1. An analysis of ITER H-mode confinement database M Valovič and ITER H-mode Confinement Working Group Acknowledgments: K Thomsen O J W F Kardaun SAS Institute ITER Expert Group Meeting on Confinement Database, Princeton, 20-23 April 1998

2. Outline _________________________________________ Characteristics of standard dataset and Log-linear regression Predictions to ITER and principal component analysis Transformation of scalings to dimensionless physics variables and Kadomtsev constraint Confidence intervals of exponents of dimensionless variables Correlation analysis of dataset in space of physics parameters. Conclusion

3. 6970 observations 12 Tokamaks: Alcator C-Mod, ASDEX, ASDEX-Upgrade, COMPASS-D, DIII-D, PBX-M, PDX, JET, JFT-2M, JT-60, TEXTOR and TCV Heating: OHM, EC, IC, NBI H-mode Confinement Database DB03V5 _________________________________________

4. Selection of standard dataset _________________________________________ SELDB3=’1111111111’ and not(PHASE=‘H’) gives: 1398 observations 11 Tokamaks PHASE=HSELM(H), HGELM(H) CONFIG= SN(L,U), DN, (IW, MAR, BOT, TOP) AUXHEAT=NONE, EC, IC, NB, NBIC

5. Composition of standard dataset _____________________________ SND iongradB->x 72% NBI+NBIC IC OH+EC 95% 3% 2% mean M eff =1.8 HSELM HGELM 41% 59% Large fraction of NBIH and HGELM

6. Software used _________________________________________ SAS 6.12 OPEN VMS 7.1 DEC Alpha at UKAEA Culham SAS Procedures: REG PRINCOMP MEANS CORR CANCORR

7. RMSE=15.8% ITER    s Log-linear regression _________________________________________ with tauc92 correction no correction: RMSE=16.48% ITER    s

8. Principal components _____________________________ ITER: I p =21MA, B T =5.68T, n 19 = 9.7, P = 178.8 MW M = 2.5, R = 8.14m, a = 2.8m,  = 1.73 PC1 PC5ITER Largest extrapolation along PC 1 ~ B 0.6 P 0.4 L 0.8 and PC 5 Formula gives very small uncertainty of prediction

9. Eigenvalues of the Correlation Matrix of engineering variables _________________________________________  principal components ‘remove’ 97% of variance.

10. ITER confinement time predicted on subsets _________________________________________ tauc correction changes the pattern, e.g. w/o correction JET is well predicted

11. Dimensionless physics variables _________________________________________ Extrapolation is measured in physics variables. Another reason is understanding.

12. Standard dataset in physics variables _____________________________ Extrapolation to ITER not only along  * NN NN * ** ITER

13. Transformation in physics variables _____________________________ z B =1 defines Kadomtsev constraint Transformation in not linear

14. Unconstrained regressions transformed to physics variables _____________________________ Kadomtsev constraint is satisfied. This is due to the presence of CMOD. When CMOD is removed the exponent of   is 1.60 without and 1.97 with tauc corrections respectively. Removing other tokamaks leaves this exponent close to unity.

15. Kadomtsev-constrained regression _____________________________ Kadomtsev constraint has negligible effect on RMSE and small effect on exponents.

16. Mapping the minima of RMSE _________________________________________  n order to investigate how well the exponents of dimensionless variables are determined we performed a systematic mapping of RMSE minima by series of constrained regressions. Regressions are executed in engineering parameters and constrained by a value of exponent of  B,  * and . As a starting point we took values obtained by free and Kadomtsev-constrained regressions. Then one exponent is varied by application of a linear constraint. Two types of scans are performed: -one exponent is varied and all others are kept at the values of RMSE minima. -one exponent is varied and others are left free giving obviously broader minimum.

17. no correction with tauc92 RMSE ____ all exponents constrained ____ y B constrained Scan of RMSE by exponent of   _________________________________________

18. Scan of RMSE by exponent of  _________________________________________ tauc92 no correction ____ all exponents constrained ____ Kadomtsev and y  * constrained

19. RMSE yy yy y  ITER /  free fit yy with tauc92 ____ all exponents constrained ____ Kadomtsev and y  constrained Scan of RMSE by exponent of  _________________________________________ w/o tauc

20. Scan of RMSE by exponent of  _________________________________________ no tauc92 correction all exponents constrained

21. Confidence intervals _________________________________________ Statistics provide a formula [1]: N eff =N/4 gives  RMSE=0.1%. From calculated minima (for all exponents fixed) we find the confidence intervals: Mapping of minima of RMSE shows that the exponents are well determined. Thus the uncertainty can not explain the discrepancy between the scaling and similarity experiments. [1] O.J.W.R Kardaun, communication, April 1998

22. Plot of dataset against formula derived from regression _________________________________________ The formula derived from regression in engineering variables does not represent well the dependencies on dimensionless variables. Correlation is low.  and * dependencies show systematic mismatch. It is not expected that the regression in engineering parameters will provide the best fit in physics parameters.    (Kadomtsev constraint, no correction)

23. Canonical Correlation Analysis _________________________________________  his method finds such linear combination ln(F) of variables ln(  *), ln(  ), ln( *), ln(M), ln(q), ln(  ) and ln(  ) which maximises the Pearson correlation coefficient: corr(ln(  *), ln(F) )=max The method treats dependent and independent variables symmetrically. Contrary to regression analysis there are no requirements on measurement errors. Selection of  *,  and * as independent variables is accepted in similarity experiments.

24.    Better correlation is obtained by mixed Bohm+GyroBohm diffusivity with weak ‘inverted’  dependence and closer to neoclassical  dependence. Such  and  dependence is close to DIIID result      C C Petty and T C Luce, 24th EPS 1994 ) and little stronger than on JET (J G Cordey at al,16th IAEA). Canonical Correlation Analysis _________________________________________

25. Comparison of Canonical Correlation and Linear Regresion (in physics variables) _________________________________________ Assisted Linear Regression shows results close to Correlation Analysis

26. Conclusions _____________________________ Log-linear regression of standard dataset of DB03V5 database has been executed. Predictions to ITER are  E = 7.2s and  E = 6.0s without and with  C92 correction resp. Changes of predicted  E when one tokamak is removed are inside the statistical error (except JET with  C92 correction). Dataset satisfy the Kadomtsev contraint. The RMSE has well localised minima as a function of exponents of main dimensionless parameters. Thus the values obtained by standard transformation of exponents power law scaling are well determined.

27. These exponents, however, give not good correlation between global thermal diffusivity and dimensionless physics parameters. Better agreement is obtained with mixed Bohm- GyroBohm  *-dependence, closer-to- neoclassical dependence of * and weak ‘inverted’  -dependence. Such * and  dependence is closer to the results of similarity experiments. At fixed *, the dependence on the geometry of magnetic field favours low q, low aspect ratio and elongated plasma. At fixed  *, scaling favours high M. Conclusions _____________________________