is because the magnetic
configuration approaches to a poloidal symmetry system
due to the torus-outward shift of the magnetic axis in
LHD [2]. It should be noted that co beam current is alittle superior to cntr. beam current as the above
mentioned. According to a theoretical prediction by
MCNBH code [9], which is based on 3-dimensional
Monte Carlo simulation, the non-inductive current
driven by the beam is approximately +5 kA. The effectof the unbalanced beam is not large here.
From the properties of the observed non-inductive
current as shown in this section, the bootstrap current isdischarge, where the above event occurs. Co NB starts

30

20
10

0

1

-10

I
E3

Ez
dt

?20
.9.

oo'

_2

100

4

10

3.8

3.9

4

2.0

4.1

2.5

3.0

time (s)R (m)Fig. 4 Time trace of (a) deposited heating power and thetoroidal plasma current, (b) the volume averaged
beta and the line averaged electron density andFig. 3 The magnetic axis position dependence of the
non-inductive current obtained experimentally under almost same beta, and the comparison be-(c) the amplitude ol mln=2/1 magnetic fluctuation

tween experimental results and the theoreticalin the discharge, where the disappearance of the
mln=2/1 mode occurs.

prediction.
r27

Watanabe K.Y . et al., Study of Time Evolution of Toroidal currenr in LHD

the injection earlier than cntr. NB, and the power of co
NB is larger than cntr. NB, which leads to large positive
toroidal cuffent. The mln=2/l mode has the frequency of
about lkHz, and it starts to increase at r=1.85s, and it
disappears at t=2.66s. The other modes of magnetic

t,*=+zs$[+*(', - '-,)],(3)

where 1o=10($,/) and /*t=I*t(S,l) are the toroidal current
and the non-inductive current inside a magnetic surface,fluctuation, for example, the mln=212 and 2/3 areand S is the toroidally averaged area enclosed by a magnetic surface, S=n2. r is the plasma minor radius. As theboundary condition, we apply the followings,

observed both before and after the disappearance of them/n=2/7 mode. When the mln=2/l mode disappears, thebeta is around 7.8Vo and plasma current is around 18kA,
which corresponds to 0.05 as the increment of I at theedge. Before and after the disappearance of the mln=2/l(s)1o(0,r)=0,

mode, the beta and the plasma current gradually
increases. Figure 5 shows the time trace of Mercier
parameter, Dy, at the t=ll2 rational surface, which is*(',-r*,)1,=".,=ocalculated with the pressure profile measured
experimentally and under the net toroidal currentlessassumption. According to currentless analysis, Dlat t=l/+n(-t"., *1,=.",.1

*-#)

(6)

As the initial condition, we adopt 1o(S,r)=1oo(S). Here weare considering bootstrap current and Ohkawa current as

2 rational surface is positive from r-1.5s to 3.0s, where
Mercier modes are expected unstable. Dq after thedisappearance of the mln=2/1 mode is larger than that
before the disappearance. The above result and thethe non-inductive current. Bootstrap current and
Ohkawa current are calculated with the measured den-sity and temperature by SPBSC code and MCNBH

strong relationship between the magnetic fluctuation andcode.

the Mercier condition show the possibility that thetoroidal current stabilizes the m/n=2/l MHD unstable

Figure 6 denotes the comparison of the time
evolution of between the observed net toroidal current
and the calculation results. The thin and thick lines
denote the experimental data and the calculation data,
respectively. Dashed line is the summation of themode. Then the most probable candidate of the reason isthe disappearance of the resonant rational surface, t=l/2,

due to the toroidal current. When we assume that thedisappearance of the magnetic fluctuation is due to thedisappearance of the resonant rational surface, we cancheck the validity of the calculation model of thetoroidal current profile evolution through thecomparison between the dynamic change of rotationaltransform theoretically calculated and the timing of theappearance and/or the disappearance of the observed
magnetic fluctuation.
The time evolution of the toroidal current iscalculated by the following diffusion equation.

bootstrap current and Ohkawa current, the magnitude ofwhich is expected in the steady state. After /=1s, theplasma stored energy and the electron density gradually

increase, and the electron temperature keeps almost

30
.p-Gt

25

_l

,Fexp

-l- - lBsilottu*u

?20
lz
Y

F15
V

o10

1

+

0.8
0.6

D,@r=0.5(p-0.5)

o- 0-44.2

00.5 11.522.533.54

0-4.2

1.4 1.6t(s)1.8

2 2.2 2.4 2.6 2.8 3

Fig.6 Time evolution of the experimentally observed

net toroidal current (thin line), the calculatedtime(s)plasma current (thick line) and the summation of

Fig.5 Time trace of Mercier parameter underthe bootstrap current and Ohkawa current(dashed line).

currentless assu mption.
128

Watanabe K.Y. et aI., Study of Time Evolution of Toroidal current in LHD

constant. The summation of Ohkawa current andchanges as the followings; peaked, flat and peaked again
with changing of bootstrap current and Ohkawa current.

bootstrap current is almost constant after t=l.0s. Thatreason is because Ohkawa current is decreasing with thedensity increasing. On the contrary, the bootstrap
current is incr:easing with the plasma stored energy

Here LIR time due to the extemal inductance is around0.9s and that LIR time due to the internal inductance isaround 0.1s, which is corresponding to the relaxation
time of current profile. Figure 8 shows the time trace ofthe central rotational transform. Before l=1.8s,the t=ll2
rational surface is expected not to be in plasma region
because the current profile is peaked. From l=1.8s to
2.7s, the t=ll2 rctional surface is expected to be inplasma region because the current profile becomes flat.
After e2.8s, the t=ll2 rational surface is expected not to
be in plasma region again because the current profile
becomes peaked. From results of Fig. 4(c) and Fig. 8,
the timing of the disappearance of the m/n=2ll magnetic
fluctuation observed experimentally fairly agrees withthe timing when the t=ll2 rutional surface theoreticallyincreasing. From t=2.2s to 2.1s, the summation ofbootstrap current and Ohkawa current is smaller than
before e2.2s because the bootstrap current is decreasing
due to flat prolile of central electron temperature. After
t=2.7s,the electron temperature profile becomes peaked,
which leads to the increase of bootstrap current and its

more peaked profile. The time evolution of thecalculated net plasma current fairly agrees with theexperimental data. The evolution of plasma current
density calculated with the diffusion equation, eq. (3), isshown in Fig. 7. The calculated plasma current profile

estimated disappears.

7060

5. Summary
In order to identify the toroidal current driving
mechanism, the time evolution of the net toroidal-*,){''t

cs0
E
g4a

I

I

current and its profile are analyzed for LHD plasmas..
We estimate the non-inductive current taking themutual coupling due to the structure surrounding theplasma, such as coils, vacuum vessel and so on, into
account. We study the properties of the non-inductive
toroidal current and find that the bootstrap current is themost probable candidate as the driving mechanism of

I

Ia
\1.
.t

A30
c

ilr

82a

t

c

I

t

910
f

oo

-10

non-inductive current in LHD experiments underbalanced NB injection. The study of non-inductive0

0.2 0.4 0.6 0.8

current in discharges under unbalanced NB injection is a
future subject.
We analyze the time evolution of toroidal current

1

normalized minor radius, pprofile in a fairly high beta LHD experiment with highFig. 7 Time evolution of calculated toroidal current density.

0.8
0.7
0.6
-o 0.5
0.4
0.3
0.2
1.4

toroidal current, where the sudden disappearance of the
mln=2ll magnetic fluctuation is observed, by using acalculation model based on bootstrap current and
Ohkawa current. The timing of the disappearance of the
mln=2ll magnetic fluctuation fairly agrees with thetiming when t=ll2 rational surface theoreticallyestimated disappears. We find that the calculation modelof the toroidal current profile proposed in this paper isquite helpful to the determination of toroidal current
profile in a LHD experiment. Our goal is to establish theCal. resultwith toroidal curret

1.6 1.8 2 2.2 2.4 2.6 2.8 3

general calculation model of the toroidal cunent profile

time(s)in LHD experiments. We should improve the calculation
model through checking the validity of neoclassicalconductivity, the estimation method of the one turnFig. 8 Time trace of the central rotational transform withthe calculated toroidal current based on the time

voltase and so on.

evolution solving.
r29

Watanabe K.Y. et al., Study of Time Evolution of Toroidal current in LHD

t6l A. Iiyoshi et al., Nucl. Fusion 39, 1245 (1999).
I7l H. Chikaraishi et al., in Proc. Int. Power Elec.

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[10] S. Sakakibara et al.,Plasma Phys. Control. Fusion

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130