Proceedings of ITC/ISHW2007

Beta and Magnetic Configuration Dependence ofLocal Transport in the LHD Plasmas
Hisamichi FUNABA1) , Kiyomasa WATANABE1) , Satoru SAKAKIBARA1) , Hiroshi YAMADA1) ,Masayuki YOKOYAMA1) , Sadayoshi MURAKAMI2) , Kazumichi NARIHARA1) ,Ichihiro YAMADA1) , Kenji TANAKA1) , Tokihiko TOKUZAWA1) , Masaki OSAKABE1) ,Kazuo KAWAHATA1) and the LHD Experimental Group1)1)2)National Institute for Fusion Science, Toki, Japan
Department of Nuclear Engineering, Kyoto University, Kyoto, Japan

In high beta plasmas on heliotron devices, the magnetic configuration changes as a result of the beta increment. Especially, the position of the magnetic axis is largely shifted outward. On the other hand, it is reported thatthe global confinement of low beta plasmas on LHD strongly depends on the magnetic configuration properties
such as the magnetic axis position. In this paper, the influence of the change of the magnetic configuration on thelocal transport is studied in low beta plasmas at first. Next, the dependence of the local transport characteristics inhigh beta plasmas on the position of the magnetic flux surface are compared with those in the low beta plasmas.
The local transport in the peripheral region of high beta plasmas is found to be more correlated with beta than theposition of the magnetic flux surface, while the local transport in the core region has almost similar characteristics
in the low beta plasmas. These results suggest that the local transport at the peripheral region in high beta plasmas
on LHD is strongly aﬀected by the anomalous transport caused by the increment in the magnitude or the gradientof beta.
Keywords: high beta, magnetic configuration, local transport, International Stellarator/Heliotron scaling
1. Introduction
The plasmas with the volume averaged beta, hβi, valueof close to 5 % have been obtained on the Large HelicalDevice (LHD) [1]. Transport analysis of such high beta
plasmas are made in this paper.
Transport study of helical plasmas has been mainly
done for low beta plasmas up to now. The activity of theInternational Stellarator/Heliotron Scaling (ISHS) has intensively progressed. Scaling laws of the energy confinement time for helical plasmas have been proposed by thisactivity (ISS95 : International Stellarator Scaling 1995)[2], ISS04 : International Stellarator Scaling 2004) [3] ).
According to the ISS04 scaling, the global confinement
property depends on the magnetic configurations as a factor. The factor is introduced as the renormalization factor.
This renormalization factor strongly depends on the magnetic axis position in vacuum or the ellipticity of the outermost magnetic flux surface in the case of LHD, although
the causes have not been clarified.
Recently, as the results of the transport analysis ofhigh beta plasmas with more than 1 % of hβi, it is foundthat the transport property is degraded in high beta plasmas compared with the gyro-reduced Bohm (GRB) modelor the ISS95 scaling. On the other hand, the magnetic flux
surfaces are shifted outside with the increment in beta dueto the Shafranov shift. The purpose of the analysis in thispaper is to clarify that the confinement degradation withauthor’s e-mail: funaba@lhd.nifs.ac.jp

β increment observed on LHD is caused by the change ofmagnetic configuration by the increment in β or it is directly caused by the increment in β or ∇β.
The energy confinement time by the ISS04 scaling,τISS04, is expressed as follows:E
τISS04= 0.134 · a2.28 R0.64 P−0.61 ne0.54 B0.84-ι0.41
E2/3 ,(1)where a, R, P, ne , B and -ι2/3 are the minor radius, the majorradius, the absorbed power, the line average electron density, the volume averaged strength of the magnetic field,
the rotational transform, -ι = ι/2π at the normalized average minor radius, ρ = 2/3, respectively. The advance fromthe previous ISS95 scaling to ISS04 is that the strong eﬀect
depending on the devices or the magnetic configurations isincluded although the dependence on the plasma parameters are similar. This eﬀect is expressed as the renormalizaton factor, fren , which represents the dependence on thedevice or the magnetic configuration. In the case of LHD,
this fren is strongly dependent on the position of the magnetic axis in vacuum, Rvac
ax , and the shape of the outermost
magnetic flux surface. In ref. [3], the dependence of fren onthe Rvac
ax is analyzed in detail and the derived fren values areISS04
shown in table 1. This fren is denoted as fren
hereafter.
ISS04
The frenvalues are evaluated in the low beta discharges.
For LHD, the beta values of the plasmas are in the range ofhβ dia i ≤ 1.5 %.
The main purpose of this study is to distinguish thecauses of the gradual confinement degradation with betafrom following two eﬀects: (1) the change of magnetic

Proceedings of ITC/ISHW2007
configuration by the increment in beta, (2) beta value or thegradient of beta. The global transport study is made taking
the change of the magnetic configuration by the increment
in beta into account. Then, analysis of the dependence ofthe local transport properties on magnetic configuration or
the beta value is made.
DeviceLHDLHDLHD

Rvac
ax [m]
3.60
3.75
3.90

Rvac
geo (2/3) [m]
3.643
3.740
3.849

ISS04
fren
0.93 ± 0.15
0.67 ± 0.06
0.48 ± 0.05

Table 1. Relation among the positions of the magnetic
axis in vacuum, Rvac
ax , the geometric center of the ρ = 2/3
ISS04
magnetic surface in vacuum, Rvac
in thegeo (2/3) and frenISS04 scaling for three magnetic configurations on LHD[3] .

[7] . Here, T i = T e and ni = f ne , are assumed, where f isdetermined by the following relation in order to make thekinetic stored energy become equal to the measured storedenergy.
∫3
W pdia =(1 + f ) ne T e dV,(2)2
where the integral is the volume integral. In this paper, thelocal transport is analyzed based on the measurements ofT e and ne . By the assumption of T i = T e , the equipartition
power between electrons and ions , Pei = 0 is assumed.
Therefore, behaviour of the eﬀective transport coeﬃcients
which are evaluated byexp
χeﬀ = (χexp
e + f χi )/(1 + f )(3)is studied.

3. Dependence of the local transport on the magnetic
configuration in the low beta regime
This paper consists of the following sections. In section 2, the method of transport analysis are described. The
dependence of the local transport on the magnetic configuration in the low beta regime is investigated in Section 3.
In Section 4, the relation between the local transport coefficients and the change of the geometric center position of
the magnetic flux surface by increment in beta is shown.
Then, its eﬀect on the confinement degradation in the highbeta region is discussed and a summary is provided in Section 5.

2. Experimental setup and the method of transport
analysis
LHD is a heliotron type device with the poloidal period number l = 2 and the toroidal period number m = 10.
The major radius position of the magnetic axis in vacuum,
Rvacax = 3.5 ∼ 4.1 m and the average minor radius a ' 0.6 m.
The magnetic field strength at the toroidally averaged magnetic axis in vacuum, B0 , is up to almost 3 T.
The value of the volume average beta, hβ dia i, is derived from the diamagnetic measurement [4]. The profilesof electron temperature, T e , are measured by the Thomson
scattering with YAG lasers. The power deposition profiles
of NBI are calculated by a three-dimensional Monte Carlosimulation code [5]. These NBI deposition profiles, which
are used in the local transport analysis, include the broadening from the birth profiles of NBI by the finite orbit effect. Here, the magnitude of the ionized power of NBI
in the calculation is set to be equal to the experimentallyestimated NBI absorbed power, which is the port throughpower minus the shine through power and contains about
10 % error [6] .
The thermal transport coeﬃcients of electrons andexp
ions, χexpe and χi , respectively are evaluated by using onedimensional transport code for helical plasmas, PROCTR

In this section, the dependence of local transport onmagnetic configurations is investigated in the low beta region first in order to clarify the causes of the degradationof the local transport property in the high beta region.
The dependence of the local transport coeﬃcients on
the magnetic configuration, which is represented by thegeometric center of the magnetic flux surface in this paper, are investigated at some minor radius positions in the
low beta plasmas. In order to evaluate this dependence,
the renormalization factor for local transport coeﬃcients,
grenχ , is introduced. And the reference local transport coeﬃcient, χISS04 , based on the global confinement scaling of
ISS04, is used in this analysis.
The energy confinement time by ISS04, τISS04, is exEpressed as follows by using the non-dimensional parameters, such as the normalized Larmor radius (ρ∗ ), νb∗ , β, A pand -ι2/3 .
1.06
τISS04= Cτ · τBohm · ρ∗−0.79 β−0.19 νb∗0.00 A0.07
p -ι2/3 , (4)E

where τBohm is the global confinement time based on theBohm type scaling and Cτ is a constant. Then, a modeled transport coeﬃcient, χISS04 , which has the same nondimensional parameter dependence as ISS04 is introducedin order to use as a reference for the local transport coeﬃcient:χISS04 = Cχ · χBohm · ρ∗0.79 β0.19 νb∗0.00 A−0.07-ι−1.06p
2/3 . (5)Here, Cχ is determined to make the average of χeﬀ /χISS04
become unity in the low beta region (hβ dia i < 1% ) in thedata set of Rvac
ax = 3.60 m (Table 2) .

Proceedings of ITC/ISHW2007
ρ0.5
0.7
0.9

Cχ
0.104
0.133
0.294

8.00

...

: global gren χ

Table 2. The normalized average minor radius, ρ, and thecoeﬃcient in χISS04 expression, Cχ .

6.00
: ρ=0.9•−1/(αP +1)−1/(αP +1) vacgISS04/ fren(Rax = 3.6 m),renχ ≡ fren(6)where αP is one of the scaling coeﬃcients when a global
scaling law is expressed as
τE = fren nαn PαP BαB aαa RαR ,(7)where n is the density and αn , αP , αB , αa and αR are exponents for each parameter. In the case of ISS04, αP =vac−0.61. The values of gISS04
renχ at the Rax = 3.60, 3.75 and3.90 m configurations are shown in Table 3.
Rvacax [m]
3.60
3.75
3.90
Table 3.

Rvac
geo (2/3) [m]
3.643
3.740
3.849

ISS04
fren
0.93
0.67
0.48

gISS04
renχ
1.00
2.33
5.48

ISS04
frenand gISS04
renχ for three magnetic configura-: ρ=0.7

gren χ

Therefore, the magnitude of χeﬀ /χISS04 represents theratio of the normalized transport coeﬃcients to the value
in the configuration of Rvac
ax = 3.6 m at each minor radius position. The ratio χeﬀ /χISS04 increases as Rvacax moves
torus-outward. This tendency of the local transport coefficients seems to agree qualitatively with the global confinement property. Then, the average value of the experimental results is defined as the renormalization factor forthe local transport coeﬃcients, grenχ , at this local position.
The factor grenχ is derived for various magnetic configurations and radial positions. The relation between grenχ andthe geometric center position of each magnetic flux surface, Rgeo (ρ), in vacuum is shown in Fig. 1. The symbols,and 4 correspond to the values at ρ = 0.9, 0.7and 0.5, respectively. The data in Rgeo (ρ) = 3.6 ∼ 3.7 m,Rgeo (ρ) = 3.7 ∼ 3.8 m and Rgeo (ρ) = 3.8 ∼ 3.9 m are forvac
the configurations of Rvac
ax = 3.60 m, Rax = 3.75 m andvac
Rax = 3.90 m, respectively.
As the geometric center of the magnetic flux surface
is shifted torus outward, grenχ increases at any minor radius positions. Moreover, it is found that the dependenceof grenχ on the geometric center position of the magnetic
surface is almost same at diﬀerent minor radial positions.
The closed circles ( ) in Fig. 1 represent the renormalization factor for transport coeﬃcients which is derived
ISS04
from the global scaling ISS04, gISS04
renχ . This grenχ is related with the global fren and it is compared with the renormalization factor for the local transport coeﬃcients, grenχ .
−1/(αP +1)
The parameter gISS04
renχ is is derived by normalizing frenvacwith the value at Rax = 3.60 m as the following,

4.00: ρ=0.5

2.00

0.00
3.60

3.70

3.80

3.90

Rgeo(ρ) [m]Fig. 1 Dependence of grenχ on the geometric center position of
the vacuum magnetic flux surface, Rgeo (ρ). fren chi.

tions on LHD.

It should be noted that an assumed spatially uniform
thermal transport coeﬃcient, hχi, in the Rvac
ax = 3.90 m configuration is about 5 times larger than that in the Rvac
ax =3.60 m configuration when it is evaluated based on ISS04.
The values of gISS04
renχ almost agree with the values of grenχwhich are evaluated from the local transport analysis fromthe results in Fig. 1. Here, as the flux surface corresponding to gISS04
renχ , the ρ = 2/3 surface is chosen because it isassumed that Rgeo (2/3) represents the property of a magnetic configuration.

4. Local transport property in the high beta regime
and its dependence on the magnetic configuration
In Fig. 2, the dependence of χeﬀ /χISS04 on Rgeo (ρ) in
the high beta regime are compared with the dependence ofχeﬀ /χISS04 on Rgeo (ρ) in the low beta regime with various
magnetic axis positions. Figs. 2, (a) (b) and (c) show theresults for ρ = 0.5, 0.7 and 0.9, respectively. The closed
circles (• ) represent the relation between χeﬀ /χISS04 andRgeo (ρ). At the all positions of ρ = 0.5, 0.7 and 0.9, themagnitude of χeﬀ /χISS04 increases with the torus outward
shift of Rgeo (ρ) of the each magnetic flux. The dependenceof increment in the ratio χeﬀ /χISS04 on Rgeo (ρ) is steeper atthe larger ρ position. One reason of this is that the magnitude of the shift of Rgeo (ρ) is smaller at larger ρ. Atρ = 0.9, an abrupt increment in χeﬀ /χISS04 is found around
Rgeo (ρ) ' 3.70 m, where hβ dia i is about 1.0 ∼ 2.5 %. The
symbols of 4,andin Fig. 2, are the same as inFig. 1, respectively. They represent the dependence of thenormalized thermal transport coeﬃcients on the geomet-Proceedings of ITC/ISHW2007

(a)

...

Raxvac = 3.60m
γ = 1.254
10 ρ = 0.5

χeff/ χeISS04

ric center position of the magnetic flux surfaces which isevaluated from the local transport analysis.
The degradation of the local transport with the increment in hβ dia i seems to be comparable with the degradationby the torus outward shift of the magnetic flux surface at
ρ = 0.7. Moreover, The degradation at ρ = 0.5 seems to
be comparable with or slightly smaller than the degradation by the torus outward shift. On the other hand, at theperipheral region of ρ = 0.9, it is observed that the degradation of the local transport is larger than that predictedfrom the torus outward shift of the magnetic flux surface.
This results shows that some eﬀects which are caused directly by the beta value or the gradient of beta may exist at
the peripheral region.

Raxvac=3.75m

1Raxvac=3.60m: local grenχ

5. Summary
3.60[1] O. Motojima, et al., Proc. of 21st IAEA Fusion Energy
Conf., OV/2-1, (Chengdu, 2006).
[2] U. Stroth, M. Murakami et al., Nucl. Fusion, 36 1063 (1996).
[3] H. Yamada, et al., Nucl. Fusion 45 1684 (2005).
[4] K.Y.Watanabe et al., Nucl. Fusion, 45 1247 (2005).
[5] S. Murakami et al., Trans. Fusion Technol.,27,256(1995).
[6] M.Osakabe, Y.Takeiri, et.al., Rev. Sci. Instrum., 72,590(2001).
[7] H.C. Howe, ORNL/TM-11521 (1990).
[8] K.Y. Watanabe et al., Plasma Phys. Control. Fusion, 49 605(2007).

3.70

3.80

3.90

Rgeo(0.5) [m]...

Raxvac = 3.60m
γ = 1.254
10 ρ = 0.7

χeff/ χeISS04(b)

Raxvac=3.90m

Raxvac=3.75m

1Raxvac=3.60m

3.60: local grenχ
3.70

3.80

3.90

Rgeo(0.7) [m](c)...

Raxvac = 3.60m
γ = 1.254
10 ρ = 0.9

χeff/ χeISS04

The following results are obtained by comparing thedependence of the local transport coeﬃcients on beta values and the geometric center positions of the magnetic
flux surfaces with the above relations between the local
transport coeﬃcients and the geometric center positions of
the magnetic flux surfaces in the low beta region . The
degradation of the local transport with the increment in hβiseems to be comparable with or slightly smaller than thedegradation by the torus outward shift of the magnetic flux
surface at ρ = 0.7 or 0.5. On the other hand, at the peripheral region of ρ = 0.9, degradation of transport coeﬃcients
which is larger than those predicted from the torus outward
shift is observed. It is considered that some eﬀects which
are directly caused by the beta value or the gradient of beta
may exist.

Raxvac=3.90m

1Raxvac=3.90m

Raxvac=3.75m

Raxvac=3.60m

3.60

3.70

: local grenχ
3.80

3.90

Rgeo(0.9) [m]Fig. 2 Dependence of the local transport coeﬃcients χeﬀ /χISS04
on Rgeo (ρ). (a) ρ = 0.5, (b) ρ = 0.7 and (c) ρ = 0.9.