J. Plasma Fusion Res. SERIES, Vol. 4 (2001) 399402

Analytical Study of Ambipolar Radial Electric Field in LHD
YoKOYAMA Masayuki*, IDA Katsumi, SANUKI Heiji, IToH Kimitaka and LHD Experimental Group
National Institute for Fusion science, oroshi-cho 322-6,Toki 509-5292, Japan(Received: 5 December 2000 / Accepted: 16 August 2001)

Abstract
The ambipolar radial electric field (E ) properties in LHD have been investigated. The E,is obtainedwith the neoclassical flux based on the analytical formulae. This approach is appropriate to clarify E,properties in a wide parameter range in a more transparent way. The comparison between
calculated Eand experimentally measured one has shown qualitative agreement. Based on this assuranceof thisapproach for LHD, wide range calculations have been performed to clarify the parameter region (Rwhere multiple E, solutions can exist. This is the region where E. domain interface can be established)torealize improved confinement as experimentally confirmed in CHS.

Keywords:
ambipolar radial electric field (E ), LHD, neoclassical flux, multiple E, solutions, E, domain interface

1. Introduction

towards improved confinement.
The analytical model for calculations is explained

Recently, significant roles of radial electric field(E ) in toroidal plasmas have been widely recognized tobriefly in Sec. 2. In Sec. 3, calculated E. is comparedwith experimental ones for LHD. The qualitative 1orachieve improved confinement modes [1]. The potential

pulsation has also been experimentally observed in CHS

even semi-quantitative) good agreement between thosevalidates the use of analytical calculation to clarify theparameter region of Rr. Finally, a summary is given in
Sec.4.

[2]. This phenomenon is physically interpreted as rhetransition between two stable states of E" with negative(ion root) and positive (electron root) values. The
electron thermal transport barrier has also been
identified in CHS [3], which contributes to the reductionof the density fluctuation associated with the structuralvariation of the E, profile. This gives the experimental
evidence of the E, domain interface as theoretically
predicted [4]. The predicted parameter (density and2. Model Description
Neoclassical particle fluxes are calculated based on

the analytical formulae [7]. The E, is then determinedwith the ambipolar condition, l" = li, where .{ denotes
the flux of plasma species with j = e for electron and j =i for ion. The use of analytical formulae stimulates toclarify E, properties in a wide parameter range in a moretemperatures) region for establishing E, domaininterface, where multiple E. solutions can exist for theambipolar condition are in semi-quantitative goodagreement with experimental results [5]. This gives theexperimental validation to consider the possibility of
improved confinement based on clarifying the plasma

uansparent way.

The magnetic configuration control is possible in
LHD with the shift of the magnetic axis position (R-) invacuum. For the Ro, = 3.75 m case, the poloidal
inhomogeneity of B ( | e,,o I I is almost equal to thegeometrical inverse aspect ratio. On the other hand, forparameter region where multiple E. solutions can exist(Rr). Based on these experimental confirmation,
parameter regions of R, are clarified for LHD [6]*C o rre sp ondin g aut ho r' s e - mai I : y o koy ama @ LH D. nifs. ac.j p399

@2001 by The Japan Society of plasma
Science and Nuclear Fusion Researchyokoyama M. et al., Analytical srudy of Ambipolar Radial Electric Field in LHD

where (4 () are (poloidal, toroidal) angles, respectively'

the R* = 3.60 m case, it is about 213 times smaller. The
contribution of satellite helicity, €1.s, becomes larger for

In the original formulae, the poloidal

inhomogeneity of B (denoted by d in ref. [7]) is simply
defined by the geometrical inverse aspect ratio, which isno longer valid. Thus, this d is replaced by I er,o | . The

Ro* = 3.60 m case even for inner region. Anothersatellite helicity, €3.1s, also becomes apparent for the R=3.60 m case with the amplitude almost equal to that ofhelicity is defined there (denoted by 6) through themodified Bessel function based on the magnetic
potential analysis [8], which is also replaced by thet1.6 at the plasma edge. Thus, the magnetic fieldstructure of LHD in vacuum - low beta conditions forthis range ofR- can be approximately expressed as
B = Boll+ €r,0 cosO + e r.ro cos(20effective helicity, es, as- l0()+ e,.o cos(9- 10() + e,,ro cos(39- 106)l ,(1),r=re1*+ el*+ €3,rc+21€,rc(€.,rc+ €,,r0)cos0+ €r,ro t:,ro cos2?f d0 l2nQ).

well and maximum E, values in the electron root andvalues for ion root in a wide range of n, are also well
reproduced. This indicates that these calculations canqualitatively or even semi-quantitatively explain theThe rotational transform is also given by equilibrium
calculation.

The neoclassical particle fluxes (and then theambipolar E,) are calculated by determining thetendency (2), which assures the analysis utilized in thispaper. It is noted that the difference of ion root values
for different flux surfaces is relatively smaller than thatof electron root. This implies that the response of the E,value on plasma parameters is more sensitive in thecoefficients, e, in eq. (1) for the model magnetic field(such as the effective helicity, ea) and equilibrium
quantities obtained from equilbrium calculations.

3. Comparison with Experimental Resultsand Existence Region ol BM in LHD
Some dependencies of E, on plasma parameters
have been identified on LHD such as (1) higherelectron root. It is also recognized that the maximum

positive E, for each flux surface increases as ptemperature (7) is required to realize electron root E, forhigher density (n), (2) E, becomes negative for higher n(by gas-puffing and/or pellet injection) and becomes? 104ii!ii

positive for lower z [9]. The tendency (2) is firstlyexamined. During the n scan experiment, n is variedwith gas-puffing with keeping electron and ion
temperatures (7" and Z;) almost unchanged both in****.iiiii---i-......... -.i---...-.--i----.- ---i------1.5 104 --x....-..
xixii :'-!i'rviiii
i":iii-11.---.......

.iAri--o"--'ioiB

ii!

:Ii

J 104 ......-....i..----------i---------i---------i--------i-:a;:r!

values and profiles. Thus, calculations are performed bytaking electron density (n") as a parameter based on nooooi'i*i^a------.- -.ixolsoooand Z profiles measured during this series ofL.,:

experiments (at 2.2 s of the discharge 13935, 13935-2.2
s with R* = 3.75 m and B = 1.5 T). The n" profile shapeis fixed in the calculations to clearly see the dependenceof E, on the z, value itself. Figure I shows E, solutions
as a function of n" ot three flux surfaces, p = 0.85 (O),0.90 (A) and 0.95 (x) associated with Fig. 1(a) in ref.
[9]. Here p is the normalized minor radius. The
calculated E, is only one negative soultion for n"> O.7 xl}te = n"G) and solutions become multiple (three) for n,a n"ra. The middle solution is unstable and one of theother two solutions is considered to be realized. As
compared with Fig. l(a) in ref. [9], ne(c) agrees rather0.

-i!
iJA--- -----i---iiiiIiiii------.. -i--- ---------i -|:-ii-.-'{-'-'--'Ii"*-ii iiiii ib-!
:---'r --..'---'i --'--'-'T-- -'"- "i-'----'-l- ---'-"'t"iI-5000-l

104

0 0-b 'l l.b 2 t.t

3

ne 11gle,n-31Fig. 1 calculated E at p = 0.85 (o).0.9 (A)and 0.95 (x)based on density and temparature profiles for13935-2.2 s by varying only the density as aparameter.

400

Yokoyama M. et al., Analytical srudy of Ambipolar Radial Electric Field in LHD

increases. The increase of VZ" is effective to realize theelectron root even for lower Z" as will be reported in a
separate paper. The VZ" increases towards the plasmaedge in this case (13935-2.2 s) associated with edge

multiplied with the amplification parameter (namp,namp = 0.5, l,2,5) with keeping its profile unchanged.

pedestal fromation tl0l.This gives enhanced

1.5 101(maximum) positive E" for outer radii even with loweriii^**qi i
I 101 :i?

7".

The difference of n"61 between cases ofR_ = 3.75
m and R- = 3.60 m is explained in ref. [9] as the resultof the difference of temperature ratio (T"lT). One mayii

io!6A;!:|i;:;iiii.i..---.--....i.--..-...-.

iI:!

;o!:i.............i........
Eooo ........ j.*:...
rio"iril:lu!
o"uii
....

consider that this difference can be explained by thedifference of the magnetic configuration. However, thisis less influential to vary n",",. Figure 2 shows calculated
E, solutions as a function of n"for three cases: solutions
for original data of temperatures (T"lTi - 1.5) at p = Q.9

in0 T:-ioii**ro- ii-5000of 13935-2.2 s with Ro, = 3.75 m (O), modified
temperature values (T"lT; = 1.0, T" is decreased

t:i+-^"i;;:::1'--trriraoo-. ;ii -"!al^, --Trrrrr !1.l:.lmd".r*",-.--.i.-*-.-.j.-'l++l.ir.iiiiji
irlalllada aaa aa-l l0{ u u.c I t.b 2 't.5 3artificially) with temperature gradients being kept thesame as those at p = 0.9 of 13935-2.2 s (A), andtemperatures (values and profiles) are kept the same as
those at p = 0.9 of 13935-2.2 s but with magnetic fielddata of Ro, = 3.60 m (x). The n"1"; does not vary for
different magnetic field data (R- = 3.60 m and 3.75 m)with the same temperature values and gradients. On theother hand, it reduces for a case with T"lTi =1.0 evenwith the magnetic field data of R* - 3 .7 5 m. Thus, it isunderstood that the density threshold for transitionsn" I l0lsm-3 J
Fig. 2 Calculated E, as a function ol n"lor three cases:
solutions for original data of temparatures (T"lTi 1 .5) at p
= 0.9 of 13935-2.2 s with 8", = 3.75 m (O ),modified temperature values lT"/Ti = 1.0, I

isdecreased artificially) with temperature gradients
being kept the same as those at p = 0.9 of 139352.2 s lAl, and temperatures (values and profiles)
are kept the same as those at p = 0.9 of 1393b-2,2
s but with magnetic field data ol R,,= 3.60 m (x).

from ion root to electron root is influenced by thetemperature ratio (7"/7,) rather than by the difference ofmagnetic configuration, which verifies the statement inref. [9]. Results for neoclassical calculation shown in3Fig. l(b) in ref. [9] are obtained with temperature
profiles withT"lTl - 1 with almost same gradients as
those of 13935-2.2 s. Actually, if one performs
calculations with T"lT,) l, n"sl becomes larger than that

2.5
2shown in that figure.

Now, the dependence of E. on temperature values0)

Jis investigated to consider the above mentioned

1.5

tendency (1). The tendency (1) can be expressed in otherwords as "electron root is possible even for higher z"
when temperatures are increased". Thus, extensiveIcalculations are performed in a wide temperature space,0_5(Ti, T"), taking n" as a parameter. Temperatures areamplified with keeping profiles to explore temperatures
up to 3 keV (at P = 0.9) both for T; and 7". Figure 3
shows the dependence of R, on n". The E, is singly
negative below the lower boundary and singly positive
beyond the upper boundary. The region between twoboundaries for each case corresponds to R,,. The
original value of n, is about 0.75 x l01e m-3, which is0T i IkeV]
Fig. 3 Dependence of R, on n parameterized by namp
(namp = O.5, 1, 2,5) based on experimental data
at p = 0.9 of 13935-2.2 s.
401

Yokoyama M. et al., Analytical Study of Ambipolar Radial Electric Field in LHD

It is seen that higher temperatures (especially T") aredemonstrated.
Based on this assurance of this approach for LHD'
wide range calculations have been performed to clarify

required to reach Ry and to rcalize the electron root forhigher n. This indicates that E, tends to become positive
as n is decreased with keeping temperatures. Thistendency well describes the experimentally observed

the parameter region for Ry, which is the region where

an E, domain interface can be established' As n isincreased, R, shifts towards higher temperatureone (tendency (1)). It can also demonstrate that E,
shows transitions from negative to positive as n" isdecreased for fixed temperatures (tendency (2)). The(especially Z,), which is consistent with above
mentioned experimental results for the transition fromion root to electron root below the threshold n value.

density scan calculations for Figs. 1 and 2 are regarded

as a slice cut of this diagram (Fig. 3) for fixed
temperatures. Thus, this diagram is valuable to grasp
general tendencies of E, on a wide range in (Ti, T")This systematic wide range calculations can give

space with density dependence. It should be noted thatthe calculations have been carried out in the density andtemperature space so as to be easily referenced in theexperiments. Theoretically, the key parameter is theelectron collisionality which is changed by n" and T"-

appropriate guidance towards improved confinement.
Further calculations for a guidance and its experimental

comprehensive understandings of experimentallyobserved tendencies of E, properties, which can give an

test will appear in a separate paper.

Acknowledgements
This work has been supported by grant-in-aid from

The weak T;-dependence of lower boundaries in Fig. 3
implies that the ion collisionality is of lesser important.
The above mentioned conditions to reach Ry and/or torealize electron root correspond to decrease electron

the Ministry of Education, Science, Sport and Culture(Monbusho), Japan.

References

collisionality.
As described above, ambipolar E, determined basedon extended analytical formulae for neoclassical particle

t1l K. Itoh and S.-I. Itoh, Plasma Phys. Control. Fusion

fluxes can be regarded to explain qualitative dependence

l2l A.Fujisawa et al., Phys. Rev. Lett. 79, lO54

of E. on temperatures and density observed in LHD.
This qualitative agreement with experimental

t3l A. Fujisawa et al., Phys. Rev. Lett. 82, 2669

observations can assure this approach to examine thepossibility of E, domain interface in LHD towards(1999), A. Fujisawa et al.,Phys. Plasmas 7, 4152
(2000).

38, 1 (1996).

ft997\.

t4l K. Itoh, H. Sanuki, S.-I. Itoh, A. Fukuyama, A.

improved confinement.

Fujisawa, M. Yagi and K. Ida, J. Plasma Fusion
Res. Suppl. 74,282 (1998) (in Japanese).

4. Summary

t5l H.Sanuki, K. Itoh, M. Yokoyamaet al., J. Phys.

The ambipolar E, properties have been examined
for LHD as a first step to explore the possibility of E,
domain interface towards improved confinement. The E,
is obtained based on the ambipolar condition with theneoclassical flux calculated by the analytical formulae.
This approach is appropriate to clarify E, properties in awide parameter range in a more transparent way.

Soc. Jpn. 69,445 (2000).

t6l A. Iiyoshi et a/., Nucl. Fusion 39,1245 (1999).
t7l L.M. Kovrizhnykh, Nucl. Fusion 24,435 (1984).
t8l A.I. Morozov and L.S. Solov'ev, Reviews ofPlasma Physics Vol.2, p.1 (ed. M.A. Leontovich)Consultants Bureau, New York (1966).
l9l K. lda et al.,IAEA-CN-77lEX9l4 presented at l8th

The calculated E, is compared with experimentallymeasured ones to check the validity of this approach.
The experimentally observed tendencies arequalitatively well reproduced. The threshold value of z

IAEA Fusion Energy Conf., Sorrento, Italy (2000).

tl0l N. Ohyabu, A. Fujisawa et al.,Phys. Plasmas 7,1802 (2000).

for the transition from ion root to electron root is also402