Proceedings of ITC/ISHW2007

Extensions of the International Stellarator Database
by High-β Data from W7-AS and LHD
A. Weller1,*, K.Y. Watanabe2, S. Sakakibara2, A. Dinklage1, H. Funaba2, J. Geiger1, J.H. Harris3,
R. Preuss1, Y. Suzuki2, A. Werner1, H. Yamada2, M.C. Zarnstorff4,
W7-AS Team1, LHD Experimental Group2
1

Max-Planck-Institut für Plasmaphysik, IPP-Euratom Assoc., D-17491 Greifswald, Germany
2
National Institute for Fusion Science, Toki, Gifu 509-5292, Japan3
Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA4Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA(Received 15 October 2007)

The ISS04 scaling of the energy confinement in stellarators/helical systems has been derived from selected dataof the International Stellarator/Heliotron Confinement Database (ISHCDB), taking only few high-β data into
account. Therefore, the basis for extrapolations to the reactor regime is insufficient. In the last years, regimes withreactor relevant beta became accessible in the Large Helical Device (LHD) and the W7-AS Stellarator. The high-β
confinement regime is close to operational boundaries determined by degradation of the equilibrium surfaces, bystability limits of pressure driven MHD modes and by available heating power. This may lead to limits of theconfinement and to modifications of scaling laws due to changes of the underlying physics. Therefore, an effort ismade to establish and to extend the high-β data subset in the ISHCDB. The data are compared with existing scalinglaws and predicted operational limits. The magnetic configuration has a significant impact on the confinement. In
particular, a deterioration of the confinement with increasing beta is found in LHD which can partially be attributed
to changes of the configuration. In order to identify the most important physical effects additional parameters arerequired to characterize the local transport and the predicted and experimental MHD properties.
Keywords: Stellarator, Helical System, Confinement, Transport, High Beta, Confinement Database, BetaLimit, Magnetic Configuration

1. Introduction
The ultimate goal of the international stellarator
program is to provide a basis for a economicallyattractive fusion energy source. The prospects of themanifold of different configurations and approaches has
to be assessed by inter-machine comparisons of theachieved global plasma parameters and local transport
properties. Likewise, the collection of reference data fromexisting machines will allow to evaluate the benefit ofconfiguration optimization as anticipated in the newW7-X [1] and NCSX [2] devices being presently underconstruction. Scalings of the global confinement havebeen established based on the analysis of different low-β
ECRH (electron cyclotron resonance heating) and NBI(neutral beam injection) scenarios investigated in several
stellarators and helical devices [3]. An extended ISHCDB
database (including a first high-β dataset from W7-AS)
resulted in the proposal of a new unified scaling of theconfinement time in stellarators based on empirical
renormalization factors depending on the configuration orcorresponding author’s e-mail: arthur.weller@ipp.mpg.dedevice [4]. In order to get a more detailed understandingof the energy transport in stellarators and helical devices
the international database effort is presently extended by
the so-called International Stellarator/Heliotron Profile
Database (ISHPDB) [5]. This activity attempts thedocumentation and analysis of 1-d data for various topics
including local energy and particle transport. With regard
to high-β physics, first studies of the effect of themagnetic configuration on the local transport in LHD
high-β discharges have been made [6]. In addition, thedefinition of an appropriate set of configuration andplasma parameters is required to characterize thedependence and impact of ideal and resistive MHD
modes on the magnetic configuration [7].
A compilation of high-β results in LHD and W7-AS
including comparisons among each other as well as totokamak results can be found in refs. [8], [9], [10], [11].
Until 2006/2007 further significant progress has beenachieved in LHD reaching volume averaged values ofβ ≈ 5% [12]. These new data are particulary important
to include in the ISHCDB/ISHPDB, since they bridge theProceedings of ITC/ISHW2007

gap to the reactor relevant β-vales. Key issues concern
the β-dependence of the confinement and the physicalmechanisms determining the stationary level of β .
The use of the database for inter-machinecomparisons depends crucially on a clear definition of aset of key parameters and on standardized analysis
procedures. However, in the high-β regime someparameters are not easily accessible, and their definition
has to be reconsidered. Most importantly, theidentification of the plasma boundary and hence thedetermination of the effective plasma radius requires amore sophisticated analysis, since the plasma edge regionof high-β plasmas is usually characterized by stochastic
field layers where a significant pressure gradient is still
maintained [13], [14], [15]. For practical reasons, themeasured pressure profiles are fitted by equilibria basedon the assumption of nested flux surfaces as calculatedwith the VMEC code [16]. In LHD, the flux contour
which contains 99 % of the measured kinetic plasmaenergy has turned out to be the most appropriate measureof the plasma edge location.
In W7-AS high-β discharges are effectively limitedby the divertor structures, and therefore the plasmaboundary is identified by the intersection of flux surfaces
with the divertor. This is achieved by using theSTELLOPT code [17] which is based on VMEC anditerates for equilibria consistent with the measured
diamagnetic energy and kinetic data [18]. The pressure
induced shift of the plasma axis can be compensated by
an approprate vertical field so that plasmas with themaximal possible plasma volume were usually
established.
In this paper, the main focus is to characterize andcompare different sets of high-β data from LHD andW7-AS and to discuss their implementation in theISHCDB. Some preliminary results on the scaling of theglobal confinement will be presented. In particular, theW7-AS high-β dataset is used to identify differences in
the confinement compared to low-β plasmas by aprobabilistic model comparison approach [19], [20].
Finally, some remarks about possible extensions towards
MHD related data will be made.

Fig.1

β from the combined W7-AS datasets versus
the time in which δ β β ≤ 10% , (normalizedto τ E ,exp. ).

plasma boundary just in contact with the divertor trough.
Most of the configuration parameters (effective plasma
radius, axis position, values of the rotational transform,volume averaged magnetic field, etc. as well as plasmadata such as β were taken from the VMEC calculation.
A second compilation was made in 2006 using asystematic W7-AS database search that was constrained tofind all high-beta cases without significant toroidal plasma
current. In a second step, only discharges were selected,
which showed a quasi-stationary behaviour (for Δt τ E ).
Using one or more time points per discharge during
quasi-stationary periods almost 400 entries in this dataset
were generated. The data cover a variety of different
configurations with the majority of cases close to theoptimum confinement at ι ≈ 1/ 2 (rotational transform).
The achieved β-values in W7-AS taken from a
combination of the 2 datasets are shown in fig. 1 plottedversus the time of the quasi-stationary period (normalized
to the confinement time). The dashed horizontal lines
indicate an equilibrium limit based on a simple model for a
critical Shafranov shift for 2 particular configurations.
Whereas in low-iota configurations the maximum achieved
beta is clearly limited by equilibrium effects, the maximum

2. W7-AS High Beta Data
A first database consisting of about 200 entries wascompiled in 2003 [8] based on cases for which dedicated
VMEC or STELLOPT calculations were made fordifferent reasons. Standard parabolic pressure profiles wereused in the VMEC calculations and (small) net-currents
were modelled with a standard current profile. In single
cases it could be shown that the experimental pressure
profiles were close to parabolic apart near the plasma edge.
The equilibria were calculated in such a way as toreproduce the measured diamagnetic energy and theFig.2 Energy confinement times normalized to ISS95(combined W7-AS datasets). The data refer todifferent configurations. The renormalization factor
according to [4] for these data is f ren = 0.86 ± 0.18 .

Proceedings of ITC/ISHW2007

beta at higher iota is mainly determined by the available
heating power. The corresponding energy confinement
times normalized to the ISS95 [3] scaling values are given
in fig. 2 as a function of β . Although a detailed
equlibrium analysis [15] revealed a deteroration of thelocal transport in the outer plasma region with increasing
beta due to the expansion of the stochastic layer, no such
evidence is found in these global confinement data.

3. LHD High Beta Data

The first comprehensive survey of high-β data from the 7thand 8th experimental campaign [10] using a similar
constrained database search was revised and extended up
to the 10th campaign (2007) during the present study. Threedatasets corresponding to 3 different configurations (3helical coil current ratio parameters γ = 1.25, 1.22 and 1.20
corresponding to aspect ratios of Ap = 5.7, 6.1 and 6.5 ataxis position Rax = 3.6 m ) with altogether about 1500
entries are compiled at the maximum of the diamagnetic
energy within each discharge. All parameters refer to thevacuum configurations. Since the rotational transform in
LHD scales as Ap ∼ ι , the Shafranov shift isΔ a ∼ 1 Ap , and hence is reduced in the γ = 1.20
configuration. This is a key to maximize the achievablebeta. The βvalues achieved in the optimum
configuration which are evaluated using parameters of thevacuum configurations are taken from the LHD databaseand are shown in fig. 3 in a similar form as the W7-AS
data. The simple estimate of the equilibrium limit showsthat the maximum beta values may already be affected byequilibrium limit effects (Note that for W7-AS the value
has been doubled due to the configuration optimization[8]). Detailed investigations with the HINT code [14]show very extended regions of field line stochastization.
For the evaluation of the confinement scaling laws, thevacuum boundary was used, which is close to the 99%
pressure contour at high beta. In contrast, VMEC gives
much lower values for the plasma radius which is notFig.3

β from the new LHD survey dataset with
Rax = 3.6 m, γ = 1.20 versus the time in whichδ β β ≤ 10% , (normalized to τ E ,exp. ).

Fig.4

Logarithmic plot of τ E ,exp. against the ISS95
scaling values. The data highlighted in blue
represent the two W7-AS high-β survey datasets.
In red, the two high-β datasets of LHD used fortransport studies are shown (ISHCDB preliminary).

consistent with the experimental data. However, thevalues for iota and the plasma position are derived fromthe available VMEC calculations.
Two more detailed datasets for studies of the local
transport behaviour and its dependence on the magnetic
configuration (γ = 1.25 and 1.22 with Rax = 3.6 m )have been compiled (about 450 entries) by the LHD team
[6][21]. These data have been added in a preliminary way
to the existing ISHCDB dataset. In Fig. 4, τ E ,exp. isplotted in comparison with the ISS95 scaling values.
The two datasets are highlighted in red together with thetwo survey datasets of W7-AS (blue) on the backgrounddata (grey) from all the existing low-beta ISHCDB data.
A closer inspection of the LHD data (fig. 5) reveals a
progressive degradation of the confinement towards highbeta, which has been found already earlier.

Fig.5 Energy confinement times normalized to ISS95( Rax = 3.6 m, γ = 1.25 , comparison of different
LHD datasets). A deterioration of the confinement
is indicated towards high-β.

Proceedings of ITC/ISHW2007

4. Discussion and Conclusions
The inclusion of high beta data from W7-AS andLHD in the ISHCDB data base provides an important testfor the validity of existing scaling laws in the high-β
regime and can lead to more reliable extrapolations to thereactor regime. In order to achieve this goal, results oflocal transport analyses have to be supplemented in theframe of the ISHPDB activity. This is currently in progress
based on selected configurations in LHD [6][22].
In addition, the quality of extrapolations from theparameter space covered by the present database will beenhanced if the observed dependencies of the data
(confinement times or local diffusivities) on the control
parameters are consistent with basic physics models. For
this purpose a Bayesian model comparison method wasapplied to W7-AS low-β and high-β global confinement
data [19][20]. The significance of the results depends
crucially on the quality of the data and the knowledge of
their errors. According to the invariance principle of basic
plasma model equations the scaling parameters are subject
to constraints depending on the model used [23]. Models
that ignore any finite-β effects were found to fit the low-β
data best, whereas finite-β models gave the best agreementwith the high-β data. Therefore, it is concluded that theglobal confinement depends on beta in this range. Since theFig.6

Range of ν * of LHD and W7-AS high-β data.
The data highlighted in blue represent the twoW7-AS survey datasets. In red, the two datasets ofLHD used for transport studies are shown(ISHCDB preliminary).

W7-AS high-β data are clustered in a high-collisionality
region well separated from the LHD data (fig. 6), a similar
model comparison analysis for LHD is required to confirm
the results from W7-AS. This will be very important inorder to assess the role of the collisionality as well. In LHD,a clear dependence of the confinement time (normalized toISS95) was found in the high collisionality regime.
Moreover, this survey indicates that LHD data even allow

for a detailled assessment of the transition from low- tohigh-β regimes.
The high-β regimes in W7-AS and LHD arecharacterized by a parameter space which is close tooperational limits depending on the configuration and on
plasma parameters. In order to clarify the role of theconfiguration dependent beta limits imposed byequilibrium and stability effects, MHD related data
including configuration parameters, data characterizing theMHD mode activity, data on local pressure profiles andresults of numerical equilibrium and stability calculations
are foreseen to include in the ISHPD in a next step, inaddition to data required for local transport studies.
This work contributes to the International Stellarator Profile
DataBase (ISHPDB) under auspices of the IEA Implementing
Agreement for Cooperation in the Development of theStellarator Concept. The first author (A. W.) would like toacknowledge his invitation by NIFS as Foreign Research Staff(Guest Professor).

References[1] G. Grieger, et al. J. Plasma Fusion Res. SERIES 1 53-56(1998)[2] M. C. Zarnstorff, et al. Plasma Phys. Control. Fusion 43
A237–A249 (2001)[3] U. Stroth, et al. Nuclear Fusion 36(8) 1063-1077 (1996)[4] H. Yamada, et al. Nucl. Fusion 45(12) 1684–1693 (2005)[5] A. Dinklage, et al. Proc. of ITC/ISHW2007, Toki, Japan
P1-037 (2007)[6] H. Funaba, et al. submitted to Plasma and Fusion Research(2006)[7] S. Sakakibara, et al. Fusion Sci. Technol. 50(2) 177-185(2006)[8] A. Weller, et al. Plasma Phys. Control. Fusion 45 (12A)
A285-A308 (2003)[9] K. Y. Watanabe, et al. Fusion Sci. Technol. 46(1) 24-33(2004)[10] A. Weller, et al. Fusion Sci. Technol. 50 158-170 (2006)[11] H. Yamada, et al. Fusion Sci. Technol. 46 82-90 (2004)[12] O. Motojima, et al. Nucl. Fusion 47 S668-S676 (2007)[13] K. Y. Watanabe, et al. Plasma Phys. Control. Fusion 49605-618 (2007)[14] Y. Suzuki, et al. Nucl. Fusion 46 L19–L24 (2006)[15] A. Reiman, et al. Nucl. Fusion 47(7) 572-578 (2007)[16] S. P. Hirshman, et al. Comp. Phys. Commun. 43(1)143-155 (1986)[17] D. A. Spong, et al. 41(6) 711-716 (2001)[18] M. C. Zarnstorff, et al. 32nd EPS Conference on Plasma
Phys., Tarragona, Spain, ECA Vol.29C P1.062 (2005)[19] R. Preuss, et al. submitted to Phys. Rev. Lett. (2007)[20] A. Dinklage, et al. Nucl. Fusion 47 1265-1273 (2007)[21] K. Y. Watanabe, et al. personal communication (2007)[22] H. Funaba, et al. Proc. of ITC/ISHW2007, Toki, Japan
P1-040 (2007)[23] J. W. Connor Plasma Phys. Control. Fusion 30(6) 619-650(1988)