Electron Density Fluctuations and Fluctuation-Induced Transport in the Reversed-Field Pinch

ELECTRON DENSITY FLUCTUATIONS AND FLUCTUATION-INDUCED TRANSPORT IN THE REVERSED-FIELD PINCH

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

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

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

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

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: 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

BP BT

r B

BT Small & Reversed at Edge

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.

d d K = A • B dV ≈ 0 (1.1) dt o dt ∫ V 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 (b˜ ) 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 = 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 = mn (1.3)

where m 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. 6 q(r)

Tokamak

~1 Dominant Resonant Surfaces ~0.2 RFP 0 radius, r 0 a

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

W ~ ˜ / q=m/n Br B

⇒

resonant magnetic surface island

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 typical island width 1,5 1,6 1,7 … q(r) (m,n)

magnetic stochasticity

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 ( Dst ) would take the form,

˜ 2 br 1 Dst ≈ π , (1.4) Bo 1 LA +1 λmfp

˜ where br 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, br 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

−4 for MST is about a meter and therefore D st ≈.3 →1.2 ×10 m. The critical aspect 9 behind this loss mechanism is that the diffusion loss rate will be proportional to the particle’s parallel velocity leading to

DLoss ∝ 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

2 ⎛ 1 ∂n 1 ∂T eEA ⎞ Γr =− Dst vT n + + , (1.6) π ⎝ n ∂r 2T ∂r T ⎠

where v T is the electron thermal velocity, n and T are the electron density and temperature, EA is the ambipolar electric field, and Dst 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,

∂n e +∇•Γ =S , (1.7) ∂t e

where Γ=ne v . Expanding Γ into its equilibrium and fluctuating components, we see that

˜ ˜ ˜ Γ=()no + n˜ (vo + v )=novo + n˜ v +nov + 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 = nov or + 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 or ≈ 0 ), leaving the fluctuation-induced transport term solely responsible for the overall radial particle flux. The fluctuation-induced particle flux is defined as

Γ fluctuation −induced = n˜ v˜ r = γ n˜ Amp v˜ r Amp cos(δ nv ), (1.10)

where n˜ Amp and v˜ rAmp are the amplitudes of the density and radial velocity

fluctuations respectively, and 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 r r 2 fluctuations. The plasma desires a flat parallel current density profile ( J • B B ), 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.

E θ,Jθ

C1 C2 C3 C4

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 MST E (a) θ 0 (V/m) 5 PPCD –10 4 b˜ 3 rms B 2 (%) 1 0 510152025 Time (ms)

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 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.

REFERENCES

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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).

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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.

The propagation of an electromagnetic wave in plasma is depicted in equation

6,7 2.1. The index of refraction (μ s, f ) for the slow and fast waves with frequency ω are

−1 2 2 2 2 2 2 ⎡ 12 ⎤ ω pe ωce sin θ ωce sin θ 2 ()μ s, f =1 − 2 1 − 2 ± 2 ()1 + F , (2.1) ω ⎢ ω 21− ω 2 ω2 ω 21− ω 2 ω2 ⎥ ⎣ ()pe ()pe ⎦

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 F is defined as

2ω ⎛ ω 2 ⎞ cosθ F = ⎜ 1 − pe ⎟ . (2.2) ω ⎝ ω 2 ⎠ sin2 θ ce

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 r r ( k ⊥ B ). 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 r r ( E || B ) becomes

12 ⎡ ω2 ⎤ μ = 1 − pe . (2.3) ord ⎢ 2 ⎥ ⎣ ω ⎦

19 r r For the extraordinary wave ( E⊥ B ), equation 2.1 simplifies to

12 ⎡ 2 2 2 ⎤ ωpe (ω − ω pe ) μ = 1 − . (2.4) ext ⎢ ω 2 ω2 − ω 2 − ω 2 ⎥ ⎣ (pe ce)⎦

2 For MST parameters, ω2 = eB m ≈ 3 ×10+20 s−2 and ω2 = e2n ε m ≈ 3 ×10 +22 s−2 . ce ()e pe e o e +12 −1 Furthermore, at the laser wavelength of 432 microns, ω ≈ 4.3 ×10 s and ω2 ≈ 2 ×10 +25 s−2 . Given that ω2 << ω2 and ω2 << ω2 , a little algebra and a ce pe pe

binomial expansion later, equation 2.4 can be simplified, yielding that μ ord ≈ μext , where

12 ⎡ ω 2 ⎤ 1 ⎛ ω 2 ⎞ μ ≈ μ ≈ 1 − pe ≈ 1 − ⎜ pe ⎟ . (2.5) ord ext ⎢ 2 ⎥ 2 ⎣ ω ⎦ 2 ⎝ ω ⎠

Recalling that k = μω c , and that ω2 = n e2 ε m where ε is the free space pe e o e o

permittivity and ne is the electron density, then the phase difference (Φ) between a wave that travels through plasma vs. air will be

⎛ ω 2 ⎞ e2 ω ⎜ pe ⎟ λ Φ= ()kvac − kplasma dz = 2 dz = 2 ne()r dz . (2.6) ∫ 2c ∫ ⎝ ω ⎠ 4πc m ε ∫ e o

Substituting in the relevant MKS values, Φ becomes

Φ=2.814 ×10−15 λ n (r)dz, (2.7) ∫ e

where λ is the FIR laser wavelength, ne is the electron density, and z 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

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.

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.

Mirrors Output Coupler

Gas Flow Out Cathode (23 kV) Gas Flow Out

Grating Anode (Ground) Gas Flow In Monitoring Beam

Anode (Ground) Piezo-electric Transducer (PZT)

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

8 states. The triatomic CO2 molecule is subject to three types of vibrational excitation – symmetric stretching, bending, and asymmetric stretching (figure

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.

OOC C O O Equilibrium Bending

OOC OOC 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

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.

(100) 6 4 5 3 1 4 2 3 Collisional Transfer of Vibrational Energy 1 2 0 1 0 10P6 Stimulated 6 Emission 5 4 (001) 9R6 3 2 6 1 5 0 4 3 (020) 2 1 0 Collisional 6 De-excitaion 5 4 (010) Excitation via 3 2 1

0 High Voltage Discharge

(000) 0 CO2 N2

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.

Wire Mesh CO Pumping Reflective Coating (10.6 μm) 2 (100 LPI) Beam

Metallic Corrugated Waveguide

Quartz Etalon 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.

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).

f=120 cm f=100 cm Power (au)

-10 -5 0 5 10 Radius (1/8's in.)

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 –

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.

P28 P43 P13 N17 N02 N32

o 5 N24 N09 P06 P21 P36

Ro = 1. 5 m a=.52m

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 Impact Parameter Toroidal Angle Chord Length (N/P) R-Ro (cm) φ (degrees) 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 ≈ 10−10 W / 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.

1000

800

600

Gain 400

200

0 0 0.5 1.0 1.5 2.0 2.5 Frequency (MHz)

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.

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

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.

offset xR()tn = AR(tn )cos[ω IF(tn )tn ]+ xR (2.8) offset xS ()tn = AS()tn cos[]ω IF ()tn tn + φ()tn + xS

We isolate φ()tn via digital complex demodulation. This technique involves

three steps: pre-processing of the reference ( xR(tn )) and signal ( xS ()tn ) data, filtering, and phase extraction. Expanding equation 2.8 into its exponential form

and removing the equilibrium offsets, xR(tn ) and xS (tn ) become,

xR()tn = []AR(tn) 2 {exp[iω IF (tn )tn ]+ exp[−iω IF (tn )tn ]} (2.9)

xS ()tn = AS ()tn 2 exp iω IF ()tn tn + iφ()tn + exp −iω IF ()tn tn − iφ()tn []{ [][]}

Additional processing is required for xR(tn ), in which the negative frequencies are filtered out (−ω IF → 0), and xR(tn ) is conjugated, forming

x t = A t 2 exp −iω t t . (2.10) R_Conj ( n ) [ R( n ) ] [ IF ( n ) n ]

Multiplying the pre-processed signals yields,

xProduct ()tn = xR_ Conj (tn )xS(tn )= [AR(tn )AS (tn ) 4] , (2.11) × exp −i2ω ()t t +−iφ()t + exp iφ()t { []IF n n n []n }

which has two components, a high frequency 2ωIF term and the desired low frequency φ()tn term. Digital filtering removes the 2ωIF term leaving

x t = A t A t 4 exp iφ t . (2.12) Filtered ( n ) [ R( n ) S ( n ) ] [ ( n )]

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

-1 φ()tn = tan {Im[xFiltered (tn )] Rex[ Filtered (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.

FWHM ≈ 0.03 radians Counts (au)

-0.10 -0.05 0.00 0.05 0.10 Phase (radians)

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 2.0

-3 1.5

1.0

(1E+13 cm ) 0.5 Electron Density Electron

0.0 0620 40 0 Time (ms)

1.0

0.8 -3 0.6

0.4 (1E+13 cm )

Electron Density Electron 0.2

0.0 15 16 17 18 19 20 Time (ms)

1.0

0.8 -3 0.6

0.4 (1E+13 cm ) Electron Density Electron 0.2

0.0 16.5 16.6 16.7 16.8 16.9 17.0 Time (ms)

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).

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).

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.

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 + Transport H

Transport Ionization

o Fueling Dissociation Γ H2 H Charge Exchange Ionization

Transport o* 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 = v N ne σv collisions , (3.4)

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

-6 Charge Exchange Recombination

-3 -1 -8 Electron -10 Impact Ionization

-12 Proton Impact Ionization -14 Radiative Recombination Log (Collision Rate (cm s )) -16 1 10 100 1000 Temperature (eV)

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.

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

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

7 hydrogen. For MST parameters, Hα production is proportional to the neutral

8,9 ionization rate. In other words, the Hα emission is given by

46 n N v , (3.7) γ Hα = α e σ ion

where ne is the electron density, 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.

Support Amici Prism Slit Detector Disc Assembly

Collimating Tube

Focusing Lens Hα Filter Photodiode 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.

1.0

0.8

0.6

0.4

0.2 Normalized Transmission 0.0 630 640 650 660 670 Wavelength (nm)

10 Figure 3.4 – The Hα filter transmission 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

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

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.

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 = 0 tearing mode activity, resonant at the reversal surface.

51 2.0 r/a=0.69 1.5 -1

-2 1.0 cm s )

α 0.5 H Photons (1Ε+18 0.0

-0.5 -20 0 20 40 60 80 Time (ms)

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

-1 3.0 -2 cm s ) 2.0 r/a=0.83 α H Photons

(1Ε+17 r/a=0.50 1.0 r/a=0.11 0.0 -2.0 -1.0 0.0 1.0 2.0 Time (ms)

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 4.0

t = 0.0 3.0 t = -1.0 -2 -1

cm s ) 2.0 α H Photons

(1Ε+17 1.0

0.0 -40 -20 0.0 +20 +40 +60 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 4.0 Standard -1 3.0 -2 α

H 2.0

1.0

(1E+17 cm s ) PPCD 0.0 Standard 8.0 -2 -1 6.0

4.0 PPCD

Electron Density 2.0 (1E+14 cm s ) Profile Control 0.0 5 10152025 Time (ms)

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 n N v , and that both α and σv are essentially γ Hα = α e σ ion ion

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,

−8 3 −1 σv ion ≈ 3.0 ×10 cm s . 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,

+7 −3 N = S n e σv ion ≈ 3.33 ×10 S n e cm , (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 Standard 2.67 -3 2.00

1.33

( 1E+11 cm ) 0.67 Neutral Density PPCD Trigger Point PPCD 0.00 5 10152025

Time (ms)

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.

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 1E+12 -3 (cm )

1E+11 Neutral Density 1E+10 0.0 0.2 0.4 0.6 0.8 1.0 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.

1E+12 -3 1E+11 (cm )

1E+10

1E+09

Neutral Density Resolution Limit 1E+08 0.0 0.2 0.4 0.6 0.8 1.0 r/a

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.

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,

P N ≈ 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

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.

-14

2 Core Neutral Temperature -15

-16 Neutral Beam Energy -17 C 6+ + H C5+ + H + Log [Cross Section (cm )] -18 0 +1+2+3+4+5 Log [Energy (eV/amu)]

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.

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.

REFERENCES

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).

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).

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

(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

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

all the terms outlined in figure 4.1 are important in typical MST discharges and simplifications can be made.

Ionization Radiative and Dielectronic Radiative and Dielectronic of i-1 state Recombination of ith state Recombination of i+1 th state

∂n i +∇•Γ = n n I − n ()n ()+I + R + D NC + n ()n ()R + D + NC ∂t i i −1 e i−1 i e i i i i i+1 e i +1 i+1 i +1

Transport Ionization Charge Exchange Charge Exchange Losses of ith state from the ith state from the i+1 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.

e- p+ + Free p Free

e- e- e- Bound Bound

a) e- e- b) e- e- Energy

Before After Before After

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).

e- Bound Free e- Bound Free γ γ

e- e- e- e- γ a) b) e- e- Energy

Before After Before After

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.

H C 6+ H+ C 5+ Free

Free γ Bound e- Bound e- Energy

Before After

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 8.0 1.0 (a) (b) 6.0 0.8 -3 -1 -3 -1 0.6 4.0 -11 -12 0.4 Radiative

Ionization Rate 2.0 1X10 (cm s )

1X10 (cm s ) 0.2 Recombination Rate 0.0 0.0 0 200 400 600 800 0 200 400 600 800 Electron Temperature (eV) Electron Temperature (eV)

8.0 3.0 (d) (c) -3 -1

-3 -1 6.0 2.0 4.0 -8 -12 1.0 2.0 Dielectronic 1X10 (cm s ) 1X10 (cm s ) Recombination Rate Charge Exchange Rate 0.0 0.0 0 200 400 600 800 0 200 400 600 800 Electron Temperature (eV) Electron Temperature (eV)

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

+19 3 −3 ne >> 10 Te (eV)(ΔEeV( )) m , (4.1)

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

i→ j i→ j εγ = nenimp σv excitation , (4.2)

i → j where nimp and ne are the impurity state and electron densities and σv excitation 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.

1.4 State Wavelength Energy (eV) a) O VII 1s 2-1s2p 1.2 C V 40.26 308 1.0 C V 34.97 355 -3 -1 0.8 C VI 33.73 368 O VIII1s-2p O VII 21.60 574 0.6 O VII 18.63 665 0.4 (1E-10 cm s ) O VIII 18.98 653 Excitation Rate 0.2 O VII 1s 2-1s3p Al XII 07.75 1600 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Electron Temperature (keV) 4.0 8.0 b) C V 1s 2-1s2p c)

3.0 -1 6.0 -3 -1 -3

2.0 VI 1s-2pC 4.0

1.0 2.0 (1E-10 cm s ) (1E-12 cm s ) Excitation Rate Excitation Rate 2 C V 1s 2-1s3p Al XII 1s -1s2p 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Electron Temperature (keV) Electron Temperature (keV)

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.

1.00 (a) (b) 1e-1 1e-2

1e-3 Transmission 1e-4 1.00 (c) (d) 1e-1 1e-2

1e-3 Transmission 1e-4 1.00 (e) (f) 1e-1

1e-2

1e-3 Transmission 1e-4 0 1020304050600 102030405060 Wavelength (Angstoms) Wavelength (Angstoms)

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 Diode Filter Absorption Edge Band Of Interest Name Thickness Material (Angstroms) (Angstroms) Al-4 1 μm Al None < 12 6 μm Mylar 5000 A Al Al/Fe 4500 A Fe 17.5 18 – 19 4000 A Al Al/Mn 5000 A Mn 19.5 20 – 23 5000 A Al Al/V 5000 V V 24.3 25 – 35 2000 A Al Al/Caf 2500 A Ag 36.0 37 – 43 1 μm CaF2 Al-3 1000 A Al 43.7 > 45 7.5 μm Polyimide

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 ≈10 nf ) which places severe constraints on the j 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.

.400 Anode

.650

.571 Cathode

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

2 ξ ≈ A A 4πS2 = .81 2 4 π 33 =4.8E − 5cm2 , (4.3) ()s d () ()

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 AW). 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) = Gξηλ∫ ()T()λ ∫ ε()x,λ dx dλ . (4.4) 0 L 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 = ∑ ηλ()i T k()λ i εi ()λi δλ(− λ i ), (4.5) i =1 where Tk ()λ is the filter transmission of the kth diode, 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

I = T • R , (4.7) 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,

ελ( ) = T −1(λ)• I ηλ( ). (4.8) [ ]

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

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

+13 -3 (Iplasma≈ 200 kA), moderate electron density (ne≈1x10 cm ) 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 1.2 a) 1.0 0.8 -3 0.6 (cm ) 0.4 Density 1E+10 0.2 O VII Density 2.5 b) 2.0

1.5 -3

(cm ) 1.0

Density 1E+9 0.5 O VIII Density 0.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Time (ms)

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

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.

4.0

3.0 -3 C V Density 2.0 State Density (1E+10 cm ) 1.0 C VI Density (Upper Bound ) 0.0 -2.5 -1.5 -0.5 0.5 1.5 2.5 Time (ms)

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

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

i→ j i→ j described by equation 4.2, which states εγ = nenimp σv excitation , the relative change in emission can be represented as

i → j i→ j i→ j i → j Δεγ εγ ≈Δne ne + Δnimp nimp + Δ σv excitation σv excitation , (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-

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 12 18-23 Angstroms O VIII Peak 10 Emission O VIII 8

4 PPCD Start 1E+11 Photons 2 O VII Peak Emission O VII 0 -5 0 5 10 15

Time (ms)

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 3 <15 Angstroms Al XII Peak Emission 2

1 PPCD Start

1E+11 Photons AL XII

0 -5 0 5 10 15 Time (ms)

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 23-38 Angstroms 4

2 PPCD Start

1E+12 Photons 1

0 -5 0 5 10 15

Time (ms)

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.

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.

2.5 38-48 Angstroms 2.0

1.5

1.0 AL XII PPCD Start 1E+12 Photons 0.5 AL XI

0 -5 0 5 10 15

Time (ms)

Figure 4.14 – Emission from the C V channel. Having burned through C V, the contributions from aluminum appear.

88 5 48-60 Angstroms 4

2 PPCD Start 1E+11 Photons 1 ? AL XIII ? 0 -5 0 5 10 15

Time (ms)

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

S = n I n − I − R n + R n + NC n − C n , (4.10) j e[]j − 1 j −1 ( j j ) j j + 1 j +1 ( j +1 j +1 j j )

where I , R , and C are the ionization, recombination (both radiative and

dielectronic), and charge exchange cross-sections, and where ne , nj and 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,

ni n e IiNn e I H = ? . (4.11)

For example, if n ine Ii Nne I H << 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 ( n i 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

+5 -1 ionization rate of hydrogen, which is ~3x10 s . Since n e Ii ne IH << 1. 0, and with

n i N ≈ 1.0 already established, n in e IiNn e I H << 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,

n i N >>1.0 , and impurity sourcing can become comparable to that of hydrogen

( n i n e IiNn e I H ~ 1). Moreover, with the concentration of neutrals depleted in the core, the recombination from charge exchange drops and equation 4.11 is no

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

with most of the bolometric power resulting from particle convection, the total bolometric power will still drop.

3 Bolometric

2 Power (MW) 1 Radiated

0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Time (ms)

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 ∂n i +∇•Γ = n n I − n ()n I + NC + n NC , (4.12) ∂t i i −1 e i−1 i e i i i +1 i+1

where ni-1, ni, and ni+1 are the densities of the i-1, i, and i+1 states, ne is the electron density, 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 (∇ • Γi → ni τi ), 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

n i −1ne Ii −1 = ni NCi + ni τ i , which yields a charge state ratio of n n I i = e i− 1 . (4.13) n i −1 NCi +1 τi

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

−1 τ = n n n I − NC . (4.14) i [( i −1 i ) e i −1 i ]

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 8

2 Confinement Time (ms) 0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 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

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.

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13 Y. Yagi, T. J. Biag, L Carraro, Y. Hirano, R. Hamada, Y. Maejima, S. Sekin, and T. Shimada, Nuclear Fusion, 37, 1775 (1997).

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

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.

1.0 P06

0.8 P21 -2

P28 0.6 (1E+15 cm ) P36 0.4 Integrated Electron Density Electron Integrated

P43 0.2 -2.0 -1.0 0.0 1.0 2.0 Time (ms)

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 1.2 a) f) 0.8

0.4 t=-2.25 ms t=-1.00 ms 1.2 b) g)

-3 0.8 0.4 t=-2.00 ms t=-0.75 ms 1.2 c) h) 0.8 0.4 t=-1.75 ms t=-0.50 ms 1.2 d) i) 0.8 Electron Density (1E+13 cm ) 0.4 t=-1.50 ms t=-0.25 ms 1.2 e) j) 0.8 0.4 t=-1.25 ms t=-0.00 ms 0.0 -40 -20 0 20 40 -40 -20 0 20 40 Radial Position (cm)

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 Current Profile Control 100 ABCDEStart ˜ 80 Low b P06 -2 60 P36 40 (1E+14 cm )

Integrated Density 20 P43 0 510152025 Time (ms)

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 0.8

0.6 E

-3 D C 0.4 B A (1E+13 cm ) Electron Density 0.2 Wall Wall 0.0 -60 -40 -20 0 20 40 60 Radial Position (cm)

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 ),

∂n e +∇•Γ= S. (5.1) ∂t

Since we measure ne and S simultaneously in each FIR chord, we can integrate the continuity equation along each chord to arrive at

L 2 L 2 L 2 L 2 ∂ ⎛ L 2 ⎞ ()∇•Γ dz = Sdz − ∂n ∂tdz= Sdz− ⎜ n dz⎟ . (5.2) ∫ ∫ ∫ e ∫ ∂t ∫ e − L 2 − L 2 − L 2 − L 2 ⎝ − L 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 ne and 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 ≈10.09. Finally, a little algebra easily isolates Γ, yielding

1 ψ ′ Γ′()ψ = ψ ξψ()dψ . (5.4) ψ ′ ∫ 0

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

1.2 (a) -3 0.8 19 0.4 (10 m ) Standard Magnetic Axis Magnetic

Electron Density PPCD 0.0 23 10 Standard 22

-1 10 21

-3 10

(m s ) 20

10 Axis Magnetic PPCD (b) Electron Source 1019 22 10 Standard 1021

-2 -1 1020 (m s ) 19 Particle Flux 10 Axis Magnetic PPCD (c) 1018 0.0 0.1 0.2 0.3 0.4 0.5 r (m)

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 = ⎜ n dV⎟ ⎜ Γ • dA ⎟ (5.5) p ∫ e ∫ ⎝ V ⎠ ⎝ A ⎠

Once again, we invoke a toroidal symmetry constraint that allows equation 5.5 to be simplified to obtain,

τ = n ψ ψ dψ a Γ a . (5.6) p ∫ e( ) ( ) ψ '

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 STAN ≈0.6 and τ PPCD ≈ 4.7 ms. (5.7) p p

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

−18 −2 −1 Pp ≈22()πRo ()2πa Γe (a)Ee ≈9.9 ×10 Γe(m s )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

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.

REFERENCES

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 20 20 15 15 10 10

5 5 a) P06 b) P13 Fluctuation Amplitude (%) 20 20 15 15 10 10

5 5 c) P21 d) P28 30 40 Fluctuation Amplitude (%) Amplitude Fluctuation 30 20 20 10 10 e) P36 f) P43 0 0 -2 -1 0 1 2-2 -1 0 1 2 Time (ms) Time (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 At Crash 30

Fluctuation 10 Amplitude (%) Away PPCD From Crash 0 -40 -20 0 20 40 60 Radial Position (cm)

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 5 r a =.11 4 r a =.54 3 r a =.83 2 Power (au) 1

0 0 1020304050 Frequency (kHz)

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 1.0 (a)

0.6 Amplitude Coherence 0.2

0.0 (b)

Baseline m=1 rad) m=1 m=0 Phase ( π -1.0 0 20 40 60 80 100 Frequency (kHz)

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 40 30 .14 k (cm ) 20 .10 φ

10 .05 -1 Number (n)

Toroidal Mode 0 .00 r/a = 0.58 -10 020406080 Frequency (kHz)

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 Δ nFWHM ~ 2 (figure 6.6a). However, at 35 kHz, the

spectrum centered near n=30 and is much broader, with Δ nFWHM ~ 39 (figure

6.6.c). The n-spectrum at 18 kHz is peaked at n=6 with a width of Δ nFWHM ~ 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 80 npeak ≈ 0 (a) 60 ΔnFWHM ≈ 2 40

Power (au) 20 0 -40 -20 0 20 40 Toriodal Mode Number (n) 0.6 npeak ≈ 6 (b)

0.4 ΔnFWHM ≈12

0.2 Power (au) 0.0 -40 -20 0 20 40 60 Toriodal Mode Number (n) 1.5 npeak ≈ 30 (c) ≈ 1.0 ΔnFWHM 39

0.5

Power (1e-2 au) 0.0 -20 020406080 Toriodal Mode Number (n)

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 2.0 Total (a) 1.5 Coherent 1.0 Incoherent 0.5 0.0 20 Total (b) 15 Coherent 10 Incoherent 5 0 20

Fluctuation Power (a.u.) Fluctuation Power Total (c) 15 Coherent 10 Incoherent 5 0 10 15 20 25 30 Frequency (kHz)

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

nr()= 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

˜ Ix( ) = Io (x)+ I (x), (6.2)

where

L 2 I˜ ()x = ∫ n˜ () r cos[]ωt + mθ + nφ + δ()r dz . (6.3) − L 2

Here, x represents the impact parameter of the chord, L is the chord’s path

length through the plasma, and z is the vertical coordinate. Equation 6.3 can be simplified to

˜ ˜ I ()x = I amp (x)cos[ωt + nφ +Δ(x)], (6.4)

where we have defined

a n˜ (r)sin[δ (r)] I˜ ()x sin Δ()x = dr (6.5) amp []∫ 2 2 x r − x and

a n˜ (r)cos[δ(r)] I˜ ()x cos Δ()x = dr . (6.6) amp []∫ 2 2 x r − x Equations 6.5 and 6.6 can be Abel1 inverted to arrive at

a ⎛ ˜ r d I amp (x)cos[Δ(x)]⎞ dx n˜ ()r cos[]δ()r =− ⎜ ⎟ (6.7) π ∫ dx 2x r2 x2 r ⎝ ⎠ − and

a ⎛ ˜ r d I amp (x)sin[Δ(x )]⎞ dx n˜ ()r sin[]δ()r =− ⎜ ⎟ . (6.8) π ∫ dx 2 x r2 x2 r ⎝ ⎠ −

122 ˜ The parameters I amp ()x and 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 bm , n . Having obtained the products ˜ ˜ I amp ()x sin[Δ()x ] and 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 1.5 Standard (a) 1.0 n=6

0.5 PPCD

1.2

-3 Standard (b) 0.8 n=7 17 (10 m ) 0.4 PPCD

1.2 Standard (c) 0.8 n=8

Electron Density Electron 0.4 PPCD 1.5 Standard (d) 1.0 n=9

0.5 PPCD 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ra 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, n˜ ()r 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.

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 (n˜ ) and the radial velocity

(v˜ r ) fluctuations is dictated by the electron continuity equation,

∂n r e +∇•()n v = S. (6.9) ∂t e

Expanding n˜ and v˜ r into their equilibrium and fluctuating components as

˜ ˜ i (k •r − ωt) f = f o + f ⇒ f o()r + f 1()r e , (6.10)

Equation 6.9 becomes ˜ i ⎡ n ∂ ⎤ no(k • v ) n˜ = v˜ •∇n + o ()rv˜ − . (6.11) ()ω − k • v ⎣ r r o r ∂r r ⎦ (ω − k • v)

To arrive at equation 6.11, we have made the usual assumption of neglecting the r zeroth order compressibility term ( ∇ • v o = 0 ), but have kept the first order term

125

( ∇• v˜ ≠ 0 ). We have also neglected the fluctuating source term ( S ˜ → 0 ). The viability of the latter assumption is supported by the correlated product of ˜ ˜ Hα and edge magnetic coil measurements ( Sb 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 n˜ and v˜ r can be different. For example, if advection or radial

compression is the cause of the density fluctuation, then n˜ and v˜ r must be 90 ˜ ˜ degrees out of phase ( i → π 2 ). However if measurements of n and v r indicate a phase difference other than π 2 , then n˜ must arise from compression within the

magnetic surface. By investigating the phase between n˜ and 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

5 4 He II 3 C V 2 1 Density (a.u.) 0 0.0 0.2 0.4 0.6 0.8 1.0 r/a

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 r r fluctuating E × B 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 v (k • Bo ) v˜ ∝ v v b˜ . (6.12) r ω − k • v r (o )

128 0.25 a)

0.15

0.05 Baseline Coherence Amplitude 0 1020304050 Frequency (kHz) 1.0 b) Phase Unresolvable radians)

π 0.0 Phase ( -1.0 0 1020304050 Frequency (kHz)

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.

0.5 0.4 No coherence 0.3 0.2 Baseline 0.1

Coherence Amplitude 0.0 0 1020304050 Frequency (kHz)

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 n˜ and 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 0.30 a)

0.20

0.10 Amplitude

Baseline 0.00 -60 -40 -20 0 20 40 60 R-Ro (cm) 1.0 b) rad) π 0.5 Phase (

0.0 -60 -40 -20 0 20 40 60 R-R (cm) o

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(δ nv ), (6.13)

where γ is the coherence amplitude and δ nv is the phase between n˜ and v˜ r . It is important to note that this term does not include all mechanisms for radial ˜ ˜ transport. For example, the contribution from J|| br 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 , n and 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 n˜ vanishes (figure 6.13b), suggesting

that n˜ and v˜ r remain out of phase much deeper into the core. This change in phase, coupled with the order of magnitude reduction of n˜ (Section 6.1.5), indicate that the fluctuation-induced transport due to core-resonant tearing modes is greatly reduced in the core.

133 1.5 a) 1.0 rad) π 0.5

0.0 Phase ( -0.5 -40 -20 0 20 40 60 Radial Position (cm)

1.5 b) 1.0 rad) π 0.5

0.0 Phase ( -0.5 -40 -20 0 20 40 60 Radial 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

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.

REFERENCES

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

⎡E Sxo ⎤ meshes. Starting from the bottom right: we have; the initial electric field E ⎣ Syo ⎦ out of the laser, the wire polarizer at the laser output ⎡10 ⎤ , and the quarter- ⎣00 ⎦

⎡ cos 2 φ + isin 2 φ sinφ cosφ(1 −i)⎤ wave plate 2 2 , where φ is the angle between the principal ⎣sin φ cosφ()1− i cos φ +i sin φ ⎦ axis of the quartz and the electric field vector. Continuing on, we have the half-

⎡cos 2ω t sin 2ω t ⎤ ()p ()p wave plate ⎢ ⎥ , rotating at an angular velocity ωp, and the two ⎣ sin() 2ω p t −cos() 2ω p t ⎦ meshes (transmission through the first and reflection off the second)

RTE 0 TTE 0 ⎡ 2 ⎤ ⎡ 1 ⎤ . Finally we have the plasma imparted Faraday rotation ⎣ 0 RTM2 ⎦ ⎣ 0 TTM1 ⎦ 142 ⎡ cosδ sinδ ⎤ , and the selection polarizer ⎡10 ⎤ , which isolates the x (toroidal) ⎣− sinδ cosδ⎦ ⎣00 ⎦ component of the electric field.

⎡ ESx ⎤ ⎡ 10⎤ ⎡ cosδ sinδ ⎤ ⎡ RTE2 0 ⎤ ⎢ ⎥ = × ⎣ ESy ⎦ ⎣ 00⎦ ⎣ −sinδ cosδ ⎦ ⎣ 0 RTM2 ⎦ ⎡ ⎤ ⎡ TTE 0 ⎤ cos() 2ω pt sin() 2ω pt 1 × A.1 0 TTM ⎢ ⎥ ⎣ 1 ⎦ ⎣ sin() 2ω pt −cos() 2ωpt ⎦ 2 2 ⎡ cos φ + isin φ sinφ cosφ()1 − i ⎤ ⎡ 10⎤ ⎡ ESxo ⎤ ⎢ 2 2 ⎥ ⎢ ⎥ ⎣ sinφ cosφ()1 − i cos φ +i sin φ ⎦ ⎣ 00⎦ ⎣ ESyo ⎦

We begin the simplification process by eliminating the polarizer terms and combining the mesh transmission and reflection matricies.

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ESx cosδ sinδ (RTE2TTE1) 0 ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ × ⎣⎢ ESy ⎦⎥ ⎣⎢ 00⎦⎥ ⎣⎢ 0 ()RTM2TTM1 ⎦⎥ A.2 ⎡ ⎤ 2 2 cos 2ω t sin 2ω t ⎡ cos φ + i sin φ 0⎤ ⎡ ESxo ⎤ ⎢ ()p ()p ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ E ⎣ sin() 2ω pt −cos() 2ω pt ⎦ ⎣⎢sin φ cosφ()1 −i 0⎦⎥ ⎣⎢ Syo ⎦⎥

Next we combine the mesh and plasma rotation to obtain,

⎡ E ⎤ ⎡ ⎤ Sx ()RTE2TTE1 cosδ (RTM2TTM1)sinδ ⎢ ⎥ = ⎢ ⎥ × ⎣⎢ ESy ⎦⎥ ⎣⎢ 00⎦⎥ . A.3 ⎡ ⎤ 2 2 cos 2ω t sin 2ω t ⎡ cos φ + i sin φ 0⎤ ⎡ ESxo ⎤ ⎢ ()p ()p ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ E ⎥ ⎣ sin() 2ω pt −cos() 2ω pt ⎦ ⎣sin φ cosφ()1 −i 0⎦ ⎣ Syo ⎦

By defining an ellipticity, ε, the quarter-wave plate / laser output combination ⎡ ⎤ cos(ω ct + φ plasma ) takes the form E ⎢ ⎥ , and equation A.3 becomes So ε sin ω t + φ ⎣⎢ (c plasma )⎥⎦ 143 E ⎡ Sx ⎤ ⎡( RTE2TTE1)cosδ (RTM2TTM1)sin δ⎤ = ESo × ⎣⎢ ESy ⎦⎥ ⎣⎢ 00⎦⎥ . A.4 ⎡ ⎤ ⎡ ⎤ cos() 2ω pt sin() 2ωpt cos()ω ct + φplasma ⎢ ⎥ ⎢ ⎥ ⎣ sin() 2ω pt −cos() 2ω pt ⎦ ⎣ ε sin()ωct + φ plasma ⎦

If we define a mesh distortion angle, θ, and an amplitude Aθ as shown in A.5,

⎡ RTM TTM ⎤ 2 2 −1⎢ 2 1 ⎥ θ = tan and Aθ = (RTE2TTE1 ) + (RTM2TTM1 ) A.5 ⎣⎢ RTE2TTE1 ⎦⎥ then we get,

⎡ ESx ⎤ ⎡ cosθ cosδ sinθ sinδ ⎤ ⎢ ⎥ = ESo Aθ ⎢ ⎥ × ⎣⎢ ESy ⎦⎥ ⎣⎢ 00⎦⎥ . A.6 ⎡ ⎤ ⎡ ⎤ ⎢ cos() 2ω pt sin() 2ω pt ⎥ ⎢ cos()ωc t + φplasma ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ sin() 2ω pt −cos() 2ω pt ⎦ ⎣ε sin()ωc t + φplasma ⎦

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

⎡ sinθ sinδ ⎤ ξ = tan−1 ⎢ ⎥ = tan −1[]tanθ tanδ A.7 ⎣ cosθ cosδ ⎦ and

2 2 Aξ = ()cosθ cosδ + ()sinθ sinδ = 1+ cos() 2θ cos() 2δ , A.8 we arrive at equation A.9. 144 ⎡ E ⎤ ⎡ ⎤ ⎡ ⎤ Sx cos 2ω t − ξ sin 2ω t − ξ cos(ωct +φplasma ) ⎢ ⎥ =E A A ⎢ ( p ) ( p )⎥ ⎢ ⎥ A.9 ⎢ E ⎥ So θ ξ ⎢ ⎥ ⎢ ⎥ ⎣ Sy ⎦ ⎣ 00⎦ ⎣ε sin(ωct + φplasma )⎦

Continuing through the multiplication we have,

⎡ cos 2ω t − ξ cos ω t +φ + ⎤ ⎢ ( p ) ( c plasma ) ⎥ E = E A A ⎢ ⎥ , A.10 Sig So θ ξ ⎢ ⎥ ⎣ ε sin() 2ω pt − ξ sin()ωc t + φplasma ⎦

and once more we employ our trick to define

⎡ ⎤ −1 ε sin( 2ω pt − ξ ) −1 ψ S = tan ⎢ ⎥ = tan [ε tan( 2ω pt − ξ)], A.11 ⎣ cos() 2ω pt − ξ ⎦

with the corresponding amplitude

A = cos2 2ω t − ξ − ε2 sin 2 2ω t − ξ ψ S ( p ) ( p ) . A.12 ()1+ ε 2 ()1 − ε2 = 1+ 2 cos() 4ω pt − 2ξ 2 ()1 + ε

This easy substitution yields a very simple result for the electric field amplitude of the signal leg incident on the corner cube diode.

E E A A A cos t Sig = So θ ξ ψ S (ωc + 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.

2 P = E + E = E2 + E2 + 2E E A.15 Sig ( Sig LO) Sig LO Sig LO

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

E2 = E2 A2 A2A2 cos2 ω t + φ −ψ Sig So θ ξ ψ S ( c plasma S )

E2 A2 A2A2 So θ ξ ψ S = 1 − cos() 2ωc t + 2φplasma − 2ψ S 2 [] . A.16 E2 A2 A2A2 E2 A2 A2 A2 So θ ξ ψ S So θ ξ ψS = − cos 2ωc t + 2()φplasma −ψ S 2 2 [ ] = after filtering → 0 146

The second term has a similar result and is derived in equation A.17.

2 2 2 ELO = ELOo cos (ωct +ω IFt) 2 ELOo = []1 + cos() 2ωct + 2ωIF t 2 A.17 E2 E 2 = LOo + LOo cos() 2ω t + 2ω t 2 2 c IF = after preamp filtering → 0

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 ωIF component (equation A.18).

E E = E E A A cos ω t + φ −ψ cos ω t + ω t Sig LO So LOo ξ ψ S ( c plasma S ) ( c IF ) ⎡ ⎤ cos()ωIF t −φ plasma +ψ S ESoELOo Aθ A Aψ ⎢ ⎥ = ξ S ⎢ ⎥ A.18 2 ⎢ ⎥ ⎣ + cos() 2ωct + ωIF t +φ plasma −ψ S ⎦

E E A A A = after preamp filtering → So LOo θ ξ ψ S cos ω t − φ +ψ 2 ( IF plasma S )

Therefore the power registered in the digitizer for the eleven channels of the FIR will have the following form,

E E A A A So LOo θ ξ ψ S PSig = cos(ωIF t −φ plasma +ψ S ), A.19 2

which by substituting in

ESo ELOo Aθ A Aψ P = ξ S , A.20 Sig _ Amp 2 147 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 ⎤ ⎡10 ⎤ ⎡ RTE1 0 ⎤ cos 2ω pt sin 2ω pt = ( ) ( ) × ⎢ E ⎥ ⎢00 ⎥ ⎢ 0 RTM ⎥ ⎢ ⎥ ⎣ Ry ⎦ ⎣ ⎦ ⎣ 1⎦ ⎣ sin() 2ω pt −cos() 2ω pt ⎦ A.22 2 2 ⎡ cos φ + i sin φ sinφ cosφ()1− i ⎤ ⎡ 10⎤ ⎡ ESxo⎤ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ sinφ cosφ()1− i cos φ + i sin φ ⎦ ⎣ 00⎦ ⎣ ESyo⎦

As in section A.2 we carry out the matrix multiplication giving

⎡ ERx ⎤ ⎡ cos 2ω t sin 2ω t ⎤ ⎡ cos(ω t) ⎤ = RTE E ( p ) ( p ) c , A.23 ⎢ E ⎥ 1 So ⎢ ⎥ ⎢ ⎥ ⎣ Ry ⎦ ⎣ 00⎦ ⎣ε sin()ωct ⎦ 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

⎡ ⎤ −1 ε sin( 2ω pt) −1 ψ R = tan = tan ε tan() 2ω pt , A.25 ⎢ cos 2 t ⎥ [ ] ⎣ ()ω p ⎦

and

A = cos 2 2ω t + ε2 sin 2 2ω t ψ R ( p ) ( p ) . A.26 ()1 + ε2 ()1− ε 2 = 1 + 2 cos() 4ωpt 2 ()1+ ε

Substitution yields the electric field incident on the mixer from the reference leg to be

E = RTE E A cos ω t − ψ , A.27 Rx 1 So ψ R [ ( c R)]

which, when including the LO leg, has the total power measured by the corner cube diode as

2 P = E + E = E 2 + E2 + 2E E . A.28 Ref ()Ref LO Ref LO Ref LO

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). 149 E2 = RTE E 2 A2 cos2 ω t −ψ Ref 1 So ψ R ( c R )

2 2 RTE1ESo Aψ = R 1 − cos( 2ω t − 2ψ ) A.29 2 [c R ] = after preamp filtering → 0

The cross term is retained,

ERef ELO = RTE1ESo ELOo Aψ R cos(ωct −ψ R )cos(ωct + ωIFt) ⎡ ⎤ cos()ω IFt +ψ R RTE1ESo ELOo Aψ ⎢ ⎥ = R , A.30 2 ⎢ ⎥ ⎣⎢ + cos() 2ωct +ωIF t −ψ R ⎦⎥

RTE1ESo ELOoAψ R = after preamp filtering → cos()ωIFt + ψ R 2 and therefore the power as measured by the digitizer is described as follows.

RTE1ESo ELOo Aψ P = R cos()ω t +ψ = P cos()ω t +ψ A.31 Ref 2 IF R Ref _ Amp IF R

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

−i (ω IFt+ψ R ) PRef _ Con = (PRef _ Amp 2)e . 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 _ Amp PRef _ Amp 4 × A.33 i ()ωIFt − φplasma+ψ S −i ()ω t +ψ − i()ω IFt −φplasma +ψ S −i()ω t +ψ []e e IF R + e e IF R

−i φ +ψ −ψ −i 2ω t − φ +ψ −ψ = P P 4 e ()plasma R S + e ()IF plasma S R ()Sig _ Amp Ref _ Amp [ ]

A low pass filter is employed to eliminate the 2ωIF term, leaving

−i(φplasma +ψ R −ψS ) PFilter_Product = (PSig _ Amp PRef _ Amp 4)e . A.34

The measured interferometry phase (Φ) becomes

⎡ Im P ⎤ −1 ()Filter_Product Φ=tan ⎢ ⎥ ⎣ Re()PFilter_Product ⎦ ⎡ ⎤ −1 −()PSig _ Amp PRef _ Amp 4 sin()φplasma +ψ R −ψ S = tan ⎢ ⎥ . A.35 ⎣ ()PSig _ AmpPRef _ Amp 4 cos(φ plasma + ψ R −ψ S )⎦

=−()φplasma +ψ R −ψ S 151

Substituting equations A.11 and A.25, we can rewrite the measured interferometry phase (Φ) to

−1 −1 Φ=−{φplasma + tan [ε tan( 2ω pt)]− tan [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=− φ − tan −1 ε tan −ξ {}plasma [ ( )]

=− φ − tan −1 ε tan −tan−1 tan θ tan δ . A.37 {plasma [([() ()])]}

⎧ TM ⎫ ≈− φ − ε tan()θ δ =−⎨ φ − εδ ⎬ {}plasma ⎩ plasma TE ⎭

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 measurements.

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

⎛ TM ⎞ εδ ≈ ()5 ()0.5 ().15 = .375 radians. A.38 ⎝ TE ⎠ 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.

RTE1ESo ELOoAψ P = R Ref _ Amp 2 A.39 1+ ε 2 1 − ε2 RTE1ESo ELOo ()() = 1+ 2 cos() 4ω pt 2 2 ()1 + ε

154 E E A A A P = So LOo θ ξ ψS Sig _ Amp 2 A.40 1 + ε2 1− ε 2 ESo ELOo Aθ ()() = 1+ cos() 2θ cos() 2δ 1 + 2 cos() 4ωpt − 2ξ 2 2 ()1+ ε

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 P = Hilbert P cos ω t −φ + ψ []Sig [ Sig _ Amp ( IF plasma S )] . A.41

= PSig _ Amp sin()ωIFt − φplasma +ψ S

2 2 By summing the square’s, {PSig } + {Hilbert[ PSig ]}= PSig _ Amp , the signal amplitude falls out easily. Using this method, we find the reference and signal envelopes are

2 2 ⎡ 2 ⎤ 2 (1 + ε )⎡ RTE E E ⎤ (1− ε ) P = P = 1 So LOo ⎢1 + cos 4ω t ⎥ , A.42 Ref _ Env Ref _ Amp ⎢ ⎥ ⎢ 2 ()p ⎥ 2 ⎣ 2 ⎦ ⎣ ()1+ ε ⎦ and

2 2 2 (1 +ε )⎡ E E A ⎤ P = P = ⎢ So LOo θ ⎥ Sig _ Env Sig _ Amp 2 ⎣ 2 ⎦

⎡ 1− ε 2 ⎤ ⎢ () ⎥ ×[]1 + cos() 2θ cos() 2δ 1 + 2 cos( 4ω pt − 2ξ). A.43 ⎣⎢ ()1+ ε ⎦⎥

= PSig _ Env _ Amp cos() 4ω pt − 2ξ

Preparing the reference as before, we convert to exponential notation, remove any equilibrium components and negative frequencies, and conjugate. This yields 155

−4iω pt PRef _ Env _ conj = (PRef _ Env _ Amp 2)e . A.44

For the signal envelope, we just remove the equilibrium components and expand to exponential form.

−i (4ω pt −2ξ) +i (4ω pt− 2ξ) P = P /2 e + e A.45 Sig _ Env _ Fil ( Sig _ Env _ Amp )[ ]

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

⎡ ⎤ P P −i ()4ω pt −2ξ +i ()4ω pt− 2ξ −4iω t = ⎢ Ref _ Env _ Amp Sig _ Env _ Amp ⎥ e + e e p ⎣ 4 ⎦[] ⎡ ⎤ P P − i()8ω pt − 2ξ = ⎢ Ref _ Env _ Amp Sig _ Env _ Amp ⎥ e−2iξ + e . A.46 ⎣ 4 ⎦[]

= filtering,8ω p term → 0

−2iξ = PProduct_Env_Ampe

The final step is to isolate the measured Faraday rotation.

⎡ Im P ⎤ −1 ⎢ ()Product_Env ⎥ −1 TM Ψ=tan ⎢ ⎥ =−2ξ =−2 tan []tanθ tanδ ≈−2 δ A.47 ⎣ Re()PProduct_Env ⎦ TE

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 factors be accurately characterized. It might also be worthwhile to investigate alternatives to the meshes, such as thin film deposited on TPX or quartz.

REFERENCES

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. 158 B.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 preprocessing, 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.

175 Laser 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. 178 pro 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 179 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 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 Negative Phase Jump 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

186 Figure 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

188 factor=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 ;;

191 input_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'

202 time_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,p eak 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 ;; ;;------206 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_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 appropriate calibration date and shot numbers.

g) Run Make_Calibration_File.pro, everything else should be automated. 212 pro 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 Code pro 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 A=14mm,B=16mm,C Edmund Scientific P45,261 Prism =15 mm Focusing Plano-Convex Lens Edmund Scientific P45,260 Lens 10cm F. L.

Hα Filter 656.3 nm, 11.5 nm Coherent-Ealing 42-5496 FWHM, 0.5” Dia. Photo-Diode .1”X.1” Active Area Advanced Photonix SD 112_452_ Detector 300 kHz Cutoff Inc. 11_221 Circular Shell Size 0 Newark Electronics JBX ER 0G04 FC Receptacle 4 Contact SDS Circular Shell Size 0 Newark Electronics JBX FD 0G04 MC Plug 4 Contact 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 2.0 a)

-2 1.5

1.0

(1E+13 cm ) 0.5 Electron Density

0.0 0.6 b)

0.4 α H (au) 0.2

0.0 1.0 c) 0.8

0.6

0.4 (unitless) SXR Ratio 0.2

0.0 150 200 250 300 350 Plasma Current (kA)

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.

1.E+0 O VIII 1E-1 O VII SXR_BE_2 SXR_BE_1 1E-2 Transmission 1E-3

1E-4 0 5 10 15 20 25 30 Wavelength (Ang.)

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