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Max-Planck-Institut für Plasmaphysik,EURATOM Association
Estimation of ne and Te with microwave diagnostics and
investigations on profile changes with RMP
Estimation of Te from ECE data
Estimation of ne from reflectometry data
Behaviour of profiles with RMP
Working title:
Working topics:
Sylvia K. Rathgeber
Motivation
ECE diagnostic: long-standing workhorse for Te analysis
Why another ECE analysis? What is different?
9/28/2010 Sylvia K. Rathgeber 2
Shine-through Current ECE analysis:
Trad = Te, ν → R
Te = 250 eV in SOL ↔Power flux density > 600 MW/m2
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 4
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 identifiedwith black-body intensity:
Assume Maxwell-distributed gyrotron velocity :
9/28/2010 Sylvia K. Rathgeber 5
em m
eBm
Rm
ReBmR
emm
00)(
)( 8
)()(23
2
TkhTkc
II BradBBB
)()( RTI eeerad TTT ,
R
RBRBBB tttot
00)(
Local thermal equilibrium
Assumption of Maxwell-distributed only valid in LTE
Non-thermal contributions might play a role
Future work
9/28/2010 Sylvia K. Rathgeber 6
erad TT ?LTE
!
Non-local measurement
Cold resonance:
non-local measurement→ emission profile broadened:• Doppler broadening: observation
not perpendicular to field line
• Relativistic effects: relativistic massincrease results in frequency shift
9/28/2010 Sylvia K. Rathgeber 7
Rm
ReBR
eX
002
2)(
?R
Shape of emissivity profile
Consider emission profile:
• Doppler broadening• Relativistic effects
9/28/2010 Sylvia K. Rathgeber 8
X
RR2
||5||
22
2||
2/322
0
22
22
2
)2
)(exp(
))cos1((
)2
()1(cossin16
)(
ddTk
cm
Tk
mn
cej
eB
e
X
eB
ee
XX
R
Interaction of radition and plasma
Absorption and reemission of radiation on ray path
9/28/2010 Sylvia K. Rathgeber 9
)()( BBII ?
Radition transport
Consider radiation transport:
9/28/2010 Sylvia K. Rathgeber 10
)()( BBII
)()(
)()(
I
I
jj
BB
)(
)()(
BBI
j Kirchhoff’s law
(valid in LTE)
)()()()(
Ijds
dI
The saving: Optical depth
Plasma optically thick:
Reabsorption narrows the observed layer
9/28/2010 Sylvia K. Rathgeber 11
)()( 0)()(
)()(
)(
BB
BB
IIII
jj
ds
dI
1)( dss 318103)( keVmTn critee
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 12
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 13
Bayesian recipe
9/28/2010 Sylvia K. Rathgeber 14
)(p
Dd
)(fD )|( dp
)( )|( )|( pdpdp )|( dp )|( dp )(p
Forward modelling of ECE data
9/28/2010 Sylvia K. Rathgeber 15
if maximized
if n
ot m
axim
ized
Calculation: jν(s), αν(s)Integration → I(ν)
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
9/28/2010 Sylvia K. Rathgeber 18
Core:high ne & Te
→ plasma optically thick
Edge:steep Te gradient & low ne
→ shine-through conditions
Modelling of Trad
9/28/2010 Sylvia K. Rathgeber 19
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
Emissivity profiles
9/28/2010 Sylvia K. Rathgeber 20
Inward-shift of emissivity maximum
Intensity reaches black-body level
Absorption < Emission → no black-body
Higher Te in observed layer than at resonance
Conventional IDA of L-mode
9/28/2010 Sylvia K. Rathgeber 21
Plasma optically thick:Te = Trad, ECE
Plasma optically thin:spline fit with edge condition
Forward modelling of L-mode
9/28/2010 Sylvia K. Rathgeber 22
Data consistent within separatrix
Plasma optically thick:Te slightly reduced
Around separatrix: Te > Trad, ECE
SOL: no data fit possible
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 23
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: 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.
9/28/2010 Sylvia K. Rathgeber 24
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 25
2/5|| TK
2/70 )(7
2T
LA
PSOL
tyconductivi:2000
platedivertor to
difference erature temp:200
platedivertor todistance :100
densityflux power :][
1,0
2
Sm
eVT
mL
WmA
P
e
SOL
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 26
GHzTB X 1002
cmRmst 10,30 cmRmst 5,1
mmRst 5,32
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
9/28/2010 Sylvia K. Rathgeber 27
GHz
GHz
LO
RF
167,133,128,101,95
18789
kHzS 25.31MHz600/600/300
• Absolute calibrated by measurements of black-body radiation from laboratory hot (773 K) and cold (77 K) sources
GHzLORFIF 182||
Radial resolution
Radial resolution dependson frequency resolution:
Frequency resolution is limited by:• Doppler broadening
(ASDEX Upgrade: 86° ≤ θ ≤ 94°)
• Relativistic effects: relativistic mass increase results in frequency shift
9/28/2010 Sylvia K. Rathgeber 28
Plasma core: RF bandwidth (ΔνRF=600MHz) matches resolution limit due to line broadening (relativistic effects dominant)
Plasma edge: resolution determined by receiver (ΔνRF=300MHz)
RR
Rm
ReBmR
em
00)(
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 29
RF
Vradrad TT
Harmonic overlap
Resonance frequencies:
160-200 GHz: depending on optical thickness, radiation consists of 2nd and 3rd harmonic
9/28/2010 Sylvia K. Rathgeber 30
mRTB
Rm
ReBmR
em
65.1,5.2
)(
00
00
Only 1st and 2nd harmonics are feasible for measurements of and from the low field side
Low density limit: optical depth
9/28/2010 Sylvia K. Rathgeber 31
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
][109.3 192 keVTn eeX
318103)( keVmTn critee
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 32
e
epCO m
en
0
2
)4(2
1 22pccCO
202 Bm
mne
mOCO
2,)1( 20 mBm
mmne
mXCO
23192 ][102 TBmn XCO
Consequence of limitations
9/28/2010 Sylvia K. Rathgeber 33
2nd harmonic X-mode is the best candidate
for ECE measurements
according to limitations due to harmonic overlap,
cut-off and optical depth
Doppler broadening
9/28/2010 Sylvia K. Rathgeber 34
Trad = Te in case of high optical depth
Trad < Te in case of low optical depth