Turbulence and Transport Measurements in Alcator C-Mod and Comparisons with Gyrokinetic Simulations by Paul Chappell Ennever B.S. Applied Physics (2009) Columbia University School of Engineering and Applied Science
Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of
Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2016 © Massachusetts Institute of Technology 2016. All rights reserved.
Author...... Department of Physics February 22nd, 2016
Certifiedby...... Miklos Porkolab Professor of Physics Thesis Supervisor
Acceptedby...... Nergis Mavalvala Professor of Physics Associate Department Head for Education
Turbulence and Transport Measurements in Alcator C-Mod and Comparisons with Gyrokinetic Simulations by Paul Chappell Ennever
Submitted to the Department of Physics on February 22nd, 2016, in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Abstract
Turbulence in tokamak plasmas is the primary means by which energy is transported from the core of the plasma to the edge, where it is lost, and is therefore the main limitation of tokamak plasma performance. Dilution of the main-ion species was found to have a stabilizing effect on ion gyroradius scale turbulence in tokamak plasmas. Dilution of deuterium tokamak plasmas is the reduction of the ratio of the deuterium ion density to the electron density, nD/ne, to less than 1.0 through the introduction of low-Z impurity species into the plasma. Controlled dilution experiments were performed on Alcator C-Mod wherein plasmas at a range of electron density and plasma current were seeded with nitrogen while a cryopump held the electron density fixed. The electron density fluctuations due to turbulence were monitored using a phase contrast imaging (PCI) diagnostic, an absolutely calibrated diagnostic that measures the line- integral of the electron density fluctuations along 32 vertical chords. In these experiments the seeding reduced the PCI density fluctuations, and had a stabilizing effect on the ion energy transport. The seeding also reversed the direction of intrinsic rotation in certain cases. Nonlinear simulations using the gyrokinetic turbulence code GYRO were performed using measured kinetic profiles from the dilution experiments both before and after the nitrogen seeding. The GYRO simulations reproduced the observed reduction in the turbulent ion energy transport with the nitrogen seeding. The GYRO simulated turbulent density fluctuations were compared to the PCI measurements using a synthetic diagnostic, and they were found to be consistent. GYRO simulations were also performed varying only the main ion dilution to explore the theoretical effects of the dilution on energy transport. Through this it was found that the dilution reduced the turbulent ion energy transport in a wide variety of cases, but primarily increased the critical gradient at low densities, and primarily reduced the stiffness of the transport at high densities. This dilution effect is related to observations of reductions in energy transport from seeding on other tokamaks, and will likely have an impact on ITER and future fusion reactors.
Thesis Supervisor: Miklos Porkolab Professor of Physics
3
ACKNOWLEDGMENTS
The work of this thesis could not have been completed alone, and there are many people whose contributions I would like to acknowl- edge. Firstly, I would like to thank my advisor, Professor Miklos Porkolab of MIT, for teaching me so much these past six years, and for supporting me and putting me in touch with others who could help me with things he could not. His help with editing the thesis as well was invaluable in making it a polished document. I would also like to thank the many people from different institutions and fields who helped to make the theoretical and experimental work in this thesis possible.
• Doctor Gary Staebler and Doctor Jeff Candy from General Atom- ics, for teaching me the GYRO and TGLF codes, as well as build- ing and maintaining them to be (relatively) simple to use for experimentalists, and answering questions I had about running the codes.
• Professor Naoto Tsujii currently at University of Tokyo, for teach- ing me how to operate the PCI system on C-Mod, analyze its data, and writing the tools necessary to run it even after he had gone.
• Doctor Eric Edlund from Princeton, for helping me with PCI during the 2014 and 2015 experimental campaigns, including testing and installing the new detector.
• Doctor J. Chris Rost of MIT, for maintaining the synthetic PCI diagnostic for GYRO, and for helping me out a lot with writing my first paper.
• Doctor Matt Reinke of Oak Ridge, for helping with the impurity analysis, as well as the ion temperature and velocity profile mea- surements, even under difficult experimental circumstances.
• All the diagnosticians of the Alcator C-Mod team, in particular Doctor Jerry Hughes for Thomson scattering, Doctor Amanda Hubbard for ECE, Doctor John Rice for HIREX, Doctor Seung Gyou Baek for Reflectometer, Doctor Jim Irby for TCI, and Doc- tor Steve Wolfe for magnetics.
• All the members of the core transport group on Alcator C-Mod, in particular Doctor Darin Ernst who helped me understand gyrokinetic theory and helped to maintain the LOKI cluster on which I worked, and Nathan Howard whose work on ETG modes helped me to understand my own results. . . .
5 In addition, I would like to extend a special thanks to my parents, Doctors John and Fanny Ennever, for setting me up for success from the very beginning, and supporting me throughout the many years of schooling that I’ve done. Finally, I would like to thank my fiance, Mei Wei Chen, for her unwavering love, support and patience. I couldn’t have made it this far without her, or her food.
6 CONTENTS
1 introduction 13 1.1 Fusion Energy 13 1.2 Tokamak Energy Transport 14 1.3 Alcator C-Mod 16 1.4 Thesis Outline 18
2 plasma turbulence 23 2.1 Linear Fluid Ion Temperature Gradient Modes 24 2.2 Gyrokinetics 26 2.3 Gyrokinetic Simulation Codes 28 2.4 The Flux-Gradient Relationship 31 2.5 Summary 33
3 phase contrast imaging 39 3.1 The Phase Contrast Imaging Technique 39 3.2 Phase Contrast Imaging on Alcator C-Mod 41 3.3 Absolute Calibration 46 3.4 Synthetic PCI Diagnostic For GYRO 47 3.5 Summary 49
4 ohmic dilution experiments 53 4.1 Motivations For Studying Dilution 53 4.2 Experimental Setup 54 4.3 Experimental Results 58 4.3.1 Effect of Nitrogen Seeding on Energy Transport 59 4.3.2 Effect of Nitrogen Seeding on Density Fluctua- tions 63 4.3.3 Effect of Nitrogen Seeding on Toroidal Rota- tion 67 4.4 Summary and Conclusions 70
5 gyro validation results 77 5.1 Linear GYRO Simulations Of Turbulent Growth Rates 78 5.2 Nonlinear Local GYRO Simulations Of Turbulence 80 5.2.1 Local GYRO Simulations at r/a = 0.8 82 5.2.2 Local GYRO Simulations at r/a = 0.6 84 5.3 TGYRO Profile Modification Using TGLF 88 5.3.1 Simulations With The Nominal Experimental Pro- files 88 5.3.2 Simulations With The TGYRO Flux-Matched Pro- files 89 5.3.3 Simulations With The Average Of TGYRO And Experimental Profiles 90
7 8 contents
5.4 Global Nonlinear GYRO Simulations Of Turbulence 92 5.5 Density Fluctuation Comparisons Between GYRO Sim- ulations And PCI Measurements 95 5.6 Summary And Conclusions 99
6 theoretical study of dilution effect with gyro 105 6.1 Effect Of Dilution On GYRO Linear Growth Rates 105 6.2 Effect of Dilution On GYRO Energy Fluxes 107 6.3 Quanitfying The Effect Of Dilution On GYRO Stiffness And Critical Gradient 112 6.4 Summary and Conclusions 115
7 concusions and future work 119 7.1 Summary of This Thesis Work 119 7.2 Conclusions and Implications 121 7.3 Future Work 122
I Appendix 127
a absolute impurity density measurements 129 a.1 Determination Of Impurity Densities From Zeff And Line Brightnesses 129 a.2 Results Of Impurity Analysis 132
b other factors considered in gyro 135 b.1 Inclusion Of ~E × B~ Shear 135 b.2 Inclusion Of Multiple Impurity Ion Species 136 LISTOFFIGURES
Figure 1 Fusion Reaction Rates 14 Figure 2 Tokamak Schematic 15 Figure 3 The LOC And SOC Regimes 16 Figure 4 GYRO 3-D Potential Fluctuations 30 Figure 5 Relationship Between Energy Flux And Tem- perature Gradient 31 Figure 6 GENE Simulations From A JET Discharge Show- ing Stiffness And Critical Gradient 32 Figure 7 PCI Schematic 41 Figure 8 Alcator C-Mod PCI System 42 Figure 9 Alcator C-Mod PCI Beam Path 43 Figure 10 Frequency Response of PCI Detectors 45 Figure 11 Example PCI Calibration Plots 47 Figure 12 Plot Of GYRO Density Fluctuations 48 Figure 13 Plot Of Synthetic PCI Spectra With And With- out ~E × B~ Drifts 49 Figure 14 Magnetic Equilibrium 55 Figure 15 Experimental Traces 56 Figure 16 Experimental Energy Confinement Times 57 Figure 17 Experimental nD/neValues 58 Figure 18 Experimental Energy Confinement Times Ver- sus neq95 60 Figure 19 Gyrobohm Normalized Flux Profiles 61 Figure 20 Example Change in Qi/QGB and a/LTi 61 Figure 21 Changes With Seeding Fluxes And Gradients At r/a = 0.6 62 Figure 22 Changes With Seeding Of Fluxes And Gradi- ents At r/a = 0.8 63 Figure 23 Experimental PCI Spectra 64 Figure 24 Change In High Phase Velocity PCI Feature 64 Figure 25 PCI Reflectometer Time Series Comparison 66 Figure 26 Toroidal Rotation Profiles 68 Figure 27 Toroidal Rotation Versus Effective Collisional- ity 69 Figure 28 Toroidal Rotation Versus nDq95 70 Figure 29 GYRO Linear Growth Rate Scan At r/a = 0.6 80 Figure 30 GYRO Linear Growth Rate Scan At r/a = 0.8 81 Figure 31 Nonlinear GYRO Simulation Example 82 Figure 32 Quantitative Comparison of GYRO and Exper- imental Energy Fluxes at r/a = 0.8 83 Figure 33 GYRO Ion Temperature Gradient Scans For Low- Denisty 0.8 MA Cases at r/a = 0.8 84
9 10 List of Figures
Figure 34 GYRO Ion Temperature Gradient Scans For High- Denisty 0.8 MA Cases at r/a = 0.8 85 Figure 35 GYRO Ion Temperature Gradient Scans For Low- Denisty 1.0 MA Cases at r/a = 0.8 85 Figure 36 Quantitative Comparison of GYRO and Exper- imental Energy Fluxes at r/a = 0.6 87 Figure 37 GYRO Ion Temperature Gradient Scans For LOC 0.8 MA Cases at r/a = 0.6 87 Figure 38 Local GYRO, TGLF, and Experimental Fluxes Using Experimental Profiles 89 Figure 39 Local GYRO, TGLF, and Experimental Fluxes Using TGYRO Profiles 90 Figure 40 Local GYRO, TGLF, and Experimental Fluxes Using Average Profiles 91 Figure 41 Ion Temperature Gradient Profiles Around r/a = 0.6 For Seeded 1.0 MA SOC Case 92 Figure 42 Comparison Of Global GYRO, Local GYRO, and Experimental Energy Fluxes Near r/a = 0.6 94 Figure 43 Linear GYRO Growth Rates Of ETG Modes 95 Figure 44 Comparison Of Synthetic PCI Computed From Global GYRO to Experimental PCI For An Un- seeded 1.0 MA SOC Plasma 96 Figure 45 Comparison Of Synthetic PCI Computed From Global GYRO to Experimental PCI For A Seeded 1.0 MA SOC Plasma 97 Figure 46 Background Subtraction To Isolate High Phase- Velocity Feature 98 Figure 47 Comparison Of Experimental And Synthetic PCI Frequency-Wavenumber Spectra 100 Figure 48 Quantitative Comparison Of Experimental And Synthetic PCI Wavenumber Spectra 100 Figure 49 GYRO Linear Growth Rate Dilution Scan For 0.8 MA LOC 106 Figure 50 GYRO Linear Growth Rate Dilution Scan For 1.0 MA SOC 107 Figure 51 Linear ETG Growth Rate Spectra With And With- out Dilution 108 Figure 52 GYRO ETG Linear Growth Rate Dilution Scan For 1.0 MA SOC At r/a = 0.6 108 Figure 53 GYRO Ion Temperature Gradient Scans For 0.8 MA LOC Case At r/a = 0.6 With And Without Dilution 109 Figure 54 GYRO Ion Temperature Gradient Scans For 0.8 MA LOC Case At r/a = 0.8 With And Without Dilution 110 Figure 55 GYRO Ion Temperature Gradient Scans For 1.0 MA SOC Case At r/a = 0.6 With And Without Dilution 111 Figure 56 GYRO Ion Temperature Gradient Scans For 1.0 MA SOC Case At r/a = 0.6 With And Without Dilution 112 Figure 57 Global GYRO Simulations For 1.0 MA SOC Case From r/a = 0.3 To r/a = 0.7 113 Figure 58 Example nD/ne And a/LTi Scans From 0.8 MA LOC Case 114 Figure 59 Quantification Of The Effect Of Dilution On Stiffness And Critical Gradient 115 Figure 60 Comparison of Zeff From Impurity Densities and Neoclassical Conductivity 131 Figure 61 Comparison Of Local GYRO With And With- out ~E × B~ Shear At r/a = 0.6 135 Figure 62 Comparison Of Local GYRO With And With- out ~E × B~ Shear At r/a = 0.8 136 Figure 63 Effect of Mulitple Ion Species 137
LISTOFTABLES
Table 1 Alcator C-Mod Parameters 17 Table 2 Summary Of Turbulent Modes 27 Table 3 Shots For Impurity Analysis 131 Table 4 Impurity Summary 132
11
INTRODUCTION 1
1.1 fusion energy
Nuclear fusion offers the possibility of unlimited clean energy for the entire world. The fusing of light nuclei into heavier nuclei pow- ers every star in the universe, and the promise of fusion energy on Earth has prompted decades of research. The fusion reaction with the largest cross-section, and therefore the one with the highest reaction rate is the deuterium-tritium reaction or D-T reaction:
D + T → He4(3.5 MeV) + n(14.1 MeV) (1a)
Figure 1 shows the cross section versus temperatures of the most commonly used fusion reactions: D-D(deuterium-deuterium), D-T, D- He3 (deuterium-helium 3), p-Li6 (proton-lithium 6), and p-B11 (proton- boron 11). It’s clear that the D-T reaction has the largest cross-section, but it peaks at an interaction energy of around 100 keV, which corre- sponds to a temperature of approximately 1 billion Kelvin. Creating and sustaining a plasma at such a high temperature is an extremely difficult scientific and engineering challenge. In addition to the very high temperatures required, it was determined early on by Lawson that the necessary conditions for a fusion reaction to produce net 21 3 energy is nTτE > 3 × 10 keVs/m [8] (the so-called "fusion triple product") where n is the plasma density, T is the plasma tempera- ture, and τE is the energy confinement time (the ratio of the stored energy to the input power). One of the best means of achieving this is to confine the plasma in a toroidal magnetic field. Currently, the most successful confinement scheme for fusion plas- mas is the tokamak [11]. The name tokamak comes from a russian acronym that translates to "toroidal chamber with magnetic coils". A schematic of the typical tokamak design is shown in Fig. 2. It con- sists primarily of a set of toroidal magnetic field coils and a central solenoid to induce a toroidal current in the plasma which will re- sult in a poloidal magnetic field. The combination of toroidal and poloidal magnetic field causes the field lines to transit the torus in both the toroidal and poloidal directions. The ratio of the number of poloidal transits to toroidal transits of a field line is referred to as the rotational transform. The rotational transform is necessary because, while the charged particles in a plasma in a perfectly straight uni- form magnetic field will remain confined perpendicular to the field, the curvature and nonuniform strength of the magnetic field in a toroidal geometry causes particles to drift outward. These drift or-
13 14 introduction
Figure 1: A plot of the fusion reaction rates versus temperature for D-D, D-T, D-He3, p-Li6, and p-B11 in Figure from Chapter 1 of Ref. [7]
2 −→ −→ mv⊥ B × ∇ B bits are given by the following: v∇B = 2qB B2 is the drift due to the nonuniform strength of the magnetic field, where m is the particle mass, v⊥ is the particle velocity perpendicular to the magnetic field, q 2 −→ −→−→ −→ mvk B · ∇ B × B is the particle charge, and B is the magnetic field; vκ = qB B3 is the drift due to the curvature of the magnetic field, where vk is the particle velocity parallel to the magnetic field. If the magnetic field was purely toroidal, the plasma would almost immediately be lost due to these drift orbits. With the rotational trans- form, these drift orbits are averaged out as particles will drift alter- nately inward and outward in minor radius as they go around the torus poloidally and toroidally. The innovation of the tokamak was to generate this rotational transform through the use of a current in the plasma, rather than through twisted external coils. This design allowed for almost complete confinement of the drift orbits and en- abled tokamaks to have by far the largest fusion triple product of any magnetic confinement configuration to date. •
1.2 tokamak energy transport
The energy confinement time of a plasma is defined as follows: τE = dW W/(PH − dt ), where W is the total plasma stored energy, PH is the 1.2 tokamak energy transport 15
Figure 2: A schematic diagram of a tokamak. From Max-Planck Institut fur Plasmaphysik.
dW total heating power applied to the plasma. In steady state dt = 0, so τE becomes just the ratio of the total plasma stored energy to the heating power. Also if the plasma is in steady state, PH is equivalent to the power that is conducted through the plasma and lost at the edge. Thus, if the energy transport can be reduced without reducing the plasma stored energy, τE will be increased. Increasing τE while maintaining high plasma temperature and density has been a key component of the world fusion program for decades. Early observations of the energy confinement time in ohmically heated tokamak plasmas (i.e. plasmas heated only by the induced current) found two characteristic confinement regimes depending on the value of the product of the electron density ne and edge safety factor q95, which is the inverse of the rotational transform and is ap- proximately equal to (r/R)(BT /Bp) where R is the major radius, r is the minor radius, BT is the toroidal magnetic field, and Bp is the poloidal magnetic field, all of which are evaluated at the edge. At low values of neq95, the energy confinement increases linearly with neq95 and this regime is referred to as the linear ohmic confinement or LOC regime. Above a critical value of neq95, the confinement saturates and is independent of neq95. This regime is referred to as the saturated ohmic confinement or SOC regime. These two regimes are illustrated in Fig. 3, which shows energy confinement times and neq95 values from Alcator C-Mod ohmic plasmas. Many other plasma physics pa- rameters vary when electron density, magnetic field, and plasma cur- rent are varied, including the electron and ion temperatures, Te and Ti, their ratio, Te/Ti, the effective charge Zeff, and ni/ne. All of these parameters influence the energy transport due to their influence on the magnitude of turbulent fluctuations. 16 introduction
50 SOC 45
40 LOC 35 (ms) e τ 30 5.2 T; 0.8 MA 25 5.2 T; 0.6 MA 5.2 T; 0.4 MA 20 0 1 2 3 4 5 6 20 -3 q95ne (10 m )
Figure 3: Plots of the energy confinement time τE versus the product of the electron density and edge safety factor neq95 from Alcator C-Mod ohmic plasmas at a range of densities and plasma currents. The dashed lines indicate the linear ohmic confinement (LOC) and saturated ohmic confinement (SOC) regimes. The different colors indicate different plasma currents, which will have different values of q95.
Collisional energy transport is insufficient to explain the level of energy transport observed in tokamak plasmas[6]. This so-called neo- classical transport includes the transport due to binary-particle colli- sions in toroidal magnetic geometry [4]. Neoclassical effects influence many areas of tokamak dynamics such as rotation and current drive, but the energy transport driven by neoclassical effects is much smaller than the observed energy transport in tokamaks. The observed en- ergy transport in tokamak plasmas that is above the neoclassically predicted levels was referred to as "anomalous transport", and has since been shown to be caused by turbulence[3]. Plasma turbulence are fluctuations in density, temperature, and potential that are driven unstable through drift orbits and plasma pressure gradients[11]. This so-called drift-wave turbulence is responsible for the vast majority of the cross-field energy transport in tokamak plasmas. A more detailed discussion of turbulence in tokamaks will be covered in Chapter 2. •
1.3 alcatorc-mod
The experiments done as part of this thesis work were performed on the Alcator C-Mod tokamak[5]. C-Mod is a compact, high-field, diverted, metal-walled tokamak that has operated at MIT’s Plasma Science and Fusion Center since 1993. The nominal parameters for C-Mod are shown in Table 1. Thanks to its small size and copper magnetic field coils, Alcator C-Mod has the highest magnetic field, 1.3 alcator c-mod 17
Parameter Value Major Radius 0.67 m Minor Radius 0.22 m Magnetic Field 5.4 T Plasma Current 1.0 MA
q95 4 Electron Density 1020m−3 Electron Temperature 5 keV Main Ion Deuterium Wall Material Molybdenum Pulse Length 2 s
Table 1: Table of typical Alcator C-Mod parameters. electron densities, and plasma pressures of any tokamak currently operating. The copper coils are cooled cryogenically to liquid nitro- gen temperatures, and the time it takes to re-cool the magnets be- tween shots (typically 10 minutes) is the limiting factor in the num- ber of shots in a day. The plasmas in Alcator C-Mod are heated by ohmic current and ion cyclotron range of frequency (ICRF) heating. The ohmic heating comes from the plasma current and the plasma’s finite resistivity. In Alcator C-Mod, the typical current is 1 MA and the typical resistivity is about 1 Ω, giving a typical ohmic heating power of 1 MW. The ICRF heating uses radio waves injected at the cyclotron frequency of a trace impurity ion species [10], either hydro- gen or helium-3 on Alcator C-Mod. The wave energy is absorbed on the impurity ions, and the impurity ions then heat the electrons and main ions through collisions. Alcator C-Mod has available up to 6 MW of ICRF heating power [2]. These heating methods provide neg- ligible input momentum to the plasma, so the rotation of the plasma in Alcator C-Mod is purely due to the intrinsic rotation. This is similar to the case of fusing reactor plasmas, which will have negligible mo- mentum input compared to their toroidal moment of inertia due to a very large size and density. The high magnetic field, plasma pressure, metal walls, and lack of external momentum input makes Alcator C-Mod the most reactor-relevant tokamak currently in operation. Alcator C-Mod has a broad program that covers areas of research important to fusion energy [9]. This program is carried out by a team of scientists, engineers, and graduate students who run the multitude of diagnostics and control systems[1]. These diagnostics monitor the plasma density, temperature, rotation, and impurity ions, as well as fluctuations in density, temperature, and magnetic field. This allows for a nearly complete accounting of the state of the plasma, which is critical for comparing experimental measurements with theory-based 18 introduction
simulations. The combination of an extensive diagnostic suite and reactor-relevant conditions makes Alcator C-Mod an ideal candidate for the validation work presented in this thesis. •
1.4 thesis outline
The remainder of the thesis is as follows: Chapter 2 describes an introduction to plasma turbulence theory, as well as the gyrokinetic codes that solve it. It covers linear turbulence theory. It then details the simulation codes GYRO, TGYRO, and TGLF which solve these nonlinear equations to determine the magnitude of the equilibrium turbulent fluctuations of the distribution functions, for a given set of input temperature and density profiles. Chapter 3 describes the phase contrast imaging (PCI) diagnostic, which is used to measure line-integrated density fluctuations in plas- mas. It covers the mechanism by which PCI images density fluctu- ations, and the theoretical response of a PCI diagnostic. It also de- scribes the phase contrast imaging system on Alcator C-Mod, which measures line-integrated density fluctuations using a 60W CO2 laser and a 32 channel cryogenically cooled photoconductive HgCdTe de- tector. It also describes the synthetic diagnostic technique used for comparing PCI measurements to GYRO simulated turbulent density fluctuations. Chapter 4 describes a series of experiments performed on Alcator C-Mod in which ohmic plasmas were seeded with nitrogen to dilute them, meaning to reduce the ratio of the deuterium and electron den- sities in the plasma, nD/ne, to less than 1. It describes the motivation for studying dilution and the experimental design. It then goes on to describe the observed effects of the nitrogen seeding on plasma energy confinement, transport, fluctuations, and rotation. Chapter 5 describes quantitative comparisons between the predicted turbulence from the gyrokinetic simulation codes GYRO and TGLF, and the measured turbulence and energy transport for both nitrogen seeded and unseeded cases in LOC and SOC regimes. Comparisons were performed both at normalized minor radius r/a = 0.8 where the experimental turbulence is well above marginal stability and at normalized minor radius r/a = 0.6 where the turbulence is close to marginal stability. Chapter 6 describes work done using GYRO to investigate the pre- dicted theoretical effects of varying nD/ne on turbulent growth rates, energy transport, and fluctuations. This allowed for isolating the ef- fects of dilution separate from the other changes caused by the nitro- gen seeding in the experiments discussed in Chapter 4. 1.4 thesis outline 19
Chapter 7 summarizes the thesis work and presents the conclu- sions. It also describes future work that could be done to extend the findings of this thesis. ?
BIBLIOGRAPHY
[1] N P Basse, A Dominguez, E M Edlund, C L Fiore, R S Granetz, A E Hubbard, J W Hughes, I H Hutchinson, J H Irby, B LaBom- bard, L Lin, Y Lin, B Lipschultz, J E Liptac, E S Marmar, D A Mossessian, R R Parker, M Porkolab, J E Rice, J A Snipes, V Tang, J L Terry, S M Wolfe, S J Wukitch, K Zhurovich, R V Bravenec, P E Phillips, W L Rowan, G I Kramer, G Schilling, S D Scott, and S J Zweben. Diagnostic systems on Alca- tor C-Mod. Fusion Science and Technology, 51(3):476–507, 2007. URL:
21 22 Bibliography
the Alcator C-Mod program. Nucl. Fusion, 45(10):S109, 2005. URL: http://stacks.iop.org/0029-5515/45/i=10/a=S09, doi: 10.1088/0029-5515/45/10/S09. 16
[6] W. Horton. Drift waves and transport. Reviews of Modern Physics, 71(3):735–778, apr 1999. URL: http://link.aps.org/doi/10. 1103/RevModPhys.71.735, doi:10.1103/RevModPhys.71.735. 16
[7] Mitsuru Kikuchi, Karl Lackner, and Minh Quang. Fusion Physics. International Atomic Energy Agency, 2012. 14
[8] J D Lawson. Some Criteria for a Useful Thermonuclear Reactor. Technical report, Atomic Energy Research Establishment, Har- well, Berkshire, U. K, 1955. 13
[9] E. Marmar, a. Bader, M. Bakhtiari, H. Barnard, W. Beck, I. Be- spamyatnov, a. Binus, P. Bonoli, B. Bose, M. Bitter, I. Cziegler, G. Dekow, a. Dominguez, B. Duval, E. Edlund, D. Ernst, M. Fer- rara, C. Fiore, T. Fredian, a. Graf, R. Granetz, M. Greenwald, O. Grulke, D. Gwinn, S. Harrison, R. Harvey, T.C. Hender, J. Hosea, K. Hill, N. Howard, D.F. Howell, a. Hubbard, J.W. Hughes, I. Hutchinson, a. Ince-Cushman, J. Irby, V. Izzo, a. Kano- jia, C. Kessel, J.S. Ko, P. Koert, B. LaBombard, C. Lau, L. Lin, Y. Lin, B. Lipschultz, J. Liptac, Y. Ma, K. Marr, M. May, R. McDer- mott, O. Meneghini, D. Mikkelsen, R. Ochoukov, R. Parker, C.K. Phillips, P. Phillips, Y. Podpaly, M. Porkolab, M. Reinke, J. Rice, W. Rowan, S. Scott, a. Schmidt, J. Sears, S. Shiraiwa, a. Sips, N. Smick, J. Snipes, J. Stillerman, Y. Takase, D. Terry, J. Terry, N. Tsujii, E. Valeo, R. Vieira, G. Wallace, D. Whyte, J.R. Wilson, S. Wolfe, G. Wright, J. Wright, S. Wukitch, G. Wurden, P. Xu, K. Zhurovich, J. Zaks, and S. Zweben. Overview of the Alcator C-Mod Research Program. Nuclear Fusion, 49(10):104014, 2009. doi:10.1088/0029-5515/49/10/104014. 17
[10] Thomas Howard Stix. Theory of Plasma Waves. McGraw-Hill Book Co., Inc., 1992. URL: http://books.google.com.hk/books?id= OsOWJ8iHpmMC. 17
[11] J Wesson and D J Campbell. Tokamaks. Oxford Engineering Sci- ence Series. Clarendon Press, 1997. URL: https://books.google. com/books?id=HCBRAAAAMAAJ. 13, 16 PLASMATURBULENCE 2
Turbulent fluctuations have long been known to be responsible for the observed anomalous energy transport in tokamak plasmas[18]. The energy flux Q driven by fluctuating fields for species j is given by[23]:
3 hE˜ θT˜ji 3 hE˜ θn˜ ji Qj = nj + Tj (2a) 2 Bφ 2 Bφ
where nj is the equilibrium species density, E˜ θ is the fluctuating poloidal electric field, T˜j is the fluctuating temperature, Bφ is the toroidal magnetic field, Tj is the equilibrium species temperature, and n˜ j is the fluctuating species density. The angle brackets refer to the time-average over time-scales much longer than the period of the fluc- tuations. This shows that it is not just the magnitude of the fluctuating quantities that are important, but the phase between the fluctuations as well. This means that in order to measure the turbulence-driven en- ergy flux experimentally, the temperature, density, and electric field fluctuations must be measured simultaneously. Such measurements are difficult in the core of tokamak plasmas, so the energy flux is usually inferred from power balance instead. This means that the measurements of fluctuations and energy fluxes can both be used to validate theoretical predictions of turbulence. The turbulent modes responsible for the anomalous energy trans- port in these plasmas are primarily drift waves[18], which are fluctu- ations with a phase velocity equal to the diamagnetic drift velocity of the particular species: vdj = (1/qjnjB)(∇pj) where the subscript j refers to the species (i.e. ions or electrons), qj is the species charge, and pj is the species pressure with pj = njTj . The modes form tur- bulent eddies with very long parallel wavelengths, and radial and poloidal wavenumbers on the order of 0.1 times the ion sound gyro- radius ρs. The fluctuations in the core density and temperature are on the order of 1% of the equilibrium values. Drift waves have been known to exist since the 1960s and work by Coppi[11], Porkolab[26], Horton[6], and others measured and char- acterized the modes in several geometries and including the effects of shear. Work by Coppi and others then extended this work to in- clude the effects of impurity ions[12] and trapped electrons[13][28]. These modes appear in all tokamaks, and are the primary limitation for energy confinement.
23 24 plasma turbulence
2.1 linear fluid ion temperature gradient modes
A magnetically confined plasma that is heated in the core and has a boundary that maintains a temperature low enough to avoid mate- rial damage, will necessarily have inhomogenous temperature, den- sity, and pressure profiles. These gradients are a source of free energy for various instabilities. In this section the linear theory of ion tem- perature gradient drift waves is explored. The derivation assumes a −→ −→ −→ slab geometry with uniform magnetic field B = B e z (where e z is the unit vector in the z direction) and a temperature gradient in the x −→ ∂Ti −→ direction: ∇Ti = ∂x e x. This will not include the effects of toroidal geometry, nor the effects of a density gradient or electron tempera- ture gradient (i.e. ∇n = ∇Te = 0). This is done for simplicity. This treatment follows similarly to the one in Chapter 2.3 of Ref. [25]. The following fluid equations for continuity, force balance, and con- servation of energy (respectively) are the basic equations for deter- mining the dynamics of the ion drift waves:
∂ ∂ δn + n δu = 0 (3a) ∂t i 0 ∂z iz ∂ ∂ 1 ∂ mi δuiz + e δΦ + δpi = 0 (3b) ∂t ∂z n0 ∂z ∂ −→ ∂ ∂ δp + +δ u · p + Γp δu = 0 (3c) ∂t i E ∂x 0 0 ∂z iz
where δni is the perturbed ion density, n0 is the equilibrium den-
sity, δuiz is the perturbed ion parallel velocity, mi is the ion mass, e is the electron charge, δΦ is the perturbed potential, δpi is the −→ perturbed ion pressure, u E is the perturbed ~E × B~ drift velocity, p0 is the background pressure, and Γ is the adiabatic exponent. In addition, the ion temperature gradient modes considered will have (ω/k) << vTe, meaning that the phase velocity of the wave is slow compared to the thermal speed of electrons. This implies that the elec- trons will have an adiabatic (or Boltzmann) response to the potential:
δne |e|δΦ = (4a) n0 Te
Finally, the fact that the mode wavelengths are all much larger than the plasma Debye length λDe means that Poisson’s equation becomes just δne = δni. This is referred to as the quasi-neutrality condition. The fluctuating quantities were then Fourier decomposed into the −→ −→ δΦ = −→ δΦ−→ ei( k · x −ωt) individual wavenumbers and frequencies: k ω k ω . If the mode amplitudes are small enough, the different wavenum- P bers and frequencies can be treated independently. When this is done, 2.1 linear fluid ion temperature gradient modes 25 and when Eq. 4a is substituted into Eq. 3b the following equations re- sult:
−iωδni + n0ikzδuiz = 0 (5a) Te ikz −iωmiδuiz + ikz δni + δpi = 0 (5b) n0 n0 kyTe ∂Ti −iωδpi − i δnn0 + Γp0ikzδuiz = 0 (5c) en0B ∂x
When these equations are combined, the following dispersion rela- tion is found:
2 2 kzcz Γ ω?Ti 1 − 2 1 + − = 0 (6a) ω τ ω
ω?Ti = (kyρi/LTi )vTi (6b) p where cs = Te/mi is the ion sound speed, τ = Te/Ti is the ratio of the electron to ion temperatures, ω?T is the ion drift frequency, i p ρi = vTi (mic/eB) is the ion thermal gyroradius, vTi = Ti/mi is the ion thermal speed, LTi = Ti/∇Ti is the ion temperature gradient scale length. In the limit of |ω?Ti | >> ω and ω >> kzcs, Eq. 6a has three roots:
2πN 1/3 2/3 3 i ω = |ω?Ti | (kzcs) e ,(7a)
with N = 0, 1, or 2. For N = 1, there is a positive imaginary compo- nent to the root and therefore the mode is unstable. In this case the mode growth rate increases with increasing kz and increasing ω?Ti . The dependence on the ion temperature gradient comes through in the ω?Ti dependence. To extend this treatment to toroidal geometry, z becomes k, x be- comes r, and y becomes θ. In addition, the curvature drift enters in through extensions to the continuity equation:
∂ δn δTi + ωdi = 0,(8a) ∂t n0 Ti
where ωdi = − (cTi)/(eBR) kθ is the sum of the ∇B and cur- vature drifts evaluated on the low field side. Combining Eq. 8a with Eq. 3c and Eq. 4a, and neglecting the parallel dynamics gives the fol- lowing equation for the frequency of the modes:
2 ω = −τωdiω?Ti (9a) 26 plasma turbulence
Given that ωdi < 0 on the low field side and ω?Ti < 0 for nor- mal temperature profiles with ∂Ti/∂r < 0 , this predicts a purely growing mode. Equations 6a and 9a both predict modes that are unstable for typical profiles and become more unstable with steeper ion temperature gradients, and both types of ion temperature gradi- ent (ITG) modes will generally exist in tokamak plasmas and drive energy transport. The other turbulence modes that are typically seen in tokamaks are trapped electron modes (TEMs), and electron temperature gradi- ent modes. A summary of the different turbulent modes and their dependencies is covered in Table 2, which is taken from Ref. [25]. The amount of energy flux driven by a particular turbulent mode 2 generally scales as γ/kr through simple random walk arguments. Thus ETG modes will not drive significant energy flux unless they have kr << kθ, meaning that they form radially elongated "stream- ers". These streamers have been predicted in simulations [19], but they require weak but non-negligible long wavelength turbulence to exist. They are also more vulnerable to being torn apart by flow shear. •
2.2 gyrokinetics
Fluid theory is useful for understanding the dependencies of indi- vidual turbulent modes, but a kinetic description of the turbulence is necessary to predict saturated turbulence amplitude. To do so, we begin with the Vlasov equations for the evolution of the distribution function f of a plasma species in six dimensions:
−→ ∂f −→ −→ q −→ v −→ −→ + v · ∇f + E + × B · ∇f = C(f) (10a) ∂t m c This six-dimensional equation can be greatly simplified if the par- −→ −→ −→ −→ ticle velocity is parametrized by v = vk e k + µ e ⊥ + ξ e ξ where 2 µ = v⊥/2B is the adiabatic invariant, ξ is the gyrophase, and C(f) is the collision operator. The position variables are also taken to be the −→ locations of the particle guiding centers R , rather than the particle positions themselves. When this substitution is performed, Eq. 10a becomes:
∂f −→ −→ q ∂ + v b · ∇−→f + v · ∇−→f + E f k R E R k ∂t m ∂vk −→ −→ ∂ qΩci ∂Φ ∂f v E · v ⊥ ∂f − Ωci f + + Ωci 2 = C(f),(11a) ∂ξ mB ∂ξ ∂µ v⊥ ∂ξ −→ where v E is the ~E × B~ velocity, and Ωci is the ion cyclotron fre- quency. For a typical magnetized plasma, the gyroradius of a particle 2.2 gyrokinetics 27
Turbulent Free Poloidal Temporal Resonance Mode Energy Spatial Scales Source Scales
Ion Tem- ∇Ti ≈ 0.3ρi ω?pi Bad curva- perature ture or neg- Gradient ative com- Mode pressibility (ITG)
Trapped ∇n or ∇Te ≈ ρi ω?ne Trapped Electron electron Mode precession (TEM) resonance (collision- less TEM), collisions between trapped and pass- ing elec- trons (dis- sipative TEM)
Electron ∇Te ≈ 0.3ρe ω?pe Bad curva- Temer- ture or neg- ature ative com- Gradient pressibility Mode (ETG)
Table 2: Table of turbulent modes that are most often seen in tokamak plasmas.
ρi = vTi (mic/eB) is the ion thermal gyroradius, ρe = vTe (mec/eB) is the electron thermal gyroradius, ω?pi = kθρscs∇p/p is the ion diamagnetic drift frequency with the pressure gradient, ω?e = kθρscs∇n/n is the electron diamagnetic drift frequency with the density gradient, and ω?pe = kθρscs∇p/p is the electron diamagnetic drift frequency with the pressure gradient. 28 plasma turbulence
is much smaller than the gradient scale-lengths of the temperature and density profiles except very near the edge. This allows the gy- romotion (and therefore the ξ dimension) to be averaged out. When this averaging is performed, Eq. 11a becomes:
∂ −→ f + v b · ∇−→ f + ∂t k R −→ q ∂ v · ∇−→ f + E f = C(f) E R k ,(12a) m ∂vk
where ... refers to quantities averaged over ξ. Equation 12a is known as the gyrokinetic equation [16]. When combined with Maxwell’s −→ −→ equations to connect f to E and B , it becomes possible to find an equation solely for f. To make the problem tractable however, it is nec- essary to make approximations based on the so-called "gyrokinetic ordering" which assumes the following about the turbulence:
ω ρi δf eδΦ | | ≈ | | ≈ | | ≈ | | << 1 (13a) Ωci R F0 Te
Which means that the mode frequency is small compared to the ion cyclotron frequency, the ion gyroradius (and therefore the wave- length of the turbulence) is small compared to the system size R, and the perturbed distribution function δf is small compared to the equi- librium distribution function F0. This allows for an expansion of the gyrokinetic equation based on this ordering into f = F0 + δf, and δf = δg − (eδΦ/T)F0. This δg is the non-adiabatic portion of the per- turbed distribution function and is what we wish to solve for. With this expansion performed, Eq. 12a becomes:
∂ e ∂ c −→ + v ∇ δg + δΦ + b × ∇ δΦ · ∇ F = C(f) ∂t k k T ∂t B 0 (14a)
When this equation is combined with Maxwell’s equations, it is possible to solve for the non-adiabatic portion of the perturbed dis- tribution function. Such a calculation is still difficult, and requires supercomputers to solve. •
2.3 gyrokinetic simulation codes
The gyrokinetic equation for realistic plasma conditions and geome- tries can only be solved through the use of supercomputer simula- tions. The development of simulation codes that can use gyrokinetics to predict turbulent transport in tokamaks has been a major effort of fusion research for the past few decades, and major progress has been 2.3 gyrokinetic simulation codes 29 made in this area thanks to significant increases in supercomputing power. There are many codes that can solve the gyrokinetic-Maxwell equations in general geometry, including GS2[22], GENE[21], and GYRO[8]. All the codes have been verified against one another in a wide array of regimes[15][24][4]. The codes differ somewhat in their collision models and their geometries, but all of them can be taken to accurately simulate the turbulence predicted by gyrokinetics if the gyrokinetic ordering is satisfied. GYRO was chosen for the work in this thesis due to its thorough documentation [3], and excellent support for multiple platforms [2]. It also shares the same input formats as the neoclassical code NEO [5], the quasilinear gyrofluid code TGLF [27], and the transport solver code TGYRO [7]. GYRO was developed by Candy and Waltz in 2003 at General Atomics [8] and was one of the first codes to be able to include global effects (i.e. the effects of radial variation of profile gra- dients). GYRO computation speed has been shown to scale with pro- cessor number up to tens of thousands of processors [9]. The history of GYRO, its development, and some of the early highlights of the code’s use can be found on [1]. Nonlinear GYRO simulates turbulence by reading in a file that con- tains the equilibrium plasma geometry, temperature, density, and ve- locity profiles. It then sets up a grid in real space, wavenumber space, and velocity space over a user-specified poloidal annulus, and applies a small perturbation to the distribution function within that domain. GYRO then evolves that perturbation to the distribution function ac- cording to the gyrokinetic equation. If the specified profiles are sta- ble, then the initial perturbation will die away. If the specified profiles are unstable, then the perturbation will grow according to the linear growth rates of the individual modes. Once the modes grow enough, they will eventually induce strong zonal flows (i.e. the modes with kθρs = 0) which have a stabilizing effect on the turbulence. These zonal flows eventually grow large enough to saturate the turbulence and it is that saturated turbulence amplitude that is then compared to the experimental quantities. Figure 4 shows the 3-D turbulent po- tential fluctuations from a particular GYRO simulation. It shows the very long parallel wavelengths (comparable to the system size), and relatively short poloidal and radial wavelengths (comparable to the ion gyroradius). This approach of assuming a quiescent background plasma and evolving a small perturbation is obviously not identical to what occurs in the experiment, but given the large separation of timescales between the turbulent growth rates (typically microsec- onds) and energy transport (typically milliseconds) it is the approach that must be taken. A much faster code would be needed to simulate the entire energy transport process. TGLF [27], the trapped-gyro-Landau-fluid code, is a much faster way to determine the energy transported by gyrokinetic turbulence. 30 plasma turbulence
Figure 4: Plot of a GYRO simulation of a DIII-D discharge. The colors indicate a higher (in red) or lower (in blue) electrostatic fluctuating potential, with green indicating the background potential of zero. The blue halo indicated the last closed flux surface of the torus. From [1].
TGLF used moments of the gyrokinetic equation to solve for the lin- ear eigenmodes for trapped ion modes, trapped electron modes, ion temperature gradient modes, electron temperature gradient modes, and kinetic ballooning modes. The eigenmodes give the growth rates and frequencies of the particular turbulent modes, and to determine the flux driven by these modes, TGLF has a quasilinear saturation rule that is calibrated against a database of nonlinear GYRO runs. TGLF treats the flux from each wavenumber mode separately, which disregards the nonlinear nature of the turbulence, but is nevertheless able to model the zonal flow saturation based on only the growth rate and wavenumber of an individual mode, as well as the curva- ture drift frequency and ~E × B~ shearing rate. The database on which the saturation rule is based covers a wide range of parameters, but it is still important to check the TGLF results against the more complete physics model in GYRO for certain cases. The speed of TGLF allows for the simulation on the energy transport timescales, and allows for more careful sensitivity analysis of the many parameters on which turbulent transport depends. Both the GYRO and TGLF computed energy fluxes are normalized ? 2 to the so-called gyrobohm unit energy flux: QGB = neTecs(ρ ) [8][27], ? where ρ = ρs/a, with a referring to the plasma minor radius. This is ? the flux that arises from a gyrobohm diffusivity: χGB = ρ csρs when the typical gradient scale length is comparable to the plasma minor radius a. This diffusivity and flux are the fundamental scaling that arises from ρs scale nonlinear drift-wave turbulence [18]. 2.4 the flux-gradient relationship 31
Figure 5: Plot of the normalized electron energy flux versus normalized electron temperature gradient for a series of DIII-D discharges with varying ECH deposition 2 profiles. The energy flux is normalized to QGB = neTecs(ρs/a) , the gyrobohm energy flux. The gradients are defined as 1/LTe = ∇Te/Te. Plot is from Ref. [17]. Different colors and symbols indicate different neutral beam configurations, which change the rotation of the plasma.
2.4 theflux-gradient relationship
The parameters that energy flux depends on most sensitively are the radial gradients of the ion and electron temperature profiles. The tur- bulent energy flux from a particular species is generally found to be an offset linear function of its temperature profile gradient. This is shown in Fig. 5 which is a plot of the normalized electron energy flux Qe/QGB for various values of the normalized electron temperature gradient 1/LTe at a particular value of the normalized minor radius ρ = 0.6. This plot is from experiments performed on DIII-D in which the temperature gradient was varied through adjusting the power deposited by two electron cyclotron heating gyrotrons [17]. For the values of the inverse gradient scale length below 2 m−1, the energy flux is independent of the gradient. As the gradient passes this value, referred to as the "critical gradient", the energy flux increases rapidly with increasing gradient. This phenomenon of the rapid increase in the turbulent energy transport with increasing gradient is known as "profile stiffness", because it was first observed as an experimental phenomenon of temperature profiles responding weakly to increases in heating power. The slope of the increase in turbulent energy flux versus gradient for gradients above the critical gradient is referred to as the "stiffness". 32 plasma turbulence
Figure 6: Plot of the relationship between the normalized ion energy flux and normalized gradient from GENE simulations of a JET discharge. The different colors indicate different values of the magnetic shear,s ˆ, and safety factor q in the simulations.
This flux-gradient offset-linear relationship is also found to be present in gyrokinetic simulations of turbulent energy transport [20][10]. Fig- ure 6 taken from Ref. [10] shows the results of nonlinear gyrokinetic simulations from the GENE code based on a JET discharge. It shows the relationship between the simulated ion energy flux and ion tem- perature gradient for different values of the magnetic shear,s ˆ, and safety factor q. Both the stiffness and critical gradient are affected by changes ins ˆ and q, and many other factors affect the values of those two parameters. The calculation of the stiffness and critical gra- dient requires nonlinear gyrokinetic simulations. Linear simulations can approximately calculate the critical gradient, however there is a well-known phenomenon known as the "Dimits Shift"[14] wherein the nonlinearly generated zonal flows will increase the critical gra- dient above the linear value. The stiffness dependent on the modes saturated amplitude, and thus also requires a nonlinear simulation to accurately compute. For reactor design and operation purposes, the energy transport is the dependent variable (because in equilibrium the energy transport will be equal to the input power) and the temperature profile gradi- 2.5 summary 33 ent is the independent variable (because the temperature profile will evolve according to the supplied input power). This makes determin- ing the exact values of the stiffness and critical gradient important for reactor operation. These two quantities are dependent on several pa- rameters, including the magnetic geometry, rotation, Te/Ti, Zeff, and as is going to be shown in this thesis work, ni/ne.
2.5 summary
The energy confinement in tokamak plasmas, and therefore their gen- eral performance, is limited by turbulent energy transport. Trans- port due to collisions between individual particles rather than col- lective turbulent fluctuations, so-called neoclassical transport, con- tributes only a small part to the total energy transport. Gyrokinetic theory can be used to predict the turbulence in tokamak plasmas that satisfy the gyrokinetic ordering in Eq. 13a, through the gyrokinetic equation in Eq. 12a. This nonlinear equation must be solved numer- ically through supercomputer codes. The GYRO code solves the gy- rokinetic equation in a general geometry, and the reduced model code TGLF can estimate the energy transport quickly. The energy transport due to turbulence is an offset-linear function of the temperature pro- file gradients. The energy transport is small and independent of the temperature gradient for values of the temperature gradient below the critical gradient, and increases rapidly with the temperature gra- dient above the critical gradient. The rate of increase of the energy transport with increasing temperature gradient is referred to as the stiffness of the transport. ?
BIBLIOGRAPHY
[1] GYRO Overview. URL: https://fusion.gat.com/theory/ Gyrooverview. 29, 30
[2] GYRO Platforms. URL: https://fusion.gat.com/theory/ Gyroplatforms. 29
[3] GYRO Website. URL: https://fusion.gat.com/theory/ Gyrousermanual. 29
[4] C. Angioni, A.G. Peeters, G.V. Pereverzev, A. Bottino, J. Candy, R. Dux, E. Fable, T. Hein, and R.E. Waltz. Gyrokinetic simulations of impurity, He ash and α particle transport and consequences on ITER transport modelling. Nuclear Fusion, 49(5):055013, 2009. URL: http://iopscience.iop. org/0029-5515/49/5/055013$\delimiter"026E30F$nhttp: //stacks.iop.org/0029-5515/49/i=5/a=055013?key=crossref. 90ec31be31bc8d5c2dcf353f6c196869, doi:10.1088/0029-5515/ 49/5/055013. 29
[5] E A Belli and J Candy. Kinetic calculation of neoclassical transport including self-consistent electron and impurity dynam- ics. Plasma Phys. Control. Fusion, 50(9):95010, 2008. URL: http: //stacks.iop.org/0741-3335/50/i=9/a=095010, doi:10.1088/ 0741-3335/50/9/095010. 29
[6] D Brock and W Horton. Toroidal drift-wave fluctuations driven by ion pressure gradients. Plasma Physics, 24(3):271– 287, 1982. URL: http://stacks.iop.org/0032-1028/24/i=3/ a=006?key=crossref.dbef5bcfc1a835db256e8df35b99f2a3, doi: 10.1088/0032-1028/24/3/006. 23
[7] J. Candy, C. Holland, R. Waltz, M. Fahey, and E. Belli. Tokamak profile prediction using direct gyrokinetic and neoclassical simulation. Physics of Plasmas, 16(6):060704, 2009. URL: http://scitation.aip.org/content/aip/journal/ pop/16/6/10.1063/1.3167820, doi:10.1063/1.3167820. 29
[8] J. Candy and R. Waltz. Anomalous Transport Scaling in the DIII-D Tokamak Matched by Supercomputer Simulation. Physical Review Letters, 91(4):045001, jul 2003. URL: http: //link.aps.org/doi/10.1103/PhysRevLett.91.045001, doi:10. 1103/PhysRevLett.91.045001. 29, 30
[9] J. Candy, R. E. Waltz, M. R. Fahey, and C. Holland. Plasma microturbulence simulation of instabilities at highly disparate
35 36 Bibliography
scales. Journal of Physics: Conference Series, 78(2007):012008, 2007. URL: http://www.iop.org/EJ/abstract/1742-6596/78/ 1/012008, doi:10.1088/1742-6596/78/1/012008. 29
[10] J. Citrin, F. Jenko, P. Mantica, D. Told, C. Bourdelle, R. Du- mont, J. Garcia, J.W. Haverkort, G.M.D. Hogeweij, T. John- son, and M.J. Pueschel. Ion temperature profile stiff- ness: non-linear gyrokinetic simulations and comparison with experiment. Nuclear Fusion, 54:023008, 2014. URL: http://stacks.iop.org/0029-5515/54/i=2/a=023008?key= crossref.5bff91ba519429a1cdd3692275d3c06f, arXiv:arXiv: 1303.2217v2, doi:10.1088/0029-5515/54/2/023008. 32
[11] B. Coppi. Instabilities due to Temperature Gradients in Com- plex Magnetic Field Configurations. Physics of Fluids, 10(3):582, 1967. URL: http://scitation.aip.org/content/aip/journal/ pof1/10/3/10.1063/1.1762151, doi:10.1063/1.1762151. 23
[12] B. Coppi, H. Furth, M. N. Rosenbluth, and R. Z. Sagdeev. Drift instability due to impurity ions. Physical Review Letters, 17(7):377, aug 1966. URL: http://link.aps.org/doi/10.1103/ PhysRevLett.17.377, doi:10.1103/PhysRevLett.17.377. 23
[13] Bruno Coppi and Gregory Rewoldt. New Trapped-Electron In- stability. Physical Review Letters, 33(22):1329–1332, nov 1974. URL: http://link.aps.org/doi/10.1103/PhysRevLett.33.1329, doi: 10.1103/PhysRevLett.33.1329. 23
[14] a. M. Dimits, G. Bateman, M. a. Beer, B. I. Cohen, W. Dor- land, G. W. Hammett, C. Kim, J. E. Kinsey, M. Kotschen- reuther, a. H. Kritz, L. L. Lao, J. Mandrekas, W. M. Nevins, S. E. Parker, a. J. Redd, D. E. Shumaker, R. Sydora, and J. Wei- land. Comparisons and physics basis of tokamak transport models and turbulence simulations. Physics of Plasmas, 7(3):969, 2000. URL: http://scitation.aip.org/content/aip/journal/ pop/7/3/10.1063/1.873896, doi:10.1063/1.873896. 32
[15] C Estrada-Mila, J Candy, and R E Waltz. Gyrokinetic simulations of ion and impurity transport. Phys. Plas- mas, 12(2):022305, 2005. URL: http://scitation.aip.org/ content/aip/journal/pop/12/2/10.1063/1.1848544, doi:http: //dx.doi.org/10.1063/1.1848544. 29
[16] E. A. Frieman. Nonlinear gyrokinetic equations for low- frequency electromagnetic waves in general plasma equilibria. Physics of Fluids, 25(3):502, 1982. URL: http://scitation.aip. org/content/aip/journal/pof1/25/3/10.1063/1.863762, doi: 10.1063/1.863762. 28 Bibliography 37
[17] J C Hillesheim, J C DeBoo, W A Peebles, T A Carter, G Wang, T L Rhodes, L Schmitz, G R McKee, Z Yan, G M Stae- bler, K H Burrell, E J Doyle, C Holland, C C Petty, S P Smith, A E White, and L Zeng. Experimental characteriza- tion of multiscale and multifield turbulence as a critical gra- dient threshold is surpassed in the DIII-D tokamaka). Phys. Plasmas, 20(5):056115, 2013. URL: http://scitation.aip.org/ content/aip/journal/pop/20/5/10.1063/1.4807123, doi:http: //dx.doi.org/10.1063/1.4807123. 31
[18] W. Horton. Drift waves and transport. Reviews of Modern Physics, 71(3):735–778, apr 1999. URL: http://link.aps.org/doi/10. 1103/RevModPhys.71.735, doi:10.1103/RevModPhys.71.735. 23, 30
[19] N T Howard, C Holland, A E White, M Greenwald, and J Candy. Synergistic cross-scale coupling of turbulence in a tokamak plasma. Phys. Plasmas, 21(11):112510, 2014. URL: http: //scitation.aip.org/content/aip/journal/pop/21/11/10. 1063/1.4902366, doi:http://dx.doi.org/10.1063/1.4902366. 26
[20] A E Hubbard, J W Hughes, E Marmar, J E Rice, J M Sier- chio, C Sung, J R Walk, D G Whyte, D R Mikkelsen, E M Ed- lund, C Kung, C Holland, J Candy, C C Petty, M L Reinke, and C Theiler. Nonlinear gyrokinetic simulations of the I-mode high confinement regime and comparisons with experiment. Physics of Plasmas, 22(5):056109, 2015. URL: http://dx.doi.org/ 10.1063/1.4921150, doi:10.1063/1.4921150. 32
[21] Frank Jenko. Massively parallel Vlasov simulation of electromag- netic drift-wave turbulence. Computer Physics Communications, 125(1-3):196–209, 2000. doi:10.1016/S0010-4655(99)00489-0. 29
[22] M. Kotschenreuther, W. Dorland, M. a. Beer, and G. W. Hammett. Quantitative predictions of tokamak energy con- finement from first-principles simulations with kinetic effects. Physics of Plasmas, 2(6):2381, 1995. URL: http://scitation.aip. org/content/aip/journal/pop/2/6/10.1063/1.871261, doi:10. 1063/1.871261. 29
[23] Paulett C. Liewer. Measurements of microturbulence in tokamaks and comparisons with theories of turbulence and anomalous transport. Nuclear Fusion, 25(5):543–621, 2011. URL: http://stacks.iop.org/0029-5515/25/i=5/a=004, doi: 10.1088/0029-5515/25/5/004. 23
[24] W. M. Nevins, J. Candy, S. Cowley, T. Dannert, A. Dimits, W. Dorland, C. Estrada-Mila, G. W. Hammett, F. Jenko, M. J. 38 Bibliography
Pueschel, and D. E. Shumaker. Characterizing electron tem- perature gradient turbulence via numerical simulation. Physics of Plasmas, 13(12):122306, 2006. URL: http://scitation.aip. org/content/aip/journal/pop/13/12/10.1063/1.2402510, doi: 10.1063/1.2402510. 29
[25] S. Kaye R.J. Goldston, R.B. White, T.S. Hahm. Physics of Confine- ment. In Fusion Physics, chapter 2, pages 59–224. International Atomic Energy Agency, 2012. 24, 26
[26] Home Search, Collections Journals, About Contact, My Iop- science, and I P Address. Plasma instabilities due to ion tem- perature gradients. Plasma Physics, 29(1):29, 1968. URL: http: //stacks.iop.org/0029-5515/8/i=1/a=005. 23
[27] G. M. Staebler, J. E. Kinsey, and R. E. Waltz. A theory-based transport model with comprehensive physics. Physics of Plasmas, 14(5):055909, 2007. URL: http://link.aip.org/link/PHPAEN/ v14/i5/p055909/s1{&}Agg=doi, doi:10.1063/1.2436852. 29, 30
[28] W. M. Tang, P. H. Rutherford, P. H. Furth, and J. C. Adam. Phys- ical review. Phys. Rev. Lett., 57(14):0–3, 1986. doi:72.15.6d,73. 60.Dt. 23 PHASECONTRASTIMAGING 3
3.1 the phase contrast imaging technique
Phase contrast imaging, or PCI, is a self-referencing interferometric technique that images perturbations in the index of refraction along the beam path. It was first developed by Fritz Zernike in the 1930s[10], and for his work he was awarded the Nobel Prize in Physics in 1953. His technique became widely utilized in biology in the form of phase contrast microscopes, which could view cellular structures in much more detail than other techniques at the time [6]. The PCI technique was first utilized to detect waves in fusion plasmas by Henri Weisen on the TCA tokamak in 1988[9]. It has since been utilized on sev- eral fusion devices, including DIII-D[1], TCV[2], LHD[8], and Alcator C-Mod[5]. The PCI technique has been used to study a variety of plasma wave phenomena, including Alfven waves, turbulent waves, and externally launched radio-frequency waves. The interaction of an infared laser beam with a fusion plasma can be viewed as coherent scattering in a dielectric medium. For elec- tromagnetic waves at frequencies ω0 much larger than the electron eB cyclotron frequency ωce = mc , the plasma dielectric is given by:
2 ωpe /0 = 1 − 2 ,(15a) ω0
p 2 where ωpe = e ne/0me is the electron plasma frequency. The r 2 p ωpe index of refraction is therefore given by N = /0 = 1 − 2 . ω0 For typical plasma parameters ωpe ≈ 100 GHz, and a CO2 laser has ω0 ≈ 100 THz, so ωpe/ω0 << 1. Therefore the index of refraction can be approximated as
2 ωpe N = 1 − 2 .(16a) 2ω0
When the laser beam passes through the plasma, it experiences a phase delay of
2 ωpe φ˜ = k (N − 1)dl = −k dl (17a) 0 0 2ω2 Z Z 0
φ˜ = −λ0re nedl,(17b) Z
39 40 phase contrast imaging
2 2 where λ0 = 2π/k0 is the laser beam wavelength, and re = e /4π0mec is the classical electron radius. If there is a perturbation in the index of refraction of the plasma caused by a density wave perpendicular ik x to the laser beam path, ne(t, x) = n0 + n˜ee n , the original wave will scatter into three components which are then combined at the detector:
δφ2 δφ δφ E = E 1 − + iE eikn + iE e−ikn ,(18a) 0 2 0 2 0 2
where E0 is the electric field of the original wave and δφ = −λ0re n˜edl << 1 is the phase perturbation from the density fluctuation. This is shown R graphically in Fig. 7, where the red beams represent the unscattered portion of the laser beam and the green beams represent the scattered portion of the laser beams. If the scattered and unscattered beams were to be combined without any modification, the resulting inten- sity would be:
2 2 2 I ∝ |E| ≈ |E0| (1 + 2δφ ) (19a)
2 I ∝ n˜e dl + const., (19b) Z which is extremely small due to how smalln ˜e/ne is in a typical fusion plasma. The way that PCI solves this problem is to introduce a π/2 phase shift to the unscattered beam relative to the scattered beam. This changes Eqn. 18a to:
δφ2 δφ δφ E = iE 1 − + iE eikn + iE e−ikn ,(20a) 0 2 0 2 0 2 where the unscattered beam is now in-phase with the scattered beam. This results in an intensity of
2 2 I ∝ |E| ≈ |E0| (1 + 2δφ) (21a)
I ∝ n˜edl + const., (21b) Z where now the intensity is linear, rather than quadratic in the line- integrated density fluctuations. This allows PCI to measure very small intensity waves. The π/2 phase shift is accomplished by having the unscattered beam take a λ/4 longer path to the detector than the scattered beams. This is accomplished through a phase-plate, which is pictured in Fig. 7 in the bottom left. The phase-plate has a λ/8 deep groove that the unscattered beam goes into, while the scattered beams do not. 3.2 phase contrast imaging on alcator c-mod 41
Figure 7: Simplified diagram of the PCI diagnostic. The red beam indicates the unscattered beam, while the green beams are the scattered beams. The square in the bottom left shows where the beams land on the phase plate (unscattered lands in the groove, while scattered lands out of the groove). Both scattered and unscattered beams are then combined at the detector.
•
3.2 phase contrast imaging on alcator c-mod
The PCI system on Alcator C-Mod was originally designed and im- plemented by Alex Mazurenko in 1998 as part of his PhD thesis[3]. It has since gone through several upgrades to the optics, detector, phase- plates, and digitizers. In the current configuration, the PCI system on −1 −1 Alcator C-Mod can measure waves with 0.5cm 6 |kR| 6 60cm , where kR refers to the wavenumber in the major radius direction of the waves, and frequencies above 1 kHz and below 2.5 MHz. The overall geometry of the system is shown in Fig. 8, with the dotted line representing the laser beam path. The laser itself is a FireStar t60 SYNRAD, Inc. 60W CO2 laser with a 10.6 µm wavelength. A laser safety shutter from Hass Laser Technologies, Inc. is in front of the laser and allows the laser to safely remain on throughout a run day, ensuring stability. The laser and shutter temperatures are controlled by a Merlin chiller to remain at 20◦C, and this chiller also cools the acousto-otic modulators (AOMs) which are part of the optical hetero- dyne system that measures the ion cyclotoron radio-frequency (ICRF) waves used to heat the plasma. To aid in alignment of the laser sys- tem, a visible HeNe laser is aligned to overlap with the infared CO2 laser on the expansion optics. The expansion optics serve to expand the CO2 laser beam, modu- late the laser beam intensity using the AOMs, overlap the CO2 and HeNe laser beams, and align the beam on the bottom steering mir- ror. They are housed in an enclosed 1 m x 3 m optics table. The ex- 42 phase contrast imaging
Figure 8: Schematic diagram of the PCI system on Alcator C-Mod. The dotted line represents the laser beam path from the bottom expansion optics, through the plasma (shown in red), up to the detector on top of the tokamak. Figure adapted from Ref. [3]
pansion of the beam is accomplished by focusing the laser onto a 1" focal length gold concave spherical mirror, and then onto an 80" off- axis parabolic mirror. These two optical components form a telescope which expands the beam radius to 8 cm. This allows for roughly uni- form laser beam amplitude over the plasma region of interest. Before being focused on the 1" spherical mirror, the beam is split in two paths and then each path is sent through an AOM to modulate the frequency by ≈ 40 MHz. One beam path is increased by 40 MHz, and the other is decreased by 40 MHz. When the two beams are then recombined, there will then be an 80 MHz oscillation of the beam amplitude, which will then enable heterodyne detection of the ≈ 80 MHz ICRF waves in the plasma. After the beams are recombined, ≈ 3% of the beam power remains. This is sufficient to get a strong signal on the detector, and too much more power can saturate the detector. The HeNe laser is made to overlap with the CO2 laser after the beams are recombined, and after the 1" spherical mirror only the HeNe laser is used for alignment. Once the laser exits the expansion optics table, it is reflected off of a 45◦ flat mirror to direct it into the tokamak. The laser beam enters the tokamak through a vacuum window at the bottom of the vacuum vessel and exits through one directly above it on the top of the vacuum vessel. It passes through windows that are centered at a major radius of 69 cm, and are each 10 cm wide ra- dially and 4 cm wide toroidally. The path of the laser beam through 3.2 phase contrast imaging on alcator c-mod 43
Figure 9: Diagram of PCI laser beam path through the plasma overlayed on a crossection of the tokamak. the plasma is shown in Fig. 9. The beam passes vertically through the center of the tokamak, and density fluctuations along the entire path are integrated according to Eqn. 21b. This means that fluctuations from the core and edge, including the top and bottom, are added together in the PCI signal. Only a portion of the entire beam width is imaged on the detector, depending on the needs of the particular experiment. Using a wider region gives better spatial coverage and wavenumber resolution, and is useful for ICRF wave measurements. Using a narrower region gives a larger maximum wavenumber de- tected, and therefore is useful for turbulence wave measurements. The imaging optics on the top of the tokamak are what determine the portion of the laser beam used. 44 phase contrast imaging
After the beam exits the tokamak, it enters a second enclosed op- tics table that is anchored to the top of Alcator C-Mod’s outer con- crete layer. It first is reflected off of a flat mirror to make the beam horizontal again, then onto another 80" focal length off-axis parabolic mirror. The beam is then focused onto a phase plate, which phase- shifts the unscattered beam by π/4 relative to the scattered beams. The phase plate is a 2" diameter flat ZnSe base window, which is coated with SiO2, titanium, platinum, and then finally gold to a thick- ness of 1.32 µm (±10%), which is one-eigth the wavelength of a CO2 laser. Only 20% of the unscattered beam power is reflected out of the groove, which increases the portion of the final signal from the scattered beam. The width of the phase plate groove will determine the lower-limit of the detectable wavenumber of the PCI system, as the scattered beam must be scattered at a large enough angle to escape the phase plate groove so as to be detectable (otherwise the scattered beam will not be phase-shifted relative to the unscattered beam). The distance d between the unscattered and scattered beams is:
kn d = F0 ,(22a) k0
where F0 = 80" (2.03 m) is the focal length of the parabolic mirror that focuses the beam onto the phase plate. This means that the mini- mum detectable wavenumber kmin for a particular phase plate groove width w is given by:
w kmin = k0 .(23a) 2F0
For the phase plate groove widths available on the Alcator C-Mod PCI system, which are 0.4, 1.1, and 4.0 mm wide, the corresponding −1 kmin values are 0.6, 1.6, and 5.8 cm respectively. The restriction on using the very smallest groove width at all times, is that it is more sensitive to vibrations and small misalignments. For the work used in this thesis, the 0.4 mm groove width phase plate is used to get the maximum detectable wavenumber range. After the laser beam is reflected off the phase plate, it passes through a 10" and a 2.5" focal length ZnSe lens. These two lenses form a telescope with a magni- fication that can be adjusted from ≈0.2 to ≈0.8 to change the width of the beam imaged into the detector, and the maximum detectable wavenumber. The limitations on the maximum wavenumber from the optical system is approximately 60 cm−1, which is far larger than what has typically been detected, so the maximum wavenumber is set by the Nyquist wavenumber of the detector. As the experiments 3.2 phase contrast imaging on alcator c-mod 45
PCI Detector Responses 1.00
New Detector Old Detector
0.10 Response
0.01 0 500 1000 1500 2000 2500 F [kHz]
Figure 10: Plots of the frequency responses for the two different detector, defined as the ratio of the square of the voltage of the output signal to the square of the voltage of the output signal from an identical amplitude signal at 15 kHz. This plot was generated using a pulsed IR LED. The same setup was used for both detectors. performed as part of this thesis work focused on turbulence, the max- imum wavenumber was set to be either 20 cm−1or 30 cm−1. Two different detectors have been used for the PCI system on Al- cator C-Mod during this thesis work. Both are 32 channel HgCdTe photoconductive liquid nitrogen cooled IR detectors with preampli- fiers with a gain of about 200. The older detector was built by Belov Technology with in-house built preamplifiers, and had rectangular detector elements that were 0.75 mm wide and 1.0 mm high which were separated by 0.1 mm. This gave it an effective channel separa- tion of 0.85 mm. The newer detector was built by Infared Systems Development, who also built the preamplifiers, and had rectangular elements that were 0.5 mm wide and 1.0 mm high which were sep- arated by 0.05 mm. This gave it an effective channel separation of 0.55 mm. The primary difference between the detectors was their fre- quency responses. Figure 10 shows the nominal frequency responses of the old detector (red) and new detector (blue). The new detector frequency response is approximately double that of the old detector. This frequency response is folded into the analysis performed, espe- cially when the measured PCI signal is compared to GYRO simula- tions. The signals from the preamplifiers are fed into a 32 channel 16- bit D-tacq digitizer, where they are digitized at 5 MHz. This gives a Nyquist frequency of 2.5 MHz, which is sufficiently above the band- width of both detectors, as Fig. 10 shows. Once digitized, the data is stored in C-Mod’s MDSPlus database server. • 46 phase contrast imaging
3.3 absolute calibration
As Eqn. 21b shows, the intensity of the combined beam at the detector is proportional to the line-integrated density fluctuation amplitude. This means that the system can be absolutely calibrated if a known amplitude wave is passed through the laser beam. On Alcator C-Mod, this is provided by an ultrasonic loudspeaker. The phase perturbation due to a sound wave in air is given by:
(N0 − 1) φ = −k pdl˜ ,(24a) 0 γp 0 Z −4 where N0 − 1 = 2.7 × 10 is the deviation from vacuum index of refraction, γ = 1.4 is the adiabatic index, p0 = 101 MPa is atmo- spheric pressure, andp ˜ is the pressure perturbation due to the sound wave that has had its amplitude and spatial structure measured with a calibrated microphone. This allows for the direct calculation of the proportionality constant to calibrate the system. The speaker is placed on the expansion optics table after the 80" parabolic mirror. A 15 kHz sound wave is used to do the calibration, which has a wavenumber of 2.75 cm−1.A 2 ms calibration shot is taken 300 ms before each plasma shot, and the calibration factor obtained from it is then applied to the plasma shot that follows. This is done for every shot to account for any small variation that occurs due to vibrations of optical compo- nents. Within a particular day the calibration factor will change by less than 1% shot-to-shot, and it will change by less than 5% between days if the optical system was not deliberately changed in between. In addition to calibrating the amplitude of the detected waves, the sound burst can also be used to directly determine the magnification M of the optical system. This is done by measuring the phase delay ∆φ between the channels:
∆φ ∆x = c ( ) (25a) s 2πF x M = ch ,(25b) ∆x where ∆x is the effective channel spacing in the plasma, F = 15 kHz is the wave frequency, cs = 344 m/s is the speed of sound in air, and xch = 0.85 or 0.55 mm is the distance between channels on the 3.4 synthetic pci diagnostic for gyro 47
Amplitude Calibration 0.015
0.010
0.005 Calibration Constant = amplitude[V] 300 x 1016 m-2/V 0.000 0 5 10 15 20 25 30 Channel Data Fit Channel Spacing Calibration 2.0 ∆ 1.5 x = 1.5 mm rad] π 1.0
0.5 phase[2 0.0 0 5 10 15 20 25 30 Channel
Figure 11: Plot of the calbration data for a particular shot using the old detector. The top plot shows the amplitude data and fit, and the amplitude calibration factor. The bottom plot shows the phase data and fit, and the effective channel spacing in the plasma. detector. This is useful because the maximum detectable wavenumber, kmax, and the wavenumber resolution, ∆k, are both functions of ∆x :
π k = (26a) max ∆x 2k ∆k = max ,(26b) nch
where nch = 32 is the number of channels of the detector. Figure 11 shows an example trace of the amplitude (top) and phase (bottom) of the signals, and their corresponding fits used to determine the amplitude calibration factor and ∆x, respectively. •
3.4 synthetic pci diagnostic for gyro
In order to compare the three-dimensional density fluctuation am- plitudes from GYRO to the line-integrated density fluctuations mea- sured by PCI, a synthetic diagnostic technique was used. The syn- thetic diagnostic was developed by J. Chris Rost who works on the DIII-D PCI system, and is detailed in Ref. [7]. The synthetic diag- nostic algorithm maps the Miller geometry[4] representation of the magnetic equilibrium used by GYRO onto real-space R and Z coordi- nates. Figure 12 shows the GYRO density fluctuations mapped into real space as done by the synthetic diagnostic for a particular simu- 48 phase contrast imaging
GYRO Density Fluctuations 0.03
0.2 0.02
0.1 0.01
0.0 n/n 0.00 δ Z(m)
-0.1 -0.01
-0.2 -0.02
-0.03 0.5 0.6 0.7 0.8 R(m)
Figure 12: Plot of the density perturbations from a GYRO simulation of an Alcator C-Mod plasma. The red lines indicate the boundaries of the region in which the PCI is detecting fluctuations.
lation of an Alcator C-Mod discharge. The synthetic diagnostic then integrates the density fluctuation amplitudes along the chords that correspond to the channels of the detector. Both the DIII-D and C- Mod PCI system geometries can be used with GYRO. In addition to being able to perform the line-integration of the density fluctuations, the synthetic diagnostic can also account for Doppler shifts due to ~E × B~ drifts. The rotation of the plasma is an important part of the resulting frequency-wavenumber spectrum, as is shown in Fig. 13. GYRO simulations account for the differences be- tween the ~E × B~ drifts in different regions of the simulation, but do not account for the overall rotation. The ~E × B~ drift has a relatively small impact on the actual simulation itself, as shown in AppendixB, so the ~E × B~ drift can be varied in the synthetic PCI without varying it in the simulations. The synthetic PCI also accounts for the real wavenumber response of the PCI system. This includes the effects of the phase plate, finite beam size, finite detector element size, and the optical geometry. All of these effects depend on the specific configuration of the system, which is input to the synthetic diagnostic before it is run. • 3.5 summary 49
Without ExB With ExB Included 1000 1000 10-2
10-3 800 800 ] 2 -4 n 10 δ 600 600 10-5
F [kHz] 400 F [kHz] 400 10-6 PCI Intensity [ 200 200 10-7
0 0 10-8 -20 -10 0 10 20 -20 -10 0 10 20 -1 -1 kR [cm ] kR [cm ]
Figure 13: Plots of synthetic PCI spectra from a GYRO simulation of an Alcator C-Mod plasma with (right) and without (left) ~E × B~ drifts included.
3.5 summary
Phase contrast imaging (PCI) is a technique used to measure line- integrated density fluctuations in plasmas. Density fluctuations cause a portion of the laser beam to scatter through a small angle, and the unscattered portion of the laser beam is phase-shifted by π/2 and then recombined with the scattered beam. The resulting intensity of the combined beams is proportional to the line-integral of the den- 2 sity fluctuations along the laser beam path: I ∝ n˜e dl + const.. The PCI system on Alcator C-Mod uses a 60W CO laser to measure 2 R line-integrated density fluctuations with wavenumbers in the range −1 −1 0.5 cm 6 |kR| 6 30 cm . The laser beam passes vertically through the core of the Alcator C-Mod tokamak and is imaged onto a 32 channel HgCdTe liquid nitrogen cooled photoconductive detector and digitized at 5 MHz. This system can be used to study a variety of plasma wave phenomena, including ICRF waves, Alfven waves, and turbulent waves. The turbulent wave measurements can be quanti- tatively compared to GYRO simulated density fluctuations, using a synthetic diagnostic that was developed by J. Chris Rost. ?
BIBLIOGRAPHY
[1] Stefano Coda. An experimental study of turbulence by phase-contrast imaging in the DIII-D tokamak. PhD thesis, Massachusetts Institute of Technology, 1997. 39 [2] a. Marinoni, S. Coda, R. Chavan, and G. Pochon. Design of a tangential phase contrast imaging diagnostic for the TCV tokamak. Review of Scientific Instruments, 77(10):10E929, 2006. URL: http://scitation.aip.org/content/aip/journal/ rsi/77/10/10.1063/1.2222333, doi:10.1063/1.2222333. 39 [3] Alex Mazurenko. Phase contrast imaging on the Alcator C-Mod tokamak. PhD thesis, Massachusetts Institute of Technology, 2001. 41, 42 [4] R. L. Miller, M. S. Chu, J. M. Greene, Y. R. Lin-Liu, and R. E. Waltz. Noncircular, finite aspect ratio, local equilibrium model. Physics of Plasmas, 5(4):973, 1998. URL: http://link. aip.org/link/PHPAEN/v5/i4/p973/s1{&}Agg=doi, doi:10.1063/ 1.872666. 47 [5] M Porkolab, J C Rost, N Basse, J Dorris, E Edlund, Liang Lin, Y Lin, and S Wukitch. Phase contrast imaging of waves and instabilities in high temperature magnetized fusion plas- mas. Plasma Science, IEEE Transactions on, 34(2):229–234, apr 2006. doi:10.1109/TPS.2006.872181. 39 [6] Oscar W Richards. Phase Microscopy 1954-56. Science, 124(3226):810–814, 1956. URL: http://www.sciencemag.org/ content/124/3226/810.short, doi:10.1126/science.124.3226. 810. 39 [7] J. C. Rost, L. Lin, and M. Porkolab. Development of a synthetic phase contrast imaging diagnostic. Physics of Plas- mas, 17(6):062506, jun 2010. URL: http://scitation.aip. org/content/aip/journal/pop/17/6/10.1063/1.3435217, doi: 10.1063/1.3435217. 47 [8] A. L. Sanin, K. Tanaka, L. N. Vyacheslavov, K. Kawahata, and T. Akiyama. Two-dimensional phase contrast interferometer for fluctuations study on LHD. Review of Scientific Instruments, 75(10):3439, 2004. URL: http://link.aip.org/link/RSINAK/ v75/i10/p3439/s1{&}Agg=doi, doi:10.1063/1.1784528. 39 [9] H. Weisen. The phase contrast method as an imaging diagnos- tic for plasma density fluctuations (invited). Review of Scien- tific Instruments, 59(8):1544, 1988. URL: http://scitation.aip.
51 52 Bibliography
org/content/aip/journal/rsi/59/8/10.1063/1.1140193, doi: 10.1063/1.1140193. 39
[10] F. Zernike. Phase contrast, a new method for the microscopic observation of transparent objects. Physica, 9:686–698, 1942. 39 OHMICDILUTIONEXPERIMENTS 4
4.1 motivations for studying dilution
The first attempts to compare the experimental energy transport in Alcator C-Mod with GYRO predictions of turbulent energy transport were done by Liang Lin, et al,[10]. It was found that GYRO over- predicted the ion energy transport and under-predicted the electron energy transport, and modifications to the profiles, within experimen- tal uncertainties, input to GYRO could not resolve this discrepancy. These experiments were performed before measurements of ion tem- perature and rotation profiles were available on Alcator C-Mod, how- ever the disagreement remained when the simulations were repeated including measured profiles. Porkolab, et al,[13] continued this work using TGLF[18], a much faster code than GYRO that uses a simplified quasilinear model, to see what could be the cause of the discrepancy. This investigation found that when a low-Z impurity species (Z = 8 in that case), rather than a high-Z impurity species like molybdenum (Z ≈ 30 in C-Mod plasmas), was assumed to be responsible for the moderate Zeff in these plasmas, the TGLF ion energy flux decreased substantially with- out a significant change in the electron energy flux. For example, a 30% decrease in nD/ne resulted in a 60% decrease in the TGLF ion energy flux. If a high-Z impurity species (Z = 30 in that case) was assumed to be responsible for the Zeff (and thus nD/ne was close to 1) there was no change in the TGLF energy fluxes versus the pure deuterium case. When the dilution from the low-Z impurities was included in GYRO simulations, small modifications to the input pro- files were sufficient to bring the experimental and simulated energy fluxes into agreement. At that time measurements of the relative con- centrations of the different impurity species in the plasma were un- available, however later measurements in similar plasmas found the average impurity charge to be Z = 8 - 10, so the assumed dilution was reasonable. In the past, there has been other experimental evidence of low-Z im- purity seeding reducing turbulent transport. One such observation is the so-called "Radiative Impurity" mode or R-I mode observations on TEXTOR[12] and DIII-D[11], where very large energy confinement in- creases were observed with impurity seeding in neutral beam-heated discharges. Larger confinement increases were observed when lower charge impurities were used, with neon (Z=10) giving the largest confinement increase. JET also observed that nitrogen seeding was
53 54 ohmic dilution experiments
needed to recover the confinement degradation seen after their switch from carbon to metallic walls[4]. They hypothesized that the dilution from the carbon impurities had been a previous hidden parameter in their confinement results. Fusion reactors will also have substantial dilution from helium ash, as well as extrinsic impurity puffing used for divertor heat flux mitigation. All these factors point to main-ion dilution being a potentially important parameter for energy transport in fusion plasmas that should be studied in detail. The remainder of the chapter is organized as follows: Sec. 4.2 de- scribes the design of the nitrogen seeding experiments performed on Alcator C-Mod to test the predicted dilution effect, and Sec. 4.3 de- scribes the observed effects of the nitrogen seeding on the energy transport, density fluctuations, and intrinsic toroidal rotation. •
4.2 experimental setup
There were many experiments performed as part of this work to study dilution: 3.5 run days in the 2012 C-Mod run campaign, and 1.5 run days in the 2015 run campaign. The runs in the 2012 and 2015 campaigns were done at different plasma currents (0.8, 1.0, and 1.2 MA) with nitrogen as the seeding impurity. The purpose of scanning the current in these plasmas was to explore the effects of changes in edge safety factor, q95, and to access a broad range of densities and temperatures. All shots were performed at a standard C-Mod toroidal magnetic field of 5.4 T and the same magnetic equilibrium shape, pic- tured in Fig. 14 as computed by EFIT[8]. Unlike most plasmas in Al- cator C-Mod, an upper-null equilibrium was used (meaning that the X-point was set to be towards the upper divertor rather than the lower divertor). This was chosen primarily so that the cryopump located at the upper divertor would be maximally effective at maintaining a con- stant density during the nitrogen seeding. The main disadvantage of this choice was that the edge Thomson scattering system would have significantly increased background noise, since the viewing optics of the system would be looking at the inner strike point. The primary interest of these experiments was the core plasma and not the edge, so this was an acceptable trade-off for density control throughout the seeding. Most of the C-Mod diagnostics were utilized during these exper- iments [2]. The electron density profile, ne, was measured using a Thomson scattering system and the data was fit using a combina- tion of splines and hyperbolic tangent functions. The electron tem- perature profile, Te was measured using a combination of Thomson scattering and electron cyclotron emission (ECE) measurements and fit in a similar manner. The sawtooth instability, present in all of these plasmas, caused the central electron temperature to change by 4.2 experimental setup 55
Figure 14: Plot of a cross-section of the magnetic equilibrium chosen for the nitrogen seeding experiments. The red line is the last closed flux surface (note the X-point above the midplane). The cyan lines are the closed flux surfaces, and the grey lines are the open field lines. The cryopump is the circular structure at the top. about 10% during a sawtooth period. The electron temperature pro- file used in the simulations and analysis was an average over many sawtooth periods. We consider this a reasonable approach, since the analysis and simulations focused on regions of the plasma far away from the sawtooth inversion radius, typically at r/a ≈ 0.3 for these plasmas. The ion temperature and toroidal velocity profiles, Ti and VT , were measured using a high-resolution X-ray crystal scattering spectrometer[15] that measures helium-like and hydrogen-like argon lines and uses the absolute width and shift of those lines to deter- mine the temperature and velocity profiles, respectively. The impu- rity brightnesses were measured with C-Mod’s extensive suite of im- purity spectroscopy diagnostics[14][9]. Radiated power profiles were measured with foil and AXUV bolometers. 56 ohmic dilution experiments
Plasma Current Foil Bolometer Radiated Power 1.4 a) 1.2 d) 1.2 1.0 1.0 0.8 0.8
[MW] 0.6 [MA] 0.6 P rad I 0.4
0.4 P 0.2 0.2 0.0 0.0 Edge Safety Factor Nitrogen VII (n=3) Line Brightness 6 b) e) 5 0.005 4 0.004
95 3 0.003 q 2 0.002 1 0.001
0 N Brightness [AU] 0.000 Line-Averaged Electron Density Central Ion Temperature
] 2.5 1.5
-3 c) 2.0 1.4 f) m 1.5 1.3 20 [keV]
1.0 i 1.2 T [10
e 0.5 1.1 n 0.0 1.0 0.6 0.8 1.0 1.2 0.6 0.8 1.0 1.2 Time [s] Time [s]
Figure 15: Time traces of experimental quantities from representative discharges at each value of plasma current used in the scan (0.8 MA in green, 1.0 MA in purple, and 1.2 MA in black). Plots of the plasma current (a), the edge safety factor q95 as computed by EFIT (b), the line-averaged electron density (c), the total radiated power as measured by the foil bolometer (d), the nitrogen VII n = 3 transition line brightness (e), and the central ion temperature (f). The spectroscopic diagnostics that measure the nitrogen brightness were changed between the 2012 campaign (when the 0.8 MA and 1.0 MA experiments were performed) and the 2015 campaign (when the 1.2 MA experiments were performed), so the relative magnitudes of the line brightnesses in (e) between the two campaigns is not necessarily indicative of a difference in nitrogen concentration.
The goal of the experiments was to probe the effect of nitrogen seeding on turbulent fluctuations and energy transport. This required plasmas where the density, temperature, and impurity content were constant for many confinement times. To accomplish this, machine parameters such as the amount of impurity seeding, plasma current, and density were held fixed during a shot and were changed shot- to-shot. Figure 15 shows time-traces of representative discharges at 0.8 MA (green), 1.0 MA (purple), and 1.2 MA (black). The panels on the left show that the magnetic geometry and electron density were held fixed throughout the nitrogen seeding, which begins at t = 0.75s and reaches a saturation between t = 1.0s and t = 1.3s as shown in the radiated power and nitrogen line brightness traces on the top and middle right panels respectively. There is a slight increase in the central ion temperature during the seeding in each case, as can be seen in the bottom right trace. The density was scanned shot-to-shot to span both the LOC and SOC regimes. The lower range of the density scans was restricted by the density at which runaway electrons began to appear (which were a complicating factor that we wanted to avoid), and the upper range of the density scans was restricted by the density above which the edge temperature became too low for any significant nitrogen seed- 4.2 experimental setup 57
0.8 MA Plasmas 1.0 MA Plasmas 1.2 MA Plasmas
LOC SOC LOC SOC LOC SOC
35
30 [ms] E τ
25
20 Before N2 Seeding Before N2 Seeding Before N2 Seeding After N2 Seeding After N2 Seeding After N2 Seeding 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.6 0.8 1.0 1.2 1.4 1.6 1.8 20 -3 20 -3 20 -3 ne [10 m ] ne [10 m ] ne [10 m ]
Figure 16: Plot of the energy confinement time versus density for plasmas at 0.8 MA (left, diamonds), 1.0 MA (middle, stars), and 1.2 MA (right, squares). The densities and confinement values before the nitrogen seeding are represented in red, while the values after the nitrogen seeding are represented in blue. The black dashed vertical lines indicate the approximate values of neLOC-SOC. ing to be applied without causing detachment and disruptions. The critical density at which the LOC-SOC transition occurs, neLOC-SOC, also increases with increasing current (through a decrease in q95). Consequently, it was possible to get plasmas at densities both above and below neLOC-SOC at every value of the plasma current. Figure 16 shows the density ranges of the scans at each current, and the corre- sponding energy confinement times. The different colors correspond to the values before the nitrogen seeding (red) and after the nitro- gen seeding (blue). The values of neLOC-SOC are shown by the vertical dashed lines, and are 0.95, 1.15, and 1.30 in units of 1020m−3 for the 0.8, 1.0, and 1.2 MA plasmas respectively. Figure 16 shows that at each current there exist several points both above and below neLOC-SOC. To get the largest variation in nD/ne, the amount of nitrogen seed- ing was chosen to be the largest that would not cause a disruption through divertor detachment[5]. This upper limit on the seeding de- pended on the plasma edge temperature, so the lower density plas- mas could tolerate more nitrogen seeding than the higher density plasmas. Higher current plasmas could also tolerate more nitrogen seeding than lower current plasmas. The amount of nitrogen seed- ing was controlled by modifying the duty cycle of the valve that controlled the nitrogen seeding, a common technique for this seed- 58 ohmic dilution experiments
Deuteron Density With And Without Nitrogen Seeding 1.0 LOC SOC 0.9
0.8 e /n D n 0.7 0.8 MA Unseeded 0.6 0.8 MA Seeded 1.0 MA Unseeded 0.5 1.0 MA Seeded 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 20 -3 neq95 [10 m ]
Figure 17: Plot of the experimental nD/nevalues for 0.8 MA (diamons) and 1.0 MA (stars) plasmas before seeding (red) and after seeding (blue) The black dashed vertical lines indicate the value of neq95 that separates the LOC and SOC regimes.
ing on Alcator C-Mod. The change in nD/ne was calculated from the Zeff and a combination of multiple line brightnesses of the intrin- sic and extrinsic impurities. The details of this calculation are in Ap- pendixA. The result is that the nitrogen seeding reduced nD/ne by 10%- 20%, with the exception of the highest density 0.8 MA case where it was only reduced by 5%. Figure 17 shows the nD/ne values before and after the seeding for representative 0.8 MA and 1.0 MA discharges. These experiments successfully produced a large data set of diluted plasmas at a range of densities and plasma currents that were steady- state through several energy confinement times. This allowed for care- ful analysis of the effects of the dilution on the turbulence and trans- port, as well as profile measurements that could be used as inputs to the gyrokinetic simulations. •
4.3 experimental results
The nitrogen seeding had several effects that are indicative of changes to the plasma turbulence, including changes to the density fluctua- tions, energy confinement, temperature profiles, energy fluxes, and toroidal rotation. The transport code TRANSP[6] was used to deter- mine the experimental ion and electron energy fluxes using a power balance calculation. TRANSP takes as inputs the temperature, rota- tion, density, and radiated power profiles, as well as the magnetic equilibrium shape and the impurity ion concentrations. In addition 4.3 experimental results 59 to performing the power-balance calculation, TRANSP output files have a complete set of the plasma’s magnetic geometry, fluxes, and kinetic profiles, on the same radial grid. This makes it simple to create GYRO input profiles from TRANSP output files.
4.3.1 Effect of Nitrogen Seeding on Energy Transport
The over-all energy confinement time was not strongly affected by the nitrogen seeding in these plasmas. Figure 18 shows no clear sys- tematic difference between the unseeded confinement times (red) and the seeded confinement times (blue). This does not necessarily mean that the seeding had no effect on the transport. One possible expla- nation is that the increase in the power dissipated through increased radiated power (See Fig. 15 d)) was comparable to the decrease in the power dissipated through a reduction in the turbulent transport. In ohmic plasmas the total power dissipated through the ion channel is smaller than the power dissipated through the electron channel, but the ion channel is predicted to be the one most strongly affected by the dilution from the seeding. Thus, the the overall confinement in ohmic plasmas was unlikely to be substantially improved by the seeding. This is the opposite of the case in neutral beam heated plas- mas in other tokamaks around the world which have Ti > Te and Qi > Qe, but is similar to the case of a fusion reactor which will have Ti < Te and Qi < Qe due to alpha heating. Despite the fact that the seeding did not change the global confinement in the plasmas in this study, the effect of the seeding on the turbulent transport could be determined from changes to the TRANSP computed energy fluxes and the measured temperature profiles. To investigate the effect of the nitrogen seeding on energy trans- port, the TRANSP computed electron and ion energy fluxes (Qe and Qi respectively) were compared before and after the nitrogen seeding. The fluxes were normalized to the gyrobohm unit energy flux (using 2 the same definition as GYRO)[3], QGB = neTecs(ρs/a) , where ne p is the electron density, Te is the electron temperature, cs = Te/mi is the sound speed, ρs = cs/(eB/(mDc)) is the ion sound gyrora- dius, and a is the minor radius. Because the gyrobohm unit flux is 5/2 proportional to neTe , it decreases rapidly towards the edge of the plasma. Figure 19 shows the gyrobohm-normalized energy flux pro- files for a representative discharge. The flux goes from 1/100th of the gyrobohm unit flux in the very core up to 100 times the gyrobohm unit flux in the edge. This change is almost entirely the change in the radial change of the gyrobohm unit flux itself, as the energy flux in absolute units only changes by a factor of 10 from the very core to the edge. The dashed lines in Fig. 19 correspond to the approximate val- ues of the normalized flux below which the turbulence is marginally stable (Q/QGB 6 1) and above which the turbulence is strongly un- 60 ohmic dilution experiments
Energy Confinement Times
LOC SOC 35
0.8 MA Unseeded 30 0.8 MA Seeded 1.0 MA Unseeded
[ms] 1.0 MA Seeded E τ 1.2 MA Unseeded 1.2 MA Seeded 25
20
2 3 4 5 6 20 -3 neq95 [10 m ]
Figure 18: Plot of the energy confinement times for ohmic plasmas before nitrogen seeding (red) and after nitrogen seeding (blue) versus the product of the line-averaged electron density ne and the edge safety factor q95. 0.8 MA plasmas are represented by the diamonds, 1.0 MA plasmas are represented by the stars, and 1.2 MA plasmas are represented by the squares. The black dashed line indicates the approximate boundary between the LOC and SOC regimes.
stable (Q/QGB > 10). It shows that the turbulence driving the flux is marginally stable for r/a < 0.5 for electrons and r/a < 0.6 for ions, and the turbulence driving the flux is strongly unstable for r/a > 0.8 for both electrons and ions. For the region of r/a < 0.3 there are saw- tooth fluctuations that could be driving energy flux well above the gyrobohm scaling. If the nitrogen seeding directly affected the stability of the turbu- lent transport, then either the stiffness or the critical gradient of the turbulent transport would be changed. To investigate whether this occurred, the change in the gyrobohm normalized energy flux with the nitrogen seeding was compared to the change in the normalized temperature gradient. If the normalized energy flux decreases while the normalized gradient increases, then that must be due to either an increase in the critical gradient or a decrease in the stiffness. If the normalized energy flux and normalized temperature gradient both increase or decrease together, that does not necessarily imply any change in either the stiffness or critical gradient of the turbulent trans- port. Thus, to determine if the nitrogen seeding had any direct effect on the turbulent transport, the change in the gyrobohm normalized energy fluxes was compared to the change in the normalized temper- ature gradients. For example, Fig. 20 shows an example case from a 1.0 MA SOC plasma near r/a = 0.8, wherein the seeding increases a/LTi while Qi/QGB decreases, which indicates that the seeding had a stabilizing effect on the turbulence at this radial location. 4.3 experimental results 61
Electron Energy Flux in GB Units Ion Energy Flux in GB Units 1000.00 1000.00
100.00 100.00 Turbulence is Turbulence is 10.00 Strongly Unstable 10.00 Strongly Unstable GB GB /Q /Q i e Q Q 1.00 1.00 Turbulence is Marginally Stable 0.10 0.10 Turbulence is Marginally Stable 0.01 0.01 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 r/a r/a
Figure 19: Plots of the gyrobohm normalized energy flux profiles for the electrons (left) and ions of a representative plasma discharge from these experiments. The dashed lines approximately correspond to the values of the normalized flux which correspond to the turbulence being marginally stable (Q/QGB 6 1) and the turbulence being strongly unstable (Q/QGB > 10).]
Normalized Ion Temperature Gradient Normalized Ion Energy Flux 8 40
6 30 GB Ti
4 /Q 20 i a/L Q 2 10 0 0 0.76 0.78 0.80 0.82 0.84 0.76 0.78 0.80 0.82 0.84 r/a Unseeded r/a Seeded
Figure 20: Plots of the profiles near r/a = 0.8 of the normalized ion temperature gradient a/LTi on the left and the normalized ion energy flux Qi/QGB on the right, for the unseeded and seeded phases of the discharge in red and blue respectively. 62 ohmic dilution experiments
Change in Electron Transport at r/a = 0.6 Change in Ion Transport at r/a = 0.6 1.0 1.0
0.5 0.5 GB GB /Q /Q i e 0.0 0.0 Q Q ∆ ∆
-0.5 -0.5
0.8 MA Plasmas 0.8 MA Plasmas -1.0 1.0 MA Plasmas -1.0 1.0 MA Plasmas -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 ∆ ∆ a/LTe a/LTi
Figure 21: Plots of the change with seeding of the gyrobohm normalized energy fluxes and temperature gradient scale lengths for electrons (left) and ions (right) at r/a = 0.6, for plasmas at 0.8 MA (diamonds) and 1.0 MA (stars). Positive values of ∆Q/QGB and ∆a/LT indicate that the quantity increased in the seeded phase, and negative values likewise indicate that the quantity decreased in the seeded phase. Points in the bottom right quadrant (i.e. where Q/QGB decreases and a/LT increases) indicate a reduction in the stiffness or an increase in the critical gradient in that case.
The results of this analysis are presented in Fig. 21 and Fig. 22 which show the change in the normalized electron and ion energy fluxes and normalized temperature gradients at r/a = 0.6 and r/a = 0.8, respectively. The left plots of Fig. 21 and Fig. 22 show that at both r/a = 0.6 and r/a = 0.8, the normalized electron energy flux and normalized electron temperature gradients both increased dur- ing the seeding, which does not necessarily imply a change in either the stiffness or the critical gradient of the turbulence driving the elec- tron transport. The right plot of Fig. 21 shows that at r/a = 0.6 the normalized ion energy fluxes decreased during the seeding, but there is no systematic change in the normalized ion temperature gradients within uncertainties. The right plot of Fig. 22 shows that at r/a = 0.8 the normalized ion energy fluxes mostly decreased with seeding, while the normalized ion temperature gradients increased with the seeding. This indicates that in this region there was either a decrease in the stiffness or an increase in the critical gradient of the ion turbu- lence, both of which indicate an overall increase in the stability of the turbulence. This region of the plasma, r/a = 0.8, is also where the tur- bulence is strongly unstable, as shown in Fig. 19, so this is consistent with the simulation result that dilution reduces the ion turbulence. In the simulations the ion energy fluxes decreased when nD/ne was decreased, while the input temperature profiles were held fixed. How- ever in the experiments the input ohmic power was constant during the seeding, so the reduction in the turbulence drive increased the temperature gradients instead. 4.3 experimental results 63
Change in Electron Transport at r/a = 0.8 Change in Ion Transport at r/a = 0.8 15 0.8 MA Plasmas 15 0.8 MA Plasmas 1.0 MA Plasmas 1.0 MA Plasmas 10 10
5 5 GB GB /Q /Q i e 0 0 Q Q ∆ ∆ -5 -5
-10 -10
-15 -15 -2 -1 0 1 2 -2 -1 0 1 2 ∆ ∆ a/LTe a/LTi
Figure 22: Plots of the change with seeding of the gyrobohm normalized energy fluxes and temperature gradient scale lengths for electrons (left) and ions (right) at r/a = 0.8, for plasmas at 0.8 MA (diamonds) and 1.0 MA (stars). Positive values of ∆Q/QGB and ∆a/LT indicate that the quantity increased in the seeded phase, and negative values likewise indicate that the quantity decreased in the seeded phase. Points in the bottom right quadrant (i.e. where Q/QGB decreases and a/LT increases) indicate a reduction in the stiffness or an increase in the critical gradient in that case.
4.3.2 Effect of Nitrogen Seeding on Density Fluctuations
The density fluctuation amplitude was measured with the phase con- trast imaging (PCI) diagnostic on C-Mod, which was described in detail in Ch. 3, Sec. 3.2. The configuration of the PCI system was such that the maximum measurable wavenumber was approximately 20 cm−1 for the experiments done in the 2012 campaign (the 0.8 and 1.0 MA plasmas) and approximately 30 cm−1 for the experiments done in the 2015 campaign (the 1.2 MA plasmas). The increase in the maximum measurable wavenumber was due to the acquisition of a new detector array that had increased sensitivity to high-frequency signals, enabling detection of high-wavenumber turbulence that had been doppler-shifted to high frequencies by the plasma rotation. PCI spectra from SOC discharges at three different currents are shown in Fig. 23. For the highest plasma current case (IP = 1.2 MA), there is a particularly high phase-velocity feature of the turbulence that is clearly separate from the lower-frequency portion of the spectrum. Figure 23 shows that after the seeding, the magnitude of this high phase-velocity feature is substantially decreased. This is a general re- sult in the 1.2 MA plasmas, and Fig. 24 shows the magnitude of the high phase-velocity feature before and after seeding for all the 1.2 MA plasmas. It shows that the amplitude of the feature is between a factor of two and a factor of four less after the nitrogen seeding than it is before the seeding. Because PCI is a line-integrated measurement, information from other diagnostics is needed to localize this observed high-phase ve- locity feature to a particular region of the plasma. C-Mod also has an 64 ohmic dilution experiments
0.8 MA Unseeded SOC 1.0 MA Unseeded SOC 1.2 MA Unseeded SOC 400 400 400 1.0000 ] -1
300 300 300 0.1000 /kHz/cm 2 ) -2 m
200 200 200 16 0.0100 Frequency [kHz] Frequency [kHz] Frequency [kHz] 100 100 100 0.0010 PCI Intensity [(10 0 0 0 0.0001 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -1 -1 -1 kR [cm ] kR [cm ] kR [cm ]
0.8 MA Seeded SOC 1.0 MA Seeded SOC 1.2 MA Seeded SOC 400 400 400 1.0000 ] -1
300 300 300 0.1000 /kHz/cm 2 ) -2 m
200 200 200 16 0.0100 Frequency [kHz] Frequency [kHz] Frequency [kHz] 100 100 100 0.0010 PCI Intensity [(10 0 0 0 0.0001 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -1 -1 -1 kR [cm ] kR [cm ] kR [cm ]
Figure 23: Plots of the PCI frequency-wavenumber spectral intensity for 0.8 MA (left), 1.0 MA (middle), and 1.2 MA (right) discharges before the nitrogen seeding (top) and after the nitrogen seeding (bottom). All discharges are in the SOC regime and have electron densities of 0.8, 1.3, and 1.8 in units of 1020m−3. The plots all have the same x and y ranges as well as the same (logarithmic) color scale. The 1.2 MA spectra had the largest change in the spectra with the nitrogen seeding.
High Vphase PCI Feature Intensity in 1.2 MA Plasmas 1.5·10-8 Intensity Before Nitrogen Seeding Intensity After Nitrogen Seeding ] 2
~ 1.0·10-8
5.0·10-9 PCI Intensity [(n/n)
0 1.2 1.4 1.6 1.8 20 -3 ne [10 m ]
Figure 24: Plot of the amplitude of the high phase-velocity feature shown in the top right plot of Fig. 23 for 1.2 MA plasmas before nitrogen seeding (red) and after nitrogen seeding (blue). The magnitude of the feature was computed by integrating the frequency-wavenumber spectrum along a fixed phase velocity (f/|k| = const) with f > 100 kHz. 4.3 experimental results 65
O-mode reflectometer[2], which measures the phase delay between a launched and reflected waves at multiple frequencies. Each wave will reflect off the cutoff layer at the outter midplane radial location √ where the wave frequency freflect = fpe = 89.8 × n20, where fpe is the electron plasma frequency, and n20 is the electron density in units of 1020 m−3. From the total phase delay between the launched and reflected wave, density profile information can be determined. From the fluctuations in this phase delay, density fluctuation information can be determined. The C-Mod reflectometer has several channels that operate at different frequencies, in particular 50, 60, 75, 88, 110, 121, and 140 GHz corresponding to densities of 0.31, 0.45, 0.70, 0.96, 1.5, 1.82, and 2.4 in units of 1020 m−3 respectively. This gives broad coverage of many radial locations in the plasma, from the edge to the core, though which channel corresponds to which value of r/a will vary with the density of the plasma. When the reflectometer fluctuation spectra were computed for these 1.2 MA shots, it was found that for channels that were looking at re- gions where 0.75 < r/a < 0.85 the fluctuations had a time history that correspond closely to the time history of the high phase-velocity fluc- tuations observed on PCI, with f > 100 kHz. Reflectometer channels that measured fluctuations outside this radial region showed little change with the seeding. This is shown in Fig. 25, which compares the fluctuation intensity for the part of the spectrum with f > 150 kHz. The reason for restricting the frequency range in this way is that the reflectometer does not measure the wavenumber of the turbulence, only the frequency. Therefore it is not possible to isolate parts of the reflectometer spectrum based on phase velocity as was done for the PCI spectrum, so instead f > 150 kHz was used because for those fre- quencies the high phase velocity feature is the only thing present in the PCI spectrum. Both the PCI and the reflectometer channel look- ing at r/a = 0.85 show a prompt decrease in the fluctuation amplitude when the nitrogen seeding begins at t = 0.75s; this seems to saturate at t = 0.9s, resulting in an overall decrease of about a factor of two with the seeding. The reflectometer channel looking at r/a = 0.65 shows a slight decrease in the fluctuation power with the seeding, but much less than the change in both the fluctuations measured by PCI and the change in the fluctuations measured by the reflectometer at r/a = 0.85. The reflectometer channels looking at r/a > 1.0 and r/a < 0.4 did not show changes in the fluctuation amplitudes with seeding. This suggests that the high phase-velocity feature that decreases with the seeding is located near r/a = 0.85 and is unlikely to be coming from outside r/a = 1.0 or inside r/a = 0.4 in this case. Other density cases at the same current also show a similar decrease in the reflec- tometer fluctuations at these r/a values. This is consistent with the results of Subsection 4.3.1, namely that the ITG modes are strongly unstable for r/a > 0.75 and the seeding is stabilizing that turbulence. 66 ohmic dilution experiments
F > 150 kHz Fluctuation Power For 1.2 MA SOC Plasma 2.5 r/a > 1.0 Reflectometer r/a = 0.85 Reflectometer 2.0 r/a = 0.65 Reflectometer r/a < 0.4 Reflectometer 1.5 PCI
1.0
0.5 Spectral Power [AU] 0.0
0.003
0.002
0.001
N VII (n=3) Brightness [AU] 0.000 0.7 0.8 0.9 1.0 1.1 1.2 1.3 t [s]
Figure 25: Plot of the total power of the F > 150 kHz fluctuations observed by PCI (black), and reflectometer channels looking at r/a > 1.0 (blue), r/a = 0.85 (green), r/a = 0.65 (gold), and r/a < 0.4 (red) on top. The spectral powers have been rescaled such that the fluctuation intensity for t ∈ [1.1s, 1.2s] is equal to 1, for ease of comparison. The bottom plot shows the nitrogen line brightness for reference, to show when the seeding happens. 4.3 experimental results 67
The 0.8 MA and 1.0 MA cases did not show as clear a change in the PCI spectrum with the seeding, but that is due to the fact that there is not as clear of a phase velocity separation of the spectrum as there is in the 1.2 MA cases. This is because there is less of a doppler shift of the turbulence at r/a ≈ 0.8, meaning that any turbulence at that location is mixed in with the low-frequency edge turbulence rather than raised above it as it is in the 1.2 MA cases.
4.3.3 Effect of Nitrogen Seeding on Toroidal Rotation
One of the most robust observed effects of the nitrogen seeding was that it was able to affect the plasma’s intrinsic toroidal rotation. The intrinsic rotation in C-Mod tokamak plasmas has been well studied[17][16], and it has been found to have two characteristic states: at low densi- ties, the intrinsic toroidal rotation profile is flat and co-current di- rected in the core, while at high densities the intrinsic toroidal rota- tion profile is hollow and counter-current directed in the core. The critical density that divides the two regimes was found to correlate closely with the LOC-SOC critical density. The nitrogen seeding was found to reverse the direction intrinsic toroidal rotation of plasmas with densities above the nominal un- seeded rotation reversal critical density and changed the profiles from flat to hollow, while it made no change to the rotation of plasmas below the critical density. Figure 26 shows examples of rotation pro- files before and after seeding for a case below the rotation reversal critical density (left) and one above it (right). In the case below the rotation reversal critical density, the seeding makes almost no change to the rotation. For the case with the density above the critical den- sity, the seeding caused the profile to change from hollow to flat and from counter-current directed to co-current directed in the core. The seeded higher density case has a similar rotation profile to the un- seeded lower density case, which suggests that the seeding was trig- gering a reversal similar to one that would be caused by lowering the density. The fact that the seeding effectively lowers the rotation reversal critical density can give some indication of the physics responsible for the intrinsic rotation. The rotation reversal had previously been found correlate to a particular value of the effective collisionality, 2 νeff ≈ RZeffne/(Te) the ratio of the electron-ion collision frequency to the curvature drift frequency, where R is the major radius in m, 20 −3 ne is the plasma density in units of 10 m , and Te is the electron temperature in keV [17]. It was observed that at fixed magnetic field and plasma current, there is a critical value of νeff above which the rotation is counter-current and below which the rotation is co-current. This comparison is shown in Fig. 27, which plots VTor against νeff for several points at different plasma currents. Figure 27 shows that both 68 ohmic dilution experiments
Rotation Velocity Profiles Before And After Nitrogen Seeding
ne < Critical ne ne > Critical ne 20 20
10 10
0 0 [km/s] [km/s] Tor Tor V -10 V -10
-20 -20
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 r/a r/a Rotation Before Nitrogen Seeding Rotation After Nitrogen Seeding
Figure 26: Plots of the profiles of the intrinsic toroidal rotation before nitrogen seeding (red) and after nitrogen seeding (blue) for a case where the density was below the rotation reversal critical density (left) and a case where the density was above the rotation reversal critical density (right). Both cases have a plasma current of 0.8 MA. Positive VTor corresponds to co-current directed rotation and negative VTor corresponds to counter-current directed rotation.
the unseeded and seeded plasma rotation shows a critical value of νeff, and that critical value of νeff increases with the nitrogen seeding. The left panel of Fig. 27 shows that for the 0.8 MA plasmas the un- seeded critical νeff is between 0.21 and 0.23, while the seeded critical νeff is approximately 0.33. The right panel of Fig. 27 shows that for the 1.2 MA plasmas the unseeded critical νeff is between 0.5 and 0.6, while the seeded critical νeff is approximately 0.63. The 1.0 MA unseeded cases do not have enough values of νeff to definitively say that there is a similar increase in the critical νeff. This suggests that νeff does not capture all of the physics responsible for the seeding-induced rotation reversal, though it does seem to apply to the unseeded data. Another model proposed by Barnes et. al.[1] and studied on MAST[7] 3/2 points to the ion-ion collisionality, ν* = q R νii/(vti ), as an im- portant parameter in the rotation reversal phenomenon. The ion tem- perature profiles were not well enough measured in all of these plas- mas to properly evaluate ν* for every case, but Fig. 28 shows that comparing the core toroidal velocity to nDq95 (which ν* is propor- tional to) demonstrates that both the seeded and unseeded plasmas at every value of the plasma current align to a single critical value of nDq95, between 2.0 and 2.5. This effect may be different from the one responsible for intrinsic rotation reversal phenomena observed in other studies, and a more thorough analysis of the rotation data 4.3 experimental results 69
0.8 MA Plasmas 1.0 MA Plasmas 1.2 MA Plasmas Before N2 Seeding Before N2 Seeding After N2 Seeding After N2 Seeding 20
10
0 (0) [km/s] Tor V -10
Before N2 Seeding After N2 Seeding -20
0.0 0.1 0.2 0.3 0.4 0.30 0.35 0.40 0.5 0.6 0.7 0.8 0.9 1.0 ν ν ν eff eff eff
Figure 27: Plot of the core toroidal rotation velocity versus effective collisionality for the different plasma currents: 0.8 MA (left), 1.0 MA (center), and 1.2 MA (right). Red symbols denote the core toroidal rotation before the nitrogen seeding, while blue symbols denote the core toroidal rotation after the nitrogen seeding. Positive VTor corresponds to co-current directed rotation and negative VTor corresponds to counter-current directed rotation. 70 ohmic dilution experiments
Core Toroidal Velocity
20
10
0.8 MA Unseeded 0.8 MA Seeded 0 1.0 MA Unseeded [km/s] T 1.0 MA Seeded V 1.2 MA Unseeded 1.2 MA Seeded -10
-20
1 2 3 4 5 6 × 20 -3 nD q95 [10 m ]
Figure 28: Plot of the core toroidal rotation velocity versus nDq95 (which is proportional to ν*) for plasmas before the nitrogen seeding (red) and after nitrogen seeding (blue). 0.8 MA plasmas are represented by the diamonds, 1.0 MA plasmas are represented by the stars, and 1.2 MA plasmas are represented by the squares. Positive VT represents co-current velocities, while negative VT represents counter-current velocities.
would be necessary to make a definitive conclusion about the mecha- nism responsible for the seeding-induced rotation reversal. The phys- ical mechanism behind intrinsic rotation remains an active area of research, and a full theoretical investigation of the rotation is beyond the scope of this work. •
4.4 summary and conclusions
Main-ion dilution is predicted by gyrokinetic simulations to have a strong stabilizing effect on ITG turbulence. Seeding experiments were performed on Alcator C-Mod to test this prediction, wherein ohmic plasmas were seeded with nitrogen while the density was held fixed, which reduced nD/neby 10%- 20%. The experiments demon- strated that the seeding reduced the ion energy transport, reduced the density fluctuations, and affected the intrinsic rotation. At r/a = 0.8, where the ITG modes are strongly unstable, the gyrobohm nor- malized ion energy flux decreased with the seeding, while the nor- malized ion temperature gradient increased, which implies that the seeding either reduced the stiffness or increased the critical gradient. The PCI density fluctuations from r/a ≈ 0.8 (again where the ITG modes are strongly unstable) also decreased significantly with the seeding in the highest plasma current cases, whereas density fluctu- ations futher inside and outside were not strongly affected by the 4.4 summary and conclusions 71 seeding. Both the changes to the density fluctuations and the ion en- ergy flux are consistent with the nitrogen seeding stabilizing the ITG turbulence. The nitrogen seeding also caused the intrinsic rotation re- versal critical electron density to decrease, and this rotation change was not due to a change in νeff but rather seemed to be related to a ? change in nDq95 which was related to νii. ?
BIBLIOGRAPHY
[1] M. Barnes, F. I. Parra, J. P. Lee, E. a. Belli, M. F. F. Nave, and a. E. White. Intrinsic Rotation Driven by Non-Maxwellian Equilibria in Tokamak Plasmas. Physical Review Letters, 111(5):055005, aug 2013. URL: http://link.aps.org/doi/10.1103/PhysRevLett. 111.055005, doi:10.1103/PhysRevLett.111.055005. 68
[2] N P Basse, A Dominguez, E M Edlund, C L Fiore, R S Granetz, A E Hubbard, J W Hughes, I H Hutchinson, J H Irby, B LaBom- bard, L Lin, Y Lin, B Lipschultz, J E Liptac, E S Marmar, D A Mossessian, R R Parker, M Porkolab, J E Rice, J A Snipes, V Tang, J L Terry, S M Wolfe, S J Wukitch, K Zhurovich, R V Bravenec, P E Phillips, W L Rowan, G I Kramer, G Schilling, S D Scott, and S J Zweben. Diagnostic systems on Alca- tor C-Mod. Fusion Science and Technology, 51(3):476–507, 2007. URL:
[3] J. Candy and R. Waltz. Anomalous Transport Scaling in the DIII-D Tokamak Matched by Supercomputer Simulation. Physical Review Letters, 91(4):045001, jul 2003. URL: http: //link.aps.org/doi/10.1103/PhysRevLett.91.045001, doi:10. 1103/PhysRevLett.91.045001. 59
[4] C. Giroud, G.P. Maddison, S. Jachmich, F. Rimini, M.N.a. Beurskens, I. Balboa, S. Brezinsek, R. Coelho, J.W. Coenen, L. Frassinetti, E. Joffrin, M. Oberkofler, M. Lehnen, Y. Liu, S. Marsen, K. McCormick, A. Meigs, R. Neu, B. Sieglin, G. van Rooij, G. Arnoux, P. Belo, M. Brix, M. Clever, I. Cof- fey, S. Devaux, D. Douai, T. Eich, J. Flanagan, S. Grunhagen, A. Huber, M. Kempenaars, U. Kruezi, K. Lawson, P. Lomas, C. Lowry, I. Nunes, A. Sirinnelli, A.C.C. Sips, M. Stamp, and S. Wiesen. Impact of nitrogen seeding on confinement and power load control of a high-triangularity JET ELMy H-mode plasma with a metal wall. Nuclear Fusion, 53(11):113025, 2013. URL: http://stacks.iop.org/0029-5515/53/i=11/a= 113025?key=crossref.b25f5cabfe1eb9a1df537a555d386b64, doi:10.1088/0029-5515/53/11/113025. 54
[5] J. A. Goetz, C. Kurz, B. LaBombard, B. Lipschultz, A. Niem- czewski, G. M. McCracken, J. L. Terry, R. L. Boivin, F. Bombarda, P. Bonoli, C. Fiore, S. Golovato, R. Granetz, M. Greenwald, S. Horne, A. Hubbard, I. Hutchinson, J. Irby, E. Marmar, M. Porkolab, J. Rice, J. Snipes, Y. Takase, R. Watterson, B. Welch,
73 74 Bibliography
S. Wolfe, C. Christensen, D. Garnier, D. Jablonski, D. Lo, D. Lumma, M. May, A. Mazurenko, R. Nachtrieb, P. OShea, J. Reardon, J. Rost, J. Schachter, J. Sorci, P. Stek, M. Umansky, and Y. Wang. Comparison of detached and radiative divertor operation in Alcator C-Mod. Physics of Plasmas, 3(5):1908–1915, 1996. URL: http://scitation.aip.org/content/aip/journal/ pop/3/5/10.1063/1.871986$\delimiter"026E30F$nhttp: //scitation.aip.org/deliver/fulltext/aip/journal/pop/ 3/5/1.871986.pdf?itemId=/content/aip/journal/pop/3/5/10. 1063/1.871986{&}mimeType=pdf{&}containerItemId=content/ aip/journal/pop, doi:10.1063/1.871986. 57
[6] R.J Goldston, D.C McCune, H.H Towner, S.L Davis, R.J Hawry- luk, and G.L Schmidt. New techniques for calculating heat and particle source rates due to neutral beam injection in axisymmet- ric tokamaks. Journal of Computational Physics, 43(1):61–78, 1981. URL: http://dx.doi.org/10.1016/0021-9991(81)90111-X, doi: 10.1016/0021-9991(81)90111-X. 58
[7] J.C. Hillesheim, F.I. Parra, M. Barnes, N.a. Crocker, H. Meyer, W.a. Peebles, R. Scannell, and a. Thornton. Depen- dence of intrinsic rotation reversals on collisionality in MAST. Nuclear Fusion, 55(3):032003, 2015. URL: http: //stacks.iop.org/0029-5515/55/i=3/a=032003?key=crossref. 5d8898a9db7caab7b45d6b63e116d164, arXiv:1407.2121v2, doi:10.1088/0029-5515/55/3/032003. 68
[8] L.L. Lao, H. St. John, R.D. Stambaugh, A.G. Kellman, and W. Pfeiffer. Reconstruction of current profile parameters and plasma shapes in tokamaks. Nuclear Fusion, 25(11):1611–1622, 1985. URL: http://stacks.iop.org/0029-5515/25/i=11/a=007, doi:10.1088/0029-5515/25/11/007. 54
[9] J K Lepson, P Beiersdorfer, J Clementson, M F Gu, M Bitter, L Roquemore, R Kaita, P G Cox, and a S Safronova. EUV spec- troscopy on NSTX. Journal of Physics B: Atomic, Molecular and Op- tical Physics, 43(14):144018, 2010. URL: http://stacks.iop.org/ 0953-4075/43/i=14/a=144018, doi:10.1088/0953-4075/43/14/ 144018. 55
[10] L Lin, M Porkolab, E M Edlund, J C Rost, M Greenwald, N Tsu- jii, J Candy, R E Waltz, and D R Mikkelsen. Studies of turbu- lence and transport in Alcator C-Mod ohmic plasmas with phase contrast imaging and comparisons with gyrokinetic simulations. Plasma Phys. Control. Fusion, 51(6):65006, 2009. URL: http: //stacks.iop.org/0741-3335/51/i=6/a=065006, doi:10.1088/ 0741-3335/51/6/065006. 53 Bibliography 75
[11] G.R. McKee, K.H. Burrell, R.J. Fonck, G.L. Jackson, M. Mu- rakami, G Staebler, D Thomas, and P. West. Impurity-induced suppression of core turbulence and transport in the DIII-D tokamak. Physical review letters, 84(9):1922–5, feb 2000. URL: http://www.ncbi.nlm.nih.gov/pubmed/11017661, doi:10.1103/ PhysRevLett.84.1922. 53
[12] A M Messiaen, J Ongena, U Samm, B Unterberg, G Van Wassenhove, F Durodie, R Jaspers, M Z Tokar, P E Vanden- plas, G Van Oost, J Winter, G H Wolf, G Bertschinger, G Bon- heure, P Dumortier, H Euringer, K H Finken, G Fuchs, B Giesen, R Koch, L Könen, C Königs, H R Koslowski, A Krämer-Flecken, A Lyssoivan, G Mank, J Rapp, N Schoon, G Telesca, R Uhle- mann, M Vervier, G Waidmann, and R R Weynants. High Con- finement and High Density with Stationary Plasma Energy and Strong Edge Radiation in the TEXTOR-94 Tokamak. Phys. Rev. Lett., 77(12):2487–2490, sep 1996. URL: http://link.aps.org/ doi/10.1103/PhysRevLett.77.2487, doi:10.1103/PhysRevLett. 77.2487. 53
[13] M Porkolab, J Dorris, P Ennever, C Fiore, M Greenwald, A Hub- bard, Y Ma, E Marmar, Y Podpaly, M L Reinke, J E Rice, J C Rost, N Tsujii, D Ernst, J Candy, G M Staebler, and R E Waltz. Transport and turbulence studies in the linear ohmic confine- ment regime in Alcator C-Mod. Plasma Phys. Control. Fusion, 54(12):124029, 2012. URL: http://stacks.iop.org/0741-3335/ 54/i=12/a=124029, doi:10.1088/0741-3335/54/12/124029. 53
[14] M L Reinke, P Beiersdorfer, N T Howard, E W Magee, Y Pod- paly, and J E Rice. Vacuum ultraviolet impurity spectroscopy on the Alcator C-Mod tokamak. The Review of scientific instruments, 81(10):10D736, oct 2010. URL: http://www.ncbi.nlm.nih.gov/ pubmed/21033927, doi:10.1063/1.3494380. 55
[15] M L Reinke, Y a Podpaly, M Bitter, I H Hutchinson, J E Rice, L Delgado-Aparicio, C Gao, M Greenwald, K Hill, N T Howard, a Hubbard, J W Hughes, N Pablant, a E White, and S M Wolfe. X-ray imaging crystal spectroscopy for use in plasma transport research. The Review of scientific instruments, 83(11):113504, 2012. URL: http://www.ncbi.nlm.nih.gov/pubmed/23206059, doi:10. 1063/1.4758281. 55
[16] M. L. Reinke, J. E. Rice, A. E. White, M. Greenwald, N. T. Howard, P. Ennever, C. Gao, A. E. Hubbard, and J. W. Hughes. Density sensitivity of intrinsic rotation profiles in ion cyclotron range of frequency-heated L-mode plasmas. Plasma Physics and Controlled Fusion, 55(1):012001, 2013. URL: http://iopscience. iop.org/0741-3335/55/1/012001$\delimiter"026E30F$nhttp: //iopscience.iop.org/0741-3335/55/1/012001/$\ 76 Bibliography
delimiter"026E30F$nhttp://iopscience.iop.org/0741-3335/ 55/1/012001/pdf/0741-3335{_}55{_}1{_}012001.pdf, doi:10.1088/0741-3335/55/1/012001. 67
[17] J. E. Rice, M. J. Greenwald, Y. a. Podpaly, M. L. Reinke, P. H. Diamond, J. W. Hughes, N. T. Howard, Y. Ma, I. Cziegler, B. P. Duval, P. C. Ennever, D. Ernst, C. L. Fiore, C. Gao, J. H. Irby, E. S. Marmar, M. Porkolab, N. Tsujii, and S. M. Wolfe. Ohmic energy confinement saturation and core toroidal rotation rever- sal in Alcator C-Mod plasmas. Physics of Plasmas, 19(5):056106, 2012. URL: http://scitation.aip.org/content/aip/journal/ pop/19/5/10.1063/1.3695213, doi:10.1063/1.3695213. 67
[18] G. M. Staebler, J. E. Kinsey, and R. E. Waltz. A theory-based transport model with comprehensive physics. Physics of Plasmas, 14(5):055909, 2007. URL: http://link.aip.org/link/PHPAEN/ v14/i5/p055909/s1{&}Agg=doi, doi:10.1063/1.2436852. 53 GYROVALIDATIONRESULTS 5
GYRO has been validated against experimental results in many regimes[3], but the plasmas studied in this work have several key differences from the plasmas in which GYRO has previously been validated. Firstly, these plasmas are close to marginal stability (i.e. Qie/QGB ≈ 1). This is an important regime for ITER, which will also be close to marginal stability[8]. Secondly, these plasmas have Te > Ti and Qe > Qi which will also be the case in any plasma that is primar- ily heated by fusion-produced alpha particles, unlike the primarily neutral-beam heated plasmas on which GYRO has previously been validated. We note that these ohmic plasmas have very low β com- pared to a fusion plasma, and are in the L-mode rather than the H- mode confinement regime and therefore electromagnetic turbulence can be neglected. Nevertheless validating GYRO (and by extension gyrokinetic theory) to predict turbulent transport in the plasma core in these nitrogen seeded ohmic plasmas would give greater confi- dence in its ability to predict the turbulence in the electron transport dominated fusing plasma regime. The GYRO simulations reported in this chapter were all done with experimental profiles as inputs. These input files were generated us- ing the profiles_gen command in GYRO from TRANSP outputs us- ing plasma state files that were generated using trxpl[9]. This is the workflow recommended by the GYRO group to ensure consistency between facilities. The simulations in this chapter include an effective impurity species, which has a charge and density chosen to match the total experimental impurity charge density and experimental Zeff as closely as possible, given that the impurity charge must be an integer. This effective impurity species was included as a kinetic species in the simulations, meaning that its distribution was evolved by GYRO in the same manner as the main ion species. As no measurement of the impurity density or Zeff profile was available, the impurity den- sity profile is assumed to have the same shape as the electron density profile (i.e. nz/ne is a constant with radius). The low temperatures in these plasmas also result in very low values of β, which results in the turbulence being almost purely electrostatic, so electromagnetic effects were not included in the simulations. These are the primary assumptions made for the simulations, and all are well justified for these cases. The GYRO simulations themselves all include modes with kθρs61, which will include ion temperature gradient modes (ITG) and trapped- electron modes (TEM) if present, but will not include the electron
77 78 gyro validation results
temperature gradient modes (ETG) which requires simulating modes with kθρs up to approximately 40. The choice of kθρs61 was primar- ily driven by practical concerns, as truly simulating the ETG turbu- lence requires simulation of both ion and electron scales simultaneously[7]. Only a few such "multi-scale" simulations have been run, and each one requires hundreds of times the computational time of the ion scale simulations and it is extremely difficult to get such simulations to converge. Thus, the simulations presented here are only able to cap- ture the ion-scale dynamics of the turbulence. While this is a limita- tion, ETG turbulence is not always a significant factor. In cases where the ITG or TEM turbulence is highly unstable, the ETG turbulence will not be very strong[5]. In cases where ITG and TEM turbulence are not very strongly unstable, the ETG turbulence can be a signifi- cant driver of energy flux. The simulations presented in this chapter focus on r/a = 0.6 (where Q/QGB ≈ 1) and r/a = 0.8 (where Q/QGB» 1). The two regions were chosen because they represent the two different operating regimes for fusion plasmas: where the turbulence is close to marginal (i.e. where a/LTi and a/LTe are close to their respective critical gradients) and where the turbulence is strongly driven (i.e. where a/LTi and a/LTe are both much greater than their respective critical gradients) respectively. The marginal regime is most relevant to the case of ITER where the turbulence will be close to marginal stability and where Q/QGB ≈ 1[8] and the heating efficiency will be most strongly influ- enced by the critical gradient for the turbulent transport. The strongly driven regime is most relevant for current tokamaks where Q/QGB» 1[6] and the heating efficiency will be most strongly influenced by the stiffness of the turbulent transport. The remainder of the chapter is organized as follows: Section 5.1 describes linear GYRO simulations of seeded and unseeded plasmas at r/a = 0.6 and r/a = 0.8; Section 5.2 describes nonlinear local GYRO simulations of plasma energy transport at r/a = 0.6 and r/a = 0.8 in both LOC and SOC regimes and comparisons with experimental measurements; Section 5.3 describes TGYRO simulations of plasma profiles for r/a < 0.9; Section 5.4 describes global simulations near r/a = 0.6 for multiple different profile shapes; and Section 5.5 describes quantitative comparisons between global and local GYRO simulated density fluctuations and PCI measurements through the use of a syn- thetic diagnostic.
5.1 linear gyro simulations of turbulent growth rates
Linear simulations with GYRO are useful for understanding what sort of modes may be unstable in a particular case, and how close to stability they may be. These linear simulations will determine the growth rate and real frequency of the most unstable mode for a par- 5.1 linear gyro simulations of turbulent growth rates 79 ticular mode number. Plasma turbulence is an inherently nonlinear problem, and modes which are linearly unstable can end up being more stable when nonlinear effects are taken into account. Neverthe- less the speed of linear computations allows for the exploration of the linear stability within a large parameter range, which can be use- ful for understanding the stability landscape. This is accomplished by doing multiple linear simulations with varying gradients. In this case the electron and ion (both main ion and impurity) temperature gradients were varied as they were the parameters to which the tur- bulence is most sensitive. The linear simulations presented also are restricted to kθρs = 0.3 because that is a representative value that plays a significant role in the nonlinear dynamics, as shown later in Fig. 31. The linear scans of representative SOC discharges with IP = 1.0 MA at r/a = 0.6 and r/a = 0.8 are shown in Figs. 29 and 30 respectively. The figures show contours of the linear growth rates of the modes as the ion and electron temperature gradients are changed, with the white line indicating the separation between modes that propagate in the electron and ion diamagnetic drift direction. Modes to the left of the white line are electron drift directed and modes to the right of the white line are ion drift directed, as determined by the sign of their real frequency. The purple crosses indicate the nominal experimental values of a/LTi and a/LTe. The simulations done at r/a = 0.6 in Fig. 29 show that ITG modes are dominant until very low values of a/LTi, both before and after the seeding. This is evidenced by the fact that the modes are ion drift directed and that their growth rates increase with increasing a/LTi. The seeding does make a small difference in the linear growth rates at this radius, shifting the dividing line between ion and electron di- rected modes to higher values of a/LTi, but the effect is small. Overall though the growth rates are nearly unchanged by the seeding in this case, and the ITG modes are dominant throughout the gradient scan, including at the nominal experimental values of the gradients. This is in contrast to the simulations done at r/a = 0.8, in Fig. 30, which shows a more complex dependence of the linear growth rates on a/LTi and a/LTe, as well as more of a difference between the seeded and unseeded cases. There are two regimes of the linear sta- bility: one at high a/LTi and low a/LTe where the growth rates de- pend primarily on a/LTi (and are therefore likely ITG modes), and one at low a/LTi and high a/LTe where the growth rates depend pri- marily on a/LTe (and therefore are likely TEMs). The line between positive and negative frequencies does not correspond exactly to the border between these regimes of the modes, but the real frequency is very close to zero near the border, so the difference is small. Com- paring the seeded case to the unseeded case shows that the seeding moved the border between the ITG and TEM regions to higher val- 80 gyro validation results
Unseeded Growth Rates At r/a = 0.6 Seeded Growth Rates At r/a = 0.6 5 5 0.35
0.30 ] s 4 4 0.25 = 0.3 [a/c s
ρ 0.20 θ Te 3 Te 3 a/L a/L 0.15
2 2 0.10
0.25
0.15
0.25
0.15
0.05 Growth Rate Of k 0.05 0.05 1 1 0.00 1 2 3 4 5 1 2 3 4 5
a/LTi a/LTi
Figure 29: The linear growth rates from GYRO for modes with kθρs = 0.3 for varying values of a/LTi and a/LTe, before seeding (on the left) and after seeding (on the right), for r/a = 0.6. The white line indicates the border between the modes that are electron diamagnetic drift directed (to the left ot the line) and ion diamagnetic drift directed (to the right of the line), as determined by the sign of the real frequency of the modes. The purple crosses show the nominal experimental values of a/LTi and a/LTe.
ues of a/LTi, and the growth rates are overall lower. This indicates that the seeding is having a stabilizing effect on the ITG turbulence in this case. The nominal experimental values of the gradients lie close to the border between the ITG and TEM dominant regimes, but the experimental value of a/LTi increases from the seeding. •
5.2 nonlinear local gyro simulations of turbulence
The linear simulations are a good initial step, but as there is no way to use them to determine the saturated amplitude of the turbulence, they cannot be quantitatively compared to steady-state plasmas. To make such quantitative comparisons, local nonlinear GYRO simula- tions were performed to compare the predicted energy fluxes and density fluctuation amplitudes to the measured values. Local GYRO assumes that the turbulence at a particular flux surface is only de- termined by the local values of the profiles and their gradients. The validity of this assumption depends on the value of ρ?, which is the ? ratio of the ion gyroradius to the system size. If ρ 61/250, then the lo- cal assumption is valid and should give the same result as the global simulations, based on work done by Candy et. al.[3]. The local simu- lations also use periodic radial boundary conditions, which removes the need for buffer zones at the edges of the simulation domain to get convergence. Local simulations can also be done nearer to the plasma edge (r/a ≈ 0.8), where global simulations do not reliably converge. 5.2 nonlinear local gyro simulations of turbulence 81
Unseeded Growth Rates At r/a = 0.8 Seeded Growth Rates At r/a = 0.8 14 14 0.5 ]
12 12 s 0.4
10 0.25 10 0.15 = 0.3 [a/c
s 0.3 ρ θ Te 8 Te 8 a/L a/L 0.2 6 6
0.1 Growth Rate Of k 0.45 4 4 0.05
0.25 0.25 0.15 0.35 0.15
2 2 0.0 2 3 4 5 6 7 2 3 4 5 6 7
a/LTi a/LTi
Figure 30: The linear growth rates from GYRO for modes with kθρs = 0.3 for varying values of a/LTi and a/LTe, before seeding (on the left) and after seeding (on the right), for r/a = 0.8. The white line indicates the border between the modes that are electron diamagnetic drift directed (to the left ot the line) and ion diamagnetic drift directed (to the right of the line), as determined by the sign of the real frequency of the modes. The purple crosses show the nominal experimental values of a/LTi and a/LTe.
Local GYRO simulations were performed on these ohmic plasmas at both r/a = 0.8 and r/a = 0.6 for both seeded and unseeded condi- tions. The simulations were run until the fluxes reached a steady-state value, and the average of the last 25% of the simulation time is taken as the predicted flux for comparison with experiments. All ion en- ergy fluxes include both the main ion and impurity ion energy fluxes, though the impurity energy flux is always less than 10% of the main ion energy flux, due to the impurity ions’ much smaller pressure. Figure 31 shows an example of one such simulation. The left plot of Fig. 31 shows the time history of the simulation from the linear growth phase that lasts the first 25 a/cs and the eventual non-linear saturation which takes another 600 a/cs. The time variation in the simulated fluxes is happening thousands of times faster than the en- ergy confinement time, so the fast variation would not be detectable in a power-balance calculation of the fluxes. Therefore the appropri- ate comparison is between the average of the simulated flux and the power-balance calculated flux, and the variation should not be thought of as an uncertainty in the simulated flux. The steady-state values of the fluxes are Qe/QGB = 24 and Qi/QGB = 7.5. The right plot of Fig. 31 shows the wavenumber spectrum of the last 25% of the simulation for both the ions and electrons. It shows that the electron flux and ion flux peak at kθρs = 0.35, and decrease to less than 10% of this peak flux for kθρs = 1, which validates the choice of kθρs 6 1 for the simulation. 82 gyro validation results
GYRO Simulation Time History GYRO Simulation Wavenumber Spectrum 60 4
50 3 40 GB 30 2 per Mode Q/Q GB 20
Q/Q 1 10
0 0 0 100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0 t [a/c ] k ρ s Ion Flux θ s Electron Flux Time-Averaged Flux Value Time-Averaging Window
Figure 31: Example GYRO simulation of an unseeded 0.8 MA SOC discharge at r/a = 0.8. The left plot shows the time-history of the simulated ion and electron energy fluxes in green and purple respectively. The black dashed lines show the time-averaging window and the black diamonds are the time-averaged flux value. The right plot shows the time-averaged wavenumber spectra of the electron and ion fluxes.
5.2.1 Local GYRO Simulations at r/a = 0.8
The results of all the local GYRO simulations done at r/a = 0.8 show generally good agreement with the experimental energy fluxes, though disagreements exist in certain cases. Figure 32 shows the GYRO sim- ulated and experimental energy fluxes for both electrons and ions at r/a = 0.8 in both the seeded and unseeded cases. The GYRO and experimental electron energy fluxes agree except for the very lowest density unseeded case (where the simulated electron energy flux is too large) and the very highest density seeded and unseeded cases, as seen in Fig. 32 a) and b). There are uncertainties in the measured profiles at these locations however, and modifications of the input temperature profiles at this location can account for much of the dis- crepancy. Section 5.3 explores the effects of profile modification in detail using TGYRO [2], and it was found that modifying the input temperature profiles within uncertainties was able to account for the discrepancy between the simulated and experimental energy fluxes. The GYRO and experimental ion energy fluxes in these cases are in agreement for the lowest density cases, but disagree in the higher density cases, as shown in Fig. 32 c) and d). The uncertainty in the ion temperature gradient is dealt with in Fig. 34. The result is that in the 1.0 MA SOC cases the disagreement between the GYRO and experimental ion energy fluxes can be accounted for within the uncer- tainties in the ion temperature gradient, but in the higher density 0.8 MA cases it cannot be. The uncertainties in the electron temperature and ion temperature profiles were explored together in Sec. 5.3 and 5.2 nonlinear local gyro simulations of turbulence 83
Unseeded r/a = 0.8 Electron Flux Seeded r/a = 0.8 Electron Flux 70 a) 70 b) 60 60 50 50 GB 40 GB 40 /Q /Q
e 30 e 30 Q 20 Q 20 10 10 0 0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 20 -3 20 -3 0.8 MA Exp. Flux neq95 [10 m ] neq95 [10 m ] 0.8 MA GYRO Flux 1.0 MA Exp. Flux Unseeded r/a = 0.8 Ion Flux Seeded r/a = 0.8 Ion Flux 1.0 MA GYRO Flux 70 70 60 c) 60 d) 50 50 GB 40 GB 40 /Q /Q i 30 i 30 Q 20 Q 20 10 10 0 0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 20 -3 20 -3 neq95 [10 m ] neq95 [10 m ]
Figure 32: Plots comparing the experimental and GYRO electron and ion energy fluxes in 0.8 MA and 1.0 MA seeded and unseeded cases at r/a = 0.8. The unseeded cases are in red while the seeded cases are in blue. The GYRO energy fluxes are in the solid circles (for the 0.8 MA cases) and the solid squares (for the 1.0 MA cases). The experimental energy fluxes are in the open diamonds (for the 0.8 MA cases) and the stars (for the 1.0 MA cases). The electron energy fluxes are in the top two plots (plots a) and b)) and the ion energy fluxes are in the bottom two plots (plots c) and d)). The LOC-SOC transition is at neQ95 = 4.3. combining changes in both profiles using TGYRO [2] yields agree- ment at r/a = 0.8. In general however, local GYRO accurately predicts the energy transport in the majority of cases at r/a = 0.8 when the un- certainty of the input temperature profiles is taken into account. The effect of the nitrogen seeding on the stiffness and critical gra- dient of the ion energy transport at r/a = 0.8 was also studied using GYRO. To determine the stiffness and critical gradient of the GYRO ion energy transport, several GYRO simulations were performed with the simulated a/LTi scanned above and below the nominal experi- mental a/LTi values. All other input parameters were held fixed. The resulting fluxes from the GYRO simulations allowed for the determi- nation of the stiffness and critical gradient of the GYRO ion energy transport. Once this analysis was performed, the effect of the seeding on the ion energy transport could be determined. The analysis was done for the 0.8 MA plasmas (with the lower density cases shown in shown in Fig. 33, and higher density cases shown in Fig. 34), as well as a 1.0 MA SOC plasma (shown in Fig. 35) in both the unseeded and seeded phases. In the 0.8 MA low density cases in Fig. 33, the seeding primar- ily increases the critical gradient of the ion energy transport. Fig- 20 −3 ure 33 a) shows that the ne = 0.5 × 10 m unseeded case now displays a clear value of the critical gradient in the ion energy trans- port. This is due to the fact that this is a very TEM dominated case, as confirmed through linear stability analysis, so the modification of a/LTi is changing the stability of the TEM turbulence that is driving the ion energy transport in this case. The seeded case at that density 84 gyro validation results
20 -3 20 -3 ne = 0.5 x 10 m Cases ne = 0.7 x 10 m Cases 30 30 a) Unseeded b) Unseeded 25 Seeded 25 Seeded 20 20 GB GB
/Q 15 /Q 15 i i
Q 10 Q 10 5 5 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 a/L a/L Ti Local GYRO Fluxes & Gradients Ti Offset-Linear Fit Critical Gradient Exp. Flux & Gradient
Figure 33: Plots of GYRO ion energy fluxes at r/a = 0.8 for different values of the ion temperature gradient in both the unseeded and seeded cases for the two lower densities in the 0.8 MA scan. The solid lines are offset-linear fits to the data, and the dashed lines indicate the critical gradients. The diamonds with the error bars are the experimental energy fluxes and gradients, and the circles are the GYRO energy fluxes and gradients.
displays the normal offset linear dependence, with a low value of Qi/QGB at a/LTi = 0, indicating that the ion energy flux is primar- ily driven by ITG modes. Figure 33 b) shows that in the ne = 0.7 × 1020m−3 case, the seeding again increases the critical gradient of the GYRO ion energy transport, but the stiffness is larger in the seeded case than it is in the unseeded case. The simulated ion energy fluxes and gradients in these low density 0.8 MA cases are quantitatively consistent with the experimental measurements. In the higher density 0.8 MA cases shown in Fig. 34, the seeding pri- marily reduces the stiffness while leaving the critical gradient largely unchanged. In both cases in Fig. 34 the GYRO predicted fluxes are much smaller than the experimental fluxes, but the trend in the fluxes with the seeding is consistent between the simulations and the exper- iment. In the 1.0 MA SOC cases in Fig. 35, the seeding reduced the stiffness and increased the critical gradient and both cases quantitatively agree with the experimental ion energy fluxes (within the uncertainties in the experimental flux and gradient). The decrease in the simulated fluxes in these cases is more substantial than the decreases in other cases, and results in both a decrease in ion stiffness and an increase in the critical ion temperature gradient. This is most likely due to the fact that the unseeded case has a nD/ne that is very close to 1, so the dilution in the seeded case is a more substantial difference.
5.2.2 Local GYRO Simulations at r/a = 0.6
In addition to local GYRO simulations performed at r/a = 0.8, where experimentally it is observed that Q/QGB» 1, simulations were also 5.2 nonlinear local gyro simulations of turbulence 85
20 -3 20 -3 ne = 0.8 x 10 m Cases ne = 1.1 x 10 m Cases 50 a) Unseeded 50 b) Seeded Unseeded 40 40 Seeded
GB 30 GB 30 /Q /Q i i
Q 20 Q 20 10 10 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
a/LTi a/LTi Local GYRO Fluxes & Gradients Offset-Linear Fit Critical Gradient Exp. Flux & Gradient
Figure 34: Plots of GYRO ion energy fluxes at r/a = 0.8 for different values of the ion temperature gradient in both the unseeded and seeded cases for the two higher densities in the 0.8 MA scan. The solid lines are offset-linear fits to the data, and the dashed lines indicate the critical gradients. The diamonds with the error bars are the experimental energy fluxes and gradients, and the circles are the GYRO energy fluxes and gradients.
20 -3 ne = 1.4 x 10 m Cases 60 Unseeded 50 Seeded 40 GB
/Q 30 i Q 20 10 0 0 2 4 6 8
a/LTi Local GYRO Fluxes & Gradients Offset-Linear Fit Critical Gradient Exp. Flux & Gradient
Figure 35: Plots of GYRO ion energy fluxes at r/a = 0.8 for different values of the ion temperature gradient in both the unseeded and seeded cases for a SOC density in the 1.0 MA scan. The solid lines are offset-linear fits to the data, and the dashed lines indicate the critical gradients. The stars with the error bars are the experimental energy fluxes and gradients, and the squares are the GYRO energy fluxes and gradients. 86 gyro validation results
performed at r/a = 0.6, where experimentally it is observed that Q/QGB≈ 1. The results of all the local GYRO simulations performed at r/a = 0.6 show a systematic over-prediction of both the ion and elec- tron energy fluxes. The results of this analysis are shown in Fig. 36. The simulated fluxes are all significantly larger than the respective experimental fluxes. The simulated energy fluxes also differ qualita- tively, because the simulated fluxes are all well above the marginally stable level (i.e. GYRO predicted energy fluxes have Q/QGB» 1) while the experimental fluxes are all below it (i.e. experimentally energy fluxes have Q/QGB≈ 1). In addition, the GYRO simulated ion energy flux is larger than the simulated electron energy flux, while in the experiment it is the reverse, and the ion energy flux is much less than the electron energy flux in every case. The fact that the ion energy flux is greater than the electron energy flux is consistent with the lin- ear results in Fig. 29, which predicts ITG modes are the only unstable low kθρs modes and that no TEM activity is present. These results are also consistent with earlier observations of GYRO simulations of C-Mod ohmic plasmas [10]. To investigate this disagreement between the simulations and ex- periment, scans of a/LTi were performed as in the previous subsec- tion to determine the stiffness and critical gradient of the local GYRO simulations. One representative case is shown in Fig. 37. The result of these scans is that the critical gradient of the simulated energy flux is much larger than the possible critical gradient of the experimental flux. The critical normalized ion temperature gradient in the GYRO simulations in Fig. 37 is approximately 1.4, while in the experiment the critical normalized ion temperature gradients must be at least 3 in order to be consistent with the energy fluxes. This suggests that there is a stabilizing mechanism present in the experiment that increases the critical gradient, but is not being accounted for in the simulation. One possible stabilizing mechanism is the effect of radially varying profile gradients, known as global effects. The influence of global ef- fects on the simulated turbulence is covered in Sec. 5.4, where it was found to have a substantial stabilizing effect on the turbulence due to the substantial radial variation of the temperature gradient and mod- erate value of ρ?. The global effects were found to have a significant stabilizing effect on the ITG turbulence, and brought the simulated ion energy flux to close to marginal stability, as seen in the experi- ment. However, this resulted in substantial under-prediction of the experimental electron energy flux, and ETG turbulence is suspected to be responsible for the electron energy flux in the experiment. • 5.2 nonlinear local gyro simulations of turbulence 87
Unseeded r/a = 0.6 Electron Flux Seeded r/a = 0.6 Electron Flux a) b) 15 15
GB 10 GB 10 /Q /Q e e
Q 5 Q 5 0 0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 20 -3 20 -3 0.8 MA Exp. Flux neq95 [10 m ] neq95 [10 m ] 0.8 MA GYRO Flux 1.0 MA Exp. Flux Unseeded r/a = 0.6 Ion Flux Seeded r/a = 0.6 Ion Flux 1.0 MA GYRO Flux c) d) 15 15
GB 10 GB 10 /Q /Q i i
Q 5 Q 5 0 0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 20 -3 20 -3 neq95 [10 m ] neq95 [10 m ]
Figure 36: Plots comparing the experimental and GYRO electron and ion energy fluxes in 0.8 MA and 1.0 MA seeded and unseeded cases at r/a = 0.6. The unseeded cases are in red while the seeded cases are in blue. The GYRO energy fluxes are in the solid circles (for the 0.8 MA cases) and the solid squares (for the 1.0 MA cases). The experimental energy fluxes are in the open diamonds (for the 0.8 MA cases) and the stars (for the 1.0 MA cases). The electron energy fluxes are in the top two plots (plots a) and b)) and the ion energy fluxes are in the bottom two plots (plots c) and d)). The LOC-SOC transition is at neq95 = 4.3.
20 -3 ne = 0.7 x 10 m Cases 25 Unseeded Seeded 20
GB 15 /Q i
Q 10
5 0 0 1 2 3 4 5
a/LTi Local GYRO Fluxes & Gradients Offset-Linear Fit Critical Gradient Exp. Flux & Gradient
Figure 37: Plots of GYRO ion energy fluxes at r/a = 0.6 for different values of the ion temperature gradient, in both the unseeded and seeded cases, for a LOC density in the 0.8 MA scan. The solid lines are the offset-linear fit to the data, and the dashed lines indicate the critical gradient. The diamonds with the error bars are the experimental energy fluxes and gradients, and the circles are the GYRO simulated ion energy fluxes and gradients. 88 gyro validation results
5.3 tgyro profile modification using tglf
The uncertainty in the measured profiles used as inputs to GYRO was investigated using the transport solver TGYRO [2]. TGYRO com- putes a set of temperature profile inputs to TGLF [11] and NEO [1] to simulate turbulent and neoclassical energy fluxes, respectively, that when combined will agree with the experimental energy fluxes. TGLF is used to simulate the tubulent energy fluxes in TGYRO because it is orders of magnitude faster than GYRO, and TGYRO requires dozens of local turbulence simulations to get convergence. The fi- nal TGYRO flux-matched profiles are checked with GYRO to ensure that the TGLF results are accurate. The neoclassical energy fluxes are all negligibly small in these cases, less than 5% of the turbulent en- ergy flux, so while they are included in TGYRO, the turbulent energy fluxes are dominant. To find flux-matching temperature profiles, TGYRO begins with a set of initial "guess" temperature profiles, then simulates the energy fluxes using those temperature profiles using TGLF and NEO at each radial location of the TGYRO run. It then steepens or flattens the temperature gradients at each radial location to increase or decrease the simulated energy fluxes to find a match the experimental values at that location. The gradients are then integrated starting from a fixed "pivot point" to get a new set of temperature profiles and the process is repeated until the simulated and experimental energy fluxes agree. The default location of the pivot point is the top of the pedestal, but as the plasmas simulated are ohmically heated and are not in H-mode, they do not have a pedestal. Instead, for this analysis the pivot point was chosen to be r/a = 0.6 because it required the smallest deviation from the experimental measurements of the temperature profiles. Three sets of temperature profiles are analyzed in this section: the nominal experimental profiles in Section 5.3.1 and Fig. 38; the TGYRO flux-matched profiles in Section 5.3.2 and Fig. 39; and the experimen- tal electron temperature profile and the average of the TGYRO flux- matched ion temperature profile and experimental ion temperature profile in Section 5.3.3 and Fig. 40.
5.3.1 Simulations With The Nominal Experimental Profiles
The nominal experimental temperature profiles input to TGYRO, as well as TGLF, GYRO, and experimental energy fluxes are plotted in Fig. 38. Figures 38 a) and 38 b) show the experimental ion and elec- tron temperature profiles respectively in black, with the error bars and the input profiles to the simulation plotted in the teal dashed lines. In this case, the input temperature profiles are the nominal ex- perimental profiles, so the curves in Figs. 38 a) and 38 b) overlay. Figures 38 c) and 38 d) show the ion and electron energy fluxes 5.3 tgyro profile modification using tglf 89
Ion Temperature Profiles Electron Temperature Profiles 2.0 a) 2.0 b) 1.5 1.5 1.0 1.0 (keV) (keV) i e T 0.5 T 0.5 0.0 0.0 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 r/a Exp. Temperature r/a Input Temperature
Ion Energy Fluxes Electron Energy Fluxes c) d) 10.00 10.00 GB GB 1.00 1.00 /Q /Q i 0.10 e 0.10 Q Q 0.01 0.01
0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 r/a Exp. Flux r/a Local GYRO Flux TGLF Flux
Figure 38: Local GYRO, TGLF, and experimental energy fluxes using nominal experimental profiles as inputs. The ion (a) and electron (b) temperature profiles used as inputs to the simulations are shown in the dashed teal lines, with the experimental temperature profiles in black with error bars. The ion energy fluxes (c) and electron energy fluxes (d) for the local GYRO (teal squares) and TGLF (teal Xs) simulations are also shown. The experimental energy fluxes are the black lines and are normalized to the value of the gyrobohm unit energy flux at each radial location. The case simulated is a 1.0 MA SOC plasma. respectively from the experiment (black lines), TGLF (teal Xs), and GYRO (teal squares). The fluxes are all normalized to the local value of the gyrobohm flux. The TGLF and local GYRO fluxes agree, except at r/a = 0.8 and r/a = 0.5 where they differ by a significant amount. In this case, TGLF reports a higher energy flux than the GYRO flux for both the ions and electrons. This is likely to be the consequence of TGLF using a linear calculation which does not include the nonlinear upsift in the critical gradient due to the zonal flows, which in these cases is expected to have a significant impact. In cases where the non- linear upshift of the critical gradient is not as large, or the turbulence is well above marginal stability, as is the case for r/a = 0.6 and 0.7 in Fig. 38, TGLF agrees with local GYRO, as expected.
5.3.2 Simulations With The TGYRO Flux-Matched Profiles
The TGYRO temperature profiles that match the TGLF predictions of the experimental energy fluxes are shown in Fig. 39. Figures 39 a) and 39 b) show the TGYRO flux-matching profiles in the purple dashed lines, and the experimental profiles are shown in the solid black lines for reference. The TGYRO flux-matching electron temper- ature profile agrees with the experimental measurements, except at 90 gyro validation results
Ion Temperature Profiles Electron Temperature Profiles 2.0 a) 2.0 b) 1.5 1.5 1.0 1.0 (keV) (keV) i e T 0.5 T 0.5 0.0 0.0 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 r/a Exp. Temperature r/a Input Temperature
Ion Energy Fluxes Electron Energy Fluxes 10.000 c) 10.000 d) 1.000 1.000 GB GB /Q
/Q 0.100 0.100 i e Q 0.010 Q 0.010 0.001 0.001 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 r/a Exp. Flux r/a Local GYRO Flux TGLF Flux
Figure 39: Local GYRO, TGLF, and experimental fluxes using TGYRO flux-matching profiles. The ion (a) and electron (b) temperature profiles used as inputs to the simulations are shown in the dashed purple lines with the experimental temperature profiles in black with error bars for reference. The ion energy fluxes (c) and electron energy fluxes (d) for the local GYRO (purple squares) and TGLF (purple Xs) are also shown. The experimental energy fluxes are the black lines and are normalized to the value of the gyrobohm unit energy flux at each radial location. The case simulated is a 1.0 MA SOC plasma.
r/a = 0.4. The TGYRO flux-matching ion temperature profile deviates significantly from the experimental measurements of the ion temper- ature. Figures 39 c) and 39 d) show the ion and electron energy fluxes, respectively, from the experiment (black lines), TGLF (purple Xs), and GYRO (purple squares). In this case the TGLF fluxes agree well with the experimental energy fluxes, indicating that the TGYRO run is well-converged. The local GYRO fluxes are significantly lower than the TGLF fluxes at all radii, which is again most likely due to TGLF not including the nonlinear upshift in the critical gradient. This result means that TGYRO is over-flattening the ion temperature gra- dient in these cases.
5.3.3 Simulations With The Average Of TGYRO And Experimental Pro- files
To investigate the possibility that there existed a set of profiles in- termediate between the experimental and TGYRO flux-matching pro- files, TGLF and GYRO runs were performed using the experimen- tal electron temperature profiles and the average of the TGYRO flux- matching ion temperature profile and the nominal experimental ion temperature profiles. The experimental electron temperature profile 5.3 tgyro profile modification using tglf 91
Ion Temperature Profiles Electron Temperature Profiles 2.0 a) 2.0 b) 1.5 1.5 1.0 1.0 (keV) (keV) i e T 0.5 T 0.5 0.0 0.0 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 r/a Exp. Temperature r/a Input Temperature
Ion Energy Fluxes Electron Energy Fluxes c) d) 10.00 10.00 GB GB 1.00 1.00 /Q /Q i 0.10 e 0.10 Q Q 0.01 0.01
0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 r/a Exp. Flux r/a Local GYRO Flux TGLF Flux
Figure 40: Local GYRO, TGLF, and experimental fluxes using the average of the experimental and TGYRO flux-matching profiles. The ion (a) and electron (b) temperature profiles used as inputs to the simulations are shown in the dashed green lines with the experimental temperature profiles in black. The ion energy fluxes (c) and electron energy fluxes (d) for the local GYRO (green squares) and TGLF (green Xs) simulations are also shown. The experimental energy fluxes are shown for reference in the black lines and are normalized to the value of the gyrobohm unit energy flux at each radial location. The case simulated is a 1.0 MA SOC plasma. was used because the TGYRO modifications to the experimental elec- tron temperature profile were small. The modelled profiles are shown in Figs. 40 a) and 40 b) with the green dashed lines, while the exper- imental profiles are shown in the black lines for reference. The mod- elled profiles are consistent with the measurements of the ion and electron temperature profiles, within error bars, but any further flat- tening of the ion temperature profile would result in disagreement with the experimental measurements. The resulting energy fluxes are shown in Figs. 40 c) and 40 d) for ions and electrons respectively from the experimental values (black lines), as well as from TGLF (green Xs), and local GYRO (green squares). The local GYRO ion energy fluxes are in much closer agreement with the experimental ion energy fluxes than in the experimental case in Fig. 38, but they are still above the ex- perimental energy fluxes at r/a = 0.6 and 0.7. As the ion temperature profile used in this case is as flat as possible, within the experimental uncertainties, to reduce the simulated ion energy fluxes any further other effects need to be taken into account. • 92 gyro validation results
Ion Temperature Gradient Profiles 6 Experimental Profile TGYRO Flux-Matching Profile 5 Average Of TGYRO and Exp. Profiles
4
Ti 3 a/L
2
1
0 0.45 0.50 0.55 0.60 0.65 0.70 0.75 r/a
Figure 41: Radial profiles of a/LTiaround r/a = 0.6 for a 1.0 MA SOC case. The teal line corresponds to the nominal experimental ion temperature profile of Fig. 38, the purple line corresponds to the TGYRO flux-matching ion temperature profile of Fig. 39, and the green line corresponds to the average of the TGYRO and nominal experimental ion temperature profiles of Fig. 40.
5.4 global nonlinear gyro simulations of turbulence
The importance of so-called global effects, namely the effects of radial nonuniformity in the profile gradients and other turbulence-relevant parameters, on turbulent energy transport and density fluctuations near r/a = 0.6 is explored in this section. The importance of these ef- ? fects is related to ρ , which is the ratio of the gyroradius ρs to the sys- tem size a. The strength of global effects has been studied in GYRO by Candy et. al. [4] using cyclone base case parameters [6] (a strongly ITG unstable DIII-D case with ne = ni and Te = Ti). They found that the critical value of ρ? needed for the local and global results to agree depended on the shape of the gradient profile. If there was only a small radial variation in the gradients, ρ?< 1/200 was needed to get agreement. If there was a more significant radial variation, ρ?< 1/400 was needed to get agreement. In these ohmic cases, ρ?is ap- proximately 1/250 at r/a = 0.6 and 1/500 at r/a = 0.8. This implies that global effects are unlikely to be important at r/a = 0.8, but they could be important at r/a = 0.6 depending on the shape of the pro- file gradients. Figure 41 shows the ion temperature profile gradients for the three cases shown in the previous section: the experimental profile in teal, the TGYRO flux-matching profile in purple, and the average of the two in green. The TGYRO profile has the least vari- ation, the average profile varies more, and the experimental profile varies the most. 5.4 global nonlinear gyro simulations of turbulence 93
Global simulations were performed on all three profiles to test how much of a role global effects might play in stabilizing the tur- bulence in these simulations near r/a = 0.6, where the local simula- tions greatly over-predict the turbulence. All three profiles have ρ?= 1/300 at r/a = 0.6 where the global GYRO simulations are centered. The results are shown in Fig. 42. The global GYRO simulation using the TGYRO flux-matching profile, shown in purple, is very close to the local value, which is to be expected from the small radial varia- tion of the temperature gradient, and both simulations give energy fluxes that are below the experimental electron and ion energy fluxes. The global GYRO simulations using the experimental temperature profile, shown in teal, has about half the ion energy flux of the local GYRO simulations, though the global GYRO ion energy flux is still larger than the experimental ion energy flux in this case. The global GYRO and experimental electron energy fluxes agree in this case, but this is misleading because the electron energy flux in this case is be- ing driven through cross-species coupling from ITG modes. This was confirmed through global simulations with increased a/LTe which did not change the value of the simulated electron energy flux. ITG modes will always drive more ion energy flux than electron energy flux, thus with only ITG modes it is not possible to match both the experimental ion and energy energy fluxes simultaneously, since the experimental electron energy flux is larger than the experimental ion energy flux. The global GYRO simulations using the average temperature pro- file, shown in green, have again about half the ion energy flux of the local GYRO simulations, and now the ion energy flux in the global simulations agrees with the experimental ion energy flux. This shows that the magnitude of the radial variation of the ion temperature gra- dient is important to how much the global effects stabilize the modes, since the case without much radial variation (the TGYRO profiles case in purple) showed no difference between the global and local energy fluxes, while the other cases with significant radial variation (the experimental and average profiles) yielded a lower global energy flux than local energy flux. Thus, through a combination of modify- ing the ion temperature profile within uncertainties and the inclusion of global effects, the ion energy flux can be brought down to the marginally stable level observed in the experiment. While the global effects can reduce the simulated ion energy flux, the simulations still predict Qi > Qe, which is in contrast to the ex- perimental energy fluxes. This is again because the electron energy fluxes in both the local and global GYRO results were being driven by ITG modes at these radial locations, and there is no electron tur- bulence in these kθρs61 simulations. This corroborates the results of the linear GYRO runs described in Sec. 5.1 at r/a = 0.6 which showed no unstable modes even at very low values of a/LTi. One mechanism 94 gyro validation results
Ion Energy Fluxes Electron Energy Fluxes 10 10
8 8
6 6 GB GB /Q /Q i e Q 4 Q 4
2 2
0 0 0.50 0.55 0.60 0.65 0.70 0.50 0.55 0.60 0.65 0.70 r/a r/a Global GYRO Flux w/ Exp. Profiles Local GYRO Flux w/ Exp. Profiles Global GYRO Flux w/ TGYRO Profiles Local GYRO Flux w/ TGYRO Profiles Global GYRO Flux w/ Average Profiles Local GYRO Flux w/ Average Profiles Experimental Energy Flux
Figure 42: Comparison of the global GYRO (colored lines), local GYRO (colored squares), and experimental (black lines) energy fluxes. The teal lines correspond to simulations done using the nominal experimental profiles, the purple lines correspond to simulations done using the TGYRO flux-matching profiles, and the green lines correspond to simulations done using the average of the experimental and TGYRO profiles.
that could be responsible for the experimental electron energy trans- port in these cases is ETG turbulence. Linear simulations done at r/a = 0.6 in these cases do show unstable modes up to kθρs= 40 (though modes with kθρs> 38 are completely stable), as seen in Fig. 43. Recent work by Nathan Howard et. al. [7] using GYRO runs that include both electron and ion scales has shown for C-Mod L-mode plasmas that in cases where the ITG turbulence is close to marginal, the ETG tur- bulence forms radially elongated streamers that drive significant elec- tron energy transport. With the ETG and ITG scales included together, the simulated and experimental energy fluxes agree. These multiscale nonlinear simulations required hundreds of millions of CPU-hours to complete, and could not be repeated for this work, but the condi- tions are similar enough that they offer a likely explanation for the under-prediction of the electron energy flux in these kθρs61 simula- tions. Profile modification and global effects are necessary, however, to bring the ITG turbulence down to the marginally stable level where ETG turbulence can begin to drive significant energy flux. If the ITG modes were truly as unstable as they were predicted to be in the nonlinear simulations using the nominal profiles in Fig. 36, the ETG turbulence would likely not be able to drive significant energy flux. 5.5 density fluctuation comparisons between gyro simulations and pci measurements 95
Linear GYRO Growth Rates For r/a = 0.6 3.0 2.5 /a] s 2.0
1.5
1.0
Growth Rate [c 0.5 0.0 0 10 20 30 40 ρ kθ s
Figure 43: Plot of Linear GYRO covering the ITG (kθρs< 1) and ETG (kθρs> 1) ranges of wavenumbers. In this simulation it was found that the modes with kθρs< 1 are ion drift directed, while the modes with kθρs> 1 are electron directed.
5.5 density fluctuation comparisons between gyro sim- ulations and pci measurements
The electron density fluctuations simulated by GYRO were also com- pared to experimental PCI measurements through the synthetic di- agnostic described in Chapter 3. This comparison was done for the global simulations between r/a = 0.3 and r/a = 0.7 that well matched the experimental ion energy flux, and local simulations of the 1.2 MA plasmas that well matched the total experimental energy flux. The electron energy flux in the global GYRO simulations was still under- predicted, presumably due to the lack of ETG turbulence in the sim- ulations. However, the comparison of synthetic to experimental PCI should still be valid, since the ETG turbulence would be at higher wavenumbers than PCI was capable of measuring during those ex- periments. Thus, the experimental signal should be comparable to the synthetic signal, even if the electron energy fluxes do not match. The comparison between the synthetic PCI spectra from global GYRO and experimental PCI spectra are shown in Fig. 44 for a 1.0 MA SOC unseeded plasma and in Fig. 45 for a 1.0 MA SOC nitrogen seeded plasma. In both figures, the frequency - wavenumber spec- tra of the experimental and synthetic PCI spectra are shown in a) and b) respectively, as well as the phase velocity window (the white lines) that is used for the quantitative comparison shown in c). The wavenumber comparison shown in Figs. 44 and 45 c) is the quanti- tative comparison of the experimental and synthetic PCI spectra. The uncertainty in the experimental spectra is primarily due to the uncer- tainty in the calibration constant of the PCI signal. This calibration constant will only modify the amplitude of the spectrum rather than 96 gyro validation results
1.0 MA SOC Unseeded Case: Global GYRO at r/a ∈ [0.3,0.7] vs Exp. Exp. PCI Spectrum Wavenumber Spectra Comparison 300 1.4 a) c) Exp. PCI
250 ]
2 Synth. PCI 0.0100 200 1.2
150 F [kHz] 0.0010 100 1.0 PCI Amplitdue [(n~) 50 ] 0.0001 2 -10 -5 0 5 10 -1 0.8 kR (cm )
Synth. PCI Spectrum 0.6 300 b) PCI Amplitdue [(n~)
250 ] 2 0.0100 0.4 200
150 F [kHz] 0.0010 0.2 100 PCI Amplitdue [(n~) 50 0.0001 0.0 -10 -5 0 5 10 -10 -5 0 5 10 -1 -1 kR (cm ) kR (cm )
Figure 44: Plots comparing the experimental PCI spectrum (a) to the synthetic PCI (b) from a global GYRO simulation spanning r/a = 0.3 to r/a = 0.7 for an unseeded 1.0 MA SOC plasma, and fixed phase-velocity comparison between the two (c). The white solid lines in (a) and (b) indicate the fixed phase velocity along which the spectrum was integrated to do the comparison in (c). The white dashed lines show the window of the integration. The solid black line in (c) is the experimental PCI and the dashed line is the synthetic PCI. The synthetic PCI spectral amplitude has been multiplied by the ratio of the experimental ion energy flux to the simulated ion energy flux.
the shape, and does not affect the difference between the seeded and unseeded experimental PCI spectra. As can be seen in Figs. 44 and 45 c), the synthetic PCI is similar in shape to the experimental PCI, but the synthetic PCI is lower in amplitude than the experimental PCI. These simulations are over a limited range of the plasma volume, while the PCI measurements are line-integrated and contain contributions from the entire plasma volume. Thus, it is to be expected that the synthetic PCI amplitude will be lower than the experimental PCI amplitude. The synthetic PCI from the seeded global GYRO simulation in Fig. 45 c) shows a lower amplitude than the synthetic PCI from the unseeded global GYRO simulation in Fig. 44 c), but there is little difference between the experimental spectra in the two cases. In addition to comparing the global GYRO synthetic PCI spectra to the overall spectrum, the local GYRO synthetic PCI of 1.2 MA plasmas can be compared to the high phase-velocity feature that was localized to r/a = 0.85, as discussed in Section 4.3.2. The 1.2 MA plasmas were studied during the 2015 experimental campaign, which had an upgraded PCI detector which was sensitive to higher frequency fluctuations. This feature was mixed in with the bulk spec- trum (which is likely coming from a different region of the plasma), 5.5 density fluctuation comparisons between gyro simulations and pci measurements 97
1.0 MA SOC Nitrogen Seeded Case: Global GYRO at r/a ∈ [0.3,0.7] vs Exp. Exp. PCI Spectrum Wavenumber Spectra Comparison 300 1.4 a) c) Exp. PCI
250 ] Synth. PCI 2 0.0100 200 1.2
150 F [kHz] 0.0010 100 1.0 PCI Amplitdue [(n~) 50 ] 0.0001 2 -10 -5 0 5 10 -1 0.8 kR (cm )
Synth. PCI Spectrum From GYRO 0.6 300 b) PCI Amplitdue [(n~)
250 ] 2 0.0100 0.4 200
150 F [kHz] 0.0010 0.2 100 PCI Amplitdue [(n~) 50 0.0001 0.0 -10 -5 0 5 10 -10 -5 0 5 10 -1 -1 kR (cm ) kR (cm )
Figure 45: Plots comparing the experimental PCI spectrum (a) to the synthetic PCI (b) from a global GYRO simulation spanning r/a = 0.3 to r/a = 0.7 for a nitrogen seeded 1.0 MA SOC plasma, and fixed phase-velocity comparison between the two (c). The white solid lines in (a) and (b) indicate the fixed phase velocity along which the spectrum was integrated to do the comparison in (c). The white dashed lines show the window of the integration. The solid black line in (c) is the experimental PCI and the dashed line is the synthetic PCI. The synthetic PCI spectral amplitude has been multiplied by the ratio of the experimental ion energy flux to the simulated ion energy flux. 98 gyro validation results
Unseeded Full Experimental Spectrum Seeded Full Experimental Spectrum 600 10-3 600 10-3 a)text b) 500 500
400 10-4 400 10-4 300 300
200 -5 200 -5 Frequency [kHz] 10 Frequency [kHz] 10 -20 -10 0 10 20 Fluctuation Intensity -20 -10 0 10 20 Fluctuation Intensity -1 -1 kR [cm ] kR [cm ]
Unseeded Sepctrum After Subtraction Seeded Spectrum After Subtraction -3 -3 600 c) 10 600 d) 10 500 500
400 10-4 400 10-4 300 300
200 -5 200 -5 Frequency [kHz] 10 Frequency [kHz] 10 -20 -10 0 10 20 Fluctuation Intensity -20 -10 0 10 20 Fluctuation Intensity -1 -1 kR [cm ] kR [cm ]
Figure 46: Plots of the experimental PCI spectra in the unseeded (a) and seeded (b) phases, and of the experimental spectra with the background subtracted isolating the high phase-velocity feature in the unseeded (c) and seeded (d) phases. The white lines indicate center of the feature window (solid line) and the boundaries of the feature window (dashed lines). All plots are on the same scales. These spectra were measured in a 1.2 MA LOC plasma.
so in order to isolate that particular feature, the background spectrum needed to be subtracted. Figure 46 shows the unseeded and seeded spectra before the background subtraction in (a) and (b), and after background subtraction in (c) and (d) respectively. The feature am- plitude in the unseeded case is much larger than in the seeded case, and it has a somewhat higher phase velocity. This high phase-velocity feature could only be isolated for the high frequency portion of the spectrum, so only frequencies above 150 kHz were compared. The ex- perimental case shown in Figs. 46, 47, and 48, was a particular LOC plasma from the 1.2 MA density scan. Experimental issues complicated the comparison between the sim- ulated and experimental spectra. There was a problem with the X- ray spectrometer used to measure the ion temperature and velocity profiles. This meant that there was significant uncertainty in the ion temperature gradients, the balance of ion and energy energy fluxes, and the ~E × B~ velocity of the turbulence. Because of these issues, local GYRO simulations that matched the total experimental energy flux were used to compare the simulated and experimental density fluc- tuation spectra. To do this, the temperature gradients were modified slightly. In addition, the synthetic PCI technique was performed mul- tiple times with different values of the ~E × B~ velocity and the one that best reproduced the experimental spectrum was used. The synthetic PCI spectra depend sensitively on the ~E × B~ velocity used in the sim- ulation, since it doppler shifts the modes and that doppler shift is the primary contributor to the observed PCI signal frequency. The exper- imental ~E × B~ velocity is well measured for the global simulations in 5.6 summary and conclusions 99
Figs. 44 and 45, but for the local simulations the value is much more uncertain. Thus, multiple values of the ~E × B~ velocity were used in the synthetic diagnostic to compare the local simulations at r/a = 0.8 with the experimental PCI spectrum, and the value of the ~E × B~ ve- locity that best matched the experimental signal was used. The best matching ~E × B~ velocity values were slightly different between the un- seeded and seeded cases, but the values are similar enough that they are consistent with the change in the line-averaged velocity (which is measured through a different diagnostic and was well measured for these plasmas). The synthetic PCI spectra are compared to the experimental fea- ture after subtraction in Fig 47. The experimental and synthetic PCI spectra have similar shapes and magnitudes. The local GYRO sim- ulations used matched the experimental total energy flux, which re- quired a 20% decrease in the electron temperature gradient from the nominal experimental value in the seeded case. This change is well within the experimental uncertainty. The two spectra are quantita- tively compared in Fig. 48, and it is clear that the two spectra agree within uncertainties, though the synthetic unseeded spectrum is not as asymmetric as the experimental unseeded spectrum. Again the main uncertainty in the experimental spectrum is the calibration fac- tor. The error bars on the synthetic spectra in Fig. 48 represent the uncertainty in the radial width of the simulated turbulence, which cannot be determined from the local simulations. The nominal width used in these simulations assumes the feature existing between r/a = 0.80 and r/a = 0.90 which is a reasonable range based on the reflec- tometer fluctuation measurements. The fact that the synthetic PCI spectrum from GYRO is consistent in magnitude with the experimental PCI spectrum when the GYRO simulated energy fluxes are also consistent with experiment provides a solid validation test of the model for these regimes. •
5.6 summary and conclusions
Linear GYRO shows that the nitrogen seeding reduced the growth rates of ITG modes at both r/a = 0.6 and r/a = 0.8. At r/a = 0.8, where the fluxes are well above the gyrobohmn scaling and there- fore the turbulence is highly unstable, nonlinear local GYRO simu- lations with kθρs< 1 agree quantitatively with experimental energy fluxes for both ions and electrons, within the uncertainty of the input temperature profiles, except in a few cases at high densities where the simulated energy fluxes are smaller than the experimental energy fluxes. At r/a = 0.8, nonlinear local GYRO also displays a decrease in the ion energy transport with the seeding, through either an in- crease in the critical gradient, or a decrease in the stiffness. This is 100 gyro validation results
Unseeded Spectrum After Subtraction Seeded Spectrum After Subtraction -3 -3 600 a) 10 600 b) 10 500 500
400 10-4 400 10-4 300 300
200 -5 200 -5 Frequency [kHz] 10 Frequency [kHz] 10 -20 -10 0 10 20 Fluctuation Intensity -20 -10 0 10 20 Fluctuation Intensity -1 -1 kR [cm ] kR [cm ]
Unseeded Synthetic PCI Spectrum Seeded Synthetic PCI Spectrum -3 -3 600 c) 10 600 d) 10 500 500
400 10-4 400 10-4 300 300
200 -5 200 -5 Frequency [kHz] 10 Frequency [kHz] 10 -20 -10 0 10 20 Fluctuation Intensity -20 -10 0 10 20 Fluctuation Intensity -1 -1 kR [cm ] kR [cm ]
Figure 47: Plots of the unseeded spectrum after subtraction in the unseeded (a) and seeded phase (b), which are the same as Fig 46 (c) and (d). Below are plots of the synthetic PCI spectrum for local GYRO simulations in the unseeded (c) and seeded (d) phase. All plots are on the same scales. The white lines indicate the averaging window used for the quantitative comparison in Fig. 48. All plots are on the same scales. These spectra were measured in a 1.2 MA LOC plasma.
Unseeded LOC Spectra PCI seeded LOC Spectra 0.030 a) 0.030 b) 0.025 0.025
0.020 0.020
0.015 0.015
0.010 0.010 PCI Amplitude PCI Amplitude 0.005 0.005 0.000 0.000 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 k (cm-1) k (cm-1) R Experimental PCI Spectrum R Synthetic PCI Spectrum
Figure 48: Plots of the experimental (solid lines) and synthetic (dashed lines) PCI wavenumber spectra in the unseeded phase (a) and seeded phase (b). The wavenumber spectra are computed by averaging between the dashed lines in Fig. 47 for frequencies between 150 kHz and 600 kHz. Both plots are on the same scales. These spectra correspond to a 1.2 MA LOC plasma. 5.6 summary and conclusions 101 consistent in all cases with the experimental changes in the ion en- ergy flux and the temperature gradient with the seeding. At r/a = 0.6, nonlinear local GYRO with kθρs< 1 substantially over-predicts the experimental ion and electron energy fluxes, and predicts that the ion energy flux is larger than the electron energy flux in contrast to the experimental energy fluxes. Through a combination of modi- fying the experimental ion temperature profile within experimental uncertainties, and accounting for the effects of radial variation of the temperature gradients (so-called global effects), this over-prediction can be reduced. The electron energy flux at r/a = 0.6 in simulations with kθρs< 1 is entirely due to ITG modes, which will mean that the simulated ion energy flux will always be greater than or equal to the electron energy flux. This implies that the electron energy flux in the experiment is likely due to short wavelength ETG modes, which can be important when the ITG modes are close to marginal stabil- ity, as they are in the experiment at r/a = 0.6. Density fluctuations from GYRO are consistent with measurements from PCI when the simulated and experimental energy fluxes agree, and in particular the local GYRO simulated density fluctuations at r/a = 0.85 and the high phase-velocity spectral feature in the experimental PCI spectra agree well within the experimental uncertainties. ?
BIBLIOGRAPHY
[1] E A Belli and J Candy. Kinetic calculation of neoclassical transport including self-consistent electron and impurity dynam- ics. Plasma Phys. Control. Fusion, 50(9):95010, 2008. URL: http: //stacks.iop.org/0741-3335/50/i=9/a=095010, doi:10.1088/ 0741-3335/50/9/095010. 88
[2] J. Candy, C. Holland, R. Waltz, M. Fahey, and E. Belli. Tokamak profile prediction using direct gyrokinetic and neoclassical simulation. Physics of Plasmas, 16(6):060704, 2009. URL: http://scitation.aip.org/content/aip/journal/ pop/16/6/10.1063/1.3167820, doi:10.1063/1.3167820. 82, 83, 88
[3] J. Candy and R. Waltz. Anomalous Transport Scaling in the DIII-D Tokamak Matched by Supercomputer Simulation. Physical Review Letters, 91(4):045001, jul 2003. URL: http: //link.aps.org/doi/10.1103/PhysRevLett.91.045001, doi:10. 1103/PhysRevLett.91.045001. 77, 80
[4] J. Candy, R. E. Waltz, and W. Dorland. The local limit of global gyrokinetic simulations. Physics of Plasmas, 11(2004):3–7, 2004. doi:10.1063/1.1695358. 92
[5] J Candy, R E Waltz, M R Fahey, and C Holland. The ef- fect of ion-scale dynamics on electron-temperature-gradient turbulence. Plasma Physics and Controlled Fusion, 49(8):1209– 1220, 2007. URL: http://stacks.iop.org/0741-3335/49/i=8/ a=008?key=crossref.03fbbed7a6f1ec4aac836366fb6983ce, doi: 10.1088/0741-3335/49/8/008. 78
[6] a. M. Dimits, G. Bateman, M. a. Beer, B. I. Cohen, W. Dor- land, G. W. Hammett, C. Kim, J. E. Kinsey, M. Kotschen- reuther, a. H. Kritz, L. L. Lao, J. Mandrekas, W. M. Nevins, S. E. Parker, a. J. Redd, D. E. Shumaker, R. Sydora, and J. Wei- land. Comparisons and physics basis of tokamak transport models and turbulence simulations. Physics of Plasmas, 7(3):969, 2000. URL: http://scitation.aip.org/content/aip/journal/ pop/7/3/10.1063/1.873896, doi:10.1063/1.873896. 78, 92
[7] N T Howard, C Holland, A E White, M Greenwald, and J Candy. Synergistic cross-scale coupling of turbulence in a tokamak plasma. Phys. Plasmas, 21(11):112510, 2014. URL: http: //scitation.aip.org/content/aip/journal/pop/21/11/10.
103 104 Bibliography
1063/1.4902366, doi:http://dx.doi.org/10.1063/1.4902366. 78, 94
[8] J.E. Kinsey, G.M. Staebler, J. Candy, R.E. Waltz, and R.V. Budny. ITER predictions using the GYRO veri- fied and experimentally validated trapped gyro-Landau fluid transport model. Nuclear Fusion, 51(8):083001, 2011. URL: http://stacks.iop.org/0029-5515/51/i=8/a= 083001?key=crossref.8333bb10eb01b7367e62587146b0d018, doi:10.1088/0029-5515/51/8/083001. 77, 78
[9] Doug McCune. trxpl. URL: http://w3.pppl.gov/{~}hammett/ work/gs2/docs/trxpl.txt. 77
[10] M Porkolab, J Dorris, P Ennever, C Fiore, M Greenwald, A Hub- bard, Y Ma, E Marmar, Y Podpaly, M L Reinke, J E Rice, J C Rost, N Tsujii, D Ernst, J Candy, G M Staebler, and R E Waltz. Transport and turbulence studies in the linear ohmic confine- ment regime in Alcator C-Mod. Plasma Phys. Control. Fusion, 54(12):124029, 2012. URL: http://stacks.iop.org/0741-3335/ 54/i=12/a=124029, doi:10.1088/0741-3335/54/12/124029. 86
[11] G. M. Staebler, J. E. Kinsey, and R. E. Waltz. A theory-based transport model with comprehensive physics. Physics of Plasmas, 14(5):055909, 2007. URL: http://link.aip.org/link/PHPAEN/ v14/i5/p055909/s1{&}Agg=doi, doi:10.1063/1.2436852. 88 THEORETICALSTUDYOFDILUTIONEFFECTWITH 6 GYRO
In addition to validating GYRO’s quantitative predictions of energy fluxes and density fluctuations, GYRO can be used as a tool to inves- tigate the theoretical effects of main-ion dilution on turbulence. The advantage of using a code like GYRO instead of an experiment, is that it allows for a carefully controlled study of the dilution inde- pendent of other changes that happen to the plasma when they are seeded as in the experiment. GYRO also allows for the study of di- lution independent of Zeff, which while not physically self-consistent, does indicate better which effect is important. This work helps to generalize these results to other cases and helps to understand which physical mechanisms are responsible for the observed reduction in the turbulence. •
6.1 effect of dilution on gyro linear growth rates
The effect of dilution on the linear grow rates of modes is studied through multiple linear GYRO runs at various values of a/LTi and Zeff, with nD/nebeing adjusted self-consistently with Zeff. These com- parisons were done for kθρs≈ 0.3, which is the peak of the ITG turbu- lence spectrum. This analysis was performed for both a 1.0 MA SOC and a 0.8 MA LOC case, and at both r/a = 0.6 and r/a = 0.8. The results of the scans in the 0.8 MA LOC case are shown in Fig. 49, with the simulations done at r/a = 0.6 shown on the left and the simulations done at r/a = 0.8 shown on the right. Both plots show that when nD/ne decreases, the linear growth rate decreases. The critical gradient of the ITG turbulence also increases with decreasing nD/ne, going from a/LTi = 1.6 to a/LTi = 4.5 as nD/ne goes from 0.96 to 0.30 for the r/a = 0.6 simulations on the left. The r/a = 0.8 simulations are highly TEM dominated except for very large values of a/LTi, so the exact critical gradient of the ITG turbulence cannot be determined. The ITG growth rate and nD/ne are approximately linearly related in the cases shown, with the value of d(γ)/d(nD/ne) ≈ 0.35 being the same for all the ITG modes for both the r/a = 0.6 and r/a = 0.8 simulations. There is also a small decrease in the TEM growth rates with the decrease in nD/ne, but the change is much smaller than the change in the growth rates of the ITG modes with d(γ)/d(nD/ne) ≈ 0.05.
105 106 theoretical study of dilution effect with gyro
0.8 MA LOC Nitrogen Seeded Case
r/a = 0.6 r/a = 0.8 0.35 0.30 0.9 0.25 0.9 0.25 0.15
0.25 0.8 0.8
/a) 0.20 s 0.7 0.7 e e /n /n
D D 0.15 n 0.6 n 0.6
0.10 Growth Rate (c 0.05 0.5 0.5 0.05 0.4 0.4 0.00 2 3 4 5 6 7 2 4 6 8
a/LTi a/LTi
Figure 49: The linear growth rates from GYRO for modes with kθρs = 0.3 for varying values of a/LTiand nD/ne, at r/a = 0.6 (on the left) and r/a = 0.8 (on the right) for a 0.8 MA LOC case. The white line indicates the border between the modes that are electron diamagnetic drift directed (to the left ot the line) and ion diamagnetic drift directed (to the right of the line), as determined by the sign of the real frequency of the modes. The purple cross is the nominal experimental value of a/LTi and nD/ne.
The results of the scans in the 1.0 MA SOC case are shown in Fig. 50, with the simulations done for r/a = 0.6 shown on the left and the simulations for r/a = 0.8 shown on the right. Both plots show that when nD/ne decreases, the growth rate decreases. The critical gradient of the ITG turbulence also increases with decreasing nD/ne, going from a/LTi = 1.5 to a/LTi = 2.5 as nD/ne goes from 0.96 to 0.66 for the r/a = 0.6 simulations on the left, and going from a/LTi = 4.5 to a/LTi = 7.7 as nD/ne goes from 0.96 to 0.66 for the r/a = 0.8 simulations on the right. For the r/a = 0.6 simulations the value of d(γ)/d(nD/ne) are 0.35 for the ITG modes and 0.05 for the TEMs, which is the same as for the 0.8 MA LOC cases at r/a = 0.6 and r/a = 0.8. For the r/a = 0.8 simulations, the value of d(γ)/d(nD/ne) is 0.9 for the ITG modes and 0.3 for the TEMs. These values are significantly larger than the values in all the other cases shown. ETG mode linear growth rate spectra were also compared at r/a = 0.6 for a 1.0 MA SOC nitrogen seeded case. Figure 51 shows the linear growth rate spectra from two simulations: one with no impu- rities (nD/ne = 1.0 in green) and one with the impurities included (nD/ne = 0.83 in black). The result is that the dilution reduces the lin- ear growth rates of the modes substantially, reducing the peak growth rate by about a factor of 3, moving the peak growth rate to lower kθρs, and completely stabilizing the highest kθρs modes. This reduction is even larger than the reduction in the low-k growth rates, which only decrease by about 30% with this dilution. The cause of this is not un- 6.2 effect of dilution on gyro energy fluxes 107
1.0 MA SOC Nitrogen Seeded Case
r/a = 0.6 r/a = 0.8 0.4 0.95 0.95 0.45 0.10 0.35
0.20 0.30 0.25
0.90 0.90 0.3 /a) s 0.85 0.85 0.15 e e /n /n
D D 0.2 n n 0.80 0.80 0.15 Growth Rate (c 0.75 0.75 0.1
0.70 0.70 0.0 1 2 3 4 5 2 4 6 8 10
a/LTi a/LTi
Figure 50: The linear growth rates from GYRO for modes with kθρs= 0.3 for varying values of a/LTiand nD/ne, at r/a = 0.6 (on the left) and r/a = 0.8 (on the right) for a 1.0 MA SOC case. The white line indicates the border between the modes that are electron diamagnetic drift directed (to the left ot the line) and ion diamagnetic drift directed (to the right of the line), as determined by the sign of the real frequency of the modes. The purple cross is the nominal experimental value of a/LTiand nD/ne. derstood at the moment, nor is the effect this may have on the result- ing energy transport. The fact that there was no substantial reduction in the electron energy flux observed in the seeding experiment sug- gests that this effect may not be as strong as the linear growth rate suggests. The effects of dilution on ETG growth rates were investigated in a scan similar to the ones in Fig. 49 and Fig. 50, but varying a/LTe and nD/neand using kθρs = 24. The results are shown in Fig. 52. Un- like in Fig. 51 where Zeff is the same in both the plotted simulations, the Zeff value is varied self-consistently with nD/ne in the results in Fig. 52. As expected from the result of Fig. 51, the dilution does sub- stantially decrease the growth rates through an increase in the critical gradient. The critical gradient goes from a/LTe = 2.1 for nD/ne = 0.98 to a/LTe = 3.9 for nD/ne = 0.68. The formula from Jenko et. al.[1] of the ETG critical gradient shows no strong dependence on nD/ne, but there is a linear dependence on Zeff through τ = ZeffTe/Ti. However this would not explain the result of Fig. 51, as the Zeff is the same in both cases. •
6.2 effect of dilution on gyro energy fluxes
Nonlinear simulations were performed to quantitatively determine the effects of dilution on the simulated turbulent energy transport. 108 theoretical study of dilution effect with gyro
Linear GYRO Growth Rates For 1.0 MA SOC at r/a = 0.6 6 nD/ne = 1.0 n /n = 0.83 5 D e /a] s 4
3
2
Growth Rate [c 1 0 0 10 20 30 40 ρ kθ s
Figure 51: Plot of the linear growth rate spectra from GYRO for a 1.0 MA SOC case at r/a = 0.6 with dilution (black) and without dilution (green). Both simulations have the same Zeff values.
ETG Growth Rates For 1.0 MA SOC Seeded Case At r/a = 0.6 10 12 2 0.95 6
/a) 10 s 0.90 8 0.85 e /n
D 6 n 0.80 4 0.75 2 Maximum Growth Rate (c 0.70 2 0 1 2 3 4 5
a/LTe
Figure 52: The linear growth rates from GYRO for modes with kθρs = 0.3 for varying values of a/LTe and nD/ne, at r/a = 0.6 for a 1.0 MA SOC case. The white line indicates the border between the modes that are electron diamagnetic drift directed (to the left ot the line) and ion diamagnetic drift directed (to the right of the line), as determined by the sign of the real frequency of the modes. The purple cross is the nominal experimental value of a/LTe and nD/ne. 6.2 effect of dilution on gyro energy fluxes 109
0.8 MA LOC seeded, r/a = 0.6 25 Ion Energy Flux For nD/ne = 1.0 Ion Energy Flux For nD/ne = 0.77 20
15 GB /Q i
Q 10
5
0 25 Electron Energy Flux For nD/ne = 1.0 Electron Energy Flux For nD/ne = 0.77 20
15 GB /Q e
Q 10
5
0 0 1 2 3 4
a/LTi
Figure 53: Plots of GYRO ion energy fluxes (top) and electron energy fluxes (bottom) at r/a = 0.6 for different values of the ion temperature gradient both with the experimental dilution (black) and no dilution (green) for a 0.8 MA LOC case. The solid lines are the offset-linear fits to the data, and the dashed lines indicate the critical gradients.
The simulations performed were restricted to kθρs < 1, as the sim- ulations in Chapter 5 were. This will show effects of dilution on the ITG modes and TEMs, but will not show the effect on the ETG modes. Simulations were performed with and without the impurity ions, with nD/ne set to 1 in the case with no impurity ions, to enforce quasineutrality. All other parameters were held fixed between the two simulations, including Zeff. This isolates the effects of nD/ne on the turbulent energy transport. Figure 53 shows the result of simulations with dilution (in black) and without dilution (in green) in a 0.8 MA LOC nitrogen seeded case at r/a = 0.6 for both the ion energy flux (top) and electron en- ergy flux (bottom) for various values of a/LTi. The result is that the inclusion of the dilution results in a doubling of the critical gradi- ent, but no noticeable change in the stiffness. Including dilution also decreases the electron energy flux in all cases, even for the lowest val- ues of a/LTi where the ITG modes are completely stable (and there is no ion energy flux). This suggests that the dilution is also having a stabilizing effect on the TEMs as well as on the ITG modes, which is consistent with the linear stability picture shown in Fig. 50 and Fig. 49. Figure 54 shows simulations of the same 0.8 MA LOC nitrogen seeded plasma but at r/a = 0.8. Again there is a decrease in the ion 110 theoretical study of dilution effect with gyro
0.8 MA LOC seeded, r/a = 0.8 80 Ion Energy Flux For nD/ne = 1.0 Ion Energy Flux For nD/ne = 0.77 60 GB
/Q 40 i Q
20
0 80 Electron Energy Flux For nD/ne = 1.0 Electron Energy Flux For nD/ne = 0.77 60 GB
/Q 40 e Q 20
0 0 2 4 6 8
a/LTi
Figure 54: Plots of GYRO ion energy fluxes (top) and electron energy fluxes (bottom) at r/a = 0.8 for different values of the ion temperature gradient both with the experimental dilution (black) and no dilution (green) for a 0.8 MA LOC case. The solid lines are the offset-linear fits to the data, and the dashed lines indicate the critical gradients.
and electron energy fluxes with dilution, but in this case it is due to a decrease in the stiffness rather than to a change in the critical gradient. This results in a smaller decrease in the energy flux with dilution when a/LTi is small, but a larger decrease in the energy flux with dilution when a/LTi is large. The electron energy flux also decreases slightly with the dilution, again indicating an effect of the dilution on the TEMs. Figure 55 shows simulations of a 1.0 MA SOC seeded plasma with dilution (in black) and without dilution (in green) at r/a = 0.6. Again there is a decrease in the ion energy flux with the dilution, and in this case the dilution decreased the stiffness much more than it increased the critical gradient (which did not appear to change at all). The elec- tron energy flux was decreased by the dilution except for the very lowest value of a/LTi, where the inclusion of dilution actually very slightly increased the electron energy flux. This is different from the case of Figs. 53 and 54, where the dilution seemed to have a small stabilizing effect on the TEMs. Instead the dilution in this case seems to have a small destabilizing effect on the TEMs, indicating that the effect of dilution on electron energy transport is not as clear-cut as it is for the ion energy transport. Figure 56 shows simulations of the same 1.0 MA SOC seeded plasma with dilution (in black) and without dilution (in green) now at r/a = 6.2 effect of dilution on gyro energy fluxes 111
1.0 MA SOC seeded, r/a = 0.6 35 Ion Energy Flux For nD/ne = 1.0 30 Ion Energy Flux For nD/ne = 0.84 25
GB 20 /Q i 15 Q 10 5 0 35 Electron Energy Flux For nD/ne = 1.0 30 Electron Energy Flux For nD/ne = 0.84 25
GB 20 /Q e 15 Q 10 5 0 0 1 2 3 4
a/LTi
Figure 55: Plots of GYRO ion energy fluxes (top) and electron energy fluxes (bottom) at r/a = 0.6 for different values of the ion temperature gradient both with the experimental dilution (black) and no dilution (green) for a 1.0 MA SOC case. The solid lines are the offset-linear fits to the data, and the dashed lines indicate the critical gradients.
0.8. Again there is a decrease in the ion energy flux with the dilution, and again it is due to a decrease in the stiffness of the ion energy transport and almost no change in the critical gradient. The change in the stiffness is even larger in this case than it is in Fig. 55, or the LOC case in Fig. 54 where the decrease in nD/ne was even larger than it is in Fig. 56. This shows how strong this dilution effect can be, and gives an indication of how much the ion energy flux can be reduced (or conversely temperature gradient increased) by even mod- erate dilution. In this case there is almost no difference in the electron energy flux for the two lower a/LTi values, indicating that the effect of the dilution on TEMs is small in this case. The effect of dilution on the global simulations was investigated for the 1.0 MA SOC case for global simulations ranging from r/a = 0.3 to r/a = 0.7. The results are shown in Fig. 57, which plots the global GYRO ion energy flux for a simulation with dilution (in black) and without dilution (in green). Again Zeff in both simulations is the same. The inclusion of dilution reduced the ion energy flux by ap- proximately a factor of two for r/a > 0.6, but makes almost no dif- ference for r/a < 0.6 where the energy flux is nearly zero. Having only a single simulation makes it difficult to exactly determine the stiffness and critical gradient, but since this is a global simulation the value of a/LTi changes with r/a and increases roughly linearly with 112 theoretical study of dilution effect with gyro
1.0 MA SOC seeded, r/a = 0.8 100 Ion Energy Flux For nD/ne = 1.0 Ion Energy Flux For nD/ne = 0.84 80
60 GB /Q i
Q 40
20
0 100 Electron Energy Flux For nD/ne = 1.0 Electron Energy Flux For nD/ne = 0.84 80
60 GB /Q e
Q 40
20
0 0 2 4 6 8
a/LTi
Figure 56: Plots of GYRO ion energy fluxes (top) and electron energy fluxes (bottom) at r/a = 0.6 for different values of the ion temperature gradient both with the experimental dilution (black) and no dilution (green) for a 1.0 MA SOC case. The solid lines are the offset-linear fits to the data, and the dashed lines indicate the critical gradients.
increasing r/a. Because there does not seem to be a large increase in the fluxes at low r/a (and therefore low a/LTi) this is an indication that the critical gradient was likely not strongly affected by dilution. This is consistent with the previous results that dilution does not strongly affect critical gradient in SOC plasmas, and shows that the global simulations do not show a significant qualitative difference in the predicted effects of the dilution on energy transport. •
6.3 quanitfying the effect of dilution on gyro stiff- ness and critical gradient
In addition to comparing pure deuterium cases to diluted cases, it is valuable to utilize GYRO to quantify the the dependence of stiff- nesses and critical gradients on nD/ne. In principle this allows one to find a maximally effective operating point wherein the turbulence- suppressing benefits of dilution do not overtake the costs through increase in radiated power and reduction of fuel ion density (and therefore fusion power). It is also necessary to understand the effects of dilution on both the critical gradient and stiffness of the trans- port because both effects are important, albeit in different regimes. When turbulent transport is well above marginal stability, as in cur- 6.3 quanitfying the effect of dilution on gyro stiffness and critical gradient 113
Global GYRO Energy Flux For 1.0 MA SOC Case 2.5 Ion Energy Flux For nD/ne = 1.0 Ion Energy Flux For nD/ne = 0.84 2.0
1.5 GB /Q i
Q 1.0
0.5
0.0 0.3 0.4 0.5 0.6 0.7 r/a
Figure 57: Plots of the ion energy flux of global GYRO simulations with dilution (in black) and witout dilution (in green) for a 1.0 MA SOC case. The global simulation extends from r/a = 0.3 to r/a = 0.7, and is centered at r/a = 0.5. rent neutral-beam heated tokamak plasmas, a reduction in stiffness will usually net larger gains in energy confinement than an increase in the critical gradient. When turbulent transport is close to marginal stability, as in the regimes predicted for ITER, a reduction in stiffness will have only a small effect on the overall energy transport, while an increase in the critical gradient will have a significant effect on the overall energy confinement. Thus to understand the practical im- plications of dilution on turbulent transport, the effects of varying nD/ne on GYRO predicted stiffness and critical gradient needed to be quantified. To perform this quantification of the effects of dilution, multiple local nonlinear GYRO a/LTi scans were performed at various values of nD/ne. The values of Zeff were changed self-consistently with the nD/ne in this analysis, unlike in the comparisons in Section 6.2, since it was simpler to do so in this case. However, the results are almost identical to the corresponding ones in the previous section, where Zeff was left unchanged. The analysis was performed at r/a = 0.6 for the 0.8 MA plasmas at a range of densities. r/a = 0.6 was chosen for this comparison in order to isolate the effect of the seeding on the ITG modes. This is because, at r/a = 0.6, ITGs are the only unstable mode with kθρs< 1 in these cases, and the ITG modes are those that are most strongly affected by dilution. Figure 58 shows the simula- tions used in one LOC case as an example. The value of nD/ne was scanned from 1.0 to 0.77, and the value of a/LTi was scanned from 114 theoretical study of dilution effect with gyro
20 -3 ne = 0.7 x 10 m (LOC), 0.8 MA Case At r/a = 0.6 25 nD/ne = 1.0 nD/ne = 0.87 20 nD/ne = 0.77
15 GB /Q i Q 10
5
0 0 1 2 3 4
a/LTi
Figure 58: Plots of three a/LTi GYRO scans varying nD/ne from 1.0 (in green), to 0.87 (in purple), to 0.77 (in blue). The solid lines are linear fits to the points, with the vertical dashed lines indicating the different values of the critical gradient. These particular points correspond to a 0.8 MA LOC seeded plasma.
0.6 to 3.8; in both cases the same offset linear fit was used to get the stiffnesses and critical gradients. This analysis was repeated for four cases of 0.8 MA plasmas, two in the LOC regime and two in the SOC regime. The full results of this analysis are shown in Fig. 59, which plots the critical gradient and stiffness of GYRO ion transport at r/a = 0.6, for four densities in the 0.8 MA density scan. The two highest density cases are in the SOC regime (in red and teal), while the two lower den- sity cases (black and yellow) are in the LOC regime. The two regimes display very different critical gradient values, with the two SOC cases having almost the same critical gradients at all nD/ne values, and the two LOC cases having almost the same critical gradients at all but the highest nD/ne values. Only the yellow (LOC) case shows a strong de- pendence of the critical gradient on the dilution, while all the other density cases show weak variation of the critical gradient with the dilution. The stiffnesses plotted on the right of Fig. 59 also show differences among the different density cases, although the stiffness uniformly increases with increasing nD/ne. The two LOC cases (in black and yellow) show less variation of stiffness with nD/ne, with the case plotted in yellow showing by far the least variation of stiffness of all cases. The two SOC cases (in red and teal) show a much stronger ef- fect of nD/ne on the stiffness of the ion transport. The higher density SOC case in black shows an extremely strong dependence of the stiff- ness on nD/ne, with the stiffness increasing by more than a factor of 4 when nD/ne changes from 0.70 to 1.0. The lower density SOC case (in red) also shows a strong dependence of the stiffness on nD/ne, 6.4 summary and conclusions 115
0.8 MA Cases At r/a = 0.6 Critical Gradients Stiffnesses )] Ti
] 3.0
Ti 12 2.5 10 2.0 )/d(a/L GB 8 /Q 1.5 i 6 1.0 4 0.5 2
Critical Gradient [a/L 0.0 0 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Stiffness [d(Q 0.70 0.75 0.80 0.85 0.90 0.95 1.00 nD/ne n = 0.5 x 1020 m-3 nD/ne e 20 -3 ne = 0.7 x 10 m n = 0.8 x 1020 m-3 e 20 -3 ne = 1.1 x 10 m
Figure 59: Plots of the effect of varying nD/ne on the GYRO predicted stiffnesses and critical gradients at r/a = 0.6 for four different density cases in the 0.8 MA scan. The two lowest densities (in black and yellow) are in the LOC regime, while the two highest densities (in red and teal) are in the SOC regime. but only between stiffness values of 0.8 and 0.9. Outside of that win- dow the stiffness is roughly independent of nD/ne. The case in red is one for which the seeding caused the intrinsic rotation to reverse di- rection, and the unseeded and seeded values of nD/ne are on either side of the bifurcation of the stiffness, so this may have some bearing on the intrinsic rotation behavior. •
6.4 summary and conclusions
In this section, the effects of dilution on turbulence were investigated using the GYRO code. The effects of varying nD/ne on linear growth rates at both low and high kθρs were studied in Sec. 6.1, where it was found that both ITG and ETG linear growth rates are strongly reduced by dilution, independent of Zeff. TEMs were also found to be somewhat affected by dilution, but much less so than were the ITG or ETG modes. The effect of dilution on nonlinear GYRO energy fluxes at r/a = 0.6 and r/a = 0.8 for both LOC and SOC cases was shown in Sec. 6.2. In all cases where ITG modes were unstable (i.e. a/LTi was above the critical gradient) the simulations that included dilution had significantly lower ion energy fluxes. Including dilution also reduced the electron energy flux through cross-species coupling from ITG modes, though the effects of dilution on electron energy flux from TEMs was mixed (increased in some cases, decreased in others). Global simulations also showed a decrease in the ion energy fluxes with dilution. Finally, in Sec. 6.3, the effect of dilution on stiffness and critical gradient was quantified at r/a = 0.6 for multiple 0.8 MA 116 theoretical study of dilution effect with gyro
density cases in both the LOC and SOC regimes. It was found that the critical gradient was not very strongly affected by dilution except for one particular LOC case, and that the stiffness was strongly affected by dilution in many cases, especially in the SOC regime. The results of this section imply that main ion dilution is the pri- mary driver of the observed reduction in the experiments discussed in Chapter 4 and the GYRO and TGLF simulations Chapter 5 due to nitrogen seeding. Even moderate reductions in nD/ne cause large changes in the ion energy transport. The results also show also shows that this effect of dilution reducing the ion energy transport is robust across multiple regimes of parameter space. These results give further confidence that this dilution effect will be present in other regimes in addition to the ohmic plasmas studied here. This means that dilution has likely been a factor in turbulent energy transport in previous ex- periments and will be a factor in transport in future reactor plasmas. ? BIBLIOGRAPHY
[1] F. Jenko, W. Dorland, and G. W. Hammet. Critical gradient for- mula for toroidal electron temperature gradient modes. Physics of Plasmas, 8(9):4096–4104, 2001. doi:10.1063/1.1391261. 107
117
CONCUSIONSANDFUTUREWORK 7
7.1 summary of this thesis work
The work presented in this thesis shows a thorough exploration of the effects of main-ion dilution on turbulent energy transport in both experiments performed on Alcator C-Mod and in theoretical predic- tions made using the gyrokinetic simulation code GYRO. Both the experimental and theoretical results show that main-ion dilution can significantly reduce turbulent ion energy transport. This dilution ef- fect has been predicted previously with TGLF[5] in work based on Alcator C-Mod plasmas, but the effect had not been verified in the experiment or studied in detail with simulations. The TGLF work motivated the experiments performed as part of this thesis work, wherein ohmic plasmas on Alcator C-Mod with a range of densities and plasma currents were seeded with nitrogen to dilute them, while a cryopump kept the electron density constant. These experiments were detailed in Chapter 4. The seeding did not significantly increase or decrease the overall energy confinement time of the plasmas, which given the fact that the radiated power went up due to the seeding means that the seeding caused a commensurate de- crease in the energy transport. The seeding caused local decreases in the ion energy transport, especially at r/a = 0.8 where the turbulence was well above marginal stability. Electron density fluctuations at that radial location also decreased in the highest plasma current cases. The seeding also affected the intrinsic toroidal rotation of higher density cases, causing seeded plasmas to have a higher critical effective col- lisionality for rotation reversals than the unseeded plasmas. The ro- tation direction was found to depend on nDq95 in both the seeded and unseeded cases, which could be related to the dilution’s effect on transport. The measured fluxes and fluctuations seen in the experiments were compared to the predictions from the nonlinear 5-D gyrokinetic code GYRO in Chapter 5. The comparisions were done at two different radial locations: r/a = 0.8 where the experimental transport is well above the marginally stable level; and r/a = 0.6 where the experimen- tal transport is close to the marginally stable level. Transport for r/a < 0.5 was not considered because the simulations predicted no unstable turbulence in this region, and transport for r/a > 0.9 was not consid- ered because there the assumptions of the gyrokinetic theory used in the GYRO simulations were not valid. The local GYRO simulations at r/a = 0.8 in general well reproduced the experimental electron
119 120 concusions and future work
and ion energy fluxes, except for the highest density cases where the fluxes were under-predicted. The local GYRO simulations at r/a = 0.6 systematically over-predicted the ion and electron energy fluxes, and predicted that the ion energy flux was larger than the electron energy flux, while in the experiments it was the opposite. It was necessary to account for the radial variation of the ion temperature gradient using more expensive global GYRO simulations to bring the modelled ion energy flux down to the experimental level. Once these global effects were taken into account, the electron energy flux was under-predicted by the simulations. This pointed to the pressence of short wavelength electron temperature gradient (ETG) modes, which have been shown in other work on C-Mod L-modes to be a significant contributor to the electron energy flux in cases where the ITG modes are close to marginal stability and the long wavelength trapped electron modes are stable [2], both of which are the case in these plasmas. The den- sity fluctuations in GYRO are consistent with the PCI measurements of line-integrated density fluctuations. Once GYRO was validated to correctly predict the energy flux due to long-wavelength modes in these cases, it could then be used as a tool to explore the effects of dilution on turbulence. These results were described in Ch. 6. In the code it was possible to vary only nD/ne, leaving all other parameters the same, unlike in the exper- iment where the seeding changed many parameters important to turbulent energy transport in addition to nD/ne. This allowed, in the modelling, for the isolation of the effects of dilution on turbu- lent energy transport. The results of that analysis were that reducing nD/ne reduced linear growth rates of both low-k (ITG and TEM) and high-k (ETG) turbulent modes. Dilution primarily reduced these growth rates through an increase in the critical gradient. The decrease in the ETG growth rates with dilution came as somewhat of a sur- prise, as there was not a significant observed reduction in the ex- perimental electron energy flux with the seeding. This means that either the other changes with the seeding affected the growth rates more than dilution, or the nonlinear ETG energy transport was not as strongly affected as the linear growth rates. In addition to the lin- ear growth rates, the effect of varying nD/ne on nonlinear critical gradient and stiffness of the ion energy transport was also analyzed. It was found that in every regime studied, the dilution had a stabiliz- ing effect on the ion energy transport. For LOC densities, the dilution affects the critical gradient more than the stiffness, while at SOC den- sities, the dilution affects the stiffness more than the critical gradient. There are many parameters that change between the LOC and SOC regimes besides the density, however, so it does not necessarily imply that the density by itself is the reason for this difference. • 7.2 conclusions and implications 121
7.2 conclusions and implications
All the work presented in this thesis shows that dilution has a stabi- lizing effect on turbulent ion energy transport. This has been shown experimentally, in GYRO simulations of experimental discharges, and through controlled parameter scans using GYRO simulations as well. This dilution effect was robust across many regimes of parameter space for the LOC and SOC regimes. While the work presented was limited to ohmic plasmas, many of the important dimensionless pa- rameters were scanned. The fact that the dilution had a significant sta- bilizing effect on the turbulent energy transport in all of the regimes gives confidence that dilution is going to have a stabilizing effect on turbulence in future reactor plasmas such as ITER. Reactor plasmas will be diluted from a variety of sources including helium ash from fusion reactions, extrinsic impurity seeding necessary to reduce di- vertor heat loads, and fast ions (which will act as a separate ion pop- ulation for the purposes of the turbulence). The presence of these diluting ion species could result in higher energy confinement and higher ion temperatures if the dilution has the predicted stabilizing effect on the turbulence. This would be especially true if the dilution increased the critical gradient of the turbulence, as the gradients in ITER are predicted to be close to marginal stability. It is unclear from the work presented whether the reduction of turbulent energy trans- port due to dilution is worth the increase in radiated power from impurities or the reduced fusion reaction rate from a decrease in the fuel ion density. However, the presence of at least some dilution is largely unavoidable for fusion plasmas so it is a positive result for fusion that this dilution should have a positive impact on the energy confinement. In addition to implications for future fusion reactors, the work in this thesis also has implications for observations in previous ex- periments. One particularly well known example is the so-called R- I mode, in which seeding impurities into neutral beam heated L- mode plasmas on DIII-D [3] and TEXTOR [4] was found to greatly increase the energy confinement time. It was found that lower charge impurity species were better for increasing energy confinement than higher charge impurities, and in addition the seeded plasmas showed a lower stiffness than the unseeded plasmas. Both of these results are consistent with the result from this thesis that the dilution from the impurities reduced the ion energy transport. The reason that those cases showed an increase in energy confinement time, whereas the ohmic plasmas in this thesis did not, was because neutral beam heated plasmas like the ones in the R-I mode have ion energy fluxes well above marginal stability and have ion energy fluxes that are larger than electron energy fluxes. Thus the dilution induced reduc- tion of the ion energy transport in the R-I mode cases resulted in 122 concusions and future work
a much larger reduction in total energy flux and therefore a much larger effect on the overall energy confinement. The impurity injec- tion in the R-I mode also caused changes to the momentum transport, resulting in larger core toroidal rotation. This effect may be related to the observations in this thesis of changes to toroidal rotation from seeding, but the effect was more pronounced in beam-heated plas- mas with external momentum input. The results of this thesis may also be related to observations of a degradation in energy confinement when tokamaks like JET [1] and ASDEX-U [6] switch from carbon to metallic walls. This confinement degradation is found to be partially recovered through the use of low-Z impurity seeding. The core transport dynamics for the ASDEX- U cases were investigated and it was found that while the pedestal height increased with seeding, a/LTi and a/LTe in the core did not change significantly with the seeding. This was found through GENE simulations to be due to the fact that the reduction in the turbulence drive from the seeding was counterbalanced by an increase in QGB be- 5/2 cause of an increase in Te from the pedestal. Since QGB scales as Te , the 10-20% increase in Te from the higher pedestal resulted in a 25- 60% increase in QGB. The effect of nitrogen seeding on pedestal trans- port could be related to dilution effects on turbulence, but further study is required to make that comparison. •
7.3 future work
Turbulence reduction due to main-ion dilution was robustly observed in a variety of regimes through the work in this thesis, but all the plasmas used as part of the analysis were ohmic L-modes. These have obvious differences from the high-performance H-mode or I- mode plasmas that will be used in fusion reactors. Therefore one of the main extensions of this work would be to do the same sort of analysis on high-performance, auxiliary-heated plasmas. This would include measuring the impurity content on a variety of similar plas- mas, and could also measure the stiffness experimentally by varying the input power and observing the resulting change in the tempera- ture profiles. The same sort of simulations would also be performed to determine the sensitivity of the stiffness and critical gradient to the dilution. There already exists a large amount of experimental data on seeding of high-performance plasmas, so it may be possible to do this sort of analysis without performing any new experiments. It would be useful to have a quantitative measure of the effect of dilution on high-performance plasma transport, as it would better inform the op- eration of future reactors. In order to extend this work to reactors with any certainty, the ef- fect of dilution on ETG turbulence would need to be investigated 7.3 future work 123 in more detail. The results described in Chapter 6 showed that the linear growth rates of ETG modes was strongly affected by dilution, but there was no strong effect of the seeding on the electron energy transport, as described in Chapter 4. This means that nonlinear sim- ulations of ETG modes would be needed to accurately simulate the turbulence. These simulations are extremely expensive, but even a two-point or three-point scan of the dilution in nonlinear simulations would be useful. It would be especially useful if the results could somehow be incorporated into the reduced-model of TGLF. Another aspect of this thesis work that invites further study is the result that the dilution can have a strong effect on the intrinsic rota- tion. This is an interesting result because it allows for a separation of the dependence of the rotation on the electron and ion collisionalities, as the electron collisionality has been shown to correlate with the intrinsic rotation, but the two species’ collisionalities are obviously closely related. Other experiments with the dilution and rotation, including ones in ICRF heated plasmas and plasmas where lower- hybrid waves are injected, could be done to further track the effects of dilution on the intrinsic rotation. In addition, these experiments would be an excellent test set to validate models of momentum trans- port. An accurate momentum transport model is necessary to predict the turbulence dynamics in a future fusion device, as the rotation of the plasma in a large fusion device will be almost entirely determined by intrinsic rotation due to its very large moment of inertia and lack of strong external momentum input compare to current devices. All of these future research directions would help to make the work of this thesis even more useful to the pursuit of the broader goal of fusion energy. ?
BIBLIOGRAPHY
[1] C. Giroud, G.P. Maddison, S. Jachmich, F. Rimini, M.N.a. Beurskens, I. Balboa, S. Brezinsek, R. Coelho, J.W. Coenen, L. Frassinetti, E. Joffrin, M. Oberkofler, M. Lehnen, Y. Liu, S. Marsen, K. McCormick, A. Meigs, R. Neu, B. Sieglin, G. van Rooij, G. Arnoux, P. Belo, M. Brix, M. Clever, I. Cof- fey, S. Devaux, D. Douai, T. Eich, J. Flanagan, S. Grunhagen, A. Huber, M. Kempenaars, U. Kruezi, K. Lawson, P. Lomas, C. Lowry, I. Nunes, A. Sirinnelli, A.C.C. Sips, M. Stamp, and S. Wiesen. Impact of nitrogen seeding on confinement and power load control of a high-triangularity JET ELMy H- mode plasma with a metal wall. Nuclear Fusion, 53(11):113025, 2013. URL: http://stacks.iop.org/0029-5515/53/i=11/a= 113025?key=crossref.b25f5cabfe1eb9a1df537a555d386b64, doi: 10.1088/0029-5515/53/11/113025. 122 [2] N T Howard, C Holland, A E White, M Greenwald, and J Candy. Synergistic cross-scale coupling of turbulence in a tokamak plasma. Phys. Plasmas, 21(11):112510, 2014. URL: http: //scitation.aip.org/content/aip/journal/pop/21/11/10. 1063/1.4902366, doi:http://dx.doi.org/10.1063/1.4902366. 120 [3] G.R. McKee, K.H. Burrell, R.J. Fonck, G.L. Jackson, M. Murakami, G Staebler, D Thomas, and P. West. Impurity-induced suppres- sion of core turbulence and transport in the DIII-D tokamak. Physical review letters, 84(9):1922–5, feb 2000. URL: http://www. ncbi.nlm.nih.gov/pubmed/11017661, doi:10.1103/PhysRevLett. 84.1922. 121 [4] A M Messiaen, J Ongena, U Samm, B Unterberg, G Van Wassenhove, F Durodie, R Jaspers, M Z Tokar, P E Vanden- plas, G Van Oost, J Winter, G H Wolf, G Bertschinger, G Bon- heure, P Dumortier, H Euringer, K H Finken, G Fuchs, B Giesen, R Koch, L Könen, C Königs, H R Koslowski, A Krämer-Flecken, A Lyssoivan, G Mank, J Rapp, N Schoon, G Telesca, R Uhlemann, M Vervier, G Waidmann, and R R Weynants. High Confinement and High Density with Stationary Plasma Energy and Strong Edge Radiation in the TEXTOR-94 Tokamak. Phys. Rev. Lett., 77(12):2487–2490, sep 1996. URL: http://link.aps.org/doi/10. 1103/PhysRevLett.77.2487, doi:10.1103/PhysRevLett.77.2487. 121 [5] M Porkolab, J Dorris, P Ennever, C Fiore, M Greenwald, A Hub- bard, Y Ma, E Marmar, Y Podpaly, M L Reinke, J E Rice, J C
125 126 Bibliography
Rost, N Tsujii, D Ernst, J Candy, G M Staebler, and R E Waltz. Transport and turbulence studies in the linear ohmic confine- ment regime in Alcator C-Mod. Plasma Phys. Control. Fusion, 54(12):124029, 2012. URL: http://stacks.iop.org/0741-3335/ 54/i=12/a=124029, doi:10.1088/0741-3335/54/12/124029. 119
[6] G Tardini, R Fischer, F Jenko, A Kallenbach, R M McDermott, T Puetterich, S K Rathgeber, M Schneller, J Schweinzer, A C C Sips, D Told, and E Wolfrum. Core transport analysis of nitro- gen seeded H-mode discharges in the ASDEX Upgrade. Plasma Physics and Controlled Fusion, 55(1):Ampegon; Australian Inst Nucl Sci & Engn; Australi, 2013. doi:10.1088/0741-3335/55/1/015010. 122 Part I
Appendix
127
ABSOLUTEIMPURITYDENSITYMEASUREMENTS A
a.1 determination of impurity densities from Zeff and line brightnesses
To measure the dilution of the main ion species in the experiments described in Chapter 4, the densities of the different impurity ion species needed to be estimated. At the time that these experiments were performed, there was no measurement of the absolute impu- rity density of the impurity species. There was however an extensive suite of spectroscopic tools that could measure line brightnesses of a wide variety of intrinsic and extrinsic impurities[1][8][6]. These di- agnostics are most often used to detect the presence of impurities and give an indication of the relative level of a particular impurity throughout a shot or between similar shots. There exists a diagnos- tic on C-Mod that measures the effective charge, Zeff, through mea- surement of bremsstrahlung radiation[2], but it was unreliable at the low densities of many of the plasmas in the experiments. The Zeff in these experiments, was instead determined through the neoclassical conductivity[4] assuming no radial variation in Zeff. This method works well in cases where all the current is inductively driven, with- out external current drive, which these plasmas did not have. The line brightnesses measured by the spectroscopic diagnostics are all related to the total amount of the particular impurity species in the plasma. The intrinsic impurities considered in this work were oxygen, fluorine, and molybdenum. The extrinsic impurities consid- ered in this work were argon and nitrogen. For each impurity species a the line brightness Ba is given by:
Ba = na ne Ca (27a)
Where na is the impurity density, ne is the electron density, and Ca is a calibration constant that will remain the same between shots so long as the plasma temperature is similar and the spectrometer geometry is not changed. To determine the values of Ca for the different impurity species, the Zeff determined from neoclassical con- ductivity was used. The formula for Zeff is:
2 Zeff = nD /ne + na /ne (Za ) (28a) a X
129 130 absolute impurity density measurements
Where Za is the charge of ion a. The impurity charge was de- termined from a coronal equilibrium model [5], which resulted in molybdenum having a charge of 30, argon having a charge of 16, and all other species being essentially fully stripped. In addition, the plasma is quasineautral, so:
ne = nD + naZa (29a) a X When equations 27, 28, and 29 are combined, they become:
2 Zeff − 1 = [Ba/(ne) (Za) ∗ (Za − 1) ∗ Ca] (30a) a X
Which depends only on experimentally measured parameters (Zeff, line brightnesses, impurity charges, and ne) and the calibration con- stants Ca. Each plasma shot will have its own set of the measured parameters, but the Ca values will be the same. By taking into con- sideration multiple shots, equation 30 becomes a matrix equation:
C = B−1Z (31a)
Where C is a vector of calibration constants for each impurity 2 species, B is a matrix where each element corresponds to the Ba/(ne) (Za) ∗ (Za − 1) values for a particular shot (the rows) and the particular im- purity species (the columns), and Z is a vector of the Zeff − 1 values for each shot. In order to compute B−1 and have confidence in the re- sult, B should be a well-conditioned matrix with as little degeneracy as possible. To do this, shots that were dominated by single impurity species were included in addition to the shots from the experiment described in this thesis, to determine the calibration constants. There was no oxygen-dominated shot available for that campaign. The shots used are shown in Table 3 below. The result is that the impurity densities from this analysis well reproduced the Zeff from the neoclassical conductivity calculation in the majority of the cases, as is shown in Fig. 60. One notable exception in this work is the intrinsic boron (which we know is present in C-Mod due to the boronization procedure), for which it proved difficult to get reliable line brightnesses at the temperatures of the plasmas in these experiments (due to very little boron remaining with bound electrons). Boron density has been di- rectly measured through charge exchange spectoscopy to be typically 1% or less of the electron density on C-Mod[3]. This means it repre- sents a fairly small contribution to the dilution and the Zeff (about 5% and 0.25 respectively). A.1 determination of impurity densities from zeff and line brightnesses 131
Shot Time Range (s) Dominant Impurity 1120131030 [1.2, 1.3] Flourine 1120224033 [1.0, 1.3] Molybdenum 1120203019 [1.1, 1.4] Argon 1120203027 [0.9, 1.1] Nitrogen 1120217022 [0.6, 1.4] Intrinsic 11120214002 [0.7, 1.0] Intrinsic 1120222017 [0.85, 1.05] Intrinsic 1120222019 [1.1, 1.3] Nitrogen 1120222021 [0.7, 0.8] Intrinsic 1120210021 [1.0, 1.3] Nitrogen 1120210003 [1.1, 1.3] Intrinsic 1120210031 [1.1, 1.3] Nitrogen 1120210007 [1.1, 1.3] Intrinsic 1120222022 [1.1, 1.3] Nitrogen
Table 3: Table of shots and time ranges for impurity analysis, as well as the dominant impurity.
Zeff Comparison 4.5
4.0
3.5
3.0
2.5
2.0 From Impurity Densities eff Z 1.5
1.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Zeff From Neoclassical Conductivity
Figure 60: Comparison of the Zeff computed from the formula for neoclassical conductivity and computed from the impurity densities for all shots in Table 3. 132 absolute impurity density measurements
ne Nitrogen Impurity Zimp nimp/ne nD/ne 20 −3 (10 m ) Puff Zeff 1.08 No 1.65 9.1 0.9% 92% 1.08 Yes 2.0 8.7 1.5% 87% 0.83 No 2.0 9.3 1.3% 88% 0.82 Yes 2.8 8.2 3.0% 75% 0.65 No 2.2 9.6 1.5% 86% 0.70 Yes 2.7 8.1 3.3% 76% 0.50 No 2.7 9.5 2.1% 80% 0.53 Yes 3.4 8.5 3.8% 68%
Table 4: Summary of impurity analysis for 0.8 MA density scan. ne refers to the line-averaged electron density. The impurity Zeff is the Zeff computed from the impurity densities and charges. Zimp and nimp refer to the charge and density of an average impurity ion species that matches the Zeff and nD/ne of the cumulative impurities.
•
a.2 results of impurity analysis
Once these shots were compiled, the calibration constants were deter- mined. From this, the concentrations of the different impurity species could be calculated from the line brightnesses and electron densities. The results of the analysis for the plasmas in the 0.8 MA scan are shown in Table 4. This was previously published by Porkolab [7]. The constants determined were valid for the 2012 experimental campaign, and thus could be used for both the 0.8 MA and 1.0 MA density scans. The 1.2 MA density scan took place during the 2015 campaign and the spectrometer geometry had been changed suffi- ciently that the previous values were no longer valid. However, the fact that the average impurity charge was reliably between 9 and 10 before seeding, and between 8 and 9 after seeding, for the ohmic plasmas, allowed for a reasonable assumption of the impurity con- centrations of the ohmic plasmas in the 2015 campaign. ? BIBLIOGRAPHY
[1] N P Basse, A Dominguez, E M Edlund, C L Fiore, R S Granetz, A E Hubbard, J W Hughes, I H Hutchinson, J H Irby, B LaBombard, L Lin, Y Lin, B Lipschultz, J E Liptac, E S Marmar, D A Mossessian, R R Parker, M Porkolab, J E Rice, J A Snipes, V Tang, J L Terry, S M Wolfe, S J Wukitch, K Zhurovich, R V Bravenec, P E Phillips, W L Rowan, G I Kramer, G Schilling, S D Scott, and S J Zweben. Diag- nostic systems on Alcator C-Mod. Fusion Science and Technology, 51(3):476–507, 2007. URL:
[2] CR Christensen. Particle transport on the Alcator C-Mod tokamak. PhD thesis, MIT, 1999. URL: http://adsabs.harvard.edu/abs/ 1999PhDT...... 133C. 129
[3] R. M. Churchill. Impurity Asymmetries in the Pedestal Region of the Alcator C-Mod Tokamak. PhD thesis, Massachusetts Institute of Technology., 2014. URL: http://hdl.handle.net/1721.1/92101. 130
[4] S.P. Hirshman, R.J. Hawryluk, and B. Birge. Neoclassical con- ductivity of a tokamak plasma. Nuclear Fusion, 17(3):611–614, 2011. URL: http://stacks.iop.org/0029-5515/17/i=3/a=016, doi:10.1088/0029-5515/17/3/016. 129
[5] I. H. Hutchinson. Principles of Plasma Diagnostics. Cambridge Uni- versity Press, 2005. URL: http://books.google.com.hk/books? id=pUUZKLR00RIC, doi:10.1017/CBO9780511613630. 130
[6] J K Lepson, P Beiersdorfer, J Clementson, M F Gu, M Bitter, L Roquemore, R Kaita, P G Cox, and a S Safronova. EUV spec- troscopy on NSTX. Journal of Physics B: Atomic, Molecular and Optical Physics, 43(14):144018, 2010. URL: http://stacks.iop. org/0953-4075/43/i=14/a=144018, doi:10.1088/0953-4075/43/ 14/144018. 129
[7] M Porkolab, J Dorris, P Ennever, C Fiore, M Greenwald, A Hub- bard, Y Ma, E Marmar, Y Podpaly, M L Reinke, J E Rice, J C Rost, N Tsujii, D Ernst, J Candy, G M Staebler, and R E Waltz. Transport and turbulence studies in the linear ohmic confine- ment regime in Alcator C-Mod. Plasma Phys. Control. Fusion, 54(12):124029, 2012. URL: http://stacks.iop.org/0741-3335/ 54/i=12/a=124029, doi:10.1088/0741-3335/54/12/124029. 132
[8] M L Reinke, P Beiersdorfer, N T Howard, E W Magee, Y Pod- paly, and J E Rice. Vacuum ultraviolet impurity spectroscopy on
133 134 Bibliography
the Alcator C-Mod tokamak. The Review of scientific instruments, 81(10):10D736, oct 2010. URL: http://www.ncbi.nlm.nih.gov/ pubmed/21033927, doi:10.1063/1.3494380. 129 OTHERFACTORSCONSIDEREDINGYRO B b.1 inclusion of ~E × B~ shear
Shear in the ~E × B~ plasma velocity is known to have a stabilizing ef- fect on plasma turbulence[1]. The plasmas studied in this work have very low rotation overall, since they are ohmically heated L-mode plasmas with no external momentum input. Thus it was not clear whether this effect will be strong enough to have a significant im- pact on the transport, so separate simulations, with and without ~E × B~ shear, were performed on test cases to determine the magnitude of those effects. Figure 61 shows the comparison of the GYRO simulated electron and ion energy fluxes, with and without ~E × B~ shear, for a 1.0 MA SOC nitrogen seeded case for r/a = 0.6, while Fig. 62 shows the corresponding results for r/a = 0.8. It’s clear from the two figures that the inclusion of ~E × B~ shear does very little to affect either the simulated electron or ion energy transport. To include the effects of ~E × B~ shear in local GYRO, the simulation must be run with non-periodic boundary conditions as is the case for global mode simulations, but with radial variation of profiles turned off. These simulations run somewhat more slowly than simulations with periodic boundary conditions, and are somewhat more difficult to set up due to the need to pick the proper buffer zone parameters. Because the ~E × B~ shear was found to have a negligible effect on the overall energy transport in these cases, it was neglected in the work described in Chapters 5 and 6. •
Ion Energy Flux At r/a = 0.6 Electron Energy Flux At r/a = 0.6 14 15 12 10 GB GB 10 8 /Q /Q i e 6 Q Q 5 4 2 0 0 0 100 200 300 400 0 100 200 300 400
t [a/cs] t [a/cs] GYRO Flux With ExB Shear GYRO Flux Without ExB Shear
Figure 61: Comparison of the local GYRO simulated ion (left) and electron (right) energy fluxes with ~E × B~ shear included (purple) and without ~E × B~ shear included (blue). The simulations were of a 1.0 MA SOC nitrogen seeded case at r/a = 0.6.
135 136 other factors considered in gyro
Ion Energy Flux At r/a = 0.8 Electron Energy Flux At r/a = 0.8 60 60 50 50 40 40 GB GB /Q
/Q 30 30 i e Q 20 Q 20 10 10 0 0 0 50 100 150 200 0 50 100 150 200
t [a/cs] t [a/cs] GYRO Flux With ExB Shear GYRO Flux Without ExB Shear
Figure 62: Comparison of the local GYRO simulated ion (left) and electron (right) energy fluxes with ~E × B~ shear included (purple) and without ~E × B~ shear included (blue). The simulations were of a 1.0 MA SOC nitrogen seeded case at r/a = 0.8.
b.2 inclusion of multiple impurity ion species
Another assumption that could be tested is whether the inclusion of an average impurity ion species would be different from including the multiple impurity ion species actually present in the plasma. The av- erage impurity ion species resulted in the same Zeff and nD/ne as the multiple impurity ion species. Figure 63 shows the effects of includ- ing multiple ion species, for a 0.8 MA SOC nitrogen seeded discharge at r/a = 0.6, on the GYRO energy fluxes. The electron, main ion, and impurity energy fluxes (on the left, middle, and right respectively) are all only slightly affected by the inclusion of multiple impurity ion species. The growth rate of the turbulence is identical for the different impurity makeups. There are small differences in the total impurity energy fluxes, but the impurity energy flux is a negligible contributor to the total energy flux in both cases. This means that if getting the impurity fluxes themselves correct is important, then including all the species individually would be important. For the work presented in this thesis, where the total energy transport was the main focus, such detailed analysis was unnecessary. ? B.2 inclusion of multiple impurity ion species 137
Electron Energy Fluxes At r/a = 0.6 Main Ion Energy Fluxes At r/a = 0.6 Impurity Ion Energy Fluxes At r/a = 0.6
15
10 GB Q/Q
5
0 0 100 200 300 0 100 200 300 0 100 200 300
t [a/cs] t [a/cs] t [a/cs] GYRO Flux With Average Impurity Species GYRO Flux With Multiple Impurity Species
Figure 63: Plots of GYRO simulations with a single average impurity species(blue) and with multiple impurity species (purple). The plots show the GYRO simulated electron energy flux (left), main ion energy flux (middle), and total impurity ion energy fluxes. This case is a 0.8 MA SOC nitrogen seeded discharge at r/a = 0.6.
BIBLIOGRAPHY
[1] W. Horton. Drift waves and transport. Reviews of Modern Physics, 71(3):735–778, apr 1999. URL: http://link.aps.org/doi/10. 1103/RevModPhys.71.735, doi:10.1103/RevModPhys.71.735. 135
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