NEUTRAL BEAM HEATING OF A REVERSED-FIELD PINCH IN THE MADISON SYMMETRIC TORUS
by
Jeff Waksman
A dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
(Physics)
at the
UNIVERSITY OF WISCONSIN–MADISON
2013
Date of final oral examination: June 18, 2013
The dissertation is approved by the following members of the Final Oral Committee: Cary Forest, Professor, Physics Jay Anderson, Associate Scientist, Physics Chris Hegna, Professor, Engineering Physics Paul Terry, Professor, Physics David Anderson, Professor, Electrical And Computer Engineering c Copyright by Jeff Waksman 2013 All Rights Reserved i
A star is drawing on some vast reservoir of energy by means unknown to us. This reservoir can scarcely be other than the sub-atomic energy which, it is known, exists abundantly in all matter; we sometimes dream that man will one day learn how to release it and use it for his service. The store is well-nigh inexhaustible, if only it could be tapped.
— Sir Arthur Stanley Eddington, 1920 ii
Abstract
Neutral beam injector (NBI) heating of electrons has been observed in the Madison Symmetric Torus (MST). This heating is consistent with a simple 1D model which includes the effect of fast ion diffusion, neutralizationlosses and classical collisional processes. This heating was achieved with a 1 MW NBI (25 kV, 40 A). Auxiliary heating of MST has been observed with Thomson scattering to be 100 50 eV in the core of 200 kA PPCD (pulsed ± poloidal current drive) plasmas. Auxiliary heating has been measured in other PPCD plasmas, but not in standard confinement plasmas. This data represents the first confirmed auxiliary heating of a reversed-field pinch (RFP) plasma. Enhanced confinement plasmas are conducive to auxiliary heating because of the improved thermal confinement. Ion temperatures were measured with a Rutherford scattering device, but investigation deter- mined that NBI fast ions are able to get into the Rutherford analyzer at a large enough rate to add a significant non-Gaussian component to the observed data. Controlling for this additional neutral hydrogen signal eliminates false detection of auxiliary heating with NBI in standard MST plasmas. Further analysis suggests that fast ions from magnetic reconnection events could also be polluting MST sawtooth Rutherford data. A 1-D model of NBI heating in MST was developed. This model takes measured and calculated inputs such as plasma temperature, plasma density and ohmic heating profiles and solves for heat diffusion coefficient profiles. The assumption underlying this calculation, that the heat diffusion coefficients do not change when the NBI is firing, is justified because the NBI does not significantly suppress mid-radius magnetic fluctuations. The base 1-D model includes fast ion deposition, fast ion diffusion and fast ion slowing down. Fast ions are lost to the plasma via thermalization and contact with the MST wall. Neutral losses iii are added to the model using NENE,` a Monte Carlo code. Finally, non-classical (resonant) fast ion diffusion is observed and modeled. This diffusion acts to limit core fast ion density, and appears as rapid bursts that occur approximately once or twice every millisecond.
The model output show that the 1-D classical model is consistent with measured Te in PPCD plasmas. Fast ion diffusion is crucial in driving a flatter heating profile to limit heat conduction- losses. Measured core Te is only possible with significant mid-radius heating. Neutralization losses are modeled, which provide a loss mechanism for fast particles near the wall. Low core neutral densities in PPCD are crucial for the measured auxiliary heating. Neutral- ization losses are significant in standard plasmas, and explain the lack of significant Te. Finally, resonant diffusion is found not to have a significant effect on auxiliary heating of MST as long as a loss mechanism (neutralization) exists for fast ions near the MST wall. The 1-D model demonstrates that auxiliary heating with NBI in PPCD can be modeled classi- cally, where fast ion diffusion is a crucial physical process. Non-PPCD plasmas require a signifi- cant loss mechanism for fast ions near the MST wall, which can be provided by neutralization. iv
ACKNOWLEDGMENTS
When I get asked years from now how I managed to survive graduate school, my answer will be that the MST team is all anybody could possibly ask for. The collaborative and cooperative spirit of every single person working on MST is a model for how science should be done. I will very much miss working in the MST control room. There are individuals within the group that deserve special thanks. First and foremost are my faculty advisors past and present, Stewart Prager and John Sarff. I also need to thank Cary Forest for stepping in at a difficult time to fill John’s shoes, saving my thesis defense. Particular thanks go to Jay Anderson and Gennady Fiksel for working with me on a daily basis, and suffering literally thousands of door knocks as I asked question after question. Gennady is the godfather of the NBI group, and it is his vision and hard work that formed the initial backbone of all of this research. Jay has done a fabulous job leading the NBI/fast ion group the past few years. Without Jay’s guidance I absolutely never would have gotten my 1-D model to start working. As Jay spent more time with me than literally anybody else on MST, I’d appreciate it if you direct any complaints about this dissertation to him. The entire NBI/fast ion group has been great, but I want to thank several in particular who have really spent a lot of time helping me along the way. Mark Nornberg provided a ton of guidance with thesis writing and coding. Deyong Liu found time during his 23 hour/day work days to do all of the TRANSP runs you see here, as well as to do a lot of the initial fast ion confinement analysis. Jon Koliner, Scott Eilermann, Josh Reusch and the Thomson scattering team all went out of their way to take and analyze a bunch of data for this dissertation. I need to thank Abdul Almagri for countless hours working on the Rutherford beam with me. I think we have tested/cleaned/fixed/replaced every single component of that beam at this point. v
MST is blessed with an incredible engineering and technical staff. Future grad students take note: If you have a technical problem of any kind, discuss it loudly near the offices of MST’s technical staff and they will come out of their offices and they will solve your problem. While every single person on staff has provided assistance of some kind or another to me, I want to thank several in particular. Steve Limbach and Steve Oliva have both spent many, many hours working on various neutral beams. The Hackerium staff of Paul Wilhite, Alex Squitieri and Andrey Levochkin has been invaluable. The electronics shop staff of Mikhail Reyfman and Dave Deicher have done a great job deciphering electronics problems, even when I don’t have drawings. I also want to thank Paul Nonn for taking the lead on the bolometer calibration setup when I was a green graduate student who didn’t know the difference between a drill press and a lathe. And I can’t forget Peter Weix, Bill Zimmerman, John Laufenberg and everybody else who kept MST running the past 7 years. Even when I blew up the L-6 inductor we were only down for about two days. I want to thank my fellow graduate students who kept me sane by playing around 8 million games of pickup basketball. I want to give special thanks/sympathy to Meghan McGarry-Unks for being my office mate for nearly my entire time in graduate school. Last, but definitely not least, I want to thank my family for everything they’ve done to make it possible for me to finish this dissertation. I got to see you far more than a person living 1000 miles away probably should (although I have a sneaking suspicion that Dad would have kept coming regularly on his own, even if I wasn’t living here, once he discovered Camp Randall, the Kohl Center and fried cheese curds). Anything I achieve in my life will be because of you. But no, you can’t call me ”Doctor Waksman”. vi
TABLE OF CONTENTS
Page
Abstract ...... ii
List of figures ...... ix
1 Madison Symmetric Torus ...... 1
1.1 The Reversed Field Pinch ...... 1 1.2 Madison Symmetric Torus ...... 5 1.3 Standard Confinement In The RFP ...... 8 1.4 The Sawtooth Cycle ...... 10 1.5 Pulsed Poloidal Current Drive ...... 15 1.6 The RFP As A Fusion Device ...... 18 1.7 Thesis Overview ...... 23
2 Experiment Diagnostics ...... 26
2.1 Neutron Detection ...... 26 2.2 Thomson Scattering ...... 27 2.3 Far-Infrared (FIR) Interferometer-Polarimeter System ...... 28 2.4 CHERS ...... 31 2.5 ANPA ...... 34 2.6 Neutral Density Profiles Via D↵ ...... 36
3 Neutral Beam Injection ...... 40
3.1 A Brief History Of Neutral Beams On Fusion Devices ...... 40 3.2 MST’s 1 MW Neutral Beam ...... 42 3.3 Classical Fast Ion Dynamics ...... 45 3.4 Measuring NBI Shine-Thru ...... 47 3.5 Fast Ion Confinement ...... 53 3.5.1 Calculating Fast Ion Confinement ...... 53 3.5.2 Classical Fast Ion Confinement Theory ...... 56 3.5.3 Fast Ion Confinement Data ...... 58 3.5.4 Resonant Fast Ion Transport ...... 63 3.6 NBI Momentum Drive ...... 68 vii
Page
3.7 NBI Current Drive ...... 73 3.7.1 Current Drive Theory ...... 73 3.7.2 Current Drive Data ...... 76
4 Rutherford Scattering ...... 80
4.1 History of Rutherford Scattering ...... 80 4.2 MST’s Rutherford Scattering Diagnostic ...... 82 4.3 Rutherford Scattering Theory: Single Particle ...... 84 4.4 Rutherford Scattering Theory For An Ion Velocity Distribution ...... 87 4.5 Rutherford Scattering Theory: Generalized For A Beam With Finite Divergence . . 88 4.6 Improved Rutherford Processing ...... 90 4.7 Rutherford Analyzer As Neutral Hydrogen Analyzer ...... 92 4.8 Rutherford Analyzer Sawtooth Analysis ...... 97
5 NBI Heating Of MST Plasmas ...... 101
5.1 Measured Auxiliary Heating Of MST Plasma ...... 103 5.2 Auxiliary Heating Data In A Variety Of Plasma Conditions ...... 107 5.3 Impact Of Current & Density On Auxiliary Heating ...... 113
6 Modeling NBI Heating ...... 116
6.1 A 1-D Classical Heating Model ...... 117 6.2 Justifying A Classical Heating Model ...... 123 6.3 Modeling With TRANSP/NUBEAM ...... 128 6.4 Anomalous Ion Heating ...... 131 6.5 Modeling The NBI ...... 133 6.5.1 Modeling The NBI Beam Path ...... 134 6.5.2 Modeling NBI Beam Deposition ...... 137 6.5.3 Modeling Fast Ion Diffusion And Slowing Down ...... 142 6.6 Model Output Comparisons To Data ...... 149 6.7 Impact Of Zeff On 1-D Model ...... 158 6.8 Impact Of Fast Ion Diffusion On Heating ...... 161 6.9 Modeling Fast Ion Charge-Exchange Losses ...... 167 6.10 Modeling Resonant Fast Ion Transport ...... 177
7 Conclusions And Future Work ...... 183
7.1 Neutral Beam Heating Of MST ...... 183 viii
Appendix Page
7.2 1-D Auxiliary Heating Model ...... 184 7.3 Future Work ...... 185
References ...... 187
APPENDIX Bolometers And XUV Detectors ...... 202 ix
LIST OF FIGURES
Figure Page
1.1 Illustration of the RFP magnetic field configution ...... 4
1.2 Photograph of MST ...... 5
1.3 Cartoon of a toroidal pinch experiment ...... 6
1.4 Directions in MST ...... 7
1.5 Typical magnetic field profiles in MST ...... 9
1.6 Modeling of stochastic magnetic fields in MST ...... 10
1.7 Example data from a standard MST discharge ...... 11
1.8 Plasma relaxation during a sawtooth crash ...... 13
1.9 Anomalous Ion Heating During A Reconnection Event ...... 14
1.10 Reduced magnetic fluctuations during PPCD ...... 16
1.11 Changes in core Te and Ti During PPCD ...... 17
1.12 Electron temperature profile changes during enhanced confinement ...... 17
1.13 MST confinement compared to scaled tokamak confinement ...... 20
1.14 MST confinement compared to modified Connor-Taylor scaling ...... 20
1.15 RFP confinement scaling in RFX-mod ...... 20
1.16 Cutaway drawing of TITAN ...... 22
2.1 A cartoon of the observation of Thomson scattering ...... 28 x
Figure Page
2.2 A drawing of MST’s FIR system ...... 29
2.3 Impact of temperature, velocity and density on CHERS measurements ...... 32
2.4 Comparison of CHERS and Rutherford PPCD data ...... 34
2.5 Design of the MST ANPA ...... 35
2.6 Photo of the MST ANPA ...... 36
2.7 NENE` calculation of neutral density profile, 200 kA PPCD ...... 37
2.8 NENE` calculation of core neutral density, 400 kA standard ...... 38
3.1 Evidence for NBI heating of TPE-RX ...... 42
3.2 CAD drawing of MST’s 1 MW NBI ...... 43
3.3 Photograph of MST’s 1 MW NBI ...... 44
3.4 A cartoon of neutral beam charge exchange ...... 45
3.5 Photograph of the NBI shine-thru detector ...... 47
3.6 Drawing of the NBI shine-thru design ...... 48
3.7 Plot of electron-impact cross section vs Te ...... 50
3.8 Plotting shine-thru measured at different NBI alignments ...... 50
3.9 Photo of the NBI smashing into the beam dump ...... 51
3.10 Shine-thru data for 200 kA low density PPCD ...... 52
3.11 Shine-thru data for 200 kA high density PPCD ...... 52
3.12 Neutron signal vs early gas ...... 53
3.13 Neutron decay vs fast ion loss time ...... 55
3.14 Measured fast ion loss times in various plasmas ...... 55
3.15 Fast ion guiding center safety factor profile ...... 57 xi
Appendix Figure Page
3.16 Neutron data for 300 kA, F=0 ...... 59
3.17 Neutron data for 200 kA, low density PPCD ...... 59
3.18 Neutron data for 200 kA, low density PPCD ...... 59
3.19 Neutron data for 400 kA, low density PPCD ...... 59
3.20 ANPA data from a 400 kA standard plasma ...... 62
3.21 Energetic particle-driven instabilities in MST plasmas ...... 64
3.22 Resonant transport of fast ions in DIII-D ...... 65
3.23 Energetic particle-driven instabilities in 200 kA low density PPCD ...... 67
3.24 ANPA data during fast-ion driven modes in 200 kA PPCD ...... 68
3.25 Impact of NBI on magnetic fluctuation magnitude and velocity ...... 69
3.26 Co-current vs Counter-current NBI data ...... 71
3.27 Neutron flux, Co- vs Counter- NBI ...... 72
3.28 Simulated fast ion density, Co- vs Counter- ...... 72
3.29 Data from DIII-D showing NBI current drive ...... 74
3.30 Data from MAST showing NBI suppression of loop voltage ...... 74
3.31 Impact of tearing modes on NBI-driven current in DIII-D ...... 76
3.32 NBI current drive data on MST ...... 77
3.33 FIR polarimetry data during NBI ...... 78
3.34 FIR polarimetry data during NBI, zoomed in on the plasma core ...... 78
4.1 Schematic of MST’s Rutherford beam and analyzer ...... 83
4.2 Photograph of MST’s Rutherford analyzer ...... 84
4.3 Cartoon of single particle Rutherford scattering ...... 85 xii
Appendix Figure Page
4.4 Small angle approximation diagram ...... 86
4.5 Geometry of a finite-width beam Rutherford scattering ...... 89
4.6 Modeled Rutherford scattering energy distribution ...... 90
4.7 Example of adjusted Rutherford gas-only data ...... 93
4.8 Example of adjusted Rutherford plasma data ...... 93
4.9 Rutherford NBI “heating” data ...... 94
4.10 Rutherford as an NPA: Raw Data ...... 95
4.11 Rutherford as an NPA: Processed Data ...... 96
4.12 Fast Ion Tail Following A Sawtooth Event ...... 97
4.13 Rutherford Temperature Output, 400 kA Standard ...... 98
4.14 Raw Rutherford Data, 400 kA Standard ...... 99
4.15 Raw Rutherford Data vs Time, 400 kA Standard ...... 99
5.1 Example plasma parameters for 200 kA low density PPCD data ...... 102
5.2 Core electron temperature with and without NBI in low density 200 kA PPCD . . . . 104
5.3 Change in core electron temperature with NBI in 200 kA low density PPCD ...... 104
5.4 Electron temperature profiles with and without NBI for low density 200 kA PPCD . . 105
5.5 CHERS data, 200 kA low density PPCD ...... 106
5.6 Core electron temperature with and without NBI in high density 200 kA PPCD . . . . 108
5.7 Change in core Te with NBI in 200 kA high density PPCD ...... 108
5.8 CHERS data, 200 kA high density PPCD ...... 108
5.9 Core electron temperature with and without NBI in low density 400 kA PPCD . . . . 109
5.10 Change in core Te with NBI in 400 kA low density PPCD ...... 109 xiii
Appendix Figure Page
5.11 Core electron temperature with and without NBI in low density 400 kA PPCD . . . . 110
5.12 Change in core Te with NBI in 400 kA low density PPCD ...... 110
5.13 Core electron temperature with and without NBI in 300 kA F=0 plasmas ...... 111
5.14 Change in core Te with NBI in 300 kA F=0 ...... 111
5.15 CHERS data, NBI On vs Off, 300 kA PPCD ...... 111
5.16 Core electron temperature with and without NBI in 400 kA standard plasmas . . . . . 112
5.17 Change in core Te with NBI in 300 kA F=0 ...... 112
5.18 Stored electron energy vs time, 200 kA low density PPCD ...... 114
5.19 Stored electron energy vs time, 200 kA high density PPCD ...... 114
5.20 Change in stored electron energy, 200 kA low density PPCD ...... 114
5.21 Change in stored electron energy, 200 kA high density PPCD ...... 114
6.1 A cartoon of the “ith” plasma volume element ...... 118
6.2 Example calculated e profiles ...... 119
6.3 Cartoon of the two-dimensional array used in the Crank-Nicolson method ...... 121
6.4 Matrix form of the Crank-Nicolson solution ...... 122
6.5 Relationship between B˜ suppression and Te(0) ...... 124
6.6 Supression of the core magnetic modes by NBI ...... 125
6.7 NBI suppression levels for all magnetic modes ...... 126
6.8 Supression of the core magnetic modes by NBI in PPCD ...... 127
6.9 NBI suppression levels for all magnetic modes in PPCD ...... 127
6.10 Typical toroidal flux profile in MST PPCD ...... 128
6.11 TRANSP fast ion deposition shape ...... 129 xiv
Appendix Figure Page
6.12 TRANSP/NUBEAM model core heating ...... 130
6.13 Impact of ion temperature assumptions on TRANSP/NUBEAM model ...... 132
6.14 Overhead view of simulated NBI path ...... 135
6.15 Distance of the NBI path from the geometric axis ...... 135
6.16 “Slice” view of simulated NBI beam ...... 136
6.17 Assumed NBI Beam Width ...... 138
6.18 NBI beamlet path ...... 138
6.19 Magnetic field “focuses” deposited fast ion ...... 140
6.20 Fast neutral density along NBI path ...... 141
6.21 Modeled fast ion density profiles ...... 143
6.22 Modeled fast ion energy deposition profile, 1-D model ...... 144
6.23 Modeled fast ion energy deposition, TRANSP/NUBEAM ...... 144
6.24 Modeled fast ion energy deposition profile, hybrid model with diffusion ...... 146
6.25 Modeled fast ion energy deposition, hybrid model without diffusion ...... 146
6.26 Comparing the modeled heat deposition profiles ...... 147
6.27 Modeled fast ion energy loss mechanisms, 1-D model ...... 148
6.28 Modeled fast ion energy loss mechanisms, TRANSP/NUBEAM ...... 148
6.29 Modeled change in core electron temperature ...... 150
6.30 Modeled electron temperature profiles ...... 150
6.31 Modeled heat diffusion losses, 1-D model ...... 152
6.32 Modeled ohmic power density, 1-D model ...... 153
6.33 Comparison of core heating components, NBI On vs NBI Off ...... 154 xv
Appendix Figure Page
6.34 Comparison of core heating components, 1-D model vs TRANSP ...... 154
6.35 Modeled core electron temperature gradient ...... 155
6.36 Impact of ion temperature assumptions on 1-D heating+deposition model ...... 156
6.37 Impact of supressed magnetic fluctuations on core Te ...... 158
6.38 Change in core Te for varied Zeff profiles in 200 kA PPCD ...... 160
6.39 Change in core Te for varied Zeff profiles in 400 kA PPCD ...... 161
6.40 1-D model output for varying TRANSP inputs ...... 162
6.41 Hybrid model core heating for varying diffusion coefficients ...... 164
6.42 1-D heating+deposition model output for varying diffusion coefficients ...... 165
6.43 Fast ion density profiles: Diffusion On vs Off ...... 166
6.44 1-D model output, varying diffusion and neutral density ...... 168
6.45 An example “reduced” neutral density profile ...... 168
6.46 Fast ion density profiles with charge-exchange ...... 170
6.47 1-D model output for 400 kA PPCD, varying neutral density ...... 171
6.48 Core neutral density vs time, 400 kA PPCD ...... 171
6.49 Fast ion density profiles with varying neutral densities, 400 kA PPCD ...... 172
6.50 1-D heating+deposition model output for 400 kA standard ...... 174
6.51 NENE` output, ensembled vs single-shot data ...... 175
6.52 Fast ion density profiles for 400 kA standard ...... 176
6.53 Simulated fast ion profiles with resonant diffusion ...... 178
6.54 Simulated core fast ion density vs time with resonant diffusion ...... 178
6.55 Simulated core Te with resonant diffusion ...... 179 xvi
Figure Page
6.56 Simualted core Te for various values of DRES ...... 179
6.57 Fast ion density profiles with and without resonant diffusion ...... 180
6.58 Calculated Te, 400 kA standard, including non-classical effects ...... 181
6.59 Simulated core fast ion density, 400 kA standard with resonant diffusion ...... 181
Appendix Figure
A.1 Cartoon of pyrobolometer calibration setup ...... 205
A.2 Bolometer calibration: optimizing the bias voltage ...... 207
A.3 AXUV detector sensitivity vs photon energy ...... 209
A.4 Photo of a bolometer/XUV pair on MST ...... 210
A.5 Typical bolometer/XUV data ...... 212
A.6 Typical bolometer/XUV data (zoom) ...... 212
A.7 Bolometer/XUV data (Standard), NBI On vs Off ...... 214
A.8 Bolometer/XUV data (PPCD), NBI On vs Off ...... 214 1
Chapter 1
Madison Symmetric Torus
1.1 The Reversed Field Pinch
Reversed Field Pinch (RFP) research is, for the most part, inspired by and funded because of the desire for fusion power. Fusion power, if harnessed and controlled on Earth, presents a nearly infinite source of clean, renewable power. The work discussed throughout this dissertation should be understood within that context. The discovery of nuclear fusion actually goes back nearly 100 years. The catalyst was the development of the mass spectrometer by Francis W. Aston in 1919, which separated out isotopes of the elements of the periodic table, allowing him to accurately measure their respective masses (a discovery for which he won the Nobel Prize in Chemistry in 1922). In 1920, Aston was able to measure the mass of hydrogen (1.008 amu) and the mass of helium (4.003 amu). Arthur Stanley Eddington, one of Aston’s colleagues at Cambridge University, speculated that same year that the fact that helium was a little less than four times the mass of hydrogen meant that combining four hydrogen together would create a helium atom along with a release of energy. Einstein’s famous formula for mass-energy equivalance (E = mc2) meant that even that tiny bit of lost mass would mean a large release of energy. At the time, the dominant theory of the Sun was Lord Kelvin’s 19th century determination that gravitational potential energy was released as heat, leading him to calculate an age for both the Sun and Earth of tens of millions of years. Eddington argued instead that the Sun was releasing 2 heat from these fusion reactions, and calculated that the Sun had 15 billion years of fuel. While Eddington’s conception of the fusion reaction being four hydrogens combining into one helium was wrong (in his defense, the neutron wouldn’t even be discovered for another decade), he was correct that fusion fueled the sun, and his estimation of 15 billion years of fuel was remarkably close to accurate. It was at a speech presenting this research, on August 24th, 1920, that Eddington gave the powerful and prescient quote that opens this dissertation. The basic theory by which hydrogen isotopes could fuse to create helium+energy was devel- oped over the next few decades, giving scientists the confidence that we could engineer fusion on Earth. The first man-made release of fusion energy was in 1952, when the hydrogen bomb was tested for the first time. Controlling fusion power in the lab proved to be a more complicated en- deavor, since the conditions for significant fusion require extremely hot (hundreds of millions of
20 3 degrees) plasmas confined at relatively large densities ( 10 m ). Research was quickly begun in magnetic confinement of high temperature plasmas, which remains the dominant fusion research in the world today. The earliest designs were mostly Z-pinches, followed by Lyman Spitzer’s fa- mous stellarator in 1951, and then the magnetic mirror. The spark that began the modern era of fusion research came in 1968, at an IAEA meeting in Novosibirsk, where Soviet researchers announceed that their new T-3 tokamak design had achieved central electron temperatures reaching 1 keV (more than ten million degrees kelvin, nearly 2,000 ⇡ times hotter than the surface of the sun). A team from Culham Laboratory in England traveled to the Soviet Union to verify the results with their own Thomson scattering device (see more on Thomson scattering in Sec. (2.2)). Their verification of 1 keV temperatures immediately made the tokamak the most popular fusion design, which is the status it still holds today. The largest magnetically-confined fusion devices today in terms of plasma current, temperature and fusion power achieved are almost all tokamaks, including JET, EAST, NSTX and DIII-D, among many others. The next step is ITER, an international tokamak of staggering size that is already under construction in Cadarache, France, with first plasma projected for 2019. ITER is designed to be the transition step to a full-scale electricity-producing fusion power plant (DEMO), which would then lead to commercial fusion power as part of the electrical grid. 3
While the tokamak is the most popular form of magnetic confinement of plasmas, there are many other confinement designs in use in large scales today. Stellarators and mirrors are still used, as are Spherical Tokamaks, a Levitated Dipole, the Reversed Field Pinch and others. The basis for the Reversed Field Pinch can be traced all the way back to the ZETA machine, which operated at Harwell Laboratory in the United Kingdom from 1957 through 1968. ZETA was a toroidal pinch device, and the largest fusion device of its era. ZETA is unfortunately famous for the wrong reasons now. It was at ZETA in 1958 that high measured neutron fluxes were mis- interpreted, and published in Nature with the blaring title: “Controlled Release Of Thermonuclear Energy” [Thonemann et al., 1958]. It was realized soon after that these high neutron signals were due to an ion acceleration process rather than fusion, and a retraction was published five months later [B.Rose et al., 1958]. Nevertheless, ZETA did play a pivotal role in the development of the RFP. It was several years after ZETA was decommissioned that John Bryan Taylor wrote his seminal paper [Taylor, 1974], based on data from ZETA operation. Taylor noted that after the initial start-up of the plasma, it would often relax into a “quiescent”, stable state. Taylor noted that for large pinch ratios, a reversed toroidal field would actually be generated near the edge. To understand what Taylor was observing we can talk about the helicity (or “knottedness”) of the magnetic field lines by defining [Moffatt, 1969]:
K = A BdV (1.1) · ZV Here, K is the magnetic helicity and A is the magnetic vector potential (B = A). Magnetic r⇥ helicity is a conserved quantity for a perfectly conducting plasma, though there will be a slight decay in helicity in a resistive plasma, as was measured in MST by Ji et al. [1995]. Over time, the plasma will “relax” into a state of minimum energy. Assuming conserved helic- ity, Taylor found the solution for this state in his 1974 paper:
B = B (1.2) r ⇥ The ideal MHD momentum balance equation is: dV ⇢ = JXB p (1.3) m dt r 4
J is the current density, p is the plasma pressure, V is the bulk plasma velocity and ⇢m is the plasma density. The equilibrium ( @ =0) form of Eq. (1.3) is JXB = p. Taken with Ampere’s` @t r Law ( B = µ J), we can say that the plasma is “force free”, since: r ⇥ o B p = J B = r ⇥ B = B B =0 (1.4) r ⇥ µo ⇥ µo ⇥
This relaxed state is known today as the “Taylor State”. Solving Eq. (1.2) for the case of a cylinder gives the magnetic field components in terms of Bessel functions [Bodin & Newton, 1980]:
B = BoJo( r) (1.5)
B✓ = BoJ1( r) (1.6)
This value is related to the pinch paramter (✓ = B (a)/B¯ 2⇡aI / ) by the expression ✓ ✓ ⇠ a/2 [Taylor, 1986]. The solution to Eq. (1.5) is a reversed toroidal field when r > 2.405. For the ZETA data Taylor was looking at, this correlated to a pinch parameter ✓ =1.202, remarkably similar to the measured ✓ 1.4 observed for the onset of the reversed field in ZETA. Thus, for ⇡ appropriate initial toroidal fields and toroidal currents, the plasma will relax into a state with a reversed toroidal field.
Figure 1.1 A cartoon of the magnetic field configuration of a typical RFP. 5
The first modern reversed-field pinch was ETA BETA II, which had its first plasma results in 1979. This was followed in the early 1980s by experiments such as TPE-1RM, ZT-40(M) and HBTX1A. The next stage of large-scale RFP experiments consisted of the Madison Symmetric Torus (MST), Reversed-Field eXperiment (RFX), RELAX, TPE-RX and EXTRAP-T2R, all of which had first plasmas around 1990. Of those five machines, all are still in operation except for TPE-RX. A new RFP, the Keda Torus eXperiment (KTX), is currently in the design and construc- tion phase in Hefei, China.
1.2 Madison Symmetric Torus
Figure 1.2 Photograph of the Madison Symmetric Torus
The Madison Symmetric Torus (MST) is a large toroidal reversed-field pinch with a major radius (R) of 1.5 m and a minor radius (a) of 0.52 m. The torus consists of a 5 cm thick aluminum shell that serves several purposes. It is a vacuum vessel, maintaining pressures 1 µTorr between ⇡ shots. It also serves as a conducting wall very close to the plasma (carbon limiters with a width ⇡ 1 cm keep the plasma off the wall) to help stabilize ideal instabilities. It also serves as a single-turn toroidal field coil to generate the start-up toroidal magnetic field. Because the torus is a conducting 6 shell, all port holes (for diagnostics, vacuum pumping, etc.) are kept small, to limit magnetic field errors.
Figure 1.3 A generic sketch of a toroidal pinch experiment [Anderson, 2001]
Typical plasma pulses in MST last for anywhere between 40 and 100 ms. Though MST can be run with several different fuels, all data used in this research were done with the standard deu-
19 3 terium. Electron density (n ) at the core is typically around 0.3 1.5 10 m , though densities e ⇥ 19 3 greater than 4 10 m have been reached during pellet injection [Wyman et al., 2008]. Core ⇥ electron temperature is typically between 0.2-2.0 keV. Toroidal plasma current (IP ) is typically between 200-600 kA. The basic operation of MST is described in Dexter et al. [1991]. A 1.3 cm insulated gap, known as the toroidal gap, is between the top and bottom half of MST, on the inboard side. A potential across this gap generates a poloidal electric current in the shell, which generates a starting BT . Next, MST behaves as a large transformer (as seen in a simplified cartoon in Fig. (1.3) [Anderson, 2001]), with the plasma (sparked by a pair of filaments) acting as a single-turn secondary. The transformer is pulsed, which generates a toroidal plasma current. This plasma current generates the poloidal field. 7
V pg + – 0° 360°
Bp v , v , Ip top view of t mode vDi MST edge p Er – p poloidal cut view core core Bt, z + V vDi tg vE x B vE x B during edge sawtooth J vmode edge Er Bp Bt, z
vt, Ip
Figure 1.4 A cartoon of the magnetic field, plasma current, pressure gradient and drift directions in MST. Note that the typical direction of the poloidal field is the opposite of the cartoon in Fig. (1.1).
The poloidal mean field MF drives current parallel to the magnetic field, which near the wall E is often nearly totally poloidal. This current drives a magnetic field in the reverse toroidal direction (see Fig. (1.4) for a drawing of the directions of these currents and fields). The reversal paramater (F ) in an RFP is defined as: B (a) F = T (1.7) B h T i This reversal parameter is a free experimental parameter for the MST operator. A larger startup
BT will mean less reversal (an F>0 mode is possible) while a smaller startup BT with a large driven poloidal current will mean deep reversal. A “standard” MST discharge has F=-0.2. A parameter that describes the amount of field line twist is the “safety factor,” q:
rB (r) q(r)= T (1.8) RBT (r)
By definition, q=0 at the reversal surface. In typical MST plasmas, q 0.2 in the core. ⇡ 8
1.3 Standard Confinement In The RFP
In any magnetic configuration, perturbations to the magnetic field can be described by a Fourier series. Working in a spherical coordinate system with minor radial component r and major ra- dius R, and assuming that the equilibrium (unperturbed) magnetic field can be described as B =
(0,B✓,B ) (B✓ and B are the general variables used for the poloidal and toroidal magnetic fields, respectively): i(k r !t) B˜(r, t)= B˜k(r)e · (1.9) Xk The wave vector k is defined as k =(kr,m/r,n/R), where m and n are the poloidal and toroidal mode numbers respectively. These magnetic perturbations are resonant when the wave vector is perpendicular to the equilibrium magnetic field:
m n 0=k B = B + B (1.10) · r ✓ R
Eq. (1.10) can solved for m/n, and for the safety factor:
m r B (r) = = q(r) (1.11) n R B✓(r)
The modes are thus resonant for any rational values of q, which represent the locations where the magnetic perturbation makes an integer number of poloidal rotations for an integral number of toroidal rotation. For example, the m=1, n=6 mode is a magnetic mode which makes exactly six poloidal rotations for every one toroidal rotation. All of the m=0 modes are resonant at q=0, which is the reversal surface. If these resonant modes are unstable then they will grow in amplitude, and the magnetic field lines can break into chains of magnetic islands at mode-rational surfaces [Biewer et al., 2003], where magnetic islands will be created by magnetic reconnection (see Rechester & Stix [1976] and Callen [1977]). The width of the magnetic islands is given as [Fiksel et al., 2005; Lichtenberg & Lieberman, 1983]:
˜bmn rmn wmn =4 (1.12) s B✓ nq mn 0 9
Standard MST Magnetic Profiles 1
0.75 B 0.5 0.25 B 0
Normalized B-Fields -0.25 0.2
0.15
0.1 /RB
0.05 q=rB 0
-0.05 0 0.2 0.4 0.6 0.8 1 r/a
Figure 1.5 Typical magnetic field profiles in MST. The top plot has normalized toroidal and poloidal magnetic fields for a typical MST plasma. The bottom plot has a typical q profile.
In Eq. (1.12), n and m are the toroidal and poloidal mode numbers, and q is the gradient of 0mn the q-profile of the m,n rational surface. The lower-n resonant surfaces (n=5,6,7) are the core-most m=1 modes in standard MST discharges, and they also occur at the the flattest part of the q-profile (see Fig. (1.5)). This means that the magnetic islands nearest the core will be largest, causing island overlap. The magnetic field lines become tangled and stochastic. Modeling of what this looks like is shown in Fig. (1.6). Overlapping magnetic islands create conditions of rapid radial transport, as the free streaming orbits of particles parallel to the magnetic field result in radial energy and particle diffusion [Biewer et al., 2003]. In standard MST plasmas this results in energy confinement times of 1 ms [Chap- ⇡ man et al., 2002]. This poor confinement is a significant problem for any potential fusion reactor, and this led to the development of an improved confinement scheme, pulsed poloidal current drive (PPCD), which is described in Sec. (1.5). 10
Figure 1.6 Modeling from Biewer et al. [2003]. The top plot includes the calculated magnetic island widths on top of the q-profile. The lower plot is a magnetic field line tracing applying the MAL code from a DEBS simulation.
1.4 The Sawtooth Cycle
The “Taylor State” described in Eq. (1.5) and Eq. (1.6) was solved for a perfectly conducting cylinder with a force-free plasma. Since MST is not a cylinder, and has finite resistivity and a non-zero pressure gradient, MST never truly reaches a relaxed state. Experimentally, the toroidal shape of MST and the magnetic shear leads to less parallel current density on the edge and greater parallel current density in the core [Terry et al., 2004]. Over time, the higher temperatures in the core lead to lower resistivities and an even larger accumulation of parallel plasma density. The existence of the plasma-wall boundary at the edge increases resistivity at the edge and also helps contribute to the peaking of the plasma current profile [Chapman, 1997]. This gradient is a source of free energy for MHD instabilities such as the tearing modes described in Sec. (1.3) is created [Biewer et al., 2003]. At some critical gradient, tearing instabilities rapidly increase, driving a rapid “relaxation” event, leading to a flattening of the profile and the pressure profile.
12
An RFP such as MST, with standard confinement, will go through this cycle repeatedly. The plasma will continue accumulating free energy, having a rapid relaxation event, and starting all over again. The data produced by this cyclical process visually looks like sawteeth, and so it is known as a “sawtooth cycle”. Each relaxation event itself is typically called a “sawtooth crash”. Data for a standard MST 400 kA plasma and sawtooth cycle can be seen in Fig. (1.7). Note that during each relaxation/sawtooth event, there is a rapid fluctuation in equilibrium toroidal current and a large increase in equilibrium magnetic field, in addition to the gigantic spike in magnetic fluctuations (on the order of 1-2% of
Figure 1.8 Data from an ensemble of relaxation events. The top plot is density data taken with the FIR interferometer described in Sec. (2.3). The vertical axis in this plot is electron density 19 3 (X10 m ). Data is from Ding et al. [2009]. The bottom two plots are electron temperature data taken with Thomson scattering (Sec. (2.2)), from Reusch [2011]. 14
Figure 1.9 Example of anomalous ion heating during a reconnection event on MST. The ion temperature data is core C+6 data from CHERS. This data is from Magee [2011].
The “anomalous” process is linked to magnetic reconnection. Magnetic field lines break apart and reconnect, changing the magnetic topology and converting magnetc energy into plasma en- ergy. This process plays an important role in the self-organization of fusion plasmas (in tokamaks, spheromaks and other devices, including the RFP), and also plays a key role in the dynamics of stars (including our sun), accretion disks and galaxies [Zweibel & Yamada, 2009]. It was noticed very early on that the level of anomalous ion heating in MST correlated with magnetic fluctuation activity [Scime et al., 1992]. Further information can be gathered by ob- serving sawtooth crashes, where ion temperature increases by 100% or more (see Fig. (1.9) for example data). Fiksel et al. [2009] determined that not only is the degree of ion heating dur- ing sawteeth a function of the magnitude of magnetic fluctuations, but it is also mass dependent ( E / E M 1/2). therm mag / i 15
1.5 Pulsed Poloidal Current Drive
The sawtooth events in MST degrade plasma confinement in two different ways [Chapman et al., 1996]. First, the increase in the strength of magnetic fluctuations drives radial transport. Second, the sawtooth crash itself drives a large injection of neutral particles, which leads to in- creased charge exchange losses. The low ( 1 ms) energy confinement times in standard MST plasmas led to a desire to suppress magnetic fluctuations to increase plasma confinement, driving the development of pulsed poloidal current drive (PPCD). Modeling by Ho [1991] and others suggested that an auxiliary parallel current driven at the plasma edge could flatten the profile and reduce the dominant magnetic fluctuations [Sovinec & Prager, 1996; Chapman, 1997]. This theory was put into practice on MST, where a fast current pulse is driven through the toroidal field winding to induce a poloidal electric field [Sarff et al., 1994]. This fluctuation reduction requires E (a) 0 [Chapman et al., 2002] where: k
E = E B/B =(E✓B✓ + E B )/B (1.13) k ·
This edge poloidal current drove magnetic fluctuation amplitudes down to record lows for MST, leading to a five-fold increase in energy confinement and record electron temperatures [Sarff et al., 1997]. With further optimization of PPCD in lower current plasmas, energy confinement of 10ms has been reached, a ten-fold improvement over standard plasmas [Chapman et al., 2002]. Representative data from a typical 400 kA PPCD shot is plotted in Fig. (1.10). The same signals from a standard (non-PPCD) 400 kA shot are plotted in comparison. In that particular PPCD shot, the enhanced confinement period begins at around 13 ms into the shot, and lasts until around 22 ms. During this period there is a dramatic reduction in magnetic fluctuations, with no relaxation events. An increase in electron density is also seen, due to the increase in particle confinement. The large decrease in magnetic fluctuations leads to a large decrease in energy diffusion (see Sec. (6.2) for data demonstrating this effect), driving an increase in core electron temperature. Core temperature data from a representative PPCD plot is shown in Fig. (1.11).
17
Figure 1.11 Core Te and core Ti measurements during 500 kA PPCD. The electron temperature is measured with soft x-ray (SXR), while the ion temperature data is central C+6 data collected with CHERS (Sec. (2.4)). Note the increase in core electron temperature, while core ion temperature stays flat. The top plot is of surface parallel electric field, which it should be noted is positive throughout the enhanced confinement period [Chapman et al., 2002].
Figure 1.12 Thomson Scattering (Sec. (2.2)) measurement of the electron temperature profile during enhanced confinement. Data is from a 380 kA PPCD discharge where the enhanced con- finement period ends at approximately 22.5 ms [Reusch, 2011].The plot on the right shows the time evolution of the electron temperature at four different radial points, with “Point: 0” being the one closest to the plasma core, and “Point: 19” being the closest to the plasma edge. 18
In Fig. (1.11), core electron temperature data was collected with the soft x-ray (SXR) diag- nostic. In this figure, CHERS data (Sec. (2.4)) was used to represent core ion temperature. This particular CHERS data was of central C+6 data, which over these time scales is an acceptable proxy for majority core ion temperature in MST (please see Sec. (2.4) and Reardon et al. [2003] for understanding why this assumption is a fair one). Core electron temperature can be seen increasing rapidly throughout the enhanced confinement period. At the same time, ion temperature at the core is not changing at all. The E (a) profile is k also shown in Fig. (1.11), where the fact that E (a) 0 throughout the enhanced confinement k period can be seen. With Thomson scattering, the full electron temperature profile throughout a PPCD shot is mea- sured with data from a representative 380 kA PPCD discharge plotted in Fig. (1.12). The rapid increase in core electron temperature can be seen clearly. Edge temperatures do not rise noticeably during PPCD, but the large change in magnetic fluctuations near the reversal surface allow a much larger mid-radius temperature gradient (see Sec. (6.2) for more on this topic). In this particular case, the enhanced confinement period ends at approximately 22.5 ms, at which point there is a very quick return to a standard temperature profile.
1.6 The RFP As A Fusion Device
The primary efforts to generate significant D-T fusion power in the laboratory so far in history have all involved tokamaks. As discussed in Sec. (1.1), the first effort to build a burning plasma experiment is currently underway in Cadarache, France. That reactor, ITER, is a tokamak. The tokamak has both advantages and disadvantages versus the RFP for generating fusion power. The tokamak has been studied for a longer period of time and on more machines than the RFP, which means that many of the key reactor components are more tested and understood. Auxiliary current drive and plasma heating are well-established, as is testing for divertors, plasma-facing components, wall materials, et cetera. Many of these issues are engineering tasks that simply must be done before a full-scale reactor can be built. That said, the biggest advantage that the tokamak 19 is perceived to have over the RFP is superior particle/energy confinement. Standard energy con- finement in MST is around 1 ms, compared to several hundred milliseconds for DIII-D, close to a full second for JET, and to a projected value of 3.7 seconds for ITER [ITER Physics Basis Editors, 1999; Shimada et al., 2007]. The stochastic nature of the RFP makes energy confinement very difficult. That said, the confinement gap between the RFP and the tokamak is not as large as it might appear at first glance. Energy confinement in tokamaks scales with basic plasma parameters. Toka- mak data scales very well with the IPB98(y,2) ELMy H-mode tokamak confinement scaling, which is [ITER Physics Basis Editors, 1999]:
0.93 0.15 0.69 0.41 0.19 1.97 0.58 0.78 ⌧E =(0.0562)I B P n M R ✏ (1.14)
The variables on the right side of Eq. (1.14) are, from left to right, toroidal plasma current, average toroidal magnetic field, loss power, line-averaged density, average ion mass, major radius, inverse aspect ratio and elongation. They are in units, where applicable, of seconds, MA, T, MW,
19 3 10 m , AMU and meters. Plugging in approximate values for MST to match the plasma current and magnetic field strength yields ”tokamak-like” energy confinement of only around 10 ms, which has been achieved in enhanced-confinement (PPCD) plasmas (Sec. (1.5)). In Fig. (1.13), MST standard and ”Improved” (i.e. PPCD) energy confinement times are com- pared to that standard tokamak scaling. It is clear in this chart both that ”tokamak-like” confine- ment has been achieved in MST, but also just how far the modern RFP is from the size and scale of the modern tokamak. Before getting to the attempts to scale confinement in the RFP, it’s necessary to introduce an important advantage of the RFP over tokamaks for generating fusion. The fact that so much of the magnetic field in the RFP is self-generated by the plasma means that the magnetic field strength needed at the magnetic coils is relatively small compared to the energy density in the plasma itself. This relationship is often described as the plasma :
Plasma Pressure nkBT = = 2 (1.15) Magnetic Pressure B /2µo 20
Figure 1.13 MST confinement compared with a scaled tokamak specified by the IPB98(y,2) ELMy-H mode empirical scaling [ITER Physics Basis Editors, 1999]. This specific figure, with the addition of MST data, is copied from Chapman et al. [2010].
Figure 1.14 MST confinement compared with Figure 1.15 Updated RFP confinement scaling modified Connor-Taylor scaling. This figure is using RFX-mod data. Plot is from Innocente et copied from Chapman et al. [2002]. The MST- al. [2009]. related dots are as follows: (a) MST 210 kA stan- dard, (b) MST 430 kA standard, (c) mid-1990s 340 kA PPCD, (d) circa-2000 390 kA PPCD, (e) circa-2000 210 kA PPCD. 21
In an engineering sense, magnetic pressure can be thought of as a proxy for machine cost, since the biggest operating cost of the reactor is the cost of generating the external magnetic fields. Plasma pressure is a proxy for fusion, since the rate of fusion reactions increases monotonically with increased plasma density and temperature. So can be seen as a proxy for fusions-per-dollar, or the financial efficiency of the machine. This makes the RFP very intriguing as a potential fusion reactor, should it be possible to keep sufficient plasma confinement. A of 26% has been measured on MST [Wyman et al., 2009], though no theoretical limit has yet been found and auxiliary heating could potentially increase further [Den Hartog et al., 2007]. 3/2 As a plasma heats up, its resistivity decreases (⌘ Te ). This means that a warmer plasma / 2 becomes increasingly difficult to heat ohmically (P⌦ ⌘j ). It has long been known that a / k tokamak must operate at an essentially ”collision-less” regime to generate sufficient fusion power, and so auxiliary heating is necessary [Golovin et al., 1970]. This conclusion led to the rapid development of the neutral beam injector (NBI) on a series of tokamaks in the early 1970s [Berry et al., 1975; Kelley et al., 1972; Menon, 1981]. The neutral beam has for decades now been the dominant source of auxiliary heating on tokamaks [Murakami et al., 1977; deGrassie et al., 2006]. A minimum of 33 MW of neutral beam power is being planned for ITER [Shimada et al., 2007]. For a more extended discussion of the history of neutral beams on tokamaks, see Sec. (3.1). The high achievable in the RFP means that a higher energy density is possible in the plasma core. This gives the RFP greater flexibility with its aspect ratio. It was found by Lawson [1977] that the RFP, if treated as a plasma column and if it had sufficiently small transport and impurity losses, could reach ignition with ohmic heating alone. Connor & Taylor [1984] theorized that confinement in the RFP was limited by resistive fluid turbulence near the reversal surface, and suggested that energy confinement could be scaled as ⌧ a2I3 /N 3/2, where N is the line- E / averaged density. Werley [1991] used that scaling to calculate that an RFP could reach ignition with ohmic heating alone with a plasma current as low as 8.1 MA. This scaling was further modified
2 3 3/2 with data existing in the mid-1990s to ⌧E =10.2(a I /N ) [Werley et al., 1996]. With this scaling in mind, an effort to design an RFP reactor (TITAN) began in 1986 [Na- jmabadi & the TITAN Research Group, 1988]. An introduction to the final report can be found 22 in Najmabadi et al. [1993]. A cutaway drawing of TITAN from that report is in Fig. (1.16). The final design for TITAN featured a major radius of 3.9 m and a minor radius of 0.6 m, as well as a plasma current of close to 18 MA. The design was theorized to produce approximately 2.3 GW of fusion power, and to generate a net electrical power of 0.97 GW. A ⌧E of approximately 150 ms was calculated. TITAN is massive compared to currently operating RFPs, but it is in fact quite small compard to ITER, which is already under construction. ITER has a major radius of 6.2 m, a minor radius of 2.0 m and a plasma current of 15-17 MA, yet will produce only 500-700 MW of fusion power [Shimada et al., 2007].
Figure 1.16 A cutaway drawing of the final TITAN design, from Najmabadi et al. [1993].
TITAN was just a design, of course, and construction has never been started on an RFP reactor. More importantly, modern RFP confinement data simply does not confirm this modified Connor- Taylor scaling. Fig. (1.14) is a plot from Chapman et al. [2002] showing confinement in MST versus this scaling. The improved confinement can be seen in the progress from standard MST plasmas to the early form of PPCD to the improved PPCD. With improved PPCD, MST actually has better confinement than is predicted by Connor-Taylor. However, this confinement does not scale as well with plasma current. More recent confinement scaling data, from RFX-mod, is plotted
sec 1 8 1,n 2 in Fig. (1.15), where br = 2 n= 15 bt (a) . ⇥ P ⇤ 23
Using modern plasma codes, improved confinement scaling for the RFP has been under- taken [Stoneking et al., 1998; Scheffel & Schnack, 2000; Scheffel & Dahlin, 2006]. None of these codes consistently match MST or RFX-mod data, however, which makes it particularly difficult to extrapolate to a reactor-sized device like TITAN. In addition to the improved , the RFP offers several other advantages for fusion over the tokamak. The fact that the field strength at the magnetic coils can be much smaller means that it is possible that superconducting coils will not be necessary. This, combined with the fact that the RFP aspect ratio is flexible and the possibility of ”single piece maintenance” (the entire first wall and blanket can be removed as a single piece), means that engineering would be much more simple for an RFP reactor than a tokamak. The concerns regarding an RFP reactor can be summed up in the word ”scale”. We simply cannot confirm at this point in time that energy confinement will scale to a reactor, and we do not know about the possibility of problems like disruptions (which are known to occur in high energy tokamaks and have been studied for many years) in a reactor-scale RFP. Because of the uncertainty with scale, it is high leverage research to press down the path previ- ously taken by tokamak researchers. Current drive, momentum drive and auxiliary heating in the RFP are very different than the tokamak. While these techniques are all well-studied in tokamaks, the surface has barely been scratched with the RFP. Auxiliary heating with an NBI is an example of this type of research. It is well-understood in tokamaks, but turns out to be quite different in the RFP. The study of auxiliary heating with an NBI on MST is the focus of this dissertation. The 25 kV fast ions studied in this dissertation are also specifically relevant for RFP fusion research. Assuming reasonable RFP reactor values of B 10 T,T 10kV, 0.1 [FES, T,CORE ⇡ e ⇡ ⇡ 2008], alpha particle Larmor orbits will be similar in radius to 25 kV hydrogen in MST (2-3 cm near the core, 10-15 cm near the edge). Further study of fast ions in MST could shed light on the dynamics of fusion products in a future RFP fusion device.
1.7 Thesis Overview
The key scientific advances covered in this thesis are: 24
Statistically significant auxiliary heating of an RFP has been measured for the first time. • A 1-D code has been developed to analyze the physics of this auxiliary heating in an RFP. • Output from the 1-D code demonstrates clearly that fast ion diffusion and mid-radius heating • are crucial to replicate the measured core Te in MST. Neutralization losses and resonant fast ion transport can be modeled. Neutralization losses • are crucial for non-PPCD plasmas. Resonant fast ion transport is found to not be significant for auxiliary heating. The remainder of this thesis is organized as follows. The measurement tools used to collect the data are discussed in Chp. (2). This chapter is a brief overview of MST diagnostics crucial to this research. The operation and physics of MST’s 1 MW NBI are discussed in Chp. (3). Classical fast ion dynamics and physics are discussed here, including both theory and data. Fast ion confinement (both classical and non-classical), slowing down, heating, momentum drive and current drive are all covered in this chapter. MST’s Rutherford scattering diagnostic is discussed in Chp. (4). Fast hydrogen ions from the NBI pollute the Rutherford analyzer enough to significantly impact the measured temperature of MST. The possibility that this physics is also affecting MST sawtooth Rutherford analysis is discussed. In Chp. (5), the heating data are presented. Statistically significant auxiliary core heating of MST is demonstrated in several different PPCD conditions. It is also shown that no significant auxiliary heating occurs in standard confinement plasmas (both F=0 and standard reversal). The latter result is due to the poor energy confinement discussed in Sec. (1.6). The 1-D model developed to model auxiliary heating in MST is discussed in Chp. (6). This is a classical 1-D model that has several layers of features. This 1-D model is a significant improvement over 0-D heating models that had been created for MST prior to the installation of the 1 MW NBI. The necessity of a 1-D model is seen in the discussion of fast ion diffusion and temperature gradients ( T ), which are found to be crucial for limiting heat losses from the MST core. This er e physics could not be detected with a 0-D model. Further, it will be shown why well-established 25 tokamak transport codes (such as TRANSP and NUBEAM) cannot currently be used to model these plasmas. The 1-D model includes fast ion deposition, classical slowing down and classical diffusion. This model is shown to be consistent with measured auxiliary heating in PPCD. To model non- PPCD MST discharges, neutralization losses are included and found to be crucial. In addition, resonant diffusion is modeled and found to be largely irrelevant to auxiliary heating. Chp. (7) is the concluding chapter of this thesis, which summarizes all of the results from the preceding chapters and also discusses possible future paths for fast ion research on MST. Finally, an appendix section (Appendix (A)) discusses the array of bolometers and XUV detec- tors on MST. This research is mostly incidental to the rest of the dissertation, but provides a useful reference to any future MST researchers that wish to use those diagnostics. 26
Chapter 2
Experiment Diagnostics
This chapter will discuss in brief the key diagnostics used for the research in this dissertation. The contributions of individual people to each diagnostic are cited as is appropriate.
2.1 Neutron Detection
Neutrons from MST are measured with a Bicron-408 plastic scintillator along with a photomul- tiplier tube (PMT). It is cylindrically-shaped, 127 mm in diameter and 127 mm in length, coupled with a 130 mm diamter photomultiplier tube [Fiksel et al., 2005]. It is a polymerized base of
22 3 polyvinyltouene doped with hydrocarbons to a density of n =5.23 10 cm [Magee, 2011]. H ⇥ The fusion that happens inside MST is predominantly D-D fusion, which has two branches that occur with approximately equal probabilities:
D + D T (1.01MeV)+p(3.01MeV) (2.1) ! He3(0.82MeV)+n(2.45MeV) (2.2) !
The cross section of a 2.45 MeV neutron with hydrogen is 2.5 barns, giving a mean free ⇡ path of =1/n 7.6 cm [Magee, 2011]. The scintillator dimensions were specifically mfp H ⇡ chosen so that most neutrons would undergo exactly one collision. The scintillator is sensitive to both neutrons and high energy photons, so it is surrounded in all dimensions by approximately 4 inches of lead, which is a good shield of x-rays while being nearly transparent to neutrons. The entire set-up is approximately 1 m from the outer wall of MST. 27
2.2 Thomson Scattering
The Thomson scattering diagnostic on MST measures electron temperature at multiple radial locations simultaneously, at multiple points in time. The physics of Thomson scattering was orig- inally discovered more than a century ago by JJ Thomson [Thomson, 1906], the same researcher who had discovered the electron just a few years prior. Thomson scattering consists of the absorp- tion and re-radiation of an electromagnetic photon by a free charged particle. This charged particle can be an ion or an electron, though for the case of the popular plasma diagnostic the charged particle measured is always the electron. The physical process is well described in the theses of previous MST graduate students who have focused on Thomson scattering, namely Den Hartog [1989]; Biewer [2002]; Stephens [2010] and Reusch [2011]. Those theses should be read for an in-depth description of the Thomson scattering process, which will just be reviewed in brief below. If we assume that the incident electron is non-relativistic (i.e. significantly slower than the photon), then the electron will only be significantly accelerated by the electric field of the photon (the acceleration due to the magnetic field will be small because the Lorentz force is a function of particle velocity). This electric field causes the electron to oscillate at the same frequency as the photon, and in the same direction as the photon. This generates dipole electromagnetic radiation with the same frequency as the absorbed photon in the reference frame of the electron. This radiation is scattered with a dipole radiation pattern. In Fig. (2.1), this is a sin( ) distribution, where is the angle between the direction of the incident photon and the direction of the scattered photon. On MST, the incoming radiation is polarized in the toroidal direction, and the collection lens is located at = 90o. The collected photon will be scattered at the same frequency of the incoming photon, in the reference frame of the electron. Assuming that the electron is in fact moving with some non- relativistic velocity (as is always the case in a laboratory plasma), the scattered photon’s frequency will be Doppler shifted. Assuming a Maxwellian distribution for the electrons in the plasma, a distribution of frequencies of scattered photons will be measured. The width of this spectrum correlates the electron temperature. 28
Figure 2.1 A cartoon of the observation of Thomson scattering, from Stephens [2010].
In practice on MST, the incoming photons come from two Spectron SL858 ND:YAG lasers that operate with a 1064 nm wavelength and a 9 ns FWHM pulse duration. The lasers can be operated in multiple modes, with each laser capable of firing 15 times during an MST shot. For most of the data in this dissertation, the lasers are fired 1 ms apart (1 kHz), giving 30 evenly-spaced time locations. Some of the data was taken when one of the two lasers was not operational, giving only 15 time points instead of 30. Scattered photons are measured at 22 radial locations. Because the Thomson scattering system is counting photons, the number of photons measured generates a statistical uncertainty. This uncertainty can be quite large for radial points near the edge of the plasma, where there is less scattering (P n ). Because this data is also often the measured / e least relevant for the research conducted in this thesis, those radial locations are often discarded (see Fig. (6.30) for an example). But since these uncertanties are purely statistical, they can be added up in quadrature. The effect of ensembling the data is to decrease the magnitude of these error bars.
2.3 Far-Infrared (FIR) Interferometer-Polarimeter System
Laser interferometry and polarimetry have long been used for a variety of physical purposes. The famous Michelson-Morley experiment, which failed in 1887 and paved the way for the Theory of Special Relatively, was an interferometry experiment. MST features a CO2 laser and 11 discrete 29 chords at approximately the same toroidal location (between 250o and 255o toroidal, on the typical MST coordinate system) [Brower et al., 2001]. The system features a time response of nearly 1 MHz, and a phase noise of only around 1o. A drawing of the system is in Fig. (2.2).
Figure 2.2 A cartoon drawing of MST’s FIR interferometer-polarimeter system, courtesy of Liang Lin.
This system can be run as an interferometer or a polarimeter in order to measure an array of equilibrium or fluctuation quantities, only two of which will be discussed here. First, as an inter- ferometer, the FIR system can produce an accurate, reliable measurement of the electron density profile. On MST, two parallel, linearly-polarized lasers can travel down two different paths: one through the plasma and a “reference” path that only passes through vacuum. At the end of those paths, the phase change in the laser across each path is then measured. Assuming that both paths have the same length, the change in phase is defined as [Ding et al., 2008]:
pa2 x2 '(x)=re ne(r)dz (2.3) pa2 x2 Z 15 In Eq. (2.3), r is the classical electron radius (2.82 10 m), is the laser wavelength, z e ⇥ is the distance along the chord, x is the impact parameter and x2 + z2 = r2. Because there are 11 30 chords, an Abel Inversion [Hutchinson, 2002] can be performed:
1 a @'(x) dx ne(r)= (2.4) 2 2 ⇡re r @x px r Z In practice, '(x) is measured for each of the 11 chords simultaenously. A numerical fit is made of the data. A spatial derivative is taken to calculate @'/@x, and then an inversion is performed to calculate an electron density profile. Throughout this thesis, electron density data will be presented in two ways. Line-averaged electron density is measured by a single-chord CO2 interferometer, while electron density profile data is measured with the FIR interferometer system. A second use of the FIR system for this dissertation is as a polarimeter to determine plasma current density measurements in the core. In this case, it is the polarization of the laser, rather than the phase, that is measured. In practice, the beam passing through the plasma and the reference beam are initially linearly polarized and orthogonal. Next, the beams are passed through a /4 wave plate, generating a counter-rotating circularly polarized beam [Brower et al., 2003]. Because the right-hand and left-hand circularly polarized waves pass through mediums with different refractive indices, the Faraday rotation angle ( ) measured will equal [Ding et al., 2003]:
2 3 2⇡ nR nL e = dz = ne(z)B (z)dz (2.5) 2 8⇡2c3✏ m2 k Z o e Z
In Eq. (2.5), B = B✓cos✓, where ✓ is the angle between the polarimeter chord and the poloidal k magnetic field. Near the magnetic axis, this term becomes localized to the plasma core since contributions to cos✓ along the chord only come from the plasma center. Brower et al. [2002] used Ampe`re’s Law to solve for the current density on axis:
2 d 1 J(0) = (2.6) µocF dx nef(r, ↵)dz
In Eq. (2.6), cF is a constant and f(r, ↵) representsR the current profile shape (IP = J(0) 2⇡rf(r, ↵)dr). This calculation of current density in the core was used to help determine whether the MSTR NBI is driving non-inductive plasma current. The data is presented in Sec. (3.7.2). 31
2.4 CHERS
Charge exchange recombination spectroscopy (CHERS) has been a popular diagnostic for mea- suring the velocity and tempurate of impurity ions in high temperature plasmas for decades [Isler & Murray, 1983; Fonck, 1985]. The basic physics of spectroscopy can be described by what hap- pens when two atoms exchange an electron. In the case of an MST plasma, imagine that a neutral hydrogen exchanges an electron with an impurity ion with atomic number A and a charge +q. The electron typically is absorbed at a high energy state. When it decays to a lower energy state the ion releases a photon:
0 +q + (+q 1) H + A H + A ⇤ (2.7) ! (+q 1) (+q 1) A ⇤ A + (2.8) !
This same physical process describes interactions between impurity ions and both background neutrals and electrons in a plasma, and can be used for a variety of different ions. Common impurities in high temperature laboratory plasmas like MST are nitrogen, oxygen, boron, carbon and aluminum. CHERS has been used on MST to measure a variety of different impurities, with the most common being C+6. The energy of the photon emitted is a function of the energy state change and the impurity type. For example, the n=7-6 transition for C+6 releases a photon with a wavelength of 343.6 nm, and with a de-excitation time of approximately 90 ns [Magee, 2011]. In a plasma, the magnitude and wavelength of the photons emitted are influenced by the ion density, flow velocity and temperature. The wavelength of the emitted photon is known for the reference frame of the ion. If the ion has a velocity relative to the observer then that photon will be Doppler shifted. And assuming that a population of ions can be described with a Maxwellian distribution, there will be a Maxwellian distribution of photons, with the width of that distribution correlating with the temperature of the ions. A greater density of impurity ions will increase the number of emitted photons as well. This effect can be seen in Fig. (2.3), where the signal for the n=7-6 transition for C+6 is plotted for varying temperatures, flow velocities and densities for a Maxwellian distribution of particles. 32
Figure 2.3 Simulation of the effect of ion temperature (left), flow velocity (center) and density (right) on the measured distribution of photons for the n=7-6 transition for C+6, from Magee [2011].
This process can be stimulated with a neutral beam in a process typically called “active-beam spectroscopy,” which offers several benefits over passive spectroscopy [Isler, 1994]. The neutral beam provides signal levels much larger than the background signal, and at wavelengths long enough for accurate measurement of the Doppler broadening and shift. In addition, the neutral beam allows for local measurements when used with an array of viewing chords. Each chord will see the signal from interactions only in the local volume encompassed by both the viewing chord and the neutral beam. The background signal can be measured with a passive chord (that sees a similar volume of plasma but does not intersect the beam), and then subtracted out in order to calculate the active signal accurately. On MST, the neutral beam used for CHERS is a 45 kV hydrogen beam. The voltage is chosen because the maximum charge exchange reaction rate (< v>) for C+6 as a function of beam energy is 45 kV. Unlike the heating beam that this dissertation focuses on (Sec. (3.2)), the ⇡ CHERS beam only has a current of around 5 A ( 250 kW) in order to limit how much the beam, ⇡ which is nominally a ”diagnostic” only, perturbs the plasma. The photons can be measured in either a poloidal or toroidal view, in order to measure either perpendicular or parallel temperature (the Doppler shift is a function of velocity relative to the 33 viewing angle). For details about the MST collection geometry, as well as a more in-depth treat- ment of how the various spectroscopic lines are separated, please see Rich Magee’s thesis [Magee, 2011]. In many cases, CHERS is used as a proxy for majority ion temperature, even though it is not actually measuring majority ion temperature. Core majority ion temperature is measured on MST with Rutherford scattering, which is described in Ch. (4). CHERS offers some advantages over MST’s Rutherford scattering diagnostic. It can make local measurements to build a temperature profile with respect to radius. CHERS also can measure over a longer period of time (the CHERS beam fires for around 20 ms, compared to the 4 ms for the Rutherford. CHERS also does not ⇡ suffer from the ”neutral particle analyzer” issues with respect to measuring a plasma with a large fast ion species - a phenomenon discussed in depth in Sec. (4.7). The theoretical argument for using CHERS as a proxy for Rutherford is that the energy equi- libration time is quite short. Using Eq. (6.35), we find that the classical energy equilibration time between ion species j and k is [Callen, 2006]:
2 2 3/2 4p2⇡nkq q ln ⇤jk m m ⌫j/k = j k j + k (2.9) eq (4⇡✏ )2m m T T o j k ✓ j k ◆
In Eq. (2.9), ln ⇤jk is defined in Eq. (3.7). For a PPCD plasma in MST very similar to the 19 3 plasmas studied for much of this dissertation (ne = 10 m , Te = 850 eV and Ti = 300 eV),
+6 +6 Reardon et al. [2003] calculated an equilibration time of ⌧ C ,D =1/⌫C ,D 100µs. They eq eq ⇡ verified this calculation with the measurements seen in Fig. (2.4). The assumption that CHERS data can be used as a proxy for majority ion temperature breaks down over short time scales. This means that CHERS cannot be used as a proxy for majority ion temperature in sawtooth analyses, for example. In addition, since this dissertation is focusing on NBI heating, it’s worth noting that impurities are heated more by fast ions than majority ions are (this fact follows from Eqs. (3.6) and (6.35)). So CHERS data can be used as a proxy for majority ion temperatures when Rutherford is not available, but in certain circumstances will not give the same output. 34
Figure 2.4 Comparison of data from CHERS and Rutherford scatterng taken in 400 kA PPCD discharges by Reardon et al. [2003].
2.5 ANPA
The advanced neutral particle analyzer (ANPA) is a diagnostic that has the capability of mea- suring the density of different ion species at a variety of energies. It is a diagnostic that has rapidly increased in complexity in modern high-energy laboratory plasma experiments. The principle of a neutral particle analyzer is a simple one. Neutrals from inside the vacuum vessel work their way into the analyzer, where they are stripped of their electron(s) and accelerated using electric and/or magnetic fields into a variety of measuring plates. What makes the ANPA ”advanced” is an improvement driven by the modern use of high-power neutral beams in plasmas. These beams created a need to measure the energy distribution of a very high-energy beam slowing down as it travels through the plasma [Medley & Roquemore, 1998]. These analyzers are now prevalent on a variety of plasma confinement devices [Afanasyev et al., 2003; Burdakov et al., 2005]. The design of the MST ANPA is drawn in Fig. (2.5). A photograph of the ANPA with the top removed can be seen in Fig. (2.6). It was designed and built by the Budker Institute of Nuclear Physics, the same laboratory that built the high-power NBI that is the focus of this dissertation. 35
Figure 2.5 A cartoon of the design of the MST ANPA. A photograph of what these parts actually look like can be seen in Fig. (2.6). This cartoon is courtesy of Scott Eilerman.
NBI ions, as well as any other ions in the plasmas at energies that can be seen by the ANPA (between approximately 5 and 30 keV) are neutralized in the plasma and head into the ANPA. The entrance of the ANPA is a carbon foil that ionizes the neutrals. They pass through a biased cylinder that focuses this beam of ions. They then pass through a permanent magnet which bends the ions into a trajectory that is a function of their energy. Then there is a vertical electric field that separates the hydrogen from the deuterium. The ions then hit channeltrons (electron multipliers) which measure particle flux and output current. Looking at the channeltrons in Fig. (2.6), the two vertical layers for the two hydrogen isotopes are clear, as is the horizontal array for the various energy levels. The MST NBI is mostly hydrogen but also contains some deuterium. MST itself is a deuterium plasma which can create high energy non-thermal ions during magnetic reconnection events. The ANPA allows for measurement of all of these different particles. An example of the usefulness of ANPA data can be seen in Fig. (3.20), which is a plot of the ANPA hydrogen channels for NBI into a standard 400 KA plasma. The slowing down of the beam fast ions, the abrupt loss of confinement at the sawtooth event, and also the NBI half-energy component can all be seen clearly. 36
Figure 2.6 A photograph of the interior of the MST ANPA. In the picture on the left, the opening to MST is the bottom right, with the lens right near the opening. The large black box contains the magnet, and the channeltrons are toward the top of the picture. The channeltrons can be seen more clearly in the picture on the right.
One other concern for interpreting data with the ANPA is the neutral density profile. The signal that the ANPA sees from any particle is a function of both the fast ion and background neutral density profiles, since the ANPA signal n n < v> . As is discussed in more detail / f n fn in Sec. (2.6), it is difficult to accurately measure neutral density profiles in MST, particularly in the most crucial areas for fast ion dynamics (the plasma core). Because the neutral density profiles are much higher near the edge than in the core, much of the ANPA signal comes from the edge of the plasma. Improved measurement of neutral density profiles will make the ANPA an even more powerful and accurate tool.
2.6 Neutral Density Profiles Via D↵
Neutral density profiles in MST are modeled with a computer code called NENE` that finds an iterative solution with spectroscopy data from an array of D↵ detectors as its main input. D↵ data comes from a similar process to the one described in Sec. (2.4), and involves a charge-exchange 37 event between neutral deuterium and an electron:
e + D e + D (2.10) ! ⇤ D D + (2.11) ⇤! The deuterium switches electrons during this charge-exchange event. The new electron gen- erally starts at a higher energy level, and as it falls to a lower energy level it releases a photon.
The most common reaction in MST is the n=3-2 transition, which produces a D↵ photon with a wavelength of 656.1 nm. This reaction happens so often in MST that it can be measured passively with an array of photodiodes at 16 locations, which each produce a line-integrated measurement .
NENE Neutral Density Profile, 200 kA PPCD 6 ) -3 5 cm 10 4
3
2 Neutral Density (10 1
0 0 0.1 0.2 0.3 0.4 0.5 Radius (m)
Figure 2.7 NENE` output of the neutral density profile at t = 21 ms for the 200 kA low-density PPCD data ensemble.
Converting these line-integrated measurements into neutral densities, given a measured elec- tron density profile and known collision cross-sections, is achieved with a code called NENE` (named after ”neutrals and neutrals”), which was developed at RFX by Fulvio Auriemma and Rita Lorenzini [Lorenzini et al., 2006], and adapted for MST by Scott Eilerman. NENE` is a Monte Carlo code which assumes the plasma to be infinite and symmetrical. Neutral particles are sourced at the plasma edge through several different processes, such as proton/wall collisions and limiter emissions. These neutrals travel into the plasma, where they are tracked until they scatter, charge-exchange or hit the wall. The code then outputs an expected D↵ emission 38
NENE Core Neutral Density, 400 kA Standard )
-3 8 cm 9 6
4
2
Core Neutral Density (10 0 8
6
4
2 Core View D_alpha (V) Core View 0 5
4
3
2
1 Bolometer Signal (MW) 0 0 10 20 30 40 Time (ms)
Figure 2.8 NENE output of core neutral density plotted versus time for a 400 kA standard shot (in this case, MST shot 1111103033). For comparison, the core D↵ data is plotted. Also, the bolometer data at 300o toroidal, 135o poloidal is plotted. 39 profile, which is compared to the measured data. The assumed neutral sources are then varied
2 and the process repeated until the error between the calculated and measured D↵ profiles ( ) is minimized. Example NENE` data is plotted in Fig. (2.7), which is the ensembled data for the 200 kA low- density PPCD case that is used so often throughout this dissertation. Here, the neutral density profile is plotted at t = 21 ms. Note that the profile is very edge-peaked, with values close to 5 x
10 3 8 3 10 cm at the edge, compared to approximately 5 x 10 cm in the core. This 100-to-1 ratio from edge-to-core is a typical output for NENE` on MST. Neutral density plotted versus time can be seen in Fig. (2.8), in which core neutral density is plotted versus time for a 400 kA standard MST shot. Because this is a standard shot there are repeated sawtooth events, which can be seen through all of the data. The raw core D↵ signal is plotted for comparison. In addition, one of the bolometer signals is plotted below. The bolometer measures all radiation out of the plasma, which in general means neutrals and radiation. This makes it a decent proxy for the neutral flux that is measured near the edge, and the signal is qualitatively very similar to the calculated neutral density. For more discussion about bolometers, see Appendix (A). The fact that the neutral density is so much larger at the edge than in the core presents a statistical problem, since all of the D↵ signals are line-averaged. This makes any computational model of neutral flux extremely insensitive to core neutral densities. NENE` output is consistent with the D↵ data for most runs, but it would still fit the data well if the core neutral density was changed significantly. In some cases, neutral density in the core could be dropped to zero and the calculated line-averaged D↵ signal would still be very close to the measured signal. While there are no published papers detailing this phenomenon, researchers on MST working with NENE` have consistently found that it seems to over-state neutral density in the core. Previous CHERS research (such as Kumar et al. [2012]) has found that NENE` output is very consistent with related data near the edge, but has a tendency to be too high in the core, particularly during PPCD. These core neutral density uncertainties, and their effects on attempting to calculate fast ion charge-exchange losses, are discussed in more detail in Sec. (6.8). 40
Chapter 3
Neutral Beam Injection
3.1 A Brief History Of Neutral Beams On Fusion Devices
Particle beams go back more than a century, and the earliest ones did not even feature artificial acceleration. For example, the beam of alpha particles used in Ernest Rutherford’s gold foil ex- periment (Sec. (4.1)) was generated only from the natural radioactive decay of a piece of radium ( 4 MeV). The earliest artificial linear proton accelerator was developed under the watchful eye ⇡ of Lord Rutherford by John Cockcroft and Ernest Walton [Cockcroft & Walton, 1930], which was inspired by the clever “resonance accelerator” designed by Rolf Wideroe¨ [Wideroe,¨ 1928]. This proton accelerator was improved over time and, in 1932, was used to “split the atom” for the first time [Cockcroft & Walton, 1933; Halpern, 2009]. It was there, at Cambridge University’s Cavendish Laboratory, that these particle accelerators were used to conduct arguably the first ever fusion experiments. They arose out of experiments be- ing done by Oliphant et al. [1934] to fire a 100 kV deuterium beam at a deuterium target. Through these experiments they discovered that some of the beam deuterons would fuse with some of the target deuterons, generating energy. Oliphant and his colleagues decided to press this further, to see if they could actually generate more energy from the fusion reactions than was necessary to produce them, but before they could really get started they were shut down by Rutherford, who viewed the potential experiment as a silly waste of time and money [Oliphant, 1972; Hora, 1984]. Lord Rutherford’s objections nonetheless, other experimenters at other labs did try to generate fusion power through these beam-target devices. It actually makes sense at first glance because of the fact that these beams can be fired at approximately the peak of the deuterium-deuterium and 41 deuterium-tritium fusion cross-sections. By the 1950s, however, it had become clear that so much of the kinetic energy was being dissipated by ionization, radiation and energy transfer to the target that such a beam-target scenario would always be unworkable as a fusion power device [Post, 1956]. By the early 1950s, fusion power research had shifted to magnetically confined plasmas. In- jecting particles into a machine with high magnetic fields requires a neutral (rather than electrically charged) beam, in order to deposit fast ions into the core of the machine and to inject current and momentum, as well as heating. Neutral deuterium beams were fired into magnetically confined plasmas by the late 1950s [Gibson et al., 1959], though significant use of neutral beams on fusion devices only began in the early 1970s [Menon, 1981]. With the tokamak becoming the most common large magnetic confinement design in the 1970s, neutral beams became popular for two reasons. First, the great increases in confinement times in the tokamak made neutral beams much more useful [Jassby, 1977] (fast ions are not very useful if they or the heat or momentum that they deposit are lost to the wall immediately). Second, it became increasingly clear that tokamaks would not be able to reach ignition with ohmic heating alone [Golovin et al., 1970; Kelley et al., 1972; Menon, 1981]. As the plasma core heats up it also
2 becomes less resistive, and since P⌦ = ⌘J it becomes increasingly more difficult to heat further. While not the only way to provide auxiliary heat to plasmas, neutral beams have become the dominant form of auxiliary heating on tokamaks [Murakami et al., 1977; deGrassie et al., 2006]. ITER, for example, is planning for a minimum of 33 MW of NBI power to achieve the temperatures necessary for significant fusion output [Shimada et al., 2007]. Neutral beams have not been as extensively studied on reversed-field pinches. As discussed in Sec. (1.6), the RFP (unlike the tokamak) can theoretically reach ignition without any auxiliary heating. Second, the poor confinement of both energy and particles in the RFP relative to the tokamak (also discussed in Sec. (1.6)) means that neutral beams will not be nearly as effective at heating or driving current as in a tokamak. In fact, prior to the research conducted for this dissertation, no conclusive evidence of auxiliary heating of an RFP had ever been demonstrated. Auxiliary heating of an RFP with an NBI was 42
Figure 3.1 Data from Sakakita et al. [2003] showing the best previous evidence for NBI heating of an RFP in TPE-RX. explored in TPE-RX with a 1.2 MW, 15 ms duration beam [Koguchi et al., 2008]. With enhanced confinement, an increased SXR signal consistent with approximately 30 eV of heating ( Te Te ⇡ 0.05) was seen, but this was within the statistical error of the SXR diagnostic, and the heating could not be confirmed with Thomson scattering [Sakakita et al., 2003]. The key graph showing this data is seen in Fig. (3.1).
3.2 MST’s 1 MW Neutral Beam
MST currently has a tangential NBI with an energy of 25 kV and a current of 40 A (1 MW of total power), with a flat-top duration of up to 20 ms following approximately 3 ms of current ramp- up. The beam spreads out with a geometric divergence of 3.35 and a random divergence (due to any perpendicular velocity of beam neutrals) of 16 milliradians. The NBI heating experiments discussed in this dissertation were all performed with a hydrogen beam, generally doped with around 3-5% deuterium. Neutron flux data suggests that there is an automatic uptake of MST pre-fill deuterium by the NBI regardless of the designed NBI fuel (this is discussed further in Sec. (3.5.1)), but a doping of 3-5% assures a healthy (but not saturated) neutron flux signal. 43
Figure 3.2 A CAD cutaway drawing of MST’s NBI and its beam path.
MST’s NBI was designed and built by the Budker Institute of Nuclear Physics in Novosibirsk, Russia. They measured the beam energy fractions as approximately 86% full energy, 10% half energy, 2% third energy and 2% E/18. The partial energies come from other molecules besides pure hydrogen atoms becoming ionized, accelerated, and dissociated back into their constituent + atoms inside the beam. For example, the half-energy comes from H2 , and the E/18 comes from water. The full energy neutrals have an energy of 25 kV. 25 kV hydrogen atoms travel at speeds of approximately 2 106 m/s. ⇥ Since we have no way on MST of measuring the neutral current, the plasma source ion current is measured instead. The ratio of ion current to neutral current can be calculated by knowing the relative cross-sections for neutralizing charged deuterium and for ionizing neutral deuterium (see Hudson [2006] for details on this calculation). Budker calculated and measured this ratio for MST’s 1 MW beam, and calculated that I /I 0.78. n i ⇡ The MST NBI resides at 55o toroidal and 17o poloidal, inclined at a downward angle of 6o relative to MST’s midplane, and at a 45o horizontal angle relative to the wall. Fig. (3.3) is a photo of MST’s 1 MW NBI. Nothing can be seen from the outside, but the arrows drawn point out the general locations of the key features. On the far left are a pair of arc discharges. Gas is pumped in and then ionized. Three grids then accelerate the positive ions to full energy and 44
Figure 3.3 A photograph of the MST’s 1 MW NBI, with the locations of some of the key compo- nents noted. The accelerating grids and arc discharges are internal, and cannot be seen directly in this photograph. focus them. The first grid sets the beam voltage, the second grid is electron suppression (around -360 V) and the third grid is grounded. The fast ions then pass through the neutralizing chamber, where most of them charge exchange with neutral gas. While it is possible to puff gas directly into the neutralizing chamber, in practice on MST there is enough gas injected through the valves near the arc discharge cathodes prior to the firing of the arc discharge that the neutralizing chamber has sufficient gas pressure from that alone. Any fast ions that do not charge exchange in the neutralizing chamber are re-directed into an ion dump by a large permanent magnet. The neutral beam then passes through a port into MST. The large cylindrical object above the neutralizing chamber in Fig. (3.3) houses a titanium gettering pump, featuring two titanium rods that will sputter titanium to absorb any extra hydrogen gas between shots. In practice, this gettering pump only needs to be used intermittently, because over time the titanium will coat the walls of the neutralizing chamber and act as a getter on its own. 45
3.3 Classical Fast Ion Dynamics
Once inside MST, most of the fast neutrals collide with the plasma electrons and ions and become ionized. The three dominant forms of ionization are drawn in cartoon form in Fig. (3.4). The likelihood of each of those three types of collisions happening (electron impact, ion impact, and ion charge exchange) are calculated in Sec. (6.5.2). The ionization, however it occurs, creates
Electron Impact
Ion Impact
Ion Charge Exchange
Figure 3.4 A cartoon showing the three main types of neutral beam charge exchange. Motion is drawn from right to left. From top to bottom, the three collision types are electron impact, ion impact, and ion charge exchange. The likelihoods of each of these three collisions happening are broken down in Sec. (6.5.2). a fast ion and an electron traveling at the same speed. These new ions and electrons will be acted on by Lorentz forces, which will send them into gyro-orbits around the magnetic field lines with a Larmor radius defined as: mv2 F = q(v B)=ma = ? ˆr (3.1) ⇥ Lorentz r mv2 r = r = ? (3.2) ! Larmor | | q v B | ⇥ | Because the new ions and electrons are moving at the same speed, it is the case that: r m Le e 1 (3.3) rLi / mi ⌧ where rLi is the ion Larmor radius and rLe is the electron Larmor radius. The reason that the ratio of Larmor radii is not exactly equal to the mass ratio is because the pitch angle of the ionized 46
particles are not precisely the same, and thus v e = v i in general, despite vi = ve for newly ? 6 ? | | | | ionized beam particles. The Larmor radius for the fast ions depends on the location of deposition since both the mag- netic field magnitude and direction are a function of radius, but it is typically around 10-15 cm near the edge of MST, and as little as 1-2 cm near the core. The primary reason for the difference in Larmor orbits is the v /v ratio, which is much larger near the core than it is near the edge. k These new charged particles will interact with the background plasma through classical Coulomb collisions. We can define the classical slowing down of a beam species s on a background species s0 by conservation of momentum: dv m s = ⌫s/s0 m v (3.4) s dt slow s s Xs0 s/s0 The variable ⌫slow in Eq. (3.4) is the slowing down collision frequency of species s on species s0, which is derived in Callen [2006] as: m + m s/s0 s/s0 s/s0 s s0 ⌫slow = ⌫o (x ) (3.5) ms0 where: 4⇡n q2q2 ln(⇤) s/s0 s0 s s0 ⌫o = 2 2 3 (3.6) (4⇡✏o) msvs 3/2 ✏oKTe ln(⇤) = ln 12⇡ns0 2 constant (3.7) ns0 e ⇡ ✓ ✓ ◆2 ◆ vs xs/s0 = (3.8) v ✓ ts0 ◆ xs/s0 s/s 2 t (x 0 )= pte dt (3.9) p⇡ Z0
In Eq. (3.8), vts0 is defined as the thermal speed of background species s0. It should also be noted that xs/s0 will tend to be either very large (if the background species is ions) or very small (if the background species is electrons), and so we can often make one of the following approximations:
lim (xs/s0 ) 1 (3.10) x !1 ⇡ 2 4 lim (xs/s0 ) ptdt(1 t + ...) x3/2 (3.11) x 0 ⇡ p⇡ ⇡ 3p⇡ ! Z 47
Assuming only two background species (ions and electrons), we can calculate a slowing down time for a species s: 1 1 ⌧ s = = (3.12) slow ⌫s s/i s/e slow ⌫slow + ⌫slow
Plugging in typical values for MST (Te = Ti =400 eV, Eb = 25 keV, Zi =2, Zb =1, ne = 19 3 n =1 10 m ) gives a fast ion slowing down time of 21.5 ms. There is now a clear picture i ⇥ of what initially happens to ionized fast neutrals. The electrons are very quickly absorbed into the plasma as fresh background electrons or lost as part of charge-exchange neutrals. The ions, however, become a brand new species of fast ions that move in large gyro-orbits at speeds far beyond that of the background ion thermal speed. For more discussion of the Larmor orbits and the various directions of the beam velocity vectors and the magnetic field vectors, please see Sec. (6.5.2).
3.4 Measuring NBI Shine-Thru
Figure 3.5 A photograph of the NBI shine-thru detector. On the right is the cone-shaped molyb- denum target sitting on the vacuum flange as it does inside MST. On the left is the aperture piece, which was removed for this photo so that everything underneath could be seen.
The ionization of beam neutrals in the plasma can be determined by measuring the attenuation of the beam with a shine-thru detector on the opposite wall of the vacuum vessel. I designed and assembled this diagnostic myself, and will it explain it here rather than in the Diagnostics chapter 48 of this thesis for easier readability and continuity. A photo of the disassembled detector is in Fig. (3.5). A drawing of the design of the shine-thru detector is in Fig. (3.6). There is a cone-shaped target made of molybdenum that is approximately 0.5 inches in length, with an aperture piece made of stainless steel that fits over the top. In the photo of the shine-thru detector, the aperture is removed so that the molybdenum target can be seen. The aperture itself is a tapered hole approximately 2 mm in diameter. The aperture is tapered so that it is smaller as it gets closer to the target. This design is so that any neutrals that hit the side of the aperture hole will deflect out rather than in.
300V - +
V
Figure 3.6 A drawing of the shine-thru detector design. The target is biased -300 V relative to the aperture. The resistive load over which the shine-thru current is measured is also drawn. This is a cartoon and is not drawn to scale.
The target and the aperture are both isolated from MST ground. A voltage bias is applied, with the target piece set to -300 V relative to the aperture piece. As neutrals come through the aperture they hit the target piece, knocking electrons free. The voltage potential sends these electrons into the aperture piece, generating a current that is measured as a voltage across a resistive load. The shine-thru detector is placed at the exact location where a straight line from the center of the beam path intersects the far MST wall (when the beam is properly aligned). The 1 MW NBI was fired into MST for the first time on February 1st, 2010. One of the first tasks was to align the beam. There are four beam position monitors (BPMs) at 90 degree angles covering the area just before the beam reaches its MST porthole, each of which is a calorimeter that measures heat load. Initially there was some uncertainty whether a perfectly aligned beam would create completely balanced BPM signals, or if magnetic effects on particles that neutralize close 49 to the permanent magnet could cause an unbalanced signal. The result was that initial attempts to align actually caused us to make alignment somewhat worse. With the aid of a camera that could image the shape and alignment of the beam, a second try at alignment succeeded. And this improvement in alignment can also be seen in the shine- thru detector signal. The shine-thru detector, because it only sees the very center of a large beam width (see Fig. (6.18) for a simulation of how wide the beam is at the shine-thru detector), is not particularly useful for exact alignment, but it is another piece of information. Data from the initial run, a run between the two alignment days, and after the final alignment day are plotted in Fig. (3.8).
Assuming a beam ionization cross-section bs and a beam velocity vb, as well as a density of plasma species s (ns), the change in the number of fast neutrals (Nb) due to ionization over a distance dx can be written: dN b = n N (3.13) dx bs s b
bsnsx The solution to Eq. (3.13) is Nb(x)=Nb(0)e , and thus we would expect a plot of fast ion deposition versus plasma density to be an exponential. For the case of ion impact and ion charge-exchange collisions, the ions can be treated as fixed objects and bi is only a function of the beam energy. For the case of electron-impact ionization the collision frequency is also a function of the velocity distribution of the electrons. This is treated by solving for the mean product of relative velocity and cross-section (< v>). The electron-impact ionization rate for a 25 kV hydrogen beam is plotted in Fig. (3.7). Because the relationship between cross-section and electron temperature is relatively flat for temperatures above 400 eV, and the fact that electron-impact ionization only accounts for around 20-25% of fast ion deposition (Sec. (6.5.2)), we can approximation shine-thru as being a function of plasma density only. And from Eq. (3.13) we would expect the relationship between shine-thru and density to be exponential. The data in Fig. (3.8) are fit to an exponential because of this. The data is also normalized to the NBI current. The data in that plot is all averaged over the length of the NBI pulse. The fact that the beam was moved out of alignment in early February and then moved into better alignment in late February is quite clear in that data. 50
Electron Impact Cross Section, 25 kV H-Beam 4
3 ) -1 s 3 cm -8 2 v> (10 < 1
0 0 1000 2000 3000 Te (eV)
Figure 3.7 A plot of the electron-impact cross-section data used here, from the EIRENE code [Ko- tov et al., 2007]. This data is for a 25kV NBI. Shine-Thru Vs Plasma Density, Various Alignments 1.2 February 3rd 1.0 February 22nd March 5th
(Arbs) 0.8 beam 0.6
0.4
Shine-thru/I 0.2
0 0.4 0.6 0.8 1.0 1.2 1.4 13 -3 Average ne During Beam Pulse (10 cm )
Figure 3.8 A plot of measured beam shine-thru versus plasma density throughout three different run days as we attempted to align the beam. All dots represent an average over the NBI pulse. All dates are from the year 2010. 51
A picture of a well-aligned beam firing into a beam dump (rather than the plasma) can been in Fig. (3.9). The neutralizing chamber is on the left. There is a beam dump on the right, which is kept in the path of the beam whenever the NBI gate valve to MST is closed. The hot spots flying off the beam dump from the 1 MW beam smashing into it are apparent.
Figure 3.9 A photo taken in 2010 of the NBI smashing into the beam dump. The neutralizing chamber can be seen on the far left. The hot spots caused by the beam smashing into the beam dump on the right can be seen clearly.
Once the NBI was aligned, the shine-thru detector was calibrated by firing the NBI through vacuum onto the opposing wall. This process has been repeated a couple of times since to make sure that the shine-thru detector hasn’t changed in any measurable way, and for more than three years the calibration has been fine. The relationship between density and beam aborption can be seen clearest with a pair of data ensembles, each of 200 kA PPCD plasmas. The two ensembles are plotted in Fig. (3.10) and
19 3 Fig. (3.11). The low density (n 0.5 10 m ) ensemble is made up of 22 shots, and the e ⇡ ⇥ 19 3 high density (n 0.75 10 m ) ensemble is made up of 29 shots. Because density changes e ⇡ ⇥ through PPCD, density here is defined at t =10ms. At all points in time, even as density increases during enhanced confinement, the “high density” ensemble has around a 50% higher density than the “low density” ensemble. Another difference between the two ensembles is that the high density 52
Beam Power, 200 kA Low Density PPCD Beam Power, 200 kA High Density PPCD 250 250 200 200 150 PPCD 150 PPCD 100 100 Ip (kA) Ip (kA) 50 50 0 0 1.5 1.5 ) ) -3 -3 1.0 1.0 cm cm 13 13 (10 (10 0.5 0.5 e e n n
0 0 1000 1000 800 800 Beam Power Beam Power 600 600 400 400 Shine-thru Shine-thru 200 200 Beam Power (kW) Beam Power (kW) 0 0 0 10 20 30 40 0 10 20 30 40 Time (ms) Time (ms)
Figure 3.10 An ensemble of 200 kA low density Figure 3.11 An ensemble of 200 kA high den- PPCD plasmas with NBI. Shine-thru is high be- sity PPCD plasmas with NBI. Beam shine-thru is cause of the plasma density, though it decreases much lower than in the low plasma density case. as plasma density increases during the enhanced confinement period. ensemble features a full length NBI pulse, while the low density case has a short ( 8 ms) beam ⇡ pulse. The lower densities and higher temperatures of PPCD plasmas in general lead to higher shine- thru than standard plasmas. In Figs. (3.10) and (3.11), it is clear that the higher-density PPCD has significantly lower shine-thru than the lower-density PPCD. Also note that shine-thru decreases with time after the enhanced confinement period begins (t 12-13 ms) as the plasma density ⇡ increases. Higher temperatures do reduce the electron impact ionization cross-section (Fig. (3.7)), but because electron-impact ionization only represents approximately 20% of all beam ionization, this effect is secondary to the reduction in shine-thru due to increased plasma density late in PPCD. This effect is true in both ensembles. 53
3.5 Fast Ion Confinement
3.5.1 Calculating Fast Ion Confinement
Once deposited in MST, some of the fast deuterium ions will fuse with background ions, gen- erating neutrons just like a beam-target fusion device. The neutron flux is measured on MST with a plastic scintillator described in Sec. (2.1). Before getting into the theory and modeling of fast ion confinement, it’s an interesting side note that the neutron signal demonstrates that there will always be some deuterium in the NBI as long as it is being fired into deuterium plasmas, even with pure hydrogen NBI fuel.
Neutron Signal vs Early Gas, February 3, 2010 0.14
0.12
0.10
0.08
0.06
0.04
0.02 V=0.109 x P + 0.016 R2=0.18 0 Average Neutron Detector Voltage (V) Neutron Detector Voltage Average 0.2 0.3 0.4 0.5 0.6 0.7 Average Neutral Pressure For 50 ms Before Beam (mTorr)
Figure 3.12 Neutron detector voltage is plotted versus the average neutral pressure for the 50ms prior to the initial firing of the NBI. Each dot represents a shot. These shots were all taken with a pure hydrogen feed into the NBI arc discharge. A best-fit line is included, with an R2 value.
In Fig. (3.12), the x-axis is the neutral pressure in MST averaged for 50 ms prior to the start of the firing of the NBI. That is a proxy for deuterium pressure inside the NBI, since there is no other way on MST to measure it. The y-axis is the average neutron detector voltage throughout the NBI pulse, noting that this signal is effectively zero in non-NBI shots. Each dot represents an average over a shot. It’s a noisy plot, but there is a clear positive relationship between early deuterium pressure in MST and the neutron signal, which tells us that enough of that deuterium is getting into the NBI to generate some beam-target fusion in MST, even with a supposedly pure hydrogen NBI. 54
We can model the neutron flux by assuming that all fusion involving fast ions will involve a fast ion fusing with a background majority ion. This is a good approximation for two reasons. First, the fast ion density is much smaller than the majority ion density. Second, the fusion cross-section increases with relative-velocity, so fast ions are more likely to collide with slow majority ions than with another fast ion traveling with a small relative velocity. If we make this assumption, and also approximate the majority ions as having no velocity, then we can model the neutron flux as:
n(t)= ni(t)nfi(t) < dd(Efi(t)vfi(t)) >dV (3.14) ZV loss Further, we can define the NBI as a fast ion source, and for a given “loss time” (⌧fi ) we have the relation: dnfi nfi = Sfi loss (3.15) dt ⌧fi
Here, the fast ion “loss time” aggregates all possible loss mechanisms, including charge- exchange with background neutrals as well as diffusive and stochastic losses. We can calculate this “loss time” in “beam blip” experiments, when a very short pulse ( 3-4 ms) is used, which ⇡ allows the approximation that all fast ions have an energy of 25 keV at the moment the beam stops firing. At this point the fast ion source term is zero, and we can calculate the fast ion slowing down from the MST density and temperature from Eq. (3.5). This means that the only unknown is the fast ion loss time. We can plot the neutron flux and fit various loss times to the data. An example from a PPCD shot is shown in Fig. (3.13). In Fig. (3.13), neutron flux predictions for various fast ion loss times are plotted. With the noise in the neutron signal and the fact that the enhanced confinement period only lasts for around 10ms, there is a limit to how accurately the fast ion loss time can be fit to the data. Over the timescales of this model, a 23 ms loss time and an loss time look very similar. This does not mean that 1 the loss times are in fact infinite, but just that they are much longer than the other time scales that dominate fast ion dynamics. Data from a broad sample of different plasma conditions is plotted in Fig. (3.14). Here,
⌧n theory is the theoretical neutron flux decay time for perfect fast ion confinement (⌧fi ). !1 55
2.8 PPCD shot 1100816093 fi=infinity NB On fi=23ms 2.1 fi=10ms fi=2ms
1.4
0.7 NEUTRON FLUX (a.u.)
0.0 10 15 20 25 30 35 40 TIME (MSEC)
Figure 3.13 A plot of neutron signal decay after NBI turns off in a PPCD shot. The modeled decay for different fast ion loss times are plotted. In this case, the neutron decay is consistent with infinite fast ion loss times.
Figure 3.14 Fast ion loss times calculated for a variety of different plasmas. The x-axis is what the neutron decay would be with infinite fast ion loss times and the y-axis is the measured neutron decay time. The horizontal error bars represent uncertainty in plasma density and temperature while the vertical error bars represent uncertainty in calculated neutron decay times. 56
The y-axis on the plot is the measured neutron flux decay time. For each data point, the error bar in the x-direction is the uncertainty in plasma density and temperature while the error bar in the y-direction is the uncertainty in fit neutron decay times. What’s interesting is that not only are fast ion loss times long ( 30 ms) in PPCD plasmas, but also in standard plasmas. Even most of the standard reversed plasmas have neutron flux decay times consistent with significantly long fast ion loss times relative to fast ion slowing down times.
3.5.2 Classical Fast Ion Confinement Theory
The magnetic safety factor q was defined in Eq. (1.11). It is a value that represents the number of toroidal orbits the magnetic field makes for every poloidal orbit, so that q =1/6 is a ratio- nal surface where a field line makes six poloidal orbits for one toroidal loop. We can choose to define a similar quantity for the fast ions, which we can call the ion guiding center (IGC) safety factor [Fiksel et al., 2005]:
qfi = rv /Rv✓ (3.16)
In Eq. (3.16), v and v✓ are the toroidal and poloidal guiding center velocities, respectively. Since it’s generally the case that v /v = B /B , it’s also generally true that q = q . The ✓ 6 ✓ fi 6 M guiding center velocity can be written as [Hudson, 2006]:
2 2 v B B v// B E B ? vGC = v b + ⇥52 + ⇥ + ⇥2 (3.17) k 2!c B !c B B | | | | | | The terms on the right side in Eq. (3.17) are, from left to right, the motion of the guiding center parallel to the magnetic field, the magnetic gradient drift, the curvature drift and ExB drift. Combining Eq. (3.16) and Eq. (3.17) was solved by Y. Tsidulko in a 2002 internal draft report for the Budker Institute of Nuclear Physics, and was first published in an Appendix to the PhD thesis of Hudson [2006]. The derivation can be read there in excruciating detail, but the answer is:
2 s 2(1 µ⌦)b rµ⌦0 q q + k ⇢ ✓ (3.18) ⇡ M b2 2Rp1 µ⌦ ✓ 57
v B In Eq. (3.18), qM is the magnetic field line safety factor, s = v·B is the direction of the k | · | guiding center velocity parallel to the magnetic field (1.0 for co-injection and -1.0 for counter- E v2 injection), ⇢ is the gyro-radius, and µ = ? = 2? . The gyro-frequency and the radial derivative Eo⌦ vo ⌦ eB e dB of the gyro-frequency are ⌦= m and ⌦0 = m dr , respectively.
qM And qfi Profiles 0.25
0.20
0.15
0.10
0.05 Safety Factor 0.00
-0.05
-0.10 0.0 0.2 0.4 0.6 0.8 1.0 r/a
Figure 3.15 The fast ion guiding center safety factor (qfi) profile plotted against the magnetic field line safety factor (qM ) profile. The magnetic island widths are plotted in purple. This was done for a 20 keV deuterium beam on MST by Hudson [2006].
The magnetic island width is defined in Eq. (1.12):
b r W =4 r (3.19) B n q s ✓ | 0| If we approximate MST as a cylinder, we can say v /V b /B, and: r ⇡ r b V v b V b B V r v r r r ✓ (3.20) B ⇡ r ! V ⇡ BV ⇡ B BV ✓ ✓ ✓ ✓ ✓ ◆ In Eq. (3.20), the term in parentheses on the right is of order unity, so we can say that v /V r ✓ ⇡ br/B✓, which means that the island width equation can be re-written as:
v r b r W =4 r 4 r (3.21) V n q0 ⇡ B n q0 r ✓ | IGC| s ✓ | IGC| 58
An important difference between the fast ion island widths and the magnetic island widths comes from the fact that the values of br are calculated at the location of the island. Because the qfi profile is generally above the qM profile for co-current NBI, the islands are at a larger radius, where the values of br,n are smaller. This makes the fast ion island widths smaller, and is why they do not overlap in Fig. (3.15). That figure is from Hudson [2006], from a 20 kV, 25 A, short-pulse (1.3 ms) beam previously installed on MST.
3.5.3 Fast Ion Confinement Data
The relationship between fast ion confinement and the MST plasma can be seen by looking at the same ensembled data used in the rest of this dissertation. In Fig. (3.16) through Fig. (3.19), neutron data is plotted for four different data ensembles. The same low density and high density 200 kA PPCD ensembles from Sec. (3.4), as well as a 400 kA standard reversal (F=-0.2) ensemble and a 400 kA low density PPCD ensemble. In all four plots, the NBI Off neutron signal is in black while the NBI On neutron signal is in red. The length of time that the NBI was fired is in green. For the PPCD plasmas, the period of enhanced confinement is noted with dashed black lines. In addition, a classical simulation of neutron flux is plotted in blue, using the 1-D heating+deposition model described in Ch. (6). The simulation includes fast ion slowing down and classical diffusion, but not any non-classical effects or neutral charge-exchange losses. In order to compare neutron scintillator data to the 1-D heating+deposition model, a calibration factor must be assumed. The scintillator has not been calibrated in its current location, but recent
10 1 1 estimates are that the signal is between 0.75-1.25 10 s V . For this data, a value of 1.0 ⇥ ⇥ 10 1 1 10 s V has been chosen. In addition, since the deuterium fraction of the beam is typically between 3 and 5%, a value of 4% is assumed here. The 400 kA low density PPCD case (Fig. (3.19)) has pure hydrogen fuel, so a 1% deuterium fraction (due to MST pre-fill) is assumed. Non-reversed plasmas are in general cooler and more collisional than PPCD plasmas, and so the neutral signal generated by the beam in those plasmas reaches its equilibrium within a few 59
Neutron Signal, 400 kA Standard Plasmas Neutron Signal, Low Density 200 kA PPCD Plasmas 3.0 2.0 NBI On NBI On NBI Off NBI On /s)
NBI Off /s) NBI On (Simulation) 10
NBI On 10 1.5 NBI On (Simulation) 2.0
1.0
1.0 0.5 PPCD Neutron Flux (x 10 Neutron Flux (x 10 0 0 0.00 0.01 0.02 0.03 0.04 0.05 0.00 0.01 0.02 0.03 0.04 0.05 Time (s) Time (s)
Figure 3.16 An ensemble of neutron flux data in Figure 3.17 An ensemble of neutron flux data in 300 kA F=0 plasmas. NBI Off data is in black 200 kA low density PPCD plasmas. The period while NBI On data is in red, and a classical sim- of enhanced confinement is noted with vertical ulation of the neutron signal is in blue. The neu- black dashed lines. The calibration factors are 10 1 1 tron signal calibration is 1.0 10 s V . The the same as Fig. (3.16). ⇥ beam deuterium fraction is assumed to be 4%.
Neutron Signal, High Density 200 kA PPCD Plasmas Neutron Signal, Low Density 400 kA PPCD Plasmas 4.0 1.5 PPCD NBI Off NBI On NBI Off NBI On NBI On /s)
NBI On (Simulation) /s) PPCD NBI On (Simulation)
10 NBI On 3.0 10 1.0
2.0
0.5 1.0 Neutron Flux (x 10 Neutron Flux (x 10 0 0 0.00 0.01 0.02 0.03 0.04 0.05 0.00 0.01 0.02 0.03 0.04 0.05 Time (s) Time (s)
Figure 3.18 An ensemble of neutron flux data in Figure 3.19 An ensemble of neutron flux data 200 kA low density PPCD plasmas. The calibra- in 400 kA low density PPCD plasmas. This was tion factors are the same as Fig. (3.16). taken with pure hydrogen fuel, meaning a signifi- cantly lower deuterium fraction for the beam and a lower neutron flux signal. For the simulation, a beam deuterim fraction of 1% was assumed. 60 milliseconds. The signal stays relatively flat until the beam turns off, when the neutral signal goes back to zero over a time scale of around 5-7 ms. For looking at the neutron signals, the most meaningful difference between the “low density” and “high density” cases is the length of the beam pulse. It is a short (8 ms) pulse in the low density case, and a full-length (23 ms) pulse in the high density case. With the short beam pulse, the neutron signal increases rapidly until the beam turns off, after which it decays very slowly. After the end of the enhanced confinement perod, the neutron signal decays much more quickly, taking around 10 ms to dissipate. This data is consistent with very good confinement (and low collisionality) of fast ions during the enhanced confinement period, followed by a decreasing temperature (and increasing collisionality) after the end of PPCD. The high density 200 kA PPCD ensemble has a long NBI pulse. In this case, the neutron signal continues to increase rapidly throughout the enhanced confinement period. The fact that the neutron signal does not plateau after 10-11 ms is consistent with a long collision time. After the end of the enhanced confinement period the neutron signal decays fairly rapidly even though the NBI is still firing, consistent with increased collisionality and fast ion losses. The rapid loss of neutron signal at the end of PPCD is primarily due to loss of fast ions from neutralization and a shortening of fast ion slowing down times as the plasma rapidly cools. This latter reason is most likely why the neutron signal drops off to zero at the end of the NBI pulse more quickly in the high density case than in the low density case. The plasma in the high density case had an additional 10 ms to cool off after the end of the PPCD period, and so was much more collisional at the moment the NBI turned off. The 400 kA PPCD ensemble also features a long NBI pulse, but has some differences from the 200 kA PPCD case. The most important difference is that pure hydrogen fuel was used for the NBI. The NBI will get some deuterium from the MST pre-fill, but the deuterium fraction is significantly lower. According to the simulated data, a 1% deuterium fraction is a reasonable fit to the data, compared to 4% for the other ensembles. The 400 kA PPCD ensemble also features a hotter (less collisional) plasma. In fact, the plasma is hot enough that the NBI Off case has some neutron production. In the three other graphs presented in this section the NBI Off neutron flux 61 data is negligible. Like the 200 kA high density PPCD case, the NBI On neutron flux increases until the end of PPCD, after which it decays away. Because the neutron scintillator is not perfectly calibrated, the simulated data is only an ap- proximated fit, but the general shape of the simulated data is a good fit for the general shape of the raw data. One final look at fast ion confinement data can be seen with the ANPA (Sec. (2.5)). Data from a shot with NBI fired into a typical 400 kA standard-reversal plasma is shown in Fig. (3.20). In Fig. (3.20) the NBI is turned on at around 14 ms and turned off at around 22 ms. This particular shot was chosen because of the convenient sawteeth timing. There is one sawtooth event at around 18 ms, and another one at around 24-25 ms. The first sawtooth event happens before most of the fast ions have been deposited, and the effect is relatively small. The loss of fast ions can be seen both in the reduction in fast ions with energies around 20 keV, and with a small downward dip in the neutron signal. The sawtooth event at around 24-25 ms is much more significant, kicking out most of the fast ions left in the plasma, seen in a dramatic reduction in both the ANPA and neutron signals. The collisionality and good confinement of even standard confinement plasmas can be seen in the period after the NBI turns off but before the big sawtooth event in Fig. (3.20). The fast ions are slowing down over a timescale of just a few milliseconds, due to the colder non-PPCD plasma. At the same time, the neutron signal is relatively flat, because the fusion cross-section doesn’t change significantly from a fast ion energy drop of a few keVs. All of the data in this section points to the same fast ion dynamics. The fast ions are very well confined in all types of MST plasmas. While collisionality is much greater in standard/cooler plasmas, fast ion slowing down dominates fast ion dynamics in both standard and enhanced- confinement plasmas. However, a rapid loss of fast ions can occur during a sawtooth event, mostly due to neutral losses. This is all explored further with fast ion modeling in Ch. (6). 62
Figure 3.20 ANPA data (hydrogen channel) from a typical 400 kA standard-reversal plasma. 63
3.5.4 Resonant Fast Ion Transport
While fast ion losses are consistent with classical theory in Sec. (3.5.3), there is evidence for non-classical fast ion transport. The non-classical transport of fast ions has been reported before. Heidbrink et al. [2007] found an anomalous flattening of the fast ion profile during Alfven´ eigenmode activity on DIII-D. Four separate diagnostics found fast ion transport generally con- sistent with classical transport for plasmas that did not have large toroidicity-induced Alfven´ eigenmodes (TAEs) or reversed-shear Alfven´ eigenmodes (RSAEs), but these same diagnostics all showed enhanced, non-classical transport when these modes appeared at significant levels. Heid- brink [2008] further found that these Alfven´ instabilities could be driven by the energetic particles themselves. Koliner et al. [2012] has reported both AE-like and energetic particle mode (EPM)-like modes during NBI operation in non-reversed MST plasmas. The EPM-like mode becomes destabilized in plasmas with a strong radial fast ion density gradient. The onset of these modes can thus be delayed by decreasing the total input NBI power, which increases the time it takes to reach the critical gradient at which these modes become destabilized. This feedback mechanism was further explored by Anderson et al. [2013], who compared results for full power NBI (1 MW) with reduced power NBI (0.6 NBI) in 300 kA, non-reversed plasmas. Here, the power is reduced only by decreasing the fast ion current, and so the fast ion energy is kept constant. The results of this analysis are plotted in Fig. (3.21). In Fig. (3.21), the red data is the full power beam while the green data is reduced power. It is seen clearly that the onset of the fast particle driven mode is delayed in the reduced power NBI case from around 18-19 ms to around 22 ms. The results that are most interesting here are the fourth and fifth plots from the top. The fourth plot from the top shows total neutron flux, while the fifth plot shows the 22 kV ANPA signal (effectively full beam energy). Note that the ANPA signal diverges rapidly for the two NBI cases until the fast particle driven mode turns on for the full power NBI case. At this point, the ANPA signal for the full power NBI case begins to decay. The reduced power NBI ANPA signal increases until the fast particle driven mode turns on in that plasma, after which the two ANPA signals are effectively identical. At the 64
Figure 3.21 Energetic particle-driven instabilities are plotted for 300 kA F=0 MST discharges. ANPA full-energy channel data and neutron flux data is also plotted. Red data is full power (1 MW) NBI while green data is reduced power (0.6 MW) NBI. Data is from Anderson et al. [2013]. 65 same time, the neutron signals for both cases continue to increase slowly throughout the pulse, with the ratio of neutron flux approximately equivalent to the ratio of NBI power. This means that fast hydrogen is driving and interacting with this instability, while fast deuterium is not.
Figure 3.22 Resonant transport of fast ions in DIII-D is demonstrated with this plot of fast ion density, from Heidbrink [2008]. The classical prediction comes from TRANSP, while the data is from fast-ion D↵ measurements in plasmas featuring fast ion-driven modes.
Fig. (3.22) shows measured fast ion profiles during fast ion-driven modes in DIII-D, from Hei- dbrink [2008]. This data shows that these modes significantly flatten the fast ion profile in the core, while leaving the fast ion population beyond r/a = 0.5 unaffected. This is consistent with what is seen in the ANPA signal in Fig. (3.21). When the fast ion population in the core (either its core density or core density gradient) hits a critical point, the EPM-like modes turn on. The lower-current NBI beam deposits fewer fast ions in the core, which is why it takes longer to excite those modes. But once those modes are excited, the core fast ion density flattens, reducing the core ANPA signal. The neutron signal, which is a proxy for global fast deuterium density, is consistent with classi- cal fast ion dynamics. This is presumably because the fast deuterium population in the core never reaches that critical level, since n /n 20 for all of these plasmas. fH fD These EPM-like modes can be seen in nearly every MST plasma with sufficient NBI power and pulse length. Data from the 200 kA low density PPCD can be seen in Fig. (3.23). Two magnetic 66 spectrograms are plotted, both of the m=1, n=5 mode. The top spectrogram is of a single shot, while the bottom spectrogram is an ensemble. In the single shot spectrogram in Fig. (3.23), the individual bursts can be seen. They appear approximately every 0.5-0.75 ms. In the ensemble, it can be seen that the modes turn on and off at approximately the same time in every shot in the ensemble. In each case, they appear at around t =11.5-12.0 ms, approximately 3.5-4.0 ms after the NBI turns on. In each case, the modes disappear as soon as the NBI stops firing at 16 ms. The effect of these fast ion-driven modes on the core fast ion population can be seen in Fig. (3.24), which features ANPA data from the same data ensemble as Fig. (3.23). Here, four energy channels are plotted from the full NBI energy to the final ANPA channel that does not include any of the half-energy component of the NBI. In Fig. (3.24), each of the ANPA channels increases from the moment the NBI turns on until the fast ion-driven modes appear. From that point until the the point that the NBI turns off at 16 ms, the highest energy channels decrease, while the lower energy channel (15.6 kV, in blue) is increasing. This result is consistent with the data in Fig. (3.21). While the bursting modes are excited, the total core fast ion density stays relatively flat. Increased spread in fast ion velocity distribution and fast ion pitch angle distribution leads to a reduction in signal of the higher energy channels. The lower-energy channels continue to increase due to increased numbers of fast ions at that density, and perhaps partially due to less interaction with the resonant instabilities than the higher energy fast ions. The conclusion of this analysis is that in all NBI plasmas of interest here, sufficient fast ion density will excite magnetic activity which will drive resonant non-classical fast ion transport near the core. This effectively sets a ceiling on core fast ion density, which presumably is a function of plasma conditions. This ceiling also only applies near the core, and it’s to be expected that fast ion density in the mid-radius and edge areas of the plasma is relatively unaffected. The impact of this fast ion transport on NBI-driven heating is discussed in Sec. (6.4). 67
Figure 3.23 Two magnetic fluctuation spectrograms for 200 kA low density PPCD is plotted. The top spectrogram is from a single shot, while the bottom spectrogram is an ensemble of shots. 68
200 kA PPCD w/ NBI, ANPA Ensemble Data 1.0 25.5 kV NBI Bursting PPCD 0.8 22.2 kV Modes 18.8 kV 15.6 kV 0.6
0.4
ANPA_H (V) ANPA_H 0.2
0.0 5 10 15 20 25 30 Time (ms)
Figure 3.24 ANPA data taken during the same 200 kA low density PPCD ensemble from Fig. (3.23). The approximate time that the bursting modes appear is overlayed.
3.6 NBI Momentum Drive
One of the impacts of a tangential NBI is momentum drive. The fast ions have a large amount of momentum, and through classical collisions we can expect some of this momentum to be deposited on the plasma. An example of this effect can be seen in Fig. (3.25). The data in Fig. (3.25) is from the 400 kA high density PPCD case, and it’s consistent with what is seen in all of the other PPCD datasets. The n=6 magnetic fluctuations are suppressed (the black curve is NBI Off while the blue curve in that panel is NBI On), while the n=7 magnetic fluctuations are not (NBI Off is brown while NBI On is red). It is also true that the n=8-12 modes are not suppressed. This effect of the NBI suppressing only the core-most magnetic mode in MST (which in the case of 400 kA PPCD is the n=6) is consistent with all of the data taken so far, and is due to the core-peaked fast ion density. The fact that the n=7 mode velocity increases substantially (as does the n=8) while the fast ion density is too small at those radial locations to significantly suppress the magnetic fluctuation magnitude is most likely due to the non-linear magnetic mode coupling, which is already known to play an important role in momentum transport in MST [Hegna, 1996; Hansen et al., 2000]. 69
400 kA High-Density PPCD
500 2500 NBI Power (KW) 400 2000 300 1500
200 1000 Ip (kA) 100 500 0 0 1.5 ) -3 PPCD 1.0 cm 13
(10 0.5 e n
0 15
) 10 G ( ˜ b 5
0
60
40
20
0
-20 Mode Velocity (km/s) Mode Velocity 0 10 20 30 40 Time (ms)
Figure 3.25 Impact of NBI on n=6 and n=7 magnetic fluctuation magnitude and velocity. In the top two panels NBI Off is black and NBI On is red. In the top panel, NBI inputed power is in brown while NBI power minus shine-thru is in blue. In the bottom two plots, NBI Off data is in black (n=6) and brown (n=7), while NBI On data is in blue (n=6) and red (n=7). 70
While NBI momentum drive is not a focus of this dissertation, there is an interesting result regarding prompt ion losses. Prompt ion losses occur in certain magnetic configurations, when a fast neutral is ionized and on its first Larmor orbit slams into the wall and is immediately lost. Due to the direction of the magnetic field in MST (see Fig. (6.16)) we would expect to have very few prompt-ion losses in “co-current” NBI, but a lot of prompt-ion losses in “counter-current” NBI (when the MST plasma current and magnetic field are reversed). Hudson [2006] calculated that prompt-ion losses for a 20 kA tangential NBI in MST would be over 50% for counter-current, and around 10% for co-current. In addition, it was calculated that counter-injected fast ions have much shorter confinement mainly due to a different ion guiding center safety factor profile. Also, the fact that the gyro- orbit direction is reversed means that the magnetic field will drag many of the fast ions away from the core, rather than dragging them into the core as with co-current NBI (see Sec. (6.5.2) for a discussion of how the magnetic field “focuses” fast ions into the core). Fast ions further from the core will suffer higher charge-exchange losses (due to higher background neutral densities) and will also have an easier time diffusing into the wall due to proximity and larger Larmor orbits. A confinement time of around 4 ms was calculated for counter-current, compared to well over 20 ms for co-current NBI. The effect of this is a significantly lower fast ion density for counter-current NBI than co-current. In Fig. (3.26), data is plotted for co- and counter-current NBI in 300 kA non-reversed plasmas. In this plot, the NBI Off data is plotted in black (for co-current NBI) and green (for counter-current NBI) while the NBI On data is plotted in red (co-) and blue (counter-). In the NBI power plot in the top panel, the solid lines are the inputed NBI power while the dotted lines are NBI power with shine-thru subtracted out. As usual, there is a significant suppression of the innermost magnetic mode (in the case of 300 kA F=0 plasmas in MST, this is the n=5 mode) for co-current NBI. However, it’s hard to detect much effect on the fluctuation amplitude at all in the counter-current case. At the same time, the co-current NBI drives a large increase in the n=5 mode velocity. The counter-current NBI significantly slows the n=5 mode velocity, but by only 30-40% as much. 71
Figure 3.26 NBI co-current and counter-current NBI data in a standard 300 kA plasma. NBI Off data is black (co-current) and green (counter-current) while NBI On data is red (co) and blue (counter). In the NBI power plot in the top panel, the solid line is the inputed NBI power while the dotted line is NBI power minus shine-thru. 72
The results in Fig. (3.26) are even more interesting when looking at the neutron flux plot in the second plot. The neutron flux in the counter-current case is only around 5% of the co-current case. The data here appears to be contradictory, both to itself (that the ratio of neutron fluxes is much larger than the ratio of momentum drive) and to Hudson [2006], which found the neutron flux for a 1.5 ms, 20 kV NBI on MST to be nearly half in the counter-current case compared to the co-current case. That data can be seen in Fig. (3.27). But these differences are due to the difference between a short-pulse and long-pulse NBI. This effect can be seen in Fig. (3.28).
Fast ion density: Co- vs Counter- Injection 3 Short-Pulse NBI Solid = Co-Current Long-Pulse NBI Dotted = Counter
2 (Arbs) fi n 1
0 0 10 20 30 40 Time (ms)
Figure 3.27 Neutron flux data for co- vs counter- Figure 3.28 Simulated fast ion density for an NBI from Hudson [2006]. This was from a pre- arbitrary plasma volume and for short-pulse and vious 1.5 ms, 20 kV NBI on MST. long-pulse NBI. Loss times of 4 ms and 20 ms are assumed for counter- and co-injection, respec- tively.
In Fig. (3.28), fast ions are deposited in an arbitrary plasma volume, with prompt losses and fast ion loss times consistent with previous calcuations. In this case, 50% prompt losses in counter- current NBI are assumed, as are fast ion loss times of 4 ms and 20 ms for counter- and co-injection, respectively. The effect of a short-pulse NBI with quick ramp-up of beam current is that a large number of fast ions are deposited before the 4 ms confinement time for counter-current fast ions can really kick in. Later in time, the ratio between co- and counter-current is much larger, as is also seen in Fig. (3.27). With slower-ramping long-pulse NBI, however, the losses of counter-current 73
NBI fast ions are apparent before a significant fast ion population is seen. This effect explains the small neutron flux in Fig. (3.26).
3.7 NBI Current Drive
One of the common uses of an NBI on tokamak devices is to drive plasma current. The current of the fast ions themselves (Ifi = qfinfiv¯fi) might be detectable. Alternatively, the fast ions can impact plasma current several other ways. The fast ions will deposit momentum on majority ions through momentum-conserving Coulomb collisions. The fast ions might also deposit momentum on electrons, or drag electrons in their path, which would serve to drown out any non-inductive current drive. An example of NBI current drive in DIII-D is given in Fig. (3.29) [Simonen et al., 1988]. There, the NBI is turned on just as the inductive current drive is beginning to drop off, driving the plasma to a higher current than was achieved before the beam. The alternative detection of non-inductive current drive is seen in Fig. (3.30), where MAST data is presented [Turnyanskiy et al., 2009]. In this case, the plasma current stays constant after the NBI is turned on, but inductive current drive is partially replaced by non-inductive current drive. This is seen experimentally by a reduction in the loop voltage.
3.7.1 Current Drive Theory
The impact of fast ions on plasma current can be visualized as a battle between two separate effects. The fast ions themselves will travel around the plasma device with some velocity and electric charge, creating a current. This current can be significant. It has been calculated with both TRANSP and the 1-D model developed in Ch. (6) that, for good fast ion confinement in MST, this fast ion current can be approximately 40 kA. At the same time, these fast ions will drag electrons along with them that will shield some of the current. Various effects can increase or decrease the amount of electron shielding of fast ion current, some of which will be discussed in this section. For a classical approach, a cylindrical plasma can be assumed, with uniform magnetic fields. Assuming that the NBI is firing long enough, a solution can be found by assuming equilibrium 74
Figure 3.29 Data from DIII-D [Simonen et al., Figure 3.30 Data from MAST [Turnyanskiy et 1988] showing NBI current drive. Plot (a) is Ip, al., 2009] showing NBI driving a reduction of where the dotted line is the NBI Off case. Plot plasma loop voltage. Total plasma current stays (b) is the NBI power. Plot (c) is the loop voltage, constant in this example, but inductive current and (d) is line-averaged electron density. drive is being replaced by non-inductive (beam) current. between the fast ions and background electrons and ions [Ohkawa, 1970]. If it is assumed that v v v , then a simple solution to the equilibrium can be found Start et al. [1978]: e fi i Z I I 1 fi (3.22) T ⇡ fi Z ✓ i ◆ What’s fascinating about Eq. (3.22) is that for a hydrogen/deuterium beam being fired into a deuterium plasma:
IT =0 (3.23)
This implies that through Coulomb collisions the fast ions are dragging electrons with them that entirely cancel out the non-inductive current, and so there is no current drive.
In reality, Zeff is not 1 in hot magnetically confined plasmas. The assumption that we have a cylindrical plasma and that we have a uniform magnetic field is false as well. In a toroidal plasma, the magnetic field is not uniform as a function of radius, and as electrons travel in their orbits they pass through areas of higher and lower magnetic field strengths. A stronger magnetic field will result in an increase in electron v . Just like in a magnetic mirror device, particles with a large ? 75 v /v will be unable to pass through the areas of highest magnetic fields, and will not be able to ? k pass all the way around the torus. They will become “trapped”, and will be unable to help cancel out the fast ion current. The physics to describe this more-complete kinetic description of the plasma is explained in [Connor & Cordey, 1974; Start & Cordey, 1980]. The solution to this problem cannot be solved analytically, but was solved numerically to first order by Start et al. [1980]:
IDRIV EN Zb 1/2 Zb vb = F =1 +1.46✏ A(Zeff , ) 1 (3.24) IBEAM Zeff Zeff ve ⇡
In Eq. (3.24), ✏ is inverse aspect ratio and A(Z , vb ) is a numerical value that can be found eff ve on a chart in Start et al. [1980]. In Eq. (3.24), the relationship from Eq. (3.22) is joined by an additional term due to electron trapping. And as stated, that term effectively wipes out all of the electron shielding of fast ions. This equation states that the driven current should be approximately equal to the fast ion current. Whatever current drive there might be after electron trapping is taken into consideration, the confinement of fast ions remains a more significant issue in RFPs than tokamaks. The existence of tearing modes and magnetic islands impact not just the confinement of fast ions but also their radial distribution. An example of this phenomenon can be seen in DIII-D data presented in Fig. (3.31). Forest et al. [1997] found that non-inductive current drive decreased significantly (often more than 50%) in DIII-D plasmas where tearing fluctuations were observed. Their modeling deter- mined that there was enhanced transport of current in the plasma, and that the reduction in current drive efficiency was due to fast ion loss. In this case, the fast ion loss would be due to tearing modes driving fast ion orbit stochasticity. This same phenomenon was observed in JT-60U by Oikawa et al. [2001]. In that case, an increase in tearing mode activity was directly correlated to a reduction in non-inductive current drive. 76
Figure 3.31 Three plots from Forest et al. [1997] showing the impact of tearing modes on NBI- dirven current on DIII-D. The modeling in these plots was all done with the ONETWO code. In Plot (a), current drive efficiency is plotted versus the neutron flux. In Plot (b) and (c), the driven current is plotted versus plasma current and NBI power, respectively. Each data point represents a 1 second long average of the experiment and a 1 second transport simulation. The represents tearing fluctuations observed on external Mirnov coils while the represents data where no measurable fluctuations were measured on the Mirnov coils.
3.7.2 Current Drive Data
Possible current drive has been looked at in every NBI data ensemble, but it has never been measurably observed. If it was going to be seen in any MST plasmas it would be in PPCD plasmas, where fast ion confinement is best and collisionality is least, but even in those cases there is no measurable change in IP . Some example data are plotted in Fig. (3.32). Loop voltage cannot be measured directly in MST. But the total poloidal magnetic flux ( ) can be modeled in a code like MSTFIT, and then related to loop voltage and the poloidal gap voltage
(Vpg): V = V (3.25) LOOP pg Assuming that the poloidal flux is slowly changing, the poloidal gap voltage can be used as a proxy for loop voltage. In Fig. (3.32), neither IP or Vpg changed measurably when the NBI was firing. No measurable change is seen in any of the other ensembles, including standard and F=0 plasmas. 77
Ip, 200kA Low-Density PPCD Plasmas Ip, 300kA F=0 Plasmas 400 250 1250 2000 Black = NBI O!
Black = NBI O! Red = NBI On Beam Power (kW)
200 1000 Bea Power (kW) Red = NBI On 300 1500
150 750 200 1000 Ip (kA) Ip (kA) 100 500
100 500 50 250
0 0 0 0 120 140 120 100 100 80 80 60 60
40 Vpg (V) Vpg (V) 40
20 20
0 0 -20 -20 0.00 0.02 0.04 0.06 0.08 0.00 0.02 0.04 0.06 0.08 Time (s) Time (s)
Figure 3.32 Ensembled NBI current drive data on MST. 200 kA low density PPCD is on the left, while 300 kA, F=0 is on the right. The inputed NBI power is plotted in brown, while the deposited NBI power is blue. In the bottom plot, poloidal gap voltage is used as a proxy for loop voltage.
Just because total plasma current doesn’t appear to change during neutral beam injection does not mean that the plasma current profile shape does not change. Liang Lin produced FIR polarime- try data for the 300 kA F=0 data ensemble, which is plotted in Figs. (3.33) and (3.34). In Sec. (2.3), it was discussed how plasma current density could be calculated from FIR po- larimetry data:
2 d 1 J(0) = (3.26) ocF dx nef(r )dz
In Eq. (3.26), J(0) is the plasma current density on axis, cF is a constant, is the faraday rotation, and f(r ) represents the current profile shape. If it is assumed that the plasma density profile is unchanged when the NBI is fired, then the following relationship then becomes true:
Jo (3.27) x x 0 78
Figure 3.33 FIR polarimetry data from 300 kA, Figure 3.34 The same FIR polarimetry data from F=0 plasma data ensembles. By looking at the Fig. (3.33), zoomed in on the three innermost ra- total plasma, there is no apparent change due to dial points. Here, there does appear to be a mea- NBI. surable impact with NBI on.
There appears to be no measurable change in the global polarimetry data in Fig. (3.33). How- ever, zooming on the three inner-most points (Fig. (3.34)) shows a change in faraday rotation. Liang Lin calculated that:
@ NBI Off: =12 1 deg/m (3.28) @x ± @ NBI On: =15 1 deg/m (3.29) @x ± According to Eq. (3.27), this means that faraday rotation data indicates an NBI-driven increase in core plasma current density by (25 10)%. ± The problem with this result is that a 25% increase in plasma current, even just within r 10 cm, should be seen globally, unless it is being partially shielded by the outer half of the plasma. So uncertainty in the FIR conclusions, combined with the fact that no significant global current drive has ever been seen, means that we cannot at this point conclude that the NBI has any signifi- cant impact on the NBI plasma current profile. 79
But current drive theory and the fact that we have seen momentum drive in the plasma core suggest that we should be able to see an increase in current density in the plasma core due to NBI. Future research here would be worthwhile. 80
Chapter 4
Rutherford Scattering
Thomson scattering data in Ch. (5) shows statistically-significant heating of electrons with NBI in MST. This data was taken with the Thomson scattering diagnostic. C+6 temperature data, taken with the CHERS diagnostic, was also shown. Due to different collision frequencies and equilibration times, there is reason to believe that auxiliary heating of majority ions during PPCD will not be the same as for impurity ions (see Sec. (2.4) for more detail on equilibration times). The Rutherford scattering diagnostic on MST measures majority ion temperature in the core of MST. The history and physics of the Rutherford scattering diagnostic are explained in this chapter. Improvements to the accuracy and understanding of the Rutherford analyzer signal are discussed. It will also be demonstrated in this chapter, however, that the Rutherford analyzer signal is polluted by NBI fast ions during beam operation. Assumptions that only Rutherford beam neutrals are being measured by the analyzer are shown to be false. This pollution signal artificially increases the “temperature” calculated. It will also be shown how this pollution signal can be calculated and pulled out of future Rutherford measurements.
4.1 History of Rutherford Scattering
Ernest Rutherford developed the theory now known as Rutherford scattering a full century ago [Rutherford, 1911]. Rutherford made his discovery when his colleagues Hans Geiger and Ernest Marsden performed their famous “gold foil experiment”, where they fired a beam of helium atoms at a thin gold foil. The theory of the atom at the time was the “Plum Pudding” model, de- veloped by J.J. Thomson (who had previously discovered the electron) a few years prior, which 81 visualized the atom as a large positively charged soup or pudding in which the negatively-charged electrons floated. If this theory were correct then one would expect every helium atom that inter- acted with the gold to scatter by some small angle. In fact, what they observed was that nearly every helium atom passed through effectively unperturbed, while approximately 1-in-8000 was scattered by a large angle. From this, Ernest Rutherford concluded that the positive charge was actually concentrated in a very small volume and that most of the volume of the atom was bereft of material. Rutherford’s research is most famous today for sparking the modern model of the atom (which is still today often called the Rutherford-Bohr model), but another very important impact was the discovery of the physics of the scattering itself - namely that it is elastic scattering. The fact that the scattering is elastic means that the energy of an atom of known mass and velocity that is scattered off a fixed target is a function of only the scattering angle and the mass ratio. The idea of using Rutherford scattering of neutral beam particles as a diagnostic on a plasma was first suggested by Abramov et al. [1972], who noted it as a solution to the problem of measur- ing the velocity distribution of a low-density medium that was reliable, accurate and that would not perturb the plasma flow. They argued that measuring the energy distribution of fast atoms scattered by a medium could, in theory, allow one to solve for the velocity distribution of that same medium. This theory was fleshed out into a practical diagnostic for a laboratory plasma by Berezovskii et al. [1980], who built a Rutherford scattering diagnostic for use on the T-4 tokamak. They noted the need for a beam of high-energy neutrals, as well as the need for the analyzer to be outside the direct path of the beam, but not so large of an angle that the Rutherford scattering cross-section became too small ( @ (csc( ✓ ))4). That paper also worked out much of the general Rutherford @⌦ / 2 scattering theory described in detail in Sec. (4.3). They concluded that a mono-energetic beam of neutral particles with energy E1 will be scattered by a plasma by a small angle ✓ with an energy spectrum described approximately by a Gaussian centered at energy E10 and with a width given by:
2 E0 = E (1 µ✓ ) (4.1) 1 1 2 = 2µE1Ti✓ (4.2) p 82
where µ = mb/mi is the mass ratio. This new diagnostic, which could measure majority ion temperatures directly, without any information about ion density or about the electron temperature or density, quickly gained use on several tokamaks, including JT-60 [Tobita et al., 1988], TEX- TOR [van Blokland et al., 1982] and JFT-2 [Takeuchi et al., 1983]. Rutherford scattering is no longer used on modern, large tokamaks. The weaknesses of Ruther- ford scattering for a fusion tokamak are twofold. First, the signal is small due to the angle- dependence of Coulomb scattering, and the signal-to-noise ratio becomes poor. Second, it is hard to engineer a beam that has enough energy and current density to not be absorbed in large numbers by the plasma. Because of this, MST is the only active fusion research device on which Rutherford scattering is still used as an ion temperature diagnostic [Den Hartog et al., 2006]. While Rutherford remains the only method of remotely measuring local majority ion temper- atures in a plasma (without knowing density or the electron temperature), there are other ways for fusion tokamaks to deduce majority ion temperatures. Charge exchange recombination spec- troscopy (CHERS) allows for localized measurements of impurity ions, from which majority ion temperatures can be deduced. For more information about the physics of CHERS and an analysis of how well CHERS data matches Rutherford data in MST, please see Sec. (2.4). CHERS diag- nostics are currently operating on machines such as NSTX [Medley et al., 2003], KSTAR [Ko et al., 2010], MAST [Conway et al., 2006] and DIII-D [Thomas et al., 1997]. Other options are neutral particle analyzers (NPAs), which provide line-average majority ion temperatures, and neutron yield measurements (NYMs), where ion temperature can be deduced if the density and fusion cross-sections are known.
4.2 MST’s Rutherford Scattering Diagnostic
MST currently has a radial Rutherford scattering diagnostic consisting of a neutral beam and a pair of analyzers. A schematic of MST’s Rutherford system can be seen in Fig. (4.1). The beam, which uses helium gas, is generally fired at an energy of approximately 16 kV, with a current of approximately 3.5 A. The beam duration is approximately 4 ms. 83
Like MST’s high-power NBI and the 45 kV DNB, the Rutherford beam was built by the Budker Institute of Nuclear Physics in Novosibirsk, Russia. And the creation of each beam pulse is effec- tively the same. Helium gas is puffed in through a pair of puff valves. Most of the gas is puffed into the cathode, which is where the plasma is formed. The plasma is then accelerated and focused through three grids for acceleration and focusing. One grid is grounded and one is charged to high voltage, with an electron suppression grid in between. The fast ion beam then passes through a neutralizing cell. The neutralizing gas can come from the anode and cathode gas valves, or can come via a third gas valve.
Figure 4.1 A schematic of MST’s Rutherford beam and analyzer. Drawing is from [Reardon et al., 2001].
Unlike MST’s high-power NBI, the neutral fraction is low in the Rutherford beam. As origi- nally constructed, the neutral fraction was approximately 50%. Only the neutral part of the beam can be used to measure the temperature of plasma ions, so a reduction of this neutral fraction reduces the signal-to-noise ratio. A photo of the Rutherford analyzers can be seen in Fig. (4.2) - they are the big yellow cylinders pointing down at the top of MST in the photo; only one is used in the analyis in this chapter. 84
Figure 4.2 A photograph of MST’s two Rutherford analyzers. They are the yellow cylinders.
Inside each Rutherford analyzer is a gas puff valve, an applied electric field, a 12 channel multi-channel plate (MCP) for detection and a corresponding 12 amplifiers. The gas puff valve fires hydrogen gas in order to strip the incoming neutral helium atoms. The ionized helium is then accelerated by the electric field into the 12 detectors, separated by energy. The energy of the particles measured by each detector is tied to the energy of the beam itself, to control for fluctuations in the beam’s high voltage supply. The ratio of the channel energies to the beam energy can be adjusted, but for most of the analysis the channels ranged from 84% to 106% of the beam energy. The reasons why it is necessary to measure particles with energies greater than the beam voltage are discussed in Sec. (4.5). The signals from each of the 12 analyzer are amplified and outputed as voltages. The analysis of these signals is described in detail in Sec. (4.4).
4.3 Rutherford Scattering Theory: Single Particle
A single instance of a Rutherford scattering can be visualized in Fig. (4.3). A fast neutral with velocity vb and mass mb scatters off a background ion with velocity vi and mass mi. The velocities 85
' v b
' v i v i
v b
Figure 4.3 Cartoon of a fast neutral launched by a Rutherford Beam scattering off of a background ion.
of the particles post-scatter are vb0 and vi0 respectively, with the beam neutral being scattered an angle ✓ from its original path. Since a Coulomb collision is an elastic collision, we can describe the collision through conservation of momentum and conservation of energy:
m v + m v = m v + m v (4.3) b b i i b b0 i i0 m v2 + m v2 = m v2 + m v2 (4.4) b b i i b b 0 i i 0
These equations will be solved by making only two assumptions. The first assumption is that the speed of the background ion is much less than the speed of the beam ion, which is true for MST (E 16 kV, E 0.1-1.0 kV). A small angle assumption is also made, and justified by the B ⇡ i ⇡ physical location of the particle analyzers ( 9 from vertical). ⇡ The first step is to begin by multiplying Eq. (4.4) by mi and rearranging terms:
m m v2 + m2v2 m m v2 = m2v2 (4.5) b i b i i b i b 0 i i 0 m2v2 = m2v2 + m m (v2 v2 ) m m (v2 v2 ) (4.6) i i 0 i i b i b b 0 ⇡ b i b b 0 86
2 2 In Eq. (4.6), the vi term is dropped because it is very small relatve to the vb term. Next, Eq. (4.3) can be rearranged and squared:
(m v + m v m v )2 =(m v )2 (4.7) b b i i b b0 i i0 m2v2 + m2v2 + m2v2 +2m m v v 2m2v v 2m m v v = m2v 2 (4.8) b b i i b b 0 b i b · i b b · b0 i b i · b0 i i0
Plugging Eq. (4.6) into the right side of Eq. (4.8) gives: m2v2 + m2v2 + m2v2 +2m m v v 2m2v v 2m m v v = m m (v2 v2 ) (4.9) b b i i b b 0 b i b · i b b · b0 i b i · b0 b i b b 0
2 Rearranging, and again dropping the vi term, gives:
m2(v2 + v2 )+m (m v2 m v2) 2m2v v +2m m v (v v )=0 (4.10) b b b 0 i b b 0 b b b b · b0 b i i · b b0
Next, the first and third terms from Eq. (4.10) can be taken separately. A small-angle approxi- mately can be used, as drawn in Fig. (4.4):
m2(v2 + v2 ) 2m2(v v )=m2(v2 + v2 2v v )=m2(v v )2 (4.11) b b b 0 b b · b0 b b b 0 b · b0 b b b0 m2(v v )2 m2(v tan(✓))2 m2v2✓2 (4.12) b b b0 ⇡ b b ⇡ b b
V ' b V b V b tan V b
V'b Vb
Figure 4.4 Diagram of the small-angle approximation used in Eq. (4.14).
Plugging Eq. (4.12) back into Eq. (4.10) and dividing by mi gives:
2 2 2 mb 2 2 (mbvb mbvb )+ vb ✓ 2mbvi (vb vb)=0 (4.13) 0 mi · 0 87
Further small angle approximations can be used to re-write Eq. (4.13):
(v v ) v tan✓ v ✓ (4.14) | b0 b |⇡ b ⇡ b 2 2 2 mb 2 2 mbvb mbvb + vb ✓ 2mbvivb✓ =0 (4.15) ! 0 mi
Eq. (4.15) can be rewritten in terms of energy, with a mass ratio defined as µ = mb : mi
E = E µE ✓2 + v ✓ 2m E (4.16) b0 b b i b b p Eb The mass ratio, the scattering angle and Eb are all known, while Eb is measured (✏ = 0 ). We 0 Eb can therefore solve Eq. (4.16) for the velocity of the ion that the beam neutral scattered off of:
E E (1 µ✓2) E ✏ (1 µ✓2) v = b0 b = b (4.17) i p 2m ✓ ✓ 2mbEb r b
4.4 Rutherford Scattering Theory For An Ion Velocity Distribution
The theory for single particle Rutherford scattering was worked out in Sec. (4.3). In a plasma, the background ions have a velocity distribution, which can be assumed to be Maxwellian in MST:
m (v v )2 i i flow f(v ) e 2Ti (4.18) i /
In Sec. (4.3), the velocity of an ion was solved for as a function of the energy of scattered Rutherford beam particles. The solution (Eq. (4.17)) is monotonic, with greater scattered energies correlating to greater plasma ion speeds. This means that a distribution of plasma ion velocities will lead to a distribution of scattered energies. This can be expressed by plugging Eq. (4.17) into Eq. (4.18):
E E (1 µ✓2) b0 b mi ✓pm E vflow f(E) exp( b b ) (4.19) / h 2Ti i 2 2 mi[Eb Eb(1 µ✓ ) mbvb✓vflow] = exp( 0 2 ) (4.20) 2Ti(mbvb✓) (E E )2 b0 MAX 2 2 f(E) e E (4.21) ! / 88
In Eq. (4.21), f(E) is the distribution of plasma ions, where EMAX is defined as the energy peak of the Rutherford analyzer signal and E is the width. By calculating the peak and width of the Rutherford analyzer signal, one can thus solve for both ion temperature and ion flow velocity. The distribution of the measured Rutherford analyzer signal can be further written in unitless quantities. The unitless velocity (˚v) and temperature (T˚) can be defined:
2 2 vi 1 mi ✏ (1 µ✓ ) 1 ✏ (1 µ✓ ) v˚i = = = (4.22) E / 2 mb ✓ p2µ ✓ b mi r r Ti T˚i = p (4.23) Eb
These new terms can be inserted into the original distribution equation (Eq. (4.18)) to describe qualitatively the distribution of scattered Rutherford neutrals in unitless terms:
2 2 m (v v ) (˚vi ˚vflow) i i flow ˚ f e 2Ti e 2Ti (4.24) / ⇡
4.5 Rutherford Scattering Theory: Generalized For A Beam With Finite Di- vergence
The final step in Rutherford scattering theory is to generalize for a realistic Rutherford diagnos- tic neutral beam with a finite beam width, and for a plasma where not all collisions occur exactly at the center of the plasma. The scattering angle can be defined as ✓. This angle matters not just because it correlates to the velocity of the scattered beam neutral, but also because Rutherford scattering has a sharp cross-sectional dependence( @ (csc( ✓ ))4). @⌦ / 2 Rutherford analyzers are designed to have views at small angles relative to the Rutherford beam because of the need to keep the signal-to-noise ratio high, and MST is no exception. This allows a small-angle approximation for the Rutherford cross-section:
@ 1 4 4 ✓ (4.25) @⌦ / sin (✓/2) ⇡
Note that there is a factor of two taken out of Eq. (4.25). In general, all of the multiplying constants will be pulled out of these equations since the Rutherford analyzer data distribution will be normalized anyway. 89
Analyzer
0
Beam
Figure 4.5 Geometry of a finite-width beam Rutherford scattering. Here, the beam is drawn at the bottom while the analyzer is at the top right. ✓ is the nominal analyzer angle (typically 9 ) 0 ⇡ while ✓ is the actual scattering angle of each beam neutral measured by the analyzer.
In Fig. (4.26), a cartoon of the Rutherford beam/analyzer geometry is drawn. The analyzer is at an angle of ✓0 relative to vertical, which is approximately 9 for all of the data in this dissertation, though it can be adjusted. The actual angle at which the fast neutral scatter at and enter the analyzer from is defined as ✓. There is some variance to this distribution ( ✓) which is directly related to the width of the Rutherford beam itself. These terms can be added to Eq. (4.24) to develop a generalized, normalized description of the energy distribution of neutrals that reach the Rutherford analyzer:
Beam Divergence Ion Velocity Distribution Scattering Cross-Section 2 2 (˜v v˜ ) (✓ ✓o) i flow ˜ 1 f = d✓( e ✓2 e 2Ti ) (4.26) ⇥ z }| { ⇥ ✓4 Z z }| { z}|{ If the beam is fired into a gas (rather than a plasma), then we can assume that the gas particle velocities are zero and the middle term in Eq. (4.26) can be dropped. This leaves an equation with two unknowns (✓0 and ✓) that can be solved for by integrating over a range of ✓ values and fitting to the data distribution. Assuming that the beam width holds for all shots taken in a day, this allows the user to have a calibrated temperature diagnostic for the rest of the run day. An example 90 of what these types of distributions look like is plotted in Fig. (4.6), where simulated distributions are plotted for an array of elements and temperatures (including “gas” data, where Ti = 0).
Rutherford Beam Simulation 1.2 Rutherford Beam 1.0 Voltage
0.8
0.6 H - 500 eV D - 300 eV
0.4 D Gas D - 500 eV 0.2 Rutherford Analyzer Voltage (Arbs) Analyzer Voltage Rutherford Al - 500 eV 0.0 12 14 16 18 Energy (kV)
Figure 4.6 Modeled Rutherford scattering energy distributions for an array of elements and tem- peratures. The assumed Rutherford beam voltage is labeled, and the 12 analyzer channels are marked with vertical, dashed gray lines. The width of the Rutherford beam is fixed and constant for each of these simulations.
Heavier elements result in curves peaked closer to the Rutherford beam voltage (a typical value for the beam voltage was chosen and labeled), while lighter gases result in curves that peak at lower energy levels. Higher temperatures mean wider distributions, though Ti=0 shots will still have width due to the finite Rutherford beam width. Note that the 12 analyzer channels are drawn with vertical, dashed gray lines. These analyzer channels can be moved by the user, but in practice are not moved unless the gas inside MST is changed. The reason for this can be seen in the plot - the 12 channels neatly cover the Gaussian of any typical deuterium plasma.
4.6 Improved Rutherford Processing
When MST’s 1 MW neutral beam was first fired into the MST plasma, the initial temperature data came from Rutherford. This data was mis-interpreted, and incorrect heating conclusions were made. It is not typical in dissertations for authors to lay out experimental mistakes, but it serves 91 a valuable purpose in this section. The reasons why the data were mis-interpreted, and how those errors were fixed, are laid out in this section. Prior to firing the 1 MW NBI into MST, some 0-D modeling of plasma temperature was done which predicted a rise in electron temperature in a standard plasma of around 40-80 eV,and perhaps half that for ions. As is laid out in Ch. (6), a 0-D model of auxiliary heating on MST is not at all sufficient to describe the effect fast ions have, but we assumed at the time that the 0-D prediction would be close to reality. When the NBI was first fired into MST plasmas, the only temperature diagnostic tool we had ready for use was Rutherford scattering, and we measured a 40 eV increase in ion temperature which we then reported back to the MST group. There were several red flags with the Rutherford data. First, the heating seemed to arrive and dissipate extremely quickly, with a time scale on both ends of the NBI discharge of around 1-2 ms. Second, the Rutherford data had the same characteristic of all Rutherford data at the time: it increased 20-40 eV throughout the 3.5 ms pulse in all plasmas, even in gas-only shots. Rutherford temperatures in gas-only shots in fact typically saw temperatures around 30 eV at the beginning of the beam pulse, rising to 60-80 eV by the end. As we know now, Rutherford was not measuring any significant heating of standard MST plasmas (heating data measured by Thomson and CHERS is plotted in Ch. (5)). Some of those reasons will be laid out in this section, while others will be the focus of Sec. (4.7). The reason why Rutherford gas-only shots showed temperatures that were too high above the real 0 eV, and the reason why Rutherford temperatures always rose 20-40 eV throughout every Rutherford pulse had to do with two bad assumptions made years ago: 1) The Rutherford beam width is constant with time. 2) The Rutherford diagnostic neutral beam spread does not change over the years. It turns out that the Rutherford beam has not only lost focus over the years, but the beam spread increases with time during each beam pulse. Because increased beam width leads to increased Rutherford analyzer signal width, the processing code interprets a widening beam as an increasing plasma temperature. 92
The way to fix this problem is to go back to the practice of taking and using a gas-only shot from each day of data, and also to modify the processing code to solve for a beam spread at each point in time. The processing code can force the particular gas-only shot to register a temperature of 0 eV at all points in time, and this will wipe out (as best as possible) the temperature errors in any other shots the rest of the day. We can call this new processing code a “continuously calibrated” code, versus a “singly calibrated” code. An example of this in practice can be seen in Fig. (4.7) In Fig. (4.7), two gas-only shots from the same day are plotted. In both shots the ion temper- ature output from the singly calibrated code is plotted in black while the output from the contin- uously calibrated code is plotted in red. The shot on the left is the shot used as the gas-only data for the day, and so by design the new code outputs a temperature of 0 eV at all points in time. In the plot on the right, the improvement of the continuously calibrated code is seen for a different shot. Since the beam spread is always changing, any future shot will never see a Ti of precisely 0 eV for any gas-only shot, but the continuously calibrated code comes very close to achieving that. The gas-only data no longer looks like it did with the singly calibrated code, where gas-only temperatures were “measured” at over 40 eV by the end of the beam pulse. The Rutherford analyzer output for a plasma shot can be seen in Fig. (4.8), which is taken from the same day as the data in Fig. (4.7). Notice that the continuously calibrated processing code is only 20 eV different from the singly calibrated processing code at the beginning of the Rutherford pulse, but significantly reduces the false increase of approximately 30 eV seen over those 4 ms.
4.7 Rutherford Analyzer As Neutral Hydrogen Analyzer
In Sec. (4.6), a clear and significant improvement to the Rutherford processing code is de- scribed. This improvement does not, however, explain the NBI “heating” measured in standard plasmas with Rutherford, which contradicts Thomson and CHERS data that show no significant heating of either electrons or impurity ions due to NBI in standard or non-reversed plasmas. In the left-hand plot in Fig. (4.9), the approximately 40 eV of “auxiliary heating” with NBI from that original run is still there, re-processed with the continuously calibrated processing code. 93
Ion Temperature: MST Shot 1100305014 40 Shot 1100305087 (Gas Shot) 80
Singly Calibrated 30 Continuously Calibrated 60
Singly Calibrated 20 Continuously Calibrated 40
Ion Temperature (eV) Temperature Ion 10
Ion Temperature (eV) Temperature Ion 20
0 0.018 0.019 0.020 0.021 0.022 0.023 0 0.042 0.043 0.044 0.045 0.046 Time (s) Time (s)
Figure 4.7 Example of adjusted Rutherford gas-only data. The two shots above are from the same day. The shot on the left is smoothed, and is the shot that is used as the day’s gas-only shot. So by definition, the new processing code sets Ti=0 throughout the shot. The shot on the right is the raw output with the old processing code versus the new, adjusted code.
Ion Temperature: MST Shot 1100305110 250
200
150 Singly Calibrated Continuously Calibrated
100 Ion Temperature (eV) Temperature Ion
50
0 0.026 0.027 0.028 0.029 0.030 Time (s)
Figure 4.8 Example of adjusted ion temperature data from Rutherford scattering. This shot is from the same day as the data in Fig. (4.7). Notice that the increase in temperature throughout the length of the Rutherford pulse is significantly reduced. 94
Rutherford Beam Measurements March 5-6, 2010 Rutherford Beam Data + Fit, 3/5/10 300 1.2 NBI On NBI Off Shot 116 (NBI On) 1.0 Black = NBI O! Shot 117 (NBI O!) Red = NBI On 200 0.8
0.6 175 eV Ti (eV) Ti
100 0.4 237 eV
0.2 Rutherford Analyzer Voltage (Arbs) Analyzer Voltage Rutherford
0.0 0 13 15 16 17 18 0.01 0.02 0.03 0.04 0.05 0.06 14 Time (s) Energy (kV)
Figure 4.9 Rutherford NBI “heating” data using the improved processing code described in Sec. (4.6). The plot on the left is the “heating” of ions that we measured in March, 2010. The plot on the right is a good example of single-shot “heating”, with the 12 analyzer data points averaged over the pulse.
The right-hand plot in Fig. (4.9) is a single NBI Off shot and a single NBI On shot from the same day, with the data averaged for the entire pulse length plotted for each of the 12 analyzer channels. Note that the difference in temperature that the code calculates for these two shots is 62 eV, although the spreading is by no means symmetric. In fact, turning the NBI On only slightly increases the data measured at the lower-energy channels while significantly increasing it at the higher energy channels.
What these data show is that the increase in processed Ti is not an actual increase in ion tem- perature, but in fact a pollution of the signal due to fast ions. The Rutherford analyzer, after all, is just a neutral particle analyzer. It measured Rutherford beam helium neutrals and operates with the assumption that too few non-Rutherford neutrals get in to significantly impact the signal, but this assumption might be wrong when the 1 MW NBI is generating a fast ion density n n /10 f ⇡ e in the MST core, some of which has a large v /v ratio and can enter the Rutherford analyzer if ? k neutralized. In Figs. (4.10) and (4.11), there is strong evidence that the Rutherford analyzer is measuring a significant fast hydrogen population, in addition to the expected fast helium population. In this 95 experiment, which consisted of 450 kA, F=0 plasmas, data were taken in six different modes. The high-power NBI was on for half of the ensemble and off for the other half. In addition, the Rutherford diagnostic neutral beam was turned on and off, and the Rutherford analyzer was turned on and off. Practically, the Rutherford analyzer is turned off by closing its shutter and blocking any particles from getting inside. This is to make sure that there is no radiation signal polluting the Rutherford output. Because the Rutherford analyzer gets its voltage from the Rutherford beam, the beam is turned off without turning the analyzer off by shutting off the Rutherford beam ignition. In this mode the Rutherford beam reaches its full voltage and puffs gas, but no arc discharge (and thus no beam) is created. Note that channel 2 of the analyzer was not operational for this data run. Omitting Channel 2 has no significant effect on the analysis because it’s at the low-energy edge of the Gaussian.
Raw Rutherford Analyzer Signal: Rutherford Beam Change In Rutherford Analyzer Signal: On vs Rutherford Beam Off vs Analyzer Valved Off 0.1 NBI On - NBI Off 0.03 Rutherford Beam+Analyzer On Rutherford Beam+Analyzer On Rutherford Beam O! Rutherford Beam O! 0.08 Rutherford Analyzer Valved O! Rutherford Analyzer Valved O! Solid = NBI O! 0.02 0.06 Dotted = NBI On
0.04 0.01 Analyzer Voltage (V) Analyzer Voltage Change In Analyzer Voltage (V) Analyzer Voltage Change In 0.02
0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 RS Analyzer Channel RS Analyzer Channel
Figure 4.10 Raw NBI On and NBI Off data averaged for a day with the Rutherford scattering diagnostic in three different modes: 1) Beam+analyzer on, 2) Beam off, analyzer on, 3) Beam on, analyzer off. Plasmas were 450 kA, F=0.
In the left-hand plot in Fig. (4.10) it is clear that the Rutherford analyzer signal is effectively zero when the analyzer is shuttered. This was the effective control for this experiment. If there was no fast hydrogen pollution then we would have expected to see no Rutherford analyzer signal with the analyzer open to the plasma but the Rutherford beam off, and this was more or less the 96
Raw Rutherford Analyzer Signal Rutherford Ion Temperature Calculation Time Averaged From 26-30 ms, 400 kA, F=0, ne=1 0.1 Time Averaged From 26-30 ms, 400 kA, F=0, ne=1 1200 NBI O! NBI O! NBI On NBI On 0.08 1000 “Adjusted” NBI On “Adjusted” NBI On
800 0.06
600
0.04
Analyzer Voltage (V) Analyzer Voltage 400 Ion Temperature (eV) Temperature Ion
0.02 200
0 0 0 2 4 6 8 10 12 26 27 28 29 30 RS Analyzer Channel Time (ms)
Figure 4.11 Processed output from the raw data in Fig. (4.10). Here the “Adjusted” Rutherford signal is the measured NBI On Rutherford signal with the “NPA signal” subtracted out. case when the NBI was off, but not at all the case when the NBI turned on. In fact, the Rutherford analyzer signal measured with the NBI on and the Rutherford beam off appears to be increasing with higher energy. Looking at the data with Rutherford beam and analyzer both on, we can see the increase in the Rutherford signal when the NBI is on, but in the plot on the right it is clear that this increase in signal really isn’t significantly different from the Rutherford beam off case. The conclusion to draw from this data is clear: the increase in Rutherford analyzer signal with the NBI On is due almost entirely to NBI fast ions getting into the analyzer, and not to any change in the plasma bulk temperature. In fact, we can call the change in signal with NBI On with the Rutherford beam off (the blue curve in the right-hand plot in Fig. (4.10)) the “hydrogen NPA signal”, since it represents the fast hydrogen signal as measured by the Rutherford analyzer. The fact that this “hydrogen NPA signal” is increasing with energy aligns with what the ANPA sees as well (see Fig. (3.20) for an example). This is also consistent with what we would expect whenever the NBI is firing, which is that the density of fast ions decreases with decreasing energy. This “hydrogen NPA signal” can be subtracted from the measured NBI On signal in order to prevent an over-estimate of the temperature in the processed fit. And it is clear in Fig. (4.11) that 97 performing this process makes the measured NBI “heating” basically disappear. In the left-hand plot, the “adjusted” NBI On signal is effectively on top of the NBI Off signal, and in the plot on the right the “adjusted” Rutherford ion temperature measurement for NBI On wipes out the original NBI On measurement of approximately 400 eV of heating. This analysis leads to two interesting conclusions. The first is that the NBI produces so many fast ions that no measurement of ion temperature with Rutherford can be done accurately without accounting for this “hydrogen NPA signal.” The second is that previous sawtooth analyses must be looked at anew.
4.8 Rutherford Analyzer Sawtooth Analysis
The fact that the Rutherford analyzer can act as a neutral particle analyzer has potential implica- tions for analysis of ion heating during a reconnection event. It is well-established that anomalous ion heating occurs during reconnection events, as magnetic energy is converted into ion thermal energy (see Fig. (1.9) for an example).
Figure 4.12 Neutral particle flux measured shortly after a reconnection event with a compact neutral particle analyzer (CNPA). The dashed line is the expectation for a thermal distribution. Plot is from Magee et al. [2011].
One interesting observation is that the ion temperature distribution in MST is not purely Maxwellian after a reconnection event. Fig. (4.12), from Magee et al. [2011], shows the neutral particle flux 98 measured with a compact neutral particle analyzer (CNPA) shortly after a reconnection event. The CNPA output is very similar to the ANPA output, but with a smaller and finer energy range. In this plot, the dashed line represents the expected ion distribution for a purely thermal distribution. As can be seen clearly, there is a non-Maxwellian tail of fast ions. There is also new ANPA data that sees this same fast ion tail out beyond 25 kV [Eilerman et al., 2012], which is well beyond the range of the Rutherford analyzer. The effect of this fast ion tail on Rutherford sawtooth analysis must be considered.
Rutherford Sawtooth Data, 400 kA, F=-0.2 500
400
300
200 Ion Temperature (eV) Temperature Ion 100
0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 Time Relative To Sawtooth (ms)
Figure 4.13 Processed data from an ensemble of sawteeth in 400 kA, F=-0.2 plasmas. Data are processed with the continuously calibrated processing code.
19 3 In Fig. (4.13), data from a 400 kA, F=-0.2 (standard), n 1 10 m plasma run are plotted, e ⇡ ⇥ re-processed with the new Rutherford processing routines. This data shows what appears to be an increase in ion temperature of approximately 350 eV. In Fig. (4.13), three time windows are drawn, each with a width of 0.1 ms. Data can be averaged over those three windows to see how the raw data shifts from before the magnetic reconnection event, during the magnetic reconnection event and after it. The raw data for the 12 Rutherford analyzer channels averaged over these three time windows is plotted in Fig. (4.14). 99
Rutherford Analyzer Raw Signals, 400 kA, F=-0.2 Rutherford Raw Signals vs Time 1.2 30
Channels 1-3 1.0 Channels 4-6 Channels 7-9 Channels 10-12 0.8 20
0.6
0.4 10 Analyzer Signal (Arbs) Analyzer Signal (mV) 0.5 ms prior 0.2 at sawtooth 0.5 ms post
0 0 13 14 15 16 17 18 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 Analyzer Energy (kV) Time Relative To Sawtooth (ms)
Figure 4.14 Raw Rutherford signals from the Figure 4.15 Raw Rutherford signals from the same 400 kA, F=-0.2 sawtooth ensemble seen in same 400 kA, F=-0.2 sawtooth ensemble seen in Fig. (4.13), at the three time windows defined. Fig. (4.13), plotted versus time. The 12 signals are plotted in terms of channel en- ergy.
The Rutherford signal is relatively low in this dataset, which is why the Gaussian is not shaped perfectly. Also, there is a tendency for the Rutherford voltage to droop slowly over the 3.5 ms that the beam stays on for. The beam voltage during each of the three beam windows is plotted in vertical dashed lines. Note that the energies of the 12 channels (the drawn diamonds) drift slowly downard over time as the beam loses voltage. The blue and green curves are laid on top of each other, while the red curve (the moment of the crash) is wider. This is what we would expect to see. However, there looks to be a much bigger spike in the lower energy channels than the higher energy channels. This is analyzed further in Fig. (4.15), where the raw data are plotted versus time. For ease of viewing, the 12 Rutherford analyzer channels are broken into four groups, with the lowest channel number referring to the lowest analyzer energy. In Fig. (4.15), it is clear that there is a very large spike during the sawtooth crash at the lowest energy channels, with much smaller spikes in the higher energy channels. This is not conclusive evidence of fast deuterium pollution, but it’s worth further investigation. This is also evidence that 100 the Rutherford signal needs to be increased to improve the signal-to-noise ratio and the shapes of the Guassians, to minimize error. Whatever the “hydrogen NPA” effect, it can be reduced by increasing the Rutherford beam strength or beam neutral fraction. When the current Rutherford beam repairs are done, it would be very valuable research to collect new Rutherford sawtooth data using the procedure laid out with the NBI in Sec. (4.7). Do a sawtooth analysis both with and without firing the Rutherford beam. If there is a large fast ion signal polluting the Rutherford analyzer during each magnetic reconnection event then it will be the only signal on the analyzer when the Rutherford beam is not firing. There is no queston that ions are heated in MST during magnetic reconnection. But it is pos- sible that previous research has over-estimated how much Ti increases for majority ions. It has been shown that anomalous heating in MST is mass dependant [Fiksel et al., 2009], so CHERS data cannot be used as a proxy for majority ion temperatures during a sawtooth crash. The only way to accurately measure majority ion temperatures during magnetic reconnection is by adjusting Rutherford data for fast ion pollution. 101
Chapter 5
NBI Heating Of MST Plasmas
Auxiliary heating of MST plasmas with NBI was definitively measured in several types of enhanced-confinement plasmas. In order to minimize statistical errors, data were ensembled. Shots were judged for the ensembles by magnetic activity, plasma current and plasma density. The clearest and strongest heating data are seen in the 200 kA low density PPCD ensem- ble. Statistically significant heating can be seen in several other PPCD plasma conditions as well. Lower current is ideal for heating because a cooler plasma is more collisional, and so there is more energy transfer between the fast ions and the background plasma. Higher density plasmas have less shine-thru and more energy deposition than low density plasmas, but the energy deposition is divided among many more background particles, and so the Te is greater in lower density PPCD. These phenomena are explained in more detail in Sec. (5.3). Data from standard plasmas showed no statistically significant heating of ions or electrons. These plasmas are more collisional than PPCD plasmas, but feature much shorter energy confine- ment times. In Fig. (5.1), a single example NBI On and NBI Off shot is plotted. The NBI Off shot is in black while the NBI On data are in red. These two example shots have identical plasma current and electron density (within statistical uncertainty). Also of note is that the neutron signal is only significant in the NBI On case. The beam power is also plotted, in the top plot in Fig. (5.1). The brown curve is the inputed NBI power while the blue curve is the absorbed beam power (beam power minus shine-thru). The beam has a ramp-up time of approximately 3-4 ms before reaching a plateau. In this particular 102
Example 200 kA PPCD Shot Data
250 1500 NBI Power (KW) 200 1200 150 900 100 Black = NBI O! 600 Ip (kA) 50 Red = NBI On 300 0 0 1.5 )
-3 PPCD 1.0 cm 13
(10 0.5 e n
0 10 8
(G) 6 7,12
= 4 n , ÷
b 2 0 2.5 2.0
1.5 1.0 0.5 0 Neutron Flux (Arbs) 0 10 20 30 40 Time (ms)
Figure 5.1 Data from an example “NBI Off” and “NBI On” shot from the 200 kA low density PPCD data set. NBI On data are in red while NBI Off data are in black. Note that in the uppermost plot, the inputed beam power is plotted in brown and the absorbed beam power (beam power minus shine-thru) is plotted in blue. 103 ensemble the beam is short-pulse, with a total length of 8 ms. The MST NBI can fire for as long as 23 ms, and some of the ensembles in this chapter will feature full-length NBI. It is also important to note that the mid-radius magnetic fluctuations are very similar between the NBI On and NBI Off cases. The relevance of this to NBI heating is discussed in Sec. (6.2).
5.1 Measured Auxiliary Heating Of MST Plasma
Thomson Scattering data were taken for the 200 kA low density PPCD ensemble (see Sec. (2.2) for a description of the Thomson scattering system). Data were taken with a 1 kHz time resolution at 22 radial locations and 30 time points. Core electron temperature data for the NBI Off and NBI On ensembles, with error bars, are plotted in Fig. (5.2). Here, “core” is defined as the five innermost Thomson radial locations (r/a 0.11). In Fig. (5.3), Te,core is plotted for this ensemble with error bars, where Te,core is just the difference in core electron temperature between the NBI On case and the NBI Off case at the same time point. In this ensemble, there is no statistically significant core heating prior to the start of PPCD.
Heating then continues throughout PPCD, even after the NBI ceases. After PPCD, Te,core drops off over a time scale of approximately 10 ms. Throughout Sec. (3.5) it was demonstrated that fast ion confinement is very long relative to the length of time PPCD is on. Because of this, the population of fast ions stays nearly intact through
PPCD, even after the NBI turns off, continuing to heat the plasma. After PPCD ends, Te,core decays for a couple of reasons. First, the fast ion population starts to decay, particularly after any large magnetic reconnection event. More importantly, plasma energy confinement times drop back to standard confinement, meaning that the electrons can no longer store much of the energy deposited on them by the remaining fast ions. It should be noted that there are multiple reasons for uncertainty in the Thomson data. There is pure statistical uncertainty because the plasmas in the ensemble are not completely identical. But also it is possible for short timescale fluctuations to be observed by Thomson scattering, including
fluctuations with a toroidal dependance. One data point that sticks out is the Te,core point at 8 104
Core Te, 200 kA PPCD Plasmas, NBI On vs Off 1000 NBI On NBI Off 800 NBI On
600 CORE Te 400
PPCD 200
0 0 10 20 30 40 Time (ms)
Figure 5.2 Core electron temperature with and without NBI in the 200 kA low density PPCD dataset. “Core” is defined as the five innermost Thomson radial locations (r/a 0.11). Change In Core Te, 200 kA PPCD Plasmas 150 NBI On
100 CORE Te
50
PPCD
0 0.00 0.01 0.02 0.03 0.04 Time (s)
Figure 5.3 Change in core electron temperature with NBI in the 200 kA low density PPCD dataset. 105 ms, which appears to show statistically significant heating prior to NBI turn on. But looking at Fig. (5.2) it is clear that this only appears at a single time point in the NBI Off ensemble, and is likely random error. The other Thomson time points taken prior to PPCD all are consistent with no statistically significant Te,core. Thomson profile data from the same dataset can be seen in Fig. (5.4). In this plot, the NBI Off and NBI On temperature profiles are plotted at four separate time points. The time points chosen are at the very beginning of the dataset (5 ms), the final time point before PPCD begins (11 ms), the moment the NBI turns off (16 ms) and the final time point before PPCD ends (21 ms).
Te Profiles, Low Density 200kA PPCD Plasmas 1000
21 ms NBI Off 800 NBI On
600 16 ms
Te(ev) 400 11 ms
200 5 ms
0 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Radius (m)
Figure 5.4 Electron temperature profiles with and without NBI for low density 200 kA PPCD. The times chosen for plotting represent, in chronological order, the beginning of the data set (5 ms), NBI on just before the onset of PPCD (11 ms), the moment the NBI turns off (16 ms) and the final time point before the end of PPCD (21 ms).
In Fig. (5.4) it can be seen clearly that the electron temperature profiles stay relatively flat for r 0.25m, even during the peak of PPCD. Also it can be seen that during PPCD the auxiliary heating occurs in the mid-radius areas as well as the core. At the end of PPCD there is statistically significant heating all the way out to r =0.25m. Note that the gradients for r 0.25m are very similar between the NBI On and NBI Off cases. 1-D modeling shows that this is a key reason why NBI heating is as large as it is (see Sec. (6.6) for more discussion about this). 106
Core C+6 Temp (CHERS Data), 200 kA Low-n PPCD 800 PPCD NBI O! NBI On 600 (eV) 400 CORE Ti
200 NBI On
0 5 10 15 20 25 30 35 Time (ms)
Figure 5.5 Core C+6 temperature measured with CHERS for 200 kA low density PPCD, with and without NBI.
C+6 temperature data, measured by CHERS, can be seen in Fig. (5.5). As discussed in Sec. (2.4), it is possible that impurity temperature is not the same as the majority ion tempera- ture. In fact, previous research has found that impurity ion temperatures and majority ion temper- atures are different in MST during PPCD [Magee, 2011]. While the equilibration time between the species in these plasmas is on the order of 100 µs, the fact that the impurity ions are more heated by the NBI than the majority ions could generate an additional disparity between impurity and majority ion temperatures. Unfortunately, for reasons discussed in Sec. (4.7), the NBI pollutes the Rutherford scattering signal, and so direct accurate measurement of majority ion temperatures in these plasmas was not possible.
In the CHERS data in Fig. (5.5), it does appear like there is some Ti,core prior to the enhanced confinement period, though the error bars do overlap. The maximum Ti,core in this data is ap- proximately 100 eV, and it occurs just at the moment that the NBI turns off. Ti,core begins to decrease before the end of PPCD. These ion temperature results do not entirely match the Thomson electron temperature data. For reasons discussed in Sec. (6.4), the unknown shape of the anomalous ion heating profile makes 107 modeling ion temperatures much more difficult than modeling electron temperatures. But despite not being able to measure quantitatively how much majority ions are heated, we can say definitively that C+6 ions are heated with NBI. One interesting observation is that this data shows decreasing core ion temperature during much of PPCD in the NBI Off case. This is a consistent result seen in all of the low density PPCD datasets. It has been long observed that ion temperature is typically flat during PPCD [Sarff et al., 1997; Chapman et al., 2002], but an ion temperature increase has been observed during enhanced
19 3 confinement with CHERS in pellet-injected plasmas with densities close to 3 10 m [Chapman ⇥ et al., 2009]. An argument for these observations is that ion temperature during PPCD is affected by two competing processes. The lack of magnetic reconnection events during PPCD means a reduction in anomalous ion heating. At the same time, the increase in electron temperature during PPCD means that there is more energy transferred between electrons and ions due to classical collisions (Pe i (Te Ti), so Pe i 0 when Te Ti). Higher density plasmas feature more ! / ! ⇡ ⇡ collisionality between electrons and ions (Pe i ni). If this is what drove higher ion temperatures ! / during enhanced confinement in high density PPCD, a small reduction in ion temperature during particularly low density PPCD is a plausible effect.
5.2 Auxiliary Heating Data In A Variety Of Plasma Conditions
Thomson data from the 200 kA high density PPCD ensemble is presented in Figs. (5.6) and (5.7). Comparing these plots to the 200 kA low density data in Sec. (5.1), the first observation is that tem- peratures are lower in the high density case. This is a typical and intuitive result.
The Te,core data in Fig. (5.7) looks qualitatively very similar to the low density data in Fig. (5.3). Auxiliarly heating is effectively zero before enhanced confinement, and then increases until the end of PPCD when it decays away, despite the NBI still being on. Te,core peaks at around 60-80 eV, which is less than the low density case. CHERS data from the 200 kA high density PPCD case is plotted in Fig. (5.8). Like the Thom- son data, the CHERS data show a smaller increase in core temperatures with NBI than the low density case. Also of note is the fact that ion temperature is higher ( 400 eV during PPCD with ⇡ 108
Core Te, 200 kA PPCD High-n Plasmas, NBI On vs Off Change In Core Te, High Density 200kA PPCD Plasmas 600 150 NBI On PPCD 500 NBI Off NBI On NBI On 400 100 CORE CORE 300 Te Te
200 50
PPCD
100
0 0 0 10 20 30 40 0 10 20 30 40 Time (ms) Time (ms)
Figure 5.6 Core electron temperature with and Figure 5.7 Change in core electron tempera- without NBI in the 200 kA high density PPCD ture with NBI in the 200 kA high density PPCD dataset. dataset.
Core C+6 Temp (CHERS Data), 200 kA High-n PPCD 1000 PPCD NBI O!
800 NBI On
600 (eV) CORE
Ti 400
NBI On
200
0 5 10 15 20 25 30 35 Time (ms)
Figure 5.8 Core C+6 temperature measured with CHERS for 200 kA high density PPCD, with and without NBI. 109 the NBI Off) than in the low density case ( 300 eV). This is again related to the fact that the ⇡ electron-ion classical heat transfer is increased in higher density plasmas. Additional NBI heating experiments were conducted at a higher plasma current, at two values of electron density. In Figs. (5.9) and (5.10), Thomson data from the 400 kA low density ensemble is plotted. If all other parameters are held equal, higher plasma currents lead to higher plasma temperatures, so electron temperature is higher in the 400 kA low density PPCD ensemble than the 200 kA low density PPCD ensemble.
Core Te, 400 kA Low-n PPCD, NBI On vs Off Change In Core Te, 400 kA Low-n PPCD 1500 200 NBI On NBI On NBI Off NBI On 150
1000 100 CORE CORE Te Te 50 500
PPCD
0 PPCD
0 -50 0 10 20 30 40 0 10 20 30 40 Time (ms) Time (ms)
Figure 5.9 Core electron temperature with and Figure 5.10 Change in core electron tempera- without NBI in the 400 kA low density PPCD ture with NBI in the 400 kA low density PPCD dataset. dataset.
The 400 kA low density Thomson ensemble is noisier than the 200 kA low density ensemble.
One of the points that stands out is the large spike in Te,core at t =22ms. But looking at Fig. (5.9), it’s clear that this spike is not due to auxiliary heating, but instead due to the fact that electron temperature falls off after enhanced confinement around 1 ms earlier (on average) than in the NBI On ensemble. The fact that the enhanced confinement plasmas ended slightly earlier with the NBI Off in this ensemble could be a statistical effect from the limited sample size, or it’s possible that there is some physics reason why a fast ion population would help prevent early termination of PPCD. 110
Thomson data from the 400 kA high density ensemble is plotted in Figs. (5.11) and (5.12). As expected, core electron temperatures are higher than the 200 kA high density case, and lower than the 400 kA low density case. And the same qualitative result is repeated: Te,core is consistent with zero prior to PPCD, increases during PPCD and peaks just as the enhanced confinement period ends.
Core Te, 400 kA High-n PPCD, NBI On vs Off Change In Core Te, 400 kA High-n PPCD 1200 140 NBI On NBI On NBI Off 1000 120 NBI On
100 800
80 CORE CORE 600 Te Te 60
400
PPCD 40
200 20 PPCD
0 0 0 10 20 30 40 0 10 20 30 40 Time (ms) Time (ms)
Figure 5.11 Core electron temperature with and Figure 5.12 Change in core electron tempera- without NBI in the 400 kA high density PPCD ture with NBI in the 400 kA high density PPCD dataset. dataset.
This Thomson analysis can be repeated for non-PPCD plasmas. Data from a standard (F=-0.2) dataset and a non-reversed (F=0) dataset will be discussed. In Figs. (5.13) and (5.14), Thomson data are plotted for the 300 kA F=0 (non-reversed) ensemble. CHERS data for the same ensemble are plotted in Fig. (5.15). Temperatures are, as expected, well lower than the PPCD cases. Tem- peratures are also constant with time due to the fact that there are no large sawtooth crashes in F=0 plasmas. The Thomson data for this ensemble show no heating of electrons. The CHERS data also show no heating outside the error bars. Thomson data for a standard plasma ensemble, 400 kA F=-0.2, are plotted in Figs. (5.16) and (5.17). Once again, the data are consistent with no significant change in electron temperature due to NBI. 111
Core Te, 300 kA F=0 Plasmas, NBI On vs Off Change In Core Te, 300kA F=0 Plasmas 400 100
NBI On NBI Off NBI On 300 50 NBI On CORE CORE 0
200 Te Te
100 -50
0 -100 10 20 30 40 50 10 20 30 40 50 Time (ms) Time (ms)
Figure 5.13 Core electron temperature with and Figure 5.14 Change in core electron temperature without NBI in the 300 kA F=0 dataset. with NBI in the 300 kA F=0 dataset.
Core C+6 Temperature (CHERS Data), 300 kA F=0 500 NBI O!
400 NBI On
300 (eV) CORE
Ti 200 NBI On
100
0 10 15 20 25 30 35 40 Time (ms)
Figure 5.15 CHERS data of core C+6 temperature for NBI On and NBI Off, for the 300 kA F=0 dataset. 112
Core Te, 400 kA F=-0.2 Plasmas, NBI On vs Off Change In Core Te, 400 kA F=-0.2 Plasmas 600 100 NBI On NBI On NBI Off NBI On 50 400 CORE CORE
Te 0 Te
200
-50
0 -100 10 20 30 40 10 20 30 40 Time (ms) Time (ms)
Figure 5.16 Core electron temperature with and Figure 5.17 Change in core electron temperature without NBI in the 400 kA F=-0.2 (standard) with NBI in the 400 kA F=-0.2 (standard) dataset. dataset.
The fact that the plasma energy confinement time is so short relative to the fast ion slowing down time in these plasmas (⌧ 1ms, ⌧ f/e 20ms) means that any auxiliary heating of non- ✏ ⇡ slow ⇡ PPCD plasmas is going to be very limited. The heat is just lost from the plasma far too quickly. The fact that these data are all consistent with zero auxiliary heating in non-PPCD plasmas does not mean that the NBI has no impact on temperatures. There is still a large amount of deposited heat, and it’s possible that with very large sample sizes (and reduced error bars) that a subtle increase in electron or ion temperature due to NBI can be measured. One experiment that could be attempted in the future would be attempting NBI in the coldest plasmas that can be created in a repeatable manner. A cooler plasma will be more collisional, and f/e 3/2 will have a decreased slowing down time (⌧ Te ). There is no reason to believe that fast ion slow / loss times or energy confinement times will be significantly reduced in these plasmas unless there is a large increase in neutral density, which is not expected, so it’s possible that the accelerated deposition of fast ion energy will generate statistically-significant auxiliary heating with NBI in non-PPCD plasmas. 113
5.3 Impact Of Current & Density On Auxiliary Heating
In Sec. (5.2) it was shown that auxiliary heating with NBI is greatest with lower density and lower current PPCD. There are physical reasons for both of these results.
The reason why auxiliary Te is reduced in higher current plasmas is that higher temperatures mean less collisionality. Higher temperatures reduce fast ion deposition slightly (see Fig. (3.7)). f/e 3/2 More importantly, ⌫ Te , which means that there is significantly less interaction between slow / fast ions and the plasma at higher temperatures. In a tokamak this is not necessarily the case because plasma energy confinement scales with plasma current (Eq. (1.14) and Fig. (1.13)). This has not proven to be the case in PPCD MST plas- mas however, where energy confinement seems to not at all scale with plasma current (Fig. (1.14)). So higher plasma currents mean less heat deposition with no decrease in heat conduction. This leads to suppressed auxiliary heating. It should be noted that the 1-D auxiliary heating modeling in Ch. (6) is consistent with these results. The results in higher densities are not so clear cut. Higher densities do lead to more fast ion deposition (see Figs. (3.10) and (3.11)) and more fast ion energy deposition (⌫f/e n ). But at slow / e the same time, higher densities mean that there is more plasma to deposit that energy on. This can be visualized by plotting total electron stored energy, rather than just electron temperatures, versus time. This is done in Figs. (5.18) and (5.19). Looking back at Figs. (5.2) and (5.6), one can see that the low density 200 kA PPCD had core electron temperatures more than 50% higher than the high density 200 kA PPCD plasmas. But in 3 Figs. (5.18) and (5.19) it can be seen that the total stored energy ( 2 neTedV ) is essentially the same in the two plasma conditions. R In Figs. (5.20) and (5.21), the change in stored energy with NBI On versus NBI Off is plotted for the two cases in red. Also, the total absorbed NBI energy ( (P P )dt) is plotted NBI shine in black. Note that the scale on the left is 100x the scale on theR right, for easier readability and comparison. 114
Stored Electron Energy, 200 kA Low-n PPCD Plasmas Stored Electron Energy, 200 kA High-n PPCD Plasmas 1200 1200
1000 NBI O! 1000 NBI O! NBI On NBI On 800 800
600 600
400 400 Stored Energy (J) Stored Energy (J)
200 200
0 0 0 10 20 30 40 50 0 10 20 30 40 50 Time (s) Time (s)
Figure 5.18 Total stored electron energy plotted Figure 5.19 Total stored electron energy plotted vs time for 200 kA low density PPCD vs time for 200 kA high density PPCD Absorbed Beam Energy, 200 kA Low-n PPCD Plasmas Absorbed Beam Energy, 200 kA High-n PPCD Plasmas 20 200 20 200 NBI On NBI On
15 150 15 150
10 100 10 100
5 50 5 50 Deposited Beam Energy (kJ) Deposited Beam Energy (kJ) Stored Energy (NBI On - NBI Off) (J) Stored Energy (NBI On - NBI Off) Stored Energy (NBI On - NBI Off) (J) Stored Energy (NBI On - NBI Off) 0 0 0 0 0 10 20 30 40 50 0 10 20 30 40 50 Time (s) Time (s)
Figure 5.20 The change in stored electron energy Figure 5.21 The change in stored electron energy is plotted vs time for 200 kA low density PPCD. is plotted vs time for 200 kA high density PPCD. Also plotted is the absorbed NBI energy, which is Also plotted is the absorbed NBI energy, which is the input NBI energy with shine-thru subtracted. the input NBI energy with shine-thru subtracted. 115
It must be kept in mind that the high density 200 kA PPCD has a significantly larger NBI pulse length than the low density 200 kA PPCD, and that the stored energy data is a little noisy. But the increase in stored electron energy in high density 200 kA PPCD is at least as large as it is in the low density case, which is consistent with the decreased NBI shine-thru and increased plasma collisionality. The reason why Te is less in the high density case is because that increased stored energy is spread amongst more total plasma particles. 116
Chapter 6
Modeling NBI Heating
In Chp. (5), a variety of data were presented that showed auxiliary heating of MST with NBI. This chapter will discuss a simple 1-D model which explains these results, and also brings greater understanding of the physics driving this increase in electron temperature. The structure of this model is developed in Sec. (6.1). In this model, heat diffusion coefficients are determined from the calculated net conducted power in the NBI Off case, solved for concentric circular volume elements. These same conduction coefficient profiles are used for the NBI On case, and are used to solve a differential equation for the time evolution of the electron temperature profile. This model can be compared to the TRANSP/NUBEAM model, as well as a couple of hybrid models, all of which are explained and defined in Sec. (6.3). The general justification for this model is discussed in Sec. (6.2). Two further key assumptions are addressed later in this chapter. In Sec. (6.10), it is shown that anomalous ion heating assump- tions are negligible in this analysis. Furthermore, a Zeff profile must be assumed, though it is shown in Sec. (6.7) that this assumption does not have a significant impact on the results either. Fast ion modeling is described in Sec. (6.5). The path the NBI fast neutrals traverse through MST is defined in Sec. (6.5.1), fast ion deposition is modeled in Sec. (6.5.2), and fast ion dynamics (slowing down and diffusion) are modeled in Sec. (6.5.3). The output of these models are compared to the Thomson heating data in Sec. (6.6). The physics underlying these results is further analyzed in Sec. (6.8). There, the importance of fast ion diffusion on both heat conduction and fast ion particle losses is explored in detail. Finally, the potential impact of charge-exchange effects is explored and discussed in Sec. (6.9). 117
6.1 A 1-D Classical Heating Model
In order to model the effect of an NBI on MST it is necessary to have a descripion of MST without NBI to which the NBI can be added. An arbitrary plasma volume with a species s has a stored energy Ws. The change in this stored energy is a heat flux in or out, which is defined as:
@ 3 A ⌃(All Heat Flux) = W˙ = n T dV (6.1) ⇥ s @t 2 s s Z In Eq. (6.1), A is the surface area of the volume element. There are many possible contributors to heat flux, but the largest are ohmic power, NBI power deposition, electron-to-ion classical energy transfer and heat conduction. For electrons, the electron-to-ion losses are very small, but they are necessary for the model because they are a large source of energy for ions. For the ions, ohmic power is too small to be included (ohmic power density ⌘J2). For ions, an anomalous ion / k heating term is also needed (see Sec: (6.4)). Radiative losses and particle diffusion losses are considered very small for both species and are dropped. This leaves as an energy balance equation for electrons: W˙ e P⌦ PBEAM e Pei = + ! + n T (6.2) A A A A e er e where: P = ⌫ein (T T ) (6.3) ei ✏ e e i Z Note that ohmic power (P⌦), beam power (PBEAM e) and electron-ion classical energy ex- ! change (Pei) are all integrated over an arbitrary plasma volume in Eq. (6.2). In addition, a sign convention has been adopted with positive e (electron energy diffusion coefficient) for negative T (electron temperature gradient). In general, the classical energy loss frequency for a particle r e of species s impacting a background species s0 is defined by Callen [2006] as:
mf ⌫s/s0 =2⌫s/s0 (xs/s0 ) ˙ (xs/s0 ) (6.4) ✏ o m ✓ s ◆
s/s0 s/s In Eq. (6.4), the collision frequency ⌫o is defined in Eq. (3.6), x 0 is defined in Eq. (3.8) and
(xs/s0 ) is defined in Eq. (3.9). This equation can be solved for MST by assuming that the plasma is a perfect torus and that everything can be described as a function of time and radius. We can 118 then cut the plasma into a finite number of circular cross-sections, with one such element drawn in Fig. (6.1).
ith volume element
ni 1 i 1Ai 1 Ti 1 ni i Ai Ti
r(i)
Figure 6.1 A cartoon of the “ith” plasma volume element. Heat conduction in and out of the volume element is shown.
Eq. (6.2) can be written for electrons for the ith volume element:
W˙ e(i)=P⌦(i)+PBEAM e Pei(i)+ e(i 1)ne(i 1) Te(i 1)A(i 1) e(i)ne(i) Te(i)A(i) ! r r (6.5) To minimize numerical errors, each of these volume-integrated power terms is integrated over the entire volume, and then the (i 1)th volume element is subtracted away from the ith volume element. For example, the total ohmic power deposited in the ith volume element is defined in the model as: r(i) r(i 1) 2 2 P⌦(i)= ⌘J Adr ⌘J Adr (6.6) k k Z0 Z0 Eq. (6.5) can be solved if the NBI is turned off (PBEAM e =0), which leaves only one ! unknown, the heat diffusion coefficient ( e(i)). In general, W˙ cannot be assumed to be zero, and is included in Eq. (6.7): i [P (i) P (i) W˙ (i)] (i)= i=0 ⌦ ei e (6.7) e n (i) T (i)A(i) P e r e 119
The e profile is linked to global electron energy confinement. The electron energy confine- ment time can be calculated: We ⌧e = (6.8) [P W˙ P ] ⌦ P e ei Example output from Eq. (6.7) is plotted in Fig. (6.2). There, profiles are plotted for low P e density 200 kA PPCD at three time points: before, during and after PPCD. Note the factor of ten reduction in e during PPCD.
e: 200 kA PPCD 104
103 /s) 2 102 ( m e
101 t = 10 ms t = 22 ms t = 35 ms
100 0 10 20 30 40 50 Radius (cm)
Figure 6.2 Calculated e profiles for 200 kA low density PPCD. The three profiles are from before, during and after PPCD. Note the factor of ten reduction in mid-radius e during PPCD.
If we assume that temperature is changing much more quickly than density (which is partic- ularly true in PPCD plasmas), we can rewrite the change in stored energy as a partial differential equation: @ 3 3 @T W˙ (i)= n T dV N (i) e (6.9) e @t 2 e e ⇡ 2 e @t Z 3 @Te ei Ne(i) P⌦(i)+PBEAM e(i) ⌫✏ Ne(i)(Te(i) Ti(i))+ ! 2 @t ⇡ ! (i 1)n (i 1) T (i 1)A(i 1) (i)n (i) T (i)A(i) (6.10) e e r e e e r e
where Ne is the total number of electrons in the ith volume element: r(i) r(i 1) N (i)= n Adr n Adr (6.11) e e e Z0 Z0 120
Eq. (6.10) can be re-written as:
ei @Te P⌦(i)+PBEAM e(i) ⌫✏ Ne(i)(Te(i) Ti(i)) + XQ(i) Te(i) = ! r (6.12) dt 1.5Ne(i)
where: X (i)=( (i 1)n (i)A(i) (i)N (i)A(i))/ T (i) (6.13) Q e e e e r e The reason for defining a heat conduction parameter X that has T in its denominator is Q r e numerical stability. It helps smooth out ripples in Te, particularly near the core, and makes the code much more robust to different types of plasma conditions. Eq. (6.12) is an equation of the form:
@T e = a + bT + c T (6.14) @t e r e
An equation of the form in Eq. (6.14) can be solved implicitly using a finite difference method. The method we use is the Crank-Nicolson method for numerical evaluation of solutions of partial differential equations [Crank & Nicolson, 1947]. In Eq. (6.14), a, b and c are all simply profiles that can be determined from data. It should be noted that some of the terms that make up those three coefficients are functions of temperature,
s/s0 such as P⌦ and ⌫✏ . In practice, these terms are adjusted for changes in temperature due to the beam. For example:
T (r, t) 3/2 P (r, t)=P (r, t) e,NBI Off (6.15) ⌦,NBI On ⌦,NBI Off T (r, t) ✓ e,NBI On ◆ The key, though, is that these terms are tabulated explicitly at each time point. This approxi- mation can be done because the rate of change in Te is small relative to the rate of change in Te during the plasma pulse, particularly in PPCD plasmas. The Crank-Nicolson method can be visualized by looking at the matrix drawn in Fig. (6.3). There, a cartoon is drawn with a discrete map of temperature values on a grid of time and radial points, where i is the index of radial values and n is the index of time values. The vertical direction represents minor radius (r(0) = r ,r(I)=r ). By definition, r(i +1) r(i)=dr. Time min max increases as one moves to the right (t(0) = tmin,t(N)=tmax). 121
r
t
Ti 1,n x Ti,n Ti,n +1
Ti+1,n dr
dt
Figure 6.3 A cartoon of the two-dimensional array over which the Crank-Nicolson method is solved. Temperature values are drawn on a grid of time and radial points.
When solving this problem in general, temperature is known at all radial coordinates at time n, and we are looking to solve for the temperature at the next discrete time point (Tn,i+1). The Crank-Nicolson method involves solving the problem at an intermediate point labeled x, which is dt at the same radial location but at the time value t(x)=t(n)+ 2 . The advantage of solving at an intermediate time point is that it is an implicit finite difference solution. This method accounts for any rapid changes in temperature, and computationally allows one to achieve a high level of accuracy with a fairly coarse data grid (dt 0.5 ms for Crank-Nicholson, though a shorter timestep ⇡ is required for processes like fast ion diffusion). The first step in the Crank-Nicolson method is to calculate all of the values and partial deriva- tives of temperature at the x point:
T + T T = i,n i,n+1 (6.16) i,x 2 @T T T = i,n+1 i,n (6.17) @t dt i,x @T Ti+1,n Ti 1,n + Ti+1,n+1 Ti 1,n+1 = (6.18) @r 4dr i,x 122
Plugging all of these values into Eq. (6.14) and separating terms gives:
Ti+1,n+1( cdt)+Ti,n+1(4dr 2bdrdt)+Ti 1,n+1(cdt)=
Ti+1,n(cdt)+Ti,n(4dr +2bdrdt)+Ti 1,n( cdt)+4adrdt (6.19)
All of the n +1terms can be put on one side, while all of the n terms can be put on the other, which accomplishes the goal of writing each time point as a function only of the previous time point. This simplifies the entire equation to the form:
↵iTi+1,n+1 + iTi,n+1 + ↵iTi 1,n+1 = i (6.20)
. . . 0 0 0 0 0 0 T0,n +1 0,n T 1 1 1 0 0 . . . 0 1,n +1 1,n T 0 2 2 2 0 0 2,n +1 2,n ...... 0 0
. . .
. .
.
. ! 0 0 ...... 0 0 0 I I TI ,n +1 I ,n
Figure 6.4 The matrix form of Eq. (6.20). Here, all of the temperature values at time n are written as a function of all of the temperatures at time n +1. All of the values in the far left matrix are numbers that come from raw data.
This equation is sketched in matrix form in Fig. (6.4). Two boundary conditions were needed.
The first used here is to fix the temperature at the outer edge ( I =1,↵I =0, I,n = Te(I)). The second boundary condition is to fix the temperature gradient of the innermost radial point ( = ↵ = 1, = T (0)dr). Note that T (0) is not going to be zero, because for o o o,n r e r e numerical reasons T is forced to be monotonic ( T is used in the denominator of Eq. (6.13)). e r e Remember also that r(0) is not equal to 0, but to dr. Since all of the values in the far left matrix 123 and the far right matrix are known data (computed heat balance terms), this matrix can be solved for the temperature at tn+1 for all radial locations. 1 This equation is of the form Ax = b, which can be solved by inverting the A matrix (x = A b), but this can take a very long time computationally. For this 1-D heating model, singular value de- composition was performed instead. In this method, the A matrix is broken into a pair of unitary matrices along with a diagonal matrix, where the diagonal values represent the “singular values” of the matrix. Both unitary and diagonal matrices are very simple for a computer to invert:
A = ULVT (6.21)
In Eq. (6.21), U and V are the unitary matrices, while L is the diagonal singular value matrix. x can thus be solved for:
1 T 1 T 1 1 1 1 T A =(ULV ) =(V ) L U = VL U (6.22)
1 1 T x = A b = VL U b (6.23)
Eq. (6.23) is an implicit solution for temperature at all radial locations at time tn+1 given the temperature at all location at time tn, and given all of the various heat fluxes in and out at all times and radial locations. The 1-D heating model thus can start with an initial temperature profile and all of the density, beam heating and ohmic power data, and can calculate the temperature at any future moment in time.
6.2 Justifying A Classical Heating Model
The most important assumption underlying the 1-D heating model is that it’s fair to solve the heat diffusion coefficients in the NBI Off model, and then to assume they’re exactly the same in the NBI On case. The assumption is that thermal conductivity of the plasma is unaffected by the NBI. Thermal conductivity of the plasma is strongly impacted by magnetic fluctuations. As dis- cussed in Sec. (1.5), a reduction in magnetic fluctuations will decrease thermal conductivity and 124 cause a plasma temperature increase. This is why PPCD, which suppresses magnetic fluctuations, causes such rapid increases in core electron temperature. Sarff et al. [2003] looked at the relationship between magnetic fluctuation suppression and core temperature. Fig. (6.5) features two key plots from that paper.
1.2 1.2
1.0 1.0
0.8 0.8 Te (0) Te (0) (keV) (keV) 0.6 0.6
0.4 0.4 0.2 0.2 0 1 2 3 4 5 0 5 10 15 15 Time-Average B˜ ˜ ˜ 2 1,6 Time-Average B rms = ∑Bn (a) n=8
Figure 6.5 Two plots showing the relationship between magnetic fluctuations and core electron temperature, from [Sarff et al., 2003]. In the right plot, the x-axis is the€ magnitude of the core- most mode (m=1,n=6).€ In the left plot, the mid-radius modes (n=8-15) are plotted. The x-axis units are Gauss.
What Fig. (6.5) shows is that while core electron temperature is very strongly correlated to decreased fluctuations of mid-radius magnetic modes, there is no apparent correlation between core electron temperature and the core-most mode (which in a MST PPCD plasma is the m=1, n=6 mode). Sarff et al. [2003] concludes that rapid transport in an isolated island near the core is not going to have a large impact on global energy confinement. If the heat can’t penetrate the mid-radius portion of the plasma, it can’t escape the plasma. This is important because in every type of MST plasma so far explored with NBI, the fast ions only suppress the core-most magnetic mode, while having no statistically significant impact on the amplitude of mid-radius or edge modes. Example magnetic suppression data is plotted in Fig. (6.6) for an ensemble of 200 kA F=0 plasmas. In these plasmas the n=5 mode is the core-most mode. Suppression is very clear only a few milliseconds after the NBI turns on, and it ramps down within 125
5 ms of the NBI turning off. Meanwhile, the n=7-12 modes are plotted to represent the mid-radius modes, and there is no significant supression. This data is consistent with all other MST plasmas explored to this point.
n=5 Amplitude, 200 kA F=0 Plasmas n=7-12 Amplitude, 200 kA F=0 Plasmas 8 4 NBI On NBI On NBI Off NBI Off NBI On 6 NBI On 3
4 2
n=5 Amplifude (G) n=5 2 1 n=7-12 Amplifude (G) n=7-12
0 0 0 20 40 60 80 0 20 40 60 80 Time (ms) Time (ms)
Figure 6.6 Magnetic fluctuation data from an ensemble of 200 kA F=0 plasmas. The core-most mode (n=5) is suppressed by a statistically significant amount by the NBI, while the mid-radius modes are not.
The data in Fig. (6.6) can be broken down by each magnetic mode. This is done Fig. (6.7). Suppression is plotted there as a percent decrease in the NBI On case relative to the NBI Off case, and it is an average of all data between 20 and 35 ms (the period before the NBI reaches its plateau phase is discarded). The error bars represent the standard deviation of the magnetic fluctuation data. An interesting note from Fig. (6.7) is that all of the modes are reduced by some amount. How- ever, there is significant (around 60%) suppression of the core-most mode (n=5), while the other modes are all suppressed by an amount ( 10%) well within the error bars. This process can be repeated for 200 kA PPCD. To see the effect more clearly, a long pulse NBI PPCD shot ensemble is used (200 kA high density PPCD). The data are plotted in Figs. (6.8) and (6.9). In Fig. (6.9), data are averaged between 12 and 20 ms. Again, only the innermost 126
Fluctuation Suppression, 200 kA F=0 100
50
0 Reduction in
-50 Magnetic Fluctuations (%) 4 6 8 10 12 14 Mode Number
Figure 6.7 Magnetic fluctuation data from the ensemble of 200 kA F=0 plasmas as Fig. (6.6). Here, all of the n=5-12 modes are plotted, with error bars. magnetic mode (n=6) is suppressed by a statistically significant amount, while the other modes are suppressed by an amount that is statistically consistent with zero. It makes sense that the NBI will only significantly impact the core-most magnetic mode. NBI density is very core-peaked (see Sec. (6.5)). These data do suggest, however, that it is possible with a larger data sample size that a statistically significant suppression of the mid-radius magnetic modes might be detectable. An interesting result is that the mid-radius modes appear to be sup- pressed more in the PPCD case than the F=0 case. This is consistent with previous research on MST which has found that much of the non-linear coupling of magnetic modes is through interac- tions with the m=0 modes at the reversal surface [Hegna, 1996; Sovinec & Prager, 1999; Hansen et al., 2000]. Non-reversed plasmas do not have a reversal surface, and so there is less significant coupling between the mid-radius magnetic modes. Because the NBI does not appear to significantly impact mid-radius or edge magnetic modes, the NBI should not be expected to impact global energy confinement. As Fig. (6.5) shows, core temperature is not impacted by the suppression of the core-most magnetic mode in PPCD. A change in global energy confinement requires a significant change in mid-radius magnetic mode 127
n=6 Amplitude, 200 kA High-n PPCD n=7-12 Amplitude, 200 kA High-n PPCD 8 5 NBI Off NBI Off PPCD 4 PPCD NBI On 6 NBI On NBI On NBI On 3 4 2
n=6 Amplifude (G) n=6 2 n=7-12 Amplifude (G) n=7-12 1
0 0 0 20 40 60 0 20 40 60 Time (ms) Time (ms)
Figure 6.8 Magnetic fluctuation data from an ensemble of 200 kA high density PPCD plasmas. The core-most mode (n=6) is suppressed by NBI while the mid-radius modes are not.
Fluctuation Suppression, 200 kA High-n PPCD 100
50
0 Reduction in
-50 Magnetic Fluctuations (%) 4 6 8 10 12 14 Mode Number
Figure 6.9 Magnetic fluctuation data from the ensemble of 200 kA high density PPCD plasmas as Fig. (6.8). All of the n=6-12 modes are plotted, with error bars. 128
fluctuations. This makes the assumption that thermal conductivity is not impacted by the NBI a fair one.
6.3 Modeling With TRANSP/NUBEAM
TRANSP is a time dependent transport analysis code developed at Princeton [Hawryluk, 1980]. Code development began more than 30 years ago, and more than 60 man-years of labor has been invested. TRANSP has several different modes of evolving the plasma, given plasma data like temperatures profiles, density profiles, location of the last closed flux surface, plasma current, gas puffing, et cetera. The TRANSP mode used in this research is the TEQ MHD evolution code originally developed at Lawrence Livermore. This code takes the pressure profiles and the initial q profile as inputs, and uses RBT at the plasma edge as a boundary condition. The primary value of TRANSP for this heating analysis is NUBEAM, an attached Monte Carlo package that performs time-dependent modeling of fast ions in an axisymmetric tokamak using a classical collision model. NUBEAM receives a plasma equilibrium from TRANSP as input, and it gives back output such as fast ion density and energy profiles, shine-through and energy deposition.
Typical Toroidal Flux Profile In MST PPCD 1.0
0.8
0.6
0.4
0.2
Normalized Toroidal Flux Toroidal Normalized 0 0 0.1 0.2 0.3 0.4 0.5 Radius (m)
Figure 6.10 A normalized toroidal flux profile from a typical MST PPCD shot. This data is actually MSTFIT output. 129
Unfortunately, the fact that TRANSP uses square root of normalized toroidal flux as its radial coordinate creates a serious problem for RFP modeling, since toroidal flux is not a monotonic func- tion for any reversed plasma. This is a particularly big problem for PPCD plasmas (see Fig. (6.10) for a toroidal flux profile from a typical PPCD shot in MST), which feature deep toroidal field reversal. Our solution to this problem was to cut off the plasma just a little before the reversal surface, to ensure a non-reversed plasma (Because TRANSP will interpolate onto its own grid, cutting off the plasma right at the reversal surface will occasionally cause TRANSP to reverse the magnetic field and crash). The NUBEAM output can then be interpolated from the “TRANSP wall” (typically at around r/a 0.6-0.7) to the real MST wall. Even though fast ion deposition is very core-peaked, ⇡ this “TRANSP wall” approximation (which eliminates approximately half of the plasma volume) creates some errors that are discussed in Sec. (6.6).
TRANSP Deposition Shape 1.2
TRANSP Output 1.0 Model Used
0.8
0.6
0.4
0.2
0.0
Normalized Fast Ion Beam Deposition 0.0 0.1 0.2 0.3 0.4 0.5
(# per volume element on grid with fixed dr) Radius (m)
Figure 6.11 Normalized fast ion deposition during 200 kA PPCD as calculated by TRANSP. The dotted line is the interpolation of the original TRANSP data out to the actual MST wall. Note that the y-axis is the total number of particles deposited in each volume element, rather than deposition density, simply because it’s easier to see shape of the fast ion deposition profile plotted this way.
By using this “TRANSP wall”, we were successful in getting TRANSP/NUBEAM to run and provide output. We also could only gave it fixed kinetic profiles. This meant that our TRANSP/NUBEAM output isn’t only missing approximately half of the plasma volume, but it had temperature and 130 density profiles that were constant with time. This is yet another source of error, since the kinetic profiles change quite rapidly during PPCD. We used kinetic profiles from the peak of the enhanced confinement period, to try to best approximate PPCD. All of these approximations and sources of error necessitated the creation of a new fast ion model designed specifically for MST (Sec. (6.5)). However, the TRANSP/NUBEAM output is still very useful as a comparison to other models. Both the fast ion deposition profiles and the heat deposition profiles were used as comparison models. The simplest model (henceforth called the “TRANSP/NUBEAM model”) uses the NUBEAM heat deposition data as the PBEAM term in Eq. (6.12). Alternatively, the fast ion deposition pro- files (such as in Fig. (6.11)) can be used. These fast ions then evolve using the NBI model from Sec. (6.5), with the option of fast ion diffusion (discussed in Sec. (6.5.3)). The model without fast ion diffusion will henceforth be called the “Hybrid model without diffusion” while the model with fast ion diffusion will be called the “Hybrid model with diffusion”.
Te,core 200 kA PPCD, TRANSP/NUBEAM vs Data 200 NBI On NBI Off Model TRANSP/NUBEAM
150 PPCD (eV) core
e, 100 T
50
0 0 10 20 30 40 Time (ms)
Figure 6.12 Core electron heating calculated with the “TRANSP/NUBEAM model” compared to the 200 kA low density PPCD data from Fig. (5.3).
The output of the TRANSP/NUBEAM model is plotted in Fig. (6.12), where it is compared to the raw data of core electron heating in 200 kA low density PPCD that was originally plotted 131 in Fig. (5.3). The TRANSP/NUBEAM model clearly does not show enough heating, particularly after the beam turns off. The reasons for this, as well as its comparison to other models, will be discussed in Sec. (6.6).
6.4 Anomalous Ion Heating
Solving the 1-D classical heating model for electrons in Sec. (6.1) required making some as- sumptions about the necessary terms. Radiation losses and heat losses due to particle diffusion were dropped because they were small. Solving the 1-D classical heating model for ions requires some different assumptions. Radiation losses and heat losses due to particle diffusion are still dropped, but so is ohmic power. The most important change to the ion version of Eq. (6.12) is the addition of anomalous ion heating. As decribed in Sec. (1.4), anomalous ion heating is an important effect on the RFP not just during sawtooth events, but also in equilibrium plasmas where it is necessary to explain why
Ti is generaly so close to Te. Quantitatively determining the anomalous ion heating profile is not an easy task, however. Ion density is not measured on MST, and ion temperature profiles were not measured in most of the plasma ensembles in this thesis, including the 200 kA low density PPCD case. In addition to estimating these values from electron data, we need to make two more crucial assumptions (the ion energy confinement time and the anomalous ion heating profile shape) that will have a lot of leverage on the ion temperature output of the 1-D heating model. But as will be shown at the end of this section, the anomalous ion heating assumptions have very little impact on the electron output of the 1-D heating model.
If the plasma has no impurities, by quasineutrality ni(r, t)=ne(r, t). Previous data show ion temperature approximately equal to electron temperature profiles at equilibrium, and also that ion temperature typically does not increase during PPCD (see Fig. (1.11)). It is also assumed that the ion energy confinement time is twice the electron energy confinement time (⌧i =2⌧e) both before and during PPCD, which is a reasonable approximation from previous calculations [Fiksel et al., 2006; Chapman et al., 2010]. The electron energy confinement time was calculated in Eq. (6.8). 132
The final major assumption, as stated previously, is the shape of the anomalous ion heating profile. The shape chosen for the 1-D heating model was a typical one for quantities in MST: r 4 P (r) 1 (6.24) ANOM / a ⇣ ⌘ With this shape and a confinement time, anomalous ion heating can therefore be solved as: W P = i i P (6.25) ANOM ⌧ ei i i i X P X
Te,core 200 kA PPCD, TRANSP/NUBEAM Varying Ti 200 NBI On NBI Off Model TRANSP/NUBEAM TRANSP/NUBEAM, Ti(r,t)=0 150 PPCD (eV) core
e, 100 T
50
0 0 10 20 30 40 Time (ms)
Figure 6.13 Core electron heating calculated with the TRANSP/NUBEAM model both with the regular ion temperature assumptions (solid line) and with Ti forced to be zero at all points in time and space (dotted line). These results are compared to the raw 200 kA low density PPCD data.
This anomalous ion heating profile matters significantly to ion temperature in the 1-D heating code. An anomalous ion heating profile shape that is very different, such as with smaller values in the core than on the edge, will have extremely high core ion energy confinement, which can create a very peaked ion temperature profile. But while this sort of uncertainty is unavoidable when calculating ion temperatures, it’s a small lever when calculating electron temperatures. This is because the only impact of ion temperature on electron temperature is via classical electron-ion collisions (Eq. (6.3)). This term is potentially significant for ions, but it is never significant for electrons. 133
The small impact of ion temperature on the 1-D heating code can be seen by forcing an un- realistically extreme ion temperature. The most extreme ion temperature that can be forced is T (r, t)=0, since this drains the maximal quantity of energy from electrons (P (T T )). i ei / e i The results of this can be seen in Fig. (6.13). Fig. (6.13) is a follow-up to Fig. (6.12), with a dotted line added to show the results with
Ti(r, t)=0. The results are not significantly different. And as previously stated, any real ion temperature profile will be a smaller drag on electron temperature since the magnitude of T T e i must be smaller when Ti(r, t) > 0. This exercise is repeated for the final “1-D heating+deposition model”, developed in Sec. (6.5), with results shown in Fig. (6.36). Again, the impact of ion tem- perature assumptions on electron temperature heating is relatively small.
6.5 Modeling The NBI
TRANSP is an extremely powerful transport code that we do not have the manpower to recreate for MST. But its severe limitations for RFP plasmas makes it insufficient to explain the electron heating observed. The fact that the plasma must be cut off at around (r/a) 0.6 0.7 means that ⇡ approximately half of the plasma volume is lost, and any fast ions deposited near the edge are lost. In addition, the existence of a “wall” at that radius means that core-deposited fast ions are lost too quickly via deposition. In addition, we could only run TRANSP with fixed kinetic profiles. These limitations necessitated the creation of a completely new code to simulate fast ion dy- namics in MST. This code calculates fast ion deposition, fast ion diffusion, fast ion losses and fast ion energy deposition. That energy deposition can then be plugged back into the 1-D heating code, creating the final “1-D heating+deposition model”, as it will henceforth be called. This code is developed in the following sub-sections. The code is based on the observations of fast ion dynamics shown in Ch. (3), particularly Sec. (3.5). Measured neutron signals are consistent with infinite fast ion loss times in both en- hanced confinement and standard confinement plasmas, where fast ion dynamics are dominated by
fi slowing down (⌧✏ <⌧fi). The exception to this rule is during a large sawtooth event, when the 134 increase in magnetic stochasticity and neutralization losses can result in a rapid loss of fast ions (see Fig. (3.20) for example data).
6.5.1 Modeling The NBI Beam Path
The first task to modeling the NBI is modeling fast ion deposition, which requires knowing the NBI beam path. The NBI geometry comes from the mechanical specifications of the device.
The beam is located at 19 poloidal (✓o) above the geometric midplane, pointed at a 6 (✓beam) downward angle. It is fired at a 45 angle horizontally relative to the MST wall ( beam). The ion source is located 1.38 m from the beam aperture. The beam aperture is 4.5 inches, which equals approximately 11.4 cm. The ion source diameter is 27.6 cm, and the beam random divergence is a very small 0.016 radians. The spread of the beam can be calculated from the focusing between the ion source and the beam aperture, which is 16.2 cm (27.6 cm - 11.4 cm). One can calculate a spread of a beam that compresses a half-width of 8.1 cm over 1.38 m:
tan(✓ )=8.1/138 ✓ =3.35 (6.26) spread ! spread
Before getting to a beam with finite width and spread it’s helpful to start by assuming a beam with no thickness, to just trace out the beam path. A discrete path length L can be defined. The starting location of the beam can also be defined as xo =(0,R+ acos o, asin o), and the change in beam location after each discrete change in path length is:
x =( Lsin , Lcos , Lsin✓ ) (6.27) beam beam beam beam
An overhead view of the beam path is plotted in Fig. (6.14). The NBI beam path is plotted in red and the geometric axis is plotted in blue. The x, y and z directions are drawn. At any given point the radial distance between the NBI path and the geometric axis can be calculated. This is plotted in Fig. (6.15). Note that the beam path is symmetric about the geometric axis. Note also that the beam path does not go directly through the geometric axis of MST, never getting closer than around 5-7 cm. When beam thickness and spreading are taken into account, however, part of the beam does pass through the MST geometric axis (see Fig. (6.18). 135
Overhead View Of NBI Path
y NBI Path
x
z a
Axis R
Figure 6.14 Overhead view of the simulated NBI beam path (red), assuming a beam with no width. The geometric axis is plotted in blue. A “slice” is taken at the dashed line, and plotted in Fig. (6.16). Radial Location Of NBI vs Path 0.6
0.5
0.4
0.3
0.2 Distance From Axis (m) Distance From
0.1
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Distance Traveled (m)
Figure 6.15 A plot of the simulated distance between the NBI beam center and MST geometric axis as a function of path length. 136
At any angle ✓, a “slice” can be taken where we can look at MST from the side. That “slice” view can be seen in Fig. (6.16). From this perspective the beam is traveling out of the page. Note that in this perspective the toroidal and poloidal directions are very easy to see, and can be drawn relative to the direction of the beam. With the beam velocity and the magnetic fields defined, a Larmor radius can be calculated: mv2 r = ? ˆr (6.28) L q v B L | ⇥ | where: v B ˆr = ⇥ (6.29) L v B | ⇥ | gives the direction from the point of beam ionizing to the center of the Larmor orbit.
“Slice” View Of NBI Path
Magnetic Axis Geometric Axis r B P
Beam B T
Figure 6.16 The “slice” view of the simulated NBI path, taken at the dashed line from Fig. (6.14). In this orientation, the NBI beam path is out of the page. Note also the directions of the poloidal and toroidal magnetic fields.
The fact that this direction almost always points inward for co-current NBI means that there are very few prompt ions losses. The opposite case is the counter-current case, where the plasma current direction is reversed and the Larmor orbit points in the opposite direction. In that case, almost every fast ion deposited within 10-15 cm of the wall orbits directly into the wall and is 137 immediately lost. This process was modeled in Hudson [2006], where it was calculated that prompt ion losses would be nearly 50% in counter-current NBI, compared to around 10% in co-current NBI. Orbit tracing is not done in this code, and so prompt ion losses are not directly calculated, but the Larmor orbits are used to calculate fast ion deposition. The exact methodology is explained in Sec. (6.5.2). To simulate a realistic (three-dimensional) NBI, the beam is broken up into an array of fast neutral “beamlets”. These beamlets are evenly spaced radially and angularly. Each beamlet has a number of fast neutrals in it, and the distribution of these fast neutrals is such that the half-width is
5 cm. These beamlets spread out as they travel through MST by an angle ✓beam. An overhead view of this calculated beam path can be seen in Fig. (6.18), where the beam is drawn at 15 cm path length increments. The shape and scope of the beamlets can best be seen in the “slice” view. A “slice” view drawing of the beamlets at three locations in the NBI path is also seen in Fig. (6.18). There, the entire path is drawn with a black line, with an arrow pointing out the direction. In this view the entire array of beamlets can be seen (in this case there are 240 beamlets representing 10 radial locations and 24 angular locations). It can also be seen that a mere 3.35 spread is significant over the large path length that the NBI travels. While the center of the beam never passes through the MST axis, the 3-D beam certainly does.
6.5.2 Modeling NBI Beam Deposition
In this model, fast ion deposition is calculated by using the known magnetic fields along with tabulated ionization cross sections from the EIRENE code [Kotov et al., 2007]. The three different types of ionization reactions considered here are drawn in cartoon form in Fig. (3.4), and are electron-impact, ion-impact and ion charge-exchange. The cross-sections are a function not just of beam energy but also (in the case of electron-impact) on the electron temperature. For the 200 kA low density PPCD case (as calculated by this model), ion charge-exchange is the most 138
Initial Fast Neutral Density vs Beam Radius 1
0.8
0.6
0.4
0.2 Normalized NBI “Beamlet” Density
0 -6 -4 -2 0 2 4 6 Distance From NBI Center (cm)
Figure 6.17 Drawing of the assumed NBI beam width, which has a half-width of 5 cm. Overhead View Of NBI Path “Slice” View Of NBI Path 4 1
Beam location drawn every 15 cm of path length NBI path 2 0.5
0 0
-2 -0.5 MST Geometric Center Distance From MST Geometric Axis (m) Geometric Distance From MST
Distance From MST Geometric Center (m) Distance From MST -4 -1 -4 -2 0 2 4 -1 -0.5 0 0.5 1 Distance From MST Geometric Center (m) Distance From MST Geometric Axis (m)
Figure 6.18 Overhead and “slice” views of the simulated NBI path for the full 3-D beam. All of the beamlets are drawn at 15 cm increments in the overhead plot, and at three separate path locations in the “slice” view (for easier viewing). The spreading of the beam throughout its path can be seen clearly. 139 powerful ionization force, accounting for approximately 60% of the NBI ionization. Electron- impact accounts for approximately 20-25% and ion-impact accounts for approximately 15-20%. The exact ratios depend on plasma temperature and beam energy, but do not change too much in typical MST plasmas. This is consistent with Hudson [2006], which also calculated that NBI ionization would primarily occur through ion charge exchange. At each discrete step along the path length, each beamlet loses some number of particles to ionization. These newly born fast ions are then deposited according to their Larmor orbits (see Eq. (6.28)). Finite Larmor orbits are approximated by calculating the radial locations encompassed by the Larmor orbit and dropping all of the fast ions evenly into each of those radial locations. For a numerically simplistic example, if there are ten fast ions deposited at r = 20 cm, and it’s calculated that fast ions deposited at that location have an initial orbit that travels in the range 15 cm Fast Ion Deposition: 200 kA PPCD, t=9ms 1 Without Magnetic Effects With Magnetic Effects 0.8 TRANSP/NUBEAM 0.6 0.4 0.2 Fast Ion Deposition Density (Normalized) 0 0 10 20 30 40 50 Radius (cm) Figure 6.19 Fast ion deposition as calculated by the 1-D model, both with and without mag- netic/Larmor effects, compared to the deposition calculated with TRANSP/NUBEAM. larger than 10-15 cm. Using orbit tracing, prompt losses in co-current NBI have previously been modeled to be approximately 10% in MST [Hudson, 2006]. Prompt losses are much more significant in counter-current NBI, where the plasma current and toroidal magnetic field are reversed relative to co-current NBI. In this case, nearly all fast ions deposited near the MST wall are lost to prompt losses. Prompt losses were previously modeled to be approximately 50% in MST [Hudson, 2006]. Another important reason that counter-current NBI is difficult to model is the fact that fast ion confinement is much poorer, and so some of the fast ion confinement assumptions made in this 1-D model are no longer accurate. For these reasons, only co-current NBI is modeled in this section. For further discussion of the difference in fast ion density between co-current and counter-current NBI, see Sec. (3.6). These fast ions, once deposited, are collected into a set of fast ion “beamlets” (which are different from the fast neutral “beamlets” discussed up to this point in this section). Beamlets are created at each radial location at every point in time, with t0 signifying the time index at which they were created. Each beamlet is born with an energy (Ef (r, t0)) equal to the full NBI energy (25 kV) and a number of fast ions (Nf (r, t0)) that is a function of the fast ion deposition profile 141 as well as the total measured current of the NBI. A 3 ms ramp-up phase of the NBI is assumed before reaching a plateau of 40 A of current, matching the experimental waveform. By particle conservation, the total number of fast ions born at any given time must be a function of the beam current and beam shine-thru: r=rmax N (r, t0)=I (t0) t(1 shinethru)/q (6.30) f BEAM f r=0 X In Eq. (6.30), t is the length of the code time step and qf is the fast ion charge. Fast Neutral Density vs Path Length Fast Neutral Density vs Path Length, Vacuum 0.2 0.2 ) ) -3 4 -3 m 4 m 17 0.1 0.1 17 0 3 0 3 -0.1 -0.1 2 2 -0.2 -0.2 1 1 Fast Neutral Density (x 10 -0.3 -0.3 Fast Neutral Density (x 10 Height Above MST Midplane (m) Above MST Height Height Above MST Midplane (m) Above MST Height 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Path Length (m) Path Length (m) Figure 6.20 The density of fast neutrals along the NBI path. On the left is for 400 kA standard reversal, while on the right is the vacuum case. Note that the density values here are taken along the central vertical axis of the beam. The beam neutral density is approximately axisymmetric about the beam path. One final way to look at the beam deposition and the spreading of the beam is to look at the density of fast neutrals along the beam path. The fast neutral density is approximately axisymmet- ric about the axis of the beam, so this data can be shown as a two-dimensional plot, as is done in Fig. (6.20). Data is taken along the vertical axis of the beam, and is plotted along the beam path. On the left is the result for a 400 kA standard reversal plasma. The plot on the right is for a vacuum shot with no beam ionization. These 400 kA plasmas have a shine-thru of only around 30%, but the spreading of the beam increases its spot size by approximately a factor of ten (see Fig. (6.18)), so it’s the beam spread that reduces fast neutral density more than beam deposition. 142 6.5.3 Modeling Fast Ion Diffusion And Slowing Down Once deposited, the fast ions are acted upon by two classical effects: diffusion and slowing down. Diffusion consists of fast ions colliding and spreading out radially. In general, fast ion diffusion consists of fast ions moving away from the core and toward the wall (assuming a core- peaked fast ion profile). Any fast ions that diffuse to the MST wall are lost. Slowing down consists of fast ions colliding with electrons and ions and depositing energy (heating). Both of these effects will be defined in this section. Diffusion in this code is assumed to follow the form of Fick’s diffusion: = D nf (6.31) ? ? 5? f/s0 2 (Te + Ti) D = ⌫ rLarmor (6.32) ? ? 2Te Xs0 where s/s0 f/s0 s/s s/s s/s (x ) 0 0 0 0 ⌫ =2⌫o (x )+ (x ) s/s (6.33) ? 2x 0 The variables in Eq. (6.33) are all defined in Eqs. (3.6), (3.8) and (3.9). Fast ion diffusion has a powerful effect in this model ( 1-5 m2/s for 25 kV fast ions), to the point that the computer grid ⇡ typically used in this model (dt 0.5-1.0 ms, dr 0.5 cm) is insufficiently large. To accurately ⇡ ⇡ model fast ion diffusion, each time step is divided into 500 sub-timesteps (dt 1 µs). The diffusion ⇡ output of the fast ion diffusion code can be seen in Fig. (6.21). In Fig. (6.21), the start and end times of the NBI are labeled. PPCD lasts from approximately 12-22 ms. Fast ion deposition is peaked, though within a few milliseconds the profile spreads out all the way to the MST wall. After the end of PPCD the fast ion density profile flattens out and starts to decrease in magnitude, due to both slowing down and diffusion to the wall. Fast ion slowing down is another source of fast ion particle losses. Fast ion energy loss to a species s0 over a code timestep t for a beamlet with Ef is: f/s0 t⌫✏ Ef s = Ef 1 e (6.34) ! 0 ✓ ◆ where the classical energy loss collision frequency is: m f/s0 f/s0 f f/s0 f/s0 ⌫✏ =2⌫o x 0 x (6.35) ms0 ⇣ ⌘ ⇣ ⌘ 143 Fast Ion Density, 200 kA PPCD Model 40 1.4 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 NBI Off ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.21 Contour plot of nf as a function of radius and time in the 1-D heating+deposition model, with fast ion diffusion included. At any given point in time, the total instantaneous heat deposition on a given plasma species is the sum of the energy exchange contribution to each species over all beamlets created up through that point in time. This relationship can be described as: t0(t) Nf (r, t, t0) Ef s0 (r, t, t0) PB s (r, t)= ! (6.36) ! 0 t t0=Xt0(0) As the fast ions slow down they eventually become thermalized, at which point there is a dramatic decrease in fast ion confinement [Hudson, 2006]. This effect is approximated in this code by “losing” any fast ions that slow down more than one slowing down time (vthermalized = vborn/e). The results for fast ion energy deposition on electrons for the same 200 kA PPCD data set are plotted in Fig. (6.22). The results of Fig. (6.22) can be compared to the fast ion densities calculated in Fig. (6.21). In general there is more heat deposition when there are more fast ions, but the plasma is also more collisional when it is colder. This means that the ratio of beam heating to fast ion density is higher near the plasma edge and after the end of PPCD. 144 PBEAM Density, 200 kA PPCD 1-D Model 40 2.0 P BEAM 35 Density (x 10 30 1.5 25 1.0 20 Time (ms) Time 5 W/m 15 0.5 3 10 ) 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.22 Contour plot of PB e as a function of radius and time in the 1-D heating+deposition ! model, with fast ion diffusion included. PBEAM Density, 200 kA PPCD TRANSP/NUBEAM 40 2.0 P BEAM 35 Density (x 10 30 1.5 25 1.0 20 Time (ms) Time 5 W/m 15 0.5 3 10 ) 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.23 Contour plot of PB e as a function of radius and time in the TRANSP/NUBEAM ! model. Note that the “wall” is at around 30-35 cm in this model. 145 For comparison, the TRANSP/NUBEAM modeled fast ion heat deposition on electrons for the same 200 kA PPCD data set are plotted in Fig. (6.23). The most important observation is that there is almost no heating beyond 30 cm, the approximate location of the MST “wall” in the TRANSP/NUBEAM case (a plot of TRANSP/NUBEAM output interpolated onto the 1-D model grid is plotted in Fig. (6.11)). Fast ions deposited near the “wall” are also lost quickly due to neutralization and fast ion diffusion. This difference in the two models in part derives from the fact that TRANSP/NUBEAM traces the orbits of the fast ions while the 1-D heating+deposition model does not. The magnetic effects are approximated in the 1-D code, but a 1-D approximation is never going to produce the same results as a 2-D orbit tracing model. But interestingly, even though the heating is flatter near the core in the TRANSP/NUBEAM model, the fact that there is no heating at all beyond r/a 0.6 ⇡ means that the overall heating profile is much more peaked in the TRANSP/NUBEAM case, which has a substantial effect on the temperature gradient and heat diffusion (see Sec. (6.6)). The modeled fast ion energy deposition on electrons for the same 200 kA PPCD data set with the two “hybrid” models are plotted in Figs. (6.24) (with fast ion diffusion) and (6.25) (without fast ion diffusion). These hybrid models use the same fast ion deposition profile and the same classical slowing down model, with the only difference being whether fast ion diffusion is included. As expected, the hybrid model without fast ion diffusion keeps a large amount of heating near the core well beyond the end of PPCD. The fast ions are stuck in the core, and deposit their heat en masse after enhanced confinement ends and plasma collisionality increases. The hybrid model with diffusion sees fewer total fast ions in the plasma (due to diffusion losses to the wall as well as more fast ion thermalization losses due to a more collisional plasma away from the core), but those fast ions are more spread out. This means that the plasma heating is less overall, but it’s flatter. This can be seen best by looking at a comparison of the four models at a single point in time (t = 22 ms) in Fig. (6.26). There are two very clear and important effects to point out in Fig. (6.26). First, that although the the 1-D heating+deposition model and hybrid model with diffusion see much less heating in the core than the other two models, they see much more heating near the edge. And the 146 PBEAM Density, 200 kA PPCD Hybrid w/ Diffusion 40 2.0 P BEAM 35 Density (x 10 30 1.5 25 1.0 20 Time (ms) Time 5 W/m 15 0.5 3 10 ) 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.24 Contour plot of PB e as a function of radius and time in the hybrid model with fast ! ion diffusion. PBEAM Density, 200 kA PPCD Hybrid w/o Diffusion 40 2.0 P BEAM 35 Density (x 10 30 1.5 25 1.0 20 Time (ms) Time 5 W/m 15 0.5 3 10 ) 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.25 Contour plot of PB e as a function of radius and time in the hybrid model without ! fast ion diffusion. 147 Beam Heating Of Electrons at t = 22ms 200 ) -3 TRANSP/NUBEAM Hybrid Model w/out Diffusion 150 Hybrid Model w/ Diffusion 1-D Heating+Deposition Model 100 50 Beam Heating of Electrons ( kW m 0 0.0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.26 Comparing the modeled heat deposition profiles for the four different 200 kA PPCD models at t = 22 ms, the time at which PPCD ends. TRANSP/NUBEAM model, despite a lot of core heating, sees very little heating outside of the core, where the plasma volume is much larger. In the case of this particular data, the volume- integrated heat deposition is 64 kW for the 1-D heating+deposition model, 66 kW for the hy- brid model with diffusion, 68 kW for the hybrid model without diffusion and 44 kW for the TRANSP/NUBEAM model. In addition, note also that diffusion appears to be strong enough to wash away most of the differences in fast ion energy deposition. Although the fast ion particle deposition profiles are quite different between TRANSP/NUBEAM and the 1-d heating+deposition model in Fig. (6.19), the energy deposition profile at the end of PPCD is nearly identical when fast ion diffusion is introduced. The total integrated heating in the TRANSP/NUBEAM model is around one-third lower than the other the models in Figs. (6.27) and (6.28), where the different fast ion energy loss components are plotted versus time. In both plots, the energy components are volume integrated over the entire plasma. The TRANSP/NUBEAM model has charge-exchange losses, which the 1-D model does 148 200 kA PPCD FI Power Loss Sources: 1D Model 500 Shine-thru 400 300 Beam Heating On Electrons Power (kW) 200 Fast ion orbit losses 100 Beam Heating On Ions 0 0 10 20 30 40 50 Time (ms) Figure 6.27 Volume integrated energy loss mechanisms for the 1-D heating+deposition model. Note that “fast ion orbit losses” in this context refer to diffusion losses to the wall. 200 kA PPCD FI Power Loss Sources: TRANSP 400 Shine-thru 300 200 Power (kW) Charge-exchange losses Beam Heating On Electrons 100 Fast ion orbit losses Beam Heating On Ions 0 0 10 20 30 40 50 Time (ms) Figure 6.28 Volume integrated energy loss mechanisms for the TRANSP/NUBEAM model. Or- bit losses include both prompt-ion losses and diffusion losses. Charge exchange losses are also included here. 149 not. It could be argued here that tthe TRANSP/NUBEAM model does not see enough change in Te because the NBI does not deposit enough beam energy plasma-wide. This is not true, however, for multiple reasons. First, it is shown later in this section that the hybrid model without diffusion sees similar temperature changes to the TRANSP/NUBEAM results despite significantly more volume- integrated heat deposition. Second, it is shown in Sec. (6.8) that the measured change in Te cannot be reproduced even if the TRANSP/NUBEAM heating is artificially increased (without a change in the normalized heating profile shape). There are two other important physics effects that can be seen in Figs. (6.27) and (6.28). First, the kinetic profiles are constant with time in TRANSP, and so shine-thru is flat. In the 1-D model, the beam is ramping up to full current from 8 ms through 11 ms, but after that the shine-thru con- tinues to increase somewhat due to the increased temperature (and thus slightly less collisional) plasmas. Second, note the ratio between the beam heating of elections and the beam heating of ions in both models. In both cases approximately 80-90% of beam heating is initially on electrons (versus ions), while dropping to close to 50% by the end of the shot. This can be seen by com- paring the energy loss frequencies. Using Eq. (6.35) and assuming that v v v (using the e f i approximations from Eqs. (3.10) and (3.11)), we can write that: 3 f/e mf f/e f/e mf 4 vf 2 vf f/e ⌫o (x ) 0(x ) f/e ⌫✏ me ⌫o me 3p⇡ ve p⇡ ve f/i = f/i mf (6.37) f/i h mf f/i f/i i ⇡ 2 ⇣ ⌘ 3 ⌫✏ ⌫o (x ) (x ) ⌫o mi mi 0 6 7 h i 4 5 This ratio is a function of electron temperature (since ve is a thermal speed) and fast ion energy, though not a function of ion temperature. That makes this ratio a function of both radius and time. Just after the NBI turns on (assuming all fast ions have an energy of 25 kV) this ratio is close to 10-to-1 in the core, and less on the edge. Over time the electrons heat up and the fast ions slow down, and both effects cause this ratio to decrease. 6.6 Model Output Comparisons To Data Output from the four different computer models are plotted in Figs. (6.29) and (6.30). In Fig. (6.29), the change in core Te due to NBI is plotted against the measured Te data. All of the 150 Te,core 200 kA PPCD, Comparing Four Models 200 NBI On Thomson Data TRANSP/NUBEAM Hybrid Model w/out Diff. Hybrid Model w/ Diff. 150 1-D Heating+Dep. Model PPCD (eV) e,core 100 T 50 0 0 10 20 30 40 Time (ms) Figure 6.29 Comparison of the modeled change in core electron temperature from the four dif- ferent models. The 1-D heating+deposition model and the hybrid model with diffusion are both consistent with the raw data during and after PPCD. The other two models are relatively consistent while the NBI is being fired, but show too little heating afterward. Te Profile, 200 kA PPCD, Four Models vs Data 1000 800 600 400 Raw Thomson Data, NBI Off Raw Thomson Data, NBI On NBI Off Model Output TRANSP/NUBEAM Electron Temperature (eV) Temperature Electron 200 Hybrid Model w/out Diff Hybrid Model w/ Diffusion 1-D Heating+Dep. Model 0 0.0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.30 Comparison of the Te profile output from the four different models at 22 ms into the shot, the same time point chosen for Fig. (6.26). Note that “NBI Off Model Output”, by definition, is exactly equal to the fit of the Thomson NBI Off data. 151 models fit the data within the error bars while the NBI is being fired, but the TRANSP/NUBEAM and hybrid model without diffusion both fall off far too quickly after the NBI is turned off. The 1-D heating+deposition model and the hybrid model with diffusion are both consistent with the data both during and after PPCD. Both of those latter two models take a full 10 ms to lose all of the auxiliary heating, which is consistent with the Thomson data. In Fig. (6.30), the electron temperature profiles are plotted for the four different models at t=22 ms, which is the same time point chosen for Fig. (6.26). Also plotted, in dark green, is the NBI Off model output. The code forces the NBI Off model to match the input NBI Off fitted Thomson data. Thomson output is fit onto a monotonic profile to improve numerical stability. Special care was taken to keep the gradient of the curves as close as possible to the Thomson output for r/a < 0.5, to get accurate core heat diffusion. The TRANSP/NUBEAM model and the hybrid model without diffusion both show significant core temperature increases in Fig. (6.30), the bigger difference between the models is actually away from the core. While the 1-D heating+deposition model and the hybrid model with diffusion both show significant heating beyond r/a=0.6 (consistent with the data), the hybrid model without diffusion sees no significant heating beyond r/a=0.4 and TRANSP/NUBEAM sees no significant heating beyond r/a=0.2. The effect of heating away from the core is a flattening of the temperature profiles. Flatter temperature profiles mean less heat diffusion, since P T and a fixed is assumed. Conduction /5 e e The diffusion power flux densities out of electrons for the 1-D heating+deposition model is plotted in Fig. (6.31). In Fig. (6.31), there is a significant decrease in heat diffusion during PPCD, when e is signif- icantly reduced. Note also that diffusion density decreases, in general, closer to the wall. This is due to the fact that the surface area of each volume element (as well as the volume of each volume element) increases with radius. The volume elements are on a fixed r grid, and so volume and surface area both increase with radius. Energy diffusion decreases during PPCD, which is to be expected because of the decreased magnetic stochasticity and plasma collisionality. Resistivity of the plasma is also reduced in hotter 152 PDiffusion Density, 200 kA PPCD 1-D Model 40 1.2 P Conduction 35 1.0 30 Density (x 10 0.8 25 0.6 20 Time (ms) Time 0.4 15 6 W/m 0.2 10 3 ) 5 0.0 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.31 Modeled heat diffusion loss density for the 1-D heating+deposition model. Note that the heat diffusion is greater at the beginning and end of the shot, when the plasma is cooler and energy confinement is worse. Note that Heat diffusion drops significantly during PPCD. 3/2 2 plasmas (⌘ Te ). This means that ohmic power (P⌦ ⌘j ) is also reduced, as can be seen in / / k Fig. (6.32), where the ohmic power density for the 1-D heating model is plotted. A similar effect is seen in the ohmic heating profiles in Fig. (6.32). There, the effect of tem- perature on ohmic power can be seen very clearly. P⌦ is a function of resistivity and the square of the current density. Plasma current is relatively constant between 5 and 40 ms (Fig. (5.1)), but temperature changes drastically due to PPCD. This is why ohmic power reduces so significantly between around 12-22 ms before returning to pre-PPCD conditions. The impact of the different models on ohmic power and diffusion losses can be seen in Fig. (6.33), which compares the 1-D heating+deposition model for NBI On and NBI Off, and Fig. (6.34), which compares the 1-D heating+deposition model to the TRANSP/NUBEAM model. In both plots, the heat fluxes are plotted for the plasma core in units of MW/m3, as a function of time, and the “core” is defined as the inner 5 cm radially. The inputs plotted are ohmic power and NBI heating, while the outputs are diffusion losses and electron-to-ion classical collisions. In Fig. (6.33) the solid lines represent NBI Off while the dotted lines represent NBI On. When the NBI turns on at 8 ms the diffusion losses rise a little to partially counter that newly deposited 153 POHMIC Density, 200 kA PPCD 1-D Model 40 8 P OHMIC 35 7 Density (x 10 30 6 25 5 20 4 Time (ms) Time 5 15 3 W/m 10 2 -3 ) 5 1 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.32 Modeled ohmic power density for the 1-D heating+deposition model. Note that ohmic heating drops significantly during PPCD due to the warmer (and less resistive) plasma. heat and diffuse it from the plasma. Over time, though, the flattening of the temperature profile reduces diffusion losses to the point that they are actually slightly less than the NBI Off case by the end of PPCD. This is what leads to a substantial increase in Te near the end of PPCD. Note also that a reduction in ohmic power can be seen due to the beam-driven temperature increase. Near the end of PPCD, T /T 110%, and the impact on ohmic power is significant. e,NBI On e,NBI Off In Fig. (6.34) the dotted lines again represent the 1-D heating+deposition model, but this time the solid lines represent the TRANSP/NUBEAM model. In this plot, note that the effect of the core heating in the TRANSP/NUBEAM model is to dramatically increase the heat diffusion. The heat diffusion after the NBI turns on is substantially greater than the 1-D heating+deposition model. The core electron temperature outputs between the two models in Fig. (6.29) do not begin to significantly diverge until near the end of PPCD, which is the same point in time at which the core heat diffusion begins to significantly diverge. In fact, core NBI heat deposition is greater in the TRANSP/NUBEAM case than the 1-D heating+deposition model for much of the shot, but this is more than compensated for by the greater heat diffusion relative to the 1-D heating+deposition model, which leads to reduced relative electron temperatures. 154 ) Core Electron Heating: NBI On vs NBI Off 3 1 Diffusion Losses NBI On NBI Heating Ohmic Power 0.8 Electron-Ion Heating Solid = NBI Off Dotted = NBI On (1-D) 0.6 PPCD 0.4 0.2 0 20 40 Core Power Density On Electrons (MW/m 0 10 30 Time (ms) Figure 6.33 Comparison of core heating components, both in and out, for the 1-D heat- ing+deposition model and for NBI Off. ) Core Electron Heating: 1-D Model vs TRANSP 3 1 Diffusion Losses NBI On NBI Heating Ohmic Power 0.8 Electron-Ion Heating Solid = TRANSP Dotted = 1-D Model 0.6 PPCD 0.4 0.2 0 Core Power Density On Electrons (MW/m 0 10 20 30 40 Time (ms) Figure 6.34 Comparison of core heating components, both in and out, for the 1-D heat- ing+deposition model and TRANSP/NUBEAM. 155 The differing core electron temperature between the two models near the end of PPCD leads to a different core ohmic power density. But the effect, while significant, is small relative to the reduction in heat diffusion losses. It can also be seen clearly in Figs. (6.33) and (6.34) just how irrelevant electron-to-ion heat exchange is for the electrons. This is why ion temperature and anomalous ion heating do not have very much leverage in this model. 200 kA PPCD Core Te, Compare Models 200 kA PPCD Core e Te, Compare 0 0 NBI On NBI On -200 PPCD -0.5 eV m/s) -400 4 Te (eV/m) Te -600 (10 Te -1.0 PPCD e -800 NBI Off NBI Off NBI On -1.5 Avg. Core Avg. NBI On -1000 Hybrid w/out Diffusion Hybrid w/out Diffusion Hybrid w/ Diffusion Core Avg. Hybrid w/ Diffusion 1-D Model 1-D Model -1200 -2.0 0 10 20 30 40 0 10 20 30 40 Time (ms) Time (ms) Figure 6.35 Comparing the core electron temperature gradients versus time for the four models, as well as for NBI Off. Here, “average core T ” is defined the line-averaged T for r/a < 0.2. r e r e The plot on the right shows, quantitatively, the amount of core heat conduction in each of the four models and the NBI Off case. To help visualize the “flattening” Te profiles, the core electron temperature gradients for the various models are plotted in Fig. (6.35). In the plot on the left the four models are compared to the NBI Off case, where “average core T ” is defined as the line-averaged T for r/a < 0.2. r e r e In the plot on the right, average core T is plotted, which gives a quantitative visualization of er e core heat conduction losses. The fact that the TRANSP/NUBEAM model and the hybrid model without diffusion see significantly greater heat diffusion after the NBI turns off is very clear in this plot. Before ending this section, two assumptions made earlier in this chapter can be discussed more quantitatively. The first of these assumptions was the shape of the anomalous ion heating profile. In Fig. (6.13) it was shown that the impact of forcing Ti=0 at all points in time and 156 Te,core 200 kA PPCD, 1-D Model Varying Ti 200 NBI On NBI Off Model 1-D Heating+Dep. Model 1-D Heating+Dep. Model, Ti(r,t)=0 150 PPCD (eV) core e, 100 T 50 0 0 10 20 30 40 Time (ms) Figure 6.36 Core electron heating calculated with the 1-D heating+deposition model (solid line), as well as with Ti forced to be zero at all points in time and space. This is a follow-up to Fig. (6.13), and shows the minimal impact of anomalous ion heating assumptions. space had a relatively small impact on the output of the TRANSP/NUBEAM model, and this exercise was repeated for the 1-D heating+deposition model, with the results plotted in Fig. (6.36). P (T T ), so forcing T to equal zero will overestimate the impact of P as much as possible, ei / e i i ei yet the effect is still too small to have a significant impact on the output. The other assumption we can discuss more quantitatively is that e is unaffected by the neutral beam. We can consider the possibile change in Te due to reductions in e values. In Sec. (6.2), it is argued that the e profiles are unchanged because only the core-most mag- netic mode is suppressed by a significant value, and suppression of this magnetic mode has not been shown to correlate with increased temperatures in PPCD. Plots like Fig. (6.9) show non-zero suppression of mid-radius magnetic fluctuations, though this suppression is clearly within the sta- tistical error bars of zero. It is worth exploring whether that small amount of suppression could have a significant impact on thermal conductivity. 157 The effect of suppressed magnetic fluctuations on electron heat transport is described in Biewer et al. [2003]. The breakup of magnetic surfaces leads to a radial excursion ofthe particle’s trajec- tory, with this thermal conductivity generally assumed to be = v DM (6.38) k In Eq. (6.38), v is the velocity along the magnetic field line and DM is the magnetic field diffu- k sivity. Rechester & Rosenbluth [1978] found that this magnetic field diffusivity can be described by the relation D ˜b2. This relationship assumes large values of the Chirikov overlap criteria, M / r or “stochasticity parameter” [Zaslavsky & Chirikov, 1972]: 1 wmn + wm n s = 0 0 (6.39) 2 rmn rm n 0 0 In Eq. (6.39), wmn is the magnetic island width for toroidal mode number n and poloidal mode number m, while m n refers to the toroidal and poloidal mode numbers of an adjacent rational 0 0 surface. Stochasticity is reduced in PPCD such that the magnetic islands do not overlap and this criteria is not met. However, we can assume that Eq. (6.38) holds in order to demonstrate the potential impact. The reduction in e is calculated by using the NBI Off profiles and scaling them along the radial magnetic fluctuation eigenfunctions from DEBS [Biewer et al., 2003]. For this simulation, the n=7-12 fluctuations were used. The results of that simulation are plotted in Fig. (6.37). The n=6 mode is not used because of the previous demonstration that suppression of the core-most mode does not impact e. Including the effect of the n=6 mode in the same analysis leads to a core Te (>200 eV) far beyond what was measured experimentally. This analysis is more confirmation that suppression of the core-most magnetic mode is not correlated with increased core temperatures. This is also confirmation that the measured Te is primarily due to fast ion heat deposition. However, it is possible that a small suppression of the mid-radius modes could be calculated with a larger data sample size, and this suppression might be responsible for a core Te increase on the order of 10-20 eV. 158 Te,core 200 kA PPCD, Adjusting e 200 NBI On Thomson Data 1-D Model Adjusted 150 e PPCD (eV) core e, 100 T 50 0 0 10 20 30 40 Time (ms) Figure 6.37 Core electron Te calculated with the 1-D heating+deposition model both without (red) and with (blue) suppressed e due to suppressed mid-radius magnetic fluctuations. Here, the n=7-12 magnetic modes were used. Because this mid-radius mode suppression is not statistically significant, and because the as- sumptions made to create Fig. (6.37) do not hold throughout the plasma pulse, e will be assumed to be the same between NBI On and NBI Off for the rest of this dissertation. If there is any NBI impact on heat conduction it is likely small and within the current levels of statistical uncertainty. However, the possibility of increased energy confinement due to NBI suppression of mid-radius magnetic fluctuations would be an interesting area for future research. 6.7 Impact Of Zeff On 1-D Model The plasma temperature cannot be modeled classically without a discussion of Zeff . Zeff is the effective ionic charge of the plasma and is defined as n Z2 Z = s s s (6.40) eff ne P For a plasma made up entirely of deuterium, Zeff = 1. A real world MST plasma, made up mostly of deuterium but also including impurities like carbon, aluminum and boron, has a Zeff value greater than 1. It is not a value that can be measured directly on MST, although there have 159 been attempts to calculate Zeff profiles indirectly in MST (see Anderson [2001] for an exam- ple). In general, Zeff will be lower in colder plasmas, where impurities are less likely to be fully stripped. The relevance of Z to this modeling is that the plasma resistivity ⌘ Z . Because ohmic eff / eff 2 power P⌦ ⌘j , P⌦ Zeff . Assuming that we already have a measured plasma current along / k / with measured temperature and density profiles all as functions of time, the effect of a larger Zeff is to increase the calculated P⌦. The effect of greater P⌦ is a larger e and decreased electron energy confinement time (Eq. (6.8)). This will mean less increase in Te for a given NBI heat deposition profile. In order to test whether Zeff is a substantial effect, heat balance calculations were performed for a variety of Zeff profiles. Temperature and density profiles were fit using data provided by Thomson and FIR. As described in Sec. (6.6), however, Thomson profiles were fit with the con- straint that they remain monotonic. To more accurately model the key parts of the plasma, matching the temperature profile gradient accurate for r/a < 0.5 was prioritized above matching the edge gradient. Both the values and profile shape of Zeff in MST have to be assumed for this analysis. It is generally believed that cold MST plasmas have a Zeff value close to 1.5 or 2, while the hot plasmas can have a Zeff of 5 or even greater in the core. To simulate this, the Zeff profiles used in this dissertation consisted of step functions. One value given for Te < 500 eV, and one for Te > 500 eV. These values are not arbitrary as plasma temperatures above 500 eV tend to see significantly greater stripping of aluminum atoms in MST. The core heating results are plotted in Fig. (6.38). There are two conclusions that can be readily made from Fig. (6.38). The first is that it’s the edge Zeff that matters more than the core value. Most of the plasma has electron tempera- ture less than 500 eV most of the time. The second conclusion that can be made is that Zeff is not a significant factor in this analysis. For a wide array of reasonable Zeff profiles, the model output is consistent with the data. In ordert to get a heating output below the Thomson data, an edge Zeff greater than 3.5 or 4 is needed, which is well above what previous MST research has predicted [Anderson, 2001]. 160 200 kA PPCD Data vs Simulation, Varying Zeff 200 NBI On Thomson Data Zeff = 2.0, 2.5 Zeff = 1.5, 2.5 Z = 1.5, 3.0 150 PPCD eff Zeff = 1.5, 3.5 Zeff = 2.0, 2.0 (eV) Zeff = 2.0, 3.0 Z = 2.0, 4.0 core eff e, 100 Zeff = 2.5, 3.5 T 50 0 0 10 20 30 40 50 Time (ms) Figure 6.38 Modeled change in core Te is plotted for a variety of Zeff profiles in 200 kA PPCD. All output is the 1-D heating+deposition model, and the two values of Zeff in the key at the upper right refer to plasma at Te < 500 eV and Te > 500 eV. This experiment can be repeated with a second dataset, the 400 kA low density PPCD ensem- ble. That output plotted in Fig. (6.39). In this ensemble, Zeff is again not a significant factor for reasonable values. The Thomson data is noisy due to the limited sample size (around 20 shots each for NBI Off and NBI On). It’s important to note that this 400 kA ensemble features NBI heating from 8-31 ms. NBI auxiliary heating dissipates after the end of PPCD, even with continuing NBI. But this modeling assumes that there are no stochastic losses of fast ions, and also that there are no charge-exchange losses of fast ions. This model breaks down for non-PPCD plasmas, particularly after a magnetic reconnection event. Charge-exchange and stochastic effects on the model are discussed in detail in Sec. (6.9). The conclusion that can be made from this analysis is that Zeff is a relatively small lever in this model. A reasonable Zeff value can be assumed and used throughout. For the 200 kA PPCD case used throughout this chapter, it is assumed that Zeff is 2 for Te < 500 eV, and 2.5 for Te > 500 eV. 161 400 kA PPCD Data vs Simulation, Varying Zeff 200 Thomson Data NBI On Zeff = 1.5, 3.0 Zeff = 1.5, 3.5 Z = 2.0, 3.0 150 eff PPCD Zeff = 2.0, 3.5 Zeff = 2.0, 4.0 (eV) Zeff = 2.5, 3.0 core e, 100 T 50 0 0 10 20 30 Time (ms) Figure 6.39 Modeled change in core Te is plotted for a variety of Zeff profiles in 400 kA PPCD. Note that the NBI is on from 8 ms to 31 ms in this ensemble. 6.8 Impact Of Fast Ion Diffusion On Heating The model output presented in Sec. (6.6) clearly shows that core electron temperature change due to NBI is influenced not just by the amount of total deposited NBI power but also by the fast ion diffusion. Fast ion diffusion serves two purposes in a 1-D model. First, it flattens the fast ion density profile by driving fast ions from the core to the edge. Second, it serves as a fast ion loss mechanism by driving edge fast ions into the MST wall. The relative impact of these two processes will be studied in this section. The impact of fast ion diffusion on heating can be quantified by using the TRANSP/NUBEAM output. It was seen in Fig. (6.29) that the TRANSP/NUBEAM predicts significantly less change in core Te than was measured after the NBI turns off as well as after PPCD turns off. It was also shown, in Fig. (6.26), that while TRANSP/NUBEAM predicts greater core heat deposition, that the volume-integrated heat deposition throughout the plasma was around 1/3 less than for the other models. This disparity is primarily due to the fact that the TRANSP/NUBEAM models features charge-exchange (which is significant near the plasma wall), while the other models do not. 162 The TRANSP/NUBEAM heat deposition output consists only of raw numbers at each point in space and time (in other words, there are no ”fast ions” that are being kept track of). As a numerical experiment, the raw power deposition numbers can be multiplied by an arbitrary number to change the amount of plasma heating. In Fig. (6.40), that very experiment is performed. The heat deposition at each point in time and space is multiplied by a factor of two and a factor of three, and the code is run again for each of those scenarios, and then compared to the baseline TRANSP/NUBEAM results as well as the 1-D heating+deposition model results. The change in core Te is plotted in the figure on the left, while the core energy budget is plotted on the right. T 200 kA PPCD, Varying TRANSP ) Core Electron Heating: NBI On vs TRANSP[x2] e,core 3 200 1 Diffusion Losses NBI On Thomson Data NBI On NBI Heating 1-D Model Ohmic Power TRANSP 0.8 Electron-Ion Heating 150 PPCD TRANSP x 2 TRANSP x 3 Solid = TRANSP[x2] Dotted = NBI On (eV) 0.6 core e, 100 PPCD T 0.4 50 0.2 0 0 0 10 20 30 40 Core Power Density On Electrons (MW/m 0 10 20 30 40 Time (ms) Time (ms) Figure 6.40 Data for the ”TRANSP/NUBEAM model” is plotted for different TRANSP heat deposition assumptions. Core electron temperature heating is plotted on the left, where brown is the calculated TRANSP/NUBEAM heat deposition output, and where purple is the output multiplied by two and blue is the output multiplied by three. On the right, core electron heating components are plotted for the regular TRANSP/NUBEAM output and for the output multiplied by two. What is key in these scenarios is that the normalized profile shape of the TRANSP/NUBEAM heating is unchanged. The heating is still very peaked, with none happening beyond the ”wall” (r/a 0.6). The total (volume-integrated) plasma heating in these scenarios is significantly more ⇡ than in the baseline 1-D heating+deposition model, and so initially there is actually more change in core Te. However there is still no significant change in Te in the mid-radius or edge areas of 163 the plasma. The result of that is a significant increase in heat diffusion out of the core that rapidly depletes this beam heating. Within a few milliseconds (the heat conduction timescale) of the NBI turn-off, these models already show too little change in core Te. There are two clear conclusions that come from this output. The first is basically a tautology: if you dump enough heat in the plasma core then (over short time scales) the plasma core will increase in temperature. But over long enough time scales ( 5 10 ms), heat diffusion will pull that heat out of the core, and that core Te will drop unless there is heating away from the core to flatten the temperature gradient. The second conclusion that can be made is that with this classical model there is no way to replicate the measured change in electron temperature in the core by simply dumping a lot of heat in the core. There must be some heating of the entire plasma to keep the temperature profiles flat enough to produce the measured results. A 0-D model of the plasma can never accurately produce predictions of NBI heating, as 1-D effects are essential. Another interesting question to answer is the effect of fast ion diffusion rates on plasma heating. The fast ion diffusion coefficient (D ) was calculated in Eq. (6.32), and it is seen in Sec. (6.6) that ? turning fast ion diffusion on or off for the hybrid model (where fast ions are deposited according to TRANSP and then slowed and diffused with the 1-D model) has a strong effect on core electron temperature. Turning on fast ion diffusion takes fast ions, which are initially deposited in a very core-peaked shape (Fig. (6.11)), and moves them toward the edge of the plasma. This flattens the Te profile, reduces heat diffusion and increases core Te. In Fig. (6.41), the impact of varied fast ion diffusion has a logical impact on core Te. The brown curve (no fast ion diffusion) and the light-blue curve (calculated D ) are just the previously- ? calculated “hybrid model without fast ion diffusion” and “hybrid model with fast ion diffusion”. The purple, dark blue and orange curves are D = D /3,D /10 and D 3, respectively. As ? ? ? ? ⇥ expected, reducing fast ion diffusion reduces the number of fast ions that work their way to the edge, which reduces fast ion heating near the edge and reduces Te in the plasma core. The most interesting result in Fig. (6.41) is the orange curve, which represents accelerated fast ion diffusion. Throughout PPCD the enhanced fast ion diffusion leads to greater core Te, but over 164 200 kA PPCD Core Heating (Hybrid), Varying FI Diff. 200 NBI On Thomson Data Diffusion Off Diffusion / 10 150 PPCD Diffusion / 3 Diffusion Diffusion x 3 (eV) core e, 100 T 50 0 0 10 20 30 40 Time (ms) Figure 6.41 Core electron temperature change data for a hybrid model with and without calculated fast ion diffusion, as well as artificially slowed-down diffusion (D = D /3,D /10) and for ? ? ? artificially sped-up diffusion (D = D 3) ? ? ⇥ . time the effect wears off and by t =30 ms the core Te is actually smaller than the non-enhanced fast ion diffusion case. In this case, the fast ion diffusion has gotten large enough that there is a significant increase in the loss of fast ions to the wall. Fast ion diffusion creates two competing effects - edge fast ion heating and edge fast ion losses - that are very much dependent on the fast ion deposition profile. This effect can be seen very clearly in Fig. (6.42), where the 1-D heating+deposition model is plotted for a variety of fast ion diffusion coefficients. The red curve is the standard 1-D heat- ing+deposition model output while the orange, yellow and blue curves represent output for varied fast ion diffusion: D = D 3,D /3 and 0, respectively. Here the results seem to contradict ? ? ⇥ ? Fig. (6.41). In this case, decreasing fast ion diffusion actually increases core Te initially, if only slightly. The big difference between these two cases is the shape of the deposited fast ions. In the hybrid case, the fast ions are deposited by TRANSP, which places the ”wall” at r/a 0.6. Fast ⇡ ion diffusion is a pre-requisite to generate any fast ion heating in the outer half of the plasma 165 T 200 kA PPCD,Varying FI Diffusion e,core ) Core Electron Heating: FI Diffusion On vs Off 200 3 1 Diffusion Losses NBI On Thomson Data NBI On NBI Heating 1-D Model Ohmic Power Diffusion x 3 0.8 Electron-Ion Heating 150 PPCD Diffusion / 3 No Diffusion Solid = Diffusion On Dotted = Diffusion Off (eV) 0.6 core e, 100 T 0.4 PPCD 50 0.2 0 0 0 10 20 30 40 Core Power Density On Electrons (MW/m 0 10 20 30 40 Time (ms) Time (ms) Figure 6.42 Output from the 1-D heating+deposition model calculated with four different fast ion diffusion coefficients (D ). In the left hand plot, the change in core electron temperature is plotted. ? In red is the calculated D , while orange is D 3, yellow is D /3 and blue is D = 0. On the ? ?⇥ ? ? right, core electron heating components are plotted for regular diffusion and for D =0. ? volume. In the 1-D heating+deposition case, however, fast ions are already deposited throughout the plasma. Fast ion diffusion is not needed to generate fast ion heating near the edge. Instead, the effect of turning off fast ion diffusion is to keep any fast ions from leaving the plasma over short time scales. Fast ions can only be lost by thermalizing, which takes a very long time in PPCD plasmas. At the peak of 200 kA PPCD plasmas in MST, the core fast ion thermalization time is on the order of 40-50 ms. Because the fast ions are confined in the core in the no-diffusion case, there is a dramatic increase in core heating of electrons, as can be seen in the right-hand plot in Fig. (6.42). Over time the heat diffusion catches up and wipes away most of that extra heat, but it is slowed because of the existence of so much fast ion heating in the outer half of the plasma volume. In Fig. (6.43), the fast ion density profiles are plotted for both the fast ion diffusion “on” and “off” cases. The difference is clear: without fast ion diffusion the fast ions that are deposited in the core stick around for the entire enhanced confinement period and are not significantly lost until the slowing down times decrease due to the cooler plasma at t 35 ms. Fast ions deposited near ⇡ 166 Fast Ion Density, 200 kA PPCD Model 40 1.4 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 NBI Off ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Fast Ion Density, 200 kA PPCD Model (Diffusion Off) 40 1.4 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 NBI Off ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.43 1-D heating+deposition model fast ion density profile results. The top plot uses the calculated fast ion density and is identical to Fig. (6.21), while the bottom plot has fast ion diffusion turned off. 167 the plasma edge show similar behavior in each case. Because the plasma is cooler (and thus more collisional) near the wall, the fast ions near the edge are lost well before the fast ions in the core. Therefore, the Te gradient increases after the end of PPCD, increasing heat diffusion. This can be seen in Fig. (6.42), where diffusion losses increase rapidly after the end of PPCD. There’s no physical reason to expect fast ion diffusion would ever be less than classical in MST. But the analysis in this section also shows that fast ion diffusion is very important as a fast ion loss mechanism beyond thermalization. Whether fast ion diffusion can substitute for charge-exchange losses is the subject of Sec. (6.9). 6.9 Modeling Fast Ion Charge-Exchange Losses Given a known neutral density profile, fast ion charge-exchange losses can be calculated in a straightforward manner. For a given fast ion “beamlet”, the charge-exchange losses are: Nloss(r, t, t0) = N (r, t, t0)n (r, t) < v> (6.41) t fi n Neutral density profiles are calculated by NENE,` which iteratively fits its model to D↵ data, through a process described in Sec. (2.6). The problem, as described in Sec. (2.6), is that the D↵ measurement is very insensitive to core emission, and tends to output core neutral densities that are too high. NENE` solves for simulated D↵ data and tries to fit it to the actual data. The fact that the line-averaged D↵ values are so dominated by the edge-peaked profiles makes core neutral density output subject to very large error bars. Previous work with NENE` on MST has generally found core neutral densities that are too high to explain other data, particularly in PPCD (where the ratio between edge and core neutral density is largest), though this has not been published anywhere. It will also be seen in this section that heating data seems consistent with smaller core neutral data than NENE` output suggests. A second consideration is that it’s possible for a fast ion to be neutralized, and then re-ionized on its path out of the machine. A fast ion with a large v /v ratio could potentially travel 2-3 m to k the wall after being neutralized, which is close to its mean free path for a re-ionization in a typical 168 MST plasma. This further contributes to the possibility that any use of NENE` charge-exchange losses will overstate the impact of core neutrals on fast ion losses. Despite the fact that we cannot accurately model the precise impact of charge-exchange losses on fast ions in the 1-D heating+deposition model, the relative impact of neutralization losses can still be studied. This can be achieved by modeling with both NENE` output and “reduced” NENE` output, with the latter being an attempt to better estimate the core neutral density. The definition of “reduced” is inevitably going to be arbitrary, but is kept consistent through this dissertation. The definition of “reduced” was chosen to be nn(r, t)=nn,NENE`(r, t)/30 in the core, nn(r, t)=nn,NENE`(r, t) near the edge, and a linear reduction in between. A plot of what this looks like can be seen in Fig. (6.45). 200 kA PPCD Core Heating, Varying Neutrals/Diff. 200 NENE Neutral Density Profile, 200 kA PPCD 1012 NBI On Thomson Data no Off no On PPCD 11 n Reduced ) 150 o 10 -3 no Reduced, D =D /3 no Reduced, D =0 (eV) 1010 core e, 100 T 109 50 Neutral Density (cm 8 10 7 0 10 0 10 20 30 40 0 0.1 0.2 0.3 0.4 0.5 Time (ms) Radius (m) Figure 6.44 Output from the 1-D heat- Figure 6.45 A plot of a NENE` neutral density ing+deposition model calculated for varying neu- profile, and the “reduced” neutral density profile tral densities and fast ion diffusion coefficients. at t = 21 ms in 200 kA low density PPCD. The result of re-running the 1-D heating+deposition model including neutral losses, for a mix of fast ion diffusion coefficients, is plotted in Fig. (6.44). The first immediate conclusion is that, as expected, electron heating is far too low without reduced neutral densities. With reduced neutral densities the predicted core Te is, for the most part, within the error bars of the data. The results of varying the fast ion diffusion coefficient should be compared to Fig. (6.42), where the diffusion coefficient is varied but there are no neutral losses included. One interesting 169 result from Fig. (6.42) was that the electron temperature increases slightly more when fast ion diffusion is turned off. The increased heat conduction was more than compensated for by the lack of a significant fast ion loss mechanism. With neutral losses included, this effect goes away. In Fig. (6.44), it is clear after the end of PPCD that greater fast ion diffusion leads to greater electron temperatures. The result can be understood by looking at the modeled fast ion density profiles, which are plotted in Fig. (6.46) for the “reduced” neutral density case. The three plots, from top to bottom, are for regular fast ion diffusion, reduced fast ion diffusion (D = D /3) and fast ion diffusion ? ? turned off. The top and bottom plots should be compared to Fig. (6.43), where the same fast ion diffusion coefficients were plotted without any neutral losses. The results of Fig. (6.46) show that core fast ion density is not particularly impacted during and immediately after PPCD by turning on a very small local neutral density. But the effect of the neutrals is significant near the edge. As fast ion diffusion is reduced, fast ions cannot be sufficiently spread to the edge to replenish those lost by neutralization, and the effect is reduced edge fast ion densities. This means reduced edge electron heating, which means increased heat diffusion and reduced core Te. The increased plasma collisionality after PPCD leads to increased fast ion diffusion, as well as slowing down. Because fast ion diffusion is a more powerful effect than slowing down, the result of increased diffusion is a rapid loss of fast ions throughout the plasma to the wall and consumption by neutralization. Fast ion density in the inner half of the plasma volume is significantly lower with increased diffusion in the first 5-10 ms after the end of PPCD. But the increased fast ion density near the wall in the top plot in Fig. (6.46) means a flatter heating profile and an increased core Te, despite the reduction in the number of total fast ions in the plasma relative to the no diffusion case. This procedure can be repeated for the 400 kA case. This ensemble is different in an important way, however, because the NBI stays on beyond the end of PPCD. We know from experimental data that a large magnetic reconnection event will lead to a rapid loss of fast ions (Fig. (3.20)). Modeling neutrals in this case will help provide an educated guess as to how much of the fast ion losses are 170 nfi, 200 kA PPCD, Neutrals Reduced, FI Diffusion On 40 1.4 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 NBI Off ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) nfi, 200 kA PPCD, Neutrals + FI Diffusion Reduced 40 1.4 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 NBI Off ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) nfi, 200 kA PPCD, Neutrals Reduced, FI Diffusion Off 40 1.4 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 NBI Off ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.46 1-D heating+deposition model fast ion density profile results with charge-exchange included with reduced neutral densities (“reduced” is defined in Fig. (6.45)). The three plots fea- tured varied fast ion diffusion. From top-to-bottom are plotted standard calculated D , D /3 and ? ? D =0. ? 171 due to the rapid influx of neutrals (particularly in the core) during a magnetic reconnection event, and how many of them are due to stochastic losses of fast ions to the wall. In order to model the effect of a rapid magnetic reconnection event, it’s necessary to use NENE` to solve a single, representative shot from the ensemble. The effect of ensembling or averaging is to wash away most of the effects of a sawtooth crash. So for the 400 kA case, a representative shot was chosen, and the NENE` output for core neutral density is plotted in Fig. (6.48). Note the reduction in core neutral density during PPCD, followed by a large influx at 22 ms, and further increases at around 27 ms and 35 ms. 400 kA PPCD Core Heating, Varying Neutrals Core Neutral Density, 400 KA PPCD (Shot 1110328050) 200 3 Thomson Data NBI On no Off n Reduced ) o -3 150 no On PPCD m PPCD 16 2 (eV) core e, 100 T 1 50 Neutral Density (x 10 0 0 0 10 20 30 0 10 20 30 40 Time (ms) Time (ms) Figure 6.47 Output from the 1-D heat- Figure 6.48 A plot of a NENE` output of core neu- ing+deposition model for the 400 kA PPCD en- tral density in 400 kA low density PPCD. This is semble. Output is plotted for neutral density on the neutral density data used in Fig. (6.47). and off, and also for “reduced” neutral density. The 400 kA ensemble results are plotted in Fig. (6.47) for three cases. These three cases are n0 =NENE,` n0 =“reduced” NENE` and n0 =0. The “reduced” case results in nearly the exact same core Te as the no neutrals case during PPCD due to the very low core neutral density, but after PPCD it is unable to reproduce the transition to no Te. The Thomson results after PPCD are limited here, and quite noisy, but it was shown repeatedly in Ch. (5) that there is no statistically significant NBI heating away from PPCD periods in MST. The simulation with the full NENE` neutral density profiles does reproduce a very small Te after the end of PPCD, but also produces far too little heating during PPCD. 172 Fast Ion Density, 400 kA PPCD Model (No Neutrals) 40 1.4 35 NBI Off 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Fast Ion Density, 400 kA PPCD Model (Reduced Neutrals) 40 1.4 35 NBI Off 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Fast Ion Density, 400 kA PPCD Model (Neutrals) 40 1.4 35 NBI Off 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.49 1-D heating+deposition model fast ion density profile results for the 400 kA PPCD ensemble. Neutral density is varied. The top plot has no neutrals, the middle plot has “reduced” neutral density and the bottom plot has NENE` neutral density. 173 The fast ion density profiles for the three cases are plotted in Fig. (6.49). Fast ion diffusion is the same in all three cases, and the effect of increased neutral density is a significant reduction in edge fast ions. In the no neutrals and “reduced” neutrals cases, there is still a very significant core fast ion population for close to 10 ms after the end of PPCD. This generates the post-PPCD Te that is not observed in the data. In the full NENE` cases, the effect of each sawtooth event is a dramatic reduction in the fast ion population. The large magnetic reconnection event that signals the end of PPCD neutralizes almost the entire fast ion population within a millisecond. This causes the small Te in this case after the end of PPCD. This fast ion profile output is consistent with previous observations which show a rapid loss of fast ions during magnetic reconnection events in MST. But the fact that the full NENE` case which produces that outcome also leads to such a large neutral density during PPCD that Te is far below the data suggests that both NENE` models are insufficient to explain what happens throughout the shot. It is possible that “reduced” NENE` is a reasonable approximation during PPCD but that full NENE` must be used after PPCD to accurately model the rapid loss of fast ions during the sawtooth crash. To further explore the impact of neutralization losses, a non-PPCD case was run through the same 1-D heating+deposition model. This was a 400 kA, F=-0.2 (standard) ensemble. NENE` was run for two different models: ensembled D↵ data and single shot D↵ data. The single shot data includes very clear magnetic reconnection events, while the ensembled data washes those out. The results of the 1-D heating+deposition model are plotted in Fig. (6.50). Both NENE` model results are included, both in the full and “reduced” neutral density cases. The no neutrals results is included as well, for comparison. Also included is the Thomson data for this case, which is consistent with zero Te throughout. In this case, the core NENE` “reduced” neutral density cases are insufficient to match the data. Both the ensembled and single shot neutral data show little variation from the no neutrals case. With the full NENE` neutral density profiles, significant heat is still seen, even in the single shot case, but it is significantly reduced. 174 The reasons for this can be seen in Fig. (6.51), where predicted core fast ion and neutral densi- ties are plotted for this dataset (for the n0 =NENE` case). The ensembled output is plotted in black while the single shot output is plotted in blue. In addition, the magnetic fluctuations are plotted for the single shot case, where it can be seen clearly that the large magnetic reconnection events all correspond with spikes in the core neutral density to as much as ten times the value away from sawtooth events. Change in Core Te, 400 kA Standard Plasma 150 NBI On Thomson Data No Neutrals Ensembled no Ensembled, reduced n 100 o Single shot no Single shot, reduced no (eV) 50 core e, T 0 -50 10 20 30 40 Time (ms) Figure 6.50 Output from the 1-D heating+deposition model calculated for the 400 kA F=-0.2 (standard) dataset. Output is plotted both with and without neutrals, and with reduced neutrals. Data is also plotted for the ensembled neutrals data and the single shot neutrals data shown in Fig. (6.51). In the top plot of Fig. (6.51), the dramatic effect of each sawtooth event on fast ion density is clear, though the NBI quickly repopulates the core with fast ions in each case. The global impact of neutralization effects are compared to the no neutrals case in Fig. (6.52). In the no neutrals case (the top plot in Fig. (6.52)), it is clear that without non-classical particle losses, the fast ions will sit in the plasma core for a very long time, as was seen in the PPCD case as well. The classical slowing down times are in this case are 20-30 ms in the core, which is too slow to significantly deplete the fast ion population until after the NBI turns off. This large population of fast ions produces the heating seen in the red curve in Fig. (6.50). 175 400 kA Standard Data, July 14, 2010 1.5 ) -3 m 18 1.0 NBI On (x 10 fi 0.5 Core n Ensembled Data Shot 1100714040 0 2.0 ) -3 1.5 m 16 Ensembled Data 1.0 Shot 1100714040 (x 10 o 0.5 Core n 0 15 10 5 Magnetic Fluctuations (G) 0 10 20 30 40 Time (ms) Figure 6.51 Predicted core fast ion and neutral densities for the 400 kA, F=-0.2 (standard) dataset with full NENE` charge-exchange effects included. The ensembled dataset is plotted in black while the single-shot dataset is plotted in blue. In the bottom plot, magnetic fluctuations ˜2 ( ( n=7,12 bn)/6) are plotted for the single shot. q P 176 nfi, 400 kA Standard, Neutrals Off 40 1.4 NBI Off 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 ) 15 NBI On 0.4 10 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) nfi, 400 kA Standard, Ensembled Neutrals 40 1.4 NBI Off 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 ) 15 NBI On 0.4 10 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) nfi, 400 kA Standard, Single Shot Neutrals 40 1.4 NBI Off 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 ) 15 NBI On 0.4 10 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.52 1-D heating+deposition model fast ion density profile results for the 400 kA F=-0.2 (standard) dataset. The top plot has no neutrals, the middle plot has ensembled NENE` output and the bottom plot has single shot NENE` output. 177 It is clear that neutralization losses with full NENE` are a much better model of the auxiliary heating data outside of PPCD periods, though not a perfect fit. Even with large sawtooth events in the single shot full NENE` model, the core fast ion population is relatively close to the no neutrals case whenever there is a long period between sawtooth events, such as around t = 17-19 ms, and t = 32-35 ms in Fig. (6.51). These periods produce electron heating that takes enough time to diffuse away as to cause significant Te in the core. It is possible that there are other non-classical loss mechanisms for fast ions. There are perhaps stochastic losses of fast ions during sawtooth events, though that is not modeled in this dissertation. Another possibility is resonant fast ion transport, as was described in Sec. (3.5.4). Modeling of that process is explored in Sec. (6.10). 6.10 Modeling Resonant Fast Ion Transport Resonant fast ion transport has been observed in MST plasmas with NBI (Sec. (3.5.4)). In Fig. (3.23), it is seen that fast ion driven instabilities appear 3.5-4 ms into the NBI pulse in even low density PPCD plasmas, where NBI shine-thru is highest. These instabilities drive non-classical transport of fast ions from the core of the plasma to the mid-radius, flattening the fast ion profile and effectively putting a ceiling on core fast ion density. While the precise physical phenomenon that drives this transport is not yet understood, its effect can be simulated. To simulate this transport, a resonant diffusion coefficient (DRES) was assumed, and this diffu- sion is turned on in the 1-D heating+deposition model whenever core fast ion density crosses a set threshhold. This resonant diffusion is only turned on for 15 µs at a time (the order of magnitude of time that fast ion driven modes tend to last), and is only at full strength for r 10 cm. The value of DRES then decreases linearly in strength until it zeroes out at r =20 cm. This resonant diffusion was simulated for the 200 kA low density PPCD ensemble. In this case, 2 18 3 D =50m /s and the resonant diffusion was turned on whenever n (r =0)> 1 10 m , RES fi ⇥ a value chosen so that the instabilities first appear 3-4 ms into the NBI pulse. The fast ion density output is plotted in Figs. (6.53) and (6.54). 178 Fast Ion Density Profiles, t=16.5 ms Core Fast Ion Density, 200 kA PPCD 2 2 ) NBI On No Resonant Diff 18 2 ) DRES = 50 m /s 18 No Resonant Diff Reduced NENE D = 50 m2/s D + Reduced NENE 1.5 RES 1.5 RES NENE DRES + NENE 1.0 1.0 0.5 0.5 Fast Ion Density (x 10 Core Fast Ion Density (x 10 0 0 0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40 Radius (m) Time (ms) Figure 6.53 Output from the 1-D heat- Figure 6.54 Core fast ion density is plotted ver- ing+deposition model with resonant diffusion for sus time for the 200 kA low density PPCD case 200 kA low density PPCD. Here, DRES = with resonant diffusion and various assumptions 50m2/s and the resonant diffusion turns on for on charge-exchange losses. 18 3 n > 1 10 m . fi ⇥ In Fig. (6.53), the fast ion density profile is plotted at the first discrete time point after the NBI turns off, for the classical case and the resonant diffusion case. As expected, the core fast ion 18 3 density is limited to n =1 10 m , and the gap between the classical and resonant diffusion fi ⇥ case goes to zero at around r =10cm. In Fig. (6.54), core fast ion density is plotted versus time both with and without resonant dif- fusion for cases without neutral losses, with “reduced” NENE` and full NENE.` It can be seen that 18 3 by setting a maximum core fast ion density of 1 10 m , the resonant diffusion kicks on right ⇥ at around 3.5 ms into the NBI pulse, which is as designed. After the NBI turns off, there are no longer any resonant diffusion events, which is again consistent with observations. The effect of resonant diffusion on core electron temperature can be seen in Fig. (6.55), where core Te is plotted for the same simulation scenarios plotted in Fig. (6.54). It can be seen that the effect of the resonant diffusion is a very narrow downward pressure on core Te, by an amount well within the margin of error. 179 In Fig. (6.56), the question of whether the value of DRES matters significantly is tested. The 2 conclusion is that it does not. For values of DRES =10, 50 and 100 m /s, the simulated core Te does not change significantly. This diffusion can be compared to classical fast ion diffusion, where D is typically on the order of 1 m2/s. ? 200 kA PPCD Simulation, Varying Neutrals & DRES 200 kA PPCD Data vs Simulation, Varying DRES 200 200 Thomson Data Thomson Data NBI On No Resonant Diff NBI On TRANSP/NUBEAM 2 DRES = 50 m /s No Resonant Diff 2 Reduced NENE DRES = 100 m /s D + Reduced NENE D = 50 m2/s 150 RES 150 RES PPCD PPCD 2 NENE DRES = 10 m /s DRES + NENE DRES + Reduced NENE (eV) (eV) DRES + NENE core core e, 100 e, 100 T T 50 50 0 0 0 10 20 30 40 0 10 20 30 40 Time (ms) Time (ms) Figure 6.55 Simulated core Te with and with- Figure 6.56 Simulated core Te for various val- out resonant diffusion, both with and without ues of DRES. It is assumed that there are no neu- neutral losses. tral losses outside of the blue and orange lines, 2 both of which use DRES = 50 m/s . In this modeling technique, resonant diffusion is only used to set a ceiling on core nfi. As long as the value of DRES is large enough to diffuse fast ions out of the core fast enough to keep nfi(r =0)at the set ceiling, the only significant impact of increasing DRES is to increase the length of time between resonant transport events. As that ceiling is lowered, the value of DRES must be increased to keep the time between resonant transport events constant, and to keep core fast ion density from bouncing around a lot near the saturated level. The overall effect of the resonant fast ion diffusion on fast ion profiles can be seen in Fig. (6.57). There, the top plot is fast ion density with “reduced” NENE,` while the bottom plot is fast ion 2 density with reduced NENE` and DRES = 50 m/s . The resonant diffusion effectively removes 18 3 areas with fast ion densities in excess of n =1 10 m from the plot. At the same time, there fi ⇥ is little effect for r>10 cm. 180 nfi, 200 kA PPCD, Reduced Neutrals 40 1.4 35 1.2 n 30 fi (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 NBI Off ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) nfi, 200 kA PPCD, Reduced Neutrals + DRES 40 1.4 35 1.2 n fi 30 (x 10 1.0 25 0.8 18 20 m Time (ms) Time 0.6 -3 NBI Off ) 15 0.4 10 NBI On 0.2 5 0 0.1 0.2 0.3 0.4 0.5 Radius (m) Figure 6.57 Simulated fast ion density profiles plotted with (bottom) and without (top) resonant fast ion diffusion. Both cases include “reduced” NENE.` 181 The fact that there are fewer fast ions in the core means that there is less heat deposition in the core. But at the same time, the flatter fast ion profiles reduces heat conduction by reducing T , r e which helps keep deposited energy in the plasma core. Those two effects cancel out for the 200 kA low density PPCD case, and leave Te relatively unchanged. The balance of these effects can be further explored in non-PPCD plasmas. In Figs. (6.58) and (6.59), simulations are run with both neutral losses and resonant diffusion losses for 400 kA standard reversal (F=-0.2) plasmas. The NENE` data used in these simulations are all single shot, rather than ensembled neutral density data. Change in Core Te, 400 kA Standard Plasma Core Fast Ion Density, 400 kA Standard 150 2.5 Thomson Data No Neutrals ) NBI On No Neutrals 18 NBI On Reduced no Reduced n 18 -3 o Reduced no, nfi_cap=1.0 x 10 m n 18 -3 o 2.0 no, nfi_cap=0.5 x 10 m 100 18 -3 Reduced no, nfi_cap=1.0 x 10 m 18 -3 Reduced no, nfi_cap=0.5 x 10 m n , n =0.5 x 1018 m-3 o fi_cap 1.5 (eV) 50 core e, 1.0 T 0 0.5 Core Fast Ion Density (x 10 -50 0 10 20 30 40 10 20 30 40 Time (ms) Time (ms) Figure 6.58 Output from the 1-D heat- Figure 6.59 Core fast ion density is plotted ver- ing+deposition model for 400 kA standard plas- sus time for 400 kA standard plasmas. A cou- mas. Both standard NENE` and “reduced” NENE` ple of different resonant diffusion cases from are included. Resonant diffusion is also included, Fig. (6.58) are included. with different “ceilings” set for the core fast ion density. In Fig. (6.58), a simulation with “reduced” NENE` but no resonant diffusion (purple) is almost perfectly on top of the classical result (red). In this case, as seen in Fig. (6.59), core fast ion 18 3 density peaks at close to 1.8 10 m . If resonant diffusion is turned on when core fast ion ⇥ 18 3 density passes 1.0 10 m (yellow) then core electron temperature actually increases a little ⇥ bit due to increased edge heating. This effect is amplified by pushing core fast ion density down 182 18 3 to 0.5 10 m (orange). That effect does not exist in the standard NENE` simulations, where ⇥ suppressing core fast ion density does not increase core electron temperature. This is an interesting contrasting result to the PPCD case, and might be further evidence that NENE` is relatively accurate for standard plasmas, despite over-estimating core neutral density in PPCD plasmas. Without sufficient neutralization losses, resonant fast ion diffusion serves to spread the fast ions out more evenly across the plasma, which suppresses temperature gradients and heat diffusion losses, thus driving up core electron temperature. With sufficient neutralization losses, however, resonant diffusion contributes to enough fast ion losses to prevent any increases in core electron temperature. The clear conclusion of these simulations is that there is a balance between classical and non- classical effects. NBI heating in PPCD plasmas can be sufficiently described with a purely classical model without neutralization losses. In non-PPCD plasmas, significant fast ion loss mechanisms are required. With sufficient neutral losses, simulations of core Te are relatively close to the measured data, both with and without resonant diffusion. If reasonable neutral density profiles can be assumed, then a complete 1-D model that includes classical effects, neutral losses and resonant diffusion losses is sufficient to explain the measured NBI heating results in MST across a wide range of plasma conditions. 183 Chapter 7 Conclusions And Future Work 7.1 Neutral Beam Heating Of MST A 1 MW NBI has been installed on MST that can use a fuel that is a mixture of deuterium and hydrogen. Typical fuel is 3-5% deuterium, though the beam will be partially composed of any gas pre-fill of MST, which is 100% deuterium for all of the plasmas studied in this dissertation. Most of the beam particles are deposited in MST, and most of those deposited particles are ionized near the core of the plasma. Neutron flux data demonstrate that fast deuterium ions can be generally treated as having classical dynamics. ANPA data suggest that a classical description is not the complete story for fast hydrogen ions. It appears as though there is a de facto ceiling on core fast ion density, and that when hydrogen fast ion density crosses that threshold in the core it drives instabilities in the plasma along with resonant transport of fast particles out of the core. The NBI suppresses core magnetic tearing fluctuations and also drives plasma momentum. It has not yet been seen to drive a significant change in total plasma current, however. The first statistically significant auxiliary heating of an RFP has been demonstrated. The 1 MW NBI has driven auxiliary heating in MST by approximately 100 eV relative to the NBI Off case in the core of low density PPCD plasmas. Heating has also been observed in higher current and higher density PPCD plasmas, though by a less significant amount. Statistically significant auxiliary heating has not yet been measured in non-reversed or standard reversal MST plasmas. It has further been observed that large fast ion populations can generate false interpretation of Rutherford scattering diagnostic data on MST. Previous results showing NBI heating of MST ions 184 in standard plasmas can be explained away entirely by subtracting this neutral hydrogen signal. It is also possible that the fast ions generated during sawtooth can pollute the Rutherford signal, and this possibility should be explored in the future. 7.2 1-D Auxiliary Heating Model A numerical model was developed to understand the physics underlying NBI heating of MST. A 1-D classical model was developed, and then additional complexities were added on to more accurately simulate fast ion dynamics and to understand more aspects of fast ion physics. The basic assumption underlying this 1-D model is that the heat diffusion coefficients do not change when the NBI is fired into MST. The argument for this is that these coefficients have only been shown to change when mid-radius magnetic fluctuations are suppressed, and there is no statistically significant suppression by NBI of any magnetic fluctuations aside from the core-most mode. Simulations demonstrate that even if the NBI decreases heat conduction coefficients then this effect will be small and within statistical error bars. The 1-D classical model includes classical slowing down and diffusion, and also solves for the initial deposition of fast ions by using numerical ionization cross-sections. This 1-D model is consistent with measured auxiliary heating in MST PPCD plasmas. Working with this classical model, it is clear that fast ion density profiles and fast ion diffusion are crucial. Deposition of energy in the core alone is not sufficient to generate the observed change in core Te. Suppression of heat diffusion is crucial, and this is achieved by having a relatively flat heat deposition, and also statistically significant heating of MST out near r/a =0.5. This mid-radius Te is also consistent with Thomson data. Neutralization losses and resonant fast ion diffusion, two additional fast ion loss processes, have been added to the 1-D model. The neutralization losses are calculated using the NENE` model, which fits D↵ data. There is reason to believe that this model over-estimates neutral density in the core, particularly in PPCD plasmas, so a “reduced” NENE` model was developed, which reduces neutral density in the core by a factor of 30. These “reduced” NENE` profiles appear to fit PPCD data well, while full NENE` profiles fit standard plasmas better. 185 Resonant fast ion diffusion has been added to the model to reproduce the observations of rapid non-classical transport events approximately every 0.5 ms once core fast ion density crosses some limit. In PPCD plasmas, the classical model alone reproduces the measured electron temperature change results. Fast ion loss effects are insignificant as long as “reduced” NENE` is used. In non-PPCD plasmas, the classical model alone is not sufficient to explain the Thomson data, which show no significant core Te. The simulated auxiliary heating can be mostly wiped out by including neutralization losses. Neutralization losses also prevent an unlikely result where resonant fast ion diffusion can actually increase Te if core fast ion density is held low enough. The conclusions from these results are that a pure classical model is sufficient to explain the notable PPCD auxiliary heating results. Non-PPCD plasmas can also be explained classically, as long as significant neutralization losses are included in the model. 7.3 Future Work There is a path for future research looking at NBI-induced plasma heating in both PPCD and non-PPCD plasmas. In PPCD plasmas, the assumption that heat diffusion coefficients are unaf- fected by the NBI should be looked at further. Data show that suppression of mid-radius magnetic modes is consistent with zero within the error bars, but it can’t be ignored that all of the mid-radius modes seem to show a very tiny amount of suppression (Fig. (6.7)). It is not inconceivable that there is some NBI-driven suppression of heat diffusion coefficients, which would increase auxiliary heating in PPCD plasmas beyond the physics modeled so far. For non-PPCD plasmas, a closer examination of core neutral density must be explored beyond NENE.` “Reduced” NENE` is a good fit for PPCD plasmas, but a more accurate model would be preferred. With accurate neutral density profiles and with a perfectly calibrated ANPA detector, it might be possible for an accurate measurement of core fast ion density in MST. 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But one key purpose of a doctoral dissertation is to aid the research of those that follow, and it’s important that future researchers know just how bolometers work, how they are calibrated, and how to understand the output from both the bolometers and XUV detectors. That all will be described in this appendix chapter. A.1 How Bolometers Work The pyrobolometer (or “bolometer”) was invented by Samuel Pierpont Langley in 1878 as a way to measure electromagnetic radiation. The bolometers used on MST, and on many other large plasma confinement devices, serve to measure total energy losses from the plasma, including both radiation and particle losses. The pyroelectric effect was actually discovered by the ancient Greeks more than 2000 years ago. The physical effect that they were noticing was that certain materials become electrically polarized when heated. In ferroelectric materials, this effect can be reversed and repeated. The crystals used for MST’s bolometers are lithium tantalate (LiTaO3), which are ferroelectric. For any pyroelectric material, the amount of electric polarization for a given temperature can be described by a pyroelectric coefficient (kpyro) defined as a function of P , the electric polarization density, and the material temperature (T ): P = kpyroT (A.1) 203 In any dielectric material, a displacement electric field (D) can be defined as a function of the electric field (E) and the polarization density: D = ✏0E + P (A.2) In Eq. (A.2), ✏0 is the standard permittivity of free space. The time rate of change of the electric displacement field is defined as the displacement current density (JD). This current density can be integrated along the width of the pyroelectric crystal ( ), and then P in Eq. (A.2) can be replaced by the right side of Eq. (A.1): @D J = (A.3) D @t @ @T J = ✏ Edx + k dx (A.4) 0 @t pyro @t ✓ Z ◆ ✓Z ◆ Defining Q as the total stored energy of the crystal and c as a heat coefficient Q = cT dx , we can write the temperature of the crystal as a function of the incident power: R dQ @ Power Power = Tdx = (A.5) dt ! @t c Z The electric field in Eq. (A.4) can be rewritten in terms of the electric current density (since V = Edx) and electrical resistivity R @ Edx = R @j , and then Eq. (A.5) can be put @t @t into Eq.R (A.4) to get the following differential equation:R @J Power J = ✏ + k (A.6) 0 @t pyro c This differential equation can be re-written and then solved: @J k Power = J + pyro (A.7) @t R✏0 ✏0Rc kpyroPower t J = + e R✏0 (A.8) ! c where is an arbitrary constant of integration. Noting that the exponential in Eq. (A.8) is of a t/RC type often seen in electric circuits, we can choose an appropriate resistivity in order to cause 204 that term to drop out of the equation over short time scales. Then, noting that I = JA, where A is the surface area of the crystal, we can conclude that: k PowerA I pyro (A.9) bolometer ⇡ c Using a crystal diameter of 0.42 inches, a thickness of 0.5 mm, a pyroelectric coefficient of 1 2 1 1 0.023 µCK cm and a specific heat of 0.06 cal g K , we can calculate a sensitivity of 6.55 8 10 Amps/Watt. ⇥ A.2 Calibrating The Bolometer In order to calibrate a bolometer, an energy source is needed that fits two requirements. First, the exact amount of energy hitting the pyrocrystal must be measurable, and the source must turn on and off rapidly to satisfy the t/RC approximation used to get from Eq. (A.8) to Eq. (A.9). In our case, this was achieved by by building an electron gun, and putting the bolometer inside a faraday cup. A cartoon of this setup is drawn in Fig. (A.1). The electron gun consists of a filament from an automobile brake light. The filament was run with 12 V across it, as designed. It was placed behind an aperture that would be biased at a voltage described by a square wave varying at 60 Hz between 0 V and -200 V. When at -200 V all electrons were repelled and the aperture was “closed”. By having a sharp square wave that is varying these voltages allowed the electron gun to turn “on” and “off” 60 times per second. When the aperture was grounded it would allow the electrons to stream out, and thus be “open”. The entire electron gun setup was then biased at -1500 V. The bolometer was placed on the receiving end of the signal, as the back end of a faraday cup. The faraday cup consisted of a cylinder with only an aperture opening facing the electron gun, to control the solid angle of electrons able to reach the cup. This signal absorber could then be run in “bolometer mode” or “faraday cup mode”. In “bolometer mode”, the electrons hitting the bolometer would heat it, creating a bolometer current which was then measured as a voltage across a resistive load between the bolometer output and the rest of the faraday cup, which was grounded. 205 0 V 0 -200 V -200 -1500V Filament Grounded Aperture Electron Beam Negative Voltage Bias Voltage Negative Secondary Suppress To & Reflected Electrons -1500 eV 0 eV Electron Beam Current Bolometer Current Figure A.1 Cartoon of the calibration setup for the pyrobolometer. The energy source (electron gun) is at the top, while the pyrocrystal is at the bottom. There is an initial aperture which limits the solid angle of the electron beam that can reach the Faraday cup which is not drawn. 206 When in “faraday cup mode” the bolometer becomes the back of the faraday cup, which is lifted from ground. The faraday cup collects all electric current entering it, and this current is measured across a resistive load between the faraday cup and ground. Because of the gold coating on the face of the pyrocrystal (see Sec. (A.3) for description of this coating), the bolometer can be electrically shorted to the faraday cup in order to produce an unbroken conducting surface to measure the current of impacting electrons. The idea behind this setup is that the electrons entering the faraday cup can be measured in two ways. In “faraday cup mode” we measure the total current of electrons (i.e. the number of electrons-per-second hitting the crystal). In “bolometer mode” these electrons heat the bolometer crystal, and we know that the “power” that the bolometer is absorbing is: Absorbed Power = Number of Electrons/second 1500 eV (A.10) ⇥ By alternating between the two modes we can measure the bolometer output signal and the power hitting the bolometer. Dividing the two gives us a measure of Amps/Watt that we can compare to the theoretical value calculated in Sec. (A.1). The key to making this measurement accurate is to make sure that no electrons are absorbed by the faraday cup that are not absorbed by the pyrocrystal, and vice versa. This is achieved with three apertures. The first aperture, as mentioned prior, physically limits the solid angle of the electron beam that is able to get into the faraday cup. This aperture is small enough that any electrons which get into it will hit the pyrocrystal. There are two further apertures that are physically large enough to not block any electrons passing the first aperture from reaching the pyrocrystal, but which are electrically biased. These apertures are drawn in Fig. (A.1). The first aperture is grounded while the second aperture is biased to a negative voltage. The grounded aperture helps to focus the electrons and prevents outside ions from entering the faraday cup. The negatively biased aperture prevents secondary electrons from escaping. If any electrons hit the pyrocrystal and generate secondary electrons, they are not allowed to leave the faraday cup. To test the efficacy of these two biased apertures, the electrical bias of the un-grounded aperture was varied from +12 V to -78 V relative to the faraday cup. The data is plotted in Fig. (A.2). The 207 two colored curves represent two different data runs (described in the figure caption), but for both runs the x-axis refers to the voltage between the faraday cup and the aperture. The y-axis is the normalized faraday cup current. Faraday Cup Current Vs Aperture Bias Voltage 1.2 1.0 0.8 0.6 0.4 0.2 Normalized Faraday Cup Current 0 -20 0 20 40 60 80 Bias Voltage (V) Figure A.2 Data from the experiment that optimized the bias voltage on the faraday cup aperture nearest the bolometer. It can be seen that for large negative voltages the signal plateaus. The two colors represent two datasets. The aperture voltage could be varied between 0 and -60 V relative to machine ground. The two datasets consisted of biasing the faraday cup -12 V relative to ground (black) and +18 V relative to ground (red). As is clear in Fig. (A.2), the faraday cup current is very small without a biased aperture. Most of the signal is being lost via secondary electrons. As the aperture bias increases there is an exponential increase in the faraday cup current, which eventually asymptotes for bias voltages above -45 V. So we can be confident that any negative potential beyond -45 V will keep all ⇡ of the secondary electrons from escaping the faraday cup, and giving an accurate measure of the power incident on the pyrocrystal. This calibration was performed on nearly ten pyrocrystals, some of which are now on MST and some of which are not. The calibration results all ended up being very close to the theoret- ical values. The bolometers all had Amp/Watt values within 10% of the theoretical value. Most were within 5%. The uncertainty in the solid angle of signal reaching the bolometer, as well as the approximation that power losses from MST are not a function of radius (discussed further in 208 Sec. (A.3)) are more significant sources of error, so the theoretical pyroelectric coefficients can be used as the actual values for bolometer arrays on MST. A.3 Using Bolometers And XUV Detectors On MST The first set of bolometers on MST was coated with 1500 Angstroms of gold on both sides, with 25 Angstroms of chrome in between the gold and the crystal to help the two materials adhere to each other. The gold is there to make a consistent conducting surface across the face of the bolometer to prevent any electromagnetic noise from inside MST from polluting the bolometer signal. An experiment that I performed found that this was not sufficient, however, and those thicknesses were increased to 5000 Angstroms of gold and 100 Angstroms of chrome. The experiment performed was to cut a cylindrical piece of boron nitride approximately 1 cm thick that can fill the entire bolometer aperture. This prevents any heating of the bolometer crystal over the timescale of an MST shot, but will allow any electromagnetic noise to pass through just fine. With only 1500 Angstroms of gold, there was a non-zero signal which did not appear in bolometers coated with 5000 Angstroms of gold. Bolometers become porous to electromagnetic noise when the gold develops microtears (in general, too small to be seen by any microscopes in the MST lab). The thicker gold made this problem go away. The bolometers are paired with XUV detectors. An XUV detector, in general, is a photodiode that measures extreme ultraviolet radiation. We used AXUV detectors from International Radiation Detectors, Inc., which are widely used on scientific projects in plasma physics, space physics and many other applications. One key advantage of these AXUV detectors is that its sensitivity is very flat across a very broad spectrum of radiation. These detectors also have very quick time resolution ( 2µs for the AXUV-20G, the model that we chose) and are insensitive to neutral ⇡ particles. This latter characteristic means that a well-calibrated AXUV signal can be subtracted from a well-calibrated bolometer signal, with the difference in power detected being the neutral losses. In Fig. (A.3), the detector sensitivity of the AXUV-100G is plotted. This data comes from Bolvin et al. [1999], who installed these detectors on Alcator C-Mod. The AXUV-20G and AXUV-100G 209 Figure A.3 AXUV-100G detector sensitivity plotted versus photon energy. Data is from Bolvin et al. [1999]. have identical characteristics, and only vary in terms of the size of the sensitive area (20 mm2 vs 100 mm2). The “G” refers to the mounting, and does not impact the crystal sensitivity. When measuring radiation across a broad spectrum, it’s essential to have a broad, flat sensitivity. In this case, it’s fair to approximate the sensitivity of the crystal as 0.3 A/W across the entire range of typical MST radiation energies ( 50 eV to 10 keV). ⇡ ⇡ The bolometers and XUV detectors are then paired up and placed on the same MST ports. On MST there are four of these pairs currently installed, at two approximate toroidal locations. At each toroidal location there is a pair on (approximately) opposite poloidal angles, so that the pairs are looking at each other. Since output assumes that the plasma power output is constant throughout the plasma volume when the MST plasma in fact is shifted several centimeters out from center, an approximate measurement can be improved by averaging the signals from the “inboard” and “outboard” bolometer/XUV pair. A photo of this setup on MST is plotted in Fig. (A.4). The green gate valve can be seen, along with the attached roughing line. The locations of the bolometer and AXUV detector are noted with blue arrows, but they cannot be seen in the photo. They are hidden from view inside cajon fittings 210 and electromagnetic shielding. Note that the shielding is connected to MST ground by a grounding strap. Figure A.4 Photo of a bolometer/XUV pair on MST. The location of the bolometer and XUV detector are noted with arrows, though neither is actually visible. Both are sitting inside a cajon fitting, and both are covered on the outside by shielding to protect the detector and the attached BNC signal cable from electromagnetic noise. The solid angle seen by each detector can be calculated by knowing the distance between the edge of the plasma and gate valve. The bolometer does not have an additional aperture - the 5 cm diameter MST port facing the plasma acts as the aperture. Because the AXUV detector is much more sensitive to power deposition, it features two apertures to limit exposure. There is a 1/64” aperture 0.5 inches from the crystal, and then a 1/32” aperture an additional 3.5 inches closer to MST. Some quick geometry finds that the XUV crystal can be treated as being at a single point 1.67 inches closer to the plasma, with a solid angle defined by the 1/32” aperture that is 2.33 inches away. The geometry can be calculated, and assuming that the power lost from MST is equally dis- tributed and istropic, a multiplier can be produced. Bolometer data can be converted to power as 9.6 MW/Volt, and XUV data is 1.49 MW/Volt. 211 A.4 Bolometer/XUV Data Using the multiplier calculated in Sec. (A.3), data can be collected. Example data from a 450 kA standard MST shot are plotted in Fig. (A.5). The bolometer and XUV detector used for that data were at 300 toroidal and 135 poloidal. Fig. (A.6) shows data from that same shot, but zoomed in to a shorter period of time so that the impact of sawtooth crashes can be seen more clearly. The data in Figs. (A.5) and (A.6) include the plasma current and the bolometer/XUV signals, as well as the toroidal gap voltage (VTG) and the alpha model ohmic power (PAlpha). The alpha model [Antoni et al., 1986] calculates the equilibrium magnetic field and current profiles from edge data [Terry et al., 2004; Deng et al., 2008]. Eq. (A.11) shows the force-free solution for the magnetic field in MST: B = B (A.11) r ⇥ In the alpha model, it is assumed that: = (1 (r/a)↵) (A.12) 0 The parameters 0 and ↵ are then uniquely determined by the experimentally measured F and ⇥ [Anderson, 2001]. In this context, the alpha model’s calculated ohmic power is useful as a reasonable approximation of the total power consumption of MST. Assuming equilibrium temperatures and densities, the total particle and radiation power losses should equal the total power consumption. The total values of power losses in MegaWatts measured by the bolometers and XUV detectors will not equal the alpha model ohmic power exactly due to inevitable uncertanties. The size of the apertures have a margin for error, as do the exact absorption characteristics of the crystals. The assumption that power losses are uniform and isotropic, and that there is no scattering into the detectors from other locations, is another source of error. But in general, the bolometer and XUV generally track the alpha model calculation. In the zoomed plot (Fig. (A.6)), the impact of magnetic reconnection can be seen much more clearly. And again, the bolometer and XUV detector appear to approximately track with the alpha model. 212 450 kA, F=-0.2 450 kA, F=-0.2 500 500 400 400 300 300 200 200 Ip (kA) 100 Ip (kA) 100 0 0 80 80 60 60 40 40 (V) 20 (V) 20 TG TG V 0 V 0 -20 -20 50 50 40 40 30 30 20 20 Alpha (MW) 10 Alpha (MW) 10 P 0 P 0 10 10 8 8 6 6 4 4 XUV (MW) 2 XUV (MW) 2 0 0 50 50 40 40 30 30 20 20 10 10 0 0 Bolometer (MW) 0 10 20 30 40 Bolometer (MW) 4 6 8 10 12 14 16 Time (ms) Time (ms) Figure A.5 Example bolometer/XUV data from Figure A.6 Zoomed-in example bolometer/XUV a 450 kA standard MST shot. data from the same shot in Fig. (A.5). At these time scales the reconnection events are much eas- ier to see. 213 The impact of the NBI on the bolometer and XUV detector can be seen in Fig. (A.7) (a 300 kA standard reversal plasma) and Fig. (A.8) (a 450 kA PPCD plasma). Both plots include one NBI Off shot (black) and one NBI On shot (red), with the total beam inputed power plotted as well. In Fig. (A.7), the sawtooth crashes are not perfectly lined up in time (there’s no reason to expect them to be), but overall there isn’t a significant difference between the NBI On and NBI Off cases. There is less signal on the bolometer and XUV detector than in the 450 kA standard reversal case plotted in Fig. (A.5), as one would expect. The NBI On and NBI Off cases are also very similar in the PPCD case in Fig. (A.8). There is a big change in bolometer/XUV during the enhanced confinement period, of course, which can be defined as the period of suppressed fluctuations in toroidal gap voltage. Besides the lack of big spikes at sawtooth events, the equilibrium bolometer signal is significantly lower than either before or after the enhanced confinement period. This is not true in the XUV signal, which actually increases a little bit during the enhanced confinement period. The results in the previous paragraph are consistent with what we already know about enhanced confinement periods. The particle confinement times increase by a factor of five or ten, while radiation levels increase from the increased temperatures. The fact that the NBI does not significantly increase the bolometer or XUV signals is not too surprising. None of the detectors have an active view of the beam. And while the fast ion can be comparable to the background in the core of the plasma, it is closer to only 5-10% of the background over the entire plasma. Considering that these fast particles have significantly increased confinement times relative to background particles, we would expect fast ions to be a very small fraction of total MST power loss. 214 300 kA Standard, NBI On vs Off 400 kA PPCD, NBI On vs Off 400 500 400 300 300 200 NBI O! 200 NBI O! Ip (kA) 100 Ip (kA) NBI On 100 NBI On 0 0 1 1 0.8 0.8 0.6 0.6 (MW) 0.4 (MW) 0.4 BEAM 0.2 BEAM 0.2 P 0 P 0 30 100 20 50 (V) 10 (V) TG TG 0 V 0 V -10 -50 50 50 40 40 30 30 20 20 Alpha (MW) 10 Alpha (MW) 10 P 0 P 0 5 5 4 4 3 3 2 2 XUV (MW) 1 XUV (MW) 1 0 0 50 50 40 40 30 30 20 20 10 10 0 0 Bolometer (MW) 0 10 20 30 40 Bolometer (MW) 0 10 20 30 40 Time (ms) Time (ms) Figure A.7 Bolometer and XUV data for NBI On Figure A.8 Bolometer and XUV data for NBI On (red) and NBI Off (block) for a 300 kA standard (red) and NBI Off (block) for a 450 kA PPCD plasma. plasma.