1. Working title:
Estimation of ne and Te with microwave diagnostics and investigations on profile changes with RMP
Working topics:
Estimation of Te from ECE data
Estimation of ne from reflectometry data
Behaviour of profiles with RMP
Sylvia K. Rathgeber

2. Motivation ECE diagnostic: long-standing workhorse for Te analysis
Why another ECE analysis? What is different?
Shine-through
Current ECE analysis:
Trad = Te, ν → R
Te = 250 eV in SOL ↔
Power flux density > 600 MW/m2
9/28/2010
Sylvia K. Rathgeber

3. Sylvia K. Rathgeber W. Suttrop, R. Fischer 9/28/2010
Estimation of Te profiles in the framework of Bayesian Probability Theory via forward modelling of ECE radiation
Sylvia K. Rathgeber
W. Suttrop, R. Fischer
9/28/2010

4. Outline Current ECE analysis
(Principle, insufficiency of assumptions, correction,
validity range)
Future ECE analysis
(Integrated Data Analysis, Bayesian Probability Theory, Forward modelling of ECE data)
Results
9/28/2010
Sylvia K. Rathgeber

5. Principle of ECE analysis
Electrons gyrate around magnetic field lines
→ emit radiation with cyclotron
frequency and its harmonics:
Tokamak:
→ each cyclotron frequency can be assigned to
the position of its resonance in the plasma
ECE intensity is identified
with black-body intensity:
Assume Maxwell-distributed gyrotron velocity :
9/28/2010
Sylvia K. Rathgeber

6. Local thermal equilibrium!
Assumption of Maxwell-distributed only valid in LTE
Non-thermal contributions might play a role
Future work ?
LTE
9/28/2010
Sylvia K. Rathgeber

7. Non-local measurement?
Cold resonance:
non-local measurement
→ emission profile broadened:
Doppler broadening: observation
not perpendicular to field line
Relativistic effects: relativistic mass
increase results in frequency shift
9/28/2010
Sylvia K. Rathgeber

8. Shape of emissivity profile
Consider emission profile:
Doppler broadening
Relativistic effects
9/28/2010
Sylvia K. Rathgeber

9. Interaction of radition and plasma?
Absorption and reemission
of radiation on ray path
9/28/2010
Sylvia K. Rathgeber

10. Radition transport Consider radiation transport: Kirchhoff’s law
(valid in LTE)
9/28/2010
Sylvia K. Rathgeber

11. The saving: Optical depth
Plasma optically thick:
Reabsorption narrows the
observed layer
9/28/2010
Sylvia K. Rathgeber

12. Outline Current ECE analysis
(Principle, insufficiency of assumptions, correction,
validity range)
Future ECE analysis
(Integrated Data Analysis, Bayesian Probability Theory, Forward modelling of ECE data)
Results
9/28/2010
Sylvia K. Rathgeber

13. Integrated Data Analysis
Combination of measured data from
different diagnostics for one joint analysis
Challenges:
Complemetary data → synergistic effects
Combined error analysis → error reduction
Resolve data inconsitensies → revelation of systematic errors
9/28/2010
Sylvia K. Rathgeber

14. Bayesian recipe Reasoning about parameter θ: prior
(uncertain) prior information
prior
distribution
+ physical model
likelihood distribution
+ (uncertain) measured data
+ Bayes Theorem
posterior distribution
9/28/2010
Sylvia K. Rathgeber

15. Forward modelling of ECE data
ne(ρ), Te(ρ) → ne(s), Te(s)
Modelling: Trad(ν)
Likelihood: TECE, rad(ν) ↔ Tmod, rad(ν)
Posterior: p(ne(ρ), Te(ρ)|dECE)
Estimates: ne(ρ)±Δne(ρ), Te(ρ)±ΔTe(ρ)
Calculation: jν(s), αν(s)
Integration → I(ν)
Prior information
if not maximized
if maximized
9/28/2010
Sylvia K. Rathgeber

16. IDA at ASDEX Upgrade ne(ρ), Te(ρ) mapping ρ(x) → ne(x), Te(x)
DLIB(ne(x), Te(x))
LIthium Beam emission profile dLIB
DDCN(ne(x))
line integra-ted DCN data dDCN
DECE(ne(x), Te(x))
ECE radiation temperature dECE
results: p(ne(ρ), Te(ρ)|dLIB, dDCN, dECE, dTS)
estimates: ne(ρ)±Δne(ρ), Te(ρ)±ΔTe(ρ)
DTS(ne(x), Te(x))
Thomson Scattering data dTS
addl. infor-mation, constraints, model parameters
9/28/2010
Sylvia K. Rathgeber

17. Outline Current ECE analysis
(Principle, insufficiency of assumptions, correction,
validity range)
Future ECE analysis
(Integrated Data Analysis, Bayesian Probability Theory, Forward modelling of ECE data)
Results
9/28/2010
Sylvia K. Rathgeber

18. Testing: Artficial profiles
Core:
high ne & Te
→ plasma optically
thick
Edge:
steep Te gradient & low ne
→ shine-through
conditions
9/28/2010
Sylvia K. Rathgeber

19. Modelling of Trad High optical depth & constant Te : Trad = Te
Low optical depth & constant Te:
Trad < Te
Low optical depth & Te gradient:
Trad > Te
→ rise too small to
explain shine-through
9/28/2010
Sylvia K. Rathgeber

20. Emissivity profiles Inward-shift of emissivity maximum
Intensity reaches black-body level
Absorption < Emission → no black-body
Higher Te in observed layer than at resonance
9/28/2010
Sylvia K. Rathgeber

21. Conventional IDA of L-mode
Plasma optically thick:
Te = Trad, ECE
Plasma optically thin:
spline fit with edge condition
9/28/2010
Sylvia K. Rathgeber

22. Forward modelling of L-mode
Data consistent within separatrix
Plasma optically thick:
Te slightly reduced
Around separatrix:
Te > Trad, ECE
SOL: no data fit possible
9/28/2010
Sylvia K. Rathgeber

23. Conclusion & Outlook Conclusion
Forward modelling of ECE radiation transport included in IDA
Slight corrections in Te profile due to finite optical depth and relativisticly broadened emssivity profile
Shine-through still unresolved
Outlook
Include Doppler broadening (consider finite acceptance angle of antenna, increase precision for general emissivity profile)
Consider non-Maxwellian velocity distribution
9/28/2010
Sylvia K. Rathgeber

24. Literature
W. Suttrop. Practical Limitations to Plasma Edge Electron Temperature Measurements by Radiometry of Electron Cyclotron Emission. Technical Report 1/306, Max-Planck- Institut für Plasmaphysik, 1997.
I.H. Hutchinson. Principles of Plasma Diagnostics. Cambridge University Press, 1987.
H.J. Hartfuss, T. Geist, and M. Hirsch. Heterodyne methods in millimetre wave plasma diagnostics with applications to ECE, interferometry and reectometry. Plasma Physics and Controlled Fusion, 39: , 1997.
A. Gelman, J.B. Carlin, H.S. Stern, and D.B. Rubin. Bayesian Data Analysis. Chapman & Hall, 1980.
R. Fischer, et. al. Probalistic lithium beam data analysis. Plasma Physics and Controlled Fusion, 50(8): (26pp), 2008.
R. Fischer, et. al. Integrated density profile analysis in ASDEX Upgrade H-modes. In 35th EPS Conference on Plasma Physics. Contributed Papers, 32D, pages P–4.010,
R. Fischer, et. al. Multiple diagnostic data analysis of density and temperature profiles in ASDEX Upgrade. In 36th EPS Conference on Plasma Physics. Contributed Papers, 33E, P–1.159, 2009.
9/28/2010
Sylvia K. Rathgeber

25. Heat conduction Parallel heat conduction strongly depends on T:
Small changes in T cause large changes in power flow
9/28/2010
Sylvia K. Rathgeber

26. Diagnostic implementation
ASDEX Upgrade:
Frequency range accessible to radio frequency (RF) receiver techniques as well as 'quasi'-optical techniques
Currently installed at ASDEX Upgrade:
Michelson interferometer:
8-channel polychromator:
60-channel heterodyne radiometer:
→ input RF signal interferes with similar signal from local oscillator
→ down-conversion to intermediate frequency
→ facilitated amplifying and filtering
9/28/2010
Sylvia K. Rathgeber

27. Diagnostic implementation
Heterodyne
radiometer:
4 antennas on
low field side
5 mixer
3 IF chains (36/12/12 channels)
IF amplifier
Band pass filter
Data acquisition
Absolute calibrated by measurements of black-body radiation from laboratory hot (773 K) and cold (77 K) sources
9/28/2010
Sylvia K. Rathgeber

28. Radial resolution Radial resolution depends on frequency resolution:
Frequency resolution is limited by:
Doppler broadening
(ASDEX Upgrade: 86° ≤ θ ≤ 94°)
Relativistic effects: relativistic mass increase results in frequency shift
Plasma core: RF bandwidth (ΔνRF=600MHz) matches resolution limit due to line broadening (relativistic effects dominant)
Plasma edge: resolution determined by receiver (ΔνRF=300MHz)
9/28/2010
Sylvia K. Rathgeber

29. Temperature resolution
Temperature resolution is limited by noise in black-body radiation emitted from the plasma (much higher than noise of receiver)
Black-body fluctuations given by radiometer formula:
High signal-to-noise ratio/ good temperature resolution needs low video bandwidth (→ long integration time) or high RF bandwidth (→ low radial resolution)
9/28/2010
Sylvia K. Rathgeber

30. Harmonic overlap Resonance frequencies:
GHz: depending on optical thickness, radiation consists of 2nd and 3rd harmonic
Only 1st and 2nd harmonics are feasible for measurements of and from the low field side
9/28/2010
Sylvia K. Rathgeber

31. Low density limit: optical depth
Te = Trad only in case of optically thick plasma (τ >> 1)
τ strongly decreases with increasing harmonic number
→ 1st harmonic O-mode and 1st and 2nd X-mode are mostly
optical thick in the bulk plasma
typical ASDEX Upgrade parameters:
measurements
9/28/2010
Sylvia K. Rathgeber

32. High density limit: Cut-off
Below eigenfrequency of plasma electromagnetic waves are completley shielded by electrons → cut-off
O-mode waves (E || B0):
X-mode waves (E ┴ B0):
Cut-off density:
9/28/2010
Sylvia K. Rathgeber

33. Consequence of limitations
2nd harmonic X-mode is the best candidate for ECE measurements according to limitations due to harmonic overlap, cut-off and optical depth
9/28/2010
Sylvia K. Rathgeber

34. Doppler broadening Trad = Te in case of high optical depth
Trad < Te in case of low optical depth
9/28/2010
Sylvia K. Rathgeber

35. Doppler & Relativistic effects
Trad = Te in case of high optical depth
Trad < Te in case of low optical depth and constant Te
Trad > Te in case of low optical depth and Te gradient
9/28/2010
Sylvia K. Rathgeber