Assessment of Global Stellarator Confinement: Status of the International Stellarator Confinement Scaling Data Base1 A.Dinklage(1,*), E.Ascasibar(2), C.D.Beidler(1), R. Brakel(1),J. Geiger(1), J.H. Harris(3), A. Kus(1), S. Murakami(4), S. Okamura(5), R. Preuss(1), F. Sano(4), U. Stroth(6), Y. Suzuki(5), J. Talmadge(7), V.Tribaldos(2), K.Y.Watanabe(5), H.Yamada(5), M.Yokoyama(5) (1) Max-Planck-Institut für Plasmaphysik, EURATOM-Association, Greifswald, Germany (2) Laboratorio Nacional de Fusión, EURATOM-CIEMAT, 28040 Madrid, Spain (3) Australian National University, Canberra, Australia (4) Kyoto University, Kyoto, Japan (5) National Institute for Fusion Science,Toki,Gifu 509-5292, Japan (6) University of Stuttgart, Stuttgart, Germany (7) University of Wisconsin, Madison, USA (*)
[email protected] Abstract The status of the International Stellarator Confinement Database is reported. Impact of configurations on confinement is confirmed and the performance close to operational limits discussed. Model comparison techniques allow for tests of physical models against data. Correlations of configuration-describing parameters against an empirical confinement enhancement are investigated.
1. Introduction Due to their flexibility, stellarator/heliotron devices cover a large magnetic configuration space. Since the ultimate goal of stellarator research aims at an alternative fusion reactor concept, the exploration of the most promising configurations requires comparative assessments of the plasma performance and how different aspects of a 3D configuration influence it. To this end, the International Stellarator Confinement Database (ISCDB) has been re-initiated in 2004 and the ISS95 [1] database has been extended to roughly 3300 discharges from nine different devices. The revision of a data set restricted to comparable scenarios led to the ISS04 scaling law [2] which confirmed ISS95 but also revealed clearly the necessity to incorporate configuration descriptive parameters. In other words, an extension beyond the set of regression parameters used for ISS95/ISS04 appears to be necessary and candidates, such as the elongation are investigated. Since grouping of data is a key-issue for deriving ISS04, basic assumptions are revised in this paper, e.g. the dependence on the heating scheme. Moreover, an assessment of statistical approaches is investigated with respect to their impact on the scaling. A crucial issue is the weighting of data groups which is discussed in terms of error-in-variable techniques. Bayesian model comparison is 1
ISCDB resources are jointly hosted by NIFS and IPP, see http://iscdb.nifs.ac.jp and http://www.ipp.mpg.de/ISS
employed for testing scaling hypotheses based on scaling invariance principles thus allowing the assessment of applicability of theory-based scaling laws on stellarator confinement. Again, subsets of ISCDB are used in order to ensure comparability of the data under investigation. 2. The ISCDB dataset and the ISS04 stellarator scaling 1
10
ATF CHS Heliotron−E Heliotron−J HSX 0
LHD
10
TJ−II Wendelstein 7−A Wendelstein 7−AS
−1
log10[τE (s)]
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AD Fig. 769 (21−11−05)
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ELMy H−mode L−mode ITER prediction (PPCF41,429(1999)) −4
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−1
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10 ISS95
log10[τ
0
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Fig. 1: Experimental confinement data of stellarators vs. ISS95 scaling. The tokamak data [3] were rephrased employing the plasma elongation and assuming a typical q profile according to [1]. For ITER, a prediction with uncertainty analysis [4] is plotted against the ISS95 scaling energy confinement value.
Revising the present global confinement datasets in figures of the ISS95 scaling law, as shown in Fig. 1, indicates the necessity to confirm at least the gross scaling of regression parameters (minor and major radius (a, R), mean density n, absorbed power P, magnetic field B and rotational transform ι). LHD data indicate enhanced performance compared to the ISS95 prediction. The tokamak data [3] in Fig. 1 are shown for comparison. A caveat for this comparison needs to be raised because of some differences in the set of regression parameters and assumptions on the rotational transform profiles [1]. Nonetheless, the ITER prediction is consistent with a log-linear extrapolation of ELMy-H mode data and the stellarator data meet with the high confinement data from tokamaks in the range of confinement times covered. Hence, there is no indication of confinement degradation – in particular from the data of the largest device, LHD.
A second conclusion can be drawn from Fig. 1: Data from different devices form clusters whose coarse tendency may differ from the general trend of ISS95. A closer inspection even shows subgroups within discharges of a single device. This observation led to the ad hoc assumption of ISS04: stellarator configurations are different, but a common scaling in the leading regression parameters is to be investigated. Technically, ISS04 assigns to each subgroup a configuration parameter which is determined as the ratio of experimentally determined confinement times and a reference scaling (here ISS95 was chosen) [2]. Regression of renormalized data results in (RMSE=0.0267):
τ EISS 04 = f (configuration) × 0.134 a 2.28 R 0.64 P −0.61 ne0.54 B 0.84ι 20./413 ∝ τ Bohm ρ *
− 0.81
ν *0.00 β −0.17 a*0
The scaling law ISS04 confirms the findings of ISS95. More details, in particular on assessments of assumptions on the regression ansatz can be found in Ref. [2]. The dependence of f on the reference scaling is vanishing if the derived scaling is used iteratively. In addition, the normalization factor ansatz is independent from the choice of reference; here ISS95 [1], Lackner-Gottardi [5] and LHD [6] scaling laws were employed as initial reference. For all references the iterative determination converges against (RMSE =0.0262)
τ EISS 04,∞ = f (configuration) × 0.137 a 2.40 R 0.64 P −0.57 ne0.55 B 0.90ι 20./093 ∝ τ Bohm ρ *
−1.22
ν *−0.11 β 0.06 a*0
This scaling law is close to ideal gyro-Bohm scaling laws [7]. The dependence on the rotational transform differs substantially from ISS95, but nevertheless the iota dependence is significant regardless which reference is initially assumed. 3. Assessment of ISCDB data
The ISS04 scaling was derived from a subset of the International Stellarator Confinement Database. Partly, the subset for scaling was restricted to data which are expected to store most of their energy as thermal energy. Therefore, density and power scans from the HSX device were not included in scaling studies. An example of such a subset is a density scan of the confinement time in the transition from normal confinement to the high density H-mode (HDH) in Wendelstein 7-AS. This transition is shown in the left frame of Fig. 2. This example shows that not a saturation of the energy confinement time appears at highest densities but a bifurcation to a different confinement mode which transforms to a density degradation of confinement for detached plasmas in divertor operation [8,9].
A second example of regression parameter variations is the comparison of the ι dependence of a W7-AS subset [10] and TJ-II data [11] shown in the right plot of Fig. 3. Taking account for the configuration factor, the ι scaling of different configurations can be compared. This approach allows one to make use of the large parameter variations of an inter-machine database. Since ISS04 corresponds to the τexp=fτISS04 line in the right plot of Fig. 3 the gross scaling of ι appears to be consistent from the inter-machine comparison. However, the variation of the data indicates possible uncertainties even in the gross scaling; for comparison the green line indicates vanishing ι dependence in ISS04. Moreover, fine structuring of the ι dependence due to low order rationals are documented and discussed in [10,11].
TJ−II − W7−AS Iota Variations (scaling agreement) 2 1.8
W7−AS ι scan [Brakel, NF 42, 903 (2002)] TJ−II (ISS04 data) ISS04 without ι dependence
1.6
τISS04 ∝ ι0.41 a2.28
τ
exp
/f×τ
ISS04
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
0.5
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ι / 2π
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Fig. 2: Left: Energy confinement time in the high-density H mode in Wendelstein 7-AS. Right: Comparison of iota scaling in W7-AS and TJ-II in figures of renormalized confinement times.
4. Assessment of models
A crucial issue in the assessment of scaling laws is the estimate of trustworthiness not only for reference purposes but ultimately for a predictive statement. One possibility to link the scaling relation with physics models is to derive dimensional constraints on the scaling exponents from invariance requirements of the basic equations describing plasma behavior connected with collisionality or plasma β [12]. In order to compare these physics models on the basis of data from the ISCDB, Bayesian probability theory is employed. For model comparison an inclusion of uncertainties is necessary. Neglecting these uncertainties means assigning equal weight to every entry in the database. However, the uncertainties of the data vary substantially being the result of different diagnostics measured in different machines during different experimental campaigns.
The error statistics taking into consideration the uncertainty both in the regression target (τE) and the machine variables (so called errors in variables approach), can be written in terms of a probability distribution, i.e. likelihood, as [13,14]: 2 ⎧ ⎛ ⎞ ⎫ y − ∑ α k xk ⎟ ⎪ ⎪ r r const r ⎪ 1 ⎜⎝ k ⎠ ⎪ p ( x ,τ E | α , σ ) = exp ⎨ − 2 2 2 ⎬ K K αk σ k ⎪ ( 2π ) 2 ∏ σ k ⎪ 2 σ τ + ∑ k k ⎩⎪ ⎭⎪
where xk refers to the ISS regression variables (a,R,P,n,B,ι). αk and σk are their respective regression exponents and errors. The denominator in the argument of the exponent makes the difference to ordinary least squares fitting. Since the present content of ISCDB covers different ranges of collisionality and plasma β, the validity of low/high collisionality and low/high beta models can be performed. A methodological summary is to be found in Ref. [15,16]. A first study focuses on the low/high β and ν∗ data from W7-AS. As can be seen in the right graph of Fig. 3, the low β data set is given by ν∗ < 23 and β < 0.004 giving a total of 223 shots, on the other hand the high β data set to ν* > 100 and β > 0.015 giving a total of 43 shots. Only shots with ι
