Bayesian modelling of a diagnostic helium beam

ADAS workshop, Armagh Observatory, October 4th, 2010

Maciej Krychowiak

M. Brix, D. Dodt, R. König, O. Schmitz, J. Svensson, R. Wolf

2

Helium beam diagnostic

Radial profiles of ne and Te at the plasma edge

ne : 1012 ... 1013 m-3

Te : 10 ... 200 eV

r 1 mm, t 1 ms

plasma

line of sight

helium beam

nozzle

Measure three spectral lines of atomic helium (typically 667, 706, 728 nm)

Compare two line ratios with CR model

Te: triplet/singlet ne: singlet/singlet

singlet triplet

3

312e cm102 n

312e cm102 n

dir

ec

tio

n o

f th

e g

as

flo

w

beam relaxation(steady-state solution)

no beam relaxation(time dependent solution)nozzle

SenkenQellen

Ionisationregungan(ab)Strahlungsregungstoßan(ab)ElektronenTransporttätStationari

eeionieeeeHed

),(d

iij

ijiij

jijij

ijiij

jijiii nnvnAnAnnvnnv

r

nv

t

n

t

trn

=0(stationary beam)

electron (de)excitation radiationtransport ionisationcharge exchange

Beam atoms penetrate the plasma, radiate, get ionised, leave the observation volume Movement in one direction → 1-dim transport equation:

CR modelling of helium beams

~ 200 uncertain rate coefficients for electron collisions, known only from calculations Collisions with protons, neutrals

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Comparisons to other diagnostics

Comparisons to TS and lithium beam (Schmitz et al. 2008):

ne 10%, Te 30%

Observed beam penetration smaller than model by 30%

Comparative measurements on TEXTOR (Schmitz, et al., Plasma Phys. Control. Fusion 50 (2008) 115004)

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CR model of the helium beam is a complex system with many uncertain

parameters

Quantitative errors in ne/Te

Probabilistic approach provides:

• Diagnostic design study: application of helium beam in the high

density divertor plasma of the stellarator W7-X

• Statements on atomic data (correction factors, uncertainties) by

analysis of (uncertain) experimental data.

Why probabilistic CR model for helium beam

RCd)|,( DTnp ee

prior knowledge

)|RC,,( DTnp ee

likelihood

Bayesian CR modelling of (relaxed) helium beam

C)R,,(C)R,,|( eeee TnpTnDp

posteriormarginalisedposterior

Take ne/Te, simulate line ratios, D

2-dim posterior(parameters of interest)

further marginalise

1-dim posteriors

01

23

45

x 1014

2

4

6

8

10

120

0.05

0.1

0.15

0.2

0.25

ne [cm-3]

Te [eV]

0 1 2 3 4 5

x 1014

0

0.5

1

1.5

2

ne [cm-3]

2 4 6 8 10 120

0.5

1

1.5

Te [eV] 6

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Model assumptions

Steady state solution (transport neglected, ne > 2×1012 cm-3)

Collisional processes included: electronic (de)excitation and ionization, no charge exchange

n = 1-5 included (29 levels)

n = 1-4: „helike_hps02he_t3.adf”, n = 5: compilation by Brix (phd)

High density, low temperature W7-X divertor plasma: ne = 1014 cm-3, Te = 5 eV

9%

4.7%

3.2%

15%

21%31%

25%

n=3-4

30%

45%

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ADAS dataset „helike_hps02he_t3.adf”: uncertainties

5%20%20% 20%

20%

n=3-4

50%

Measure 2 line ratios

Relatively large ne/Te errors

Diagnostic design study

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ne=128%

Te=45%

Measure 3 absolute line intensitiesbeam density (attenuation) uncertain: +/- 50%

Strongly reduced ne/Te errors

10

ne=66%

Te=8%

Fit 3 line intensitiesMeasurement error: 5%

ne = 66% Te = 8%

Enlarge signal noiseIncrease number of spectral lines

Measurement errors 5 → 10%

Te: 8.7%

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Fit one additional line (501.6 nm)

Te: 8.5% Fit two more lines(492.2, 504.8 nm)

Te: 7.8%

ne [cm-3] ne [cm-3]

ne [cm-3]

ne = 122%

ne = 103%ne = 107%

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0 5 10 15

x 1012

0

0.005

0.01

0.015ne, sigma = 1.89E+012, (49.392 %), relErrorNe: 0.5%, relErrorTe: 0.5%,samples: 30000

ne [cm-3]

ne=50%

10 20 30 40 50 600

0.002

0.004

0.006

0.008

0.01Te, sigma = 8.34, (26.221 %), relErrorNe: 0.5%, relErrorTe: 0.5%,samples: 30000

Te [eV]

Te=26%

Use comparisons to other diagnostics at TEXTOR: ne = 10%, Te = 30%

Te = 32 eV, ne = 4×1012 cm-3

Run Bayesian analysis using RCs and their uncertainties from „helike_hps02he_t3.adf”

Refining the beam excitation model

Some RC uncertainties in „helike_hps02he_t3.adf” are overestimated !

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Refining the beam excitation model

Priors:

• RCs from „helike_hps02he_t3.adf” as before• New: ne, Te: Gauss profile, width of 10% and 30% respectively (observation)

Marginalise over ne, Te and all rate coefficients except for the ones of interest

ee21 dTdnRCd)|RC,(RC Dp

prior knowledge

)|RC,,( DTnp ee

likelihood

C)R,,(C)R,,|( eeee TnpTnDp

posteriormarginalisedposterior

Result:

• RC (11S-31S) 9.9% (11% in ADAS)

• RC (31S-31P) 18.2% (30% in ADAS)

But:• The model is not complete

Principle suitability of Bayesian analysis for judging atomic data

Thank you for your attention