Heating and stability of Columbia Neutral Torus stellarator plasmas

Kenneth C. Hammond

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2017 c 2017

Kenneth C. Hammond All Rights Reserved ABSTRACT

Heating and stability of Columbia Neutral Torus stellarator plasmas

Kenneth C. Hammond

This thesis describes physics research carried out at the Columbia Neutral Torus (CNT) stellara- tor after its adaptation from a non-neutral plasma experiment to a device relevant to magnetic fusion energy research. Results are presented in the areas of plasma heating and related topics (microwave-assisted plasma start-up, overdense heating, inversion of stellarator images), as well as to stellarator stability and related topics (high β, error fields). This thesis also describes the engineering improvements which enabled the said adaptation of CNT. The first step of that process involved the installation of a low-power, pulsed 2.45 GHz magnetron. In those initial experiments it was found that the simultaneous use of microwave start-up and of an emissive hot cathode re- sulted in non-linearly increased electron densities, implying a synergy between the two start-up methods. Then, a 10 kW, 2.45 GHz heating system was commissioned including a custom-designed transmission line and launch antenna. Highly overdense plasmas (a factor of 4 above the cutoff density) were obtained with this system, both for O-mode and X-mode polarization. The analy- sis of Langmuir probe profiles of density and temperature required the accurate mapping of the minor radius in the plasma, which motivated a study of CNT error fields. This resulted in a new numerical method for inferring coil misalignments from flux surface measurements. The improved knowledge of the actual magnetic field geometry of CNT permitted to develop and successfully apply an inversion technique to experimental plasma images. This technique (“onion peeling”) reconstructs radial emissivity profiles, and can be considered a 3D generalization of Abel inversion. Finally, simulations of high-β plasma equilibria in different CNT magnetic configurations indicate that (1) ballooning stability limits should be accessible at volume-averaged β as low as 0.9% and (2) ballooning-stable β values as high as 3.0% should be attainable with heating powers as low as 40-100 kW and 1-3 MW respectively, according to stellarator energy confinement scaling laws. Contents

List of Figures v

List of Tables xvi

1 Introduction 1 1.1 Magnetic fusion energy ...... 1 1.1.1 Fusion requirements ...... 2 1.1.2 Tokamaks and Stellarators ...... 3 1.1.3 Stellarator research priorities ...... 8 1.2 CNT as a fusion experiment ...... 9 1.2.1 Potential for high β at low power ...... 11 1.2.2 Low aspect ratio ...... 12 1.2.3 Simple coil configuration ...... 12 1.2.4 Experimental flexibility ...... 12 1.3 Thesis overview ...... 14

2 Error field diagnosis 17 2.1 Flux surface measurements ...... 21 2.1.1 Experimental setup ...... 21 2.1.2 Results ...... 22 2.2 Photogrammetry ...... 22 2.3 Numerical study of ι sensitivity ...... 27 2.4 Optimization ...... 30 2.4.1 Optimization procedure ...... 30

i 2.4.2 Numerical considerations ...... 31 2.4.3 Verification ...... 33 2.4.4 Inferring coil displacements from experimental data ...... 35 2.5 Discussion ...... 40 2.5.1 Optimizing for cross-sections with islands ...... 40 2.5.2 Extension to more complex devices ...... 43

3 Onion-peeling inversion of stellarator images 45 3.1 Method ...... 46 3.2 Forward problem ...... 48 3.3 Inversion of experimental images ...... 48 3.3.1 Image processing ...... 48 3.3.2 Reconstructions of glow discharges ...... 51 3.3.3 Reconstruction of an ECH discharge ...... 53

4 First ECH plasmas 54 4.1 Microwave plasma heating ...... 55 4.1.1 Waves in plasmas ...... 55 4.1.2 Cyclotron damping ...... 60 4.2 Heating and diagnostics ...... 63 4.2.1 Low-power magnetron ...... 63 4.2.2 Diagnostics ...... 64 4.2.3 Langmuir probes ...... 65 4.2.4 Probe signal analysis in CNT ...... 67 4.3 Electron temperature and density profiles as functions of background pressure . . . . 71 4.4 Afterglow and decay time measurements ...... 74 4.5 Synergy between ECH and thermoelectrons ...... 75 4.5.1 Interpretation: possible synergy mechanisms ...... 77

5 Overdense plasma heating 79 5.1 Guided waves ...... 80 5.2 Heating and diagnostics in CNT ...... 83

ii 5.2.1 Heating system ...... 83 5.2.2 Diagnostics ...... 85 5.3 Profile measurements: evidence of overdense heating ...... 87 5.3.1 Dependence on heating power ...... 88 5.3.2 Dependence on magnetic field ...... 91 5.4 Discussion: mode conversion, UHR accessibility, and heating mechanisms ...... 95 5.4.1 SX-B mode conversion ...... 95 5.4.2 FX-B conversion ...... 95 5.4.3 O-X-B conversion ...... 95 5.4.4 Collisional absorption at the UHR ...... 97 5.4.5 Wave damping mechanisms ...... 97 5.4.6 Earlier experiments in other devices ...... 99 5.5 Initial full-wave calculations ...... 100

6 High-β equilibria and ballooning stability 103 6.1 Equilibrium and stability theory ...... 105 6.1.1 The ideal MHD model ...... 105 6.1.2 Validity of ideal MHD in CNT plasmas ...... 106 6.1.3 MHD stability ...... 107 6.1.4 Ballooning instabilities ...... 109 6.2 Fixed-boundary VMEC solutions for CNT geometry ...... 110 6.3 Input parameters ...... 112 6.4 Methods for equilibrium and stability calculations ...... 113 6.5 Free-boundary and stability results ...... 114 6.6 Heating power requirements ...... 119

7 Summary and conclusions 124

8 Future research directions 128

Bibliography 131

Appendices 145

iii A Parametrization of Poincar´ecross-sections 146

B Determination of X for Poincar´edata 148 B.0.1 Treatment of magnetic islands ...... 150

iv List of Figures

1.1 Schematics of coils and plasmas for three toroidal magnetic confinement concepts. (a) tokamak (reprinted from Ref. [4]); (b) classical stellarator (reprinted from Ref. [4]); (c) modular-coil stellarator (reprinted from Ref.[5])...... 3 1.2 Cut-away drawing of a set of toroidal, nested flux surfaces, shown in alternating red and white. Along with each surface, a single field line is shown, traced for several laps around the torus. Note that each field line remains essentially bound to its respective surface along its trajectory...... 5 1.3 Poincar´eplot for a cross-section of the W7-X stellarator in Greifswald, Germany. In the center (core) region, where the field lines conform to flux surfaces, the sets of (red) puncture points from each field line are neatly separated in concentric loops (i.e., cross-sections of flux surfaces). The green puncture points form a magnetic island chain outside the LCFS. The blue points form a stochastic region outside the LCFS. The pink curve represents the vacuum vessel wall. From Ref. [10]: Y. Suzuki and J. Geiger, Plasma Phys. Control. Fusion 58, 064004 (2016). c IOP Publishing.

Reprinted with permission. All rights reserved...... 6 1.4 Examples of magnetic fields that confine trapped particles. Upper images show three-dimensional flux surfaces in which the color indicates the B . Lower contours | | show a flux surface flattened into a grid of the poloidal angle θ and toroidal angle φ with contours of B overlain with magnetic field lines in red. (a) Quasi-helical | | symmetry, as in the HSX stellarator in Madison, WI. (b) Quasi-axisymmetry, as in the NCSX stellarator concept. (c) Quasi-isodynamicity, as in W7-X. Adapted from Ref. [14]...... 7

v 1.5 (a) Schematic of the CNT plasma and coil set. (b) Photo of the CNT vacuum vessel. The two larger coils in the schematic are visible in light blue in the photo...... 10 1.6 Magnetic field lines (in black) overlain on contours of B of a typical CNT flux | | surface. From B it can be seen that CNT has two field periods; i.e., the field goes | | through two identical cycles in the toroidal angle φ. θ = 0 is on the outboard side and θ π is on the inboard side...... 11 ± 1.7 Comparison of the shapes of the magnetic confining volume produced in the three

different θtilt configurations in CNT...... 14

1.8 Profiles of ι in CNT’s three θtilt configurations. Reprinted from Ref. [38]...... 15

2.1 Schematic of the CNT interlocked coils and last closed flux surface. The translucent patch on the right-hand side represents the cross-section at φ = 90◦ in which Poincar´e measurements were taken as described in Section 2.1 The Cartesian axes are also shown on the lower right as a reference...... 19 2.2 Setup for flux surface measurements. (a) electron gun; (b) diagram of phosphor rods and actuator mechanism, reprinted from Ref. [19]; (c) long-exposure photograph revealing a flux surface cross-section...... 20 2.3 Comparison of numerical Poincar´eplots (black dots) with experimental results (ad- jacent surfaces are shown in alternating red and blue for clarity). Numerical data

were determined assuming that the coils were perfectly aligned. (a) IIL/IPF = 3.68;

(b) IIL/IPF = 3.18...... 23 2.4 Plots of ι profiles for selected current-ratios in CNT in the 78◦ IL coil configuration.

In the red curves, which are labeled with respective values of IIL/IPF , each dot represents a closed flux surface and dashed segments indicate island chains. Some low-order rational numbers are shown as horizontal dashed lines. Note that current- ratios of 3.18 and above are not predicted to contain ι = 1/3, contrary to measurements. 24

vi 2.5 Rendering of the CNT vacuum vessel and PF coils overlaid with arrows showing the displacements of the coils from their design positions. Each arrow corresponds to one photogrammetric marker. Arrow lengths are to scale with one another but are exaggerated relative to the vacuum vessel and coils. The largest arrow shown has a length of 44 3 mm. Blue (red) arrows represent deviations in the same (opposite) ± direction of the vector normal to the surface where they originate. Reproduced from Ref. [57]...... 25 ◦ 2.6 (a) Numerically generated Poincar´eplots at the φ = 90 cross-section with IIL/IPF = 3.18 from a configuration incorporating the measured PF coil displacements (blue) and from the design configuration (black). (b) Calculated profiles of ι for this con- figuration...... 26

2.7 Derivatives of ιaxis with respect to the magnitude of different classes of coil displace- ments. (a) Translations of the coils in the Cartesian directions illustrated in Fig. 2.1. (b) Tilts of the coils in the directions of their respectivea ˆ and ˆb axes. Thea ˆ and ˆb axes for the IL coils are illustrated in Fig. 2.8; for both PF coils,a ˆ =x ˆ and ˆb =y ˆ. A more detailed description of each displacement class is given in the text...... 28 2.8 Schematic of the two IL coils along with their respective axes of symmetry (red arrows),a ˆ vectors (green arrows), and ˆb vectors (blue arrows) as described in the text. Thex ˆ,y ˆ, andz ˆ directions as defined in Fig. 2.1 are shown in yellow, magenta, and cyan respectively...... 29 2.9 Contour plot of χ2 and the path taken by the optimizer in a verification test with two

free parameters. The initial guess p0 is (0, 0) and the target parameters are denoted by the red . Contours are on a logarithmic scale with solid contours representing × powers of 10...... 34 2.10 Schematic illustrating the definitions of the a and b components of the angular dis- placement parameters of the coils. In this image, the IL1 coil is used as an example. The blue surface represents the plane of the unperturbed coil, and the red arrow represents a unit vector along the axis of the perturbed coil...... 35

vii 2.11 Results of optimizations in which the interlocked coils were free to move in 10 pa- rameters to match Poincar´edata generated from made-up coil perturbations. (a)

Parameter evolution toward targets of zIL1 = 5 mm and the rest of the parameters

zero. (b) Parameter evolution toward targets of bIL1 = 0.002, bIL2 = 0.004, and the rest zero. (c) Descent of χ2 for both optimizations...... 36 2.12 (a) Evolution of the displacement parameters of the IL coils during an optimization using experimental Poincar´edata from the 3.68 current-ratio. (b) Descent of χ2 associated with the Poincar´edata resulting from the displacements...... 38 2.13 (a) Experimental data (red and blue dots) plotted alongside numerical data (black)

for a field line trace that assumed no coil displacements, both at IIL/IPF = 3.68, identical to what is plotted in Fig. 2.3a. (b) The same experimental data as in (a) exhibited better agreement with the numerical data that accounted for the coil ∗ displacements inferred from the optimization (p3.68), again at IIL/IPF = 3.68. . . . 39

2.14 Effect of using, in calculations for (a) IIL/IPF = 3.50 and (b) IIL/IPF = 3.18, the ∗ coil displacements inferred for IIL/IPF = 3.68 (i.e., p3.68). Measurements for the respective current ratios are shown for comparison...... 41 2.15 Comparison of ι profiles calculated for selected current-ratios in the nominal and

optimized coil configurations. (a) Profiles for the nominal IL coil positions (p0) and assuming no PF coil displacements, identical to what is plotted in Fig. 2.4. (b) ∗ Profiles computed using the optimized IL coil configuration (p3.68) and the measured PF coil displacements. Each dot represents a closed flux surface, so while the curves

for IIL/IPF = 3.04 and 2.92 in plot (b) do not pass through ι = 1/3, they contain large three-island chains on their edges. Note that, in the optimized configuration

(b), current-ratios IIL/IPF less than 3.5 contain ι = 1/3, consistent with observations. 42

viii 3.1 Schematic illustration of the onion-peeling method. The red lines represent lines-of-

sight of two camera pixels whose brightness is denoted by p1 and p2. The lines-of- sight both travel through one or more layers of plasma, whose borders are shown as alternating blue and white surfaces. The lengths of the segments within the respective surfaces constitute the components of the L matrix as defined in the text.

Note that some components (like L21 here) are actually due to the sum of two or

more segments due to multiple crossings of a layer. The terms ei are the emissivities of each layer...... 47 3.2 Photograph of a typical argon plasma in CNT heated with 1 kW ECH with CNT’s two in-vessel coils in clear view...... 49 3.3 Simulated images of three synthetic profiles. (a) profiles; (b) image of hollow profile; (c) image of uniform profile; (d) image of peaked profile. The upper halves of of (b)-(d) approximate the field of view of the photo in Fig. 3.2...... 49 3.4 (a) Schematic of the key objects in the camera’s field of view. (b) Raw image of a typical glow discharge (Sect. 3.3.2). (c) The same image, after noise subtraction, with boxes indicating the regions where pixel data were used for reconstructions (Sect. 3.3.1)...... 50 3.5 Reconstructed profiles and images of glow discharges. (a) Profiles with the emis- sive filament located on fiv different flux surfaces with effective minor radii given in the legend and as vertical dashed lines. Solid curves are fits to cubic smoothing splines, which are used for the image reconstructions. (b)-(c) Image and reconstruc- tion for the 3.3 cm filament position in the regions where pixel data were used for reconstructions (Fig. 3.4b). (d)-(e) Image and reconstruction for the 5.7 cm filament position...... 52 3.6 Reconstruction of a 1 kW ECH discharge. (a) Emissivity profile with a cubic spline fit. (b) Photographic data used for the reconstruction. (c) Synthesized image based on the reconstructed profile...... 53

ix 4.1 Topology of cutoffs and resonances for O- and X-mode heating in a toroidal plasma. Each circle represents a poloidal cross-section in which B decreases from left to right and n decreases from center to edge. The shaded regions in each diagram do not admit propagation for the respective mode. (a) O-mode heating at the first

harmonic (ω = ωce) Provided the shaded region (where n > nco) does not overlap

the cyclotron resonance (ω = ωce), a beam launched from the right (the dashed line) will propagate, unobstructed, toward the cyclotron resonance. (b) X-mode heating at the first harmonic. In this case, a beam launched from the right will be reflected off of the RH (fast X) cutoff, never reaching the resonance. A beam launched from the left, however, will reach the resonance, provided it avoids the LH (slow X) cutoff. (c) X-mode heating at the second harmonic. In this case, launch from the right hand side is unobstructed by the fast X cutoff. Adapted from Ref. [73]: R. Prater, Phys. Plasmas 11, 2349 (2004), with the permission of AIP Publishing...... 59 4.2 Typical form of a Langmuir I-V curve for a cold, quasineutral plasma. The plasma

potential, Vplasma, occurs at the sharp corner in the black curve at V 5 V. The ≈ red dashed curve illustrates the softening that often occurs in the region near the plasma potential. Reproduced from Ref. [77]: R. L. Merlino, Am. J. Phys. 75, 1078 (2007), with the permission of the American Association of Physics Teachers. . . . . 66 4.3 Raw data collected during a typical ECH plasma shot. The RF leakage shown in the bottom plot is collected by the “sniffer” antenna placed outside the vacuum vessel and is used to determine when the microwave pulses are occurring...... 68 4.4 Sample Langmuir probe data corresponding to 3.27 ms after the start of a microwave pulse (using the time origin defined Fig. 4.5), before and after processing. Each point plotted is the average of four measurements of current at the same probe bias. (a) Total probe current, linear fit to the ion saturation region, and the remaining electron current after subtracting the fit curve. (b) Electron current replotted on a logarithmic scale. Colored fit curves represent exponential fits to the hot electron population (red), bulk electron population (blue), and electron saturation region

(green). The dashed vertical line indicates Vfloat. The intersection between the blue

and green curves is taken to be Vplasma...... 69

x 4.5 Typical plasma parameters measured during and after a microwave pulse. The pulse duration is indicated by the average RF leakage measurement in the bottom plot. . . 70 4.6 Profiles of temperature and density obtained at the 3.00 current ratio in nitrogen gas at a background neutral pressure of 1.0 10−5 Torr...... 72 × 4.7 Electron density (blue squares) and temperature (red circles) measured at an in- termediate minor radius and coherently averaged over the middle 3 ms of each mi-

crowave pulse. (a) N2; (b) He ...... 73 −5 4.8 Afterglow parameters for N2 with pressure 1.0 10 torr for IIL/IPF = 3.00. (a) × Recombination coefficient α at different probe positions. (b) Decay time τ at different

probe positions; (c) Example comparison of a fit curve with the form of nag (Eq. 4.19) to data for probe position 7 cm...... 74 4.9 Profiles of electron density as a function of emitter bias with a background of nitrogen −5 gas at 1.0 10 Torr at IIL/IPF = 3.00. Reference cases using only the magnetron × (blue) and only the emissive filament with a -200 V bias (black) are also shown for comparison. The vertical gray stripe marks the location of the emissive filament. . . 76

5.1 Vector plots showing the orientation and relative magnitude of the electric field for the fundamental mode of (a) a rectangular waveguide and (b) a circular waveguide. The cross-sections are scaled to have the same cutoff frequency. Note that the tangential component at the wall is always zero...... 82 5.2 (a) Schematic of the microwave heating system as described in the text. Drawing prepared by Y. Wei. (b) Detail of the waveguide launcher. Adapted from a drawing by Nor-Cal Products. (c) Photograph of the twistable, flexible waveguide leading into the rectangular-to-circular taper. (d) Photograph of the launch antenna inside the CNT vessel...... 84

xi 5.3 (a) Launch antenna, X-mode Gaussian beam (green), cross-section of the CNT flux surfaces (black), and electron cyclotron harmonics for 2.45 GHz (red). The beam boundary corresponds to field intensities of 1/e2 relative to the values on the central axis. Note that, in this plane, the O-mode beam would be wider. The x axis is coaxial with the waveguide and approximately intersects the magnetic axis at x = 0. The y axis is perpendicular to the magnetic field (and x) at y = 0. (b) Three-dimensional schematic of the configuration inside the vacuum vessel, including the LCFS, the launch antenna, the Langmuir probe, and the IL coils. The translucent black rect- angle indicates the cross-section in (a) with the x and y directions represented by the green and blue arrows...... 86 5.4 Photographs of the process for aligning the waveguide to the magnetic field. (a) an Ar discharge localized to the magnetic axis; (b) the same discharge, viewed through the waveguide window via the taper. The taper is rotated until the axis aligns to the rectangular cross-section with the desired angle...... 87 5.5 Profiles of Ar for O-mode launch at various power levels, at a backfill pressure of −5 (1.4 0.2) 10 torr. (a)-(b) Langmuir probe measurements of (a) ne and (b) Te, ± × projected on the x axis (as defined in Fig. 5.3). The curves are 7th-order polynomial fits to the data. The vertical gray line denotes the LCFS. The horizontal dashed line is the cutoff density for O-mode propagation. (c)-(d) Contours of cutoffs and resonances in the xy plane (Fig. 5.3) for (c) 0.5 kW heating power and (d) 8.0 kW. Regions overdense to O-mode propagation are shaded; those portions that could be accessible after O-X conversion are shaded lighter. The solid green line indicates the axis of the vacuum microwave beam, whereas the dashed and dotted green curves are contours of 1/e and 1/e2 of the on-axis beam intensity, respectively. The star in (c) is a reference for the discussion in Sec. 5.4.5. The vacuum wavelength is shown in (d)...... 89 5.6 Similar to Fig. 5.5, but for X-mode launch. The shaded regions in (c) and (d) indicate regions of evanescence for the X-mode...... 90

xii 5.7 Dependence of plasma parameters on power from the experiments described in Sec. 5.3.1. (a) Volume averaged electron density, (b) Electron temperature on the flux surface at 90% of the effective minor radius, (c) Plasma energy, (d) Volume- averaged β, (e) Energy confinement time...... 92 5.8 Similar to Fig. 5.5, but for the scan of magnetic field strengths in the case of O-mode launch. Each plasma coupled to 1 kW of heating power...... 93 5.9 Similar to Fig. 5.8, but for X-mode launch. Each plasma coupled to 1 kW of heating power, except for the B = 88 mT case, which coupled to 2 kW...... 94 5.10 Schematic of how the X-mode may access the UHR from the high-field side. O- waves (originating from the launcher or wall reflections) may pass unobstructed through regions evanescent to X-mode (gray), reflect as X-waves off the in-vessel coils, and reach the UHR. In addition, X-waves from the launcher or wall reflections may partially transmit through the outermost evanescent layer and reach the UHR either directly or after reflecting off the coils. Denser plasmas make the evanescent regions thicker, thus restricting wave access. Shown near the top of the diagram are

references for λ0 and the attenuation lengths λatt,O and λatt,SX for the O- and slow 17 −3 X-mode for ne = 1.5 10 m and B = 70 mT. Note that the launcher is actually × | | located above the cross-sectional plane shown here and does not intersect the LCFS. 96 5.11 Accessibility of the plasma to first-pass cyclotron damping for cases of O-mode launch at different magnetic field strengths, corresponding to Fig. 5.8c-d. The green stripes represent portions of the vacuum microwave beam intersecting the cyclotron reso- nance (red) within the LCFS. Note that a large fraction of the beam must travel through the evanescent region (gray) inside the O-mode cutoff (blue). The darker green stripe is the portion of the beam which reaches the resonance before encoun- tering the O-mode cutoff...... 99 5.12 Contours of normalized E in plasma cross-sections for the scan of B with O- h| |i | | mode launch (Fig. 5.8). Dashed and solid lines correspond to resonances and cutoffs as indicated. Normalizations are not the same in the different plots. (a) B = 80 | | mT; (b) B = 88 mT; (c) B = 97 mT. Plots prepared by A. K¨ohn...... 101 | | | |

xiii 6.1 Schematic picture of a ballooning instability. The two arc-shaped boxes are cross- sections of adjacent flux tubes. The upper tube has a higher pressure; hence p ∇ points upward. The magnetic field B is assumed to point out of the page with curvature κ pointing upward such that the “gravitational” force is downward. The dashed line is a perturbation ξ to the interface between the two flux tubes. The bad curvature illustrated here will cause the perturbation to grow...... 109 ◦ 6.2 Results of a series of fixed-boundary simulations of CNT with θtilt = 78 , IIL/IPF =

2.5, Ip = 0, and a centrally peaked pressure profile as shown in Fig. 6.3. (a) Flux surface comparison between calculations with β = 0% and β = 6.6%, with the h i h i vacuum last closed flux surface (LCFS) as a reference; (b) Relative Shafranov shift as defined in the text...... 111 6.3 Normalized profiles of pressure used as inputs for the calculations in this chapter. During the β scans, each of these profiles was scaled according to the desired β value.112 6.4 (a) Highest volume-averaged β, (b) bootstrap current, and (c) relative Shafranov

shift as functions of the coil current ratio (IIL/IPF ) and θtilt for the three different configurations of pressure and field strength discussed in the text. Here, “highest β” refers to the highest value before the plasma becomes ballooning-unstable (filled symbols) or the highest value for which it was possible to compute a stable equilib- rium, without however encountering the ballooning stability limit yet (open symbols; see also Fig. 6.5c). Some markers overlap one another due to similar results...... 115 6.5 Evolution of key paramaters during the β scan carried out for the high-β magnetic ◦ configuration (θtilt = 78 , IIL/IPF = 2.75, B = 0.04 T, hollow pressure profile) and ◦ for the configuration that became ballooning-unstable at the lowest β (θtilt = 88 ,

IIL/IPF = 2.75, B = 0.08 T, peaked pressure profile). The vertical dashed line indicates the value of β at which the least stable configuration first becomes bal- looning unstable. (a) bootstrap current; (b) relative Shafranov shift; (c) maximum calculated ballooning growth rate within the plasma volume. Note that the growth rate never becomes positive for the high-β configuration...... 116

xiv 6.6 Contour plot of ballooning stability in (β, IIL/IPF ) parameter space for the peaked- ◦ profile configurations with θtilt = 88 , B = 0.08 T. The shaded regions are unstable

(γmax > 0)...... 117 6.7 Maximum stable β plotted against aspect ratio for each of the test configurations. Symbols are as defined in the legend in Fig. 6.4...... 118 ◦ 6.8 Properties of the high-β equilibrium attained with IIL/IPF = 3.25, θtilt = 78 , and the hollow pressure profile. (a) Flux surfaces for the equilibrium with β = 3.0%. (b) Flux surfaces for the same configuration with β = 0%. (c) Boundaries for the equilibrium with β = 3.0% (solid lines) compared with boundaries calculated with β = 0 (dashed lines) for three different poloidal cross-sections. (d) Profile of toroidal current (i.e., the calculated bootstrap current). (e) Profile of rotational transform with β = 3.0% (solid line) and β = 0% (dashed line)...... 118

6.9 Like Fig. 6.8, but for the least stable equilibrium (β = 0.9%), attained with IIL/IPF ◦ = 2.75, θtilt = 88 , and the peaked pressure profile...... 119 6.10 Accessible β at different levels of heating power according to stellarator scaling laws for (a) the high-β configuration and (b) the least stable configuration. Values of β above the solid line are considered inaccessible because the plasma density exceeds the Sudo limit. Values below the dashed line correspond to plasmas underdense to 2.45 GHz ECH in the extraordinary mode; values between the solid and dashed lines

could be attainable by overdense heating. The red lines are contours of Te = 30 eV in the (P, β) parameter space. The red circles indicate the power levels needed to attain the maximum β in each configuration...... 123

B.1 (a) Experimental Poincar´edata obtained for the current-ratio 3.68 (black dots) and fit curves to selected flux surfaces (blue lines). (b) The same experimental data overlaid with level curves from the fitted geometric parameters in red. Radially extending red lines are curves of constant θ; closed red loops are curves of constant ρ.149

xv List of Tables

6.1 Values of the ideal MHD validity criteria (Eq. 6.3) for the high-β CNT equilibria to be simulated in this chapter for two possible ion species...... 107 6.2 Comparison of present CNT experimental parameters with those of the least stable and high-β configurations...... 122

A.1 The first few polynomials Ps(ρ) as discussed in the text...... 147

xvi Acknowledgments

This thesis would not have been possible without the help of many people, both in the Columbia Plasma Physics Laboratory and beyond. A number of undergraduates have made invaluable contributions to this work. Scott Massidda collaborating on my first research project at Columbia, set the stage for the earliest microwave experiments on CNT, and showed me around New York City when I first moved in. I am also appreciate of Alek Anichowski, a deft tracer; Rui Diaz-Pacheco for his work with the fast camera; and Yumou Wei for helping to make the 10 kW microwave system a reality. Many graduate students have also helped me through the years. I am grateful to Bill Cappecchi for working with Scott and me in my early days here. I have also had the privilege of working with Tony Clark, Claudia Caliri, Michel Doumet, and Wilkie Choi. I count myself as very lucky to have met Ryan Sweeney and Max Roberts through this program, from whom I have learned so much and whom I value as friends and colleagues. Many others at Columbia helped to make this thesis work go smoothly. Mike Mauel and Andrew Cole have been always been wonderful sources of support and advice. Montse Fernandez-Pinkley, Dina Amin, and Mike Garcia all work tirelessly to make things run smoothly for graduate students like me. And I am especially grateful to Nick Rivera and Jim Andrello for all that they taught me about electronics and vacuum systems, and all the times they prevented me from blowing things up. I am very fortunate to have continuing ties with former members of the CNT group. Peter Traverso, among other things, took time out of his fifth-year reunion at Columbia to teach me how to fix a Langmuir probe array. I have also received valuable advice and guidance from Paul Brenner and Xabier Sarasola. I owe a debt of gratitude to Thomas Sunn Pedersen, the former CNT PI, both for collaborating with the error-field work and for offering me my next job.

xvii I have also received much support from collaborating research facilities over the years. Alf K¨ohn of IPP Garching contributed full-wave simulations that greatly helped with the interpretation of the overdense plasma results. From PPPL, Dick Majeski and Bob Kaita generously donated their fast camera. Roger Raman guided me through an interesting project on NSTX. And Sam Lazerson has been a consistent source of advice and insight for everything I set out to do with the field line tracing code, VMEC, and many other codes that I relied on for this thesis work. I am also grateful to my parents for consistently supporting me through all my endeavors, through good times and bad. They have always been there for me, and I feel very fortunate to have them in my life. And finally, I owe a debt of gratitude to my advisor, Francesco Volpe. He has taught me most of what I know about plasma physics, guided me through challenging projects, and helped me grow as a person and a scientist. Over the last five years since I first stumbled into his lab, he has shaped me into the PhD candidate I am today.

xviii Chapter 1

Introduction

1.1 Magnetic fusion energy

One of the great science and engineering challenges of this century will be to replace fossil-fuel-based energy sources with more sustainable ones that do not produce greenhouse gases. A number of renewable concepts have already been implemented, including geothermal energy, hydroelectricity, wind turbines, photovoltaics, and nuclear fission. Nevertheless, carbon-based energy sources still make up more than 80% of the world’s energy production as of 2016 [1]. While all of the aforemen- tioned energy sources have much potential for expansion, incorporating additional solutions will help to expedite the transition to carbon-free energy production. One promising concept is nuclear fusion. Fusion occurs when two atomic nuclei join together to form a heavier nucleus. If the original nuclei are lighter than Fe56, the reaction will release energy. In the case of deuterium (D) and tritium (T) fusing to form helium (He) plus a neutron, this energy is imparted as kinetic energy in the reaction products (3.5 MeV for the helium particle and 14 MeV for the neutron). The most straightforward way to harness this energy is to convert it to heat; for example, by allowing the neutrons to strike a wall. The heat can then be used to generate electricity through a system of turbines, essentially the same as what is used to convert the heat from coal, natural gas, or oil combustion. Fusion offers a number of advantages over other renewable sources. Unlike wind and solar energy, it requires little land and does not depend on the weather; therefore, it is not intermittent and does not require batteries for energy storage. It does not require the construction of dams (as

1 with hydroelectricity), which can disrupt local ecosystems and release methane, a greenhouse gas, as a result. And unlike nuclear fission, it poses no risk of a catastrophic meltdown and does not necessitate long-term storage of radioactive waste.

1.1.1 Fusion requirements

For a fusion reactor to be viable as a power plant, its reactants must be very hot and dense. The reason for this is that, in order to fuse, the reactant nuclei must overcome the electrostatic repulsion forces that arise from both nuclei having positive charge. The high-temperature (T ) requirement arises from the need for particles to have enough energy to overcome this barrier; the high-density (n) requirement arises from the need for enough particles to be present for reactions to occur. An additional requirement is that the energy produced by the reactions must be contained for a long enough time (τ) within the fuel to to drive additional reactions (i.e., the reaction must be burning, or self-sustaining). These three requirements together can be expressed quantitatively through the fusion triple-product [2]:

nT τ F (T ) (1.1) ≥ In other words, the product nT τ must exceed a certain minimum value F (T ) that varies with temperature. For deuterium and tritium, F (T ) has a minimum at T 20 keV [3], which is more ≈ than 200 million degrees Celsius.1 In order to sustain high-enough values of the triple product, a fusion reactor must confine its reactants. The most well-known fusion reactor today, the Sun, employs gravitational confinement: its gravitational force is strong enough to hold its reactants in a hot, dense mass. This is not feasible for a power plant, however, because the Sun’s gravity is not reproducible on Earth. Two leading concepts for terrestrial fusion power plants are those of inertial confinement and magnetic confinement. In inertial confinement, a pellet of fuel is subject to rapid and ultra-high compression, which heats and condenses the fuel so as to temporarily raise nT τ to the minimum. In magnetic confinement, a magnetic field is used to trap the reactants (which are hot enough to be dissociated as a plasma of charged ions and electrons which are subject to magnetic forces).

1One electron-volt (eV) is the amount of energy an electron gains if it is accelerated by one volt, equal to 1.6 × 10−19 J. A thermal plasma with a mean particle energy of 1 eV has a temperature of 11, 600◦ C. In this thesis, as is common in the fusion field, plasma temperatures are expressed in eV.

2 (a) (b) (c)

Figure 1.1: Schematics of coils and plasmas for three toroidal magnetic confinement concepts. (a) tokamak (reprinted from Ref. [4]); (b) classical stellarator (reprinted from Ref. [4]); (c) modular-coil stellarator (reprinted from Ref.[5]).

The magnetized plasma must then be heated through one or more mechanisms until it reaches the minimum nT τ.

1.1.2 Tokamaks and Stellarators

In today’s magnetic confinement research, two of the most extensively studied concepts for fusion reactors are the tokamak (Fig. 1.1a) [3] and the stellarator (Fig. 1.1b-c) [6]. Both are similar in that they create toroidal, or doughnut-shaped, magnetic fields. Toroidal magnetic topology is advantageous (and, perhaps, necessary) because the particles, heavily constrained perpendicularly to the magnetic field due to the Lorentz force, are essentially free to move parallel to the magnetic field. Thus, if the magnetic field does not wrap around on itself the way it does in a torus, particles will quickly escape wherever the magnetic field lines intersect the wall of the device. It is not sufficient for the magnetic field to be purely toroidal, however. “Purely toroidal” here means that the magnetic field lines exist in planes perpendicular to the axis of symmetry. This is due to the drifts that charged particles tend to undergo across magnetic fields. In a purely toroidal field, the curvature of the field lines will cause positively and negatively charged particles to drift in opposite directions along the axis. This separation of charge will create an electric field, which in turn will cause all of the particles to drift outward away from the axis [7]. Although drifts across the field are slow compared to the unimpeded motion of the particles along field lines, toroidal confinement will fail in the absence of a way to counter the drifts.

3 One solution to this conundrum is to have the magnetic field lines twist as they travel around the torus, such that particles that would otherwise drift out through the top of the torus are quickly swept to the bottom and vice versa. This twist is referred to as rotational transform, ι, defined as the number of poloidal revolutions (the short way around the torus) a magnetic field line makes in the process completing one toroidal revolution (the long way around). The fundamental difference between tokamaks and stellarators is in the way they generate rotational transform. Tokamaks use electromagnetic coils to generate a toroidal field and rely on a large toroidal current in the plasma to add the poloidal component. Tokamaks typically also employ poloidal field coils that contribute to the poloidal field, but they alone are not sufficient to generate the rotational transform. Stellarators, on the other hand, generate fields with rotational transform using coils alone – no plasma current required. While stellarator coil sets are typically more complicated to build, stellarators offer some advantages over tokamaks. For one, the lack of a large plasma current makes it easier to operate in a steady state and reduces the risk of plasma disruptions that could damage the device [8]. There are many ways to design a stellarator coil set [9]. Earlier designs tended to employ helical coils that wrapped around the entire torus (Fig. 1.1b). If the currents in these coils flow in alternating directions, the configuration is designated as a classical stellarator; if they all flow in the same direction, the configuration is a torsatron. More modern experiments tend to employ modular coils, which individually only wind around a small part of the torus but collectively simulate the effect of helical coils. The shapes of these modular coils can be optimized to improve plasma experimental conditions, particularly for confinement. A basic requirement for the magnetic field in a stellarator (and for any other toroidal confinement device) is that it must form nested flux surfaces. In other words, if a streamline (field line) is traced tangentially to the magnetic field, it must remain approximately confined to a toroidal surface (Fig. 1.2). This property is important for plasma confinement, as particles flow quickly along field lines. If a single field line from the center (core) of the plasma were to connect to the edge, particles and energy would quickly flow out of the plasma. Toward the interior of the toroidal confining volume, flux surfaces converge to a single curve referred to as the magnetic axis. As one moves toward the edge of the confining volume, eventually field lines will cease to conform to toroidal surfaces. The behavior of the field lines outside this last

4 Figure 1.2: Cut-away drawing of a set of toroidal, nested flux surfaces, shown in alternating red and white. Along with each surface, a single field line is shown, traced for several laps around the torus. Note that each field line remains essentially bound to its respective surface along its trajectory.

closed flux surface (LCFS) depends on the configuration. In some cases the field lines will form a stochastic region in which they remain confined to a toroidal region of finite thickness (but not a surface). In other cases they will form a chain of magnetic islands, or tubes of flux surfaces separate from the main confining volume. These features can be visualized in a Poincar´eplot (Fig. 1.3), a cross-section of the torus displaying puncture points from a set of field lines that make multiple toroidal revolutions. For any edge topology, as one moves further outward, eventually the field lines will strike the vacuum vessel wall or other plasma-facing components. While the presence of flux surfaces is crucial for good particle confinement, the properties of the flux surfaces can significantly impact the confinement ability. One important characteristic is how the strength of the magnetic field, B is distributed across each surface. This is due to the presence | | of trapped particles, which “bounce” between regions of high B [7]. If B is not designed carefully, | | | | the trapped particles will quickly drift across flux surfaces and out of the plasma. One way to keep the particles confined is to make the field quasi-symmetric, in which B is periodic along any field | | line [11]. Tokamaks inherently have this symmetry, but stellarators do not necessarily. In the case of quasi-helical symmetry, contours of constant B wrap helically around the torus (Fig. 1.4a). | |

5 Figure 1.3: Poincar´eplot for a cross-section of the W7-X stellarator in Greifswald, Germany. In the center (core) region, where the field lines conform to flux surfaces, the sets of (red) puncture points from each field line are neatly separated in concentric loops (i.e., cross-sections of flux surfaces). The green puncture points form a magnetic island chain outside the LCFS. The blue points form a stochastic region outside the LCFS. The pink curve represents the vacuum vessel wall. From Ref. [10]: Y. Suzuki and J. Geiger, Plasma Phys. Control. Fusion 58, 064004 (2016). c IOP

Publishing. Reprinted with permission. All rights reserved.

6 (a) (b) (c)

HSX NCSX W7-X

modB surfaces

θ θ θ

25 φ φ φ 25

Figure 1.4: Examples of magnetic fields that confine trapped particles. Upper images show three- dimensional flux surfaces in which the color indicates the B . Lower contours show a flux surface | | flattened into a grid of the poloidal angle θ and toroidal angle φ with contours of B overlain with | | magnetic field lines in red. (a) Quasi-helical symmetry, as in the HSX stellarator in Madison, WI. (b) Quasi-axisymmetry, as in the NCSX stellarator concept. (c) Quasi-isodynamicity, as in W7-X. Adapted from Ref. [14].

In the case of quasi-axisymmetry, contours of constant B move only toroidally, as in a tokamak | | (Fig. 1.4b). Another strategy for confining trapped particles is to make the field quasi-isodynamic, H such that the quantity mvkd`, the integral of the particle’s mass m times its velocity vk parallel to the field through a full bounce cycle, is uniform across a flux surface [12].2 This is illustrated in Fig. 1.4c. A detailed explanation of why quasi-symmetry and quasi-isodynamicity confine trapped particles may be found, for example, in Ref. [13].

2Quasi-helical symmetry and quasi-axisymmetry are in fact special cases of quasi-isodynamicity [12].

7 1.1.3 Stellarator research priorities

While the stellarator concept is a promising candidate for a fusion power plant, a number of scientific and engineering questions remain to be addressed to determine its viability. A report issued in 2016 by the National Stellarator Coordinating Committee [15] summarized the principal areas of interest as follows:

1. Optimized magnetic configurations. The three classes of field quasi-symmetries identified above each have different advantages and disadvantages. For example, quasi-axisymmetric fields result in the formation of significant bootstrap current (a toroidal plasma current which forms due to interactions between trapped and passing particles), whereas the quasi- isodynamic field of W7-X sustains little to no bootstrap current [16]. An extensive experi- mental and theoretical comparison of the various configurations will be required.

2. Turbulence and transport optimization. While quasi-symmetric configurations are all capable of confining trapped particles, further refinements can be made to suppress undesired phenomena, such as turbulence, and improve plasma performance.

3. Divertor design and optimization. Divertors are surfaces placed outside the confining volume to interrupt the trajectories of particles on their way out of the plasma. Hence, they must withstand significant heat loads, which depend strongly on the magnetic field. It is believed that certain stellarator edge field configurations may lead to reduced heat loads, especially as compared to tokamak divertors.

4. Plasma material interactions. In parallel to efforts to minimize divertor heat loads, it is imperative to develop materials that can withstand the heat loads without degrading or sputtering impurities into the plasma.

5. Energetic particle confinement and transport. D-T fusion reactions will produce high- energy alpha particles, which may interact with the plasma to excite instabilities (Alfv´en eigenmodes). Their numerical modeling must be further developed.

6. High β stability. Meeting the minimum nT τ product for fusion necessitates a high plasma

pressure, p = nkBT , where kB is Boltzmann’s constant. Due to engineering limits on feasible magnetic field strengths, this will necessitate a high β,

8 p β = 2 , (1.2) B /2µ0 | |

the ratio of p to the magnetic pressure. Here µ0 is the permeability constant. Magnetohydro- dynamics (MHD) dictates that plasmas with high β are vulnerable to instabilities that can, in turn, degrade confinement [17].

7. Impurity transport and accumulation. When atoms that are not deuterium or tritium enter a sufficiently hot plasma, they will lose their electrons and become highly-charged (“high-Z”) ions. High-Z impurities undergo more frequent and violent collisions with bulk plasma particles due to their stronger electrostatic forces. This causes the hot plasma particles to radiate away their kinetic energy, resulting in significant power losses [18]. Additionally, impurities do not take part in fusion reactions and dilute the fusion fuel. With this in mind, stellarators must be designed to avoid accumulating impurities, through optimizing the magnetic field and/or by using plasma materials that limit impurity production.

8. Coil simplification. Stellarator coils must have complicated non-planar shapes in order to produce optimized magnetic fields. However, complicated coils are difficult to manufacture and restrict access to the plasma vessel for reactor maintenance. It is hence important to improve coil codes to design coilsets that can generate optimized fields while also being feasible from an engineering point of view.

One underlying objective of the work in this thesis is to establish the Columbia Neutral Torus (CNT) experiment as a contributor to these research objectives. The next section will describe the CNT experiment and its potential to address the issues described in the areas of divertor design, high β, and coil simplification.

1.2 CNT as a fusion experiment

CNT (Fig. 1.5) is a stellarator remarkable for its simplicity and for its compactness. Whereas most stellarators are built with complicated non-planar coils, CNT’s magnetic field is generated by just four circular, planar coils (Fig. 1.5a). In addition, it has the lowest aspect ratio (the ratio of major radius to minor radius; i.e. the “thin-ness” of the torus) ever attained in a stellarator [19].

9 (a) PF coils (b)

θtilt

IL coils

Figure 1.5: (a) Schematic of the CNT plasma and coil set. (b) Photo of the CNT vacuum vessel. The two larger coils in the schematic are visible in light blue in the photo.

The configuration is also highly resilient to misalignments in the coil set, as will be shown in this thesis. In terms of the configuration styles discussed in Sec. 1.1.2, CNT’s interlocked coils most closely resemble a torsatron with two “helical” coils of poloidal/toroidal mode number m/n = 1/1 (thus, effectively, circular). While CNT’s coils were not optimized for quasi-symmetry or quasi- isodynamicity, its field most closely resembles quasi-helical symmetry (Fig. 1.6 and, for comparison, Fig. 1.4). Upon its construction in 2004 as the Columbia Non-neutral Torus [20], CNT’s original re- search focus was on pure-electron and partially-neutralized plasmas. Among the accomplishments in its first decade of operation are the successful formation and diagnosis of a pure-electron plasma [21, 22], confirmation of the Poisson-Boltzmann model of pure-electron equilibria [23, 21], character- ization of instabilities driven by minority ion populations [24], achievement of electron confinement times exceeding 100 ms with the aid of a radial electric field [25], and characterization of dynamics within plasmas with arbitrary degrees of neutrality [26]. While CNT was designed with non-neutral plasma science in mind, it is also well-suited to make

10 Fieldlines and |B| on flux surface 1

3 0.1

2 0.09

1 0.08 0.07

θ 0 0.06

−1 0.05

−2 0.04 Magnetic Field Strength (T) 0.03

−3 0 1 2 3 4 5 6 φ

Figure 1.6: Magnetic field lines (in black) overlain on contours of B of a typical CNT flux surface. | | From B it can be seen that CNT has two field periods; i.e., the field goes through two identical | | cycles in the toroidal angle φ. θ = 0 is on the outboard side and θ π is on the inboard side. ± contributions to magnetic fusion research. CNT exhibits a number of unique characteristics that make it attractive as a fusion research device, as described in this section.

1.2.1 Potential for high β at low power

CNT’s coils consist of copper windings that employ 10 kW of water cooling. This permits steady- ≈ state operation at fields ranging from 0.025 to 0.05 T on the magnetic axis. Higher fields can be obtained for short periods of time, limited by the heating of the coils. CNT’s typical operating field, which ranges from roughly 0.09 to 0.16 T on the magnetic axis, can be held for 30 to 40 seconds. These fields are more than an order of magnitude lower than what is typical for large fusion experiments. However, as mentioned in Sec. 1.1.3, much of the fluid dynamics within the plasma 2 hinge not upon the absolute plasma pressure p or magnetic pressure B /2µ0, but rather on the | | ratio between the two, β. Owing to CNT’s weak magnetic field and small volume, the amount of heating power required to attain a high β value ( 3%) should be much less than what would be required in a larger, higher- ≈

11 field device. Hence, CNT has the potential to study plasma dynamics and stability of relevance to fusion experiments. Chapter 6 examines these prospects in more detail.

1.2.2 Low aspect ratio

While conceptual designs exist for stellarators with even lower aspect ratios than CNT [27], CNT has the lowest aspect ratio among existing or ever built devices. CNT thus has the potential to be at the forefront of low-aspect ratio experimental research, which is of great interest to both tokamaks and stellarators. Low aspect ratios are considered favorable for multiple reasons. First, good confinement requires a large minor radius a but depends weakly on the major radius R. In addition, fusion power correlates crudely with plasma volume, which depends quadratically on a and linearly on R. Furthermore, low aspect ratios are known to promote plasma stability in tokamaks [28]. This may be true for stellarators as well [27, 29], although work to be presented in this thesis does not support that hypothesis. High values of β have been achieved in spherical tokamak experiments such as START [30] and NSTX [31]. CNT has the potential to test analogous limits in a stellarator configuration. Projections for stability at high β are described in Chapter 6.

1.2.3 Simple coil configuration

CNT has a remarkably simple coil set by stellarator standards, as is evident from Fig. 1.5. While the simplicity comes at the expense of optimized transport properties, it has proven advantageous for the development of error-field diagnostic techniques. Error fields arise from misalignments in the coil set, and can have deleterious effects on plasma stability and confinement [32, 33]. Chapter 2 describes the development of a numerical method for diagnosing coil misalignments. The procedure has the potential to be scaled up to diagnose stellarators with much more complex coil sets.

1.2.4 Experimental flexibility

Highly shaped coils tend to restrict access to the vessel for necessary procedures such as maintenance and diagnostic installation. Although coil sets were recently designed that meet optimization requirements while also being tokamak-like on the outboard side of the torus [34], no such designs have been yet been constructed. CNT, however, does not face these restrictions. The interior of the

12 vessel is easily accessible through multiple ports distributed around the chamber, and restriction by the coils is minimal, resulting in wide-angle camera views and diagnostic access. In addition, the vacuum vessel, rather than conforming tightly to the plasma boundary as is common in most stellarators, extends well beyond the plasma. This makes it relatively easy to design, install, and move a variety of experimental components. To date, these have included Langmuir probes for plasma diagnosis (Chapter 4), a phosphor rod for flux surface measurements (Ch. 2), and a microwave launcher for plasma heating (Chapter 5). Potential future experiments made possible by the large chamber might include:

1. In-vessel trim coil. Many tokamaks and stellarators incorporate trim coils to correct for field errors or to suppress instabilities in the plasma. These are typically placed outside the vessel, and therefore ac currents have to be relatively strong for the corresponding fields to penetrate the vessel walls and affect the plasma. Placing such a coil inside the vacuum vessel and adjacent to the plasma would allow for more efficient coupling. A single, movable perturbing coil and a set of magnetic probes and other diagnostics could enable the first experimental equivalent of Boozer’s “sensitivity matrix” analysis [35] of the impact of different field errors on transport, confinement, equilibrium stability, and other properties.

2. Island divertor. As discussed in Sec. 1.1.3, divertors are crucial components in a fusion reactor. While CNT was not designed with a divertor, it would be straightforward to construct and install one within the available space in the vessel. With sufficient heating power, it would be possible to measure the heat flux on the divertor plate with, for example, an infrared camera. Given that CNT is a low-field stellarator, measuring the divertor “strike point” would be important for validating theories (e.g., Ref. [36]) and experiments [37] suggesting that it is perhaps too narrow in tokamaks but not in stellarators.

Another flexible aspect of CNT is its magnetic configuration. As seen in Fig. 1.5a, CNT has four coils: two interlocked (IL) coils located inside the vessel, and two poloidal field (PF) coils located outside. The tilt angle, θtilt, between the IL coils is adjustable to three different settings: 64◦, 78◦, and 88◦. Each of these settings will produce a different plasma shape, as shown in Fig. 1.7. ◦ The different θtilt configurations also offer different regimes of rotational transform profiles: the 64 configuration yields profiles of ι that increase with minor radius; the 78◦ configuration yields nearly

13 (a) 64º (b) 78º (c) 88º

Figure 1.7: Comparison of the shapes of the magnetic confining volume produced in the three different θtilt configurations in CNT.

flat profiles, and the 88◦ configuration yields profiles that decrease with minor radius (Fig. 1.8). Each type of ι profile comes with advantages and disadvantages. Profiles with high positive shear (the rate of change of ι from one flux surface to another) tend to be more stable to MHD activity, whereas low shear on the edge is more advantageous for island divertors because it permits the formation of larger magnetic islands on the edge [15].

In practice, θtilt is determined by the brackets on the CNT vacuum vessel wall that support each IL coil. Each bracket has three bolt holes corresponding to one of the three possible θtilt settings. The θtilt setting is changed by opening the vessel and lifting each coil into a different set of bolt holes. The profile of ι may also be finely tuned on a shorter time scale (i.e., between plasma discharges) by changing the ratio of the currents in the IL coils relative to the PF coils.

1.3 Thesis overview

This thesis describes the first efforts undertaken to switch the focus of CNT from non-neutral plasmas to quasi-neutral plasmas of relevance to magnetic fusion energy. These efforts have resulted in contributions to our understanding of (1) stellarator stability, through error-field diagnosis and calculations of high-performance equilibria in the CNT configuration, and (2) plasma heating,

14 0.7

0.6 64 deg 78 deg 0.5 88 deg

0.4

0.3

Rotational Transform Rotational 0.2

0.1

0.0 0.00 0.02 0.04 0.06 0.08 0.10 Distance From Magnetic Axis (m)

Figure 1.8: Profiles of ι in CNT’s three θtilt configurations. Reprinted from Ref. [38].

through the observation of a synergy between microwave and hot-cathode heating, diagnosis of overdense microwave-heated plasmas, and the development of a technique to study the plasma’s visible emissivity in response to heating. Chapter 2 describes the development of a numerical technique designed to infer misalignments in the CNT coils that explain previously-observed discrepancies between the predicted and observed magnetic field geometry. In addition to identifying a set of coil displacements that agree well with the measurements, the results provided a detailed understanding of the magnetic geometry resulting from the field errors, permitting more accurate alignment of diagnostic instruments and analysis of plasma behavior. Chapter 3 demonstrates an application of this improved understanding of the magnetic geometry to the development of an image-processing technique. With the help of three-dimensional models of the plasma flux surfaces, the intensity of light emitted from each surface is inferred from a single wide-angle camera image of the plasma. While the emission measurements in this thesis were integrated across the visible spectrum, the technique may in principle be used to track profiles of emission in visible, ultra-violet, and x-ray bands or for specific lines. These profiles would reveal information about plasma properties such as temperature and impurity content.

15 Chapter 4 describes the first microwave-heating experiments conducted in CNT by means of a low-power (< 1 kW magnetron), as well as the development of diagnostic tools and data analysis routines to efficiently obtain time- and space-resolved measurements through the course of the plasma pulses. In addition, the combined usage of microwave heating with a hot-cathode emitter placed in the plasma is observed to create a synergistic effect that results in markedly higher plasma densities over what is observed with either source alone. Chapter 5 describes the commissioning of a 10 kW microwave heating system and measurements of the first plasmas made with this system. The incorporation of a transmission line and launch antenna accommodate more localized deposition of the microwave heating power. Plasmas are found to have high density, sometimes four times as high as the highest (“cutoff”) density permitting propagation of electromagnetic (EM) waves at the heating frequency. Density profiles are shown to only be sensitive to the magnetic field when the launched wave’s electric field is parallel to the background magnetic field (O-mode). Possible overdense heating mechanisms are discussed. The observation that the present heating system is not limited by density cutoffs helps build the case for CNT’s high-β plasma capabilities. It also motivates computational studies in the relatively little-explored area of wave interactions with plasmas comparable in size with the wavelength. Finally, Chapter 6 describes numerical predictions for plasmas that might be attained if the heat- ing power is increased by one to two orders of magnitude. To this end, simulations are conducted of plasmas in CNT with correspondingly higher β. These configurations were also evaluated for a particular plasma instability (the ballooning mode) that may limit the achievable β. Ballooning- stable configurations with volume-averaged β as high as 3.0% are identified; other configurations are predicted to become ballooning-unstable at β as low as 0.9%. Stability did not exhibit a clear dependence on aspect ratio. The most unstable configuration had a second region of stability at higher β. These predictions suggest that, with heating power as low as 40-100 kW, CNT has the potential to experimentally identify stellarator β limits.

16 Chapter 2

1 Error field diagnosis

Error fields (EFs) have been and are the subject of intense study in tokamaks [32, 33] where relative errors as small as 10−4 or even 10−5 are known to affect stability, cause disruptions, and degrade confinement and plasma rotation. Error fields are also very important in modern transport- optimized stellarators [40, 41], whose performances rely on carefully optimized 3D magnetic fields. The specially shaped coils that generate such fields are numerically optimized and, typically, are built and positioned with very high precision. In other words, errors are minimized at the construction stage. Vacuum fields are then experimentally characterized by a standard technique involving an electron beam and a fluorescent rod [42, 43, 44, 19]. Incidentally, it should be noted that, since this technique requires flux surfaces to exist in a vacuum, it is not applicable to devices such as tokamaks that require plasma current to generate flux surfaces. The measured surfaces are usually confirmed to be in good agreement with the desired, optimal configuration. However, if not in agreement, a comparison of computed and measured flux surfaces can shed light on possible sources of errors. At that point one can either (1) correct the error “at the source” (reposition one or more coils) [45] or (2) apply EF corrections by means of a separate set of dedicated coils. This latter approach is quite common in tokamaks [32, 33, 46]. As for stellarators, EF corrections were applied in LHD by means of Resonant Magnetic Perturbation (RMP) coils, also deployed in other MHD studies [47]. W7-AS used “special” and “control” coils to vary the toroidal mirror term and

1The work described in this chapter was published in Ref. [39]: K. C. Hammond, A. Anichowski, P. W. Brenner, T. S. Pedersen, S. Raftopoulos, P. Traverso, and F. A. Volpe, “Experimental and numerical study of error fields in the CNT stellarator,” Plasma Phys. Control. Fusion 58, 074002 (2016). c IOP Publishing. Reproduced with permission. All rights reserved.

17 boundary island geometry and study their effect on plasma properties [48], and W7-X has used its “trim coils” for similar purposes [49, 50, 51]. Earlier possibilities for EF correction in W7-X were discussed in Ref. [52]. Stellarators are considered complicated to build. It has been suggested that their attractiveness as reactors could increase if their construction is simplified, or construction tolerance relaxed, without significantly degrading the plasma properties [53]. At this point it is highly hypothetical, but a possible route to simplification could consist in: i) slightly relaxing the tolerance and ii) correcting the errors a posteriori, either by approach (1) or (2) listed above. In either approach, it is useful to experimentally quantify the displacements, tilts and, possibly, deformations of the actual coils, compared with design values. This is useful anyway, even if stricter tolerance is adopted, to confirm that the construction imperfections are indeed smaller. As a first step in exploring such route, we have complemented the well-established experimen- tal technique mentioned above with metrology measurements and with a numerical method that “inverts” the flux surface errors into error sources such as imperfections in coils’ positions and tilts. Error field resiliency was an especially important consideration for CNT due to the permissive tolerances used in the construction of the coils and the vacuum vessel. Whereas present-day stel- larators are typically built to tolerances on the order of 10−3 to 10−4 [54, 55], CNT’s tolerances were of order 10−2 to 10−3 to minimize the cost and complexity of construction. ◦ From 2005-2010, the interlocked coils were kept in the θtilt = 64 configuration. The flux surfaces for this configuration were measured experimentally and found to agree very well with numerical ◦ predictions [19]. More recently, CNT’s coils were switched to the θtilt = 78 configuration. This configuration is predicted to have less magnetic shear than the previous one. As a result, EFs that resonate with rational surfaces within the ι profile are expected to have more significant effects on the magnetic geometry, leading to larger islands and equilibrium deformation. A detailed understanding of the magnetic geometry in a central objective for any magnetic fusion device. For CNT, it will allow for more precise alignment of diagnostics and improve the accuracy of equilibrium modeling and reconstruction. It will also assist in the design of equipment whose shaping and placement are heavily dependent on the field, such as island divertors.

This chapter describes the first detailed measurements of the flux surface geometry of the θtilt = 78◦ configuration, which exhibits noticeable disagreements with predictions (Sec. 2.1). Multiple

18 Figure 2.1: Schematic of the CNT interlocked coils and last closed flux surface. The translucent patch on the right-hand side represents the cross-section at φ = 90◦ in which Poincar´emeasurements were taken as described in Section 2.1 The Cartesian axes are also shown on the lower right as a reference.

19 (a)

(b) (c)

Figure 2.2: Setup for flux surface measurements. (a) electron gun; (b) diagram of phosphor rods and actuator mechanism, reprinted from Ref. [19]; (c) long-exposure photograph revealing a flux surface cross-section.

efforts were then made to determine the sources of the EFs under the assumption that the main sources of error are displacements of the coils from their design positions. These efforts included (1) photogrammetric measurements of the positions of the PF coils (Sec. 2.2), (2) studies of the effects of different classes of displacements on the rotational transform ι (Sec. 2.3), and (3) design of an optimization algorithm to find the most likely displacements of the IL coils that lead to the observed flux surface geometry (Sec. 2.4). The potential broader applicability of the optimization algorithm to more complex coil configurations is discussed in Sec. 2.5.

20 2.1 Flux surface measurements

2.1.1 Experimental setup

The flux surface geometry in CNT is measured with a standard technique involving an electron beam and a fluorescent rod [19, 44]. Electrons are emitted from an electron gun (Fig. 2.2a) at energies of roughly 80 eV. The electrons travel along a field line until they strike an obstacle, either a part of the vessel (if an open field line) or an aluminum rod that extends into the confining region. The aluminum rod is coated with ZnO:Zn fluorescent powder that emits blue-green light (505 nm) when struck by electrons. In CNT, two fluorescent rods are positioned at the mid-plane of the vacuum vessel, which coincides with the toroidal cross-section at φ = 90◦ (Fig. 2.1). The rods can be rotated across this surface with an external actuation mechanism described in Ref. [19] and illustrated in Fig. 2.2b. Background pressures are maintained at the base pressure of the CNT vacuum vessel (< 10−8 torr) to minimize collisions with neutral atoms and molecules. If a long-exposure (5-10 s) photograph is taken of the fluorescent rods as they are scanned across the φ = 90◦ plane during the emission of an electron beam, the resulting image will show a cross-section of the flux surface on which the beam was emitted (Fig. 2.2c). Certain parts of the cross-section may not appear in the image due to shadowing by the electron gun or the limited extent of the fluorescent rods. For the measurements described in this chapter, a digital camera was positioned outside the vacuum vessel to face the plane of the cross-section through a fused-silica viewport. Images were acquired using ten seconds of exposure time. In addition to the fluorescent glow from the rods, the images contained some regions of stray light. This was due to blackbody emission from the electron gun filament reflecting off components in the vacuum vessel. Regions of stray light were eliminated manually from each image before processing.2 The relative positions of each pixel were determined based on the length of the rod in the image. The absolute positions of each pixel were then determined by comparing the position of the rod to the position of a fixed landmark in the chamber. The lens axis of the camera was confirmed to be within one degree of perpendicular to the φ = 90◦ plane through the use of a leveling tool. This uncertainty could contribute to up to 3 mm of displacement of the resulting image, as the camera was placed roughly one meter away from the cross-section.

2A. Anichowski assisted with the flux surface measurements and image processing.

21 2.1.2 Results

Two examples of experimentally measured Poincar´ecross-sections are shown in Fig. 2.3, overlaid with numerically computed Poincar´edata for the corresponding current-ratios. The numerical data, determined using a Biot-Savart field line tracer described in Ref. [50], were generated assuming that the coils were perfectly aligned. Note that the qualitative agreement is poor: the outboard side of the surfaces at IIL/IPF = 3.68 (Fig. 2.3a) are flatter and have more vertical elongation than the experimental surfaces. In addition, the experimental data for IIL/IPF = 3.18 (Fig. 2.3b) have prominent islands that are not predicted numerically. In addition to the geometric disagreements in the Poincar´esections at toroidal angle φ = 90◦, the measured rotational transform ι has been observed to differ from numerical predictions. While detailed profiles of ι have not been measured in CNT, it is possible to identify low-order rational surfaces through field line visualizations [56]. In this technique, the electron gun is operated while the vacuum vessel is back-filled to a neutral pressure between 10−5 and 10−4 Torr. At these pressures, electron-neutral collisions are frequent enough that the path of the electron beam emits a glow that is visible to the naked eye. In the vicinity of a low-order rational surface, the beam can be seen to strike the back side of the electron gun after a finite number of toroidal transits.

For example, for IIL/IPF < 3.5, a region exists in which the electron beam is observed to strike the electron gun after three toroidal transits, indicating the presence of a surface (or island chain) with ι = 1/3. On the other hand, numerically calculated ι profiles (Fig. 2.4) only contain ι = 1/3 for IIL/IPF < 3.18. Evidently, an EF is causing a systematic offset of the rotational transform. The observed discrepancies between experiment and calculations motivated a more detailed study of the possible sources of EFs.

2.2 Photogrammetry

Prior to the start of the work in this thesis, displacements of the PF coils (but not the IL coils) from their design positions were measured directly using photogrammetry [57]. These measurements are shown in Fig. 2.5, in which each arrow represents the displacement of a point on the coil from its design location. Certain regions of the PF coils are separated by as much as 44 3 mm from their ± intended locations, which is well outside of the 10 mm tolerance specified for construction. It is

22 Figure 2.3: Comparison of numerical Poincar´eplots (black dots) with experimental results (adja- cent surfaces are shown in alternating red and blue for clarity). Numerical data were determined assuming that the coils were perfectly aligned. (a) IIL/IPF = 3.68; (b) IIL/IPF = 3.18

23 0.42 4.38

0.4 2/5 4.12

3.89 5/13 0.38 3/8 3.68 4/11 3.50 0.36 5/14 3.33 Rotational transform 0.34 3.18 3.04 1/3 2.92 0.32 0 2 4 6 8 10 12 14 16 Minor radius (cm)

Figure 2.4: Plots of ι profiles for selected current-ratios in CNT in the 78◦ IL coil configuration. In the red curves, which are labeled with respective values of IIL/IPF , each dot represents a closed flux surface and dashed segments indicate island chains. Some low-order rational numbers are shown as horizontal dashed lines. Note that current-ratios of 3.18 and above are not predicted to contain ι = 1/3, contrary to measurements.

24 Figure 2.5: Rendering of the CNT vacuum vessel and PF coils overlaid with arrows showing the displacements of the coils from their design positions. Each arrow corresponds to one photogram- metric marker. Arrow lengths are to scale with one another but are exaggerated relative to the vacuum vessel and coils. The largest arrow shown has a length of 44 3 mm. Blue (red) arrows ± represent deviations in the same (opposite) direction of the vector normal to the surface where they originate. Reproduced from Ref. [57].

noteworthy that CNT’s coils still produce good flux surfaces despite having displacements of this magnitude. The observed misalignments of the PF coils are believed to have arisen during a procedure to ◦ ◦ change the IL coil configuration from θtilt = 64 to θtilt = 78 . The PF coils must be tilted out of the way to permit access to the IL coils, and they sit on hinged support structures for this purpose. By the end of the procedure, the structures had visibly deformed due to internal stresses arising from the tilting, likely affecting the positions of the PF coils themselves. To determine the effects of the measured PF coil displacements on the flux surfaces, new field line traces were conducted with the PF coils offset according to the photogrammetry data. For these calculations, the PF coils were still assumed to be circular and planar but were tilted and translated so as to best fit the displacement vectors. According to these fits, the northern (z > 0) and southern (z < 0) PF coils were translated 5 mm and 22 mm respectively, and their axes were tilted by 1.0◦ and 1.3◦ respectively. A comparison of the numerical Poincar´eplots with and without the displacements is shown in

25 ◦ Figure 2.6: (a) Numerically generated Poincar´eplots at the φ = 90 cross-section with IIL/IPF = 3.18 from a configuration incorporating the measured PF coil displacements (blue) and from the design configuration (black). (b) Calculated profiles of ι for this configuration.

26 Fig. 2.6 using IIL/IPF = 3.18. Although the plots are not identical (in particular, the magnetic axis has shifted slightly outboard and field lines initiated at the same locations do not overlap perfectly), the effects of the PF coil displacements do not appear to be sufficient to explain the observed experimental discrepancies. In particular, the slight increase in the ι profile is actually the opposite effect of what would be needed to explain the ι = 1/3 island chain seen at an intermediate minor radius in Fig. 2.3 b. Since the PF coil displacements were not enough to explain the observed differences in Poincar´e cross-sections, additional sources of field error were sought.

2.3 Numerical study of ι sensitivity

In the search for additional sources of field error, the most likely candidate was thought to be displacements to the IL coils. Previous work [38] has shown that a displacing of either IL coil has a significantly greater effect on the magnetic geometry than translating either PF coil. (For this reason, the tolerance for the IL coils was set at 2 mm, one-fifth of the PF coil tolerance.) To identify the possible causes of the systematic offset in ι discussed in Section 2.1, a series of numerical field line traces was computed for different classes of IL coil displacements. Field lines were also traced for similar classes of PF coil displacements for comparison. The classes of displacements were sorted into symmetric (equal coil movement in opposite directions) and antisymmetric (equal coil movement in the same direction). For each class of displacement, ι profiles were calculated for a series of magnitudes of the displacement. To represent the calculated trends in ι, Fig. 2.7 shows derivatives of ιcore (i.e., the limit of ι as the minor radius approaches zero) with respect to each type of displacement. Derivatives were estimated by second-order finite differences about the nominal positions of the coils at IIL/IPF = 3.68. The displacements plotted in Fig. 2.7 collectively represent the twenty degrees of freedom that the four coils have for rigid motion (five for each coil; rotation of a coil about its axis is ignored due to symmetry). The translational displacements shown in Fig. 2.7a have fairly intuitive interpretations. A positive “ILC co-x” displacement, for example, involves motion of both the IL coils in the positive x direction as indicated in Fig. 2.1. An “ILC counter-x” displacement, on the other hand, involves motion in opposite ways in the x direction, with positive displacement indicating the IL coils moving apart and negative displacement indicating the IL coils moving toward one another.

27 (a) ILC co-x Positive ILC co-y Negative ILC co-z ILC counter-x ILC counter-y ILC counter-z PFC co-x PFC co-y PFC co-z PFC counter-x PFC counter-y PFC counter-z

10-7 10-6 10-5 10-4 10-3 10-2 dι /dx (mm-1) core (b) ILC co-a-tilt ILC co-b-tilt ILC counter-a-tilt ILC counter-b-tilt PFC co-a-tilt PFC co-b-tilt PFC counter-a-tilt PFC counter-b-tilt

10-6 10-5 10-4 10-3 10-2 10-1 dι /dθ (deg-1) core

Figure 2.7: Derivatives of ιaxis with respect to the magnitude of different classes of coil displace- ments. (a) Translations of the coils in the Cartesian directions illustrated in Fig. 2.1. (b) Tilts of the coils in the directions of their respectivea ˆ and ˆb axes. Thea ˆ and ˆb axes for the IL coils are illustrated in Fig. 2.8; for both PF coils,a ˆ =x ˆ and ˆb =y ˆ. A more detailed description of each displacement class is given in the text.

28 Figure 2.8: Schematic of the two IL coils along with their respective axes of symmetry (red arrows), aˆ vectors (green arrows), and ˆb vectors (blue arrows) as described in the text. Thex ˆ,y ˆ, andz ˆ directions as defined in Fig. 2.1 are shown in yellow, magenta, and cyan respectively.

The angular displacements shown in Fig. 2.7b represent tilts of the coil in two orthogonal directions denoted bya ˆ and ˆb. Thea ˆ and ˆb vectors for the IL coils are illustrated in Fig. 2.8. “ILC co-b-tilt” displacement, for example, refers to both IL coils tilting by some angle θ such that their axes rotate toward their respective ˆb unit vectors. A change in ILC co-b-tilt is equivalent to an ◦ adjustment of θtilt (nominally 78 ). “ILC counter-a-tilt” displacement, as another example, refers to the IL coils tilting such that IL1 tilts its axis in the positivea ˆIL1 direction and IL2 tilts its axis ˆ in the negativea ˆIL2 direction. For both PF coils,a ˆ =x ˆ and b =y ˆ. From Fig. 2.7, it is evident that the two classes of displacements that have the greatest influ- ence on ι are the separation of the IL coils (along the x axis) and the tilt angle between the IL coils. Another less-prominent influence arises from the separation of the PF coils. This is roughly equivalent to changing the PF coil current, as it strengthens or weakens the Helmholtz field created by the coils, and illustrates how adjusting the PF current relative to the IL current serves as a fine adjustment to the ι profile. The relative sensitivity of ι to the different classes of coil displacements in Fig. 2.7 suggests that,

29 if the offset of the ι profile is indeed caused by coil displacements, it is very likely that displacements of the IL coils play a significant role. This observation motivated a more detailed study of IL coil displacements to be described in the following section.

2.4 Optimization

2.4.1 Optimization procedure

The optimization method employed here tweaks the coil positions in small increments until the cal- culated Poincar´edata resulting from the perturbation matches with the experimentally obtained Poincar´ecross-sections. The procedure is conceptually similar to methods used in some plasma equilibrium reconstruction codes such as EFIT and V3FIT [58, 59], which determine plasma equi- librium parameters that fit to diagnostic signals. In this procedure, Poincar´ecross-sections take the role of the diagnostic signals, whereas coil displacements have the role of the equilibrium pa- rameters. For the optimizations described in this chapter, the observed Poincar´ecross-sections are mapped into a vector X of discrete geometric parameters, which may be thought of equivalently as physics parameters derived from flux surface topology. The definitions of these parameters are provided in Appendix A, and the procedure for determining them from experimental and numerical Poincar´e data is given in Appendix B. The coil misalignments were encapsulated in a vector p of displacement parameters (i.e., engineering parameters). The work in this chapter used two variants of p, both of which will be described later in this section. An optimization begins with target geometric parameters X∗ for the desired Poincar´ecross- section geometry and an initial guess p0 of coil displacement parameters. The geometric parameters associated with p0 (or any set of displacement parameters p) are then X(p). The discrepancy F(p) between X∗ and X(p) is defined as

∗ Fi(p) = Xi(p) X (2.1) − i The optimization algorithm attempts to find p∗ such that F = 0 using the Newton-Raphson method [60]. Expanding F about a coil configuration p that differs from p∗ by δp, i.e.,

30 F(p + δp) = 0 = F(p) + Jδp + O δp2
, (2.2) the iterative Newton step δp is given by a solution to the equation

F(p) = Jδp. (2.3) − Here, J is the Jacobian,

∂Fi Jij = (2.4) ∂pj In general, p is not of the same dimension as F. We thus find the optimal δp through linear least-squares. The Best Linear Unbiased Estimator (BLUE) for δp is that which minimizes χ2 [61]:

χ2 = FT C−1F (2.5)

 ∗ ∗ ∗ where Cij = cov Xi ,Xj is the covariance matrix for the target parameters X . This estimator for δp is given by

−1 δp = J T C−1J
J T C−1F. (2.6) −

2.4.2 Numerical considerations

The covariance matrix

For experimental Poincar´edata, the covariance matrix C is estimated by evaluating X for multiple samples of the pixels obtained from composite flux surface images. For each sample, pixels are selected randomly with replacement (bootstrapping) [62]. The number of pixels available to sample from a particular flux surface in the images used for this analysis ranged from 300 to more than 10,000. Each sample consists of 400 pixels per flux surface. For the verification studies conducted in Sec. 2.4.3, the optimizer was programmed to fit the coils to numerically determined Poincar´edata (i.e., a manufactured solution). To obtain a covariance matrix for numerical Poincar´edata, multiple Poincar´eplots are obtained by tracing field lines from randomized initialization points. Each sample consists of a trace of 200 field lines followed for 200

31 toroidal revolutions. (A certain percentage of the field lines will terminate if they are initialized outside of the last closed flux surface; these are ignored.) As indicated in Eq. 2.6, C must be inverted for the analysis. This can introduce significant numerical error if C is ill-conditioned. To mitigate this risk, rather than directly inverting C, its inverse is approximated through singular value decomposition:

C = UΣU T (2.7)

Here, the orthogonal matrices U to the left and right of Σ are the same due to the inherent symmetry of C. Σ is a diagonal matrix of the singular values. If C is ill-conditioned, the lowest singular value(s) will be much less than the greatest singular value. As a workaround, a cutoff is imposed such that all singular values of C less than a factor 1/κco of the greatest singular value are set to zero. The pseudoinverse of this approximation is given by

C−1 UΣ−1U T , (2.8) ≈ where Σ−1 is a diagonal matrix in which the nth nonzero diagonal element is equal to the inverse th of the n nonzero diagonal element of Σ. For the work described in this chapter, κco was chosen empirically to be 108 for the calculations in this chapter as the lowest value for which the Newton direction was not noticeably compromised.

The Jacobian

The elements of the Jacobian in this implementation are computed by second-order finite differences:

Fi (p + ∆pj) Fi (p ∆pj) Jij = − − (2.9) ∆pj | | ∗ Note that, due to its dependence on X via F, Jij contains random error and may be correlated with other matrix elements. Because of this, Eq. 2.6 does not, strictly speaking, give the BLUE for δp, since one of the underlying assumptions is that J is non-random. Nevertheless, we have found that this estimator is sufficient in many cases as long as the finite differencing interval ∆p is sufficiently large.

32 Line-search

If F depends nonlinearly on p, there is a risk that the Newton step δp will overshoot a local minimum in χ2. To rectify such occurrences, a line-search algorithm checks the Newton step after each iteration to ensure that the average rate of decrease in χ2 over the interval δp is at least as great as the gradient of χ2 evaluated at the starting point p of the iteration. If this condition is not met, the algorithm samples χ2 at shorter distances along the direction of the Newton step to identify the local minimum. More details on this algorithm are given in Ref. [60].

2.4.3 Verification

To verify the performance of the algorithm, some tests were conducted in which the code was used to solve for a known coil displacement p∗ using target parameters X∗ generated from a field-line trace that used p∗ as its input. One early test was meant to determine whether the algorithm could distinguish two classes of displacements that produced qualitatively similar outcomes. One such pair of displacement classes is (1) separation of the coils along the x axis and (2) adjustment of θtilt. Incidentally, these two classes were also found to have the greatest impact on rotational transform (Fig. 2.7), a strong determiner of the cross-section geometry. A p vector was thus defined with just two components:

  1  2c (∆xIL1 ∆xIL2)   x −  p =   (2.10)  1  ∆θtilt cθ

Here, ∆xIL1 and ∆xIL2 are the displacements along the machine x axis of the first and second IL coil, respectively, from their nominal positions. ∆θ is the displacement of the coil angle from its ◦ nominal value of 78 . cx and cθ are scaling constants equal to 1 m and 1 radian, respectively. The ∗ −3 target parameters X were generated using p1 = 0 and p2 = 8 10 ; i.e., no discrepancy in coil × ◦ separation and a decrease of 0.46 in θtilt. The initial guess p0 was (0, 0) and the finite differencing interval for both components was 10−3. The results of this test are shown in Fig. 2.9. Note that the contours of χ2 exhibit a shallow valley along a line consisting of a family of displacements that would result in similar Poincar´e geometry. Nevertheless, the algorithm succeeded in moving the components of p to within one

33

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4

−0.5 χ2 contours Parameter path

Discrepancy in ILC tilt angle (deg) −0.6 Target

0 5 10 Discrepancy in ILC x separation (mm)

Figure 2.9: Contour plot of χ2 and the path taken by the optimizer in a verification test with two free parameters. The initial guess p0 is (0, 0) and the target parameters are denoted by the red . × Contours are on a logarithmic scale with solid contours representing powers of 10.

finite differencing interval of the target values on the fourth iteration. Subsequent verifications tested the ability of the code to identify similar coil displacements when more coil parameters were free. In these tests, the IL coils were permitted to undergo rigid rotations and transformations; hence, each coil was allowed five degrees of freedom: three translational and two angular. The translational parameters, x, y, and z, are simply Cartesian displacements along the respective unit vectors indicated in Fig. 2.1. The angular parameters are illustrated in Fig. 2.10.

As shown in the diagram, aIL and bIL are orthogonal projections of a unit vector representing the perturbed coil axis onto the plane of the unperturbed coil. The aIL1 and bIL1 components correspond ˆ to the directionsa ˆIL1 and bIL1 illustrated in Fig. 2.8. The polar displacement angle of the perturbed p 2 2 axis of the first IL coil can thus be computed as arcsin aIL1 + bIL1 . Fig. 2.11 shows the outcomes of two calculations that optimized the ten IL coil parameters as described above to a target X∗ vector generated numerically from a chosen set p∗ of displacements. ∗ In Fig. 2.11a, the only nonzero component of p was zIL1, which was set to 5 mm, a simple ∗ translation of the first IL coil. In Fig. 2.11b, there were two nonzero components of p : bIL1 =

34 Figure 2.10: Schematic illustrating the definitions of the a and b components of the angular dis- placement parameters of the coils. In this image, the IL1 coil is used as an example. The blue surface represents the plane of the unperturbed coil, and the red arrow represents a unit vector along the axis of the perturbed coil.

ˆ 0.002 and bIL2 = 0.004; i.e., both coils were tilted toward their respective b vectors, but by different amounts. Both optimizations descended by multiple orders of magnitude in χ2 in the first three steps (Fig. 2.11c), but afterward, χ2 flattened out: the optimizations did not converge any further toward p∗. One possible explanation is that a relatively long finite differencing interval ∆p was used to determine the Jacobian in each step: all translational (x, y, and z) components used an interval of

1 mm, and the angular components (a and b) used an interval of 0.001. All of the final values pi ∗ of the components are well within ∆pi of the target values pi (Fig. 2.11a-b). Another contributing factor may be the numerical uncertainty of F.

2.4.4 Inferring coil displacements from experimental data

After the verification tests, the algorithm was applied to experimental Poincar´edata. The target geometric parameters X∗ were determined from the cross-section data for the current-ratio 3.68 (Fig. B.1). This dataset was chosen over datasets from other current-ratios for its abundance of fully-characterized flux surfaces and for its lack of magnetic islands, two attributes that facilitate accurate calculations of X∗ (Sec. 2.5.1). The PF coils were held fixed with displacements determined

35 agt of targets evolution Parameter param- (a) 10 in of perturbations. move targets coil to toward made-up free from were generated coils Poincar´e data interlocked match the to which eters in optimizations of Results 2.11: Figure b IL 1 0 = z IL . 002, 1 madters fteprmtr eo b aaee vlto toward evolution Parameter (b) zero. parameters the of rest the and mm 5 = b IL (b) (a) (c)

2 2

0 = χ 10 10 10 10 Displacement parameter Displacement parameter -2 -2 11 0 2 4 6 0 2 4 6 . 5 7 9 0,adters eo c ecn of Descent (c) zero. rest the and 004, 10 8 6 4 2 0 36 Iteration x b a z y x b a z y x b a z y x b a z y IL IL IL IL IL IL IL IL IL IL IL IL IL IL IL IL IL IL IL IL 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 Plot b Plot a × × × × × × × × (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) 10 10 10 10 10 10 10 10 3 3 3 3 3 3 3 3 χ 2 o ohoptimizations. both for by a best-fit to the photogrammetry data discussed in Sec. 2.2. All ten parameters associated with the IL coils were free. The initial guess p0 was the nominal IL coil configuration. As in the 10- parameter verifications conducted in Sec. 2.4.3, the finite differencing interval was 1 mm for all translational motion and 0.001 for all angular displacements. The set of displacements obtained in ∗ the final iteration will be referred to hereafter as p3.68. The shifts in IL coil positions during the fifteen steps of the optimization are shown in Fig. 2.12a, with the accompanying descent of χ2 shown in Fig. 2.12b. The largest translational moves from the starting positions occurred in the negativey ˆ direction for both coils, averaging to -22 mm. Thez ˆ displacement was nearly identical at around 6 mm for both coils, and thex ˆ displacement averaged to about 3 mm with a 3 mm “counter-” component (cf. the displacement classes in Fig. 2.7). It should be noted that, while the “co-” translational motion of the coils may indicate that the IL coils are indeed off of their nominal positions, it may also reflect misalignment of the fluorescent rod. The largest angular shift was of the first IL coil, which tilted about 1◦ away from its nominal ◦ axis alonga ˆIL1, effectively decreasing θtilt to 77 . With the results of Fig. 2.7 in mind, this shift, combined with the 3 mm of counter-motion in thex ˆ direction, is likely responsible for most of the downward offset in ι observed experimentally. During the optimization, the value of χ2 (Fig. 2.12b) for the Poincar´edata associated with the displacements decreased by a factor of more than 100. Most of this decrease occurred during the first five iterations, after which improvement was insignificant. The final value of χ2 is lower for this optimization than in the verifications (Fig. 2.11c) by about an order of magnitude; however, this is primarily due to the fact that the target parameters X∗ determined here from the experimental data have greater uncertainty than those of the manufactured solutions. ∗ The qualitative improvement in the numerically predicted Poincar´eplots generated using p3.68 versus p0 is shown in Fig. 2.13. Fig. 2.13a, identical to Fig. 2.3a, refers to IIL/IPF = 3.68 and compares the experimental data to a numerical prediction generated from p0 (i.e., coil displacements were neglected). Note, again, the disagreements in the vertical elongation on the outboard side, the concavity on the inboard side, and in the position of the magnetic axis. Most of the numerical cross-sections intersect multiple experimental ones. ∗ Fig. 2.13b shows the same experimental data overlaid on numerical data computed from p3.68.

37 sn xeietlPicredt rmte36 urn-ai.()Dsetof optimization displacements. Descent an the (b) during from resulting coils Poincar´e current-ratio. data IL the 3.68 the the of from parameters Poincar´e data displacement experimental the using of Evolution (a) 2.12: Figure (b) (a) χ2 Displacement parameter 10 10 10 10 -30 -20 -10 10 20 0 4 5 6 7 015 10 5 0 b a z y x b a z y x IL IL IL IL IL IL IL IL IL IL 2 1 2 1 2 1 2 1 2 1 × × × × (mm) (mm) (mm) (mm) (mm) (mm) 10 10 10 10 3 3 3 3 Iteration 38 χ 2 soitdwith associated Figure 2.13: (a) Experimental data (red and blue dots) plotted alongside numerical data (black) for a field line trace that assumed no coil displacements, both at IIL/IPF = 3.68, identical to what is plotted in Fig. 2.3a. (b) The same experimental data as in (a) exhibited better agreement with ∗ the numerical data that accounted for the coil displacements inferred from the optimization (p3.68), again at IIL/IPF = 3.68.

The incorporation of the displacements in the comparison has dramatically improved the qualitative agreement. The discrepancies in magnetic axis, as well as the shaping of the inboard and outboard sides, has essentially vanished, and there far fewer cases of numerical cross-sections intersecting multiple experimental ones.

Figs. 2.14a-b show comparisons at other current-ratios (IIL/IPF = 3.50 for 2.14a and IIL/IPF = ∗ 3.18 for 2.14b) in which the numerical Poincar´edata were generated from p3.68. The agreement is not as good as in Fig. 2.13b, which is to be expected because the error vector F used in the optimization was based exclusively on the geometry of the IIL/IPF = 3.68 cross-section. It is noteworthy, however, that for IIL/IPF = 3.18 the optimization nonetheless predicts three islands near the edge of size and position comparable to what is observed. Note also the improvement in ∗ agreement for IIL/IPF = 3.18 using p3.68 in Fig. 2.14b over Fig. 2.3b in which the numerical data were calculated from the nominal coil positions. Although the data in Fig. 2.14a have prominent gaps near the axis, these are not interpreted as

39 islands but rather are a shadowing effect resulting from the electron beam striking the back of the gun after the third transit. Note that the concentric datasets (alternating in red and blue) near the axis each contain only three discrete points. To confirm the presence of islands, the data points would need to wrap continuously around the islands as they do in 2.14b. With only three discrete points, however, islands cannot be distinguished from good flux surfaces. ∗ Finally, calculated values of ι at various current-ratios IIL/IPF using p3.68 are shown in Fig. 2.15b.

Note that, for IIL/IPF < 3.5, the profiles of ι contain the value 1/3, which is consistent with the observations described in Sec. 2.1.2. Recall that the ι profiles that had been predicted for the nominal coil positions (Fig. 2.4) failed to account for the presence of ι = 1/3 for IIL/IPF 3.18 ≥ and that, furthermore, the PF coil displacements alone were insufficient to explain this disagree- ment (Fig. 2.6). Hence, the optimization of the IL coil displacements has resolved the discrepancy between the observations and predictions.

2.5 Discussion

2.5.1 Optimizing for cross-sections with islands

For the optimization conducted for the CNT IL coils in this chapter, the target parameters in ∗ X came exclusively from a single current-ratio topology (IIL/IPF = 3.68) that had no significant magnetic islands. In principle, a cross-section containing magnetic islands should also be useable for an optimization if adequate experimental surface data are available, in the sense that many puncture points are distributed evenly around the cross-section of the surface. Such data were not available from CNT cross-sections featuring prominent islands, largely as a result of two factors. The first was that cross-sections containing the large ι = 1/3 island chain were cut off at the inboard side due to the finite extent of the fluorescent rod. On the other hand, had the rod extended further inboard, it would have collided with the IL coils during rotation. Thus, as seen in Fig. 2.14b, there are no complete surfaces recorded outside the island chain. This factor may be unique to CNT as it relates to the particular coil configuration. The second factor, more likely to arise in other stellarators, is the shadowing of near-rational surfaces in the vicinity of island chains. This is a result of the electron beam striking the back side of the gun before reaching the fluorescent rod. This effect is visible in surfaces near the magnetic

40 Figure 2.14: Effect of using, in calculations for (a) IIL/IPF = 3.50 and (b) IIL/IPF = 3.18, the coil ∗ displacements inferred for IIL/IPF = 3.68 (i.e., p3.68). Measurements for the respective current ratios are shown for comparison.

41 (a) Nominal coil positions (b) Inferred coil positions 0.42 5/12 4.38

0.4 4.12 2/5 3.89 4.38 5/13 0.38 3.68 4.12 3/8 4/11 0.36 3.50 3.89 5/14 3.33 3.68 3.18 0.34 3.50 3.04 1/3 Rotational transform 2.92 3.33 0.32 3.18 3.04 4/13 0.3 2.92 3/10

0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 16 Average minor radius (cm) Average minor radius (cm)

Figure 2.15: Comparison of ι profiles calculated for selected current-ratios in the nominal and optimized coil configurations. (a) Profiles for the nominal IL coil positions (p0) and assuming no PF coil displacements, identical to what is plotted in Fig. 2.4. (b) Profiles computed using ∗ the optimized IL coil configuration (p3.68) and the measured PF coil displacements. Each dot represents a closed flux surface, so while the curves for IIL/IPF = 3.04 and 2.92 in plot (b) do not pass through ι = 1/3, they contain large three-island chains on their edges. Note that, in the optimized configuration (b), current-ratios IIL/IPF less than 3.5 contain ι = 1/3, consistent with observations.

42 axis in Fig. 2.13a-b and Fig. 2.14a, in which only three dots appear, corresponding to the first three toroidal transits of the electron beam. In this case, the shadowing was not a major concern because only a small portion of the magnetic surfaces was shadowed. But in low-shear profiles with one or more rational surfaces at intermediate minor radii, large portions of the cross-section may be shadowed, thereby obscuring a large amount of geometric information and preventing the determination of accurate geometric parameters. However, if the above factors are absent or limited in extent, we expect that cross-sections with island chains should be useable for optimization. Poincar´edata from field lines within a chain of large islands, identifiable by large empty intervals in the poloidal angle θ, are ignored in the parametrization procedure (Appendix B.0.1). Thus, only the closed flux surfaces on either side of the island chain will determine X, and, in turn, the Netwon steps δp in an optimization. Although the information about the magnetic geometry within the islands themselves is lost, the kinking of nearby closed flux surfaces adjacent to the island chain is retained. It is hypothesized that the closed flux surfaces with island-induced kinking should be sufficient for the optimization.

2.5.2 Extension to more complex devices

Although the vector p of coil parameters used for the optimization in this chapter had only ten components, this vector can in principle be expanded to contain arbitrarily many parameters char- acterizing non-rigid displacements for multiple coils. One logical extension for CNT would be to allow for elliptical compression of the IL coils, which would add two more parameters for each coil (one for ellipticity and one for phase). More general deformations may be treated, for example, as Fourier series in which each additional m number would require the addition of two parameters (coefficients) if the coil is assumed to stay planar. With the addition of even more parameters, the coil could be allowed to deform off of its nominal plane. It follows that, for an arbitrary stellarator, the size of p would scale as the number of coils to be optimized times the number of degrees of freedom for rotation, translation and deformation afforded to each coil. Of course, each coil need not be given the same number of degrees of freedom. Assuming X consists of geometric data from only one current-ratio (which need not be the case), each column of the Jacobian can be computed with a single call to the field line tracer.

43 Different columns, however, corresponding to perturbations to different components of p, require separate field line traces. Since the field line traces are the most demanding part of the optimization procedure, the computation will scale linearly with the size of p. It should be noted, however, that since the columns of the Jacobian may be determined in parallel with one another, the time need not scale linearly if multiple processors are available.

44 Chapter 3

Onion-peeling inversion of stellarator 1 images

As CNT moves toward higher-β plasma regimes, its diagnostic requirements will change. In partic- ular, the high-β plasmas studied in this thesis were diagnosed primarily with Langmuir probes as described in Chapters 4 and 5. As Langmuir probes are invasive instruments that extend into the core of the plasma, they will not be compatible with higher plasma temperatures and densities. A diagnostic system better suited for high β would be one that doesn’t contact the plasma directly but instead makes inferences based purely on radiation from the plasma. Plasma radiation comes in many forms—including microwaves, visible light, x-rays, and, if it is a DT fusion plasma, neutrons—and many different tools are available for each. The focus in this chapter is on visible emission: in particular, a method is developed for reconstructing the radial emissivity profile based on 2D fast-camera images. A number of well-established techniques exist for inferring plasma emission profiles based on line-of-sight measurements. The measurements are interpreted as line integrals of emission and are typically acquired in 1D or 2D arrays. This poses a question of “inverting” the acquired datasets into 2D or 3D maps, respectively, of plasma emissivity of the particle or wavelength of interest. Tomography has succeeded at this in tokamaks [64] and stellarators [65], primarily for X-ray

1The work described in this chapter was published in Ref. [63]: K. C. Hammond, R. R. Diaz-Pacheco, Y. Ko- rnbluth, F. A. Volpe, and Y. Wei, “Onion-peeling inversion of stellarator images,” Rev. Sci. Instrum. 87, 11E119 (2016). Reproduced with the permission of AIP Publishing.

45 and visible emission [66, 67]. An advantage of tomography is that it does not require knowledge of the flux surface geometry, although the geometry can constrain the tomography for better results. A disadvantage, though, is that it requires several cameras around the plasma. Under the assumption of the flux surface geometry being perfectly known, a single camera is sufficient to invert a diagnostic image and infer the emission profile. If, additionally, the flux surfaces are axisymmetric, one can use the Abel transform [68, 69]. This is appropriate in quiescent tokamak and spherical tokamak plasmas. In stellarators it is reasonable to assume a good knowledge of the flux surfaces, which are nearly entirely determined by external currents and are magnetohydrodynamically more quiescent than in tokamaks and sperical tokamaks. This is especially true when the ratio of kinetic pressure to magnetic pressure, β, is low: in that case, stellarator equilibria are easier to compute, faithful to experiments and magnetohydrodynamically stable. However, stellarator flux-surfaces are obviously not axisymmetric. The onion peeling algorithm [69] can be considered a generalization of the Abel inversion to non-axisymmetric problems. Onion peeling is used here to invert wide-angle visible images of a stellarator plasma for the first time. The work takes advantage of the error field study presented in Chapter 2, giving good confidence in the knowledge of CNT flux surfaces. The method is described in Sec. 3.1. Sec. 3.2 is devoted to the “forward problem”: toy models of the emissivity profile are used to generate simulated camera images. The synthetic image cor- responding to an edge-peaked profile is in best qualitative agreement with the actual experimental images. The inverse problem is solved in Sec. 3.3. The method is first tested with a glow discharge plasma for validation and then applied to a microwave-heated plasma.

3.1 Method

The inversion method relies on two principal assumptions. The first is that the plasma can be modeled as a set of nested discrete layers, each of which has a uniform emissivity. In the context of a toroidal magnetic confinement device, these layers are bounded by flux surfaces (Fig. 3.1). The second assumption is that the emissive layers contribute to the brightness of each camera pixel in proportion to the distance that the pixel’s line-of-sight travels through each layer. With these assumptions in place, the brightness p of each pixel is related to the emissivity of

46 L11

L22

to camera L21

p 1 p 2

e1 e2 e3 e4 e5

Figure 3.1: Schematic illustration of the onion-peeling method. The red lines represent lines-of- sight of two camera pixels whose brightness is denoted by p1 and p2. The lines-of-sight both travel through one or more layers of plasma, whose borders are shown as alternating blue and white surfaces. The lengths of the segments within the respective surfaces constitute the components of the L matrix as defined in the text. Note that some components (like L21 here) are actually due to the sum of two or more segments due to multiple crossings of a layer. The terms ei are the emissivities of each layer. each layer by a system of linear equations,

bp = Le, (3.1)

th where pi is the brightness of the i pixel, b is a constant of proportionality, Lij is the total distance th th th traveled by the i pixel’s line-of-sight through the j layer, and ej is the emissivity of the j layer. The elements of the matrix L can be calculated based on knowledge of (1) the flux surfaces, (2) the camera’s position, orientation, and field of view, and (3) the positions of any obstacles (such as CNT’s in-vessel coils) obstructing the camera’s view of portions of the plasma. For the low-β plasmas studied in CNT for this work, any differences from the measured vacuum flux surfaces were neglected. The pixel data tends to be noisy and can exhibit nontrivial correlations. Hence, if one is to reconstruct the emissivity profile based on pixel data, it is best to have many more pixels than

47 layers to reconstruct. An unbiased estimator of the emissivity profile can then be expressed as [61]

−1 e LT C−1L
LT C−1 bp, (3.2) ≈ where C is the covariance matrix for the pixel noise. When the emissivity profile e is calculated in this way, it is informative to plug it back into Eq. 3.1 to synthesize an image p∗ for comparison with the original image.

3.2 Forward problem

As an initial proof-of-concept and a test of the reliability of the calculations of L, synthetic images were generated based on profiles that were specified a priori. L was calculated to correspond to a camera view similar to that of the photograph shown in Fig. 3.2. The test profiles were given three basic functional forms: hollow (linearly increasing from core to edge), uniform, and peaked (linearly decreasing from core to edge). These profiles and the resulting synthetic images are shown in Fig. 3.3. Of the three test profiles, the hollow one (Fig. 3.3b) has the best qualitative agreement with the photograph in Fig. 3.2, suggesting that the plasma in the photo (which is typical of the 1 kW electron cyclotron ≈ heated (ECH) discharges studied in CNT) has greater visible emission at the edge than in the core. This is consistent with a higher rate of recombination reactions occurring in the edge.

3.3 Inversion of experimental images

3.3.1 Image processing

Images used for profile reconstructions were acquired by a high-speed CMOS camera manufactured by Canadian Photonics. The camera was placed outside the vacuum chamber with a view through a fused silica window. The field of view is shown in Fig. 3.4. As Eq. 3.1 does not account for external sources of light, sheets of low-outgassing black foil manufactured by Acktar were fixed to the internal vessel walls to prevent reflection in the camera’s lines-of-sight. To account for pixel noise, a large number of frames (greater than or equal to the total number of pixels to be analyzed) was acquired with no light sources present. The mean value of the noise

48 Figure 3.2: Photograph of a typical argon plasma in CNT heated with 1 kW ECH with CNT’s two in-vessel coils in clear view.

(a) 1 Hollow 0.8 Uniform 0.6 Peaked 0.4 0.2

Emissivity (A.U.) 0 0 2 4 6 8 10 12 14 Effective minor radius (cm) (b) (c) (d)

Figure 3.3: Simulated images of three synthetic profiles. (a) profiles; (b) image of hollow profile; (c) image of uniform profile; (d) image of peaked profile. The upper halves of of (b)-(d) approximate the field of view of the photo in Fig. 3.2.

49 (a) (b) (c)

plasma coil

coil port nozzle

Figure 3.4: (a) Schematic of the key objects in the camera’s field of view. (b) Raw image of a typical glow discharge (Sect. 3.3.2). (c) The same image, after noise subtraction, with boxes indicating the regions where pixel data were used for reconstructions (Sect. 3.3.1).

acquired for each pixel was then subtracted from the corresponding pixel in a plasma image. The result of this subtraction is shown in Fig. 3.4c. The background frames were also used to compute the pixel noise covariance matrix C for the inversion (Eq. 3.2). Optical noise originating from the th chamber was neglected. The propagated error σnoise,j for the j layer in the profile was calculated as the square root of the jth diagonal element of the posterior covariance matrix [61], given by LT C−1L
−1. For each plasma image, ten independent inversions were conducted using disjoint subsets of the pixels (shown as blue boxes in Fig. 3.4b). The subsets used in this work contained between 560 and 860 pixels. To be included in a subset, a pixel’s line-of-sight needed to terminate against black foil. The ten different profiles e obtained from each of the subsets were then averaged to obtain a final reconstructed profile. This was done to partially cancel the effects of small errors in the camera alignment, which would otherwise cause some contributions to some pixels to be misattributed to layers not within their lines-of-sight. The standard deviation among these measurements will be notated as σalign. The total uncertainty of the emission from the jth layer was then computed from the quadrature sum of the alignment error σalign and the noise error σnoise averaged over N pixel subsets:

50 v 2 u 2 N ! uσalign,j X σnoise,nj σ = t + (3.3) j N N n

3.3.2 Reconstructions of glow discharges

Field line and flux surface visualizations are often utilized in CNT for diagnostic alignment [56]. These are discharges maintained by a heated, biased filament. They tend to emit a visible glow that is localized to the flux surface where the filament is located. In the case of rational or near-rational surfaces, the glow is restricted to the flux tube connecting one side of the filament to the other. On non-rational surfaces, however, the entire surface tends to be emissive. While this glow is not perfectly uniform, it is nonetheless expected that an inversion of such a glow would result in an emission profile that is peaked around the layer where the emitter is located. Hence, inversions of glow discharges can serve as tests of the method’s validity. Quiescent glow discharges were filmed at 25 frames per second with 15 ms exposures.2 These were taken at the camera’s lowest gain setting (400, on a scale of 400 to 975) to minimize noise and pixel saturation. Fig. 3.5a shows emissivity profiles from glow discharges from several filament locations at successively larger effective minor radii. As expected, each profile is peaked. Addi- tionally, the peak locations agree well with the locations of the filament, which were measured independently through field line mapping procedures similar to those described in Refs. [19, 44]. Glow discharges on surfaces with larger minor radii tend to have lower calculated emissivity overall; this is consistent with visual inspection of the discharges. Fig. 3.5b-e show raw and reconstructed images corresponding to two of the profiles from Fig. 3.5a. While the agreement is qualitatively good in many respects, there are some features from the photos that are not replicated in the reconstructions. Perhaps the most noticeable are the bright, narrow streaks that are visible in the middle of the plasmas in the center and left of Figs. 3.5b,d. The streak is a nonuniformity on its surface (even on non-rational surfaces, regions with short connection lengths to the filament tend to glow brighter than the rest of the surface). Such a nonuniformity is not expected to carry over in the inversion process, due to the assumption of uniform emissivity within a layer discussed in Sec. 3.1.

2R. Diaz-Pacheco assisted with the video acquisition and image processing in this chapter.

51 (a) 2.5 2.3 cm 3.3 cm 2 4.6 cm 5.7 cm 7.0 cm 1.5

1 Emissivity (A.U.) 0.5

0 0 2 4 6 8 10 12 14 Effective minor radius (cm) (b) (c)

Processed image Reconstruction (3.3 cm) (3.3 cm)

(d) (e)

Processed image Reconstruction (5.7 cm) (5.7 cm)

Figure 3.5: Reconstructed profiles and images of glow discharges. (a) Profiles with the emissive filament located on fiv different flux surfaces with effective minor radii given in the legend and as vertical dashed lines. Solid curves are fits to cubic smoothing splines, which are used for the image reconstructions. (b)-(c) Image and reconstruction for the 3.3 cm filament position in the regions where pixel data were used for reconstructions (Fig. 3.4b). (d)-(e) Image and reconstruction for the 5.7 cm filament position.

52 (a) 1 (b) (c)

0.8

0.6

0.4

Emissivity (A.U.) 0.2

0 0 2 4 6 8 10 12 14 Effective minor radius (cm)

Figure 3.6: Reconstruction of a 1 kW ECH discharge. (a) Emissivity profile with a cubic spline fit. (b) Photographic data used for the reconstruction. (c) Synthesized image based on the recon- structed profile.

3.3.3 Reconstruction of an ECH discharge

1 kW ECH discharges lasting for roughly 7 ms (characterized in more detail in Chapter 4) were filmed at 500 frames per second with 2 ms exposures at the lowest gain setting. Fig. 3.6 shows a profile, image data, and reconstructed image from a typical frame. Note that, except for the outermost layer, the profile follows a roughly linear trend similar to the hollow test profile in Fig. 3.3. In contrast to the image comparisons from Fig. 3.5, in which the image data had fine structure that did not appear in the reconstructions, the reconstruction for the ECH discharge in Fig. 3.6c has fine structure that is not visible in the image data (Fig. 3.6b). Much of the structure in the reconstruction results from the jump in emissivity at the edge as seen in the profile, which would result in bright spots wherever the outermost layer is nearly tangent to lines-of-sight. It is not fully understood why the jump in emissivity appears in the calculated profile despite not appearing experimentally. It is conjectured that it may be a spurious effect resulting from emission outside the closed flux surfaces; i.e., in the scrape-off layer.

53 Chapter 4

First ECH plasmas

During its first years of operation, the (non-neutral) plasmas in CNT were sourced by heated filaments that emitted electrons on or close to the magnetic axis. The electrons would fill the confining volume through transport perpendicular and parallel to the magnetic field. Ions would also be present in the plasma in a quantity dependent primarily on the background neutral pressure [26]. As CNT’s research focus moved from low-energy non-neutral plasmas to higher-energy, fusion- relevant plasmas, a higher-power source was needed. Electron cyclotron heating (ECH) was chosen as the new heating mechanism. ECH is employed universally on large fusion experiments for many purposes including (1) assisting breakdown in tokamaks [70, 71, 72], (2) acting as the primary breakdown source in stellarators, (3) localized heating in tokamaks and stellarators [73], and (4) stabilization of tokamak instabilities [74]. CNT’s ECH abilities are undergoing a sequence of power upgrades, beginning with low power (a 1 kW source) to test the concept and to develop diagnostic capabilities and ending with high power (> 100 kW) sufficient to attain high β. This chapter will describe the first phase of microwave heated plasmas; Chapter 5 will present results from the second-phase ECH source (10 kW); Chapter 6 will present numerical projections for the high-β plasmas expected in the third phase. The present chapter is structured as follows: Sec. 4.1 reviews the physics of ECH and wave propagation through plasmas, both crucial subjects in microwave heating. Sec. 4.2 describes the first-generation heating system on CNT as well as the diagnostic hardware and analysis techniques. Sec. 4.4 reviews a quantification of particle losses through analysis of the decaying phase between

54 heating pulses. Sec. 4.3 presents the dependence of plasma parameters on the species and pressure of the working gas. Sec. 4.5 describes a synergistic effect observed when the low-power ECH system is combined with an electron emitter.

4.1 Microwave plasma heating

4.1.1 Waves in plasmas

In fusion applications, microwaves are typically launched toward the plasma from an external source. Hence, for efficient absorption, they must propagate through the plasma without excessive reflection or refraction. Thus, when designing a microwave heating system, it is crucial to understand the physics of wave propagation in plasmas. In this section, a number of expressions will be quoted governing the propagation of waves through infinite, uniform (although not necessarily isotropic) media. While real plasmas are never perfectly uniform or infinitely extending, many of the results for infinite uniform media still hold to close approximation, and permit the development of simple but informative analytic expressions. For the microwave frequencies of interest in this thesis, ions are typically too massive to respond to the waves but electrons will significantly affect the propagation. Propagation in a cold plasma

(electron temperature Te = 0) with stationary ions with a uniform magnetic field B0 in the ˆz direction has the following dispersion relation [75]:

P (n2 R)(n2 L) tan2 θ = − − − (4.1) (Sn2 RL)(n2 P ) − − In this expression, θ is the angle between the propagation vector k and the background magnetic

field B0 = B0ˆz, n is the index of refraction defined as c/vp = ck/ω, and:

55 ω2 R = 1 pe (4.2a) − ω(ω + ωce) ω2 L = 1 pe (4.2b) − ω(ω ωce) 1 − S = (R + L) (4.2c) 2 ω2 P = 1 pe (4.2d) − ω2

These expressions are given in terms of the electron cyclotron frequency ωce and electron plasma frequency ωpe:

eB0 ωce = me s (4.3) 2 e ne ωpe = me0

Here e is the electronic charge, me is the electron mass, 0 is the vacuum permittivity, and ne is the electron density. In this case of propagation perpendicular to the magnetic field (i.e., θ = 90◦), it is apparent from Eq. 4.1 that at least one of the factors in the denominator must be zero. These two possibilities correspond to two branches of the dispersion relation: the ordinary (O) mode, in which n2 P = 0, − and the extraordinary (X) mode, in which Sn2 RL = 0. − The case of O-mode propagation with n2 P = 0 implies, according to the definition in Eq. 4.2d: −

c2k2 = ω2 ω2 (4.4) − pe It is evident from this equation that the wavenumber k will be imaginary if the frequency is less than the plasma frequency. This corresponds to an exponentially decaying wave mode, which means that the wave will evanesce. Put in another way, if the electron density ne exceeds the O-mode cutoff density,

m  n = e 0 ω2, (4.5) co e2

56 a wave of frequency ω cannot propagate. Physically, this means that if the electron density exceeds the cutoff, the electrons can respond quickly enough to cancel out the wave fields. This phenomenon explains why light reflects off a metallic surface: the free electrons in the metal exceed the cutoff density for visible light, allowing them to shield the interior of the metal from the waves and causing the light to be reflected. Note also that the phase velocity approaches infinity as ω decreases toward

ωpe. In general, singularities in the dispersion relation corresponding to a vanishing refractive index component (thus, vp = ) are designated as cutoffs. ∞ In O-mode, the wave electric field is parallel to B0 and the wave magnetic field is perpendicular to B0. Hence, there is no “competition” between the Lorentz forces exerted on the particle by the wave and the Lorentz force from B0. For this reason, it makes intuitive sense that the O-mode dispersion relation in Eq. 4.4 does not depend on ωce or B0. In the case of X-mode propagation, where Sn2 RL = 0 (Eq. 4.1) the opposite is true: the − wave electric field is perpendicular to B0 and the wave magnetic field is parallel to B0. Hence, the electrons in the plasma experience counteracting pushes and pulls from the wave and the background field. The dispersion relation correspondingly becomes a function of B0:

c2k2 (ω2 ω2 )(ω2 ω2 ) = − R − L , (4.6) ω2 ω2(ω2 ω2 ) − UH with ωR, ωL, and ωUH defined as

1  q  ω = ω + ω2 + 4ω2 (4.7a) R 2 c ce pe q 2 2 ωUH = ωpe + ωce (4.7b) q 1  2 2  ωL = ωc + ω + 4ω (4.7c) 2 − ce pe

From this dispersion relation, it is evident that there are two cutoffs: the right-handed cutoff, where

ω = ωR, and the left-handed cutoff, where ω = ωL. In addition, the phase velocity is zero when

ω = ωUH; this corresponds to the upper hybrid resonance (UHR). Note that, from Eqs. 4.7a-4.7c,

ωL ωUH ωR. ≤ ≤ From Eq. 4.6, it can also be seen that there are two frequency bands in which propagation is permissible (i.e., k is real): ω > ωR and ωL < ω < ωUH. A wave in the former (latter) band is some-

57 times said to be in the fast (slow) X mode due to their respective phase velocities. Correspondingly,

ωR and ωL are alternatively known as the fast-X and slow-X cutoffs. One additional property of the slow X mode is that the wave becomes longitudinal (hence, electrostatic) as ω ωUH. → Although the electron cyclotron resonance (Sec. 4.1.2) and the UHR are both called “reso- nances,” they refer to different phenomena. The former inherently involves the exchange of energy between waves and particles. The latter is purely a wave phenomenon, which does not necessarily involve energy exchange. In practice, however, waves are often strongly damped at resonances in plasmas with finite collisionality. The effect of collisionality on a plasma wave is similar to the effect of frictional drag on a pendulum: collisions extract a certain amount of energy from the wave per oscillation. Hence, damping is much stronger near the UHR than elsewhere in a plasma because the slow phase velocity cause the wave to undergo more oscillations per unit distance. In plasma heating applications, in which the microwave frequency is typically fixed, it can be more helpful to think of the cutoffs and resonances in terms of densities. Analogous to the derivation of the O-mode cutoff density in Eq. 4.5, Eqs. 4.7 may be solved for density as follows:

m0 2
nR = ω ωωce (4.8a) e2 − m0 2 2
nUH = ω ω (4.8b) e2 − ce m n = 0 ω2 + ωω
(4.8c) L e2 ce

Unlike the O-mode cutoff (Eq. 4.5), these densities vary with the magnetic field through ωce. Formulated this way, these equations imply that a heating frequency ω cannot propagate in regions of the plasma where n > nL or nUH > n > nR. Note that R cutoffs and UHRs will never appear in

“high-field” regions where ω < ωce. Fig. 4.1 shows the typical topology of cutoffs and resonances in a fusion plasma. Each shaded region is inaccessible to propagation. The borders of the shaded regions are contours of nco, nR, nUHR, and nL. The layout of these regions determines which heating methods are available. Note that in the case of X-mode heating at the first harmonic (Fig. 4.1b), it is impossible to reach the resonance from the low-field side. In devices with especially high β or especially low B (or both), the cyclotron resonance may be | |

58 Phys. Plasmas, Vol. 11, No. 5, May 2004 Heating and current drive by electron cyclotron waves 2351

͑see Sec. II G for exceptions͒. Summarizing the treatment presented by Stix31 here for completeness, in this limit the electrons and ions are zero-temperature frictionless charged fluids in a constant magnetic field. The dispersion relation can be written in the form 2 2 2 ␻ p ⍀e P͑n ϪR͒͑n ϪL͒ D n, , ,␪ ϭtan2 ␪ϩ ϭ0, ͩ ␻2 ␻ ͪ ͑Sn2ϪRL͒͑n2ϪP͒ ͑1͒ where nϭc͉kជ͉/␻ is the index of refraction, ␻ is the fre- 2 1/2 quency of the wave, ␻ pϭ(nee /␧0me) is the electron plasma frequency, ␪ is the angle between the wave vector kជ and the magnetic field Bជ , and in the limit that the ion mass is much greater than the electron mass, 2 Pϭ1Ϫ͑␻ p /␻͒ ,

Rϭ͑PϪ⍀e /␻͒/͑1Ϫ⍀e /␻͒, ͑2͒

Lϭ͑Pϩ⍀e /␻͒/͑1ϩ⍀e /␻͒, and Sϭ(RϩL)/2 and Dϭ(RϪL)/2. From Eq. ͑1͒ for ␪ ϭ␲/2 ͑perpendicular propagation͒ the solutions are RL n2ϭP and n2ϭ , ͑3͒ S which transform smoothly as ␪ 0 ͑parallel propagation͒ into → n2ϭL and n2ϭR. ͑4͒ These two solutions to the dispersion relation form the nor- mal modes of wave propagation in the plasma. The mode associated with n2ϭP for perpendicular propagation and 2 Phys.with Plasmas,n Vol.ϭ 11,L No.for 5, May parallel 2004 propagation is Heating called and current the drive ordinary by electron cyclotron waves 2351 mode ͑O-mode͒ and the other mode is the extraordinary ͑see Sec. II G for exceptions͒. Summarizing the treatment presentedmode by͑ StixX-mode31 here for͒ completeness,. in this limit the electrons and ions are zero-temperature frictionless charged fluids inThe a constant propagation magnetic field. The limits dispersion of relation these modes are given by the 2 2 cancutoffs be written in (n theϭ form0) and resonances (n ϱ). At the physical 2 2 2 → ␻ p ⍀e P͑n ϪR͒͑n ϪL͒ locationD n, , of,␪ aϭtan cutoff2 ␪ϩ the waveϭ is0, reflected. The cutoffs can be ͩ ␻2 ␻ ͪ ͑Sn2ϪRL͒͑n2ϪP͒ easily seen by writing the dispersion͑1͒ relation in the alternate 31 whereformnϭc͉kជ͉/␻ is the index of refraction, ␻ is the fre- 2 1/2 quency of the wave, ␻ pϭ(nee /␧0me) is the electron plasma frequency,An4ϩ␪Bnis the2 angleϩPRL betweenϭ the0, wave vector kជ ͑5͒ ជ and the magnetic field B, and in the limit that the ion mass is ω = ωce ω = ωce ω = 2ωce muchwhere greater thanA theand electronB mass,are algebraic functions of P,R,L, and ␪. ϭ Ϫ 2 FromP 1 ͑␻ Eq.p /␻͒ ,͑5͒, cutoff takes place where the product PRL Rϭ͑PϪ⍀ /␻͒/͑1Ϫ⍀ /␻͒, ͑2͒ equals zero.e Thesee are shown in Fig. 1͑a͒ for the O-mode in Lϭ͑Pϩ⍀e /␻͒/͑1ϩ⍀e /␻͒, 31 andtheSϭ( customaryRϩL)/2 and Dϭ Clemmow–Mullaly–Allis(RϪL)/2. From Eq. ͑1͒ for ␪ ͑CMA͒ diagram, ϭ␲/2 ͑perpendicular propagation͒ the solutions2 are 2 in which the abscissa is ␻ p/␻ ͑proportional to electron den- RL 2 2 2 sityn2ϭPn and) andn2ϭ the, ordinate is ⍀ /␻͑3͒ ͑proportional to B ). For e S e whichthe transform O-mode smoothly the as cutoff␪ 0 ͑parallel is P propagationϭ0, which͒ is dependent only on into 2 →2 the density, ␻ pϭ␻ . For ␻ϭᐉ⍀e , where ᐉ is the harmonic n2ϭL and n2ϭR. ͑4͒ 20 Ϫ3 number, this cutoff density nco is given by nco(10 m ) st nd These two2 solutions2 to the dispersion relation form the nor- (a) O-mode (b) 1 harmonic X-mode (c) 2 harmonic X-mode malϭ modesᐉ B ofT/10.3, wave propagation where in theB plasma.T is The the mode magnetic field in units of associated with n2ϭP for perpendicular propagation and 2 2 2 ␻ ␻ withTesla.n ϭL for Propagation parallel propagation from is called the the ordinary outside of the plasma ( p/ FIG. 1. ͑a͒ CMA diagram for the O-mode. ͑b͒ Typical ray trajectory for the modeϭ0)͑O-mode to͒ theand the cyclotron other mode isresonance the extraordinary or its harmonicsFigure 4.1: Topologyis possible of cutoffs and resonances for O- and X-mode heating in a toroidal plasma. mode ͑X-mode͒. O-mode in a schematic tokamak geometry in which the chain-dashed line fromThe propagation either limits the of low-fieldthese modes are sidegiven by or the the high-fieldEach circle represents side of the a poloidalrepresents cross-section the axis in which of symmetryB decreases and the from magnetic left to right field and fallsn inverselydecreases with cutoffs (n2ϭ0) and resonances (n2 ϱ). At the physical major radius. ͑c͒ CMA diagram for the X-mode showing the right-hand and locationplasma of a cutoff to the the wave cyclotron is reflected.→ The resonance cutoffs can be providedfrom center the region to edge. with The shaded regions in each diagram do not admit propagation for the respective easily2 seen by2 writing the dispersion relation in the alternate left-hand cutoffs and the upper hybrid resonance for ␪ϭ60 deg ͑upper ␻31 у␻ is avoided. This is illustrated for schematic tokamak form p mode. (a) O-mode heating atchain-dashed the first harmonic line͒ and (ω 90= degωce)͑lower Provided chain-dashed the shaded line region͒. ͑d͒ Typical (where ray geometryAn4ϩBn2ϩPRL inϭ0, Fig. 1͑b͒. In practice,͑5͒ wave refraction becomes trajectories for the fundamental and ͑e͒ for the second harmonic. In all cases n > nco) does not overlap the cyclotron resonance (ω = ωce), a beam launched from the right (the whereimportantA and B are for algebraic waves functions with of P,␪R,*L,␲ and/2␪. as the density approaches the dashed arrows represent possible ray trajectories. From Eq. ͑5͒, cutoff takes place where the product PRL dashed line) will propagate, unobstructed, toward the cyclotron resonance. (b) X-mode heating at This articleequals zero.is copyrighted These are shown as in Fig.indicated 1͑a͒ for the in O-mode the abstract. in Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: the customary Clemmow–Mullaly–Allis ͑CMA͒ diagram,31 2 2 the128.59.144.175 first harmonic. On: In this Mon, case, 02 a Dec beam 2013 launched 19:26:52 from the right will be reflected off of the RH (fast in which the abscissa is ␻ p/␻ ͑proportional to electron den- 2 2 2 sity ne) and the ordinate is ⍀e /␻ ͑proportional to B ). For X) cutoff, never reaching the resonance. A beam launched from the left, however, will reach the the O-mode the cutoff is Pϭ0, which is dependent only on 2 2 the density, ␻ pϭ␻ . For ␻ϭᐉ⍀e , where ᐉ is the harmonic 20 Ϫ3 resonance, provided it avoids the LH (slow X) cutoff. (c) X-mode heating at the second harmonic. number, this cutoff density nco is given by nco(10 m ) 2 2 ϭᐉ BT/10.3, where BT is the magnetic field in units of ␻2 ␻2 In this case, launch from the right hand side is unobstructed by the fast X cutoff. Adapted from Tesla. Propagation from the outside of the plasma ( p/ FIG. 1. ͑a͒ CMA diagram for the O-mode. ͑b͒ Typical ray trajectory for the ϭ0) to the cyclotron resonance or its harmonics is possible O-mode in a schematic tokamak geometry in which the chain-dashed line from either the low-field side or the high-field side of the representsRef. the axis [73]: of symmetry R. Prater, and the magneticPhys. fieldPlasmas falls inversely with11, 2349 (2004), with the permission of AIP Publishing. plasma to the cyclotron resonance provided the region with major radius. ͑c͒ CMA diagram for the X-mode showing the right-hand and 2 2 left-hand cutoffs and the upper hybrid resonance for ␪ϭ60 deg ͑upper ␻ pу␻ is avoided. This is illustrated for schematic tokamak chain-dashed line͒ and 90 deg ͑lower chain-dashed line͒. ͑d͒ Typical ray geometry in Fig. 1͑b͒. In practice, wave refraction becomes trajectories for the fundamental and ͑e͒ for the second harmonic. In all cases important for waves with ␪*␲/2 as the density approaches the dashed arrows represent possible ray trajectories. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.144.175 On: Mon, 02 Dec 2013 19:26:52

59 completely inaccessible to O- and X-modes. In this case, microwave heating may still be possible via the Bernstein mode, a third propagation mode that exists in finite-temperature plasmas and has no upper density limit. This will be discussed in more detail in Chapter 5.

4.1.2 Cyclotron damping

The microwave heating systems implemented in this thesis were based on the concept of cyclotron damping. Cyclotron damping occurs when charged particles—electrons, in our case—in a magnetic

field are exposed to fields oscillating at integer multiples of the particles’ cyclotron frequency, ωc. For waves propagating perpendicular to the magnetic field, the resonance condition for the wave frequency ω is:

`qB ω = , (4.9) mγ where ` is an integer (the harmonic number), B is the magnetic field strength, q and m are the particles’ charge and mass, respectively, and γ is the relativistic factor given by 1/p1 v2/c2, − where v is the particle’s speed. For the cold, non-relativistic plasmas studied in this thesis, γ = 1 and will be dropped for all analysis. (It becomes significant, however, in fusion plasmas with electron temperatures Te 1 keV or more.) ≈ The mechanism of cyclotron damping may be understood on a single-particle level by considering the equations of motion of an electron in a static magnetic field B0 in the presence of an oscillating electric field. The equations, in turn, arise from the Lorentz force law:

F = m ˙v = qE + qv B (4.10) ×

Assume B0 = B0ˆz and that the wave field is propagating in the x direction with wavenumber k and frequency ω: E = E0 exp(ikx iωt). Neglecting the wave magnetic field and using ωce eB0/me, − ≡ Eq. 4.10 may be written as:

60 eEx ikx−iωt v˙x + ωcevy = e (4.11a) − me eEy ikx−iωt v˙y ωcevx = e (4.11b) − − me eEz ikx−iωt v˙z = e (4.11c) − me

Following the approach of Prater (Ref. [73]), one may change the velocity variables vx and vy to a rotating frame with a right-handed velocity v− vx ivy and a left-handed velocity v+ vx +ivy. ≡ − ≡ Additionally, define left- and right-handed components of the wave field: E± = Ex iEy. Since ± the natural perpendicular motion of electrons in the magnetic field is right-handed, we restrict our attention to the right-handed and parallel (z) equations of motion:

eE− ikx−iωt v˙− + iωcev− = e (4.12a) − me eEk ikx−iωt v˙k = e (4.12b) − me

If the wave field E is assumed to be small enough so as not to perturb the electron’s orbit significantly during a single gyro-period, the spatial coordinate x in the expression for E may be approximated as the x motion of the electron in the absence of a wave; i.e., x = rLe sin(ωcet), where rLe is the electron Larmor radius:

eE− v˙− + iωcev− = exp [ikrLe sin(ωcet) iωt] (4.13a) − me − eEk v˙k = exp [ikrLe sin(ωcet) iωt] (4.13b) − me −

Utilizing a Bessel function identity, these equations may be rewritten as infinite series:

61 ∞ eE− X v˙− + iωcev− = Jj(krLe) exp[ijωcet iωt] (4.14a) − m − e j=−∞ ∞ eEk X v˙ = Jj(krLe) exp[ijωcet iωt] (4.14b) k − m − e j=−∞

The equations are now in a form with an intuitive interpretation. Eq. 4.14a is that of a simple harmonic oscillator driven by an infinite series of sinusoidally oscillating forces. The oscillator will absorb the most energy from resonant force, with frequency ωce. This means that if the wave frequency ω is set to the `th harmonic of the cyclotron frequency, the resonant term in the series will be j = ` 1. Similarly, the resonant term in Eq. 4.14b is j = `,(i.e., the driving force with no − oscillations). Multiplying the amplitude of the resonant force term by the velocity will give a rough estimate of the power transferred:

P− ev−E−J`−1 (krLe) (4.15a) ≈ −

P ev E J` (krLe) (4.15b) k ≈ − k k

Note that Eq. 4.15a corresponds to the X-mode because the electric field has a right-hand polarized component. Correspondingly, Eq. 4.15b corresponds to the O-mode because the electric

field is parallel to B0. In this single-particle treatment, whether the electron absorbs or loses power depends on its phase relative to the driving field. In typical heating applications, the phases of the particles are randomized due to collisions and due to entering and leaving the resonant area at irregular intervals. Due to the proportionality of power absorption to particle velocity, an accelerating particle with an initial velocity v0 gains more energy per unit time than what a decelerating particles with the same v0 loses [7]. Hence, the resonance results in a net transfer of energy from the waves to the plasma. Eq. 4.15 illustrates how the absorption mechanism differs for different wave modes. Consider

first the case of a cold plasma, in which rLe = 0. In this case, absorption is only possible in the

62 modes ` corresponding to the J0 (the only Bessel function that is finite with zero argument). In the O-mode case (Eq. 4.15b), absorption is only possible at the “zeroth” harmonic, ` = 0 (i.e., electrons will be accelerated by a dc electric field parallel to B0). In the X-mode case, absorption at the first harmonic, ` = 1, is possible. possible. The efficiency of first-harmonic X-mode at arbitrarily low temperatures makes it an attractive option for initiating a plasma discharge. In cases of finite rLe, absorption becomes possible in the first-harmonic O-mode and the second-harmonic X-mode. As a closing remark, it should be noted that the heuristic derivation in this section only consid- ered the case of wave propagation perpendicular to the magnetic field (in the x direction). However, in general cyclotron damping may occur at oblique propagation angles. One additional consider- ation for oblique propagation is that parallel-streaming electrons will see a Doppler shift due to the parallel component of the wave’s motion. The resonance condition (Eq. 4.9) generalized for Doppler shifting is [76]

`eB ω = + kkvk, (4.16) γme where kk is the parallel component of the wave vector and vk is the parallel component of the electron’s velocity. Hence, damping will not occur precisely where ω = `ωce. In a thermal plasma, this causes the absorption layer to broaden around the nominal cyclotron resonance according to 1 the distribution of vk.

4.2 Heating and diagnostics

4.2.1 Low-power magnetron

The first microwave heating source employed in CNT was a low-power, 2.45 GHz magnetron. The magnetron was extracted from a standard microwave oven and mounted on the vessel so that its antenna was positioned close to a viewing window with a 15 cm diameter.2 The setup was ≈ enclosed in a box lined internally with a microwave-absorbing material to reduce external leakage for reasons of safety and to avoid electromagnetic (EM) interference with diagnostics and with the

1 As seen from the γ term in Eq. 4.16, broadening may also occur even when kk = 0 [76]; however, this effect is negligible for the cold plasmas studied here. 2The extraction and mounting was done by S. Massidda.

63 data acquisition system and, thus, ultimately, reduce noise. Magnetrons are steady-state microwave sources driven internally by an electron-emitting fil- ament with a DC bias. In practice, the bias of the filament (and, hence, the microwave power output) is not always constant. The output of the magnetron described here is pulsed at 60 Hz, the frequency of the power mains. Each cycle produces a microwave pulse lasting roughly 6.5 ms. The total microwave output from the magnetron is on the order of 1 kW. However, only a fraction of this power couples with the plasma. Due to the rudimentary setup, a significant portion of power does not pass through the window and is instead deposited into the absorbing box, as confirmed by “hot spots” on the box. In addition, part of the microwaves which effectively entered the vessel made multiple reflections on its inner resistive walls and were absorbed by them, or leaked out of the vessel through a few unshielded ports. While the amount of power coupled to the plasma was not measured, based on comparing the densities obtained with this magnetron to those obtained with the 10 kW magnetron (Chapter 5), the coupled power is believed to be well under 500 W.

4.2.2 Diagnostics

The Langmuir probe has been the primary instrument for diagnosing ECH plasmas in CNT. The plasma-facing components are essentially the same as used to diagnose non-neutral plasmas in CNT [22]. The probe tips are made from halogen light bulbs in which the glass has been removed to expose the tungsten filament. The bulbs are mounted in boron nitride sockets embedded in an alumina rod that extends from the chamber wall into the plasma. The alumina rod can be moved longitudinally by means of an edge-welded bellows drive. An recently installed electronic moving slide controls the position with sub-millimeter precision.3 A formed bellows permits to tilt the whole setup both vertically and horizontally, by a limited range of angles. In this way the Langmuir probe can be scanned along different chords, aiming more toward the plasma core or edge. The flux coordinate of a probe can be determined using the field line mapping technique de- scribed in Sec. 2.1.1. Instead of an electron gun, however, the probe itself is used as the emitter. The filament is heated with a 6 V battery and biased to 80-100 V to emit electrons with the ≈ 3Y. Wei assisted with the installation.

64 chamber pumped down to ultrahigh vacuum. The flux-surface cross-sections recorded for the fila- ment are then compared to cross-sections from the numerical model, from which the flux coordinate (and flux labels, such as the effective minor radius) can be determined. The work in this Chapter was conducted prior to the error-field study in Chapter 2, so the flux coordinates presented here are subject to systematic errors of 10-20%. The bias of the probe is set by a bipolar power supply whose output is controlled by waveforms programmed by the user. Current is measured by a series resistor, and bias is measured with a voltage divider. Microwave leakage outside the vacuum vessel is detected by a radio-frequency (RF) “sniffer” probe. This consists of an antenna connected to a diode mounted near the magnetron. The antenna receives microwaves and delivers them to the input of the diode, which rectifies the 2.45 GHz signal and outputs a DC voltage. The voltage output is roughly proportional to the square root of the power received by the antenna. While the recorded leakage power cannot be used as an absolute measure of power coupled to the plasma, it serves as a precise indicator of when the microwave pulses are occurring and is hence utilized to synchronize the probe signals.

4.2.3 Langmuir probes

Langmuir probes are common diagnostics employed in cold, tenuous plasmas. The probe is essen- tially a small metal tip held on an insulating rod which is inserted into the plasma. The relationship between the electrostatic potential of the tip and the current flowing through it (into the plasma) can then be used to reveal properties of the plasma. One common Langmuir probe method involves using an external power supply to sweep the bias of the tip relative to the wall of the vacuum vessel. Assuming the probe does not significantly perturb the plasma, the current flowing through the probe will depend only on the plasma properties and on the shape and size of the probe. A typical relationship between the probe bias and current for quasi-neutral plasma consisting of electrons and positive ions is shown in Fig. 4.2 and can be explained as follows. When the probe potential is higher than the local plasma potential, Vplasma, the probe attracts the electrons and repels the ions. Electrons will tend to concentrate around the probe, forming a sheath that has the effect of shielding the surrounding plasma from the potential of the probe. The electron

65 Figure 4.2: Typical form of a Langmuir I-V curve for a cold, quasineutral plasma. The plasma potential, Vplasma, occurs at the sharp corner in the black curve at V 5 V. The red dashed curve ≈ illustrates the softening that often occurs in the region near the plasma potential. Reproduced from Ref. [77]: R. L. Merlino, Am. J. Phys. 75, 1078 (2007), with the permission of the American Association of Physics Teachers.

66 current is determined by how many electrons enter the sheath. In principle this current should be independent of the bias of the probe; however, due to the finite temperature of the electrons, the sheath will expand as Vbias increases. The electron flux through the sheath edge will increase with its surface area, resulting in an increasing trend in I(V ).

When the probe potential is equal to Vplasma, the plasma particles experience no force from the probe. It follows that, as V is reduced to Vplasma, the sheath will shrink and ultimately vanish. The electron current to the probe is thus determined by how many electrons strike the probe surface.

This current, defined as the electron saturation current, Ies, is related to the probe area A and electron thermal velocity ve,th as follows [77]:

1 I = en v A (4.17) es 4 e e,th

When the probe potential is less than Vplasma, electrons are repelled by the probe. As a result, the electron concentration around the probe will decrease, which in turn causes a shielding effect analogous to what occurs when V > Vplasma. Due to the repulsive force, however, not all electrons incident on the sheath edge will reach the probe surface. Instead, the probe will only absorb electrons whose kinetic energy exceeds e(V Vplasma). If the electrons are in thermal equilibrium − − (i.e., their energies have a Maxwellian distribution), the electron flux to the probe will decay exponentially as V decreases from Vplasma. From this decay rate, one can determine the electron temperature, Te.

The probe current I also takes a small contribution from ions, notably in the region V < Vplasma where ions are attracted to the probe. The ion flux is typically much smaller than the electron flux due to the greater mass and lower thermal velocities of the ions. The ion contribution to I(V ) can be seen in Fig. 4.2 for V < 20 V, where the electron contribution has decayed to zero. The − current is seen to decrease slowly as V becomes more negative, indicative of a sheath expansion effect analogous to what was described for V > Vplasma.

4.2.4 Probe signal analysis in CNT

The plasma potential, density, and temperature are measured by obtaining a curve of current versus probe-potential relative to the chamber (an I(V ) curve). In steady-state plasmas, it is generally sufficient to sweep the voltage linearly to obtain an I(V ) curve. The plasmas considered in this

67 oeta,bti sasmdhr htteiacrc nteincreti ninfiatrltv to plasma relative insignificant the is to current rises ion the bias in probe inaccuracy the the that as here accurate assumed is This less it 77]. becomes but [78, current potential, Refs. ion in described the the procedures of of to probe approximation portion similarly the ion-saturation characteristic, to the entire contribution to the electron from fitted the subtracted is isolate curve To linear 4.4a). a (Fig. then, Volts current, 100 to 50 by potential plasma of set a shows 4.4 Fig. bin. bin. voltage per point data stairstep one several containing After An 4.3c). the pulse. (Fig. a seconds, within period of time pulse elapsed period the a as over long obtained as are lasts sweeps step each which in 4.3a) (Fig. supplies. power scales the time of on capabilities vary voltage-sweeping to the parameters by plasma resolved the be causes cannot which that Hz, 60 at pulsed are however, Chapter, to used is and vessel the vacuum occurring. in the are shown outside pulses leakage microwave placed RF the antenna The when “sniffer” determine shot. the plasma by ECH collected typical is a plot during bottom collected data Raw 4.3: Figure h o auaincreti bevdt rwlnal stepoeba erae rmthe from decreases bias probe the as linearly grow to observed is current saturation ion The waveform voltage stairstep a output to programmed instead is supply power the reason, this For

-15 -10 dBm mA V 10 14 18 22 -5 0 5 0 1 2 3 I 1 03 05 07 80 70 60 50 40 30 20 10 0 Typical Langmuirprobedata ( V RF leakage Probe current Probe bias uv ste osrce o aheasdtm i,ec curve each bin, elapsed-time each for constructed then is curve ) Time (ms) 68 I ( V aaaebne codn ovlaeand voltage to according binned are data ) I ( V aafroeelapsed-time one for data ) I ( V hrceitcand characteristic ) 1 10-2 (a)Total current (b) Vplasma Electron current 0.8 Fit Vfloat 10-3 0.6

0.4 10-4 0.2 Current (mA) -5 0 Electron current (A) 10

-0.2 10-6 -100 -50 0 -40 -20 0 20 Probe bias (V) Probe bias (V)

Figure 4.4: Sample Langmuir probe data corresponding to 3.27 ms after the start of a microwave pulse (using the time origin defined Fig. 4.5), before and after processing. Each point plotted is the average of four measurements of current at the same probe bias. (a) Total probe current, linear fit to the ion saturation region, and the remaining electron current after subtracting the fit curve. (b) Electron current replotted on a logarithmic scale. Colored fit curves represent exponential fits to the hot electron population (red), bulk electron population (blue), and electron saturation region

(green). The dashed vertical line indicates Vfloat. The intersection between the blue and green curves is taken to be Vplasma.

the electron current. Fig. 4.4b shows the electron current on a logarithmic scale. In a tenuous Maxwellian plasma, as discussed in Sec. 4.2.3, electron current exhibits exponential growth for V < Vplasma and would thus appear linear on a logarithmic scale. In CNT’s microwave-heated plasmas, however, this portion of the curve typically contains two linear sections with different slopes, suggesting the presence of two populations of electrons with different temperatures. The second, warmer electron population is often observed in ECH discharges [78, 79, 80, 81]. To obtain traces of plasma parameters as a function of time elapsed during and after a microwave pulse, algorithms were developed to identify regions of interest within each I(V ) curve. Regions of interest included the two linear portions for Vprobe < Vplasma and the “knee” occurring at the

69 uaini niae yteaeaeR ekg esrmn ntebto plot. bottom pulse the The in measurement pulse. leakage microwave RF a average the after by and indicated during is measured duration parameters plasma Typical 4.5: Figure

dBm 1015 m-3 Ensemble-averaged pulsecharacteristics -10 2.5 7.5 eV V 10 15 20 10 15 20 -5 0 5 0 5 0 5 0 5 01 14 12 10 8 6 4 2 0 Ela p sed time 70 RF leakagedetected Plasma potential ( Electron density Hot elec.temp. Bulk elec.temp. ms ) plasma potential. First, the data were smoothed by means of cubic splines to the extent that the first and second derivatives were not dominated by noise. The characteristic portions of the curve were then identified as follows:

The floating potential Vfloat was identified as the zero in total current. (Note: all further • analysis will concern a smoothed electron current with the ion current subtracted out.)

The approximate plasma potential Vplasma was identified as the value where the second deriva- • 2 2 tive d I/dV reached its minimum to the right of Vfloat (the “knee” in the curve).

A noise floor is defined based on the sensitivity of the analog-to-digital converter that digitizes • the current signal. This noise-floor potential, Vnoise is recorded.

The region between Vnoise and the approximate Vplasma is expected to vary exponentially, and • is inspected for the presence of a warm electron population. If that exists, a peak in second derivative is expected to exist where the slow growth-rate of the warm population meets the rapid growth-rate of the bulk population. If such a peak exists, an exponential fit is made to

the raw data in the region between the Vfloat and the said peak. A second exponential fit is

made between the peak and Vplasma to determine the bulk temperature. In the absence of a

peak, the full region between Vfloat and Vplasma is fitted as the bulk population.

The electron saturation current Ie,sat and a more precise Vplasma are identified. For this, the • fit for the bulk electron population is intersected with an additional exponential fit of the

curve on the right of the approximate Vplasma. The latter portion of the curve is not actually expected to vary exponentially, but can be treated as such for the purposes of determining

Vplasma and Ie,sat.

4.3 Electron temperature and density profiles as functions of back- ground pressure

The typical plasma evolution as determined from a set of I(V ) curves is shown in Fig. 4.5. Vplasma typically rises quickly to 20 V during the microwave pulse and drops to 5 V at the end. The ≈ ≈ bulk temperature remains at approximately 5 V throughout the discharge. Such low temperatures

71 i.46 rfie bandi h o-oe xeiet edt hwflttmeaueprofiles, temperature flat show to tend experiments low-power the in obtained Profiles 4.6. Fig. in resulting the during energy than confined in better error probe. are of particles source that evident principal is The it afterglow. pulse, the of end the at ately SNR. low a microwave the to when are due temperature sources similarly cold errors is the Both off in is scatter current. The pulse probe systematic. total not the statistical, from be from to current ratio expected ion signal-to-noise the lower subtract the to (1) the used to of model due portion is “warm-population” temperatures the warm N in the for in (SNR) is eV scatter (16 neutrals The potential of [83]. ionization ionization data) the electron-impact this above because electrons ionized for fully sink not energy are an that plasmas of typical are gas nitrogen in ratio current 1 3.00 of the pressure at neutral obtained background density a and at temperature of Profiles 4.6: Figure oesol.Cmaigti otefco-ftodo nbl temperature bulk in drop factor-of-two the to this Comparing slowly. more h lcrndensity electron The h esrmnsdsrbds a eerpae o ieetlctoso h agurprobe, Langmuir the of locations different for repeated were far so described measurements The T e and n e rfie.A xml fapol bandfor obtained profile a of example An profiles. n

e 16 -3

nrae ail ttebgnigo h irwv us u decays but pulse microwave the of beginning the at rapidly increases Electron density (10 m ) Electron temperature (eV) 0.2 0.4 0.6 0.8 10 15 20 25 0 5 0 1 ...... 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Hot population Bulk population . 0 × 10 n − e 5 esrmnsi h netit nteae fthe of area the in uncertainty the is measurements Torr. 72 I ( ρ V uv n 2 h mrcso ftelinear the of imprecision the (2) and curve ) I IL 2 /I 8] h okn a for gas working the [82], F P T e 3 = curn immedi- occurring . 0i hw in shown is 00 1.2 12 1.2 12 (a) (b) ) )

-3 1 10 -3 1 10 m m 16 16 0.8 8 0.8 8

0.6 6 0.6 6

0.4 4 0.4 4 Electron temperature (eV) Electron temperature (eV) Electron density (10 Electron density (10

0.2 2 0.2 2 5 10 15 20 25 30 35 40 45 0 50 100 150 200 250 Pressure (µtorr) Pressure (µtorr)

Figure 4.7: Electron density (blue squares) and temperature (red circles) measured at an interme- diate minor radius and coherently averaged over the middle 3 ms of each microwave pulse. (a) N2; (b) He

indicative of a broad region of microwave energy deposition throughout the plasma. This is sup- ported by the facts that (1) the second harmonic of the cyclotron resonance passes through nearly every flux surface, (2) the density is well below the cutoff for 2.45 GHz waves (7.5 1016 m−3), and × (3) the unfocused microwave beam is very broad (much broader than the plasma) and impinges on the plasma from different directions after multiple reflections in the vessel. These circumstances imply that the waves can easily access the resonance at all locations in the plasma.

Density and temperature varied significantly with the working gas utilized (N2 or He here) and its pressure, as shown in Fig. 4.7. Measurements in this figure were performed at an intermediate minor radius using IIL/IPF = 3.50. For both gases, the lowest pressure plotted corresponds to the lowest pressure (to within 2-5 µtorr) at which breakdown could be attained reliably with ECH.

Te was in general higher for He than for N2. This is consistent with the higher ionization energy of helium (25 eV [84]), which would permit higher-energy electrons to thermalize. The reduction in temperature with neutral gas is also expected, as the mean free path for ionization and other inelastic electron-neutral conditions decreases.

73 (a) (b) (c) 7 cm Data 6 6 16 Fit

) 10 ) -1 -3 s 3

4 4 m m 16 (ms) -13 15 τ 10 (10 e (10 2 2 n α

0 0 1014 0 10 20 30 40 0 10 20 30 40 0 5 10 15 Probe displacement (cm) Probe displacement (cm) t (ms)

−5 Figure 4.8: Afterglow parameters for N2 with pressure 1.0 10 torr for IIL/IPF = 3.00. (a) × Recombination coefficient α at different probe positions. (b) Decay time τ at different probe positions; (c) Example comparison of a fit curve with the form of nag (Eq. 4.19) to data for probe position 7 cm.

4.4 Afterglow and decay time measurements

As seen in Fig. 4.5, the electron density drops quickly at the end of the microwave pulse but persists well into the gap between pulses (the afterglow), suggestive of particle confinement on the order of milliseconds. To quantify the decay time, the afterglow density data were fit to a simple model for density decay [85, 86]:

dn n = αn2 (4.18) dt − − τ The terms on the right-hand side constitute two electron loss mechanisms: recombination, propor- tional to both the ion and electron densities (both assumed to equal n in a quasineutral plasma), and diffusive decay proportional to only the electron density. Thus, α is the recombination coeffi- cient and τ is the exponential decay time. The differential equation may be solved through direct integration:

n e−t/τ n (t) = 0 (4.19) ag −t/τ
1 + ατn0 1 e − −5 Afterglow parameters for an N2 discharge at pressure 1.0 10 torr for IIL/IPF = 3.00 are × shown in Fig. 4.8. Each data point in Fig. 4.8a-b represents a fit to the ensemble-averaged density in the afterglow phase for its respective probe position. An example of a fit, for probe position

74 7 cm, is shown along with the data in Fig. 4.8c. The fits were qualitatively good, and 89% had reduced χ2 less than 1.5. Assuming the rate coefficients remain the same during the pulse-on phase and there are no additional significant loss mechanisms, they may be used to estimate the number of ionizations required to keep the density at a steady state (i.e., dn/dt = 0 in Eq. 4.18). Using the pulse flat-top density of 8 1015 m−3, α = 1.25 10−13 m3 s−1, and τ = 3 ms, one finds a total loss × × rate of 1.1 1019 m−3 s−1, dominated by the recombination term. To match this loss rate would × require about 1400 ionizations per electron per second on average. However, it should be noted that for Te = 5 eV, the majority of electrons do not have the minimum energy required for impact ionization; therefore, the ionization frequency for the energetic electrons would be much higher.

4.5 Synergy between ECH and thermoelectrons

As mentioned at the beginning of this chapter, previous studies of non-neutral and quasi-neutral plasmas in CNT used heated, biased filaments as the sole plasma and energy source. The filament was typically located near the magnetic axis and biased to 150 V to 200 V relative to the − − chamber. The emitted electrons would fill the magnetic surfaces, and a finite ion content would arise depending on the background neutral density. At the base pressures of CNT ( 10−9 Torr), ≈ the ion density was negligible; at base pressures of 10−6 to 10−5 Torr, the ion density would be sufficient to render the plasma quasi-neutral [26]. An experiment was carried out to determine the effect of the emitter bias on the density profiles of plasmas initiated by the emitter alone, with the newly available microwave startup, and by a combination of the two. For this experiment, a second probe was installed and used as an electron emitter. This permitted the first probe array to scan the plasma for profile measurements while the emission location remained constant. The minor radial location of the emitter could be determined in two ways: (1) visually, by initiating a glow discharge (flux surface visualization) with the emissive probe on the second array and adjusting the position of the first array until its measuring probe coincided with the illuminated flux surface, and (2) by checking for a small spike in temperature in the profile obtained by the scanning probe. −5 A series of density profiles was then obtained in N2 gas at a neutral pressure of 1.0 10 torr × with the emitter bias set at different values. Results are shown in Fig. 4.9. As the figure indicates,

75 3.5 No beam, ECH only )

-3 3 50 eV beam

m 100 eV beam

16 2.5 200 eV beam 200 eV, no ECH 2

1.5

1

0.5 beam Electron density (10 surface 0 0 10 20 30 Probe displacement (cm)

Figure 4.9: Profiles of electron density as a function of emitter bias with a background of nitrogen −5 gas at 1.0 10 Torr at IIL/IPF = 3.00. Reference cases using only the magnetron (blue) and × only the emissive filament with a -200 V bias (black) are also shown for comparison. The vertical gray stripe marks the location of the emissive filament.

76 bias adjustments to the localized emitter significantly influenced the global density profiles. Fur- thermore, the density profile at the highest bias tested (-200 V) exceeds the densities attained using either source individually by a factor of nearly three. Evidently, the densities when both sources are used simultaneously (e.g., the red curve in Fig. 4.9) are higher than the sum of the individual contributions (blue and black curves in Fig. 4.9).

4.5.1 Interpretation: possible synergy mechanisms

Several candidate mechanisms were proposed to explain the observed synergy. The first proposed mechanism is driven by the efficiency of ECH absorption. In the density and temperature regimes observed in these experiments, the optical thickness (i.e., the fraction of incident wave power absorbed by the plasma in one pass) for both O- and X-mode in the first and 2 second harmonic is proportional to rLe, where rLe is the electron Larmor radius [76]. For a typical microwave-only N2 plasma with Te = 5 eV, this corresponds to an average rLe = 60 µm (120 µm) at the first- (second-) harmonic resonance. However, if the emissive filament is turned on with a bias of, for example, -200 V, it will introduce a population of electrons with rLe = 400 µm (800 µm) that would absorb the microwave power more efficiently. Furthermore, enhanced single-pass absorption would reduce the number of passes of the microwave beam through the plasma, thereby reducing power losses due to wall absorption or leakage through unshielded ports. The mechanisms just described illustrate how the heated, biased probe could enhance microwave start-up. However, the reverse is also true: microwaves could assist plasma formation by the emissive probe. For instance, microwaves break down the gas in between the two electrodes (the emissive probe and the vacuum vessel). In this way, they introduce free electrons that add up to the thermoelectrons injected by the filament and, like them, are accelerated by the electric field and generate secondary electrons by collisions with the gas molecules. As soon as a plasma has formed, however, this mechanism will be restricted to the plasma sheath around the probe and around the vessel. Elsewhere, the applied electric field will be shielded by the plasma. Another proposed mechanism is an enhancement of the Schottky emission by the biased filament in the presence of the oscillating microwave field. The emission current Iem of a heated filament has the functional form [87]:

77 p 3 ! 2 Φ e E/4π0 Iem T exp + (4.20) ∝ −kBT kBT

Here, T is the temperature of the metal, Φ is the metal’s work function, and E is the electric field on the surface of the metal. When the plasma is sustained by the filament alone (no microwaves), E is constant and is determined by the probe’s bias voltage and the thickness of the sheath around the probe. If microwaves are added, however, they will contribute a sinusoidally oscillating term to

E. Due to the nonlinear dependence of Iem on E, the time average of Iem may increase as a result of the oscillations. The increased Iem would result in a greater population of energetic electrons in the plasma. Since the cross-section for electron impact ionization of N2 is peaked near 100 eV [88], a greater presence of electrons with energies of 50-200 eV would lead to more frequent impact ionizations and therefore a denser plasma. An important difference with tokamaks, which often employ emissive filaments to assist in the startup process, should be noted: during start-up, and before a plasma current has fully formed, the tokamak field lines are open, and connect the electron-emitting cathode with the vessel or components therein, acting like an anode. In a stellarator, instead, no plasma current is needed, and the field lines inside the LCFS are closed. Hence, drifts and collisions need to be invoked to explain how, eventually, the electrons reach the vessel. In particular, the thermoelectrons injected experience an E B drift due to the applied electric field E (which will be rapidly Debye-shielded × everywhere except near the probe and the vessel). In addition, the applied E alters the spatial distribution of Vplasma and, therefore, the electric field in the plasma away from the probe and vessel wall. This was already shown to have a significant impact on drift orbits in previous experimental and theoretical work on pure-electron CNT plasmas. In particular, engineering Vplasma to be nearly uniform on flux surfaces was predicted to reduce open drift orbits [89] and, experimentally, made an order-of-magnitude improvements in particle confinement over the previous Vplasma [25]. It is possible that the oscillating microwave field, which would result in a polarization drift, alters particle orbits in an manner favorable for confinement.

78 Chapter 5

Overdense plasma heating

In Chapter 4, the viability of ECH in CNT was proven in experiments with a low-power magnetron. In order to meet CNT’s research goal of high β plasmas for fusion-relevant research, however, more heating power will be necessary. In addition, since both ne and Te must increase, the plasma will move into a regime in which the electron plasma frequency ωpe will greatly exceed the cyclotron frequency ωce. In this regime, the plasma is overdense, preventing electromagnetic waves from propagating to the electron cyclotron resonance (Sec. 4.1.1). One strategy employed in fusion experiments to penetrate overdense regions is to facilitate mode conversion of waves incident on the plasma. These techniques often have the goal of exciting electron Bernstein waves (EBWs), electrostatic waves that cannot propagate in a vacuum but can propagate in a hot plasma with no upper limit on density [90]. In large tokamak and stellarator experiments, three techniques are commonly employed to attain this mode conversion. In the SX-B scheme, implemented on experiments such as WT-3 [91] and LHD [92], a wave launched from the high-field side in the slow (S) extraordinary (X) mode mode converts to an EBW upon reaching the upper hybrid resonance (UHR). In the FX-B scheme, a fast X-mode wave launched from the low-field side couples mode-converts to an EBW at the UHR after “tunneling” through a narrow region in which the fast X-mode is evanescent. In the O-X-B scheme [93], a wave launched from the low-field side in the ordinary (O) mode impinges on the O-mode cutoff with a special oblique incidence, converts to the slow X-mode, and eventually converts to an EBW at the UHR. This has been implemented on devices including

79 W7-AS [94] and TJ-II [95]. Overdense heating has also been achieved in smaller devices such as CHS [96], TJ-K [97], and WEGA [98]. In these experiments, the magnetic field is sufficiently low that the vacuum wavelength

λ0 associated with ωce is on the order of the plasma minor radius a. While the wave propagation and damping are more difficult to analyze in this regime, full-wave modeling was used to verify the successful implementation of O-X mode conversion in WEGA [98], and confirmed UHR heating as the dominant mechanism in TJ-K [97].

CNT [20] can be viewed as a further extension of this long-wavelength regime in which λ0 is 12.2 cm for the heating frequency of 2.45 GHz, which amounts to about one-third of the major radius R = 30 cm. On this scale, the broad launched microwave beam will strike a broad region of the plasma at a range of incident angles and polarizations. This variation has been limited to some extent in CNT by placing a waveguide very close to the plasma edge. No focusing element was used in this initial study. Also, the CNT launch system is not yet optimized for any particular overdense heating mechanism. In spite of this, as will be shown in this chapter, plasmas overdense to O-mode propagation by factors of more than 4 have been attained in CNT. Sec. 5.1 briefly reviews the electrodynamics of guided waves, which are utilized in this work to launch microwaves into the plasma. In Sec. 5.2, the heating and diagnostic systems employed for this work are described in detail. Sec. 5.3 presents experimental density and temperature profiles during heating for O- and X-mode launch. Their dependence on power and magnetic field is also examined. Finally, possible heating mechanisms are discussed in Sec. 5.4.

5.1 Guided waves

Hollow pipes were adopted for CNT, as well as for most EC and EBW heating systems in fusion, due to their superior power-handling capability compared to coaxial cables, striplines, and other types of waveguides. The propagation of electromagnetic waves through a hollow waveguide is governed by Maxwell’s equations with the standard boundary conditions for conducting surfaces (i.e., no parallel electric field, no perpendicular magnetic field). Consider first the case of a rectangular waveguide with width a and height b in the x and y dimensions, respectively. Waves will propagate through the guide in the z direction. The uniformity of the system in z and in time t permits the

80 waves to be Fourier-transformed in those dimensions, giving wave modes with an exp(iωt ikz) − dependence. Here ω is the wave frequency and k is the wavenumber, related to the wavelength λ by λ = 2π/k. If the electric field is transverse (i.e., it has no component in the direction of propagation), then it can be expressed as a sum of eigenmodes as follows [99]:

  (n/b) cos(mπx/a) sin(nπy/b) X iωµ0πc   E(x, y, z, t) = Emn  (m/a) sin(mπx/a) cos(nπy/b) exp (iωt ikz) (5.1) ω   m,n mn −  − 0

In the above expression, m and n are horizontal and vertical mode numbers, µ0 is the vacuum permeability, c is the vacuum speed of light, ωmn is the cutoff frequency of the (m, n) mode, defined as

r m2 n2 ω = πc + , (5.2) mn a b and the propagation constant k is related to ω through the dispersion relation,

2 1 2 2
k = ω ωmn (5.3) c2 − Note that if the microwave frequency ω is less than the cutoff frequency, the wavenumber k will be imaginary and the mode will be evanescent. As a result, in a rectangular waveguide where a > b, there exists a frequency band in which only one transverse-electric (TE) mode can propagate. (It can also be shown that no transverse-magnetic modes can propagate at this frequency.) This mode, which has m = 1 and n = 0, is referred to as the fundamental TE mode. Its transverse electric field is plotted in Fig. 5.1a. Note that it is uniformly polarized parallel to the short edge. The cutoff effect is advantageous because it promotes mode “purity”: any higher-order modes excited by nonuniformities or imperfections in the waveguide will be suppressed if their respective cutoff frequencies are greater than the wave frequency. Hence, if waves are injected into a guide at a frequency that only passes in the fundamental mode, the electric field will retain the polarization shown in Fig. 5.1a throughout the guide. Further, if the waveguide functions as an antenna (in which one end is open and it launches waves into vacuum), the orientation of the waveguide effectively

81 (a) (b)

Figure 5.1: Vector plots showing the orientation and relative magnitude of the electric field for the fundamental mode of (a) a rectangular waveguide and (b) a circular waveguide. The cross-sections are scaled to have the same cutoff frequency. Note that the tangential component at the wall is always zero.

controls the polarization of the launched waves. This property will be utilized in the launch system described in Sec. 5.2.1. If the waveguide has a circular cross-section, the electric field may be expressed as an analogous two-dimensional sum of m and n modes with characteristic cutoff frequencies. The fundamental mode is transverse-electric, and its transverse electric fields are illustrated in Fig. 5.1b. Similarly to the rectangular fundamental mode, the intensity is greatest in the center of the guide. In contrast to the rectangular mode, which is everywhere polarized at the same angle, the circular mode exhibits a dominant polarization direction with variations that increase toward the walls. This is a result of the boundary condition forbidding electric fields tangential to conducting surfaces. Nevertheless, if a circular waveguide carrying its fundamental mode is used as a launch antenna as described above, most of the launched wave energy will be in the dominant polarization. Another difference from the rectangular case is the axial symmetry of the circular waveguide, implying that the angle of the dominant polarization is arbitrary. This property is used to our advantage in the microwave launch system described in Sec. 5.2.1, which permits rotation of the linear polarization.

82 5.2 Heating and diagnostics in CNT

5.2.1 Heating system

The microwave heating system, which was designed and installed as part of this thesis work,1 is shown schematically in Fig. 5.2a. The microwave source is a 10 kW, 2.45 GHz magnetron manufactured by Muegge. Its output is CW in its main mode of operation, but reducing the input power will result in pulse-modulated output of roughly 5 kHz. The magnetron launcher excites the

TE10 mode (Fig. 5.1a) in the rectangular waveguide at the beginning of the transmission line. A three-stub tuner is employed to control the percentage of the magnetron output that couples to the plasma, with reflected power absorbed in a water-cooled isolator upstream of the tuner (not shown). The power injected in the vessel is determined by the difference of the forward and reflected power as measured by a pair of Schottky diodes fixed to a dual directional coupler. This net injected power is an upper bound for the power actually deposited in the plasma. The rest is dissipated on the resistive walls or leaks out of the vessel through unshielded ports after multiple reflections. Downstream of the tuner is a twistable, flexible section of rectangular waveguide followed by a rectangular-to-circular taper—custom-fabricated by Space Machine & Engineering—that can rotate freely on the flange on its circular side (Fig. 5.2c). Together, these permit the arbitrary rotation of the polarization of the guided waves. The polarization is linear in the TE10 mode (Fig. 5.1a), tapered into the TE11 mode (Fig. 5.1b). By choosing the correct angle relative to the magnetic field, the antenna may be oriented to couple primarily to the O-mode or X-mode in the plasma. Following the taper is the launch antenna, custom-built by Nor-Cal Products (Fig. 5.2b,d): a section of circular waveguide that leads into the vacuum vessel up to near the plasma edge. The launch antenna is effectively a re-entrant port, held at atmospheric pressure to avoid unwanted high-order harmonic excitation within the waveguide. Such excitation would break down the gas and create a plasma that would partly absorb and/or reflect the high-power microwaves. Both effects are undesired. A quartz window at the end of the launch antenna functions as the vacuum break. A schematic of the orientation of the launch antenna relative to the plasma is shown in figure 5.3. An illustration of a 2.45 GHz Gaussian beam coupled to the launcher is also shown, as well as

1Y. Wei assisted with the installation and commissioning.

83 Twist-flex flange, flange, flange, (a) Rotatable (b) waveguide unsealed Conflat o-ring seal Taper

Tuner air Oblique launcher air vacuum 10 kW, 2.45 GHz CW magnetron Fused-quartz quartz window window (c) (d) twist-flex waveguide taper

Figure 5.2: (a) Schematic of the microwave heating system as described in the text. Drawing prepared by Y. Wei. (b) Detail of the waveguide launcher. Adapted from a drawing by Nor-Cal Products. (c) Photograph of the twistable, flexible waveguide leading into the rectangular-to- circular taper. (d) Photograph of the launch antenna inside the CNT vessel.

84 contours marking the first and second harmonics of the electron cyclotron frequency ( B = 87.5 | | mT and 43.8 mT, respectively). Due to the lack of access ports on the plasma midplane (which is actually vertically oriented in the laboratory frame), the antenna enters at an oblique angle from a port slightly above the toroidal midplane, calculated such that its axis intersects the vacuum magnetic axis at a toroidal angle φ = 90◦. Alignment of the microwave polarization is accomplished on CNT by maintaining a hot-cathode discharge on the magnetic axis [56]. This effectively illuminates the magnetic axis, revealing the direction of the magnetic field (Fig. 5.4a). The axis can be seen by looking down the taper and through the launch window (Fig. 5.4b). The taper is then rotated so that the axis of its rectangular end makes the desired angle with the magnetic axis. For example, for O-mode polarization, the taper is rotated so that the magnetic axis is parallel with the short walls of the rectangular cross- section. Note that the field’s pitch angle, when viewed through the waveguide, changes by less than 20◦ between the axis and the edge.

5.2.2 Diagnostics

The primary plasma diagnostic used in this chapter was an array of Langmuir probes as described in Chapter 4. The probe array intersects the plasma between toroidal angle φ = 180◦ and φ = 215◦, far from the line-of-sight of the launch antenna which aims at φ = 90◦ (Fig. 5.3b). Electron temperature and density were determined from probe I(V ) characteristics as in Chap- ter 4. The main difference was that the heating in this chapter was approximately constant, not pulsed. Hence, I(V ) data for a given probe location was obtained by sweeping V several times during a 0.25 - 0.5 s time period. Since there were no pulses, it was not necessary to bin the data according to time. The measurements in this chapter were taken after the completion of the error-field study in Chapter 2; hence, the flux coordinates of the probe locations, as well as the geometry of the

flux surfaces, are known with greater accuracy than in Chapter 4. Electron temperature Te and density ne are assumed to be constant on flux surfaces; hence, a one-dimensional scan of probe measurements let to a characterization of the Te and ne everywhere within the last closed flux surface (LCFS). For positions outside the LCFS, Te and ne were assumed to remain constant as a function of distance from the LCFS and were hence deduced from the outermost measurements

85 (a) Flux surfaces 20 Cycl. harmonics Vacuum beam second 10 harmonic (cm)

y 0

10 first harmonic 20 20 10 0 10 20 30 40 x (cm) (b) coil

probe

y

coil

x

launcher LCFS

Figure 5.3: (a) Launch antenna, X-mode Gaussian beam (green), cross-section of the CNT flux surfaces (black), and electron cyclotron harmonics for 2.45 GHz (red). The beam boundary corre- sponds to field intensities of 1/e2 relative to the values on the central axis. Note that, in this plane, the O-mode beam would be wider. The x axis is coaxial with the waveguide and approximately intersects the magnetic axis at x = 0. The y axis is perpendicular to the magnetic field (and x) at y = 0. (b) Three-dimensional schematic of the configuration inside the vacuum vessel, including the LCFS, the launch antenna, the Langmuir probe, and the IL coils. The translucent black rectangle indicates the cross-section in (a) with the x and y directions represented by the green and blue arrows.

86 (a) (b)

Figure 5.4: Photographs of the process for aligning the waveguide to the magnetic field. (a) an Ar discharge localized to the magnetic axis; (b) the same discharge, viewed through the waveguide window via the taper. The taper is rotated until the axis aligns to the rectangular cross-section with the desired angle.

available.

The behavior of ne close to the launch window presents an additional source of uncertainty. Models of magnetized plasma sheaths near solid surfaces, however, predict that a magnetic pre- sheath will extend into the plasma by a distance ωci/cs, the ion cyclotron frequency divided by the sound speed [100]. Near the window, this distance is of order 1 cm for singly ionized Ar. The thin layer in which such a density drop occurs has little effect on wave propagation due to its thinness compared to the wavelength and other scales of interest. Additionally, that layer is highly underdense to O-mode.

5.3 Profile measurements: evidence of overdense heating

Electron temperature and density profiles were obtained for different heating powers and magnetic field strengths. For each profile, a number of discharges were conducted with the same heating and pressure parameters while the probe position was scanned through the plasma and scrape-off layer. The discharges lasted for 8-10 s. In a typical discharge, the heating power gradually increases over the first four seconds as the magnetron warms up. The data for the profiles that follow were taken at t = 4-5 s of each discharge, during the flat-top of heating power.

87 5.3.1 Dependence on heating power

Fig. 5.5a-b show ne and Te profiles for Ar plasmas heated with O-mode-launched waves at different heating powers P . Also shown are 7th-order polynomial smoothing fits as a guide to the eye. The −5 backfill pressure was (1.4 0.2) 10 torr in each case. Profiles of Te (Fig. 5.5b) were each hollow, ± × and variations of Te with P were localized to the edge; both of these observations suggest that the power is mostly deposited in the edge.

Profiles of ne (Fig. 5.5a) were peaked at intermediate minor radii. Note that nearly the entire plasma was overdense to O-mode propagation at all power levels, including the areas where Te was peaked, indicating that overdense heating likely accounts for a significant portion of the power input to the plasma. Figs. 5.5c-d show contours of cutoffs and resonances in the xy cross-section (defined in Fig. 5.3) calculated from the O-mode profiles at the lowest and highest power levels. Also shown is the launch window and the “vacuum microwave beam” (i.e., the propagation axis and width of the microwave beam as they would look in the absence of plasma). As indicated by the O-mode cutoff curves, both cross-sections are almost entirely overdense to O-mode propagation. However, some O-X conversion may occur in the lighter-shaded regions at the “turning points” where the cutoffs of the slow X- and O-modes are locally degenerate. Fig. 5.6 shows results for a power scan with X-mode launch for the same Ar gas pressure.

The shapes and trends in the ne and Te profiles were similar to the O-mode case; again, heating seems to occur predominantly at the edge of the plasma. The low, medium, and high power levels were obtained using the same stub tuning as the corresponding power levels for the O-mode scan; however, the values of P were markedly different. This indicates that the coupling of the plasma to X-mode was different from that of O-mode, which would result from an expected difference in plasma reflectivity. This is consistent with the contours in Figs. 5.5cd and 5.6c-d. In all cases, the fast X cutoff, which reflects X-mode but is transparent to O-mode, is very close to the launch window.

In generating the contours in Figs. 5.5c,d and 5.6c-d, ne and Te were extrapolated outside the LCFS as discussed in 5.2.2. As a result, these contours are less accurate than measurements inside the LCFS. In the case of O-mode launch, this inaccuracy should not impact the analysis because ne is usually well below cutoff outside the LCFS; hence, refraction will be negligible and O-mode

88 4 (a) 0.5 kW (c) 0.5 kW 4.0 kW 20 8.0 kW 3 ) Prev. system

−3 O Cutoff 10 LCFS m Beam axis

17 2 Beam 1/e

(cm) 2 y 0 Beam 1/e

(10 Cycl. resonance e O cutoff n Upper hybrid 1 Fast X cutoff −10 Slow X cutoff Evansc. for O−mode Access. by O−X conv. 0 8 (b) 20 (d) 8.0 kW 6 10 4 (eV) (cm) e y T 0 2

LCFS λ −10 0 0 0 4 8 12 16 −30 −20 −10 0 10 20 x (cm) x (cm)

Figure 5.5: Profiles of Ar for O-mode launch at various power levels, at a backfill pressure of −5 (1.4 0.2) 10 torr. (a)-(b) Langmuir probe measurements of (a) ne and (b) Te, projected on ± × the x axis (as defined in Fig. 5.3). The curves are 7th-order polynomial fits to the data. The vertical gray line denotes the LCFS. The horizontal dashed line is the cutoff density for O-mode propagation. (c)-(d) Contours of cutoffs and resonances in the xy plane (Fig. 5.3) for (c) 0.5 kW heating power and (d) 8.0 kW. Regions overdense to O-mode propagation are shaded; those portions that could be accessible after O-X conversion are shaded lighter. The solid green line indicates the axis of the vacuum microwave beam, whereas the dashed and dotted green curves are contours of 1/e and 1/e2 of the on-axis beam intensity, respectively. The star in (c) is a reference for the discussion in Sec. 5.4.5. The vacuum wavelength is shown in (d).

89 4 (a) 0.7 kW (c) 0.7 kW 5.0 kW 20 6.5 kW 3 ) FX cutoff

−3 UHR 10

m SX cutoff LCFS

17 2 Beam axis (cm) Beam 1/e y 0 2 (10 Beam 1/e e Cycl. resonance n O cutoff 1 Upper hybrid −10 Fast X cutoff Slow X cutoff Evansc. for X−mode 0 8 (b) 20 (d) 6.5 kW 6 10 4 (eV) (cm) e y T 0 2

LCFS λ −10 0 0 0 4 8 12 16 −30 −20 −10 0 10 20 x (cm) x (cm)

Figure 5.6: Similar to Fig. 5.5, but for X-mode launch. The shaded regions in (c) and (d) indicate regions of evanescence for the X-mode.

90 propagation would not be sensitive to variations in ne. The analysis is less clear for X-mode launch because ne outside the LCFS is on the order of the UHR and FX cutoff densities. In this case, an error in the density estimate could switch the medium from overdense to underdense for X-mode propagation. Yet, even if the region in front of the launcher becomes evanescent for the X-mode, as in the case of Fig. 5.6c (as opposed to Fig. 5.6d), finite tunnelling across it is still possible, provided the region is not too thick. The dependence of additional plasma parameters on power for both the O-mode and X-mode scans is shown in Fig. 5.7. Data points are also included from the B scan described in Sect. 5.3.2 | | for the cases with the same B as the experiments in this section. Volume-averaged density ne was | | h i R calculated using the smoothed ne profiles as (1/V ) ne(ρ)dV , where ρ is the flux surface coordinate.

The edge temperature, Te,90, is defined as the value of the smoothed Te profile on the flux surface R at 90% of the effective minor radius. Stored energy Wplasma was calculated as ne(ρ)kBTe(ρ)dV , R 2 volume-averaged β as (1/V ) ne(ρ)kBTe(ρ)/( B /2µ0)dV , and energy confinement time τE was | | estimated as Wplasma/P . As P is an upper bound for the coupled power (Sec. 5.2.1), τE is really a lower bound for the confinement time.

The marked increase in volume-averaged density ne with power (Fig. 5.7a) indicates that the h i plasma is not fully ionized. While Te,90 increases slightly with power, it remains below 10 eV at all power levels (Fig. 5.7b). Cold plasmas with Te in this range tend to suffer large radiative losses as discussed in Sec. 4.3 and in Ref. [83]. One driver of radiative losses, electron-ion recombination, 2 increases roughly in proportion with ne and ni and thus in proportion with ne in a quasineutral plasma. Hence, it is not surprising that τE decreases with ne (Fig. 5.7e). If further power upgrades attain higher Te (i.e., > 50eV ), radiative losses should diminish, allowing for even higher Te and improved τE.

5.3.2 Dependence on magnetic field

A series of experiments was also conducted in which the heating power was kept roughly constant but the strength of the magnetic field, B was varied. The intermediate B was the same as for | | | | the power scan in Sec. 5.3.1. The higher and lower values correspond to the greatest permissible increase or decrease in field, respectively, such that neither the fundamental nor the second harmonic resonant surfaces intersect the launch window (this reduces the risk of damage to the window.) The

91

) 3

−3 (a)

m 2 17 (10

〉 1 O−mode

e

n X−mode

〈 0 8 (b)

(eV) 6 e,90 T 4 30 (c) 20 (mJ)

10 plasma W 0 8 (d) ) 6 −5

(10 4 β

2 (e) 20 s) µ (

E 10 τ

0 0 2 4 6 8 10 Power (kW)

Figure 5.7: Dependence of plasma parameters on power from the experiments described in Sec. 5.3.1. (a) Volume averaged electron density, (b) Electron temperature on the flux surface at 90% of the effective minor radius, (c) Plasma energy, (d) Volume-averaged β, (e) Energy con- finement time.

92 3 (a) 80 mT (c) 80 mT 88 mT 20 97 mT

) O Cutoff

−3 2 10 LCFS m Beam axis

17 Beam 1/e

(cm) 2 y 0 Beam 1/e

(10 Cycl. resonance e 1 O cutoff n Upper hybrid Fast X cutoff −10 Slow X cutoff Evansc. for O−mode Access. by O−X conv. 0 8 (b) 20 (d) 97 mT 6 10 4 (eV) (cm) e y T 0 2

LCFS λ −10 0 0 0 4 8 12 16 −30 −20 −10 0 10 20 x (cm) x (cm)

Figure 5.8: Similar to Fig. 5.5, but for the scan of magnetic field strengths in the case of O-mode launch. Each plasma coupled to 1 kW of heating power.

values of B given in the legends are for the origin of the xy plane. For all experiments in this | | scan, the stub tuning was the same. These resulted in net power coupling of 1 0.2 kW in all cases ± except that of X-mode at 87.5 mT which had a net coupling of about 2 kW. The results are shown in Fig. 5.8 for O-mode launch and in Fig. 5.9 for X-mode launch. While the ne and Te profiles were insensitive to changes in B for X-mode launch, both exhibited significant | | variation with B for O-mode launch. In particular, the core ne halved as B increased, and Te | | | | became increasingly hollow.

93 3 (a) 80 mT (c) 80 mT 88 mT 20 97 mT

) FX cutoff

−3 2 UHR 10

m SX cutoff LCFS

17 Beam axis (cm) Beam 1/e y 0 2 (10 Beam 1/e e 1 Cycl. resonance n O cutoff Upper hybrid −10 Fast X cutoff Slow X cutoff Evansc. for X−mode 0 8 (b) 20 (d) 97 mT 6 10 4 (eV) (cm) e y T 0 2

LCFS λ −10 0 0 0 4 8 12 16 −30 −20 −10 0 10 20 x (cm) x (cm)

Figure 5.9: Similar to Fig. 5.8, but for X-mode launch. Each plasma coupled to 1 kW of heating power, except for the B = 88 mT case, which coupled to 2 kW.

94 5.4 Discussion: mode conversion, UHR accessibility, and heating mechanisms

5.4.1 SX-B mode conversion

Since the launch occurs from the low-field side, first-pass SX-B mode conversion is not possible. However, because the beam encounters both the O-mode cutoff and the FX cutoff within 15 cm of the launch window, a portion of the incident power, whether polarized in O- or X-mode, is reflected into the chamber. After multiple reflections off the chamber walls, “polarization scrambling” will occur [101], causing incident O-mode to be reflected as X-mode and vice versa. Some of the reflected X-mode may access the UHR from the high-field side, resulting in some SX-B conversion (Fig. 5.10). It should be noted that much of the core plasma is overdense to slow X waves (for example, (Figs. 5.6c-d). This restricts the wave access to the UHR (see arrow in Fig. 5.10), thereby partly inhibiting the SX-B conversion. This mechanism could be more viable in low-density plasmas. However, the fact that the case of least-restricted access was also the case with the lowest core density and temperature (Fig. 5.8) indicates that SX-B conversion does not play a significant role in heating the plasma.

5.4.2 FX-B conversion

First-pass FX-B conversion should be possible in the case of X-mode launch. In this scheme, the mode conversion efficiency depends strongly on the thickness ∆x of the layer between the fast X cutoff and the UHR [102]. The power transmission T 2 through this layer is approximately exp( πη), where η = 2π∆x/λ0 [75]. For example, the X-mode launch at B = 97 mT (Fig. 5.9), − | | | | with ∆x = 0.5 cm gives T 2 = 45%. However, if the density is overestimated outside the LCFS by 25%, ∆x would increase to 5 cm, giving T 2 < 0.1%.

5.4.3 O-X-B conversion

For O-X-B conversion, devices with higher magnetic fields tend to impose stringent requirements on launch angle and polarization ellipticity for efficient O-X coupling at the O-mode cutoff layer. CNT’s launch antenna, on the other hand, was not placed at an angle optimized for O-X coupling and can only output linearly polarized waves. The beam may be optimized in the future, for

95 λatt,O λatt,SX 60 air λ vacuum 0

40 reflected O-mode reflected coil X-mode 20

SX access to UHR

(cm) 0 Z

−20 launcher

−40 LCFS UHR Fast X cutoff 60 − Slow X cutoff Evanesc. for X−mode −60 −40 −20 0 20 40 60 R (cm)

Figure 5.10: Schematic of how the X-mode may access the UHR from the high-field side. O-waves (originating from the launcher or wall reflections) may pass unobstructed through regions evanescent to X-mode (gray), reflect as X-waves off the in-vessel coils, and reach the UHR. In addition, X- waves from the launcher or wall reflections may partially transmit through the outermost evanescent layer and reach the UHR either directly or after reflecting off the coils. Denser plasmas make the evanescent regions thicker, thus restricting wave access. Shown near the top of the diagram are references for λ0 and the attenuation lengths λatt,O and λatt,SX for the O- and slow X-mode for 17 −3 ne = 1.5 10 m and B = 70 mT. Note that the launcher is actually located above the × | | cross-sectional plane shown here and does not intersect the LCFS.

96 example, with a refocusing mirror, which would collimate the O-mode beam at the O-mode cutoff at the optimal angle. Nevertheless, significant O-X coupling should be possible in CNT due to its relatively low value of Ln/λ0 at the location of the O-mode cutoff (0.3 to 0.4 for the O-mode profiles shown here). Here,

Ln = n/(dn/dx). In this parameter range, the requirements on propagation angle are expected to relax [103, 104].

5.4.4 Collisional absorption at the UHR

The combination of high electron-ion collisionality (νei = 1-10 MHz) with low temperatures in CNT can stifle the conversion from slow-X waves to EBWs. In order for a slow-X mode to convert to an EBW, its perpendicular wavenumber k⊥ (perpendicular to B) must grow significantly. This is because EBWs typically have wavelengths on the order of the electron Larmor radius, much shorter than the vacuum wavelength [75]. For Te = 5 eV and B = 70 mT, rLe = 80 µm. 2.45 GHz waves | | 5 with λ = 80 µm propagate with phase velocity vp = fλ = 2 10 m/s. To couple to an EBW, × then, the slow-X mode must decelerate until its phase velocity reaches this level. Assuming the wave propagates 10 cm along the UHR before conversion with an average phase velocity of 4 105 × m/s, this would take on the order of 3 10−7 s. Since this is on the same order as the mean time × between collisions between electrons (1/νei), the slow X wave would damp rapidly, depositing its energy at the UHR without mode-converting. Hence, conversion to EBWs is not likely to occur on a significant scale.

5.4.5 Wave damping mechanisms

As can be seen from Fig. 5.3, the fundamental cyclotron resonance, where ωrf = ωce, crosses through most flux surfaces. Portions of this resonant surface are directly accessible to the vacuum microwave beam—for example, at coordinates (9.5 cm, -6 cm) in Fig. 5.5c, indicated by the star symbol. In principle, cyclotron damping of electromagnetic O- and X-waves should be possible in these areas. The optical depth τ, essentially the fraction of incident beam power absorbed at the resonance in one pass,2 is given by Table XII in Ref. [76]:

2If τ exceeds 1, the plasma absorbs all incident beam energy and is said to be optically thick.

97 2  2 2 2
4 2 ωpe vt  1 + 2 cos θ sin θ LB τO1 = π N 3 (5.4) ωce c (1 + cos2 θ) λ0 !2 2  2 2 2 5 ωpe ωce vt  2 LB τX1 = π N 1 + 2 cos θ (5.5) ωce ωpe c λ0

Here, N is the index of refraction, θ is the angle between the propagation vector and B, vt is the thermal velocity, and LB = B /(d B /dx) is the scale length of variation in B along the beam | | | | | | trajectory. The O1 subscript denotes first-harmonic O-mode; X1 denotes first-harmonic X-mode.

These quantities tend to be quite low in the regimes of Te and ne seen here; for example, at the −4 −7 location mentioned above, τO1 < 10 and τX1 < 10 .

These quantities tend to be quite low in the regimes of Te and ne seen here; for example, at −4 −7 the location mentioned above, τO1 < 10 and τX1 < 10 . Fusion plasmas, which are typically optically thick to first-harmonic O-mode, exceed CNT parameters by three orders of magnitude in 2 vt and two orders in 1/λ0. It is also noted that some cyclotron damping may also occur in overdense regions, as O- and X-modes incident on cutoffs will penetrate a finite distance into the overdense region and reach the Doppler-broadened EC resonance with finite wave amplitude. This penetration depth depends 17 −3 on ne. For example, in the 0.5 kW-heated O-mode case (Fig. 5.5), when ne 1 10 m , ' × (and assuming a location where B 90 mT, θ = 90◦), the dispersion relation (Eq. 4.1) gives | | ' wavenumber k = 30i for O-mode propagation; hence, the electric field decays to 1/e of its incident value within 3 cm of the cutoff layer. The penetration depth would be lower in denser plasmas; hence, this effect would decrease with density. Regions of the cyclotron resonance where cyclotron damping might occur in underdense and overdense regions are illustrated schematically in Fig. 5.11 for the low-field and high-field cases with O-mode launch. Evidently the high-field plasma, in which the plasma was less dense, is accessible to a greater amount of cyclotron damping by a non-evanescent portion of the beam in the first pass. In addition, recall from above that the high-field case would be susceptible to more damping in the overdense region due to its larger penetration depth. Thus, for both underdense and overdense regions, cyclotron damping is expected to occur in greater amounts in the high-field case, in which the plasma was less dense overall. However, the greatest levels of heating appear to occur in the

98 (a) 80 mT (b) 97 mT

Figure 5.11: Accessibility of the plasma to first-pass cyclotron damping for cases of O-mode launch at different magnetic field strengths, corresponding to Fig. 5.8c-d. The green stripes represent portions of the vacuum microwave beam intersecting the cyclotron resonance (red) within the LCFS. Note that a large fraction of the beam must travel through the evanescent region (gray) inside the O-mode cutoff (blue). The darker green stripe is the portion of the beam which reaches the resonance before encountering the O-mode cutoff.

low-field case, as evidenced by the greater ne. Due to this and the low optical thickness, cyclotron damping of O- and X-mode electromagnetic waves is not expected to play a significant role in the plasma heating. For this reason, and due to the possibility of mode conversion discussed above, it is more likely that the dominant power deposition mechanism is collisional damping of X-mode waves near the upper hybrid resonance.

5.4.6 Earlier experiments in other devices

Similar mechanisms have been observed on other devices with similar plasma parameters. Full-wave simulations and modulated power measurements for 2.45 GHz plasmas in TJ-K both indicated that power deposition occurred primarily at the UHR for both O- and X-mode heating [97]. In WEGA, simulations and experiments both pointed to significant deposition of EBWs at the cyclotron res- onance [98]. The TJ-K experimental setup is perhaps more similar to that of CNT because of the steep density gradients in the vicinity of the upper hybrid resonance. The WEGA setup, by contrast, was designed (1) to place the UHR in a region with a low ne gradient and (2) made use of

99 an antenna optimized for O-X coupling. Power modulation measurements on CHS indicated signif- icant levels of EBW heating for both O- and X-mode (linearly polarized), injected both normally and obliquely to the O-cutoff [96].

5.5 Initial full-wave calculations

To better understand the interactions between the microwaves and the plasma, some of the density cross-sections presented in this chapter were analyzed by the full-wave code IPF-FDMC [105].3 The IPF-FDMC code solves Maxwell’s equations coupled with the electron equation of motion in a nonuniform magnetized plasma. Hence, it accounts for effects such as O-X mode conversion and tunneling of the X-mode through the evanescent region between the UHR and fast X cutoff. −5 −3 A damping term corresponding to a collision frequency ν within the range 10 < ν/ωrf < 10 is assumed, which damps the slow X-mode in the vicinity of the upper hybrid layer [97]. The total damped power is not sensitive to the value of ν within this range. In general this damping term describes a combination of collisional damping and linear conversion to EBWs. However, as discussed in Sec. 5.4.4, it is expected that collisional damping dominates over conversion to EBWs in the CNT plasma parameters. Preliminary results of full-wave calculations for the scan of B in the case of O-mode launch | | are shown in Fig. 5.12. The plots show contours of normalized E , the time-averaged amplitude h| |i of the microwave electric field. Note that the normalization is not the same from one plot to another. In addition, resonances and cutoffs are shown as dashed and solid lines, respectively. The simulations were conducted in two dimensions in the xy cross-section as shown, for example, in Fig. 5.8c-d. The launcher is located on the right of the computational domain. Perfectly absorbing conditions were assumed at the domain boundaries; hence, the plotted E refers to the first pass h| |i of microwaves through the plasma but neglects any additional power re-impinging on the plasma after reflections off the vessel walls. In each case, a region of enhanced E is seen in a narrow region around the UHR, correspond- h| |i ing to a wave resonance. Because only the X-mode is sensitive to this resonance, but these cases all refer to O-mode launch, it is evident that O-X mode conversion has occurred. Furthermore,

3The full-wave calculations in this section were conducted by collaborator A. K¨ohn, Max Planck Institute for Plasma Physics, Garching, Germany.

100 (a)

(b)

(c)

Figure 5.12: Contours of normalized E in plasma cross-sections for the scan of B with O- h| |i | | mode launch (Fig. 5.8). Dashed and solid lines correspond to resonances and cutoffs as indicated. Normalizations are not the same in the different plots. (a) B = 80 mT; (b) B = 88 mT; (c) | | | | B = 97 mT. Plots prepared by A. K¨ohn. | | 101 regions of field enhancement near the UHR are associated with high levels of collisional damping. Qualitatively, the extent of the region of field enhancement along the UHR appears to increase as B increases from 80 mT (Fig. 5.12a) to 97 mT (Fig. 5.12c), suggesting that overall first-pass | | power absorption also increases as B increases. However, the measured ne, and, therefore, total | | power absorption in the partly ionized CNT plasma, decreases with increasing B (Fig. 5.8a). | | This suggests that waves reflected off the vessel walls (and possibly off the coils) play a significant role in heating the plasma—enough to negate the trends in first-pass absorption—and cannot be neglected.

102 Chapter 6

High-β equilibria and ballooning 1 stability

Much of the foregoing work in this thesis has had the outcome of preparing CNT for high-β research. The error-field diagnosis in Chapter 2 produced a detailed understanding of the vacuum magnetic field, a necessity for accurate analysis of plasma equilibria. The onion-peeling technique developed in Chapter 3 has the promise to serve as a non-invasive diagnostic for plasmas that are too hot and dense to permit Langmuir probe measurements. And the measurements of microwave heated plasmas in Chapters 4 and 5 have established that overdense heating is possible with 2.45 GHz in CNT, indicating that attainable values of β will not necessarily be restricted by the low density cutoffs imposed by the low magnetic field. In this chapter, it is shown that CNT has the potential to make valuable contributions to experimental research on the existence of β limits imposed by ballooning instabilities. This is a topic of great interest to stellarator fusion research, as stellarator β limits have yet to be identified experimentally [107, 108]. However, simulations and stability calculations of high-β CNT equilib- ria presented in this chapter indicate that, in some magnetic configurations, CNT may become ballooning unstable at β as low as 0.9%. This would allow for both an experimental validation of ballooning instability theory at a much lower β than on larger devices and would serve as a test of

1The work described in this chapter was published in Ref. [106]: K. C. Hammond, S. A. Lazerson, and F. A. Volpe, “High-β equilibrium and ballooning stability of the low aspect ratio CNT stellarator,” Phys. Plasmas, in press. Reproduced with the permission of AIP publishing.

103 whether the ballooning instability is in fact β-limiting. The potential to reach high β at low power is partly due to CNT’s low magnetic field: in general 2 B < 0.3 T, but B < 0.1 T was adopted for this study. As a result, the magnetic pressure B /2µ0 is very low, and more amenable plasma pressures (2500 times lower than in a 5 T reactor, if not smaller) suffice to reach high β. The need for high plasma pressure will require heating at high density. In this regard, overdense plasma heating was demonstrated in CNT in Chapter 5. Increas- ing the heating power would result in high power densities in the small CNT plasma (V 0.1 m3). ≈ On the other hand, the small size of CNT is co-responsible for poor energy confinement. Even so, scaling-law calculations to be presented in this chapter suggest that 40-100 kW of microwave power might be sufficient to reach the lowest β limit of 0.9%. The potential for instabilities to occur at β as low as 0.9% arises in part from the fact that CNT is a classical, non-optimized stellarator. In particular, it was not optimized for high stability, making its stability limit lower and easier to access. An interesting competing effect might arise from the CNT low aspect ratio, A 1.9. This characteristic, relatively under-explored in stellarators, ≥ increased the stability limit in spherical tokamaks and favored the achievement of higher β compared to tokamaks. It is interesting to verify whether the low aspect ratio has a similar beneficial effect on stellarator stability, although evidence presented here suggests this not to be the case, at least not for CNT. The structure of this chapter is as follows: Sec. 6.1 provides a brief overview of the concepts from ideal magnetohydrodynamic (MHD) theory relevant to ballooning stability. Sec. 6.2 summarizes considerations for calculating VMEC equilibria in the CNT configuration and presents some fixed- boundary results. Sec. 6.3 reviews the assumptions used for calculations of equilibrium parameters, bootstrap current, and stability. The methods for the calculations are then described in Sec. 6.4. Sec. 6.5 presents the main free-boundary equilibrium and stability results. Sec. 6.6 describes scaling- law calculations to predict how much power will be necessary to attain the equilibria in Sec. 6.5.

104 6.1 Equilibrium and stability theory

6.1.1 The ideal MHD model

The general approach applied in this Chapter toward identifying possible high-β plasma configu- rations relies on the theory of ideal magnetohydrodynamics (MHD). Ideal MHD models the plasma as a fluid whose particles are subject to electromagnetic forces and are assumed to be in thermal equilibrium. Specifically, the large-scale behavior of the plasma is modeled in the following system of equations [2]:

dρ + ρ v = 0 (6.1a) dt ∇ · dv ρ = J B p (6.1b) dt × − ∇ E + v B = 0 (6.1c) × d  p  = 0 (6.1d) dt ργ ∂B E = (6.1e) ∇ × − ∂t

B = µ0J (6.1f) ∇ × B = 0 (6.1g) ∇ ·

Eq. 6.1a expresses conservation of mass in terms of mass density ρ and fluid velocity v. Eq. 6.1b governs momentum conservation in terms of ρ, v, current density J, magnetic field B, and pressure p. Eq. 6.1c, introducing the electric field E is a generalized form of Ohm’s law for a non-resistive magnetized plasma. Eq. 6.1d expresses conservation of energy in terms of p, ρ, and the adiabatic constant γ. Eqs. 6.1e-6.1g are Maxwell’s equations in which the ∂E/∂t is neglected in Ampere’s law. When the plasma is in a state of equilibrium, all time derivatives vanish, and Eqs. 6.1 reduce to seven equations:

105 p = J B (6.2a) ∇ ×

B = µ0J (6.2b) ∇ × B = 0 (6.2c) ∇ ·

Note that Eq. 6.2a directly implies that the plasma pressure p is constant on a flux surface, as its gradient is everywhere perpendicular to B. We can therefore specify p(ψ) as a function of the flux ψ enclosed by its respective surface. Another important surface function is the rotational transform, ι (Sec. 1.1.2), which can be expressed in terms of ψ and enclosed poloidal flux φ as dφ/dψ. The central purpose of the VMEC code [109] employed in this chapter is to identify a plasma equilibrium by solving Eqs. 6.2 subject to constraints on p, ι, and ψ.

6.1.2 Validity of ideal MHD in CNT plasmas

The ideal MHD equations (Eq. 6.1) can be derived from Maxwell’s equations and a full kinetic de- scription of the forces governing the plasma particles if the plasma satisfies the following conditions [17]:2

r Small ion Larmor radius: Li 1 (6.3a) a  r 2  a  rm Low resistivity: Li e 1 (6.3b) a vTiτii mi 

Here, rLi is the ion Larmor radius, τii is the ion-ion collision time, and a is the plasma minor radius. The first condition (Eq. 6.3a) essentially states that the plasma must be magnetized; i.e., particle transport perpendicular to the magnetic field must be constrained by the Larmor gyration on the spatial scale of the experiment. The second condition (Eq. 6.3b) ensures that resistive diffusion occurs slowly relative to the time scale of interest to MHD stability ( vTi/a). ≈ Values of the validity criteria (Eq. 6.3) for the CNT equilibria to be simulated later in this

2The derivation of the MHD equations also makes use of a third assumption; namely, that overall collisionality is high (but not so high as to render Eq. 6.3b invalid). This is actually not met by most fusion plasmas, but other properties tend to uphold the validity of the MHD equations regardless. For details, see Chapter 2 of Ref. [17].

106 Species Deuterium Argon n (1017 m−3) 50 200 50 200 rLi/a 0.08 0.08 0.4 0.4 2   q rLi
a me 4 10−5 2 10−4 0.003 0.01 a vTiτii mi × ×

Table 6.1: Values of the ideal MHD validity criteria (Eq. 6.3) for the high-β CNT equilibria to be simulated in this chapter for two possible ion species.

Chapter are shown in Table 6.1. Values are shown for two different density regimes each in the cases of deuterium and argon as the working gas. The low-resistivity requirement is easily satisfied in all cases. The small-Larmor-radius condition is also met in both cases, although it is marginal for argon due to its greater mass. Hence, the ideal MHD equations used to model equilibrium and stability in this chapter should describe light-ion plasmas well.

6.1.3 MHD stability

The mere existence of a MHD equilibrium (Eq. 6.2), does not imply that it will be attainable exper- imentally. In particular, if the equilibrium is unstable, it cannot be sustained. An instability arises if the plasma’s potential energy can be reduced by moving away from its equilibrium configuration. In other words, a stable equilibrium will experience restorative forces if it is perturbed, whereas an unstable equilibrium will experience forces that amplify the perturbation. Some of the most deleterious plasma instabilities can be identified through linear analysis within the MHD model. In linear stability theory, a small displacement ξ˜ to the plasma is modeled as v ∂ξ˜/∂t. The displacement is then postulated as a set of normal modes, ξ˜ = ξ exp (iωt). It ≡ can be shown that, if the plasma boundary is fixed by a conducting wall,3 the change in potential energy δW arising from this perturbation is equal to [17]:

Z 1 n 2 2 2 2 δW = dV [ (ξ⊥ B)]⊥ + B ξ⊥ + 2ξ⊥ κ + µ0γp ξ 2µ0 ∇ × × ∇ · · |∇ · | (6.4) ∗ ∗ o 2µ0 (ξ p)(ξ κ) µ0J ξ b [ (ξ B)] − ⊥ · ∇ ⊥ · − k ⊥ × · ∇ × ⊥ × ⊥

3In cases where the plasma is surrounded by vacuum and therefore has a deformable boundary, additional terms will be introduced to the integrand of Eq. 6.4. However, the ballooning modes studied in this chapter are internal modes that do not affect the plasma boundary, so the extra terms have been omitted for brevity.

107 Here, the and subscripts denote components parallel and perpendicular to B, respectively; b k ⊥ is the unit vector B/ B ; κ is the magnetic field curvature, b b; and the integral is taken over | | · ∇ the plasma volume V . All quantities in the integrand other than ξ are the equilibrium quantities from the solution to Eqs. 6.2. If the perturbation ξ˜ results in a positive change in energy δW , the plasma forces will be restorative and the equilibrium will be stable to ξ˜. Conversely, if δW is negative, the perturbation is unstable. Note that the first three terms in the integrand of Eq. 6.4 always contribute positively to δW ; hence, they correspond to stabilizing responses to ξ˜. The last two terms, however, may be destabilizing. Due to the proportionality of the penultimate term to p, instabilities that take ∇ their main negative contribution from this term are referred to as pressure-driven instabilities. Correspondingly, instabilities that take their main negative contribution from the last term are referred to as current-driven instabilities. The lower the plasma pressure is relative to the magnetic field, the more likely the first two terms in the integrand will compensate for negative contributions from the fourth term. Hence, lower-β (Eq. 1.2) plasmas are less likely to suffer pressure-driven instabilities. Note also that the sign of the pressure-driven term in Eq. 6.4 depends on the direction of the pressure gradient p ∇ relative to the magnetic curvature κ. In cases of “good” curvature, in which p and κ are in ∇ opposite directions, the term is stabilizing; in the opposite case the curvature is “bad” and the term is destabilizing. A toroidal plasma usually contains regions of both good and bad curvature, with good curvature on the inboard side and bad curvature on the outboard side if the pressure profile is centrally peaked. Some physical intuition for why bad curvature is destabilizing may be attained by through an analogy to the Rayleigh-Taylor instability that occurs at the interface between a dense liquid sitting on top of a lighter liquid [7]. Consider two flux tubes of plasma on the outboard side of a torus (a region of bad curvature) on adjacent flux surfaces as illustrated in Fig. 6.1. The “gravitational” force from the two-liquid analogy arises from the centrifugal force that tends to throw the plasma particles outward as they stream along the flux tubes. Hence, the gravitational force is toward the outboard side (opposite the field curvature vector), and so the flux tube on the inner surface is “on top of” the flux tube on the outer surface. Furthermore, assuming the pressure gradient points toward the plasma core (i.e., the profile is peaked), the “upper” flux tube has a higher

108

109

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κ Δ have a short wavelength perpendicular to B and a long wavelength parallel to B. It can be shown that, under these assumptions, ξ˜ can be modeled in terms of normal modes of a linear, second-order ordinary differential equation in which the variable is the displacement along a field line [110, 111]. The eigenvalues associated with these modes correspond to the growth rates of the modes; a positive growth rate indicates linear instability and a negative growth rate indicates stability. Hence, to determine the ballooning stability of a location in the plasma, it suffices to identify the mode with the highest growth rate. This is the purpose of the COBRAVMEC code employed in this chapter [112]. Because of the tendency of a ballooning instability to rapidly transport particles across the magnetic field and out of the plasma, ballooning instabilities are believed to impose a limit on the maximum achievable β in stellarator configurations [17]. To date, the highest plasma beta among stellarators—about 5%—have been obtained in W7-AS [108] and in LHD [113]. Neither plasma was found to be unstable, implying even higher stability limits in those devices. Therefore, to date analytical and numerical investigations of ballooning modes [114, 115, 116, 117, 118] and other high-β instabilities [119] have only received partial validation by experiment: experiments confirmed certain values of β to be stable, but could not verify whether even higher values were unstable [107, 108]. Experimental access to the stellarator β limit and experimental characterization of instabili- ties would finally enable comparison with theory and improve our understanding and predictive capability. Access to higher β (and, yet, stability) could also lead to more compact and efficient stellarator reactor designs, currently assuming volume-averaged beta β = 3-6% [120, 121]. In fact, h i the main optimization criterion in the HELIAS reactor is to maintain a stability limit β > 4% h i while reducing the Pfirsch-Schl¨utercurrents and Shafranov shift [120].

6.2 Fixed-boundary VMEC solutions for CNT geometry

VMEC may assume a fixed or free plasma boundary depending on the purpose of the calculation. In fixed-boundary mode, the plasma boundary is assumed to be known ab initio and the full magnetic field is determined in the calculation without knowledge of the external coils. In free-boundary mode, the coil configuration and the field that it generates are known and the plasma boundary is determined using an energy principle [122]. Free-boundary mode was used in this work because

110 10 (a)

5

0 Z (cm)

-5 Vacuum LCFS VMEC, β=0% -10 VMEC, β=6.6% 10 15 20 25 30 35 40 45 R (cm) 0.2

h (b) 0.1 R/a ∆ 0 0 1 2 3 4 5 6 7 hβi (%)

◦ Figure 6.2: Results of a series of fixed-boundary simulations of CNT with θtilt = 78 , IIL/IPF = 2.5,

Ip = 0, and a centrally peaked pressure profile as shown in Fig. 6.3. (a) Flux surface comparison between calculations with β = 0% and β = 6.6%, with the vacuum last closed flux surface h i h i (LCFS) as a reference; (b) Relative Shafranov shift as defined in the text.

the shape of the plasma boundary was expected to vary with β and plasma current. As an initial test of concept, however, a number of high-β calculations were performed in fixed- boundary mode for CNT-like configurations. One example is shown in Fig. 6.2. In this case, the boundary was set to conform to the LCFS obtained from vacuum field line calculations with ◦ θtilt = 78 and IIL/IPF = 2.5. For this choice of θtilt and IIL/IPF , fixed-boundary equilibria were obtainable for β up to 6.6% (Fig. 6.2). The relative Shafranov shift ∆R/ah, defined according to h i Ref. [123] as the horizontal shift ∆R in the magnetic axis over the horizontal minor radius ah, is seen to increase linearly with β, as expected (Fig. 6.2b). While these results are not particularly relevant to future comparisons with experiment, they are useful nonetheless as a verification that VMEC yields reasonable results at high beta and low aspect ratio, comparable to calculations made for a spherical stellarator concept in Ref. [27].

111 1

0.8

0.6

0.4

Normalized pressure 0.2 Hollow Peaked 0 0 0.2 0.4 0.6 0.8 1 Normalized flux

Figure 6.3: Normalized profiles of pressure used as inputs for the calculations in this chapter. During the β scans, each of these profiles was scaled according to the desired β value.

6.3 Input parameters

CNT’s field strength for ECH plasmas, and possibly electron Bernstein wave heated (EBWH) plasmas, is effectively constrained by the requirement for B = 87.5 mT for heating at the first | | electron cyclotron harmonic or B = 43.7 mT for the second harmonic at 2.45 GHz. With B fixed, | | | | the only two degrees of freedom controlling the vacuum-field configuration are θtilt and IIL/IPF . The size of the free boundary is determined by the total enclosed magnetic flux. This was chosen to approximately equal the flux enclosed in the vacuum-field LCFS for the respective configuration. Pressure profiles were assumed to have one of the two functional forms shown in Fig. 6.3. The hollow profile corresponds to a typical Langmuir probe measurements of ECH plasmas in CNT. The peaked profile shown was also considered for comparison, because peaked profiles cannot be ruled out from future experiments. As CNT does not have a solenoid transformer and does not deploy any form of current drive (EC, EBW, or other), the toroidal current density was assumed to be equal to the bootstrap current density, Jbs. Profiles of Jbs were calculated for each equilibrium using the BOOTSJ code, which evaluates bootstrap currents in nonaxisymmetric plasma configurations using the drift kinetic equation [124]. Since BOOTSJ requires the electron density and the electron and ion temperatures as inputs, these quantities were estimated as follows. For equilibria with the hollow (experimentally obtained) pressure profiles, the electron temperature profile was chosen to resemble the corresponding tem-

112 perature measurements. For equilibria with the peaked pressure profiles, since no corresponding experimental data are available, temperature profiles were assumed to be flat. As CNT does not employ any direct ion heating mechanisms, the ion temperature was assumed to be 0.3 times the electron temperature at all locations. The electron temperature was constrained to not exceed 30 eV (hence, increases in β were mostly driven by increases in density). The desire for low tempera- ture and high density arises from the energy confinement scaling (Eq. 6.6). We impose a lower limit of 30 eV to avoid excessive radiative losses associated with peak radiation from hydrogen isotopes, other working gases such as noble gases, and common impurities such as oxygen and carbon [18].

6.4 Methods for equilibrium and stability calculations

The maximum achievable β for each configuration (defined by θtilt, IIL/IPF , B , and pressure | | profile) was determined using the following procedure:

1. Conduct a free-boundary VMEC calculation with β = 0 and zero toroidal current.

2. Using the results of the previous step as an initial guess, determine a free-boundary equilib- rium with the pressure incrementally increased in magnitude (while maintaining the profile shape in Fig. 6.3).

3. Calculate the bootstrap current profile, dIbs/dψn, from the result of the previous step. ψn is the normalized toroidal magnetic flux, which is used as a surface coordinate.

4. Re-calculate the equilibrium in step 2, incorporating the output from step 3 to obtain a self-consistent result.

5. Repeat steps 2-4 until either: (1) VMEC fails to descend robustly to an equilibrium solution or (2) the solution is not stable to ballooning instabilities, as determined below.

Each self-consistent equilibrium was evaluated for ballooning stability by the COBRAVMEC code, which computed growth rates on a grid of 1620 locations (15 toroidal 18 poloidal 6 × × radial) throughout the plasma volume. The maximum among these values was defined as γmax.

γmax was recorded for each equilibrium (i.e., for each β increment) such that γmax could be plotted as a function of β. The maximum ballooning-stable β was then defined as the highest value of

113 β for which γmax < 0. This was obtained through interpolation of γmax(β) (as in, for example, h i Fig. 6.5c). This procedure was carried out in three different plasma parameter regimes. The first used the experimental hollow pressure profile (Fig. 6.3) and a magnetic field strength B appropriate for | | first-harmonic ECH (denoted hereafter by B 0.08 T). The second used the same pressure profile ≈ but half the field strength as is appropriate for second-harmonic ECH (denoted by B 0.04 T). ≈ The third used the peaked pressure profile and B 0.08 T. ≈

6.5 Free-boundary and stability results

Maximum volume-averaged β values for attainable tilt angles and current ratios are plotted in Fig. 6.4 alongside other quantities of interest. The highest volume-averaged β not vulnerable to ballooning instability was 3.0%. This was obtained in two configurations, one with B 0.04 T ≈ ◦ and IIL/IPF = 2.75; the other with B 0.08 T and IIL/IPF = 3.25. Both had θtilt = 78 and ≈ the hollow pressure profile. The latter is expected to be easier to attain experimentally due to the favorable scaling of confinement time with B (Eq. 6.6) and will be referred to hereafter as the high-β configuration. The highest stable β values attained in the other two tilt angles were 2.5% ◦ ◦ for θtilt = 88 and 1.8% for θtilt = 64 . Open markers in Fig. 6.4 represent configurations in which the VMEC code did not find equilibria that were ballooning-unstable. In other words, the actual β limit for the configurations denoted by open symbols could be even higher than shown in Fig. 6.4. One such configuration is the high-β configuration described above: its β limit could in fact be higher than 3%. The lowest β threshold for ballooning stability (0.9%) was found in a configuration with ◦ θtilt = 88 and the peaked pressure profile. This will be referred to hereafter as the least sta- ble configuration. The evolution of some key parameters for this and the high-β configuration during the β scan are shown in Fig. 6.5. The nonlinearity in relative Shafranov shift ∆R/ah

(Fig. 6.5b) is due to the change in the shape of the plasma with β, resulting in a non-constant ah. The configuration with the lowest ballooning-unstable β will be referred to hereafter as the least stable configuration. Note that while the least stable configuration initially becomes unstable at β = 0.9%, it exhibits a second region of stability for 1.1% < β < 1.5% (Fig. 6.5c), probably due to the high bootstrap

114 oe ybl;seas i.65) oemresoelpoeaohrdet iia results. similar to due another one overlap yet markers limit Some stability 6.5c). ballooning Fig. the also encountering see however possible symbols; was without (open it equilibrium, which stable for value a highest compute the to or “highest symbols) Here, (filled ballooning-unstable text. becomes the plasma in discussed strength field and ( ratio current coil the of functions volume-averaged Highest (a) 6.4: Figure

∆R/a I (kA) h bs Max β (%) -0.3 -0.2 -0.1 θ θ θ -2 tilt tilt tilt 0 2 4 0 0 1 2 3 4 . . 4.5 4 3.5 3 =88 =78 2.5 =64 2 (c) (b) (a) ° ° ° , 0.08T,hollow , 0.08T,hollow , 0.08T,hollow I IL /I F P β b otta urn,ad()rltv hfao hf as shift Shafranov relative (c) and current, bootstrap (b) , and ) Current ratioI θ θ θ tilt tilt tilt =88 =78 =64 115 θ ° ° ° , 0.04T,hollow , 0.04T,hollow , 0.04T,hollow tilt Open: ballooninglimitnotfound Filled: maximumballooning-stable o h he ieetcngrtoso pressure of configurations different three the for IL β /I eest h ihs au eoethe before value highest the to refers ” PF θ θ θ tilt tilt tilt =88 =78 =64 ° ° ° , 0.08T,peaked , 0.08T,peaked , 0.08T,peaked β httegot aenvrbcmspstv o h high- the Note for volume. positive plasma relative becomes never the (b) rate within current; growth rate the growth bootstrap that ballooning (a) calculated maximum unstable. (c) ballooning shift; Shafranov becomes first configuration stable least ofiuainta eaeblonn-ntbea h lowest the 0 at ballooning-unstable became that configuration iue65 vlto fkyprmtr uigthe during ( paramaters key configuration of Evolution 6.5: Figure . 8T ekdpesr rfie.Tevria ahdln niae h au of value the indicates line dashed vertical The profile). pressure peaked T, 08 θ tilt 78 = -1 ∆R/a Max γ (s ) h -0.4 -0.2 -0.3 -0.2 -0.1

0.2 0.4 I (kA) ◦ bs , 0 0 0 1 2 3 4 4 3 2 1 0 I (c) (b) (a) IL /I F P 2 = . 75, B 116 β 0 = (%) . β 4T olwpesr rfie n o the for and profile) pressure hollow T, 04 cncridotfrtehigh- the for out carried scan β High- Least stableconfiguration configuration. β β configuration ( θ tilt unstable 88 = ◦ , I IL /I F P β twihthe which at 2 = β magnetic . 75, B = 2.75

2.5 PF /I IL I 2.25

2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 β (%)

Figure 6.6: Contour plot of ballooning stability in (β, IIL/IPF ) parameter space for the peaked- ◦ profile configurations with θtilt = 88 , B = 0.08 T. The shaded regions are unstable (γmax > 0).

current (Fig. 6.5a) and consequently high shear. Second regions of ballooning stability have been the subject of theoretical research for both tokamak and stellarator configurations (for example, Refs. [125, 116, 117]). Fig. 6.6 shows that this region could be accessed by a proper “trajectory” in a two-dimensional space spanned by heating power (roughly proportional to β and to the pressure gradient) and coil-current ratio (controlling the ι profile, hence magnetic shear).

The total bootstrap-currents Ibs (Fig. 6.4b) are low in comparison with the coil-currents (40 to 90 kA-turns in the IL coils). For the configurations with hollow pressure profiles, this is partly a result of Jbs < 0 near the axis and Jbs > 0 near the edge. These current profiles also lead to negative Shafranov shifts for many of the configurations (Fig. 6.4c). Fig. 6.7 shows the maximum ballooning-stable β values for each configuration considered in Fig. 6.4, plotted against the aspect ratio. The figure does not show configurations for which a stability threshold was not found. The maximum stable β did not exhibit a clear trend—growing for some configurations and decreasing for others. This is in contrast with tokamaks, where lower aspect ratios correlate with greater stability to ballooning modes and ideal kinks [30]. Characteristics of the high-β equilibrium are shown in Fig. 6.8. Fig. 6.8a-c compare the plasma geometry for β = 3.0% to the vacuum configuration (β = 0%). Due to the sign change of the bootstrap current profile (Fig. 6.8d), the radial shifts of the axis and the boundary are in opposite directions (Fig. 6.8c). The toroidal current significantly affects the rotational transform profile near the axis (Fig. 6.8e) but less so near the edge.

117 3

2.5 (%)

β 2

1.5

Max stable 1

0.5 1.5 2 2.5 3 3.5 4 Aspect ratio

Figure 6.7: Maximum stable β plotted against aspect ratio for each of the test configurations. Symbols are as defined in the legend in Fig. 6.4.

40 10 (a) (c) φ=0° (d) φ=46° ) 2 2 5 30 φ=90° 0 0 20 (kA/m Z (cm) -5 bs j -2 -10 β = 3.0% 10 0 1 10 (b) Z (cm) (e) 5 -10 0.5 0 ι -20 Axis, β=3.0% Z (cm) -5 Axis, β=0% β = 3.0% -10 LCFS, β=3.0% 0 β = 0% -30 LCFS, β=0% β = 0% 20 30 40 50 20 40 60 0 0.2 0.4 0.6 0.8 1 Normalized flux ψ R (cm) R (cm) n

◦ Figure 6.8: Properties of the high-β equilibrium attained with IIL/IPF = 3.25, θtilt = 78 , and the hollow pressure profile. (a) Flux surfaces for the equilibrium with β = 3.0%. (b) Flux surfaces for the same configuration with β = 0%. (c) Boundaries for the equilibrium with β = 3.0% (solid lines) compared with boundaries calculated with β = 0 (dashed lines) for three different poloidal cross-sections. (d) Profile of toroidal current (i.e., the calculated bootstrap current). (e) Profile of rotational transform with β = 3.0% (solid line) and β = 0% (dashed line).

118 h etn oe eurmnsfrteseaisotie ntepeiu eto a eroughly be can section previous laws. scaling the stellarator in follows outlined as scenarios estimated the for requirements power heating The requirements power Heating 6.6 abscissa. the along 6.4 Fig. with in associated shown profiles plots transform the rotational of the of offset an of principal be setting The to each shown symmetry. was stellarator errors two-field-period these the of misalign- break significant effect that have errors coils field CNT’s exhibit 2, and Chapter ments in shown as However, simplicity. computational of excursion pressure. lower smaller the the with as consistent well are as the 6.9e), 6.9d), (Fig. from (Fig. 6.9e) current (Fig. bootstrap transform the of rotational and 6.9a-c) (Fig. high- geometry surface flux in differences ( equilibrium stable least the for 2.75, but 6.8, Fig. Like 6.9: Figure Z (cm) Z (cm) -10 -10 10 10 -5 -5 eestimate We for here calculations the for assumed was symmetry stellarator perfect that noted be should It The 6.9. Fig. in shown are configuration stable least the of characteristics Corresponding 0 5 0 5 β β β (b) (a) θ =0% =0.9% tilt ofiuainrsl rmrl rmtedifferent the from primarily result configuration 03 050 40 30 20 88 = R (cm) ◦ n h ekdpesr profile. pressure peaked the and , I IL β /I s[7] as F P ec,icroaigCTsfil rosi xetdt euti noffset an in result to expected is errors field CNT’s incorporating Hence, .

Z (cm) -30 -20 -10 10 20 30 40 0 (c) β 04 60 40 20 = n R (cm) e k B B 119 LCFS, LCFS, Axis, Axis, T e 2 / + β β 2 β β =0% =0.9% =0% =0.9% φ φ φ µ n =90 =46 =0 0 i k ° ° ° B θ T tilt i

. 2 ι j (kA/m ) 0.5

and bs β -2 0 1 0 2 . . . . 1 0.8 0.6 0.4 0.2 0 0 = (e) (d) I IL . %,atie with attained 9%), /I F P Normalized flux ι h oe magnitude lower The . rmisvcu values vacuum its from ψ n I IL β β /I =0% =0.9% F P (6.5) = Here, ne and ni are the electron and ion densities (assuming a single dominant ion species), Te and Ti are the electron and ion temperatures, kB is Boltzmann’s constant, B is the magnetic field, and µ0 is the vacuum permeability. We assume a quasi-neutral plasma with ne = ni and, as in

Sec. 6.3, Ti = 0.3Te. Hence, the numerator of Eq. 6.5 simplifies to 1.3nekBTe. This, in turn, may be re-expressed in terms of heating power P as 1.3(τEP/V ), where V is the plasma volume and where τE is the energy confinement time.

We estimate τE using the ISS04 scaling law [126]:

2.28 0.64 −0.61 0.54 0.84 0.41 τE,ISS04 = 0.134frena R P n¯e B ι2/3 (6.6)

Here R and a are the plasma major and minor radius in meters, B is the magnetic field in Tesla, and ι2/3 is the rotational transform evaluated at two-thirds the minor radius of the LCFS. The heating power P (in MW in this formula) is treated as an independent variable. Incidentally, the ISS04 dataset involved several heliotrons/torsatrons and one device with circular coils (the TJ-II heliac). Also note that CNT is essentially a heliotron/torsatron (not a classical stellarator) with two ` = 2 “helical” coils of poloidal number m = 1 and toroidal number n = 1 which, in effect, are circular. It should be noted that the values of R, B, and line-averaged electron densityn ¯e (in units 19 −3 of 10 m ), are lower in CNT than in other devices which τE,ISS04 is based upon. Said otherwise, the scaling law is being extrapolated here. Furthermore, τE,ISS04 is determined based on current- free or nearly current-free plasmas, whereas the high-β, low-aspect-ratio equilibria evaluated in this chapter contain a small but finite bootstrap current.

The renormalization coefficient, fren, is a device-specific fitting parameter. Among the stel- larators in the ISS04 database, this value ranged from 0.25 for TJ-II to unity for a configuration of W7-AS. This parameter has not been calculated for CNT. However, it was observed that the parameter was inversely correlated with toroidal effective ripple, eff. This tendency follows from the association of greater ripple with a larger population of helically trapped particles. In partic- ular, eff(2/3) (evaluated at two-thirds of the minor radius) was found to relate to fren roughly as −0.4 fren eff(2/3) for values of eff(2/3) between 0.02 and 0.4 (see Fig. 7 in Ref. [126]). For CNT, ∝ ◦ a value of eff(2/3) of 1.6 was calculated for the configuration θtilt = 64 , IIL/IPF = 4.22. [127].

Extrapolating the relationship observed in the ISS04 database to the range of eff for CNT, we −0.4 posit fren (0.25 0.05)eff(2/3) , leading to an estimate of fren = 0.21 for CNT with upper ≈ ±

120 and lower bounds of 0.25 and 0.17, respectively.

τE is evaluated in two main regimes of electron density densityn ¯e: The first is the highest attainable before the plasma radiatively collapses, determined through a formula derived by Sudo [128],

r PB n = 2.5 . (6.7) Sudo a2R

A similar formula was derived at W7-AS [129] and yields similar results, not shown for brevity. The second is the cutoff density above which microwaves for ECH cannot propagate. For heating at the fundamental harmonic in the ordinary mode at frequency ωrf = 2π 2.45 GHz, this is (Eq. 4.5): ×

 m n = 0 e ω 2. (6.8) co,O e2 rf

For heating at the second harmonic in the extraordinary mode using the same heating frequency (and, therefore, half the field), the highest density at which propagation may occur throughout the plasma corresponds to the right-handed cutoff, effectively half of the ordinary-mode cutoff:

1  m n = 0 e ω 2; (6.9) co,X2 2 e2 rf

Results of these calculations for the high-β and least-stable configurations are shown in Fig. 6.10. The three black curves shown in each plot, from top to bottom, give the value of β determined using Eqs. 6.5-6.6, with densities from Eqs. 6.7, 6.8, and 6.9, respectively. Thus, the blue-shaded region below the curve for β(nco,O) is accessible with underdense plasmas, and the red-shaded region between β(nco,O) and β(nSudo) corresponds to β values that are accessible to overdense microwave heating.

The red lines in Fig. 6.10a-b are (P, β) contours corresponding to Te = 30 eV, determined by inverting Eqs. 6.5 and 6.6. To the left of these lines, Te is lower; to the right, Te is higher. Thus, to obtain β = 3.0% in the high-β configuration while maintaining a minimum Te of 30 eV, about 1.8 MW of power will be needed (Fig. 6.10a). To obtain β = 0.9% in the least stable configuration at Te = 30 eV, about 60 kW will be needed. It should be noted, however, that these estimates are sensitive to fren (Eq. 6.6). If these estimates are redone using the upper and lower bounds mentioned above (0.17 < fren < 0.25), a range of 1 to 3 MW is established for the high-β configuration and

121 Parameter Present Least stable High-β θtilt (deg) 78 88 78 a (m) 0.14 0.12 0.13 R (m) 0.31 0.30 0.32 P (kW) 0.5-8 40-100 1,000-3,000 17 −3 ne (10 m ) 0.5-3 50 200 B (T) 0.08 0.08 0.08 ι2/3 0.36 0.62 0.09 Te (eV) 4-8 30 30

Table 6.2: Comparison of present CNT experimental parameters with those of the least stable and high-β configurations.

40 to 100 kW for the least stable configuration. Both configurations fall within the overdense region for heating with 2.45 GHz and will therefore both require overdense microwave heating. The parameters of these configurations are compared with present experimental conditions in Table 6.2. Note that ι2/3 for the least stable configuration exceeds that of the high-β configuration by a factor of nearly 7. This corresponds to a factor of

2 increase in τE (Eq. 6.6) due to ι2/3 alone. This, combined with the reduced β, explains why so much less power is required for the least stable configuration.

122 -1 10 (a) high-β inaccessible 10-2 overdense β

-3 10 n = n Sudo T = 30 eV n = n underdense co,O n = n co,X2 10-4 10-2 10-1 100 101 Power (MW) -1 10 (b) least stable inaccessible

10-2 overdense

β T = 30 eV

10-3 underdense

10-4 10-2 10-1 100 101 Power (MW)

Figure 6.10: Accessible β at different levels of heating power according to stellarator scaling laws for (a) the high-β configuration and (b) the least stable configuration. Values of β above the solid line are considered inaccessible because the plasma density exceeds the Sudo limit. Values below the dashed line correspond to plasmas underdense to 2.45 GHz ECH in the extraordinary mode; values between the solid and dashed lines could be attainable by overdense heating. The red lines are contours of Te = 30 eV in the (P, β) parameter space. The red circles indicate the power levels needed to attain the maximum β in each configuration.

123 Chapter 7

Summary and conclusions

CNT, formerly an experiment for non-neutral plasma physics, has now been established as a fusion- oriented machine with a demonstrated ability to address topics of interest to the fusion community. The work in this thesis has contributed to our abilities to diagnose stellarator error fields, an important topic for devices whose magnetic fields may be compromised by small misalignments in the coils. It has developed an image-based technique for diagnosing emission from three-dimensional plasmas, a valuable asset in a field that prizes non-invasive and space-saving diagnostics. It has also demonstrated overdense microwave heating, a capability that is often necessary to initiate and sustain a high-β fusion plasma. In addition, this thesis has shown that CNT is fit for analysis by widely-used simulation codes such as VMEC. Initial results from these codes indicate that CNT has the potential to seek—and perhaps even reach—a stellarator β limit, a phenomenon that to this day has not been found experimentally. Toward the goal of diagnosing error fields, stellarator coil misalignments were inferred from flux surface measurements. The measurements consisted of photographs of Poincar´e cross-sections obtained using a standard stellarator technique with an electron beam and a phosphor- coated rod. An algorithm was developed to extract the geometric properties of the cross-section from these photographs and compare them to simulated cross-sections from a field line tracing code. The Newton-Raphson method was then employed to iteratively perturb the positions of the coils used by the field line tracing code until the code’s output agreed with the flux surface mea- surements. This technique has the potential to be generalized to any stellarator device in which Poincar´ecross-section measurements are possible.

124 The improved understanding of CNT’s own field errors was crucial for multiple subsequent projects, including diagnostic alignment and three-dimensional reconstructions. For one, an onion- peeling technique was developed to determine radial profiles of plasma emission based on single wide-angle images. The onion-peeling method makes the assumption that the plasma can be approximated as a set of discrete, nested, uniformly-emitting layers, which is reasonable for a stellarator equilibrium. In addition, the brightness of each pixel in the plasma image is assumed to take contributions from each layer in proportion to the layer’s emissivity as well as the length of the line-of-sight segment within the layer. These assumptions together imply a straightforward system of linear equations relating the layer emissivity to pixel brightness, which can be solved using linear least-squares. This was implemented on CNT with a fast camera after the installation of non- reflective covers in the vacuum vessel to ease the analysis. The concept was tested using hot-cathode glow discharges that approximately illuminated single layers of the plasma. The reconstructed profiles exhibited the expected features, indicating that the onion-peeling method was working as desired. This technique should be useable on any magnetic fusion device with nested flux surfaces whose geometry is well understood. The first generation of microwave plasma studies in CNT was conducted with a low-power, 2.45 GHz, 60 Hz-pulsed magnetron fixed to a viewport. During this time, probe circuitry and data analysis techniques were adapted to suit the pulsed microwave plasmas. In particular, a set of functions were developed to acquire time-resolved, swept Langmuir probe traces based on ensembles of a few hundred pulses. These measurements led to the observation of non-Maxwellian electron distributions exhibiting at least two characteristic temperatures. In addition, it was found that incorporating a hot, biased (emissive) cathode filament into a microwave discharge permitted marked increases in plasma density such that the density using both plasma sources together exceeded the sum of what could be attained using either source individually, suggesting a synergy between two commonly-employed plasma start-up methods. The microwave heating system was then upgraded to inject powers of up to 10 kW into the plasma. Along with the power upgrade, a launch antenna consisting of an open circular waveguide was designed. The antenna enters the chamber through an access port at an oblique angle chosen so that it aims at the magnetic axis. In addition, the antenna extends from the chamber wall almost to the edge of the plasma, allowing for better localization of the power deposition. The transmission

125 line upstream of the antenna permitted arbitrary rotation of the polarization angle of the electric field, permitting comparisons between plasmas heated by O-mode-launched and X-mode-launched waves. The plasmas produced by this new system were found to exceed the O-mode cutoff density by a factor of more than 4 with 8 kW of injected power. Overdense plasmas were seen with heating powers as low as 0.5 kW. An analysis of the response of profiles of temperature and density to changes in heating power and magnetic field strength led to the the working hypothesis that the most likely heating mechanism is the collisional damping of slow X waves at the UHR. These results have provided a motivation for the development and deployment of advanced codes for analyzing wave dynamics in three-dimensional plasmas with length scales comparable to the heating wavelength. Ultimately, it was clear from these experiments that the O-mode and X-mode cutoffs do not impose hard limits on the attainable density in CNT plasmas. This indicates that the density (and temperature) may be increased much more if enough additional power is injected to overcome the radiation barrier, clearing the way for experiments at high β. Finally, ideal MHD simulations and stellarator scaling laws predicted that a β limit as low as 0.9% may be reached in the CNT configuration with less than 100 kW of heating power. This conclusion was reached with the aid of widely-used stellarator codes. High- β plasma parameters were extrapolated for a range of magnetic configurations using an iterative procedure to calculate equilibria with self-consistent bootstrap current profiles. These equilibria, in turn, were evaluated for ballooning stability to find the configuration with the highest attainable β at which plasmas were expected to be stable, as well as the configuration with the lowest β at which a ballooning stability limit might be reached.

The magnetic configuration—θtilt and IIL/IPF —as well as the pressure profile had a strong ◦ influence on ballooning stability. A configuration with θtilt = 78 , IIL/IPF = 3.25 and a hollow pressure profile was found to be ballooning stable for β = 3% with no ballooning stability limit ◦ identified. On the other hand, a configuration with θtilt = 88 , IIL/IPF = 2.75, and a peaked pressure profile became ballooning-unstable at β = 0.9%. An additional interesting feature of this configuration was a second region of stability for 1.1% < β < 1.5%, which may be accessible by ramping up the heating power using a lower initial IIL/IPF and could serve as an interesting test

126 of theory. Energy-balance calculations using the ISS04 stellarator scaling predicted that that plasmas in the highly stable configuration (β 3.0%) should be attainable with 1-3 MW of heating power, ≥ whereas the unstable configuration may reach its ballooning limit (β = 0.9%) with less than 100 kW of heating power.

127 Chapter 8

Future research directions

CNT is a versatile device that offers opportunities to contribute to magnetic fusion research in many different ways. Some ideas for future research directions, enabled or inspired by the investigations in this thesis, are presented below.

Improvements to the error field algorithm. The optimization algorithm used to deduce coil misalignments in Chapter 2 has room for improvement. Some upgrades might include (1) simultaneously optimizing to data from multiple current-ratios using F vectors that include geo- metric parameters for multiple Poincar´ecross-sections, (2) operating the field line tracer at greater numerical precision to reduce the uncertainty in numerically generated Fourier coefficients and al- low for smaller finite difference intervals for the computation of the Jacobian, (3) expanding the p vector to include other sources of error including coil deformations, uncompensated coil leads, and displacements of the PF coils, and (4) investigating ways of generalizing the X vector to in- clude information about Poincar´edata inside islands, possibly analogous to generalizations of 3D toroidal equilibria found in codes like PIES [130], SIESTA [131], and SPEC [132]. Many of these improvements will be more demanding computationally, although much of the algorithm can be parallelized (in particular, the calculation of the covariance and Jacobian matrices).

Incorporation of the scrape-off layer in the onion-peeling model. As mentioned in Chapter 3, reconstructions of microwave discharges tended to ascribe unexpectedly high values to the outermost layer of the plasma. Since the onion-peeling model currently does not allow for visible emission outside the last closed flux surface, it was conjectured that the high edge emissivity was a spurious result of fitting an image with a bright scrape-off layer to a model that assumes a

128 dark scrape-off layer. One way to incorporate the scrape-off layer would be to add external layers to the plasma model through the calculation of “ghost surfaces,” or toroidal surfaces that conform as best as possible to the magnetic field in regions where true flux surfaces do not exist. Some definitions for such surfaces are proposed in Ref. [133]. Once defined, these surfaces could be easily incorporated to the onion peeling technique as layers.

Time- and spectrally-resolved emissivity profile studies. While the development of the onion-peeling model in Chapter 3 concentrated on single images of plasmas, the camera’s fast- framing capabilities would permit the evaluation of a succession of profile inversions at nearby time-points in the plasma evolution. In addition, the placement of optical filters in front of the fast camera lens would allow profiles from different frequency bands or emission lines to be distinguished.

Island divertor studies. As mentioned in Sec. 1.2.4, island divertor configurations are of great interest in the magnetic fusion field because of their predicted tendency to produce broad heat-flux patterns on plasma facing components, which are put less stress on the materials than the narrow patterns typical of tokamak divertors. As first observed in Ref. [57] and understood via the coil misalignments inferred in Chapter 2, CNT—due to its field error—can access a magnetic configuration with a large island chain on the edge of the confining region (Fig. 2.14b). Such a divertor would be relatively simple to install in CNT. It would permit comparisons of the strike point pattern, diagnosed with an infrared camera, to theoretical predictions.

Full-wave, 3D simulations of overdense heating. The preliminary 2D full-wave calcula- tions of first-pass microwave heating shown in Chapter 5 were not able to fully explain observed trends in the overdense plasmas (in particular, the decrease in ne with increasing B ). It is expected | | that microwaves reflected off the vessel walls (and possibly the IL coils) play a significant role in heating the plasma and cannot be neglected, as was the case with overdense heating at 2.45 GHz in TJ-K [97]. To accurately model reflections in the complex geometry of the CNT vessel, 3D calcula- tions will be necessary. CNT’s plasmas would hence offer a demanding—and useful—experimental test to validate new or existing 3D codes.

Further synergy studies. Studies of the synergy between microwaves and thermoelectrons described in Chapter 4 are in their infancy and there are many opportunities for further experiments to further characterize the interaction. For example, how does the synergy affect the minimum gas

129 pressure requirements for reliable plasma start-up? Does the effect change if different working gases are used? Is the effect still present as microwave power is ramped up? All of these questions would help to address the extent to which the synergy may be useful for CNT and for other devices.

High-β studies. In Chapter 6 it it was predicted that volume-averaged β of 3% or greater should be attainable in the CNT configuration, and that certain configurations should become ballooning-unstable at β as low as 0.9%. A second region of stability was also observed in the most-unstable configuration. Upgrading CNT to test these predictions experimentally would be of great interest to all stellarator experiments, as β limits have not yet been observed on stellarators. Testing these limits would require a significant power upgrade: 100 kW for β = 0.9%, and 3 MW for β = 3%. However, conducting these studies on CNT would accomplish major objectives within the fusion research community.

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144 Appendices

145 Appendix A

Parametrization of Poincar´e cross-sections

The level of agreement between two sets of Poincar´edata is determined by fitting the nested flux surface cross-sections to a discrete set of geometric parameters X. The components of X are the coefficients of a linear combination of orthogonal functions consisting of a Fourier series in the poloidal angle θ and a polynomial series in the normalized minor radius ρ. For a flux surface characterized by some particular ρ, the R and Z coordinates of the point with poloidal angle θ are expressed as

M X R(ρ, θ) = R0(ρ) + Rcm(ρ) cos(mθ) m=1

M X + Rsm(ρ) sin(mθ) (A.1) m=1

M X Z(ρ, θ) = Z0(ρ) + Zcm(ρ) cos(mθ) m=1

M X + Zsm(ρ) sin(mθ) (A.2) m=1 As indicated in the above representation, the coefficients of each Fourier mode are themselves functions of ρ. This dependence is represented as a linear combination of polynomials Ps(ρ); for example:

146 s Ps(ρ) 0 1 1 √3 (2ρ 1) − 2 √5 6ρ2 6ρ + 1
− 3 √7 20ρ3 30ρ2 + 12ρ 1
4 3 70ρ4 −140ρ3 + 90ρ2 − 20ρ + 1
− −

Table A.1: The first few polynomials Ps(ρ) as discussed in the text.

Rc1(ρ) = Rc10P0(ρ) + Rc11P1(ρ) + ... + Rc1SPS(ρ) (A.3)

The polynomials Ps(ρ) are chosen to be orthonormal in the inner product defined by

Z 1 Pi,Pj = Pi(ρ)Pj(ρ)dρ = δij, (A.4) h i 0 and the first few polynomials in this set are listed in Table A.1. The vector X of geometric parameters, then, is just a list of these coefficients:

X = R00, ..., R0S,Rc10, ..., Rc1S, ..., RcMS, {

Rs01,RsMS,Z00, ..., ZsMS (A.5) } To ensure that the set of coefficients is unique for a particular set of Poincar´edata, it is necessary to use a precise definition for ρ and θ. In this work, ρ for a particular cross-section is defined as p A/Amax, where A is the enclosed area and Amax is a reference area not to exceed the area enclosed by the last closed flux surface. In this way, ρ = 0 at the magnetic axis and ρ = 1 on the edge of the region of the Poincar´ecross-section to be parametrized. The poloidal angle θ is defined to advance in direct proportion to the arc length along a flux surface cross-section and is set to zero on the outboard side where Z is equal to Z(ρ = 0).

147 Appendix B

Determination of X for Poincar´edata

An example of a fit of coefficients to experimental data is shown in Fig. B.1. As discussed in Section 2.1, the experimental data were separated according to flux surface. For each flux surface, a curve was fit iteratively to the Poincar´epoints in such a way as to minimize the disagreement with the data points. An example of a set of fit curves is shown in Fig. B.1a. When the fitting iterations are complete, the curve was parametrized in θ as described in A and the ith measured surface was as- signed a ρi value based on the enclosed area. Each fit curve (R (ρi, θ) ,Z (ρi, θ)) was then projected M onto a Fourier series, yielding one set of coefficients Rcm (ρi) ,Rsm (ρi) ,Zcm (ρi) ,Zcm (ρi) { }m=0 for each value ρi. Sets of corresponding Fourier coefficients (e.g., Rc1 (ρi) i) were then fit by linear { } least-squares onto polynomials in ρ of degree S, from which the polynomial coefficients as shown in Eq. A.3 were obtained. The calculations described in this chapter used M = 14 and S = 7. Because experimental data points are often limited near the magnetic axis, certain constraints are enforced in the least-squares fitting in order to avoid spurious oscillations in the ρ polynomials. In particular, all polynomials for Fourier coefficients of m 1 are constrained to be zero at the ≥ axis. In addition, all coefficients of m 2 are constrained to have a first derivative of zero at the ≥ axis. Furthermore, coefficients with m = 0 are only expanded to second order in ρ (i.e., requiring s 2). ≤ Once coefficients are calculated for a set of Poincar´edata, they are arranged into a vector X containing (S + 1)(4M + 2) elements. Because of the constraints described in the above paragraph, 2(S 2) + 8M 4 of those components are redundant, and are therefore removed for subsequent − − analysis.

148 Figure B.1: (a) Experimental Poincar´edata obtained for the current-ratio 3.68 (black dots) and fit curves to selected flux surfaces (blue lines). (b) The same experimental data overlaid with level curves from the fitted geometric parameters in red. Radially extending red lines are curves of constant θ; closed red loops are curves of constant ρ.

149 Ideally, the coefficients X should be unique for a given set of Poincar´edata. However, in practice the coefficients will vary slightly depending on which Poincar´epoints are used for this fit. Specifically, for experimental data, this depends on which surfaces are measured and which of the subsequent pixels are used for the analysis. For numerical data, the dependence is on the locations where the field line traces are initialized. Hence, each component of X will have an associated uncertainty that may be correlated with that of other components. Sec. 2.4.1 and 2.4.2 will describe how these correlations are accounted for in the optimization.

B.0.1 Treatment of magnetic islands

If a chain of large islands exists within a cross-section, it will be identified during the curve-fitting process by poloidal gaps in the Poincar´edata. Since the parametrization described in A only permits closed flux surfaces, the islands are simply ignored during in the fits of Fourier coefficients to the polynomials Ps(ρ) described above. Since some Fourier coefficients may change abruptly from the axis-facing side of the island to the edge-facing side, it may be necessary to incorporate higher-order polynomials Ps(ρ) in the fits, leading to more components in the X vector. If one were to plot the curves of constant ρ specified from the geometric parameters X (as in Fig. B.1b), one would observe a continuous deformation from the shape of the last closed flux surface on the axis-facing side of the island chain to the first closed surface on the edge-facing side. These deformed curves in the island region clearly do not reflect the actual island geometry and are simply a result of interpolation between the data for the core-facing side and the edge-facing side. In other words, the laminar flux surfaces occurring in place of the islands reflect the fact that the X vector contains information only about the flux surfaces on either side of the island chain.

150