An analysis of ITER H-mode confinement database M Valovič and ITER H-mode Confinement Working Group...

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

_____________________________


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*


BETAN

0

1

2

3

4

NUSTAR

0.001 0.010 0.100 1.000 10.000


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


____ 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


____ 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.


HCHI

1

10

100

RHOSTAR

0.001 0.010 0.100


HCHI

1

10

100

BETA

0.001 0.010 0.100


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


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


CHISTAR

0.0001

0.0010

0.0100

0.1000

CHIFIT6

0.0001 0.0010 0.0100 0.1000


HCHI

0.1

1.0

10.0

RHOSTAR

0.001 0.010 0.100


HCHI

0.1

1.0

10.0

BETA

0.001 0.010 0.100


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

_____________________________

An analysis of ITER H-mode confinement database M Valovič and ITER H-mode Confinement Working Group...

Documents

Transcript of An analysis of ITER H-mode confinement database M Valovič and ITER H-mode Confinement Working Group...