Post on 18-Dec-2015
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
4
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
7
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