Max-Planck-Institut für Plasmaphysik,
EURATOM Association
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
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
2
Max-Planck-Institut für Plasmaphysik,
EURATOM Association
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
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
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Principle of ECE analysis
 Electrons gyrate around magnetic field lines
→ emit radiation with cyclotron   m eB
m
me
frequency and its harmonics:
BR
 Tokamak: Btot  Bt  Bt ( R)  0 0
R
eB0 R0
 m  m ( R )  m
me R
→ each cyclotron frequency can be assigned to
the position of its resonance in the plasma
 ECE intensity is identified
2
I ( )  I BB ( )  3 2 k BTrad
with black-body intensity:
8 c
(h  k BT )
 Assume Maxwell-distributed gyrotron velocity : Trad  Te,  Te
I ()  Te ( R)
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5
Local thermal equilibrium
!?
 Trad  Te
 Assumption of Maxwell-distributed only valid in LTE
LTE
 Non-thermal contributions might play a role
Future work
9/28/2010
Sylvia K. Rathgeber
6
Non-local measurement
?
 R
2eB0 R0
 Cold resonance: 2 X ( R) 
me R
 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
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Shape of emissivity profile
 R

R  R

2 X
 Consider emission profile:
me 3 / 2
e 2 c 222X
2
2
j2 X ( ) 
sin  (cos   1)ne (
) 
16 0
2k BTe
   ((1  || cos ) 
 exp(
2 X
)

me c 2 (  2  || )
2k BTe
)  5 d  d||
• Doppler broadening
• Relativistic effects
9/28/2010
Sylvia K. Rathgeber
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Interaction of radition and plasma
?
 I ( )  I BB ( )
 Absorption and reemission
of radiation on ray path
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Radition transport
 I ( )  I BB ( )
 Consider radiation transport:
dI ( )
 j ( )   ( ) I ( )
ds
j ( )
 j ( ) 
I ( )
I BB ( )
j ( )
 ( ) 
I BB ( )
9/28/2010
Kirchhoff’s law
(valid in LTE)
Sylvia K. Rathgeber
10
The saving: Optical depth
Plasma optically thick:       ( s )ds 1  (neTe )crit  31018 keVm3
dI ( )
j ( )
 ds  j ( )  I ( ) I ( )  0
BB

I ( )  I BB ( )
 Reabsorption narrows the
observed layer
9/28/2010
Sylvia K. Rathgeber
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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
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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
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Sylvia K. Rathgeber
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Bayesian recipe
Reasoning about parameter θ:
(uncertain) prior information
+ physical model
D  f ( )
+ (uncertain) measured data
d  D 
+ Bayes Theorem
9/28/2010
prior
distribution
p ( )
p(d |  )
p( | d )  pp((dd ||))  p ( )
Sylvia K. Rathgeber
likelihood
distribution
posterior
distribution
14
Forward modelling of ECE data
if not maximized
ne(ρ), Te(ρ)
→ ne(s), Te(s)
Calculation: jν(s), αν(s)
Integration → I(ν)
Prior
Likelihood:
information
TECE, rad(ν) ↔
Tmod, rad(ν)
Posterior:
p(ne(ρ),
Te(ρ)|dECE)
9/28/2010
Modelling:
Trad(ν)
if maximized
Sylvia K. Rathgeber
Estimates:
ne(ρ)±Δne(ρ),
Te(ρ)±ΔTe(ρ)
15
IDA at ASDEX Upgrade
ne(ρ), Te(ρ)
mapping ρ(x) → ne(x), Te(x)
DLIB(ne(x), Te(x))
DDCN(ne(x))
DECE(ne(x), Te(x))
DTS(ne(x), Te(x))
LIthium Beam
emission
profile dLIB
line integrated DCN
data dDCN
ECE radiation
temperature
dECE
Thomson
Scattering
data dTS
addl. information,
constraints,
model
parameters
results: p(ne(ρ), Te(ρ)|dLIB, dDCN, dECE, dTS)
estimates: ne(ρ)±Δne(ρ), Te(ρ)±ΔTe(ρ)
9/28/2010
Sylvia K. Rathgeber
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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
17
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
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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
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Emissivity profiles
 Inward-shift of emissivity maximum
 Absorption < Emission → no black-body
 Intensity reaches black-body level
 Higher Te in observed layer than at resonance
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Sylvia K. Rathgeber
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Conventional IDA of L-mode
 Plasma optically
thick:
Te = Trad, ECE
 Plasma optically
thin:
spline fit with
edge condition
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Sylvia K. Rathgeber
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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
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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
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Literature
 W. Suttrop. Practical Limitations to Plasma Edge Electron Temperature Measurements
by Radiometry of Electron Cyclotron Emission. Technical Report 1/306, Max-PlanckInstitut 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: 1693-1769, 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): 085009 (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,
2008.
 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.
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Heat conduction
 Parallel heat conduction strongly depends on T: K||  T 5/ 2
 Small changes in T cause large changes in power flow

PSOL 2 0

(T ) 7 / 2
A
7L
PSOL
[Wm  2 ] :
A
L  100m :
T  200eV :
power flux densit y
dist ance t o divert orplat e
t emperat uredifference
t odivert orplat e
 0,e  2000Sm1 : conduct ivit y
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Diagnostic implementation
 ASDEX Upgrade: B  T  2 X  100GHz
 Frequency range accessible to radio frequency (RF) receiver
techniques as well as 'quasi'-optical techniques
 Currently installed at ASDEX Upgrade:
• Michelson interferometer: t  30ms, R  10cm
• 8-channel polychromator: t  1ms, R  5cm
• 60-channel heterodyne radiometer: t  32s, R  5mm
→ input RF signal interferes with similar signal from local oscillator
→ down-conversion to intermediate frequency
→ facilitated amplifying and filtering
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Diagnostic implementation
 Heterodyne
radiometer:
• 4 antennas on
low field side
• 5 mixer
 RF  89  187GHz
 LO  95,101,128,133,167GHz
• 3 IF chains (36/12/12 channels)  IF | RF  LO | 2 18GHz
 IF amplifier
 Band pass filter   300 / 600 / 600 MHz
 Data acquisition S  31.25kHz
• Absolute calibrated by measurements of black-body radiation from
laboratory hot (773 K) and cold (77 K) sources
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Radial resolution
m ( R )  m
eB0 R0
me R
 Radial resolution depends

R  R
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)
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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:
Trad  Trad
 V
 RF
 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
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Harmonic overlap
 Resonance frequencies:
eB0 R0
m ( R )  m
me R
B0  2.5T , R0  1.65m
 160-200 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
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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:  2 X  3.9 1019  neTe [keV ]
measurements  (neTe )crit  31018 keVm3
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High density limit: Cut-off
 Below eigenfrequency of plasma electromagnetic waves are
completley shielded by electrons → cut-off
 O-mode waves (E || B0): CO
ne e 2
 p 
 0 me
1
2
2
2
X-mode waves (E ┴ B0): CO  (c  c  4 p )
mO
2
 Cut-off density: nCO  m
0
me
B2
mX
nCO
 m(m  1)
9/28/2010
0
me
2X
B 2 , m  2  nCO
 2 1019 m3 B[T ]2
Sylvia K. Rathgeber
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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
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Doppler broadening
 Trad = Te in case
of high optical
depth
 Trad < Te in case
of low optical
depth
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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
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