Experimental Observation of Fluctuation-Driven Mean Magnetic Fields in the Madison Dynamo Experiment

EXPERIMENTAL OBSERVATION OF FLUCTUATION-DRIVEN MEAN MAGNETIC FIELDS IN THE MADISON DYNAMO EXPERIMENT

Erik J. Spence

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

Doctor of Philosophy

(Physics)

at the

UNIVERSITY OF WISCONSIN–MADISON

2006 c Copyright by Erik J. Spence 2006

To my beautiful wife, Susanne. ii

ACKNOWLEDGMENTS

Many are the people that deserve thanks for the help they have given me over the years of my graduate career. Enumerating them all is a difficult endeavour and it is certain that not all will be thanked to the level they deserve. For this I apologize. Since this is inevitable it is my hope that this section will be the most incomplete part of this work. First and foremost, I express my thanks and gratitude to my advisor, Professor Cary Forest, for hiring me as a research assistant and bearing with me all these years. Obviously, this work would not exist without him. (since neither would the experiment!) This glaring point aside, his influence on my growth and maturation as a scientist has been noteworthy. Though I have failed him numerable times, he has been patient with me and encouraged me to reach the potential of which I am capable. His greatest lesson, “if it doesn’t look right it probably isn’t,” is a gift that will be put to great use in my future career as a scientist. My partner in research, Mark Nornberg, is a great man; I have nothing but praise for him. His kindness, patience, forbearance, piety, prudence, humour and dedication are inspirational. Many has been the time that I have been unjust to him, and he has repaid me with charity. He is truly a better laboratory partner than I deserve and I am a better man for him sharing his life with me. Mark’s contributions to this work are not limited to his superlative personality. His intelligence is significant, and his ability to learn exceeds my own. He has consistently helped me find the weaknesses in my work and to better my understanding of the science with which I have struggled. My eternal thanks to him for walking this journey with me. Another to whom I am very grateful is Roch Kendrick. His tireless dedication to the project has led to the construction of the leading experiment of its kind in the world, without which, of course, this thesis would never have happened. He is uncompromising on quality and is one of iii most pragmatic and adaptable people I have known. My thanks to him for all he has taught me about, but certainly not limited to, machining, plumbing, electrical wiring, metallurgy, chemistry, generators, motors, heating, cooling, insulating, and how to prepare a “killer” pork shoulder. While the previous three contributed by far the most academically to my effort to reach this point, many others have also had a hand in getting this thesis to completion. Craig Jacobson has worked extensively on the water model of the experiment, especially since it was relocated to the Tantalus facility. Adam Bayliss has helped me to develop my understanding of the physics of the experiment, both through his simulations and through our conversations. Carlos Parada, our latest team member, has been a great help in taking much of the data presented herein. Many others have helped with construction of the water and sodium versions of the experiment: Hal Canary, Michael Fix, Brian Grierson. Rob O’Connell, the first postdoctoral researcher for the experiment, did much of the development that brought the water into functioning form. Helping him in that effort was Jonathan Goldwin, who worked on the analysis of the 30 cm version of the water model, helping to develop the final impeller design. My thanks to all of them for their contribution to this work. But not all contributions to this work have been of an academic stripe. I have had the privilege of enjoying the company of many a fellow graduate student and friend on this journey, and I raise a glass to all of them in thanks: Stephie, Dan, Jim, Mike, Dave, Jodi, Steve, Bob, Olivia, Jenny, Hilarie, Yoyi, and many others. In particular I raise a cold one to my good friend Christian Ast, whose company I had the honour of sharing for many years as his roommate. His companionship and counsel continue to be a comfort to me, and I am sure that I would not have survived this adventure without him. Complementing the support of my friends has been the never-wavering support of my family. My nutty sisters, God bless them, continue to be a refreshing source of silliness and humour, even after all these years together. The support of my parents has been unquestionable, and their love never-ending. I will always be indebted to them. Since long before I took her as my wife, Susanne has been a continuous source of support during the many trials, failures and eventual successes of this project. I could never possibly earn iv

the love that she has shown me, but offer it she does, each and every day. The choice of dedication of this work is an easy one to make. One further expression of thanks is necessary. Throughout my graduate career the presence of Jesus the Christ has been a source of strength and perseverance for me like none other. The Holy Spirit has been generous with guidance and wisdom and peace, and because of this the thesis that you are reading exists. Thus, if there is any good in this work then let it be for the glory of God.

ËÓÐi DeÓ GÐÓÖia vi

TABLE OF CONTENTS

Page

LIST OF TABLES ...... ix

LIST OF FIGURES ...... x

NOMENCLATURE ...... xiii

ABSTRACT ...... xiv

1 Introduction ...... 1

1.1 MHDandDynamoExperiments ...... 4 1.2 MotivationfortheMadisonDynamoExperiment ...... 8 1.3 TheRoleofTurbulence...... 11 1.4 ThesisOutline...... 13

2 Experimental Apparatus ...... 15

2.1 WaterModel ...... 16 2.1.1 DescriptionoftheWaterApparatus ...... 16 2.1.2 LaserDopplerVelocimetry...... 21 2.1.3 WaterDataRuns ...... 25 2.2 SodiumExperiment...... 26 2.2.1 DescriptionoftheSodiumApparatus ...... 26 2.2.2 HeatingandCoolingtheExperiment...... 30 2.2.3 LaboratorySodiumSafety ...... 34 2.2.4 FillingtheSphere...... 37 2.2.5 MagneticFieldsandDataAcquisition ...... 38 2.2.6 SodiumDataRuns ...... 43

3 Determination of the Mean Velocity Field ...... 45

3.1 LDVMeasurements...... 46 3.2 FittingtheVelocityField ...... 47 3.2.1 SphericalHarmonicExpansion...... 49 3.2.2 VelocityFieldFits ...... 50 vii

Page

4 Predicting the Induced Field ...... 57

4.1 MagneticInductionEquation ...... 57 4.1.1 InteractionTerms...... 59 4.1.2 GauntandElsasserIntegrals ...... 60 4.1.3 MagneticBoundaryConditions ...... 61 4.2 CodingtheProblem...... 62 4.2.1 RadialProﬁlesandOperators ...... 62 4.2.2 CalculatingtheInducedField ...... 63 4.2.3 TestingtheCode ...... 63 4.3 DeterminingtheAppliedField ...... 65 4.4 PredictedMagneticFields ...... 67

5 Measurements of the Induced Magnetic Field ...... 71

5.1 MeasuringInducedMagneticFields ...... 72 5.1.1 ExternalMagneticFields...... 73 5.1.2 InternalMagneticFields ...... 76 5.2 FittingInternalMagneticFields ...... 78 5.2.1 FittingInternalPoloidalFields ...... 80 5.2.2 FittingInternalToroidalFields ...... 81

6 Evidence for Fluctuation-Driven Currents ...... 87

6.1 PoloidalInducedFields...... 88 6.1.1 ExternalPoloidalFields ...... 88 6.1.2 InternalPoloidalFields...... 91 6.1.3 InducedDipoleMoment ...... 91 6.2 ToroidalInducedFields...... 95 6.3 Fluctuation-InducedFields ...... 96

7 Discussion and Summary ...... 102

7.1 ComparisonwithSimulation ...... 102 7.2 FormsoftheTurbulentEMF ...... 103 7.3 ImplicationsfortheDynamo ...... 105 7.4 Summary ...... 106 viii

Page

LIST OF REFERENCES ...... 108

APPENDICES

AppendixA: HallProbeInputVoltageDrift...... 114 AppendixB: FINDFrequencyShifting ...... 118 AppendixC: MagneticFieldBoundaryConditions ...... 121 AppendixD: Advectionand DiffusionMatrixOperators ...... 123 Appendix E: Axisymmetric Velocity and Magnetic Fields Cannot Induce Dipole Mo- ments...... 131 Appendix F: Calculation of the Magnetic Field due to the EMF ...... 135 ix

LIST OF TABLES

Table Page

1.1 ExperimentalSpeciﬁcations ...... 11

2.1 Parameter values used to calculate the radial LDV measurementposition ...... 24

3.1 ParametervaluesusedwiththeFINDsoftware ...... 45

5.1 Inducedexternalpoloidalharmonicstatistics ...... 77

6.1 Energy in the measured and predicted induced magnetic ﬁeld...... 88

A.1 Circuitboardelementvalues ...... 114

A.2 Probe circuit board voltages as a function of magnetic ﬁeldstrength ...... 116

B.1 LDVlaserlightcharacteristics ...... 118

B.2 Minimum shift frequency values needed by the FIND software...... 119 x

LIST OF FIGURES

Figure Page

1.1 SchematicoftheRigadynamoexperiment ...... 6

1.2 SchematicoftheKarlsruhedynamoexperiment ...... 7

1.3 Dudley and James’ t2s2 ﬂow...... 10

1.4 Growth rate versus Rm forDJﬂow ...... 11

2.1 Kinematic viscosities of water and sodium versus temperature ...... 17

2.2 Watermodelexperimentschematic ...... 18

2.3 Photographofanimpeller ...... 20

2.4 Photographofthewatermodel...... 23

2.5 SchematicoftheMadisonDynamoExperiment ...... 27

2.6 Cover of the February 2006 issue of Physics Today ...... 28

2.7 Impellerrotationrateversusmotorpower ...... 29

2.8 Photograph of the Madison Dynamo Experiment during construction ...... 33

2.9 Electrical conductivity of sodium versus temperature ...... 34

2.10 Sodiumsafetygear ...... 36

2.11 ScreenshotoftheLookoutcontrolscreen ...... 39

3.1 Watermodelmeasurementpositions ...... 47

3.2 Poloidal water model velocity measurement time series ...... 48 xi

Figure Page

3.3 Toroidal water model velocity measurement time series ...... 48

3.4 Impellermodeldatapositions ...... 52

3.5 VelocityﬁeldﬁttoLDVdata...... 55

3.6 Profiles used in the fit to the LDV data, and resulting velocityfield...... 56

4.1 Noiseanalysisforthepredictedinducedﬁeld ...... 64

4.2 Dipoleappliedﬁeldstreamlines ...... 67

4.3 Predictedtotalmagneticﬁelds ...... 68

4.4 Spatial distributionof the predicted Stuart number ...... 69

5.1 Timeseriesofsurfacemagneticﬁelds ...... 74

5.2 Timeseriesofexternalinducedmagneticmodes ...... 75

5.3 Internalmagneticmeasurementpositions ...... 78

5.4 Toroidalinternalmagneticﬁeldmeasurements ...... 79

5.5 Fitpoloidalmagneticﬁeldandradialproﬁles ...... 82

5.6 Poloidal magnetic field fit to internal field measurements ...... 83

5.7 Fittoroidalmagneticﬁeldandradialproﬁles ...... 84

5.8 Toroidal magnetic field fit to internal field measurements ...... 85

5.9 Spatial distributionof the measuredStuart number ...... 86

6.1 Inducedpoloidalstreamfunctioncomparison ...... 89

6.2 Induced external poloidal magnetic ﬁeld comparison ...... 90

6.3 Induced internal poloidal magnetic ﬁeld comparison ...... 92

6.4 Induceddipolemomentversusappliedﬁeld ...... 94

6.5 Induced dipole moment versus impeller rotation rate ...... 95 xii

Figure Page

6.6 Inducedtoroidalmagneticﬁeldcomparison ...... 97

6.7 Induced internal toroidal magnetic ﬁeld comparison ...... 98

6.8 MagneticﬁeldinducedbytheEMF ...... 101

A.1 Hallprobecircuitdiagram ...... 115

A.2 CircuitdiagramfortheHallprobeoutputvoltage ...... 115

D.1 Illustrationofthediffusionmatrix ...... 127

D.2 Illustrationoftheadvectionmatrix...... 129 xiii

NOMENCLATURE

Dynamo a system whereby kinetic energy is converted to magnetic energy at a rate which maintains a magnetic ﬁeld against Ohmic diffusion.

MHD the acronym for magnetohydrodynamics, the study of the dynamics of electrically-conducting ﬂuids.

DJ theDudleyandJames t2s2 ﬂow, the velocity ﬁeld which inspired the Madi- son Dynamo Experiment.

LDV the acronym for Laser Doppler Velocimetry, the technique used to measure the velocity ﬁeld of the water model of the sodium apparatus.

RPM theacronymforrevolutionsperminute,theunitmostoften used to describe the rotation rate of the impellers.

EMF the acronym for electromotive force, the force induced by the movement of conductors across magnetic ﬁeld lines.

G Gauss,aunitofmeasureformagneticinductance.

VDC theacronymforvoltsdirectcurrent,thetypeofvoltage needed to power the magnetic probes. xiv

ABSTRACT

A spherical liquid sodium experiment has been constructed at the University of Wisconsin- Madison with the goal of exploring magnetohydrodynamic dynamo action and turbulence. First sodium was achieved in August 2004, and a number of milestones have since been accomplished. These include realizing magnetic Reynolds numbers, based on impeller tip speed, of over 100, observation of advection of an applied magnetic field by flowing sodium, and observation of the ω-effect. An identical-scale water model of the experiment has also been constructed, allowing determination of the mean velocity field of the sodium. Magnetic fields, induced by the flowing sodium when exposed to an applied magnetic field, are presented. The induced fields are easily measurable and have significant fluctuations, consistent with magnetic field being advected by a turbulent flow. The mean induced fields are compared to magnetic fields predicted to be induced by the mean flow of sodium, as determined by fitting a spherical harmonic expansion to mean velocity field measurements made on the water model. Features of the measured field are well-described by the fields predicted, but some features cannot be explained. An external dipole moment is measured which cannot be generated by the mean axisymmetric velocity field, proof of which is given. The measured toroidal and poloidal induced magnetic fields within the sphere are significantly weaker than the predicted field. These effects are attributed to large scale currents generated by the action of turbulence: a turbulent electromotive force. 1

Chapter 1

Introduction

The earliest serious suggestion that astrophysical magnetic fields might be caused by the motion of electrically-conducting fluids was made by Sir Joseph Larmor in 1919 [38]. His suggestion was made to explain the magnetic field of the Sun, which had been detected by the measurement of Zeeman splitting in the spectral lines of sun-spots [29]. Though hesitantly presented at the time, his idea is now the accepted means by which stars, planets, and some moons [34], generate their magnetic fields [62]. The source of energy for this fluid motion is thermal convection, caused by nuclear reactions in stars and metallic freezing in planets. Briefly described, the process by which magnetic fields are generated by fluid flow is as follows. As highly-conducting fluid flows across magnetic field lines, an electromotive force is induced within the fluid which results in the generation of currents. These currents cause the magnetic field lines to be effectively ‘frozen’ to their respective fluid elements, resulting in the flow amplifying the magnetic field and stretching it in the direction of the flow [2]. Diffusion eventually acts to break the field lines and reorganize the magnetic field topology. If the resulting magnetic field is oriented such that it reinforces the original field, and if there is sufficient amplification of the field to overcome Ohmic losses, then the magnetic field will grow in time; a process known as the magnetohydrodynamic (MHD) dynamo [12]. This process converts kinetic energy of the fluid into magnetic energy. Flows which generate magnetic fields that do not reinforce the original field, or which do not have sufficient amplification, will not result in a growing magnetic field. The exact mechanism by which any particular flow generates a growing magnetic field is very dependent upon the geometry of the flow, which in turn is dependent upon the sources of free energy for the 2

fluid, boundary conditions, fluid viscosity, et cetera. As such, no catholic theory of dynamo action exists. Once it was appreciated that flows of conducting fluid could generate magnetic fields, attention was focused on determining which flows would magnetically self-excite. Several anti-dynamo the- orems were soon discovered, excluding the possibility of certain velocity-field categories from self- exciting: purely toroidal flows [9], purely radial flows [46] and plane two-dimensional flows [44] cannot maintain a steady magnetic field. The most important anti-dynamo theorem is due to Cowling [15], who proved that axisymmetric velocity fields are incapable of sustaining axisymmetric current distributions. Stated another way, axisymmetric velocity fields are incapable of generating growing magnetic fields that are both coaxial to the velocity field and axisymmetric. Since the Earth has an essentially-axisymmetric magnetic field, efforts were put forth to find non- axisymmetric velocity fields that might generate such a magnetic field. Early work in this regard was dominated by linear magnetic stability analyses of flows in a spherical geometry, to determine whether a specific velocity field would magnetically self-excite, and if so what its growth rate would be. Extending the work of Elsasser [19], the pioneers in this field were Bullard and Gellman [9], who established the formalism of the problem and were the first to test a velocity field for linear stability. Lacking sufficient computing power, however, they incorrectly concluded that the velocity field they had proposed would self-excite [25, 40, 54]. Later authors would improve upon this work, and find velocity fields that are linearly unstable to magnetic perturbations [60, 28, 54, 37, 16]. These analyses merely determined if a flow was magnetically unstable. More advanced physics, such as exploration of the back-reaction of the magnetic field on the velocity field, or the resulting saturated state, was not accessible to these studies. With the advent of modern computing resources, three-dimensional fully-self-consistent dynamic simulations of MHD dynamos became possible. Simulations of the Earth’s core reproduce the dipole-dominated structure of the observed field, the westward drift of magnetic anomalies, and even pole reversals [27, 36, 33, 13]. These simulations provide ready access to physical regimes that are difficult to explore experimentally or observationally due to large distances, large scales, and extreme conditions. 3

But like the linear calculations before them, these simulations have shortcomings. The simulations are incapable of resolving the extremely small values of many of the physical parameters of the Earth, such as the Rossby number or the Ekman number [61]; inaccurate parameter values are routinely used to lighten the computational burden. The Earth’s fluid Reynolds number is also typically inaccurate in these simulations, since the large Reynolds number results in very fine turbulence, requiring an extremely dense grid to resolve. These grids are computationally expensive; lower fluid Reynolds numbers, resulting in lower levels of turbulence, are customarily used in these simulations. This is a significant deficiency, since astrophysical bodies are often turbulent, and the role of fluctuations in these systems is not well understood. Whether dynamos exist in some situations despite turbulence, or perhaps because of it, is a question that simulations have not addressed satisfactorily due to the computational limits involved.. In contrast, astrophysical turbulence levels are easily generated in some liquid-metal experiments, allowing access to astrophysical regimes of physics in a relatively controlled environment. Liquid-metal experiments are one of the leading means by which MHD turbulence is being explored. This thesis presents data measured on the Madison Dynamo Experiment, a liquid-sodium experiment constructed to explore the physics of MHD turbulence and dynamos. By applying external magnetic fields to mean flows of sodium that are stable to magnetic perturbations the role of turbulence in magnetic field production is elucidated. It has been found that fluctuations induce large scale magnetic fields as-strong or stronger than the fields induced by the mean flow. This result is significant, as it indicates that fluctuations can potentially play a major role in the development of astrophysical dynamos. The rest of this chapter is organized as follows. First, a brief review of the history of dynamo experiments in general, and liquid-metal experiments in particular, is given in Section 1.1. The velocity field which motivated the construction of the Madison Dynamo Experiment, the Dudley and James t2s2 flow, is described in Section 1.2. The possible effects turbulence might play in the experiment, and the classical way of describing those effects, are discussed in Section 1.3. An outline of the thesis is given in Section 1.4. 4

1.1 MHD and Dynamo Experiments

The earliest liquid-metal experiments set out to confirm the basic MHD physics upon which dynamo theory ultimately rests. The earliest such experiments were built by Hartmann and La- zarus [30], who explored the modifications to a flow of mercury caused by an externally-applied magnetic field. Another early experiment, built by Lehnert, was a cylinder of stirred liquid sodium within an externally-applied magnetic field [39]. Azimuthal induced magnetic fields, orthogonal to the axial applied field, were observed, confirming that high-conductivity fluids advect magnetic field in the direction of the flow. The generation of toroidal magnetic fields in this manner, by differential toroidal flow stretching poloidal applied field, is known as the ω-effect. Further con- firmation of MHD theory came from the liquid-sodium experiment of Steenbeck et al. [68]. This experiment forced liquid sodium through a series of braided channels. The complex fluid motion twisted an applied field and generated a measurable voltage difference between the ends of the experiment, indicating that the flowing sodium had induced an electromotive force, as expected. The first experiments to generate spontaneous growth of magnetic fields were the experiments by Lowes and Wilkinson [41, 42]. These experiments consisted of two iron cylinders, oriented at an angle with respect to each other, spinning within an iron block. The intent was to have the cylinders amplify magnetic fields orthogonal to each other, with each cylinder amplifying a field orthogonal to the other. The cylinders were wetted to the block with thin layers of mercury. The experiments are noteworthy in that excited magnetic fields and magnetic field reversals were measured. But in contrast to astrophysical situations, the experiment lacked homogeneity due to the presence of mercury at the boundaries of the cylinders. Also complicating the comparison between the experiment and astrophysical bodies was the use of solid iron, a material that has non-linear magnetic properties at higher magnetic field strengths. Further experimentation led to the first liquid-metal experiment to demonstrate self-excitation, located near Riga, Latvia. A schematic of the apparatus can be seen in Figure 1.1. The apparatus consists of liquid sodium propelled down a long vertical cylinder in a twisting pattern, with the sodium returning to the top of the cylinder through a coaxial outer cylinder. Stationary sodium 5

sits outside the returning sodium within a second outer coaxial cylinder. The velocity field of the flowing sodium was carefully prescribed so as to magnetically self-excite [21]. Not only was a magnetic eigenmode excited by this experiment, but the growth of the mode lasted long enough for the back reaction to occur (the Lorentz force modified the velocity field to halt magnetic field growth), resulting in a saturated magnetic state [22]. Following the successful dynamo at Riga, an experiment in Karlsruhe, Germany, also reported a self-excited magnetic field [73]. This experiment, presented in Figure 1.2, consists of a series of pipes, each of which is surrounded by a second pipe which contains helical baffles. The pipes are arranged vertically in a rectangular pattern, and sodium is pumped through the pipes. The net result is a periodic flow pattern, which is expected to dynamo, of a type first suggested by Roberts [59], and further developed by Busse [10, 11]. This flow develops a growing magnetic field without the use of an externally-applied field to start it, using the ambient background as a seed field. The sodium in the Karlsruhe and Riga experiments is confined to a non-simply-connected geometry and as such these experiments are unnatural flow systems. Both of the experiments impose a scale separation upon the flow, confining it to a scale smaller than the overall size of the device. This scale separation prevents fluctuations from developingup to the full size of the experiment, re- pressing what might otherwise be an important physical effect. If allowed to develop to the largest scales, fluctuations can have non-trivial consequences, potentially generating magnetic fields on the scale of the system [67, 6]. Experiments which allow this to occur, while better analogies to astrophysical systems, have yet to observe growing magnetic fields. Following the Karlsruhe and Riga experiments, liquid-sodium experiments with simply-connected geometries were constructed. The open geometry of these experiments allows fluctuations to develop to the size of the device. These experiments all use rotating impellers, disks or spheres within the fluid to generate a turbulent velocity field; no baffles or pipes are used to constrain the flow. At Swarthmore University, for example, a 15 cm spherical container filled with liquid sodium was agitated with an impeller, generating a vigorous and turbulent flow. The conductivity of the sodium was measured as a function of rotation rate, and a turbulence-induced reduction in the conductivity of sodium was measured [57]. 6

Figure 1.1 A schematic of the Riga dynamo experiment. The sodium is propelled downward by an impeller (1) at the top of the innermost cylinder (2), and returns upward through an outer shell (3). Stationary sodium sits in the outermost layer (4). Figure taken from [23]. 7

Figure 1.2 Schematic of the Karlsruhe dynamo experiment. Sodium is pumped through pipes containing helical bafﬂes arranged in a periodic pattern. Figure taken from [73].

Another spherical sodium experiment, at the University of Maryland, studied the role of turbulence in the approach of a spherical flow to self-excitation [53]. The sodium was stirred by a pair of impellers, and when exposed to an external magnetic field the flow was shown to induce extremely turbulent magnetic fields. The authors speculated that the approach of the experiment to magnetic self-excitation would be characterized by intermittent magnetic field excitations, an effect that has been observed in the Madison Dynamo Experiment [49]. The largest liquid-metal experiment, other than the Madison Dynamo Experiment, is the von Kármán sodium experiment, a cylindrical experiment in Cadarache, France. The velocity field of the sodium in this experiment is generated using a pair of discs, one at each end of the cylinder. The experiment has observed magnetic field advection by the flows of liquid metal [52], as well as large scale magnetic fields generated by helical flows [8]. Fluctuation-induced magnetic fields may have been observed in this device [55]. This experiment has come the closest, other than the 8

Madison Dynamo Experiment, to magnetic self-excitation, as measured by the magnetic Reynolds number. The Madison Dynamo Experiment, the experiment which supplied the data for this thesis, is currently the largest liquid-sodium experiment in the world. With its open flow geometry turbulence develops to the largest scales, making it an ideal device to use to explore the role of fluctuations in magnetic field generation.

1.2 Motivation for the Madison Dynamo Experiment

The evolution of a magnetic field in the presence of an electrically-conducting fluid, with velocity field v, is governed by the magnetic induction equation: ∂B 2B = (v B)+ ∇ , (1.1) ∂t ∇ × × µσ where µ and σ are the magnetic permeability and conductivity of the fluid, respectively. This equation is the result of combining Maxwell’s equations, without the displacement current, with the generalized form of Ohm’s law:

J = σ (E + v B) . (1.2) × The earliest computational studies of dynamos examined the kinematic regime, in which the velocity field is unaffected by the presence of the magnetic field. The benefit of this assumption is that equation 1.1 becomes linear in B, resulting in a significant simplification of the problem. The kinematic assumption implies that the strength of the Lorentz force on the fluid, J B, is weak × compared to inertial forces. Equation 1.1 is dedimensionalized, with time scaled to the resistive

2 timescale (τσ = µ0σa , where a is a characteristic length of the system), ∂B = Rm (v B)+ 2B. (1.3) ∂t ∇ × × ∇

The dimensionless quantity Rm = µ0σav0, where v0 is a characteristic speed of the system, is known as the magnetic Reynolds number. It characterizes the magnitude of advection relative to diffusion in the system. The time dependence of the magnetic ﬁeld is taken to be eλt, giving

λB = Rm (v B)+ 2B, (1.4) ∇ × × ∇ 9 an eigenvalue equation for B. A search is now performed for velocity fields such that (λ) > 0, ℜ indicating that the velocity field is linearly unstable to small perturbations in B. The common theme of the earliest velocity fields which were explored, with the exception of the flows examined by Gubbins [28], was non-axisymmetry, the underlying assumption being that simple flows were incapable of generating a dynamo. But in 1989 Dudley and James [16] proposed a very simple velocity field (henceforth known as DJ) which they demonstrated to be capable of magnetic self-excitation at relatively modest magnetic Reynolds numbers. This velocity field is characterized by an axisymmetric double-vortex flow consisting of a poloidal cell in each hemisphere counter-rotating in the toroidal direction. The poloidal cells roll inward at the equator and outward at the poles. A graphical depiction of the flow is given in Figure 1.3. Since the flow is in a spherical geometry it is described using spherical harmonics; this flow is known as the t2s2 flow, since both the toroidal and poloidal components are described using the ℓ = 2, m = 0 spherical harmonics (this will be described in more detail in Section 3.2.1). Presented in Figure 1.4 is the real part of the growth rate, λ, of the fastest growing eigenmode versus Rm, for the DJ flow. At Rm = 50 the growth rate becomes positive and the flow magnetically self-excites. The eigenmode is described by a dipole field oriented to pass through the equator of the sphere. As a result, even though the flow is axisymmetric, Cowling’s theorem [15] is not violated since the excited magnetic field is not axisymmetric. While an axisymmetric magnetic field is more interesting from the perspective of most, if not all [47, 48], astrophysical examples, generation of a dynamo in any simply-connected geometry would lead to a significant increase in the understanding of the physics of self-excitation. The beauty of the flow presented by Dudley and James is not only its remarkable simplicity, but that it is not beyond reason to suggest that one could build an experiment that might generate such a velocity field. A single impeller in each hemisphere, counter-rotating with respect to each other, thrusting fluid toward the poles, should generate a flow that is at least similar to the one described. An Rm of about 50 can be attained in a reasonably-sized experiment; the specifications of the Madison Dynamo Experiment can be seen in Table 1.1. A half-metre radius sphere of liquid sodium, with six-inch impellers rotating at achievable rotation rates, can easily generate magnetic 10

v pol

v φ

-0.5 -0.3 0.0 0.3 0.5 Speed [arb]

Figure 1.3 The Dudley and James t2s2 flow. The axis of rotation is horizontal. In the upper half are the contours of the poloidal stream function, in black, and the colour contours of the magnitude of the poloidal field. In the lower half are the contours of the toroidal flow. The arrows indicate the direction of the poloidal flow.

Reynolds numbers, based on impeller tip speed (Rmtip = µ0σavtip), of over 100. Of course, generating a growing magnetic field is not merely a matter of generating a high enough magnetic Reynolds number, the ability of a flow to self-excite is closely coupled to the geometry of the flow. The geometry needs to be carefully prescribed to be confident that the system will self-excite. Also, the DJ flow is assumed to be laminar, the effects of turbulence have been ignored in the calculation of the data presented in Figure 1.4. Nonetheless, the potential of the experiment to develop a growing magnetic field was sufficient to justify construction of the Madison Dynamo Experiment. Its initial goal has been to generate a simply-connected liquid-metal dynamo, and explore the MHD physics involved therein. 11

20 40 60 0

-5 Real Growth Rate -10

-15 0 20 40 60 80 Rm

Figure 1.4 Growth rate for the fastest growing eigenmode for DJ ﬂow versus Rm. This eigenmode has a positive growth rate for a relatively low Rm.

1.3 The Role of Turbulence

The role that large scale fluctuations might play in the generation of magnetic fields in astrophysical bodies or liquid-metal experiments is an active field of research. The previous experiments which have succeeded in achieving dynamo action used severely-constricted flow geometries which suppressed the development of large scale fluctuations. Liquid sodium has a relatively low Prandtl number, P m = µ σν 10−4 (where ν is the kinematic viscosity of sodium), and as 0 ∼ such any velocity field with a significant magnitude is expected to be very turbulent. The Madison Dynamo Experiment, with its open flow geometry, allows this turbulence to develop to the largest

Madison Dynamo Experimental Speciﬁcations

−1 µ [H/m] a [m] σ [Ωm] RPM [rev/min] Impeller radius [m] vtip [m/s] Rmtip 4π 10−7 0.5 1.05 107 1800 0.15 28.7 189.3 × × Table 1.1 Speciﬁcations of the Madison Dynamo Experiment. 12

scales. A complete model of the experiment must include the effects of turbulent fluctuations on the generation of magnetic fields. The standard way of addressing the role of turbulence is through mean field theory [35]. In this treatment, the velocity and magnetic fields are separated into their mean and fluctuating components,

v = v + v˜, (1.5) h i B = B + b˜, (1.6) h i

where the average is an ensemble average and v˜ = b˜ = 0. Inserting this expansion into equation 1.3 and averaging gives ∂ B h i = Rm v B + v˜ b˜ + 2 B . (1.7) ∂t ∇ × h i×h i × ∇ h i The new term, as compared to equation 1.3, is the term of concern. This is the turbulent electromotive force (EMF), = v˜ b˜ . Should turbulent fluctuations of v and B interact coherently E × magnetic fields can be generated.D E As will be seen later in this work, this term can result in the production of large scale magnetic fields. The typical way of attempting to make sense of this term is to expand it in terms of the mean magnetic field: = v˜ b˜ = α B + β B + γ B + .... (1.8) E × h i ∇×h i ×h i D E α is related to kinetic helicity in the turbulence, and results in current being generated parallel to the mean magnetic field. The α-effect explicitly requires scale separation between the turbulence and mean field; the Karlsruhe experiment is based on a ‘laminar α-effect,’ meaning large scale helicity is forced on the fluid by the presence of baffles in the pipes, resulting in currents being generated parallel to B . β is related to the energy in the turbulence, and causes anomalous diffusion of h i the magnetic field. γ is caused by gradients in the magnitude of the turbulence, and results in flux being expelled from regions of high turbulence to low turbulence, causing a diamagnetic effect. Whether it is valid to express the turbulent EMF as equation 1.8, or whether one should expect to express any physics related to turbulence in terms of the above expansion coefficients, remains 13 to be seen. The expansion is not a typical expansion, in the sense that there is no small parameter which characterizes the importance of the terms: there is no reason to assume this expansion converges. As such, the terms aid in the development of intuition, but should not be expected to represent what is actually occurring within the experiment.

1.4 Thesis Outline

The original goal of this work was to develop a self-consistent model of the mean magnetic and velocity fields of the Madison Dynamo Experiment. It was na¨ıvely assumed that the effects of turbulence would be small and that the induced magnetic fields would be well-described by the inductive action of the mean velocity field, as was the case for the Karlsruhe and Riga experiments. This was found to be incorrect. In the experiment’s open geometry the fluctuations generate large scale magnetic fields with a magnitude that is a significant fraction of the magnitude of the fields induced by the mean flow. The mean velocity field was thus found to be insufficient to explain all of the measured magnetic fields. Consequently, the new goal of this work is to demonstrate that there exist fluctuation-driven currents in the Madison Dynamo Experiment that result in large scale magnetic fields. This may also be true for astrophysical systems; if so then the role that fluctuations play in these systems must also be understood. The algorithm used to demonstrate the presence of turbulence-induced magnetic fields is as follows. First the mean velocity field of the flowing sodium is determined. Because it is difficult to measure velocity fields in liquid metals the velocity field is measured in an identical-scale water model of the sodium experiment. Once the velocity field is known, the magnetic field that should be induced by said velocity field, given an applied field, is determined. The same external field is applied to the sodium apparatus, with the impellers rotating at the same rate as in the water model, and the magnetic field induced by the turbulent flow of sodium is measured. Any discrepancies between the predicted induced field, due to the mean velocity field, and the mean fields actually measured must be due to the turbulence. 14

Chapter 2 describes the apparatuses that were used to generate this work. Two main devices were used, a sodium experiment, as described above, and a water model of the sodium apparatus. The water model is used to determine the velocity field of the sodium. The details of the major components which are important to the devices are given, including descriptions of the data acquisition systems, laboratory sodium safety, and limitations of the devices. Chapter 3 describes the data taken from the water model of the experiment, and the fitting that is done to determine the mean velocity field of the sodium. This is used to determine the mean magnetic field that should be induced by the mean flow. The manner by which this field is calculated and the format of the problem is described in Chapter 4. Also in this chapter can be found the description of the applied field, and the manner by which it is determined. The field predicted to be induced by the mean velocity field is given at the end of the chapter. The internal and external magnetic field measurements taken on the sodium apparatus are presented in Chapter 5. The mode decompositions of the external magnetic fields are presented, as are the fits of the poloidal and toroidal magnetic fields to the internal data. These data are then compared to the predicted values in Chapter 6, wherein is presented the evidence that a turbulent EMF is active in the sodium apparatus: the detection of an induced dipole moment that cannot be generated by the mean velocity field, and a general weakness in the measured induced internal poloidal and toroidal magnetic fields. Finally, discussion and summary of the results are given in Chapter 7. 15

Chapter 2

Experimental Apparatus

There are two main experimental apparatuses that have been used to produce this work: a liquid-sodium experiment and an identical-scale water model of the same. The sodium apparatus is the main experimental device in this study. Sodium, having a high electrical conductivity, is an appropriate fluid for exploring mean-field magnetohydrodynamics and MHD turbulence. Sodium is not the easiest fluid to handle as it reacts violently with water and thus poses a significant health and safety risk. Much of the technology for handling large volumes of sodium was developed for the cooling of fast breeder nuclear reactors, and this experiment has borrowed much of this expertise. The mean velocity field of the flowing sodium is required to determine the role of fluctuations in the experiment. However, measuring the velocity field of a flowing liquid metal can be difficult. Liquid metals are opaque and typically chemically corrosive or excessively hot, thus excluding optical techniques from measuring the velocity field, as well as many probes which can be directly inserted into the fluid. Two techniques, however, have had some success in determining liquid- metal velocity fields. First, ultrasound Doppler velocimetry (UDV) systems have been used with some success [18, 66]. With this technique ultrasonic waves are launched into the flowing liquid metal using an ultrasonic transducer. Particles flowing with the fluid reflect these waves; these reflections are then measured by the same ultrasonic transducer. If the particle has a component of motion in the direction of the ultrasonic wave then the wave reflected by the particle will be Doppler shifted and the component of the velocity field in that direction can be determined. A UDV system is currently under development for the Madison Dynamo Experiment. This will 16 allow direct measurement of the velocity field of the sodium. The development of this diagnostic is not yet complete, however; no UDV data is presented in this document. The second technique which has been used to determine the velocity field of flowing liquid metals is velocity field inversion. This technique involves inverting the fluid’s velocity field using magnetic field [4, 7, 71, 70] and electric potential [72] measurements, in the low-Rm limit. A review of this technique is given by Stefani and Gerbeth [69]. Attempts were made to use a similar technique to ascertain the velocity field of the sodium in the Madison Dynamo Experiment. These attempts failed, however, due to the production of large scale magnetic fields by fluctuations, the topic of this work. Because the fluctuations induce large scale magnetic fields, both the velocity field and the fluctuations need to be modelled to successfully invert the velocity field. Unfortu- nately, there does not yet exist a model for such fluctuations and so the inversion was not possible. Lacking either a direct or indirect means of determining the mean velocity field of the flowing sodium, water is used to model the flows of the liquid metal. An identical-scale water model of the sodium apparatus has been constructed for this purpose.

2.1 Water Model

Water, obviously, does not have the electrical conductivity needed to perform MHD experiments. However at the appropriate temperatures water and sodium have the same kinematic viscosity [58, 64, 65]. The range of temperatures that match the kinematic viscosities of water and sodium is shown in the blue rectangle of Figure 2.1. Thus, at the correct temperatures, the two fluids are hydrodynamically identical and will behave identically under the same conditions. Mea- suring the velocity field of water in a given apparatus is equivalent to measuring the velocity field of the sodium in the same apparatus. The water model is used for this purpose.

2.1.1 Description of the Water Apparatus

Measuring the velocity field of flowing water is significantly easier than measuring the velocity field of flowing sodium, for the simple reason that optical methods can be employed. To this end an identical-scale water model of the sodium apparatus was constructed. A schematic of this model 17

Sodium Temperature [oC] 100 120 140 160 180 1.2

/s] 1.0 2 H O 2 0.8 Na 0.6

0.4

0.2 Kinematic Viscosity [cm

0.0 20 40 60 80 100 Water Temperature [oC]

Figure 2.1 Kinematic viscosities of water and sodium versus two ranges of temperature. At the temperatures within the blue rectangle the sodium and water can have identical kinematic viscosities.

can been seen in Figure 2.2. The sphere is 42 inches in diameter (about 1 metre), and is made of stainless steel. Shafts and impellers, identical to those used in the sodium experiment except for the handedness of the impellers, enter the sphere through each pole. The shafts are held in place by two bearing sets, both external to the sphere. The shafts are driven with two 45 kW (60 HP) motors, which are independently controlled with variable frequency drives (VFD). The motors can generate a maximum impeller rotation rate of 1200 RPM; the rotation rates of the impellers are

monitored using rotation rate encoders. The sphere has ﬁve 1-inch-thick Pyrex R 1 windows at different polar positions along the top of the sphere; these windows deﬁne the φ =0 longitude of the sphere. Laser Doppler Velocimetry (LDV) is used to measure two components of the velocity

ﬁeld, vθ and vφ, at different polar and radial positions. These measurements are used to reconstruct the mean velocity ﬁeld of the water, as will be described in Chapter 3.

1Pyrex is a registered trademark of Corning Incorporated. 18 Experiment. Five windows in the sphere allow an LDV . A traverse varies the polar and radial position of the laser φ v olume. and θ v Figure 2.2 Schematic of the water model of the Madison Dynamo system to measure two components of the velocity ﬁeld, ﬁberoptic probe, changing the position of the measurement v 19

The rotating impellers generate an axisymmetric velocity field in the water. A photograph of one of the impellers can be seen in Figure 2.3. The impellers are stainless steel, square-pitched boat propellers which have been modified to generate a velocity field which is closer to the DJ velocity field and closer to a field which will magnetically self-excite [51, 20]. The radius of the propellers has been reduced to 6 inches, and a 4 inch deep stainless steel ring has been added to the propeller’s outer edge, a ring known as a Kört nozzle. This nozzle eliminates the radial thrust generated by the spinning impeller. To increase the toroidal flow, three 1/2 inch bars have been added to the outer edge of the Kört nozzle. The impellers have been dynamically balanced to improve their rotational stability. Spinning impellers are known to generate cavitation due to the rapid pressure drop which occurs on the impeller’s trailing edge. Cavitation damages the impellers as the bubbles collapse on the impeller’s surface, and the presence of bubbles in the fluid modifies the mean flow. Cavita- tion also reduces the drag of the impellers on the flow, thus reducing the momentum transferred to the water and interfering with the velocity field generated. To prevent cavitation the sphere is pressurized whenever the impellers are rotating. The pressurization is accomplished using a pressurized bladder tank. The pressurization needed to suppress cavitation is determined empirically; a pressure of 80 psi is used for a rotation rate of 1000 RPM. The cavitation level is monitored by a transducer which measures the vibrations generated by the collapsing bubbles. The water experiment is controlled by National Instruments Corporation’s2 LookoutTM 3 software, an industrial control software well-suited to the needs of running a medium-sized physics experiment. The software interfaces with National Instruments’ FieldPointTM 4 input/output modules, as well as other control cards. FieldPoint modules monitor temperatures and pressures, open and close valves and relays, and perform other monitoring and control tasks. Lookout also interfaces with the VFDs to control the rotation rates of the motors. Because the VFDs do not have feedback on the motor rotation rates the set points of the VFDs often need to be adjusted to at- tain the desired rate. This process is now automatically implemented by Lookout, for the water

211500 North Mopac Expressway, Austin, TX 78759-3504, U.S.A. 3Lookout is a trademark of National Instruments Corporation. 4FieldPoint is a trademark of National Instruments Corporation. 20

Figure 2.3 Photograph of one of the impellers used in the water and sodium experiments. The K¨ort nozzle is added to reduce radial thrust. The small 1/2 inch bars on the outside of the nozzle increase the toroidal ﬂow. 21

model, by monitoring the rotation rates of the motors and adjusting the VFD set points accordingly. Rotation rates for both of the motors now agree within 1 RPM. ± The water’s temperature is monitored by Lookout via a thermocouple inserted into one of the water model’s access ports. At higher impeller rotation rates the water in the experiment requires periodic cooling to prevent the water’s temperature from straying too far from the optimal temperature of 34 ◦C (the water is run at the low end of the blue range (see Figure 2.1) because the sodium is usually run at 98-105 ◦C). Consequently, the experiment is run in a duty-cycle mode, meaning that when the temperature exceeds the speciﬁed temperature range the motors are stopped and the water is actively cooled until the water’s temperature is at the bottom of the acceptable range. The range used is 32-36 ◦C. The active cooling of the water is accomplished using a side cooling loop attached to the sphere. A pump circulates water from the sphere into a water-water heat exchanger. The cooling side of the heat exchanger is cooled using a Neslab reservoir chiller.

2.1.2 Laser Doppler Velocimetry

The LDV system is a package system manufactured by TSI Incorporated5. A photograph of the LDV system in action can be seen in Figure 2.4. The LDV system begins with the laser, a Spectra-Physics6 Series 2000 Argon-Ion laser. The laser is continuous wave and has a maximum output of 5 Watts. The laser comes as a package system, complete with its own power supply. The power supply requires 208 Volt AC power and continuous water cooling. The power supply also comes with built-in interlocks, which have been connected to the interlock system of the laboratory. Should someone attempt to enter the laboratory while the laser is running the interlocks will be activated and the laser turned off.

The laser beam exits the laser cavity and enters a Model 9201 ColorBurst R 7 Multicolor Beam

Separator. Within this device the laser light is separated into the two colours that are used for the velocity measurements, green (514.5 nm) and blue (488.0 nm). Data concerning the beams are listed in Table B.1. These two beams are then each split, for a total of four beams. One beam of

5500 Cardigan Road, Shoreview, MN 55126, U.S.A. 61335 Terra Bella Avenue, Mountain View, CA 94043, U.S.A. 7ColorBurst is a registered trademark of TSI Inc. 22

each colour then passes through a Bragg cell, causing the frequency of the beam to be shifted by

8 40 MHz, based upon a signal sent to the ColorBurst device by the Model 9230 ColorLink R Plus

Multicolor Receiver. The four beams exit the Beam Separator and enter fiberoptic cables which lead to the fiberoptic probe. The fiberoptic probe is a TSI model 9253-350. This probe focuses and angles the four beams such that they all cross at a single point, defining a ‘measurement volume,’ the volume being de-

termined by the frequency of the laser beams. The green beams measure vθ and the blue beams

measure vφ. The laser fiberoptic probe is mounted to a traverse which is based upon a converted X-ray mount; the traverse changes the fiberoptic probe’s polar position by rotating about an axis which passes through the equator of the sphere and is horizontally level with the base of the experiment. The fiberoptic probe is moved radially by moving the fiberoptic probe’s base along the length of the traverse. Both the polar and radial positions of the fiberoptic probe are controlled by stepper motors. As seen in Figure 2.2, the fiberoptic probe always points in the negative radial direction. The changes in the fiberoptic probe position allow the velocity field of different positions to be measured. The traverse’s positions are specified by a ‘traverse file,’ that is used by the FINDTM 9 program (discussed below) to set the traverse to the desired positions. To get the radial position of the measurement volume correct one must account for the index of refraction of both the windows and the water, and the angle of the laser beams as they leave the fiberoptic probe. The equation which describes this relationship is

tan θ tan θ tan θ F = F air + t 1 Pyrex + d 1 air , (2.1) 0 tan θ − tan θ − tan θ H2O H2O H2O

where F is the distance from the ﬁberoptic probe to the measurement volume, F0 is the distance from the ﬁberoptic probe to the measurement volume without any water or glass present, t is the thickness of the glass, and d is the distance from the probe to the surface of the glass. The θ values are measured with respect to the angle normal to the window. The values that are used to calculate the radial position, as a function of d, are given in Table 2.1. The table also lists the indices of refraction, n, that are important to the problem.

8ColorLink is a registered trademark of TSI Inc. 9FIND is a trademark of TSI Inc. 23

Figure 2.4 Photograph of the water model, along the axis of rotation. The laser beams can be seen within the water volume. 24

LDV Measurement Position Parameter Values

F0 t θair θPyrex θH2O nair nPyrex nH2O 35.0 cm 2.54 cm 0.0713 radians 0.0485 radians 0.0536 radians 1.0 1.47 1.33

Table 2.1 Parameter values used to calculate the radial LDV measurement position, as described

by equation 2.1. The n values are the various indices of refraction. The values of θPyrex and θH2O are not independent, they are a consequence of θair and the indices of refraction.

The two intersecting laser beams of each colour create an optical interference pattern within the measurement volume. Reflecting particles, made of small flakes of titanium-coated mica (Kalliro- scope Corporation10 part number PM-01), are mixed with the water. The mica particles are approximately neutrally buoyant: they stay suspended within the fluid when the impellers are rotating and are assumed to be flowing with the same velocity as the water. As the mica particles pass through the measurement volume they reflect the interference pattern created by the laser beams. The fiberoptic probe sends this reflected light down a fiberoptic cable to the ColorLink Plus Multicolor Receiver. The motion of the particles Doppler shifts the interference pattern, and this Doppler shift is used to determine the velocity of the passing particle. Because the frequency of one of each colour of laser beam has been offset by 40 MHz the interference pattern is not static, but rather moves relative to the measurement volume. This motion of the interference pattern defines a direction for the measurement; without this motion a static interference pattern would exist and it would not be possible to tell whether a particle passing though the volume was going in the positive or negative direction. The light detected by the fiberoptic probe is received by the ColorLink Plus Multicolor Re- ceiver. This device separates the returning light into its green and blue components. Photomul- tiplier tubes then convert the light into an electrical signal. This signal is then high-pass filtered to remove any DC component to the signal. The resulting signal is then mixed with a sine wave of known frequency corresponding to 40 MHz plus-or-minus the frequency shift specified in the FIND software (see Table 3.1 for the values used in this work). This new signal is a combination

10264 Main Street, P.O. Box 60, Groton MA 01450, U.S.A. 25

of frequencies, the signal frequency plus-or-minus the sine wave frequency. The signal is low-pass ﬁltered to remove the high frequencies and then sent to the Digital Burst Correlator. Frequency shifting is implemented by FIND in the ColorLink Plus to distinguish between positive and negative particle velocities. The shift frequency is chosen so as to give a range of speeds that encompasses those expected to be measured, both in the positive and negative directions. This is discussed in more detail in Appendix B. The IFA 755 Digital Burst Correlator analyses the signal generated by the ColorLink Plus. It calculates the velocity components of the reﬂecting particles based on the Doppler shift frequencies of the two signals, one for each colour. The Burst Correlator also correlates the two signals to determine if two data points are ‘simultaneous,’ meaning that the two signals occurred within some time window (see Table 3.1). The resulting data is then returned to the FIND software on the computer controlling the experiment.

2.1.3 Water Data Runs

A typical experimental water data run proceeds as follows. First the experimental equipment is turned on and its operation is confirmed. This includes powering the motors, turning on the VFDs, turning on the laser and the laser cooling, and turning on the water cooling for the water experiment’s mechanical seals. The sphere is filled with mica-laden water (if the sphere has been previously emptied), and the sphere is pressurized using nitrogen. The motors are started and the rotation rates are confirmed. A few tens of seconds are allowedto pass to allowthe flow to stabilize. FIND then begins data collection. The ColorBurst, fiberoptic probe traverse, ColorLink Plus, and Burst Correlator are all run by the FIND software. The software interfaces with the devices to control them, give them their various parameter settings and to record the data. The software also moves the traverse between measurement positions. When Lookout determines that the water temperature is too high and that cooling of the water is needed, Lookout sends a ‘data inhibit’ (low TTL) signal to the Burst Correlator. This signal prevents the Burst Correlator from sending data back to FIND for processing. The motors are then stopped by Lookout, the valve which closes the cooling loop is opened, and the pump which runs 26

the external cooling loop for the experiment is started. Once the water reaches the bottom of the acceptable temperature range the cooling pump is stopped, the cooling loop valve is closed, and the motors are restarted. After a brief pause the data acquisition resumes. Once all the data for a run has been taken, the data is converted from the raw format output by FIND to storage in a Common Data Format (CDF) ﬁle. This ﬁle is then accessed for the data manipulation and analysis.

2.2 Sodium Experiment

The sodium apparatus is the main experimental device. Sodium is signiﬁcantly more difﬁcult to handle and use than water, and as such the sodium apparatus requires considerably more infrastructure than the water model. The sodium experiment, in contrast to the water model, requires a holding tank to store the sodium when not in use, oil heating and cooling for the sphere, electrical heating for the sodium transfer lines, specially-designed mechanical seals to contain the sodium, and oil heating and cooling for the mechanical seals.

2.2.1 Description of the Sodium Apparatus

The main experimental apparatus is a 42 inch diameter (about 1 metre), 5/8 inch thick, stainless steel sphere. A schematic of the apparatus is given in Figure 2.5; a photo of the experiment is given in Figure 2.6. Like the water experiment, two shafts enter the sphere at each pole, and at the end of each shaft is an impeller. The impellers are identical to the water experiment impellers, except for handedness. Each impeller is driven by an 100 HP motor which is controlled by the same VFDs that control the water experiment motors. Each VFD records its motor’s torque and power information. The power used by the 100 HP motors, when stirring the sodium, follows a cubic relationship with rotation rate as shown in Figure 2.7. The 100 HP motors are expected to be able to rotate the impellers up to at least 1500 RPM [50]. Unlike the water model which supports the shafts with only two bearing sets, three bearing sets support the shafts in the sodium apparatus, one of which is contained within the sphere and is wetted to the sodium. Two mechanical cartridge seals seal the shafts where they enter the sphere. 27

Figure 2.5 Schematic of the Madison Dynamo Experiment. The velocity field is generated by two counter-rotating impellers. Two sets of coils, one coaxial and one transverse to the drive shafts, are used to apply various magnetic field configurations. The magnetic field induced by the flow is measured using Hall-effect sensors both on the surface of the sphere and within tubes that extend into the flow.

These seals contain three sealing layers. The first layer is a mechanical seal made of a sodium- compatible ceramic. Behind this seal is an oil reservoir. The oil circulates through the seal body to an external oil heating and cooling system. The oil is a hydro-cracked mineral oil; the oil heats, cools, and lubricates the ceramic seals. The oil is kept at a pressure 30 psi above the pressure of the sodium. Should the first mechanical seal fail the oil will flow into the sodium versus the sodium flowing into the seal body. Behind the oil reservoir is a second ceramic mechanical seal. Outside this seal is a buffer region containing argon gas. The argon flows through the buffer region and exits the seal body. The argon prevents the oil from oxidizing. Should sodium find its way to the buffer region, passing through the two ceramic seals, the flow of gas would stop, leading to a buildup of pressure in the argon line. 28

Figure 2.6 Cover of the February 2006 issue of Physics Today, featuring the Madison Dynamo Experiment. The sphere is in the centre, and is covered in white insulation to help maintain the temperature of the sphere when hot. One of the motors which drive the impellers can be seen in the bottom-left corner of the photograph. The sphere expansion manifold, the insulated tubes leaving the top of the sphere, can be seen at the top of the photograph. The sphere heating system’s oil manifolds, the vertical tubes covered in metallic-covered insulation, can be seen at either side of the sphere. 29

200

150

100

Rm (based on tip speed) Motor 1 Motor 2 0 0 20 40 60 80 Power [kW]

Figure 2.7 The motor power used by the 100 HP motors, when stirring the sodium, follows the characteristic cubic relationship with the impeller rotation rate, which is proportional to Rm. The curve is extrapolated to the maximum motor power available, yielding Rmmax = 150 (1500 RPM).

Pressure sensors have been installed to monitor the inflowing argon pressure, allowing detection of a seal failure. Behind the argon buffer is a static Grafoil R 11 packing. Should the sodium find its way to the argon buffer it will be contained in that region by the packing. When not in use the experiment’s sodium, about 275 U.S. gallons, is stored liquid in a holding tank which sits in a steel-lined vault below floor level. The holding tank is kept below floor level so that if the experimental vessel needs to be suddenly emptied of sodium, or if power is lost to the experiment, gravity can be used to return the sodium to the holding tank. The holding tank is electrically heated by 240 Volt Calrod R 12-type heaters strapped to the outside of the tank.

Kaowool R 13 mineral wool insulation covers the tank to reduce heat loss. The temperature of

the tank is sampled using type K thermocouples. The output of the heaters is controlled using a

11Grafoil is a registered trademark of GrafTech International. 12Calrod is a registered trademark of General Electric Company. 13Kaowool is a registered trademark of Thermal Ceramics, Inc. 30

proportional-integral-derivative (PID) pulse-width modulated stand-alone temperature controller. The tank is kept at 120 ◦C and can hold up to 335 gallons. The holding tank sits upon 4 load cells which continuously weigh the tank, allowing the inventory of sodium to be known up to 0.5 pound. The load cells are simple scales based upon strain ± gauge technology. Three electrical contact switches (‘level detectors’), at three different heights in the holding tank, give direct conﬁrmation of the presence of sodium at a particular tank height. These switches consist of a single lead, held at 24 VDC, whose length is protected from the sodium by a sodium- compatible ceramic. Only the tip of the lead is exposed to sodium. When sodium is in contact with the lead the 24 VDC potential of the lead is shorted to ground, indicating the presence of sodium at that tank height. The expansion tank contains a similar set of three level detectors.

2.2.2 Heating and Cooling the Experiment

Sodium freezes at 98 ◦C, and as such the experiment must be heated from room temperature before the experiment can be charged with sodium. The heating process takes 5 hours: 4 hours to reach 120 ◦C followed by a further hour of maintaining the experiment at temperature to stabilize the system. The heating rate is kept relatively low so as to minimize heating and expansion stress on the experiment. There are many systems that are used to heat the experiment. The sodium transfer lines, the pipes that run from the holding tank to the sphere and from the sphere to the expansion tank, are electrically heated. The expansion tank, the main sodium valve, the sphere’s 16 inch ﬂange necks and ﬂange faces are similarly electrically heated. Like the holding tank, Calrod-type heaters are strapped to the outside of these components, which are then covered with stainless steel foil and then mineral wool insulation to minimize thermal losses. The temperature of the components is sampled using thermocouples and the heater output is controlled using the same type of PID controller used on the holding tank. Unlike all of the other non-holding-tank components, the main sodium valve is kept at 120 ◦C when the experiment is not in use, because the valve has a small volume of sodium trapped within it. 31

240 Volt self-regulating heating tape is wrapped around the many flanges of the experiment that are exposed to sodium: at the bottom of the expansion tank, on the top and bottom of the sphere, where the transfer lines meet the main sodium valve, and at the top of the holding tank. This heating tape consists of two parallel wires separated by an electrically-conducting polymer. The polymer has a relatively high resistivity; when an electric potential is applied between the two wires, electricity flows through the polymer and heat is generated. As the tape gets close to 120 ◦C, the ‘set point’ temperature of the tape, the conductivity of the polymer drops and the tape stops heating. In this way the tape can be wrapped around itself, and insulated, without overheating, a common problem with constant-wattage heating tapes. This self-regulating tape is also used to heat the sphere’s expansion manifold (the expansion manifold can be seen in Figure 2.6: the white insulated lines rising up from the sphere). This expansion manifold consists of stainless steel tubes that rise from the sphere’s access ports. These tubes give the argon backfill gas a path to escape from the access ports as the sphere is filled with sodium. Without an escape path large bubbles of argon would form in the ports, potentially causing problems if these bubbles started wandering around the sphere. The bearings in the bearing-shaft assemblies (not the bearing set contained within the sodium) are heated and lubricated with an oil system that is shared between the two assemblies. Not being exposed to sodium, these bearings do not need a high temperature and are heated to 60 ◦C. The cartridge seals which seal the sphere at the shaft entry points are heated and cooled using a hydro-cracked mineral oil. Because the first ceramic seal is exposed to sodium, the seal body is maintained at 120 ◦C when the experiment is in operation. Each seal has its own circulation system, composed of a reservoir of oil in which is contained an 3000 Watt heater, a pump and an oil-air heat exchanger. Because very little heat is deposited to the seals, the excess heat of the oil is deposited to the air rather than to some other fluid. The sphere itself is not only heated, like the other components of the system, but also actively cooled. The sphere needs cooling because, like the water model, when the impellers are rotating at higher rotation rates a significant amount of heat is added to the sodium. This heat needs to be extracted to maximize the conductivity of the sodium (this will be discussed in more detail in 32

Chapter 5) and to keep the magnetic probes from overheating. The sphere is heated and cooled using an oil-based heat transfer system. Copper wedge-shaped plates have been bonded to the surface of the sphere using a heat-transfer putty. These copper pads can be seen in a photograph of the experiment during construction, Figure 2.8. Copper tubing has been soldered to these copper plates, through which flows the heat-transfer oil. Mineral wool insulation covers the copper plates to reduce ambient heat loss. A large insulated reservoir contains the bulk of the system’s oil. This oil is heated using a 12 kW in-line heater and cooled using a 75 kW oil-water heat exchanger, located outside of the experimental lab. The oil is pumped from the reservoir, through the in- line-heater chamber, then through a three-way valve. The three-way valve sends a fraction of the oil to the water-oil heat exchanger, if the system needs cooling, and the rest of the oil goes to the manifolds which distribute the oil to the copper pads on the sphere’s surface. These vertical manifolds near the sphere can be seen, covered in metal-coated insulation in Figure 2.6, and bare in Figure 2.8, on either side of the sphere. The smaller white-insulation-covered tubes leaving the manifold, in Figure 2.6, go to each copper pad. The output of the heater, and the amount of oil sent for cooling by the three-way valve, is controlled by a PID controller, based on a thermocouple temperature measurement of the surface of the sphere. Unlike the PID controllers used for heating the other components in the system, as described above, this PID controller has two outputs: one for the output of the heater and one for the three-way valve. Cooling of the sphere is necessary to keep the sodium in the desired temperature range. When stirring the sodium with two 100 HP motors only a short duration of time passes before the temperature of the sodium begins to rise. The temperature of the sodium needs to minimized for two reasons. First, the Hall probes which measure the magnetic fields in the experiment will be dam- aged if the temperature of the probes exceeds 150 ◦C. Second, the electrical conductivity of liquid sodium varies with temperature [14]. As seen in Figure 2.9, the conductivity of sodium has a maximum value of 1.05 107(Ωm)−1 near the freezing point of sodium (98 ◦C) and decreases by about × 4% for every 10 ◦C. Since Rm σ, the sphere is kept at a temperature of 98-105 ◦C to optimize ∝ Rm without freezing the sodium. Fortunately the viscosity of sodium doesn’t change much in this temperature range, as seen in Figure 2.1. 33 n n be field ratory during construction of the experiment. The sphere ca hich distribute the heat transfer oil to the copper plates ca heat-transfer plates on its surface. The external magnetic Figure 2.8 Photograph of the Madison Dynamo Experiment labo be clearly seen onprobes the can right be side seen of between the the photo, copper with plates. the The copper manifolds w seen standing vertically on either side of the sphere. 34

1.1·107 ] -1

1.0·107

9.0·106 Conductivity [(Ohm m)

8.0·106 100 110 120 130 140 150 Temperature [oC]

Figure 2.9 Electrical conductivity of sodium versus temperature. The sodium’s temperature is minimized when trying to maximize Rm.

2.2.3 Laboratory Sodium Safety

Liquid sodium, especially large volumes of liquid sodium, can be very dangerous if not handled carefully. Sodium reacts violently with water, even with moisture in the air or in the skin. This reaction releases sodium oxide, which is very caustic if inhaled, and hydrogen gas, which can cause hydrogen explosions at high enough concentration. Sodium safety is taken very seriously in the Madison Dynamo Experiment laboratory, and a great deal of time and effort has been put forth to make the laboratory a safe place to work with sodium. The first step to having a sodium-safe environment is to minimize the likelihood of anyone being exposed to liquid sodium. To this end researchers are very rarely in the laboratory when the experiment is being run. However, occasionally experimenters must be in the laboratory when the sodium is not in the holding tank, in particular when the sodium is being transferred to and from the sphere. To maximize the safety of the experimenters, safety equipment is worn by any experimenters in the laboratory whenever the sodium is not in the holding tank, or whenever sodium- contaminated equipment is being handled. A photograph of an experimenter wearing the safety 35 equipment can be seen in Figure 2.10. The safety equipment consists of fire-proof Nomex R 14 undergarments, socks, pants and jackets. Steel-toed boots are worn, as well as protective breathing masks and face shields. Leather spats, gloves and aprons are also worn. The purpose of the safety equipment is to offer a protective boundary between sodium and the experimenter should sodium leak out of the experiment or caustic smoke get into the air. Should the experiment have a breach or leak, steps have been taken to prepare the laboratory and alert the experimenters. The first line of defence is the storage environment of the liquid sodium, the holding tank. The holding tank sits in a steel-lined vault. Should the sodium leak from the holding tank it would be contained in the vault, and if it caught fire it would quickly cool due to the relatively high thermal conductivity of the steel. The bottom of the holding tank’s vault is lined with four ‘sodium detectors,’ which are pairs of exposed wire that run parallel to each other, one of which is at an electrical potential of 20 Volts DC. If sodium were to fall onto one of these detectors it would complete the electrical circuit between these two wires, an event that would be detected by Lookout and activate an alarm. The holding tank is also sitting upon load cells; if the holding tank began to leak its weight would drop. This event would eventually be noticed by the experimenters since the weight of the sodium is recorded before and after each experimental run campaign. To increase the safety of the sodium transfer lines, all joints and fittings in the sodium transfer lines have been welded; no threaded fittings have been used. All flanges are flat, raised-face flanges which use a Grafoil gasket, a sodium-compatible wound graphite gasket. Stainless steel trays are placed below the sphere and main sodium valve to prevent sodium from reaching the concrete floor in the event of a leak. Sodium will extract the water out of concrete and as such concrete is not sodium compatible. These drip trays slope downward to guide any escaped sodium into a region out from underneath the experiment, where experimenters might be able to extinguish a fire. These drip trays have enough volume to contain the entire inventory of sodium contained within the experiment and, like the holding tank vault, are lined with sodium detectors.

14Nomex is a registered trademark of DuPont. 36

Figure 2.10 Sodium safety gear worn by experimenters while in the laboratory, when the sodium is not in the holding tank. 37

Should sodium escape from the experiment and start to burn, various liquid-metal firefighting equipment is located in the laboratory. There are two main fire extinguishing components: soda ash (calcium carbonate dust), and class D fire extinguishers. The soda ash is shovelled onto burning sodium to suffocate the fire, should it be contained within a drip tray. The class D fire extinguisher is needed because standard fire extinguishers are not safe to use on liquid-metal fires. Burning sodium expels sodium oxide into the air. If this is inhaled the sodium oxide reacts with the water in the body to produce sodium hydroxide, a powerful base. To address the problem of sodium oxide-contaminated air, an air Venturi scrubber system has been constructed and installed in the laboratory. The scrubber system works by sucking large amounts of air from the room and injecting it into a spinning cone of water. The water reacts with any sodium oxide in the water to make sodium hydroxide. This sodium hydroxide stays in solution, while the air is expelled from the building. The water is continuously pumped into the cone from a holding tank below the cone. The pH of the water in the system is tested after each scrubber use. If the pH rises too high the water in the system is replaced.

2.2.4 Filling the Sphere

Once the experiment has been brought to temperature the liquid sodium is transferred pneu- matically from the holding tank to the sphere using pressurized argon. Argon is used for all pressurization in the sodium apparatus, as it is inert and does not react with sodium. As the holding tank is pressurized the sodium exits the holding tank through a long dip tube which extends from the top to near the bottom of the holding tank. The sodium passes through a sodium transfer line, through the main sodium valve, through more sodium transfer line, and then enters the bottom of the sphere. The sodium fills the sphere from the bottom until the sphere is full. The sodium passes through yet another sodium transfer line until it reaches the bottom of the expansion tank. The expansion tank is filled until the liquid level rises to an appropriate height, monitored by the weight of the sodium which remains in the holding tank and the electrical level detectors within the expansion tank. The expansion tank, being only partially filled, accommodates changes in the volume of the sodium due to variations in temperature. 38

The sodium expansion manifold, consisting of the stainless steel lines which leave the sphere’s access ports, is not heated when the sodium is transferred from the holding tank to the sphere. As the sodium ﬁlls the sphere the expansion manifold allows the access ports to be degassed of argon, but the stainless steel lines also ﬁll with sodium. With only a 1/2 inch diameter and the heating turned off, the sodium in these lines freezes before traveling very far up the lines. The sodium is allowed to freeze in these lines so as to form a plug, which prevents sodium from being pushed up the lines due to centrifugal forces acting on the spinning sodium. Like the water experiment, the sodium experiment is pressurized to suppress cavitation. Pres- surized argon is applied to the sodium through the top of the expansion tank. Cavitation is monitored with an ultrasonic transducer. Again, the amount of pressurization is determined empirically. In general, however, less pressurization of sodium is needed for a given impeller rotation rate, since the vapour pressure of sodium is much lower than the vapour pressure of water. Like the water experiment, the sodium apparatus is controlled using National Instruments Cor- poration’s Lookout industrial control software. The control interface of the software can be seen in Figure 2.11. The interface displays the major components of the experiment, as well as the important buttons and switches. Windows containing more detailed descriptions of individual components can be brought to the fore by pressing one of many buttons.

2.2.5 Magnetic Fields and Data Acquisition

Two pairs of coils generate magnetic fields which are used to study the inductive effects of the flow. One pair is coaxial with the drive shafts, the other transverse, as shown in Figure 2.5. The coils can produce fields which are dominated by the dipole, quadrupole, transverse dipole, and transverse quadrupole field components. Switching between coil configurations is easily accomplished using a switchboard beside the sphere. The coils which make a pair are run in series to guarantee that the two coils pass the same current. The coils are water-cooled on a closed-loop system, with the water chiller located outside the sodium laboratory. A DC power supply provides the coils with up to 600A of current to generate fields up to 150 G on axis, depending on magnetic configuration. The coil current is measured using a current sensor. The coil power supply is run in 39 sodium experiment. Figure 2.11 Screenshot of the Lookout control screen for the 40

constant-voltage mode, meaning that a constant voltage is applied across the coils. This can lead to small variations in the applied current over the course of a data run, due to changes in the resistance of the coils due to heating. These changes are small, < 3%, and are not considered important for the results presented in this work. The magnetic fields in the experiment are measured using Hall-effect probes (Analog Devices15 AD22151 Linear Output Magnetic Field Sensors) on integrated circuits with internal temperature compensation. These integrated circuits are mounted onto circuit boards with the appropriate infrastructure to run the probe. The magnetic field saturation level of the probes is determined by the values of resistors on the circuit board. These values and the layout of the circuit board can be found in Appendix A. In the current configuration, the probes saturate at about 170 G. The ± probes are powered by 5 Volts DC. The main source of noise on the Hall probe signals is thermal noise from the Hall probe itself. To reduce the noise levels, the probes are low-pass filtered at about 100 Hz. Since the characteristic time associated with the magnetic field is the resistive time of the experiment, τ 3 seconds, sig- σ ∼ nals that are of significantly higher frequency than the resistive dissipation scale, Rm/τ 50 Hz, σ ∼ are not important [50]. Worth noting is the fact that the low-pass filtration cutoff is below the skin frequency of the sphere, the frequency above which signals will not pass through the sphere’s stainless steel shell. For a 5/8 inch thick stainless steel shell (δ = 1.6 cm), with a resistivity of η =1/σ =7.2 10−7Ω m, the frequency is f =(πµ σδ2)−1 = 712 Hz. × skin 0 Quarter-inch studs are welded to the sphere 0.35 inches from the desired probe position. A 1.75 inch diameter stainless steel ring is then placed on the sphere’s surface around the stud. A stainless steel disk of the same diameter, which contains a 0.25 inch mounting hole, is placed on the ring with the stud passing through the mounting hole. A nut is then tightened on the stud, securing both the ring and the disk to the sphere. The Hall probe circuit board is mounted to 0.5 inch plastic standoffs which are mounted to the disk using brass screws, on the sphere side of the disk. Two BNC feed-throughs pass through the disk, from outside the disk-cylinder to within. To these two BNC connectors are connected coaxial cables that supply the 5 VDC to the probe and

15One Technology Way, P.O. Box 9106, Norwood MA 02062, U.S.A. 41

return the output voltage of the circuit board to the data acquisition system. Some of these cables can be seen in Figure 2.6. The 5 VDC that is sent down the coaxial cable is single ended, meaning that the outer sheath of the coaxial cable is floating. This is done to prevent a ground loop from developing between the ground of the 5 V power supply and the ground of the probe circuit board. The ground connection for these two signals is made at the data acquisition system. The probes that are placed on the surface of the sphere are bench calibrated before mounting. A 10.5 inch long, 2.5 inch diameter solenoid, consisting of an acrylic tube wrapped in three layers of fine magnetic wire, was constructed for this purpose. The probe being calibrated is mounted on its stainless steel disk and ring, and a stainless steel bolt fastens the probe fixture to a plastic cylinder. This plastic cylinder slides into the calibration solenoid, resting such that the probe is at the centre of the solenoid, both radially and lengthwise. The probe is mounted to the cylinder such that it measures the z-component of the solenoid’s magnetic field. The probe is powered with the requisite 5 VDC, and the output voltage of the probe is charted as a function of both positive and negative solenoid current. A Gaussmeter is then given the same treatment, with the Gaussmeter’s probe being mounted at the center of the solenoid. Magnetic field generated at the center of the solenoid, as a function of solenoid current, is then charted. The combination of these two relations gives the output voltage of the probe as a function of measured magnetic field. The output voltage of the probes is sensitive to the 5 Volt input voltage used to power the probes. Unfortunately, the regulated 5 Volt power supply used to power the probes can drift in voltage. To address this issue, the voltage of the power supply is noted during probe calibration and adjustments are made to the measured voltage values, to account for any change in input voltage, before they are converted to magnetic field values. The adjustments that need to be made to the data are small, and the details of how this is calculated are found in Appendix A. Sixty-six probes are positioned on a grid on the surface of the sphere (6 positions in θ, 11 positions in φ), mounted as described above. (some of these probes can be seen mounted to the sphere in Figure 2.8.) These probes measure the radial component of the magnetic field. A further 42

8 probes are similarly mounted to the bearing-shaft assemblies near the poles of the sphere, measuring the z component of the field (in cylindrical coordinates). These probes point predominantly in the radial direction, but do contain a small component in the θ direction. These 74 probes external to the sodium can resolve spherical harmonic components up to a polar order of ℓ = 7 and an azimuthal order of m =5. Linear arrays of probes in stainless steel tubes enter the sphere through the sphere’s access ports. The ports are terminated with Grayloc R 16 flanges. To mount the stainless steel tubes within the ports, matching Grayloc flanges were welded to female Swagelok R 17 connectors. The stainless steel tubes have had matching Swagelok connections swaged to them, allowing the stainless steel tubes to enter the sphere while sealing the tubes from the outside environment. The depths of the stainless steel tubes depend upon the location of the tube. The stainless steel tubes near the poles of the sphere only extend about 5 inches into the sodium; if they extended much further they would be dangerously close to the impeller. Those tubes away from the poles extend about 10 inches into the sodium. The short tubes near the poles are not deep enough to place a full array of probes into the sodium; only the innermost 6 probes are within the sphere. The actual locations of the internal probes can been seen in the data presented in Chapter 5. The stainless steel tubes which encase the internal sensor arrays vibrate like cantilevered reeds when the impellers are driven at rotation rates above 900 RPM. The amplitude of the vibrations increases with flow speed, so experiments are limited to the lower rotation rates to prevent damaging the tubes and risking a breach of the experiment. The linear Hall probe arrays consist of 10 single-component probes that can be oriented to measure either axial or toroidal magnetic fields. Seven lines of probes are presented in this work. Six lines of probes are at the same θ positions as the external probes. The seventh line of probes is at the equator. The 5 VDC power is brought to, and magnetic signals taken away from, the probes through two unshielded CAT-5 cables. This is not the best cable for transmitting measured signals since it is rather prone to picking up electromagnetic noise. It was used for reasons of economy.

16Grayloc is a registered trademark of Oceaneering International, Inc. 17Swagelok is a registered trademark of Swagelok Company. 43

The outer 4 probes on each line have visibly-more-noisy signals than the inner 6; this is due to the fact that the electrical ground for the line of probes runs on the CAT-5 cable that transmits the measured signals for the innermost 6 probes. The outer 4 probes have no electrical ground running through their cable. Calibration of the linear arrays of Hall probes is more difficult to accomplish in a bench situation than the surface Hall probes. Rather than calibrate these probes on the bench, these probes are calibrated in situ. The probes are mounted to their stainless steel enclosures, and fields are applied to the probes by the external magnetic field coils while there is no sodium in the sphere. Since the sphere is empty, there are no currents within the sphere and the external Hall probes measurements can be used to reconstruct the field within the sphere (this will be described in more detail in Sec- tion 4.3), and calculate the field value at the internal probe positions. This gives a relation between the output voltage of the linear Hall probes and the measured magnetic field. The voltage of the surface of the sphere is sampled at 17 locations, 16 measurement positions plus another equatorial measurement to act as a reference. The voltages are measured from stainless steel leads that have been spot-welded to various positions on the surface of the sphere. The signals from these leads pass through isolators to electrically isolate the data acquisition system from the sphere. Like the magnetic field measurements these data are sampled by 16-bit digitizers at a rate of 1 kHz per channel. No sphere voltage measurements will be presented in this document.

2.2.6 Sodium Data Runs

A typical sodium run day is a full day of work. The day begins by starting up all of the heating equipment to bring the experiment up to operating temperature. An extensive checklist is used to make sure that no steps are missed. Five hours later, after the experiment is hot, safety equipment is donned by the experimenters. This safety gear is worn by anyone who is in the laboratory while the sodium is not being held in the holding tank. The experimenters then enter the laboratory and transfer the sodium to the experiment. The coil power supply is turned on, as is the water cooling for the coils. The experimenters then leave the laboratory and go to the control room to perform experiments. 44

The procedure for taking a typical sodium dataset is as follows. First, the desired applied field is selected by setting the applied-field switchboard to the appropriate setting. This requires entering the laboratory, and so the experimenter who switches the applied field must be wearing the appropriate safety gear. The external magnetic field is then applied to the sphere, and the motors are started. All electrical heating of the sphere (flange necks, flange faces and heating tape on the sphere’s top and bottom flanges) is turned off, as the signals from the heating equipment are detectable by the Hall probes. After a brief pause to allow the fluid to come to equilibrium, data is acquired. After several minutes of acquiring data, depending on the run, the data acquisition stops. The sphere’s electrical heating is turned back on, and the motors are stopped. National Instruments Corporation’s LabVIEWTM 18 data acquisition software is used to acquire the data and control the data acquisition cards. Once the data has been saved to file by LabVIEW, much like the water data, the raw voltage data from the probes is converted to magnetic field signals and is saved to a CDF file. Other information is also saved to the CDF, including surface voltage measurements, coil current, timestamps, probe positions and directions, and other information. This CDF file is accessed to study the data. Once the day’s data has been taken, the sodium is returned to the holding tank. The sphere’s expansion manifold is heated to melt the sodium in those lines to allow that sodium to drain to the holding tank as well.

18LabVIEW is a trademark of National Instruments Corporation. 45

Chapter 3

Determination of the Mean Velocity Field

The water model of the experiment is used to determine the mean velocity field of the sodium. The mechanics of doing a water data run were described in Section 2.1.3. Careful consideration must be given to the FIND software to make sure that the data being collected is being measured correctly. There are many settings that can be specified to run FIND; a list of the important parameters and the values used for the data presented are given in Table 3.1. The most important parameters to be specified by the experimenter are the shift frequencies and the frequency ranges, the two parameters that are used by the ColorLink Plus to increase the resolution of the velocity measurement. These parameters must be specified carefully; if the data is outside the range of the measurements specified by the frequencies then aliasing of the data will occur. A detailed explanation of how the shift frequency and frequency range are chosen is given is Appendix B. Once the data are collected, a spherical harmonic expansion of the velocity field is fit to the mean values of the poloidal and toroidal time series measured at each position, subject to appropriate boundary conditions and other restrictions. This fit is the mean velocity field used with the applied magnetic field to predict the induced magnetic field due to the mean flow of sodium.

Frequency Frequency PMT Coincidence Number of Shift Range Voltage Window Data points Green Channel 5 MHz 1-10MHz 1500 Volts 100 µs 3000 Blue Channel 5 MHz 1-10MHz 1620 Volts

Table 3.1 Parameter values used with the FIND software. 46

3.1 LDV Measurements

The amount of time spent by FIND at a measurement position is determined by one of two criteria: either the specified number of data points is taken (typically 3000), or a timeout elapses. The latter rarely occurs. Once a time series has been taken for a particular measurement position, FIND saves the data in a file, moves the traverse to the next measurement position, and proceeds to take data at that position. All of the data points are taken at a particular window before moving to the next window. An important bug in the FIND software is that FIND does not stop taking data while the traverse is moving. This is not an important effect if the traverse is only moving radially, as the radial move only takes about 2 seconds. However, if the traverse is moving from one window to the next the delay can be significant. To deal with this flaw the first measurement position at each window is repeated in the traverse file, and thus repeated during the data acquisition run. The first data point is taken while the traverse is moving and is later discarded during data processing. The second measurement at that measurement position begins after the traverse has finished moving, and is the time series used for that measurement position. The water velocity field is measured at 99 positions, after redundant and inaccurate data points are removed. The locations of the measurement positions can be seen in Figure 3.1. The positions are chosen so as to sample the bulk flow. This means using measurement positions that are not too close to the impellers or the sphere’s surface. In practice this results in measurements that are not within about a centimetre of these surfaces. The maximum depth that can be measured by the LDV system is limited by the focal length of the fiberoptic probe. The mean values of the water measurement time series are used to determine the velocity field. An example of a time series of the poloidal velocity field can be seen in Figure 3.2; the toroidal component of the velocity field is given in Figure 3.3. This is not a large time series compared to the sodiumdata. Its small size is due to the fact that someof the data positions are rather difficult to measure. The deeper into the sphere the measurement position, in general, the weaker the reflected signal that reaches the probe head and thus the more difficult the measurement. Conversely, the 47

Figure 3.1 Cross section of the water model. The axis of symmetry runs horizontally, with θ =0 at the rightmost pole. The volumes occupied by the impellers are indicated with rectangles. The water model measurement positions are indicated with dots. The colours correspond to the measurements presented in Figure 3.5.

closer the measurement position is to the window the stronger the reflected signal the higher the number of data counts. The same number of counts are taken at each position, for consistency. One of the concerns that has been raised about the experiment is the question of axisymmetry, a fundamental assumption of the experiment. Close examination of Figure 3.2 reveals what appears to be the presence of a periodic signal in the data, though the signal overall has a well-defined mean. This periodic behaviour, however, is lacking in the toroidal component. Whether the overall velocity field is axisymmetric in a mean sense cannot be determined without measuring the velocity field at multiple positions in φ simultaneously. This is not possible with the current configuration of the experiment. Nonetheless, as will be discussed in Chapter 6, the mean measured induced magnetic field is predominantly axisymmetric, indicating that the mean velocity field is also axisymmetric.

3.2 Fitting the Velocity Field

The velocity field of the water model can only be measured at discrete points. To calculate the magnetic field induced by the flow, however, requires knowing the velocity field everywhere 48

Poloidal Speed [m/s] -2

-4 0 5 10 15 20 Time [s]

Figure 3.2 Time series of a water data measurement for vθ.

Toroidal Speed [m/s] 0

-2 0 5 10 15 20 Time [s]

Figure 3.3 Time series of a water data measurement for vφ. 49

within the sphere. To accomplish this a spherical harmonic expansion of the velocity ﬁeld is ﬁt to the measured data, subject to a variety of constraints.

3.2.1 Spherical Harmonic Expansion

The geometry of the system is spherical, and thus it is natural to express the velocity field in terms of a spherical harmonic expansion. The first step in this expansion is to break the velocity field into two pieces: v = t + s, (3.1)

where t is the toroidal component of the ﬁeld, and s is the poloidal component. The velocity ﬁeld is assumed to be incompressible, v =0, (3.2) ∇· and as such these components can be expanded using the formalism of Bullard and Gellman [9],

t = tα = [tα(r)Yα(θ,φ)rˆ] , (3.3) α α ∇ × X X s = sα = [sα(r)Yα(θ,φ)rˆ] , (3.4) α α ∇×∇× X X

where the φ dependence of the spherical harmonics is expressed as either sin(mαφ) or cos(mαφ),

and the sums over α are sums over all valid combinations of ℓα, mα, sine and cosine, for ℓα > 0 and 0 m ℓ . Negative values of m are not used since they result in degeneracies in ≤ α ≤ α α the spherical harmonic basis set when the spherical harmonics are real, as they are in this case.

(Because axisymmetry is assumed, the summation over all α is actually just a summation over ℓα, but the full expansion is listed here for completeness.) The spherical harmonics are normalized,

sin(mαφ) 1 2ℓα +1 (ℓα mα)! mα Yα(θ,φ)= − P (cos θ) , (3.5) 2 (ℓ + m )! ℓα   (1 + δmα,0)π s α α cos(m φ)  α  p where mα is the associated Legendre polynomial and is the Kronecker delta function. The Pℓα (x) δa,b  

factor of 1/ (1 + δmα,0)π is due to the normalization of the φ part of the spherical harmonic. This part usuallyp takes the form 1/√2π when the spherical harmonic is fully complex, but when the φ 50

part of the spherical harmonic is expressed in terms of sines and cosines the normalization takes the form speciﬁed.

The tα and sα satisfy the following orthogonality relations:

2π π t t sin θ dθdφ = N δ t (r)t (r), (3.6) α · β α α,β α β Z0 Z0 2π π p s (r)s (r) ∂s (r) ∂s (r) s s sin θ dθdφ = N δ α α β + α β , (3.7) α · β α α,β r2 ∂r ∂r Z0 Z0 2π π t s sin θ dθdφ =0, (3.8) α · β Z0 Z0

where Nα = pα = ℓα(ℓα + 1). This terminology is obviously redundant, since Nα = pα, but is included for consistency with other works, since other authors [9, 54, 63] use different spherical harmonic normalizations, resulting in N = p . α 6 α 3.2.2 Velocity Field Fits

Inserting equations 3.3 and 3.4 into equation 3.1, the expansions give

ℓ (ℓ + 1) v = α α s (r)Y (θ,φ), (3.9) r r2 α α α X 1 ∂s (r) ∂Y (θ,φ) v = α α , (3.10) θ r ∂r ∂θ α X t (r) ∂Y (θ,φ) v = α α , (3.11) φ − r ∂θ α X where axisymmetry has been assumed. Reconstruction of the velocity field from the mean measured values requires fitting the radial profiles, sα(r) and tα(r), to the values of vθ and vφ measured on the water model, via equations 3.10 and 3.11 [20]. The radial profiles are modelled using splines, subject to the single boundary condition vr(a)=0 (the fluid is contained within

the sphere), where a is the radius of the sphere, meaning sα(a)=0. The no-slip condition is not adopted, as the boundary layers in the system are quite thin, due to the system’s large Reynolds number, Re = av /ν 105. The toroidal radial proﬁles also require an outer boundary condition. 0 ∼ 51

2 ∂ tα(r) The natural boundary condition for splines is used: ∂r2 = 0. There is no physical meaning r=a associated with this condition; it is selected merely because some boundary condition is required.

Near the centre of the sphere, the radial profiles must go to zero to prevent the magnitude of the velocity field from going to infinity. As can be seen from equations 3.9 and 3.11, the minimum radial dependence needed to prevent the velocity field from diverging is s (r) r2 and α ∼ t (r) r. However, regularity at the origin is also required, meaning the velocity field is required α ∼ to not suddenly change direction as the flow crosses the origin. The radial dependence required to enforce regularity is s (r) rℓα+1 and t (r) rℓα+1 [63, 16, 17]. This radial dependence is α ∼ α ∼ forced upon the fit radial profiles near the origin. To model the velocity field in an area that cannot be measured, the toroidal velocity within the regions occupied by the impellers is specified at nine positions, in a three-by-three grid. The outermost points are on the outer impeller surface and the innermost points are one third of the way from the axis of rotation, in a cylindrical sense. The layout of the data positions for an example impeller is given in Figure 3.4. Along the length of the impeller the data points are positioned at the centre of the impeller and one inch on either side of the centre. The toroidal speed assigned to

these positions is given by vφ = ωρ, where ω is a rotation rate, and ρ is the distance from the axis of rotation to the data position. The toroidal speed is taken to be same for all positions of identical ρ (a cylindrical rigid rotor). Two such sets of model data are added to the toroidal data being fit, one model in each hemisphere, with positive and negative values of ω to model the impellers’ counter- rotation. The value of ω used depends on the data being fit, of course. The ω which gives a toroidal speed consistent with those measured within the experiment is typically much lower ( 30%) than ∼ the rotation rate of the impellers. This, again, can be attributed to the thin boundary layer near the edge of the impeller. It is desirable to have an analogous model for the poloidal speed within the impeller region. Unfortunately it is not clear how to implement such a model, or what model would be physically reasonable. As a result, only the toroidal velocity is modelled in the impeller region. As stated above, the poloidal and toroidal radial profiles are modelled using splines. The first free spline point (the first spline point that can be manipulated by the fitting routine) is located 52

Impeller

Model data positions axis Figure 3.4 Schematic of half of a cross section of an impeller, showing the axis of rotation and the toroidal impeller model data positions. The impeller is assumed to be axisymmetric. The impeller’s location within the sphere is given in Figure 3.1.

at the radial location of the innermost data point plus 0.1 (recall that the radius of the sphere is a =0.533 m). The value of 0.1 is arbitrary; its purpose is to offset the innermost spline point from the innermost data point so that the spline point is in an area where there is data to constrain it, rather than an area where there is not. The rest of the free spline points are equally spaced out to either the outer edge in the case of toroidal proﬁles, or to one position back from the outer edge, in the case of the poloidal proﬁles. The last poloidal spline point is not free; it is set to zero to

force the outer boundary condition, sα(a)=0. The number of spline points used for the proﬁles is variable and is set by the researcher.

Research Systems, Incorporated’s1 IDL R 2 software is used to do the ﬁtting of the radial pro-

files to the measured data, as well as most of the data analysis presented in this work. IDL is a flexible interactive data analysis and visualization software. The program used to fit the radial profile to the data is a modification of IDL’s ‘curvefit’ program, which in turn is based upon the algorithm suggested by Marquardt for performing least-squares fitting of nonlinear parameters [43]. The choice of which spherical harmonic components to use in the fit to the data is an important one. The dominant components to the flow, since the flow is based on the DJ velocity field, are the

ℓα =2, mα =0 (axisymmetric) components. But due to the fact that the impellers are essentially cylindrical objects that have been forced into a spherical geometry, a whole spectrum of velocity ﬁeld components are expected. The toroidal velocity ﬁeld is well-described by even-numbered

1Research Systems, Incorporated is a wholly owned subsidiary of ITT Industries. 2IDL is a registered trademark of Research Systems, Incorporated. 53

flow components. Adding odd-numbered flow components to the fit does not reduce the χ2 of the fit to the data. With five windows the data is capable of resolving five different spherical harmonic components. Unlike the fit to the toroidal velocity measurements, the poloidal velocity fit requires an odd- numbered flow component to adequately model the data. In particular, an ℓα = 1 component is needed. The source of this component is not well-understood. It could be due to geometric imbalances within the sphere. It could be more simply due to instabilities in the shear layer at the equator, resulting in sloshing of the flow from one hemisphere to the other (some evidence of this can be seen in the poloidal LDV data presented in Figure 3.2). Nonetheless, the velocity field is well-described by relatively few spherical harmonic components, and the magnitude of the ℓα =1 component is small compared to the other flow components. One standard deviation of the time series of the poloidal and toroidal water data are used to scale the data used in the fit, thus giving relative weight to the data based on fluctuation level. The actual uncertainty in the data, of course, is far smaller than one standard deviation, since the standard error is the standard deviation divided by the square root of the number of points. A fit to the mean values of LDV time series, for the impellers rotating at 1000 RPM, with the toroidal fitting parameter 2πω/60 = 330 RPM, is given in Figure 3.5, with the fit radial profiles and resulting flow presented in Figure 3.6. The fit to the data is good, though it is again worth noting that the maximum speed of the data, in the toroidal direction, is significantly weaker than the tip speed of the impeller, which is about 16 m/s. The impeller is not particularly efficient at generating bulk toroidal flow; most of the toroidal flow is in the immediate vicinity of the impeller. The fit toroidal flow is described by even-numbered spherical harmonic components, resulting in two hemispherical cells which flow in opposing directions, as expected. The very rapid drop in the toroidal speed with increasing distance from the impeller, as seen in Figure 3.6, is in contrast to the DJ velocity field where the peak toroidal speed is found near the center of the main poloidal cell. The fit flow is consistent with simulations which model this experiment with much higher Prandtl numbers [6]. As expected, the fit poloidal flow is characterized by a large vortex in each 54

hemisphere which rolls inward at the equator and outward at the poles. As can be seen in Fig- ure 3.6, there is some asymmetry in the fit poloidal flow across the equator, caused by the presence of the s1 spherical harmonic component in the fit, though this component’s contribution is small.

The velocity field in equation 1.3 has been normalized to some v0, and the derivatives used to calculate its sα(r) and tα(r) radial profiles normalized to a. The radial profiles just calculated, however, were found using real data and thus have units. To use the velocity field radial profiles in equation 1.3 the profiles must be converted to unitless form. This is accomplished using the conversion ′ 2 sα(r) = sα(r)/a , ∂s′ (r) 1 ∂s (r) α = α , (3.12) ∂r a ∂r ′ tα(r) = tα(r)/a, where the primed quantities have units of m/s. The profiles are scaled by v0 to remove these units. The opposite transformation is used to convert the unitless induced magnetic field radial profiles to dimensional form, which can then be used to calculate actual magnetic field values. 55

Poloidal Toroidal 5

-5 θ = 0.60 θ = 0.60 5

-5 θ = 1.05 θ = 1.05 5

Velocity [m/s] -5 θ = 1.50 θ = 1.50 5

-5 θ = 1.95 θ = 1.95 5

-5 θ = 2.41 θ = 2.41 0.20 0.30 0.40 0.50 0.20 0.30 0.40 0.50 Radial Position [m] Radial Position [m]

Figure 3.5 Velocity field fit to water data, for the impellers rotating at 1000 RPM. The fit is the solid line. The colours correspond to the colours of the data point positions given in Figure 3.6. The error bars represent fluctuation levels and are one standard deviation of the data. The actual uncertainties in the data are smaller than the data point diamonds. 56

v pol

vφ

-4.9 -2.5 0.0 2.5 4.9 Speed [m/s]

T2 T4 0.5 T6 T8 Toroidal 0.0 0.10 S1 S2 S4 0.05 Poloidal 0.00 0.1 0.2 0.3 0.4 0.5 Radial Position [m]

Figure 3.6 Top: Velocity field fit to the water data presented in Figure 3.5, with the poloidal flow in the upper hemisphere and the toroidal in the lower. The axis of symmetry runs horizontally. The measurement positions are indicated with coloured dots, and the regions occupied by the impellers indicated with rectangles. The arrows indicate the direction of the poloidal flow. Bottom: Radial profiles which fit the data presented in Figure 3.5. 57

Chapter 4

Predicting the Induced Field

The necessity of knowing the mean velocity field of the sodium is driven by the need to know what parts of the measured induced magnetic field are due to the mean velocity field, and what parts are due to fluctuations. Once the mean velocity field has been fit to the mean LDV data, as described in Chapter 3, the magnetic field induced by that flow, given the applied field, can be calculated. This chapter presents the format by which this problem is constructed, and develops the algorithm which calculates the answer. It also describes the manner by which the applied field is represented and computed. The predicted induced field for the velocity field found in Chapter 3 is also presented.

4.1 Magnetic Induction Equation

Consider a spherical container, of radius a, that contains an electrically-conducting fluid. The velocity field of the fluid, v, is assumed to be constant in time and axisymmetric. The problem is started with the dedimensionalized magnetic induction equation, equation 1.3, with time scaled to the resistive timescale (τ = µ σa2 3 seconds in the Madison Dynamo Experiment), σ 0 ∼ ∂B = Rm (v B)+ 2B. (1.3) ∂t ∇ × × ∇

It is assumed that the fluid has the magnetic permeability of the vacuum (µ = µ0), an assumption that is approximately true for sodium [58]. The expansion of the magnetic field in terms of spherical harmonics follows the analogous expansion of the velocity field in Section 3.2.1. The magnetic field is first broken into two pieces:

B = T + S, (4.1) 58

where T is the toroidal component of the field, and S is the poloidal component. The magnetic field is solenoidal, B = 0, and as such can be expanded in spherical harmonics in the same ∇· manner as the velocity field:

T = T = [T (r)Y (θ,φ)rˆ] , (4.2) β ∇ × β β β β X X S = S = [S (r)Y (θ,φ)rˆ] , (4.3) β ∇×∇× β β β β X X where the spherical harmonics are those defined by equation 3.5. Expanding the velocity and magnetic fields in this manner reduces the expression of the vector fields to two sets of radial scalar profiles, s (r), t (r) and S (r), T (r) . Expressed explicitly in terms of these profiles, { α α } { β β } B becomes

ℓ (ℓ + 1) T (r) ∂Y (θ,φ) 1 ∂S (r) ∂Y (θ,φ) B = rˆ β β S (r)Y (θ,φ) + θˆ β β + β β r2 β β r sin θ ∂φ r ∂r ∂θ β X T (r) ∂Y (θ,φ) 1 ∂S (r) ∂Y (θ,φ) + φˆ − β β + β β . (4.4) r ∂θ r sin θ ∂r ∂φ This is the magnetic ﬁeld version of equations 3.9, 3.10, and 3.11, where the non-axisymmetric parts of the equation have been included. The expanded equations of the velocity and magnetic ﬁelds components, equations 3.3, 3.4, 4.2 and 4.3, are now inserted into the magnetic induction equation, equation 1.3. The equation is

dotted with a pseudo-vector ﬁeld whose radial proﬁle is unity everywhere, Tγ or Sγ , and integrated over the unit sphere. Using the orthogonality relationships for Tβ and Sβ (see equations 3.6, 3.7, and 3.8), the evolution equations for Tγ(r) and Sγ (r) are found (see [9] for details):

∂T Rm γ = 2 T + [(s S T )+(t S T )+(s T T )+(t T T )] (4.5) ∂t ∇ℓγ γ r2 α β γ α β γ α β γ α β γ α,β X ∂S Rm γ = 2 S + [(s S S )+(t S S ) +(s T S )+(t T S )] (4.6) ∂t ∇ℓγ γ r2 α β γ α β γ α β γ α β γ α,β X where ∂2 ℓ (ℓ + 1) 2 = γ γ (4.7) ∇ℓγ ∂r2 − r2 59

and the time derivative should not be confused with the toroidal velocity radial profile, t (r). 2 α ∇ℓγ represents the diffusive part of the equations, while the terms in () brackets represent interactions between velocity-field and magnetic-field modes, and depend upon the values of α, β, γ, the structure of the velocity and magnetic field radial profiles in question, and their derivatives. Equa-

tions 4.5 and 4.6 represent an inﬁnite set of differential equations for the evolution of Tγ (r) and

Sγ(r), which are coupled through the interaction terms given above. Evaluation of the equations requires truncating the expansion at some harmonic.

4.1.1 Interaction Terms

The interaction terms are listed below. Their derivation is long and complicated. The results from Bullard and Gellman [9] have been reproduced here with appropriate changes in normalization, but have been conﬁrmed elsewhere [63].

K ∂S ∂s (s S S )= αβγ p c s β p c α S α β γ N α α α ∂r − β β ∂r β γ L ∂2s 2 ∂s ∂s p s ∂S ∂2S (s S T )= − αβγ p α α S 2 c α + α α β + p s β α β γ N β ∂r2 − r ∂r β − γ ∂r r ∂r α α ∂r2 γ Lαβγ (sαTβSγ )= pαsαTβ Nγ K ∂s 2s ∂s ∂T (s T T )= αβγ p c α α + p c α T + p c s β (4.8) α β γ N α α ∂r − r γ γ ∂r β α α α ∂r γ Lαβγ (tαSβSγ )= pβtαSβ Nγ K ∂S ∂t 2t (t S T )= − αβγ (p c + p c )t β + p c α α S α β γ N γ γ β β α ∂r β β ∂r − r β γ (tαTβSγ )=0

Lαβγ (tαTβTγ )= pγtαTβ Nγ

where Kαβγ is the Gaunt integral [24] and Lαβγ is the Elsasser integral [19]. Use has also been made of c =(p p p )/2. α α − β − γ These interaction terms can be understood as velocity ﬁeld components generating new mag-

netic ﬁeld components through advection. The (tαSβTγ ) term, for example (the term responsible 60

for the ω-effect) can be read as: “a toroidal velocity field component, tα, interacts with a poloidal magnetic field component, Sβ, to generate a toroidal magnetic field component, Tγ.” In this sense it can be understood why the (tαTβSγ ) term is always zero—it is not possible for a toroidal velocity field component to interact with a toroidal magnetic field component to generate a poloidal magnetic field component.

4.1.2 Gaunt and Elsasser Integrals

The interaction terms each depend on either the Gaunt1

2π π

Kαβγ = Yα(θ,φ)Yβ(θ,φ)Yγ(θ,φ) sin θ dθdφ (4.9) Z0 Z0 or Elsasser

2π π ∂Y (θ,φ) ∂Y (θ,φ) ∂Y (θ,φ) ∂Y (θ,φ) L = Y (θ,φ) β γ β γ dθdφ (4.10) αβγ α ∂θ ∂φ − ∂φ ∂θ Z0 Z0 integral. These integrals are only non-zero for speciﬁc combinations of α, β and γ. These combinations form a set of selection rules, analogous to allowed state transitions in quantum mechanics, which restrict which modes interact with one another . Both the Gaunt and Elsasser integrals are non-zero only if

m m m =0 • α ± β ± γ ℓ ℓ ℓ ℓ + ℓ • | α − γ| ≤ β ≤ α γ For non-zero Gaunt integrals it is also required that

ℓ + ℓ + ℓ is even • α β γ the number of cos(mφ) harmonics is odd • The Elsasser integrals take non-zero values only if

1This integral is called the Gaunt or Adams-Gaunt integral by the dynamo community, but is called the Adams integral by others. It was initially examined by Adams [1], but merely as a mathematical curiosity. 61

ℓ + ℓ + ℓ is odd • α β γ the number of cos(mφ) harmonics is even • all the harmonics are different • Other obscure selection rules, based on symmetries which occur in the integrals with low frequency, have been noted by Gjellestad [26] and James [32]; these are not listed here. Evaluation of the interaction terms in equations 4.5 and 4.6 requires knowing the values of the Gaunt and Elsasser integrals. These integrals are computed using two steps. First, the φ part of the integral is calculated by Fourier transforming the φ-dependent part of the integrand. The θ integral is then evaluated using 2ℓmax + 1 point Gauss-Legendre integration, where ℓmax is the ℓ value of the highest order harmonic in the expansion. The computed values of the integrals have been compared to values calculated by Moon [45] and values calculated using IDL to good agreement once differences in spherical harmonic normalization were removed.

4.1.3 Magnetic Boundary Conditions

Solving equations 4.5 and 4.6 requires accounting for the magnetic ﬁeld boundary conditions. The toroidal magnetic ﬁeld boundary conditions are fairly intuitive. To prevent the θ and φ com-

ponents of equation 4.4 from diverging at the origin, Tγ (0) = 0. Because the sphere is simply- connected currents cannot be generated within the sphere which generate toroidal ﬁelds outside the sphere. As a result the toroidal magnetic ﬁeld must go to zero at the sphere edge, meaning

Tγ (a)=0. This conclusion is also demonstrated, based on a lack of toroidal currents outside the sphere, in Appendix C. The poloidal magnetic radial profiles must satisfy the boundary condition ∂S (r) ℓ S (a) γ + γ γ =0 (4.11) ∂r a r=a at the surface, which is found by matching the normal and tangential components of the internal magnetic field to the vacuum magnetic field at the sphere’s surface (see Appendix C for details). Assuming that the tangential component is continuous across the boundary implies that currents generated at the sphere’s surface exist for less than τσ. 62

4.2 Coding the Problem

The equations that describe the evolution of the magnetic field modes are continuous. The values of the function and equations are approximated by discretizing the functions and operators in the radial direction. For the work presented herein, the number of radial points is 100. This is taken to be a sufficient radial resolution, since increasing the number of radial points does not significantly change the answers found.

4.2.1 Radial Proﬁles and Operators

The magnetic and velocity radial proﬁles are discretized in the radial direction, with 0

field vector is therefore dropped, leaving the set of Sγ (1) as the last entries in the magnetic field vector. The diffusion and advection terms in equations 4.5 and 4.6 are also discretized, becoming matrix operations on the vector B. ∂B = DB + AB (4.12) ∂t where D is the diffusion term, 2 , and A is composed of the mode-interaction terms. To keep ∇ℓγ the matrices finite, the set of Sγ and Tγ is truncated to a maximum value of ℓγ = 7, the upper limit of the resolution of the external Hall probe array (see Section 2.2.5). Non-axisymmetric modes are not included in the analysis presented herein; since the system is essentially axisymmetric including higher azimuthal-order modes does not significantly affect the results. Derivatives are accomplished using second-order finite differencing. The operators D and A are banded, but not symmetric, since they relate different magnetic modes to one another. A complete description of the implementation of the discretization and construction of B, A and D can be found in Appendix D. 63

4.2.2 Calculating the Induced Field

To calculate the induced magnetic field, the total magnetic field is separated into two parts, the externally-applied field, B0, and the field induced by B0 interacting with the flow, b, such that B = B0 + b. The applied field is generated by currents external to the sphere, and as such B = 0 for r 1. The external field is not affected by the diffusion term in equation 1.3, ∇ × 0 ≤ since 2B = B =0, and thus DB =0. Inserting this expanded field into equation ∇ 0 −∇ × ∇ × 0 0 4.12 one finds ∂B ∂b 0 + = A (B + b)+ Db (4.13) ∂t ∂t 0 The time dependence of the magnetic field is now removed from the problem by assuming that both the applied and induced fields are constant in time. Rearranging the resulting equation gives

[A + D] b = AB (4.14) − 0

To isolate b, LU factorization is performed on the operator on the left side of equation 4.14, and this factorized matrix is then inverted using standard LAPACK libraries [3].

− b = [A + D] 1 AB (4.15) − 0

This gives the induced magnetic ﬁeld, expressed as [Sγ, Tγ]. To compare b to data, however, it must be converted to actual ﬁeld values,

− d = Fb = F [A + D] 1 AB (4.16) − 0

where F is an operator that maps the induced field radial profiles to field values at the probe positions, both within and without the sphere. This allows direct comparison of the calculated induced field to probe measurements.

4.2.3 Testing the Code

The stabilityof equation 4.15 is open to question. The condition number of the operator [A + D] is 104 105, depending on the speciﬁc parameters used to create it. An operator with such a ∼ − 64

Average Normalized Difference [%] 0 0 5 10 15 Noise Level [%]

Figure 4.1 Ensemble average of (b b˜)/b for noise levels of 0-15% of the elements of B , i − i i 0 versus noise level. An axisymmetric dipole ﬁeld was applied. The data are for a 0.0 Hz applied

ﬁeld. The result is approximately linear, indicating that equation 4.15 is stable to low levels of noise.

high condition number is likely to amplify high frequency noise residing in B0, resulting in an unsatisfactory result.

To test the stability of equation 4.15, normally-distributed random noise was added to B0, ˜ with standard deviations ranging from 0-15% of the elements of B0. b was then calculated, using ˜ equation 4.15, for this noisy vector B0. Various levels of noise were applied and the average of the elements of (b b˜)/b was found for each noise level, where b is the ith element of b. i − i i i

An ensemble average of 100 such calculations was performed for each noise level; the results

can be found plotted in Figure 4.1, as a function of noise level. The results presented are for an axisymmetric DC uniform applied field, on a typical flow for the Madison Dynamo Experiment. The results are approximately linear as a function of the level of applied noise, indicating that the variation in b˜ is caused by the applied noise and not by amplification by the operator. 65

This calculation was repeated for different maximum values of ℓ, different radial spacings, different applied field geometries, different values of Rm, and different measured velocity profiles. All the results have been approximately linear functions of noise level. While not definitive, this empirical experiment gives confidence that no regularization is needed for the inversion of [A + D], and that equation 4.15 is stable to low levels of noise in B0. The implementation of the discretization of the radial profiles and the operators was also bench- marked against other codes. In particular, the discretized version of equation 1.4 was calculated,

λB = [A + D] B, (4.17) and used to determine the magnetic field growth rates for the flow. The first test was the free decay case, Rm = 0 (A = 0), which has the well-known growth rate of π2 [44]. The code agrees − with this value, as can be seen in Figure 1.4. The code has also been tested against published results [28, 16] to good agreement: Figure 1.4 agrees with the results published by Dudley and James for the t2s2 flow.

4.3 Determining the Applied Field

The applied fields, B0, are those fields which can be produced by the two pairs of external field coils surrounding the Madison Dynamo Experiment’s main experimental vessel. As discussed in

Section 2.2.5, the coils are capable of producing ﬁelds which are dominated by the dipole (ℓβ =1, mβ = 0), quadrupole (ℓβ = 2, mβ = 0), “transverse dipole” (ℓβ = 1, mβ = 1, cosine) and

“transverse quadrupole” (ℓβ =2, mβ =2, cosine) field components. Because the sphere is simply- connected, only poloidal fields can be applied, or measured, from outside the sphere. Probes must be placed within the sphere to measured the sphere’s toroidal magnetic field.

To construct an applied field vector, B0, the magnetic field must be known throughout the volume of the sphere. Because the Hall probes on the sphere’s surface measure the radial component of the magnetic field the magnetic field within the sphere can be reconstructed completely when the sphere is empty. A DC field is thus applied to the sphere and measured at the external probe 66

positions; the background ﬁeld is subtracted from these measurements, giving a vector of mea- ∗ sured ﬁeld values, [B ]i = Br(ri, θi,φi). Since the sphere is empty there are no currents therein and 2Φ =0 for r a, where Φ is the magnetic scalar potential (B = Φ ). In a spherical ∇ m ≤ m −∇ m geometry the solution for Φm is well known (see, for example, Jackson [31]) ∞

ℓβ −(ℓβ+1) Φm(r,θ,φ)= Cβr + Dβr Yβ(θ,φ). (4.18) β X For the region including the origin Dβ = 0 (this is the opposite of the treatment used in Ap- pendix C, for the case of currents within the sphere). This gives the expansion for the radial magnetic field: − B (r,θ,φ)= C ℓ rℓβ 1Y (θ,φ). (4.19) r − β β β β X C B∗ EC C To find the expansion coefficients, [ ]j = Cβj , the matrix equation = is solved for , ℓ −1 where [E] = ℓ r βj Y (θ ,φ ) is the design matrix for the system. E is inverted using a ij − βj i βj i i truncated Singular Value expansion [56]. Once the coefficients are known they are used to find

Sβ(r), for all r, through the radial component of equation 4.4. Again, since only poloidal ﬁelds can be applied from outside the sphere, Tβ(r)=0 for B0.

The applied magnetic field radial profiles, Sβ(r) and Tβ(r), are based upon data, and thus are dimensional. Like the velocity field radial profiles of Chapter 3, these profiles are scaled using equation 3.12 to remove the units due to non-normalized derivatives. The opposite transformation is applied to b before using it to calculate magnetic field values.

An example of the result of building the applied field vector, B0, from magnetic field measurements is given in Figure 4.2. Plotted are the streamlines of the axisymmetric dipole-dominated (uniform) applied field. The streamlines are given by

r ℓβ +1 ∂Y (θ,φ) sin θS (a) β r a (external currents) − β a ∂θ ≥   a ℓβ ∂Y (θ,φ) Ψ(r, θ)=  β (4.20)  sin θSβ(a) r a (internal currents) − r ∂θ ≥  ∂Yβ(θ,φ) sin θSβ(r) r a − ∂θ ≤  where the case r a depends upon whether the currents are external to the sphere, as in the ﬁeld ≥ coils, or internal to the sphere, as in generated by the ﬂowing sodium. 67

Figure 4.2 Contours of the stream function of the uniform (dipole-dominated) applied field, reconstructed from magnetic field measurements. The applied field is axisymmetric, and so only a cross-section of the sphere is shown. The axis of symmetry is horizontal.

4.4 Predicted Magnetic Fields

The total magnetic field predicted for the 1000 RPM mean velocity field determined in Chap- ter 3, when exposed to a uniform applied field, is given in Figure 4.3. The figure presents the contours of both the poloidal stream function and the toroidal magnetic field. The magnitude of the poloidal field is indicated by the colour in the upper half of the plot. Several features are noteworthy. First, the total poloidal magnetic field exhibits the behaviour expected, with the field lines of the applied field being drawn inward at the equator by the poloidal flow (see Figure 4.2 for an illustration of the shape of the applied field). The poloidal field is compressed at the core of the sphere and experiences significant amplification, up to several times the magnitude of the applied field. While it appears that there is no external induced poloidal field, in fact there is, however it is not visible since the plot is dominated by the applied field. The external induced field will be discussed further in Section 6.1. Since the sphere is simply-connected no toroidal magnetic field can be applied to the sphere; all toroidal field is due to the advection of the applied field by the velocity field. The induced toroidal magnetic field, like the fit to the measured toroidal field described in Figure 5.7, predicts a strong 68

B pol

Bφ

-4.9 -2.5 0.0 2.5 4.9 B/B app

Figure 4.3 Predicted total magnetic field, scaled to the uniform applied field, for the mean velocity field presented in Chapter 3. Top: Contours of the poloidal magnetic stream function. The colours indicate the magnitude of the poloidal field. Bottom: Contours of the induced toroidal magnetic field. The dots indicate the measurement positions of the probes for the toroidal measurement run. The axis of symmetry is horizontal. 69

0.0 11.7 23.3 35.0 46.6 Stuart Number [%]

Figure 4.4 Spatial distribution of the predicted Stuart number, based on the velocity ﬁeld of Fig- ure 3.6 and the magnetic ﬁeld of Figure 4.3.

ω-effect, with an induced toroidal magnetic field significantly stronger than the applied poloidal field. An important assumption in the calculation of the induced field by equation 4.15 is that the equation is linear, meaning that the applied field is sufficiently weak that the velocity field is not modified by the Lorentz force. The strength of the Lorentz force relative to the fluid’s inertial forces is characterized by the Stuart number [74] (also called the MHD interaction parameter),

2 N = σaB0 /ρv0, where B0 is a characteristic magnetic field magnitude. For flow speeds of 5m/s (as seen in Figure 3.6), and an applied field of B = 60G, the Stuart number is N 4%, which 0 ≃ means Lorentz forces acting on the fluid should be weak relative to inertia. The magnetic field should be passively advected by the flow. However, since the velocity and total magnetic fields are known everywhere, the spatial distribution of the Stuart number can be calculated to determine a posteriori if the Lorentz force is important. Figure 4.4 shows the Stuart number as a function of position, where the magnitudes

of the velocity and magnetic ﬁelds at every position are used for the values of v0 and B0 in the 70

calculation of N. As can been seen, if the field in the experiment is as strong as is predicted then N 0.5 in the core. This might be a problem, since it violates the assumption that the magnetic ≃ field is weak so as to not affect the flow (the kinematic regime). Whether or not this is also the case for the experiment will be examined in Section 6.2. 71

Chapter 5

Measurements of the Induced Magnetic Field

The details of the mechanics of a sodium run were described in section 2.2.6. In short, once the sphere is full of sodium an external magnetic field is applied to the sphere and the motors are started. The ensuing magnetic fields are measured. The field isapplied to amean flow that isknown to not self-excite, as evidenced by the lack of growing magnetic fields when the flow is generated and no external fields applied. All of the data presented in this work is due to axisymmetric uniform applied fields. The magnetic fields of the experiment are measured using Hall probes, as described in Sec- tion 2.2.5. Data from the magnetic probes are sampled by 16-bit digitizers on PC-based data acquisition cards at a rate of 1 kHz per channel; the cards are software-controlled using LabVIEW. The cards are multiplexed, meaning that each card only possesses a single analog-to-digital con- verter. As a result, the 32 channels on each card are not sampled simultaneously. Rather, the channels are sampled in equally-spaced intervals within the sampling period. For example, if the sampling frequency is f, then the data is actually sampled at 32f. Channel 0 is sampled at some

time t0, channel 1 at t0 +1/(32f), channel 2 at t0 +2/(32f), et cetera. Channel 0 is then sampled

again at time t0 +1/f. This results in all channels being sampled at frequency f, with the actually sampling time of each channel being offset by 1/(32f) from the previous channel. Though the data is not strictly simultaneous it is treated as simultaneous since the offsets in time of each channel are very small relative to the resistive time of the experiment. 72

5.1 Measuring Induced Magnetic Fields

The data of interest is the magnetic field induced by the flowing sodium. If there is an applied magnetic field then the quantity measured by the probes is not the induced field but rather the total magnetic field, induced plus applied. To find the induced magnetic field, the applied field must be removed from each probe signal. To determine the relationship between the applied field current and the applied field measured by each probe, a slowly-ramping field is applied to the empty sphere, and the magnetic field is measured by each probe. The relationship between the coil current and the measured field, for each probe and each coil configuration, is then determined. Thus, measurement of the coil current, and knowing the applied field configuration, allows for the removal of the applied field from each probe signal. The experimental laboratory also contains ambient background magnetic field that must be removed from the probe signals to arrive at the field induced by the flowing sodium. This background field comes from many sources. The largest component is the Earth’s magnetic field, which passes at a 40◦ angle, as measured from vertical, through the equator of the sphere (the experiment runs east-west lengthwise). The other main sources of background magnetic field are the welds in the experiment. Though stainless steel is not a magnetic material, when pieces of the metal are welded together the location of the weld often becomes magnetic. Annealing of the object is required to remove the magnetism. The sphere was not annealed after construction since annealing an object the size of the sphere would be very expensive. Such an annealing is also unnecessary, since the sphere is not shielded from the Earth’s magnetic field, so the background field must be removed from the probe signals anyway. At the beginning of each run campaign the background magnetic field of the laboratory is measured by the probes. The mean value of the background field measurements taken by each probe is subtracted from the signals later measured. The background magnetic field, like the externally-applied field, is advected by the flowing sodium. In an ideal situation there would be no background field and all magnetic fields induced by the flowing sodium would be due to the advection of the applied field (in the non-dynamo regime). This would remove any ambiguity as to the source of the field which was advected by 73

the sodium. Nonetheless, advection of the background magnetic field is not considered to be an important part of the induced field since the applied field is an order of magnitude larger than the background field. By far the largest induced fields are due to the advection of the applied field.

5.1.1 External Magnetic Fields

An example of data measured on the sodium experiment is given in Figure 5.1. The applied field is uniform (dipole-dominated), as described in Figure 4.2. The magnetic field measurements from six probes on the surface of the sphere, positionedat different values in θ, are plotted. Because the applied field is axisymmetric and the mean flow is axisymmetric the mean induced field is also axisymmetric, and so these time series are representative of all probes of the same θ values. At the beginning of the time series the sodium is not moving, and so the magnetic field measured by the probes is merely the applied field. At t = 7 s the motors are started and the flowing sodium induces its own field. This induced field can be seen in the offsets of the magnetic field signals from their Rm =0 values. The offsets have a non-zero mean and are easily detected by the probes, but have noticeably strong fluctuations. Fluctuations are particularly strong at the two probes near the equator, θ = 1.39 and θ = 1.76. The equator is a region with large amounts of shear in the flow, as the two hemispherical cells grind against each other at that location. Consequently, the magnetic field measured in this region is very turbulent. Once the induced field has been determined for each probe, the external induced magnetic field is decomposed into its spherical harmonic components. This decomposition is done analogously to the determination of the applied field vector from magnetic field measurements in Section 4.3. The difference in this case is that instead of including the origin in the calculation the field at infinity is included, resulting in Cℓ,m =0 in equation 4.18 (this is the same as the case considered in Appendix C). This decomposition is done for every time step of the data, giving each magnetic mode a time series. Time series of the induced magnetic field modes, for a uniform applied field, are presented in Figure 5.2. This time series is the mode decomposition of the data presented in Figure 5.1 with the applied and background fields removed. At the beginning of the time series the impellers are 74

150

tip 100

Rm 50 Motor 1 0 Motor 2 60 θ = 0.65 θ = 1.02 40

20 θ = 1.39 0 θ = 1.76 Field [G] -20 θ = 2.12 -40 θ = 2.49 -60

0 10 20 30 Time [s]

Figure 5.1 Time series of radial magnetic fields measured on the surface of the sphere, at six positions in θ and a single value in φ, for a uniform applied field. When the time trace starts the sodium is stationary, revealing the magnitude of the applied field. At t = 7 s the motors ramp to 1300 RPM (Rmtip = 130). The induced field measurements have strong broad-band fluctuations indicative of turbulence. 75

0 Field [G]

-20

Applied (3,0) -40 (1,0) (4,0) (2,0) (5,0) 0 10 20 30 Time [s]

Figure 5.2 Time series of external induced magnetic field modes, as reconstructed from external magnetic field measurements, of which the time series in Figure 5.1 are a part. Because these are induced modes the various time series are zero until the motors are turned on at t = 7 s. The spherical harmonic (ℓ, m) indices are in the lower left corner. The applied field is constant throughout the plot. 76

not rotating, and so the induced field is zero. Once the impellers begin turning a whole spectrum of modes develop; the most significant of these are plotted. The modes fluctuate much like the measured probe signals and, like the probe signals, have well-defined means. The plotted modes are axisymmetric. The non-axisymmetric induced modes fluctuate wildly and have close-to-zero means. This can be seen in Table 5.1, wherein is presented some of the characteristics of the mode time series. The energy of a given field mode is calculated by evaluating 2µ E = B 2 d3x, 0 β | β| where Bβ is the magnetic field due to a specific mode, R

2 (ℓβ + 1)D β r a 2µ a2ℓβ +1 ≥  0 Eβ =  a a (5.1)  2 2 2 2 ℓ (ℓβ + 1) S (r) ℓ (ℓ + 1) ∂S (r)  β β β β β 2 2 dr + 2 + Tβ(r) dr, r a 2µ0 r 2µ0 " ∂r # ≤  Z0 Z0   where Dβis the expansion coefficient of the magnetic field, as described in equation 4.18, Sβ(r) and Tβ(r) are the magnetic radial profiles of equations 4.2 and 4.3, and the internal magnetic field has been assumed axisymmetric. The integrals are over the appropriate volumes, meaning 0 r a for r a and a r for r a. It is encouraging to note that the energy of ≤ ≤ ≤ ≤ ≤ ∞ ≥ the measured external field peaks at ℓβ =5 and then rapidly drops off, indicating that the external modes with the most energy are being resolved. Of particular importance is the presence of an induced dipole moment in Figure 5.2. This mode cannot be generated by the mean flow of sodium and must be due to the action of a turbulent electromotive force [67]. This component to the induced magnetic field will be discussed in greater detail in Chapter 6.

5.1.2 Internal Magnetic Fields

The internal probes measure the magnetic fields within the sphere. Unfortunately, like the Hall probes outside the sphere, these probes are only capable of measuring a single magnetic field component. They are oriented in either the θ or φ direction. The locations of the probes used to measure the toroidal component of the field for some runs are indicated in Figure 5.3. The depth of 77 the probes is limited by the how deep the stainless steel tubes can descend into the turbulent flow. Those tubes near the poles of the sphere cannot enter very far, lest they contact the impellers. An example of a set of toroidal field measurements is presented in Figure 5.4. The signals from the deepest of each of the six lines of probes shown in Figure 5.3 are presented. These data are from the same dataset as the data shown in Figure 5.4 (uniform applied field, motors rotating at 1300 RPM). Before the motors begin rotating the probes detect very little magnetic field, even though a field is being applied (see Figure 5.2). This is expected since the probes are oriented in the φ direction, the direction orthogonal to the applied field. When the motors are started the toroidal motion of the sodium sweeps the applied magnetic field into the toroidal direction, generating toroidal magnetic field through the ω-effect. The rotation of the sodium is very efficient at generating toroidal magnetic field from poloidal field. As can be seen, the toroidal field has a very large mean, easily larger than the applied field in some spots. The mean induced field is strongest for the probes nearest the poles of the sphere and

Harmonic (ℓ, m) Energy Br,max RMS Fluctuation 1, 0 (dipole) 309.0 mJ 15.6G 15.8G 2, 0 7.1 3.7 5.7 3, 0 31.6 10.8 11.2 4, 0 15.1 9.5 10.2 5, 0 56.5 22.2 22.5 6, 0 0.8 3.1 6.3 7, 0 0.1 1.4 4.1 1, 1 0.3 1.0 7.8 2, 1 0.1 0.8 3.4

Table 5.1 Mean energy in the largest measured induced external poloidal harmonics, maximum mean radial field on the sphere’s surface, and field fluctuation level for several spherical harmonic components, for Rmtip = 130 and an applied field of 47G. 78

Figure 5.3 Cross section of half of the sodium apparatus, with toroidal field measurement locations indicated with dots. The axis of symmetry is horizontal, with θ = 0 at the rightmost pole. The volumes occupied by the impellers are indicated with rectangles. The colours correspond to θ positions of the measurements presented in Figure 5.4. weakest near the equator. Like the sodium data presented in Figure 5.1, the fluctuations are largest at the equator, with root-mean-square values several times the mean value of the magnetic field.

5.2 Fitting Internal Magnetic Fields

The fitting of the internal magnetic field to the internal magnetic field measurements is very similar to the fitting of the velocity field to the velocity field measurements as presented in Sec- tion 3.2. Like the velocity field, the internal magnetic field is assumed to be axisymmetric since the velocity field is axisymmetric and the applied field is axisymmetric. The magnetic field within the sphere can be reconstructed from magnetic field measurements if there is high enough spatial resolution to do so. With six or seven lines of probes, six or seven spherical harmonic components can be resolved. The equations used to fit the magnetic field to the magnetic field measurements are two of the axisymmetric components of equation 4.4: 1 ∂S (r) ∂Y (θ,φ) B = β β , (5.2) θ r ∂r ∂θ β X T (r) ∂Y (θ,φ) B = β β . (5.3) φ − r ∂θ β X 79

100 HP140 0 -100 r = 0.43 θ = 0.65 100 HP130 0 -100 r = 0.31 θ = 1.02 100 HP150 0 -100 r = 0.28 Field [G] θ = 1.39 100 HP160 0 -100 r = 0.28 θ = 1.76 100 HP110 0 -100 r = 0.31 θ = 2.12 100 HP100 0 -100 r = 0.43 θ = 2.49 0 10 20 30 Time [s]

Figure 5.4 Time series of measured internal toroidal magnetic fields, at the deepest position of each line of probes presented in Figure 5.3. A 47G dipole field is applied, and the motors rotate at 1300 RPM when turned on at t = 7 s. Because the probes are oriented in the φ direction, they detect essentially no applied field. The field induced by the flowing sodium is large and very turbulent. 80

Like the velocity field fit, the radial profiles, Sβ(r) and Tβ(r), are modelled using splines. The values that are used to represent the magnetic field data at the probe positions are chosen carefully. At issue is the fact that the probability distribution functions for the magnetic field time series are not always Gaussian. In fact in some cases the distribution functions can have multiple peaks. Rather than using the mean, therefore, the most probable value of the time series is used for the value of the data to which the field is fit. The algorithm used to estimate the most probable value estimates the rate of an inhomogeneous Poisson process by jth waiting times [56].

5.2.1 Fitting Internal Poloidal Fields

The most significant difference between fitting the poloidal velocity field and fitting the internal poloidal magnetic field lies in the polar boundary conditions at the sphere’s surface (vr(a)=0 versus equation 4.11). The measurements of the magnetic field outside the sphere give an accurate value of Sβ(a) and its derivative, resulting in a powerful outer boundary condition for the poloidal fit. To guide the poloidal fit to a physically-reasonable answer, model data points are also added to the fit data, in this case between the outer-most radial data points and the sphere’s surface, for the five lines of probes that are fully immersed into the sodium. These model data points prevent the fitting program from finding a solution that shunts the poloidal flux to the edge of the sphere, rather than the core where the flow should be pushing it. The internal magnetic field fit to poloidal magnetic field measurements is presented in Fig- ure 5.5, for the impellers rotating at 1000 RPM with a uniform applied field. As can be seen, the internal poloidal field is dominated by a dipole induced field in opposition to the applied field. This is expected, of course, based on Figure 5.2, but the extent to which this is true is surprising. The fit is also interesting due to the presence of X-points (points of zero poloidal field) in the internal poloidal magnetic field, a feature lacking in the predicted magnetic field (see Figure 4.3). These X-points, unfortunately, are an extrapolation of the data. Direct measurement of these important features would be preferable to having the fit predict their presence; this is a goal of an upgraded set of diagnostics, which will include a line of probes that pass through the centre of the sphere. 81

The data itself, measured at seven chords of internal probes, and the resulting ﬁt, are presented in Figure 5.6.

5.2.2 Fitting Internal Toroidal Fields

Unlike the fitting of the toroidal velocity field where there is no physical toroidal boundary condition due to the high fluid Reynolds number, the toroidal magnetic field is subject to the condition that Tβ(a)=0. Other than this difference the technique for fitting the toroidal magnetic data is exactly analogous. The magnetic field fit to toroidal magnetic field data, and the corresponding radial profiles, are presented in Figure 5.7. Again, a uniform field was applied, and the impellers had a rotation rate of 1000 RPM. The fit has been scaled to the magnitude of the applied field to demonstrate the strength of the ω-effect. The fit is dominated by odd-numbered radial profiles, which is expected for this flow with a uniform applied field. Also noted is a strong peak in the induced toroidal magnetic field near the impellers. The data used for this fit, measured with the six chords of Hall probes shown in Figure 5.3, and the fits to those data, are presented in Figure 5.8. As can be seen, the fit to the toroidal field measurements is good. These data will later be compared to the magnetic field expected for the mean flow measured in Chapter 3. Having fit both the poloidal and toroidal magnetic fields to the internal magnetic field data, the strength of the magnetic field in the sphere relative to the inertial forces acting on the fluid can be determined. Figure 5.9 presents the Stuart number as a function of position, based upon the velocity field fit of Section 3.2.2 and the magnetic fields of Figures 5.5 and 5.7, plus the applied field. The Stuart number is small for most of the volume of the sphere. However, as with Figure 4.4, the core of the sphere has a field strength large enough to expect that the assumption of a velocity field unaffected by the applied field may not be valid. This will be addressed further in Section 6.1.3. 82

B pol

0.0 1.1 2.2 3.3 4.4 B/B app

10 S1 S2 5 S3 S4 S5 Poloidal 0

-5 0.1 0.2 0.3 0.4 0.5 Radial Position [m]

Figure 5.5 Top: Poloidal magnetic field fit to the internal magnetic field measurements in Fig- ure 5.6. The data has been scaled to the magnitude of the applied field. The axis of symmetry is horizontal. The measurement positions are indicated with coloured dots, and the regions occupied by the impellers indicated with rectangles. Bottom: Radial profiles which fit the data presented in Figure 5.6. 83

Poloidal 1.0 0.5 0.0 θ = 0.65 1.0 0.5 0.0 θ = 1.02 1.0 0.5 0.0 θ = 1.39 1.0

app 0.5

B/B 0.0 θ = 1.57 1.0 0.5 0.0 θ = 1.76 1.0 0.5 0.0 θ = 2.12 1.0 0.5 0.0 θ = 2.49 0.30 0.35 0.40 0.45 0.50 Radial Position [m]

Figure 5.6 Poloidal magnetic field fit to internal Hall probe measurements, for the impellers rotating at 1000 RPM, with a dipole applied field. The fit is the solid line. The error bars represent one standard deviation of the data. The actual data uncertainty is about the size of the diamonds. 84

Bφ

-2.0 -1.0 0.0 1.0 2.0 B/B app 5 0

-5 -10 T1 Toroidal T2 -15 T3 -20 T5 0.1 0.2 0.3 0.4 0.5 Radial Position [m]

Figure 5.7 Top: Toroidal magnetic field fit to the internal magnetic field measurements in Fig- ure 5.8. The data has been scaled to the magnitude of the applied field. The axis of symmetry runs horizontally. The measurement positions are indicated with coloured dots, and the regions occupied by the impellers indicated with rectangles. Bottom: Radial profiles which fit the data presented in Figure 5.8. 85

Toroidal 0.5 0.0 -0.5 -1.0 -1.5 θ = 0.65 0.5 0.0 -0.5 -1.0 -1.5 θ = 1.02 0.5 0.0 app -0.5

B/B -1.0 -1.5 θ = 1.39 0.5 0.0 -0.5 -1.0 -1.5 θ = 1.76 0.5 0.0 -0.5 -1.0 -1.5 θ = 2.12 0.5 0.0 -0.5 -1.0 -1.5 θ = 2.49 0.30 0.35 0.40 0.45 0.50 Radial Position [m]

Figure 5.8 Toroidal magnetic field fit to internal Hall probe measurements, for the impellers rotating at 1000 RPM. The fit is the solid line. The colours correspond to the colours of the data point positions given in Figure 5.3. The error bars represent one standard deviation of the data. The actual data uncertainty is about the size of the diamonds. 86

0.0 15.9 31.7 47.6 63.4 Stuart Number [%]

Figure 5.9 Spatial distribution of the measured Stuart number, based on the velocity ﬁeld of Fig- ure 3.6 and the magnetic ﬁelds of Figures 5.5 and 5.7. 87

Chapter 6

Evidence for Fluctuation-Driven Currents

The goal of this work was to determine the magnetic fields induced by fluctuations within the experiment. This goal has been accomplished by determining the magnetic field that should be induced by the mean flow of sodium when exposed to an external magnetic field and comparing that field to the field actually measured in the experiment; any discrepancies between the two must be due to fluctuations. The predicted magnetic fields were presented in Chapter 4 and the measured induced magnetic fields were presented in Chapter 5. The comparison of the two, and presentation of the evidence for fluctuation-driven currents, is the topic of this chapter. Two pieces of evidence are supplied: an induced dipole moment that cannot be generated by the mean flow, and a general weakening of the measured induced magnetic field relative to the predicted field. The magnetic field induced by the fluctuations is also presented. To facilitate the comparison of the predicted and measured induced magnetic fields, the energy content of the fields, as a function of spherical harmonic mode, is presented in Table 6.1, where the energy has been calculated using equation 5.1. The energy of the internal fields differ markedly, with two major differences. First, the predicted internal magnetic field has an order of magnitude more energy in it than the measured field. This is predominantly due to the fact that the internal induced magnetic field measured in the experiment is weak compared to the predicted field; this will be discussed in this chapter. Second, the energy in the measured magnetic field is distributed somewhat evenly between the various modes, in contrast to the predicted field where almost all of the energy is in the ℓβ = 1 and ℓβ = 3 modes. That the energy is more evenly spread could be due to odd-numbered velocity field components, or could be due to other causes, such as a fluctuation-driven EMF. 88

6.1 Poloidal Induced Fields

The magnitude of the induced external poloidal field is less than half the magnitude of the applied field. To more directly compare the predicted and measured magnetic fields induced by the flowing sodium, the applied fields are removed from total field to examine the induced fields alone. The poloidal stream lines for these two cases can be seen in Figure 6.1. It can immediately be noted that the overall geometry of the two induced fields differs significantly.

6.1.1 External Poloidal Fields

As seen in Figure 6.1, the predicted external induced magnetic ﬁeld is dominated by a hexapolar

(ℓβ = 3) structure, while the measured ﬁeld is predominantly dipolar (ℓβ = 1). This discrepancy is also seen in Table 6.1, in the energies of the external magnetic ﬁelds. The predicted external

field has most of its energy in the ℓβ =3 mode. The measured field has a richer spectrum of mode content, but most of the energy is in the dipole component. The predicted external field has no energy whatsoever in the external dipole component.

Field Energy Harmonic (ℓ, m) Internal External Measured Predicted Measured Predicted 1, 0 (dipole) 6.8 J 43.52 J 164.6 mJ 0.0 mJ 2, 0 1.0 0.07 19.6 1.7 3, 0 7.9 17.58 50.1 107.8 4, 0 4.6 0.02 8.9 1.8 5, 0 2.2 0.93 39.4 67.3 6, 0 N/A 0.00 0.6 0.9 7, 0 N/A 0.24 0.4 7.3

Table 6.1 Mean energy in the measured and predicted induced magnetic ﬁeld for several spherical harmonic components, for Rmtip = 100 and an applied ﬁeld of 47G. 89

0.0 3.3 6.6 9.8 13.1 B/B app

Figure 6.1 Contours of the poloidal stream function for the measured (top) and predicted (bottom) induced fields. The upper figure is the same as the upper part of Figure 5.5, but has been scaled to the predicted field. The colour contours indicate the magnitude of the poloidal field. The axis of symmetry is horizontal. 90

0.4

0.2

app 0.0 B/B -0.2 B -0.4 r

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Theta [radians]

Figure 6.2 Measured and predicted external radial magnetic ﬁeld values versus θ. The predicted values are indicated with the green line. The error bars represent ﬂuctuation levels and are one standard deviation of the data. Recall that the outer two θ values have four probes, while the inner six have eleven.

With such gross differences in the external magnetic structure it comes as no surprise that the predicted and measured induced external magnetic field data do not agree. Figure 6.2 presents the measured and predicted external radial magnetic field values at the probe positions. Because the measured external magnetic field is predominantly axisymmetric, the most-probable probe values are plotted versus θ. The predicted field is also axisymmetric. As can be seen, there is spread in the probe measurements of constant θ, but for the most part the measurements are well-centered about a particular value. Of note are the probe values at θ =1.75, which have rather large standard deviations. This location represents the θ value with the largest fluctuations (see Figure 5.1), resulting in probability distributions that are not Gaussian. The agreement between the predicted values and the measured values in Figure 6.2 is poor, with the overall structure of the predicted magnetic field in general disagreement with the measurements. The predominant reason for this disagreement is the lack of external dipole component in the predicted field. 91

6.1.2 Internal Poloidal Fields

Though the energies contained in the internal predicted and measured magnetic fields (see Ta- ble 6.1) are the total internal magnetic energy (poloidal plus toroidal), rather than just the poloidal, they give an indication of which internal poloidal modes may be most important. Since the dominant modes of the measured and predicted internal induced fields are very different, the internal fields of the two cases are topologically dissimilar, as seen in Figure 6.1. The predicted field is comprised of simple loops of flux, with only a single O-point per closed magnetic field line. The measured field, in contrast, has multiple O-points within the larger dipolar loops, resulting in the formation of X-points in the internal and external fields. The amplification of the poloidal magnetic field at the centre of the sphere, observed in the predicted field, is not reproduced by the fit to the internal poloidal magnetic measurements. The fit predicts an amplification of the field in the region near the impellers, but the centre of the sphere is an area of weak magnetic field since there is an X-point in that area. Like the external magnetic field measurements, the agreement between the internal measured and predicted poloidal field values at the probe positions is poor. This comparison can be seen in Figure 6.3. The values are in fairly good agreement with the probes at the equator, but everywhere else the measured values are a third to a half of the value expected. Instinctively one would expect that the internal poloidal magnetic field measurements would be larger than the predicted field, rather than smaller, since the induced external dipole moment results in a field in the negative z direction. That this is not true implies that there is some other effect at work that is causing the net field to be significantly smaller than the field expected from the mean flow. This weakening of the poloidal field is caused by the action of fluctuation-induced currents.

6.1.3 Induced Dipole Moment

One of the major symptoms of the disagreement between the measured and predicted poloidal magnetic ﬁelds is the presence of an external induced dipole moment. The induced dipole component exists for the predicted ﬁeld but is expected to go to zero at the sphere’s surface, resulting in 92

Poloidal 1.5 1.0 0.5 0.0 θ = 0.65 1.5 1.0 0.5 0.0 θ = 1.02 1.5 1.0 0.5 0.0 θ = 1.39 1.5

app 1.0 0.5 B/B 0.0 θ = 1.57 1.5 1.0 0.5 0.0 θ = 1.76 1.5 1.0 0.5 0.0 θ = 2.12 1.5 1.0 0.5 0.0 θ = 2.49 0.30 0.35 0.40 0.45 0.50 Radial Position [m]

Figure 6.3 Measured and predicted internal poloidal induced magnetic ﬁeld values. The predicted values are the solid lines. The colours correspond to the coloured dots in Figure 6.1. The error bars represent ﬂuctuation levels and are one standard deviation of the data. The actual data uncertainty is about the size of the diamonds. 93

no dipole moment. The reason the dipole moment is absent from the predicted field is because it is impossible for an axisymmetric mean velocity field and an axisymmetric applied field to induce an external dipole moment [67]. The first discovered proof of this, to the author’s knowledge, is given in Appendix E, along with another proof. The first published proof of this is given here. When both the velocity field and magnetic field are axisymmetric the only non-trivial component of the dipole moment, µ x J d3x, is oriented along the symmetry axis and results from ≡ × currents flowing in the toroidal direction,R

3 µz = sJφ d x, (6.1) Z where a cylindrical coordinate system, (s,φ,z), has been adopted and the symmetry axis is aligned with the z axis. These currents can only be generated by the v B force due to the mean ﬁelds. × B is expressed in terms of an axisymmetric ﬂux function, Ψ(s, z),

B = Ψ φ + B (s, z)φˆ. (6.2) ∇ × ∇ φ

Using Ohm’s law then gives

sJ = sσ [v ( Ψ φ)] φˆ φ × ∇ × ∇ · = σ v Ψ (v Ψ) φˆ φˆ φ∇ − · ∇ · = hσ (vΨ) , i (6.3) − ∇· where use has been made of Ψ φˆ =0, the fluid has been assumed incompressible, v =0, ∇ · ∇· and the electric field in Ohm’s law is zero since the system is axisymmetric. Inserting equation 6.3 into equation 6.1 and making use of Gauss’ theorem and v nˆ = 0, where nˆ is the unit vector · normal to the vessel’s surface, one finds that µz =0. It is interesting to notethat it is only the dipole moment that vanishes; moments which include different powers of s in equation 6.1 are non-zero in general. This conclusion is also independent of geometry; any simply-connected axisymmetric system gives the same result. The induced dipole moment certainly has large fluctuations, but has a non-zero mean (see Figure 5.2). The initial assumption upon measuring this moment was that it was a calibration 94

1.5 ] 3 1.0

0.5

0.0

-0.5

-1.0 Dipole Moment [G m -1.5 -100 -50 0 50 100 Field [G]

Figure 6.4 Induced dipole moment versus applied field, for a dipole applied field and the impellers rotating at 1000 RPM. A linear fit is added to guide the eye. The error bars represent fluctuation levels and are one standard deviation of the data. The actual data uncertainty is about the size of the diamonds.

error. Further research revealed that it depends upon the magnitude and direction of the applied field (Figure 6.4) and the rotation rate of the impellers (Figure 6.5), the latter demonstrating that the effect is not a calibration problem. An interesting observation is that the moment is always in opposition to the applied field. The fact that the induced dipole moment is linear in applied field, as seen in Figure 6.4, is comforting. As has already been noted, the core of the sphere may have a high Stuart number for the fields typically applied to the sphere, meaning that the magnetic field within the sphere may be strong enough to modify the flow. While the linearity of the induced dipole moment with respect to applied field is not a definitive test, the lack of observed saturation with applied field is consistent with the velocity field of the sodium being relatively unaffected by the applied field. The dominant external induced field is also linear in applied field, giving further support to the kinematic assumption. Since the induced dipole moment cannot be generated by the axisymmetric mean flow, the possibility of a non-axisymmetric mean flow must be addressed. The presence of the internal probes in the sphere, in particular, could break the axisymmetry of the flow. While a possibility, 95 ]

3 0.0

-0.5

-1.0 Dipole Moment [G m

0 20 40 60 80 100 120 140 Rmtip

Figure 6.5 Induced dipole moment versus impeller rotation rate, expressed as Rmtip. The moment’s rotation rate dependence begins parabolic but then becomes linear. The error bars represent fluctuation levels and are one standard deviation of the data. The actual data uncertainty is about the size of the diamonds. this is not the case. A comparison of data from experiments before the internal probe arrays were installed with data from experiments with the tubes present indicates that the disturbance in the flow due to the tubes has negligible effect on the large scale induced external magnetic field. The induced field is predominantly axisymmetric in both cases (see Table 5.1). If there were a significant non-axisymmetric mean flow then a whole spectrum of external non-axisymmetric magnetic fields should be induced, which are not observed. If the induced dipole moment is not due to the mean flow, either axisymmetric or non-axisymmetric, then it must be due to a turbulent EMF of some form. Fluctuations are inducing a large scale magnetic field in the experiment.

6.2 Toroidal Induced Fields

Unlike the poloidal fields, the toroidal magnetic field is orthogonal to the axisymmetric applied field and so all measured toroidal magnetic field is induced by the flowing sodium; no applied field needs to be removed from the measured signal to compare with the predicted field. The 96

predicted and measured toroidal magnetic fields are presented in Figure 6.6, where the toroidal fit of Section 5.2.2 has been adjusted to the scale of the predicted toroidal field. As can be seen, both the measured and predicted toroidal fields demonstrate the ω-effect, with peaks in the toroidal field in each hemisphere. However, as is clear from the figure, the measured magnetic field is significantly weaker than the field that is predicted. The predicted and measured toroidal field data are presented in Figure 6.7. These data confirm that which is presented in Figure 6.6. The induced toroidal field is much weaker than is expected from the mean flow alone. A turbulent EMF is generating currents that are weakening the induced field, a measurement consistent with simulations of the experiment [6].

6.3 Fluctuation-Induced Fields

Since the mean magnetic fields in the poloidal and toroidal direction have been determined, and the mean velocity field of the experiment is known, it is possible to determine the magnetic fields induced by fluctuations in the experiment. One might na¨ıvely think that the field due to the fluctuations could be determined by merely subtracting the predicted induced field from the measured field. However, if the goal is to determine the fields solely due to the fluctuation-driven EMF then this is not the case. The measured magnetic field is a combination of many fields:

Bmeasured = Bapp + Bmean flow+app + Bfluc + Bmean flow+fluc, (6.4)

where Bapp is the applied field, Bmean flow+app is the field induced by the mean flow interacting with

the applied field and itself, Bfluc is the field due to the fluctuation-driven EMF, and Bmean flow+fluc is the field induced by the mean flow interacting with the field due to the fluctuation-driven EMF and itself. Subtracting the predicted induced field from the measured field (which has already had its applied field removed) results in the quantity Bfluc + Bmean flow+fluc. This is not the quantity desired; the effect of the mean flow advecting the field due to the fluctuation-driven EMF must be accounted for. 97

-4.9 -2.4 0.0 2.4 4.9 B/B app

Figure 6.6 Measured (top) and predicted (bottom) toroidal magnetic fields. The measured fit is the fit presented in Figure 5.7, but has been adjusted to the scale of the predicted field. 98

Toroidal 0

-2

-4 θ = 0.65 0

-2

-4 θ = 1.02 0

app -2 B/B -4 θ = 1.39 0

-2

-4 θ = 1.76 0

-2

-4 θ = 2.12 0

-2

-4 θ = 2.49 0.30 0.35 0.40 0.45 0.50 Radial Position [m]

Figure 6.7 Measured and predicted internal toroidal induced magnetic ﬁeld values. The predicted values are the solid lines. The colours correspond to the coloured dots in Figure 6.6. The error bars represent ﬂuctuation levels and are one standard deviation of the data. The actual data uncertainty is about the size of the diamonds. The axis of symmetry is horizontal. 99

This is accomplished by reconsidering Ohm’s law,

J = σ (E + v B) . (1.2) × By using the measured mean velocity and magnetic fields the currents generated by the mean velocity field can be determined. The magnetic field resulting from these currents, which is equivalent to Bmean flow+app + Bmean flow+fluc can then be calculated. Subtracting this quantity from the measured field gives Bfluc. The calculation of J due to the action of the mean flow is accomplished in two parts. The easier of the two is the calculation of Jφ, since Eφ =0 in axisymmetric systems. This gives

J = σ (v B) . (6.5) φ × φ Calculation of the poloidal component of the current is a little trickier, since in general the poloidal component of the electric ﬁeld is not zero in axisymmetric systems. This problem is solved by invoking the MHD hypothesis, J = 0, which essentially says that currents prevent charges ∇· from building up in any one location, a claim made clearer by stating the continuity equation:

∂ρ + J =0, (6.6) ∂t ∇· where ρ is the charge density. Applying this assumption to Ohm’s law gives

J = σ (E + v B) ∇· ∇· × = σE + σ (v B) ∇· ∇· × = σ 2Φ+ σ (v B) − ∇ ∇· × =0, where Φ is the electric potential. To calculate the poloidal component of the electric ﬁeld the potential is found using − Φ= 2 1 (v B) , (6.7) ∇ ∇· × subject to boundary conditions at the centre and edge of the sphere. At the centre the condition Φ(r = 0) = 0 is used. At the outer boundary a more complicated boundary condition, based 100

upon the value of v B at the outer edge, is used. Details of this boundary condition are given in × Appendix F. The poloidal component of the current is then given by

− 2 1 Jpol = σ 1 (v B) (6.8) − ∇ ∇ ∇· × pol nh i o The magnetic fields which result from the poloidal and toroidal currents are then subtracted from the measured fields to get Bfluc. The poloidal and toroidal fields induced by the EMF are presented in Figure 6.8. The poloidal field is dominated by a dipolar component which, as mentioned previously, the mean flow is incapable of inducing. The toroidal magnetic field, as expected, is in opposition to the field induced by the mean flow in the areas near the poles and in the core near the impellers. This helps explain the general weakening of the measured toroidal magnetic field. Also worthy of attention is the magnitude of the fields. Relative to the fields expected from the mean flow, Figure 4.3, the fields due to the EMF are about the same magnitude. The field induced by the EMF is not small. 101

B pol

Bφ

-5.0 -2.5 0.0 2.5 5.0 B/B app

Figure 6.8 Poloidal (top) and toroidal (bottom) magnetic ﬁelds induced by the turbulent EMF. 102

Chapter 7

Discussion and Summary

Comparing the mean measured induced magnetic field to the field predicted from the mean velocity field as measured in the water model indicates that the two fields disagree in important ways. That the mean velocity field does not satisfactorily explain the induced fields implies the existence of a fluctuation-driven EMF acting in the sodium. There are two main symptoms which motivate this conclusion. First, there is an induced external dipole moment which the axisymmetric mean flow is not capable of inducing when exposed to an axisymmetric applied field. Since the moment is not induced by the mean flow it must be due to fluctuations. Second, there is a general weakness to the induced magnetic fields within the sphere; both the poloidal and toroidal fields have significantly lower magnitudes than the fields predicted from the mean flow. The turbulent EMF appears to result in a diamagnetic effect which resists the field induced by the mean flow. These effects have not been noted in other experiments with open flow geometries [55, 53], though these experiments lack the diagnostics to measure the effects presented herein. The open geometry of the experiment has led to fluctuations on the scale of the apparatus, which has led to a fluctuation-driven EMF generating currents on the largest scales. These fluctuation-induced fields could have significant astrophysical importance and warrant further study.

7.1 Comparison with Simulation

Adam Bayliss, a recent graduate student of the University of Wisconsin-Madison, has developed a three-dimensional self-consistent MHD simulation code that has been used to model the experiment [5]. The simulation includes a model of the impellers that generate the ﬂow, but suffers 103

from the turbulence-resolution problems that plague other simulations. Nonetheless, turbulent velocity fields are generated in the simulations, though not as turbulent as the experiment. Turbulent magnetic fields are also induced when the simulated flow is exposed to an external magnetic field. These simulated induced fields lend credence to the results presented in Chapter 6. Turbulent simulations of the experiment indicate that a dipole moment should be induced by a turbulent EMF, and that the form of the internal poloidal magnetic field induced by the EMF is qualitatively similar to that presented in Figure 6.8. The simulations also indicate that the fluctuation-driven currents result in an overall weakening of induced fields compared to laminar simulations, a result consistent with the measurements presented in this work. And again, the structure of the toroidal magnetic field induced by the EMF in the simulations is qualitatively consistent with the measurements in the experiment, as per Figure 6.8. Concern was raised in Section 5.2.2 that the Stuart number might be too high in the core of the sphere, resulting in a large Lorentz force acting on the fluid and violation of the kinematic assumption. The fact that the simulations qualitatively confirm the results of the measurements is taken as reassurance that while the kinematic assumption may be violated the turbulent EMF is still present, as the kinematic assumption is certainly not violated in the simulations [6].

7.2 Forms of the Turbulent EMF

What form or distribution of turbulence might generate such large scale magnetic fields? At this time, without direct measurements of the fluctuating components of v and B there are few reasons to include or exclude any particular cause over another; all of the terms in the mean field expansion of the turbulent EMF, equation 1.8, indicate physics that might generate the observations. Nonetheless, several potential processes merit discussion. The expansion of the turbulent EMF in terms of the mean magnetic field is recalled:

= v˜ b˜ = α B + β B + γ B + .... (1.8) E × h i ∇×h i ×h i D E A toroidal α-effect could produce large scale toroidal currents by interacting with the observed ω-effect. Helical eddies may be shed by the impellers in the toroidal direction, generating the 104 needed helicity for the α-effect. Such currents could be responsible for the induced dipole moment. The greatest weakness in this idea is the fact that the experiment is a very low magnetic Prandtl number system. The resistive dissipation scale of the experiment is about 15 cm [50]. Any eddies that might be responsible for the α-effect would need to be at least this size. A symptom of a possible α-effect is the parabolic dependence of the induced dipole moment on the impeller rotation rate, as shown in Figure 6.5. The argument that an α-effect might behave in such a way is as follows. It is first noted that B RmB , where B is the magnitude of the φ ∝ 0 0 poloidal applied field. If the toroidal currents which generate the induced dipole moment are due to an α-effect then J αB . But α v˜ ( v˜) τ v˜2τ /ℓ v˜ Rm, where τ is φ ∝ φ ∝ · ∇ × cor ∝ cor ∝ ∝ cor the correlation time of the eddy and ℓ is the eddy size. This leaves J Rm2B , the parabolic φ ∝ 0 dependence on rotation rate observed. These scaling arguments, of course, are merely qualitative, but they do help to motivate the type of fluctuations to look for in the experiment. The β-effect leads to turbulent reductions of the fluid conductivity [35]. The physics of the effect is simple: the high-conductivity small scale eddies twist magnetic fields into tight bundles, causing large gradients in the field that quickly diffuse away, resulting in a loss of magnetic field. A non-uniform β-effect could cause uneven distributions of current that generate the dipole moment when otherwise there might not be one. A uniform β-effect would cause a drop in the magnitude of the induced magnetic field, which could explain the unexpectedly low poloidal and toroidal magnetic fields. Since the turbulence levels vary throughout the sphere, getting stronger near the core as measured on the water model, it is likely that if there is a significant β-effect it is non- uniform. A third possible turbulence-induced effect is the expulsion of magnetic flux from regions of high turbulent intensity to low turbulent intensity, known as the γ-effect [35]. This effect results in diamagnetism, with γ v2. The intensity of the turbulence within the sphere varies with ∼ −∇ position, as measured in the water model; the regions near the impellers have the highest levels of turbulence with the intensity dropping outward towards the sphere edge. Given this fact, the quantity v2 is a non-zero quantity, pointing somewhat in the radial direction (in the sense of −∇ a cylindrical coordinate system). Depending on the nature of the magnetic field in the sphere this 105

effect might have the correct form to generate either the dipole moment, or magnetic fields which oppose the field induced by the mean flow. Expanding the EMF in terms of the mean magnetic field may not be appropriate, since the largest fluctuations in the magnetic field do not satisfy the scale-separation and homogeneity requirements usually imposed in the expansion of the mean-field EMF [35]. The largest turbulent magnetic fluctuations are m = 1. Their Gaussian probability distribution is centered at zero, consistent with a passively-advected magnetic field in a turbulent cascade of velocity fluctuations. These m = 1 fluctuations in B could, in principle, interact with m = 1 fluctuations in the flow and average to give a net toroidal current that would generate the anomalous dipole moment. However, m = 2 velocity fluctuations are the dominant fluctuations found in simulations of the experiment [5].

7.3 Implications for the Dynamo

The role of the turbulent EMF is either good or bad, depending on the point of view of opiner. If viewed from the paradigm of desiring a dynamo based upon the expectations of the laminar calculations, then the turbulent EMF is a detriment. The induced dipole moment interferes with the structure of the internal magnetic field, and the induced magnetic field in general is weaker than is expected. Both of these effects will likely increase the critical magnetic Reynolds number needed for self-excitation, since the velocity field will need to be stronger to induce a strong enough magnetic field to overcome diffusion. Alternatively, the turbulent EMF may be a source for very interesting physics since it experimentally demonstrates that axisymmetric dynamos may be possible with axisymmetric mean flows; the turbulent EMF allows a path by which this can occur without violating Cowling’s theorem. Access to this regime probably requires a very significant magnetic Reynolds number [6], but perhaps may be a region which can be explored. 106

7.4 Summary

A spherical liquid-sodium experiment has been constructed to explore MHD dynamo theory and turbulence. The liquid sodium is given an axisymmetric velocity field by two impellers, attached to shafts which enter the sphere through each pole. An axisymmetric magnetic field is applied to the sphere by external field coils. The magnetic fields of the system, both those applied and those induced by the flowing sodium, are measured by probes on the surface of the sphere, and probes within the volume of the sodium. To determine the role that fluctuations play in inducing magnetic fields, the induced fields due to solely the mean flow of sodium are needed. The velocity field of the sodium experiment is determined by measuring the velocity field in an identical-scale water model of the sodium apparatus. The water model is outfitted with windows which allow Laser Doppler Velocimetry to be used to measure the velocity field of the water. An expansion in spherical harmonics is then fit to these measured values, determining the velocity field of the sodium throughout the sphere. A simple finite-difference code has been developed to calculate the magnetic field that should to be induced by the mean velocity field of the sodium, as measured on the water model, given the velocity field in question and the applied magnetic field. The behaviour of the predicted induced magnetic field is consistent with the current understanding of the physics of the experiment, demonstrating compression of poloidal magnetic flux near the core of the sphere, and an induced toroidal magnetic field indicating the presence of the ω-effect. The induced external magnetic field is decomposed into its spherical harmonic components. A significant dipole component is present. This dipole component cannot be induced by an axisymmetric velocity field coaxial with an axisymmetric applied magnetic field. Proof of this is given. Since the dipole component is not induced by the mean flow it must be due to the action of fluctuations. A turbulent EMF has been demonstrated. Spherical harmonic expansions are fit to the poloidal and toroidal magnetic fields induced by the flowing sodium within the sphere. The values of the magnetic fields at the internal probe 107

positions are compared to those values calculated to be induced by the mean flow of sodium. Sig- nificant discrepancy between the two is noted. Both the poloidal and toroidal measured magnetic fields are significantly weaker than the predicted magnetic field values. While the topology of the measured toroidal magnetic is essentially the same as the predicted, just weaker, the topology of the measured poloidal field differs significantly from that which is predicted. The presence of a large scale dipole in the sphere changes the magnetic topology of the internal poloidal magnetic field, generating magnetic field O-points and X-points where they are not expected. The anomalous weakening of the both the poloidal and toroidal fields relative to that expected based on the mean flow, must be due to fluctuation-driven currents. This is the first observation of a turbulent EMF in a laboratory sodium dynamo experiment. Explicit characterization of the EMF is impossible without more detailed knowledge of the form of the turbulence and direct measurement of the fluctuating components of v and B. Future work will be directed towards identifying the characteristics of the fluctuations responsible for producing the induced dipole moment, and the weakening of the induced fields. 108

LIST OF REFERENCES

[1] J. C. Adams. On the expression of the product of any two Legendre’s coefficients by means of a series of Legendre’s coefficients. Proc. Roy. Soc. London, 27:63–71, 1878. [2] H. Alfvén. On the existence of electromagnetic-hydrodynamic waves. Ark. Mat. Astron. o. Fys., 29:1–7, 1943. [3] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Green- baum, et al. LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999. [4] J. Baumgartl, A. Hubert, and G. Müller. The use of magnetohydrodynamic effects to inves- tigate fluid flow in electrically conducting melts. Phys. Fluids A: Fluid Dyn., 5(12):3280– 3289, 1993. [5] R. A. Bayliss. Numerical simulations of current generation and dynamo excitation in a mechanically-forced, turbulent flow. PhD thesis, University of Wisconsin-Madison, 2006. [6] R. A. Bayliss, C. B. Forest, M. D. Nornberg, and P. W. Terry. Numerical simulations of magnetic field generation in a mechanically-forced spherical dynamo. Submitted to Phys. Rev. E, 2006. [7] D. V. Berkov and N. L. Gorn. Reconstruction of the velocity distribution in conducting melts from induced magnetic field measurements. Comp. Phys. Comm., 86(3):255–263, 1995. [8] Mickaël Bourgoin, Louis Marié, François Pétrélis, Cécile Gasquet, Alain Guigon, Jean- Baptiste Luciani, Marc Moulin, Frédéric Namer, et al. Magnetohydrodynamics measurements in the von Kármán sodium experiment. Phys. Fluids, 14(9):3046–3058, 2002. [9] E. C. Bullard and H. Gellman. Homogeneous dynamos and terrestrial magnetism. Philos. Trans. R. Soc. London, Ser. A, 247(928):213–278, 1954. [10] F. H. Busse. A model of the geodynamo. Geophys. J. Roy. Astro. Soc., 42:437, 1975. [11] F. H. Busse, U. Müller, R. Stieglitz, and A. Tilgner. A two-scale homogeneous dynamo, an extended analytical model and an experimental demonstration under development, in Springer Proceedings in Physics, volume 69. Springer-Verlag, New York, 1992. 109

[12] S. Childress and A. Gilbert. Stretch, Twist, Fold: The fast dynamo. Springer-Verlag Telos, Berlin, 1995.

[13] U. R. Christensen, J. Aubert, P. Cardin, E. Dormy, S. Gibbons, G. A. Glatzmaier, E. Grote, Y. Honkura, et al. A numerical dynamo benchmark. Physics of the Earth and Planetary Interiors, 128:25–34, 2001.

[14] J. G. Cook and G. Fritsch. Thermal conductivity in the liquid phase, Chapter 7.2, in Hand- book of Thermodynamic and Transport Properties of Alkali Metals, R. W. Ohse, editor. In- ternational Union of Pure and Applied Chemistry Chemical Data Series. Blackwell Science Publications, Oxford, 1985.

[15] T. G. Cowling. The magnetic ﬁeld of sunspots. Mon. Not. R. Astron. Soc., 94:39–48, Novem- ber 1933.

[16] M. L. Dudley and R. W. James. Time-dependent kinematic dynamos with stationary ﬂows. Proc. R. Soc. London, Ser. A, 425(1869):407–429, 1989.

[17] Martin Lee Dudley. A numerical study of kinematic dynamos with stationary ﬂows. PhD thesis, University of Sydney, 1988.

[18] S. Eckert and G. Gerbeth. Velocity measurements in liquid sodium by means of ultraound doppler velocimetry. Experiments in Fluids, 32:542–546, 2002.

[19] W. Elsasser. Induction effects in terrestrial magnetism Part I. theory. Phys. Rev., 69(3– 4):106–116, 1946.

[20] C. B. Forest, R. A. Bayliss, R. D. Kendrick, M. D. Nornberg, R. O’Connell, and E. J. Spence. Hydrodynamic and numerical modeling of a spherical homogeneous dynamo experiment. Magnetohydrodynamics, 38(1/2):107–120, 2002.

[21] A. Gailitis, O. Lielausis, S. Dement’ev, E. Platacis, A. Cifersons, G. Gerbeth, T. Gundrum, F. Stefani, et al. Detection of a ﬂow induced magnetic ﬁeld eigenmode in the Riga dynamo facility. Phys. Rev. Lett., 84(19):4365–4368, 2000.

[22] A. Gailitis, O. Lielausis, E. Platacis, S. Dement’ev, A. Cifersons, G. Gerbeth, T. Gundrum, F. Stefani, et al. Magnetic ﬁeld saturation in the Riga dynamo experiment. Phys. Rev. Lett., 86(14):3024–3027, 2001.

[23] A. Gailitis, O. Lielausis, E. Platacis, G. Gerbeth, and F. Stefani. Laboratory experiments on hydrodynamic dynamos. Rev. Mod. Phys., 74(4):973–990, 2002.

[24] J. A. Gaunt. The triplets of helium. Philos. Trans. R. Soc. London, Ser. A, 122(790):513–532, 1929. 110

[25] R. D. Gibson, P. H. Roberts, and S. Scott. The Bullard–Gellman dynamo, Chapter 3, pages 577–602, in The Application of Modern Physics to the Earth and Planetary Interiors, S. K. Runcorn, editor. Wiley-Interscience, New York, 1969.

[26] Guro Gjellestad. Note on the deﬁnite integral over products of three Legendre functions. Proc. Nat. Acad. Sci. USA, 41(11):954–956, 1955.

[27] G. A. Glatzmaier and P. H. Roberts. A three-dimensional convective dynamo solution with rotating and ﬁnitely conducting inner core and mantle. Physics of the Earth and Planetary Interiors, 91:63–75, 1995.

[28] D. Gubbins. Numerical solutions of the kinematic dynamo problem. Philos. Trans. R. Soc. London, Ser. A, 274(1241):493–521, 1973.

[29] G. E. Hale. On the probable existence of a magnetic ﬁeld in sun-spots. Astron. J., 28:315– 343, 1908.

[30] J. Hartmann and F. Lazarus. Hg dynamics II. experimental investigations on the ﬂow of mercury in a homogeneous magnetic ﬁeld. Kgl. Dan. Vid. Selsk. Mat.-Fys. Medd., 15(7):1– 45, 1937.

[31] John David Jackson. Classical Electrodynamics. John Wiley & Sons, Inc., New York, third edition, 1999.

[32] R. W. James. The Adams and Elsasser dynamo integrals. Philos. Trans. R. Soc. London, Ser. A, 331:469–478, 1973.

[33] A. Kageyama and T. Sato. Generation mechanism of a dipole ﬁeld by a magnetohydrodynamic dynamo. Phys. Rev. E, 55:4617–4626, 1997.

[34] M. G. Kivelson, K. K. Khurana, C. T. Russell, R. J. Walker, J. Warnecke, F. V. Coroniti, C. Polanskey, D. J. Southwood, et al. Discovery of Ganymede’s magnetic ﬁeld by the Galileo spacecraft. Nature, 384:537–541, 1996.

[35] K. Krause and K. H. R¨adler. Mean-ﬁeld Magnetohydrodynamics and Dynamo Theory. Pergammon Press, New York, 1980.

[36] Weijia Kuang and Jeremy Bloxham. A numerical model of the generation of the Earth’s magnetic ﬁeld. Nature, 389:371–375, 1997.

[37] S. Kumar and P. H. Roberts. A three-dimensional kinematic dynamo. Proc. R. Soc. London, Ser. A, 344(1637):235–258, 1975.

[38] Joseph Larmor. How could a rotating body such as the sun become a magnet? Rep. Br. Assoc. Adv. Sci., pages 159–160, 1919. 111

[39] B. Lehnert. An experiment on axisymmetric flow of liquid sodium in a magnetic field. Ark. Fys., 13(10):109–116, 1957. [40] F. E. Lilley. On kinemtic dynamos. Proc. R. Soc. London, Ser. A, 316(1525):153–167, 1970. [41] F. J. Lowes and I. Wilkinson. Geomagnetic dynamo: A laboratory model. Nature, 198(4886):1158–1160, 1963. [42] F. J. Lowes and I. Wilkinson. Geomagnetic dynamo: An improved laboratory model. Nature, 219:717–718, 1968. [43] D. W. Marquardt. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math., 11(2):431–441, 1963. [44] H. K. Moffatt. Magnetic field generation in electrically conducting fluids. Cambridge Uni- versity Press, Cambridge, England, 1978. [45] Wooil Moon. Numerical evaluation of geomagnetic dynamo integrals (Elsasser and Adams– Gaunt integrals). Comp. Phys. Comm., 16:267–271, 1979. [46] T. Namikawa and S. Matsushita. Kinematic dynamo problem. Geophys. J. Roy. Astro. Soc., 19:395–415, 1970. [47] Norman F. Ness, Mario H. Acuña, Kenneth W. Behannon, Leonard F. Burlaga, John E. P. Connerney, Ronald P. Lepping, and Fritz M. Neubauer. Magnetic fields at Uranus. Science, 233(4759):85–89, 1986. [48] Norman F. Ness, Mario H. Acuña, Leonard F. Burlaga, John E. P. Connerney, Ronald P. Lepping, and Fritz M. Neubauer. Magnetic fields at Neptune. Science, 246(4936):1473– 1478, 1989. [49] M. D. Nornberg, E. J. Spence, R. D. Kendrick, C. M. Jacobson, and C. B. Forest. Intermittent excitation of a magnetic eigenmode in a turbulent flow of liquid sodium. Submitted to Phys. Rev. Lett., 2006. [50] M. D. Nornberg, E. J. Spence, R. D. Kendrick, C. M. Jacobson, and C. B. Forest. Measure- ments of the magnetic field induced by a turbulent flow of liquid metal. Phys. Plasmas, May 2006. (to be published). [51] R. O’Connell, R. Kendrick, M. Nornberg, E. Spence, A. Bayliss, and C. Forest. On the possibility of an homogeneous MHD dynamo in the laboratory, in Dynamo and Dynamics, a Mathematical Challenge, P. Chossat, D. Ambruster, and I. Oprea, editors, volume 26 of NATO Science Series II, Mathematics, Physics and Chemistry, pages 59–66, Dordrecht, 2000. NATO Science Programme, Kluwer Academic Publishers. [52] P. Odier, J.-F. Pinton, and S. Fauve. Advection of a magnetic field by a turbulent swirling flow. Phys. Rev. E, 58(6):7397–7401, 1998. 112

[53] N. L. Pefﬂey, A. B. Cawthorne, and D. P. Lathrop. Toward a self-generating dynamo: The role of turbulence. Phys. Rev. E, 61(5):5287–5294, 2000.

[54] C. L. Pekeris, Y. Accad, and B. Shkoller. Kinematic dynamos and the Earth’s magnetic ﬁeld. Philos. Trans. R. Soc. London, Ser. A, 275:425–461, 1973.

[55] F. Pétrélis, M. Bourgoin, L. Marié, J. Burguete, A. Chiffaudel, F Daviaud, S. Fauve, P. Odier, et al. Nonlinear magnetic induction by helical motion in a liquid sodium turbulent flow. Phys. Rev. Lett., 90(17):174501–174505, 2003.

[56] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B.P. Flannery. Numerical Recipes in C: The Art of Scientiﬁc Computing. Cambridge University Press, Cambridge, England, second edition, 2002.

[57] A. B. Reighard and M. R. Brown. Turbulent conductivity measurements in a spherical liquid sodium ﬂow. Phys. Rev. Lett., 86(13):2794–2797, 2001.

[58] Robert C. Weast, Ph.D. CRC Handbook of Chemistry and Physics. CRC Press, Inc., Boca Raton, FL, sixty-sixth edition, 1986.

[59] G. O. Roberts. Spatially periodic dynamos. Philos. Trans. R. Soc. London, Ser. A, 266(1179):535–558, 1970.

[60] G. O. Roberts. Dynamo theory of geomagnetism (geomagnetism theory of dynamos in homogeneous ﬂuid masses, considering Rikitake self reversing, kinematic and hydromagnetic dynamo problems), in World magnetic survey 1957–1969, A. J. Zmuda, editor, pages 123– 131, Paris, 1971. International Union of Geodesy and Geophysics Publication Ofﬁce. Re- ported by P. H. Roberts.

[61] Paul H. Roberts and Gary A. Glatzmaier. Geodynamo theory and simulations. Rev. Mod. Phys., 72(4):1081–1123, 2000.

[62] G¨unther R¨udiger and Rainer Hollerbach. The Magnetic Universe: Geophysical and Astro- physical Dynamo Theory. Wiley, New York, 2004.

[63] Graeme Ross Sarson. Kinematic Dynamo Calculations for Geomagnetism. PhD thesis, Uni- versity of Leeds, 1994.

[64] E. Shpil’rain, K. A. Yakimovich, V.A. Fomin, S. N. Skovorodjko, and A. G. Mozgovoi. Den- sity and thermal expansion of liquid alkali metals, Chapter 6.3.3, in Handbook of Thermody- namic and Transport Properties of Alkali Metals, R. W. Ohse, editor. International Union of Pure and Applied Chemistry Chemical Data Series. Blackwell Science Publications, Oxford, 1985. 113

[65] E. Shpil’rain, K. A. Yakimovich, V. A. Fomin, S. N. Skovorodjko, and A. G. Mozgovoi. Dy- namic and kinematic viscosity of liquid alkali metals, Chapter 7.3, in Handbook of Thermody- namic and Transport Properties of Alkali Metals, R. W. Ohse, editor. International Union of Pure and Applied Chemistry Chemical Data Series. Blackwell Science Publications, Oxford, 1985.

[66] Daniel R. Sisan, Nicol´as Mujica, W. Andrew Tillotson, Yi-Min Huang, William Dorland, Adil B. Hassam, Thomas M. Antonsen, and Daniel P. Lathrop. Experimental observation and characterization of the magnetorotational instability. Phys. Rev. Lett., 93(11):114502, 2004.

[67] E. J. Spence, M. D. Nornberg, C. M. Jacobson, R. D. Kendrick, and C. B. Forest. Observation of a turbulence-induced large scale magnetic ﬁeld. Phys. Rev. Lett., 96(5):055002, 2006.

[68] M. Steenbeck, I. M. Kirko, A. Gailitis, A. P. Klyavinya, F. Krause, I. Ya. Laumanis, and O. A. Lielausis. Experimental discovery of the electromotive force along the external magnetic ﬁeld induced by a ﬂow of liquid metal (α-effect). Sov. Phys.–Doklady, 13:443–445, 1968.

[69] F. Stefani and G. Gerbeth. Velocity reconstruction in conducting ﬂuids from magnetic ﬁeld and electric potential measurements. Inverse Problems, 15(3):771–786, 1999.

[70] F. Stefani and G. Gerbeth. A contactless method for velocity reconstruction in electrically conducting ﬂuids. Meas. Sci. & Tech., 11(6):758–765, 2000.

[71] F. Stefani and G. Gerbeth. On the uniqueness of velocity reconstruction in conducting ﬂuids from measurements of induced electromagnetic ﬁelds. Inverse Problems, 16(1):1–9, 2000.

[72] F. Stefani, T. Gundrum, and G. Gerbeth. Contactless inductive ﬂow tomography. Phys. Rev. E, 70:056306, 2004.

[73] R. Stieglitz and U. M¨uller. Experimental demonstration of a homogeneous two-scale dynamo. Phys. Fluids, 13(3):561–564, 2001.

[74] J. T. Stuart. On the stability of viscous ﬂow between parallel plates in the presence of a co-planar magnetic ﬁeld. Proc. R. Soc. London, Ser. A, 221(1145):189–206, 1954. 114

Appendix A: Hall Probe Input Voltage Drift

The output voltage of the Hall probes used to measure the magnetic ﬁelds in the experiment is sensitive to the its 5 Volt DC input power, Vcc. If this value drifts then the output of the probe will correspondingly drift. Consequently, knowing how to account for changes in the magnetic ﬁeld signal due to changes in the input voltage is very important. This appendix describes how one compensates for the drift in supply voltage.

A.1 Probe Circuitry

The diagram for the circuit board that holds the Hall probe integrated circuit is given in Fig- ure A.1. The values of the resistors and capacitors on the board, are given in Table A.1. The value of the resistor R1 affects the temperature compensation of the probe. The values of R2 and R3 are chosen so as to have a measurement range balanced about B = 0 ( 170 G in this case). C ± 1 is chosen so as to set the low-pass filter cutoff at about 100 Hz. C2 filters high frequency noise from the input power. A simplified circuit diagram describing how the Hall probe output voltage is processed is given in Figure A.2. This diagram is a simplification of the actual circuit as it doesn’t distinguish between components that are within the integrated circuit and components that are on the circuit board.

The ﬁrst step in determining how Vin gets processed is to determine the current running from

Vcc/2 to Vout. Assuming that there is no time dependence in the problem C1 can be ignored and

Circuit Board Element Values

R1 R2 R3 C1 C2 Vcc 15 kΩ 3.3 kΩ 100 kΩ 10 nF 0.1 µF 5V

Table A.1 Resistor and capacitor values used on the Hall probe circuit board. The elements are chosen to allow a measurement range of 170 G and are low-pass ﬁltered at about 100 Hz. ± 115

Vcc

NC 1 8

R1 R2 2 7

C 3 6 2

C1 R3

4 5 Vout

Figure A.1 Circuit diagram for the Hall probe integrated circuit, as recommended by the Hall probe manufacturer, for bipolar detection. R1 is chosen for optimum temperature change compensation. Changing the values of R2 and R3 changes the measurement range of the circuit. C1 affects the ﬁltration of the output signal.

Vcc/2

R2 C1

− Vout Vin +

Figure A.2 Circuit diagram describing the processing of the Hall probe’s output signal, Vin. The operational ampliﬁer and the source of the Vcc/2 power are inside the integrated circuit, while the two resistors and capacitor are outside the integrated circuit (see Figure A.1 for details). This circuit is based on the documentation supplied by the probe manufacturer. 116

Approximate Voltages Magnetic Field Vout Vin

3Vcc Maximum Vcc 4 Vcc Vcc None 2 2 Vcc Minimum 0 4 Table A.2 Various values for the voltages seen in Figure A.2, as a function of the magnetic field strength. The values of V are based on empirical observations, for R R . The values of V out 2 ≃ 3 in are those that must generate Vout based on the circuit diagram in Figure A.2. It is interesting to note that the voltage generated by the Hall probe, Vin, does not range from 0 to Vcc. the current is given by V /2 IR IR = V , (A.1) cc − 2 − 3 out where it has been assumed that the current is running from Vcc/2 to Vout and that the operational amplifier is not drawing any current. This gives V /2 V I = cc − out . (A.2) R2 + R3 Recalling the first rule of operational amplifiers, the amplifier does whatever is necessary to make the two inputs have the same voltage, one notes

Vin = Vout + IR3. (A.3)

Rearranging this equation, and plugging in equation A.2, gives R + R R V V = 2 3 V 3 cc (A.4) out R in − R 2 2 2 Thus, as Vcc is moved the output signal will move with it as well. Consequently, if the probe was

calibrated at a Vcc which differs from the Vcc being supplied to the probe then the probe will have an incorrect offset, at the very least.

A.2 Field Calculation Modiﬁcations

Accounting for the dependence of Vout on Vcc requires accounting for the dependence of Vin on

Vcc, as seen in equation A.4. The approximate values that Vin and Vout take, depending on the state 117

of the magnetic ﬁeld, are listed in Table A.2, based on empirically-observed values of Vout. From these values, and given that the response of the probe is very linear, it is reasonable to generate a generic linear relationship between Vin, Vcc and the magnetic ﬁeld strength:

V V = AB + cc (A.5) in 2

where A is some factor, and B is the strength of the magnetic ﬁeld. If this equation is then plugged into equation A.4 one ﬁnds that

R + R V R V V = 2 3 AB + cc 3 cc out R 2 − R 2 2 2 R + R R + R R V = 2 3 AB + 2 3 3 cc R R − R 2 2 2 2 R + R V = 2 3 AB + cc . (A.6) R 2 2

This is very interesting, since all of the resistor dependence has fallen away from Vcc. For a power supply voltage that differs from the calibration voltage VccCal by some small amount δ, ′ Vcc = VccCal + δ, one ﬁnds that δ V ′ = V + . (A.7) out out 2 Thus, the procedure for correcting for the change in power supply voltage is to calculate δ/2 = (V ′ V )/2 and then add this quantity to the signal voltage. This is done when the raw magnetic cc− ccCal ﬁeld data is converted to CDF format. 118

Appendix B: FIND Frequency Shifting

FIND uses frequency shifting to distinguish between positive and negative particle velocities. The idea is to take the Doppler-shifted 40 MHz interference pattern that is returned to the Color- Link Plus and extract the Doppler-shifted frequency, meaning the difference between the returned frequency and 40 MHz, while at the same time distinguishing between positive and negative velocities. This is done by adding a 40-MHz-plus-shift-frequency signal to the returned signal, after it has been converted to an electrical signal, and low-pass ﬁltering the signal to extract out the Doppler frequency (see Section 2.1.2 for details of the LDV hardware). The extra shift frequency is added to distinguish between positive and negative particle motion. Without the shift the negative velocities would generate negative frequencies, which are indistinguishable from positive frequencies. The goal in selecting a frequency shift is to select a frequency range that contains both the positive and negative expected velocities. Such a selection is based upon whether positive, negative or mixed velocity values are expected in the data, what the range of those values might be, and the physical characteristics of the laser light. Details of the laser light are listed in Table B.1. Since our velocity ﬁeld changes sign as the traverse is moved from one hemisphere to the other, both positive and negative velocity values, of approximately the same magnitude, are expected, both for the poloidal and toroidal measurements. An absolute upper bound on the velocity magnitude is given by the tip speed of the impeller. For the impellers running at 1000 RPM the tip speed is slightly less than 16 m/s. However, experience making measurements on the water model indicates

Measurement Volume Fringe Number of Minimum # of fringes Wavelength Diameter Length Spacing Fringes required by processor Green 514.5 nm 90.5 µm 1.31 mm 3.73 µm 24 16 Blue 488.0 nm 85.8 µm 1.24 mm 3.54 µm

Table B.1 Characteristics of the laser light, taken from the ﬁberoptic probe documentation. 119

that the maximum speeds for this rotation rate, including fluctuations, never exceed 10 m/s for the toroidal measurements, and 8 m/s for the poloidal. These values will be used to determine the correct shift frequency for the green and blue beams. The determination of the frequency shift and range needed is a matter of simple arithmetic. As described in the ColorLink Plus documentation, the shift frequency, fshift, should be at least twice the Doppler frequency for the maximum negative velocity, which in this case is the same as the maximum positive velocity, 2vmax fshift > 2fDoppler = (B.1) df where df is the fringe spacing for the given frequency and vmax is the maximum speed being considered. Given this calculation, the minimum values required for proper frequency shifting are listed in Table B.2. The effective shift frequencies are predetermined by the FIND software; the effective shift frequency closest to the minimum is chosen. While the 5 MHz shift for the blue channel is slightly lower than the minimum, no behaviour reminiscent of aliasing has been seen in the data, and so this shift has been taken to be acceptable. Once a minimum and effective shift frequency has been determined which has a sufficient range to include the expected measurements it must be confirmed that this shift supplies the processor with a sufficient number of fringes to process the measurement. This is calculated by f N = N 1+ effective , (B.2) s f Doppler where Ns is the number of fringes generated, N is the number of fringes in the measurement volume, feffective is the effective shift frequency and fDoppler has a sign depending on whether it

Fringe Maximum Minimum Effective # of Negative Spacing Speed Shift Shift Shifted Fringes Green Channel 3.73 µm 8 m/s 4.3 MHz 5 MHz 32 Blue Channel 3.54 µm 10 m/s 5.6 MHz 5 MHz 19

Table B.2 Minimum required shift frequency values for use with the FIND software, as calculated using equation B.1, the selected effective shift, and the number of fringes generated when the particle is traveling in the same direction as the fringes. 120

is a positive or negative ﬂow being considered. The number of fringes generated by the passing particle, using these effective shifts, can be seen in Table B.2. As can be seen, the shift frequencies generate enough fringes to be processed by the hardware. One further test is yet required: the frequencies involved must not get ﬁltered by the low-pass

filtering that is done within the ColorLink Plus. This is done by comparing the sum feffective + fDoppler to the low-pass filter frequency. The situation at hand falls into the 0-23 MHz range of frequencies. feffective + fDoppler is well below the cutoff of 23 MHz, and so these shifts should be effective choices. 121

Appendix C: Magnetic Field Boundary Conditions

This appendix describes how the magnetic field boundary condition on the poloidal radial profile, in equation 4.11, is derived. For a system where currents are generated only within a sphere of radius a, centered at r =0, the region r a is current-free. As such this region satisfies B =0 ≥ ∇× and the magnetic field can be expressed as the gradient of a scalar potential, B = Φ . The −∇ m scalar potential, of course, solves Laplace’s equation in the region r a: ≥ 2Φ =0. (C.1) ∇ m The solution to this equation, in spherical coordinates, was already presented in Section 4.3:

ℓβ −(ℓβ+1) Φm(r,θ,φ)= Cβr + Dβr Yβ(θ,φ). (4.18) β X For the region not including the origin Cβ = 0 (this is the opposite of the treatment found in Section 4.3, for the case of magnetic ﬁeld due to currents outside the sphere). The magnetic ﬁeld is thus

−(ℓβ+2) Br(r,θ,φ)= Dβ(ℓβ + 1)r Yβ(θ,φ), (C.2) β X − ∂Yβ(θ,φ) B (r,θ,φ)= D r (ℓβ+2) , (C.3) θ − β ∂θ β X − 1 ∂Yβ(θ,φ) B (r,θ,φ)= D r (ℓβ +2) . (C.4) φ − β sin θ ∂φ β X Matching the equations for the magnetic ﬁeld within the sphere, equation 4.4, to equations C.2 and C.3 at the surface of the sphere (equation C.4 does not add any new information), it is found that

ℓβ(ℓβ + 1) (ℓβ + 1)Dβ Sβ(a)Yβ(θ,φ)= Yβ(θ,φ) (C.5) a2 a(ℓβ +2) T (a) ∂Y (θ,φ) 1 ∂S (r) ∂Y (θ,φ) D ∂Y (θ,φ) β β + β β = β β (C.6) a sin θ ∂φ a ∂r ∂θ − a(ℓβ +2) ∂θ r=a where the summations have been dropped, since the expansions are matched mode-to-mode. These two equations reveal:

Tβ(a)=0 (C.7) 122

Dβ ℓβSβ(a)= (C.8) aℓβ 1 ∂S (r) D β = β . (C.9) a ∂r −a(ℓβ +2) r=a

Equation C.7 was already known, and follows from the fact that the system is simply-connected, and that there are no toroidal currents outside the sphere. Combining equations C.8 and C.9 so as to eliminate Dβ gives equation 4.11:

∂S (r) ℓ S (a) β + β β =0, (4.11) ∂r a r=a

the boundary condition for the poloidal radial profiles at the sphere’s surface. Equation C.6 requires some comment. Strictly speaking, the tangential component of the magnetic field is not continuous across the sphere’s surface if there are surface currents. For equation C.6 to be valid the assumption is made that any surface currents that develop last for less than a resistive time, τσ, and thus are not important for the dynamics of the magnetic field. 123

Appendix D: Advection and Diffusion Matrix Operators

This appendix gives an overview of how the matrix operators A and D are implemented. This code was originally written by Rob O’Connell, the ﬁrst postdoctoral researcher for the Madison Dynamo Experiment, and so this implementation is to his credit. This appendix is much more detailed than is really necessary. The detail is put here for the education of those who will be using this code in the future.

D.1 Setting up a Banded System

The induced magnetic ﬁeld, b, is calculated using equation 4.14:

[A + D] b = AB , (4.14) − 0 where both the magnetic ﬁelds and the operators are to be discretized. The problem is linear in B, and so can be set up as a linear system of equations via a radial discretization. The radial direction is split into n equally-spaced segments, r = (i + 1)/n, where i = 0, 1, 2,...,n 1 (r = 1 is i − the surface of the sphere since the code is written in normalized units). The velocity radial proﬁles

sα(r) and tα(r) for the poloidal and toroidal ﬂow are then represented by si, et cetera, where si

corresponds to sα(ri). The magnetic ﬁeld and its derivatives are then speciﬁed by:

S (r ) S (D.1) β i → i ∂S (r) S S − β i+1 − i 1 (D.2) ∂r → 2h r=ri 2 ∂ Sβ(r) Si+1 2Si + Si−1 − (D.3) ∂r2 → h2 r=ri

where second-order ﬁnite differences are used for the derivatives. There are several ways to set up the system; if the system is set up carefully, however, the matrices will become banded and far-off- diagonals are zero. This eases the computation signiﬁcantly since there exist special algorithms to deal with matrices of this sort. If there are n radial points, and p spherical harmonic modes, then in general the total size of the eigenvector is 2np, where the poloidal and toroidal magnetic 124

modes have been counted separately. Consequently there are (2np)2 elements in the diffusion and advection matrices. This can be very large. Hence the necessity of a banded system. If the eigenvector is setup such that

B = [Sall p, Tall p,Sall p, Tall p,Sall p, Tall p,...,Sall p, Tall p,Sall p, Tall p,Sall p ] (D.4)

i=0 i=1 i=2 i=n−3 i=n−2 i=n−1

then the system| will{z be} banded.| {z Note} | the{z lack} of Tall|p, i={zn−1 elements.} | {z These} |{z elements} are not

included because Tβ(r =1)=0 and so including these elements does not add any information. This implementation of the eigenvector has (2n 1)p elements, rather than the more general 2np − discussed above. The eigenvector is organized in blocks, one block for each radial position. Within each of these blocks are the radial proﬁles for each spherical harmonic mode evaluated at that radial position. The poloidal modes are listed ﬁrst, followed by the toroidal modes. This construction results in banded operators because the interaction terms which relate different modes relate only the radial position of any given mode to its nearest radial neighbours, thus restricting the non-zero elements to a band around the diagonal. The matrices are stored in LAPACK band format [3]. This format reduces the full banded matrix to a compact form. To construct the matrix in band format the number of columns (k) and

the number of upper (nu) and lower (nl) diagonals of the original matrix are determined. The matrix is then stored as a k (n + n + 1) matrix. Consider the example matrix in equation D.5, × u l which has one upper diagonal (nu =1) and 2 lower diagonals (nl =2):

a11 a12 0 0 0   a21 a22 a23 0 0   a31 a32 a33 a34 0  . (D.5)      0 a42 a43 a44 a45      0 0 a53 a54 a55     125

In band format this matrix is reduced to

0 a12 a23 a34 a45   a11 a22 a33 a44 a55 . (D.6)   a21 a32 a43 a54 0      a31 a42 a53 0 0      Recall that, for this system, k = (2n 1)p, and as will be discussed in Sections D.1.1 and D.1.2, − n =3p and n =4p. Since n = 100 for most implementations, k (n + n + 1), and so band u l ≫ u l matrix format saves a signiﬁcant amount of space.

D.1.1 The Diffusion Matrix

The means by which the matrix operators relate different spherical harmonic components is most easily illustrated with a ﬁgure. Figure D.1 shows the general layout of the diffusion matrix, the simpler of the two operators. The ﬁgure assumes that there are 10 radial positions being

considered. Sj refers to the block of all poloidal radial proﬁles evaluated at radial position j, and

similarly for Tj. Recall that equations 4.5 and 4.6 involve summations over α and β for any given γ magnetic mode. Because the velocity field radial profiles are assumed to be known, the summation over α is implicit and is not important to the structure of the matrices. The indices of the β modes are listed across the top of the matrix and the indices of the γ modes are listed down the left side (the diffusion operator does not consider the β indices in its operation, this is just mentioned for consistency with the advection matrix). Recall that the functional form of the diffusion operator, D, is described by ∂2 ℓ (ℓ + 1) 2 = γ γ . (D.7) ∇ℓγ ∂r2 − r2 The second derivative in this operation is performed as per equation D.3. The second term in the operator is a matrix contribution that applies directly to the mode in question, γ. The two terms provide contributions to the matrix at the radial position of the mode, and the radial positions on either side of that mode. As an illustration of this, consider the entry labelled as “Standard Contribution” in Figure D.1. Here one of the toroidal radial profiles is being considered, the 126

m=1 Tℓ=2 mode say, at the third radial position. The entries in the matrix due to this contribution are indicated with dots. The second term in equation D.7 gives a contribution to the matrix on the diagonal ( 2(2+1)/(3/10)2). The first term also has a contribution on the diagonal ( 2/(1/10)2), − − with h =1/10. The first term also has contributions to the matrix for the same mode at the radial positions on either side of the diagonal (1/(1/10)2). The horizontal distance between the diagonal contribution and the contributions on either side are 2p, where p is the number of poloidal and toroidal modes. All of the entries follow the form for a Standard Contribution except for the entries in the last radial block. Worth noting is that on the inner edge (r = 1/n) for both the poloidal and toroidal and the outer edge for the toroidal (r = (n 1)/n) the Standard Contribution also applies. The − reason that it works for these edges is that the contribution to the central difference which is off the matrix is zero, S(0) = T (0) = T (1) = 0 and so the form of the central difference, as given in equation D.3, is exactly correct. A problem arises, however, for the poloidal profile at r =1. One cannot do central differencing at the outer edge of the sphere, since there are only known points on one side of the edge, and S(r > 1) =0 in general. To escape this problem the diffusion operator, as given in equation D.7, 6 is abandoned at the outer edge. In its place the boundary condition on the poloidal radial profile, equation 4.11, is used. Obviously a derivative taken in the form of equation D.2 will still not work, so second order backward differencing is used instead of central differencing. Using backward differencing, the first derivative takes the form

∂S (r) 3S 4S − + S − β n − n 1 n 2 (D.8) ∂r → 2h r=1

The poloidal boundary condition, using backward differencing, is the entry labelled “Endpoint Contribution” in Figure D.1. The bandwidth that the diffusion matrix needs would have been 4p, as indicated by the diagonal lines going through the Standard Contribution (2p above the diagonal and 2p below). However, the need for backward differencing at the outer boundary increases the needed bandwidth for the diffusion matrix to 6p. 127

S1 T1 S2 T2 S3 T3 S4 T4 S5 T5 S6 T6 S7 T7 S8 T8 S9 T9 S10 B

T1 Standard Contribution

S5 Endpoint Contribution

S10

Figure D.1 Illustration of the diffusion matrix. 10 radial positions have been assumed. Sj refers to the block of all poloidal magnetic radial proﬁles evaluated at radial position j, and similarly for Tj. Elements accessed to make a typical entry in the matrix are indicated with dots and are labelled “Standard Contribution.” The contribution due to the poloidal entries at r = 1 are exempliﬁed by the points labelled “Endpoint Contribution.” The bandwidth of the matrix is indicated by the outer diagonal dashed lines. 128

D.1.2 The Advection Matrix

The advection matrix is significantly more complex than the diffusion matrix. The most important difference between the two matrices is that the diffusion matrix is always symmetrically- banded about the diagonal for r = 1, meaning that β = γ. This is not the case for the advection 6 matrix, resulting in a matrix with much more structure. The physics of the advection matrix is expressed in the interaction terms found in equation 4.8. An illustration of how the various interactions contribute to the advection matrix is given in Figure D.2. The empty squares in the figure represent the β mode being considered, while the dots represent the matrix contributions. These contributions are discussed below, in ascending order of complexity. Naturally, these interactions apply for all appropriate modes at all radial positions, not just the modes and positions displayed in the figure.

The simplest contributions to the advection matrix are given by the (tαSβSγ ) and (tαTβTγ ) interaction terms. As seen in Figure D.2, these terms contribute to a γ mode of the same type, poloidal or toroidal, as the β mode, at the same radial position. This is why the contributions are in the same grid box as the associated square. Because this interaction contains no derivatives there is only a single entry for the contribution. A slightly more complicated contribution to the advection

matrix is due to the (sαTβSγ) term. This term also adds a single-entry contribution to the matrix, however in this case it is a toroidal β mode that is interacting with a poloidal γ mode. This results in the contribution being in a different grid box than the β contribution in Figure D.2. The rest of the interaction terms all contain derivatives, and as such they contribute off-diagonal elements at locations farther from the diagonal than the terms discussed above. The (sαSβSγ) and

(sαTβTγ) terms are the simplest of these, since their β and γ magnetic modes are of the same type, and thus the center of their central differences are within the same grid box in Figure D.2. In a sense the contributions due to these interaction terms are the most similar to the Standard Contribution of the diffusion matrix, though these terms contribute ﬁrst derivatives rather than the second derivatives of the diffusion matrix.

The most complicated additions to the advection matrix are due to the (sαSβTγ ) and (tαSβTγ )

terms. These terms are not unlike the (sαTβSγ ) term, in that the β and γ magnetic modes are of 129

S1 T1 S2 T2 S3 T3 S4 T4 S5 T5 S6 T6 S7 T7 S8 T8 S9 T9 S10 B

S1 (sαSβSγ) T1

T2 (sαSβTγ) and (tαSβTγ)

T3 (sαTβSγ) S4

T7 (sαTβTγ) S8

T8 (tαSβSγ) S9

T9 (tαTβTγ) S10 Figure D.2 Illustration of the advection matrix. Like Figure D.1, 10 radial positions have been assumed. Sj is the block of all poloidal magnetic radial proﬁles evaluated at radial position j, and similarly for Tj. The indices of the β modes are listed across the top and the γ modes along the left. Examples of entries due to the interaction terms are given throughout the matrix, with contributions to the matrix indicated with dots. Empty squares indicate the diagonal element associated with a contribution, but is not a contribution itself. A dotted line connects the diagonal element with the associated matrix contribution. The bandwidth of the matrix is indicated by the diagonal dashed lines. 130 different types. In this case the β mode, located in a different grid box than the γ mode, is also the center of derivatives. The (sαSβTγ) is the only interaction term that contains second derivatives, as seen in equation 4.8. Unlike the diffusion matrix, no modiﬁcations are made to the interaction term evaluations at r = 1. In fact, the interaction terms are not evaluated at r = 1 at all when equation 4.14 is implemented. All standard advection and diffusion operations are evaluated from 0

Appendix E: Axisymmetric Velocity and Magnetic Fields Cannot Induce Dipole Moments

Proof that a magnetic dipole moment cannot be generated by an axisymmetric flow when exposed to an axisymmetric magnetic field has been supplied by Spence et al. [67]. The first published proof, to the author’s knowledge, is given in Section 6.1.3. Two other proofs are given here, the first is a less-general proof which only applies to a spherical geometry. The second is presented in a cylindrical geometry, and is closely related to the published proof.

E.1 Proof 1

It is certainly possible for a dipole poloidal component to be induced within the sphere, and

ℓγ =1 does satisfy all mode-interaction selection rules for ℓα =2 and ℓβ =1 (see Section 4.1.2), but this induced field is expected to go to zero at the sphere’s surface, and thus no net dipole moment should be produced. The proof of this is as follows. It is first noted that both the velocity field and applied field are axisymmetric, and thus the induced field is also axisymmetric. This fact is intuitive, since there is nothing to break the symmetry, and is also expressed in the selection rules.

The Elsasser integrals are always zero for this case, since mα = mβ = mγ =0 means that cosine modes are used for all the spherical harmonics, resulting in an odd number of cosines. The only

non-zero interaction term in equation 4.6 is thus (sαSβSγ). Consequently, in its time-independent, axisymmetric form, equation 4.6 is

R K ∂S (r) ∂s (r) ∂2S (r) p S (r) m αβγ β α γ γ γ (E.1) 2 pαcαsα(r) pβcβ Sβ(r) + 2 2 =0, pγ r ∂r − ∂r ∂r − r α,β X

where (sαSβSγ) has been expanded into its functional form and, as already noted, pα = ℓα(ℓα +1) and c = (p p p )/2. The speciﬁc case of the induced dipole, ℓ = 1, is now examined. α α − β − γ γ Multiplying by r2, it is noted that

∂2S (r) d ∂S (r) r2 1 2S (r)= r2 1 2rS (r) . (E.2) ∂r2 − 1 dr ∂r − 1 132

Integrating from 0 to r, one ﬁnds

r R K ∂S (r) m αβ1 (p c + p c ) s (r) β dr 2 α α β β α ∂r − α,β X Z0 R K p c ∂S (r) m αβ1 β β s (r)S (r)+ r2 1 2rS (r)=0, (E.3) 2 α β ∂r − 1 α,β X where integration by parts has been performed, and use has been made of the boundary condition s (0) = 0. Observing that the selection rules for K = 0 require that ℓ = ℓ 1, it is easy to α αβ1 6 β α ± 2 ∂S1(r) show that (pαcα + pβcβ)=0. Setting r = a leaves a ∂r 2aS1(a)=0, using the boundary r=−a

condition sα(a)=0. The boundary condition on the poloidal magnetic ﬁeld, equation 4.11, is

recalled, leaving S1(a)=0. The dipole component of the induced ﬁeld goes to zero at the sphere’s surface. This is also true in the presence of an applied ﬁeld.

Proof that S1(a)=0 implies that there is no external dipole moment is now required. The definition of the dipole moment, as expressed in a spherical geometry when the velocity field and magnetic field are both axisymmetric, is given by

µ µ = 0 J r3 sin2θ drdθdφ. (E.4) z 4 φ Z The current is found by using µ J = B. Applying this to equation 4.4 gives the expansion for 0 ∇ × the axisymmetric toroidal current. Inserting this into equation E.4 gives,

1 1 ∂2S (r) p S (r) ∂Y (θ,φ) µ = γ γ γ γ r3 sin2θ drdθdφ. (E.5) z 4 r ∂r2 r3 ∂θ γ − X Z Considering the angular integral ﬁrst, one ﬁnds

4 3π ℓγ =1 ∂Yγ (θ,φ) 2 − sin θ dθdφ =  q (E.6) ∂θ  Z 0 ℓ =1 γ 6  One can quickly see where this is going. The radial integral is now considered,

a 1 ∂2S (r) p S (r) 1 1 1 r3 dr = 3aS (a), (E.7) r ∂r2 − r3 − 1 Z0 133

where only the ℓγ = 1 case is considered, since the angular integral has removed the necessity to

consider the other cases. Consequently, proving that S1(a)=0 does indeed imply that µz = 0. It

is interesting to note that this treatment only applies to the dipole (ℓγ =1) case; other induced ﬁeld components are not so affected.

E.2 Proof 2

Consider a bounded, steady-state, axisymmetric system of magnetic and velocity ﬁelds. The velocity ﬁeld is described in terms of an axisymmetric stream function, Φ(s, z),

v = Φ φ + v (s, z)φˆ, (E.8) ∇ × ∇ φ

where a cylindrical coordinate system, (s,φ,z), has been adopted. As noted in Section 6.1.3, the only non-trivial component to the dipole moment, µ x J d3x, is oriented along the symmetry ≡ × axis and results from currents ﬂowing in the toroidal directR ion,

3 µz = sJφ d x. (6.1) Z These currents can only be generated by the EMF due to the mean flow of fluid, v B. With the × velocity and magnetic fields described by equations E.8 and 6.2, Ohm’s law becomes

J = σ ( Φ φ) ( Ψ φ) φˆ φ ∇ × ∇ × ∇ × ∇ · Φ Ψ = σ ∇ × ∇ φˆ. (E.9) s2 ·

In magnetic flux coordinates, the volume element in equation 6.1 can be expressed as d3x = s ′ ′ ′ dΨ dL/Bp, where Bp is the magnitude of the poloidal field, Ψ(s, z)=2π 0 Bz(s , z)s ds is the magnetic flux through a disk of radius s at position z, and L is a distanceR along the flux contour. Thus, equation 6.1 becomes

Φ Ψ µ = σ ∇ × ∇ φˆ dL dΨ. (E.10) z sB · Z p The velocity stream function is now expressed in terms of magnetic coordinates Φ(Ψ, L) such that Φ can be separated into terms parallel and perpendicular to the ﬂux surface. Noting that ∇ 134

Ψ =2πsB and that Φ Ψ φˆ = Ψ ∂Φ/∂L, the integral becomes |∇ | p ∇ × ∇ · |∇ | ∂Φ µ =2πσ dΨ dL. (E.11) z ∂L ZΨ Zℓ The limits of integration depend upon the nature of the magnetic flux surfaces. Two types of magnetic surfaces exist: closed flux surfaces and those with open field lines which penetrate the surface of the fluid. On closed flux surfaces, one integrates around the entire surface, giving

L(∂Φ/∂L) dL =0. When field lines pass through the fluid’s surface the condition that the fluid is h contained,H id est Φ=0 on the vessel wall, results in (∂Φ/∂L) dL = Φ(h) Φ(g)=0, where ∇ g − h and g are the points where the field lines enter and exitR the fluid. Hence, µz =0. Once again, it is noted that it is only the dipole moment that vanishes; moments which include different powers of s in equation 6.1 are non-zero in general. This conclusion is also independent of geometry; any simply-connected axisymmetric system gives the same result. 135

Appendix F: Calculation of the Magnetic Field due to the EMF

As noted in Section 6.3, determining the magnetic field due to the fluctuation-driven currents requires calculating all fields that are due to the advective action of the mean velocity field. To that end, Ohm’s law, J = σ (E + v B) , (1.2) × is used to calculate the current generated by the mean velocity field interacting with the mean measured magnetic field. Once this current is found, the resulting magnetic field can be calculated, and from this the fluctuation-driven magnetic fields. The current, given by µ J = B′, can be 0 ∇ × calculated from equation 4.4:

′ p 1 ∂T (r) ∂Y (θ,φ) µ J = rˆ γ T ′ (r)Y (θ,φ)+ θˆ γ γ 0 r2 γ γ r ∂r ∂θ γ X ′ 1 ∂2S (r) p ∂Y (θ,φ) + φˆ γ γ S′ (r) γ , (F.1) r ∂r2 − r3 γ ∂θ where one will recall that pγ = ℓγ(ℓγ + 1), and axisymmetry of the magnetic field has been assumed. The primes indicate that these radial profiles are for the field being calculated using the current, rather than the mean magnetic field measured in the experiment. This expansion can be ′ ′ used to bypass the current and go directly to the radial profiles, Sγ (r) and Tγ(r), that are needed to build the magnetic field.

F.1 Azimuthal Current

As noted in Section 6.3, calculating the azimuthal current, and thus the poloidal radial proﬁles, is accomplished using equation 6.5:

J = σ (v B) . (6.5) φ × φ 136

Expanding out v B in terms of the vector components, and then radial proﬁles, one ﬁnds × µ J = µ σ (v B) 0 φ 0 × φ ′ 1 ∂2S (r) p ∂Y (θ,φ) γ γ S′ (r) γ = µ σ (v B v B ) r ∂r2 r3 γ ∂θ 0 r θ θ r γ − − X 1 ∂2 p ∂Y (θ,φ) µ σ ∂S (r) ∂Y (θ,φ) γ S′ (r) γ = 0 p s (r) β Y (θ,φ) β r ∂r2 − r3 γ ∂θ r3 α α ∂r α ∂θ γ α,β X X ∂s (r) ∂Y (θ,φ) p α S (r) α Y (θ,φ) (F.2) − β ∂r β ∂θ β ∂Y (θ,φ) Equation F.2 is multiplied through by r3 ǫ and the orthogonality relation ∂θ

2π π ∂Y (θ,φ) ∂Y (θ,φ) γ ǫ sin θdθdφ = p δ (F.3) ∂θ ∂θ ǫ γ,ǫ Z0 Z0 is then invoked, leaving

2π π 2 2 ∂ ′ µ0σ ∂Sβ(r) ∂Yβ(θ,φ) ∂Yγ (θ,φ) r 2 pγ Sγ(r)= pαsα(r) Yα(θ,φ) sin θdθdφ ∂r − pγ  ∂r ∂θ ∂θ α,β X Z0 Z0  2π π ∂s (r) ∂Y (θ,φ) ∂Y (θ,φ) p α S (r) α Y (θ,φ) γ sin θdθdφ . (F.4) − β ∂r β ∂θ β ∂θ  Z0 Z0  To calculate the radial proﬁles for the current, the operator on the left is inverted,

2π π 2 −1 ′ 2 ∂ µ0σ ∂Sβ(r) ∂Yβ(θ,φ) ∂Yγ (θ,φ) Sγ (r)= r 2 pγ pαsα(r) Yα(θ,φ) sin θdθdφ ∂r − pγ  ∂r ∂θ ∂θ α,β X Z0 Z0 2π π  ∂s (r) ∂Y (θ,φ) ∂Y (θ,φ) p α S (r) α Y (θ,φ) γ sin θdθdφ , (F.5) − β ∂r β ∂θ β ∂θ  Z0 Z0  ′ subject to the usual boundary conditions for the poloidal magnetic radial profiles, Sγ(0) = 0 and equation 4.11. This gives the radial profiles for the poloidal magnetic field induced by the mean velocity field. The operator which was inverted is constructed in a manner similar to the diffusion matrix in Appendix D. The angular integrals on the right are evaluated using Gauss-Legendre quadrature. 137

F.2 Poloidal Current

Calculating the poloidal current is a little more difficult than the toroidal current, since the poloidal electric field must first be found, which requires the electric potential. As noted in Sec- tion 6.3, taking the divergence of Ohm’s law gives

2Φ= (v B) . (F.6) ∇ ∇· ×

The potential is expressed as an expansion in spherical harmonics,

Φ= fq(r)Yq(θ,φ). (F.7) q X In a spherical geometry, equation F.6 then becomes,

2Φ= (v B) ∇ ∇· × 1 ∂ ∂ p 1 ∂ r2 q f (r)Y (θ,φ)= r2 [v B v B ] r2 ∂r ∂r − r2 q q r2 ∂r θ φ − φ θ q X 1 ∂ + (sin θ [v B v B ]) r sin θ ∂θ φ r − r φ ∂ ∂ ∂ r2 p f (r)Y (θ,φ)= r2 [v B v B ] ∂r ∂r − q q q ∂r θ φ − φ θ q X 1 ∂ + (sin θ [v B v B ]) , (F.8) r sin θ ∂θ φ r − r φ where it has been recalled that the spherical harmonic is an eigenfunction of the angular derivitives of the Laplacian in a spherical geometry. Further manipulation of the right side gives

∂ t (r) ∂S (r) ∂s (r) T (r) ∂Y (θ,φ) ∂Y (θ,φ) r2 (v B)= r2 α β α β α β ∇· × ∂r r ∂r − ∂r r ∂θ ∂θ α,β X p s (r)T (r) ∂Y (θ,φ) ∂Y (θ,φ) + α α β α β p Y (θ,φ)Y (θ,φ) r2 ∂θ ∂θ − β α β p t (r)S (r) ∂Y (θ,φ) ∂Y (θ,φ) β α β α β p Y (θ,φ)Y (θ,φ) , − r2 ∂θ ∂θ − α α β 138

where Legendre’s differential equation has been used. Multiplying equation F.8 by a spherical harmonic, and integrating over the unit sphere gives

∂ ∂ ∂ t (r) ∂S (r) ∂s (r) T (r) r2 p f (r)= r2 α β α β ∂r ∂r − q q ∂r r ∂r − ∂r r × α,β X 2π π ∂Y (θ,φ) ∂Y (θ,φ) α β Y (θ,φ) sin θdθdφ ∂θ ∂θ q Z0 Z0 p s (r)T (r) + α α β r2 × 2π π ∂Y (θ,φ) ∂Y (θ,φ) α β p Y (θ,φ)Y (θ,φ) Y (θ,φ) sin θdθdφ ∂θ ∂θ − β α β q Z0 Z0 pβtα(r)Sβ(r) − r2 × 2π π ∂Y (θ,φ) ∂Y (θ,φ) α β p Y (θ,φ)Y (θ,φ) Y (θ,φ) sin θdθdφ. ∂θ ∂θ − α α β q Z0 Z0

As with the toroidal current, the operator on the left is then inverted to give fq(r). The boundary conditions on fq(r) require some discussion. The reference voltage for the calculation is taken to be at the origin, meaning fq(0) = 0. The outer radial boundary condition is more subtle. To begin, the divergence of Ohm’s law is taken:

J =0= σ [ E + (v B)] . (F.9) ∇· ∇· ∇· × Integrating over the volume of the sphere,

E d3x = (v B) d3x, (F.10) ∇· − ∇· × Z Z and using the divergence theorem gives,

Φ rˆ dΩ = (v B) rˆ dΩ, (F.11) ∇ · × · I I where time independence has been assumed. Equating the integrands gives

∂Φ = v B . (F.12) ∂r − φ θ|r=a r=a

139

Expanding out the various components, multiplying through by a spherical harmonic, and integrating over the unit sphere leaves

2π π ∂f (r) ℓ t (a)S (a) ∂Y (θ,φ) ∂Y (θ,φ) q = β α β α β Y (θ,φ) sin θdθdφ, (F.13) ∂r − a3 ∂θ ∂θ q r=a α,β Z Z X 0 0

where equation 4.11 has been used. If the no-slip condition were being assumed then tα(a)=0 and the right side of equation F.13 would be zero.

With the inner and outer boundary conditions on fq(r) speciﬁed Φ is determined. The next step is to calculate the poloidal component of the current. Since both the r and the θ components of J ′ in equation F.1 depend on Tγ (r), only one of the two components are needed to determine the toroidal magnetic ﬁeld. The radial component is chosen since its structure is simpler:

′ pγT (r) ∂f (r) µ J = γ Y (θ,φ)= µ σ q Y (θ,φ) 0 r r2 γ 0 − ∂r q γ " q X X t (r) ∂S (r) ∂s (r) T (r) ∂Y (θ,φ) ∂Y (θ,φ) + α β α β α β . r ∂r − ∂r r ∂θ ∂θ α,β # X Multiplying though by a spherical harmonic and integrating over the unit sphere leaves

′ pγT (r) ∂f (r) γ = µ σ γ r2 0 − ∂r 2π π t (r) ∂S (r) ∂s (r) T (r) ∂Y (θ,φ) ∂Y (θ,φ) + α β α β α β Y (θ,φ) sin θdθdφ . r ∂r − ∂r r ∂θ ∂θ q  α,β X Z0 Z0 ′  Thus giving Tγ (r), the required radial profile. ′ ′ Using the Sγ(r) and Tγ (r) profiles calculated from the measured velocity and magnetic fields, the magnetic field induced by the action of the mean velocity field can be calculated. Subtracting this field from the measured field leaves the magnetic field generated by the fluctuations, as shown in Figure 6.8.