ITER Expert Group Meeting on Confinement Database, Princeton, 20-23 April 1998 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
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
H-mode Confinement Database DB03V5 _________________________________________ 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
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
Composition of standard dataset _____________________________ • SND iongradB->x • 72% • NBI+NBIC IC OH+EC • 95% 3% 2% • mean Meff =1.8 • HSELM HGELM • 41% 59% • Large fraction of NBIH and HGELM
Software used _________________________________________ SAS 6.12 OPEN VMS 7.1 DEC Alpha at UKAEA Culham SAS Procedures: REG PRINCOMP MEANS CORR CANCORR
Log-linear regression _________________________________________ with tauc92 correction RMSE=15.8% ITER tE=6.0 s no correction: RMSE=16.48% ITER tE=7.2 s
Principal components _____________________________ ITER: Ip =21MA, BT =5.68T, n19 = 9.7, P = 178.8 MW M = 2.5, R = 8.14m, a = 2.8m , k = 1.73 PC5 ITER PC1 Largest extrapolation along PC1 ~ B0.6 P0.4 L0.8 and PC5 Formula gives very small uncertainty of prediction
Eigenvalues of the Correlation Matrix of engineering variables _________________________________________ 5 principal components ‘remove’ 97% of variance.
ITER confinement time predicted on subsets _________________________________________ tauc correction changes the pattern, e.g. w/o correction JET is well predicted
Dimensionless physics variables _________________________________________ Extrapolation is measured in physics variables. Another reason is understanding.
Standard dataset in physics variables _____________________________ bN ITER r* bN ITER n* Extrapolation to ITER not only along r*
Transformation in physics variables _____________________________ zB=1 defines Kadomtsev constraint Transformation in not linear
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 cB is 1.60 without and 1.97 with tauc corrections respectively. Removing other tokamaks leaves this exponent close to unity.
Kadomtsev-constrained regression _____________________________ Kadomtsev constraint has negligible effect on RMSE and small effect on exponents.
Mapping the minima of RMSE _________________________________________ In 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 cB, r* and b. 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.
Scan of RMSE by exponent of cB _________________________________________ ____ all exponents constrained ____ yB constrained RMSE no correction with tauc92
Scan of RMSE by exponent of r* _________________________________________ ____ all exponents constrained ____ Kadomtsev and yr* constrained no correction tauc92
tITER/tfree fit yr yn yb Scan of RMSE by exponent of b _________________________________________ ____ all exponents constrained ____ Kadomtsev and ybconstrained RMSE w/o tauc with tauc92 yb
Scan of RMSE by exponent of n* _________________________________________ all exponents constrained no tauc92 correction
Confidence intervals _________________________________________ Statistics provide a formula : Neff=N/4 gives dRMSE=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.  O.J.W.R Kardaun, communication, April 1998
Plot of dataset against formula derived from regression _________________________________________ (Kadomtsev constraint, no correction) r* n* b The formula derived from regression in engineering variables does not represent well the dependencies on dimensionless variables. Correlation is low. b and n* dependencies show systematic mismatch. It is not expected that the regression in engineering parameters will provide the best fit in physics parameters.
Canonical Correlation Analysis _________________________________________ This method finds such linear combination ln(F) of variables ln(r*), ln(b), ln(n*), ln(M), ln(q), ln(e) and ln(k) which maximises the Pearson correlation coefficient: corr(ln(c*), ln(F) )=max The method treats dependent and independent variables symmetrically. Contrary to regression analysis there are no requirements on measurement errors. Selection of r*, b and n* as independent variables is accepted in similarity experiments.
Canonical Correlation Analysis _________________________________________ r* n* b Better correlation is obtained by mixed Bohm+GyroBohm diffusivity with weak ‘inverted’ b-dependence and closer to neoclassical n* -dependence. Such b and n* -dependence is close to DIIID result c~b-0.15 n*0.37 (C C Petty and T C Luce, 24th EPS 1994 ) and little stronger than on JET (J G Cordey at al,16th IAEA).
Comparison of Canonical Correlation and Linear Regresion (in physics variables) _________________________________________ Assisted Linear Regression shows results close to Correlation Analysis
Conclusions _____________________________ Log-linear regression of standard dataset of DB03V5 database has been executed. Predictions to ITER are tE= 7.2s and tE= 6.0s without and with tC92 correction resp. Changes of predicted tE when one tokamak is removed are inside the statistical error (except JET with tC92 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.
Conclusions _____________________________ These exponents, however, give not good correlation between global thermal diffusivity and dimensionless physics parameters. Better agreement is obtained with mixed Bohm-GyroBohm r*-dependence, closer-to-neoclassical dependence of n* and weak ‘inverted’ b -dependence. Such n* and b dependence is closer to the results of similarity experiments. At fixed n*, the dependence on the geometry of magnetic field favours low q, low aspect ratio and elongated plasma. At fixed r*, scaling favours high M.