ELECTRON DENSITY FLUCTUATIONS AND FLUCTUATION-INDUCED
TRANSPORT IN THE REVERSED-FIELD PINCH
by
Nicholas E. Lanier
A dissertation submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
(Physics)
at the
University of Wisconsin–Madison
1999
i
ELECTRON DENSITY FLUCTUATIONS AND FLUCTUATION-INDUCED TRANSPORT IN THE REVERSED-FIELD PINCH
Nicholas E. Lanier
Under the supervision of Professor Stewart C. Prager
At the University of Wisconsin–Madison
An extensive study on the origin of density fluctuations and their role in particle
transport has been investigated in the Madison Symmetric Torus reversed-field pinch. The
principal physics goals that motivate this work are: investigating the nature of particle
transport in a stochastic field, uncovering the relationship between density fluctuations and
magnetic field fluctuations arising from tearing and reconnection, identifying the
mechanisms by which a single tearing mode in a stochastic medium can affect particle
transport.
Following are the primary physics results of this work. Measurements of the radial
electron flux profiles indicate that confinement in the core is improved during pulsed
poloidal current drive experiments. Correlations between density and magnetic fluctuations
demonstrate that the origin of the large amplitude density fluctuations can be directly
attributed to the core-resonant tearing modes, and that these fluctuations are advective in the
plasma edge; however, these fluctuations appear compressional in the core, provided the
nonlinear terms are small. Correlations between density and radial velocity fluctuations
indicate that although the fluctuations from the core-resonant modes dominate at the edge,
their relative phase is such that they do not cause transport there, consistent with the
expectation that core modes do not destroy edge magnetic surfaces. This is not the case in the
plasma core, where the density and radial velocity fluctuations are in phase, indicating that
ii these fluctuations couple to induce transport. Measurements during PPCD discharges show a
large reduction in density fluctuations associated with the core-resonant modes. Furthermore,
the phase of these fluctuations in the core changes to be π/2 relative to the radial velocity
fluctuations, indicating these fluctuations no longer couple to induce transport.
iii
Acknowledgements
Although my defense was only two hours, it represented the culmination
of a long and challenging path. In pursuing my degree, there have been many
noteworthy individuals that have offered support and direction, and although I
have done the work, they have made this possible, and I wish to acknowledge
their efforts.
Prior to my graduate career, five individuals stand out as being very
influential in my progress in physics. Mr. Larry Dean, my high-school physics
teacher who started my formal training in physics, Russ Coverdale, the
academic advisor at Purdue, who stuck me in the Honors curriculum and forced
me to swim. Still as an undergraduate, my first real world work experience was
obtained with Dr. John Molitoris, may he always have a place to sit, and Dr.
Paul Springer, who showed the faith in my leadership skills by sending me to
Russia to run some great physics experiments. Finally I’d like to thank Dr. C.
Choi, who introduced me to plasma physics and opened the door to my coming to
Wisconsin.
iv
My years at Wisconsin have been the most enjoyable of my life and the
MST group has been a principal reason for that. Faculty such as Sam Hokin,
Paul Terry, James Callen, and Chris Hegna (if not he should be) have really
worked to expand my plasma physics knowledge. I am especially grateful for the
efforts of my advisor Stewart Prager, Cary Forest, and Darren Craig (who will
be faculty someday, no doubt about it). MST staff like John Sarff, who
introduced me to PPCD, Dan “former vacuum man now diversifying into
computer repair” Den Hartog, Genady “come with an envelope leave with a
solution” Fiksel have really fostered my experimental talents. Not to be
underestimated are the benefits gained from working with David “the Texan”
Brower and Yong “lip smackin’ good” Jiang. Finally, I thank Dale, Larry, Paul,
Mikey, Kay, John, Don and the rest of the MST support crew for helping to turn
my ideas into reality.
By far the most outstanding aspect of MST life are the graduate students.
In my six years here, students like, James “Jimbo” Chapman, Carl “the only
man I’ve seen argue (and win) with Callen” Sovinec, Jay “the Mason” Anderson,
Ted “Ironman” Biewer, Brett “the big lovable vacuum Nazi” Chapman, Ching-
Shih “LT” Chaing, Alex “BA” Hansen, Derek “the Bavenator” Baver, Paul
“Wrong glass sir” Fontana, Cavendish “the Dishman” Mckay, Susanna
“nickname pending” Castillo, and of course Neal “it’ll happen someday” Crocker,
have made my career here unforgettable. I have no wish to leave such a
remarkable set of individuals, but my development as a physicist requires it.
Finally I’d like to thank those outside my work life, my parents who not so
jokingly quote that I was bred for science, my sister Catherine, my friends, Scott
v
Kruger, Paul Ohmann, Brian Totten, others that have been supportive of my
efforts here. I have been truly blessed.
In memory of
Katherine Nicole Lanier (December 20, 1996)
vi
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Introduction 1
1.1 The Reversed Field Pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Magnetic Island Formation and Stochasticity . . . . . . . . . . . . . . . . . . 6
1.3 Stochastic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Fluctuation-Induced Radial Particle Flux . . . . . . . . . . . . . . . . . . . . . 10
1.5 Controlling Fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Overview of Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
vii
2 The Far-Infrared Laser System 17
2.1 Plasma Interferometry Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The Far-Infrared Laser Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Diagnostic Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 The CO2 Pumping Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 The Twin Far-Infrared Laser . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 Power Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.5 Detection Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Digital Phase Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Neutral Hydrogen Density In MST 39
3.1 Hydrogen Fueling in MST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.1 The Fueling Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.2 Franck-Condon Neutrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.3 Neutral Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.4 Measuring Neutral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 The Hα Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 Alignment and Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Hα Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
viii
3.3.1 Hα Behavior in Standard Discharges . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Hα Behavior in PPCD Discharges . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Neutral Particle Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Neutral Particle Profiles in Standard and PPCD Discharges . . 55
3.4.2 Neutral Particle Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.3 Neutral Particle Population and CHERS . . . . . . . . . . . . . . . . . . 59
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Impurity Behavior In MST 64
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Radiative and Dielectronic Recombination . . . . . . . . . . . . . . . 68
4.2.3 Charge Exchange Recombination. . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Charge State Equilibrium (Coronal or LTE) . . . . . . . . . . . . . . . . . . . . 71
4.4 Electron Impact Excitation and Line Emission . . . . . . . . . . . . . . . . . . 72
4.5 The ROSS Filtered Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5.1 Filter Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5.2 The Soft X-ray Diodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5.3 Diagnostic Geometry and Light Collection. . . . . . . . . . . . . . . . 77
ix
4.5.4 Deciphering Impurity Line Emission . . . . . . . . . . . . . . . . . . . . 78
4.5.5 Line Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 Impurity Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6.1 Impurity Concentration in Standard Discharges . . . . . . . . . . . 80
4.6.2 Impurity Concentration in PPCD Discharges . . . . . . . . . . . . . 83
4.6.3 Electron Sourcing From Impurities . . . . . . . . . . . . . . . . . . . . . 88
4.6.4 Impurity Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.7 Estimating Impurity Confinement Times . . . . . . . . . . . . . . . . . . . . . . 91
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Radial Electron Flux Profile Measurements 96
5.1 Equilibrium Electron Density Behavior . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.1 Density Profiles in Standard Discharges . . . . . . . . . . . . . . . . . 97
5.1.2 Density Profiles During PPCD . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2 Radial Particle Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 Extracting Radial Particle Flux . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.2 Radial Particle Flux in Standard and PPCD Discharges . . . . . 104
5.2.3 Particle Confinement Times . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Convective Power Loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
x
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Fluctuations and Fluctuation-Induced Particle Transport 110
6.1 Electron Density Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1.1 Chord-Integrated Fluctuation Amplitude . . . . . . . . . . . . . . . . . 112
6.1.2 Frequency Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.1.3 Wave Number Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.4 Correlation Between Density and Magnetic Fluctuations . . . . 119
6.1.5 Local Density Fluctuation Profiles . . . . . . . . . . . . . . . . . . . . . 120
6.2 Origin of Density Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.1 The Electron Continuity Equation . . . . . . . . . . . . . . . . . . . . . . 124
6.2.2 Measurements of the Radial Velocity Fluctuations . . . . . . . . . 125
6.2.3 Nature of Density Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 Fluctuation-Induced Particle Transport . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7 Conclusions 136
A Polarimetry / Interferometry Discussion 140
A.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
xi
A.2 Derivation of Measured Signal Power . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.3 Derivation of Reference Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.4 Digital Extraction of Interferometer Phase . . . . . . . . . . . . . . . . . . . . . . 149
A.5 Extracting the Polarimetry Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B FIR Density Codes and Analysis Procedures 157
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.2 Processing FIR Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.2.1 General Code Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B.2.2 The FIR Processing Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2.3 Pre-Inspection of Processed Data . . . . . . . . . . . . . . . . . . . . . . . 173
B.2.4 Inspection Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.2.5 Manual Removal of Phase Jumps. . . . . . . . . . . . . . . . . . . . . . . . 184
B.2.6 The Manual Processing Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
C FIR Polarimety Codes and Analysis Procedures 200
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
C.2 Processing Polarimetry Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
C.2.1 The Polarimetry Processing Code . . . . . . . . . . . . . . . . . . . . . . 201
C.3 Mesh Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
xii
D Hα, CO2 and other Processing Codes 215
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
D.2 The Hα Processing Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
D.3 The CO2 Processing Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
E Hα Array Components List 227
E.1 The Hα Parts List. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
F The SXR Ratio – What Does It Really Mean? 228
F.1 Dispelling the Myth Behind the SXR Ratio . . . . . . . . . . . . . . . . . . . . . 228
xiii
List of Tables
2.1 FIR Chord Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 The FIR Mesh Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1 Lines Monitored By ROSS Spectrometer . . . . . . . . . . . . . . . . . . . . . 73
4.2 ROSS Filter Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
E.1 The Hα Parts List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
xiv
List of Figures
1.1 The magnetic field configuration of the RFP . . . . . . . . . . . . . . . . . . . 4
1.2 The RFP q profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Tearing mode island formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Magnetic island overlap in RFP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 The PPCD circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Fluctuation reduction during PPCD. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 The Far-Infrared Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 The CO2 pumping laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 CO2 mode of vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 The CO2 lasing cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 The twin FIR laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 FIR beam profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 FIR chord locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.8 Preamplifier gain curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.9 Phase resolution histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
xv
2.10 Chord-integrated density for r ~ -24 cm . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 The MST fueling cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Collision rates for atomic hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 The Hα detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Hα filter transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Chord-averaged Hα trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Hα emission over sawtooth crash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.7 Radial profile of chord-integrated Hα emission . . . . . . . . . . . . . . . . . . 53
3.8 Chord-integrated Hα in standard and PPCD plasmas . . . . . . . . . . . . . . 54
3.9 Chord-averaged neutral density in standard and PPCD plasmas . . . . . 56
3.10 Neutral density profile in standard discharge . . . . . . . . . . . . . . . . . . . 58
3.11 Neutral density profile in PPCD discharge . . . . . . . . . . . . . . . . . . . . . 58
3.12 Charge exchange cross-sections for CHERS . . . . . . . . . . . . . . . . . . . . 61
4.1 Impurity state density continuity equation . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Impact ionization cartoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Radiative and dielectronic recombination cartoons. . . . . . . . . . . . . . . 69
4.4 Charge exchange recombination cartoon . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Collision rates for O VII and O VIII . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Excitation rates for core impurity states of C, Al ,and O . . . . . . . . . . 73
4.7 ROSS filter transmission curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8 The AXUV-100 diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
xvi
4.9 O VII and O VIII densities over sawtooth . . . . . . . . . . . . . . . . . . . . . 81
4.10 C V and C VI densities over sawtooth . . . . . . . . . . . . . . . . . . . . . . . . 82
4.11 O VII and O VIII emission in PPCD . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.12 ROSS emission (High energy channel) . . . . . . . . . . . . . . . . . . . . . . . 86
4.13 ROSS emission ( C VI channel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.14 ROSS emission ( C V channel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.15 ROSS emission ( B IV channel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.16 Bolometric vs. radiated power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.17 O VIII confinement time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 Chord-integrated density over crash . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Electron density profiles over sawtooth crash . . . . . . . . . . . . . . . . . . 99
5.3 Chord-integrated density during PPCD . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Electron density profiles during PPCD . . . . . . . . . . . . . . . . . . . . . . . . 102
5.5 Total radial particle flux in standard and PPCD discharges . . . . . . . . 105
6.1 Chord-integrated density fluctuations over sawtooth crash . . . . . . . . . 113
6.2 Chord-integrated density fluctuation profiles . . . . . . . . . . . . . . . . . . . 114
6.3 Chord-integrated density fluctuation frequency spectrum . . . . . . . . . . 115
6.4 Density fluctuation m behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.5 Average toroidal mode spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.6 Toriodal mode spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.7 Density fluctuation coherence with core-resonant modes . . . . . . . . . . 120
xvii
6.8 Radial density fluctuation profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.9 Computed C V and He II profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.10 Coherence between density and radial velocity of He II . . . . . . . . . . . 128
6.11 Coherence between density and radial velocity of C V . . . . . . . . . . . . 129
6.12 Coherence profile between density and radial velocity of He II . . . . . 131
6.13 Coherence phase profile in standard and PPCD discharges . . . . . . . . . 133
B.1 Example of a phase jump missed by FIR processing code . . . . . . . . . . 177
B.2 Example of incorrect offsetting of the FIR data . . . . . . . . . . . . . . . . . . 178
B.3 Graphic interface of MAN_FIX_FAST.PRO. . . . . . . . . . . . . . . . . . . . . 185
B.4 Missed phase jump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
B.5 Zoomed in view of phase jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.6 Example of a “GOOD” density trace . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
F.1 The SXR ratio vs. Plasma Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
F.2 The transmission curves of the BE_1 And BE_2 foils . . . . . . . . . . . . . . 230
1
1: Introduction
Three physics goals motivate this thesis. They include: investigating the
nature of particle transport in a stochastic magnetic field, uncovering the
relationship between density fluctuations and magnetic field fluctuations arising
from tearing and reconnection, identifying the mechanisms by which a single
tearing mode in a stochastic medium can affect particle transport.
These issues are particularly relevant to the reversed-field pinch1 (RFP)
because improving confinement continues to be the primary obstacle in
advancing the RFP as a fusion concept. Recent theoretical understanding
predicts that large magnetic tearing modes resonant in the core are responsible
for particle and energy transport2 in the RFP, and has led to the idea that
confinement can be improved by reducing these fluctuations. Magneto-
Hydrodynamics (MHD) modeling indicates that these tearing modes are driven
by gradients in the parallel current density gradient, and can be reduced
through auxiliary current drive.3 These predictions are supported by recent
experimental evidence showing that during pulsed poloidal current drive
2
(PPCD), which in an experiment designed to flatten the edge parallel current
density gradient, can halve the magnetic fluctuations while increasing the global
energy confinement fivefold.4
Understanding fluctuations and their role in confinement continues to be
a primary research focus of the MST group. Past experiments, limited to the
extreme plasma edge, have explored both magnetic and electrostatic fluctuation-
induced particle5,6 and energy7 transport. These experiments led to two
conclusions about transport in the RFP. The fluctuation-induced particle
transport experiments showed that electrostatic transport dominates over the
magnetic component in the edge, but further in; the magnetic fluctuations play a
larger role. The second conclusion was that although particle transport from
magnetic fluctuations was small, energy transport was not.
This work aims to improve our understanding of the transport processes
over the entire RFP plasma cross section. This is conducted in two parts: by
quantitatively investigating the equilibrium particle transport through
simultaneous measurement of the electron density and source profiles (from
both hydrogen and impurities), and exploring the fluctuations and fluctuation-
induced particle transport by examining the relationship between electron
density fluctuations and radial velocity fluctuations. Five experimental tools
enabled this study in the MST8 reversed-field pinch. They include a fast multi-
chord far-infrared laser interferometer to measure equilibrium and fluctuating
electron density throughout the plasma, a multi-chord Hα radiation diagnostic to
quantify the electron sourcing from ionization of neutral hydrogen, a thin-film
multi-foil diode spectrometer to estimate the electron source from impurities, a
fast Doppler spectrometer to monitor impurity ion radial velocity fluctuations,
3
and inductive current profile control (known as PPCD) to alter the fluctuation
and particle transport characteristics.
This work reports three primary conclusions. First, through
measurements of the radial electron flux profile, we have determined that
pulsed poloidal current drive, or PPCD, experiments improve particle
confinement in the reversed-field pinch (RFP) core. Second, most of the large
amplitude density fluctuations are directly attributed to the core-resonant
tearing modes, and that these density fluctuations are compressional in the core
and advective (i.e. resulting from the radial motion of the equilibrium density
gradient) in the edge. Finally, we demonstrate for standard discharges, that the
density fluctuations associated with the core-resonant tearing modes do cause
particle transport in the core but do not cause transport in the RFP edge, but
when magnetic fluctuations are reduced (during PPCD), particle transport from
these core-resonant modes also drops.
In this introductory section we revisit some basic principles of the MST
RFP as well as a heuristic description of magnetic tearing modes and their
relevance to particle transport. We also briefly discuss the inductive current
profile control capability that has proved very useful in examining the
relationship between magnetic fluctuations and confinement in the RFP. In the
final section we present an overview of the thesis.
1.1 The Reversed-Field Pinch (RFP)
The RFP is a toroidally axisymmetric current-carrying plasma where the
toroidal magnetic field amplitude is of the same order as the poloidal magnetic
field. An interesting feature of the RFP is that upon startup the plasma
4
naturally relaxes to its preferred state where the toroidal field reverses
direction, hence the name ‘reversed’-field pinch (figure 1.1). This relaxation
mechanism, sometimes referred to as the ‘Dynamo’, is responsible for the
sustainment of the RFP discharge; however, in carrying out this task, the
dynamo degrades the particle and energy confinement of the plasma.
Conducting Shell Surrounding Plasma
BT Small & Reversed at Edge
BTBP
r B
Figure 1.1 – The magnetic field configuration of the RFP. The toroidal field is about the same magnitude as the poloidal field and reverses direction near the plasma edge.
The preferred RFP state was first derived by Taylor9 in 1974 and was
based on the conjecture that the magnetic helicity ( Ko ) integrated over the entire
plasma volume would be conserved.
ddt
Ko =ddt
A • B V∫ dV ≈ 0 (1.1)
By minimizing the magnetic energy with respect to the magnetic helicity, Taylor
arrived at a preferred magnetic configuration described by
∇ × B = λB , (1.2)
5
where λ is a constant. Equation 1.2 describes the RFP minimum energy state in
which the dynamo works to maintain.
The dynamo mechanism has been the subject of a number of exhaustive
studies. In 1998, spectroscopic measurements performed by Chapman10 reported
that the correlated cross product between the magnetic and velocity fluctuations
( ˜ v × ˜ b ) was sufficient to balance parallel ohm’s law in the RFP core and sustain
the RFP discharge. More recently, the measurements of Fontana11 reached a
similar conclusion about parallel ohm’s law balance in the RFP edge, confirming
the earlier Langmuir probe results measured by Ji12 in 1992.
The magnetic field fluctuations ( ) that contribute to the dynamo term
result from resistive tearing instabilities within the plasma. Unlike the
tokamak, the low toroidal field of the RFP leads to a safety factor (q
˜ b
= aBT RoBP )
that is much less than 1.0, where q monotonically decreases and changes sign
where the toroidal field reverses (figure 1.2). As a consequence, this magnetic
configuration has a large number of closely packed low-order resonant surfaces.
Resonant surfaces occur at radial locations where q is rational, or in other
words,
q = m n (1.3)
where and n are integers. Rational surfaces are undesirable because
magnetic field tearing and reconnection is permitted at these radial locations,
making them susceptible to the formation of magnetic islands.
m
6
radius, r
q(r)
RFP
Tokamak
~1~0.2
0 a0
Dominant Resonant Surfaces
Figure 1.2 – The q profile of the RFP has many low-order, closely packed resonant surfaces. These surfaces are susceptible to tearing mode formation.
1.2 Magnetic Island Formation and Stochasticity
With tearing and reconnection permitted at a rational surface, magnetic
islands can form (figure 1.3). These islands, often referred to as modes, are
undesirable because they allow heat and particles to rapidly traverse the radial
extent of the island and thereby degrade confinement.
7
q=m/n
resonant surface
⇒
magnetic island
W ~ ˜ B r / B
Figure 1.3 – Rational surfaces permit tearing and reconnection of the magnetic field to occur, allowing islands to form. Magnetic islands degrade confinement by allowing rapid transport across the island’s width.
The situation outlined above is compounded in the RFP because as islands
form and begin to grow on the many closely packed rational surfaces, they can
overlap. When islands overlap, the magnetic field becomes stochastic, and the
field lines can wander freely throughout the overlap region. If a large number of
islands are overlapping, large stochastic regions can form in the plasma, and
instead of rapidly transporting heat and particles just across an island width,
the confinement is degraded over the entire stochastic region (figure 1.4).
8
q(r)
magnetic stochasticity
1,51,6
1,7 …
typical island width
(m,n)
Figure 1.4 – If a large number of magnetic islands overlap, a large area of the plasma can become stochastic, further enhancing the particle and energy transport.
1.3 Stochastic Transport
Rechester and Rosenbluth,13 who modeled electron heat transport via
parallel conduction along wandering field lines, addressed the fusion relevance
of transport in a stochastic magnetic field in 1978. They conjectured that the
stochastic diffusion coefficient ( ) would take the form, Dst
Dst ≈ π˜ b rBo
21
1 LA + 1 λmfp
, (1.4)
where ˜ b r Bo is the fluctuation to mean field ratio for the magnetic field, λ mfp is
the electron collision mean free path, and LA is the autocorrelation length. In
MST, ˜ b r Bo is typically about 1-2% and the collisional mean free path is long,
on the order of tens of meters. The autocorrelation is basically a fudge-factor and
for MST is about a meter and therefore Dst ≈.3 →1.2 ×10 −4 m . The critical aspect
9
Loss
behind this loss mechanism is that the diffusion loss rate will be proportional to
the particle’s parallel velocity leading to
D ∝ Dst v || . (1.5)
The implications of a velocity dependent diffusion rate are far reaching in that
by preferentially transporting particles of higher energy, one leads to the
possibility of current or momentum diffusion and non-maxwellian distribution
functions. It was the idea of current diffusion that lead Jacobson and Moses14,15
(1984) to propose the kinetic dynamo theory (KDT) as a means for sustaining the
RFP discharge; however, this mechanism has yet to be observed (although one
might argue we haven’t looked very hard). In defense of the MHD dynamo, self-
consistent calculations conducted by Terry and Diamond16 (1990), indicate that
the current transport from the KDT is insufficient to explain the dynamo.
The concept of stochastic diffusion was applied to particle transport by
Harvey17 (1981), who proposed that if particle diffusion were weighted by
parallel velocity, the electrons would be transported more rapidly inducing an
ambipolar radial electric field. Assuming that the local distribution functions did
not deviate substantially from Maxwellian, the radial particle flux would be
described as
Γr = −
2π
Dst vT n1n
∂n∂r
+1
2T∂T∂r
+eEA
T⎛ ⎝
⎞ ⎠ , (1.6)
where is the electron thermal velocity, n and vT T are the electron density and
temperature, EA st is the ambipolar electric field, and D is the stochastic diffusion
coefficient described in equation 1.4. The result is that particle diffusion is not
driven solely by gradient in density as predicted in the Fick’s Law case, but that
10
gradients in electron temperature and the ambipolar electric field would also be
important. In this report, we do not address the validity of Harvey’s suppositions
or apply equation 1.6 to our radial particle flux measurements. However, in
chapter 5, we compare our measured total radial particle flux with the particle
transport modeling conducted for RFX discharges,18 and equation 1.6 is vital to
those results. With the profile measuring capabilities of the FIR interferometer,
Thomson Scattering system, and Heavy-Ion Beam Probe (HIBP), it is hoped that
experiments to validate equation 1.6 will be undertaken by MST.
1.4 Fluctuation-Induced Radial Particle Flux
Experimentally, we extract Γ from the electron continuity equation by
simultaneously measuring the electron density and source. In this section, we
expand Γ to isolate the fluctuation-induced particle flux term, and identify the
measurable quantities.
The equilibrium particle flux (Γ ) is defined in the electron continuity
equation in the balancing term between the change in electron density and the
electron source,
∂ne
∂ t+∇ • Γ = Se , (1.7)
where Γ = nev . Expanding Γ into its equilibrium and fluctuating components, we
see that
Γ = no + ˜ n ( ) v o + ˜ v ( )=nov o + ˜ n ̃ v +no˜ v + ˜ n v o . (1.8)
11
Imposing toroidal axisymmetry on the equilibrium quantities and averaging
over a flux surface, the two cross terms integrate to zero leaving the radial
particle flux as
Γr = novor + ˜ n ̃ v r = Γequilibrium + Γ fluctuation− induced (1.9)
With the classical E×B inward pinch velocity small and assuming no anomalous
pinch effects, the equilibrium radial velocity is negligible (v ), leaving the
fluctuation-induced transport term solely responsible for the overall radial
particle flux. The fluctuation-induced particle flux is defined as
or ≈ 0
Γ fluctuation −induced = ˜ n ̃ v r = γ ˜ n Amp ˜ v r Amp cos δ nv( ), (1.10)
where and are the amplitudes of the density and radial velocity
fluctuations respectively, and
˜ n Amp ˜ v rAmp
γ and δ nv are the coherence amplitude and phase
between the density and velocity fluctuations. In the linear ideal
magnetohydrodynamics (MHD) description, the velocity fluctuations that cause
fluctuation-induced particle transport are directly linked to the magnetic
fluctuations.
1.5 Controlling Fluctuations
As mentioned earlier, the large magnetic fluctuations are a result of the
plasma attempting to fluctuate itself back to its preferred energy state. Although
this process sustains the RFP discharge, the magnetic fluctuations degrade
confinement. One of the principal research goals of MST has been to develop
ways to control magnetic fluctuations in the RFP; pulsed poloidal current drive
(PPCD) has been very successful at accomplishing this.
12
PPCD is based on the premise that any work done to aid the plasma in
reaching its preferred state, means that less work is required of the magnetic
fluctuations. The plasma desires a flat parallel current density profile ( rJ •
rB B2 ),
but the ohmic heating applied to MST is very inefficient at driving parallel
current in the plasma edge. As a result, gradients in the parallel current density
profile form and resistive tearing modes (magnetic islands) become unstable and
begin to grow. As the modes grow, they flatten the current density profile but
also degrade confinement. PPCD is designed to drive current in the RFP edge,
thereby reducing the need for the magnetic fluctuations.
The experimental setup is elegantly simple (figure 1.5). Current is driven
poloidally around the conducting shell thereby changing the toroidal magnetic
field. From Faraday’s law (∇ × E = − ∂B ∂t ), the change in the toroidal magnetic
field creates a poloidal electric field that drives poloidal current. As the field at
the RFP edge is principally poloidal, the current driven is parallel to the
magnetic field and works to flatten the parallel current density gradient.
Eθ,Jθ
C1 C2 C4C3
Figure 1.5 – The PPCD circuit. Current driven in the shell changes the toroidal magnetic field, thereby producing a poloidal electric field that works to flatten the edge parallel current density gradient.
13
The results from PPCD have been very encouraging. Measurements to
date have shown that the magnetic fluctuations are halved and that global
energy confinement increases fivefold.4 Shown in figure 1.6, are the poloidal
electric field pulses and the subsequent reduction in magnetic fluctuation
amplitude.
5 10 15 20Time (ms)
1
234
(%)
˜ b rmsB
250
–10
5
0
5
PPCD
(V/m) Eθ
(a)MST
Figure 1.6 – The poloidal electric field (top) and the magnetic fluctuations (bottom). Note the substantial reduction in fluctuation percentage.
Beyond obtaining overall confinement improvements, PPCD has proven to
be invaluable in studying the RFP. PPCD offers the ability to turn the
fluctuation levels in MST up or down, allowing a more complete investigation of
the role of magnetic fluctuations in particle and energy confinement in the RFP.
1.6 Overview of Thesis
In this work we have, measured the radial particle flux profile in both
standard and PPCD discharges, characterized and quantified the large-scale
14
density fluctuations over the entire plasma cross section, and measured
(qualitatively in the core, quantitatively in the edge) the fluctuation-induced
particle flux from the global core-resonant tearing modes. The chapters that
discuss this work are organized as follows. Chapter 2 introduces the Far-
Infrared (FIR) Laser Interferometer* system that was employed to measure both
the equilibrium and fluctuating components of the electron density profiles. The
discussion focuses on theory of operation, diagnostic hardware, and the phase
analysis technique that has greatly expanded the diagnostic’s time response.
Chapter 3 describes the multi-chord Hα detector array used in measuring the
electron source for the ionization of neutral hydrogen. This chapter also
addresses some very important secondary issues, such as core neutral
population and power loss via neutral transport, that have been uncovered
during this investigation. The measurements of the electron source from high-Z
impurities are discussed in chapter 4. We introduce the ROSS multi-foil diode
spectrometer used in determining the impurity concentrations of oxygen, carbon
and aluminum. Building on the electron source information discussed in
chapters 3 and 4, chapter 5 presents the results of the radial particle flux
measurements. We also investigate the general behavior of the electron density
profiles during standard and PPCD discharges and what their features state
about the particle confinement properties of MST. Finally, we address the
question of fluctuation-induced transport. In chapter 6 we characterize the
large-scale density fluctuations by examining their amplitude, frequency
spectra, spectral content, and relation to both magnetic and radial velocity
*Developed in collaboration with the University of California at Los Angeles
Plasma Diagnostic Group.
15
REFERENCES
fluctuations. From the measurements reported in chapters 5 and 6, we conclude
that, PPCD improves particle confinement in the MST core, the large-scale
density fluctuations are directly attributed to the global core-resonant tearing
modes and are compressional in the core but advective in the edge, and finally
we state that the core-resonant tearing modes do cause transport in the RFP
core but not in the edge.
1 H. A. Bodin and A. A. Newton, Nuclear Fusion, 19, 1255 (1980).
2 D. D. Snack, et. al., Proceedings of Fourteenth International Conference on Plasma Physics and Controlled Nuclear Fusion Research, IEIA, Wurzburg, Germany (1992).
3 Y. L. Ho, Nuclear Fusion 31, 341 (1991).
4 J. S. Sarff, N. E. Lanier, S. C. Prager, M. R. Stoneking, Physical Review Letters, 78, 62 (1997).
5 M. R. Stoneking, S. A. Hokin, S. C. Prager, G. Fiksel, H. Ji, and D. J. Den Hartog, Physical Review Letters, 73, 549 (1994).
6 T. D. Rempel, C. W. Spragins, S. C. Prager, S. Assadi, D. J. Den Hartog, and S. Hokin, Physical Review Letters, 67, 1438 (1991).
7 G. Fiksel, S. C. Prager, W. Shen, and M. R. Stoneking. Physical Review Letters, 72, 1028 (1994).
8 R. N. Dexter, D. W. Kerst, T. W. Lovell, S. C. Prager, and J. C. Sprott, Fusion Technology 19, 131 (1991).
9 J. B. Taylor, Physical Review Letters, 33, 1139 (1974).
10 J. T. Chapman, Ph.D. Thesis (1998).
11 P. W. Fontana, Ph.D. Thesis (1999).
12 H. Ji, A. F. Almagri, S. C. Prager, and J. S. Sarff, Physical Review Letters, 73, 668 (1994).
16 13 A. B. Rechester and M. N. Rosenbluth, Physical Review Letters, 40, 38 (1978).
14 A. R. Jacobson and R. W. Moses, Physical Review Letters, 52, 2041 (1984).
15 A. R. Jacobson and R. W. Moses, Physical Review A, 29, 3335 (1984).
16 P. W. Terry and P. Diamond, Physics of Fluids, B2, 428 (1990).
17 R. W. Harvey, M.G. McCoy, J.Y. Hsu, and A. A. Mirin, Physical Review Letters, 47, 102 (1981).
18 D. Gregoratto, L. Garzotti, P. Innocente, S. Martini, A. Canton, Nuclear Fusion, 38, 1199, (1998).
17
2: The Far-Infrared Laser System
In collaboration with the University of California at Los Angeles Plasma
Diagnostics Group, we have developed a high time response, multi-chord far-
infrared (FIR) laser interferometer1 to measure the equilibrium and fluctuating
density profiles. The vertical viewing heterodyne system is capable of measuring
electron density fluctuation behavior, up to 500 kHz, simultaneously in eleven
chords.2 Furthermore, the system has recently been upgraded to allow poloidal
field measurement capability;3 however, this work is still in progress and
unrelated to the physics goals presented in this report. In this chapter we will
describe the far-infrared laser system (FIR), theory of operation (Section 2.1),
and principle components (Section 2.2). We also will introduce the digital phase
extraction technique (Section 2.3) that has been instrumental in increasing the
diagnostic’s time response and phase resolution,4,5 and present some typical
data.
2.1 Plasma Interferometry Theory
The underlying principle behind plasma interferometry is that an
electromagnetic wave will propagate through plasma and air at different speeds.
18
The propagation of an electromagnetic wave in plasma is depicted in equation
2.1.6,7 The index of refraction (μ s, f ) for the slow and fast waves with frequency
ω are
μ s, f( )2
=1 −ω pe
2
ω2 1 −ωce
2
ω 2sin2 θ
2 1 − ω pe2 ω2( )±
ωce2
ω2sin2 θ
2 1 − ω pe2 ω2( ) 1 + F2( )1 2⎡
⎣ ⎢
⎤
⎦ ⎥
−1
, (2.1)
where ω pe and ω ce are the electron plasma and cyclotron frequencies with θ
being the angle between the wave propagation vector and the magnetic field in
the plasma and is defined as F
F =
2ωωce
1 −ω pe
2
ω 2
⎛
⎝ ⎜ ⎞
⎠ ⎟ cosθ
sin2 θ. (2.2)
We can see from the complexity of equations 2.1 and 2.2 that a rigorous solution
for a wave propagating through a magnetized plasma, where θ is continually
changing, would quickly get frighteningly complicated. As always in plasma
physics, we strive to avoid complexity while including the required amount of
physics, and this case is no exception. To first order we can examine the special
case where the wave propagates perpendicular to the background magnetic field
( rk ⊥
rB ). With θ = π 2 , the index of refraction for the ordinary wave, defined when
the electric field vector of the wave in parallel to the background magnetic field
( rE ||
rB ) becomes
μord = 1 −
ω pe2
ω2
⎡
⎣ ⎢ ⎤
⎦ ⎥
1 2
. (2.3)
19
For the extraordinary wave ( rE ⊥
rB ), equation 2.1 simplifies to
μ ext = 1 −
ωpe2 ω 2 − ω pe
2( )ω 2 ω2 − ω pe
2 − ω ce2( )
⎡
⎣ ⎢
⎤
⎦ ⎥
1 2
. (2.4)
For MST parameters, ωce2 = eB me( )2
≈ 3 ×10+20 s−2 and ω pe2 = e2ne εome ≈ 3 ×10+22 s−2 .
Furthermore, at the laser wavelength of 432 microns, ω ≈ 4.3 ×10+12 s −1 and
ω2 ≈ 2 ×10 +25 s−2 . Given that ωce
2 << ωpe2 and ω pe
2 << ω2 , a little algebra and a
binomial expansion later, equation 2.4 can be simplified, yielding that μ ord ≈ μext ,
where
μord ≈ μext ≈ 1 −
ω pe2
ω 2
⎡
⎣ ⎢ ⎤
⎦ ⎥
1 2
≈ 1 −12
ω pe2
ω2
⎛
⎝ ⎜ ⎞
⎠ ⎟ . (2.5)
Recalling that k = μω c , and that ω pe2 = nee
2 εo me where ε o is the free space
permittivity and n is the electron density, then the phase difference e Φ( )
between a wave that travels through plasma vs. air will be
Φ = kvac − kplasma( )∫ dz =
ω2c
ω pe2
ω 2
⎛
⎝ ⎜ ⎞
⎠ ⎟ ∫ dz =
λe2
4πc2meεo
ne r( )∫ dz . (2.6)
Substituting in the relevant MKS values, Φ becomes
Φ = 2.814 ×10−15 λ ne r( )∫ dz , (2.7)
where λ is the FIR laser wavelength, n is the electron density, and is the
coordinate along the length of the chord through the plasma. From equation 2.7,
as the beam passes through the plasma, the presence of electrons along the path
length slows the propagation, thus causing its phase to be shifted from that of
e z
20
the reference beam. Thus a measurement of this imparted phase shift is a
measure of the number of electrons along the beam’s line of sight.
2.2 The Far-Infrared Laser Interferometer
We have constructed a multi-chord far-infrared laser interferometer to
measure the phase shift described in equation 2.7. The FIR system, outlined in
figure 2.1, consists of a high-powered, continuous operation, CO2 laser, two
optically pumped FIR lasers, dielectric waveguide and wire grid mesh
assemblies, and twelve independent FIR detector assemblies. In this section we
present a general diagnostic overview, detailed descriptions of the principal
components, and typical operating parameters for the FIR laser system.
2.2.1 Diagnostic Overview
The FIR system is a vertical viewing heterodyne system that is capable of
measuring electron density behavior with a high degree of speed and accuracy.
The system functions by using a high-power CO2 laser to pump the twin FIR
cavities producing two independent FIR laser beams. The two cavities are
adjusted to operate at slightly different frequencies so that when mixed, produce
a modulated signal. The peaks of this modulated signal provide the benchmarks
from which a relative phase between chords is measured.
21
22
2.2.2 The CO2 Pumping Laser
The heart of the FIR laser system is the continuous power, CO2 pumping
laser (figure 2.2). Designed by Apollo Laser Corporation, the Model 150 is a
continuous flow, tunable gas laser that is capable of steady state operation at
powers of 125-150 Watts depending on the line of interest. The laser consists of
two water-cooled, gas-filled discharge tubes, a partially reflective (80%) ZnSe
output coupler, and a gold coated blazed grating. The grating is grooved at 135
lines per inch blazed for 10.6 μm (Hyperfine part # ML-303-0-1X0.825), and
allows the CO2 laser to be tuned to the appropriate FIR pumping line. For
continuous operation the gas mixture of choice is 6 % CO2, 18 % N2 and 76 % He.
Gas Flow In
Gas Flow Out
Grating
Output Coupler
Piezo-electric Transducer (PZT)
Mirrors
Cathode (23 kV)
Anode (Ground)
Gas Flow Out
Anode (Ground)Monitoring Beam
Figure 2.2 – The CO2 pumping laser primarily consists of two co-linear discharge tubes, a grating for tunability and a partially reflective mirror (output coupler) that allow continuos operation.
Unlike shorter wavelength lasers whose principal transitions are atomic,
the CO2 lasing transitions result from changes between vibrational energy
states.8 The triatomic CO2 molecule is subject to three types of vibrational
excitation – symmetric stretching, bending, and asymmetric stretching (figure
23
2.3). Vibrational energy is transferred to the CO2 molecule by collisions resulting
in an excited state. When the molecule relaxes to a lower vibrational state, the
energy is dissipated as a photon, as is the case for atomic transitions. Although
both processes result in the emission of a quantized photon, the vibrational
energy levels are more plentiful and closely packed then their low n atomic
counterparts. This results in laser emission that is more like a continuum. To
obtain the monochromatic emission required for the efficient pumping of the FIR
laser, a grating is used to isolate the particular vibrational transition of interest.
O OC
O OC
O OC
O OC
Equilibrium Bending
Symmetric Stretching Asymmetric Stretching
Figure 2.3 – The CO2 molecule is subject to three types of vibration: bending, symmetric stretching, and asymmetric stretching.
To ensure that the population of vibrationally excited CO2
molecules in the discharge tubes is sufficient for high-powered lasing,
additional gases are introduced to enhance excitation. The process of
continually exciting (pumping) and de-exciting (lasing) the CO2 molecule
is displayed in (figure 2.4). Nitrogen, which is diatomic, has only one
degree of vibrational freedom (symmetric stretching) and is easily excited
by collisions in the discharge tube. Since vibrationally excited N2 is
similar in energy to the CO2 excited state, N2 can efficiently transfer its
energy to a CO2 molecule during a collision. Stimulated emission occurs
24
and the CO2 molecule begins to radiate its energy. To minimize the
amount of re-absorption, helium is added to enhance the collisional de-
excitation of the CO2.
1
65432
0
1
65432
0
1
65432
0
1
65432
0
Collisional Transfer of Vibrational Energy
10P6
9R6
(010)
(020)
(001)
(100)
CO2 N2(000)
1
432
0
0
1
Exc
itatio
n vi
a H
igh
Volta
ge D
isch
arge
Stimulated Emission
Collisional De-excitaion
Figure 2.4 – The CO2 lasing cycle. Collisions within the high-voltage discharge tube excite the N2 molecules. The excited N2 molecules transfer their energy to CO2 molecule which then relaxes via stimulated emission. Although not a part of this cycle, Helium is added to enhance the collisionality within the discharge tube.
2.2.3 The Twin Far-Infrared Laser (FIR)
The FIR, displayed in figure 2.5, is an optically pumped system that
converts the near 10 micron output of the CO2 into two semi-independent beams
of much longer wavelength.9 The wavelength of operation can range from 100
microns to several millimeters and is solely governed by the choice of laser gas.
On MST, Formic Acid (HCOOH) is used to yield an output wavelength of 432.5
microns (≈ 700 GHz); however, the system can be run with methanol (CH3OH) or
25
difluoromethane (CH2F2) which can yield output wavelengths of 119 and 184
microns respectively. Tuning around the Formic Acid transition is achieved with
a wire mesh/quartz plate combination that forms a Fabry-Perot etalon that is
adjusted to maximize output power. Though the input pumping power is over
100 W, the FIR output is only about 30 mW per laser cavity. Once optimized for
power the cavity length mirrors can be positioned independently to vary the
interference frequency between the lasers.
Metallic Corrugated Waveguide
CO Pumping Beam
2
Quartz Etalon
Reflective Coating (10.6 μm)Wire Mesh (100 LPI)
TPX Output Windows Cavity Length Mirrors
(Gold Coated)
Figure 2.5 – The twin FIR laser system. The entire chamber is filled with 200 mT of Formic acid vapor. The CO2 pumping beam is focused into the corrugated tubes where FIR lasing occurs. Tunability is achieved by adjusting the spacing between the wire mesh and the quartz etalon. The interference frequency between the twin FIR lasers is dictated by the placement of the cavity length mirrors.
26
A principal advantage of pumping both FIR cavities with the same CO2
laser is that any fluctuation in CO2 power will be equally distributed among the
FIR lasers. Issues such as reflections back into the laser cavity (termed laser
feedback), vibrations, variations in temperature, and power line noise can cause
a laser’s output power to fluctuate. However, with this configuration, even if
these issues reduce the stability of the CO2 power and the FIR power fluctuates
the modulated signal will still be very stable.
2.2.4 Power Distribution
The output of each FIR laser is focused through a polyethylene plano-
convex lens into a dielectric waveguide that carries the beam to the vacuum
vessel. The waveguides are air-filled plexi-glass tubes, which have an inner
diameter of 3.5 inches, and help channel the beam in a manner that preserves
the mode symmetry and reduces power loss. The effectiveness of the waveguide
is highly sensitive to the input beam size, so to ensure optimum transmission, a
number of lenses were tested to focus the beam into the waveguide entrance.
The results show the 120 cm focal length lens was best suited for preserving a
small beam through the waveguide (figure 2.6).
27
-10 -5 0 5 10Radius (1/8's in.)
f=120 cmf=100 cm
Pow
er (a
u)
Figure 2.6 – The FIR signal beam profile out of the waveguide, incident on the meshes above the vacuum vessel. The 120 cm focal length lens provides the tightest beam waist of about 2.4 cm.
The size of the beam is an important issue for the MST interferometer
because the entrance holes in the aluminum tank are drilled separately and
deliberately made small to minimize field errors. With an inner diameter of only
3.5 centimeters, a large FIR beam can be greatly attenuated by the small
entrance holes, thereby reducing the laser power through the tank. More
importantly, especially for polarimetry, a large beam can reflect off the inner
walls of the entrance tubes and contaminate the measured phase. To address
this latter issue, two sets of threaded inserts were constructed, one set with 48
threads per inch (TPI) and the other with 20 TPI. These inserts are installed in
both entrance and exit holes and help ensure that any laser power impacting the
inner walls will be scattered as opposed to coherently reflected.
The eleven FIR chords are separated into two arrays that are toroidally
displaced by five degrees. The chords view impact parameters range from r/a of –
28
0.62 to +0.83. The toroidal displacement, shown in figure 2.7, was originally
designed to minimize the field errors that would be associated with an array of
closely packed holes in the conducting shell. Although unplanned, this
arrangement has some significant advantages when examining density
fluctuations, which will be addressed in later chapters. Additional information
on the relevant chord parameters is outlined in table 2.1.
5o
=Ro 1.5m=a . 52m
P43
P36
P28
P21
P13
P06
N02
N09
N17
N24
N32
Figure 2.7 – The 11 chords a separated into 2 arrays, displaced by 5.0 degrees toroidally. They view impact parameters (R-Ro) of -32, -24, -17, -09, -02, +06, +13, +21, +28, +36, and +43 cm.
29
Chord Name (N/P)
Impact Parameter R-Ro (cm)
Toroidal Angle φ (degrees)
Chord Length L (cm)
N32 -32 255 81.97 N24 -24 250 92.26 N17 -17 255 98.29 N09 -09 250 102.4 N02 -02 255 103.9 P06 +06 250 103.3 P13 +13 255 100.7 P21 +21 250 95.14 P28 +28 255 87.64 P36 +36 250 75.04 P43 +43 255 58.48
Table 2.1 – The impact parameters, toroidal location, and chord lengths of the 11 FIR chords.
The laser power is distributed among the various FIR chords by an array
of thin metallic wire grid meshes. Manufactured by Buckbee/Mears of St. Paul,
MN, the meshes are electroformed out of a nickel substrate and can be obtained
with a variety of line densities. A number of exhaustive tests were conducted
and only five mesh types have proven suitable for the FIR system. The geometric
features of these meshes are outlined in table 2.2.
It is important to note that the meshes continue to be the fundamental
weakness in the FIR system, with regard to obtaining accurate polarimetry
results. Although specifically chosen to minimize polarization distortions, the
cumulative effect of propagating through as many as six meshes on the beam
polarization introduces enough error that the polarimeter measurements are
unable to adequately constrain the toroidal current density profile. A number of
possible solutions are still being explored; these include meshes with more exotic
30
grid geometries or perhaps partially reflecting thin coated quartz or TPX (poly-4-
methylpentene-1) mirrors.
Lines Per Inch Space (In.) Wire (In.) Part # 50 0.01732 0.00268 MN-13
125 0.00645 0.00155 MN-26 150 0.00570 0.00097 MN-28 200 0.00406 0.00094 MN-31 500 0.00154 0.00046 MN-41
Table 2.2 – The principal characteristics of the five mesh types used in distributing the laser power among the 11 chords.
2.2.5 Detection Electronics
Once through the vacuum vessel and combined with the local oscillator
laser (Reference Beam), the modulated interference beam is measured with a
specially fabricated diode/preamplifier assembly. The diode, which as a
Gallium/Arsenide (GaAs) Schottky corner-cube mixer,10 offers both a very low
noise-equivelent-power (NEP) of ≈ 10−10W / Hz and a time response of up to a
few MHz, ideal for far-infrared detection. The principal disadvantage of the
corner-cube mixer is that measurement efficiency is very sensitive to incident
angle, and this can be problematic in cases of high-density or high-fluctuation
where refraction effects tend to steer the FIR beam around.
The mixer sensitivity to beam input angle also places stringent
requirements on alignment. The FIR beam is focused onto the mixer with a
31
plano-convex polyethylene lens that has a focal length of 8 cm. The detector
assembly for each channel is directly mounted on a rotating stage which is then
affixed to three orthogonally arranged translation stages, thus allowing absolute
freedom in detector placement. The procedure for alignment consists of
iteratively adjusting mixer angle and placement until the signal is maximized.
This tedious process of alignment is conducted independently for all 12 channels
(11 chords + reference) and should be repeated about every two months.
A low noise, high speed preamplifier is directly connected to the output of
the corner-cube detector. The preamp, designed by Dr. Don Holly, amplifies and
filters the mixer signal, removing any low (< 300 kHz) and high (> 3 MHz)
frequency components that may be present. The preamp gain, displayed in
figure 2.8, is typically around 103 for frequencies near the laser interference
frequency.
0
200
400
600
800
1000
0 0.5 1.0 1.5 2.0 2.5Frequency (MHz)
Gai
n
Figure 2.8 – The preamplifier response function. The preamplifier bandpass ranges from about 350 kHz to near 2.0 MHz. General operation has the IF at 750 kHz and yields a maximum bandwidth of 400 kHz.
32
The output of the preamplifiers is fed into a variable amplifier that allows
the signal levels to be adjusted independently before being sent to the digitizers.
This final stage allows modification of the signal amplitude to obtain optimal
resolution from the digitizer. Typically the signal amplitudes into the digitizers
will require adjustment three or four times a day due to the tendency of the laser
power to drift in time.
Although the phase measurement is inherently amplitude independent,
proper management of the signal amplitude can greatly enhance the
interferometer’s performance. Often, on good days, the FIR signal has sufficient
power to saturate the mixer preamplifiers, causing a non-sinusoidal output. This
distortion severely contaminates the phase measurement. This problem is
addressed by inserting small pieces of paper or cardboard in front of the mixers,
which attenuates the incident beam.
2.3 Digital Phase Extraction
Direct digitization of the amplifier output stores the raw data directly and
allows post processing of interferometer phase. This approach offers three
important advantages. First, fast time resolution is obtained without the need
for complex high-speed analog comparators. Second, the freedom offered by
digital processing increases the accuracy of the phase calculation and reduces
the susceptibility to noise. Finally, by allowing examination of the raw data prior
to phase extraction, confidence in the measurement is enhanced.
The 12 channels (11 chords + reference) are digitized by two Joerger 612’s
that have been modified for a maximum input voltage of ±2.5 Volts. The
33
amplifiers are adjusted so that the input signal levels are about 3V peak-to-peak
and the laser IF is centered at 750 kHz. The signals are digitized at 1 MHz
which undersamples the IF and produces an aliased IF signal at 250 kHz. Since
the Nyquist frequency is 500 kHz, the maximum bandwidth for this
arrangement is 250 kHz. If higher bandwidth is desired, a digitization rate of 3
MHz can be employed and the IF can be adjusted to 875 kHz. Though not
intuitively obvious, the change in IF is necessary because of the low frequency
cutoff characteristics of the mixer preamplifiers (figure 2.8). With the above
modifications, the bandwidth of the interferometer can be improved to greater
than 500 kHz, although this has diminishing gains since the chord-integrated
nature of the measurement severely attenuates the smaller scale, high-
frequency fluctuations.
The raw data for the reference and signal beams takes the form outlined
in equation 2.8. Both are sinusoidal, oscillating at the IF frequency of ω IF , where
φ tn( ) represents the shift in phase from the plasma electron density.
xR tn( )= AR tn( )cos ω IF tn( )tn[ ]+ xR
xS tn( )= AS tn( )cos ω IF tn( )tn + φ tn( )[ ]+ xSoffset
offset
(2.8)
We isolate φ tn( ) via digital complex demodulation. This technique involves
three steps: pre-processing of the reference ( xR tn( )) and signal ( ) data,
filtering, and phase extraction. Expanding equation 2.8 into its exponential form
and removing the equilibrium offsets,
xS tn( )
xR tn( ) and xS tn( ) become,
34
xR tn( )= AR tn( ) 2[ ] exp iω IF tn( )tn[ ]+ exp −iω IF tn( )tn[ ]{ }xS tn( )= AS tn( ) 2[ ] exp iω IF tn( )tn + iφ tn( )[ ] + exp −iω IF tn( )tn − iφ tn( )[ ]{ }
(2.9)
Additional processing is required for xR tn( ), in which the negative frequencies
are filtered out (−ω IF → 0 ), and xR tn( ) is conjugated, forming
xR_Conj tn( )= AR tn( ) 2[ ]exp −iω IF tn( )tn[ ]. (2.10)
Multiplying the pre-processed signals yields,
xProduct tn( )= xR_ Conj tn( )xS tn( )= AR tn( )AS tn( ) 4[ ]× exp −i2ω IF tn( )tn + −iφ tn( )[ ]+ exp iφ tn( )[ ]{ }
, (2.11)
which has two components, a high frequency 2ω IF term and the desired low
frequency φ tn( ) term. Digital filtering removes the 2ω IF term leaving
xFiltered tn( )= AR tn( )AS tn( ) 4[ ]exp iφ tn( )[ ]. (2.12)
Finally, the ratio of the imaginary and real parts of equation 2.12 removes any
amplitude dependence, allowing an inverse tangent to extract the phase, as
φ tn( )= tan-1 Im xFiltered tn( )[ ] Re xFiltered tn( )[ ]{ }. (2.13)
Digital complex phase extraction has been very successful on MST. This
method has increased the accuracy of the phase determination and has
dramatically increased the time response of the interferometer. Figure 2.9 shows
a histogram plot of the digitally extracted phase for a vacuum discharge. In an
ideal world, this should be a delta function centered at zero; however, laser
fluctuations, vibrations, and electronic noise from the mixers and preamplifiers
35
all contribute to noise in the measured phase. From this plot we determine the
minimum resolvable line-averaged density to be around 3.5x10+10 cm-3.
-0.10 -0.05 0.00 0.05 0.10
FWHM ≈ 0.03 radians
Phase (radians)
Cou
nts
(au)
Figure 2.9 – The histogram of the digitally extracted interferometer phase for vacuum discharge. The resolution limit is about 0.03 radians which corresponds to a line-averaged density of ≈ 3.5x10+10 cm-3, or about 0.4% of the equilibrium density.
The fast time response is clearly visible in figure 2.10, which displays a
typical chord-averaged time trace. In the past, the analog comparators limited
the time response to less than 10 kHz; however, employing the digital phase
extraction technique allows the tearing mode fluctuations to be resolved. This
single improvement has dramatically increased the utility for the FIR
interferometer by allowing the physics of high-frequency density fluctuations to
be comprehensively investigated.
36
16.5 17.016.6 16.7 16.8 16.90.0
1.0
0.2
0.4
0.6
0.8
Time (ms)
Ele
ctro
n D
ensi
ty
(1E
+13
cm
)-3
0 620 400.0
1.0
2.0
0.5
1.5
0Time (ms)
Ele
ctro
n D
ensi
ty
(1E
+13
cm
)-3
15 2016 17 18 19Time (ms)
Ele
ctro
n D
ensi
ty
(1E
+13
cm
)-3
0.0
1.0
0.2
0.4
0.6
0.8
Figure 2.10 – The digital phase extraction technique allows the high-frequency density fluctuations to be resolved. In the bottom plot, the large 17 kHz fluctuation (which arises from the m=1, n=6 tearing mode) is clearly visible.
37
2.4 Summary
We have constructed a high-speed multi-chord far-infrared laser
interferometer to quantitatively measure the equilibrium and fluctuating
density profile behavior. By implementing a digital phase extraction technique,
the system is now capable of resolving fluctuations up to 500 kHz with a phase
resolution of ~0.03 radians. The eleven chords are separated into two arrays,
toroidally displaced by 5 degrees, and span impact parameters ranging from
r/a=-0.61 to r/a=+0.83.
REFERENCES
1 S. R. Burns, W. A. Peebles, D. Holly, and T. Lovell, Review of Scientific Instruments, 63, 4993 (1992).
2 Y. Jiang, N. E. Lanier, D. L. Brower, Review of Scientific Instruments, 68, 703 (1999).
3 N. E. Lanier, J. K. Anderson, D. L. Brower, C. B. Forest, D. Holly, and Y. Jiang, Review of Scientific Instruments, 68, 718 (1999).
4 D. W. Choi, E. J. Powers, R. D. Bengtson, G. Joyce, D. L. Brower, W. A. Peebles, and N. C. Luhmann Jr., Review of Scientific Instruments, 57, 1989 (1986).
5 Y. Jiang, D. L. Brower, L. Zeng, and J. Howard, Review Scientific Instruments, 68, 902 (1997).
6 S. E. Segre, Plasma Physics, 20, 295 (1978).
7 M. A. Heald and C. B. Wharton, Plasma Diagnostics with Microwaves, (Academic Press, New York, 1979).
8 K. Chang, Handbook of Microwave and Optical Components Vo.l 3, (John Wiley & Sons Inc., New York, 1990).
9 T. Lehecka, R. Savage, R. Dworak, W. A. Peebles, and N. C. Luhmann, Jr. and A Semet, Review of Scientific Instruments, 57, 1986 (1986).
38
10 H. R. Fetterman, P. E. Tannenwald, B. J. Clifton, C. D. Parker, and W. D. Fitzgerald, and N. R. Erickson, Applied Physics Letters, 33(2), 151 (1978).
39
3: Neutral Hydrogen Density in MST
The neutral hydrogen population is important for two principal reasons.
First, when ionized, they provide a source of electrons that must be considered
when examining transport phenomena. Second, charge exchange with neutral
hydrogen is the dominant recombination mechanism for high-charge state
impurities and thus very important in determining the relative abundances of
impurity charge states. Therefore before one can examine electron transport
characteristics, it is imperative that the issue of the neutral population be
addressed.
We have developed a novel multi-chord Hα array to quantitatively
measure the neutral population in MST. We have determined that fueling in
MST is dominated by transport induced wall recycling. Measurements show the
neutral density profiles in standard MST discharges are hollow with core
densities of order 1x10+10 cm-3, while during PPCD, the neutral density in the
core drops dramatically. This reduction is attributed to the higher confinement
of PPCD decreasing wall interactions, hence lowering the hydrogen influx.
40
In this chapter we outline the principles behind wall fueling, electron
sourcing, and neutral penetration into the plasma (Section 3.1). In Section 3.2
we introduce the multi-chord Hα array, which is used to quantitatively measure
neutral density. Section 3.3 outlines the general physics results obtained in
standard and PPCD discharges. In the latter subsections of 3.3, we briefly
discuss some secondary issues regarding the neutral population in MST, such as
neutral particle loss and the role of the background neutral hydrogen density as
it applies to the Charge Exchange Recombination Spectroscopy diagnostic
(CHERS) currently under development.
3.1 Hydrogen Fueling in MST
MST fueling is primarily accomplished through wall recycling and as any
experienced operator will tell you, this serves as both a blessing and a curse. A
principal advantage of wall recycling is that fueling is achieved much more
uniformly around the plasma. However this process is strongly dependent on
wall condition, and with one inopportune locked shot or wall interaction, the
delicate balance you have slaved all afternoon to attain has just been scrapped.
3.1.1 The Fueling Cycle
The fueling cycle in MST consists of five processes and is outlined in
figure 3.1. Particles lost to the wall release molecular hydrogen into the plasma.
Upon emerging, collisions with electrons force them to dissociate. The neutrals
that are not directly lost back to the wall begin to penetrate the plasma where
they either undergo charge exchange with thermal ions or are ionized via
electron collisions, hence fueling the plasma.
41
H2 Ho
Ho*
Γ DissociationFueling
Ionization
Charge Exchange
Ioni
zatio
n
Transport
Transport
Transport
H+
Figure 3.1 – The MST fueling cycle. Particles lost from the plasma (Γ) bombard the wall and introduce molecular hydrogen (H2). The molecules dissociate forming neutrals (H0) that undergo ionization (H+) or charge exchange (H0*).
At the plasma boundary, dissociation of molecular hydrogen results
primarily from collisions with thermal electrons.1,2 The two most likely
processes are electron impact ionization and electron impact dissociation. In
electron impact ionization, an electron collides with the molecular hydrogen,
imparting enough energy to both dissociate the molecular hydrogen and ionize
one of the dissociated neutrals.
e− + H2 → H + H + + 2e− (3.1)
Since the binding energy of molecular hydrogen is about 4.5 eV, and the
ionization energy for neutrals is 13.6 eV, the process requires electron energies
around 18 eV. This constraint on energy indicates that electron impact
ionization is most prominent at higher electron temperatures.
At lower temperatures, electron impact dissociation dominates. In this
process, an electron collides with a hydrogen molecule and imparts enough
energy to dissociate the H2, and possibly excite a resulting neutral, but not
enough to ionize the hydrogen.
42
e− + H2 → H + H * + e− (3.2)
Here, H* represents an atomic hydrogen in an excited state. Since this process
does not require ionization, the electron threshold energy is only the molecular
hydrogen binding energy (4.5 eV); thus this process dominates at electron
energies between 4.5 and 18 eV.
3.1.2 Franck-Condon Neutrals
While low temperature electrons, less than 4.5 eV, cannot dissociate
molecular hydrogen by themselves, they do play an important role in the neutral
hydrogen fueling characteristics. An electron that collides with a hydrogen
molecule can still transfer energy to it without initiating a dissociative process.
This is achieved by vibrationally exciting the diatomic hydrogen. A multitude of
collisions can continue to excite the molecular hydrogen until it dissociates into
two excited neutrals, with typical energies of about 4 eV.
e− + H2 → e− + H2* →. ..→ 2H* + e− (3.3)
These excited neutrals, referred to as Franck-Condon neutrals, are important
because their high energy allows them to penetrate deeper into the plasma
before being ionized. When addressing neutral penetration and electron and
proton sourcing, the Franck-Condon’s are the dominant contributors.
3.1.3 Neutral Penetration
The mean free path between collisions determines the neutral penetration
depth. Defined as
λ N = vN ne σv collisions , (3.4)
43
the mean free path is the ratio of the neutral velocity to the electron collision
frequency. The collision rates ( σv ) for neutral hydrogen are displayed in figure
3.2, and are computed by integrating the product of the cross section (σ ) and the
relative velocity (v) between particles over the Maxwellian velocity distribution.
For plasma temperatures less than 10 eV, charge exchange stands alone as the
dominant process for atomic hydrogen.3 However, as temperature increases, the
probability of electron impact ionization increases to a level on par with charge
exchange.4 For all relevant MST temperatures, the effects of radiative
recombination and proton impact ionization are negligible.5
1 10 100 1000
-6
-8
-10
-12
-14
-16
Charge Exchange Recombination
Electron Impact Ionization
Radiative Recombination
Proton Impact
Ionization
Temperature (eV)
Log
(Col
lisio
n R
ate
(cm
s
))
-3-1
Figure 3.2 – The collision rates vs. electron temperature for atomic hydrogen. At temperatures above 10 eV, charge exchange and electron impact ionization rates are comparable. Radiative recombination and proton impact ionization are negligible processes.
44
In typical low current discharges, measurements conducted near the wall
with Langmuir probes indicate a plasma electron temperature near 20 eV, and
an electron density of about 2.0x10+12 cm-3. Measurements of ion temperature
are less well known and are assumed to be around 10 eV at the plasma
boundary. These quantities rise quickly, and at a depth of 15 cm in from the
wall, approach an electron temperature and density near 200 eV and 9.0x10+12
cm-3 respectively. With the final assumption that ion and electron densities are
equivalent, these conditions dictate that the ionization mean free path of a 2 eV
neutral will be about
λ ion ≈ 2E + 4 / 7E +12 × 2.2E − 8( ) ≈13cm , (3.5)
and a charge exchange mean free path of
λ cx ≈ 2E + 4 / 7E + 12× 3.0E −8( ) ≈ 10cm . (3.6)
For a Franck-Condon neutral, which has an energy near 4 eV, these mean free
paths increase by a factor of 2 , meaning λ ion ≈18cm and λ cx ≈14 cm . In both
cases, the neutral is more prone to charge exchange than ionization.
Recognizing that charge exchange is a prevalent process in MST is the key
to understanding the neutral population throughout the entire plasma. As
stated above, a neutral that propagates into the plasma will either be ionized or
undergo charge exchange. If ionized, the newly formed ion will continue to exist
until it is lost via transport, or converted back to a neutral by charge exchange.
Recall that for MST plasmas, radiative recombination is rare for all but the
outer centimeter. However, a neutral that undergoes charge exchange will
transfer its electron to another ion, forming a new neutral with a temperature
equivalent to the local ion temperature. This is very important in that as a
45
neutral propagates deeper into the plasma, successive charge exchanges have
the effect of increasing the neutral’s temperature.6 As the temperature of the
neutral increases, so does the mean free path, thus enabling it to penetrate deep
into the plasma core, or escape out to the wall.
Since charge exchange is so prominent in MST, the neutral population in
the core is quite large. Moreover, because of the successive charge exchanges
required for the neutrals to reach the core, the neutral temperature profile
should be similar to that of the ions. The importance of this latter issue will be
addressed later in the discussion of the feasibility of Charge-Exchange
Recombination Spectroscopy (CHERS).
3.1.4 Measuring Neutral Density
Measuring the neutral density profile offers a number of experimental
and interpretive challenges. The neutral density is highly susceptible to wall
interactions, and can have very asymmetric and localized characteristics.
Moreover, the neutral profile tends to be very hollow, making it difficult for
inversion techniques to reconstruct local profiles from chord-integrated
measurements. However, by applying some novel engineering, the neutral
density diagnostic on MST has become one of the most robust measurements
currently being employed.
The neutral particle density is extracted from measurements of the Hα
photon emission. An Hα photon has a wavelength of 656.3 nanometers and is
emitted during an electronic transition between the n=3 and n=2 levels of atomic
hydrogen.7 For MST parameters, Hα production is proportional to the neutral
ionization rate.8,9 In other words, the Hα emission is given by
46
γ Hα= αneN σv ion , (3.7)
where is the electron density, ne N is the neutral particle density, σv ion
represents the electron impact ionization rate, and α is the proportionality
constant between Hα excitation and ionization. Over the range of MST
discharges, the α parameter varies very little (α ≈ 0.08 → 0.09 ), and is assumed
constant. However, the impact ionization rate can have a strong dependence on
electron temperature, (recall figure 3.2). Therefore, extraction of the neutral
hydrogen density requires the simultaneous measurement of Hα flux, and both
electron temperature and density.
3.2 The Hα Array
To measure Hα emission profiles; a novel monochromator system was
developed. The multi-chord system is built around nine compact filtered diode
assemblies that were designed with simplicity in mind. Consisting only of a
focusing lens, prism, optical filter, and photodiode (figure 3.3), these
monochromators take advantage of the dominance of the Hα line by using a
narrow bandpass filter to obtain spectral resolution. The filter, whose
characteristics are outlined in figure 3.4, has a bandpass region centered near
657 nanometers, with a full-width-half-max (FWHM) of about 11 nm. The diode
detector is an advanced photonix internally amplified photodiode with a gain of
105 and maximum frequency of 300 kHz.
47
Focusing Lens
Amici Prism
H Filterα
Slit
Photodiode
Support Disc
Collimating Tube
Detector Assembly
Output Plug
Figure 3.3 – Component schematic for the Hα detector. The space conserving design consists of a focusing lens, bandpass filter, and a photodiode.
0.0
0.2
0.4
0.6
0.8
1.0
630 640 650 660 670Wavelength (nm)
Nor
mal
ized
Tra
nsm
issi
on
Figure 3.4 – The Hα filter transmission10 measured with a calibration sphere. Peak transmission is a 657 nm with a FWHM of 11 nm.
The novelty of this system is its co-linear arrangement with the far-
infrared interferometer (figure 2.1). By configuring the system in this way, two
48
key problems are averted. First, since the Hα emission is most prominent in the
edge, it is extremely sensitive to wall interactions. By employing this co-linear
method, the Hα detectors can be focused through the vacuum vessel without
viewing any of the interior walls; thereby ensuring that wall contamination is
minimized. The second issue is that Hα emission can be very asymmetric, both
poloidally and toroidally. By simultaneously measuring both Hα emission and
electron density in the same location, the uncertainties in comparing toroidally
displaced measurements are eliminated. An additional key point that will be
discussed later is that this technique allows the electron radial particle flux to be
obtained from a single inversion of the difference between the chord-integrated
Hα and electron density quantities.
3.2.1 Alignment and Calibration
To successfully quantify the Hα emission profile, proper alignment and
calibration are critical. The alignment procedure amounted to replacing the
photodiode detector with a high intensity light emitting diode (LED) and
adjusting the detector orientation so that the image of the slit is centered on the
FIR focusing lens underneath the vacuum vessel. By aligning the system in this
manner, we ensure co-linearity with the FIR chords and reduce the
susceptibility to wall interactions.
The calibration procedure is also quite simple; however, it must be
repeated every couple of months. With the detectors in place, a small calibrating
sphere with another Hα bandpass filter is used to cross calibrate the multi-chord
system. The extra Hα filter is necessary because the calibrating sphere emits
uniformly over the entire visible spectrum and it was determined that one filter
49
was insufficient in removing these broadband contributions. This is not an issue
for the plasma case in which only a few lines dominate the visible spectrum. It is
important that the calibration be conducted with the FIR meshes in place and
should be repeated if any of the meshes are removed or changed. Even if no
changes are made, frequent calibrations are recommended because, in time, the
FIR meshes will collect dust, thus changing their transmission properties.
Finally, the TPX windows of the FIR system are not baffled during PDC
operation and as a result become coated over time. The coating is most serious
for the outer chords and over the span of six months can reduce the transmission
of Hα radiation by 50%. For best results these windows should be removed,
cleaned, and if necessary, replaced twice a year or prior to any serious run
campaign.
Having obtained a relative calibration for the nine-chord Hα array, we
employ an absolutely calibrated dedicated Hα radiation monitor. This monitor
was configured to view a chord with impact parameter of 20.5 cm, which was
specially baffled such that wall interactions were small and the window coating
was minimized. The detector had been calibrated on a test bench using a well-
characterized light source of known intensity. This detector provides an absolute
measure of Hα photon flux and converts the relative calibration to an absolute
one. This calibration introduces the most uncertainty into the Hα profile
measurements. Primarily resulting from the toroidal displacement of the
measurements, this error is systematic and can be as high a 20%. However, this
error will only affect the absolute magnitude of the Hα profile measured and not
the profile characteristics.
50
3.3 Hα Emission
Having discussed the Hα diagnostic, and recalling that Hα emission is
described by equation 3.7, we now turn our attention to Hα behavior. In the
subsequent sections we discuss the Hα emission and the implications for
standard and PPCD discharges.
3.3.1 Hα Behavior in Standard Discharges
The time variation of Hα emission is very uniform over the entire MST
operational range. Figure 3.5 displays the temporal behavior of the chord-
integrated Hα emission during a standard low current discharge for impact
parameter of r/a = 0.69. In general, emission spikes to large levels early in the
discharge when the plasma is still forming. Emission then drops to a reasonably
steady state value during the flat-top phase. Throughout the discharge, bursts in
the emission occur regularly with magnetic relaxation events, or sawteeth,
indicating the sensitivity of Hα radiation to plasma wall interactions. These
bursts also correlate well with the small-sawteeth that are often associated with
the m tearing mode activity, resonant at the reversal surface. = 0
51
-20 0 20 40 60 80
-0.5
0.0
0.5
1.0
1.5
2.0r/a=0.69
Time (ms)
H P
hoto
nsα
(1Ε+
18 c
m
s )
-2-1
Figure 3.5 – The chord-averaged Hα emission vs. time for a standard low current discharge.
The high correlation between Hα emission and sawteeth is most easily
observed by ensemble averaging over many sawtooth events. Figure 3.6 displays
the chord-averaged Hα emission for three impact parameters ranging from
r/a=0.11 to r/a=0.83, ensembled over 600 events. Away from the event, the
emission is constant until about 0.25 ms prior to the sawtooth where it rises
sharply to its maximum value at the crash. Emission then decays at a much
slower rate, of order 1.0 ms, to its pre-crash value. During a crash, Hα increases
of a factor of two are typical but this can be much larger, of order 10, in high
current, high-density discharges.
52
4.0
3.0
2.0
1.0
0.0-2.0 -1.0 0.0 1.0 2.0
Time (ms)
H P
hoto
nsα
(1Ε+
17 c
m s
)-2
r/a=0.83
r/a=0.50
r/a=0.11
-1
Figure 3.6 – Hα emission increases dramatically at the sawtooth crash. This is an indicator of increased neutral density population.
All chords show an increase in the Hα emission at the crash with the most
dramatic increases at the edge. Away from the crash, the chord-averaged Hα
emission profile indicates values of 5.0x10+16 cm-2 s-1 in the core, rising to
1.5x10+17 cm-2 s-1 in the edge (figure 3.7). At the peak of the crash, these values
increase two-fold in the core and 2.7 times in the edge. Although the rise in Hα
emission is observed in all chords, it is the edge parameters that are dictating
the Hα behavior. At the crash, increases in edge electron density, electron
temperature, and neutral hydrogen concentration couple together to increase the
Hα emission.
53
H P
hoto
nsα
(1Ε+
17 c
m s
)-2
-1
t = 0.0t = -1.0
-40 -20 0.0 +20 +40 +60
1.0
2.0
3.0
4.0
0.0
Minor radius (cm)
Figure 3.7 – Radial profile of chord-integrated Hα emission. The edge peaked emission profile increases in all chords at the crash.
3.3.2 Hα Behavior in PPCD Discharges
During PPCD discharges, when particle and energy confinement is
enhanced, Hα emission drops while both electron density and temperature
increase. We infer from this behavior that the neutral particle density is being
reduced. Figure 3.8 outlines the chord-averaged Hα emission and the chord-
integrated electron density for 200 kA standard and PPCD discharges. Both
cases represent multi-shot ensembles, 267 standard and 136 PPCD discharges.
For the standard discharges, the ensemble windows were chosen to be fixed,
ranging from 5 ms to 25 ms. However, during PPCD discharges, the current
drive pulse is generally triggered off a sawtooth, and thus the fire time will vary
from shot to shot. For these discharges, the ensemble windows are set around
the triggering sawtooth, thus enabling the ensemble to accurately superimpose
different PPCD shots without washing out any important characteristics.
54
PPCD
Standard
PPCD
Standard
Profile Control
5 10 15 20 2Time (ms)
0.0
1.0
2.0
3.0
4.0
0.0
4.0
8.0
6.0
2.0
H α(1
E+1
7 cm
s
)-2
-1(1
E+1
4 cm
s
)-2
-1E
lect
ron
Den
sity
5
Figure 3.8 – The chord-averaged Hα emission and chord-integrated electron density for impact parameter of r/a=0.69.
For the ensembled data, the Hα emission in the standard discharges
steadily decreases from 5 to 20 ms, while the electron density is reasonably flat
from 10 to 20 ms. In the PPCD case, both the density and Hα emission are lower
prior to the onset of PPCD. This was planned such that during this time of
optimum confinement (between 18-20 ms), the electron densities would be
comparable. While both discharge types show similar Hα / electron density ratios
prior to the onset of PPCD, during the time of optimum confinement (18-20 ms),
the densities are comparable but the Hα emission is much lower for the PPCD
case. Recalling that γ H =α
αneN σv ion , and that both α and σv ion are essentially
constant for electron temperatures above 20 eV, the drop in Hα emission must
result from a decrease in neutral particle density.
55
3.4 Neutral Particle Density Measurements
In the last section we investigated the behavior of Hα emission in both
standard and PPCD discharges. As explained earlier, quantifying the Hα
emission is equivalent to measuring the electron source from ionization of
neutral hydrogen. In this section we will address the extraction of the actual
neutral profile. We present profiles obtained during both standard and PPCD
discharges and will discuss the relevance of neutrals in convective energy
transport as well as overall complications arising from large neutral
concentrations in MST.
3.4.1 Neutral Particle Profiles for Standard and PPCD Discharges
Before discussing the local neutral density profiles, we take a moment to
examine the chord-averaged neutral behavior. To accomplish this, we note that
the electron impact ionization rate is essentially constant for electron
temperatures above 20 eV. This leads to the very useful approximation,
σv ion ≈ 3.0 ×10−8 cm3s−1. With this in mind, we can define a chord-averaged
neutral density which will be weighted by the electron density as just the ratio of
the ionization rate, obtained from the Hα emission, and the electron density. In
other words,
N = S n e σv ion ≈ 3.33 ×10+7S n e cm−3 , (3.8)
where S and n e are the chord-averaged ionization rate (Hα-emission/α) and
electron density respectively. For fixed density profile, this quantity will be
proportional to the chord-averaged neutral density. This is where the co-linear
arrangement between the FIR and Hα arrays really pays off. Since both
56
measurements are conducted simultaneously and in the same plasma location,
the weighted chord-averaged neutral density is obtained without conducting a
spatial inversion, and without the added uncertainties an inversion process
introduces. Figure 3.9 displays the chord-averaged neutral particle density for
both standard and PPCD discharge at an impact parameter of r/a=0.69.
3.33
0.67
1.33
0.00
2.00
2.67Standard
PPCDPPCD Trigger Point
5 10 15 20 2Time (ms)
Neu
tral D
ensi
ty -3( 1
E+1
1 cm
)
5
Figure 3.9 – Chord-averaged neutral particle density vs. time for low current (200 kA) standard and PPCD discharges. The specific chord shown above is located at impact parameter of r/a=0.69, and clearly shows the dramatic reduction in neutral particle density during the enhanced confinement period of PPCD.
We see that during PPCD, chord-averaged neutral particle density is
reduced nearly an order of magnitude. It is important to recognize that this
reduction in neutral population is not a temperature effect; recall that the
ionization cross section changes very little with temperature. This reduction is
solely the result of enhanced confinement minimizing the plasma wall
interaction; hence the neutral influx is greatly reduced.
57
Utilizing the MSTFIT reconstruction program, the chord-averaged Hα and
electron density measurements were inverted to yield local profiles of each
quantity. With the electron temperature profile, acquired courtesy of the multi-
chord Thomson Scattering system, the radial profiles of the neutral density
[ N(r) ] can be extracted.
The neutral density profiles confirm the chord-averaged results. Obtained
from the MSTFIT reconstruction code, the neutral density profile, shown in
figure 3.10, is principally peaked at the edge. During standard low current
discharges, the neutral density ranges from about 1 to 2x10+10 cm-3 in the core to
greater than 5x10+12 cm-3 at the plasma boundary. However during PPCD, the
neutral density profile drops below resolvable limits in the core and develops a
more edge peaked nature (figure 3.11). Limited by the diagnostic, we can only
place an upper bound on the core neutral density during PPCD, which is around
8x10+8 cm-3. At the edge, the neutral density does increase to the same levels
seen in standard discharges, however, the upturn is much sharper and occurs
farther out in radius.
58
0.0 0.2 0.4 0.6 0.8 1.01E+10
1E+11
1E+12
Neu
tral D
ensi
ty(c
m
)-3
r/a
Figure 3.10 – Neutral particle density during standard discharges. Upper and lower bands represent the uncertainties in the inverted profile. Errors represent statistical uncertainty and do not include uncertainties due to calibration.
Resolution Limit
0.0 0.2 0.4 0.6 0.8 1.0
Neu
tral D
ensi
ty(c
m
)-3
r/a
1E+10
1E+11
1E+12
1E+09
1E+08
Figure 3.11 – Neutral particle density during PPCD discharges. Upper and lower bands represent the uncertainties in the inverted profile. Errors represent statistical uncertainty and do not include uncertainties due to calibration.
59
3.4.2 Neutral Particle Losses
Neutrals are very efficient at transporting energy and particles out of the
plasma. Since they can move freely in magnetic fields, the only inhibitors to
being lost to the outer wall are collisions. For typical MST plasmas, the collision
times for ionization and charge exchange are between two to four microseconds
(2-4 μs). A neutral in the MST core with energy of 100 eV, has a thermal velocity
of 140 km/s, which translates to an escape time (a vth ) of three microseconds and
a mean free path of about 40 to 60 cm. With the mean free path on the same
order as the plasma radius, on average, half the neutrals in the core will be lost
directly to the wall. This can be a sizeable energy loss. From the measurements
presented above, we assume an average neutral particle density of 4x10+10 cm-3
at an average temperature of 20 eV. If every three microseconds, half of these
particles are lost to the wall, this yields a power loss of,
NP ≈ γNVT N / τ L ≈ 170 kW , (3.9)
where γ is the loss fraction, N is the neutral particle density, V is the plasma
volume, T N neutral temperature, and τ L represents the loss time. During PPCD,
neutral power loss should fall at the same rate as the neutral population. Given
that the radiated power for low current standard discharges is of order two
megawatts, the power lost via neutrals is not negligible.
3.4.3 Neutral Population and CHERS
With the implementation of a neutral beam diagnostic for the purpose of
conducting Charge Exchange Recombination Spectroscopy (CHERS),11 a
renewed interest in the neutral density population in MST has developed. The
60
CHERS diagnostic has already proven to be a very versatile tool in acquiring
localized plasma behavior,12 however recent measurements attempting to
examine the charge exchange recombination behavior of impurity carbon ions in
MST has presented some interesting challenges. The present goal was to
examine the recombination process,
C 6 + + N → C 5+ + p , (3.10)
where when this process occurs, the excited C VI (C5+) atom decays into its
ground state by emitting a photon that can be spectroscopically measured. The
underlying principle is that if one can inject a focused beam of high-energy
neutrals through the plasma, this process will be enhanced locally along the
path of the beam. Therefore light collected by a spectrometer, aligned
perpendicular to the beam, will be dominated by emission from the beam
location. However this requires that the recombination rate be much larger in
the beam than the background plasma.
The charge exchange recombination rate is defined as the product of
neutral density and the cross section. The cross section is highly dependent on
neutral temperature13, ,14 15 and is depicted in figure 3.12. We see that as long as
the background neutral population is very cold, say less than 10 eV, the
exchange rate for the beam will be orders of magnitude greater than that of the
background neutral. However, since most of the neutrals in the core have
undergone charge exchange prior to reaching the core, their temperature will be
much greater than 10 eV. Moreover, the neutrals in the core are likely to be
excited, meaning when they undergo charge exchange, the electron will be
deposited in the higher n atomic states.16 The bottom line is that the high
61
temperature and the excitation of the core neutrals will produce such a large
background that the charge exchange resulting from the beam will be difficult to
extract in standard discharges.
0 +1 +2 +3 +4 +-18
-17
-16
-15
-14
Log [Energy (eV/amu)]
Log
[Cro
ss S
ectio
n (c
m )
]2
5
C + H 6+ C + H 5+ +
Neutral Beam
Energy
Core Neutral Temperature
Figure 3.12 – Charge exchange recombination cross section vs. neutral particle energy for the recombination of C6+ to C5+. While the cross-section for beam neutrals is 1000 times larger for Franck-Condons, it is only 3 to 5 times as large for thermalized neutrals in the core.
3.5 Summary
Fueling in MST is dominated by transport induced wall recycling. The
molecular hydrogen introduced into the plasma quickly dissociates and the
subsequent neutrals penetrate into the plasma. This initial penetration is short-
lived as the neutral is ionized via electron impact or undergoes charge exchange,
in which the electron is just transferred to a locally thermalized ion. The high
rate of charge exchange allows neutrals to penetrate deep in the core, all the
while increasing in temperature.
62
REFERENCES
During a sawtooth crash, the neutral density is increased as enhanced
plasma wall interactions introduce more influx from the walls. This increase is
transitory, decaying away in about 1 ms. The enhanced confinement of PPCD
shows a dramatic reduction in neutral particle density, which is consistent with
the reduction of plasma wall interaction. Neutral density profiles are very
hollow, ranging from 1-2x10+10 cm-3 in the core up to 5x10+12 cm-3 at the plasma
boundary, for low current standard discharges. PPCD reduces the neutral
density in the core below the diagnostic resolution (~8x10+8 cm-3), and increases
the gradient at the edge. Finally, the large neutral densities in MST can be
responsible for a sizable fraction of the total radiated power.
1 H. S. W. Massey, Electronic and Ionic Impact Phenomena, (Oxford University Press, New York, 1969).
2 D. J. Rose and M. Clark Jr., Plasmas and Controlled Fusion (John Wiley & Sons Inc., New York, 1961).
3 R. L. Freeman and E. M. Jones, Analytic Expressions for Selected Cross-Sections and Maxwellian Rate Coefficients, UKAEA Internal Report CLM-R 137, (1974).
4 W. Lotz, Astrophysics Journal Supplements, 14, 207 (1967).
5 Y. N. Dnestrovskii and D. P. Kostomarov, Numerical Simulation of Plasmas, (Springer-Verlag, New York, 1985).
6 S. Hokin, C. Watts, and E. Scime, Proton and Impurity Ion Temperature Profiles (Bull. Of Amer. Phys. Soc. 36, 8 November (1994).
7 W. L. Wiese, M. W. Smith, and B. M. Glennon, Atomic Transistion Probabilities, (National Bureau of Standards, 1966).
8 L. C. Johnson, E. Hinnov, Journal of Quantitative Spectroscopy and Radiative Transfer, 13, 333 (1973).
63
9 I. H. Hutchinson, Principles of Plasma Diagnostics (Cambridge University Press, New York, 1994).
10 Figure courtesy of S. Castillo, 10 July (1997). 11 R. J. Fonck, R. J. Goldston, R. Kaita, and D. Post, Applied Physics Letters, 42, 239 (1983).
12 R. J. Fonck, D. S. Darrow, and K. P. Jaehnig, Physical Review A, 29, 3288 (1984).
13 A. Salop and R. E. Olson, Physical Review A, 16, 1811 (1977).
14 H. Ryufuku and T. Watanabe, Physical Review A, 20, 1828 (1979).
15 R. A. Phaneuf, Physical Review A, 24, 1138 (1981) 16 R. J. Fonck, Private Communications, (1999).
64
4: Impurity Behavior in MST
Having addressed the issue of electron sourcing from neutral hydrogen,
we turn our attention to the electron source contributions from plasma
impurities. As mentioned earlier, quantitative knowledge of the electron source
profiles is critical to obtaining the total radial particle flux (Γ ). Impurity
sourcing is particularly important at higher plasma temperatures, where
hydrogen is ionized, but higher Z impurities continue to contain bound electrons.
Our measurements indicate that during standard discharges, electron sourcing
from impurities is small and can be neglected; however, in high confinement
PPCD discharges where the core neutral population drops, impurity sourcing
can be appreciable and should be considered. In this chapter, we discuss the
impurity concentration measurements of carbon, oxygen, nitrogen, and
aluminum in both standard and PPCD discharges. Furthermore, we introduce a
multi-channel filtered diode spectrometer designed to monitor the highly-ionized
core charge states of the fore mentioned impurities. The chapter layout is as
follows, general introduction (Section 4.1), review of atomic processes (Sections
4.2 and 4.3) and line emission (Section 4.4), and the ROSS filtered spectrometer
65
(Section 4.5). The principal impurity concentration results for standard and
PPCD discharges are presented in Section 4.6 along with some secondary
conclusions about impurity radiation and impurity confinement times (Section
4.7).
4.1 Introduction
Magnetically confined plasmas always contain some fraction, however
small, of high Z impurities. This is undesirable because highly stripped
impurities are very efficient at radiating energy from the plasma.1 Processes like
electron capture or collisional excitation and emission convert electron energy
into radiation, and higher charge states can capture and radiate energy from
higher temperature electrons. For example, the capture threshold for ionized
hydrogen is about 14 eV, but the threshold for fully stripped aluminum exceeds
2.3 keV. This inequity shows that even a very small impurity fraction (order 1%)
can be very problematic in fusion plasmas.
For particle counting, knowledge of impurity concentrations is important
in properly assessing electron source contributions. Unlike hydrogen, which has
a low ionization potential (13.6 eV), the high charge states of aluminum have
ionization potentials of > 1 keV, indicating impurities can continue to be a strong
source of electrons at high temperatures when hydrogen has already burned
through. Impurity sourcing is of particular importance during PPCD discharges
when the neutral hydrogen population decreases by two orders of magnitude in
the core.
Absolute spectroscopic measurements of characteristic emission lines
provide a means for identifying and quantifying the impurity concentrations in
66
MST. Past work in the extreme-ultraviolet (EUV) and visible (VIS) spectral
range have identified the presence of carbon and oxygen. Unfortunately, these
experiments were unable to provide absolute flux measurements. Moreover,
since the lines in the EUV and VIS range result from higher n (principal
quantum number) transitions, their transition probabilities are less accurately
known making it virtually impossible to quantify state densities from these line
intensities.
Quantitative measurement of K shell transitions offers the most reliable
method to extract impurity state densities. These transitions (1s-2p or 1s2-2p)
are well characterized and result from hydrogen-like (H-like) and helium-like
(He-like) impurity charge states that are prevalent in the MST core.
Unfortunately, these transitions emit in the x-ray-ultraviolet (XUV) range and
require more delicate measurement techniques. For this task, we have
constructed a multi-channel filtered soft x-ray spectrometer to quantitatively
measure the emission from H-like and He-like states of carbon, oxygen, and
aluminum.
4.2 Atomic Processes
In a plasma discharge, an impurity species can consist of a multitude of
different charge states, from singly ionized to fully stripped, depending on the
plasma temperature and density. A balance among ionization, particle loss, and
recombination rates determines the relative densities of each charge state. The
interdependencies between charge states, displayed in figure 4.1, illustrate the
complexity in extracting state densities. Typically, a rigorous time dependent
solution requires solving for all the state densities simultaneously. However, not
67
all the terms outlined in figure 4.1 are important in typical MST discharges and
simplifications can be made.
∂ni
∂t+ ∇ •Γ i = ni −1neIi−1 − ni ne Ii + Ri + Di( ) + NCi( ) + ni+1 ne Ri +1 + Di+1( ) + NCi +1( )
Ionization of i-1 state
Radiative and Dielectronic Recombination of i th state
Radiative and Dielectronic Recombination of i+1 th state
Transport Losses
Ionization of i th state
Charge Exchange from the i+1 th state
Charge Exchange from the i th state
Figure 4.1 – The impurity state density continuity equation. The density of the i th charge state is maintained by a balance among, ionization, transport and recombination processes.
4.2.1 Ionization
The principal ionization processes are electron and proton impact
ionization (figure 4.2). In each case, a free electron or proton collides with a
partially stripped ion and excites an electron into an unbound state. For plasma
temperatures below 10 keV, the proton impact cross section is negligible
compared to that from the electrons and is typically ignored in MST plasmas.
68
Ene
rgy
FreeBound
Before After
a)
e-
e-
e- e-
e-
FreeBound
Before After
b)
p+p+
e- e-
e-
Figure 4.2 – Electron (a) and Proton (b) impact ionization. Both cases involve a transfer of energy from the free particle into the atom that then gets passed into the bound electron, which is expelled.
4.1.2 Radiative and Dielectronic Recombination
Radiative recombination (figure 4.3a) occurs when a free electron collides
with an ion and is captured. The energy lost by the electron is emitted as a
photon and because this is a free-bound transition, the photon energy is not
quantized. Dielectronic recombination (figure 4.3b) can occur when a free
electron collides with a partially ionized (i.e. not fully stripped) ion. Often the
electron will simultaneously excite the bound electron before being captured.
Provided the excited electron relaxes before auto-ionization occurs, this process
produces two photons, one unquantized (resulting from the electron capture) and
one quantized (from de-excitation).
69
Before After
γe-
e-
e-
e-
e-γ
e-
FreeBound
Before After
γe-
e-
Ene
rgy
FreeBound
a) b)
Figure 4.3 – Radiative (a) and Dielectronic (b) recombination. Radiative recombination involves a simple electron capture process, however dielectronic recombination occurs as a simultaneous electron capture and ion excitation. Both the processes are negligible for the highly ionized charge states of carbon, oxygen and aluminum.
4.2.3 Charge Exchange Recombination
The final process, charge exchange, results from a collision between a
neutral or partially ionized atom with another partially ionized or fully stripped
ion (figure 4.4). Often, this collision will result in an electron being exchanged
from one atom to the other, hence the term charge exchange. In standard MST
discharges, the large concentration of neutral hydrogen in the core makes charge
exchange the dominant recombination process for high Z impurities. This is
clearly evident in figure 4.5, which displays the ionization and recombination2,3
rates for balance between O VII and O VIII. It is important to note that although
radiative and dielectronic4 recombination are negligible for H-like and He-like
high Z impurities, these processes can be significant in determining lower charge
state densities.
70
e-Free
Bound
Before After
γ
Ene
rgy
e -
H C 6+ C 5+H+
Free
Boun
d
Figure 4.4 – Charge exchange recombination involes the transfer of a bound electron from one atom to another. This process is the dominant recombination process for the highly stripped impurity ions in MST.
71
1X10
(c
m s
)
1X10
(c
m s
)1X
10
(cm
s )
1X10
(c
m s
)
0 200 400 600 8000.0
2.0
4.0
6.0
8.0Io
niza
tion
Rat
e-3
-1-1
1
0 200 400 600 8000.0
2.0
4.0
6.0
8.0
Cha
rge
Exc
hang
e R
ate
-3-1
-8
0 200 400 600 800
1.0
0.8
0.6
0.4
0.2
0.0
Rad
iativ
e
Rec
ombi
natio
n R
ate
-3-1
-12
0 200 400 600 8000.0
1.0
2.0
3.0
Die
lect
roni
c R
ecom
bina
tion
Rat
e-3
-1-1
2
Electron Temperature (eV)
Electron Temperature (eV)
Electron Temperature (eV)
Electron Temperature (eV)
(a) (b)
(c)(d)
Figure 4.5 – Ionization of O VII (a), radiative recombination of O VIII (b), dielectronic recombination of O VIII (c), and charge exchange of O VIII (d) rates. Note that dielectronic and radiative recombination rates are small indicating that the state ratios are maintained by the balance of ionization, transport and charge exchange.
4.3 Charge State Equilibrium (Coronal or LTE)
When electron collisions dominate the radiative processes, the plasma is
said be in Local Thermodynamic Equilibrium (LTE). The LTE condition requires
that
ne >> 10+19 Te eV( ) ΔE eV( )( )3m −3 , (4.1)
72
where Te is the electron temperature, and ΔE is the transition energy. However,
the density in MST is far too low to satisfy this condition for VIS, EUV, and XUV
transitions and a more appropriate model should be employed.
In lower density plasmas, where radiative processes dominate collisional
ones, the coronal equilibrium model is used. In coronal conditions, a neutral or
partially ionized atom resides in its ground state until a collision excites it. Once
excited, the atom immediately returns to its ground state and emits a photon.
The coronal equilibrium assumption is well suited for the K shell transitions of
high Z impurities, where the state lifetimes are in the range of tens of
picoseconds, and the collision times are on order of milliseconds.
4.4 Electron Impact Excitation and Line Emission
Electron impact excitation is the dominant excitation process in MST.
Very similar to the ionization process described above, electron impact excitation
occurs when an electron deposits energy, via collision, to a neutral or partially
stripped atom. If the energy transferred is greater than the threshold for
excitation, the atom can be excited from its ground state. The atom immediately
begins to radiate photons as the electron in the excited atom cascades down to
its ground state. For H-like and He-like high Z impurities, the excited state
lifetimes are so short that virtually all excitations are followed by a single
radiative decay back to the ground state. This balance leads to an emissivity
from the i → j transition of
εγi→ j = nenimp σv excitation
i→ j , (4.2)
73
where and n are the impurity state and electron densities and nimp e σv excitationi → j is
the excitation rate for the i → j transition.
A principal advantage in monitoring the K shell transitions of H-like and
He-like ions is that the electron excitation rates are well characterized. Figure
4.6a-c outlines the relevant excitation rates for C V, C VI, O VII, O VIII, and Al
XII. These rates, reported by Mewe,5 should be accurate to within 30-50%.
Therefore, by measuring electron density, temperature, and photon emission
from a well-characterized transition, equation 4.2 can be used to extract the
impurity state density.
Exc
itatio
n R
ate
(1E
-10
cm s
)-3
-1
1s-2pC VI
1s -1s2p2C V
1s -1s3p2C V0.0
1.0
2.0
3.0
4.0
Electron Temperature (keV)0.0 0.2 0.4 0.6 0.8 1.0
1s -1s2p2O VII
1s -1s3p2O VII
1s-2pO VIII
Exc
itatio
n R
ate
(1E
-10
cm s
)-3
-1
1.4
1.0
0.6
0.2
1.2
0.8
0.4
0.0
Electron Temperature (keV)0.0 0.2 0.4 0.6 0.8 1.0
Exc
itatio
n R
ate
(1E
-12
cm s
)-3
-1
Electron Temperature (keV)0.0 0.2 0.4 0.6 0.8 1.0
0.0
2.0
4.0
6.0
8.0
1s -1s2pAl XII 2
C V C V C VI O VII O VII O VIII Al XII
40.26 34.97 33.73 21.60 18.63 18.98 07.75
308 355 368 574 665 653
1600
State Wavelength Energy (eV) a)
b) c)
Figure 4.6 – The electron excitation rates for the principal transitions for H-like and He-like (a) oxygen, (b) carbon, and (c) aluminum. The table outlines the photon wavelength and energy from these transitions.
74
4.5 The ROSS Filtered Spectrometer
To monitor the transitions outlined in figure 4.6, a low-cost robust multi-
foil filtered spectrometer was developed. Multi-foil spectrometers have been
previously used on RFX6 and MST7 to routinely monitor high charge state
impurities. Based on previous designs and similarly named, the ROSS filtered
spectrometer consists of a six channel diode array, where each diode is coated
with a thin-film multi-layer filter designed to accentuate a narrow portion of the
XUV spectrum. By completely redesigning the thin-film filters, directly coating
the diode surface, and calibrating on a synchrotron source, the diodes are
capable of making absolute flux measurements, and with some assumptions
about the emission characteristics of the XUV range, the line intensities for the
dominant core impurities can be quantified. This information, coupled with
independent electron temperature and density measurements allows the
impurity state densities to be resolved.
4.5.1 Filter Characteristics
The multi-layer filter characteristics are outlined in figure 4.7a-f and
table 4.2. The principal concept in designing a thin-film multi-layer filter is to
properly choose materials that have natural absorption edges in the region of
interest. To isolate the O VII (~18, and ~21 Angstroms) and O VIII (~18
Angstroms) emission, we use iron, manganese, and vanadium, which have
absorption edges at 17.5, 19.5 and 24.3 Angstroms respectively. To separate the
C V (~40 Angstroms) and C VI (~33 Angstroms) impurity states, the fluorine
edge at 36 Angstroms is used. The Polyimide (C22H10N2O5) filter has a very
strong carbon edge at ~43.7 Angstroms which isolates C V from the lower energy
75
emissions from B IV. The Mylar (C10H8O4) is used simply to attenuate
wavelengths above 50 Angstroms where the aluminum becomes transparent.
Tran
smis
sion
1e-4
1e-3
1e-2
1e-1
1.00
Tran
smis
sion
1e-4
1e-3
1e-2
1e-1
1.00
Tran
smis
sion
1e-4
1e-3
1e-2
1e-1
1.00
0 10 20 30 40 50 60Wavelength (Angstoms)
0 10 20 30 40 50 60Wavelength (Angstoms)
(a)
(c)
(e) (f)
(b)
(d)
Figure 4.7 – The transmission characteristics for the (a) Al/Fe, (b) Al/Mn, (c) Al/V, (d) Al/CaF, (e) Al-3, and (f) Al-4 channels.
The thin-film filters were designed using the XCAL8 software package
and measurements of their transmission properties were conducted at the
National Synchrotron Light source at Brookhaven National Laboratory.9 The
transmission characteristics were right in line with the theoretical predictions
made by XCAL.
76
Filter Diode Name Thickness Material
Absorption Edge (Angstroms)
Band Of Interest (Angstroms)
Al-4 1 μm 6 μm
Al Mylar None < 12
Al/Fe 5000 A 4500 A
Al Fe 17.5 18 – 19
Al/Mn 4000 A 5000 A
Al Mn 19.5 20 – 23
Al/V 5000 A 5000 V
Al V 24.3 25 – 35
Al/Caf 2000 A 2500 A 1 μm
Al Ag
CaF236.0 37 – 43
Al-3 1000 A 7.5 μm
Al Polyimide 43.7 > 45
Table 4.2 – ROSS filtered diode array filter characteristics.
4.5.2 The Soft X-ray Diodes
The focal point of the six-channel diode array is the AXUV-100 silicon
detector.10 The diode has an active area of 1 cm2 and is virtually 100% efficient
for photons above 10 eV. The large active area makes it ideal for situations
where low light levels are an issue. However, this comes at the expense of a
large junction capacitance (C j ≈10 nf ) which places severe constraints on the
time response of the detector. With the goal of measuring equilibrium impurity
behavior, the frequency response limitation of around 40 kHz was not a concern.
Cathode
Anode
.571
.650
.400
Figure 4.8 – The AXUV 100 Diode. All dimensions in inches.
77
4.5.3 Diagnostic Geometry and Light Collection
The diodes are isolated with respect to each other in a tightly packed
hexagonal arrangement allowing them to all fit in a single 2.75 inch flange.
Their collection angles are individually collimated with long aluminum tubes
that are configured so that each diode samples the same chord through the
plasma. The entrance and exit slits on the tubes are 0.40 inches in diameter and
separated by 13 inches. This geometry leads to a solid angle collection (ξ ), or
etendue, of
ξ ≈ AsAd( ) 4πS2 = .81( )2 4 π 33( )2
=4.8 E − 5cm2 , (4.3)
where As and Ad are the entrance and exit slit areas respectively, and S is the
distance between the slits. As mentioned earlier, the diodes have a near perfect
conversion efficiency ( η ), producing one electron/hole pair for every 3.63 eV
incident on the detector ( η ≈ 0.27 A W ). The small current from the diodes is fed
into an amplifier with a gain (G ) of 1.5x10+6 V/A before being stored by the
digitizer.
Given some emissivity [ε x,λ( )] that is assumed to only vary along the line
of sight of the spectrometer, the signal level measured by the diode can be
expressed as
Sig (Volts) = Gξ η λ( )
0
∞
∫ T λ( ) ε x,λ( )L∫ dx dλ . (4.4)
A rigorous solution for ε x,λ( ) would require an infinite set of diodes each
sensitive to a different wavelength. Quite obviously this solution is impractical
and we are left to explore ways to simplify this problem.
78
4.5.4 Deciphering Impurity Line Emission
It is believed that the region between 5 and 50 angstroms is dominated by
line emission and the spectrum can be approximated by a collection of delta
functions with differing amplitudes, each representing a narrow band of
emission. The filters of the ROSS spectrometer are designed to isolate the bands
outlined in table 4.2. If we make this assumption, the current measured in each
diode can be approximated as a linear combination of the products of band
emission and filter transmission. In other words the measured diode current is
Ik = η λi( )T k λ i( )ε i λi( )δ λ − λ i(
i =1
n
∑ ), (4.5)
where Tk λ( ) is the filter transmission of the diode, kth ε λ( ) is the band
emissivity at wavelength λ , and η λ( ) is the photon conversion efficiency. For
these wavelengths, the diodes are virtually 100% efficient, meaning the number
of electrons per photon of wavelength λ is
η electrons( ) ≈ 3417 λ Angstroms( ). (4.6)
The goal of the ROSS spectrometer is to obtain the emissivity [ε λ( )]. A closer
inspection of equation 4.5 shows that the diode current ( I ) is just a matrix and
can be written as
, (4.7) I T R= •
where we have defined R = η λ( )ε λ( ). By inverting the filter transmission matrix
(T ), we can directly solve for R and extract ε λ( ) as follows,
ε λ( ) = T −1 λ( )• I[ ] η λ( ). (4.8)
79
With the simultaneous measurements from the ROSS Spectrometer six-diode
array, the emission amplitudes of the six principal bands that dominate the
spectrum can be extracted.
Once the emission is known, the extraction of the line intensities can
begin. For example, let us examine the 20-23 Angstrom band. Emission in this
region results primarily from the 1s2-1s2p transition of O VII. With the photon
intensity known, the state density of O VII can be estimated as outlined in
equation 4.2. The emission in the 18-19 Angstrom band, will be dominated by
two lines, the 1s-2p line of O VIII and the 1s2-1s3p transition of O VII. By using
the electron excitation information outlined in figure 4.6, we can obtain the
relative contribution of the O VII and O VIII lines. With the density of O VII
obtained from the 20-23 angstrom channel, the state density of O VIII can be
determined.
4.5.5 Line Contamination
The ROSS filtered spectrometer has proven very effective in isolating
emission from the charge states of oxygen and aluminum. However, it has not
been as successful with carbon. This is attributed to the large contributions of
aluminum and nitrogen, which have not been previously measured in this
region. The emission of H-like and He-like nitrogen dominates the region
between 24 and 30 angstroms and as a result the nitrogen emission
contaminates the measurement of C VI. A more interesting and unexpected
result is the large presence of aluminum. The common lore among ancient
MST’ers was that carbon is the dominant impurity and should overshadow
everything else. A principal result from the ROSS spectrometer is that
80
aluminum, not carbon, is the principal contributor to emission in this region,
especially at higher plasma temperatures. In an attempt to see the bright side,
the inability of the ROSS to quantify the carbon presence has actually led to a
greater understanding of impurity concentrations in MST
4.6 Impurity Effects
Using the data obtained from the ROSS spectrometer, we can estimate
the impurity fractions of oxygen, and less accurately aluminum and carbon. The
data presented in section 4.6.1 was collected during standard low current
(Iplasma≈ 200 kA), moderate electron density (ne≈1x10+13 cm-3) discharges and was
ensemble-averaged over about 400 sawtooth events. Section 4.6.2 presents the
PPCD results, which represent an ensemble over 137 events. The parameters for
these discharges were configured such that at peak confinement the plasma
current and density were similar to that of the standard case.
4.6.1 Impurity Concentration in Standard Discharges
The results for oxygen in standard discharges, displayed in figure 4.9,
indicate that the chord-averaged density of O VII is on the same level as the
neutral hydrogen density in the core and the O VIII density is smaller by a
factor of five. At these plasma currents, the electron temperature is measured to
around 220 eV, and at this temperature virtually all the oxygen will reside in
the He-like O VII and the H-like O VIII states. This leads to an estimate of the
total impurity fraction from oxygen being between 0.11% to 0.14%.
81
a)
b)
O VIII Density
O VII Density
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
2.5
0.0
1.0
2.0
0.5
1.5
0.4
0.8
1.2
0.2
0.6
1.0
Time (ms)
Den
sity
1E
+9
(cm
)
-3D
ensi
ty 1
E+1
0 (c
m
)-3
Figure 4.9 – Chord-averaged impurity state densities for (a) O VII and (b) O VIII ensembled over 400 sawteeth events in standard discharges. The gray traces represent the uncertainties in the density arising from the error in the calculated excitation rates. These densities translate to an overall oxygen fraction between 0.11 % to 0.14 %.
Measurements of aluminum and carbon are more difficult to interpret.
First, at low temperatures, the excitation of the AL XII 1s2-1s2p transition is
rare, and the flux measured by the ROSS spectrometer is very close to the lower
resolution limit which make a determination of the temporal behavior
unreliable. We can however assign an upper limit to the aluminum
concentration which is ≈ 0.25%. The principal challenge in extracting carbon
densities is that of contamination. As discussed in section 4.5.5, the carbon
channels can be contaminated with nitrogen and aluminum. While removing the
nitrogen component from the C VI channel is a hopeless endeavor, at low
82
electron temperatures, the aluminum contamination of the C V channel is very
small. Therefore, for low temperature, low current discharges, the C V
measurement should be reasonably accurate.
For the same discharges outlined above, we have calculated the state
densities for C V and C VI. For the C VI channel, we have assumed no nitrogen
content, which should lead to a serious over-estimate of C VI concentration, but
should be useful in serving as an upper bound. The chord-averaged state
densities, outlined in figure 4.10, show a C V concentration two to three times
that of neutral hydrogen in the core, with similar values for C VI.
C V Density
C VI Density (Upper Bound )
0.0
1.0
2.0
3.0
4.0
-2.5 -1.5 -0.5 0.5 1.5 2.5Time (ms)
Sta
te D
ensi
ty
(1E
+10
cm
)-3
Figure 4.10 – Chord-averaged impurity state densities for C V and C VI ensembled over 400 sawteeth events in standard discharges. The gray traces represent the uncertainties in the density arising from the error in the calculated excitation rates. The C VI trace represents an upper bound on the state density. We estimate an overall carbon fraction less than 0.50 %.
Measurements from the ROSS spectrometer indicate that the overall
impurity concentration in low current standard discharges is less than 1%.
While carbon appears to be the most abundant (≤ 0.50%), it does not dominate
83
over oxygen (≤ 0.15%) and aluminum (≤ 0.25%). Moreover, the carbon fraction of
~0.50% includes any contributions from nitrogen, indicating the carbon fraction
is probably much lower.
4.6.2 Impurity Concentration in PPCD Discharges
The impurity concentration data for PPCD discharges is much more
complicated to interpret. The transient nature of the PPCD discharge renders
the steady state assumption completely invalid. Moreover, all the quantities that
determine charge state balance and x-ray emission, such as electron density,
electron temperature, and neutral density, are dramatically changing, making it
virtually impossible to accurately back out the impurity state temporal behavior.
With this in mind, we have to settle for obtaining rough estimates on how the
impurity concentrations are changing during PPCD. Given that line emission is
described by equation 4.2, which states εγi→ j = nenimp σv excitation
i→ j , the relative change
in emission can be represented as
Δεγi → j εγ
i→ j ≈ Δne ne + Δnimp nimp + Δ σv excitationi→ j σv excitation
i → j , (4.9)
provided the cross terms in Δ are small. Therefore by quantifying the relative
changes in ionization rate, electron density, and line emission we can estimate
the change in impurity state density.
The data presented in the subsequent pages was obtained during the
January, 1999 confinement run. The initial discharge conditions were low
current (200 kA), low temperature (220 eV), and low electron density (~4x10+12
cm-3 chord-averaged). At the optimum of PPCD the electron temperature was
measured to be ~ 500 eV with a chord-averaged electron density of 7.0x10+12 cm-
84
3. The first sawtooth after 8 ms into the shot served as the PPCD trigger point.
The PPCD capacitor banks were then fired at some time after the initial trigger
(usually 1.5 ms). The ROSS data presented below is ensemble-averaged over 137
good PPCD shots, where the sawtooth was the reference point. Prior to the
confinement run, a concerted effort to boronize and condition the machine took
place. As a result, the initial impurity densities were much lower, almost an
order of magnitude, than those presented in section 4.6.1.
A robust feature of a good PPCD discharge is the huge increase in soft x-
ray emission. Figure 4.11 displays ROSS data from the two oxygen channels. We
see that before PPCD takes effect, the O VII emission dominates that from O
VIII. As confinement improves, the O VIII emission continues to rise even after
O VII burns through. The shift toward O VIII becoming the dominant state is an
indicator that the recombination and transport rates of O VIII are shrinking
relative to ionization of O VII. Taking into account the changes in temperature
and density, these emission amplitudes suggest that the O VIII density is
increasing by, about a factor of three, while the O VII concentration is actually
dropping. The overall oxygen concentration doesn’t change much for these
discharges.
85
0
2
4
6
8
10
12
O VIII
O VII
O VIII Peak Emission
O VII Peak Emission
PP
CD
Sta
rt
-5 0 5 10 15
1E+1
1 P
hoto
ns
Time (ms)
18-23 Angstroms
Figure 4.11 – The emission from O VII and O VIII. Unlike standard discharges, O VIII dominates during PPCD.
Emission from Al XII shows the most dramatic increase during PPCD.
Shown in figure 4.12, the emission increases a factor of 30, peaking about 10 ms
after the PPCD trigger time, which is also after the time at which the O VIII
emission peaks indicating that even O VIII is burning through. Most of this
emission is from the increase in temperature. Acounting for the temperature and
density increases, the overall aluminum concentration increases a factor of two,
± 60 %.
The emission between 23 and 38 angstroms is displayed in figure 4.13.
In this region, N VI, N VII and C VI are all contributing to emission. With all
three states mixed together, determination of any particular state density is
impossible. We can say that all these states burn through and any carbon or
nitrogen remaining in the plasma is fully stripped.
86
Al XII Peak Emission
0
1
2
3
-5 0 5 10 15
1E+1
1 P
hoto
ns
Time (ms)P
PC
D S
tart
AL XII
<15 Angstroms
Figure 4.12 – The emission from 1.5 KeV Al XII line. Overall emission increases 30 fold, but the state density increase is estimated to be only a factor of two or three.
5
4
3
2
1
0-5 0 5 10 15
1E+1
2 P
hoto
ns
Time (ms)
PP
CD
Sta
rt
23-38 Angstroms
Figure 4.13 – The emission from the C VI channel. This channel measures contributions from C VI as well as N VI, and N VII. All of these states burn through.
87
The last two channels from the ROSS are displayed in figures 4.14 and
4.15. Designed primarily to look at lower charge states of carbon and boron, at
high temperatures when these states have burned through, these channels
become sensitive to emission from high charge states of aluminum.
Although we are unable to make any quantitative statements about the
carbon and nitrogen content during PPCD, it is reasonable to assume that the
sourcing characteristics for these impurities should be similar to those of
aluminum and oxygen. Having ascertained that the total concentrations of
aluminum and oxygen do not drastically change, it is highly unlikely that carbon
and nitrogen would behave differently.
AL XI
AL XII
-5 0 5 10 15
1E+1
2 P
hoto
ns
Time (ms)
0
0.5
1.0
1.5
2.0
2.5
PP
CD
Sta
rt
38-48 Angstroms
Figure 4.14 – Emission from the C V channel. Having burned through C V, the contributions from aluminum appear.
88
? AL XIII ?
5
4
3
2
1
0-5 0 5 10 15
1E+1
1 P
hoto
ns
Time (ms)P
PC
D S
tart
48-60 Angstroms
Figure 4.15 – Emission from the 48 to 60 Angstrom region. Again we see the unmistakable features of the high aluminum charge states.
In summary, during PPCD electron temperature and density rise and as a
result x-ray emission increases emormously. Measurement from the ROSS
indicated that overall aluminum concentration increases about twofold, the
increase in oxygen is within the error. Though not directly measured, we infer
that the concentrations of carbon and nitrogen show similar trends.
4.6.3 Electron Sourcing From Impurities
Quantitatively measuring the electron source from impurities remains
one of the most complicated problems in experimental plasma physics. To
correctly account for electron sourcing requires a complete understanding of the
ionization, recombination, and charge exchange rates for every impurity state
present in the plasma. A mathematical description of the impurity electron
source from a charge state j is
Sj = ne Ij − 1nj −1 − I j − Rj( )nj + Rj + 1nj +1[ ]+ N Cj +1nj +1 − Cjnj( ), (4.10)
89
where I , R , and C are the ionization, recombination (both radiative and
dielectronic), and charge exchange cross-sections, and where n , and e nj N , are
the electron, impurity state, and neutral densities. Keep in mind that equation
4.10 is just the electron source from the jth state, and the total source requires a
sum be taken over all states of all impurities.
We can simplify the problem by examining how the ionization rate for the
dominant impurity states compares with the ionization rate of neutral hydrogen,
H = ? . (4.11) n ni eIi eNn I
For example, if nineIi Nne IH << 1 , then the hydrogen ionization completely
dominates any impurity contributions. It is important to note that we are
neglecting all recombination processes, which, if included, would further reduce
impurity electron sourcing. In the core, the dominant measured impurities are C
V, C VI, O VII, O VIII, and AL XII. For the most part in standard discharges, all
of these state densities are on the same order as the neutral hydrogen
population ( ni N ≈ 1.0). However the ionization rates are much lower for the
impurities, ranging from ~1000 s-1 for C V down to ~10 s-1 for Al XI, than the
ionization rate of hydrogen, which is ~3x10+5 s-1. Since neIi ne IH << 1.0, and with
in N ≈ 1.0 already established, n i en Ii eNn IH << 1 , and electron sourcing for these
states can be neglected.
However, during PPCD discharges, N drops dramatically while the high Z
charge states increase in abundance. When the neutral density drops,
ni N >> 1.0 , and impurity sourcing can become comparable to that of hydrogen
( n ni eIi eNn IH ~ 1). Moreover, with the concentration of neutrals depleted in the
core, the recombination from charge exchange drops and equation 4.11 is no
90
longer an overestimate of the impurity sourcing. Hence, during PPCD, impurity
may become important and must be considered.
4.6.4 Impurity Radiation
An interesting feature about PPCD discharges is that the soft x-ray
measurements show a dramatic increase in emission, but the overall bolometric
power always drops.11 In an effort to quantify the radiated power more
accurately, an uncoated AXUV-100 diode was installed to measure the plasma
power loss via photons. The diode, was placed about 40 cm from the plasma
boundary, and was collimated with two 1 mm diameter slits, 13 cm apart. This
configuration guaranteed that no charged particles would impact the diode,
making it sensitive only to radiation of photon energy greater than ~3 eV. The
overall total bolometric power would continue to be monitored by the pyro-
electric crystal bolometer,12 which is a heat measuring diagnostic, sensitive to
both radiation and particles.
The results from this experiment, displayed in figure 4.16, present an
interesting clue to energy and particle confinement in MST. The data shown was
taken from the same parameters mentioned above, low current and moderate
plasma density. Again a sawtooth ensembling (~400 events) technique was used
to obtain the average behavior over the crash. The total bolometric power
averaged over the crash is measured to be ~ 1.7-2.0 MW, but the radiated power
from photons is only ~ 300 kW. This definitively states that most of the radiated
power is a result of convective losses through particle transport. While PPCD
increases overall photon radiation, particle loss is dramatically reduced, and
91
with most of the bolometric power resulting from particle convection, the total
bolometric power will still drop.
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.00
1
2
3
4
Pow
er (M
W)
Time (ms)
Bolometric
Radiated
Figure 4.16 – Comparison between total bolometric power (particles + photons), measured with crystal pyro-bolometer, and total radiated power (photons above 3 eV), obtained with a surface barrier diode. The bolometric power is almost 10 times larger.
4.7 Estimating Impurity Confinement Times
From figure 4.1 the impurity state balance in cases when the radiative
and dielectronic recombination are small can be expressed as ∂ni
∂t+ ∇ • Γ i = ni −1neIi−1 − ni neIi + NCi( ) + ni +1NCi+1 , (4.12)
where ni-1, ni, and ni+1 are the densities of the i-1, i, and i+1 states, n is the
electron density, e
N is the neutral hydrogen density, and I and C are the
ionization and charge exchange rates respectively. Examining the steady state
(∂ni ∂ t → 0 ) case, where transport at the plasma surface is approximated as
state density over confinement time (∇ • Γ i → ni τ i ), the density of each charge
92
state is controlled by a balance between ionization, charge exchange, and
transport. Substituting these approximations into equation 4.12, we find that
ni −1ne Ii −1 = niNCi + ni τ i , which yields a charge state ratio of
ni
ni −1
=neIi− 1
NCi + 1 τ i
. (4.13)
We see that the state ratio has a strong dependence on both neutral fraction and
confinement time but a weaker temperature dependence, which is imbedded in
the ionization and charge exchange rates. Solving for impurity particle
confinement time, equation 4.13 becomes
τ i = ni −1 ni( )neIi −1 − NCi[ ]−1. (4.14)
Using the rates outlined in figure 4.5 and the measurements of O VII and O VIII
density concentrations obtained in standard discharges (Section 4.6.1), we are
able to estimate the confinement time of O VIII.
The impurity ion confinement time for the O VIII charge state is
estimated to be between 2 and 6 milliseconds away from the sawtooth crash
(figure 4.17). This is slightly longer then the ≈ 1 ms measured for the electrons
as might be expected for the slower moving, heavier impurity O VIII ions. This
result is similar to the measurements in TPE-1RM20 which showed the
confinement time of boron to be 1.5 times that of majority species.13 An
important result of this measurement is the realization that the confinement
time is on order of the charge exchange recombination time ( 1 NC8 ≈ 3.5 ms ). This
implies that transport plays as much of a role in the reduction of O VIII
concentration as charge exchange and thus cannot be neglected.
93
0
2
4
6
8
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
Con
finem
ent T
ime
(ms)
Time (ms)
Figure 4.17 – The impurity particle confinement for the O VIII charge state. The upper and lower plots represent the systematic error, resulting primarily from the uncertainty in the state densities. The shaded regions are near the sawtooth crash and indicate where the steady state assumptions break down.
4.8 Summary
By implementing a low-cost, robust, multi-foil filtered spectrometer, we
have determined that the dominant impurities in MST are carbon, oxygen,
nitrogen, and aluminum. In standard, low current, moderate density discharges,
the total impurity fraction has been measured to be less than one percent, of
which ~0.14% results from oxygen, ~0.25% from aluminum, ~0.50% from carbon
and nitrogen together. After conditioning and boronization, this fraction has
been observed to drop an order of magnitude.
In the core, the dominant charge states are C V, C VI, N VI, N VII, O VII,
O VIII, and Al XII, where the densities are determined by the balance of electron
impact ionization with transport and charge exchanges losses. Radiative and
dielectronic recombination processes are negligible. Electron sourcing from these
94
REFERENCES
impurities is measured to be small in standard discharges, but may become
important during PPCD when the source from hydrogen falls in the core.
By investigating the impurity density ratio of the O VII and O VIII charge
states, we estimate the impurity confinement in standard, low current, moderate
density discharges to be between 2 and 6 milliseconds. This is substantially
longer than the confinement measured for the electrons (~1 ms), but on the same
order as the charge exchange recombination time which implies that direct
particle losses play an important role in reducing the population of high Z charge
states in the plasma core. Finally, we comment that the bolometric radiated
power is dominated by convective particle losses and not by photon radiation,
which only amounts to about 300 kW.
1 C. Breton, C. De Michelis, M. Mattoli, Nuclear Fusion, 16, 6 (1976).
2 T. P. Donaldson and N. J. Peacock, Journal of Quantitative Spectroscopy and Radiative Transfer, 16, 599 (1976).
3 A. Salop and R. E. Olson, Physical Review A, 19, 1921, (1979).
4 A. Burgess, Astrophysics Journal Letters, 139, 776 (1964).
5 R. Mewe, Astronomy and Astrophysics, 20, 215 (1972).
6 L Marrelli, P. Martin and A. Murari, Measurements in Science and Technology, 6 ,1690, (1995).
7 S. Hokin, R. J. Fonck, and P. Martin, Review of Scientific Instruments 63, 5039 (1992).
8 S. Mrowka, Oxford Research Corp., Richmond, CA.
9 J. Seely, R. Korde, F. Hanser, J. Wise, G. E. Holland, J. Weaver, and J. C. Rife, Characterization of Silicon Photodiode Detectors with Multilayer Filter Coatings for 17-150 A, SPIE Meeting, 18-23 July (1999).
95
10 International Radiation Detectors, Torrance, CA.
11 J. S. Sarff, S. A. Hokin. Hi Ji, S. C. Prager, C. R. Sovinec, Physical Review Letters, 72, 3670 (1994).
12 G Fiksel, J. Frank, and D. Holly, Review of Scientific Instruments, 64, 2761 (1993).
13 Y. Yagi, T. J. Biag, L Carraro, Y. Hirano, R. Hamada, Y. Maejima, S. Sekin, and T. Shimada, Nuclear Fusion, 37, 1775 (1997).
96
5: Radial Electron Flux Profile Measurements
With the FIR interferometer and Hα array monitoring the electron density
and source profiles simultaneously, the radial electron flux profile can be
extracted. By employing PPCD to reduce the magnetic fluctuations and
measuring the total radial flux profile, we are able to move beyond statements of
global confinement parameters and make local assessments of how confinement
is changing. Measurements in standard discharges indicate that the radial
electron flux increases with radius, ranging from ~1-3x10+20 (m-2s-1) in the core
to ~3.5x10+21 (m-2s-1) at the edge. These edge values are consistent with
previously measured fluctuation-induced particle transport1 and are similar to
those obtained by modeling in RFX.2 During PPCD, the radial flux profile
decreases by an order of magnitude with the core showing a more dramatic
reduction, providing the first definitive evidence that PPCD improves core
confinement. Having already discussed the electron source profile measurements
from neutral hydrogen (Chapter 3) and impurities (Chapter 4), in this chapter
we examine the equilibrium electron density behavior in both standard and high
97
confinement PPCD discharges (Section 5.1). We then move to measurements of
the radial particle flux, again, in both standard and PPCD cases (Section 5.2).
Finally we touch upon the secondary issues of particle confinement time (Section
5.2.3), radiative power balance, and convective power loss (Section 5.3).
5.1 Equilibrium Electron Density Behavior
Every successful plasma experiment, big or small, has devoted both time
and energy to measuring equilibrium electron density. On the surface,
monitoring the equilibrium electron density for fusion studies bounds the overall
particle content and is essential for making quantitative statements about
energy and particle confinement times. Upon closer inspection, one notes that
the electron density profile itself hides valuable clues to the plasma’s source and
transport characteristics. Although both source and transport effects couple to
form the density profile, amplitude and gradient changes in the density profile
provide strong hints as to how the electron source and transport are changing.
In the subsequent subsections, we examine the behavior of the chord-integrated
and inverted profile measurements of the electron density in both standard and
PPCD discharges.
5.1.1 Density Profiles in Standard Discharges
To examine the density behavior in standard discharges, we once again
employ the sawtooth ensembling technique. Data from low current, moderate
density discharges was segmented into 4 ms windows, centered at the sawtooth
crash time. This particular ensemble contained 271 events spread out over 87
shots. The temporal behavior of the chord-integrated density over the sawtooth
98
crash is displayed in figure 5.1 for the outboard chords located at impact
parameters of r= 6, 21, 28, 36, 43 cm.
P06
P43
P21
P28
P36
-2.0 -1.0 0.0 1.0 2.0
1.0
0.8
0.6
0.4
0.2
Time (ms)
Inte
grat
ed E
lect
ron
Den
sity
(1
E+1
5 cm
)
-2
Figure 5.1 – Chord-integrated electron density over the sawtooth crash for impact parameters of 6, 21, 28, 36, and 43 cm.
Away from the crash, the central chords steadily rise in density, reaching
a peak at a half millisecond before the crash. Moving out in radius, this change
in density becomes less significant to where in the outermost chords, the density
actually decreases between sawteeth. The modifications in the density
measurements begin to appear about 250 microseconds before the actual crash
time as the core chords decrease while the edge measurements increase. Over
the crash, reductions of 10% in the core and increases of 40% in the edge are
typical. The chord-integrated data indicates the density reaches its flattest
profile some 100-200 microseconds after the crash before beginning to peak back
up.
99
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
t=-2.25 ms
t=-2.00 ms
t=-1.75 ms
t=-1.50 ms
t=-1.25 ms
t=-1.00 ms
t=-0.75 ms
t=-0.50 ms
t=-0.25 ms
t=-0.00 ms
1.2
0.8
0.4
1.2
0.8
0.4
1.2
0.8
0.4
1.2
0.8
0.4
1.2
0.8
0.4
0.0-40 -20 0 20 40 -40 -20 0 20 40
Radial Position (cm)
Ele
ctro
n D
ensi
ty (1
E+1
3 cm
)
-3
Figure 5.2 – The inverted electron density profiles over the sawtooth crash. In general, the profiles are flat in the core with steep edge gradients.
The density profiles of the chord-integrated data discussed above are
displayed in figure 5.2. We applied the profile inversion technique outlined by
Park3 and conducted inversion every 0.25 ms leading up to the crash. All the
profiles exhibit an overall flatness over the core, with very steep edge gradients,
which seems to be a general trend in standard MST discharges, indicating that
the RFP plasma is predominantly edge confined. It should be noted that these
profiles are similar to those observed in RFX4 at similar densities. Approaching
the crash, the density in the core rises slightly (~5%); however, at the crash, the
density profile broadens and the overall electron content decreases. The profile
100
redistribution and global reduction in particle count are interpreted as an
overall confinement degradation in the core during the sawtooth crash.
5.1.2 Density Profiles During PPCD
For the most part, the electron density profiles in standard discharges
change very little over a sawtooth cycle. The profiles are broad, perhaps slightly
hollow, and with the exception of the crash time itself, usually about the same
amplitude throughout the sawtooth cycle. During PPCD discharges, when the
magnetic fluctuations are reduced and the confinement is improved, the density
profile can change much more dramatically. The profile grows in amplitude and
develops much more structure.
The temporal behavior of the chord-integrated electron density, for impact
parameters of 6, 36, and 43 centimeters, during a typical high-confinement
PPCD discharge is displayed in figure 5.3. This particular shot was run with low
initial density and a PPCD start time around 9 ms into the discharge. We see
that the density in the central-chord (P06) starts around 4x10+15 cm-2 and
increases two fold at the time of peak confinement. The P36 chord shows a slight
increase prior to the onset of PPCD but remains relatively stable until ~17 ms,
when it begins to rise. Finally, the outer-most chord actually drops in the early
stages of the low magnetic fluctuation period, but for the most part changes very
little.
101
55 10 15 20 2
Low ˜ b P06
P36
P430
20
40
60
80
100 Current Profile Control StartA B C D E
Time (ms)
Inte
grat
ed D
ensi
ty
(1E
+14
cm
)
-2
Figure 5.3 – The chord-integrated electron density during PPCD for impact parameters of 6, 36, and 43 cm. The A, B, …E represent time slices for which the electron density profiles are computed and displayed in figure 5.4.
The inverted electron density profiles for times outlined in the previous
figure are displayed in figure 5.4. We see that prior to the onset of PPCD, the
density profile is broad, perhaps slightly hollow, with a large edge gradient
(Trace A). At 10.5 ms (Trace B), the edge gradient becomes more steep as the
overall profile increases in amplitude. Trace C, which is at ~12.6 ms into the
discharge and ~1 ms into the period of low magnetic fluctuations, shows a
similar profile as seen with trace B but a larger amplitude. By 15 ms (Trace D),
the amplitude growth is slowing and the profile is beginning to develop a
hollowness. At peak chord-integrated density, 17.3 ms (Trace E), the increase in
the core density has slowed relative to the intermediate radii, and a clear
hollowness develops. By 17 ms, the overall electron content has increased nearly
60%.
102
Wal
l
Wal
l
AB
C
D
E
0.2
0.4
0.6
0.8
0.0-40 -20 0 20 40-60 60
Radial Position (cm)
Ele
ctro
n D
ensi
ty
(1E
+13
cm
)
-3
Figure 5.4 – The electron density profiles for discharge displayed in figure 5.3 at times (A) 8.2, (B) 10.5, (C) 12.6, (D) 15, (E) 17.3 ms. From 8 to 17 ms, the overall electron content has increased by 60%.
We interpret this profile behavior as follows. Before the onset of PPCD (~8
ms), the profile is similar to a standard discharge where density is flat over the
core region with steep edge gradients. As PPCD begins (10-12 ms), the plasma
compresses (which is a symptom of removing toroidal flux, hence forcing the
plasma deeper into reversal), wall interactions are reduced, and confinement
begins to improve. The improvement in confinement means that electrons begin
to collect in the core, once they are ionized from either hydrogen or impurities.
The electron sourcing in the core begins to fall off (~15 ms) as impurity states
burn through and the reduced wall interactions slow the replenishment of
neutral hydrogen reaching the core. By 17 ms, the electron source in the core has
been virtually depleted thereby inhibiting the density rise. However sourcing at
the edge is still quite large and a hollowness develops in the profile because,
with the increase in core confinement, the particle’s inward diffusion is greatly
slowed. This increase in core confinement is examined in more detail in the next
section.
103
5.2 Radial Particle Flux
The novel co-linear arrangement of the Hα array and FIR interferometer
is a tremendous advantage when measuring the radial particle flux (Γ ). By
simultaneously monitoring the electron density and source in the same chords,
the radial particle flux can be extracted with a single spatial inversion, thereby
greatly enhancing the accuracy of the measurement. In this section we introduce
the mathematical technique for the extraction of the radial particle flux (Section
5.2.1). Radial flux measurements for both standard and PPCD cases are
discussed in Section 5.2.2, and finally we address the issue of particle
confinement (Section 5.2.3).
5.2.1 Extracting Radial Particle Flux
The particle flux (Γ ) is defined in the electron continuity equation
(equation 5.1), where the divergence of Γ is the balancing term between the
electron source (S ) and the temporal change in the electron density (∂ne ∂ t ),
∂ne
∂ t+ ∇ • Γ = S . (5.1)
Since we measure n and e S simultaneously in each FIR chord, we can integrate
the continuity equation along each chord to arrive at
∇ • Γ( )
− L 2
L 2
∫ dz = Sdz− L 2
L 2
∫ − ∂ne ∂t dz− L 2
L 2
∫ = S dz− L 2
L 2
∫ −∂
∂ tne dz
− L 2
L 2
∫⎛
⎝ ⎜
⎞
⎠ ⎟ . (5.2)
We see from equation 5.2 that the integral of the divergence of the flux will
simply be the difference between the chord-integrated electron source (which is
proportional to the chord-integrated Hα emission) and the time derivative of the
chord-integrated FIR signal. Hence, by measuring n and e S simultaneously in
104
the same location, the change in chord-integrated density can be subtracted
directly from the chord-integrated Hα, eliminating the need to invert the density
and source profiles independently.
If we assume that both density and electron source are flux functions,
then we can invert equation 5.2 to isolate the divergence term and arrive at
∇ •Γ ψ( ) = INV αIHα ψ( ) − ∂Ine ψ( ) ∂t{ }≡ ξ ψ( ), (5.3)
where we have defined IHα and Ine to be the chord-integrated Hα emission and
electron density, ψ is the flux coordinate and ξ ψ( ) is an arbitrary function
representing the output of the inversion. Recall from chapter 3 that α ≈ 1 0.09 .
Finally, a little algebra easily isolates Γ , yielding
Γ ′ ψ ( )=
1′ ψ
ψ0
′ ψ
∫ ξ ψ( )dψ . (5.4)
5.2.2 Radial Particle Flux in Standard and PPCD Discharges
The electron density and ionization source were measured for 267
standard discharges, all at low current and moderate density. Ensembles were
conducted over sawteeth and profiles were computed over a 1 ms time window
starting from 1.5 ms and ending 0.5 ms before the crash. During this time, the
change in electron density is very small and the flux is dominantly determined
by the electron source. The electron density and ionization source profiles are
displayed in figure 5.5a-b. The corresponding radial particle flux is outlined in
figure 5.5c. In standard discharges, the radial electron flux in the core is ~ 1-
3x10+20 (m-2s-1) and gradually increases with radius to a value of ~3x10+21 (m-2 s-
1). These flux measurements at the edge are consistent with those presented by
105
Rempel1 (1991), who found the electrostatic fluctuation-induced particle
transport to be 3.1(±1.2)x10+21 (m-2 s-1). Moreover, the measured flux profile is
very similar to the modeled flux profiles of high (I/N) discharges in RFX.2
Mag
netic
Axi
sM
agne
tic A
xis
0.0 0.1 0.2 0.3 0.4 0.5
(m
s )
-3-1
Ele
ctro
n D
ensi
ty(1
0
m
)19
-3(a)
Ele
ctro
n S
ourc
e
(b)
(c)Par
ticle
Flu
x (m
s
)-2
-1
2310221021102010
Standard
PPCD
0.0
0.4
0.8
1.2
StandardPPCD
Standard
PPCD
22102110201019101810
r (m)
Mag
netic
Axi
s
1910
Figure 5.5 –The electron (a) density, (b) source, and (c) radial flux profiles for standard and enhanced confinement PPCD discharges. The gray bands represent the error in profiles as determined by a Monte Carlo perturbation technique.
The electron density, source, and radial flux for enhanced confinement
PPCD discharges are also outlined in figure 5.5a-c. The ensemble for the PPCD
case consisted of 136 discharges with the averaging window chosen to be from 6
to 8 ms after the PPCD bank firing time. During this particular experiment, the
initial density in the PPCD discharges was lowered so at the ensemble times,
the densities of the standard and PPCD cases would be similar. As a result, both
106
density profiles are roughly equivalent in overall amplitude, but the PPCD case
shows more structure, such as gradient formation in the core and a steeper
gradient in the edge.
With the PPCD electron density surreptitiously manipulated to match the
standard case, the change in particle transport manifests itself as a reduction in
the electron source required to maintain the given density profile. During PPCD,
the electron source drops more than an order of magnitude, with the core
showing the most dramatic reduction. In fact, the drop in electron source from
hydrogen in the core is so considerable that it is likely that impurity sourcing is
not negligible.
The radial particle flux shows the same substantial reductions as the
electron source (figure 5.5c). The radial particle flux by tenfold in the edge and
nearly hundredfold in the core. The drop in radial particle flux coupled with the
appearance of density gradients in the core definitively state that PPCD has a
direct effect on particle confinement in the MST core.
5.2.3 Particle Confinement Times
The reduction in the radial particle flux is a clear indication that the
confinement properties of the plasma are being enhanced. However, when
discussing issues of confinement, it is customary speak in terms of a
“confinement time”. In laymen’s terms, the particle confinement time (τ p ) is
defined as the time it would take for the plasma to escape to the wall if all
sourcing were turned off. Mathematically this is described in equation 5.5, as the
ratio of the total particle content over the particle loss rate at the plasma
boundary.
107
τ p = ne dVV∫
⎛
⎝ ⎜ ⎞
⎠ ⎟ Γ • dA
A∫
⎛
⎝ ⎜ ⎞
⎠ ⎟ (5.5)
Once again, we invoke a toroidal symmetry constraint that allows equation 5.5
to be simplified to obtain,
τ p = neψ '∫ ψ( )ψ dψ a Γ a( ). (5.6)
We have calculated the particle confinement times for the standard and
PPCD cases discussed in the last section. In accordance with equation 5.6, we
integrated the density profiles over ψ yielding radial particle contents of
~8.8x10+17 m-1 for the standard case, and ~8.5x10+17 m-1 during PPCD. The radial
flux at the plasma boundary (Γ a( )) is ~2.7x10+21 and ~3.5x10+20 m-2 s-1 for
standard and PPCD respectively. These measurements lead to the computed
particle confinement times of
τ pSTAN ≈0.6 and τ p
PPCD ≈ 4.7 ms. (5.7)
These numbers are similar to the measured energy confinement times, which
were 0.93 ms and 7.1 ms for standard and PPCD discharges respectively.
5.3 Convective Power Loss
An interesting digression that stems naturally from the radial particle
flux measurement is the estimation of the convective power loss in MST. In
Section 3.4.2 we noted that direct neutral loss could be on order several hundred
kilowatts. With the bolometric and radiated power measured in Section 4.6.4, we
found that of the ~1.7 MW of total power, the radiation could account for only
~200 kW. Having measured the radial particle flux, if we estimate the average
108
temperature of the lost particle, and assume ambipolarity, the convective power
lost can be computed in accordance with
Pp ≈2 2πRo( ) 2πa( )Γe a( )Ee ≈9.9 ×10−18 Γe m −2 s−1( )Ee eV( ). (5.8)
Since we estimate neutral loss and radiative power to account for ~300-400 kW
of the total bolometric power, the power balance requires that the convective loss
be ~1.3-1.4 MW. With the measured flux in the edge being ~3.5x10+21 (m-2s-1) in
standard discharges, the average particle energy required to balance the power
is ~35-40 eV. This temperature might be on the high side, but it is certainly
within reason, indicating that the measured flux yields a convective power loss
value that is consistent with the overall radiative power balance requirements.
5.4 Summary
The density profiles in standard low current discharges are roughly flat
across the core with steep gradients in the edge. With the exception of just after
the sawtooth crash, when the overall particle content drops, these profiles show
very little change in amplitude or shape over the sawtooth cycle. During
confinement enhanced PPCD discharges, the overall particle content has been
observed to increase as much as 60%. Moreover, in the latter stages, the
confinement improvement in the core coupled with a more edge-peaked source
profile produces hollow electron density profiles.
The radial electron fluxes were measured for both standard and PPCD
discharges. In both cases the radial flux is observed to increase with radius;
however, the overall profile amplitude during PPCD is tenfold lower than in
standard plasmas. The profile in the standard case ranges from ~1-3x10+20 (m-2s-
109
REFERENCES
1) in the core to ~3.5x10+21 (m-2s-1) at the edge, matching previous edge
measurements on MST1 and displaying a strong similarity to those obtained via
particle transport modeling on RFX.2 During PPCD, the flux drops to ~1-3x10+18
(m-2s-1) in the core rising to ~2.5x10+20 (m-2s-1) at the edge. These flux
measurements during PPCD irrefutably demonstrate an increase in overall
particle confinement and a definitive change in the transport characteristics in
the core. For the conditions examined in this thesis, the particle confinement
time is measured to increase from 0.6 ms in the standard discharges to about 5
ms for the PPCD case which are both approximately equal to the measured
energy confinement times. Finally, we note that the radial particle fluxes
measured for the standard discharges are sufficient in amplitude to
accommodate the radiated power balance, given reasonable estimates for the
average energy per particle lost.
1T. D. Rempel, C. W. Spragins, S. C. Prager, S. Assadi, D. J. Den Hartog, and S. Hokin, Physical Review Letters, 67, 1438 (1991).
2 D. Gregoratto, L. Garzotti, P. Innocente, S. Martini, A. Canton, Nuclear Fusion, 38, 1199, (1998).
3 H. Park, Plasma Physics and Controlled Fusion, 31, 2035 (1989).
4 S. Martini, V. Antoni, L. Garzotti, P. Innocente, and G. Serianni, Controlled Fusion and Plasma Physics, 18, 454 (1994).
110
6: Fluctuations and Fluctuation-Induced Particle Transport
The problem of fluctuation-induced transport in magnetically confined
plasmas involves three principal elements: identifying the origin of fluctuations,
understanding the link between these fluctuations and transport, and
developing ways to control the fluctuations that cause transport. In this chapter,
we investigate the cause of the large-scale density fluctuations over the entire
plasma cross-section, their role in particle transport, and the reduction of these
fluctuations and particle transport during current profile experiments. We have
found that the large-scale density fluctuations can be directly attributed to the
core-resonant magnetic tearing modes. In the outer region, the fluctuations
result from the advection of the equilibrium density gradient and do not cause
transport in this region. However in the core, we find these fluctuations are
compressional in nature, and could cause substantial particle transport. During
current profile control experiments (PPCD), the large-scale density fluctuations
dramatically decrease in amplitude, concurrent with similar reductions in the
measured equilibrium radial particle flux.
111
This chapter consists of three sections. In section 6.1 we report on the
character of the electron density fluctuations, addressing the amplitude,
frequency spectra, wave number content, and relationship with the core-
resonant magnetic fluctuations. We also present the inverted local fluctuation
profiles of the density fluctuations coherent with the core-resonant n=6→9
tearing modes. Section 6.2 seeks to identify the origin of the density fluctuations
by investigating the relationship between the density and radial velocity
fluctuations. Finally, in Section 6.3, we discuss the transport implications of the
density fluctuations in both standard and PPCD discharges.
6.1 Electron Density Fluctuations
The large amplitude magnetic fluctuations typically observed in the RFP
pale in comparison to the fluctuations in density. While magnetic fluctuation
amplitudes are on order of a few percent, local measurements of the density
fluctuations in the plasma edge can exceed 50% in some conditions. Generally,
large fluctuations are undesirable in fusion experiments because of their
tendency to degrade particle and energy confinement. However, in the RFP, it
appears that many of the fluctuations in density are a result of a fluctuating
magnetic field radially displacing an equilibrium density gradient, and if this is
the case, these fluctuations become much less important in the overall
confinement question. As a prelude to the particle confinement issue, in this
section we quantitatively investigate the electron density fluctuations observed
in MST, by identifying principle aspects of their character, such as amplitude,
frequency and wave number content, and relation to magnetic and electrostatic
fluctuations.
112
6.1.1 Chord-Integrated Fluctuation Amplitude
In standard, low current, moderate density discharges, the chord-
averaged fluctuation amplitudes observed by the 11 chord FIR interferometer
range from about 15% in the core to about 30-35% at the edge. Sawtooth
ensembled data show that the fluctuation amplitudes rise sharply at the crash.
Displayed in figure 6.1a-d, the chord-integrated fluctuation amplitudes
ensembled over 421 sawteeth show only slight increases in the core while the
edge rise is much more substantial. The outermost chord (P43) also shows a
residual peak after the crash indicating an increase in the particle influx from
the wall. At higher currents, when the wall interaction during the crash becomes
more violent, this secondary influx is much more dramatic.
113
0 0
20
10
15
5
30
20
10
20
10
15
5
20
10
15
5
20
10
15
5
40
30
10
20
P06 P13
P21 P28
P36 P43
a) b)
c) d)
e) f)
Fluc
tuat
ion
Am
plitu
de (%
) Fluctuation Am
plitude (%)
-2 -1 0 1 2Time (ms)
-2 -1 0 1 2Time (ms)
Figure 6.1 – The chord-averaged density fluctuation amplitudes for impact parameters (a) +6 cm, (b) +13 cm, (c) +21 cm, (d) +28 cm, (e) +36 cm, and (f) +43 cm. This data represents an average over 421 low current (200 kA), moderate density (~0.9x10+13 cm-3) standard discharges.
During PPCD, these fluctuations decrease threefold in the core, as shown
in figure 6.2. It is important to note that these plots include all fluctuations, and
that the overall fluctuation power will be dominated by the low frequency (< 10
kHz) components, such as changes in the equilibrium. We will see later that
although overall fluctuation amplitudes drop only a factor of 3, reductions at
other frequencies, namely those associated with the core-resonant tearing
modes, can be much more dramatic.
114
40
30
20
10
0-40 -20 0 20 40 60
Radial Position (cm)
Fluc
tuat
ion
A
mpl
itude
(%)
At Crash
PPCDAway
From Crash
Figure 6.2 – The chord-integrated fluctuation amplitude vs. impact parameter at and away from the sawtooth crash, and during PPCD.
6.1.2 Frequency Spectrum
The frequency spectra of the observed chord-integrated density
fluctuations are strongly dependent on impact parameter. For the center-most
chords, the density fluctuations appear small, and the power spectrum decreases
monotonically with frequency (figure 6.3). As impact parameter increases, a
large peak between 10 and 20 kHz develops in the spectrum and dominates the
fluctuation power. The edge-most chord still shows this peak, although it no
longer dominates due to a uniform increase in power over the entire frequency
spectrum. We will see that the large peak near 15 kHz results from the core-
resonant magnetic tearing modes. This peak has an m=1 nature, which explains
why it is not seen in the central chord. The high frequency power, observed in
the edge chord, results from smaller-scale fluctuations with toroidal mode
numbers greater than 20.
115
r a =.11
r a =.54
r a =.83
0 10 20 30 400
1
2
3
4
5
Frequency (kHz)
Pow
er (a
u)
50
Figure 6.3 – The chord-averaged density fluctuation power spectra for impact parameters of 0.11, 0.54, and 0.83.
6.1.3 Wave Number Content
The novel design of the multi-chord FIR system allows the resolution of a
density fluctuation’s poloidal and toroidal mode numbers. By correlating
between radially displaced chords, the poloidal structure (m spectrum) of the
fluctuation can be extracted, while correlation between two toroidally displaced
chords provides a toroidal mode number (n) spectrum.
Fluctuations observed in inboard and outboard chords are highly
coherent. The coherence amplitude, figure 6.4a, shows especially high coherence
for the 15 kHz peak and demonstrates that even the small-scale, high frequency
fluctuations are coherent well above the baseline. The associated phase, figure
6.4b, indicates that the coherent fluctuations below 10 kHz are dominantly m=0,
while the fluctuations between 10 and 20 kHz and >30 kHz exhibit an m=1
character.
116
0 20 40 60 80 100
0.0
-1.0
1.0
0.6
0.2
m=0
m=1 m=1
Frequency (kHz)
Coh
eren
ce
Am
plitu
dePh
ase
(π ra
d)
(a)
(b)
Baseline
Figure 6.4 – Coherence (a) amplitude and (b) phase between r/a = 0.62 inboard and r/a = 0.83 outboard chords. Both chords are at the same toroidal angle. Note that the fluctuations below 10 kHz have an m=0 nature while those above 10 kHz are m=1 like.
The average n-spectrum (figure 6.5), obtained from correlations between
two toroidally displaced chords, shows that the fluctuation power below 10 kHz
results from the n=1→4, while fluctuations with n>30 are the principle
components above 30 kHz. The density fluctuations between 10 and 30 kHz are
a product of n=5 to 20, where most of the power is from n=6 to 10.
117
.14
.10
.05
.00
k (cm )
φ-1
Toro
idal
Mod
e
Num
ber (
n)
0 20 40 60 80Frequency (kHz)
r/a = 0.580
10
20
30
40
-10
Figure 6.5 – The average toroidal mode number and wave number spectrum for impact parameter r/a=0.58. Here the average mode number is defined as the average of the measured n-spectrum at a given frequency.
In addition to the rise in average n, the n-spectrum broadens considerably
at higher frequencies (figure 6.6). At 3 kHz, the n-spectrum is very peaked
around the n=0 with a ΔnFWHM ~ 2 (figure 6.6a). However, at 35 kHz, the
spectrum centered near n=30 and is much broader, with ΔnFWHM ~ 39 (figure
6.6.c). The n-spectrum at 18 kHz is peaked at n=6 with a width of ΔnFWHM ~ 12
(figure 6.6b) and is consistent with the expectation that the density fluctuations
in the low frequency range result from core-resonant magnetic tearing modes.
118
Pow
er (a
u)
80
60
40
20
0-20 0-40 20 40
FWHMΔn ≈ 2npeak≈ 0
Toriodal Mode Number (n)
(a)
Pow
er (a
u)0.6
0.4
0.2
0.0-20 0 20 40-40 60
FWHMΔn ≈12npeak≈ 6
Toriodal Mode Number (n)
(b)P
ower
(1e-
2 au
) 1.5
1.0
0.5
0.0 0 20 40 60-20 80
FWHMΔn ≈39npeak≈ 30
Toriodal Mode Number (n)
(c)
Figure 6.6 – The toroidal mode number (n) spectrum for (a) 3 kHz, (b) 18 kHz, and (c) ~35 kHz at an impact parameter of r/a = 0.56.
With the wave number information, we can characterize the density
fluctuations as follows. The density fluctuations below 10 kHz are low n
(n~1→4), m=0 perturbations that are resonant at the reversal surface, while the
fluctuations between 10 and 30 kHz, result from m=1, n=5→15 core-resonant
tearing modes. The high frequency fluctuations (> 30 kHz), that are coherent,
are also m=1, but result from n>25 and are resonant just inside the reversal
surface.
119
6.1.4 Correlation Between Density and Magnetic Fluctuations
To determine the spatial harmonic content of the density fluctuations
which arise from the global magnetic fluctuations, we correlate the chord-
integrated FIR measurements with the Fourier harmonics of the magnetic
fluctuations that were obtained from the 64-position toroidal coil array. The
density fluctuation power that is coherent with the m=1, n=5→15 core-resonant
tearing modes is displayed in figure 6.7a-c, along with the total and incoherent
fluctuation power for impact parameters of r/a = 0.11, 0.54, and 0.83. We see
that the center-most chord is poorly coherent with the m=1 magnetic
fluctuations (figure 6.7a), as would be expected since the central chords are
relatively insensitive to m = odd perturbations. At larger impact parameters,
virtually all of the power between 10 and 20 kHz is coherent with the n=5→15
modes (figure 6.7b). In the plasma edge, the density fluctuations are less
coherent with the core-resonant tearing modes as the relative contribution from
smaller scale, higher frequency magnetic and electrostatic fluctuations
increases.
120
0.00.51.01.52.0
05
1015
20
05
10
1520
10 15 20 25 30Frequency (kHz)
Fluc
tuat
ion
Pow
er (a
.u.)
(a)
(b)
(c)
CoherentIncoherent
Total
CoherentIncoherent
Total
CoherentIncoherent
Total
Figure 6.7 – The total, coherent, and incoherent power between the chord-integrated density fluctuations and the m=1, n=5→15 core-resonant magnetic tearing modes at impact parameters (a) 0.11, (b) 0.54, and (c) 0.83.
6.1.5 Local Density Fluctuation Profiles
The radial density fluctuation profile of a particular harmonic can be
obtained by inverting the correlated component of the chord-averaged
measurements. For an m=0 mode, the inversion proceeds as for the equilibrium
density, which invokes up/down symmetry. For an m=1 mode we perform the
inversion as follows. Let the total density be described as
n r( ) = no r( ) + ˜ n r( )cos ωt + mθ + nφ + δ r( )[ ], (6.1)
121
where φ and θ are the toroidal and poloidal angles, and ˜ n r( ) and δ r( ) are the
radial functions of the amplitude and phase of the density fluctuation. A chord-
integrated measurement of this perturbation can be written as
I x( ) = Io x( )+ ˜ I x( ) , (6.2)
where
˜ I x( ) = ˜ n r( )cos ωt + mθ + nφ + δ r( )[ ]
− L 2
L 2
∫ dz . (6.3)
Here, x represents the impact parameter of the chord, L is the chord’s path
length through the plasma, and is the vertical coordinate. Equation 6.3 can be
simplified to
z
˜ I x( ) = ˜ I amp x( )cos ωt + nφ + Δ x( )[ ], (6.4)
where we have defined
˜ I amp x( )sin Δ x( )[ ]=
˜ n r( )sin δ r( )[ ]r2 − x2
x
a
∫ dr (6.5)
and
˜ I amp x( )cos Δ x( )[ ]=
˜ n r( )cos δ r( )[ ]r2 − x2
x
a
∫ dr . (6.6)
Equations 6.5 and 6.6 can be Abel1 inverted to arrive at
˜ n r( )cos δ r( )[ ]= −
rπ
ddx
˜ I amp x( )cos Δ x( )[ ]2x
⎛
⎝ ⎜
⎞
⎠ ⎟
r
a
∫dx
r2 − x2 (6.7)
and
˜ n r( )sin δ r( )[ ]= −
rπ
ddx
˜ I amp x( )sin Δ x( )[ ]2 x
⎛
⎝ ⎜
⎞
⎠ ⎟
r
a
∫dx
r2 − x2. (6.8)
122
The parameters and ˜ I amp x( ) Δ of a specific m and n structure are isolated
by correlating the fluctuating part of the integrated density with a Fourier
component of the magnetic fluctuations ˜ I ˜ b m, n . Having obtained the products
and ˜ I amp x( )sin Δ x( )[ ] ˜ I amp x( )cos Δ x( )[ ] for each chord, an Abel inversion is
conducted, and the radial functions ˜ n r( ) and δ r( ) are easily extracted.
The local radial density fluctuation profiles [ ˜ n r( )] have been measured in
both standard and improved confinement PPCD discharges. Displayed in figure
6.8a-d, the fluctuation profiles in standard discharges for the m=1, n=6→9
helicities are broad, with amplitudes ~1.0%. An interesting feature is that as
toroidal mode number increases, the peak in the density fluctuation profile
moves outward in radius. This is consistent with the expectation that, for a
constant density gradient, the density fluctuation arising from a magnetic
tearing mode should be largest near its resonant surface.
123
(10
m
)-3
17E
lect
ron
Den
sity
0.5
1.0
1.5
0.4
0.8
1.2
0.0
0.5
1.0
1.5
0.4
0.8
1.2
0.0 0.2 0.4 0.6 0.8 1.0r a
PPCD
PPCD
PPCD
PPCD
(c)
(d)
(b)
(a)n=6
n=7
n=8
n=9Standard
Standard
Standard
Standard
Figure 6.8 – The radial density fluctuation profiles for m=1, (a) n=6, (b) n=7, (c) n=8, and (d) n=9 helicities for standard and PPCD discharges. The gray bands are error bars from the Abel inversion.
During enhanced confinement PPCD discharges, when both the magnetic
tearing mode and chord-integrated density fluctuations are reduced, drops
more than an order of magnitude and becomes more edge peaked (figure 6.8a-d).
Local amplitudes range from ~0.05% in the core to about 0.1% near the edge.
The location of the peak is near the toroidal field reversal surface (r/a~0.85) and
seems to be independent of toroidal mode number (n), indicating that the very
steep edge density, formed during PPCD, is playing a strong role in these
fluctuations.
˜ n r( )
124
6.1 Origin of Density Fluctuations
We have established above that the dominant density fluctuations are
associated with core-resonant tearing modes (with the exception of the small-
scale fluctuations in the extreme edge). In this section, we report measurements
of the impurity ion flow velocity, which, when combined with measurements of
density and magnetic field fluctuation, allow us to deduce whether the flow is
compressional or advective, and whether it is consistent with the predictions of
magnetohydrodynamics (MHD). We begin by examining the relationship
between the density and velocity fluctuations via the electron continuity
equation (Section 6.2.1). Section 6.2.2, exhibits the results of the velocity
fluctuation measurements and the subsequent inferences about the cause of the
density fluctuations are presented in Section 6.2.3.
6.2.1 The Electron Continuity Equation
The relationship between the electron density ( ) and the radial velocity
(
˜ n
˜ v r ) fluctuations is dictated by the electron continuity equation,
∂ne
∂ t+ ∇ • ne
r v ( )= S . (6.9)
Expanding and ˜ n ˜ v r into their equilibrium and fluctuating components as
f = f o + ˜ f ⇒ f o r( )+ ˜ f 1 r( )ei k •r − ωt( ), (6.10)
Equation 6.9 becomes
˜ n =i
ω − k • v ( )˜ v r • ∇rno +
no
r∂∂r
r˜ v r( )⎡ ⎣
⎤ ⎦
−no k • ˜ v ( )ω − k • v ( ). (6.11)
To arrive at equation 6.11, we have made the usual assumption of neglecting the
zeroth order compressibility term ( ∇ •rv o = 0 ), but have kept the first order term
125
( ∇ • ˜ v ≠ 0 ). We have also neglected the fluctuating source term ( ). The
viability of the latter assumption is supported by the correlated product of
H
˜ S → 0
α and edge magnetic coil measurements ( ˜ S ̃ b θ ) which shows no significant
coherence at the core tearing mode frequencies. The final caveat is that the
nonlinear terms are assumed to be small.
From equation 6.11 we see that density fluctuations can arise from three
processes: advection of the equilibrium gradient (the first term), radial
compression (the second term), and compression within the magnetic surface
(the third term). Another important feature of equation 6.11 is that depending
on which process is governing the electron density fluctuations, the phase
between and ˜ n ˜ v r can be different. For example, if advection or radial
compression is the cause of the density fluctuation, then and ˜ n ˜ v r must be 90
degrees out of phase ( i → π 2 ). However if measurements of and ˜ n ˜ v r indicate a
phase difference other than π 2 , then must arise from compression within the
magnetic surface. By investigating the phase between and
˜ n
˜ n ˜ v r we can identify
which terms in the continuity equation are contributing to the density
fluctuations.
6.2.2 Measurements of the Radial Velocity Fluctuations
For the measurement of ion radial flow fluctuations, a custom designed
Doppler spectrometer, with high light throughput was employed. Named the Ion
Dynamics Spectrometer2, ,3 4(IDS), this diagnostic is capable of measuring chord-
integrated ion temperature and flow fluctuations with a time resolution of ~10
μs. The IDS offers three collection geometries, each designed to isolate a specific
126
component of ion flow.* To resolve the radial component of the flow fluctuations,
we used the 4.5 inch diameter radial viewport which was located at 210 degrees
toroidally, 22.5 degrees poloidally. This placement was just ~45 degrees away
(toroidally) from the FIR interferometer.
Although a chord-averaging diagnostic, the radial localization of the IDS
measurement can be enhanced by monitoring different impurities and charge
states. Typically He II (He1+) and C V (C4+) are the impurities of choice. The
density profiles for these states, as predicted from the Multi-Ion Species
Transport Code (MIST)5 are displayed in figure 6.9. The He II is most abundant
at the edge, above r/a~0.6, which provides excellent enhancement of the radial
velocity fluctuations in this region. C V, which exhibits a much broader profile, is
dominant in the plasma core. With this in mind, observations of C V will
measure the average of the flow fluctuations over the plasma interior
(0.0<r/a<~0.8).
He IIC V
0.0 0.2 0.4 0.6 0.8 1.00
2
45
3
1
r/a
Den
sity
(a.u
.)
Figure 6.9 – The state density profiles for He II and C V as predicted by MIST for low current, moderate density discharges.
*For a detailed description of the IDS measurement capabilities, the reader is once again referred to J. T. Chapman’s Ph.D. thesis.
127
To interpret the IDS results, we assume that the electron radial flow
velocity equals the impurity ion flow velocity, as occurs if the flow arises from a
fluctuating rE ×
rB drift. This assumption follows from the MHD modeling of the
RFP and is consistent with the probe measurements conducted at the extreme
plasma edge.6,7
The edge radial velocity fluctuations, measured via He II emission, are
coherent with the core-resonant tearing modes (figure 6.10a). The phase
difference between ˜ v r and ˜ b r (measured at the plasma boundary) is ~ 0 radians
(figure 6.10b), in agreement with linear ideal MHD which predicts
˜ v r ∝
vk •
vB o( )
ω −v k • v v o( )
˜ b r . (6.12)
128
Phase Unresolvable
0 10 20 30 40Frequency (kHz)
0 10 20 30 40Frequency (kHz)
0.0
1.0
-1.0
Pha
se (π
radi
ans)
0.25
0.15
0.05C
oher
ence
Am
plitu
de
Baseline
a)
b)
50
50
˜
Figure 6.10 – The coherence (a) amplitude and (b) phase of the correlated product between the radial velocity fluctuations (v r ), measured by He II, and the radial magnetic field fluctuation (b ˜ r ) for the m=1, n=6 helicity.
Information on the core ˜ v r is obtained from the C V emission. We find
that ˜ v r , averaged over the interior region is small and shows no measurable
coherence with the core-resonant tearing modes (figure 6.11). Since the
measurement with He II, described earlier, established the presence of the
radial velocity fluctuations in the outer portion of the plasma, the nearly null
result of the chord-averaged C V signal implies a radial velocity fluctuation
whose phase flips sign in the core. This π phase shift is consistent with the ideal
MHD expectation that ˜ v r , due to a given tearing mode, reverses across the
129
mode’s resonant surface. From equation 6.12, this effect arises from the k • Bo
term, which flips sign across the rational surface.
Although consistent with the linear ideal MHD interpretation, predictions
of the radial velocity fluctuation profile made by DEBS,8 a nonlinear MHD
simulation code, does not predict the phase flip across a resonant surface.
Historically, DEBS has been accurate at predicting the radial magnetic
fluctuation profiles and this discrepancy with the experimental observations
remains a mystery that warrants deeper exploration.
0 10 20 30 40Frequency (kHz)
Coh
eren
ce A
mpl
itude
Baseline
50
0.5
0.3
0.1
0.2
0.4
0.0
No coherence
Figure 6.11 – The coherence amplitude of the correlated product between the radial velocity fluctuations (v ˜ r ), measured by C V, and the radial magnetic field fluctuation (b ˜ r ) for the m=1, n=6 helicity. Note there is no significant coherence.
The phase flip of ˜ v r across a tearing mode resonant surface has been
observed for the m=0 modes. Local measurements of the impurity ion radial
velocity fluctuations, conducted with the Ion Dynamics Spectroscopic Probe9
(IDSP), have verified that the phase of ˜ v r resulting from the low n, m=0 tearing
modes does indeed flip sign across the reversal surface.10
130
6.2.3 Nature of Density Fluctuations
The velocity fluctuations of the He II ions, measured by the IDS, are also
coherent with the large-scale density fluctuations seen by the FIR
interferometer. The coherence amplitude, at 18 kHz, between the radial velocity
fluctuations of He II and the chord-integrated density fluctuations obtained from
the FIR interferometer is displayed in figure 6.12a. The peak coherence ranges
from ~0.10 at the edge rising to ~0.28 near r/a~0.6 before falling again in the
core. The phase is outlined in figure 6.12b, and indicates that and ˜ n ˜ v r in the
edge are π/2 out of phase. This phase shift, coupled with the information in
equation 6.12, indicates that the large-scale density fluctuations in the edge
result from either advection of the equilibrium density gradient ( ˜ v r∇r no ) or
radial compression of the plasma. Based on the large equilibrium density
gradient in the outer region (gradient scale length ~ 0.2a), and the expectation
that the radial gradient in ˜ v r for a tearing mode is slowly varying away from its
rational surface, we conjecture that the advective term dominates in the edge.
Hence, the large-scale density fluctuations appearing in the edge are merely a
result of an advecting equilibrium density gradient caused by a fluctuating
magnetic field as in an ideal MHD plasma.
In the core, where the equilibrium gradient vanishes, the large-scale
density fluctuations are compressional. Furthermore, the phase of the density
fluctuations is shifted by π/2 relative to the edge (figure 6.12b). This phase shift
coupled with the constraints on ˜ v r from the C V measurement indicates that the
density and radial velocity fluctuations are in phase in the core. Therefore, the
large-scale coherent density fluctuations in the plasma core must result from the
compression described by the third term of equation 6.11.
131
Pha
se (π
rad)
1.0
0.5
0.0
Baseline
a)0.30
0.20
0.10
0.00
Am
plitu
de
-60 -40 -20 0 20 40 60
-60 -40 -20 0 20 40 60
R-R (cm)o
R-R (cm)o
b)
Figure 6.12 – The coherence (a) amplitude and (b) phase between the radial velocity fluctuations of He II and the chord-integrated density fluctuations obtained from the FIR interferometer.
6.2 Fluctuation-Induced Particle Transport
The phase relation between density and radial velocity fluctuations also
provides key information on the fluctuation-induced particle transport. The
fluctuation-induced radial particle flux is
Γr = ˜ n ̃ v r = γ ˜ n ˜ v r cos δnv( ), (6.13)
where γ is the coherence amplitude and δ nv is the phase between and ˜ n ˜ v r . It is
important to note that this term does not include all mechanisms for radial
transport. For example, the contribution from ˜ J || ˜ b r is not included; however, in
the edge, this term is measured to be small.
132
Since we have established that δ nv ~ π 2 in the outer region of the plasma
(r/a > 0.6), the fluctuation-induced particle flux from the dominant core-resonant
modes is measured to be small. Therefore, although the core-resonant modes are
relatively large in the edge, they do not cause particle transport. This result is
consistent with the expectation that such modes do not destroy edge magnetic
surfaces.11,12 Such is not the case in the plasma core, where the density
fluctuations exhibit a �/2 phase shift relative to the edge. With δ nv ~ 0 , and ˜ n ˜ v r
couple efficiently to drive radial particle loss. Although the magnitude of ˜ v r in
the core is unknown, estimates suggest that the ˜ n ̃ v r could be enough to account
for all the particle transport inside r/a < 0.40.
A remarkable feature that appears during improved confinement PPCD
discharges is that the radial phase shift in vanishes (figure 6.13b), suggesting
that and
˜ n
˜ n ˜ v r remain out of phase much deeper into the core. This change in
phase, coupled with the order of magnitude reduction of (Section 6.1.5),
indicate that the fluctuation-induced transport due to core-resonant tearing
modes is greatly reduced in the core.
˜ n
133
-40 -20 0 20 40 60
Pha
se (π
rad)
Radial Position (cm)
1.5
-0.5
0.0
0.5
1.0
1.5
-0.5
0.0
0.5
1.0
Pha
se (π
rad)
a)
b)
-40 -20 0 20 40 60Radial Position (cm)
Figure 6.13 – The phase shift between the chord-integrated density and edge radial velocity fluctuations for (a) standard and (b) PPCD discharges. During PPCD the density fluctuations change phase resulting in the vanishing π/2 shift in the core.
6.3 Summary
In summary, simultaneous measurements of the fluctuating density,
radial plasma velocity, and magnetic field elucidate the cause of the density
fluctuations and particle transport in the RFP. We find that most of the density
fluctuations result from core-resonant tearing modes. Furthermore, these
fluctuations are advective in the edge (consistent with ideal MHD predictions)
and compressional in the core. Direct measurements of the fluctuation-induced
particle flux, in the outer region of the plasma reveals that the core-resonant
134
REFERENCES
tearing modes do not cause transport at the edge. However, inferences from
chordal measurements of the radial velocity indicate that these modes do cause
transport in the core. During PPCD discharges, in which auxiliary current drive
is applied to reduce transport, the radial particle flux decreases dramatically
(Chapter 5). Furthermore, the density fluctuations decrease, and the region of
vanishing fluctuation-induced particle flux extends deeper into the core. An
important caveat is that the chord-averaged nature of the density and velocity
fluctuation measurement limits the spatial resolution, and more localized
fluctuations which may drive transport are not addressed here.
1 W. M. Barr, Journal of Optical Society of America 52, 885 (1962).
2 D. J. Den Hartog and R. J. Fonck, Review of Scientific Instruments, 65, 3238, (1994).
3 J. T. Chapman and D. J. Den Hartog, Review of Scientific Instruments, 68, 285, (1996).
4 J. T. Chapman, Ph.D. Thesis (1998).
5 R. A. Hulse, Nuclear Technology/Fusion 3, 259 (1983).
6 H. Ji, A. F. Almagri, S. C. Prager, and J. S. Sarff, Physical Review Letters 72, 668 (1994).
7 P. W. Fontana, G. Fiksel, Bulletin of American Physical Society 44, 10 November (1999).
8 C .R. Sovinec, Ph.D. Thesis (1995).
9 G. Fiksel, D. J. Den Hartog, and P. W. Fontana, Review of Scientific Instruments, 69, 2024 (1998).
10 P. W. Fontana, Ph.D. Thesis (1999).
135
11 M. R. Stoneking, S. A. Hokin, S. C. Prager, G. Fiksel, H. Ji., and D. J. Den Hartog, Physical Review Letters, 73, 549 (1994).
12 G. Fiksel, S. C. Prager, W. Shen, and M. R. Stoneking, Physical Review Letters, 72, 1028 (1994).
136
7: Conclusions
Diagnostic Developments
We have developed a high-speed multi-chord far-infrared (FIR) laser interferometer
to measure equilibrium and fluctuating electron density. A principal advancement of this
system has been the implementation of a digital phase extraction technique, which has
enhanced the time response and phase resolution, allowing measurement of the density
fluctuations associated with the core-resonant tearing modes. To measure the equilibrium
electron source profile from ionization of neutral hydrogen we have designed, constructed,
and implemented a multi-chord Hα array. Its colinear arrangement with the FIR
interferometer allows the extraction of the total particle flux with a single inversion, thereby
enhancing the accuracy of the measurement. Finally, we have developed an impurity
monitoring diagnostic for the purpose of estimating the electron source from high Z
impurities. Named the ROSS filtered spectrometer, this diode spectrometer is capable of
making absolute measurements of line emission from the highly ionized states of carbon,
oxygen, and aluminum.
137
Primary Physics Results
This work reports three primary physics results. First, through measurements of the
radial electron flux profile, we have determined that pulsed poloidal current drive
experiments improve confinement in the reversed-field pinch core. In standard discharges the
total radial electron flux profile is measured to be about 1-3x10+20 (m-2s-1) in the core;
however, when PPCD is enabled, the radial particle flux in the core drops almost
hundredfold, strongly indicating a confinement enhancement in the core.
Second, we have shown that the origin of many large amplitude density fluctuations
is directly attributed to the core-resonant tearing modes, and that these fluctuations are
advective in the RFP edge but are compressional in the core (subject to the nonlinear terms
being small). The correlation phase between density and radial velocity fluctuations in the
RFP edge is measured to be ~π/2 indicating the density fluctuation is formed from an
advecting equilibrium density gradient or the radial compression of the plasma. With the
steep edge density gradient and the radial velocity fluctuation amplitude slowly varying away
from the resonant surface, the advective term dominates. In the core, we deduce that the
density and velocity fluctuations are in phase indicating the density fluctuations result from
compression within the magnetic surface, provided the nonlinear terms are small.
Finally, we have demonstrated that the density fluctuations associated with the core-
resonant tearing modes do not cause transport in the RFP edge, but can be responsible for
transport in the core during standard discharges; however, when PPCD is employed to reduce
the core-resonant magnetic fluctuations, transport from these modes drop, and confinement
in the core is improved. Since the density and velocity fluctuations are ~π/2 out of phase in
the edge, these fluctuations do not couple to cause transport; however in the core, where they
are in phase, these density fluctuations can cause transport. During PPCD, the relative phase
138
between the density and radial velocity fluctuations are observed to change to ~π/2,
indicating the fluctuations are no longer coupling to produce radial particle transport.
Secondary Physics Results
On the path to characterizing the electron density and source behavior for the
measurements presented above, a number of secondary physics results have been realized.
We have found that the neutral concentration in the core for standard low current discharges
is quite high (~1-2x10+10 cm-3). Because of this large concentration the dominant electron
source is from the ionization of neutral hydrogen, and charge exchange recombination is the
dominant recombination process for high charge state impurities.
We have measured the overall impurity concentration to be less than one percent
where carbon, aluminum, oxygen, and nitrogen concentrations are all roughly equivalent.
While the overall concentrations can vary greatly (order of magnitude) depending on
machine conditioning, the impurity fraction does not appear to change appreciably during
PPCD. With respect to radiative losses, comparison of the bolometric versus radiated power
indicates that nearly 85% of the dissipated power results from energy convection via direct
particle loss.
Future Work
The most essential avenue to pursue in the future is to work on enhancing the
localization of the MST measurement capability. A successful CHERS diagnostic could, in
principle, provide highly localized measurements of radial velocity fluctuations. With the
localized density fluctuations obtained via the FIR, the fluctuation-induced particle transport
from the core-resonant tearing modes can be quantitatively measured. A localized measure of
equilibrium and fluctuating plasma potential (φ) is essential for mapping out the radial
139
electric field. This measurement, combined with the profile capabilities of the Thomson
Scattering system and the FIR interferometer, could be used to examine the validity of the
idea of stochastic particle transport presented by Harvey (1982). While we have been able to
qualitatively assess the existence of fluctuation-induced transport in the RFP core;
quantitative measurements await advancement in localization.
140
A: Polarimetry/Interferometry Discussion
A.1 Introduction
Although this work has not discussed in any detail the FIR Polarimetry
upgrade, much effort in this area has been conducted. The MST polarimeter
system was first constructed and employed on the Microwave Tokamak
Experiment (MTX) by Rice.1,2 Later, the equipment was relocated to Texas
where it proved very successful for poloidal field measurements on the TEXT-U3
tokamak. In the summer of 1996, with the cooperation of the UCLA Plasma
Diagnostics Group, plans were formalized to install this system on the MST.
The principal components of the system on MST were to be exactly the
same as those employed on TEXT-U, with one notable exception. The RFP
requirement for a close conducting shell for ideal MHD stability necessitated the
use of small access holes for the FIR beams. This constraint required the
substitution of individual wire meshes for the large parabolic mirror that was
used previously. These wire meshes remain the primary impediment towards
achieving accurate polarimetry data on MST.
141
The principal problem with the wire grid meshes arises from their
asymmetric reflectivity properties. For the polarimeter to function properly, it is
critical that the polarization of the FIR beam be maintained throughout the
system. However, with each mesh changing the beam polarization, accurately
measuring the polarization change from the plasma becomes very difficult.
A.2 Derivation of Measured Signal Power
In an effort to better understand how the wire grid meshes affect the
electron density and poloidal magnetic field measurements, we present a
complete derivation for the interferometer and polarization phases as measured
by the MST FIR system. We employ the Jones matrix representation, where
each 2 x 2 matrix corresponds to the effect of one optical element in the FIR
system.
We start with equation A.1 on the following page, which describes the
modification of the electric field vector as it propagates through the FIR system.
This derivation assumes only a dual mesh system, but is easily generalized to n
meshes. Starting from the bottom right: we have; the initial electric field ESxo
ESyo
⎡ ⎣
⎤ ⎦
out of the laser, the wire polarizer at the laser output , and the quarter-
wave plate
1 00 0
⎡ ⎣
⎤ ⎦
cos 2 φ + isin 2 φ sinφ cosφ 1 − i( )sin φ cosφ 1− i( ) cos2 φ + i sin2 φ
⎡ ⎣
⎤ ⎦ , where φ is the angle between the principal
axis of the quartz and the electric field vector. Continuing on, we have the half-
wave plate cos 2ωpt( ) sin 2ω
pt( )
sin 2ω p t( ) −cos 2ω p t( )⎡
⎣ ⎢
⎤
⎦ ⎥ , rotating at an angular velocity ωp, and the two
meshes (transmission through the first and reflection off the second)
. Finally we have the plasma imparted Faraday rotation RTE2 00 RTM2
⎡ ⎣
⎤ ⎦
TTE1 00 TTM1
⎡ ⎣
⎤ ⎦
142cosδ sinδ−sinδ cosδ
⎡ ⎣
⎤ ⎦ , and the selection polarizer , which isolates the x (toroidal)
component of the electric field.
1 00 0
⎡ ⎣
⎤ ⎦
ESx
ESy
⎡ ⎣ ⎢
⎤ ⎦ ⎥ =
1 00 0
⎡ ⎣
⎤ ⎦
cosδ sinδ−sinδ cosδ
⎡ ⎣
⎤ ⎦
RTE2 00 RTM2
⎡ ⎣
⎤ ⎦
×
TTE1 00 TTM1
⎡ ⎣
⎤ ⎦
cos 2ω pt( ) sin 2ω pt( )sin 2ω pt( ) −cos 2ωpt( )
⎡
⎣ ⎢
⎤
⎦ ⎥ ×
cos2 φ + isin2 φ sinφ cosφ 1 − i( )sinφ cosφ 1 − i( ) cos2 φ + i sin2 φ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
1 00 0
⎡ ⎣
⎤ ⎦
ESxo
ESyo
⎡ ⎣ ⎢
⎤ ⎦ ⎥
A.1
We begin the simplification process by eliminating the polarizer terms and
combining the mesh transmission and reflection matricies.
ESx
ESy
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=cosδ sinδ
0 0
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
RTE2TTE1( ) 00 RTM2TTM1( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
×
cos 2ω pt( ) sin 2ω pt( )sin 2ω pt( ) −cos 2ω pt( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
cos2 φ + i sin2 φ 0sinφ cosφ 1 − i( ) 0
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
ESxo
ESyo
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
A.2
Next we combine the mesh and plasma rotation to obtain,
ESx
ESy
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=RTE2TTE1( )cosδ RTM2TTM1( )sinδ
0 0
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
×
cos 2ω pt( ) sin 2ω pt( )sin 2ω pt( ) −cos 2ω pt( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
cos2 φ + i sin2 φ 0sinφ cosφ 1 − i( ) 0
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
ESxo
ESyo
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
. A.3
By defining an ellipticity, ε, the quarter-wave plate / laser output combination
takes the form ESo
cos ω ct + φ plasma(ε sin ω c t + φ plasma(
⎡
⎣ ⎢ ⎢
⎤
⎦
))⎥ ⎥ , and equation A.3 becomes
143ESx
ESy
⎡
⎣ ⎢ ⎤
⎦ ⎥ = ESoRTE2TTE1( )cosδ RTM2TTM1( )sin δ
0 0⎡
⎣ ⎢ ⎤
⎦ ⎥ ×
cos 2ω pt( ) sin 2ωpt( )sin 2ω pt( ) −cos 2ω pt( )
⎡
⎣ ⎢
⎤
⎦ ⎥
cos ω ct + φplasma( )ε sin ωct + φ plasma( )
⎡
⎣ ⎢
⎤
⎦ ⎥
. A.4
If we define a mesh distortion angle, θ, and an amplitude Aθ as shown in A.5,
θ = tan −1 RTM2TTM1
RTE2TTE1
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥ and Aθ = RTE2TTE1( )2
+ RTM2TTM1( )2 A.5
then we get,
ESx
ESy
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
= ESoAθ
cosθ cosδ sinθ sinδ0 0
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
×
cos 2ω pt( ) sin 2ω pt( )sin 2ω pt( ) −cos 2ω pt( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
cos ωct + φplasma( )ε sin ωct + φplasma( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
. A.6
Here we have used ωc to represent the far-infrared laser frequency (≅ 700 GHz
for λ≅ 432.6 μm) and φ plasma to be the phase shift in the signal beam from the
electrons present in the plasma, i.e. the interferometry phase. Using the same
trigonometry trick shown above, we define the parameters
ξ = tan−1 sinθcosθ
sinδcosδ
⎡ ⎣ ⎢
⎤ ⎦ ⎥ = tan −1 tanθ tanδ[ ] A.7
and
Aξ = cosθ cosδ( )2 + sinθ sinδ( )2 = 1+ cos 2θ( )cos 2δ( ) , A.8
we arrive at equation A.9.
144
ESx
ESy
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=ESoAθ Aξcos 2ωpt − ξ( ) sin 2ω pt − ξ( )
0 0
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
cos ωct +φplasma( )ε sin ωct + φplasma( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥ A.9
Continuing through the multiplication we have,
ESig = ESo Aθ Aξ
cos 2ω pt − ξ( )cos ωct +φplasma( )+
ε sin 2ω pt − ξ( )sin ωct + φplasma( )
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
, A.10
and once more we employ our trick to define
ψ S = tan −1 ε sin 2ω pt − ξ( )cos 2ω pt − ξ( )
⎡
⎣ ⎢
⎤
⎦ ⎥ = tan −1 ε tan 2ω pt − ξ([ )], A.11
with the corresponding amplitude
Aψ S= cos2 2ω pt − ξ( )− ε2 sin2 2ω pt − ξ( )
=1+ ε 2( )
21+
1 − ε2( )1 + ε2( )cos 4ω pt − 2ξ( )
. A.12
This easy substitution yields a very simple result for the electric field amplitude
of the signal leg incident on the corner cube diode.
ESig = ESo Aθ Aξ Aψ Scos ωct + φ plasma − ψS( ) A.13
The electric field vector incident on the corner cube diodes from the local
oscillator (LO) beam is much simpler. Since the beam polarization is linear, it
can be written without derivation and is shown in equation A.14. The LO electric
field consists of an arbitrary amplitude that is dependent on the beam
propagation efficiency throughout the system, and a sinusoidal term that
145
oscillates about the laser light frequency (ωc ) that is shifted by the interference
frequency (ωIF ).
ELO = ELOo cos ωct +ω IFt( ) A.14
The FIR power measured in the diodes is going to be the square of the
vector sum of incident electric field components presented in equations A.13 and
A.14.
PSig =
E Sig +
E LO( )2
= ESig2 + ELO
2 + 2ESigELO A.15
The preamplifier on the mixer output has a bandpass filter that ranges from
about 250 kHz to about 2.5 MHz (See Chapter 2, Section 2.2.5). The effect of this
filtering on the measured mixer power is examined by expanding each term on
the right side of equation A.15. Looking at the first term, we have
ESig2 = ESo
2 Aθ2 Aξ
2Aψ S
2 cos2 ωct + φplasma −ψ S( )
=ESo
2 Aθ2 Aξ
2Aψ S
2
21 − cos 2ωct + 2φplasma − 2ψ S( )[ ]
=ESo
2 Aθ2 Aξ
2Aψ S
2
2−
ESo2 Aθ
2 Aξ2 AψS
2
2cos 2ωct + 2 φplasma −ψ S( )[ ]
= after filtering → 0
. A.16
146
The second term has a similar result and is derived in equation A.17.
ELO2 = ELOo
2 cos2 ωct +ω IFt( )
= ELOo2
21 + cos 2ωct + 2ω IF t( )[ ]
= ELOo2
2+ ELOo
2
2cos 2ωct + 2ω IF t( )
= after preamp filtering → 0
A.17
The final term is a cross term, and consists primarily of two harmonics, one at
the IF frequency, one a twice the laser frequency. As expected the 2ω c term is
filtered out leaving only the ω F component (equation A.18). I
ESigELO = ESoELOo Aξ Aψ Scos ωct + φplasma −ψ S( )cos ω ct + ω IFt( )
=ESoELOo Aθ Aξ Aψ S
2
cos ω IF t −φ plasma +ψ S( ) + cos 2ωct + ω IF t +φ plasma −ψ S( )
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
= after preamp filtering →ESoELOo Aθ Aξ Aψ S
2cos ω IFt − φplasma +ψ S( )
A.18
Therefore the power registered in the digitizer for the eleven channels of the FIR
will have the following form,
PSig =ESoELOo Aθ Aξ Aψ S
2cos ω IF t −φ plasma +ψ S( ), A.19
which by substituting in
PSig _ Amp =ESoELOo Aθ Aξ AψS
2, A.20
147
equation A.19 becomes
PSig = PSig _ Amp cos ωIFt −φplasma +ψ S( ). A.21
A.3 Derivation of Reference Power
Recall that the phase, be it interferometry or polarimetry, is determined
by comparing the phase of the 11 signal channels to a reference channel. This
reference channel consists of most of the same components with the exception
that it does not propagate through the plasma. In equations A.1 through A.21 we
examined the measured FIR power of the signal channels. We now turn our
attention to the reference leg.
We start with equation A.22, which is exactly the same as A.1 with the
exception that the plasma matrix is missing and only one mesh matrix is
present. (If you recall, equation A.1 described transmission through mesh #1 and
reflection off of mesh #2.)
ERx
ERy
⎡
⎣ ⎢ ⎤
⎦ ⎥ =1 00 0
⎡ ⎣ ⎢
⎤ ⎦ ⎥
RTE1 00 RTM1
⎡ ⎣ ⎢
⎤ ⎦ ⎥
cos 2ω pt( ) sin 2ω pt( )sin 2ω pt( ) −cos 2ω pt( )
⎡
⎣ ⎢
⎤
⎦ ⎥ ×
cos2 φ + i sin2 φ sinφ cosφ 1− i( )sinφ cosφ 1− i( ) cos2 φ + i sin2 φ
⎡
⎣ ⎢ ⎤
⎦ ⎥ 1 00 0
⎡ ⎣ ⎢
⎤ ⎦ ⎥
ESxo
ESyo
⎡
⎣ ⎢ ⎤
⎦ ⎥
A.22
As in section A.2 we carry out the matrix multiplication giving
ERx
ERy
⎡
⎣ ⎢ ⎤
⎦ ⎥ = RTE1ESocos 2ω pt( ) sin 2ω pt( )
0 0
⎡
⎣ ⎢ ⎤
⎦ ⎥ cos ωct( )
ε sin ωct( )⎡
⎣ ⎢ ⎤ , A.23
⎦ ⎥
which is further simplified to
148
ERx = RTE1ESo cos 2ω pt( )cos ωct( )+ ε sin 2ω pt( )sin ωct( )[ ]. A.24
Once again this form lends itself to using the trigonometry trick discussed
previously, where we define
ψ R = tan −1 ε sin 2ω pt( )cos 2ω pt( )
⎡
⎣ ⎢
⎤
⎦ ⎥ = tan −1 ε tan 2ω pt( )[ ], A.25
and
Aψ R= cos2 2ω pt( )+ ε2 sin2 2ω pt( )
=1 + ε2( )
21 +
1− ε 2( )1+ ε 2( )cos 4ωpt( )
. A.26
Substitution yields the electric field incident on the mixer from the reference leg
to be
ERx = RTE1ESoAψ Rcos ωct − ψ R( )[ ], A.27
which, when including the LO leg, has the total power measured by the corner
cube diode as
PRef =
E Ref +
E LO( )2
= ERef2 + ELO
2 + 2ERef ELO. A.28
Once again we must consider the filtering of the pre-amplifier by examining the
terms on the right side of equation A.28. The first term is completely filtered out,
as is the second term (recall equation A.17).
149ERef
2 = RTE1ESo2 Aψ R
2 cos2 ωct −ψ R( )
=RTE1ESo
2 Aψ R
2
21 − cos 2ω ct − 2ψ R([
= after preamp filtering → 0
)] A.29
The cross term is retained,
ERef ELO = RTE1ESoELOo Aψ Rcos ωct −ψ R( )cos ωct + ω IFt( )
=RTE1ESoELOo Aψ R
2
cos ω IFt +ψ R( )
+ cos 2ωct +ω IF t −ψ R( )
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
= after preamp filtering →RTE1ESoELOoAψ R
2cos ω IFt + ψ R( )
, A.30
and therefore the power as measured by the digitizer is described as follows.
PRef =RTE1ESoELOo Aψ R
2cos ω IFt +ψ R( )= PRef _ Amp cos ω IFt +ψ R( ) A.31
A.4 Digital Extraction of the Interferometer Phase
Having obtained the measured FIR power for the signal and reference
channels, described by equations A.21 and A.31, we are now ready to extract the
interferometry and polarimetry phase. Recall that
PSig = PSig _ Amp cos ωIFt −φplasma +ψ S( ) A.21
and
PRef = PRef _ Amp cos ω IFt +ψ R( ). A.31
150
We begin the phase extraction by preparing the reference channel as outlined in
Chapter 2, Section 3. Expressing in complex notation, filtering out the negative
frequencies, and taking the complex conjugate, equation A.31 becomes
PRef _ Con = PRef _ Amp 2( )e−i ω IFt+ψ R( ). A.32
Multiplying the Ref_Con term (Equation A.32) with the signal channel power
(Equation A.21), we form the Product term, shown below.
PProduct = PSig × PRef _Con
= PSig _ AmpPRef _ Amp 4( )×
ei ωIFt − φplasma+ψ S( )e−i ω IFt +ψR( ) + e− i ω IFt −φplasma +ψ S( )e−i ωIFt +ψ R( )[ ]= PSig _ AmpPRef _ Amp 4( ) e−i φplasma +ψ R −ψS( ) + e−i 2ω IFt − φplasma+ψ S −ψ R( )[ ]
A.33
A low pass filter is employed to eliminate the 2ω F term, leaving I
PFilter_Product = PSig _ AmpPRef _ Amp 4( )e−i φplasma +ψ R −ψS( ). A.34
The measured interferometry phase (Φ) becomes
Φ = tan−1Im PFilter_Product( )Re PFilter_Product( )
⎡
⎣ ⎢
⎤
⎦ ⎥
= tan −1 − PSig _ AmpPRef _ Amp 4( )sin φplasma +ψ R −ψ S( )PSig _ AmpPRef _ Amp 4( )cos φ plasma + ψ R −ψ S( )
⎡
⎣ ⎢
⎤
⎦ ⎥
= − φplasma +ψ R −ψ S( )
. A.35
151
Substituting equations A.11 and A.25, we can rewrite the measured
interferometry phase (Φ) to
Φ = − φplasma + tan −1 ε tan 2ω pt( )[ ]− tan−1 ε tan 2ω pt −ξ( )[ ]{ }. A.36
Equation A.36 shows that during operation of the polarimeter, the total
interferometer phase measured is a combination of the desired quantity, φplasma ,
which results from the plasma electrons, and two contamination terms
attributable to the rotating of the beam’s elliptical polarization.
An interesting, and very important, exercise is to examine the case where
the spindle bearing is either turned off or removed. In this case ωp = 0 , however
the quarter-wave plate is still installed. During large campaigns, this is often
how the interferometer is run because of the reduced effort in realigning the
spindle bearing / quarter-wave plate assemblies when the polarimetry
measurement is again needed. Although ωp = 0 , the quarter-wave plate is still
producing an elliptical beam polarization, and equation A.36 becomes
Φ = − φplasma − tan −1 ε tan −ξ( )[ ]{ }= − φplasma − tan −1 ε tan − tan−1 tan θ( )tan δ( )[([{ ])]}≈ − φplasma − ε tan θ( )δ{ }= − φplasma − TM
TEεδ
⎧ ⎨ ⎩
⎫ ⎬ ⎭
. A.37
With this arrangement, we find that the desired interferometry phase is
now contaminated with a term that goes as the product of the mesh distortion
152
ratio (TM/TE), the beam ellipticity (ε), and the Faraday rotation angle (δ). This is
important for three reasons:
a) The mesh distortion ratio (TM/TE), although constant throughout the
shot, will vary from channel to channel complicating the extraction of
profiles.
b) The Faraday rotation angle (δ) is not constant during the shot and this
produces a time dependent error in the electron density measure-
ments.
c) Most importantly, the Faraday rotation angle (δ) changes sign across
the magnetic axis. This means that on the high-field (inboard) side, the
systematic error works to reduce the measured interferometry phase
shift, while on the low-field (outboard) side, the error increases the
phase measurement. This will erroneously exaggerate the outward
shift of the plasma and is a possible explanation for the why MSTFIT
often has difficulty fitting the FIR points down to the accuracy of the
measurement.
We can estimate the amplitude of this error term by recognizing that the
mesh distortion ratio can range from 0.3 to as much as 5.0 in some chords. With
an ellipticity of 0.5 (which is standard), and a medium current discharge (350
kA) producing a maximum Faraday rotation angle ~0.2 radians, the error term
can be as high as
TMTE
⎛ ⎝
⎞ ⎠ εδ ≈ 5( ) 0.5( ) .15( ) = .375 radians. A.38
153
This angle corresponds to an error in density of ~2.9E+11 cm-3. In some channels
and plasma parameters, this might be as high as 10%. The bottom line, it is best
to remove the quarter-wave plate (force ε → 0 ) when attempting to accurately
measure the electron density. Moreover, the additional term in equation A.37
should be implemented into the MSTFIT density inversion code. This final note
is that any ellipticity present in the FIR beam will contaminate the density
phase and if hyper-accurate density measurements are desired, additional
polarizers placed after the distributing meshes (these are the meshes placed
above the tank) and before the vacuum vessel might help with this problem.
A.5 Extracting the Polarimetry Phase
Once again we are forced to recall equations A.21 and A.31,
PSig = PSig _ Amp cos ωIFt −φplasma +ψ S( ) A.21
PRef = PRef _ Amp cos ω IFt +ψ R( ) , A.31
The polarimetry phase is extracted from the shift between the modulated
amplitudes of the signal and reference beams. The expressions for the reference
and signal amplitudes are shown in equations A.39 and A.40 respectively. In
both cases the modulation arises from a sinusoidal term, oscillating at four times
the spindle bearing rotation frequency.
PRef _ Amp =
RTE1ESoELOoAψ R
2
= RTE1ESoELOo
21+ ε 2( )
21+
1 − ε2( )1 + ε2( )cos 4ω pt( )
A.39
154
PSig _ Amp =ESoELOo Aθ Aξ AψS
2
= ESoELOo Aθ
21+ cos 2θ( )cos 2δ( )
1 + ε2( )2
1 +1− ε 2( )1+ ε 2( )
cos 4ωpt − 2ξ( ) A.40
The amplitudes are isolated by using a Hilbert transformation, which shifts the
signal by π/2, effectively changing the cosine to a sine. For example,
Hilbert PSig[ ]= Hilbert PSig _ Amp cos ω IFt −φ plasma + ψS( )[ ]= PSig _ Amp sin ω IFt − φplasma +ψ S( )
. A.41
By summing the square’s, PSig{ }2+ Hilbert PSig[ ]{ }= PSig _ Amp
2 , the signal amplitude
falls out easily. Using this method, we find the reference and signal envelopes
are
PRef _ Env = PRef _ Amp2 =
1 + ε 2( )2
RTE1ESoELOo
2⎡ ⎣ ⎢
⎤ ⎦ ⎥
2
1 +1− ε 2( )1+ ε 2( )cos 4ω pt( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ , A.42 ⎥
and
PSig _ Env = PSig _ Amp2 =
1 +ε 2( )2
ESoELOo Aθ
2⎡ ⎣ ⎢
⎤ ⎦ ⎥
2
× 1 + cos 2θ( )cos 2δ( )[ ] 1 +1− ε 2( )1+ ε 2( )cos 4ω pt − 2ξ(
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
= PSig _ Env _ Amp cos 4ω pt − 2ξ( )
) . A.43
Preparing the reference as before, we convert to exponential notation, remove
any equilibrium components and negative frequencies, and conjugate. This
yields
155
PRef _ Env _ conj = PRef _ Env _ Amp 2( )e−4iω pt . A.44
For the signal envelope, we just remove the equilibrium components and expand
to exponential form.
PSig _ Env _ Fil = PSig _ Env _ Amp / 2( )e−i 4ω pt −2ξ( ) + e
+i 4ω pt− 2ξ( )[ ] A.45
Multiplying equations A.44 and A.45, our product exhibits two principal terms,
one component is oscillating around 8ω p , and contains the ξ term that we are
trying to isolate. Filtering the product to remove the 8ω p term we arrive at
PProduct_Env = PRef _ Env _ conj × PSig _ Env _ Fil
=PRef _ Env _ AmpPSig _ Env _ Amp
4
⎡
⎣ ⎢
⎤
⎦ ⎥ e−i 4ω pt −2ξ( ) + e+i 4ω pt− 2ξ( )[ ]e−4iωpt
=PRef _ Env _ AmpPSig _ Env _ Amp
4⎡ ⎣ ⎢
⎤ ⎦ ⎥ e−2iξ + e
− i 8ω pt − 2ξ( )[ ]= filtering,8ω p term → 0
= PProduct_Env_Ampe−2iξ
. A.46
The final step is to isolate the measured Faraday rotation.
Ψ = tan−1Im PProduct_Env( )Re PProduct_Env( )
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ = −2ξ = −2 tan−1 tanθ tanδ[ ]≈ −2
TMTE
δ A.47
Equation A.47 again points out that the mesh distortion factor (TM/TE)
couples with the Faraday rotation angle (δ) as reported earlier.4 To obtain
reasonable polarimetry measurements, it is imperative that the mesh distortion
156
REFERENCES
factors be accurately characterized. It might also be worthwhile to investigate
alternatives to the meshes, such as thin film deposited on TPX or quartz.
1 B. W. Rice, Review of Scientific Instruments, 63, 5002 (1992). 2 B. W. Rice, Ph.D. thesis, University of California-Davis, CA 1992, UCLR-LR-
111863. 3 D. L. Brower, L. Zeng, and Y. Jiang, Review of Scientific Instruments, 68, 419
(1997). 4 N. E. Lanier, J. K. Anderson, C. B. Forest, D. Holly, Y. Jiang, and D. L. Brower
Review of Scientific Instruments, 70, 718 (1997).
157
B: FIR Density Code Listings and Analysis Procedures
B.1 Introduction
As outlined in Chapter 2, direct digitization of the FIR signals enables a
more accurate determination of the interferometry phase. The cost of this benefit
is the requirement of more complex phase extraction and data processing
techniques. In this appendix, we commit to print the computer codes that are
utilized during extraction of the interferometer data and outline the procedures
for data analysis.
B.2 Processing FIR Data
Processing of the FIR interferometry data is conducted in three steps,
these being,
i) Phase Computation
ii) Visual Inspection
iii) Manual Reconstruction.
158B.2.1 General Code Notes
The phase is computed, as outlined in Chapter 2, by the fir_proc.pro
program. The code is completely automated and conducts the initial pre-
processing, extraction of the interferometer phase, and recording of important
parameters such as laser power, interference frequency, and bandwidth. The
output parameters are written to the F level of the database and are discussed
below.
1) FIR_FAST_* (* refers to N32,N24,…,P36,P43,REF)
These signals are the CHORD-AVERAGED electron density
measurements. The units are in particles per cm2. The suffixes refer to the
chord’s radial impact parameter in centimeters. For example N32 is –32 cm, P21
is +21 cm, and so on.
2) FIR_LASER_IF
Stored in the variable ‘FIR_LASER_IF’ is the interference
frequency (IF) of the laser, in units of kHz. Typical operation has the IF at 750
kHz for a digitization rate (DR) of 1 MHz. However, if faster time response is
desired, the recommended values are an IF of 875 with a digitization rate of 3
MHz. In this case, it is important to remember not to run with an IF of 625
because the preamp filter response will serve to limit the bandwidth.
3) FIR_BANDWIDTH
This is the computed measurement bandwidth, also in units of kHz.
This is determined by the minimum frequency difference between the Nyquist
and the aliased IF or just the aliased IF and zero. During standard operation (IF
159
of 750 kHz and DR of 1 MHz), the maximum bandwidth is 250 kHz, however the
processing code filters down to 200 kHz. This can be adjusted if desired, however
the chord averaged nature of the measurement defeats the purpose of operating
at very high frequencies.
4) FIR_LASER_POWER
This signal is the peak voltage difference of the Reference channel.
Given in units of Volts, FIR_LASER_POWER has no real meaning other than
providing an indicator for the trustworthiness of the data. If this value is less
than a volt or so, one should scrutinize the data very carefully.
5) JUMP_STATS
Once the processing is done and the code has removed most, if not
all of the π phase jumps, the computer will store the difference between the
number of positive and negative π phase jumps for each channel. In an ideal
world, this number should always be zero, but often it is not. JUMP_STATS is
an 11 element array in which this value is stored, where element 0 corresponds
to N32, 1 to N24 and so on. If the stored value is different than zero, odds are the
code has missed a jump and it must be manually fixed. We will discuss more of
the usefulness of JUMPS_STATS later.
B.2.2 The FIR Processing Code
pro fir_proc,date,shot ;;----------------------------------------------------------------------- ;; ;; Nicholas E. Lanier ;; ;; 20-apr-1997 Original Version ;; ;; 21-Jan-1999 Modified for PROC Code ;;
160;; This program will digitally extract the LINE_AVERAGED interferometry ;;phase from raw data digitized by the two TR612's. The input signal names ;;all have the prefix 'FIR_612_' with the suffix's of N32,N24,...,P43,REF. ;;The processed phases are stored in the F level of the "mst$data" database ;;under the prefix of 'FIR_FAST_'. All chords share the same time trace ;;stored under the name 'FIR_FAST_TM'. ;; This program also stores for each shot the information about laser ;;frequency, laser power and maximum bandwidth. The signal names are ;;FIR_LASER_IF, FIR_LASER_POWER, and FIR_BANDWIDTH. Furthermore, an 11 ;;elements array is also stored called JUMP_STATS. The number (one for each ;;chord) is an indication of the quality of the data. The lower the value ;;the better. This number is derived in the processing as the number of ;;calculated Pi shifts minus the number of -Pi shifts. If the difference is ;;large, the code has not properly removed the phase jumps from the raw ;;data so the processed data will be unsuitable for use. ;; ;;----------------------------------------------------------------------- ;; ;;---------------------------Setting Up---------------------------------- ;; set_db,'mst$data' ;Setting to proper database x_pos=[-32.,-24.,-17.,-09.,-02.,6.,13.,21.,28.,36.,43.] ;chord locations (cm) z_path=2*sqrt(52.*52.-x_pos*x_pos)/100. ;chord path length (m) shot,shot ;Setting Shot date,date ;Setting Date ;; ;dummy=set_errors('none') ;Set Errors to Quite Mode ;; red_factor=4 ;Store Smaller Array Size ;; time_store_name='F.FIR_FAST_TM' ;Time Array Store Name ;time_store_name='P.FIR_FAST_TM' ;Use When Running as PROC Code time_units='ms' ;Time Stored Units dens_units='cm^-2' ;Density Stored Units ;; jump_store_name='F.JUMP_STATS' ;jump stats store name jump_diff_tot=fltarr(11) ;This quantity is the difference ;between up jumps and down jumps. ;it relates to the quality of ;data. ;; conversion_factor=12.16 ;conversion factore comes from ;the interferometer phase eq. ;= (lambda*e^2)/ ; (4*PI*c^2*m_e*eps_not) ;lambda = 432.5E-6 m ;e = 1.602E-19 e/C ;c = 2.997E+8 m/s ;m_e = 9.11E-31 kg ;eps_not= 8.85E-12 F/m ;Data stored in units of 10^13 ;particles / cm^3 ;; ;;----------------------------------------------------------------------- ;; ;; ;; Begin phase extract, download and check for time information
161;; get_fir_time_info,time,array_size,dig_speed,abort_shot_1 ;; ;; Check and prepare reference signal ;; prepare_reference,reference,laser_power,abort_shot_2 ;; ;; Defing abort variable ;; abort_shot=min([abort_shot_1,abort_shot_2]) ;; ;; Abort shot if abort_shot=0 ;; if (abort_shot ne 0) then begin ;; ;; Resize time data and write data into database ;; resize_data,array_size,red_factor,time,time_thn write_data,time_store_name,time_thn,time_units ;; ;; Compute bandwidth, power, dig_speed, and preamp response function ;; preamp,array_size,dig_speed,laser_power,reference,response,bandwidth ;; ;; Incorperate the bandwidth into the signal filtering ;; filter_pt=bandwidth*array_size/float(dig_speed) ;; ;; Prepare reference signal ;; conjugate_reference,array_size,response,reference,conj_reference ;; ;; Main loop for chord processing ;; for chord=0,10,1 do begin ;; ;; Get names and preprocess signal ;; get_names,chord,signal_name,store_name prepare_signal,signal_name,signal,abort_channel print,signal_name ;notify user of progress
162 ;; ;; If sigan is OK then continue ;; if (abort_channel ne 0) then begin filter_channel,response,signal,signal_fil ;; ;; Computing and filtering the product ;; product=conj_reference*signal_fil signal_fil=0 filter_product,array_size,filter_pt,product,product_fil ;; ;; Calculating interferometer phase ;; phase=atan(imaginary(product_fil),float(product_fil)) product_fil=0 ;; ;; Remove the Phase jumps ;; remove_phase_jumps,array_size,phase,jump_diff jump_diff_tot(chord)=jump_diff ;; ;; Rebining data to smaller size ;; resize_data,array_size,red_factor,phase,phase_thn ;; ;; Subtracting offset ;; off=fix(dig_speed/1000.) ;offset average width phase_thn=phase_thn-avg(phase_thn( (array_size- $ 1000)/red_factor:array_size/red_factor-1)) ;NOTE: Must have a spare millisecond left at end of ;shot for offset. ;; ;; Conversion to electrons/cm^2 ;; conversion=(conversion_factor*z_path(chord)) ;; ;; Store the data into database ;; write_data,store_name,phase_thn/conversion,dens_units endif
163 endfor ;; ;; Store the jump number totals ;; write_data,jump_store_name,jump_diff_tot,'Jumps' endif ;; ;;----------------------------------------------------------------------- end pro get_fir_time_info,time,array_size,dig_speed,abort_shot_1 ;;----------------------------------------------------------------------- ;; ;; Subroutine downloads the time array and checks for errors. ;; ;; Variable Name Definition ;; ;; INPUTS: none ;; ;; OUPUTS: time time array in ms ;; ;; dig_speed digitization speed ;; ;; abort_shot_1 abort shot indicator ;; ;;----------------------------------------------------------------------- ;; time=data('fir_612_ref_tm') ;Downloading time array dummy=size(time) ;Checking size abort_shot_1=dummy(0) ;Checking for data if (abort_shot_1 ne 0) then begin dig_speed=(dummy(1)-1)/(time(dummy(1)-1)-time(0)) ;Getting digitization speed time=1000*time ;Coverting to ms array_size=dummy(1) ;Getting array_size dummy=0 ;Saving virtual memory endif ;; ;;----------------------------------------------------------------------- end pro prepare_reference,reference,laser_power,abort_shot_2 ;;----------------------------------------------------------------------- ;; ;; Subroutine dowloads reference signale and checks for errors. ;; ;; Variable Name Definition ;; ;; INPUTS: none ;; ;; OUTPUTS: reference Raw Reference signal ;; ;; laser_power Laser Power in (A.U.)
164;; ;; abort_shot_2 Abort shot indicator ;;----------------------------------------------------------------------- ;; reference=data('fir_612_ref') ;Downloading raw data dummy=size(reference) ;Checking size abort_shot_2=dummy(0) ;Checking for errors ;; ;;-------Checking if there is reasonable signal------- ;; if (abort_shot_2 ne 0) then begin laser_power=max(reference)-min(reference) ;; ;;---------Max Amplitude must be ge .2-------- ;; if (laser_power le .2) then begin abort_shot_2=0 ;Skip shot print,'Shot Skipped due to Laser Low Power' endif endif dummy=0 ;Saving virtual memory ;; ;;----------------------------------------------------------------------- end pro get_names,chord,read_name,store_name ;;------------------------------------------------------------------------- ;; ;; Subroutine returns raw signal names and store names ;; ;; Variables Name Definition ;; ;; INPUTS: chord chord counter ;; ;; OUTPUTS: read_name name of data to be read ;; ;; store_name name of store data name ;; ;;------------------------------------------------------------------------- ;; read_name_prefix='fir_612_' ;read name prefix ;store_name_prefix='P.FIR_FAST_';Use for PROC code store_name_prefix='F.FIR_FAST_' ;Store location prefix name_suffix=['N32','N24','N17','N09','N02','P06', $ 'P13','P21','P28','P36','P43'] ;Suffix Array ;; ;;-------------------Defining Names----------------------- ;; read_name=strtrim(read_name_prefix+name_suffix(chord),2) store_name=strtrim(store_name_prefix+name_suffix(chord),2) ;; ;;------------------------------------------------------------------------- end pro prepare_signal,signal_name,signal,abort_channel
165;;----------------------------------------------------------------------- ;; ;; Subroutine dowloads reference signal and checks for errors. ;; ;; Variable Name Definition ;; ;; INPUTS: signal_name Signal name to be read ;; ;; OUTPUTS: signal Raw signal ;; ;; abort_channel Abort channel indicator ;;----------------------------------------------------------------------- ;; signal=data(signal_name) ;Downloading raw data dummy=size(signal) ;Checking size abort_channel=dummy(0) ;Checking for errors dummy=0 ;Saving virtual memory ;; ;;----------------------------------------------------------------------- end pro resize_data,array_size,reduction_factor,data_in,data_out ;;------------------------------------------------------------------------- ;; ;; Subroutine rebins an input signal to a more appropriate size ;;based on the time resolution of the diagnostic. Default reduction ;;is about 4 for data stored at 1 MHZ ;; ;; Variables Name Definition ;; ;; INPUTS: array_size Array Size ;; ;; reduction_ Reduction factor ;; factor ;; ;; data_in Input data to be reduced ;; ;; OUTPUTS: data_out Reduced output data ;; ;;------------------------------------------------------------------------- ;; data_out=rebin(data_in,array_size/reduction_factor) ;; end ;;------------------------------------------------------------------------- pro write_data,store_name,store_data,units ;;------------------------------------------------------------------------- ;; ;; Subroutine writes data to the database and checks to see if ;;properly written. ;; ;; Variable Name Definition ;; ;; INPUTS: store_name Name dat is to stored as ;; ;; store_data Data to be stored ;; ;; OUTPUTS: none
166;; ;;------------------------------------------------------------------------- ;; status=put_data(store_name,store_data,units) ;Writing data ;; ;;-----If not properly written,print error message ;; if (status lt n_elements(store_data)) then begin print,'Error in Storing',store_name endif ;;------------------------------------------------------------------------- end pro preamp,array_size,dig_speed,laser_power,reference,response,max_bandwidth ;;------------------------------------------------------------------------- ;; ;; Subroutine computes the Laser IF and Bandwidth of the shot. ;;The IF, Bandwitdh, and Laser power are store in the database under ;;the names----FIR_LASER_IF,FIR_BANDWIDTH, and FIR_POWER. Using the ;;calculated bandwitdh, the preamplifer response function is computed. ;; ;; Variable Name Definition ;; ;; INPUTS: array_size array size ;; ;; dig_speed digitization speed ;; ;; laser_power FIR laser Power ;; ;; reference raw reference signal ;; ;; OUTPUTS: response preamp response function ;; ;;------------------------------------------------------------------------- ;; default_bandwidth=2.0E+5 nyquist=dig_speed/2. ;; ;; Stored lader power is defined as the refernce signal ;;voltage level. ;; laser_power=max(reference)-min(reference) ;; ;; Compute LASER_IF from the peak in the frequency spectrum ;; ref_fft=fft(reference,-1) peak_location=where( float(ref_fft) eq $ max(float(ref_fft(100:array_size/2-100))))+100 laser_if=(float(peak_location(0))/array_size)*dig_speed+nyquist ;; ;; Maximum bandwidth is defined as the minimum frequency difference
167 ;;between zero and IF or the Nyquist and the IF. ;; max_bandwidth=min([(float(peak_location(0))/array_size $ *dig_speed),(nyquist-(float(peak_location(0)) $ /array_size)*dig_speed),default_bandwidth]) window_size=float(max_bandwidth)/dig_speed*array_size ;; ;; Response function for signal filtering is computed with the ;;appropriate bandwidth ;; response=fltarr(array_size) response(peak_location(0)-window_size+1:peak_location(0) $ +window_size-1)=1 response((array_size-1)-(peak_location(0)+window_size-1): $ (array_size-1)-(peak_location(0)-window_size+1))=1 ;; ;; Write quantities into database ;; write_data,'F.FIR_LASER_IF',laser_if/1000.,'kHz' write_data,'F.FIR_BANDWIDTH',max_bandwidth/1000.,'kHz' write_data,'F.FIR_POWER',laser_power,'a.u.' default_bandwidth=0 ;saving virtual memory peak_location=0 ;saving virtual memory window_size=0 ;saving virtual memory laser_power=0 ;saving virtual memory laser_if=0 ;saving virtual memory nyquist=0 ;saving virtual memory ref_fft=0 ;saving virtual memory ;; ;;----------------------------------------------------------------------- end pro conjugate_reference,array_size,response,reference,conj_reference ;;----------------------------------------------------------------------- ;; ;; Subroutine prepares the reference by zeroing the imaginary ;;frequency components and conjugating, is a sense converting the COS ;;to and EXP. ;; ;; Variable Names Definition ;; ;; INPUTS: array_size duh! ;; ;; response preamp response function ;; ;; reference reference data ;; ;; OUTPUTS: conj_refernce conjugated reference ;; ;;----------------------------------------------------------------------- ;; ;; ;; Transform into frequency space
168 ;; ref_fft=fft(reference,-1) ;; ;; Remove equilibrium components and imaginary frequencies ;; ref_fft(0)=0 ref_fft(array_size/2:*)=0 ;; ;; Conjugate and return to time domain ;; conj_reference=conj(fft(response*ref_fft,1)) ref_fft=0 ;saving virtual memory ;; ;;----------------------------------------------------------------------- end pro filter_channel,response,signal_in,signal_out ;;----------------------------------------------------------------------- ;; ;; Subroutine filters the input signal as dictated by the response ;;function computed in the preamp subroutine. ;; ;; Variable Name Definition ;; ;; INPUTS: response filtering response function ;; ;; signal_in input signal to be filtered ;; ;; OUTPUTS: signal_out output of filtered signal ;; ;;----------------------------------------------------------------------- ;; ;; Transform to frequency domain ;; signal_fft=fft(signal_in,-1) ;; ;; Filter signal ;; signal_fft(0)=0 signal_out=fft(response*signal_fft,1) signal_in=0 ;saving virtual memory signal_fft=0 ;saving virtual memory ;; ;;----------------------------------------------------------------------- end pro filter_product,array_size,filter_point,product_in,product_out ;;----------------------------------------------------------------------- ;; ;; Subroutine filters product according to the limitations dictated ;;by the bandwidth.
169;; ;; Variable Name Definition ;; ;; INPUTS: array_size Duh! ;; ;; filter_point filter point calculated from ;; bandwidth ;; product_in input product ;; ;; OUTPUTS: product_out output product ;; ;;----------------------------------------------------------------------- ;; ;; Transforming into frequency domain, filtering, then transforming ;;back into time ;; product_fft=fft(product_in,-1) product_fft(filter_point:array_size-filter_point-1)=0 product_out=fft(product_fft,1) product_in=0 ;saving virtual memory ;; ;;----------------------------------------------------------------------- end pro remove_phase_jumps,array_size,phase,jump_diff ;;----------------------------------------------------------------------- ;; ;; This subroutine is the most important of the digital phase extraction ;;technique. Here we remove the phase jumps that contaminate the extracted ;;phase. The difficulty in this procedure is that the random noise in the ;;phase measurement makes the phase jumps non-uniform, ie some are greater ;;than Pi and some are less then Pi. In chords where beam refraction ;;greatly increases the signal noise level, the phase jumps can be ;;indistinguishable from plasma fluctuations and have to be inspected by ;;eye. ;; ;; Variables Name Definition ;; ;; INPUTS: array_size for the last time....duh ;; ;; phase input phase to be modified ;; ;; OUTPUTS: phase modified output phase ;; ;; jump_diff statitstics on jumps ;; ;;----------------------------------------------------------------------- ;; ;; Sort phase jumps in ascending order ;; dphase=temporary(phase-shift(phase,1)) sort_order=sort(dphase) sorted_dphase=dphase( sort_order ) ;; ;; Isolatethe number up and down jumps ;;
170 up_jump_locs=where( sorted_dphase ge !pi ) down_jump_locs=where( sorted_dphase le -!pi ) max_jump_num=max( [ n_elements(up_jump_locs), $ n_elements(down_jump_locs) ] ) min_jump_num=min( [ n_elements(up_jump_locs), $ n_elements(down_jump_locs) ] ) ;; ;; compute the difference between up and down jumps ;;in and ideal world this should always be zero ;; jump_diff=max_jump_num-min_jump_num ;; ;; If there are jumps then fix the phase ;; if ((up_jump_locs(0) ne -1) and (down_jump_locs(0) ne -1)) then begin ;; ;;Define path integral to be the integral of the ;;density trace over time. The goal is to minimize ;;this value because extraneous jumps will increase ;;this value. ;; path_integral=sorted_dphase(0:(array_size/2-1))+ $ reverse(sorted_dphase(array_size/2:*)) dphase=0 ;saving virtual memory ;; ;; Find number of jumps that minimizes the path_integral ;; min_location=where( $ min(path_integral(min_jump_num:max_jump_num)) $ eq path_integral(min_jump_num:max_jump_num) ) $ + min_jump_num - 1 path_integral=0 ;saving virtual memory ;; ;; Sort jumps according to the time they occur ;; up_jumps=sort_order(0:min_location(0)) down_jumps=sort_order(array_size-min_location(0)-1 : $ array_size-1) min_location=0 ;saving virtual memory sort_order=0 ;; '' ;; ;; total number of jumps and their apprapriate sign ;;(up or down) ;;
171 total_jumps=[up_jumps,down_jumps] total_sign=[make_array(n_elements(up_jumps),value=+1.0),$ make_array(n_elements(down_jumps),value=-1.0)] ;; ;; Compute and print number of jumps ;; jump_order=sort(total_jumps) total_jumps=total_jumps(jump_order) total_sign=total_sign(jump_order) print,n_elements(total_jumps),' Jumps',jump_diff,' Diff' jump_order=0 ;; ;; Fix the phase ;; fix_phase,array_size,total_jumps,total_sign,phase ;thn=findgen(6550)*10 for troubshooting purposes ;stop total_jumps=0 ;saving virtual memory total_sign=0 ; '' endif ;; ;;----------------------------------------------------------------------- end pro fix_phase,array_size,jump_locs,sign,phase ;;----------------------------------------------------------------------- ;; ;; This subroutine modifies the phase when given a jumps location and ;;polarity. ;; ;; Variable Name Definition ;; ;; INPUTS: jump_locs jumps locations ;; ;; sign jump polarity ;; ;; OUTPUTS: phase modified phase ;; ;;----------------------------------------------------------------------- ;; ;; Setting up jump location array ;; num=n_elements(jump_locs) jump_locs=[jump_locs,array_size] sum=0 ;; ;; Modify phase loop ;; for i=0,num-1,1 do begin
172 sum=sum+sign(i) phase(jump_locs(i):jump_locs(i+1)-1)=phase(jump_locs(i) $ :jump_locs(i+1)-1) + 2*!pi*sum endfor ;; ;;----------------------------------------------------------------------- end
B.2.3 Pre-Inspection of Processed Data
Having completed the automated processing, we now begin to inspect the
FIR data. The principal objective is to classify the data quality as either GOOD,
FIXABLE, or CRAP (for lack of a better technical term). This is the most tedious
aspect of the FIR analysis and requires that all eleven chords of each shot be
visually inspected and categorized.† At first sight this task may seem
impossible, however my years of experience on this matter coupled with my
inherent laziness has come up with a system that is quite efficient.
We begin the visual inspection process by utilizing the get_stats.pro.
This program, displayed on the following pages, will download the laser and
jump statistics for a given range of shots and write them in a text file. This file
can then be printed and provides important information on whether the FIR
data will be viable.
† An example of the tremendous tedium, the Great Chapman Run of ‘99 required visual
inspection of over 12,000 signals.
173
Program GET_STATS.PRO
;;------------------------------------------------------------------------- ;; ;; Nicholas E. Lanier ;; ;; 20-apr-1997 Original Version ;; ;; 05-jan-2000 Modified Version ;; ;; The program downloads the laser statistics for a set of shots ;;and stores them into a text file. ;; ;;------------------------------------------------------------------------- ;; dt=' ' & info=' ' ;initial set-up start=0 & end_shot=0 read,'Input date (No quotes) => ',dt ;getting date print,' ' ;; read,'Input start shot => ',start ;getting starting shot print,' ' ;; read,'Input end shot => ',end_shot ;getting end shot ;; set_db,'mst$data' ;setting database ;; s=set_date(dt) ;setting date ;; save_name=strupcase('stats_'+dt+'.dat') ;stat file name ;; jump_num=fltarr(11) ;jump array ;; get_lun,lun ;get lun number ;; ;;------------------------------------------------------------------------- ;; openw,lun,save_name ; open file ;; ;; Write the file header ;; printf,lun,'Laser Satistics for data taken on' printf,lun,' ' printf,lun,dt printf,lun,' ' ;; ;; Begin main shot loop ;; for i=start,end_shot,1 do begin s=set_shot(i) ;setting shot print,i ;user information ;; ;; Downloading power, bandwidth, laser ;;
174 pow=strtrim(string(data('fir_power')),2) band=strtrim(string(data('fir_bandwidth')),2) las=strtrim(string(data('fir_laser_if')),2) ;; ;; Downloading jump statistics ;; jump_num(0)=data('jump_stats') jumps=strtrim( string( fix(jump_num(*)) ) ,2) jump_str=' ' for k=0,10,1 do begin jump_str(0)=jump_str(0)+' '+jumps(k) endfor sht=strtrim(string(i),2) ;shot number string
info='Shot= '+sht+' Band= '+band+' IF= '+las+' Pwr= '+pow+’ $ Jumps'+jump_str
;; ;; Printing to file ;; printf,lun,info printf,lun,' ' endfor close,lun ;close the file free_lun,lun ;free the lun number ;;--------------------------------------------------------------------------- end
The output file that is written by the get_stats program will be named
“STATS_DD-MMM-YYYY.dat”. For example, STATS_31-DEC-1999.dat will have
the laser statistics from December 31, 1999. The stored values in the STATS file
are shot number, laser bandwidth, laser interference frequency, laser power, and
the jump statistics. A sample of the statistics output file is displayed on the
following page.
175Laser Statistics for data taken on
31-dec-1999
Shot= 34 Band= -1 IF= -1 Pwr= -1 Jumps -1 0 1 0 0 0 0 0 0 0 0
Shot= 35 Band= 4.31824 IF= 504.318 Pwr= 1.47217 Jumps 122 114 116 90 91 124 116 85
125
Shot= 36 Band= 200.000 IF= 777.615 Pwr= 3.92212 Jumps 0 0 0 0 1 0 3 1 0 1 1
Shot= 37 Band= -1 IF= -1 Pwr= -1 Jumps -1 0 1 0 0 0 0 0 0 0 0
Shot= 38 Band= 200.000 IF= 764.865 Pwr= 3.87817 Jumps 1 0 0 2 0 1 0 1 1 0 0
With this information at our disposal, we can limit the number of shots we
are going to spend effort on examining. Based on the data in the file above, I
would make the following interpretations.
Shot 34 Laser not yet on Shot 35 Laser improperly tuned (or still warming up), no bandwidth, a lot of
jumps (> 5 per channel). This shot unsalvageable. Shot 35 Bandwidth and Power good, jumps look good. Inspect this one. Shot 37 No data, could be a storage problem...Skip shot. Shot 38 Bandwidth and Power good, jumps look good. Inspect this one.
Therefore based on the information above, I would not waste any time
inspecting shots 34, 35 and 37.
B.2.4 Inspection Code
Shot inspection is done using the code inspect.pro. Given a date and
shot number, inspect.pro will plot the processed FIR data for visual inspection.
The delay time between plotting each channel is variable, but the default setting
is 0.2 seconds. (I like to run this on Versaterm because I can sit back, and cycle
very fast looking for anomalies. If I see one, I just scroll back on the plots and
mark the appropriate channel on my shot stat list.) While running this code you
are looking for three items:
176
a) Phase jumps. Not all phase jumps are caught in the automatic
analysis routine. Any residual jumps must be manually removed.
The figure B.1 shows what a phase jump would look like.
Phase Jump
Figure B.1 – An example of a phase jump missed by the automated analysis routine. This jump must be removed manually.
b) Offset problems. Sometimes the trace looks good, but the offset
after the shot is not zero (this usually results from phase jumps
that occur very early in the shot (<3 ms)). Often to fix these
problems, I just insert a phony phase jump early in the shot to
make up any difference. This is acceptable because we never use
the data before 5 ms.
177
Figure B.2 – Offset problems appear when the missed phase jumps occur early in the shot (< 5.0ms). If the data for t > 5.0 ms looks good, then a phase jump of correct polarity is manufactured at some point before 5.0 ms, so that the baseline after the shot is zero.
c) Flipped phase. Depending on whether the reference laser leads or
lags the signal laser, the computed density trace would appear
upside down. This is a simple problem and has no deep meaning. If
the density traces for a shot are inverted, inspect.pro presents an
option, called “FLIP”. After the last channel for a shot has been
displayed, you will have an option to enter either (q) for quit,
(return) for display the next shot, or (f) for flip. If flip is selected,
inspect will read the shot data, invert it, rewrite the inverted data
to the database, and display the inverted data again for inspection.
178pro inspect ;;------------------------------------------------------------------------- ;; ;; Nicholas E. Lanier ;; ;; 07-Jan-2000 Original Version ;; ;; 11-Jan-2000 Modified Version ;; ;; This program reads and displays the fast processed data for ;;user inspection. Although elimination of residual phase jumps must be ;;conducted with "man_fix_fast.pro", this program can read and flip the ;;data in cases when this action is appropriate. ;; ;;------------------------------------------------------------------------- ;; ;; ;; Call user input routine ;; user_input,dat,start_shot,end_shot,wait_time ;; ;; Show the shots specified by user ;; show_set,dat,start_shot,end_shot,wait_time ;; ;;-------------------------------------------------------------------------- end pro user_input,dat,start_shot,end_shot,wait_time ;;------------------------------------------------------------------------- ;; ;; Subroutine allows the user to specify date, shots, and ;;plotting delay time. ;; ;; Variable Name Definition ;; ;; INPUTS: none ;; ;; OUTPUTS: dat user specified date ;; ;; start_shot first shot to inpect ;; ;; end_shot last shot to inspect ;; ;; wait_time delay time between plotting ;; ;;------------------------------------------------------------------------- ;; ;; ;; User input of date ;; input_date: dat=' ' ;initialize date variable read,'Enter Date of Interest(No Quotes)-> ',dat
179print,' ' ;printing blank line date_length=strlen(dat) ;extract string length if (date_length ne 11) then begin ;check to see if length print,'Error in Date Entry' ;is appropriate, goto,input_date ;if not then repeat entry endif ;; ;; User input of shots to inspect ;; input_shots: end_shot=0 ;initializing start_shot=0 ;shot variables valid_shots=0 ;valid input flag while (valid_shots eq 0) do begin on_ioerror,bad_number read,'Enter First Shot To Inspect -> ',start_shot print,' ' ;printing blank line on_ioerror,bad_number read,'Enter Last Shot To Inspect -> ',end_shot print,' ' ;printing blank line valid_shots=1 ;inputs are OK, set flag bad_number: ;if entry error then repeat if NOT valid_shots then print,'Shots must be numbers.' endwhile ;; ;; More error checking, last shot must be larger than first ;; start_shot=fix(start_shot) & end_shot=fix(end_shot) if (start_shot gt end_shot) then begin print,'Last shot less than First' goto,input_shots ;repeat if error endif ;; ;; Enter plot delay time ;; wait_time=0.0 ;Initialize wait variable valid_time=0 ;valid input flag while (valid_time eq 0) do begin on_ioerror,bad_time read,'Enter Plot Delay Time (I Suggest .20)-> ',wait_time print,' ' valid_time=1 ;all OK, set flag
180 bad_time: ;repeat if error if NOT valid_time then print,'Time must be number' endwhile set_db,'mst$data' ;set database ;; ;;----------------------------------------------------------------------- end pro show_set,dat,start_shot,end_shot,wait_time ;;----------------------------------------------------------------------- ;; ;; This subroutine is the program's main body. It loops over all ;;shots specified by the user and calls the display routine, called ;;'show_fast_shot'. Upon inspection, the user can also opt to flip the shot ;;by entering 'f'. This command calls the 'flip_fast' routine which then ;;flips the fast data and re-displays for user inpection. ;; ;; Variable Name Definition ;; ;; INPUTS: dat date ;; ;; start_shot first shot to display ;; ;; end_start last shot to be inspected ;; ;; wait_time time between plotting ;; ;; OUTPUTS: none ;; ;;----------------------------------------------------------------------- stall_command=' ' ;initializing stall command ;; ;; Main shot loop ;; for shot_number = start_shot, end_shot, 1 do begin ;; ;; Display the shot data ;; show_fast_shot,dat,shot_number,wait_time ;; ;; Prompt for command ;; print,' ' ;print blank line print,' ' ;print blank line read,' (Return) for next shot, (f) to flip, '+ $ '(q) to quit. -> ',stall_command ;; ;; Check for flip or quit command ;; if (strupcase(stall_command) eq 'F') then begin
181 flip_fast,dat,shot_number shot_number=shot_number-1 endif if (strupcase(stall_command) eq 'Q') then begin shot_number=end_shot endif endfor ;; ;;----------------------------------------------------------------------- end pro show_fast_shot,dat,shot_number,wait_time ;;----------------------------------------------------------------------- ;; ;; Subroutine plots the fir_fast data for user inspection. ;; ;; Variable Name Definition ;; ;; INPUTS: dat date ;; ;; shot_number shot number to be displayed ;; ;; OUTPUTS: none ;; ;;----------------------------------------------------------------------- date,dat ;set date shot,shot_number ;set shot number chrd_suffix=['N32','N24','N17','N09','N02','P06','P13','P21', $ 'P28','P36','P43'] ;defining chord suffix s=set_inc(100) ;set read increment to ;every 100 points tm=data('fir_fast_tm') ;download time array if (n_elements(tm) gt 1) then skip_ind=0 else skip_ind=1 ;if data then set skip ;indicator while (skip_ind eq 0) do begin ;if NOT skip then do name=' ' ;initialize name variable !ytitle='1E+14 cm^-2' ;define axis' labels !xtitle='ms' set_xy,0,70,0,2 ;set plot parameters ;; ;; Begin main display loop ;; for chord=0,10,1 do begin name='fir_fast_'+chrd_suffix(chord) ;define signal name dens_data=data(name) ;download the data wait,wait_time ;plot delay option
182 !mtitle=string(shot_number)+' '+dat+' '+chrd_suffix(chord) ;defining main title plot,tm,dens_data ;plot the data dens_data=0 ;saving virtual memory endfor ;continue with next chord tm=0 ;saving virtual memory skip_ind=1 ;shot done, se indicator endwhile s=set_inc(1) ;reset read increment to ;every point ;; ;;----------------------------------------------------------------------- end pro flip_fast,dat,shot_number ;;----------------------------------------------------------------------- ;; ;; Subroutine flips the fir_fast data and stores back into database. ;; ;; Variable Name Definition ;; ;; INPUTS: dat date ;; ;; shot_number shot number to be displayed ;; ;; OUTPUTS: none ;; ;;----------------------------------------------------------------------- date,dat ;set date shot,shot_number ;set shot number name=' ' ;initialize name variable s=set_inc(1) ;set read increment to ;every point chrd_suffix=['N32','N24','N17','N09','N02','P06','P13','P21', $ 'P28','P36','P43'] ;defining chord suffix ;; ;; Begin main flip loop ;; for chord=0,10,1 do begin name='fir_fast_'+chrd_suffix(chord) ;define signal name temp=data(strtrim(name,2)) ;download the data size_test=size(temp) ;check for real data ;; ;; If data is there then write back to database ;; if (size_test(0) ne 0) then begin putmds,'f.'+name,'e+13cm^-3',-temp ;write the flipped signal
183 ;back into the database endif temp=0 ;saving virtual memory endfor ;continue with next chord ;; ;;----------------------------------------------------------------------- end
D.2.5 Manual Removal of Phase Jumps
Once you have gone through and compiled a list of shots requiring manual
repair, phase jumps can be removed using the man_fix_fast.pro routine. The
code will ask the user for a date and a shot number. It will then present a user
interface window much like that first developed by Jim Chapman in the
sawselect.pro.
Positive Polarity Phase JumpNegative
Polarity Phase Jump
Figure B.3 – Above is an example of the graphic interface of man_fix_fast.pro. The menu offers six commands, “Manual”, “Quit”, “Next Chord”, “Zoom In”, “Write Data”, and “Zoom Out”. Dotted lines point out the most probable phase jumps of each polarity.
The function of the six command buttons, seen above and below the graph,
are outlined below.
184
a) Quit – Exits the man_fix_fast routine.
b) Next Chord – Downloads and displays the data from the next FIR
chord WITHOUT storing the current chord into the database.
c) Write Data – Writes the data into the database and moves onto the
next chord. This is used in cases where the data has been modified.
d) Zoom In – Allows the user to Zoom In on the data for closer inspection.
This function is utilized by first clicking the cursor on “Zoom In”
button, then moving the cursor to the point of interest on the graph
and clicking again.
e) Zoom Out – Zooms Out. Duhh.☺
f) Manual – This is the most sensitive command. It allows the user to
remove a phase jump that he/she thinks is there, but the computer
does not recognize.
The last function allowed by the code does not utilize a command button.
As displayed in Figure B.3, the graph of the FIR data is overlaid with two
vertical dotted lines. These lines indicate where the computer thinks the most
likely phase jumps are. Often these do not agree with the user’s opinions.
However, on rare occasions they do, and the phase jump can be removed by
simply moving the cursor to the dotted line of choice and clicking. In cases where
the computer does not identify the proper jumps, they must be removed using
the “manual” function button.
185
Phase Jump
Figure B.4 – A phase jump that was missed by the computer is clearly visible around 16 ms.
For instructional purposes, let us work through the procedure of a test
case in which we modify some processed FIR data. Let us assume we have the
data as shown in figure B.4. We see that the automated routine has missed a
phase jump around 16 ms. Before removing the jump, we zoom in for a better
look. By clicking the “Zoom In” button and then clicking on our suspected phase
jump, the computer replots the FIR data around our selected point of interest.
Place Cursor Here Then Click
186Figure B.5 – After Zooming in, the existence of the phase jump is confirmed. We click “Manual”, then click at jump location to remove the jump.
Having ascertained that the phase jump is indeed real, we remove it by
first clicking on the “Manual” button, then clicking at the location of the phase
jump. Based on the slope at the selected cursor location, the computer will
automatically decide the appropriate polarity of the modification. After the
cursor location has been identified by the computer, the jump is removed, the
program zooms out, and replots the data for inspection, as shown in figure B.6.
Figure B.6 – The jump is removed, everything looks good. Ready to write the data and move on to the next chord.
Once the visual inspection is complete and everything looks fine, we click
the “Write Data” button to store the modified signal into the database. Then we
move on to the next chord.
187
For those that are interested, I list the man_fix_fast.pro program for
visual inspection.
D.2.6 The Manual Processing Code
pro fix_fast ;;------------------------------------------------------------------------- ;; ;; Nicholas E. Lanier ;; ;; 04-apr-1997 Original Version ;; ;; 10-jan-2000 Modified Version ;; ;; This program is designed to offer the FIR user a manual override ;;option to processing the Fast FIR data. Often the FIR processing code ;;is incomplete in its extraction of phase jumps and it is necessary ;;to manually process the data. To aid in the speed at which shots can be ;;processed, this program is designed for operation with an X window ;;compatible system. ;; ;;------------------------------------------------------------------------- ;; ;; Prompt user for date and shot information ;; user_input,dat,shtn ;; ;; Initialize general variables ;; !noeras=1 ;disable erase crd_sfx=['N32','N24','N17','N09','N02','P06','P13', $ 'P21','P28','P36','P43'] ;define chord suffix x_pos=[-32.,-24.,-17.,-09.,-02.,6.,13.,21.,28.,36.,43.] ;define radial chord positions z_path=2*sqrt(52.*52.-x_pos*x_pos)/100. ;calculate path length chan=0 ;initializing channel ;indicator shot: ;main loop marker ;; ;; Defining plot labels, title, xtitle, and ytitle ;; !mtitle='Shot = '+strtrim(string(fix(sht)),2)+' '+dat $ +' Chord = '+crd_sfx(chan) !ytitle='' & !xtitle='' & fancy=2
188factor=12.16*z_path(chan) ;conversion factor between ;line-averaged density to phase shot,sht ;set to appropraite shot name=strtrim('FIR_FAST_'+crd_sfx(chan),2) ;defining signal name ;; ;; Downloading data ;; dens_tm=data('fir_fast_tm') dens=data(name) array_size=n_elements(dens) ;define data array size if (array_size gt 1) then begin ;if data then continue find_jumps: zm_ind=0 ;define zoom indicator ;; ;; Call find jumps subroutine ;; find_jumps,dens,dens_tm,jumps,jump_loc,plot_data_tm,plot_data plot: ;; ;; Call plot subroutine ;; plot_template,plot_data_tm,plot_data,jumps,p_pos,zm_ind ;; ;; Ask for cursor command ;; cursor,xc,yc,4,/normal ; dummy=" " & read,dummy ;include dummy line if working ;in TEK, exclude for X term ;; ;; Main logic case statement ;; case 1 of (yc lt .12):case 1 of ;LOWER command line (xc lt .35):begin ;"ZOOM IN" ;; ;; Call Zoom_In subroutine ;; zoom_in,dens,dens_tm,plot_data_tm,plot_data,zm_ind goto,plot ;plot again after zoom end (xc lt .65):goto,write ;"WRITE" (xc lt .95):begin ;"ZOOM OUT"
189 ;; ;; Call Zoom Out subroutine ;; zoom_out,dens,dens_tm,plot_data_tm,plot_data,zm_ind goto,plot ;plot again after zoom end endcase (yc lt .87):goto,pick_jumps ;jump selected, go and fix it (yc lt .95):case 1 of ;UPPER command line (xc lt .35):begin ;"MANUAL" overide ; Manual selection of jumps only works ;is zoom has been selected. Must check that ;zoom has been conducted. if (zm_ind eq 1) then begin happy=1 ;zoom ok ;; ;; Carry out manual adjustment ;; manual,dens,dens_tm,fix_loc,ind endif else begin happy=0 ;zoom not ok endelse case 1 of (happy eq 0):goto,plot ;zoom NOT ok, just plot again (happy eq 1):goto,modify_jumps ;zoom OK, fix the jump endcase end (xc lt .65):goto,quit ;"QUIT" (xc lt .95):goto,next_chord ;"NEXT CHORD" endcase endcase pick_jumps: ;; ;; Call pick jumps subroutine ;; pick_jumps,p_pos,xc,yc,jumps,jump_loc,fix_loc,ind,plot_ind if (plot_ind eq 1) then goto,plot ;if new jumps found, then ;plot again modify_jumps: ;;
190 ;; Call modify jumps subroutine ;; modify_jumps,dens,fix_loc,ind goto,find_jumps ;selected jumps have been fixed ;find new jumps and repeat write: ;; ;; Write the data to the F level of the database ;; putmds,strtrim('F.'+name,2),'10^13 cm^-3',dens/factor factor=0 goto,next_chord ;data written, goto next chord endif next_chord: ;; ;; If NOT last channel the change chord, else change shot and ;;reset channel ;; if (chan lt 10) then begin chan=chan+1 ;next chord endif else begin sht=sht+1 ;next shot chan=0 ;reset channel endelse goto,shot ;repeat for next shot quit: ;quit selected ;; ;;------------------------------------------------------------------------- end pro user_input,dat,start_shot ;;------------------------------------------------------------------------- ;; ;; Subroutine allows the user to specify date, shots, and ;;plotting delay time. ;; ;; Variable Name Definition ;; ;; INPUTS: none ;; ;; OUTPUTS: dat user specified date ;; ;; start_shot first shot to inpect ;; ;;------------------------------------------------------------------------- ;; ;; ;; User input of date ;;
191input_date: dat=' ' ;initialize date variable read,'Enter Date of Interest(No Quotes)-> ',dat print,' ' ;printing blank line date_length=strlen(dat) ;extract string length if (date_length ne 11) then begin ;check to see if length print,'Error in Date Entry' ;is appropriate, goto,input_date ;if not then repeat entry endif ;; ;; User input of shot to modify ;; input_shots: start_shot=0 ;initializing shot variable valid_shots=0 ;valid input flag while (valid_shots eq 0) do begin on_ioerror,bad_number read,'Enter First Shot To Inspect -> ',start_shot print,' ' ;printing blank line valid_shots=1 ;inputs are OK, set flag bad_number: ;if entry error then repeat if NOT valid_shots then print,'Shots must be numbers.' endwhile set_db,'mst$data' ;set database date,dat ;set date ;; ;;------------------------------------------------------------------------- end pro find_jumps,dens,dens_tm,jumps,jump_loc,plot_data_tm,plot_data ;;------------------------------------------------------------------------- ;; ;; This subroutine is resposible for finding the two most likely ;;jumps, one of each polarity. ;; ;; Variable Name Definition ;; ;; INPUTS: dens electron density data array ;; ;; dens_tm electron density time array ;; ;; OUTPUTS: jumps the two most likely jumps in ;; time space ;; jumps_loc locations of these jumps in ;; array space ;; plot_data data to be plotted ;; ;; plot_data_tm plot data time array ;;
192;;------------------------------------------------------------------------- ;; diff=dens-shift(dens,1) ;define difference array jump_loc=[where(diff eq min(diff(10:*))), $ where(diff eq max(diff(10:*)))] ;most likely jumps occur when ;abs value of diff is largest jumps=[dens_tm(jump_loc(0)),dens_tm(jump_loc(1))] ;find when in time these jumps ;occur plot_data_tm=dens_tm ;re-define plot data time trace plot_data=dens ;re-define plot data ;; ;;------------------------------------------------------------------------- end pro plot_template,plot_data_tm,plot_data,jumps,p_pos,zm_ind ;;------------------------------------------------------------------------- ;; ;; Subroutine plot the control button template and the electron ;;density trace. ;; ;; Variable Name Definition ;; ;; INPUTS: plot_data electron density data array ;; ;; plot_data_tm electron density time array ;; ;; jumps locations of the two most likely ;; jumps ;; p_pos main viewport window parameters ;; ;; zm_ind zoom indicator ;; ;; OUTPUTS: none ;; ;;------------------------------------------------------------------------- ;; label=[' Manual !3',' Quit!3 ','Next Chord!3', $ ' Zoom In!3 ','Write Data!3',' Zoom Out!3 '] ;control button labels xx=[.05,.05,.95,.95,.05,.05,.95,.95,.05,.35,.35, $ .65,.65,.95,.95,.65,.65,.35,.35] yy=[.05,.95,.95,.05,.05,.87,.87,.12,.12,.12,.05, $ .05,.12,.12,.87,.87,.95,.95,.87] ;x and y positions for control ;buttons xpos=[.20,.50,.80,.20,.50,.80] ;y position for labels ypos=[.89,.89,.89,.07,.07,.07] ;x position for labels p_pos=[.09,.17,.94,.82] ;viewport dimensions for main ;plotting window erase ;erase before plotting
193 !type=96 ;set plot type !psym=0 ;plot lines plots,xx,yy,/normal ;plot control buttons ;; ;; plot labels ;; for i=0,5,1 do begin xyouts,xpos(i),ypos(i),strtrim(label(i),2) $ ,/normal,charsize=2,alignment=.5 endfor !type=12 ;set plot type if (zm_ind eq 1) then !psym=-4 else !psym=0
;if in zoom mode, ;accentuate each point
plot,plot_data_tm,plot_data,position=p_pos,/normal ;plot the density data oplot,[jumps(0),jumps(0)],[!cymin,!cymax],linestyle=5,thick=2 oplot,[jumps(1),jumps(1)],[!cymin,!cymax],linestyle=2,thick=2
;plot location of most likely ;jumps
;; ;;------------------------------------------------------------------------- end pro zoom_in,dens,dens_tm,plot_data_tm,plot_data,zm_ind ;;------------------------------------------------------------------------- ;; ;; Subroutine modifies the plot_data array so the a smaller time ;;window is displayed, thus allowing a closer inspection of the phase ;;behavior. The zoom location is selected via cursor. ;; ;; Variable Name Definition ;; ;; INPUTS dens electron density ;; ;; dens_tm electron density time ;; ;; OUTPUTS plot_data data to be plotted ;; ;; plot_data_tm data time array ;; ;; zm_ind zoom indicator ;; ;;------------------------------------------------------------------------- ;; zm_ind=1 ;set zoom indicator ;; ;; Await user command for zoom location ;; cursor,x_zm,y_zm,4,/data ; dummy=" " & read,dummy ;include dummy line if working ;in TEK, exclude for X term
194 t=fix(10*x_zm)/10. ;find zoom point zm_loc=where(dens_tm ge t-.3 and dens_tm le t+.3) ;zoom location in array space plot_data_tm=dens_tm(zm_loc) ;re-define plot data plot_data=dens(zm_loc) ;re-define plot data zm_loc=0 ;saving virtual memory t=0 ;saving virtual memory ;; ;;------------------------------------------------------------------------- end pro zoom_out,dens,dens_tm,plot_data_tm,plot_data,zm_ind ;;------------------------------------------------------------------------- ;; ;; Subroutine modifies the plot_data array to zoom out and display ;;the entire time trace of the electron density. ;; ;; Variable Name Definition ;; ;; INPUTS dens electron density ;; ;; dens_tm electron density time ;; ;; OUTPUTS plot_data data to be plotted ;; ;; plot_data_tm data time array ;; ; zm_ind zoom indicator ;; ;;------------------------------------------------------------------------- ;; zm_ind=0 ;set zoom indicator plot_data_tm=dens_tm ;re-defining plot data plot_data=dens ;re-defining plot data ;; ;;------------------------------------------------------------------------- end pro pick_jumps,p_pos,xc,yc,jumps,jump_loc,fix_loc,ind,plot_ind ;;------------------------------------------------------------------------- ;; ;; Subroutine prompts user for cursor imput that selects which of ;;the preselected phase jumps are to be fixed. ;; ;; Variable Name Definition ;; ;; INPUTS p_pos center viewport dimensions ;; ;; xc cursor x position ;; ;; yc cursor y position ;; ;; jumps suspected jump locations ;; in time ;; ;; jump_loc jump locations in array space ;;
195;; OUTPUTS fix_loc location of selected jump ;; ;; ind jump polarity ;; ;; plot_ind replot indicator ;; ;;------------------------------------------------------------------------- ;; check=min([xc-p_pos(0),yc-p_pos(1),p_pos(2)-xc,p_pos(3)-yc]) ;check to see if cursor point lies ;in main window if (check le 0) then begin ;and if NOT then replot plot_ind=1 endif else begin ;else find location of jump plot_ind=0 xy_new=convert_coord(xc,yc,/normal,/to_data) ;convert cursor from screen ;data coordinates dist=abs([xy_new(0)-jumps(0),xy_new(0)-jumps(1)]) ;figure out which jump was selected ind=where(dist eq min(dist)) fix_loc=jump_loc(ind(0)) ;solve for location in array space endelse ;; ;;------------------------------------------------------------------------- end pro manual,dens,dens_tm,fix_loc,ind ;;------------------------------------------------------------------------- ;; ;; This subroutine allow user to select with a cursor a site where ;;a residual phase jump is suspected of being. The location and polarity ;;of the jump extracted and sent on to be modified. ;; ;; Variable Name Definition ;; ;; INPUTS dens electron density ;; ;; dens_tm electron density time trace ;; ;; OUTPUTS fix_loc jump location ;; ;; ind jump polarity ;; ;;------------------------------------------------------------------------- ;; ;; ;; Await user specification of jumps location ;; cursor,x_m,y_m,4,/data ; dummy=" " & read,dummy ;include dummy line if working ;in TEK, exclude for X term jump_loc=min(where(dens_tm ge x_m)) ;find selected jump location
196 fix_loc=jump_loc(0) ;define jump location slope=dens(fix_loc)-dens(fix_loc-1) ;use slope to compute jump ;polarity if (slope le 0) then ind=[0] else ind=[1] ;define indicator array accordingly ;; ;;------------------------------------------------------------------------- end pro modify_jumps,dens,fix_loc,ind ;;------------------------------------------------------------------------- ;; ;; Subroutine removes a selected phase jump. ;; ;; Variable Name Definition ;; ;; INPUTS dens electron density ;; ;; fix_loc location of jump to be fixed ;; ;; ind up or down jump indicator ;; ;; OUTPUTS dens fixed electron density ;; ;;------------------------------------------------------------------------- ;; sign=[1,-1] ;sign of phase jump dens(fix_loc)=dens(fix_loc:*)+sign(ind(0))*!pi ;fixing the phase jump ;; ;;------------------------------------------------------------------------- end
So … You are now an expert in the inner workings of the FIR. Enjoy !
Big Nick
200
C: FIR Polarimetry Code Listings and Analysis Procedures
C.1 Introduction
I am a strong proponent of digital analysis. It simply rules! The drawback
of delayed results is more than made up for by the power and freedom allowed
by the digital computation technique. This is especially true for the polarimeter,
where the spindle bearing (the root of all evil) and wire meshes serve to
seriously contaminate the desired polarimetry phase. Most importantly, the
existing polarimetry hardware is unable to effectively deal with some of these
contaminants, such as the 2ω p peak from the asymmetry in the spindle bearing.
C.2 Processing Polarimetry Data
The polarimetry measurement is a thousand times more technically
challenging than that of the interferometer. The sensitivity of the measurement
to the beam polarization and diagnostic vibration make resolving the small
Faraday rotation phase shift a daunting task. On a brighter side, because the
phase shifts are very small (<0.20 radians), phase jumps are virtually non-
201
existent and hence the extensive effort required to remove the jumps from the
electron density signals is not necessary.
The processing code entrusted to extract the measured polarimetry phase,
called pol_proc.pro, is very similar to that used for the FIR interferometer.
Some notable differences include an envelope extraction technique (using
Hilbert transformations), specialized notch filters to remove the contaminant
harmonics of the spindle bearing, and a lookup table (or Calibration File) that
allows removal of the mesh distortion factors. I will not discuss any of these
items in detail for I feel that in my absence, the mandate for digital processing of
polarimetry data is non-existent. However, I present the raw codes in hopes that
the truth will someday come to light and these codes will be useful.
C.2.1 The Polarimetry Processing Code
pro pol_proc,date,shot ;;------------------------------------------------------------------------- ;; ;; Nicholas E. Lanier ;; ;; 29-mar-1999 Modified ;; ;; 21-Jan-1999 Originally written ;; ;; Program is designed to process and store digital polarimetry data. ;; ;;------------------------------------------------------------------------- ;; ;;------------------------Preliminary Definitions-------------------------- ;; set_db,'mst$data' ;Setting database shot,shot ;Setting Shot date,date ;Setting Date ;; dummy=set_errors('none') ;Set Errors to Quite Mode ;; peak=0 ;location of modulation freq pol_cal_factor=fltarr(11) ;poloidal calibration numbers reduction_factor=32 ;Store Smaller Array_size ;; time_store_name='F.FIR_FPOL_TM' ;Time Array Store Name ;time_store_name='P.FIR_FPOL_TM';Use When Runnung as PROC Code cal_store_name='F.POL_CAL_FACTORS'
202time_units='ms' ;Time Stored Units ;; ;;------------------Downloading Time And Reference Arrays------------------ ;; get_pol_time_info,time,array_size,dig_speed,abort_shot_1,zero_pt prepare_reference,reference_envelope,abort_shot_2 ;; abort_shot=min([abort_shot_1,abort_shot_2]) ;; ;;----------------------Checking For Goodness of Shot---------------------- ;; if (abort_shot le 1) then begin ;Check For Bad Shot ;; ;;--------------------Upload Calibration and Store Time-------------------- ;; upload_calibration,cal_angle,cal_phase stop ;; resize_data,array_size,reduction_factor,time,time_thn write_data,time_store_name,time_thn,time_units ;; ;;---------------------Preprocessing Reference Channel--------------------- ;; filter_envelope,array_size,dig_speed,reference_envelope,ref_env_fil,peak conjugate_reference,array_size,ref_env_fil,conj_reference ;; ;;---------------------------Begin Main Loop------------------------------ ;; for chord=0,10,1 do begin ;Channels 0 Thru 10 ;chord=2 ;; ;;--------Retrieve Store Name and Prepare Signal---------- ;; get_names,chord,read_name,store_name prepare_signal,read_name,signal_envelope,abort_channel print,read_name if (abort_channel le 1) then begin ;Check For Signal ;; ;;--------------Filter Signal and Product------------------ ;; filter_envelope,array_size,dig_speed,signal_envelope,sig_env_fil signal_envelope=0 ;Saving Virtual Memory product=conj_reference*sig_env_fil sig_env_fil=0 ;Saving Virtual Memory filter_product,array_size,dig_speed,product,filtered_product,peak product=0 ;Saving Virtual Memory ;; ;;----------------Computation Of Phase-------------------- ;; phase=atan(imaginary(filtered_product),float(filtered_product))
203 offset=avg(phase(0:zero_pt(0))) ;Phase Offset compute_factor,offset,cal_angle(*,chord),cal_phase(*,chord),factor pol_cal_factor(chord)=factor phase_out=(phase-offset)/factor phase=0 ;; ;;--------------Resizing and Storing Array---------------- ;; resize_data,array_size,reduction_factor,phase_out,phase_new write_data,store_name,phase_new,'radians' endif ;Error Checking Block endfor ;End Main Loop write_data,cal_store_name,pol_cal_factor,'unitless' endif ;Error Checking ;; ;;------------------------------------------------------------------------- end pro upload_calibration,calibration_angle,calibration_phase get_lun,n openr,n,'calibration_file' new_size=2500 calibration_angle=fltarr(new_size,11) calibration_phase=fltarr(new_size,11) dummy_name=' ' n_size=0.0 for i=0,10,1 do begin readf,n,dummy_name readf,n,n_size delta_temp=fltarr(n_size) phase_temp=fltarr(n_size) readf,n,delta_temp readf,n,phase_temp x=findgen(new_size)/float(new_size)* $ (delta_temp(n_size-1)-delta_temp(0))+delta_temp(0) calibration_angle(0,i)=x calibration_phase(0,i)=interpol(phase_temp,delta_temp,x) delta_temp=0 phase_temp=0 n_size=0 x=0 endfor close,n
204 free_lun,n end pro get_pol_time_info,time,array_size,dig_speed,abort_shot_1,zero_pt ;;------------------------------------------------------------------------- ;; ;; Subroutine downloads the time array and checks for errors. ;; ;; Variable Name Definition ;; ;; INPUTS: none ;; ;; OUTPUTS: time time array in ms ;; ;; dig_speed digitization speed ;; ;; abort_shot_1 Abort Shot Indicator ;; ;;------------------------------------------------------------------------ ;; time=data('fir_612_ref_tm') ;Downloading Time Array dummy=size(time) ;Checking Size abort_shot_1=dummy(0) ;Checking for data if (abort_shot_1 ne 0) then begin dig_speed=(time(dummy(1)-1)-time(0))/(dummy(1)-1) ;Getting Digitazation Speed time=1000*time ;Converting to ms zero_pt=min( where (time ge 0) ) array_size=dummy(1) ;Getting Array Size dummy=0 ;Saving virtual memory endif ;;------------------------------------------------------------------------- end pro prepare_reference,reference_envelope,abort_shot_2 ;;------------------------------------------------------------------------- ;; ;; Subroutine downloads reference signal, extracts the modulated ;;envelope, and checks for errors. ;; ;; Variable Name Definition ;; ;; INPUTS: None ;; ;; OUTPUTS: Reference_ Reference Envelope ;; envelope ;; ;; abort_shot_2 Abort Shot Indicator ;; ;;------------------------------------------------------------------------- ;; reference_raw=data('fir_612_ref');Downloading raw data dummy=size(reference_raw) ;Checking size abort_shot_2=dummy(0) ;Checking for errors dummy=0 ;Saving virtual memory
205 ;; ;;------------Extracting Modulated Envelope-------------- ;; if ( abort_shot_2 eq 1) then begin extract_envelope,reference_raw,reference_envelope endif ;;------------------------------------------------------------------------- end pro prepare_signal,signal_name,signal_envelope,abort_channel ;;------------------------------------------------------------------------- ;; ;; Subroutine downloads Signal data, extracts the modulated ;;envelope, and checks for errors. ;; ;; Variable Name Definition ;; ;; INPUTS: signal_name Signal name to be read ;; ;; OUTPUTS: signal_ Signal Envelope ;; envelope ;; ;; abort_channel Abort channel indicator ;; ;;------------------------------------------------------------------------- ;; signal_raw=data(signal_name) ;Downloading raw data dummy=size(signal_raw) ;Checking size abort_channel=dummy(0) ;Checking for errors dummy=0 ;Saving virtual memory ;; ;;------------Extracting Modulated Envelope-------------- ;; if (abort_channel eq 1) then begin extract_envelope,signal_raw,signal_envelope endif ;;------------------------------------------------------------------------- end pro extract_envelope,dummy,dummy_out ;;------------------------------------------------------------------------- ;; ;; Subroutine extracts modulated envelope. ;; ;; Variables Name Definition ;; ;; INPUTS: dummy raw input signal ;; ;; OUTPUTS: dummy_out amplitude modulation of ;; input signal ;; ;;------------------------------------------------------------------------- ;; dummy_out=dummy^2+hilbert(dummy)^2 ;Extract envelope dummy=0 ;Saving virtual memory ;; ;;-------------------------------------------------------------------------
206end pro get_names,chord,read_name,store_name ;;------------------------------------------------------------------------- ;; ;; Subroutine returns raw signal names and store names ;; ;; Variables Name Definition ;; ;; INPUTS: chord chord counter ;; ;; OUTPUTS: read_name name of data to be read ;; ;; store_name name of store data name ;; ;;------------------------------------------------------------------------- ;; read_name_prefix='fir_612_' ;read name prefix ;store_name_prefix='P.fir_fpol_';Use for PROC code store_name_prefix='F.fir_fpol_' ;Store location prefix name_suffix=['N32','N24','N17','N09','N02','P06', $ 'P13','P21','P28','P36','P43'] ;Suffix Array ;; ;;-------------------Defining Names----------------------- ;; read_name=strtrim(read_name_prefix+name_suffix(chord),2) store_name=strtrim(store_name_prefix+name_suffix(chord),2) ;; ;;------------------------------------------------------------------------- end pro filter_envelope,array_size,dig_speed,dummy_in,dummy_out,peak ;;------------------------------------------------------------------------- ;; ;; Subroutine filters both reference and signal around the 4kHz ;;modulated peak. Default settings are .5-7.5 kHz pass filtering. This ;;limits the bandpass to 3.5 kHz. This window can be reduced to 3-5 kHz ;;if and overall phase response is limited to 1kHz. ;; ;; Variables Name Definition ;; ;; INPUTS: array_size Array Size ;; ;; Dig_speed Digitization Speed ;; ;; dummy_in signal to be filtered ;; ;; OUTPUTS: dummy_out filtered signal ;; ;;------------------------------------------------------------------------- ;; ;; ;;-----Define Low Cut Frequency ;; low_cut_freq=500. ;Hz low_cut_point=fix(low_cut_freq*array_size*dig_speed)
207 ;; ;;-----Define High Cut Frequency ;; high_cut_freq=7500. ;Hz high_cut_point=fix(high_cut_freq*array_size*dig_speed) ;; ;;-----Begin Filtering ;; dummy_fft=fft(dummy_in,-1) dummy_fft(0:low_cut_point)=0 dummy_fft(high_cut_point:array_size-high_cut_point-1)=0 dummy_fft(array_size-low_cut_freq-1:*)=0 dummy_out=fft(dummy_fft,1) peak_loc=where( abs(dummy_fft(low_cut_point:high_cut_point)) eq $ max( abs(dummy_fft(low_cut_point:high_cut_point)) )) $ + low_cut_point peak=peak_loc(0) dummy_fft=0 ;Saving virtual memory dummy_in=0 ;Saving virtual memory ;; ;;------------------------------------------------------------------------- end pro conjugate_reference,array_size,ref_env_fil,conj_reference ;;------------------------------------------------------------------------- ;; ;; Subroutine turns the filtered reference envelope in a complex ;;function by eliminating the negative frequency spectrum. This step is ;;necessary for the complex phase decomposition calulation. ;; ;; Variables Name Definition ;; ;; INPUTS: array_size Array Size ;; ;; ref_fil_env filtered reference envelope ;; ;; OUTPUTS: conj_reference complex conjugate of filtered ;; reference ;; ;;------------------------------------------------------------------------- ref_fft=fft(ref_env_fil,-1) ;Taking fft ref_fft(array_size/2:*)=0 ;Removing negative frequencies conj_reference=conj(fft(ref_fft,1));Conjugating ref_fft=0 ;Saving virtual memory ;; ;;------------------------------------------------------------------------- end pro filter_product,array_size,dig_speed,dummy_in,dummy_out,peak ;;------------------------------------------------------------------------- ;; ;; Subrouting conducts the final filtering of the output phase.
208;;Default setting is 1Khz. Frequency response can be increased to ;;>3.5 kHz provided that ;; ;; 1) THE BANDPASS FILTERING CONDUCTED ABOVE DOES NOT ;; LIMIT THE BANDWITDH, ;; ;; 2) THE NOTCH FILTERS BE USED TO ELIMANATE THE HARMONICS ;; IN THE PHASE AT 1KHZ AND 2KHZ THAT ARISE FORM THE ASYMMETRY ;; IN THE SPINDLE BEARING HALF-WAVE PLATE. ;; ;; Variables Name Definition ;; ;; INPUTS: array_size Array Size ;; ;; Dig_speed Digitization Speed ;; ;; dummy_in signal to be filtered ;; ;; OUTPUTS: dummy_out filtered signal ;; ;;------------------------------------------------------------------------- ;; ;; ;;-------Defining High Cut Frequency ;; high_cut_freq=3500 ;Hz high_cut_point=fix(high_cut_freq*array_size*dig_speed) ;; ;;-------Filtering ;; dummy_fft=fft(dummy_in,-1) dummy_fft(high_cut_point:array_size-high_cut_point-1)=0 ;; ;;-------Notch filtering ;; peak_1khz=1*fix(peak/4) wind_1khz=2 dummy_fft(peak_1khz-wind_1khz:peak_1khz+wind_1khz)=0 dummy_fft(array_size-1-peak_1khz-wind_1khz : $ array_size-1-peak_1khz+wind_1khz)=0 peak_2khz=fix(peak/2) wind_2khz=3 dummy_fft(peak_2khz-wind_2khz:peak_2khz+wind_2khz)=0 dummy_fft(array_size-1-peak_2khz-wind_2khz : $ array_size-1-peak_2khz+wind_2khz)=0 peak_3khz=3*fix(peak/4) wind_3khz=2 dummy_fft(peak_3khz-wind_3khz:peak_3khz+wind_3khz)=0 dummy_fft(array_size-1-peak_3khz-wind_3khz : $ array_size-1-peak_3khz+wind_3khz)=0 dummy_out=fft(dummy_fft,1) dummy_fft=0 ;Saving virtual memory dummy_in=0 ;Saving virtual memory
209 ; faxis=findgen(array_size)/array_size/dig_speed ; stop ;; ;;------------------------------------------------------------------------- end pro compute_factor,offset,calibration_angle,calibration_phase,factor location=min(where(calibration_phase ge offset)) dangle=deriv(calibration_angle) dphase=deriv(calibration_phase) factor=dphase(location)/dangle(location) location=0 dangle=0 dphase=0 end pro resize_data,array_size,reduction_factor,data_in,data_out ;;------------------------------------------------------------------------- ;; ;; Subroutine rebins an input signal to a more appropriate size ;;based on the time resolution of the diagnostic. Default reduction ;;is about 32 for data stored at 1 MHZ ;; ;; Variables Name Definition ;; ;; INPUTS: array_size Array Size ;; ;; reduction_ Reduction factor ;; factor ;; ;; data_in Input data to be reduced ;; ;; OUTPUTS: data_out Reduced output data ;; ;;------------------------------------------------------------------------- ;; data_out=rebin(data_in,array_size/reduction_factor) ;; end ;;------------------------------------------------------------------------- pro write_data,store_name,store_data,units ;;------------------------------------------------------------------------- ;; ;; Subroutine writes data to the database and checks to see if ;;properly written. ;; ;; Variable Name Definition ;; ;; INPUTS: store_name Name dat is to stored as ;; ;; store_data Data to be stored ;;
210;; OUTPUTS: none ;; ;;------------------------------------------------------------------------- ;; status=put_data(store_name,store_data,units) ;Writing data ;; ;;-----If not properly written,print error message ;; if (status lt n_elements(store_data)) then begin print,'Error in Storing',store_name endif ;;------------------------------------------------------------------------- end
C.3 Mesh Calibration
It is highly recommended that the polarimeter system be calibrated prior
to any serious run campaign. Each chord is calibrated by placing a spinning half-
wave plate on top of the tank and measuring the resulting phase shift. The set
up for the calibration procedure is as follows.
a) Start up the Polarimeter, making sure of satisfactory tuning,
alignment, and power level.
b) Set up analog comparators and check “TP’s” appropriately.
c) Use the TR612’s. Set the digitization frequency to 50 kHz and set the
channel memory to 131072. This should digitize for about 2.6 seconds.
With the calibration plate spinning around 4 Hz (on a new 9V
battery), there should be many revolutions resolved.
d) Digitize the “TP” signals. BE SURE TO REMOVE THE RC PHASE
LAG INSTALLED IN THE REFERENCE CHANNEL!
211
e) In succession, take a zero shot, then place rotating wave plate on N32,
store data, N24, store data , etc.
f) Edit Make_Calibration_File.pro accordingly by inserting the ap-
propriate calibration date and shot numbers.
g) Run Make_Calibration_File.pro, everything else should be auto-
mated.
212pro make_calibration_file ;; ;; date='21-jan-1999' ;Date of Calibration ;; ;; n32,n24,n17,n09,n02,p06,p13,p21,p28,p36,p43 ;; shot_cal=203 ;Calibration Shot shots=[ 204,205,206,207,208,209,210,211,212,215,214] ;Shot list ;; ;;-------------------open calibration file--------------------- ;; openw,1,'calibration_file' ;; chord_sfx=['n32','n24','n17','n09','n02','p06','p13','p21','p28','p36','p43'] ;; ;;---------------------Main shot loop------------------------- ;; for i=0,10,1 do begin sig_nam=strtrim('fir_612_'+chord_sfx(i),2) cal_temp,date,shots(i),shot_cal,sig_nam,p,seg,off,rat,delta num=n_elements(seg) printf,1,sig_nam printf,1,num printf,1,delta,seg endfor close,1 end pro cal_temp,date,cal_shot,off_shot,channel,cal_phase,seg_phase,offset, ratio,delta ; ;------------------Set Up-------------------- ; label=['Calibration Shot','Zero Point Shot'] set_db,'mst$data' shots=[cal_shot,off_shot] date,date dig_speed=50000. ;50 kHz dt=1./dig_speed ;sec ; ;-----------------Main Loop------------------- ; for j=0,1,1 do begin print,label(j) shot,shots(j) ;---------------Downloading Data--------------- ;
213 sig=data(channel) ref=data('fir_612_ref') array_size=n_elements(sig) ; ;---------------Preparing Signal--------------- ; sig=sig-avg(sig) sig_amp=sig^2+hilbert(sig)^2 sig_new=sig/sqrt(sig_amp) ; ;---------Finding Wave Plate Frequency--------- ; temp=fft(sig_amp,-1) peak_loc=30+where( abs(temp(30:50)) eq max(abs(temp(30:50))) ) temp=0 factor=dig_speed/array_size plate_frequency=factor*peak_loc(0) print,'Calibration Plate Frequency',plate_frequency ; ;-------------Preparing Reference-------------- ; ref_amp=ref^2+hilbert(ref)^2 ref_new=ref/sqrt(ref_amp) ; ;-----------Finding Spindle Frequency---------- ; temp=fft(ref_new,-1) & temp(0)=0 peak_loc=10000+where( abs(temp(10000:11000)) eq $ max(abs(temp(10000:11000))) ) spindle_frequency=factor*peak_loc(0) print,'Spindle Bearing Frequency',spindle_frequency ; ;------Filtering Around Spindle Frequency------ ; window_size=fix(2*plate_frequency/factor) temp(0:peak_loc(0)-window_size)=0 temp(peak_loc(0)+window_size:*)=0 ref_con=conj(fft(temp,1)) ref_amp=0 ref_new=0 temp=0 ; ;--------------Complex Product----------------- ; prod=ref_con*sig_new sig_new=0 ref_con=0 ; ;-------------Filtering Product---------------- ; temp=fft(prod,-1) cut_point=fix(2*plate_frequency/factor) temp(cut_point:array_size-cut_point-1)=0 prod_fil=fft(temp,1) temp=0 ;
214;--------------Extracting Phase---------------- ; phase=atan(imaginary(prod_fil),float(prod_fil)) ; ;---------Segmenting Measured Phase------------ ; if ( j eq 0) then begin jump_loc=where( (phase-shift(phase,1)) le -!pi ) num=n_elements(jump_loc) sum=fltarr(jump_loc(3)-jump_loc(2)+1) for i=2,num-3,1 do begin sum=sum+phase(jump_loc(i):jump_loc(i+1)-1) endfor seg_phase=sum/(num-4) seg_phase_size=n_elements(seg_phase) delta=findgen(seg_phase_size) $ *2*!pi*plate_frequency/dig_speed/2. cal_phase=phase jump_loc=0 sum=0 num=0 endif ; ;---------------Phase Offset------------------- ; offset=avg(phase(10000:121072)) phase=0 ; ;-------------TM/TE Extraction----------------- ; if ( j eq 1) then begin location=where( seg_phase ge offset ) omega_seg_phase=deriv(seg_phase) omega_seg_phase_avg= $ avg(omega_seg_phase(location(0)-3:location(0)+3)) ratio=omega_seg_phase_avg / (2*!pi* $ plate_frequency)*dig_speed location=0 endif endfor ;j loop end
215
D: Hα , CO2, and Other Processing Codes
D.1 Introduction
In early 1999, there was a big push to update our proc codes to be IDLv5.0
compatible. I ended up rewriting BR_PROC.pro, PC_PROC.pro, and CO2_PROC.
They have all been thoroughly tested and can be found in the [LANIER.PROC]
directory. In this appendix, I only present (without discussion) the Hα and the
CO2 codes. The others are readily available for those that are interested.
D.2 The Hα Processing Code pro hal_proc,date,shot ;;--------------------------------------------------------------------------- ;; ;; Nicholas E. Lanier ;; ;; 28-mar-1999 Modified Last ;; ;; 28-mar-1999 Originally Written ;; ;; Program downloads and processes H_alpha data from the ;;H_alpha array ;; ;;--------------------------------------------------------------------------- ;; ;;-------------------------------Setting Up----------------------------------
216 ;; time_units='seconds' data_units='1/s cm^2' calibration_units='I/cm^2s' set_db,'mst$data' ;Setting database shot,shot ;Setting shot date,date ;Setting date ;; ;;----------------------------Calibration Info------------------------------- ;; ;; Currently we have 2 h_alpha systems on MST, the single chord H_ALPHA ;;and the H_ALPHA array. The single chord h_alpha detector was originally ;;calibrated by Dimitri (circa 1995). I have since redone the calibration and ;;think the results appear more reasonable. For the record, his number was ;; ;; 712.*95.3*1E13*H_alpha_2.3 total ionizations in MST per sec ;; ;; The above calibration routinely gave particle confinement times a factor ;;of ten greater than those for energy. ;; ;; As stated before I have redone this calibration (9/98). I found that ;; the signal chord h_alpha gives about 1V/5.35e17 excitations. ;; final_calibration = 5.35e17 ;exc/s cm^2 ;; ;; We then introduce another calibration factor 'array_to_sc' which is ;;the calibration between the single chord h_alpha and the h_alpha array ;;chord with the similar impact parameter. array_to_sc = 5.0 ;unitless ;; ;; The calibrations were conducted with an amplifier gain of 1000. If ;;this has been changed, you must change the 'amp_gain_setting amp_gain_setting = 1000. amp_adjustment = amp_gain_setting/1000. ;; ;; The cross-chord calibration array is given below. ;; cross_calibration=[1.0,1.0,.973,.715,.965,.876,.847, $ .925,.935,1.096,1.015] ;unitless ;; ;; For plasma temperatures in MST the ratio of ionizations to excitations ;;is about 11./1. So we introduce ;; exc_to_ion_ratio = 1./11. ;unitless ;;
217 ;; The total calibration numbers for the h_alpha array is total_calibration=exc_to_ion_ratio*final_calibration* $ array_to_sc*amp_adjustment/cross_calibration ;ion / s cm^2 ;; ;; This final calibration number gives the total number of ionizations ;;occuring along the line of sight of the h_alpha array. In other words the ;;chord integrated ionizations occuring per second. ;; ;;---------------------------------------------------------------------------- ;; ;; ;; Defining the read and store names ;; read_name_prfx='HAL_A12_' store_name_prfx='F.HAL_FIN_' chord_sfx=['N32','N24','N17','N09','N02','P06','P13','P21','P28','P36','P43'] ;; ;; Download time trace and check for data ;; time=data(read_name_prfx+'P06_tm') dummy=size(time) abort_shot=dummy(0) ;; ;; If all OK than begin main loop ;; if (abort_shot ne 0 ) then begin ;; ;; Write time trace ;; write_data,'F.HAL_FIN_TM',time,time_units write_data,'F.HAL_FIN_CALIB',total_calibration,calibration_units ;; ;; Main loop ;; for i=2,10,1 do begin ;; ;; Download raw data and subtract offset ;; raw_data=data(read_name_prfx+chord_sfx(i)) raw_data=raw_data-avg(raw_data(0:100)) ;; ;; Process data ;; store_data=raw_data*total_calibration(i)
218 ;; ;; Write data ;; store_name=store_name_prfx+chord_sfx(i) write_data,store_name,store_data,data_units raw_data=0 ;Saving virtual memory endfor endif ;; ;;------------------------------------------------------------------------- end pro write_data,store_name,store_data,units ;;------------------------------------------------------------------------- ;; ;; Subroutine writes data to the database and checks to see if ;;properly written. ;; ;; Variable Name Definition ;; ;; INPUTS: store_name Name data is to stored as ;; ;; store_data Data to be stored ;; ;; OUTPUTS: none ;; ;;------------------------------------------------------------------------- ;; status=put_data(store_name,store_data,units) ;Writing data ;; ;;-----If not properly written,print error message ;; if (status lt n_elements(store_data)) then begin print,'Error in Storing ',store_name endif ;;------------------------------------------------------------------------- end
D.3 The CO2 Processing Codepro co2_proc,date,shot ;;--------------------------------------------------------------------- ;; ;; Nicholas E. Lanier ;; ;; 28-mar-1999 Last modified ;; ;; 01-feb-1999 Original version ;; ;; Program computes and stores the single chord CO2 data into ;;the database. ;; ;;---------------------------------------------------------------------
219 ;; ;;----------------------------Setting Up------------------------------- ;; set_db,'mst$data' ;setting database date,date ;setting date shot,shot ;setting shot ;; ;;-----------------------Parameter Definitions-------------------------- ;; m_to_cm=1E-6 ;Convert m^3 t cm^3 lambda_co2=10.58E-6 ;wavelength of co2 (m) lambda_hene=3.39E-6 ;wavelength of Hene (m) c=2.997E+8 ;speed of light (m/s) e=1.602E-19 ;electron charge (C) m_e=9.11E-31 ;electron mass (kg) epsilon=8.85E-12 ;permativity (F/m) path_length=4*.52*2 ;Laser Path Length (m) ;; ;; Interferometer conversion factor from phase to density ;; factor= e^2 / ( 4 * !PI * c^2 * m_e * epsilon ) ;conversion factor ;; ;; Signal names of raw data ;; signal_names=['co2_sin_r','co2_cos_r','hene_sin_r','hene_cos_r'] ;; ;; Download co2 time trace and check for valid signal ;; get_co2_time_info,time,array_size,dig_speed,abort_shot ;; ;; Begin main loop ;; if (abort_shot ne 0) then begin abort_ind=0 ;reset abort indicator signals=fltarr(array_size,4) ;initialize signal array ;; ;; Reading in and checking for valid data ;; for name=0,3,1 do begin prepare_signal,signal_names(name),temp_signal,abort_channel signals(0,name)=temp_signal abort_ind=abort_ind+abort_channel temp_signal=0 endfor ;; ;; If all signals 'good' continue on ;; if (abort_ind eq 4 ) then begin
220 ;; ;; Calulate HeNe and Co2 Phase ;; co2_phase=atan(signals(*,0),signals(*,1)) hene_phase=atan(signals(*,2),signals(*,3)) signals=0 ;; ;; Remove te pahse jumps ;; remove_phase_jumps,array_size,co2_phase remove_phase_jumps,array_size,hene_phase ;; ;; Calculate appropriate wvaelength ratio ;; compute_ratio,co2_phase,hene_phase,ratio wavelength_factor=1./(1.-ratio^2) ;; ;; Extract final density ;; density=( 1./ ( factor * lambda_co2 * path_length )) * $ ( co2_phase - ratio * hene_phase )*m_to_cm * $ wavelength_factor ;; ;; Subtracting the offset ;; density=temporary(density-avg(density(array_size-2000:*))) hene_phase=0 ;Saving virtual memory co2_phase=0 ;Saving virtual memory ;; ;; Write the data to the database ;; store_data,time,density endif endif ;; ;;----------------------------------------------------------------------- end pro get_co2_time_info,time,array_size,dig_speed,abort_shot_1 ;;----------------------------------------------------------------------- ;; ;; Subroutine downloads the time array and checks for errors. ;; ;; Variable Name Definition ;; ;; INPUTS: none ;;
221 ;; OUPUTS: time time array in ms ;; ;; dig_speed digitization speed ;; ;; abort_shot_1 abort shot indicator ;; ;;----------------------------------------------------------------------- ;; time=data('co2_cos_r_tm') ;Downloading time array dummy=size(time) ;Checking size abort_shot_1=dummy(0) ;Checking for data if (abort_shot_1 ne 0) then begin dig_speed=(dummy(1)-1)/(time(dummy(1)-1)-time(0)) ;Getting digitization speed time=1000*time ;Coverting to ms array_size=dummy(1) ;Getting array_size dummy=0 ;Saving virtual memory endif ;; ;;----------------------------------------------------------------------- end pro prepare_signal,signal_name,signal,abort_channel ;;----------------------------------------------------------------------- ;; ;; Subroutine dowloads interferometer signals and checks for errors. ;; ;; Variable Name Definition ;; ;; INPUTS: signal_name Signal name to be read ;; ;; OUTPUTS: signal Raw signal ;; ;; abort_channel Abort channel indicator ;;----------------------------------------------------------------------- ;; signal=data(signal_name) ;Downloading raw data dummy=size(signal) ;Checking size abort_channel=dummy(0) ;Checking for errors dummy=0 ;Saving virtual memory ;; ;;----------------------------------------------------------------------- end pro remove_phase_jumps,array_size,phase ;;----------------------------------------------------------------------- ;; ;; This subroutine finds the number and location of any present ;;phase jumps and calls "FIX_PHASE" for there removal. ;; ;; ;; Compute phase difference between each point dphase=temporary(phase-shift(phase,1)) ;; ;; Reorder phase differences
222 sort_order=sort(dphase) sorted_dphase=dphase( sort_order ) ;; ;; Find number of points where dphase ge or le !pi ;; These are phase jumps that must be remove ;; up_jump_locs=where( sorted_dphase ge !pi ) down_jump_locs=where( sorted_dphase le -!pi ) ;; ;; Find the maximum and minimum number of jumps ;; max_jump_num=max( [ n_elements(up_jump_locs), $ n_elements(down_jump_locs) ] ) min_jump_num=min( [ n_elements(up_jump_locs), $ n_elements(down_jump_locs) ] ) ;; ;; If there are jumps the begin ;; if ((up_jump_locs(0) ne -1) or (down_jump_locs(0) ne -1)) then begin ;; ;; Compute path integral ;; path_integral=sorted_dphase(0:(array_size/2-1))+ $ reverse(sorted_dphase(array_size/2:*)) dphase=0 ;; ;; Find where the path length would be minimized ;; this should be where the optimum number of ;; phase jumps have been removed ;; min_location=where( $ min(path_integral(min_jump_num:max_jump_num)) $ eq path_integral(min_jump_num:max_jump_num) ) $ + min_jump_num - 1 path_integral=0 ;; ;; Find where the phase jumps are ;; up_jumps=sort_order(0:min_location(0)) down_jumps=sort_order(array_size-min_location(0)-1 : $ array_size-1) min_location=0 sort_order=0
223 ;; ;; Define the final jump locations (total_jumps) and ;; whether they are + or - jumps (total_sign) ;; total_jumps=[up_jumps,down_jumps] total_sign=[make_array(n_elements(up_jumps),value=+1.0),$ make_array(n_elements(down_jumps),value=-1.0)] ;; ;; Reorder from lowest to greatest array position ;; jump_order=sort(total_jumps) total_jumps=total_jumps(jump_order) total_sign=total_sign(jump_order) jump_order=0 ;; ;; Remove the actual phase jumps ;; fix_phase,array_size,total_jumps,total_sign,phase total_jumps=0 total_sign=0 endif ;; ;;----------------------------------------------------------------------- end pro fix_phase,array_size,jump_locs,sign,phase ;;----------------------------------------------------------------------- ;; ;; This subroutine removes the phase jumps. ;; ;; ;; Find how many jumps ;; num=n_elements(jump_locs) jump_locs=[jump_locs,array_size] sum=0 ;; ;; Time to remove jumps ;; for i=0,num-1,1 do begin ;; ;; Keeping track of total shift ;; sum=sum+sign(i) ;; ;; Modigfying the phase array ;;
224 phase(jump_locs(i):jump_locs(i+1)-1)=phase(jump_locs(i) $ :jump_locs(i+1)-1) + 2*!pi*sum endfor ;; ;;----------------------------------------------------------------------- end pro compute_ratio,co2_phase,hene_phase,ratio ;;----------------------------------------------------------------------- ;; ;; Before every shot, the co2 moves the upper mirror back and forth ;;some distance. This change in path length appears as a phase shift in both ;;the HeNe and CO2 phases. Since the path length change is the same for both ;;lasers, the measured phase shift will be proportional to their wavelength ;;factors. ;; ;; The subroutine just measure the ratio of the slopes in the measered ;;phases during this change in path length. ;; ;; ;; Start and finish of mirror motion ;; start_pt=1000 end_pt=10000 ;; ;; Finding slope. ;; a=poly_fit(hene_phase(start_pt:end_pt),co2_phase(start_pt:end_pt),2) ;; ;; Slope = ratio ;; ratio=a(1) ;; ;;----------------------------------------------------------------------- end pro store_data,time,density ;;----------------------------------------------------------------------- ;; ;; This subroutine prepares the data for storage. ;; ;; ;; Resize data ;; store_size=8192 store_start=min(where(time ge -10. )) store_location=findgen(8192)+store_start(0) ;; ;; Writing the data ;;
225 ; write_data,'P.N_CO2_TM',time(store_location),'ms' ; write_data,'P.N_CO2',density(store_location),'cm-3' store_location=0 ;Saving virtual memory store_start=0 ;Saving virtual memory store_size=0 ;Saving virtual memory ;; ;;------------------------------------------------------------------------- end pro write_data,store_name,store_data,units ;;------------------------------------------------------------------------- ;; ;; Subroutine writes data to the database and checks to see if ;;properly written. ;; ;; Variable Name Definition ;; ;; INPUTS: store_name Name dat is to stored as ;; ;; store_data Data to be stored ;; ;; OUTPUTS: none ;; ;;------------------------------------------------------------------------- ;; status=put_data(store_name,store_data,units) ;Writing data ;; ;;-----If not properly written,print error message ;; if (status lt n_elements(store_data)) then begin print,'Error in Storing',store_name endif ;;------------------------------------------------------------------------- end
227
E: Hα Array Components List
E.1 Hα Parts List
As a matter of public record I thought it useful to document the
components of the Hα Array. See figure 3.3 (page 47) for placement information.
Hα ALPHA PARTS LIST Item Description Supplier Stock Number Amici Roof Prism
A=14mm,B=16mm,C=15 mm
Edmund Scientific P45,261
Focusing Lens
Plano-Convex Lens 10cm F. L.
Edmund Scientific P45,260
Hα Filter 656.3 nm, 11.5 nm FWHM, 0.5” Dia.
Coherent-Ealing 42-5496
Photo-Diode Detector
.1”X.1” Active Area 300 kHz Cutoff
Advanced Photonix Inc.
SD 112_452_ 11_221
Circular Receptacle
Shell Size 0 4 Contact
Newark Electronics JBX ER 0G04 FC SDS
Circular Plug
Shell Size 0 4 Contact
Newark Electronics JBX FD 0G04 MC SDS
Table E.1 – The Hα parts list.
228
F: The SXR Ratio. What does it really mean?
F.1 Dispelling the Myth Behind the SXR Ratio
A number of years ago, dating back to the days of the Gulf War, some
eager young graduate student placed two soft x-ray detectors on MST (No…it
wasn’t me, Brett or even Neal). These detectors were just surface barrier diodes
each with a beryllium filter of differing thickness. For those of you keeping score
at home, the thicknesses were 0.3 mils (7.6 μm) and 0.6 mils (15.2 μm). As time
passed, these detectors were assimilated into that elite group known as the
operations diagnostics and are still to this day stored for every MST shot. This in
and of itself is not a problem, however, somewhere back before the existence of
PPCD, the idea formed that the ratio of these two measurements would be an
indicator of electron temperature. After all, the ratio does drop at the sawtooth
crash, when the electron temperature is known to fall. When PPCD began to
show favorable results, the SXR ratio miraculously increased to levels never
before observed, further cementing this temperature myth into law.
229
0.0
0.2
0.4
0.6
0.8
1.0
150 200 250 300 350Plasma Current (kA)
0.0
0.5
1.0
1.5
2.0
0.0
0.2
0.4
0.6
Ele
ctro
n D
ensi
ty(1
E+1
3 cm
)
-2(a
u)H
αS
XR
Rat
io(u
nitle
ss)
a)
b)
c)
Figure F.1 – The (a) chord-averaged electron density, (b) the Hα central chord, and (c) SXR ratio for standard discharge. Note the flat behavior of the SXR ratio as current increases. The SXR ratio appears to be insensitive to election temperature.
The myth clashes with reality when one observes that the SXR ratio does
not vary with plasma current. Figure F.1 displays the chord-averaged electron
density, chord-integrated Hα, and SXR ratio for an ensemble of 200 shots at
varying currents. By holding the electron density and Hα (neutral concentration)
fixed and ramping up the current, the only free parameters are the electron
temperature and the particle confinement time. When varied from 150 to 350
230
kA, the SXR ratio shows no change, while the electron temperature has surely
risen.
This is not that surprising when one examines more closely what the SXR
diodes are really measuring. The beryllium foils are, for the most part,
broadband high pass x-ray filters, where the 0.3 mil passes photons of energy
greater than 400 eV, and the 0.6 mil foil begins to transmit at 600 eV (see figure
F.2). For these energies, the dominant emission is from O VII and O VIII and as
fate would have it, the two foils are excellent at separating between the two. In
other words, the SXR ratio is simply a measure of the O VIII emission over the
combined emission from both O VII and O VIII.
SXR_BE_1SXR_BE_2
1.E+0
1E-1
1E-2
1E-3
1E-4
Tran
smis
sion
0 5 10 15 20 25 30Wavelength (Ang.)
O V
III
O V
II
Figure F.2 – The transmissions of SXR_BE_1 (0.3 mil Beryllium), and SXR_BE_2 (0.6 mils of Beryllium). The principal lines of O VII and O VIII are at 21.6 and 18.8 Angstroms respectively.
Recalling equation 4.12 (back on page 91), the state ratio between O VII
and O VIII is determined by the balance between ionization, charge exchange,
231
and transport. During PPCD, electron temperature rises increasing ionization,
neutral population falls decreasing charge exchange recombination, and particle
confinement is improved thereby reducing direct loss of high charge state
impurities. All these factors work together to increase the O VIII / O VII fraction
and increase the SXR ratio. Although the electron temperature does play a role,
the change in the SXR ratio results more from the depleted neutral density and
an improved particle confinement.
So, as I step off my soapbox, I would like to express my gratitude for
having the opportunity to get this off my chest and I would like to reiterate how
much I have enjoyed my graduate career at Wisconsin. For those remaining,
enjoy your time there, there is no other place quite like it. ☺
“Big” Nick Lanier
