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
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
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
_________________________________________
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
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
TAUTH
0.001
0.010
0.100
1.000
10.000
TAUPRED
0.001 0.010 0.100 1.000 10.000
RMSE=15.8%
ITER tE=6.0 s
E I p BT ne PL M R a k 0 036 0 95 0 07 0 42 0 64 0 22 1 72 0 25 0 68. . . . . . . . .
Log-linear regression
_________________________________________with tauc92 correction
no correction:
RMSE=16.48% ITER tE=7.2 s
E I p BT ne PL M R a k 0 043 0 82 0 24 0 45 0 64 0 17 1 78 0 42 0 24. . . . . . . . .
Principal components
_____________________________
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
PRIN5
-2
-1
0
1
2
PRIN1
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
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
2
41
1
8 22 0158
1398 47 30 12
N
pc
stdj
ITERj
j
.. %
PC1
PC5 ITER
Largest extrapolation along PC1 ~ B0.6 P0.4 L0.8
and PC5
Formula gives very small uncertainty of prediction
ln(ip) ln(B) ln(n) ln(p) ln(m) ln(R) ln(a) ln(k) std pc(iter)/stdpc1 0.47 0.17 0.03 0.45 0.29 0.36 0.44 0.36 2.05 4.21pc2 0.05 0.30 0.69 0.03 0.31 -0.44 -0.28 0.22 1.34 0.07pc3 0.02 0.84 -0.05 0.01 -0.12 0.13 0.03 -0.51 0.99 3.07pc4 0.10 0.01 0.33 0.23 -0.89 0.01 0.03 0.21 0.76 0.72pc5 -0.32 -0.30 0.38 0.56 0.13 0.30 -0.08 -0.48 0.49 3.88pc6 0.32 -0.19 0.49 -0.62 0.00 0.33 0.27 -0.26 0.32 2.79pc7 -0.23 0.16 0.04 -0.18 0.02 0.67 -0.53 0.39 0.28 1.35pc8 -0.71 0.16 0.14 -0.14 0.02 0.01 0.60 0.25 0.15 0.09
Eigenvalues of the Correlation Matrix of engineering variables
_________________________________________
proportionpc1 0.52pc2 0.22pc3 0.12pc4 0.073pc5 0.029pc6 0.013pc7 0.013pc8 0.003
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
no correction
5 6 7 8
w/o TCV
w/o PDX
w/o PBXM
w/o JT60U
w/o JFT2M
w/o JET
w/o D3D
w/o COMPASS
w/o CMOD
w/o AUG
w/o ASDEX
HSELM
IGRADB=1
tauc92 correction
5 6 7 8
w/o TCV
w/o PDX
w/o PBXM
w/o JT60U
w/o JFT2M
w/o JET
w/o D3D
w/o COMPASS
w/o CMOD
w/o AUG
w/o ASDEX
HSELM
IGRADB=1
TauE_ITER (sec)
Dimensionless physics variables
_________________________________________
a
EB
kT
eBkT
W
n V
Mm kT
eBa
nkT
B
qR qRn
T
M M
qaB
R I a
a
R
T T N
p
eff
Gr
2
3
22
2
2
20
3 2 3 2 2
0
,
*
* / /
Atomic physics: (Lackner), (Greenwald Hugill)*
Extrapolation is measured in physics variables. Another reason is understanding.
Standard dataset in physics variables
_____________________________
Extrapolation to ITER not only along r*
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
BETAN
0
1
2
3
4
NUSTAR
0.001 0.010 0.100 1.000 10.000
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
BETAN
0
1
2
3
4
RHOSTAR
0.001 0.010 0.100
bN
bN
n*
r*
ITER
ITER
Transformation in physics variables
_____________________________
zB
z
z
z
zM
zq
z
z
x
xB
xn
xP
xM
xR
xa
x
xP
C
I
1
1
E
I xI BxBnxnPxPMxMRxRaxa x
1 1
E B
B
M q
z z z zM
zqz z z
* *
zB=1 defines Kadomtsev constraint
Transformation in not linear
no correction:
with tauc92 :
BM q
BM q
1 04 110 0 38 012 1 01 119 117 214
110 0 69 0 53 010 0 95 0 61 2 76 2 55
.*. .
*. . . . .
.*. .
*. . . . .
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
_____________________________no correction:
RMSE 16.5 % 16.5 %
with tauc92 :
RMSE 15.8 % 15.8 %
B
B
M q
M q
*. .
*. . . . .
*. .
*. . . . .
115 0 37 012 1 02 116 117 213
0 83 0 50 010 0 97 0 55 2 72 2 52
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.
0.155
0.16
0.165
0.17
0.175
0.18
0.185
0.19
0 0.5 1 1.5 2
B
B M qy y y yM
y y yqy
* *
no correction
with tauc92
yB
RMSE
____ all exponents constrained
____ yB constrained
Scan of RMSE by exponent of c B
_________________________________________
Scan of RMSE by exponent of * r_________________________________________
0.155
0.16
0.165
0.17
0.175
0.18
0 0.5 1 1.5
yrho*
RM
SE
tauc92
no correction
____ all exponents constrained
____ Kadomtsev and yr* constrained
B
M qy y y
My y y
qy
* *
RMSE
yb
B
M qy y y
My y y
qy
* *
-0.5
0
0.5
1
1.5
2
-0.5 0 0.5 1 1.5 2
yr
yn
tITER/tfree fit
yb
with tauc92
____ all exponents constrained
____ Kadomtsev and yb constrained
Scan of RMSE by exponent of b_________________________________________
0.14
0.16
0.18
0.2
-0.5 0 0.5 1 1.5 2
w/o tauc
Scan of RMSE by exponent of * n_________________________________________
B
M qy y y
My y y
qy
* *
no tauc92 correction
all exponents constrained
0.164
0.165
0.166
0.167
0.168
0.169
0.17
0.171
0 0.05 0.1 0.15 0.2 0.25
y_nu*
RM
SE
Confidence intervals
_________________________________________
Statistics provide a formula [1]:
RMSE
RMSE
4
2Neff
Neff=N/4 gives dRMSE=0.1%.
From calculated minima (for all exponents fixed) we find the confidence intervals:
yB
y
y y
01 015
01 0 05
. .
. .
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
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. 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.
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
HCHI
1
10
100
RHOSTAR
0.001 0.010 0.100
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
HCHI
1
10
100
BETA
0.001 0.010 0.100
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
HCHI
1
10
100
NUSTAR
0.001 0.010 0.100 1.000 10.000
*r b *n
*
F
F M q *. .
*. . . . .11 0 37 012 1 0 12 12 21
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
CHISTAR
0.0001
0.0010
0.0100
0.1000
CHIFIT1
0.00001 0.00010 0.00100 0.01000
(Kadomtsev constraint, no correction)
*
B
corr ln *,ln . F 0 75
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.
*
F M q 01900 49 0 26 0 42 0 55 1 4 0 77 0 76.*. .
*. . . . .
*
F
*r b *n
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
CHISTAR
0.0001
0.0010
0.0100
0.1000
CHIFIT6
0.0001 0.0010 0.0100 0.1000
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
HCHI
0.1
1.0
10.0
RHOSTAR
0.001 0.010 0.100
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
HCHI
0.1
1.0
10.0
BETA
0.001 0.010 0.100
TOK ASDEX AUG CMOD COMPASSD3D ITER JET JFT2MJT60U PBXM PDX TCV
HCHI
0.1
1.0
10.0
NUSTAR
0.001 0.010 0.100 1.000 10.000
Better correlation is obtained by mixed Bohm+GyroBohm diffusivity with weak ‘inverted’ -bdependence 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).
Canonical Correlation Analysis
_________________________________________ corr ln *,ln . F 0 90
Comparison of Canonical Correlation and Linear
Regresion (in physics variables)
_________________________________________
Assisted Linear Regression shows results close to Correlation Analysis
LinReg CanCorr
y_rho* 0.44 0.49
y_beta -0.23 -0.26
y_nu* 0.38 0.42
y_M -0.49 -0.55
y_q 0.68 0.76
y_eps 1.24 1.39
y_kappa -0.69 -0.77
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.
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.
Conclusions
_____________________________
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