Measurements of Two-Fluid Relaxation in the Madison Symmetric Torus

MEASUREMENTS OF TWO-FLUID RELAXATION IN THE MADISON SYMMETRIC TORUS

Joseph C. Triana

A dissertation submitted in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

(Physics)

at the

UNIVERSITY OF WISCONSIN–MADISON

2017

Date of ﬁnal oral examination: 4/21/17

The dissertation is approved by the following members of the Final Oral Committee: John S. Sarff, Professor, Physics Abdulgader F. Almagri, Senior Scientist, Physics Carl R. Sovinec, Professor, Engineering Physics Paul W. Terry, Professor, Physics Jan Egedal, Associate Professor, Physics c Copyright by Joseph C. Triana 2017 All Rights Reserved i

ACKNOWLEDGMENTS

I would like to thank Abdulgader Almagri for all of the guidance he provided for making the probes, programs, and presentations that form the basis for this thesis research. And for Kringles. Those are delicious. I would also like to thank John Sarff for calmly and patiently describing the physics I was studying in the greater context of plasma physics in general. A special thanks to Josh Sauppe and Carl Sovinec whose work in NIMROD and subsequent discussions proved to be very complimentary to the research conducted here. To all my friends in the Physics Department, thank you for making my time learning and working here enjoyable. Does anyone want to watch Avatar? And to all my friends in Madison, thank you for all the fun things I’ve gotten to do. Whether it’s camping out for tickets to the Great Taste or helping sand Bondo in the sweltering heat of summer, I have throughly enjoyed my time in Madison. To beer and ACen. To all my family, wherever you may live now, thanks for all the memories growing up and making new ones as we grow older. To my parents, thanks for having somewhere warm to visit during the depths of a Midwestern winter. It was nice to be able to thaw out in the sun. And last but not least, thank you Nyla. You’re a bad girl in the best way possible. Your pet human Kendra isn’t too bad either. But in all seriousness, thank you so much Kendra. You have been there for me whenever I needed you and helped me stay focused whenever I started to get distracted. You have introduced me to so many people and activities that I would have never done on my own. Because of you, I was able to enjoy my time as a grad student. And because of you, I know I will be able to enjoy this new chapter in my life too. DISCARD THIS PAGE ii

TABLE OF CONTENTS

Page

LIST OF FIGURES ...... v

ABSTRACT ...... viii

1 Introduction ...... 1

1.1 Magnetic Relaxation ...... 2 1.2 RFP Sustainment ...... 4 1.3 Tearing Modes ...... 5 1.4 Ohm’s Law ...... 7 1.4.1 Momentum Equation ...... 10 1.5 Overview of Results ...... 10

2 Experimental Design and Diagnostics Used ...... 12

2.1 The Madison Symmetric Torus ...... 12 2.1.1 MST Directions ...... 14 2.1.2 Mode Decomposition ...... 14 2.1.3 Controllable Experimental Parameters ...... 16 2.1.4 Multiple Helicity Operation ...... 17 2.1.5 Available Diagnostics ...... 18 2.2 The Deep Insertion Hall Probe ...... 19 2.2.1 Bench Calibration ...... 24 2.2.2 In Situ Calibration ...... 25 2.2.3 Measurement Uncertainties ...... 25 2.2.4 Uncertainty Propagation ...... 27 2.3 Complementary Diagnostics ...... 28 2.3.1 Far Infrared Interferometry ...... 28 2.3.2 Thompson Scattering ...... 28 2.3.3 Beam Based Diagnostics ...... 29 2.4 Summary ...... 29 iii

Page

3 Analysis Methodology ...... 31

3.1 Probe Perturbation ...... 31 3.2 Port Hole Perturbation ...... 32 3.3 Flux Surface Average ...... 34 3.4 Equilibrium Reconstruction ...... 36 3.5 Shafranov Shift Correction ...... 40 3.6 Shot to Shot Variation ...... 41 3.7 Pseudospectrum ...... 45 3.8 Summary ...... 48

4 Probe Measurements ...... 49

4.1 Magnetic Field Measurements ...... 49 4.2 Equilibrium Reconstruction ...... 51 4.3 Correlated Lorentz Force Calculation ...... 53 4.4 Measurements of Mean-Field Ohm’s Law ...... 56 4.4.1 ηJk ...... 56 4.4.2 Ek ...... 58 D E 1 ˜ ˜ 4.4.3 n e J × B ...... 60 o k D E 4.4.4 v˜ × B˜ ...... 60 4.5 Current Relaxation ...... 62 4.6 Momentum Evolution Measurements ...... 65 4.7 Reversing Plasma Current and Momentum ...... 68 4.8 Measurements of the Tearing Mode Structure ...... 69 4.9 Conclusion ...... 73

5 Simulation Comparison ...... 74

5.1 NIMROD Background ...... 74 5.2 NIMROD Results ...... 76 5.3 Comparison with Experimental Measurements ...... 78 5.4 Summary ...... 84

6 Conclusions ...... 87

6.1 Deep Insertion Hall Probe ...... 87 6.2 Two-Fluid Contributions to Relaxation ...... 88 6.3 Suggestions for Future Work ...... 89 iv

Appendix Page

LIST OF REFERENCES ...... 91

APPENDICES

Appendix A: Supplemental Derivations ...... 98 Appendix B: DIHP Operation ...... 105 Appendix C: Physical Directions on MST ...... 110 DISCARD THIS PAGE v

LIST OF FIGURES

Figure Page

1.1 Illustration of resonant surfaces in typical MST plasmas ...... 6

2.1 Schematic of directions on MST ...... 15

2.2 Illustration of coil form used for DIHP construction ...... 21

2.3 View of coil windings without particle shield ...... 22

2.4 Illustration of radial locations of prior measurements ...... 23

3.1 Two plasma discharges with the DIHP inserted(black) and withdrawn(red) ...... 33

3.2 Plot of the parallel edge electric ﬁeld at different points during ensemble averaging . . 36

3.3 The phase of the m = 0, n = 1 mode during relaxation for the events used in this ensemble ...... 37

3.4 Poloidal plane directions used in Grad-Shafranov derivation ...... 39

3.5 An illustration of the magnetic ﬂux surface projected onto the probe geometry. . . . . 42

3.6 Effect of δ on the measurement of the Hall EMF ...... 45

p r 3.7 Histogram showing the measured Vj of the Hall EMF evaluated at a = 0.65 . . . . 46 4.1 Five ensembles taken at different radii, overplotted, measured at t = 0 ...... 50

4.2 Equilibrium Magnetic Field during relaxation at t = 0.0 ms ...... 51

4.3 q proﬁle at t = −0.5 ms and t = 0.0 ms relative to a relaxation event ...... 52

4.4 Toroidal magnetic ﬁeld from MSTFit reconstruction with probe data overplotted . . . 53

4.5 Induced electric ﬁeld from MSTFit reconstruction with probe data overplotted . . . . 54 vi

Figure Page

4.6 FIR interferometry data inverted with MSTFit reconstruction ...... 54

4.7 Parallel current before and during a relaxation event ...... 57

4.8 ηJk measurements with η calculated from FIR interferometry and TS data ...... 58

4.9 Calculated induced parallel electric ﬁeld from changing magnetic ﬂux ...... 59

4.10 Hall dynamo measurement from DIHP ...... 61

4.11 The inferred MHD dynamo term from a balance of two-ﬂuid Ohm’s law ...... 62

4.12 Normalized lambda proﬁle from the alpha model with DIHP data overplotted . . . . . 64

4.13 Change in core plasma momentum inferred from the m=1, n=6 tearing mode . . . . . 66

4.14 Measured Maxwell stress from the DIHP during relaxation ...... 67

4.15 Experimental data with alpha model overplot with positive lambda ...... 70

4.16 Terms in Ohm’s law during a positive lambda run campaign ...... 71

4.17 Two-ﬂuid contributions from both λ scenarios ...... 71

4.18 Calculated structure of the m = 0, n = 1 mode on the top, the m = 1, n = 6 mode on the left, and the m = 1, n = 10 on the right during relaxation ...... 72

5.1 NIMROD simulation of MHD and Hall dynamos[62] ...... 77

5.2 Time averaged Hall and MHD emfs from the second event in Fig 5.1, courtesy J. Sauppe 79

5.3 Time averaged Maxwell and Reynolds stresses from the second event in Fig 5.1, courtesy J. Sauppe. The sign of the Maxwell stress has been inverted for amplitude comparison with the Reynolds stress ...... 80

5.4 Comparison of DIHP and NIMROD EMF proﬁles ...... 81

5.5 Illustration of the inferred global proﬁle of Ohm’s law from previously measured data on MST ...... 83

5.6 Parallel plasma velocity evolution measured at several radii [55] ...... 85 vii

Appendix Figure Page

A.1 Directions of Eφ calculation in MST ...... 99

A.2 Directions of Eθ calculation in MST ...... 101

B.1 Illustration of the assembly of the probe parts discussed ...... 105 viii

ABSTRACT

This is a dissertation for the completion of a Doctorate of Philosophy in Physics degree granted at the University of Wisconsin-Madison.

Recent measurements and extended MHD simulations expose the importance of two-fluid physics in the relaxation and self-organization of the current and momentum profiles in RFP plasmas. A hallmark of relaxation is that the inductive electric field is not balanced by resistive dissipa-

tion, prompting the study of fluctuation-induced emfs in the generalized Ohm’s law, Ek − ηJk = D E D E ˜ 1 ˜ ˜ − v˜ × b + ne j × b , where k is with respect to hBi and the two terms on the right known k k as the MHD and Hall dynamo terms, respectively. The Hall emf is measured in the outer half of the MST plasma minor radius using an armored deep-insertion Hall probe. The emf matches pre- r r viously measurements in the edge ( a > 0.8) but in the new region examined (0.6 < a < 0.8) it is much larger than E − ηJ, implying the MHD dynamo must also be large and oppositely directed. Recent nonlinear simulations that include two-fluid effects using the extended-MHD NIMROD code show complex radial structure for the emf terms, but the size of the measured Hall emf is much larger than predicted by the simulations. In the two-fluid model, the Hall dynamo couples to the parallel momentum as the mean-field Maxwell stress. The simulations predict relaxation of the parallel flow profiles that is also qualitatively consistent with measurements in MST plasmas. 1

Chapter 1

Introduction

The progress in nuclear fusion research is advanced by numerous experiments across the globe, each lending a unique viewpoint on some of the problems that a viable reactor faces. One of the most fundamental issues is plasma confinement and the instabilities associated with it. By understanding those instabilities, a commercially viable fusion reactor can be created, alleviating many of the energy related issues that plague the world today. At the University of Wisconsin-Madison exists one of the world’s largest Reversed Field Pinches (RFP) [17]. The RFP is a long studied fusion device that utilizes high magnetic shear to contain the plasma. It serves as a testbed for researching various plasma phenomena important for progressing our understanding of high temperature plasmas. One essential subject studied is the spontaneous rearrangement of magnetic field topology, creating a more stable configuration. This process is part of a larger class of phenomena called magnetic self-organization and allows the RFP to operate for much longer times than would otherwise be permissible due to resistive dissipation of the magnetic fields. The RFP refers to a particular arrangement of magnetic fields used to contain a hot toroidal plasma [4]. The toroidal and poloidal magnetic fields are comparable in magnitude, differing from the Tokamak which has a much larger toroidal field relative to its poloidal field. The RFP has developmental origin in the ambitious project to create a viable fusion reactor. A large amount of research today in fusion is focused on the Tokamak but studies have shown that the RFP is an alternative method for achieving this goal. The TITAN reactor study [13] exhibits the capabilities of an RFP fusion reactor although several engineering and physics problems remain, many of which are present for Tokamak fusion reactors. RFP research bolsters our ability to overcome the 2 hurdles of achieving fusion ignition by providing an auxiliary reactor design and elucidating the behavior of thermonuclear plasmas in new parameter spaces. The work in this thesis aims to understand the physics during magnetic relaxation, measuring the relative strength of terms that dictate the behavior of the plasma. Only recently have localized measurements shown that two-fluid effects, representing the separation of electron and ion dynamics, are necessary for the accurate portrayal of plasma relaxation in the RFP [19, 43]. Quasilinear theory predicts that theses effects are confined to narrow regions in the plasma [50]. However, recent nonlinear simulations have shown that these narrow regions are significantly broadened across the plasma, leading to global profile modification [39]. One of the key results of this thesis is the measurement of two-fluid effects from the wall to the plasma mid-radius, showing the impact of these effects is global in nature rather than constrained to a narrow region, agreeing with simulations. We also find that these effects are responsible for the relaxation of the current density and momentum profiles. Refined knowledge of these terms help inform and constrain computational models striving to recreate the physics seen in experiment, leading to more accurate simulations. Additionally these measurements act as a benchmark, alerting researchers to areas where different physical effects are important. This thesis will serve as a guide to understand the plasmas that are measured and how they relate to other plasmas of interest. Background into the measurements and methodology will be given to ensure a fruitful transmission of information.

1.1 Magnetic Relaxation

The RFP is rich in complex plasma phenomena that are the subject of numerous papers and theses. This thesis will focus on the hallmark observation in an RFP, magnetic relaxation. Mag- netic relaxation refers to a toroidal plasma’s tendency to approach a lower energy state through a redistribution of the magnetic topology. The theory of magnetic relaxation employs a global approach and does not invoke specific internal mechanisms to relax the plasma. A toroidal plasma can relax into a lower energy state if there is sufficient toroidal field and toroidal current. This has been observed in multiple experiments and has been studied for the last 50 years but contemporary 3 measurements and simulations have finally exposed the importance of two-fluid effects during this process. If one assumes the global magnetic helicity is an invariant of plasma motion, minimizing the magnetic energy at a fixed global helicity, then the resultant configuration, called a Taylor state, is predicted [74]. The magnetic helicity is defined, K ≡ H A · B dV , where A is the magnetic vector potential given by the expression B = ∇ × A. Taylor argued that if the global helicity is robustly conserved, the magnetic field relaxes into configurations satisfying the equation

∇ × B = λB (1.1) where λ is a constant. On the Madison Symmetric Torus, the magnetic helicity is observed to be conserved more than the energy, supporting Taylor’s hypothesis [36]. Two-fluid simulations also find helicity to be conserved better than magnetic energy during relaxation events [64]. Coupling Eq. 1.1 with Ampere’s law, a Taylor state is a configuration in which the current is parallel to the magnetic field. Additionally these solutions have a reversed toroidal field near the edge if critical values for the toroidal current and field are exceeded. Reversed field pinches operate on this principal and relaxation of the magnetic field routinely occurs. In RFPs, magnetic relaxation is found in two regimes. 1) During the initial formation of the RFP. The plasma undergoes large changes in the magnetic field profiles that result in the typical reversed toroidal magnetic field the experiment is known for [54, 47]. 2) During plasma sustainment. As the toroidal field resistively decays, there are discrete events that cause the plasma to return to a Taylor-like state [5, 16, 76]. The initial formation of the RFP involves continuously evolving plasmas that are extremely difficult to measure with reasonable precision. The second type of relaxation is marked by events that are highly reproducible as we are able to control the plasma equilibrium to a much greater degree than plasma startup. These events are also responsible for the sustainment of the RFP plasma. 4

1.2 RFP Sustainment

From the name alone, there are multiple physics implications from the RFP. A reversed field in a steady state equilibrium means that a location exists in which the toroidal field is equal to zero but the derivative is not, allowing a poloidal current to exist via Ampere’s law. This poloidal current, parallel to the magnetic field by definition, implies that a parallel electric field must exist if resistive Ohm’s law is in balance, Ek = ηJk, where η is the plasma resistivity. The electric field will be induced by changing toroidal flux and a system with a reversed field cannot be in a steady state equilibrium. Therefore the plasma will decay away at a rate determined by the resistivity. Examining the magnetic diffusion equation

∂B η = ∇2B (1.2) ∂t µo and substituting values for the discharges studied in this thesis, we find that the plasma has a decay time of about 20 ms; however, reversed discharges are observed for over 50 ms. A one-dimension modeling of plasmas on the ZT-40 device [79] have shown that the reversed experimental discharges last for times much longer than the modeling predicts [7]. Experiments on the RFX-mod [68] have demonstrated that reversed discharges can be sustained indefinitely, as long as enough plasma current is supplied [49]. For the RFP to persist against resistive dissipation, a regeneration of toroidal flux needs to occur. This flux generation is referred to as an RFP dynamo as it bears resemblance to astrophysical dynamos. The RFP dynamo converts the externally induced poloidal flux into toroidal flux, pushing the plasma towards a Taylor state before the toroidal field decays again. These events are impulsive and are precipitated by the tearing instability in the RFP [81, 11]. It should be noted that the cyclic process of flux generation and decay is referred to as sawteeth in the RFP community due to signals resembling sawtooth oscillations in the tokamak community [80]. However, we wish to emphasize the connection to magnetic relaxation and will continue to refer to the process as such. 5

1.3 Tearing Modes

The tearing mode is a resistive instability and as the name implies, tears apart magnetic field lines and reconnects them in new topologies. This process is part of a larger class of phenomena known as magnetic reconnection and occurs in many laboratory experiments [37] and astronom- ical bodies [41]. Study of this phenomena is crucial to understanding the processes that govern magnetic relaxation on the RFP. Tearing modes are the dominant instability in RFPs. The resonant condition for the tearing instability occurs in locations where k·B = 0. A perturbation can be written as x˜ = x(r)ei(mθ+nφ), where m and n are the poloidal and toroidal numbers, respectively. Considering the equilibrium part of the magnetic field in a straight-field coordinate system, we can apply the criteria of tearing m n resonance to arrive at the equation 0 = kθBθ + kφBφ = r Bθ + R Bφ. Rearranging the equation yields the expression m rB = − φ ≡ q (1.3) n RBθ This states that the safety factor, or q, needs to be a rational number for a mode to be resonant, the safety factor being a measure of magnetic pitch. Physically this corresponds to locations where the magnetic field returns to its starting position after m poloidal transits and n toroidal transits. In locations where resonance is not satisfied, following one field line indefinitely covers an entire toroidal surface. The tearing instability is driven by gradients in the parallel current density [28]. If there is finite resistivity and tearing resonance is satisfied, the field lines in the region can tear and reconnect, forming an x-point, similar to what is seen in Sweet-Parker reconnection [72, 52]. In periodic geometries, the x-point can connect to itself or adjacent x-points forming o-points, also referred to as magnetic islands. These islands can grow as the tearing mode saturates. Multiple tearing modes are resonant in RFPs, as seen in a typical q profile on MST in Fig. 1.1, and islands can grow large enough to overlap nearby islands. Particles flow freely along field lines and when islands overlap, rapid radial transport can occur and a loss of heat and particles results. 6

Figure 1.1 Illustration of resonant surfaces in typical MST plasmas 7

Another feature of the tearing modes is that they can nonlinearly interact with each other through three wave coupling [61]. Core resonant tearing modes, m = 1 modes for MST, are linearly unstable [51, 60] and will couple nonlinearly to the m = 0 tearing modes resonant in the edge. Eventually the tearing modes saturate and the plasma relaxes, removing the drive for the tearing modes. An externally applied loop voltage generates toroidal plasma current which Ohmi- cally heats the plasma, preferentially peaking the parallel current density on-axis. This provides the free energy source for the tearing modes and the process begins anew. Tearing modes create fluctuations in the magnetic field and plasma flow profiles through the reconnecting fields [22]. These fluctuations can coherently interact with each other to induce electromotive forces (EMFs) that act on the current and flow profiles. The EMFs have the result of flattening the profiles, pushing the system towards a Taylor-like state and reducing the drive for the tearing modes. Quasilinear work done in slab geometry has shown that the two-fluid effects are localized to the resonant surface with large amplitudes [50]. However, small distances away from the resonant surface, the amplitudes decrease greatly, leaving only the single-fluid terms. It is thought that nonlinear effects play an important role in broadening the layer in which the two-fluid effects are important, as seen in extended-MHD nonlinear computations where these effects are included [40]. Measurements of two-fluid effects in the core [19] and edge [43] of MST show that these effects are important to the relaxation of RFP plasmas.

1.4 Ohm’s Law

Ohm’s law can be derived by using the equations for the electron and ion momentum evolution. The two equations are combined after multiplying each by the ratio of species charge and mass, and rewriting in terms of current density, J, and plasma center-of-mass velocity, v, in the new equation. Once these equations are combined and reduced, we arrive at an expression that is typically referred to as a generalized Ohm’s law, given by

1 me ∂J E + v × B = ηJ + (J × B − ∇pe) + 2 (1.4) ene nee ∂t 8

where pe, ne, and η is the electron pressure, plasma electron density, and plasma resistivity, respectively [26]. Due to quasi-neutrality, the e subscript on the density is typically omitted. The last term in the equation is the electron inertia and is measured to be very small during discharges on MST. To determine the effect on plasma equilibria during relaxation, we utilize mean-ﬁeld electrodynamics [42] which states that quantities can be written as X = hXi + X˜ with the properties that

hhXii = hXi (1.5) D E X˜ = 0 (1.6) D E X˜ hY i = 0 (1.7)

hX hY ii = hXi hY i (1.8)

This treatment allows us to simplify complex local behaviors into mean-ﬁeld quantities that can then be measured. Applying the mean-ﬁeld approach to Ohm’s law and rearranging yields

1 1 hEi = hηJi − hv × Bi + J × B − ∇p (1.9) ne ne e

n˜ The two terms on the right are inversely proportional to the density. Assuming hni 1, the quantities follow the form X ≈ hXi − hnX˜ i . The Lorentz force term is then given by n hni hni2

D˜ ˜ E D ˜E D ˜ E D ˜ ˜ E J × B hJi × hBi J × B n˜J × hBi hJi × n˜B n˜J × B ≈ + − − − (1.10) en e hni e hni e hni2 e hni2 e hni2

We are ultimately interested in the dynamics parallel to the equilibrium magnetic field, so the first and third terms are eliminated. The plasma is nearly force free, with J × B = ∇p ≈ 0, therefore hJi ∝ hBi and the fourth term is eliminated as well. The last term is a triple product of fluctuations is expected to be small. Density fluctuations are largest in the edge and are dominated by slow changes in the equilibrium density [45]. The maximum contribution from density fluctuations occurs when they are perfectly coherent and in phase with J˜ × B˜ and are estimated to be less than 10% in the edge where the Lorentz force is measured to be smallest. Similarly, density fluctuations are not expected to contribute to the electron pressure gradient term. The equilibrium 9

pressure gradient is perpendicular to the mean magnetic ﬁeld and removing other perpendicular components from Eq. 1.9 yields

D ˜E D ˜ E 1 D˜ ˜ E hEik = hηi hJik + η˜J − v˜ × B + J × B (1.11) k k hni e k

hBi where we have used the notation hXik ≡ hXi · |B| to signify the parallel projection onto the equilibrium magnetic field. The magnitude of the term involving plasma resistivity fluctuations is estimated to be very small and will be omitted. Simplifying the notation to be consistent with other presentations of mean-field Ohm’s law, we arrive at the final expression of

D ˜ E 1 D˜ ˜ E Ek ≈ ηoJk − v˜ × B + J × B (1.12) k noe k

where we have omitted the angled brackets for the singular quantities and subscripted the equilib-

rium scalar quantities as Xo. The second and third terms on the right hand side are the fluctuation- induced EMFs referred to as the MHD and Hall EMFs, respectively. The Hall EMF arises from the inclusion of two-fluid effects in mean-field Ohm’s law. Both of these terms can generate magnetic flux to sustain the reversed discharge against resistive dissipation. Original measurements of the MHD EMF in the core [15] and edge [25] of MST found this term large enough to explain the RFP dynamo in those regions. Single-fluid MHD simulations using the DEBS code [66] at experimentally relevant plasma parameters were able to simulate plasmas exhibiting dynamo activity that sustained the discharge indefinitely [59]. However measurements of the MHD EMF near the reversal surface saw the term diminish in amplitude and was D J˜×B˜ E no longer able to support the RFP dynamo. The Hall EMF, ne , was thought to increase in amplitude to balance mean-field Ohm’s law and was observed near the reversal surface [43, 78]. Furthermore, measurements of the m=1, n=6 component of the Hall EMF are significant in the core [19]. Two-fluid effects are now known to be important during relaxation in RFPs and extensive profile measurements across the plasma radius are required to illuminate regions where the effects are important. The effects also couple into the dynamics of the plasma momentum through the Lorentz force acting on the plasma and momentum transport is observed during relaxation [32]. 10

1.4.1 Momentum Equation

A analogous approach to calculating mean-field Ohm’s law can be used to calculate the mean- field plasma momentum equation. The expression for momentum evolution is given by ∂v m n + m n (v · ∇) v = J × B − ∇p − ∇ · Π (1.13) i ∂t i The terms on the left hand side result from the advective derivation of the plasma velocity. The right hand side are the center-of-mass force densities, the Lorentz force, the pressure gradient, and the viscous stress tensor. For similar reasons in the calculation of the mean-field Ohm’s law, the pressure gradient term is negligible in the parallel mean-field momentum equation. Separating each term into its mean-field and fluctuating parts and taking the flux surface average, we arrive at ∂V D E k ˜ ˜ 2 mino ≈ −mino h(v˜ · ∇) v˜ik + J × B + minoν∇ Vk (1.14) ∂t k where we have used a simple collisional viscosity for the far right term. The two fluctuation- induced terms acting on the plasma momentum are the Reynolds and Maxwell stresses, respec- D E tively. The correlated Lorentz force, J˜ × B˜ appears as the Maxwell stress and as part of the k Hall EMF in Ohm’s law, indicating a coupling of current relaxation dynamics to momentum relaxation dynamics. The force due to viscosity is not expected to be large during plasma relaxation based on estimations of the plasma viscosity [1, 27].

1.5 Overview of Results

The physics of magnetic relaxation is complex and requires two-fluid effects to properly describe the behavior. On the Madison Symmetric Torus, we have the opportunity to develop a robust insertable probe to study the relaxation phenomena in depth. The probe measures all three components of the magnetic field, roughly doubling the radial resolution and insertion depth of previous measurements, creating the most detailed measurements of the plasma equilibrium to date. Our measurements agree well with MSTFit toroidal equilibrium reconstructions and the alpha model, lending more confidence to both descriptions of plasma equilibria. We have also made measurements of the structure of prominent tearing modes, providing more accurate tearing mode 11 profiles for correlation analyses. Measurement of the the correlated Lorentz force requires radial derivatives of magnetic fluctuations and we have developed methodology to ensure uncertainties are minimized. Our measurements indicate significant two-fluid contributions during relaxation near the plasma mid-radius, affecting both current density and momentum dynamics. Numeri- cal simulations of plasmas similar to the ones studied in this thesis show strong two-fluid effects throughout the plasma volume, in agreement with our measurements. We are then able to infer global profiles of the mean-field Ohm’s law to compare with past localized measurements for a thorough understanding of relaxation dynamics. 12

Chapter 2

Experimental Design and Diagnostics Used

The Madison Symmetric Torus is a Reversed Field Pinch in the Physics Department at the University of Wisconsin-Madison [17]. One of the goals of the experiment’s construction is to study the RFP dynamo, one of the key subjects of this thesis. A focus on the aspects of design relevant for the work done in this thesis will be provided here. The operational parameters of the studied discharges will also be discussed.

2.1 The Madison Symmetric Torus

MST is a moderately sized toroidal device with a major radius of 1.5 m and a minor radius of 0.52 m. The plasma is contained by a 5 cm thick aluminum shell that serves as the vacuum vessel and a single turn toroidal field winding. The use of a thick conducting shell has the added benefit of stabilizing resistive wall modes for the duration of typical plasma discharges. However, cuts need to be created for the externally created magnetic fields to penetrate into the interior of the device. The vertical cut, allowing the poloidal field to enter the vessel, is referred to as the poloidal gap. Similarly, the toroidal gap is the horizontal cut that allows the toroidal field to enter the vessel. The gaps created for magnetic field penetration can be a source of large error fields. To minimize the effect of these fields on the plasma, an extensive series of coils have been added to measure and counteract the error field [1]. The resultant field allows the device to retain its high degree of symmetry, the S in the device’s acronym. Ample diagnostic access to the plasma has also been built into the device. Care has been taken to ensure the symmetry and that the diagnostic port holes present minimal perturbation to 13 the equilibrium field. Port holes have been drilled into the device from −45◦P to 225◦P, where 0◦P is the outboard poloidal gap as seen in Fig. 2.1. The remaining poloidal area is covered by vacuum pump holes and blank holes to maintain the toroidal symmetry of the device. The unique design of the toroidal field (TF) and poloidal field (PF) winding systems has allowed ample diagnostic access to the plasma. As previously mentioned, the vacuum vessel serves as the TF winding. The PF winding system consists of a 2 Wb iron-core transformer with the plasma acting as a single turn secondary. Creative design of the PF system allows it to be located at a single toroidal angle while minimizing the error-field created by doing so. Inside the vacuum vessel, several carbon limiters are arranged to protect diagnostics, implanted into the wall of MST, from the plasma. As a result of this, the plasma boundary occurs at a radial location of 50-51 cm, dependent on the poloidal angle due to toroidal effects. The plasma discharge is initiated by creating a toroidal seed magnetic field before the start of the plasma. Then a series of filaments inject electrons into the device to enhance the breakdown of the deuterium gas inside the vacuum vessel. A series of capacitor banks are discharged through the primary winding of the iron-core transformer, inducing a toroidal electric field through the changing magnetic flux. The electric field accelerates the filament sourced electrons, ionizing the neutral gas via collisions. More charged particles are accelerated by the electric field in an avalanche process, creating toroidal plasma current which then creates a poloidal field that helps to confine the plasma. After the formation of the plasma, it relaxes to a reversed field configuration when the bulk of studies done on MST are conducted. Once the transformer exhausts the available flux, the plasma can no longer be supported and extinguishes. The amount of gas puffed during the shot and the capacitor bank voltage dictate the plasma density and current. Additional capacitors can be discharged, altering the discharge duration or amount of reversal, leading to different plasma regimes such as Pulsed Parallel Current Drive (PPCD) or Quasi-Single Helicity (QSH) plasmas. The plasmas studied for this thesis are known as “standard” low current plasmas in the MST group. 14

2.1.1 MST Directions

The initial fields created by the TF and PF windings set the field directions on MST. Due to the magnetic geometry of an RFP, the magnetic field on axis is peaked and purely toroidal. This direction will be the positive toroidal direction, counter-clockwise when viewed from above. The radial direction will be outward from the geometric axis. In a right handed coordinate system (RHCS), the poloidal direction will be up on the inboard side of the toroid. A schematic of the directions in MST are provided in Fig. 2.1. Due to the way the TF and PF windings are set up, the induced toroidal current is in the negative toroidal direction, which is anti-parallel to the magnetic field in the core. In the nomenclature of magnetic relaxation, MST has negative values of lambda, J·B λ = µo B2 , a measure of parallel current in an RFP. For those familiar with MST and its associated signals, the sign of a particular signal may not coincide with the physical interpretation of that signal in the RHCS presented above. For example, the voltage measured at the toroidal gap is stored to have a positive number at the beginning of data recording to verify that the electrical system is working properly. The sign of this signal should be inverted to calculate the physical direction of the edge electric field. A comprehensive list of operational signals and their associated signs used in this thesis are provided in Appendix C.

2.1.2 Mode Decomposition

On MST, an array of magnetic ﬁeld measurements is located at the wall on the lower inboard side of the vacuum vessel. There are 64 coil locations, each measuring all three components of the magnetic ﬁeld, spaced equally around the machine at a single poloidal angle. Formally, this allows for only the toroidal mode number to be decomposed from the array. However, we are able to infer the poloidal mode number from the toroidal mode number (i.e., m = m(n)). The resonance condition for the dominant instability in MST is given by m = − rBφ and equilibrium n RoBθ m reconstructions show that n < 0.2 for the plasmas studied in this thesis. Therefore resonant tearing modes must have m = 0 for n ≤ 5. Additionally, there is a small vacuum gap between the plasma ˜ ˜ and the conducting wall leading to Br = Jr = 0 at the wall, allowing us to derive the expression 15

Figure 2.1 Schematic of directions on MST 16

˜ Bφ(a) an ˜ ˜ = . For m = 0 modes, Bθ(a) = 0. Measurements from the toroidal array show that Bθ(a) Rm ˜ ˜ Bφ(a) Bθ(a) during relaxation for n ≤ 5 modes and we can still approximate them as m = 0 modes. The 6 ≤ n ≤ 8 modes have m = 1 and measurements of the modes from the toroidal array agree with the calculated mode amplitude ratios from the previous equation [10, 12, 77]. Higher n modes are lower amplitude and are assumed to be m = 1 as m = 2 modes are shown to be stable in an MHD stability analysis of RFP equilibria [3]. A phase velocity can be calculated for each mode and the phase velocity of core-most resonant mode is strongly correlated with spectroscopic measurements of impurity ion ﬂow [14], allowing us to use it as a proxy for plasma velocity.

2.1.3 Controllable Experimental Parameters

MST has the capability of running a multitude of different experimental regimes. The work in this thesis involves what are called “standard discharges” and the relaxation events are more preva- lent during this operation. Other more specialized modes of operation, such as external current drive or enhanced conﬁnement, are not examined in this thesis as these modes typically employ methods to reduce the effect of tearing modes. The various “knobs” for changing experimental parameters will be described here and motivation for the values used in the studied discharges will be provided. Plasma discharges on MST usually operate in the range of 200-500 kA plasma currents. As of the writing of this thesis, the upper limit of plasma current is around 600 kA but is an equipment limitation rather than a physical one. The lower limit of reversed discharges that display large magnetic relaxation events is around 150 kA. MST is able to run at lower currents but the charac- ter of the events begins to change, with the events becoming muted and more frequent. Reversed discharges naturally occur once Θ ≡ Bθ(a) exceeds some threshold, Θ > 1.2 from MHD stability hBφi

analysis [4], with Bθ(a) determined from total toroidal plasma current. When the plasma current is lowered with the toroidal ﬂux held constant, discharges are no longer able to self-reverse, transitioning into a regime in which different physics are at play. As the work in this thesis wishes to explore the physics of magnetic relaxation, lower currents are not examined in detail but provide an avenue for interesting future work. The bulk of data in this thesis uses plasma discharges that 17 have currents of 200 kA. This current was chosen to maximize probe longevity and RFP dynamo strength. Currents too high reduce the lifetime of the probe and currents too low have events with weaker dynamo activity, decreasing the signal to noise ratio.

19 1 MST typically produces plasmas with line averaged electron densities of 1 · 10 m3 . The upper density limit on MST appears to follow the empirical Greenwald density limit [8], which is given by I [MA] 1 · 1020 hn i = p (2.1) G πa2 m3 19 1 for a circular cross-section [30]. For probe relevant discharges, nG ∼ 3 · 10 m3 . The lower range of plasma densities is limited by concerns for the device construction. At low densities, the plasma has a higher likelihood of terminating early, transferring the stored energy in the capacitor bank through the PF winding, exceeding the safe operating limits and damaging the PF winding and other associated systems. At higher densities, this risk is alleviated. However, this is primarily a concern at higher plasma currents when more energy is stored in the capacitor banks. Additionally, the probe itself can act as a source of fuel for the plasma by sputtering during the discharge and low density plasmas are less reproducible. As a result, running with plasma densities of about

19 1 1 · 10 m3 allows for reproducible plasmas with the added beneﬁt of favorable comparability to prior measurements.

2.1.4 Multiple Helicity Operation

Standard operation on MST observes multiple active tearing modes during relaxation of the plasma, a state sometimes called Multiple Helicity to differentiate it from other states such as PPCD or QSH. These states can be modified through control of the reversal parameter, F, and is defined as the toroidal field at the wall divided by the average toroidal field, F ≡ Bt(a) . It hBti is a dimensionless measure of how reversed the plasma is and helps to characterize the types of discharges on MST. Negative values denote reversed discharges. Experimentally, this value is controlled by the voltage applied to the TF circuit after the plasma has formed. A standard value for MST is F = −0.2. Tearing modes can be suppressed by modifying the edge current profile, and thus the drive for the instability, in a state called PPCD [9] by deepening reversal as 18 the discharge progresses. Tearing modes are observed to become weaker and more frequent as F values approach zero. Eventually the plasma enters a regime in which the m = 0 rational surface is outside the plasma, greatly reducing the nonlinear coupling to the core tearing modes. Under the correct conditions, a single tearing mode can become much larger than the other modes and the plasma departs from the traditional multiple tearing mode case, referred to as a QSH state [48]. F values that are more negative are associated with higher magnetic shear and larger amplitudes of the core tearing modes. Additionally, deeply reversed plasmas can make the m = 0 mode become unstable, altering the nonlinear coupling to other tearing modes [11]. This thesis aims to study the magnetic relaxation that is mediated by the interaction of multiple tearing modes. Data in this thesis was collected from discharges with F = −0.2 as these plasmas produced robust relaxation events to study. However, plasmas that enhance or suppress tearing modes could provide an interesting avenue to explore changes to magnetic relaxation.

2.1.5 Available Diagnostics

The Madison Symmetric Torus provides ample access to many diagnostics, allowing the plasma phenomena strongly exhibited in RFPs to be thoroughly analyzed. It is capable of producing a wide array of interesting discharges but the main focus is producing plasmas that will allow for probe insertion while still exhibiting magnetic relaxation. The primary diagnostic used in this thesis is the Deep Insertion Hall Probe, developed to access new regions of the plasma to explore the importance of two-ﬂuid during magnetic relaxation. The majority of the remainder of this chapter will be devoted to describing this probe and the analysis that accompanies the interpretation of the raw data. Additional sources of data are used to supplement the Deep Insertion Hall Probe measurements, such as diagnostics to measure plasma density and temperature, and will be discussed at the end of the chapter. The data collected from these diagnostics will have a chapter dedicated to the discussion of the results and how it ﬁts into the larger picture of magnetic relaxation. 19

2.2 The Deep Insertion Hall Probe

The primary purpose of this probe is to measure the parallel component of the correlated Lorentz force, which appears as the Hall EMF in Ohm’s law and the Maxwell stress in the momentum equation. Using a cylindrical, low-ω approximation to Ampere’s law to infer the current, we arrive at

D E 2 ∂ D E B 1 ∂ D E B µ J˜ × B˜ = + B˜ B˜ θ θˆ + + B˜ B˜ φ φˆ (2.2) o r ∂r r θ B r ∂r r φ B

The full derivation is available in Appendix A.4. Measurement of the correlated Lorentz force only requires a radial derivative of the total measured magnetic field. The Deep Insertion Hall Probe (DIHP) is designed to allow for the simultaneous measurements of the total magnetic field at multiple radial positions. The probe is constructed out of a ceramic called boron nitride. It is widely used for probe applications due to its excellent ability to withstand thermal shock and its vacuum compatibility. It is also easy to machine allowing precise probe geometries to be constructed. There are essentially three parts to the construction of the probe: 1) a form for winding loops of wire around to create flux loops, 2) a coil housing to position the separate loops of wire, and 3) some form of shielding to encase the entire apparatus to protect it from the plasma. With the exception of the wire, all pieces are made from boron nitride to allow for small tolerances when the probe is assembled as differential expansion would need to be accounted for when various materials are used. The coil forms are cubes of boron nitride with deep grooves cut into the faces of the cube to serve as guides for the wires. The coils are then hand wound around these grooves, the design allowing for 20 loops for each direction measured. As the normal of each individual loop cannot be guaranteed to be identical, this creates an inherent uncertainty of the direction of the normal of the entire coil, albeit a small one. Each winding will have a separate normal that is ideally in line with the geometric coordinate it is measuring. However the sum of all windings will yield a coil with the majority contribution coming from the intended direction and minor contributions from the other directions. This uncertainty can be corrected for during the calibration and alignment of 20 the probe. In Fig. 2.2 we have an illustration of the coil form used. Each color refers to a particular normal direction and each loop can be slightly misaligned with respect to loops of the same color. A coil housing is made using a cylindrical stock of boron nitride. A groove is machined down the length of the cylinder for placement of the coil forms. This groove prevents the coils from shifting with respect to adjacent coils, allowing for high precision measurements of the location of each coil. The DIHP is cylindrical with the experiment’s radial direction lying coincident with the probe’s axis. A series of probe protections are then added to the exterior of the coil housing. First a paste of boron nitride and sauereisen cement, an adhesive designed for high temperature ceramic applications, is used to cover the exposed magnet wire. Then a thin layer of silver paint is added for electrostatic shielding of the probe. The coil housing is inserted into a 3.2 mm thick cylindrical shell of boron nitride, referred to as a particle shield, to shield the probe from the plasma. The entire assembly is attached to a retractable arm which also serves as the vacuum boundary to the outside. The retractable arm is manual, requiring the experimenter to radially position the probe using an attached ruler. The retractable arm is connected to a narrow indicator to eliminate any parallax when recording the insertion depth. In Fig. 2.3 a view of the internal wiring of the probe is seen, before it is encased in the series of protections against the harsh plasma environment. Sixty twisted leads are seen in the picture, each corresponding to a separate magnetic field measurement. The number sixty was chosen as it allows a doubling of the radial insertion depth while doubling the radial resolution. It also utilizes all available digitizer and integrator channels dedicated to probe work on MST. Insulated magnet wire with a gauge of 32 is used to allow for a robust coil design as thinner wire is prone to damage during construction. The design of the probe allows for extensive radial profiles to be obtained. In Fig. 2.4 several radial locations are illustrated. Past measurement locations are shown relative to the plasma size and reversal surface. The DIHP is also shown, vastly increasing the extent of the measured area. Special care has been taken to thoroughly examine the sources of uncertainty of the probe measurements. The reason is that this probe was developed to measure small fluctuations in the 21

Figure 2.2 Illustration of coil form used for DIHP construction 22

Figure 2.3 View of coil windings without particle shield 23

Figure 2.4 Illustration of radial locations of prior measurements 24 magnetic field, the radial direction in particular can be subject to errors that are hard to compen- sate for or verify. Additionally, calculation of the correlated Lorentz force requires precise radial gradients of the magnetic field fluctuations to be measured. This process amplifies any imperfections that may exist, leading to incorrect measurements. A careful examination will yield greater confidence in the measurements and calculations conducted.

2.2.1 Bench Calibration

The DIHP is calibrated on the bench using a Helmholtz coil setup. A Helmholtz coil is a particular configuration of two identical circular coils to create a nearly uniform magnetic field over a large volume that is coincident with the planes of the coils. The two coils have a separation equal to their radius. The uniform magnetic field created is in the direction normal to the coils and is given by the equation 3 4 2 nµ I B = o (2.3) 5 R where n, I, and R are the number of loops in each coil, the current, and the radius of the coils, respectively [31]. To utilize a Helmholtz coil for calibrating the magnetic probe, a waveform generator applies a sinusoidal current through the coil, creating a time varying magnetic field that can be measured by the magnetic probe via Faraday’s law

I ∂ I E · dl = − B · dA (2.4) ∂t

The waveform generator can be swept in frequency, allowing the probe’s frequency response to be measured, which is constant for the tearing mode dynamics that the probe is designed to measure. The voltage measured by the probe is then proportional to the effective coil area since it is static. The effective coil area is the area projected along the magnetic field measured. Ideally it would be equal to the area but in reality there can be some projection of the coil normal onto another magnetic field direction. This effective coil area is then used during plasma discharges to calculate the magnetic field from the voltage measurements gathered by the probe. 25

2.2.2 In Situ Calibration

On MST, we have the capability of producing a toroidal field without plasma. We use this field as an auxiliary method to verify the Helmholtz coil calibration for the toroidally aligned coils on the probe. We then rotate the probe 90◦ to verify the poloidal calibrations. There is no secondary check of the radially aligned coils but we can use a composite correction factor calculated from the toroidal and poloidal facing coils if any discrepancy is found. The vacuum calibration has the additional benefit of using the same hardware that will be used during plasma discharges. Any systematic error that would cause changes in the measured voltage on MST when compared to the bench calibration can be accounted for during this check. The equation used to calibrate the probe follows. S is the field quantity of interest and V is the value of the voltage as stored in the database.       Sr k11 k12 k13 Vr              Sθ  =  k21 k22 k23   Vθ  (2.5)       Sφ k31 k32 k33 Vφ An aligned, idealized probe would have the diagonal elements inversely proportional to the area of the coils and the off diagonal elements would be zero. When properly calculated, the elements in the array will give us information about the alignment and calibration of each coil in the probe. However, due to the limitations of the vacuum alignment we are only able to create 6 equations

for the 9 unknowns. Using the small angle approximation, we can calculate k22 and k33 from the

vacuum ﬁeld. This gives us 2 more equations and we will set k11 equal to the value calculated from the Helmholtz calibration. The in situ calibration then preempts the Helmholtz calibration factors in the cases where the toroidal ﬁeld can be used.

2.2.3 Measurement Uncertainties

A calculation of the uncertainties in the magnetic field measurements is essential for correct interpretation of the data. Using the uncertainties in the Helmholtz calibration, we have estimated an error of 3% in the calculated areas of the coils. This was confirmed using the work from Brandon Dzuba which involved a precisely machined single turn loop within the Helmholtz coil 26 setup. Using vacuum shots as an additional method of calibration, this error can be significantly reduced for the toroidal and poloidal coils. However, there is no radial field to act as a secondary calibration method so the 3% must be accounted for. In addition, the uncertainty introduced during probe alignment must also be accounted for. There are two sources of alignment uncertainty within the experimental data. The first is the misalignment of any coils within the probe. The second is a misalignment when mounting the probe to MST. Both sources of misalignment can be accounted for in the same correction factor. The coils and coil housing were machined with tolerances of 0.13 mm, allowing extremely accurate spatial knowledge of the coils. However, the coils were handwound, leaving some room for error in the direction of the normal for each individual loop. The measured signal will then be a combination of the field along the intended direction of the normal that I will refer to as the parallel direction and a contribution from the other orthogonal directions that will be referred to as the perpendicular direction. Mathematically, this can be expressed as B = Bk cos(δ) + B⊥ sin(δ), where δ is a the small angle subtended by the normal of the coil to the intended direction. Due to the geometry of the coil form used in the DIHP, this angle is very small and this error will tend to average out since many loops of wire are used in a single coil and the perpendicular projection is randomly distributed. In addition to the coil alignment, the probe rotation must be factored into the equation for proper calibration and uncertainty analysis. The probe is inserted into MST via a 90◦ poloidal port. The angle of rotation is controlled by rotating the probe with respect to the fixed probe rail. An angular caliper is used to ascertain the exact angle of rotation. By utilizing the vacuum toroidal field on MST, we have a well known field to measure with the probe. Coupling this knowledge with rotation of the probe, we are able to calculate a rotation matrix to figure out any nonorthogonal orientation of the coils. This rotation matrix will simultaneously correct for any coil misalignment described earlier. The residual uncertainty comes from how well the rotation matrix can correct for any misalignment, which is a function of the calibration of the coils. An uncertainty of 0.5◦ has been estimated for perpendicular contributions to the 27 signal. For example, toroidal field measurements have an uncertainty of ±10 Gauss due to the minute projection of the poloidal field onto the toroidal coil.

2.2.4 Uncertainty Propagation

The data presented in this thesis can come from direct measurements or calculations based on several sources of data. The calculation of new quantities using other data, such as calculation of the electric ﬁeld by spatial integration and temporal differentiation of the measured magnetic ﬁeld, can introduce additional sources of uncertainty. To account for these new uncertainties, we need to propagate the uncertainties through the calculation. In general, we can use the following equation

2 X ∂f σ2 = σ (2.6) f(x1,x2,x3,...) ∂x xi i i This equation assumes the uncertainties sum in quadrature and each variable is independent of each other [75]. For the data presented in this thesis, this is a valid assumption. We will be calculating quantities that are parallel to the magnetic field, which will involve an assortment of varying quantities with individual uncertainties. For example, we would like to know Eφ√·Bφ+Eθ·Bθ the uncertainty for the equation Ek = 2 2 . We will be calculating several quantities of this Bφ+Bθ form so we would like to find the error without using any assumptions that are unique to a specific

situation. Using Equation 2.6 above, we calculate the uncertainty for Ek to be s B 2 B 2 1 B2 1 B2 σ = φ σ + θ σ + E2 − φ σ2 + E2 − θ σ2 Ek Eφ Eθ φ 3 Bφ θ 3 Bθ B B B B 2 B B 2

−Vtg 1 R a 0 ˙ 0 However the calculations for E are based upon measurements of B. Since Eθ = 2πa + r r r Bφdr , σ2 2 2 Vtg 2 ∂ 1 R a 0 ˙ 0 we can calculate the uncertainty of Eθ to be σE = 2 2 +σ ˙ ˙ r Bφdr . An additional θ 4π a Bφ ∂Bφ r r

calculation for Eφ is also required. As these equations for uncertainty can become algebraically in- tensive, further calculation of the uncertainty will not be explicitly derived in this thesis. However, all data presented will include error bars derived from this method. 28

2.3 Complementary Diagnostics

In addition to the insertable probe developed for this work, measurements from other diagnostics have been used and will be brieﬂy described here. This overview is not meant to be comprehensive but to provide a brief refresher for the physical mechanisms that these diagnostics rely on and to mention the speciﬁcs as they apply to MST.

2.3.1 Far Infrared Interferometry

The Far Infrared Interferometry (FIR) system [6] is an 11 chord interferometer, utilizing the difference in the refractive index of the plasma compared to vacuum to calculate an integrated line plasma density. The profile can then be inverted to produce a density profile. The FIR system used on MST can measured the dynamic evolution of the density profile on a time scale faster than the tearing mode activity. This allows us to obtain evolving profiles during magnetic relaxation to calculate density dependent quantities, such as the plasma resistivity or the Hall dynamo.

2.3.2 Thompson Scattering

The Thompson Scattering (TS) system [58] depends on the Thompson scattering of photons to determine the temperature of the plasma. A high intensity laser is pulsed to provide temporally discrete bursts of coherent radiation to propagate through the plasma. This causes the free electrons in the plasma to oscillate due to the electric ﬁeld of the incident laser light. The oscillating electron then radiates in the same frequency of the incident light and is collected away from the plasma. The frequency of the light will undergo Doppler broadening due to the thermal motion of the plasma, allowing for a plasma temperature measurement to be inferred from the width of a Gaussian ﬁtted to the light spectra. Radial resolution can be determined by the amount of scattered light collected from the plasma as well as the collection optics. For the TS system used on MST, this amounts to 21 separate radial positions that spans most of the entire plasma radius. 29

2.3.3 Beam Based Diagnostics

The ChERS and Rutherford Scattering diagnostics provide measurement of impurity and bulk ion flows, respectively, using neutral beam based diagnostic methods. We will discuss the two systems here since they provide a measurement of plasma velocity which is used in the determination of plasma flow relaxation. The ChERS is a Charge-Exchange Recombination Spectroscopy [46] diagnostic. A mono- energetic beam of ions is used to bombard the plasma ion background. The resultant collision can result in the exchange of charges between the beam species and a charged ion. The charged ion is now neutral and will be measured by an Ion Doppler Spectroscopy system, determining the original energy of the ion. This provides a localized measurement of ion flow, allowing for a profile to be built up. The Rutherford Scattering (RS) system [56] uses a mono-energetic beam of ions whose energy is much higher than the bulk plasma energy. Some of these energetic ions will undergo small angle Coulomb scattering, also called Rutherford scattering. The energy spectra of the scattered ions are measured and the ion velocity distribution is calculated. If a Maxwellian distribution is assumed, then the ion temperature is inferred from the width of the measured energy spectrum. The shift of the energy spectrum can be used to calculate the bulk ion flow. The RS system on MST is able to measure the poloidal flow. In this thesis, we will use data from each of these diagnostics for plasma velocity profile information on MST. Prior measurements using these diagnostics will be shown in conjunction with DIHP measurements to help determine the global structure of Ohm’s law and the momentum equation.

2.4 Summary

The Madison Symmetric Torus is an unique device that has plenty of opportunity to study many plasma phenomena. The Deep Insertion Hall Probe is a new probe developed to access unmeasured regions of the plasma. The intent is to measure small fluctuations in the magnetic field to infer the 30 correlated Lorentz force which appears in Ohm’s law as the Hall EMF and the momentum equation as the Maxwell stress. To fully understand the measurements and their implications on the dynamics of plasma relaxation, special care needs to be taken to account for any sources of uncertainty. These measurements are also presented along side measurements of the density and temperature profiles captured by the FIR and TS systems, respectively. Finally, velocity measurements can be gathered from other optical based diagnostics, allowing us a fuller picture of the dynamics present. 31

Chapter 3

Analysis Methodology

The previous chapter outlined the data available for the work in this thesis on the Madison Symmetric Torus. However signiﬁcant work is required to interpret and transform these data into measurements of relaxation dynamics. This chapter outlays the methodology and analysis used in the interpretation of these data, establishing the necessary framework to realize the results presented later in this thesis. The plasma equilibrium, and thus all of our measurements of it, are functions of time. In the following analyses, we have omitted the explicit time dependence of the measurements to simplify the notation and discussion of the particular analysis technique. However, the analysis is conducted at each moment in time relative to the magnetic relaxation event. For example, the Shafranov shift and ensemble average are functions of time and separate calculations are performed for each moment in time.

3.1 Probe Perturbation

There is always the question of whether or not an insertable probe perturbs the plasma. A more accurate phrasing of the question for this research is whether or not an insertable probe affects the dynamics that are the subject of study. The former question can be difﬁcult to prove absolutely and even the mere presence of the probe is evidence of a perturbation, i.e. plasma no longer exists in the location of the probe. The latter question is better suited for examination. 32

The simplest test for evidence of problematic probe perturbations is comparing the dynamics of the plasma with and without the probe inserted. Of course the internal dynamics cannot be compared directly since the probe provides the internal measurements but large scale dynamics can be easily measured outside of the plasma through toroidal flux measurements, tearing mode velocities and amplitudes, and other plasma parameters used to categorize the discharges. In Fig. 3.1 we have two discharges shown. In black, the discharge is with the probe inserted and in red is with the probe withdrawn. The magnetic activity and toroidal flux are indications of relaxation phenomena. The level of magnetic activity is comparable in the two discharges, the specific signal used is the edge resonant m = 0, n = 1 tearing mode amplitude. The amplitude and duration of the toroidal flux events are also similar, meaning that the RFP dynamo we aim to study is also similar. The primary difference is the timing of the spikes in magnetic activity, which is independent of probe insertion as it is a highly nonlinear, impulsive event. Furthermore, we know that the tearing mode is a long wavelength instability, typically on the order of a meter for MST discharges. The wavelength is much larger than the probe diameter so we would not expect it to have an effect on tearing mode dynamics.

3.2 Port Hole Perturbation

An additional perturbation that needs to be discussed is the port hole perturbation. To access the interior of MST, diagnostic access ports need to be drilled into the conducting shell of MST. This disrupts the image currents flowing in the shell, allowing magnetic flux to expand into the port hole. The design of the probe allows it to be fully shielded from the plasma that can accompany the expanding flux. However, any measured magnetic field will be reduced when compared to a region without the port hole. The perturbed field falls off to about 2% of the non-perturbed field within a diameter of the port hole [24]. The expanded magnetic flux also presents another problem for data analysis. Cylindrical geometry is used to approximate the equilibrium current density from the magnetic field measurements. As one approaches the port hole, the approximation breaks down as the expanded flux is more representative of a sphere and the current density can no longer be calculated with the probe geometry. For this reason, measurements involving current are typically 33

Figure 3.1 Two plasma discharges with the DIHP inserted(black) and withdrawn(red) 34 omitted from our analysis if made within a diameter width of the porthole. For the data presented r in this thesis, this corresponds to a location of a > 0.95.

3.3 Flux Surface Average

To calculate the mean-field component of a quantity, we need to find a way to remove the non- axisymmetric contribution to the quantity. This can be accomplished by averaging it over the entire flux surface. The quantity X is equal to a mean-field component, hXi, and a non-axisymmetric component, X˜. When we average over a flux surface, the non-axisymmetric component evaluates to zero and we are left with the mean-field component, seen by the following equation

1 Z 1 Z 1 Z XdS = hXi + X˜ dS = hXi dS = hXi (3.1) R dS R dS R dS

We see that the flux surface average of a quantity is equal to the mean-field component of that quantity. To calculate the flux surface average, one would have a dense array of measurements throughout the plasma to sample the entire flux surface. However, this is impractical and a simpler solution can be found. Experiments on MST have historically used an assumption referred to as the “Random Phase Approximation”. Measurements are typically made at a single location in the plasma which clearly does not constitute a flux surface average. However the plasmas examined have rotating helical structures present due to the tearing modes, allowing one to sample different locations on the flux surface with a fixed diagnostic. The large scale plasma phenomena are similar, event to event, if experimental conditions remain roughly constant (i.e. plasma current, density, reversal parameter, etc.). To evenly sample the entire flux surface, an flat phase distribution is required. Since MST was designed to be toroidally symmetric, a sufficiently large ensemble of random events will have an approximately uniform phase distribution. If these assumptions hold, we can take an ensemble of 1 P our fixed measurements in lieu of a toroidal and poloidal array of measurements, hXi ≈ n Xi. Measurements in this thesis take advantage of a probe developed to reach new radial locations. Due to the size of the plasma exposed surface of the probe, an increase in discharge variability is observed compared to measurements made at lesser insertion depths. An experimental campaign 35 was run to ascertain the insertion depths for obtaining reproducible plasmas. A conditioning protocol was developed to insure similar plasmas for ensembling data. This was achieved by slowly inserting the plasma into low current discharges, allowing the probe to reach a quasi-equilibrium state in which the plasma fueling requirements and impurity radiation levels remain constant. We use the residual gas analyzer to observe the boron line emission and the total fuel used during the discharge to determine when a probe is conditioned. The fuller description of the conditioning protocol is provided in Appendix B. We are interested in identifying events to incorporate into a flux surface average measurement. On MST there is a long history of analyzing data using a sawtooth ensemble. A sawtooth is the colloquialism for an individual relaxation event and a sawtooth ensemble describes the averaging procedure for the data. The goal of a sawtooth ensemble is to identify similar dynamics in the plasma discharge and normalize them to the same time axis. The events are then averaged around the same time, allowing a distribution to be built up to increase the signal to noise ratio. For the work in this thesis, we are able to use the results of a sawtooth ensemble to approximate a flux surface average, assuming the criteria laid out in the previous section are satisfied. In Fig. 3.2, the top two plots are the edge electric field during a plasma discharge. The edge parallel electric field is induced by the change in total toroidal flux and spikes in the signal mark times of large flux generation. We will use these spikes to identify plasma relaxation events, defining the peak to be t = 0 for the ensemble. The identified events are shown with a vertical, dashed red line in the middle plot. We want to ensure that the events ensembled are similar to each other. One way of achieving this is to set bounds on other measured signals, such as core tearing mode velocity, total toroidal plasma current, plasma density, etc. Additional bounds can be set at different times during the event to account for pulse shape discrepancies. For the data presented in 1 this thesis, all events ensembled have Ip = 195±10 kA, ne = 1.0±0.15 m3 , and F = −0.2±0.025. Any events satisfying the criteria selected have nearly identical equilibria and relaxation behavior should be comparable. The bottom plot in Fig. 3.2 is the ensemble averaged electric field. We observe that the higher frequency fluctuations during the discharge has been removed and a clearly defined pulse shape is visible. This procedure is applied to all measurements presented in this 36

Edge electric field 4 2 0 −2

V/m −4 −6 −8 −10 10 15 20 25 30 35 40 4 2 0 −2

V/m −4 −6 −8 −10 10 15 20 25 30 35 40 2 0 −2

V/m −4 −6 −8 −2 −1 0 1 2 Time (ms)

Figure 3.2 Plot of the parallel edge electric ﬁeld at different points during ensemble averaging thesis. In Fig 3.3, we have plotted the phase distribution of the m = 0, n = 1 mode for the events used in the ensemble during relaxation. As we can see, the distribution of phases does not center around a particular phase and the random phase approximation is satisﬁed for the data presented in this thesis.

3.4 Equilibrium Reconstruction

Equilibrium reconstruction is a powerful tool to use in conjunction with our flux surface averaged data. The equilibrium reconstruction routine used on MST is the MSTFit code [2]. It is a nonlinear Grad-Shafranov toroidal equilibrium reconstruction that utilizes available diagnostics on MST to constrain the equilibrium. The model fits functions for F and P , which are measures of the toroidal field and pressure profiles, respectively. The routine also has the capability of utilizing internal magnetic field measurements to constrain equilibrium reconstructions, which has been done for the data presented in this thesis. The plasma equilibrium can be solved for by examining a force balance in a toroidal geometry. Plasmas are subject to additional forces when bent into a toroidal geometry. The surface area on the 37

Histogram Density 4

2 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radians Figure 3.3 The phase of the m = 0, n = 1 mode during relaxation for the events used in this ensemble 38 inboard side of a torus is less than the outboard side; this imbalance causes an excess of pressure on the inboard side. Additionally the toroidal plasma current expands outward from the Lorentz force exerted when the plasma is bent into a ring. Mathematically, these forces on the plasma can be examined through the Grad-Shafranov equation [29, 67]. A brief derivation of the equation will be provided here. The poloidal magnetic ﬁeld can be written in terms of the poloidal ﬂux, ψ

1 B = ∇ψ × ∇φ (3.2) p 2π

the 2π appears as a normalization factor and the total magnetic ﬁeld can be written as

1 B = B + B φˆ = ∇ψ × ∇φ + F ∇φ (3.3) p φ 2π

φˆ where we have substituted F = RBφ and ∇φ = R . We are also using a cylindrical geometry with the azimuthal component being the toroidal component. A poloidal cross-section is then composed of the R and Z directions as seen in Figure 3.4. We can use Ampere’s law to expand the current in terms of ﬂux functions, given by 1 ∇ψ ˆ µoJ = ∇ × B = ∇F × ∇φ − R∇ · 2 φ 2π R (3.4)

= ∇F × ∇φ + µoJφR∇φ

1 ∇ψ where we have reduced the equation by substituting the toroidal current, µoJφ = − 2π R∇ · R2 . We will assume a force-free equilibrium, or J×B = ∇P . Using the chain rule, we can rewrite ∂ derivatives in terms of ψ so that ∇ = ∇ψ ∂ψ . Plugging in the equations for current and magnetic ﬁeld, we arrive at 0 JφR FF ∇ψ 0 − 2 = P ∇ψ (3.5) 2π µo R 0 ∂ where refers to ∂ψ . Rearranging the equation to solve for Jφ yields

0 2πF F 0 Jφ = + 2πRP (3.6) µoR This equation depends on two quantities, F (ψ) and P (ψ), which completely describe the plasma equilibrium. 39

Figure 3.4 Poloidal plane directions used in Grad-Shafranov derivation 40

The work in this thesis treats the F and P profiles as cubic splines with 4 and 8 degrees of freedom, respectively. It will also fit magnetic data available from the Deep Insertion Hall Probe. The internal constraints provided by the probe allow for better fits to be acquired and multiple time points can be fitted, allowing for a dynamic evolution of the modeled plasma. The constraints on the equilibrium reconstruction are provided by measurements from Thompson scattering, Far Infrared Spectroscopy, flux loops, internal magnetic field measurements, and an array of magnetic field measurements located at the wall of MST. Density profiles from the reconstruction are used in the calculation of the Hall EMF when combined with the correlated Lorentz force. In the past, some measurements of the Hall EMF assumed a flat density profile. While this is not true in the edge, the correlated Lorentz force is measured to be very small in the edge, yielding correct values for the Hall EMF, within error bars. Langmuir probes have also provided density information in the past. However, they only provided a single radial measurement per discharge and require longer times between discharges due to the need for cooling of the plasma exposed metal tip. The use of FIR constrained density profiles from MSTFit allow for accurate density information to be used, especially closer to mid-radius where accessibility via Langmuir probes would be severely limited.

3.5 Shafranov Shift Correction

Due to the toroidal geometry of the experiment, there is an outward force on the plasma that causes the magnetic axis to shift from the geometric axis. The forces are captured by Eq. 3.6 and we are able to directly measure their effect on the equilibrium magnetic field measured. Since MST has a close-fitting conducting shell and a circular cross section, we would expect that the flux surfaces are approximately circular in shape. The equilibrium magnetic field, by definition, is constrained to a flux surface; meaning there should be no perpendicular component surface in flux coordinates. We also do not expect a toroidal distortion to the flux surfaces since MST is toroidally symmetric. Therefore any equilibrium radial field measured, Sr, is due solely to the magnetic field being shifted relative to the geometric coordinates. We use this information to convert the measured magnetic field in probe coordinates into the flux coordinates. This conversion applies a 41 rotation matrix given by       Bρ cos(θ) −sin(θ) 0 Sr              Bp  =  sin(θ) cos(θ) 0   Sθ  (3.7)       Bt 0 0 1 Sφ where the quantity S is measured by the probe, and we solve for B. The indicies refer to the direction measured, with r, θ, φ referring to the probe geometry and ρ, p, t referring to the flux geometry. The p and t are the poloidal and toroidal components, respectively. From the definition of a flux surface, the radial-like coordinate Bρ will be zero during a flux surface average,

hBρi ≡ 0 = cos(θ) hSri − sin(θ) hSri (3.8) so we solve for the angle θ by utilizing flux surface averaged measurements. Once this correction is implemented, we know magnitudes of the magnetic field in flux geometry. This correction is carried through to other calculations that involve the magnetic field, such as the Hall EMF. An example of the shift is provided in Fig. 3.5, where the shifted flux surface has been exaggerated for clarity. The distance from the geometric center is given by r and the distance to the flux surface center is ρ. For small values of ∆, the two distances are the same and there is no shift. From DIHP r measurements, the shift linearly increases to values of about 6cm at a ∼ 0.65 and plateaus during relaxation. This is also observed with equilibrium reconstructions.

3.6 Shot to Shot Variation

A large effort was made to interpret and estimate uncertainties caused by what is called “shot to shot” variation. Simply put, shot to shot variation encapsulates any measurement variation observed between plasma discharges with nearly identical external experimental parameters. For example, Nyla makes measurements in low current plasmas. A week later, Joe makes the same measurements in the same plasmas. They compare equilibrium measurements of the magnetic ﬁeld and the observed variation is small. However, when they compare a ﬂuctuation based measurement, such as the Hall EMF, larger variation is observed. These variations should be encompassed into the calculation and presentation of uncertainties of a measurement. A proper estimate of the 42

Figure 3.5 An illustration of the magnetic flux surface projected onto the probe geometry. 43 shot to shot variation should also represent the confidence of reproducibility of the same measurement. We will examine shot to shot variability as it pertains to the data in this thesis, primarily the result of ensemble averaging hundreds of events together and equating that to the flux surface average. There are three possible sources of shot to shot variation examined in this thesis. 1) Low signal to noise ratio. This variation can be minimized through the collection of more events used in an ensemble and can be accounted for with accurate uncertainty quantification of the measurement. The other sources for shot to shot variation are more difficult to quantify. 2) The minute differences in plasma equilibria produced during similar discharges using the same experimental controls. When these minute differences manifest themselves in observable ways, such as the transition of a plasma to an enhanced confinement mode, they can be accounted for in an ensemble using stringent criteria for event selection and are no longer considered shot to shot variations. However, differences that cause slight changes in the tearing mode activity cannot be selected for during an ensemble but will still cause a variation in the measurement of the RFP dynamo. 3) The ensemble not exactly equaling a flux surface average. If the ensemble does not sample the flux surface evenly, then some locations are oversampled relative to others. We will calculate a composite term that incorporates the last two uncertainty sources. The reason we are concerned with the estimation of shot to shot variation is the effect it has on the calculation of fluctuations. Our measured signal is defined to be B ≡ hBi + B˜, where the angled brackets refer to a flux surface average and the tilde refers to the non-axisymmetric component. An ensemble average is defined as

n n n 1 X 1 X 1 X [B] ≡ B = hBi + B˜ (3.9) n i n i n i i=1 i=1 i=1 where the subscript denotes individual events. When we choose the events used in our ensemble to satisfy the conditions in Sec. 3.3, the term on the right is very small compared to the ensemble average and formally vanishes when events are sampled evenly from a single ﬂux surface. Our 44

ﬂuctuation measurement is then given by

n n ! 1 X 1 X X˜ ≡ B − [B] = hBi + B˜ − hBi + B˜ i i i i n j n j j=1 j=1 n n 1 X 1 X (3.10) = B˜ + hBi − hBi − B˜ i i n j n j j=1 j=1 | {z } | {z } δave δflux

where we have used X˜ to differentiate it from the non-axisymmetric fluctuation B˜ we are at- tempting to measure. From the second line in the equation, we see that our measurement of the fluctuation is equal to the non-axisymmetric fluctuation plus two additional terms. The first term,

δave, results from the difference between the flux surface average and the ensemble average. The second term, δflux, is the average fluctuation. Both terms vanish in the limit that the events are sampled evenly from a single flux surface. However, the ensembles are constructed out of hundreds of events spanning many similar plasma equilibria and both terms may contribute to the small fluctuation amplitude measured. Additionally, terms like the Hall EMF depend on the correlation of magnetic fluctuations and δ is expected to be partially correlated with the tearing fluctuations as both are dependent on the plasma equilibrium. We are unable to calculate δ directly and must include its effect in our analysis. One way to achieve this is to divide the ensemble in half and perform our measurements on each half. Any difference between our measurements is the result of δ by definition. δ is a function of the particular ensemble average and a distribution of ensembles is required to properly account for its effect. In Fig. 3.6 we have randomly divided our ensemble in half and measured the Hall EMF for each half, plotting the results in blue. We then redivide our ensemble and remeasure the Hall EMF, plotting the results in red. The process is repeated once more and plotted in green. We observe that the general profile shape of the Hall EMF is preserved but the actual amplitudes of the EMF vary. This effect is due to δ and minimal variation is observed for measurements of the equilibrium magnetic field. We will calculate a variance to quantify the effect of δ on any 2 ([X]1−[X]2) particular measurement. Mathematically we define our ensemble variance as Vj = 2 , where the ensemble average is calculated for each half denoted by 1 and 2. To calculate the 45

−20 V/m

−40

−60

0.6 0.7 0.8 0.9 1.0 Normalized Radius

Figure 3.6 Effect of δ on the measurement of the Hall EMF

variance for the entire ensemble, and thus the effect of δ, we calculate the average variance from PN j VJ halving the ensemble, given by Vave = N . We calculate the variance from 100 divisions for p the data presented in this thesis, plotting Vj in Fig. 3.7 for the measurement of the Hall EMF r at a = 0.65. The average difference measured is the quantity used to describe the shot to shot variation for the Hall EMF at that location.

3.7 Pseudospectrum

The Madison Symmetric Torus has an array of magnetic ﬁeld measurements located at the wall of the experiment at a single poloidal angle. Previous measurements on MST have found that the magnetic tearing modes have poloidal mode numbers that are a function of its toroidal mode number [12], allowing us to decompose the ﬁeld measurements into its poloidal and toroidal mode numbers using only the single array. Additionally the mode spectrum can be correlated with internal probe measurements. A technique called pseudospectral analysis utilizes this correlation 46

0 0 5 10 15 20 Difference in V/m

p r Figure 3.7 Histogram showing the measured Vj of the Hall EMF evaluated at a = 0.65 47 to infer the mode spectrum of the probe measurements. An in depth description of this technique can be found in the thesis of Dr. Tim Tharp [77] but a brief overview will be presented here. The Fourier components of our magnetic field measurement X, and the magnetic field measured at the wall B, can be defined as follows

∞ X 1 i(m0θ+n0φ) X = x 0 e (3.11) 2 n n0=−∞

Bn = |bn| cos(mθ + nφ − δbn ) (3.12)

i(mθ+nφ) i(mθ+nφ) xne ≡ cnbne (3.13)

where we have adopted the notation used in Ref. [77] and m and n are the poloidal and toroidal mode numbers, respectively. The toroidal mode number implies the poloidal mode number, as per the discussion in Sec. 2.1.2. The last equation states that the magnetic spectrum at our measurement

location, xn, is related to the magnetic spectrum measured by the toroidal array, bn, through a

complex constant of proportionality, cn. Since magnetic mode structures are radially broad in MST, this is a valid assumption. Taking advantage of the orthogonality of the Fourier basis set

used, we can extract a speciﬁc mode if we are able to integrate XBn over the toroidal and poloidal angles. We can take the ﬂux surface average of that quantity and using the formalism presented by Tharp, we decompose our signal through the equation

† hXBni + i XBn xn = (3.14) 2 hBni

which is calculated from three ﬂux surface averaged quantities. This equation allows the mode decomposition of the measured signal to be calculated by correlation with the magnetic array located at the edge of the plasma. This can be calculated for any mode number but we are restricted to the number of modes that are resolved by the array of measurements located at the edge. For MST, mode numbers up to 15 can be resolved but in practice only modes up to 8 are calculated due to the increased uncertainty present due to low signal. We are interested in calculating the uncertainty in the pseudospectrum calculation. For simplicity, we will deﬁne a new quantity q , given by q ≡ hXBni . The uncertainty in q will be x x hBni x 48 proportional to the uncertainty in the pseudospectrum.

2 2 2 σhXBni hXBni σqx = + 2 σhBni hBni hB i n (3.15) 2 σ2 σX Bn 2 ∝ + 2 X + qx N N hBni From the ﬁrst term, we see that the uncertainty in this measurement is at least the uncertainty of the signal being decomposed. As the mode spectra are typically low amplitude, uncertainties are quite large for the majority of modes calculated. However we are still able to glean information from the dominant modes, such as the m = 0, n = 1 tearing mode resonant near the plasma edge.

3.8 Summary

The onerous task of properly analyzing magnetic data is essential for the accurate interpretation of it and all sources of uncertainties need to be assessed to be able to comment correctly on measurements. In this chapter, we have explained the background analysis that the results in this thesis depend on, allowing us to comment freely on the implications of our measured and calculated terms. 49

Chapter 4

Probe Measurements

The Deep Insertion Hall Probe developed and discussed in depth in Chapter 2 has allowed for extensive profile measurements of the magnetic field to be acquired. Thorough analysis of the magnetic field data, outlined in Chapter 3, allow us to utilize these data in the assessment of the relaxation dynamics on the Madison Symmetric Torus.

4.1 Magnetic Field Measurements

The Deep Insertion Hall Probe allows us to measure extensive profiles of the equilibrium magnetic field. In Fig. 4.1 we have overplotted five ensembles from different radial positions with similar plasma conditions. These data were collected during a scoping run to determine the extent that the DIHP can be inserted without deleterious effects to the plasma. The data is averaged over a 150 µs window around t = 0 with respect to the relaxation event, defined to be the the period of maximum toroidal flux generation. The continuity of the magnetic profiles from separate radial insertions shows that the plasma equilibria are very similar, as expected from our assumptions when we performed the flux surface average analysis. We are able to collect useful equilibrium data up r to a ∼ 0.4. However, measurements of fluctuation-induced EMF for the deeper insertions have larger uncertainties, likely due to the increased variability of the RFP dynamo structure since we have tried to keep the plasma equilibria as similar as possible. The observed variability is part of the motivation for the thorough analysis of the shot to shot variation discussed in Sec. 3.6, and the cal- r culated uncertainties prevent accurate EMF measurements at deeper insertions, 0.4 ≤ a ≤ 0.6. As r a result, we will focus on measurements taken from the ensemble with radial insertion of a ≥ 0.6 50

1200 1000 800

600 Bφ −B 400 θ B (Gauss) 200 0 −200 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Normalized Radius

Figure 4.1 Five ensembles taken at different radii, overplotted, measured at t = 0

for the majority of data in this thesis as these data can be collected through a single insertion depth and does not require the extrapolation of data from deeper insertions. We can comment on the equilibrium data plotted in Fig. 4.1. At the maximum DIHP insertion r of a ∼ 0.4, we see that the toroidal and poloidal fields are equal in magnitude. The poloidal r magnetic field maximum occurs around a ∼ 0.6, which is coincident with the insertion depth used for measurements presented in the remainder of this thesis. Near the wall, we begin to see the effects from port hole field error through the steeping of the gradient in the magnetic field, highlighted by the light gray region. The dark gray region presents measurements taken beyond the wall of MST, showing the continuation of the flux expansion into the port. We are unable to calculate the current density in the gray regions and data will be omitted when current density measurements are presented. We will select one of the ensembles from Fig. 4.1 to examine in depth. In Fig. 4.2 the values of the magnetic field are averaged over the same window as previously. The reversal of the toroidal r field is observed at a = 0.84, defined to be the location of the resonant surface for all m = 0 modes. 51

1200

1000 Bφ

−Bθ 800

600

400 B (Gauss)

200

−200 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Radius

Figure 4.2 Equilibrium Magnetic Field during relaxation at t = 0.0 ms

This location is referred to as the reversal surface. Near the deepest radial location, the poloidal field is close to the maximum value measured from the scoping run. We can compare the profiles from before and during a relaxation event through the q profile, defined as q ≡ − rBφ , where we RoBθ have used the cylindrical approximation and Ro is the major radius. In Fig. 4.3, we have plotted two time slices representing dynamics before, at t = −0.5 ms, and during the plasma relaxation, at t = 0.0 ms. We see that the reversal surface moves outward and the toroidal field increases in amplitude near the wall, deepening reversal during the relaxation event. Not shown is the return of the profile to prerelaxation levels after the event. This tends to be a gradual process and continues until the next relaxation event. Since before and after relaxation event levels are very similar, we will only show before and during time slices for visual clarity.

4.2 Equilibrium Reconstruction

Results from an MSTFit reconstruction are used in some of the results presented in this thesis. The ﬁtting routine also provides an inversion of the line averaged density proﬁle measured by 52

Figure 4.3 q proﬁle at t = −0.5 ms and t = 0.0 ms relative to a relaxation event 53

0.25

0.20

0.15

0.10 −0.5 ms

(Tesla) 0.0 ms ϕ B 0.05

0.00

−0.05 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Radius

Figure 4.4 Toroidal magnetic field from MSTFit reconstruction with probe data overplotted the FIR system. In Fig. 4.4 we see the fits of the toroidal magnetic field before and during a relaxation event, the DIHP data denoted by individual points in the edge. Before the event, we see good agreement between the fit and the probe data. However, during the event we see the fit slightly deviates from the probe data towards the core of MST. This discrepancy becomes amplified when we perform radial integrals to calculate the induced electric field, plotted in Fig. 4.5. The disagreement with the inferred electric field from the DIHP is indicative of the need for increased degrees of freedom for the fitting of the F profile near the edge during relaxation. This has little effect on the fitting of the pressure profile, which the temperature and density profiles used in this thesis depend on, allowing accurate profiles to be constructed without an exact fit to magnetic data. The density profiles used to calculate the Hall EMF from the correlated Lorentz force are presented in Fig. 4.6.

4.3 Correlated Lorentz Force Calculation

One of the main goals of this thesis is to calculate two-ﬂuid contributions to plasma relaxation. We know from previous experiments that the largest two-ﬂuid contribution results from the Hall 54

10 −0.5 ms 0.0 ms 5

0 (V/m) || E

−5

−10 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Radius

Figure 4.5 Induced electric ﬁeld from MSTFit reconstruction with probe data overplotted

1.4

1.2

1.0

0.8 −0.5 ms 0.0 ms 0.6 Density*1e19/m^3 0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Radius

Figure 4.6 FIR interferometry data inverted with MSTFit reconstruction 55

EMF in Ohm’s law and the Maxwell stress in the momentum equation. Both of these terms in- D E volve the correlated Lorentz force, given by J˜ × B˜ . This requires measurement of the current and magnetic field simultaneously. However we can derive an expression that yields the parallel component of the correlated Lorentz force in terms of measurable magnetic fluctuations. The DIHP was designed to measure extensive profiles of these fluctuations. Using Ampere’s law we can rewrite the Lorentz force, which we will define as F ≡ J × B, solely in terms of the magnetic field. We will assume cylindrical symmetry for this derivation.

F = J × B ˆ = (JzBr − JrBz) θ + (JrBθ − JθBr)z ˆ ˆ 1 1 ∂(rBθ) 1 ∂Br 1 ∂Bz ∂Bθ F· θ = − Br − − Bz µo r ∂r r ∂θ µo ∂θ ∂z 2B B ∂(B B ) ∂(B B ) 1 ∂(B2 + B2 + B2) µ F = r θ + r θ + θ z − r θ z o θ r ∂r ∂z 2r ∂θ 1 ∂Bz ∂Bθ 1 ∂Br ∂Bz F· zˆ = − Bθ − − µo ∂θ ∂z µo ∂z ∂r B B ∂(B B ) 1 ∂(B B ) 1 ∂(B2 + B2 − B2) µ F = r z + r z + θ z − r θ z o z r ∂r r ∂θ 2 ∂z

As we see from the above equations, the Lorentz force is dependent on spatial derivatives of the magnetic field. However, these expressions can be simplified since we are examining a mean-field H XdS approach to the relaxation dynamics. Taking the flux surface average, defined as hXi ≡ H dS , the equation for the parallel component of the correlated Lorentz reduces to just radial dependencies of the magnetic field fluctuations, given by the equation D˜ ˜ E 2 ∂ D ˜ ˜ E ˆ 1 ∂ D ˜ ˜ E µo J × B = + BrBθ θ + + BrBz zˆ (4.1) k r ∂r r ∂r as the derivatives in θ and z are complete derivatives. This formula for the correlated Lorentz force can be measured by a radial array of magnetic triplets, influencing the design of the DIHP. An expanded derivation of this term is found in Appendix A.4. 56

4.4 Measurements of Mean-Field Ohm’s Law

The previous analysis of magnetic data from the DIHP will provide a way to measure multiple components from mean-field Ohm’s law, which can be thought of as describing the dynamics of the more mobile electrons. In each following four subsections, a term of mean-field Ohm’s law will be discussed along with any assumptions or additional measurements required to calculate the term. The equation for two fluid Ohm’s law is discussed in Sec. 1.4 and is given by

D ˜ E 1 D˜ ˜ E Ek − ηJk = − v˜ × B + J × B (4.2) k ne k

4.4.1 ηJk

The equilibrium current density is calculated by using Shafranov shift corrected cylindrical symmetry. The equations for the equilibrium current density are

µo hJi = ∇ × hBi (4.3) ∂ hB i µ hJ i = − z (4.4) o θ ∂r 1 ∂(r hB i) µ hJ i = θ (4.5) o z r ∂r where the derivatives with respect to θ and φ are assumed to be zero due to the symmetry of the equilibrium magnetic field. In Fig. 4.7 we plot the calculated parallel current profiles before and during a relaxation event. During the relaxation event, we see a reduction in the current towards the core and a generation of current between the wall and reversal surface. The overall effect is that of current profile relaxation. We can also look at the resistive decay component of Ohm’s law by calculating the plasma resistivity. Here we have used Spitzer resistivity, consistent with measurements from Kuritsyn and Tharp. However the contribution to Ohm’s law remains small when compared to the other terms during the relaxation so the exact choice of resistivity model is not as important as the other terms. For the plasma resistivity we use temperature and density profiles calculated from MSTFit reconstructions, constrained by measurements from the FIR interferometry and Thomson Scattering systems, respectively. The estimated profile shows current generation peaking around 57

−100 −0.5 ms 0.0 ms )

3 −200 (kA/m ||

J −300

−400

−500 0.6 0.7 0.8 0.9 1.0 Normalized Radius

Figure 4.7 Parallel current before and during a relaxation event 58

0.0 −0.5 ms 0.0 ms −0.2

−0.4 (V/m) || J η −0.6

−0.8

−1.0 0.6 0.7 0.8 0.9 1.0 Normalized Radius

Figure 4.8 ηJk measurements with η calculated from FIR interferometry and TS data

r a = 0.9. Closer towards the core, we observe very little change in the current. The amplitudes are also much smaller than other measured terms in Ohm’s law, before and during relaxation events.

4.4.2 Ek

∂hBi The electric field is inferred from the magnetic induction equation, ∂t = −∇ × hEi. Here we again use Shafranov shift corrected cylindrical geometry and can calculate path integrals that surround locations where we have measured the change in magnetic flux. The full derivation is available in App. A. When calculated, the electric field is found to be as follows

Z a 0 0 Vtg 1 r ∂ hBφ(r )i 0 hEθ(r)i = − + dr (4.6) 2πr r r ∂t Z a 0 Vpg ∂ hBθ(r )i 0 hEφ(r)i = − − dr (4.7) 2πRo r ∂t where Vtg and Vpg are loop voltages measured at the toroidal and poloidal gaps, respectively. The loops are located at the edge of the plasma and serve as boundary conditions for our electric ﬁeld calculation. 59

0 (V/m) || E

−5 −0.5 ms 0.0 ms

−10 0.6 0.7 0.8 0.9 1.0 Normalized Radius

Figure 4.9 Calculated induced parallel electric ﬁeld from changing magnetic ﬂux 60

In Fig. 4.9 we have plotted the parallel electric field before and during a relaxation event. Before the event, we see that the electric field is negative with relatively small amplitude. During the event, we see an increase in the electric field near the edge, corresponding to a generation of magnetic flux. Near mid-radius we see that the parallel electric field goes to zero. The values for the parallel electric field are much larger than what is measured for the resistive dissipation during the event, but before the event the values are small and comparable, leading to an observed balance of mean-field Ohm’s law during this time between the two terms discussed already. Additional terms are required for balance during the relaxation event. D E 4.4.3 1 J˜ × B˜ noe k As seen before, the Hall EMF can be decomposed into a function of radial derivatives allowing the DIHP to measure the term. We have used the density profile in Fig. 4.6, created by line- integrated FIR density measurements inverted by a MSTFit equilibrium reconstruction constrained by our data. Combining the density profile with the calculated Lorentz force, given by Eq. 4.1, we can calculate the Hall EMF. In Fig. 4.10 we have plotted the Hall EMF before and during a relaxation event. Our measurements are quantitatively similar to that of previous measurements in the edge of MST by Kuritsyn and Tharp. The Hall EMF in the edge is observed to be small until r the the plasma reversal surface, a = 0.84, growing in amplitude during relaxation and balancing

Ek − ηJk. Deeper into the plasma, the Hall EMF grows to a large negative amplitude, being much

larger than the measured Ek − ηJk at the plasma mid-radius. The implication of our measurements is that the MHD EMF term must be very large as well to balance mean-ﬁeld Ohm’s law. Before the event, the calculated Hall emf is close to zero with the exception of the most interior measurements.

D E 4.4.4 v˜ × B˜

Although not directly measured, we can infer the behavior of the single-ﬂuid term, the MHD EMF, from an assumed balance of mean-ﬁeld Ohm’s law. As mentioned before, the large amplitude of the Hall EMF near mid-radius implies a large MHD EMF to balance it. In Fig. 4.11 we have 61

−10

−20 −0.5 ms 0.0 ms

Parallel Hall EMF (V/m) −30

−40

0.6 0.7 0.8 0.9 1.0 Normalized Radius

Figure 4.10 Hall dynamo measurement from DIHP 62

−20 E||−η J|| /n e

Ohm’s law contributions (V/m) o −, Inferred −40 0.6 0.7 0.8 0.9 1.0 Normalized Radius

Figure 4.11 The inferred MHD dynamo term from a balance of two-ﬂuid Ohm’s law taken the values of the terms in Ohm’s law during a relaxation event to determine the proﬁle of the MHD EMF. From previous measurements on MST, we know that the MHD EMF balances

Ek − ηJk near the wall and tapers off to zero around the reversal surface [25]. This behavior matches qualitatively with the inferred MHD EMF from the measurements, assuming a two-ﬂuid Ohm’s law describes the dynamics in question. In addition, previous measurements utilizing the CHERS diagnostic have seen hints at large MHD EMF activity mid-radius [21]. However, large increase in signal noise associated with tearing mode reconnection prevent precise measurements during a relaxation event. Future attempts to measure the MHD EMF using more sophisticated techniques may illuminate the mid-radius behavior of the MHD EMF.

4.5 Current Relaxation

The DIHP allows us to measure three out of the four terms present in two-fluid Ohm’s law, affording a nearly complete picture of the relaxation process in MST plasmas. It has been observed that the parallel current profile undergoes a flattening during bursts of magnetic activity, as seen 63 in Fig. 4.12. We can compare our measurements of this flattening with Taylor’s theory of plasma relaxation, where the quantity λ is predicted to be constant across the plasma radius, with λ defined J·B as λ ≡ B2 [74]. Taylor’s theory of relaxation is based on minimizing the magnetic energy with respect to global R helicity, Ko ≡ A·B dVo. The solutions to this are states in which the parallel current is constant across the plasma radius, or ∇ × B = λB. However, this implies that the parallel current at the boundary is nonzero and assumes that the plasma is in a force-free equilibrium. To describe plasma equilibria that are not fully relaxed, we can include the effect of finite pressure gradients and prescribe a radially varying λ profile. The model is called the alpha model, given by the equation B × ∇p ∇ × B = λ(r)B + (4.8) 2B2 Experimentally, the λ profile is found to have a radial dependence given by

r α λ(r) = λ 1 − (4.9) o a

The values for λo and α can be determined only from measurements of Ip, hBti, and Bt(a) since the alpha model fully describes the equilibrium magnetic field. MHD stability analysis of these profiles predict excitement of current driven m = 1 instabilities that rearrange the current profile to more stable configurations [3]. In Fig. 4.12 we have plotted the a normalized lambda profile, aλ, where a is the minor radius of MST. The plotted lines are from Eq. 4.9 and points with error bars are measurements from the DIHP. Before relaxation, we have good agreement with the alpha model, with α = 2.4, and the alpha model is a good description of the plasma equilibrium. During relaxation, a discrepancy between the model and measurements is observed, with α = 4.4. This difference occurs during the peak of the RFP dynamo activity, a period of time where the MHD and Hall EMFs generated are large and acting on the current profile. Our measurements show a stronger flattening of the λ profile not captured by the alpha model due to these EMFs. We can also estimate the value of lambda on axis through the scaling of measurements conducted on higher current discharges.

Using data from [5], we calculate a value of λo ∼ −2.9. This agrees with the picture of stronger 64

−1

λ −2 t =−0.50 ms a t = 0.00 ms

−3

−4 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Radius

Figure 4.12 Normalized lambda proﬁle from the alpha model with DIHP data overplotted 65

ﬂattening from our measurements. During relaxation the plasma is exposed to a large Hall EMF that our probe has measured, and the simple alpha model no longer approximates the plasma at this time. Part of the Hall EMF appears as the Maxwell stress in the mean-ﬁeld momentum equation and we will examine the forces working on the plasma momentum during relaxation.

4.6 Momentum Evolution Measurements

The momentum equation governs the motion of the ions as they are much more massive than the electrons. A two-ﬂuid approach can also be used to determine the evolution of the plasma momentum. The mean-ﬁeld averaged momentum equation is discussed in Sec. 1.4.1 and given by

∂Vk D˜ ˜ E min = −min h(v˜ · ∇) v˜ik + J × B (4.10) ∂t k

We see that the last term, the Maxwell stress, can be measured by the Deep Insertion Hall Probe and provides a natural coupling between the momentum equation describing the ion dynamics and Ohm’s law describing the electron dynamics. The change in plasma momentum is given by the left hand side of Eq. 4.10. The change in density is relatively small compared to the change in velocity during a relaxation event so its contribution is typically omitted. In Fig 4.13 we use the phase velocity of the innermost resonant tearing mode, the m = 1, n = 6, as a proxy for plasma ﬂow measurements at the mode’s resonance r location, a ∼ 0.3. This is a valid assumption based on spectroscopic ﬂow measurements of the plasma compared to tearing mode phase velocities [14]. The density used is determined by the MSTFit inversion of FIR measurements. We see in Fig 4.13 that the contribution from the change

∂(ρVk) in density is observed to be much smaller than the change in velocity contribution and ∂t = ∂Vk ∂ρ ∂Vk ρ ∂t + Vk ∂t ≈ ρ ∂t . The change in plasma momentum is in the positive toroidal direction, which is against the direction of plasma ﬂow. This corresponds to a force acting on the plasma to slow it down during relaxation events in the core, either due to the Maxwell stress or the Reynolds stress. The Maxwell stress appears as a correlated Lorentz force in the momentum equation. This term provides a coupling between the current and momentum equations, allowing us to examine the relaxation in both quantities. The magnetic energy is much larger than the kinetic energy in RFPs 66

2.0 ) 3 1.5

ρ∂tv 1.0 v∂tρ

0.5

0.0

−0.5 Total momentum change (N/m

−1.0 −2 −1 0 1 2 Time (ms)

Figure 4.13 Change in core plasma momentum inferred from the m=1, n=6 tearing mode 67

40 )

3 20

−20

−40

−60 Maxwell Stress (N/m −80 −100 0.6 0.7 0.8 0.9 1.0 Normalized Radius

Figure 4.14 Measured Maxwell stress from the DIHP during relaxation and momentum dynamics will have negligible feedback onto current dynamics. The Maxwell stress term then behaves as a source term acting on the total change in momentum. In Fig. 4.14 we have plotted the measured parallel Maxwell stress during a relaxation event. This large force density causes the plasma to respond by the change in plasma inertia and the development of a Reynolds stress. However, the magnitude of the Maxwell stress is much larger than the observed plasma inertia, which is only a few N/m3 and an equally large Reynolds stress in the opposite direction must develop to combat the large stress acting on the plasma. The Maxwell stress mid- radius is acting in the opposite direction of the plasma inertia in the core, implying an inversion of the stress in the core to drive the flow change there. One can take Taylor’s theory of relaxation and apply it to two species by conserving electron and ion helicities separately [33, 70]. The minimization of electron energy yields the previous results of flattening the λ profile. By minimizing the ion energy, one obtains a state in which ρV·B the parallel momentum profile is flattened, λi = B2 , where we end up with a similar form to J·B the aforementioned λ profile, λe = B2 . Flattening of the parallel momentum profile has been 68 previously observed on MST but the forces driving relaxation have only been measured in the edge during events. Our probe measurements of the Maxwell stress can illuminate some the the relaxation behavior mid-radius. The Maxwell stress interacting with the Reynolds stress, resulting in flow change is observed on MST in the edge using probes [43], in quasilinear MHD simulations [20], and in nonlinear extended MHD simulations [40]. Measurements of the Reynolds stress are unavailable further into the plasma due to the difficulty of obtaining localized radial velocity fluctuation measurements. Our probe measurements show a large Maxwell stress acting on the plasma mid-radius, implying that there should be a large change in plasma momentum in that location assuming that the Maxwell stress is the impetus for flow change. Previous Rutherford measurements of the plasma velocity do in fact show a large change in the velocity in that region, which will be discussed in Sec. 5.3. However we estimate that the magnitude of the Maxwell stress is about an order of magnitude larger than the change in plasma momentum mid-radius. The inference is that a large Reynolds stress is present, nearly balancing the Maxwell stress. It is also observed that the direction of the measured stress is consistent with that required to change the observed plasma flow, albeit at a much larger amplitude. To relax the core plasma flow, an equally large Maxwell stress is predicted, with the direction reversed with respect to the measured term mid-radius.

4.7 Reversing Plasma Current and Momentum

An experimental condition we can alter is the direction of the inductive electric field. This will in turn change the direction of the plasma current and plasma velocity relative to the magnetic field. Since tearing modes are current gradient driven instabilities, we would expect that the EMFs working to restore the current profile would reverse with the switch in current profile direction. In Fig. 4.15 we have plotted the alpha model along with DIHP measurements. As expected, the λ profile is reversed with respect to Fig. 4.12 and the profile is flattened during the event. We observe in Fig. 4.16 that the Hall EMF reverses direction along with the inductive electric field, corresponding to the change in direction required for relaxation of the current density profile. We also observe that the change in core plasma momentum also reverses. The experimental change in 69

λ also serves as a secondary check of the measurements since amplitudes of the measured terms are not expected to change as long as the plasma parameters are kept as similar as possible. Ad- ditionally, this change in direction allows for more direct comparison to past simulations as they are typically computed using a positive λ equilibrium. We see in Fig. 4.17 that the profile shapes are quantitatively similar. The observed differences in the profiles are due to the increased variance in the discharges from running the experiment in a positive λ configuration and a decreased sample size when compared to the negative λ data. MST is seldom run in this configuration so additional experimental data is currently unavailable to verify the direction of other terms related to relaxation.

4.8 Measurements of the Tearing Mode Structure

Through the analysis provided in Sec. 3.7, we can construct the pseudospectrum during relaxation events, which is done by correlating the magnetic field measurements with edge array measurements. The uncertainties present in conducting this analysis are larger than typical pseudospectral mode amplitudes, with the exception of the largest modes. The m = 0, n = 1 mode is associated with the facilitation of the nonlinear coupling to the core resonant tearing modes and the m = 1, n = 6 mode is the core-most resonant mode for the discharges studied in this the- r sis. We have also examined the m = 1, n = 10 mode, which is resonant around a ∼ 0.6, just outside the extent of the DIHP insertion. The mode amplitudes evaluated during the relaxation event are plotted in Fig. 4.18, blue and green representing the toroidal and poloidal components, respectively. Measurement of the tearing mode spectrum provides an excellent opportunity to test the models that are employed by other researchers that infer the magnetic structure. However the data used for the analysis in this thesis creates large uncertainties for the tearing mode spectrum, likely due to the increased variation observed during calculation of the shot to shot uncertainty discussed in Sec. 3.6. Additional work to refine the measurements is possible, perhaps through the development of a smaller probe or the collection of more data. Additionally, more stringent criteria for the selection of events used in an ensemble could reduce the uncertainties in the measurement, however the collection of more data is required to have a sufficiently large enough sample size. 70

λ 2 t = −0.5 ms a t = 0.0 ms

0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Radius

Figure 4.15 Experimental data with alpha model overplot with positive lambda 71

E||−η J|| /noe 20

0 Terms in Ohm’s law (V/m)

−20

0.6 0.7 0.8 0.9 1.0 Normalized Radius

Figure 4.16 Terms in Ohm’s law during a positive lambda run campaign

40 (−1)*Negative Hall EMF Positive Hall EMF

−20 Parallel Hall EMF (V/m) −40

0.6 0.7 0.8 0.9 1.0 Normalized Radius Figure 4.17 Two-ﬂuid contributions from both λ scenarios 72

100

Bφ

Bθ 50

0 B (Gauss)

−50

−100 0.6 0.7 0.8 0.9 1.0 Normalized Radius

100 100

Bφ Bφ

Bθ Bθ 50 50

0 0 B (Gauss) B (Gauss)

−50 −50

−100 −100 0.6 0.7 0.8 0.9 1.0 0.6 0.7 0.8 0.9 1.0 Normalized Radius Normalized Radius

Figure 4.18 Calculated structure of the m = 0, n = 1 mode on the top, the m = 1, n = 6 mode on the left, and the m = 1, n = 10 on the right during relaxation 73

4.9 Conclusion

We have seen that the correlated Lorentz force, in the form of the Hall EMF and Maxwell stress, affects the relaxation of the current and momentum profiles. The newly measured regions in the plasma mid-radius predict EMFs with comparable magnitudes, leading to little dynamo activity there. However the two-fluid EMF couples to the momentum evolution and is able to drive the flow relaxation measured in experiment. This relaxation causes the plasma to move towards a Taylor-like state, in which the current density and momentum profiles are flatter. We have also inferred that two-fluid effects must exist in the core region to drive the observed plasma flow measured. This behavior implies that two-fluid effects, in addition to single-fluid effects, are important throughout the plasma and is similar to what is observed in extended MHD simulations. Additional measurements are needed to diagnose the global behavior of the plasma during relaxation. Obtaining core measurements is currently beyond the capability of insertable probes on MST but non-invasive diagnostics, such as the FIR polarimetry system, can measure some components of the Hall dynamo [19]. The ChERS diagnostic can measure velocity fluctuations which can then be combined with magnetic fluctuations to calculate the MHD EMF profile. Finer radial resolution measurements of the plasma velocity can verify the Maxwell stress dependence on flow relaxation. A fuller understanding of plasma behavior during relaxation events will help guide future experiments and simulations. 74

Chapter 5

Simulation Comparison

As computer processing power increases, scientists are able to utilize high performance com- putation to create physically relevant simulations of plasma dynamics. In essence, the goal is to create a virtual experiment in which minute details of the plasma behavior can be thoroughly examined. Simulation results can then be checked against measurements to the benefit of both. At the University of Wisconsin-Madison, we have a close relationship with computational physicists and theorists that allow fruitful discussion of simulation results and how it relates to experiment. Past simulations using the DEBS code [66] have captured some tearing mode dynamics assuming only single fluid contributions on MST [57]. However measurements of the two-fluid dynamics in the core of MST [18] have shown that they play an important role in plasma relaxation and should be included for a more complete understanding of plasma dynamics. Measurements presented in this thesis and past measurements in the edge of MST have also confirmed the role of two-fluid terms [43, 78] in plasma relaxation, further cementing the needed inclusion. Incorporation of two- fluid effects into simulations would be a natural step to take. The NIMROD code [69], an initial value nonlinear code, has performed runs that utilize an extended MHD model to include two-fluid physics, yielding a more accurate description of the relaxation phenomena that occur in RFPs.

5.1 NIMROD Background

The Non-Ideal Magnetohydrodynamics with Rotation - Open Discussion (NIMROD) code is an initial value solver used to simulate nonlinear plasma effects in many different plasma scenarios. Recently, this code has been used to simulate RFP dynamics by including two-ﬂuid terms and ion 75

finite-Larmor radius effects [39, 40, 63, 65]. The inclusion of these terms have lead to noticeable changes in the plasma behavior. The current profile is relaxed by the competition of the MHD and Hall EMFs, whereas previously the relaxation was done solely by the MHD EMF. The inclusion of ion gyroviscosity has a stabilizing influence on linear tearing modes [39]. The equations used to create the simulation results presented in this thesis include additional terms in Ohm’s law and the momentum evolution equation that were not included in the mean- field analysis presented earlier in the thesis. The additional terms tend to be small but still play an important role in the physics that govern plasma evolution. The equations are J × B ∇p m ∂J E = −v × B + Λ ηJ − − e + e (5.1) e ne ne ne2 ∂t ∂v m n = −m v · ∇v + J × B − ∇p − ∇ · Π − Λ ∇ · Π (5.2) i ∂t i iso i gyr

The markers, Λe and Λi, are binary and are used to indicate when certain effects are included in

the simulation. Λe causes two-fluid effects to enter Ohm’s law and Λi causes finite Larmor radius (FLR) effects to enter the momentum equation. The NIMROD simulation includes contributions from pressure gradient effects and electron inertia in Ohm’s law, although it is observed that the latter term has negligible influence on the evolution of the electric field. In the momentum equation, pressure gradient effects are included along with the viscous stress tensors. The first viscous stress tensor results from a simple collisional viscosity. The second tensor represents gyro-orbit frequency shifts and ellipticity arising from ∇E [38] (i.e., FLR effects). Additional equations govern the evolution of density and temperature profiles but will not be discussed here for simplicity. The simulation results discussed in this thesis include both two-fluid and ion FLR effects but we will mention the differences observed when the effects are omitted in simulation. One way to examine the contributions of the different terms is to rewrite Ohm’s law in a dimensionless form. The equation then becomes

1 δ β δ δ 2 ∂J 0 E0 + v0 × B0 = J 0 + i J 0 × B0 − i ∇0p0 + e (5.3) S a 2 a e a ∂t0

0 τR 2µop where X refers to the normalized quantities. The dimensionless quantities S ≡ , β ≡ 2 , τA B 2 2 δi c δe c ≡ 2 , and ≡ 2 are the Lundquist number, the ratio of ﬂuid pressure to magnetic pressure, a aωi a aωe 76 the normalized ion skin depth, and the normalized electron skin depth. For the discharges studied

5 δi δe in this thesis, the values are S ∼ 5 · 10 , β ∼ 0.1, a ∼ 0.2, and a ∼ 0.003. Simulations are run in dimensionless form and these parameters afford us a quick look to compare the relative strength of terms. The simulation results presented uses the following parameters: 1) β = 0.1, 2) the magnetic Prandtl number, the ratio of resitive to viscous diffusion times, P ≡ τR = 1.0, 3) S = 2 · 104, and m τν

di 4) a = 0.173. These parameters are on axis values and are similar to the studied MST discharges with the exception of the Lundquist number. Typical discharges on MST are S ∼ 106 however the work presented in this thesis employs lower current and temperature plasmas, lowering the Lundquist number to values closer to current simulations. Simulation results with experimental Lundquist numbers are unavailable as it would require computational resources far beyond what is currently feasible. A more complete description of the simulation framework is presented in [62].

5.2 NIMROD Results

In Fig 5.1 we have reproduced a plot of the dynamo EMFs during a NIMROD simulation,

labeled as Case C in [62] (Λe = Λi = 1), to differentiate it from the simulations that do not include two-fluid effects and ion gyro-viscosity. The top plot is the MHD EMF and the bottom is the Hall EMF. The vertical dashed lines represent the relaxation events we are interested in. The time periods are chosen such that the event begins when magnetic fluctuation energy greater than 5% is distributed in the non-dominant modes. This ensures that the events selected involve multiple active modes which is typical of relaxation in RFPs. Also shown by the solide color traces are the resonant surfaces for a few large tearing modes. For this particular simulation, three events occur before the simulation terminates. The simulation begins in a paramagnetic pinch equilibrium and the system evolves into a sat- urated single-helicity state involving the m = 1, n = 6 core resonant tearing mode. The first relaxation event in the simulation disrupts this single-helicity state. This event is accompanied with the MHD and Hall EMFs working in unison to relax the current profile, reversing the edge toroidal field and bringing the system closer to a relaxed state. The subsequent events show additional tearing modes that are of similar amplitude to the dominant mode during relaxation. Current 77

Figure 5.1 NIMROD simulation of MHD and Hall dynamos[62] 78 relaxation is then predominantly done by the MHD dynamo in the core with the Hall dynamo working in opposition, albeit at a smaller amplitude. All simulations with multiple events show the same difference between the first and subsequent events. As the first event features a single-helicity state resultant from the initial conditions, it should be considered separately from the subsequent events. The subsequent events start from a broader mode spectrum that is more representative of typical relaxation dynamics in standard MST discharges. We will focus on the second event in the simulation. In Fig 5.2 the EMFs are averaged during the relaxation event shown by the vertical dashed lines in Fig 5.1. The units have been converted for comparison to MST discharges similar to those studied in this thesis. The plot shows both EMFs active across the majority of the plasma radius. In the edge, the Hall EMF reduces to zero r around a ∼ 0.8 while the MHD slowly tapers off to zero at the wall. It is observed that the interplay of both terms are responsible for current relaxation; the larger MHD EMF relaxes the current in the core. In Fig 5.3 we have the Maxwell and Reynolds stress profiles which act on the plasma momentum. We see that the two stresses nearly balance each other throughout the plasma radius. Sig- nificant changes to the parallel flow around relaxation events is observed, mirroring the Maxwell

stress profile. This is not observed when two-fluid effects are omitted from the simulation (Λe = 0). The Maxwell stress is the largest force density acting on the plasma, driving the flow change in experiment and the other stresses act to suppress the Maxwell stress.

5.3 Comparison with Experimental Measurements

The data in this thesis have used lower current plasmas to improve the longevity of the probe and increase plasma reproducibility. A side beneﬁt of these plasmas is lower plasma temperature and lower magnetic ﬁelds. This leaves us with a Lundquist number that is almost an order of magnitude lower than standard discharges on the experiment, allowing our data to be more comparable to simulation than previous experiments. 79

100

50 ] m

/ 0 V [

E 50 V˜ B˜ −h × i 1 ˜ ˜ n e J B 100 h i h × i 0.0 0.2 0.4 0.6 0.8 1.0 r/a

Figure 5.2 Time averaged Hall and MHD emfs from the second event in Fig 5.1, courtesy J. Sauppe 80

100

50 ] 3 0 m / N [

F 50

J˜ B˜ −h × i 100 m nV V −h i ·∇ i 0.0 0.2 0.4 0.6 0.8 1.0 r/a

Figure 5.3 Time averaged Maxwell and Reynolds stresses from the second event in Fig 5.1, courtesy J. Sauppe. The sign of the Maxwell stress has been inverted for amplitude comparison with the Reynolds stress 81

Figure 5.4 Comparison of DIHP and NIMROD EMF proﬁles

In Fig. 5.4 we have rescaled the plot of the measured Hall EMF and the MHD EMF, inferred from an assumed balance in the mean-field Ohm’s law, for easier visual comparison with the NIM- ROD EMF profiles. The structure of the EMF profiles from simulation bears qualitative similarity to what is seen in experiment. The amplitudes of the EMFs are similar to what is measured on MST in the outer half of the plasma. However, the radial location of the local extrema when comparing experiment to simulation appear in different locations. The Hall EMF is measured to first peak at the reversal surface of MST whereas the NIMROD profiles show the peaking occurring r further into the plasma at a ∼ 0.6. The overall profile shape of the simulated EMFs appears con- tracted towards the core when compared to experimental values. One possible explanation for this difference is that the measured data is constructed from hundreds of different relaxation events but the simulation shows the flux surface from a single event. The same behavior may be present in experiment but is not typical for the plasma relaxation. Our measurements of the Maxwell stress allows us to compare to some of the momentum evolution results from the NIMROD simulation. The simulations observe that the change in plasma momentum is induced by the Maxwell stress, and show positive parallel flow change in the core. This direction is in agreement with the measured plasma inertia in Fig. 4.13. If the Maxwell stress drives the changes in the parallel plasma momentum, as seen in simulation, our measurement of the stress near the plasma mid-radius infers a negative change in parallel momentum there. Since our measured Maxwell stress is much larger than the estimated change in parallel momentum, a 82 large Reynolds stress needs to develop and nearly balance the Maxwell stress for the mean-field momentum equation to be in balance, similar to what is seen in Fig. 5.3. We have measurements of three of the four terms in the mean-field Ohm’s law from the edge to near the plasma mid-radius. We also have equilibrium reconstructions of the non-EMF terms,

Ek and ηJk, across the entire plasma radius. Since we have a measurement of the plasma inertia in the core, we can infer the direction of the Maxwell stress, and thus the Hall EMF in the core. Using the information available to us, we can construct radial profiles of mean-field Ohm’s law and compare it to the simulation. We will assume that any radial structure will be smooth and broad as strong radial gradients will manifest themselves through generation of magnetic flux and the measured and reconstructed equilibrium magnetic profiles are relatively smooth. In Fig. 5.5 we have pieced together our hypothesis of the global balance of mean-field Ohm’s law. The shaded region represents the undiagnosed region for the discharges studied in this thesis. In the unshaded region, we have the Hall dynamo in red and Ek − ηJk in black. The blue is the inferred MHD dynamo profile from a balance of mean-field Ohm’s law. We know from MSTFit equilibrium reconstructions as well as FIR polarimetry measurements in higher temperature plasmas that the core electric field is large and negative during relaxation events [76]. We also know that ηJk is much smaller than the electric field during events. As a result, we expect the black curve to be dominated by the parallel electric field structure, smoothly changing signs from the edge to the core. From the momentum equation, we expect the Maxwell stress to follow the direction of

∂Vk min ∂t . In the core of MST plasmas, this direction is positive and the Hall EMF must be positive as well. For a balance of mean-field Ohm’s law, we require that the MHD EMF be in the same direction as the core electric field. For mid-radius, we need the core estimates to connect to the probe measurements in that location. Comparing our estimation of the global EMFs in Fig 5.5 with the simulation in Fig 5.2, we see that the EMF profiles are qualitatively similar, showing that the simulation is able to reproduce features observed in experiment, such as the reversal of signs for both EMFs going from the plasma mid-radius to the core. We can also compare our estimated profiles of mean-field Ohm’s law with localized measurements made in previous experiments on MST. The large Hall EMF in the plasma core and 83

Figure 5.5 Illustration of the inferred global profile of Ohm’s law from previously measured data on MST 84 mid-radius infer large changes in plasma momentum. In Fig 5.6, we have replotted measurements of the parallel plasma flow evolution during a relaxation event from [55]. These data were made using a combination of the Rutherford scattering diagnostic to measure the poloidal flow and tearing mode velocities to estimate the toroidal flow. Measurements of the parallel flow at the plasma mid-radius, shown in blue and green, decrease sharply during a relaxation event, corresponding to a negative change in the parallel plasma momentum, in agreement with our inference through the measured Maxwell stress. The Maxwell stress is much larger than the change in parallel plasma momentum, requiring a large Reynolds stress via the mean-field momentum equation. Previous measurements of the plasma velocity fluctuations in the edge using an insertable ion dynamics spectroscopy probe (IDSP) [23] and a Mach probe [44] have shown that the Reynolds stress grows to a large amplitude, nearly balancing the large Maxwell stress measured at the reversal surface, leading to a small net change in plasma momentum [43], consistent with the simulation results in Fig. 5.3. The estimated Hall EMF profile also infers a MHD EMF profile, consistent with previous measurements of the MHD EMF in the core [15].

5.4 Summary

Simulations provide an invaluable tool in the analysis of relaxation in the RFP. We have constructed global profiles of the terms in mean-field Ohm’s law based on experimental measurements that show remarkable agreement with the profiles from simulation. The observed differences between simulation and experiment in the edge might be able to be addressed with detailed edge modeling, since the coupling to edge resonant tearing modes is essential to relaxation and the effect of ion gyroviscosity is observed to be largest in the edge. Additionally, simulation results are the flux surface averaged values of a single event, whereas the experiment averages many discrete events together to approximate a flux surface average. In both experiment and simulation, variation between individual events is observed and may account for the differences in the calculated EMF structure. Simulations have also provided predictions of modal contributions to relaxation, presenting a future avenue for pseudospectral measurements. Unfortunately, measurements from 85

Figure 5.6 Parallel plasma velocity evolution measured at several radii [55] 86 the DIHP are incapable of accurately measuring these components but upgraded probe design and analysis may be able to calculate these terms. 87

Chapter 6

Conclusions

The importance of two-fluid physics in the relaxation of reversed-field pinch plasmas has been examined experimentally in the Madison Symmetric Torus. We have found that large two-fluid contributions exist throughout the plasma volume during plasma relaxation. The correlated Lorentz force interacts with the MHD EMF by means of the Hall EMF to relax the current profile and drives flow relaxation through the Maxwell stress in the momentum equation. These terms peak during relaxation and recede between events. Recent nonlinear simulations using the NIMROD framework have also shown this behavior using plasma parameters that approximate MST’s experimental conditions.

6.1 Deep Insertion Hall Probe

We have developed a robust magnetic probe to measure magnetic field fluctuations to depths approaching the plasma mid-radius. This probe improves on previous iterations by measuring all components of the magnetic field in a centered, cylindrical geometry and provides the most extensive measurements of the equilibrium magnetic field to date. It was also designed with a significant shield to allow the probe to reach the desired depths and radial resolution sufficient to see finer details in the structure of the fluctuation profiles. In the gathering and analysis of data, we have conducted an extensive study on the source and propagation of uncertainties as they apply to magnetic field fluctuation analysis. The methodology applied in interpreting data from deeply inserted probes can be applied to future iterations of probe work on MST. Additionally, the shot-to-shot variation technique is pertinent for other types 88 of measurements to explore potential reasons for unexpected measurement variation. The probe survived thousands of discharges and work can be extended into higher current discharges. As noted in Appendix B, changes to the current probe design need to be made to extend measurements further into the plasma. We have also employed equilibrium reconstructions to determine the density and resistivity profiles to accurately measure the terms in Ohm’s law.

6.2 Two-Fluid Contributions to Relaxation

We have measured multiple terms in Ohm’s law to examine the evolution of the current density profile. We have found that the Hall EMF balances the resistive dissipation of the electric field near the reversal surface, in agreement with prior measurements in that location. Near the plasma mid-radius a large Hall EMF is measured and we infer that a large MHD EMF exists to balance mean-field Ohm’s law, meaning an interplay of single- and two-fluid effects is at work in this region. Measurements of the correlated Lorentz force also allows us to investigate the plasma momentum evolution through the Maxwell stress. Previous flow measurements using Rutherford scattering have shown relaxation of the momentum profile. We have observed the Maxwell stress peaking to a large amplitude mid-radius where the measured change in plasma momentum is large, consistent with the Maxwell stress being the impetus for flow change in the bulk of the plasma. Recent NIMROD simulations have conducted runs with parameters similar to the MST discharges studied in this thesis. Work has been done examining the effect two-fluid physics has on the behavior of relaxation through these simulations. The inclusion of two-fluid effects mani- fests itself predominantly through the changes in the plasma flow during relaxation driven by the Maxwell stress. During current profile relaxation they have observed a similar interplay in the Hall and MHD EMFs, with magnitudes comparable to what is measured on MST. We have estimated global structures of the terms in mean-field Ohm’s law based on measurements and have found remarkable qualitative agreement with the EMF structure from simulation. NIMROD simulations of the plasma show contributions to the EMFs from multiple modes across the plasma radius. Experimentally, we can alter the structure of the tearing modes, either 89 by enhancement of the core resonant m = 1, n = 6 tearing mode in single-helicity discharges or suppression of the nonlinear coupling to the edge resonant m = 0 modes. We can then compare the measured EMF profiles and their mode contributions to simulation results, allowing us to examine the effect individual modes have on relaxation.

6.3 Suggestions for Future Work

We have obtained extensive profiles of multiple terms in Ohm’s law during magnetic relaxation on MST, yielding a more complete picture of the plasma behavior. Future work towards the fuller understanding of relaxation would entail measurements of the MHD EMF profile and the plasma momentum profile. With the addition of extensive MHD EMF profiles, the two-fluid Ohm’s law presented in this thesis will be complete. Our measurements of the Hall EMF near the plasma mid-radius predict a large MHD EMF where no refined measurements currently exist. Current progress with an upgrade to the IDS system can calculate local velocity fluctuations and may be correlated with the DIHP magnetic field measurements to infer the MHD EMF. Similarly, a run campaign with the ChERS diagnostic, similar to what was done by Ennis [21, 22], can illuminate this predicted large term. Furthermore, a capacitive probe has been recently developed to measure D ˜ E the combination of the Hall and MHD EMFs, otherwise known as the total dynamo, v˜e × B [73]. As this term is the combination of both EMFs, it alone should balance Ohm’s law if no additional physics is at work. During the probe’s development, total dynamo was measured to D ˜ E balance mean-field Ohm’s law near the reversal surface [71], v˜e × B = Ek − ηJk, as expected from the measurement of the terms separately. This probe can utilize analysis methodology from the DIHP to accurately measure extensive profiles up to the plasma mid-radius. Measurements of the plasma momentum profile during relaxation should mirror the Maxwell stress profile shape if it is the impetus for flow relaxation. An upgrade of the power supply system is currently underway and provides a way to achieve reproducible discharges at much lower currents [34]. This allows us to decrease the Lundquist number, making comparisons to simulation more favorable and perhaps transitioning into a regime where two-fluid effects are reduced. The inclusion of these proposed measurements will yield insightful data covering what the DIHP was not able to address. 90

We have exposed the importance of two-fluid effects throughout the plasma during relaxation on the RFP. A similar phenomena is observed in Tokamaks, under the name of “Flux Pumping” that occurs in “Hybrid” discharges [53, 35]. Our measurements of large two-fluid effects indicate that these effects could be important in the observed stationary states in Tokamak discharges. The two-fluid effects could also couple to momentum relaxation in the Tokamak through the Maxwell stress, pointing to another avenue in which these effects could be studied. The work in this thesis provides a fuller understanding of the two-fluid dynamics that are essential to the relaxation of reversed-field pinch plasmas and can help describe similar phenomena in the Tokamak. It is hoped that the work provided here will guide future experiments and simulations for years to come. 91

LIST OF REFERENCES

[1] AF Almagri, S Assadi, SC Prager, JS Sarff, and DW Kerst. Locked modes and magnetic field errors in the madison symmetric torus. Physics of Fluids B: Plasma Physics, 4(12):4080– 4085, 1992. [2] JK Anderson, CB Forest, TM Biewer, JS Sarff, and JC Wright. Equilibrium reconstruction in the madison symmetric torus reversed field pinch. Nuclear fusion, 44(1):162, 2003. [3] V Antoni, D Merlin, S Ortolani, and R Paccagnella. Mhd stability analysis of force-free reversed field pinch configurations. Nuclear Fusion, 26(12):1711, 1986. [4] HAB Bodin and AA Newton. Reversed-field-pinch research. Nuclear fusion, 20(10):1255, 1980. [5] DL Brower, WX Ding, SD Terry, JK Anderson, TM Biewer, BE Chapman, D Craig, CB For- est, SC Prager, and JS Sarff. Measurement of the current-density profile and plasma dynamics in the reversed-field pinch. Physical review letters, 88(18):185005, 2002. [6] DL Brower, Y Jiang, WX Ding, SD Terry, NE Lanier, JK Anderson, CB Forest, and D Holly. Multichannel far-infrared polarimeter-interferometer system on the mst reversed field pinch. Review of Scientific Instruments, 72(1):1077–1080, 2001. [7] EJ Caramana and DA Baker. The dynamo effect in sustained reversed-field pinch discharges. Nuclear fusion, 24(4):423, 1984. [8] Kyle Caspary. Density and Beta Limits in the Madison Symmetric Torus Reversed-Field Pinch. PhD thesis, University of Wisconsin - Madison, 2014. [9] BE Chapman, JK Anderson, TM Biewer, DL Brower, S Castillo, PK Chattopadhyay, C-S Chiang, D Craig, DJ Den Hartog, G Fiksel, et al. Reduced edge instability and improved confinement in the mst reversed-field pinch. Physical review letters, 87(20):205001, 2001. [10] James Chapman. Spectroscopic Measurement of the MHD Dynamo in the MST Reversed Field Pinch. PhD thesis, University of Wisconsin - Madison, 1998. [11] S Choi, D Craig, F Ebrahimi, and SC Prager. Cause of sudden magnetic reconnection in a laboratory plasma. Physical review letters, 96(14):145004, 2006. 92

[12] Seung-Ho Choi. Origin of Tearing Mode Excitation in the MST Reversed Field Pinch. PhD thesis, University of Wisconsin - Madison, 2006. [13] RW Conn and F Najmabadi. The titan reversed-field pinch fusion reactor study. In Proc. 12th Symp. on Fusion Engineering, 1987. [14] DJ Den Hartog, AF Almagri, JT Chapman, H Ji, SC Prager, JS Sarff, RJ Fonck, and CC Hegna. Fast flow phenomena in a toroidal plasma. Physics of Plasmas, 2(6):2281–2285, 1995. [15] DJ Den Hartog, JT Chapman, D Craig, G Fiksel, PW Fontana, SC Prager, and JS Sarff. Measurement of core velocity fluctuations and the dynamo in a reversed-field pinch. Physics of Plasmas (1994-present), 6(5):1813–1821, 1999. [16] BH Deng, WX Ding, DL Brower, AF Almagri, KJ McCollam, Y Ren, SC Prager, JS Sarff, J Reusch, and JK Anderson. Internal magnetic field structure and parallel electric field profile evolution during the sawtooth cycle in mst. Plasma Physics and Controlled Fusion, 50(11):115013, 2008. [17] RN Dexter, DW Kerst, TW Lovell, SC Prager, and JC Sprott. The madison symmetric torus. Fusion Science and Technology, 19(1):131–139, 1991. [18] WX Ding, DL Brower, D Craig, BH Deng, G Fiksel, V Mirnov, SC Prager, JS Sarff, and V Svidzinski. Measurement of the hall dynamo effect during magnetic reconnection in a high-temperature plasma. Physical review letters, 93(4):045002, 2004. [19] WX Ding, DL Brower, BH Deng, AF Almagri, D Craig, G Fiksel, V Mirnov, SC Prager, JS Sarff, and V Svidzinski. The hall dynamo effect and nonlinear mode coupling during sawtooth magnetic reconnection. Physics of Plasmas (1994-present), 13(11):112306, 2006. [20] F Ebrahimi, VV Mirnov, SC Prager, and CR Sovinec. Momentum transport from current- driven reconnection in the reversed field pinch. Physical review letters, 99(7):075003, 2007. [21] DA Ennis. Local Measurements of Tearing Mode Flows and the MHD Dynamo in the MST Reversed-Field Pinch. PhD thesis, University of Wisconsin - Madison, 2008. [22] DA Ennis, D Craig, S Gangadhara, JK Anderson, DJ Den Hartog, F Ebrahimi, G Fik- sel, and SC Prager. Local measurements of tearing mode flows and the magnetohydrodynamic dynamo in the madison symmetric torus reversed-field pinch. Physics of Plasmas, 17(8):082102, 2010. [23] G Fiksel, DJ Den Hartog, and PW Fontana. An optical probe for local measurements of fast plasma ion dynamics. Review of scientific instruments, 69(5):2024–2026, 1998. [24] PJ Fimognari, AF Almagri, JK Anderson, DR Demers, JS Sarff, V Tangri, and J Waksman. Port hole perturbations to the magnetic field in mst. Plasma Physics and Controlled Fusion, 52(9):095002, 2010. 93

[25] P. W. Fontana, D. J. Den Hartog, G. Fiksel, and S. C. Prager. Spectroscopic observation of ﬂuctuation-induced dynamo in the edge of the reversed-ﬁeld pinch. Phys. Rev. Lett., 85:566– 569, Jul 2000.

[26] Jeffrey P. Freidberg. Plasma Physics and Fusion Energy. Cambridge University Press, 2007.

[27] Richard Fridstrom,¨ Stefano Munaretto, Lorenzo Frassinetti, Brett E Chapman, Per R Brun- sell, and JS Sarff. Tearing mode dynamics and locking in the presence of external magnetic perturbations. Physics of Plasmas, 23(6):062504, 2016.

[28] Harold P Furth, John Killeen, and Marshall N Rosenbluth. Finite-resistivity instabilities of a sheet pinch. Physics of Fluids (1958-1988), 6(4):459–484, 1963.

[29] Harold Grad and Hanan Rubin. Hydromagnetic equilibria and force-free ﬁelds. Journal of Nuclear Energy (1954), 7(3):284–285, 1958.

[30] Martin Greenwald, JL Terry, SM Wolfe, S Ejima, MG Bell, SM Kaye, and GH Neilson. A new look at density limits in tokamaks. Nuclear Fusion, 28(12):2199, 1988.

[31] David J Grifﬁths. Introduction to electrodynamics. Prentice Hall, 1999.

[32] AK Hansen, AF Almagri, D Craig, DJ Den Hartog, CC Hegna, SC Prager, and JS Sarff. Mo- mentum transport from nonlinear mode coupling of magnetic ﬂuctuations. Physical review letters, 85(16):3408, 2000.

[33] CC Hegna. Self-consistent mean-ﬁeld forces in turbulent plasmas: Current and momentum relaxation. Physics of Plasmas (1994-present), 5(6):2257–2263, 1998.

[34] DJ Holly, JR Adney, KJ McCollam, JC Morin, and MA Thomas. Programmable power supply for mst’s poloidal ﬁeld. In Fusion Engineering (SOFE), 2011 IEEE/NPSS 24th Sym- posium on, pages 1–4. IEEE, 2011.

[35] SC Jardin, N Ferraro, and I Krebs. Self-organized stationary states of tokamaks. Physical review letters, 115(21):215001, 2015.

[36] H Ji, SC Prager, and JS Sarff. Conservation of magnetic helicity during plasma relaxation. Physical review letters, 74(15):2945, 1995.

[37] Hantao Ji, Masaaki Yamada, Scott Hsu, and Russell Kulsrud. Experimental test of the sweet- parker model of magnetic reconnection. Physical Review Letters, 80(15):3256, 1998.

[38] Allan N Kaufman. Plasma viscosity in a magnetic ﬁeld. The Physics of Fluids, 3(4):610– 616, 1960.

[39] JR King, CR Sovinec, and VV Mirnov. First-order ﬁnite-larmor-radius effects on magnetic tearing in pinch conﬁgurations. Physics of Plasmas, 18(4):042303, 2011. 94

[40] JR King, CR Sovinec, and VV Mirnov. First-order ﬁnite-larmor-radius ﬂuid modeling of tearing and relaxation in a plasma pincha). Physics of Plasmas (1994-present), 19(5):055905, 2012.

[41] RA Kopp and GW Pneuman. Magnetic reconnection in the corona and the loop prominence phenomenon. Solar Physics, 50(1):85–98, 1976.

[42] Fritz Krause and K-H Radler.¨ Mean-ﬁeld magnetohydrodynamics and dynamo theory. Perg- amon Press, 1980.

[43] A Kuritsyn, G Fiksel, AF Almagri, DL Brower, WX Ding, MC Miller, VV Mirnov, SC Prager, and JS Sarff. Measurements of the momentum and current transport from tearing instability in the madison symmetric torus reversed-ﬁeld pinch. Physics of Plasmas, 16:055903, 2009.

[44] A Kuritsyn, G Fiksel, MC Miller, AF Almagri, M Reyfman, and JS Sarff. Probes for measuring fluctuation-induced maxwell and reynolds stresses in the edge of the madison symmetric torus reversed field pinch a. Review of Scientific Instruments, 79(10):10F127, 2008.

[45] NE Lanier, D Craig, JK Anderson, TM Biewer, BE Chapman, DJ Den Hartog, CB For- est, SC Prager, DL Brower, and Y Jiang. An investigation of density ﬂuctuations and electron transport in the madison symmetric torus reversed-ﬁeld pinch. Physics of Plasmas, 8(7):3402–3410, 2001.

[46] RM Magee, DJ Den Hartog, G Fiksel, STA Kumar, and D Craig. Toroidal charge exchange recombination spectroscopy measurements on msta). Review of Scientiﬁc Instru- ments, 81(10):10D716, 2010.

[47] W Mao, BE Chapman, WX Ding, L Lin, AF Almagri, JK Anderson, DJ Den Hartog, J Duff, J Ko, STA Kumar, et al. Physics and optimization of plasma startup in the rfp. Nuclear Fusion, 55(5):053004, 2015.

[48] L Marrelli, P Martin, G Spizzo, P Franz, BE Chapman, D Craig, JS Sarff, TM Biewer, SC Prager, and JC Reardon. Quasi-single helicity spectra in the madison symmetric torus. Physics of Plasmas (1994-present), 9(7):2868–2871, 2002.

[49] S Martini, M Agostini, C Alessi, A Alﬁer, V Antoni, L Apolloni, F Auriemma, P Bettini, T Bolzonella, D Bonﬁglio, et al. Active mhd control at high currents in rfx-mod. Nuclear Fusion, 47(8):783, 2007.

[50] VV Mirnov, CC Hegna, and SC Prager. A hall dynamo effect driven by two-ﬂuid tearing instability. Plasma Physics Reports, 29(7):566–570, 2003.

[51] William A Newcomb. Hydromagnetic stability of a diffuse linear pinch. Annals of Physics, 10(2):232–267, 1960. 95

[52] Eugene N Parker. Sweet’s mechanism for merging magnetic fields in conducting fluids. Journal of Geophysical Research, 62(4):509–520, 1957. [53] CC Petty, ME Austin, CT Holcomb, RJ Jayakumar, RJ La Haye, TC Luce, MA Makowski, PA Politzer, and MR Wade. Magnetic-flux pumping in high-performance, stationary plasmas with tearing modes. Physical review letters, 102(4):045005, 2009. [54] JA Phillips, DA Baker, RF Gribble, and C Munson. Startup of reversed field pinches and current ramping using dynamo action. Nuclear fusion, 28(7):1241, 1988. [55] SC Prager, AF Almagri, DL Brower, D Craig, DJ Den Hartog, and W Ding. Momentum transport from tearing instability. In Proceedings of the 22nd IAEA Fusion Energy Confer- ence, 2008. [56] JC Reardon, G Fiksel, CB Forest, AF Abdrashitov, VI Davydenko, AA Ivanov, SA Ko- repanov, SV Murachtin, and GI Shulzhenko. Rutherford scattering diagnostic for the madison symmetric torus reversed-field pinch. Review of Scientific Instruments, 72(1):598–601, 2001. [57] JA Reusch, JK Anderson, DJ Den Hartog, F Ebrahimi, DD Schnack, HD Stephens, and CB Forest. Experimental evidence for a reduction in electron thermal diffusion due to trapped particles. Physical review letters, 107(15):155002, 2011. [58] JA Reusch, MT Borchardt, DJ Den Hartog, AF Falkowski, DJ Holly, R OConnell, and HD Stephens. Multipoint thomson scattering diagnostic for the madison symmetric torus reversed-field pinch). Review of Scientific Instruments, 79(10):10E733, 2008. [59] Joshua Reusch. Measured and Simulated Electron Thermal Transport in the Madison Sym- metric Torus Reversed Field Pinch. PhD thesis, University of Wisconsin - Madison, 2011. [60] DC Robinson. High-β diffuse pinch configurations. Plasma Physics, 13(6):439, 1971. [61] JS Sarff, S Assadi, AF Almagri, M Cekic, DJ Den Hartog, G Fiksel, SA Hokin, H Ji, SC Prager, W Shen, et al. Nonlinear coupling of tearing fluctuations in the madison symmetric torus. Physics of Fluids B: Plasma Physics (1989-1993), 5(7):2540–2545, 1993. [62] Joshua Sauppe. Extended magnetohydrodynamic modeling of plasma relaxation dynamics in the reversed-field pinch. PhD thesis, University of Wisconsin - Madison, 2015. [63] Joshua Sauppe. Extended mhd modeling of tearing-driven magnetic relaxation. In APS Meeting Abstracts, 2016. [64] JP Sauppe and CR Sovinec. Two-fluid and finite larmor radius effects on helicity evolution in a plasma pinch. Physics of Plasmas, 23(3):032303, 2016. [65] JP Sauppe and CR Sovinec. Extended mhd modeling of tearing-driven magnetic relaxation. Physics of Plasmas, 24(5):056107, 2017. 96

[66] DD Schnack, DC Barnes, Z Mikic, DS Harned, EJ Caramana, and RA Nebel. Numerical simulation of reversed-ﬁeld pinch dynamics. Computer Physics Communications, 43(1):17– 28, 1986.

[67] VD Shafranov. Plasma equilibrium in a magnetic ﬁeld. Reviews of Plasma Physics, 2:103, 1966.

[68] P Sonato, G Chitarin, P Zaccaria, F Gnesotto, S Ortolani, A Buffa, M Bagatin, WR Baker, S Dal Bello, P Fiorentin, et al. Machine modiﬁcation for active mhd control in rfx. Fusion engineering and design, 66:161–168, 2003.

[69] C.R. Sovinec, A.H. Glasser, T.A. Gianakon, D.C. Barnes, R.A. Nebel, S.E. Kruger, S.J. Plimpton, A. Tarditi, M.S. Chu, and the NIMROD Team. Nonlinear magnetohydrodynamics with high-order ﬁnite elements. J. Comp. Phys., 195:355, 2004.

[70] LC Steinhauer and A Ishida. Relaxation of a two-specie magnetoﬂuid. Physical review letters, 79(18):3423, 1997.

[71] Douglas Stone. Magnetic relaxation during oscillating ﬁeld current drive in a reversed ﬁeld pinch. PhD thesis, University of Wisconsin - Madison, 2013.

[72] PA Sweet. Electromagnetic phenomena in cosmical physics. In iAU Symp, volume 6, pages 123–34, 1958.

[73] Mingsheng Tan, Douglas R Stone, Joseph C Triana, Abdulgader F Almagri, Gennady Fiksel, Weixing Ding, John S Sarff, Karsten J McCollam, Hong Li, and Wandong Liu. A multichannel capacitive probe for electrostatic fluctuation measurement in the madison symmetric torus reversed field pinch. Review of Scientific Instruments, 88(2):023502, 2017.

[74] J. B. Taylor. Relaxation of toroidal plasma and generation of reverse magnetic ﬁelds. Phys. Rev. Lett., 33:1139–1141, Nov 1974.

[75] J.R. Taylor. An Introduction to Error Analysis. University Science Books, 1982.

[76] SD Terry, DL Brower, WX Ding, JK Anderson, TM Biewer, BE Chapman, D Craig, CB For- est, R OConnell, SC Prager, et al. Measurement of current proﬁle dynamics in the madison symmetric torus. Physics of Plasmas (1994-present), 11(3):1079–1086, 2004.

[77] TD Tharp. Measurements of Nonlinear Hall Reconnection in the Reversed Field Pinch. PhD thesis, University of Wisconsin - Madison, 2009.

[78] TD Tharp, AF Almagri, MC Miller, VV Mirnov, SC Prager, JS Sarff, and CC Kim. Mea- surements of impulsive reconnection driven by nonlinear hall dynamics. Physics of Plasmas, 17:120701, 2010. 97

[79] KS THOMAS, DA BAKER, JN DIMARCO, JN DOWNING, and A HABERSTICH. Zt- 40 experiment. In BULLETIN OF THE AMERICAN PHYSICAL SOCIETY, volume 24, pages 939–940. AMER INST PHYSICS CIRCULATION FULFILLMENT DIV, 500 SUN- NYSIDE BLVD, WOODBURY, NY 11797-2999, 1979.

[80] Stodiek Von Goeler, W Stodiek, and N Sauthoff. Studies of internal disruptions and m= 1 oscillations in tokamak discharges with softx-ray tecniques. Physical Review Letters, 33(20):1201, 1974.

[81] RG Watt and RA Nebel. Sawteeth, magnetic disturbances, and magnetic ﬂux regeneration in the reversed-ﬁeld pinch. The Physics of Fluids, 26(5):1168–1170, 1983. 98

Appendix A: Supplemental Derivations

The purpose of this appendix is to expand on the calculations used throughout my research. Previous measurements have included derivations [71, 77], however they speciﬁcally applied to the probe used (single radial measurement or no radial coil). The Deep Insertion Hall Probe used throughout this thesis has a large array of radially displaced coils, allowing a more generic function to be used for ﬁeld calculations. The derivations in this appendix will use the right handed coordinate system outlined in Section 2.1.1

A.1 Current Density

Using the magnetic field, we can calculate the current profiles by assuming axisymmetry. A low frequency approximation to Ampere’s law as well as a large aspect ratio approximation is used to calculate the current from the magnetic field.

µoJo = ∇ × Bo ∂B 1 ∂(rB ) = − φ θˆ + θ φˆ ∂r r ∂r Toroidal effects are not expected to greatly change this calculation. However, ﬂux expansion near the porthole causes this approximation to break down, creating large uncertainties in the calculation. Fortunately, this only occurs within a porthole diameter from the wall and does not affect the majority of measurements.

A.2 Electric Field

H ∂ H The electric field on MST can be calculated from Faraday’s law, given by E · dl = − ∂t B · dA. To calculate the toroidal electric field, we need to enclose a region of poloidal flux where the perimeter lies along areas of constant electric field. We will use a surface bounded by the magnetic axis and a line of constant minor radius, given by the shaded region in Figure A.1. We will use a cylindrical approximation to complete the integrals, meaning the bounded surfaces will be rectangles instead of annuli. 99

Figure A.1 Directions of Eφ calculation in MST 100

I I E · dl = Eφdl = Eφ(0)Ro − Eφ(r)Ro

I Z r Z Ro Z r ∂ ∂ 0 ∂ 0 0 − B · dA = − Bθdr dRo = −Ro Bθ(r )dr ∂t ∂t 0 0 ∂t 0 Rearranging, we arrive at

Z r 0 ∂Bθ(r ) 0 Eφ(r) = Eφ(0) + dr 0 ∂t However, this equation requires knowledge of the electric field at the core and the profile of the magnetic field at inner radii. For probe measurements, we have the profiles from the edge to the radius of interest and we can change the limits of integration to reflect this.

Z a 0 Z r 0 Z a 0 ∂Bθ(r ) 0 ∂Bθ(r ) 0 ∂Bθ(r ) 0 Eφ(a) = Eφ(0) + dr = Eφ(0) + dr + dr 0 ∂t 0 ∂t r ∂t Z a 0 ∂Bθ(r ) 0 Eφ(a) = Eφ(r) + dr r ∂t

0 R a ∂Bθ(r ) 0 Therefore, Eφ(r) = Eφ(a) − r ∂t dr . During MST discharges, the value of the voltage at the wall is measured. We know that V = − R E · dl so using MST nomenclature and the cylindrical 0 approximation again, V = −E (a)2πR , giving us E (r) = − Vpg − R a ∂Bθ(r ) dr0. As a MST pg φ o φ 2πRo r ∂t specific aside, the sign of Vpg stored in the database does not necessarily correspond to a sign that is consistent with the RHCS layed out in the beginning. Further information on this topic is discussed in App C. We can perform a similar analysis to calculate the poloidal electric field. The surface is now a circle that encloses the toroidal field, as seen in Figure A.2. I I E · dl = Eθdl = Eθ(r)2πr

I Z r Z 2π Z r 0 0 ∂ ∂ 0 0 0 r ∂Bφ(r ) 0 − B · dA = − r Bφ(r )dr dθ = −2π dr ∂t ∂t 0 0 0 ∂t 101

Figure A.2 Directions of Eθ calculation in MST 102

0 0 1 R r r ∂Bφ(r ) Rearranging, we arrive at Eθ(r) = − r 0 ∂t . Again, we need the formula in terms of probe measurable quantities.

Z a 0 0 Z r 0 0 Z a 0 0 1 r ∂Bφ(r ) 0 1 r ∂Bφ(r ) 0 r ∂Bφ(r ) 0 Eθ(a) = − dr = − dr + dr a 0 ∂t a 0 ∂t r ∂t Z a 0 0 1 r ∂Bφ(r ) 0 = − −rEθ(r) + dr a r ∂t Z a 0 0 a 1 r ∂Bφ(r ) 0 Eθ(r) = Eθ(a) + dr r r r ∂t

For MST, toroidal gap voltage is given by Vtg = −Eθ(a)2πa. Plugging into the poloidal electric 0 0 Vtg 1 R a r ∂Bφ(r ) 0 ﬁeld equation yields Eθ(r) = − 2πr + r r ∂t dr .

A.3 Pressure

Assuming that Jo × Bo = ∇po, we can calculate the pressure gradient in MST discharges. We will also assume axisymmetry of the magnetic ﬁelds.

∇po = Jo × Bo

∇po · rˆ = Jo × Bo · rˆ

= JθBφ − JφBθ −1 ∂Bφ 1 ∂(rBθ) = Bφ − Bθ µo ∂r µor ∂r 2 −1 ∂Bφ 1 ∂Bθ = − Bθ + r Bθ 2µo ∂r µor ∂r 2 −1 ∂ 2 2 Bθ = Bφ + Bθ − 2µo ∂r µor Z r −1 Z r ∂ 1 Z r B2 ˆ0 0 2 2 0 θ 0 ∇po · r dr = 0 Bφ + Bθ dr − 0 dr 0 2µo 0 ∂r µo 0 r Z r Z r Z r 2 ∂po 0 −1 ∂ 2 2 0 1 Bθ 0 0 dr = 0 Bφ + Bθ dr − 0 dr 0 ∂r 2µo 0 ∂r µo 0 r 2 2 2 2 Z r 2 Bθ (r) + Bφ(r) Bθ (0) + Bφ(0) 1 Bθ 0 po(r) − po(0) = − + − 0 dr 2µo 2µo µo 0 r B2(r) − B2(0) Z r 2 k k 1 Bθ 0 po(r) = po(0) − − 0 dr 2µo µo 0 r 103

Here we have a function for the pressure. As before, rearrangement is neccessary to get the equation in probe measureable terms.

B2(a) − B2(0) Z a 2 k k 1 Bθ 0 po(a) = po(0) − − 0 dr 2µo µo 0 r B2(a) − B2(0) Z r 2 Z a 2 k k 1 Bθ 0 Bθ 0 = po(0) − − 0 dr + 0 dr 2µo µo 0 r r r B2(a) Z a 2 B2(0) Z r 2 k 1 Bθ 0 k 1 Bθ 0 po(a) + + 0 dr = po(0) + − 0 dr 2µo µo r r 2µo µo 0 r Plugging this into our pressure equation we get

B2(a) − B2(r) Z a 2 k k 1 Bθ 0 po(r) = po(a) + + 0 dr 2µo µo r r This calculation is strongly dependent on the proﬁles near the edge and

A.4 Correlated Lorentz Force

One of the main measurements of this thesis is the ﬂuctuation-induced electromotive force, known as the Hall dynamo. This term, as well as the Maxwell stress, is found by calculation of the correlated Lorentz force. Using a low-ω Ampere’s law approximation, we can rewrite the term 104 using only the magnetic ﬁeld

F = J × B ˆ = (JzBr − JrBz) θ + (JrBθ − JθBr)z ˆ ˆ 1 1 ∂(rBθ) 1 ∂Br 1 ∂Bz ∂Bθ F · θ = − Br − − Bz + Bθ ∇ · B µo r ∂r r ∂θ µo ∂θ ∂z | {z } 0 2B B ∂(B B ) ∂(B B ) 1 ∂(B2 + B2 + B2) µ F = r θ + r θ + θ z − r θ z o θ r ∂r ∂z 2r ∂θ hB B i ∂(B B ) ∂(B B ) 1 ∂(B2 + B2 + B2) µ hF i = 2 r θ + r θ + θ z − r θ z o θ r ∂r ∂z 2r ∂θ hB B i ∂ µ hF i = 2 r θ + hB B i o θ r ∂r r θ 1 ∂Bz ∂Bθ 1 ∂Br ∂Bz F · zˆ = − Bθ − − + Bz ∇ · B µo ∂θ ∂z µo ∂z ∂r | {z } 0 B B ∂(B B ) 1 ∂(B B ) 1 ∂(B2 + B2 − B2) µ F = r z + r z + θ z − r θ z o z r ∂r r ∂θ 2 ∂z hB B i ∂(B B ) 1 ∂(B B ) 1 ∂(B2 + B2 − B2) µ hF i = r z + r z + θ z − r θ z o z r ∂r r ∂θ 2 ∂z hB B i ∂ µ hF i = r z + hB B i o z r ∂r r z

We have used the property of ∇·B = 0 to simplify the expression in terms of total derivatives. We H XdS have also applied the flux surface average to the Lorentz force, defined as hXi ≡ H dS , and the total derivatives with respect to θ and z have vanished under flux surface averaging. Recombining the equation we arrive at

2 ∂ 1 ∂ µ hFi = + hB B i θˆ + + hB B i zˆ o r ∂r r θ r ∂r r z

The remaining equation allows us to measure the correlated Lorentz force with a radial array of magnetic ﬁeld measurements. 105

Appendix B: DIHP Operation

The intent of this appendix is to have a preserved document outlining probe design and notes for use in future data collection. This will also serve as a way to include rationale for probe design features and to pass down probe usage tips and tricks.

B.1 Probe Speciﬁcs

The probe consists of 4 parts. The the particle shield, the coil housing, the probe coupler, and the probe rail. On MST, the probe rail is standardized and acts as a way to translate the probe into the plasma. The double o-ring on the shaft acts as the vacuum barrier and an angular caliper gives rotational precision. Since I did not design the rail, I will focus on the other probe parts. A cartoon of the probe is seen in Fig B.1 with some feature exaggerated for clarity.

B.1.1 Particle Shield

The particle shield acts as a device to protect the inner workings of the probe from the harsh plasma environment. Ideally, the plasma exposed surface should be one piece to eliminate the possibility of plasma breaching the interior. It should also have good thermal and vacuum compatibility as the experiment is a pulsed device under moderate vacuum. The material should also be

Figure B.1 Illustration of the assembly of the probe parts discussed 106

low-Z to reduce core radiation and its contribution to plasma Zeff . Boron Nitride ﬁts this description. The grade used is Saint-Gobain Combat Solids AX05. Previous iterations of probes have use HP grade boron nitride. However, AX05 is constructed without the use of an organic binder, reducing the likelyhood of contamination into the plasma. Using the AX05, I have noticed a reduction in the time it takes to condition a probe but have not experimentally veriﬁed the reduction. The particle shield is 16” long, the maximum length for premade cylinders of AX05, and 1” thick. A 3/4” diameter hole is bored through the cylinder to leave 1/8” thickness shell of boron nitride to protect the probe housing. The bottom thickness is 1/4” to reinforce the bottom of the probe since it is a common location for failure.

B.1.2 Coil Housing

This part can go by other names but it primarily serves as a way to secure the actual measurement apparatus, the magnetic triplet coils, in a known location. The material is also boron nitride to minimize differential expansion/contraction between it and the particle shield. There is no need to use the more expensive AX05 material and HP grade is used for the DIHP coil housing. The coil housing is cylindrical and should have a little bit of clearance to fit into the particle shield. Once the coils are fixed into the coil housing, a thin layer of silver paint is applied, used for electrostatic shielding. Thus the diameter of the housing should be a little less than the inner diameter of the particle shielding. Additionally, the cylindrical stock used should be one size larger than the final diameter. This will ensure that the finished housing is true since the boron nitride stock may have imperfections. The design of the DIHP has a deep groove carved into the length of the cylinder. The deep groove allows the coils to be concentric with the housing axis. This is important as a perturbation to the radial magnetic field is created when the plasma current diverts around the probe. Radial coils that are centered do not see this radial field perturbation. The length of the groove is enough for the amount of coils required. Then a smaller channel is carved to allow for passage of the wires from the coils. The grooves need to be deep enough for a layer of sauereisen cement to help seal the coils and wires in place. It is suggested that the housing should be inserted into the 107 particle shield to check the fit. If the fit is very tight, sand down the housing until a looser fit is achieved. Then a thin layer of silver paint covers the entire coil housing and should fit snuggly into the particle shield.

B.1.3 Probe Coupler

The probe coupler allows the physical probe to be attached to the probe rail which is attached to MST. It is typically a single piece of aluminum that directly attaches to a stainless steel tube that exists in MST high vacuum and outside the machine, which is part of the probe rail design. However the connection to the boron nitride probe body is the design aspect that will be described here. As the DIHP is much longer and heavier than a typical probe, proper design of the coupler needed to take place. Boron nitride is strong but it is brittle so a friction clamp should not be used. Through holes were drilled on the particle shield and the shield slides over the coupler. Threaded holes were tapped onto the coupler and a screw acts as a location to hang the shield off of. This allows the boron nitride to expand at a different rate than the coupler without damaging the method of holding it in place. A negative copy is machined into the top of the coil housing to allow the coupler to surround the top of it, securing it in place. The design of the coupler is shown in Fig B.1 and how it ﬁts together with the other parts. The red shape are the screws that hold everything in place.

B.2 Conditioning Protocol

To reduce the effects of probe outgassing and impurity leaching into the plasma, a protocol has been developed to ensure the probe is clean and plasmas are reproducible. We have observed that inserting the probe into the plasma too quickly leads to discharges that are lower current and duration, similar to those seen after a vent. Following this procedure reduces the likelihood of these poor discharges. 108

B.2.1 Probe Cleaning

Once the boron nitride is machined to designed speciﬁcations, gloves need to be worn to handle the probe. Oils present on skin can be absorbed onto the particle shield, possibly contributing to the outgassing and impurity content of the plasma. Fingerprints can even be seen on the probe surface after hundreds of discharges on MST. The outer shield should be washed with alcohol before and after probe assembly. After the probe is washed, it needs to be placed into an autoclave ﬁtted with an attachment for the probe vacuum jacket. The temperature should be set between 100C-200C to boil off water without melting the insulating covering on the wires that make up the pickup coils. Ideally, it should bake for 12 hours before being attached to MST. Follow the current vacuum protocol for opening valves to MST, typically available in the document library on the plasmawiki as of the writing of this thesis.

B.2.2 Plasma Conditioning

The night before a planned run, the probe should be inserted until the boron nitride shield is fully exposed to the PDC plasma. The probe coupler, typically made of aluminum, should still be at or behind the machine wall. If a new particle shield is being used, several nights of exposure to PDC plasmas should be used. There is no established duration for conditioning the probe with PDC plasmas, but I have used 3 nights of exposure with good results. For conditioning the probe with RFP discharges, one to two days should be planned depending on the intended insertion depth of the run and how recently the probe was used. For running plasmas with a new particle shield, it may take longer to follow the insertion protocol. For the first 15 cm of insertion, we have done insertions in 3 cm increments. To increase insertion, we have found that having 5 consecutive discharges that display normal RFP behavior is sufficient for conditioning the newly exposed surface. For deeper insertions, we move to 2 cm increments. Once at final insertion depth, conditioning plasmas should be run until the signal for the BIV levels have stabilized. Signals increase for every insertion but reduce to some threshold for that depth after a few discharges. Once levels have stabilized, we have found plasma discharges to be very 109 reproducible and data collection can begin. As for the time between discharges, we have used 3 minute cycle times for insertion levels up to 10 cm, 3.5 minutes for 10-20 cm, and 4 minutes for 20-30 cm insertions. We have not been able to extend insertion depths past 30 cm with the DIHP. Longer cycle times are used to allow for cooling of the probe between discharges as the magnet wire insulation can melt at temperatures above 200 C.

B.3 Deep Insertion Hall Probe Speciﬁcs

The current iteration of the DIHP has been color coded with blue, green, and red to identify the different coil signals, representing the toroidal, poloidal, and radial magnetic field, respectively. The signals have been divided between two LEMO connection sites, odd number coils at the side site and even coils at the top site. Coils are numbered starting at the most interior measurement. The highest number coil, 20 for the DIHP, will be the furthest away from the plasma core. Some measurement locations have been damaged and data are not available there. The most likely cause is a break in the magnet wire at the base of the coil and cannot be fixed without replacing the entire coil assembly. The currently installed particle shield has minimal wear and tear. It is highly suggested that the DIHP only be used at 90P port hole locations as significant torque can be placed on the probe due to the extended probe rail and the probe assembly may sag, causing large uncertainties in the location of the measurements. 110

Appendix C: Physical Directions on MST

In the theory of magnetic relaxation, a quantity called λ is used as a measure of field aligned J·B current, defined as λ ≡ B2 . This quantity is dependent on the direction of the parallel current. For standard discharges on the Madison Symmetric Torus, λ < 0 due to the original construction of the experiment. The experiment can be reconfigured to have λ > 0 by either changing the direction of B or the induced J. The comparison of data from a multitude of sources can obfuscate the discussion of terms in Ohm’s law and in the momentum equation. There is no expected difference in the two types of discharges other than the absolute sign. However, presentation of data may change to keep in line with an established precedent and data may be republished without reference or knowledge of this precedent. This section offers a thorough evaluation of the signs for any data presented in this thesis and records it for posterity. As the magnetic field is purely toroidal in the core of an RFP, it is logical to define this direction as positive. For a right-handed coordinate system, this sets the signs for the other components in a toroidal geometry since the radial direction is outward, or towards the wall of the device. In a λ < 0 discharge, typical for MST plasmas, the induced toroidal current is in the negative toroidal direction. This creates a poloidal magnetic field in the negative theta direction. The induced toroidal current is caused by an electric field applied in the negative toroidal direction. Other directions in MST will be dictated by plasma phenomena. During a relaxation event, the total toroidal flux increases sharply, inducing a positive poloidal edge electric field. The core electric field is negative both during and between relaxation events. The parallel current is negative across the entire plasma radius both during and between relaxation events. Core plasma flow has been measured to be in the same direction of the parallel current density there, negative for λ < 0 discharges. This appendix uses some of the nomenclature of MST and its data taking system, such as the MDSPlus, a software package for data acquisition. The following is meant to be a guide and we 111

strongly suggest for anyone following it to verify and understand the directions for themselves rather than blindly following it.

C.1 Magnetic Field Directions

In the introduction, we have already described the directions of the magnetic ﬁeld. Following are some quantities that are stored in the database.

Quantity MDSPlus Storage RHCS Sign

Bφ(0) n/a +

Bφ(a) - -

Bθ(a) + -

C.2 Electric Field Directions

Since the signs of the magnetic field are known, we are able to determine the sign of the ∂ electric field with some known information. Faraday’s law tells us that Eθ(a) ∝ − ∂t Φt and ∂ Eφ(a) ∝ − ∂t Φp. During a sawtooth crash, the total toroidal flux increases so we expect that Eθ(a) is negative during this time. At plasma startup, there is no poloidal field. Then poloidal field is

induced by the toroidal current and grows to a large negative value. We would expect Eφ(a) to be positive during this time since the total poloidal ﬂux is decreasing.

Quantity MDSPlus Storage RHCS Sign

Eφ(a) during plasma startup n/a +

Eθ(a) during sawtooth crash n/a -

C.3 Voltages

The voltages of the toroidal and poloidal gaps are frequently used during data analysis so these signs should be known. The deﬁnition of electric potential is V = − H E · dl. Since the

equilibrium electric ﬁeld is toroidally and poloidally symmetric, we expect that Vtg ∝ −Eθ(a) 112

and Vpg ∝ −Eφ(a). We have already determined the signs of the electric ﬁeld so the signs of the voltages are known.

Quantity MDSPlus Storage RHCS Sign

Vpg during plasma startup + -

Vtg during sawtooth crash + +

C.4 Other Operational Measurements

We know that the core toroidal plasma flow is in the direction of the parallel current there. Therefore, flow is in the negative direction. The toroidal flux generated is in the positive direction since it sustains the toroidal field and that is in the positive direction.

Quantity MDSPlus Storage RHCS Sign

Vφ(0) + - Φ + +