J. Plasma Fusion Res. SERIES, Vol.4 (2001) 54-60

Dynamic Behavior Caused by Bifurcation Property of Radial
Electric Field in a Toroidal Helical Plasma of CHS
KENJi.
FUJISAWA AKihide*, IGUCHI HaruKazu, MINAMI TAKAShi, YOSHIMURA YASUO, TANAKA
Masayuki,
YOKOYAIvTA
ITOH Kimitaka, SANUKI Heiji, KOJIMA Mamoru, ITOH Sanae-I.1,
MASAKi'OKAMURA ShOiChi, IDA KAtSUMi, ISOBE MitSUtAKA, NISHIMURA ShiN' OSAKABENOMURA IZUMi, SHIMIZU AKihiTO, TAKAHASHI ChihiTO, MATSUOKA KEiSUKE'

HAMADA Yasuji and FUJIWARA Masami
National Institute for Fusion Science, Toki, Japan' 509-5292
lRIAM, Kyushu IJniversity, Kasuga 816-8580 Japan(Received: 5 December 2000 / Accepted: 17 August 2001)

Abstractprobe in a
Bifurcation properties in radial electric field have been studied using a heavy ion beambifurcationoftheoverview
toroidal helical plasma of Compact Helical System. This paper presents anphenomena observed in potential structure and its dynamics, such as potential patterns with transportof thisbarrier formation, and self-organized oscillation termed electric pulsation. Detailed analysis
densityoscillatory state provides an interesting result of spatio-temporal evolutions of potential andprofile during a pulseJike event.

Keywords:
radial electric field, bifurcation and transition, electric pulsation, internal transport barrier, CHS,

HIBP'

neoclassical theory, pattern formation, E -shear

theories, radial electric field is highly nonlinear,

1. Introduction
Plasma is a non-equilibrium matter with richespecially in a low collisional regime, and solitary wave
solutions were found in evolution of radial electric field[9]. Actually, a number of bifurcation phenomena havebeen observed in the experiments of a toroidal helicalnonlinearity, where bifurcation phenomena may be
expected. In fact, the discovery of the H-mode in the
ASDEX tokamak [1] proves the bifurcation nature oftoroidal plasmas. There, the bifurcation in transport

plasma in Compact Helical System (CHS) UOl.

This paper describes patterns of potential profiles
observed in electron cyclotron resonance (ECR) heatedplasmas. The appearance of the patterns is expressed ina bifurcation diagram on a plane of central potential anddensity. A dynamic bifurcated state of nonlinear
oscillation is introduced with its analysis of spatiotemporal evolutions of potential and density profiles'
The experimental bifurcation property is discussed inproperty results in a sudden change of its structure. Also
nonlinear oscillations, edge localized modes (ELMs)'

have been observed and considered to be repetitivetransitions between two distinct structures, L- and Hmodes. Many theories and experiments show that these
phenomena should be ascribed to the bifurcation of theradial electric field (E) t2-61.
The bifurcation nature of toroidal helical plasmas
has been expected with neoclassical theories including

association with the neoclassical property.

helical ripple effects on the transport [7,8]. In such*

@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

C o r r e sp ondin g aut ho r' s e - mail : fuj i s aw a @ nifs. ac.j p

54

Fujisawa A. et al., Dynarnic Behavior Caused by Bifurcation Propertyof Radial Electric Field in a Toroidal Helical ptasma

2. Experimental Devicesof CHS(a)In the Compact Helical System (CHS) tlll, thestructure of internal electrostatic potential has been
investigated using a heavy ion beam probe (HIBp). The

#' r

f
t-r

-Y

CHS is a heliotron/torsatron device whose major andminor radii are 1.0 m and 0.2 m, respectively. The
aspect ratio of approximately 5 is the lowest in toroidal
helical devices at present. The helical ripple coefficient

e

bell

rt
fT

J;

at the edge is eg - 0.3, that is comparable to the inverse
aspect ratio of about 0.2.

The CHS device is equipped with an HIBP whose

beam energy is up to 200 keV. The HIBp has, inprinciple, the capability in simultaneous measurementsof potential, density, magnetic field and their fluctuation|21. In this HIBP, an additional beam sweep system is
set in front of the energy analyzer as well as on theaccelerator side. By using two sweep systems, thecomplicated beam trajectory in the magnetic field of a
toroidal helical device is managed to extend theobservable range widely over almost the whole plasma

0.5

(b)

region [3].
Cesium beam of 72keY is used for the configuration where the presented experiments are performed. In
the HIBP measurements, potential profile is obtained bysweeping the beam trajectory continuously. Timeevolution of potential can be obtained in every a few
millisecond at the fastest. On the other hand, dynamics
or fluctuation of plasma can be detected when the beam
orbit, or observation point, is fixed. A high temporal
resolution up to approximately 500 kHz is attainable bytuning the gain of current-voltage converters.

Bel

-i

J

I

O

I

a

"l
d-

bif urcation

bif urcation

Do
I

Hil

Mexican hat

00.5

3. Bifurcation Characteristics

1n" (1x1o13cm-31

3.1 Stationary Bifurcated Branches

In the CHS plasmas, the observed potential has
been classified into five identical patterns. Figure lashows the typical potential profiles for these fivepatterns. The patterns are termed bell, dome, hill,
Mexican-hat and, well in order from the higher to theFig. 1 (a) Representative shapes of potential profiles thathave been ever observed in CHS. (b) Dependenceof the central potential on the line-averaged
density in the case of ECR heating power of 100
kW. In low density regimes, bifurcations of thepotential profile are observed at two criticallower central potential value.
The Neutral Beam Injection (NBl)-heated plasmas

values of density. Characteristic shapes ofpotential profilesare identical for five regimes.

only show well-pattern, while the electron cyclotron
resonance (ECR) heated plasmas do every patterns
according to the line-averaged density and the ECRheating power. As collisionarity becomes lower byincreasing the heating power or decreasing the lineaveraged density, the central potential becomes higher.
In other words, positive potential patterns of hill, dome
and bell turn to be more probable. These three areconsidered as bifurcated patterns in a low density

regime, since transitions between bell, dome and hill
patterns (or discrete changes in patterns) have been
observed [14,15]. On the other hand, the hill, Mexicanhat and well profiles can change continuously as theline-averaged density alters.
Figure lb shows dependence of the pattern on theline-averaged density. In the diagram, the change of the55

Fujisawa A. et al., Dynamic Behavior Caused by Bifurcation Propertyof Radial Electric Field in a Toroidal Helical Plasma of CHS

pattern is expressed in terms of the central potential
values for ECR-heated plasmas, where its heating poweris fixed at approximately 100 kW. The central potential

heating power (P"6r11 = 300 kW) is applied to the NBIsustained plasma. The solid lines in Fig. 2 show
waveforms of central potential and at an outer radius of

decreases with an increase in the line-averaged density

p = O.59 in the CHS plasmas.
In this discharge, the self-organized oscillation isalong three bifurcated branches; hill, dome and bellbranches. The diagram indicates two critical values ofdensity where bifurcation in potential profile patternoccurs. At the first critical density of n" - 0.5 x l0tl

maintained for a period (l = 50 - 90 ms) in a combinedphase of the ECR and NBI heating. In that period, thewaveform at the center exhibits negative pulses with its
amplitude of -0.6 kV quasi-periodically, while that at p
= 0.59 shows quasi-periodic positive pulses with the

cm-3, the potential pattern bifurcates into hill and dome

branches. As density decreases further and reaches thesecond critical value of n. = o.3 x 1013 cm-3, thesame interval. In both waveforms the pulse-like event

occurs in approximately every 2 ms, or with frequencyof -0.5 kHz. The timescale of a pulse (- a few dozen
microseconds) is much faster than the diffusive one (- afew milliseconds). The oscillation can be recognized as

potential pattern bifurcates into dome and bell-branches.
The change from dome to bell pattern can becharacterized by outward movement of the maximum

E-shear radius from P = 0.3 to p - 0.5. In this low
ECR-heating power (PsspH = 100 kW), the realization of
the hill branch is most probable amongst the three

repetitive transitions between quasi-equilibrium states(see the next section), although the life time of the lower
potential state is much shorter than that of higher
potential state. The other plasma parameters are also
pulsating being synchronized with the potential pulses.
At the same ECR-heating power as the case of Fig'
2, the amplitude and frequency alter with the line-

bifurcated branches in that low density region [15].

3.2 Oscillatory States - Emergence of Electric
Pulsation
As the EcR-heating power increases, branches of

averaged density. The pulsation frequency increases andthe pulse amplitude decreases, with an increase in thedome and bell become more probable to be realized thanthe hill branch. Furthermore, an increase in the ECRheating power realizes a dynamic oscillatory state, thatis termed electric pulsation [6]. The most dynamic
example is shown in Fig. 2. This examples are fromhydrogen plasmas to which almost maximum ECR-

density. By changing the ECR-heating power, we haveobserved a number of variation of pulsation patterns[10]. At the power threshold, the repetitive transitions

are observed to occur between two branches, forexample, dome and hill. There the life time of hill anddome are almost similar [15].

4. Transport Barrier Formation
In the five pattems of profile profiles, the dome andbell-profiles have a rather large E -shear at radii of p =0.3 and p = 0.5, respectively.According to the present
paradigm of E -shear reduction of fluctuation, transportbarrier formation is expected at those radii. In fact,
electron thermal transport barrier has been confirmed forplasmas with the dome pattern profile [7].
Figure 3a shows E -structure around the transportbarrier, using tanh[r/a]-function. The position of the

(b)
I

E

o
c

o

#63366maximum d-shear is located at the normalized radius of

x

50

60

70

P = O.3. The maximum value is estimated as
approximately 40 Ylcmz, and the full width of half90t (rns)maximum of E -shear is approximately 1.4 cm. Figure
3b illustrates the density fluctuation power; in that low
density regime of rn" = 9.5 x 1013 cm-3, the fluctuation in
the detected beam current 616(r) well reflects local
fluctuation of density at the ionization point [18]. The
fluctuation reduction is reduced by 48 Vo at E -shear

Fig. 2 (a) Pulsating behavior of the potentials at thecenter and at an outer radius of p = 0.59 in the
CHS plasmas. This phenomenon is referred to aselectric pulsation. The example is observed in a
combined ECH + NBI heating phase. (b) The time
evolution of the line averaged electron density
measured with an HCN interferometer.
56

Fujisawa A. et al., Dynamic Behavior Caused by Bifurcation Propertyof Radial Electric Field in a Toroidal Helical plasma of CHS
(a)

E

o

E

o t=O ps
a t=150 ps
o t=200 ps
I t=250 ps
D t=3oo ps

(a)

(\l

o

J
trJ

"tr.

o
\t

I\

'.7J\
'l

Jv

(b)

T -\

+

o
3
o
(L

o

t'.
:...r. ---;_...r.-.{.
!
.u..-a- "'4'-"o"'f
0.'l

0.2

0.3

r.

^t.

o.4

0.2

Fig. 3 (a) Fine structure of radial electric field and its

shear around the transport barrier. (b) Density
fluctuation l6n"ln.l integrated from 5 kHz to 70 kHz
around the barrier measured with the HIBP. Here,circles represents the fluctuation for the dome
state with the transport barrier. The fluctuation (inaubitary unit) in the hill state is shown by the

(b)

1

0.4

p

n.I-r..
0.6 0.8

rrrrrrt=150pS
t=25Ops

"fE
()

N

-o

dashed line as a reference.

I0.

t I

.j

go

maximum radius. The result supports the present
paradigm that E -shear should reduce the fluctuation,and should contribute to the formation of transportbarrier. In case of the plasmas with bell-feature, themaximum E-shear is found approximately 6QY/cm2 at
the radial position of p = 0.5, therefore, the fluctuation0

0.2 0.4 p 0.6 0.8

1Fig. 4 (a) Evolution of reconstructed potential profile
during electric pulsation.The potential profile in
the high potential state at t = 0 is represented by

reduction and formation of transport barrier are
expected. The similar feature of electron temperature
profile has been recently observed in Wendelstein 7-AS

the closed circles. The potential profiles after thetransition are shown at f = 150 ps, t=250 /rs and t= 300 rc. (b) Evolution of reconstructed density
profile during electric pulsation using the detectedbeam current signals. The bold line indicates thedensity profile with Abel-inversionin the highpotential state.This suggests the density profile
after transition from the high to the low potential
state becomes a centrally-peaked one.

and TJ-II stellarators tl9,z0l.

5. Barrier Front Dynamics
5.1 Reconstruction of Potential Profile
Evolution During Pulsation

A detailed analysis of the previous example ofelectric pulsation indicates that the electric pulsation is
not simply recognized as repetitive transitions between
two distinct states. The good correlation between soft xray signal and potential pulse allows to reconstruct theat the high potential state (t = 0 1ts), that is the initialpotential profile with the bell feature. The first transition

spatio-temporal evolution of potential profile during
pulsation by taking the central chord of soft x-ray signal
as a reference clock. In a discharge, more than a dozen

happens in radial electric field around p = O.5, and thepotential profile changes into a shape represented by thepulses are availableto produce their statisticallyaveraged waveforms of potential.
Figure 4a shows the evolution of the reconstructed

open circles at t =150 ps. During the next hundred
microseconds until / = 250 1ts, the radial electric field at
the periphery becomes positive, and the potential profile

potential profiles during transitions, using these timeaveraged waveforms. Closed circles represent the profile

alters its shape a little slowly. After the plasma shows
the profile indicated by the closed squares (t = 250 1ts),
57

Fujisawa A. et aI., Dynamic Behavior Caused by Bifurcation Propertyof Radial Electric Field in a Toroidal Helical Plasma of CHS

another transition occurs around the core at p - 0.3.
Then the plasma takes the intermediate potential profile
with the dome feature, represented by the open squares,
at / = 300 ps. The potential profile rather gradually
recovers to the initial state. In other words, the footpoint of a strong shear of the radial electric field moves

should, therefore, play a role in the reformation ofdensity profile. A possible candidate is convective
fluxes induced by poloidal asymmetry in potential
during the pulsation. The detailed analysis clearly
demonstrates that more interesting detail exists to beinvestigated in the electric pulsation.

outwards. In approximately 2 ms later, almost the same

6. Neoclassical Property of Radial Electric

process is repeated.

Field

The detected beam current also exhibits a goodcorrelation with the soft X-ray signals and the potential
signal in a wide range of minor radius. The density ofthe discharges is sufficiently low for us to ascribe thechange in the detected beam current to a local densitychange. This allows to deduce a transient change ofdensity profile during the pulsation except the plasma

The bifurcation nature of radial electric field, thatshould be related with the phenomena introduced in thispaper, has been predicted by the neoclassical theory.
Here, we briefly present an example of the neoclassicalbifurcation characteristics as follows.
A neoclassical calculation for hydrogen is shown in
Fig. 5. Here, we use the Hastings formula to solve theedge [8]. The density profile can be reconstructedusing the time-averaged values of interferometer signals
including pulses. The density profile obtained in thisprocess should mainly reflect high potential state owing
to its longer life time, although some small contribution
in the low potential state is involved.
Similarly to the previous treatment of potential

ambipolar condition li'(E,) =W(E,) [7,8], the assumed
parameters are shown in the Figure caption. We choose
sr, = 0.063 and q- 0.16, where e1 and q are helicalripple coefficient and toroidal coefficient (or inverse
aspect ratio), respectively. Figure 5a show equi-contoursof E on tbe n.-7" plane, and the painted crescent region
indicates an area where two stable and an unstable roots
exist. The contour of E, = 0 is indicated by the bolddashed line. The other contours of constant electric field
are also shown by the thin dashed lines.

signals, statistical averages of the beam current signals
during a pulse are obtained by taking the central chordof soft X-ray as a reference clock. Assuming that thepath integral effect is completely neglected, the ratio ofbeam current before and after transitions Is(/)/Is(l = 0)
are supposed to purely reflect the ratio of the local
density change n"(t)ln"(t = 0). Hence, evolution of thedensity profile during the pulsation can be inferred bymultiplying the ratio to the Abel inverted density profilein the high potential state.

Figures 5b and 5c present changes ofradial electricfield along lines crossing the crescent region. Bothcurves have a bifurcation nature of the radial electricfield. However, as is shown in Fig. 5a, the electric field
is uniquely defined above n" - 1.5 x l0r3 cm-r. No
bifurcation phenomenon, hence, is expected above thecritical density. This feature should be associated with

Figure 4b shows evolution of the reconstructed

the experimental observation that the bifurcation

density profiles. The ratios and the reconstructed density
profiles are presented at three points of time, / = 150 ps,

phenomena have not been observed in a higher densityt = 250 ps and r = 300 ps. At the time of r = 150 /rs, aclear boundary ofthe change is seen at the location ofp

regime at a fixed ECR-heating power.
The ripple coefficients of this example correspond- 0.4. The density increases inside the boundary, while

to p = 0.5 of the CHS plasma. Similar diagrams ofneoclassical bifurcation can be drawn for other radial

it decreases outside. Then, the density profile recovers

positions of CHS. The experimental results obtained todate have not shown any contradiction to neoclassicalexpection in terms of transition time scale, and plasma

to the initial values of the high potential state after thedensity profile modification becomes to be localizedaround the boundary.

parameters where the bifurcation phenomena occur,within the present precision.

The increase in density from / = 0 ps to t = 150 ps
is from 4(0) = 0.4 x 1013 cm-3 to 4(0) = 0.7 x l0r3 cm-3.
The amount ofchange is quite large. In order to explain
this increase in density, necessary change of particle
s't at p - 0.4.
fluxes is approximately 6f - 3 x 19re"--z
The neoclassical fluxes accompanied with the absolutevalue of radial electric field, however, is predicted to be
only about 6'l - l0r5 cm-2 s-r. Another mechanism

7. Discussion and Conclusion
The spatio-temporal patterns of potential introduced
here can be interpreted on the basis of the neoclassicalbifurcation property. A neoclassical calculation can
produce the patterns observed in the experiments [21].
58Fujisawa A. et al., Dynamic Behavior Caused by Bifurcation Propertyof Radial Electric Field in a Toroidal Helical Plasma of CHS' E;0 (kV/m)(a) 0.8
E,='10 (kYlm)'-----H-,€0€r represent the viscous coefficient and the dielectric
constant of the plasma, respectively [22]. Assuming that-j. is a cubic function of E. (corresponding to theneoclassical expectation), the above diffusion equation
gives a solution whose function form is tanh[r/A] with A

0.6*,[lttott\E/Ny, where AE. is the difference betweeno5o o.+

t

.-

F

,r'

(c)

1

E
E

[!

1.5

1

n" {x1013cm-3)(b)rrespectively, and Aj| corresponds to the difference
between the maximum radial currents during the backand forth transitions between the radial electric fields.
The difference of the radial current is experimentally
deduced as Aj| - 5 Nmz t231. The difference of radial
electric field AE is - 6 kV/m. The dielectric consrantsg€r is estimated as 1.3 x l0-8 F/m. Then, the viscouscoefficient p is - 5 mzls, which is comparable to theelectron thermal diffusivity. 'lhe anomaloas viscouscoefficient can be affected by E -structure, as is similar
to the thermal diffusivity. Consequently, the E -structurearound the barrier may have a mutual link, that results ina more complicated E -structure than the simple form oftanh[r/A]. Finer structural measurements around the.-'.u E= -4 (kV/m)

0.5

J

the radial electric fields inside and outside the barrier,E; -3 (kV/m)

E'

,1r-'

F
-Y

I

E'

uJ

I

a

t ti

I
I

(

att'

barrier are an interesting future work.

E.

o.02

E' L
0.04

n" {x1oracm3)

E'

0.5

During the electric pulsation, the change of theradial electric field can be regarded as a limit cycle in

0.6

the bifurcation curves, if the electric pulsation isT"(r)observed at a radius; for example, p = 0.3, the foot-point

Fig. 5 An example of neoclassical calculation. The
assumptions used in the calculation are f(p) = 300
eY, Z.o = 2, dln n.lplldp = 0 and ?ln T"(pll)p = dln
T,lOl/dp = -1.1. We choose rr, = 0.063 and q = 0.16,
which corresponds to p - 0.5 of the CHS hydrogenplasma. (a) Region where multiple steady statesof the barrier for dome-profile. However, the detailed
analysis of the electric pulsation in Fig. 2 shows that thereality should be more complicated. It indicates thatduring the pulse the potential profile changes rather
slowly from dome to the bell pattern. Several processes

exist in n"-[ plane (painted region). Contours ofconstant electric field are shown by dashed lines.
(b) Bifurcation curve of radial electric field as acan be deduced during the pulse-like event; first,
transition from bell to hill-like profile, second, transition
from hill to dome-profile, finatly the profile changes

function of density. 7"(p) = 0.5 keV is assumed. (c)Bifurcation curve of radial electric fielo as afunction of temperature. n.(pl = 5 x 1012 cm-3 isfrom dome to bell profile rather gradually.
An asymmetrical change in potential is suggested
to explain the rapid change in density profile during thepulsation. There non-local effect may play an important

assumed,In potential profile of dome (or bell), the radial electricfield inside P = 0.3 (or p - 0.5) belongs to higher field
branch, while the outside field still remains in low field
branch. In the connection layer at which two bifurcatedd-branches converge, rather strong E-shear is created
to form the transport barrier. Better transport property ofpositive d should partially contribute to improving theconfinement in the core region.
The Er-structure around the barrier allows us toestimate the momentum diffusivity, or the viscouscoefficient, using the diffusion equation for radial
electric field pesei2E,lO* + j,(E,) = 0, where p androle on the structural reformation. A number ofexperimental observations exist to show that an event
happening at a point propagates to other points more

rapidly than in diffusive time scales t24,251. This is aquite unique feature with the structural formation inplasmas, compared to many other examples ofdissipative structures, or self-organized systems. In
order to pursue this non-local effect in plasmas, it isessential to measure physics quantities at different radii

using diagnostics with high spatio-temporal resolution.
This is an important future work.
In conclusion, we have presented several examples
59

Fujisawa A. et al., Dynamic Behavior Caused by Bifurcation Propertyof Radial Electric Field in a Toroidal Helical Plasma of CHSof bifurcation phenomena observed in toroidal helicaltlll K. Matsuoka, S. Kubo, M. Hosokawa et al., inplasmas produced in CHS. In plasmas, thermal variables

Plasma Physics and Controlled Nuclear Fusion
Research 1988 (Proc. 12th Int. Cozl Nice, 1988),

such as temperature and density are nonlinearly coupledwith electric and magnetic fields, and its structure isself-consistently determined by mutual links between

International Atomic Ene r gy Ag ency,Y ienna, 1989,

these variables; for instance, E.-structure changes
transport, then transport changes thermal variables

t12l T.P. Crowley, IEEE Trans. on Plasma Phys. Sci.

(temperature, density and so on), finally temperature andt13l A. Fujisawa, H. Iguchi, S. Lee, T.P. Crowley, Y.
Hamada, S. Hidekuma and K. Kojima, Rev. Sci.

Yol.2,4ll.
22,29r (1994).

density determine radial electric field again. The

Instrum. 67, 3099 (1996).

structure of the plasma cannot be determined not only

by local property but also by long range interaction(non-local effect). As is shown in the present

t14l A. Fujisawa, H. Iguchi, T. Minami et al., J. ofexperiments of CHS, the plasmas provides us, therefore,[15] A. Fujisawa, H. Iguchi, T. Minami et al., Plasma
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t18l A. Fujisawa, H. Iguchi, S. Lee and Y. Hamada,Rev. Sci. Instrum. 68,3393 (1997).

Plasma and Fusion Res. 75, 604 (1999).

an newly interesting object to be investigated as anonlinear and non-equilibrium system [26].

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60