Dynamics of Rotation and Magnetism in the Sun’s
Convection Zone and Tachocline
byLoren Isaac Matilsky
A.B., Cornell University, 2015
M.S., University of Colorado Boulder, 2018

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Astrophysical and Planetary Sciences
2022
Committee Members:
Juri Toomre, Chair
Bradley W. Hindman
Steven R. Cranmer
Maria D. Kazachenko
Keith Julien

ii
Matilsky, Loren Isaac (Ph.D., Astrophysical and Planetary Sciences)
Dynamics of Rotation and Magnetism in the Sun’s Convection Zone and Tachocline
Thesis directed by Prof. Juri Toomre

In this thesis, we assess the theoretical dynamics achieved in the solar interior, with particularfocus on the solar tachocline. We use the open-source Rayleigh code on parallel supercomputers tosimulate 3-D, rotating spherical shells of convection. These shells cover much of the solar convection
zone and in the tachocline models, a portion of the underlying radiative interior. This thesis divides
solar dynamics into two distinct classes: The hydrodynamic (HD) Sun (which explores convectionin the presence of differential rotation) and the magnetohydrodynamic (MHD) Sun (which explores
how a self-excited solar dynamo interacts with convection and rotation).
In the HD Sun, we discuss how the Near-Surface Shear Layer (NSSL) might be generated by
fast downflow plumes. We also identify a physical mechanism whereby the Sun might establish aninternal latitudinal temperature gradient and thus achieve isorotation contours significantly tilted
with respect to the rotation axis. In the MHD Sun, we focus on the global magnetism and rotation
profiles achieved in self-excited dynamo simulations. We first describe how dynamos in convectionzone-only shells display remarkable bistability: Two distinct magnetic cycles—each reminiscent ofobserved behavior in the solar cycle—are supported by the convection simultaneously.
Finally, we present an MHD simulation achieving a solid-body-rotating radiative interior
and differentially rotating convection zone. This shear layer, similar to the solar tachocline, isdynamically maintained by magnetic torques acting against viscous torques. Our work is thus thefirst to identify a “magnetic tachocline confinement scenario” operating in a fully 3-D, nonlinear
global simulation. Furthermore, the magnetism is produced by dynamo action, even below theregion of convective overshoot. Rather than the classical “abyssal deep”—i.e., a largely motion-freereservoir that accumulates magnetism pumped in from above—we argue that the Sun’s radiativeinterior may contain inertial oscillations that couple to the dynamo.

Dedication

To my mother, who taught me that work meant dedication and perseverance.
To my father, who taught me that work meant force times distance.

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Acknowledgements

First and foremost, I thank my thesis advisor Juri Toomre for his unwavering support of myresearch and career goals, as well as his mature scientific perspective, which shaped the content
and structure of this thesis. Working with Juri gave me the opportunity to attack big-ticket items
in solar-physics research (e.g., the magnetic tachocline) that I might not have been exposed tootherwise. I also owe thanks for the Epicurean dinners, desserts, and classroom snacks created by
Juri and his wife Linda, as well as the constant supply of Costco biscotti and dried mango in LCD.
Bradley Hindman really served as my “co-advisor” during this thesis (although he has informed me that there is no such thing). As the second author of the majority of my publications
to date, he has always provided invaluable guidance, support, and a common-sense approach toscience that has made my writing, figures, and theoretical ideas much clearer. His no-nonsense,
extremely thorough, red editing pen has torn apart most the pages that follow and allowed me to
put them back together in a substantially improved state.
A wide range of scientific colloborators and acquaintances have informed my research overthe last few years. At the risk of invariably neglecting someone, I thank: Connor Bice, Nicholas
Featherstone, Catherine Blume, Mark Miesch, Sacha Brun, Kyle Augustson, Antoine Strugarek,
Keith Julien, Ryan Orvedahl, Rachel Howe, Axel Brandenburg, Steven Tobias, Gustavo Guerrero,
Benjamin Brown, Nicholas Brummell, and Pascale Garaud. I also thank the members of my thesis
committee: Juri Toomre, Bradley Hindman, Maria Kazachenko, Keith Julien, and Steven Cranmer.
I owe a special thanks to Gwen Dickinson, who made all things in the “Toomre lab” runsmoothly while following the complex rules designed by NASA, JILA, and the University of Col-v
orado. She also read my proposals and papers, taught me interesting and unfamiliar grammar rules,
and was a lovely office-neighbor to chat with. I thank JILA’s expert administrative staff (Agnieszka
Lynch, Cindy Torres, Kim Monteleone, and Beth Kroger) and technical staff (Jim Mckown and J.R.
Raith). I also thank the graduate program assistant of the Astrophysical and Planetary Sciences
Department (Michelle Hamernik, followed by Troy Latta) for their immense help in navigating theburaucracy of the Graduate School.
This thesis would not have been possible without the support of my family and friends. My
mother Ruth, father Terry, siblings Matt, Athena, Jake, and Sara, and siblings-in-law Jeff, Page,
and Michelle, have all fielded many phone calls from a stressed-out grad student over the years, andhave always responded with sage wisdom and never exasperation :) For teaching me the importanceof hide-and-seek, watching movies, and running around outside, I thank my adorable nieces andnephews: Ben, Jem, Eliza, Ivy, Kai, Wilder, and Michael.
To my many roommates over the years, thank you for keeping the kitchen clean (sometimes),
the trash empty (sometimes), and the rent low (always). I owe a special thanks to Mario Castaneda,
my house-mate of now almost four years, who suffered me in the best and worst of times, always got
me into the mountains, and taught me that running long distances in inhospitable environments
isn’t always miserable, as long as there’s a good view. Our cute apartment on Lee Hill Driveand extremely thorough approach to recycling kept me sane during the first two years (so far,
unfortunately) of the Covid-19 pandemic. Finally, I thank Rocky Mountain Runners (RMR) for
giving me the inspiration to wake up at 5 a.m. in the bitter cold on Wednesdays, for making me
run up Green Mountain on Mondays, and for rescuing me from my computer to adventure outsidearound the Great American West. The subset of RMR known only as “Group” via WhatsApp (youknow who you are) were some of the best friends anyone has ever had. Thanks for the stethoscope;
after seven years working on my doctorate, I finally feel like a doctor!
I was supported directly during this work by the Future Investigators in NASA Earth andSpace Sciences Technology (FINESST) award 80NSSC19K1428, by a George Ellery Hale Graduate Fellowship, and by a University of Colorado Boulder Chancellor Fellowship. This research

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was additionally supported by the NASA grants NNX13AG18G, NNX17AG22G, 80NSSC18K1127,
80NSSC17K0008, 80NSSC18K1125, 80NSSC19K0267, NNX16AC92G, and NNX14AC05G. Computational resources were provided by the NASA High-End Computing (HEC) Program throughthe NASA Advanced Supercomputing (NAS) Division at Ames Research Center. The Rayleighcode is hosted by and receives support from the Computational Infrastructure for Geodynamics(CIG), which is supported by the National Science Foundation awards NSF-0949446 and NSF1550901.

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Contents

Chapter
1 Brief Introduction to the Global Solar Dynamo
1.1

1.2

1.3

1.4

Solar Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Cyclic Behavior in Sunspots and Solar Magnetism . . . . . . . . . . . . . . .

3

1.1.2

Helioseismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.1.3

The Internal Rotation Rate of the Sun . . . . . . . . . . . . . . . . . . . . . . 15

Solar Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.1

Modeling of Near-Surface Convection and Active Regions . . . . . . . . . . . 20

1.2.2

Global Spherical-Shell Dynamo Simulations . . . . . . . . . . . . . . . . . . . 22

1.2.3

Insights into the Solar Differential Rotation from Global Simulations . . . . . 28

1.2.4

Global Models Including a Tachocline . . . . . . . . . . . . . . . . . . . . . . 33

Our Approach to Global Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.3.1

The Anelastic MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.3.2

Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.3.3

The Rayleigh Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2 The Role of Downflows in Establishing Solar Near-Surface Shear
2.1

1

46

Introduction: Our Theoretical Understanding of the Solar Near-Surface Shear Layer(NSSL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

viii
2.2

Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3

Global Flow Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4

Busse Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.5

Downflow Plumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.6

Torque Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.7

Angular Momentum Transport by Upflows and Downflows . . . . . . . . . . . . . . . 75

2.8Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.9More Recent Modeling of the NSSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3 Revisiting the Sun’s Strong Differential Rotation along Radial Lines

88

3.1

Introduction: The Sun’s Tilted Isorotation Contours . . . . . . . . . . . . . . . . . . 89

3.2

Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.3

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.4

Thermal wind balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.5

Poleward energy transport from Busse columns . . . . . . . . . . . . . . . . . . . . . 107

3.6

Effect of outer thermal boundary condition . . . . . . . . . . . . . . . . . . . . . . . 110

3.7

Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.8

Diagnostic Parameters for the Simulation Suite . . . . . . . . . . . . . . . . . . . . . 121

4 Exploring Bistability in the Cycles of the Solar Dynamo through Global Simulations

123

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.2

Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.3

Bistable Magnetic Cycles in the Dynamo Cases . . . . . . . . . . . . . . . . . . . . . 131

4.4

Polar Caps of Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.5

Evolution of the Fourfold-Wreath Cycle . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.6

Evolution of the Partial-Wreath Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.6.1

4.7

Partial-Wreath Cycle Period and Hemispheric Asymmetry . . . . . . . . . . . 149

Bistability Trends with Higher Magnetic Prandtl Number (Prm ) . . . . . . . . . . . 153

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4.8

4.9

The Dynamical Origins of Each Cycling Mode . . . . . . . . . . . . . . . . . . . . . . 159
4.8.1

Dynamics of the Fourfold-Wreath Cycle . . . . . . . . . . . . . . . . . . . . . 159

4.8.2

Dynamics of the Partial-Wreath Cycle . . . . . . . . . . . . . . . . . . . . . . 165

Bistability in the Context of Solar Observations . . . . . . . . . . . . . . . . . . . . . 167

4.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.11 Definitions of Non-Dimensional Numbers and Diffusion Times . . . . . . . . . . . . . 172
5 Confinement of the Solar Tachocline by Dynamo Action in the Radiative Interior

174

5.1

Radiative Spread of the Solar Tachocline . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.2

Tachocline Instabilities and the Fast Confinement Scenario . . . . . . . . . . . . . . 181

5.3

The Slow MHD Tachocline Confinement Scenario . . . . . . . . . . . . . . . . . . . . 185

5.4

Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

5.5

Solid-Body Rotation in the MHD Case’s Radiative Interior . . . . . . . . . . . . . . 191

5.6

Dynamical Maintenance of the MHD Case’s Tachocline . . . . . . . . . . . . . . . . 195

5.7

Tachocline Confinement by Cycling Partial Wreaths . . . . . . . . . . . . . . . . . . 204

5.8

Inertial Modes and Dynamo Action in the Radiative Interior . . . . . . . . . . . . . 211

5.9

Magnetic Tachocline Confinement in Global Simulations . . . . . . . . . . . . . . . . 216

5.10 Revisiting Magnetic Tachocline Confinement in the Sun . . . . . . . . . . . . . . . . 219
5.11 Supplement: Numerical Specifics and Equation Identities . . . . . . . . . . . . . . . 222
5.11.1 Background Thermodynamic State . . . . . . . . . . . . . . . . . . . . . . . . 222
5.11.2 Induction Equation in Spherical Coordinates . . . . . . . . . . . . . . . . . . 224
6 Concluding Remarks

227

Bibliography

232

Appendix

x

TablesTable
2.1

Model Parameters for the NSSL Experiment . . . . . . . . . . . . . . . . . . . . . . . 55

3.1

Model Parameters for the “Fixed-Flux/Fixed-Entropy” Experiment

3.2

Diagnostic Parameters for Each Thermal Boundary Condition . . . . . . . . . . . . . 122

4.1

Common Input Parameters for the Bistability Experiment . . . . . . . . . . . . . . . 128

4.2

Fluid Parameters for the Different Cases in the Bistability Experiment . . . . . . . . 156

4.3

Kinetic and Magnetic Energies in the Bistability Experiment . . . . . . . . . . . . . 157

5.1

Diffusive Time-Scales in the Radiative Interior of the Sun and Our MHD Case . . . 180. . . . . . . . . 93

xiFiguresFigure1.1

Sunspot Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Magnetic Butterfly Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

The Internal Rotation Rate of the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4

Magnetic Wreaths in Global Dynamo Simulations . . . . . . . . . . . . . . . . . . . . 27

1.5

Spherical-Shell Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1

Helioseismically Inferred Rotation Rate (with Errors) . . . . . . . . . . . . . . . . . . 49

2.2

Influence of Density Stratification on Flow Structures . . . . . . . . . . . . . . . . . 52

2.3

Influence of Density Contrast on Rotation Profiles along Radial Lines . . . . . . . . 56

2.4

Influence of Density Contrast on Rotation Frequency . . . . . . . . . . . . . . . . . . 57

2.5

Influence of Density Contrast on Meridional Circulation . . . . . . . . . . . . . . . . 59

2.6

Radial Energy Fluxes for High Density Contrast . . . . . . . . . . . . . . . . . . . . 60

2.7

Influence of Density Contrast on Rossby Number . . . . . . . . . . . . . . . . . . . . 62

2.8

Busse Columns on Equatorial Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.9

Following the Movement of Busse Columns . . . . . . . . . . . . . . . . . . . . . . . 65

2.10 Following the Movement of Intense Downflow Plumes. . . . . . . . . . . . . . . . . 66

2.11 Following the Vertical Structure of Downflow Plumes . . . . . . . . . . . . . . . . . . 67
2.12 Torque Balance for High and Low Density Contrast . . . . . . . . . . . . . . . . . . 70
2.13 Equilibrium of the Steady-State Differential Rotation . . . . . . . . . . . . . . . . . . 71

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2.14 Torque Balance in High- and Low-Latitude Ranges . . . . . . . . . . . . . . . . . . . 72
2.15 Angular Momentum Flux Due to Upflows and Downflows . . . . . . . . . . . . . . . 73
2.16 Coupling of Thermal Boundary-Layer Depth to Rotational Constraint . . . . . . . . 77
2.17 Angular Momentum Transport by Downflow Plumes . . . . . . . . . . . . . . . . . . 79
2.18 Reynolds-Stress Correlation in Upflows and Downflows for High and Low Density
Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.19 The Relation Between Meridional Circulation, Rotation Rate, and Reynolds Stressin the NSSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.1

The Solar Rotation Profile in the Meridional Half-Plane (Again) . . . . . . . . . . . 90

3.2

Simulated Rotation Contrasts for Each Thermal Boundary Condition . . . . . . . . 98

3.3

Simulated Latitudinal Rotation Profiles for Each Thermal Boundary Condition . . . 99

3.4

Simulated Full Rotation Profiles for Each Thermal Boundary Condition . . . . . . . 100

3.5

Schematic of the Different Types of Contour Tilt . . . . . . . . . . . . . . . . . . . . 102

3.6

Latitudinal Tilt-Angle Profiles for Each Thermal Boundary Condition . . . . . . . . 102

3.7

Thermal Wind Balance: The Pressure Perspective . . . . . . . . . . . . . . . . . . . 104

3.8

Thermodynamic Profiles for Each Thermal Boundary Condition . . . . . . . . . . . 106

3.9

Curl of Thermal Wind Balance: The Entropy Perspective . . . . . . . . . . . . . . . 108

3.10 Heat Transport by Busse Columns: Laminar and Turbulent . . . . . . . . . . . . . . 111
3.11 Geostrophic Balance within Busse Columns: Laminar and Turbulent . . . . . . . . . 112
3.12 Colatitudinal Energy Transport for Each Thermal Boundary Condition . . . . . . . 115
3.13 The Sun’s “Thermal Wind” Temperature Profile . . . . . . . . . . . . . . . . . . . . 117
3.14 Total Busse-Column Heat Transport for Each Thermal Boundary Condition . . . . . 117
3.15 The “Leaky Poles” of Fixed-Entropy Simulations . . . . . . . . . . . . . . . . . . . . 119
4.1

Turbulent Flows in a Hydrodynamic Dynamo Progenitor . . . . . . . . . . . . . . . . 130

4.2

Two Magnetic-Field Structures in a Bistable Dynamo . . . . . . . . . . . . . . . . . 132

4.3

Extended Evolution of Bistable Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . 133

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4.4

Polar Caps of Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.5

Flux-Transport of Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.6

Walking though a Regular Polarity Reversal . . . . . . . . . . . . . . . . . . . . . . . 140

4.7

Regular Fourfold-Wreath Cycles

4.8

Spatial Scales of the Toroidal and Poloidal Magnetic Fields . . . . . . . . . . . . . . 142

4.9

Changing Symmetry of Magnetic Field about the Equator . . . . . . . . . . . . . . . 143. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.10 Irregular Partial-Wreath Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.11 Walking through a Partial-Wreath Cycle . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.12 Modulation of Active Longitudes during a Partial-Wreath Cycle. . . . . . . . . . . 148

4.13 Periodogram of the Fourfold-Wreath Cycle . . . . . . . . . . . . . . . . . . . . . . . . 151
4.14 Temporal Evolution of Hemispheric Asymmetry . . . . . . . . . . . . . . . . . . . . . 152
4.15 Fourfold Wreaths at Different Levels of Turbulence . . . . . . . . . . . . . . . . . . . 154
4.16 Bistability in a More Turbulent Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.17 Inductive Generation of the Fourfold Wreaths . . . . . . . . . . . . . . . . . . . . . . 162
4.18 Mean-Shear Production of the Fourfold Wreaths . . . . . . . . . . . . . . . . . . . . 163
4.19 Inductive Generation of the Poloidal Magnetic Field . . . . . . . . . . . . . . . . . . 163
4.20 Inductive Generation of Partial Wreaths during a Cycle . . . . . . . . . . . . . . . . 166
5.1

A Simulated Radiative Interior Magnetically Forced into Solid-Body Rotation . . . . 193

5.2

Flow and Field Strengths in the MHD case . . . . . . . . . . . . . . . . . . . . . . . 194

5.3

Non-Axisymmetric Partial Wreaths Forcing Solid-Body Rotation . . . . . . . . . . . 196

5.4

Magnetic Torque Maintaining the MHD Case’s Tachocline . . . . . . . . . . . . . . . 200

5.5

Dynamo Action in the MHD Case’s Radiative Interior . . . . . . . . . . . . . . . . . 203

5.6

Full Morphology of the MHD Case’s Dynamo . . . . . . . . . . . . . . . . . . . . . . 205

5.7

Time-Radius Diagram of the Confining Dynamo . . . . . . . . . . . . . . . . . . . . 206

5.8

Confining Magnetism as Partial Wreaths . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.9

Cycling Frequencies of the Partial Wreaths . . . . . . . . . . . . . . . . . . . . . . . 210

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5.10 Magnetic Torque from Each Cycling Frequency . . . . . . . . . . . . . . . . . . . . . 210
5.11 Rossby Waves and Other Low-Frequency Inertial Oscillations . . . . . . . . . . . . . 213
5.12 Gravity Waves Below the Tachocline . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Chapter 1

Brief Introduction to the Global Solar Dynamo

The dynamical origin of the solar magnetic field remains one of the outstanding fundamental
problems of stellar astrophysics. The Sun’s observed surface magnetism, locally visible at thesurface as sunspots and active regions, waxes, wanes, and reverses its polarity sense fairly regularly
every ∼11 years. This cycling behavior indicates periodic changes in the Sun’s global magnetic fields(the magnetism at very large length-scales). The cycling must be caused by regenerative dynamo
action—the self-excited production of magnetic energy from kinetic energy—in the interior. Muchis still unclear regarding what sets the dynamo’s cycle period, the overall amplitudes of individual
cycles, and where in the interior the dynamo operates (for recent reviews, see, e.g., Hathaway 2015;
Charbonneau 2020).
The solar dynamo, as the source of sunspots and active regions, leads to surface magnetic
activity (flares and coronal mass ejections—or “space weather”) that profoundly influences life on
Earth. Space weather causes satellite damage, power-grid interruptions, and changes to atmosphericchemistry and global climate (e.g., Webb & Howard 2012; National Research Council 2012; Benz
2016; and references therein). One of the central goals of solar physics is thus to provide a complete
physical description of the dynamo. A satisfactory theory would reproduce the observed cycling
behavior—which is the strongest observational constraint—and predict particularly active cycles
and individual space-weather events before they occur.
In this thesis, we focus on the global solar dynamo. “Global” here means we investigate thelarge-scale spatial structure and long-time-scale temporal behavior of the Sun’s interior magnetic

2fields and fluid flows. We necessarily simplify the physical processes near the photosphere, sincethese require local simulations that are specially designed to treat the 3-D radiation field and complex equation of state in the near-surface layers. “Local” here means a Cartesian box straddlinga localized region of the solar near-surface layers, often with sphericity and global-scale flows ignored. We describe local simulations more fully in Section 1.2.1. We discuss certain observations
that are fundamentally governed by the near-surface physics, like sunspots and active regions, but
only inasmuch as they inform us about the global properties of the Sun.

1.1

Solar Observations
In the past few decades, significant progress has been made in observing the Sun. The adventof space telescopes has provided an unprecedentedly detailed view of solar active regions (e.g., vanDriel-Gesztelyi & Green 2015), the chromosphere (e.g., Carlsson et al. 2019), and the solar wind(e.g., Cranmer & Winebarger 2019). Both ground-based and space-based telescopes have also
allowed helioseismology—the study of acoustic wave-modes as they appear at the solar surface—toprecisely map interior solar thermodynamic structure and differential rotation. The differential
rotation plays a key role in the dynamo through shearing poloidal magnetic field into toroidal
field (the “Ω-effect”; e.g., Parker 1955; Rädler 1986; Pipin & Kosovichev 2011). A theoretical
dynamo model thus needs to explain not only the observed cycles and polarity reversals, but also
the differential rotation.
In this section, we provide a general background to the relevant observational aspects of the
global dynamo problem. We focus on the observations that are believed to probe the global structureof the Sun’s interior magnetism and differential rotation, and discuss the topics that are directly
relevant to the investigations presented in this thesis. These topics are the near-surface shear layer(NSSL) at the top of the convection zone (Chapter 2), the significant tilt of the isorotation contours
away from the rotation axis in the bulk of the convection zone (Chapter 3), the cyclic behaviorsof the global magnetic field (Chapter 4), and finally, the tachocline of shear at the base of theconvection zone (Chapter 5).

3
1.1.1

Cyclic Behavior in Sunspots and Solar Magnetism

Sunspots are the visual indicators of active regions, which are areas of the solar surface where
intense magnetism erupts (Figure 1.1). Typically, sunspots arrive in pairs (bipolar active regions),
with each spot forming the foot-point of a magnetic arc. Sunspot pairs appear against a backgroundof tiny granular cells (individual cells about ∼1,000 km across), which are the convective motions
at the solar surface known as granulation. Magnetized sunspots appear dark because the magnetic
field locally suppresses the convective heat transport in the surrounding fluid and thus the radiative
output of the photosphere. Sunspots are not only important as the main “output” of the solar
dynamo (which, in turn, affects life on Earth via space weather), but likely also influence the dynamo
itself. Their collective areas, field amplitudes, topologies, and lifetimes are quantities measured tohigh precision with space telescopes and have rich distributions (e.g., van Driel-Gesztelyi & Green2015 and references therein).
An individual sunspot pair is joined by lines that fall roughly along east–west parallels. The
bipolar nature of the pair (in Figure 1.1, opposite-sign line-of-sight magnetic-field components)
indicates that sunspots are formed when a largely toroidal magnetic flux rope buoyantly rises tothe photosphere. According to Joy’s law, these emerging flux ropes are not purely toroidal, but are
tilted with respect to the east–west direction. As shown schematically in Figure 1.1, the leading(east-most) sunspot is preferentially closer to the equator than the trailing (west-most) one. The
polarity of the leading spot defines the “polarity sense” of the bipolar active region. Positive
polarity sense is defined as positive radial magnetic field for the leading spot, and similarly for thenegative polarity sense. Note that positive polarity sense means a flux rope with magnetic fieldlines directed in the negative azimuthal direction.
Figure 1.1 shows snapshots of the line-of-sight (∼radial) magnetic field at the solar surface in
Cycles 22 and 23. In any given cycle, sunspot pairs in the North have polarity sense opposite to thepairs in the South. This is known as Hale’s polarity law. The tilt angles of individual sunspot pairs
are in reality widely distributed about the ∼7◦ -degree characteristic Joy’s-law tilt (e.g., Stenflo &4

Figure 1.1: Sunspot Schematic. (Upper left) Sunspots are formed when magnetic field lines
erupt through the photosphere. The arrows show how the magnetism flows out; in the center ofthe spot (net upflow), the field lines erupt outward, and surrounding the spot (net downflow), thefield lines to bend back toward the surface. (Lower left) Sunspot pairs possess a characteristic tilt,
known as Joy’s law. The line joining two spots (tilt direction) is inclined with respect to east–west
parallels. (Right) Line-of-sight magnetic field on the solar disc for two adjacent sunspot cycles. In
solar-based spherical coordinates, line-of-sight field is basically radial field near disc-center. Thesesolar-disc “magnetograms” illustrate Hale’s polarity law: Sunspot groups have opposite polarity
sense in the Northern and Southern Hemispheres and each hemisphere’s polarity sense reverses
from one cycle to the next. Image credit: David Hathaway, solarcyclescience.com.

5
Kosovichev 2012; Li 2018) and a non-negligible fraction (∼10%) of bipolar active regions thus havepolarity senses opposite to what Hale’s polarity law would predict. We discuss some implicationsof these “anti-Hale” active regions in Chapter 4.
Sunspots vary in number and strength every ∼11 years—a periodicity in surface magnetic
energy known as the sunspot cycle. Hale’s polarity law, in addition to stating that the opposite
hemispheres have on average opposite polarity senses, states that the polarity sense in each hemisphere reverses from one cycle to the next. Because of the polarity reversal, a full Hale polarity
cycle, 22-year cycle, or simply, solar cycle, is ∼22 years. Several such solar cycles are shown in
Figure 1.2. Sunspots first appear at mid-latitudes, and then at sites (active latitudes) closer tothe equator as the cycle progresses, in a pattern of equatorward propagation. This pattern also
resembles butterfly wings, so time-latitude plots like Figure 1.2 are called butterfly diagrams. We
note that in Figure 1.2, the sunspots themselves are not propagating equatorward, just the active
latitudes of field emergence.
Sunspots decay slowly into small-scale flux filaments (e.g., McIntosh 1981; Wang et al. 1991;
van Driel-Gesztelyi et al. 1999; Hathaway & Choudhary 2008). Opposite-polarity filaments mostly
cancel, but some residual flux of a preferred polarity remains due to Joy’s law. This residual flux
is observed to be transported poleward (Figure 1.2) and is believed to form the seeds of the next
cycle in the flux-transport, or Babcock-Leighton, dynamo theories (Babcock, 1961; Leighton, 1969).
According to Figure 1.2, the poleward-propagating flux is strongest at cycle minimum. Unlike
equatorward “propagation” from the emergent sunspot pattern, poleward propagation likely does
represent moving flux, advected by the meridional circulation (e.g., Howe et al. 2013; Petrie 2015;
and references therein). After the flux is swept poleward, the meridional circulation (or possibly
downward advection by convective plumes) pumps the flux into the tachocline, where it can get
sheared by the Ω-effect into the opposite-polarity toroidal field of the next cycle (e.g., Dikpati &Gilman 2009; Karak et al. 2014; Charbonneau 2020; and references therein).
We have reasonably frequent sunspot sketches and counts for the last ∼400 years. These
historical sketches, or sunspot numbers, which were standardized after about 1849 by defining

6

Figure 1.2: Magnetic Butterfly Diagram. Photospheric surface line-of-sight (∼radial) magnetic field is shown in time-latitude space. Each temporal pixel set (strip parallel to y-axis) is alongitudinal average of the field over the observed solar disc. Hale’s polarity law is clearly visible,
as is the migration of the average line-of-sight magnetic field to the equator (the butterfly wings).
The polar regions have opposite polarity senses that switch sign from one cycle to the next near
the time of solar maximum. The flux at the poles peaks near solar minimum. Image credit: David
Hathaway, solarcyclescience.com.

7
the International Sunspot Number (Wolf, 1861), give a rough proxy of the Sun’s magnetic-fieldstrength.1 For as long as sunspot numbers have been recorded, the ∼11-year periodicity is clearlypresent (Hathaway, 2015). Other proxies—the abundances of the radioisotopes 10 Be and 14 C intree rings and glacial ice cores—yield even rougher magnetic-field activity proxies, but nonethelessclearly show the presence of the primary 11-year sunspot cycle since the beginning of the Holocene
period, about 11,700 years ago (e.g., Usoskin 2013 and references therein). The isotope proxies also
confirm the presence of the multi-cycle Gleissberg modulation, which has a period of ∼100 years(e.g., Gleissberg 1939; Peristykh 2003). The Gleissberg cycle is clearly a statistically significant
feature, but different analyses yield different assessments of its period (e.g., Hathaway et al. 1999;
Ogurtsov et al. 2002).
Another important long-term behavior of the solar cycle is the tendency toward “grandminima.” These are long intervals when multiple cycles are much weaker than average. The mostprominent example is the Maunder Minimum (e.g., Maunder 1890; Eddy 1976), which clearly shows
record-low sunspot numbers during an extended period from 1645–1715. The grand minima also
show up in the isotope proxies; 27 statistically significant minima have thus been identified during
the Holocene period (Usoskin et al., 2007). All in all, the Sun appears to exist in a magneticallyquiet state, corresponding to a grand minimum, about one-sixth of the time (Usoskin, 2013).
We also point out the active longitudes observed from the sunspot record, which we discuss at
length in Chapter 4. Active longitudes have long been hypothesized to explain a persistent ∼27-day
periodicity in the distribution of active regions in longitude and time (e.g., Svalgaard & Wilcox
1975; Bogart 1982; Balthasar & Schussler 1983; Antonucci et al. 1990; Neugebauer et al. 2000;
Henney & Harvey 2002; Kitchatinov & Olemskoi 2005; Ivanov 2007). The observed periodicity isnear the solar rotation rate and would indicate that there are preferential solar longitudes (longitude
measured in a rotating frame) that are more active than average for long intervals. These longitudes
may also correspond to increased amplitude in other activity indicators, like solar flares (e.g., Bai
1

See Hathaway 2015 and references therein for an in-depth discussion of the historical sunspot record and its
complicated correlation with the Sun’s magnetic field.

8& Sturrock 1987; Gyenge et al. 2016). How long active longitudes typically persist is the sourceof debate, but they may last up to century-long time-scales (e.g., Berdyugina & Usoskin 2003).
Interestingly, the active longitudes in each hemisphere may be decoupled, with slightly different
periodicities (both near ∼27 days) in the North and South (e.g., Bai 1990). This would suggestthat active longitudes may be related to the Sun’s hemispheric asymmetry, which we now discuss.
There is a long-observed statistically significant hemispheric asymmetry with respect to mostmeasures of magnetic activity, which we discuss more fully in Chapter 4. The level of asymmetry(“North minus South” over “North plus South”) is typically around ∼20% (e.g., Norton & Gallagher
2010). It shows up in almost all activity proxies, including sunspot number, total sunspot area,
prominences, chromospheric emission, and flare occurrence (e.g., Newton & Milsom 1955; Roy1977; Swinson et al. 1986; Chang 2009; Norton et al. 2014; Hathaway 2015; Deng et al. 2016). It islargely non-cyclic, having multiple “periodicities” (peaks in period-space with large widths) ranging
between ∼10 and ∼50 years (e.g., Duchlev & Dermendjiev 1996; Ballester et al. 2005; Deng et al.
2016; and references therein). The source of the hemispheric asymmetry is still not well understood,
but it is usually thought to arise from the nonlinearity inherent to the dynamo system. This resultsin coupling between even (equatorially symmetric) and odd (equatorially antisymmetric) dynamo
modes (e.g., Kleeorin & Ruzmaikin 1984; Tobias 1997; Hotta & Yokoyama 2010; Weiss & Tobias
2016). This same coupling may lead to long periods of weak dynamo amplitudes reminiscentof grand minima. Additionally, stochastic fluctuations in realizations of the convective flows arethought to lead to “mixed-mode” solutions (typically a dipole eigenmode of the dynamo equations,
mixed with a quadrupole eigenmode). There can then be frequency beating between the two modes(e.g., Moss et al. 1992; Charbonneau 2007; Usoskin et al. 2009; Belucz & Dikpati 2013; Passos &Charbonneau 2014; Brun et al. 2015), which recently has been hypothesized to yield hemisphericasymmetry with the observed periodicities (Schüssler & Cameron, 2018).

9
1.1.2

Helioseismology

Helioseismology is the study of the acoustic wave-modes that are trapped below the photosphere (e.g., Christensen-Dalsgaard 2002; Howe 2009; and references therein). Because the waves
travel deep into the solar interior, they can be used to infer the internal structure of the Sun. Helioseismic data consists of two quantities, each spatially resolved across the solar disc: Doppler shiftsof the wavelengths of various spectral lines (full-disc “Dopplergrams”) and full-disc continuumintensity maps. The Dopplergrams can be interpreted as the line-of-sight component of the fluid
velocity at the photosphere (strictly, at the formation height of the given spectral line).
The velocity and intensity at the solar surface fluctuate on the timescale of ∼5 minutes. Thiswas first reported by Leighton et al. (1962) as the “five-minute oscillations.” Initially, the fiveminute oscillations were thought to be resonant acoustic modes trapped in the solar atmosphere(e.g., Howard 1967; Frazier 1968; Tanenbaum et al. 1969). The conclusion that these waves were
not surface phenomena, but rather, global normal modes of the Sun, was first discussed in Ulrich(1970) and Leibacher & Stein (1971). This interpretation was confirmed by Deubner (1975) and
Rhodes et al. (1977), who showed that that the wavenumber-frequency power spectrum of the
five-minute oscillations was concentrated at discrete temporal frequency (f ) “ridges” for a given
spherical-harmonic-degree wavenumber (`). This equal spacing in frequency was predicted by thetheory of global oscillations (e.g., Cox 1980; Unno et al. 1989; Gough & Kosovichev 1993). The
driving mechanism for the five-minute oscillations is believed to be the highly turbulent convection
near the photosphere, which in turn is driven by radiative cooling (e.g., Goldreich & Keeley 1977).
Although Deubner (1975); Rhodes et al. (1977) identified clear ridges in wavenumber-frequency
space, they could not distinguish between individual normal modes of oscillation. This was firstachieved by Claverie et al. (1979), who used Dopplergrams integrated across the whole solar discto identify the low-degree (i.e., large-wavelength) normal modes. Helioseismology as we know it
currently (according to Basu 2016) began when Duvall & Harvey (1983) measured the frequencies
of a large number of modes that covered a wide range of horizontal wavenumbers. The frequencies

10of several millions of individual modes can be measured and are used to probe the solar interior(e.g., Libbrecht et al. 1990; Thompson et al. 2003).
To precisely determine normal-mode frequencies, the acoustic power spectra must have highresolution in frequency space. This requires helioseismic data to be continuously collected forlong time intervals, which can be accomplished either with a space telescope or with groundbased telescopes located at multiple points around the globe.2

One of the most prominent

ground-based observatories is the Global Oscillation Network Group (GONG; Harvey et al. 1996),
managed by the National Solar Observatory (NSO). GONG collects solar-disc images at 5-arcsecond
diffraction-limited resolution. The original GONG images were 512×512 pixels (corresponding to
the telescope’s diffraction limit), but are now 1,024×1,024 pixels since detector upgrades in 2001.
GONG has continuously observed medium-degree (` . 300) acoustic modes since 1995. 3
Higher-resolution images (composed of higher-quality data because of the absence of atmospheric seeing effects) can be obtained by space telescopes. NASA/ESA’s Michelson-Doppler
Interferometer (MDI) aboard the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995)
collected 4-arcsecond full-disc images (1,024×1,024 pixels) from 1996–2011. A followup mission,NASA’s Heliospheric and Magnetic Imager (MDI) aboard the Solar Dynamics Observatory (SDO;
Scherrer et al. 2011) has collected the highest-resolution images (1 arcsecond or 4,096×4,096 pixels)
and has been operational since 2011, with the SDO mission itself extended until 2030. GONG and
HMI (and in the past, MDI) both collect images with a cadence of ∼1 minute, allowing the precise
identification of high-frequency modes.
Global helioseismology refers to the study of the Sun’s global (normal) acoustic modes (e.g.,
Howe 2009; Basu 2016; and references therein). For a given normal mode, properties of solar structure in the regions where the mode has high amplitude perturb the theoretical mode frequency.
The measured frequency perturbation thus senses different parts of the Sun. When many frequency
2

Another way to get continuous observations is to observe from near the South Pole (e.g., Jefferies et al. 1988;
Duvall et al. 1993; Braun et al. 1996). This data yielded the first forays into time-distance helioseismology (briefly
described below), and also suggested that sunspots are efficient absorbers of acoustic-mode energy.
3
Converting angular resolution to spatial resolution on the solar disc (ignoring curvature) yields 1 arcsecond ∼700 km (about the size of a granule).

11
perturbations are collected, they can be inverted to yield a spatial map of the perturbing quantity in the solar interior. Spherically symmetric solar structure (e.g., hydrostatic stratification,
adiabaticity, and temperature) can be obtained precisely from roughly 0.06R

to 0.96R

via 1-Dinversions (e.g., Basu 2016). The spherically asymmetric rotation rate (as described below) canbe obtained as a function of radius (∼0.5R

to ∼1R ) and latitude (equator to ∼75◦ ) via 2-D

inversions. There are many unique inversion techniques, each with their own assumptions, whichyield slightly different results, errors, and spatial resolution (see examples of 1-D inversions in, e.g.,
Gough 1985; Däppen et al. 1991; Basu et al. 2009). All inversions are similar, however, in that
their spatial resolution in radius becomes much finer as the photosphere is approached from below.
This is because the high-degree acoustic modes are trapped in the near-surface layers. There aremany more high-degree modes than low-degree ones (decreasing the inversion error) and their radial
wavelengths are shorter (making the inversion resolution finer).
The high precision of global inversions (∼1% for solar structure) yielded one of the major
triumphs of helioseismology in helping to resolve the solar neutrino problem. This problem canbe stated as: “The number of neutrinos coming from reactions in the Sun’s thermonuclear core,
as measured by the first neutrino detectors, was too small by a factor of ∼3 compared to whatstandard solar models would predict.” This discrepancy was first realized by Raymond Davis andJohn Bahcall in the 1960s. They designed the first neutrino detector, which consisted of a large∼100,000-gallon tank of perchloroethylene (C2 Cl4 ; i.e., dry-cleaning fluid) at the Homestake Mine inSouth Dakota (e.g., Davis 1964; Bahcall 1964; Davis et al. 1968). The tank was installed 4,900 feetunderground in an old mineshaft, along with instrumentation to collect and count the radioactive
37 Ar atoms produced by the weak reaction, ν + 37 Cl → e− + 37 Ar.
eSequestering the tank likethis shielded it from cosmic rays. See a detailed review of the “Homestake experiment” and its
significance in Davis (2003). A solar neutrino would interact with the chlorine about 15 times every
month. Note that Homestake was only sensitive to the 8 Be neutrinos with energies > 0.81 MeV,
which come from the reaction 8 B → 8 Be + e+ + νe in the Sun’s thermonuclear core (this reaction
takes place on the p–p III branch, which is least dominant of all the p–p branches in the Sun). The12experiment was insensitive to the more fundamental fusion reaction 1 H + 1 H → 2 D + e+ + νe thatis the first step in all the p–p branches.
Davis and Bahcall measured a neutrino flux of ∼3 SNU,4 while the standard solar modelat the time predicted ∼20 SNU (Bahcall & Shaviv, 1968). The solar models were refined and theHomestake experiment’s measurements improved, but the discrepancy persisted. Final values were2.56 ± 0.16 (statistical) ± 0.16 (systematic) SNU measured by Homestake (Cleveland et al., 1998)
and solar-model predictions of ∼8 SNU (e.g., Bahcall et al. 2001). Additional neutrino experiments
using other techniques were devised over the years, notably an underground tank of water at the
Kamioka Mine in Japan (e.g., Suzuki 1995; Koshiba 2003) and an underground tank of gallium in
Italy (the GALLEX experiment; e.g., Anselmann et al. 1995. GALLEX measured the first flux fromneutrinos produced by the fundamental 1 H + 1 H fusion reaction. All these experiments consistently
measured significantly fewer neutrinos than predicted by solar models, usually fewer by a factor of∼3. For their pioneering neutrino measurements, Raymond Davis and Masatoshi Koshiba sharedthe 2002 Nobel Prize in Physics, along with Riccardo Giacconi for his work in X-ray astronomy.
Nuclear physics predicts a neutrino flux that is highly sensitive to the temperature Tc of thesolar core (scaling like Tc18 for the 8 Be neutrinos measured in the original Homestake experiment).
After the initial results from Homestake, there were many attempts to “tweak” various solar models
to make the core temperature lower (for a review, see Christensen-Dalsgaard 2021 and references
therein). Such attempts involved a range of ideas: A rapidly rotating solar core to reduce therequired temperature for hydrostatic balance (e.g., Bartenwerfer 1973; Demarque et al. 1973);
substantial mixing of the core to increase the core hydrogen abundance and thus decrease therequired Tc for solar energy production (e.g., Bahcall et al. 1968; Ezer & Cameron 1968; Dilke &Gough 1972); and even the presence in the core of the weakly interacting massive particles (WIMPs)
invoked to explain dark matter, which could contribute to the solar energy transport and lessen therequired radiative temperature gradient (e.g., Steigman et al. 1978; Spergel & Press 1985; Faulkner
A solar neutrino unit (SNU) corresponds to 10−36 solar neutrino reactions per second per chlorine nucleus; hence
the very large tank!
4

13& Gilliland 1985). To remain consistent with the helioseismically inferred Tc , which was constrainedto ∼1% precision and agreed well with the standard solar models, most of these tweaked models had
to be ruled out (e.g., Elsworth et al. 1990; Dziembowski et al. 1990; Christensen-Dalsgaard 1991).
It became increasingly clear that if the solar models were right, and the neutrino experiments werereliable, then the underlying assumptions of particle physics had to be revisited. The resolution,of course, was bequeathing some mass to the neutrino, which would then oscillate between threedifferent types (or flavors) while on transit to Earth (e.g., Haxton et al. 2013). The first detectors
had been designed to detect only one of these flavors (the electron-neutrino νe ) and thus had beenmeasuring only ∼1/3 of the total solar-neutrino flux. Definitive proof of neutrino oscillations camefrom the heavy-water (2 H2 O) solar-neutrino detector at the Sudbury Neutrino Observatory (SNO)in Canada (e.g., McDonald 2016). Arthur McDonald shared the 2015 Nobel Prize in Physics withTakaaki Kajita, who detected the oscillations of another neutrino flavor (the muon-neutrinos νµ ),which are created by cosmic-ray interactions in the upper atmosphere (Kajita, 2016).
We note that the Sun’s complicated equation of state plays a large role in any comparisonof solar models with global helioseismology. Near the outer surface, partial ionization regions,kinetic effects associated with the non-collisional plasma, and non-equilibrium thermodynamics
all become important. In the deep interior, the equation of state for the opacity must assume
some uncertain input physics, like nuclear reaction rates, elemental abundances, and atomic crosssections. For maximum accuracy and reduced computational overhead, most solar models use pretabulated equations of state. Two popular choices are the Livermore-based OPAL tables (e.g.,
Rogers et al. 1996) and the international Opacity Project (OP) tables (e.g., Berrington 1995). The
latter solves the Mihalas-Hummer-Däppen (also referred to as “MHD,” unfortunately) equation ofstate (Mihalas et al., 1988; Hummer & Mihalas, 1988; Däppen et al., 1988; Mihalas et al., 1990).
State-of-the-art tables like OPAL and OP were instrumental in bringing solar models more into
accord with helioseismology, as happened during the solar-neutrino experiments (e.g., ChristensenDalsgaard et al. 1996; Guzik & Swenson 1997; Basu et al. 2000).55Significant issues remain,

The equation of state assumed by a particular solar model substantially affects the predicted oscillation frequen-

14
however, which can be summed up as the “solar modeling problem” (see the review by Buldgen
et al. 2019). Quantities like the base of the convection zone, radial profiles of the sound-speed
and density, and the helium abundance are precisely measured by helioseismology and disagree
somewhat with those predicted by standard solar models including the most up-to-date equationsof state (e.g., Basu & Antia 2008; Asplund et al. 2009; Nordlund et al. 2009; Palme et al. 2014;
Asplund et al. 2021; and references therein).
Global helioseismology is the best tool for measuring solar structure and the internal rotation
rate (see the next Section 1.1.3), but it is insensitive to other quantities like the meridional circulation and non-axisymmetric flows. Local helioseismology (e.g., Duvall 1998; Gizon & Birch 2005;
and references therein) is the study of the short-lived acoustic waves that do not have time to travel
all the way around the Sun. Because they do not have time to form standing waves horizontally,
these waves are not normal modes and do not show up at discrete wavenumbers in the global power
spectra. The waves will nonetheless travel several horizontal wavelengths before decaying and canbe observed over localized horizontal regions of the solar disc. Because the waves are still resonant
in radius, observing them in a given localized region for a long enough time interval can yield a
3-D view of the solar interior underneath that region.
Local helioseismology is really a combination of techniques for measuring the non-normalmode acoustic power. In ring-diagram analysis, local power spectra are formed from placing asmall “tile” on the solar disc, collecting a continuous stream of high-cadence “tiled” Dopplergrams,
and computing a 3-D Fourier transform of the resulting data cube (e.g., Hill 1988; Schou et al.
1998; Haber et al. 2000, 2002). The Fourier transform of this cube reveals nearly circular “rings”
in a kx -ky diagram at fixed f (where kx and ky are the components of the horizontal wave vector).
Much as in global helioseismology, perturbations in the measured frequencies are sensitive to theproperties of the Sun below the tile, and can be inverted to yield the perturbing quantities. In timedistance helioseismology, the travel times of acoustic modes are measured between two locations on
cies, and thus changes the level of agreement between the model and helioseismology. See Basu (2016); her Figure6.

15
the solar surface, connected by a theoretically known ray path. Differences between the theoreticaland measured wave travel times are sensitive to the internal fluid properties along the connecting
ray path (e.g., Duvall et al. 1993; Jensen & Pijpers 2003; and references therein). Again, measured
anomalies from many modes can be inverted to infer the spatial structure of the underlying anomaly.
Local helioseismic analyses have given subsurface flow maps of the horizontal velocity field(e.g., Hindman et al. 2004; Zhao & Kosovichev 2004; Haber et al. 2004; Hanasoge et al. 2012; Greeret al. 2015, 2016). These flow maps have been used to measure the Rossby number and coherencedepth of supergranulation in the outer layers (Hanasoge et al., 2012; Greer et al., 2015, 2016), thepresence of diverging flows and complex vortices below a sunspot (e.g., Zhao & Kosovichev 2004;
Haber et al. 2004), and the radial dependence of the recently observed solar Rossby waves (e.g.,Löptien et al. 2018; Gizon et al. 2020b; Proxauf et al. 2020; Gizon et al. 2021). As described in
Chapter 2, subsurface flow maps may prove helpful in constraining the interior Reynolds stresstorque that is believed to drive near-surface shear. This Reynolds stress has yet to be measured.
Local helioseismology has also probed the Sun’s internal meridional circulation (e.g., Giles
et al. 1997; Haber et al. 2004; Hathaway 2012; Chen & Zhao 2017; Mandal et al. 2018; Gizon et al.
2020a; Braun et al. 2021), often with conflicting results. For example, there seems to be a growing
body of evidence that the Sun may have multiple circulation cells in each hemisphere (e.g., Chen &Zhao 2017; Mandal et al. 2018). It should be noted that global simulations have consistently produced solar-like differential rotation (fast equator, slow poles) only with multiple circulation cells,
and anti-solar differential rotation with single circulation cells (e.g., Featherstone & Miesch 2015).
Part of the problem is that meridional circulation is inherently difficult to measure helioseismically,
because it perturbs the acoustic-mode frequencies at second-order, unlike the differential rotation,
which perturbs the frequencies at first-order (e.g., Braun & Birch 2008; Zhao et al. 2012).

1.1.3

The Internal Rotation Rate of the Sun

Throughout this thesis, we denote the interior angular velocity of the Sun (and later, of
simulations), by Ω, which we also call the rotation rate. We define Ω ≡ hvφ i /r sin θ + Ω0 , where16
hvφ i is the longitudinally and temporally averaged azimuthal velocity in a frame rotating at Ω0 . In
simulations, Ω0 is the chosen “reference” rotation rate and for the Sun, we define Ω0 ≡ Ω
the sidereal Carrington rate (Ω= 2.87 × 10−6 rad s−1 or Ω /2π = 457 nHz or 2π/Ω

to be

= 25.38days). Here, r and θ are the standard spherical coordinates (radius and colatitude, respectively).
We report the rotation rate in rad s−1 . In keeping with helioseismic tradition, we also refer to the“rotation frequency” Ω/2π, which we report in Hz.
The helioseismically inferred rotation frequency of the Sun is shown in Figure 1.3. Closest
to the photosphere is the near-surface shear layer (NSSL). This is a layer of thickness ∼ 0.05R ,wherein the rotation frequency drops by about 5% with increasing radius. This effect is roughly
latitude-independent. A 5% reduction in rotation frequency over 5% of the solar radius implies
that Ω ∼ 1/r in the NSSL. This dependence is notable, since a region of homogeneous angularmomentum density (as the NSSL was originally thought to be; e.g., Foukal & Jokipii 1975) would
have Ω ∼ 1/r2 . The radial shear in the NSSL (although likely weaker than the shear in thetachocline at the base of the convection zone; discussed below) is quite strong. Given the NSSL’s
proximity to sunspots, it may play a more fundamental role in the solar dynamo cycle than is usuallyassumed. Brandenburg (2005) in particular has advocated for a “distributed dynamo shaped bynear-surface shear” (see also Pipin & Kosovichev 2011). The maintenance of the NSSL is still mostly
unexplained. We present our attempts to systematically model the NSSL using global simulationsin Chapter 2.
Going deeper into the interior, the isorotation contours are tilted with respect to the solar
rotation axis in the bulk of the convection zone. This tilt was a surprise for two reasons. First,
mixing-length theory predicted that the motions in most of the convection zone should be strongly
rotationally constrained, meaning a Taylor-Proudman-like constraint (i.e., invariance along therotation axis) should hold (e.g., Pedlosky 1987; Kitchatinov & Ruediger 1995; Durney 1999; Rüdiger& Hollerbach 2004). Second, early nonlinear global simulations mostly yielded cylindrically aligned
isorotation contours (e.g., Gilman 1977; Glatzmaier 1984; Miesch et al. 2000; Brun & Toomre
2002; Brun et al. 2004). Miesch et al. (2006) found that the Taylor-Proudman constraint in global

17

Figure 1.3: The Internal Rotation Rate of the Sun. The helioseismically inferred rotation
frequency is shown, averaged in both longitude and time and assuming an equatorially symmetricprofile. Rotation frequency Ω/2π is plotted in units of nHz. For geometrical reference, the baseof the convection zone is r/R ∼ 0.72 and we mark the approximate locations of the two shear
layers in each panel. (a) Rotation frequency in the meridional plane. The x-axis corresponds to thesolar equator and the y-axis to the axis of rotation. In the colorbar, the isorotation contours areequally spaced 10 nHz apart. The dashed lines are inclined by 25◦ with respect to the rotation axis
and roughly match the tilt of the isorotation contours at mid-latitudes. (b) Same solar rotation
frequency as depicted in panel (a), but plotted along radial lines constant solar latitudes. The
thickness of the curves represents the error in the helioseismic inversion; error increases with depth.
Figure adapted from Howe 2009.

18
simulations could be broken by imposing a latitudinal entropy gradient at the base of the convection
zone. We discuss our progress in identifying a physical mechanism to drive such a latitudinal entropy
gradient (and thus potentially tilt the Sun’s isorotation contours) in Chapter 3.
Possibly the most striking feature of the helioseismically measured rotation profile is thetachocline, a thin, largely radial boundary layer in which the strong differential rotation in theconvection zone transitions to nearly solid-body rotation in the stably stratified radiative interior(e.g., Brown et al. 1989; Thompson et al. 2003). We note that a radial shear layer in the Sun(though not the tachocline) was postulated as early as Dicke (1964). Dicke noted that if the Sun
were slightly oblate due to the core rotating much more rapidly than the observed surface rotation(with an implied radial shear layer in between), the solar gravitational potential could be modified.
For a rapidly rotating core (which could be caused by the solar wind torquing the outer layersof the Sun before the inner layers), classical Newtonian gravitation could explain the precessionof Mercury’s perihelion instead of general relativity. Dicke & Goldenberg (1967) attempted tomeasure the solar oblateness—(Req − Rpol )/R , where Req and Req are the Sun’s equatorial andpolar radii, respectively—finding a value of 5 × 10−5 . This, in turn, would imply a core rotating
about 20 times faster than the surface, which would be enough to explain the observed precessionof Mercury’s perihelion.

6

There have been numerous attempts to seismically measure therotation rate of the core, with most evidence indicating approximately solid-body rotation between∼0.2R

and ∼0.7R(i.e., the base of the tachocline; see Howe 2009, the references in her Table1). Helioseismology thus largely supports Einstein (and not Dicke), although rapid rotation below∼0.2R cannot be completely ruled out.
Because only the long-wavelength (low-`) acoustic modes penetrate to the base of the convection zone (and because there are fewer of them), the spatial resolution of the global inversions in
the tachocline region is fairly coarse. Different inversion techniques yield slightly different results
6

Dicke (1964)’s initial suggestion of a rapidly rotating core incited a slew of controversy and had the upside of
sparking significant interest in the Sun’s internal rotation rate. Subsequent attempts to measure the solar oblatenessyielded values much lower than Dicke’s first measurement (e.g., Roxburgh 1967; Kraft 1967; Goldreich & Schubert
1968; Fivian et al. 2008), as was also confirmed by Dicke himself (Dicke et al., 1986, 1987). As of today, the solar
oblateness still has not been precisely measured, but the situation may improve with next-generation space telescopes(see Damiani et al. 2011; Rozelot et al. 2019; and references therein).

19
for tachocline thickness, but generally all lie around ∼0.05R (e.g., Howe 2009; her Table 2). Thatvalue likely reflects the resolution of the inversions at ∼0.72R , and the solar tachocline may infact be much thinner. For example, an alternative estimate of the tachocline’s thickness was given
in Elliott & Gough (1999). The helioseismically measured sound-speed is discontinuously higher
than the sound-speed predicted by standard solar models in the tachocline (this is known as the“tachocline anomaly”; e.g., Basu & Antia 2008). Elliott & Gough (1999) invoked enhanced mixing
caused by the supposed tachocline’s meridional circulation (Gough & McIntyre, 1998) to efficiently
mix the tachocline layer in addition to the convection zone. This would lessen the amount of heliumin the tachocline compared to standard solar models (which include effects from helium settlingand the assumption that the convection zone is well mixed), lower the mean molecular weight, andthus increase the sound-speed. Using this simplified tachocline-mixing model in comparison withhelioseismic structure inversions, Elliott & Gough (1999) inferred a tachocline width of ∼0.02R .
We also note that there may be a significant prolateness to the tachocline, with the equatorial
thickness smaller than the high-latitude thickness (difference in thicknesses of about 0.02R ; e.g.,
Charbonneau et al. 1999a; Basu & Antia 2003). Finally, Howe et al. (2000) reported a ∼1.3-year
periodicity in the equatorial rotation rate near the tachocline using early GONG and SOHO/MDI
data (1995–1999). This claim received considerable attention, since tachocline variations could
yield dynamo and magnetic activity variations at a similar period, which might, in turn, explain
the observed ∼1-year periodicity measured from heliospheric and geomagnetic data (e.g., Silverman& Shapiro 1983; Richardson et al. 1994).
Finally, both global and local helioseismology have confirmed significant torsional oscillationsin the solar differential rotation, originally detected from the synoptic velocity maps at the Mount
Wilson 150-foot tower (Howard & Labonte, 1980). See also the analysis by Ulrich (2001) using
Mount-Wilson surface-velocity data. Small (±0.5%) variations in the subsurface rotation rate
propagate equatorward with the active sunspot latitudes, indicative of deep-seated flow patterns
dynamically coupled to the solar cycle. Global inversions (e.g., Howe et al. 2000; Vorontsov et al.
2002) have shown that the torsional oscillations extend deep into the convection zone, especially at20high latitudes, where they may reach all the way to the tachocline. At low latitudes, the inversions
support a picture wherein the torsional-oscillation pattern originates in the convection zone andpropagates both upward and equatorward (e.g., Vorontsov et al. 2002; Basu & Antia 2003). At
high latitudes, by contrast, fast and slow rotation bands propagate poleward (e.g., Antia & Basu
2001; Schou 2003), suggesting they are related to the flux-transport inferred from the solar butterfly
diagram (Figure 1.2).

1.2

Solar Modeling
The observational data just described imposes strong constraints on theories of the solar

dynamo. A successful theory must reproduce: (1) The temporal behavior of the magnetic field(e.g., the 22-year solar cycle, the Gleissberg modulation, and the grand minima) and (2) the solar
differential rotation (e.g., the tilted isorotation contours and the two shear layers). Theoretical
calculations (both analytical and numerical) have made much progress in the last few decades, but
still fall short of describing some of the major observed behaviors. In this section, we briefly goover recent progress in theoretical calculations, particularly the global simulations. We describe
the major triumphs simulations have achieved and the fundamental mysteries that remain.

1.2.1

Modeling of Near-Surface Convection and Active Regions

Simulating sunspots and active regions generally involves utilizing the equations of magnetohydrodynamics (MHD) while simultaneously treating the radiation field (radiative MHD, or
RMHD). As the photosphere is approached from below, the mean-free-path of a photon becomescomparable to the dynamical length-scales associated with the convection (in fact, the coolingcaused by photons is what drives the convection). Calculating the radiation field accurately thus
becomes a formidable problem, since 3-D, non-gray radiative transport must be taken into account.
Because of the heavy computational overhead, near-surface simulations are usually local, treatinga small Cartesian box straddling the photosphere.
The radiation field is treated by sorting it into a small number of groups in frequency space

21
and then solving the time-dependent radiative transfer equation (e.g., Vögler et al. 2004; Rempel
et al. 2009). The “R” in RMHD then boils down to calculating a realistic equation of state for the
opacities as functions of density and temperature. Usually this equation of state involves tables,
like OPAL or OP, mentioned in Section 1.1.2. Several numerical tools for simulating the radiatively
driven near-surface “magneto-convection” in the Sun have been developed. Comparisons of thesecodes appear to agree quite well with respect to observable properties like emergent intensity andgranular convective topology, but have small differences that are attributable to different treatmentsof the radiation field and/or the equation of state (e.g., Beeck et al. 2012).
One of the triumphs of the local simulations has been the reproduction of active regions, solar
flares, and coronal mass ejections, with magnetic-field strengths and flow properties remarkably
similar to observations (e.g., Cheung et al. 2010; Stein & Nordlund 2012; Rempel & Cheung 2014;
Rempel 2016; Chen et al. 2017). Granulation is also reproduced quite well, with the observed fluxexpulsion of granules (the sweeping of small-scale magnetic flux into inter-granular lanes) correctly
captured by the simulations (e.g., Stein & Nordlund 2000, 2006; Rempel et al. 2009). The local
models of active regions usually treat an individual sunspot as it emerges in the outermost layers of
the Sun (i.e., the outer ∼30,000 km or so). We note, however, that recently very large calculations
have captured the rise of a buoyant flux rope through a whole convection zone from base to surface,
including sophisticated near-surface physics. These simulations tracked the flux rope through its
eruption as a bipolar active region, with results remarkably similar to observations (see Hotta et al.
2019; Hotta & Iijima 2020).
With the exception of the “hero” calculations just mentioned, the main difficulty local simulations face is the absence of self-consistently generated large-scale flows and magnetic fields. Theirboundary conditions (what they are “fed”) are thus inconsistent with realistic stellar circulations,differential rotation, and global magnetic-field topology. Conversely, the global models described
below suffer from unrealistic treatments of near-surface physics. In this regard, progress has been
made by “patching” a local near-surface simulation onto a global simulation. For example, Chenet al. (2017) simulated the buoyant rise of a flux rope to form an erupting active region, using

22
an initial magnetic configuration taken directly from a global simulation (Fan & Fang, 2014). We
speculate that such patching of local models onto global models may be the future path by which
numerical modeling makes detailed contact with solar dynamo observations. Such efforts are currently underway as part of the large multi-university “Whole Sun” collaboration, which has been
funded for six years by an ERC Synergy Grant (e.g., Brun et al. 2017; Strugarek et al. 2017; Brunet al. 2022). This project aims to use advances in supercomputing power to develop next-generation
numerical solar simulations (including the radiative interior, convection zone, chromosphere, andcorona) to run on future exascale machines.

1.2.2

Global Spherical-Shell Dynamo Simulations

The first global dynamo models were based on mean-field theory, which is still the speaking
language of the global simulation community. In light of the solar cycle, a key goal of meanfield theory is to describe the physical processes leading to a “dynamo loop,” or the achievementof a polarity reversal. We previously described one such loop, the Babcock-Leighton, or fluxtransport, dynamo. Another commonly invoked dynamo loop is the class of αΩ-type dynamos(e.g., Parker 1955). Here, the toroidal magnetic field is produced from poloidal field via mean
shear by the differential rotation (the already-mentioned “Ω-effect”). The poloidal magnetic field isthen produced from toroidal magnetic field by the electromotive force (emf) associated with helicalconvection (the “α-effect”; e.g., Moffatt 1963; Brandenburg 2005). Helical motions (or “cyclonic
events,” following the terminology of Parker 1970) can locally twist the toroidal magnetic field lines
into loops. Each loop is associated with an elemental current, and thus an emf. If the magnetic
diffusion is low enough, the net current from these cyclonic events can convert toroidal into poloidal
field (see Figure 7.2 of Moffatt 1963).
In the case of the Sun, because the Coriolis force tilts the toroidal field slightly (consistent with
Joy’s law), the α-effect creates poloidal field in the opposite direction to its previous orientation,
thus creating a polarity reversal and completing the dynamo loop.7 While the nonlinearity of the7

In comparison to the flux-transport dynamos, the α-effect here takes the place of the transport of small-scale

23
problem is oversimplified, processes like the Ω- and α-effects consistently help the interpretations ofsimulations (e.g., Brown et al. 2010, 2011; Racine et al. 2011; Augustson et al. 2013; Warnecke et al.
2014; Augustson et al. 2015; Guerrero et al. 2016a; Matilsky & Toomre 2020a; Bice & Toomre 2020;
Brun et al. 2022). Pipin (2012); Brandenburg (2018) give excellent reviews of mean-field theory,
especially how it has recently evolved in light of ever-more-powerful numerical simulations.
In the so-called interface dynamo paradigm, the solar tachocline is invoked as the main siteof magnetic field amplification through the Ω-effect (e.g., Parker 1993; Charbonneau & MacGregor
1997). The tachocline is also in a privileged position since it lies directly above the stably stratified
radiative interior. The stable stratification strongly inhibits radial motion and is thought to lead toconditions for which tachocline-produced magnetic fields are stored stably for long time-intervals(e.g., van Ballegooijen 1982; Moreno-Insertis et al. 1992; Ferriz-Mas & Schuessler 1994; Gilman
2000a). In the interface dynamo paradigm, the tachocline acts as a magnetic-field generator andthe radiative interior acts as a storage reservoir.8 When concentrated bundles of magnetism within
the radiative interior become strong enough for buoyancy instabilities to set in, they rise, traversing
the convection zone to erupt as sunspot pairs (e.g., Cline et al. 2003; Jouve & Brun 2009; Weberet al. 2013; Fan & Fang 2014).
The concepts of mean-field theory just described have been instructive in understanding thebasic operation of the solar 22-year cycle. Mean-field theories fall short, however, of addressing thefully turbulent, nonlinear processes like the helical convection (emf) that maintains the poloidal
magnetic field and the Reynolds stress that maintains the differential rotation. For more insight,
we must turn to global simulations, which are fully 3-D, non-axisymmetric, and self-consistent(each field is coupled to the other fields). The nonlinear effects are calculated explicitly in a global
model, rather than parameterized. Finally, in contrast to the local near-surface models, global
models capture the spherical geometry and large-scale transport processes of the star.
residual flux from decayed sunspots. Since the Coriolis force is believed to tilt emergent magnetic field in a mannerconsistent with Joy’s law, the two types of α-effect are in some ways equivalent.
8
The practice of treating the solar radiative interior as a quiescent “abyssal deep” (to borrow the oceanographicterm) is longstanding. One of the central findings of this thesis is that stellar radiative interiors may be filled withstrong horizontal motions that significantly affect the dynamo. We return to this point in our conclusions (Chapter
6).

24
To capture the long time-scales and large length-scales associated with the global solar dynamo, the complicated near-surface physics necessary for local simulations (Section 1.2.1) must be
simplified. Typically, global simulations have a smooth, impenetrable outer boundary that replaces
radiative cooling with thermal conduction or an artificial cooling layer. The exact form of the MHD
equations solved, the numerical algorithm used to solve them, and the geometry of the domain aretreated differently by different codes. In this thesis, we use the Rayleigh MHD code (Featherstone & Hindman, 2016a; Matsui et al., 2016; Featherstone, 2018; Featherstone et al., 2021), whichsolves the anelastic (sound-wave-filtering) MHD equations (see Equations (1.1)) in full sphericalshells. Rayleigh uses a pseudo-spectral approach to evolve the equations in time, in which frequenttransformations are made between “spectral space” (fluid quantities decomposed into Chebyshev
polynomials radially and spherical harmonics horizontally) and “physical space” (fluid quantities
described as functions of position).
We note that Rayleigh is one of many global MHD codes, some of which we briefly describe here. The pseudo-spectral Anelastic Spherical Harmonic (ASH) code is the precursor toRayleigh, but employs a different form of the radiative heating and conductive heat flux (e.g.,
Clune et al. 1999; Brun & Toomre 2002; Brun et al. 2011). The Pencil code is a purely finitedifference partial differential equation solver (e.g., Brandenburg & Dobler 2002) and is modular,able to be used for a variety of astrophysical applications. In the context of solar models, Pencil has
been used to simulate convection in spherical-shell wedges (e.g., Warnecke et al. 2014; Käpylä et al.
2017). The Reduced Speed of Sound Technique (RSST) code (e.g., Rempel 2005, 2006; Hottaet al. 2015) solves the fully compressible MHD equations in spherical shells, but with a continuity
equation that effectively becomes anelastic in the deeper layers, reducing the sound-speed where
it would become extremely large. RSST has given rise to a new code, Radiation and RSST for
Deep Dynamics (R2D2) that can realistically treat the near-surface radiation field and essentiallysimulate the whole convection zone (in a large Cartesian box) on short time-scales (e.g., Hottaet al. 2019; Iijima et al. 2019; Hotta & Iijima 2020). Finally, the Eulerian-Lagrangian- (EULAG-)
MHD code (e.g., Smolarkiewicz & Prusa 2004; Prusa et al. 2008) solves partial differential equations

25
via an advection scheme (specifically, the “Multidimensional positive definite advection transport
algorithm,” or MPDATA) that introduces numerical diffusivity wherever the simulation would benumerically unstable. MPDATA effectively introduces diffusion only on the smallest length-scales,
and also allows EULAG-MHD simulations to remain stable even on grids with modest resolution.
EULAG-MHD can be used for multiple geometries, and recently has been used to simulate solar andstellar dynamos in spherical shells (e.g., Racine et al. 2011; Guerrero et al. 2016a, 2019).
In many global simulations, convection is excited by a solar luminosity conducted through theinner spherical shell. This is essentially a spherical version of the “thermal instability of a layer of
fluid heated from below”—i.e., classical Rayleigh-Bénard convection (see Chandrasekhar 1961; his
Chapter II). In this thesis, luminosity-injection is distributed throughout the convection zone using
a heating function, but the heating is still concentrated in bottom parts of the convection zone. At
the top of the domain, the luminosity is carried out by thermal conduction.9 One common featureof ignoring radiative cooling is that none of the global models have convection driven exclusively
from above, or at the small granular scales. 10
With significant advances in computing power over the last few decades, global models are
now able to generate large-scale, coherent, and cycling magnetic fields. Brown et al. (2010, 2011)
found that coherent magnetic “wreaths” of toroidal field, wrapping all the way around the sphere,
could exist in turbulent simulations of a convection zone with no tachocline. This was initiallya surprise, given the dominant role attributed to the tachocline in the classical interface dynamo
paradigm, and the general belief that highly turbulent flows in the convection zone would shredapart any large-scale magnetism (e.g., Parker 1975). Large-scale wreaths have since been confirmed
as robust features of multiple MHD codes in F-, G-, K-, and M-type stars (e.g., Nelson et al. 2013b;
Augustson et al. 2013; Passos & Charbonneau 2014; Augustson et al. 2015; Yadav et al. 2015;
Guerrero et al. 2016a; Matilsky & Toomre 2020a; Bice & Toomre 2020, 2022; Brun et al. 2022).
9

Note that in the RSST and R2D2 codes, the luminosity is shed instead via an artificial cooling layer near the top
boundary.
10
The EULAG-MHD code is slightly different than the others; the convection is driven by dragging the thermal
perturbations to a pre-defined slightly superadiabatic background state. It is thus unclear exactly where the luminosity
enters and leaves the spherical shell, or even what “luminosity” means in this context.

26
In all global models, any potential wreaths are necessarily maintained by fully nonlinear dynamo
processes. Nevertheless, the differential rotation and helical convection still play fundamental roles,
suggesting that the basic characteristics of a mean-field αΩ-type dynamo may survive, even at
higher levels of turbulence.
Figure 1.4 shows an example of coherent magnetic wreaths in the simulations of Matilsky &Toomre (2020a) (see also Matilsky & Toomre 2020b and Chapter 4). The wreaths are strongest near
mid-latitudes, and their cores lie deep in the convective shell, near the inner boundary. Although
the wreaths are quite prominent and have connectivity all the way around the sphere, they areclearly advected and shredded by the turbulent convection. At the instant shown in Figure 1.4, thetoroidal field is symmetric about the equator (unlike in the solar cycle). Furthermore, there are four
wreaths total (two per hemisphere), whereas the butterfly diagram of Figure 1.2 would suggest two
wreaths in the actual Sun (one per hemisphere, each having opposite polarity). We note, however,that the torsional oscillations (e.g., Howe et al. 2000) propagate equatorward in two bands per
hemisphere, like the wreaths in Figure 1.4. It is possible that wreath-like structures, if they indeed
are present in the solar convection zone, form an additional “‘magnetic reservoir,” distinct from thereservoir in the radiative interior that is invoked in the interface dynamo paradigm. Both reservoirs
could produce buoyant magnetic loops that break off the wreath and rise to form sunspot pairs,with individual pairs bearing some type of signature from their parent reservoir.
The dynamos in global models have also been able to achieve some striking cycling behavior.
In the early wreath-building simulations of Brown et al. (2011) and Nelson et al. (2013b), thecycles were fairly irregular and the fields sometimes non-axisymmetric. Passos & Charbonneau(2014), by contrast, achieved highly regular polarity-reversing cycles (with large-scale axisymmetricwreaths) that remained stable for over 1650 years (∼40 polarity reversals at ∼40 years per cycle).
The long-term behavior of the cycles was reminiscent of the Gleissberg modulation, and evennumbered cycles had different amplitude than odd-numbered cycles, as is observed for the Sunas the Gnevyshev-Ohl rule (Gnevyshev & Ohl, 1948). Notably, Passos & Charbonneau (2014)
included a tachocline of shear between the stable and unstable layers (more on this in Section 1.2.4).

27

Figure 1.4: Magnetic Wreaths in Global Dynamo Simulations. (Left) Toroidal magnetic
field (Bφ ) on a spherical slice in the deep convection zone of a global dynamo simulation (Matilsky& Toomre, 2020a). Red tones indicate field in the prograde direction (Bφ > 0) and blue tones
indicate field in the retrograde direction (Bφ < 0). The field is plotted in full Mollweide view (alllongitudes are shown) and is composed of four toroidal “wreaths” of strong magnetism—two ineach hemisphere. Parallels and meridians are shown every 30 degrees, with the equator, sphereboundary, and central meridian lines plotted slightly thicker. (Right) Longitudinally averaged
magnetic field hBφ i at the same instant as the Mollweide slice on the left, shown in the meridional
plane. The four wreaths have maximum amplitude close to the base of the convection zone. The
curved boundaries of the meridional plane represent the inner and outer radii of the spherical shell,
the radial level of the Mollweide slice is shown as a dotted black curve, and radial lines (lines of
constant θ) are marked every 15 degrees. The top labels indicate the toroidal magnetic field, theradial level in the shell, and the simulation-time sampled in rotations.

28
Augustson et al. (2015) also achieved remarkably regular cycles, which displayed equatorward
propagation of wreaths at low latitudes and poleward propagation of small-scale magnetism athigh latitudes. The high-latitude magnetic field had average polarity opposite to that of the lowlatitude wreaths—highly reminiscent of the solar butterfly diagram (Figure 1.2). Furthermore,
the simulation in Augustson et al. (2015) stopped cycling for an extended interval, similar to asolar grand minimum, and then resumed cycling afterward. Finally, Matilsky & Toomre (2020a)
found a “bistable” dynamo, consisting of two types of cycle operating simultaneously. One was aregularly-reversing equatorward-propagating system of axisymmetric wreaths (Figure 1.4), and theother was a longitudinal modulation of non-axisymmetric (azimuthal wavenumber m = 1) “partial
wreaths,” similar to those presented in Brown et al. (2011) and Nelson et al. (2013b). The partial
wreaths in Matilsky & Toomre (2020a) were further recognized to possess striking similarities to theobserved active longitudes and hemispheric asymmetry on the Sun. The bistable simulations aredescribed in detail in Chapter 4. Recently, Bice & Toomre (2022) made progress in understanding
the generation of non-axisymmetric magnetic cycling via so-called patchy convection (active nestsof strong convection localized to a particular longitude—e.g., Brown et al. 2008).

1.2.3

Insights into the Solar Differential Rotation from Global Simulations

All dynamos in global modeling so far encountered are intimately coupled to the achieveddifferential-rotation profile via the Ω-effect. Here, we discuss some of the successes and shortcomingsof global models in explaining the helioseismically inferred solar rotation rate. At zeroth-order,
solar-like differential rotation consists of the rapidly rotating equator and slowly rotating polar
regions. The maintenance of solar-like differential rotation is now widely believed to arise fromangular momentum transport by interior convective rolls, also known as “Busse columns,” “Taylor
columns,” “convective Rossby waves,” and/or “banana cells” in the literature. We usually refer tothese rolls as Busse columns in this thesis (Taylor columns in Chapter 4). The possibility for Bussecolumns to drive the solar differential rotation was first suggested in the pioneering work of Busse(1970a,b), Gilman (1972), and Busse (1973). The presence of Busse columns (and their ability to29maintain differential rotation) was subsequently confirmed in nonlinear 3-D convection simulations(e.g., Glatzmaier 1984; Gilman & Miller 1986; Elliott et al. 2000; Brun & Toomre 2002).
An example of a spherical-shell convection simulation is shown in Figure 1.5. The radial
velocity acts as a fairly good tracer of the flow. Between about ±30 degrees latitude, the convection consists of fairly coherent columnar rolls aligned with the rotation axis, which are the Bussecolumns. Above the mid-latitude regions, the convection appears more isotropic and vortical, although some of the downflow lanes are clearly the ends of Busse columns. The flow field shown in
Figure 1.5 is typical of reasonably turbulent global simulations. Figure 1.5 also shows the asymmetry between upflows (broad and slow) and downflows (thin and fast) that arises from the density
contrast across the shell, and is not present in Rayleigh-Bénard convection.
Busse columns arise in regions of strong rotational constraint, which in this thesis is definedby low Rossby number (ratio of rotation period to convective turnover time) and low Ekman
number (ratio of the rotation period to viscous diffusion time).11 In other words, strong rotational
constraint occurs when the rotation is so fast that it influences all other dynamical processes. The
solar NSSL (Figure 1.3) presented the first indication that rotational constraint may break downnear the photosphere, where the velocities are high and the convective turnover time is short. As
described in Section 1.1.3, the rotation rate decreases with radius in the NSSL. In fact, this behavior
was inferred even before helioseismology by observing that magnetic tracers—which are believed
to be anchored below the photosphere—rotate faster than the surrounding plasma (e.g., Howard &Harvey 1970; Wilcox et al. 1970; Foukal 1972). Foukal & Jokipii (1975) proposed that high-velocity,
rotationally unconstrained fluid in the NSSL could homogenize angular momentum in radius, thus
yielding Ω ∼ 1/r2 . Helioseismology, of course, revealed that Ω ∼ 1/r in the NSSL, so clearly Foukal& Jokipii (1975)’s mechanism did not entirely work. Nevertheless, the general idea that near-surface
turbulence maintains the NSSL remains prevalent (e.g., Miesch & Hindman 2011; Hotta et al. 2015;
Matilsky et al. 2019. As we will discuss in-depth in Chapter 2, some recent numerical simulations
All simulations in this thesis have low Ekman numbers ∼10−4 –10−3 , so we need only technically consider the
Rossby number when assessing rotational constraint. However, it is still useful to invoke the Ekman number when
discussing certain aspects of rotational constraint.
11

30

Figure 1.5: Spherical-Shell Convection. Snapshot of the radial velocity vr field near the outersurface a convection simulation in a rotating spherical shell (Matilsky & Toomre, 2020a). The
velocity is plotted in color (red tones indicate upflow, blue tones indicate downflow) on a sphericalsurface just below the outer boundary in orthographic projection. The sphere is viewed from above
the 20th parallel in the Northern Hemisphere. Meridians and parallels are shown as black curves
every 30 degrees, and the equator, sphere boundary, and central meridian are plotted slightly
thicker. Where the meridians meet at the top of the figure is the North Pole, which also lies on
the rotation axis. The top labels indicate the radial velocity, the radial level in the shell, and thesimulation-time sampled in rotations.

31
have attempted to capture the NSSL with an outer layer of rotationally unconstrained fluid (Hottaet al. 2015; Matilsky et al. 2019). These attempts have so far not succeeded.12 Some phenomenonin addition to the NSSL—likely the near-surface meridional circulation or magnetic fields—is likely
not being faithfully captured by the global simulations.
As already mentioned, the substantial tilts of the isorotation contours have been difficultfor global simulations to capture. Rempel (2005) suggested that the cylindrical contours usuallyachieved in global simulations were the result of too much rotational constraint (or equivalently,
Taylor-Proudman constraint). A“thermal wind,” or an equator-to-pole entropy gradient, can break
the Taylor-Proudman constraint. Rempel (2005) suggested that such a thermal wind might arise
from the convection zone’s coupling to the tachocline. Inspired by this idea, Miesch et al. (2006)
imposed an equator-to-pole entropy gradient at the bottom boundary in a global simulation. The
result was significant contour tilt, with the tilt angle basically adjustable in proportion to thestrength of the imposed entropy gradient. It remains unclear, however, how a thermal wind in
the tachocline would couple to the solar convection zone (the thermal conductivity is very low in
the Sun) and also what would drive this thermal wind in the first place. In Matilsky et al. (2020)(Chapter 3), we show that Busse columns transport heat poleward, in addition to transporting
angular momentum outward. This gives a physical mechanism to drive a thermal wind and tilt theSun’s isorotation contours.
We finally note that all global models currently suffer from a fundamental problem withthe differential rotation as simulations are pushed to more turbulent regimes. This “convective
conundrum” may be stated following (O’Mara et al., 2016): “The convective velocities required
to transport the solar luminosity in global models of solar convection appear to be systematicallylarger than those required to maintain the solar differential rotation and those inferred from solar
observations.” More specifically, if simulations are pushed to increasingly turbulent regimes (while
keeping the rotation rate and luminosity fixed at the solar values), the velocities at large scales grow
12

The exception are some quite solar-like NSSLs achieved (at high latitudes only) in some of the MHD simulationsof Guerrero et al. (2016a,b, 2019). Guerrero et al. (2016a,b) invoke rotationally unconstrained turbulence as causing
the NSSLs, but do not perform any further dynamical analysis.

32larger than helioseismology would predict. The large-scale flows lose their rotational constraint,
and thus their columnar nature (Figure 1.5), failing to transport angular momentum outward. The
simulation as a whole becomes “anti-solar,” with fast polar regions and a slow equator. We notethat the physical origin of the anti-solar states is still not well understood (e.g., Featherstone &Hindman 2016b; Hindman et al. 2020; and references therein). It should also be noted that only
some helioseismic studies (e.g., Hanasoge et al. 2012) yield small velocities on the large scales. Other
helioseismic investigations (Greer et al., 2015, 2016) yield substantially higher flow velocities, andso it is not entirely clear how unrealistic the simulations’ large-scale flow velocities are.
In any case, it is likely that the flows achieved in global models are somewhat unrealistic (ingeneral) because of the way the system is driven. A solar convection zone is required to carry out thesolar luminosity via convective heat transport, which depends on the flow structures’ temperature,
speed, and size. The efficiency of the transport also depends on the thermal conductivity, since agiven fluid parcel can only transport energy for as long as it can retain its hotness or coldness. In theSun, the energy leaves the system via radiative cooling, which drives the convection and occurs at
much different length-scales than those set by the thermal conductivity. In most global simulations,by contrast, the luminosity exits via conduction, implying that flow structures are driven at thesame scales as they dissipate heat. We speculate that using more realistic near-surface physics, inwhich the buoyant driving occurs at scales totally separate from the scales of dissipation, may help
to solve the convective conundrum.
Practically, global simulations could achieve a reduction in large-scale flow amplitudes via a
reduction in the buoyant driving at the large scales. O’Mara et al. (2016) suggest that this could
be accomplished via small-scale magnetic field acting to enhance the viscosity, but not the thermal
conductivity. This effective “high-Prandtl-number” regime would lead to small-scale downflow
plumes carrying more thermal content (i.e., driving the small scales harder), and require a weakervelocity at large scales to carry out the solar luminosity. Some recent very-high-resolution global
simulations (`max ∼ 4,096) lend support to the magnetically enhanced viscosity, yielding solar-like
differential rotation at the solar rotation rate and luminosity, with large-scale flows inhibited by a

33
small-scale dynamo (Hotta & Kusano, 2021).

1.2.4

Global Models Including a Tachocline

The helioseismically inferred tachocline (why it is observed to be so thin, at ∼0.05R ) presents
a major puzzle for solar physics. Spiegel & Zahn (1992) first recognized that left to its own devices,
the tachocline should have spread to a thickness (at minimum) of ∼0.4R by the current age of theSun. This “thermal” or “radiative” spread occurs when the tachocline’s thermal wind (latitudinal
entropy gradient) and accompanying meridional circulation imprint downward via radiative diffusion. The burrowing meridional circulation transports angular momentum and thus carries with it
the differential rotation (the phenomenon of radiative spread is discussed extensively in Chapter5). In the Sun, there must be a torque that forces the radiative interior to remain in solid-bodyrotation and thus prevent the spread of the tachocline. The origin of this torque is the subjectof various tachocline confinement scenarios (e.g., Miesch 2005; Brun & Strugarek 2019; and references therein). The two most commonly invoked scenarios are hydrodynamic (HD; e.g., Spiegel& Zahn 1992) and magnetohydrodynamic (MHD; e.g., Gough & McIntyre 1998). In Chapter 5,
we clearly demonstrate for the first time that magnetism can force solid-body rotation in the radiative interior of a fully 3-D, nonlinear, global simulation. In contrast to previously hypothesizedmagnetic confinement scenarios (Gough & McIntyre, 1998; Forgács-Dajka & Petrovay, 2001), theconfining magnetism in our simulation is primarily non-axisymmetric. Furthermore, the magnetism
is produced by induction, even below the layer of convective overshoot. Our work thus hints thatthe solar radiative interior may be dynamically active, rather than the usually supposed “abyssal
deep.” This is one of the central findings of this thesis, as we discuss at length in Chapters 5 and 6.
With the exception of Brun & Zahn (2006) and Strugarek et al. (2011a,b), global models
have not focused on addressing the tachocline confinement problem.
13

13

Instead, they have

In the Brun and Strugarek models, a poloidal magnetic field was put into the radiative interior as an initialcondition in a 3-D simulation. This was meant to assess whether the poloidal field could confine the tachocline—
as in Gough & McIntyre (1998)—with null results. The initially confined poloidal field spread diffusively into theconvection zone and imprinted the differential rotation inward along magnetic-field lines, seeming to follow Ferraro’s
law of isorotation (Ferraro, 1937).

34
implemented a tachocline artificially and explored its effects on the differential rotation, global
dynamo, and MHD waves and instabilities. This was first done in Browning et al. (2006), who used
an explicit forcing term to create a steady-state tachocline in a global, MHD ASH simulation. In
the more recent simulations, tachocline spread is controlled diffusively. In the ASH and Rayleighmodels (e.g., Brun et al. 2011; Alvan et al. 2014, 2015; Brun et al. 2017; Bice & Toomre 2020),
the sub-grid-scale diffusions (which appear in the equations as enhanced Newtonian diffusions) are
explicitly dropped by ∼3–4 orders of magnitude in the radiative interior. A similar lowering of the
sub-grid-scale diffusions is implemented in the EULAG-MHD simulations (e.g., Passos & Charbonneau
2014; Guerrero et al. 2016a; Beaudoin et al. 2018; Guerrero et al. 2019). In EULAG-MHD, there
are no explicit diffusions, but implicit numerical diffusions associated with the advection operator,
MPDATA. These numerical diffusions roughly scale with the convective velocities, and since thevelocities are so small in the radiative interior, the diffusions are effectively very low. However
the diffusions are lowered, the same effect is accomplished: the tachocline still spreads inward, butvery slowly. For the time-scales over which the simulations can be run, the tachocline is effectively
stationary and its influence on the dynamics can be assessed.
Browning et al. (2006) were the first to show that the tachocline could produce large-scale,coherent magnetic structures in the radiative interior in a 3-D simulation. This lent support tonotion of the radiative interior as a magnetic-field reservoir that could act to stabilize the global
dynamo. Furthermore, although the early convection-zone-only simulations of Brun et al. (2004)(upon which Browning et al. 2006’s tachocline simulation was based) possessed largely incoherent
magnetic field at many scales and irregular polarity reversals of the mean field, the introduction of
the tachocline yielded steady polarity of the mean poloidal field, even in the convection zone. It
was clear that even in turbulent 3-D simulations, the magnetic field in the convection zone could“feel” the influence of the tachocline.
The initial results of Browning et al. (2006) were also supported by EULAG-MHD simulations(e.g., Ghizaru et al. 2010; Racine et al. 2011; Passos & Charbonneau 2014). The simulation of
Ghizaru et al. (2010) contained both coherent magnetism (field strengths peaking just below the35tachocline) and regular polarity reversals. Racine et al. (2011) ran Ghizaru et al. (2010)’s simulation further and identified significant torsional oscillations. Racine et al. (2011) also fitted theturbulently generated emf to a full α-tensor. This mean-field analysis suggested that the dynamo
cycle was driven primarily by the torsional oscillations. Furthermore, the presence of strong magnetic field significantly reduced the differential rotation contrast by a factor of ∼3, an effect wealso observe in the MHD simulation with magnetically confined tachocline discussed in Chapter
5. Finally, Passos & Charbonneau (2014) ran a simulation similar to that of Ghizaru et al. (2010)
and Racine et al. (2011), but with the addition of explicit numerical diffusion for the purpose ofstabilizing the solution. This “Millennium Simulation” was run for an extended interval, achieving stable, regular cycling behavior for over 1650 years. Passos & Charbonneau (2014) identified
several behaviors similar to the solar cycle: A “sunspot” cycle (with a period of ∼40 years), a
Gleissberg modulation (with a long-term periodicity of ∼100 years), and a Gnevyshev-Ohl rule(adjacent cycles alternating in strength; Gnevyshev & Ohl 1948).
Magneto-shear (or magneto-rotational) instabilities (Balbus & Hawley, 1991) and magnetobuoyancy instabilities are likely important effects in the solar tachocline, where concentrationsof toroidal magnetic field exist amidst strong gradients in the rotation rate. These effects canmake magnetic fields at certain horizontal wavenumbers unstable, possibly contributing to the
low latitudes of sunspot emergence and the high latitudes of ephemeral active regions (Gilman,
2018a,b). Lawson et al. (2015) searched the Millennium Simulation for such MHD instabilities,
using a slightly different version of the simulation that included an additional friction term at thebase of the stable layer to damp out numerically unstable gravity modes (this was also done in
Alvan et al. 2014, 2015). Lawson et al. (2015) calculated the phasing between various componentsof the magnetic and kinetic energies in the tachocline, and identified the same non-axisymmetricmagneto-shear instability (azimuthal wavenumber m = 1, 2) of Miesch (2007).
More EULAG-MHD tachocline simulations were presented in Guerrero et al. (2016a,b, 2019),
with results largely in-line with previous work. In Guerrero et al. (2016a,b), the simulations with atachocline were shown to produce larger-scale, longer-cycle, more coherent, and stronger magnetic

36
fields than their convection-zone-only counterparts. Similar results were also obtained by Bice &Toomre (2020), who used the Rayleigh code to simulate M dwarfs with and without a tachocline,
by Augustson et al. (2012, 2013), who used ASH to simulate dynamos in F-type stars, and by Brunet al. (2022), who used ASH to simulate dynamos in G- and K-type stars. The models of Guerrero
et al. (2019) were unique in that they reproduced the observed increase of dynamo cycle period(e.g., Noyes et al. 1984; Böhm-Vitense 2007, whereas the opposite trend (decrease in cycle periodwith rotation period) was reported in the prior investigation of Strugarek et al. (2017). It should
finally be noted that many of the simulations in Guerrero et al. (2016a,b, 2019) achieved somestriking solar-like differential rotation, including a tachocline (as was expected from their sub-gridscale diffusion technique), tilted isorotation contours, and the most solar-like NSSL achieved todate in a global model. The tilt of the isorotation contours was further observed in the EULAG-MHD
models of Beaudoin et al. (2018) and attributed to the build-up of an equator-to-pole temperature
contrast caused by rotationally influenced convective overshoot.

1.3

Our Approach to Global Simulations
Here, we briefly describe the numerical attributes of the global simulations presented in thisthesis. Our models were all run with the Rayleigh code and shared many of the same properties,
like boundary conditions, background states, and geometry. We describe these common properties
here, to avoid repeating them in the subsequent Chapters when possible. In all cases, we consider
a spherical shell of fluid, representative of the solar interior, governed by the one-fluid equations of(magneto)hydrodynamics, or (M)HD. The shell extends from a deep inner radius (ri ∼ 0.49R orri ∼ 0.72R ) to a near-surface outer radius ro ∼ 0.95R , and thus covers ∼75% of the convection
zone (by radius), and sometimes a portion of the radiative interior. We describe the shell using thestandard spherical coordinates r (radius), θ (colatitude), and φ (azimuth angle), and the corresponding unit vectors êr , êθ , and êφ . We also use the cylindrical coordinates (λ, φ, z) = (r sin θ, φ, r cos θ)and unit vectors (êλ , êφ , êz ). We use the following terms interchangeably: “vertical” ↔ “radial,”“toroidal” ↔ “φ component,” and “poloidal” ↔ “r and θ components.”

37
1.3.1

The Anelastic MHD Equations

In stellar interiors, the plasma is less dense further from the core, and this density contrast (orstratification) is an essential feature. For example, there is pronounced asymmetry between upflows
and downflows in stratified solar-like convection (e.g., Miesch et al. 2000; Brun & Toomre 2002;
also see Figure 1.5), whereas there is up-down symmetry in classical Rayleigh-Bénard convection(e.g., Chandrasekhar 1961). In the anelastic approximation (e.g., Gough 1969; Glatzmaier 1984;
Jones et al. 2011), compressibility is kept in the sense that the density (ρ) can vary weakly in space
and time so that the overall radial stratification is captured. However, the rapid fluctuations inρ as occur in sound waves are filtered out. This allows the simulations to have long time-steps
and run for decades-long time-scales (thus capturing dynamo behavior), which is not possible if
the simulations are required to track the fast sound waves. The approximation is appropriate
for a low-Mach-number flow, wherein the velocities are much smaller than the sound-speed andthe sound waves are thus decoupled from the rest of the dynamics. Mathematically, the anelastic
approximation simply corresponds to setting the time-derivative of density (∂ρ/∂t) in the continuity
equation to zero.
We denote the total density, temperature, specific entropy, and pressure of the fluid by ρ,
T , S, and P (respectively) and the fluid velocity and magnetic fields by v and B (respectively).
The anelastic approximation is then ∇ · (ρv) = 0. Additionally, it is assumed (as in the Boussinesq
equations—e.g., Spiegel & Veronis 1960) that perturbations in the thermodynamic variables awayfrom the 1-D background are small. Each thermodynamic variable is thus written as, e.g., ρ = ρ+ ρ̂,
where ρ is the “background,” or “reference” density (which can vary slowly in space, and technicallyalso in time) and the “perturbed” density ρ̂ is small, so |ρ̂|  ρ. In the Rayleigh code (and all theother global codes we are aware of), slow rotation, well below break-up, is assumed. The Sun isthus treated as spherically symmetric to first-order and the centrifugal force is neglected. We also
assume that the reference thermodynamic state is time-independent and spherically symmetric—i.e., ρ = ρ(r) and ∇ρ = (dρ/dr)êr , and similarly for the other reference thermodynamic variables

38
P (r), T (r), and S(r).
In stellar radiative interiors, energy is transported solely by radiation, which can be regarded
as a thermal diffusion (e.g., Eddington 1916; Rosseland 1930; Collins 1989). In global MHD codes,
the complicated thermodynamics governing the interior opacity is usually neglected. In ASH (e.g.,
Brun & Toomre 2002) and Pencil (e.g., Brandenburg 2016), the radiative diffusive coefficient istreated as a simple function of radius. In Rayleigh, the “radiative diffusion” does not operate like aconductive flux, and instead is treated as a source of heat, Q(r). The heat source is fixed a priori andis independent of density and temperature perturbations, chemical composition, etc. Q(r) → 0 near
the outer surface so that instead of radiative cooling, a thermal flux F cond carries away the solar
luminosity. More specifically, Q(r) is chosen to have the fixed radial profile Q(r) = α[P (r) − P (ro )](nonzero only in the convection zone), with the normalization constant α chosen such that a solar
luminosity L = 3.846 × 1033 erg s−1 is forced through the domain (see Featherstone & Hindman
2016a). This prescription for the internal heating function coincides well with the radiative heating
calculated by more sophisticated solar models (see model S described in Christensen-Dalsgaard
et al. 1996, for example). In particular, Q(r) is concentrated roughly in the bottom third of theconvection zone and thus resembles (to some degree) the “heating from below” in Rayleigh-Bénard
convection and in other global simulations.
With these assumptions, the MHD equations reduce to


∂v
P̂
ρ(r)
= − ρ(r)v · ∇v − 2ρ(r)Ω0 × v − ρ(r)∇
∂t
ρ(r)
 
Ŝ
1
+ ρ(r)
g(r)êr + ∇ · D +(∇ × B) × B,
cp
4π
and

ρ(r)T (r)(1.1a)∂ Ŝ
dS= − ρ(r)T (r)v · ∇Ŝ − ρ(r)T (r)vr∂tdr+ ∇ · [ρ(r)T (r)κ(r)∇Ŝ] + Q(r) + D : ∇v +∂B
= ∇ × [v × B − η(r)∇ × B],∂tη|∇ × B|2 ,4π(1.1b)(1.1c)∇ · [ρ(r)v] = 0,

(1.1d)∇ · B = 0,(1.1e)

and39
which are the momentum, entropy (equivalently, energy or heat), and induction equations, followedby the continuity equation and “no magnetic monopoles” (or “solenoidal”) condition, respectively(e.g., Clune et al. 1999; Featherstone & Hindman 2016a). Here, D = ρ(r)ν(r)[∇v + ∇v T −(2/3)(∇ · v)I], I is the identity tensor, cp = 3.5 × 108 erg g−1 K−1 is the specific heat at constantpressure, ν(r), κ(r), and η(r) are the momentum, thermal, and magnetic diffusivities (respectively),and g(r) is the body force. All simulations in this thesis have a thermal Prandtl number ofunity (Pr ≡ ν/κ = 1, or ν = κ), though the magnetic Prandtl number (Prm ≡ ν/η) takes onother values (Prm is nevertheless never far from unity). All diffusivity profiles have the sameradial shape, so each Prandtl number is constant. Strictly, g(r) should be the “effective” gravity,
including both the gravitational and centrifugal accelerations. Since the centrifugal term is ignored,g(r) = ∇(GM /r), where G is the gravitational constant and M ≡ 1.9989 × 1033 g is the massof the Sun (the self-gravity of the shell is ignored). We also denote the magnitude of gravitational
acceleration by g(r) ≡ |g(r)| = GM /r2 .
In a real star, the molecular diffusivities are almost entirely negligible, and so here ν(r),κ(r), and η(r) must be regarded as sub-grid-scale representations of unresolved turbulence. For
simplicity, they appear in Equations (1.1) as turbulently enhanced values with the same form as
the Newtonion diffusivities. The actual turbulent transport is not a diffusive process, however,and the choice of enhanced diffusivities is made primarily for computational tractability. For fullconsistency with classical theory, ν(r), κ(r), and η(r) would be chosen based on various mixinglength-theory assumptions for the Sun (e.g., Brandenburg 2016 and references therein). In ourwork, however, we simply choose the diffusivities to be arbitrary functions of density. We note thatthe heat conduction diffuses the entropy instead of the temperature: the conductive heat flux is
F cond = −ρ(r)T (r)κ(r)∇Ŝ, which is meant to capture the unresolved turbulent mixing of heat. A
moving fluid parcel carries heat energy that is proportional to its entropy content, Ŝ, which initiallyis the same as the entropy of the background fluid. If the fluid parcel moves adiabatically and thenmixes its heat energy with the surrounding fluid after traveling a small distance, there will be an
associated unresolved (or “eddy”) thermal-energy flux proportional to ∇Ŝ.

40
Our models employ a perfect gas for the equation of state. Combined with the first lawof thermodynamics and the assumption of small thermodynamic perturbations, the perfect gas
equation takes the form
ρ̂
T̂
P̂
Ŝ
P̂
− =
− ,
=
ρ
cp
P
T
γP

(1.2)

where γ = cv /cp (cv is the specific heat at constant volume, or equivalently, constant density). We
assume full ionization and three translational degrees of freedom, yielding γ = 5/3 throughout theSun. We note that although the thermodynamics is linearized in Equation (1.2), the velocity andmagnetic fields are fully nonlinear in Equations (1.1).
The reference state is required to be in hydrostatic balance,dP= −ρg,
dr

(1.3)

1 dP
1 dρ
1 dS
=
−
,
cp dr
ρ dr
γP dr

(1.4)

P = ρRT ,(1.5)

satisfy the first law of thermodynamics,

and be a perfect gas,

where R = (γ − 1)cp /γ is the gas constant. We note that strictly R = kB /µmH (with µ the mean
molecular weight, kB Boltzmann’s constant, and mH the mass of the hydrogen atom) should beradially varying, especially between the well-mixed convection zone and slowly-circulating radiativeinterior. For simplicity, we do not consider compositional mixing and treat µ = 0.59 as a constant inboth space and time, its value determined from the canonical mass fractions for hydrogen (X = 0.75)
and helium (Y = 0.25). We ignore the nonzero metallicity and assume full ionization.14

This inturn yields cp = 3.5 × 108 erg g−1 K−1 .
Once Equations (1.3)–(1.5) are assumed, the background state can be specified entirely by
the choice of g(r) (we have chosen g(r) = GM /r2 ) and dS/dr (see Chapter 5 for details). All of
14

Recall that µ1 = 2X + 34 Y for a fully ionized gas with no metals.

41
the simulations in this thesis include a convection zone, which we enforce by choosing dS/dr = 0.
This stratification is equivalent to a polytrope with index n = 1.5 (see Jones et al. 2011 fordetails). Although the background state is thus technically not unstable (but marginally stable),the system still convects, driven both by the internal heating (concentrated in the lower layers)
and the outer conductive cooling layer (concentrated in the upper lowers). Once the convection
is established, the spherically symmetric part of ∂ Ŝ/∂r becomes < 0, i.e., the convection zone
becomes slightly superadiabatic. Thus, our simulations act conversely to classical mixing-length
theory: the convection drives the superadiabatic gradient.
In writing Equation (1.1a), the Lantz-Braginsky-Roberts (LBR) approximation (Lantz, 1992;
Braginsky & Roberts, 1995) is used. Equations (1.3)–(1.5) show that the buoyancy force is   1
P̂Ŝρ̂P̂ 1 dS− ∇P̂ − g êr = −∇+êr .
g êr +
ρρρcp
ρ cp dr(1.6)

The LBR momentum equation (1.1a) thus consists of ignoring the term due to the backgroundentropy gradient in the buoyancy, which will be exact for an adiabatic (dS/dr = 0) reference
state—in our models, the convection zone. Of course, for simulations including a tachocline andlayer of strong stable stratification (dS/dr > 0), the LBR approximation should be inapplicable.
Research on this topic is far from conclusive, but simulations of small Cartesian boxes of fluid
show that LBR may be the best form of the anelastic approximation in stable layers, despite
being mathematically incorrect. LBR correctly conserves energy in the excited gravity-wave field,
whereas anelastic formulations that keep the entropy-gradient term in the buoyancy do not (Brown
et al., 2012; Vasil et al., 2013). Notably, these Cartesian investigations only explored an isothermal
atmosphere, and so the energy-conserving properties of LBR in our stable layers, which have radiallyvarying T (r), are unknown.

1.3.2

Initial and Boundary Conditions

Equations (1.1), combined with Equations (1.2), form a system of nonlinear partial differential
equations in eight variables (Ŝ, P̂ , and the three components each of v and B) that is first-order42
in time and second-order in space. The system thus requires eight initial conditions (one for eachvariable) and sixteen boundary conditions (one at both the inner and outer boundaries for eachvariable; see Equation (1.7) below). The coefficients of the system are non-constant but only dependon the radial coordinate r.
In the HD variables (Ŝ, P̂ , and the three components of v), we initialize the system withrandomly distributed small (∼10 erg g−1 K−1 ) fluctuations in Ŝ (equivalent to Ŝ/cp ∼ 10−8 ) and set
the other fluid variables to zero. These thermal perturbations quickly yield nonzero fluctuations inv and P̂ , and then become convectively unstable, reaching amplitudes of Ŝ ∼ 103 erg g−1 K−1 (andcorresponding buoyantly-driven radial velocities of vr ∼ 100 m s−1 ) in the most turbulent cases. If
the simulation is MHD (including B), small (∼1 G) magnetic perturbations B 0 are included as anqinitial condition in parallel with the thermal perturbations (equivalent to (B 0 2 /8π)/P ∼ 10−8 ).
Dynamo action (exponential amplification of the initial seed field to amplitudes of ∼104 G for thestrongest dynamos) is possible, but not guaranteed.
Each boundary of the shell is treated as an impenetrable sphere that is stress-free. In the
MHD simulations, the magnetic field is matched onto a potential field at both boundaries. At the
inner boundary, there is no conductive heat flux, and at the outer boundary, there is either an
outward conductive flux of the solar luminosity L , or else fixed entropy. All simulations in thisthesis thus share the following boundary conditions:∂ v 
θ∂r
B = ∇Φ

andr

vr = 0∂ vφ 
==0∂r r

at ri and ro(impenetrable),(1.7a)

at ri and ro(stress-free),(1.7b)∇2 Φ = 0

at ri and ro(match to potential field),(1.7c)

whereŜ = 0

or

∂ Ŝ
=0
at ri
(fixed flux),
∂r
∂ Ŝ
L
=−
at ro
2
∂r
4πro ρ(ro )T (ro )κ(ro )(fixed flux or fixed entropy).

(1.7d)(1.7e)The outer boundary condition on the entropy (either one) ensures that as the system equilibrates, a43
sharp gradient in the spherical mean of Ŝ—i.e., a thermal boundary layer—develops near the outersurface, which carries out the solar luminosity via thermal conduction. Recall that in a stellar
convection zone, by contrast, the luminosity is ultimately carried out by radiative cooling.

1.3.3

The Rayleigh CodeThe Rayleigh code (Featherstone & Hindman, 2016a; Matsui et al., 2016; Featherstone et al.,2021) solves the LBR anelastic equations of (M)HD. The underlying serial algorithm of Rayleighclosely follows that of the Anelastic Spherical Harmonic (ASH) code (e.g., Clune et al. 1999;
Brun et al. 2004) but is open-source and has a highly efficient parallel algorithm, scaling well on
up to half a million CPU cores. To strictly enforce the anelastic approximation, ∇ · (ρv) = 0, andsolenoidal condition, ∇ · B = 0, the vector velocity field is decomposed into two streamfunctions
and the vector magnetic field into two flux-functions:ρv = ∇ × ∇ × (W êr ) + ∇ × (Z êr ),(1.8a)

B = ∇ × ∇ × (C êr ) + ∇ × (Aêr )(1.8b)

This formalism satisfies Equations (1.1d)–(1.1e) by construction, and further reduces the other
Equations (1.1a)–(1.1b) to a system in six unknowns (Ŝ, P̂ , W , Z, C, and A), requiring six initialconditions and twelve boundary conditions. Physically, the streamfunctions W and Z are related to
the radial velocity and vorticity, respectively (C and A are related to the radial magnetic field andPcurrent density, respectively). For example, if we decompose W (r, θ, φ, t) = `m W`m (r, t)Y`m (θ, φ),where the Y`m (θ, φ) are the spherical harmonics (and similarly for vr ), we find that the spectral
coefficients are related by (vr )`m = [`(` + 1)/r2 ]W`m .
Each variable is expanded on a discretized grid, using spherical harmonics in the horizontal
directions (directions tangent to spherical surfaces) and Chebyshev polynomials in the vertical(radial) direction. The vertical grid is composed of one or more stacked domains of Chebyshev
collocation points, which cluster near the ends of each domain. One Chebyshev domain is used
for the convection-zone-only models and multiple domains are used for the models including a44radiative interior. Both the Chebyshev polynomials and the Legendre polynomials are dealiased byapproximately one-third the number of collocation points. For example, if the vertical grid has Nrradial (Chebyshev) collocation points in a given domain, 2Nr /3 Chebyshev polynomials are usedin the vertical expansion for that domain. Similarly, if the horizontal grid has Nθ colatitudinal(Legendre) collocation points, spherical harmonics of maximum degree `max = 2Nθ /3 are used in
the expansion. The number of longitudinal (Fourier) collocation points is Nφ = 2Nθ so that theazimuthal order m of the highest-degree spherical harmonics ranges from −`max to +`max − 1.
The linear terms (e.g., diffusion and buoyancy) in Equations (1.1) are evaluated in spectral(Chebyshev and spherical-harmonic) space, while the nonlinear terms (e.g., advection, induction,and the Coriolis force even though it is technically linear) are evaluated in physical space. The
time is advanced using a semi-implicit Crank-Nicolson scheme for the linear terms and an explicit
Adams-Bashforth scheme for the nonlinear terms. The use of both spectral and physical spaceat each time step makes the code “pseudo-spectral.” See, e.g., Clune et al. (1999) for details onRayleigh’s anelastic, pseudo-spectral algorithm, and Featherstone et al. (2021) for details on its
parallel structure.

1.4

Structure of this Thesis
In the subsequent four Chapters, we describe the research results of this thesis. Chapters 2–4

are essentially restatements of the three publications, Matilsky et al. (2019) (see also the conference
proceedings paper, Matilsky et al. 2018), Matilsky et al. (2020), and Matilsky & Toomre (2020a) (seealso the conference proceedings paper, Matilsky & Toomre 2020b), respectively. The work presented
in Chapter 5 is currently being prepared for an upcoming publication, Matilsky et al. (2022) andwas briefly described in the conference proceedings paper, Matilsky & Toomre (2021). We broadly
divide our research into two classes: dynamics of the Sun without magnetism (HD) and dynamics
with magnetism (MHD). In Chapter 2 (first HD Chapter), we present our findings on the torquebalance achieved in the solar Near-Surface Shear Layer (Matilsky et al., 2018, 2019). In Chapter
3 (second HD Chapter), we present our substantial progress in understanding the cause of the45
helioseismically observed isorotation contour tilts (Matilsky et al., 2020). In Chapter 4 (first MHD
Chapter), we describe in detail the remarkable phenomenon of “bistability,” in which the global
solar dynamo is found to be capable of supporting two very different cycles simultaneously (Matilsky& Toomre, 2020a,b). In Chapter 5 (second MHD Chapter), we present and dynamically analyze an
MHD simulation that appears to yield a magnetically confined tachocline self-consistently (Matilsky& Toomre, 2021; Matilsky et al., 2022). Finally, in Chapter 6, we present some concluding remarks
and directions of future research.

Chapter 2

The Role of Downflows in Establishing Solar Near-Surface Shear

As discussed in Section 1.1.3, the fine structure of the internal rotation of the Sun is intimately
tied to the solar dynamo. The dynamo therefore cannot be understood without explaining thesalient features of the differential rotation. Because the two problems are coupled through meanshear induction (Ω → B) and the Lorentz force (B → Ω), it is likely that a complete understandingof each is only accessible through reproducing both the butterfly diagram and the differential
rotation simultaneously. We begin the research portion of this thesis by describing our work treating
the differential rotation separately—i.e., in a purely hydrodynamic (HD) setting.
Often, the tachocline is invoked as the main source of the interior shear that must create
large-scale solar toroidal magnetic field. However, as pointed out in Brandenburg (2005), thedynamo may in fact be “distributed,” operating in both the tachocline and Near-Surface Shear
Layer (NSSL). The NSSL, though it has weaker shear than the tachocline does, is closer to thesurface and thus may play a larger role in sunspot emergence than is usually assumed. The NSSL isalso the oldest known feature of the Sun’s internal rotation rate. Before helioseismology confirmed
its presence (and mapped its latitudinal structure), it was discovered through tracking sunspots
over 50 years ago. Sunspots were believed to be “magnetically anchored” below the photosphere
and were observed to move with the solar rotation faster than the surrounding gas. This impliedthat the rotation rate increased with depth close to the surface (Howard & Harvey, 1970; Wilcox
et al., 1970). The magnitude of the shear estimated back then (rotation rate falling with radius by∼5% over the outer ∼5% of the Sun) still agrees with current helioseismic deductions.

47
We have attempted to self-consistently reproduce the solar NSSL in numerical simulations ofa solar-like convection zone by increasing the density contrast across rotating, 3-D spherical shells.
We explore the hypothesis that high density contrast leads to near-surface shear by creating a rotationally unconstrained layer of fast flows near the outer surface. Although our high-contrast models
do have near-surface shear, it is confined primarily to low latitudes (between ±15◦ ). Two distinct
types of flow structures maintain the shear dynamically: rotationally constrained Busse columns
that transport angular momentum outward, and rotationally unconstrained downflow plumes that
deplete angular momentum from the outer fluid layers. The plumes form at all latitudes, and infact are more efficient at transporting angular momentum inward at high latitudes. The presence ofBusse columns at low latitudes thus appears essential to creating near-surface shear in our models.
We conclude that a solar-like NSSL is unobtainable from a rotationally unconstrained outer fluidlayer alone. In numerical models, the shear is eliminated through the advection of angular momentum by the meridional circulation. Therefore, a detailed understanding how the solar meridional
circulation is dynamically achieved will be necessary to elucidate the origin of the Sun’s NSSL.
This Chapter is primarily a restatement of the publication Matilsky et al. (2019) (also see
the related conference proceedings paper, Matilsky et al. 2018). However, we add some additional
perspectives informed by recent NSSL studies over the last few years. As the primary author of
the paper, I conducted the simulations, performed their analyses, created the Figures, and wrote
the text. My two coauthors Juri Toomre and Bradley Hindman provided advice and guidance
throughout the process, gave detailed text edits, and suggested schematic designs of some of the
Figures. Special thanks is owed to Nicholas Featherstone, who formulated the idea of creating arotationally unconstrained near-surface fluid layer in simulations by increasing the density contrast,
and also for providing me with a pre-release version (0.9.1) of the Rayleigh code.

48

2.1

Introduction: Our Theoretical Understanding of the Solar Near-Surface
Shear Layer (NSSL)Helioseismology has revealed the presence of two boundary layers of shear at the top andbottom of the solar convection zone (CZ). In the tachocline at the bottom, differential rotation in
the CZ transitions sharply to solid-body rotation in the radiative zone below (see Figure 2.1). At
the top of the CZ, there is a 5% reduction in rotation rate with increasing radius over a radial
distance of ∼35,000 km, which is largely a uniform feature at all latitudes. This latter feature
is known as the Near-Surface Shear Layer (NSSL). Both boundary layers may play a significant
role in the solar dynamo, since rotational shear creates toroidal magnetic field from poloidal field
through the Ω-effect. However, the dynamical origins of these boundary layers are still not wellunderstood.
The solar differential rotation (from Figure 2.1, the equator rotates about 30% faster than thepoles) is believed to be the result of convectively driven Rossby waves, which manifest in the solar
CZ as Busse columns, also known as banana cells in the literature (e.g., Gilman 1972; Busse 2002;
Brun & Toomre 2002; Nelson et al. 2018). Busse columns are convective rolls of fluid aligned withthe rotation axis. Each roll has a cross-sectional tilt in the equatorial plane such that upflows moveprograde and downflows move retrograde, the net result being the transport of angular momentum
away from the rotation axis (see Figure 6 of Busse 2002). The Busse columns thus tend to spin up
the equator (which is far from the rotation axis) compared to the polar regions, which are close to
the rotation axis.
By contrast, in the NSSL (see Figure 2.1), the surface layers rotate 5% slower than the layers
just below. Foukal & Jokipii (1975) hypothesize the following explanatory mechanism for theformation of the NSSL: Fluid parcels conserve their specific angular momentum L in the outermost
layers of the Sun as they move in the radial direction. Thus, for a steady-state system, the angularmomentum profile L(r) should be constant with radius. Since specific angular momentum is related
to the local fluid rotation rate Ω through L = Ωr2 sin2 θ, this would imply Ω ∝ 1/r2 along a radial

49

Figure 2.1: Helioseismically Inferred Rotation Rate (with Errors). Temporally and longitudinally averaged solar rotation rate Ω/2π obtained from an inversion of GONG frequency splittings.
Rotation rate is plotted as a function of the fractional solar radius at different latitudes. The dashedlines represent 1σ error bounds for a single inversion. Figure credit: Howe et al. (2000).

50
line in the outermost fluid layers—a decrease in rotation rate with radius.
In order to successfully homogenize angular momentum, the flows need to be free from rotational constraint so that they do not get captured by Busse-column rolls. The degree of rotational
constraint is parameterized by the Rossby number Ro, which is the ratio of rotation period toconvective overturning time. Thus, according to Foukal & Jokipii (1975), the Sun must possess aregion near the outer surface in which
Ro ≡

v0
& 1,
2Ω0 L

(2.1)

where v 0 is a typical velocity of the convective flow structures (that may vary with radius), L istheir typical length scale and Ω0 is the background rotation rate. Thus, a rotationally unconstrained
fluid structure is one that is either fast, small-scale, or both. In the models explored here, we findthat only the downflows (in particular, structures we call downflow plumes) are sufficiently fast andsmall-scale to be rotationally unconstrained.
Previous numerical simulations of rotating, spherical-shell convection that captured aspects ofthe solar NSSL have all possessed a layer of rotationally unconstrained fluid near the outer surface.
Guerrero et al. (2013) saw an NSSL arise near the equator of their models due to the mixingof angular momentum by fast, high-Rossby-number convective motions. In modeling the banded“zonal jets” observed in longitudinal flows on the gas and ice giants, Gastine et al. (2013) found thathigh density contrast across the spherical shell (∼150) enabled low-Rossby-number Busse columns
to exist in the deep fluid layers and high-Rossby-number, small-scale convection to occur near theouter surface. Hotta et al. (2015) have come closest to a self-consistent reproduction of an NSSL ina model with density contrast across the shell of ∼613. They successfully achieved a Rossby-number
transition with a thin rotationally unconstrained outer layer, and reported prominent near-surface
shear at high latitudes.
Features of near-surface shear thus seem to appear in models with high density contrast. Thisis because increasing the density contrast across the spherical shell—while keeping the density at
the inner surface fixed—decreases the near-surface density scale-height, which is thought to be a51
good representative length-scale of the convection. The luminosity pumped into the system is also
shed near the outer surface; if the fluid parcels there are much less dense, they will be endowed witha much greater temperature deficit and be accelerated higher (downward) speeds via the buoyancy
force. Both effects serve to increase the Rossby number near the outer surface.
In this work, we systematically investigate the nature of rotationally unconstrained nearsurface layers across a range of density contrasts in rotating spherical shells. We explicitly analyze
the simulations at the two ends of this range, with contrasts of ∼20 and ∼150. We find the lowcontrast simulation is entirely dominated by Busse columns at low latitudes, as expected. The
high-contrast simulation, on the other hand, exhibits rotationally unconstrained fast flows near
the outer surface, as in previous work. However, our analysis reveals that only the downflows arerotationally unconstrained and transport angular momentum inward, creating shear. The upflows
still exist in Busse columns and transport angular momentum outward, even near the outer surface.
In Section 2.2, we describe the parameter space we explore. In section 2.3, we discuss theglobal character of the flows achieved, both instantaneously and averaged over time. In Sections
2.4 and 2.5, we describe the structure and evolution of Busse columns and downflow plumes,
respectively. In Section 2.6, we discuss the dynamical balance of torques in our models and its
relation to the simulated features of near-surface-shear. In section 2.7, we examine in detail the
Reynolds stress from Busse columns and downflow plumes, which manifests in the separate upflowand downflow-contributions to the angular-momentum flux. In Section 2.8, we discuss our resultsin the general context of meridional force balance.

2.2

Numerical Model
As described in Section 1.3, we numerically evolve a rotating, stratified shell of ideal gas

using the Rayleigh code, here the pre-release version 0.9.1 (Featherstone & Hindman, 2016a;
Matsui et al., 2016; Featherstone, 2018). The computational domain of the layer here consists of
the CZ only: A spherical shell with inner radius ri = 5.000 × 1010 cm = 0.719 R and outer radiusro = 6.586 × 1010 cm = 0.947 R , where R≡ 6.957 × 1010 cm. Recall that Rayleigh makes

Figure 2.2: Influence of Density Stratification on Flow Structures. Mollweide projections of the radial velocity vr on sphericalsurfaces for cases N3 and N5. Red tones indicate upflows (vr > 0) and blue tones indicate downflows (vr < 0). Two radial levels are
sampled: a near-surface level (r/ro = 0.988) and a level mid-way through the domain (r/ro = 0.881). In panel (c), the colorbar isbinormalized to highlight the asymmetry between upflows and downflows near the outer surface for case N5. The dashed 40◦ × 40◦ box at
the centers of panels (a), (c) and (d) show the regions that are magnified in later Figures.

52

53
use of the anelastic approximation to increase the maximum allowable time step by removing
sound waves. Specifically, Rayleigh here solves Equations (1.1) with no magnetic terms and noreference-entropy-gradient term. The temporally steady and spherically symmetric reference state
has adiabatic stratification (dS/dr = 0; this is equivalent to a polytrope with index n = 1.5; see
Jones et al. 2011 for a complete description), but develops a slightly superadiabatic spherical meanin Ŝ due to the convection. Setting the values of the density at the inner boundary and the overall
density contrast across the shell fully determines the background state.
Recall that for all simulations described in this thesis, a solar luminosity L is driven throughthe layer via a fixed internal heating function Q(r) and is carried out via thermal conduction at
the top boundary. Furthermore, all simulations have impenetrable, stress-free bottom and top
boundaries, fixed-flux (∂ Ŝ/∂r = 0) bottoms. Here, the simulations have fixed-entropy (Ŝ = 0) tops,
they rotate at three times the solar Carrington rate (3Ω ) to ensure that a solar-like differential
rotation (fast equator and slow polar regions) is attained,1and they have a Prandtl number(Pr ≡ ν/κ) of unity (ν and κ are spatially constant, each equal to 2 × 1012 cm2 s−1 throughout theshell; all simulations in this thesis have a Prandtl number of unity). In this work, we compare twomodels with different density contrasts. The relevant model parameters are shown in Table 2.1.
Here, Nρ refers to the number of density scale heights across the domain. The density contrast
from the inner to the outer boundaries (also shown in Table 2.1) is related to the number ofscale heights by ρi /ρo = exp(Nρ ), where ρi and ρo refer to the values of ρ at the inner and outerboundaries, respectively. The thermal diffusion time Tdiff (equivalently, the viscous diffusion time)
estimates how long it takes for heat (momentum) to diffuse across the full spherical shell. The
averaging time refers to the time interval used in the temporal averages of fluid quantities—e.g.,differential rotation, meridional circulation and Reynolds stress. This interval coincides with thetime from equilibration (total energy flux constant with radius) to the end of the simulation. All
In this Chapter only, Ω = 2.6 × 10−6 rad s−1 , which is the value of the solid-body rotation of the radiativeinterior according to Spiegel & Zahn (1992). It was thus often used in the early ASH models (e.g., Brun & Toomre
2002; Browning et al. 2006; Brown et al. 2010). In subsequent Chapters, Ω = 2.87 × 10−6 rad s−1 , which is thesidereal Carrington rate.
1

54
input parameters are fixed in the two models, except for Nρ , which takes on the values 3 and 5 (fordensity contrasts across the shell of ∼20 and ∼150, respectively). We refer to the resulting models
as cases N3 and N5, respectively.

2.3

Global Flow Properties
Figure 2.2 shows the radial velocity of the flow on spherical surfaces near the outer boundary

and at mid-depth for cases N3 and N5. At low latitudes, the alternating lanes of upflow anddownflow are parallel to the rotation axis and indicate the locations of Busse-columns pairs. The
columns at mid-depth have connectivity to their counterparts in the near-surface layers, but they
are significantly slower and the downflow lanes are thicker, which is a consequence of the density
stratification. At high latitudes, there is less noticeable alignment of the fluid structures withthe rotation axis, although this may in part be due to angular distortion effects of the sphericalprojection. We note that in the near-surface layers of case N5, there are downflow lanes oriented
north–south (parallel to meridians) and lanes oriented east–west (parallel to lines of latitude). The
places where the two types of lanes cross we call interstices. We shall see that the interstices are
sources of prominent downflow plumes, which evolve independently from the Busse columns andtransport angular momentum in the opposite direction.
We turn next to the mean-flow properties of our models. The average radial profile of rotation
rate at various latitudes for cases N3 and N5 is shown in Figure 2.3 and the full rotation rate in themeridional plane is shown in Figure 2.4. Case N3 exhibits the stronger differential rotation, with avariation of ∆Ω = 155 nHz from equator to pole. This corresponds to a differential rotation fractionof ∆Ω/Ω0 = 0.125. As the density contrast across the layer increases, the overall differential rotation∆Ω decreases, along with the differential rotation fraction. Case N5 has a differential rotation fromequator to pole of only 81 nHz and a differential rotation fraction of ∆Ω/Ω0 = 0.065. We notethat both these values of fractional differential rotation are significantly smaller than the value∆Ω /Ω ∼ 0.3 observed in the Sun.
While the rotation rate in case N3 increases monotonically in both radius and latitude, case

55

Table 2.1: Model Parameters for the NSSL Experiment. Input Model Parameters for Cases
N3 and N5. All parameters are displayed to 4 significant digits. Nr , Nθ , and Nφ refer to thenumber of grid points in the radial, colatitudinal, and longitudinal directions (respectively) in
the computational domain. The global Ro refers to the Rossby number defined in Equation 2.1,
averaged over the full spherical shell. We take the typical convective length scale to be L = Hρ (r) =−1/(d ln ρ/dr) (the local density scale height) and the typical speed v 0 (r) to be the rms amplitudeof the convective velocity (the velocity with the longitudinally averaged component subtracted),
with the mean in the rms referring to an average in time and over a spherical surface of radius r.
Parameter
ri
ro
cp
ν
κ
γ
ρi
Ω0
Ω0 /2πP0 ≡ 2π/Ω0
Tdiff ≡ (ro − ri )2 /κNρρi /ρo(Nr , Nθ , Nφ )Global Ro
Averaging timeN3

N5
5.000×1010 cm
6.586×1010 cm
3.500×108 erg K−1 g−1
2.000×1012 cm2 /s
2.000×1012 cm2 /s
1.667
0.1805 g/cm3
7.800×10−6 rad/s
1,241 nHz
9.323 days
1,456 days (3.985 yr)
3.000
5.000
20.10
148.4
(128, 768, 1536) 256, 1152, 2304)
0.13450.479353.78 yr
34.07 yr
2,107 P0
1,335 P0
13.50 Tdiff
8.550 Tdiff

56

1375
1350

latitude [deg]1325/2 (nHz)

45
60
75

0
15
30

1300
1275
1250
1225
1200
1175

r/ro

(a) N31300/2 (nHz)

1280
1260
1240
1220
1200
1180

(b) N5

0.80

0.85

r/ro

0.90

0.95

1.00

Figure 2.3: Influence of Density Contrast on Rotation Profiles along Radial Lines. Temporally and longitudinally averaged rotation rate as a function of radius at various latitudes forcases N3 and N5 (compare to the Sun’s rotation rate in Figure 2.1). The location of the prominent
dip in rotation rate in the equatorial region of case N5 is indicated by the vertical dashed line atr/ro = 0.941 in panel (b). The amplitude of the dip is indicated by the two horizontal dashed lines
separated by 34.1 nHz.

57

Figure 2.4: Influence of Density Contrast on Rotation Frequency. Temporally and longitudinally averaged rotation frequency in the meridional plane for cases N3 and N5, respectively.
The frame rotation frequency Ω0 /2π has been subtracted off, so that positive values (red tones)
indicate prograde rotation, while negative values (blue tones) indicate retrograde rotation. In eachpanel, there are 15 dashed contours, corresponding to rotation-frequency values that evenly divide
the range indicated by the colorbar.

58
N5 exhibits features reminiscent of near-surface shear. The most prominent of these is a dip in
the rotation rate at low latitudes near the outer surface. Here, the rotation rate decreases withradius by ∼2.7%. The effect is comparable in magnitude to the Sun’s NSSL, which is characterized
by a ∼5% decrease in rotation rate near the top of the CZ. This low-latitude dip has been arobust feature of other work (e.g., Brun & Toomre 2002; Brandenburg 2007; Guerrero et al. 2013;
Gastine et al. 2013; Hotta et al. 2015) and is also referred to as a dimple in the literature. At
high latitudes in case N5, there are some signs of shear as well, although the overall effect is much
weaker, corresponding to a reduction in angular velocity of only ∼0.5%. Furthermore, the shear
has both a negative and positive radial gradient. Hotta et al. (2015) saw this phenomenon as well.
The meridional circulation (here, represented by the magnitude of the longitudinally averaged
mass flux, with circulation streamlines overplotted) for cases N3 and N5 is shown in Figure 2.5.
At low latitudes in the Northern Hemisphere (for both cases), there are three cylindrically stacked
circulation cells. The inner and outer cells are counterclockwise and the sandwiched inner cellis clockwise. The sandwiched clockwise cell is large in case N3, but small in case N5, while theopposite is true for the counterclockwise cell near the outer surface. Furthermore, case N5 has two
additional clockwise cells at high latitudes, one near the outer boundary and one near the innerboundary. For both cases, the large counterclockwise cell at high latitudes is concentrated in a thinband near the outer surface, where there is strong poleward flow. In the Southern hemisphere foreach case, the circulation patterns are the same, but with the clockwise/counterclockwise sense ofeach cell reversed.
Figure 2.6 shows the breakdown of radial energy fluxes for case N5. These fluxes are defined

59

Figure 2.5: Influence of Density Contrast on Meridional Circulation. Vector-magnitudeof the temporally and longitudinally averaged mass flux—| hρvpol i |—in the meridional plane forcases N3 and N5. The vector-magnitude of the mass flux corresponds to the density of meridional
circulation streamlines, several of which are denoted by the dashed contours. Here vpol = vr êr +vθ êθ
refers to the poloidal part of the velocity. Red (positive values) indicates clockwise circulation, whileblue (negative values) indicates counterclockwise circulation.

60

Integrated radial energy flux

1.25

4 r2× energy flux/L

1.00
0.75
0.50
0.25
0.00
0.25
0.50

Ftotal
Fenth

Fcond
Frad

0.80

0.85

Fvisc
FKE

r/ro

0.90

0.95

1.00

Figure 2.6: Radial Energy Fluxes for High Density Contrast. Decomposition of the spherically integrated radial energy flux for case N5, as defined in Equation (2.2). The location of thebase of the thermal boundary layer is shown as the dashed vertical green line, chosen by eye tocoincide with the near-surface local minimum (in radius) of the conductive energy flux.

61(e.g., Featherstone & Hindman 2016a) to be
DE
Fenth (r) ≡ ρ(r)cp vr T̂ ,*Fcond (r) ≡ −κρ(r)T (r)(2.2a)
∂ Ŝ
∂r

+

1
FKE (r) ≡ ρ(r) v 2 vr ,2

andFvisc (r) ≡ − hv · Di ,Z
1 roFrad (r) ≡ 2
Q(x)x2 dx.
r r,(2.2b)(2.2c)(2.2d)(2.2e)

Here, the angular brackets denote spherical averages. For the most part, four main fluxes contributeto the outward transport: the radiative flux Frad , the conductive flux Fcond , the enthalpy flux Fenth(which represents the convective transport of heat), and the kinetic energy flux FKE . In the bottom
layers (r/ro . 0.85), energy is transported primarily by the radiative flux and about 25% of theenergy is transported by the conductive flux. As the radiative flux decreases with radius, theenthalpy flux begins to take over. Around r/ro ∼ 0.97, the convective heat flux is dominant.
Finally, near the outer surface, the boundary conditions on the velocity and entropy force all fluxes
to vanish except for the conductive flux, which carries a solar luminosity out of the domain in anarrow thermal boundary layer. The extreme flatness of the total energy flux in case N5 indicatesa mature state of statistical equilibrium for the energy transport.

2.4

Busse Columns
We recall that Busse columns are convective rolls of fluid aligned with the rotation axis.

Adjacent rolls have opposite senses of spin, so that each columnar downflow lane traces the region
in between two Busse-column rolls. In Figure 2.8, we show the equatorial cross section of radial
velocity for case N3. The prograde tilt of the downflow lanes is obvious: the portions of the laneclose to the outer surface are at a higher longitude than the portions of the lane close to the innersurface. As a general rule, the columns extend in depth all the way through the layer; however,
several structures (especially the downflows in the upper half of the layer) only extend through

62

101

N3
N5

Ro

100

10 1

10 2

0.80

0.85

r/ro

0.90

0.95

1.00

Figure 2.7: Influence of Density Contrast Rossby Number. Radial profile of the local Rossby
number defined in Equation (2.1) for cases N3 and N5. The spherically averaged rms velocity v 0 (r)and the typical length scale L = Hρ (r) are defined as in the caption to Table 2.1. The dashed line
shows the critical value of the Rossby number Ro = 1, which is reached only for case N5.

63

Figure 2.8: Busse Columns on Equatorial Cuts. Instantaneous profile of the radial velocity vr
on an equatorial cut of the computational domain for case N3. The view is from the North Pole,so the longitude φ increases in an counterclockwise sense. The radial velocity has been divided by
its rms value at each radius and the colorbar is binormalized to show the asymmetry in the upflowand downflow-speeds with respect to the rms.

64
about half the layer or less.
The character of the Busse columns depends on the level of rotational constraint in thesimulations. We show the radial profile of Ro for cases N3 and N5 in Figure 2.7. The near-surface
outer layers of case N5 are rotationally unconstrained (Ro > 1), in contrast to those of case N3(see also Figures 2.2(a, c)). In the deep layers, on the other hand, both cases are rotationallyconstrained, with low Rossby numbers and definitive Busse-column structure (Figures 2.2(b, d)).
In the deep layers of the shell, the Busse-column structure is clearest (e.g., Figures 2.2 and2.8). We estimate the longitudinal wavenumber of the Busse columns in each case by finding thepeak of the power spectrum of the sectoral spherical harmonics (first averaging the spectrum intime and in radius over the inner half of the shell). We find a clear peak in each power spectrum,with mpeak ∼ 28 for case N3 and mpeak ∼ 24 for case N5. For both cases, a typical Busse column
thus has a longitudinal extent (2πri /mpeak ∼ 1.3 × 1010 cm) comparable to the depth of the shell(∼1.6 × 1010 cm). Featherstone & Hindman (2016b) show that the wavenumber of Busse columns
scales with the Rossby number like mpeak ∼ Ro−1/2 . Near the inner surface, the ratio of the Rossby
numbers between cases N3 and N5 is ∼ 0.76 (see Figure 2.7). This corresponds to a ratio of Bussecolumn wavenumbers of (0.76)−1/2 ≈ 1.15, in agreement to lowest order with our estimates of mpeakfrom the power spectra (28/24 ∼ 1.17). The comparable values of the Busse-column wavenumbers
in cases N3 and N5 illustrate the fact that although the near-surface layers of the two cases havevery different flow structures, the deep layers of the two simulations (where the Rossby numbers
are only slightly different) have similar flow structures.
Figure 2.9 shows the temporal evolution of the near-surface flow field in case N3. To bettersee the evolution, we magnify a 40◦ × 40◦ patch centered at the equator, indicated by the dashed
box in Figure 2.2(a). Some downflow lanes (which trace the regions in between adjacent Bussecolumn rolls) have been labeled with capital letters to indicate how they are advected and distortedby the flow. Although the downflow lanes are rather long-lived (lane A, for instance, maintains
its structure for several rotation periods before getting absorbed by another lane), they are notsimply advected passively by the flow. They frequently merge, disappear, and reappear, indicating

65

Figure 2.9: Following the Movement of Busse Columns. Temporal evolution of a patch ofradial velocity vr for case N3. Spherical cuts of the full velocity field are taken at the near-surface
radial level r/ro = 0.988. Each panel shows a 40 ◦ × 40◦ patch of the spherical surface centeredat the equator, with successive patches equally spaced in time by roughly a quarter of a rotation
period. The frame of the patch is rotating at the local equatorial rotation rate in order to see thesuper-rotation of the columns.

66

Figure 2.10: Following the Movement of Intense Downflow Plumes. Similar to Figure 2.9,
but for case N5. Spherical cuts are taken at r/ro = 0.988 and each panel shows a 40◦ × 40◦ patchof the full surface centered at the equator, equally separated in time by roughly a quarter of arotation period. The frame of the patch is rotating at the local equatorial rotation rate.

67

Figure 2.11: Following the Vertical Structure of Downflow Plumes. Magnified view of the40◦ ×40◦ equatorial patch of radial velocity denoted by the dashed box in Figure 2.2(c). Each panel
shows the same patch at a common time for successively deeper layers, evenly spaced throughout
approximately the upper half of the spherical shell. The saturation values (in m/s) for the colorbar
are shown to the right of each panel. Several downflow plumes (beginning as interstices in panela) have been traced in depth by small black circles. These plumes/interstices are labeled in panel(a) by capital letters A–F. Interstices A–C correspond to the same interstices identified in Figure
2.10. The near-surface plume locations and labels are shown in gray in panels (b–d).

68that no single Busse column lives for a protracted interval. Furthermore, the lanes (and hence, thecolumns) only extend coherently to about ±15◦ ; beyond this latitude range, the lanes move more
slowly, eventually breaking off and joining the swirling small-scale flow at high latitudes. We notethat the differential rotation of Figure 2.3(a) correspondingly occupies mainly the narrow latitude
band between ±15◦ . Each lane is advected by several degrees over the whole rotation period,
corresponding to a pattern speed of the Busse columns that is ∼40 m/s faster than the backgroundrotation rate. In other words, the Busse columns super-rotate with respect to the background flow.

2.5

Downflow Plumes
Figure 2.10 shows the evolution of the near-surface flow field for case N5, with several in-

terstices (regions where the north–south and east–west downflow lanes cross) labeled by capitalletters. We see that each upflow is surrounded by a more-or-less polygonal network of downflow
lanes and thus may be regarded as a cell. The cells are stacked in the axial direction such that
there are 10 or so downflow lanes that connect throughout the whole patch north–south, analogous
to the downflow lanes between pairs of Busse-column rolls in case N3 (Figure 2.9). On average, theinterstices move in the prograde direction, indicating that they are super-rotating like the Bussecolumns. The interstices have much shorter lifetimes than the Busse columns, splitting up andmerging several times over the course of a rotation period. In panel (b), for example, interstice A
has split into two interstices A1 and A2 , while in panel (c), the two interstices have merged again.
At any given instant of time, the interstices shown in Figure 2.10 are the sources of downflow
plumes. The plumes can be seen by following the interstices down in depth. In Figure 2.11, we
consider the patch from Figure 2.10(a) and examine the connectivity of the downflows from thenear-surface layers to mid-depth. As the patches get successively deeper, the downflow associatedwith each interstice intensifies in amplitude and becomes more localized. We refer to the entire
structure (interstice to localized downflow near mid-depth) as a downflow plume. It is important
to note that the plumes do not coincide with the trajectory of a fluid parcel launched downward
from the interstice. Since the interstices move prograde in time, each radial point on the plume69
corresponds to a fluid parcel that was launched when the parent interstice (top of the plume) was
positioned retrograde from its present location.
The coherence of the north–south downflow lanes in Figure 2.11 increases with depth and theplumes slowly fade. No single plume extends in depth more than ∼0.1 ro (or about 2/5 the depthof the layer) from the near-surface layer shown in panel (a). This is consistent with the ephemeral
nature of the interstices; over the radial extent of a plume, the fastest speeds are on average ∼400m/s in the plume core, while the lifetime of an individual interstice is ∼2 days. Thus, the total
depth-extent of the plumes should be ∼ (400 m/s) × (2 days) ∼ 0.1 ro .

2.6

Torque Balance
The steady-state distribution of angular momentum (and by extension, the differential rota-tion) can be understood in terms of the angular momentum transport, or torque, due to various
aspects of the flow. In equilibrium, the torque balance (e.g., Elliott et al. 2000; Brun & Toomre
2002; Miesch & Hindman 2011) is expressed asτrs + τmc + τv ≡ 0,

(2.3)

where

and

0
τrs ≡ −∇ · [ρr sin θ v 0 φ vpol],

(2.4a)

τmc ≡ − hρvpol i · ∇L,(2.4b)

τv ≡ ∇ · [ρνr2 sin2 θ∇Ω].

(2.4c)

Here vpol ≡ vr êr + vθ êθ is the poloidal part of the fluid velocity and L ≡ r sin θ(Ω0 r sin θ + hvφ i) =
Ωr2 sin2 θ is the fluid’s specific angular momentum in the non-rotating lab frame. Angular brackets
indicate a combined temporal and longitudinal average and the primes indicate deviations from theaverage.
Figure 2.12 shows the balance of torques for cases N3 and N5. At each point in the meridional
plane, the magnitude of the sum of the torques is a factor of ∼100 smaller than the magnitude ofFigure 2.12: Torque Balance for High and Low Density Contrast. Temporally and longitudinally averaged torque balance in themeridional plane for cases N3 and N5. In each panel, overlapping plots show the torque densities due to Reynolds stress, meridional
circulation, and viscosity from left to right. Positive torque, which tends to increase angular momentum in the axial direction êz (orequivalently, tries to accelerate the fluid in the longitudinal direction êφ ) is shown in red and negative torque in blue.
70

KE (106 erg cm 3)

71

3
2

N3

1
0

0

20

40

60

time (rotation periods P0)

N5
80

100

Figure 2.13: Equilibrium of the Steady-State Differential Rotation. Temporal variation of
the kinetic energy density in the differential rotation (KEφ ≡ (1/2)ρ hvφ i2 ) for cases N3 and N5
over 100 rotation periods. Here, the kinetic energy has been averaged over the entire shell. Timeis measured from the beginning of equilibration for each case.

any of the three torques individually. Furthermore, following equilibration in each simulation, thekinetic energy in the differential rotation (KEφ ) varies by only a few percent from its temporally
averaged value, as shown in Figure 2.13. We have also confirmed that there is no secular drift inthe kinetic energy with time over several thousand rotation periods, indicating a mature state ofequilibrium in the torque balance.
For each case, the torque is roughly constant on cylinders near the equator, whereas it isroughly constant on spheres at high latitudes. The main feature that sets case N5 apart fromcase N3 is the strong band of negative Reynolds-stress torque near the outer surface of case N5,
extending across all latitudes; in case N3, there is only negative Reynolds-stress torque at highlatitudes, and it is significantly weaker than in case N5.
D
E
D
E
D
E
0
A decomposition of vφ0 vpol
into its two component-correlations vr0 vφ0 and vθ0 vφ0 reveals
that the Reynolds-stress torque is almost entirely dominated by the radial turbulent angularmomentum flux
Fr ≡ ρr sin θ vr0 vφ0(2.5)

for both cases N3 and N5. We now argue that this radial flux is produced by the two types of flow
structures mentioned previously, Busse columns and downflow plumes. 22Note that here in Chapter 2, we keep the notation of Matilsky et al. (2019). We use a capital Roman “F ”
to denote the spherically integrated radial energy flux (units of luminosity) and a curly “F” to denote the angularTorque (g cm 1 s 2)

r/ro0.80 0.85 0.90 0.95 1.00(a) N3, low-latitude

r/ro

v

total

0.80 0.85 0.90 0.95 1.00

rsmc(b) N3, high-latitude

r/ro

0.80 0.85 0.90 0.95 1.00(c) N5, low-latitude

r/ro

0.80 0.85 0.90 0.95 1.00(d) N5, high-latitude

Figure 2.14: Torque Balance in High- and Low-Latitude Ranges. Torque balance averaged over low latitudes (between ±15◦ ) andhigh latitudes (between ±45◦ and ±60◦ ) for cases N3 and N5. Each curve shows a torque averaged over its particular latitude region andplotted as a function of radius.

1.0

0.5

0.0

0.5

1.0

×107

72

r (g s 2)

r/ro0.80 0.85 0.90 0.95 1.00

0.80 0.85 0.90 0.95 1.00r/ro(b) N3, high latitudes

upflow
downflowtotal(a) N3, low latitudes

×1016

r/ro

0.80 0.85 0.90 0.95 1.00(c) N5,
low latitudes

r/ro0.80 0.85 0.90 0.95 1.00(d) N5, high latitudes

Figure 2.15: Angular Momentum Flux Due to Upflows and Downflows. Radial angular-momentum flux from the Reynolds stress(Fr ) averaged over low latitudes (between ±15◦ ) and high latitudes (between ±45◦ and ±60◦ ) for cases N3 and N5. The flux has been
divided into the component from the upflows (vr > 0) and the component from the downflows (vr < 0). The sum of these components isequal to the total flux.

2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5

73

74
The cross-sectional tilt of the Busse columns (Figure 2.8) tends to give upflows (vr0 > 0)a positive vφ0 and downflows (vr0 < 0) a negative vφ0 . Within a Busse column, both upflows anddownflows thus yield a positive (radially outward) angular-momentum flux. The net result is thatangular momentum is taken away from the inner layers (negative torque) and deposited in the outerlayers (positive torque). Figure 2.14(a) shows the low-latitude torque balance for case N3. The
Reynolds-stress torque is positive in roughly the upper quarter of the domain and negative in thelower three-quarters, indicating that Busse columns dominate the Reynolds stress at all depths. In
case N5, however, the Reynolds-stress torque is negative near the outer surface (Figure 2.14(c)),
indicating the influence of an additional transport mechanism.
The downflow plumes do not follow the Busse column tilt. There are only a few other physicalmechanisms to create correlations in vr0 and vφ0 , and the most obvious of these is deflection by the
Coriolis force. For downflow plumes (vr0 < 0), the deflection is prograde (vφ0 > 0), corresponding toa negative (radially inward) transport of angular momentum. This picture is consistent with the
Reynolds-stress torque at high latitudes for both cases (Figures 2.14(b, d)): the Reynolds-stresstorque is negative in the outer layers and positive in the inner layers, corresponding to the inward
transport of angular momentum from the outer surface to the deeper layers.
At low latitudes, the presence of downflow plumes is most prominent in case N5 near the outersurface, where the Reynolds-stress torque is negative. Correspondingly, there is a dip in rotation
rate at low latitudes for case N5, but not for case N3. This lends substantial support to the argumentof Foukal & Jokipii (1975), at least at low latitudes: downflow plumes are Coriolis-deflected (orequivalently, they conserve their angular momentum), transporting angular momentum radiallyinward and creating near-surface shear.
The torque from the meridional circulation is determined by the alternating pattern of clockwise and counterclockwise cells in Figure 2.5, coupled with the fact that angular momentum in
our models is roughly constant on cylinders (λ = constant). From Equation (2.4), this means thatmomentum flux. In Chapter 3, we use “F” to denote the non-integrated energy flux (units of luminosity per unit
area) and “Ir ” to denote the spherically integrated energy flux.

75
the torque has a sign opposite to that of hρvλ i. Near the equatorial boundaries of the circulation
cells, the meridional flow is dominated by the radial component hρvr i = hρvλ i. Consequently, τmcchanges sign in Figures 2.14(a, c) at the radial locations of the north–south cell boundaries.
At high latitudes, a decomposition of the meridional circulation into its radial and latitudinalcomponents reveals that only the latitudinal term − hρvθ i (∂L/r∂θ) contributes significantly tothe torque. Geometrically, this is due to the shape of the near-surface counterclockwise cell of
meridional circulation in both cases, which is most intense in a region highly elongated in thelatitudinal direction. The resultant poleward flow carries high-angular-momentum fluid from theequator and brings it to high latitudes, thereby creating a positive near-surface torque.
The viscous flux of angular momentum −ρνr2 sin2 θ∇Ω is proportional to the negative gradient of the rotation rate. This is similar to a Fickian diffusivity (acting on the rotation rate, notthe specific angular momentum), meaning that viscosity simply tends to bring the rotation rate toa constant value (i.e., it eliminates shear). In this sense, the viscous torque is “passive,” responding
only to counteract the shear produced by the Reynolds stress and meridional circulation. Thus, inFigure 2.14, the viscous torque is simply negative or positive according to the sign of the combined
Reynolds stress and meridional circulation torques.

2.7

Angular Momentum Transport by Upflows and Downflows
Figure 2.15 shows the Reynolds-stress angular-momentum flux Fr broken up into upflow-

and downflow-components for cases N3 and N5 in the high- and low-latitude regions. For case N3
at low latitudes, both components of the flux have the same sign—positive in most of the domain
except for a narrow region of very weak negative flux near the outer boundary. The strong positive
fluxes are consistent with the tilts of the Busse columns in the equatorial plane, which makes theDEcorrelation vr0 vφ0 positive for both upflows and downflows. In the middle layers, the magnitudeof the upflow-flux is about twice as large as that of the downflow-flux. This asymmetry may be
attributed to the fact that although most downflows are part of a Busse column, there are also
downflows whose speeds are large enough that they are really plumes that lack a negative vφ0 . The

76
DE
overall positive correlation vr0 vφ0 is thus weaker for the downflows.
For case N5, the low-latitude upflow-flux looks rather similar to the upflow-flux in caseN3, except that it peaks in the middle of the layer, as opposed to peaking in the upper half.
The downflow-flux in the bottom half of the layer (r/ro . 0.87) is positive, consistent with theorganization of both upflows and downflows into Busse columns in the deeper layers. The top halfof the shell, by contrast, has negative downflow-flux, whose maximum magnitude is over twice as
great as the maximum magnitude of the positive downflow-flux. Thus, at low latitudes, the inward
transport of angular momentum comes only from the downflow plumes; the upflows are dominatedby Busse columns.
When the low-latitude upflow and downflow-fluxes are added, there is a positive slope in theradial profile of the total angular-momentum flux Fr . The effect on the Reynolds stress torque,which scales like the radial derivative of r2 Fr , is to produce the narrow region of negative torque near
the outer boundary as seen in Figure 2.14(c). This negative torque is responsible for maintaining
the low-latitude near-surface shear against viscosity.
At high latitudes in both cases, the upflow- and downflow-fluxes have the same sign: negativein most of the fluid layer, except near the shell boundaries. Near the outer surface, there issignificant asymmetry in the flux magnitudes between cases N3 and N5, and separately between
the upflow- and downflow-flux of case N5. We argue that this arises mostly from the asymmetry in
upflow- and downflow-speeds. Figure 2.16 shows the rms radial speeds of upflows and downflows
in cases N3 and N5 as functions of radius. In case N5, the downflows are about twice as fast as the
upflows near the outer surface. The Reynolds stress, which scales like vr2 , is correspondingly larger
for the downflows than the upflows. Similarly, since both the upflows and downflows are about
twice as fast in case N5 than in case N3 near the outer surface, both the upflow- and downflow-fluxes
are greater in case N5.
We note here that the Rossby number associated with the rms radial downflow speed at thebase of the thermal boundary layer is ∼0.2 for case N3 and ∼1.2 for case N5. Here we define the
0 (r)/2Ω H (r), where v 0 (r) is the rms convective speed of thedownflow Rossby number as vrms
0 ρ
rms

77

250

N3, upflow
N3, downflow
N5, upflow
N5, downflow

200

100
50
0.80

0.85

r/ro

0.90

0.95

RMS Velocity (m/s)

150

0
1.00

Figure 2.16: Coupling of Thermal Boundary-Layer Depth to Rotational Constraint.
Spherically and temporally averaged radial convective velocity amplitudes for cases N3 and N5.
The rms amplitudes are computed separately for the upflows and downflows. The vertical magentalines indicate the location of the base of each thermal boundary layer for cases N3 and N5 (compare
to Figure 2.6).

78
downflows, averaged over spherical surfaces of radius r. Hence, the downflows in case N5 are, on
average, rotationally unconstrained according to Equation (2.1). This agrees with the fact thatthe downflow-flux is negative in the outer layers of case N5; the downflow plumes conserve theirangular momentum and thus transport it radially inward.
We now verify that it is only the downflow plumes that carry angular momentum inward at
low latitudes, and not the slower downflow lanes associated with Busse columns. Figure 2.17 showsa similar “dive” through the fluid layer as in Figure 2.11, but this time illustrates the instantaneous
angular-momentum flux in the patch. We see that at r/ro = 0.957 (which is close to r/ro = 0.942—
the extremum of the flux from the downflows in Figure 2.15(b)), the central region of nearly every
plume is associated with a positive vφ0 and thus a negative angular-momentum flux. The notable
exception is plume D, which has the deepest extent of any of the plumes. However, plume D has
negative flux in the deepest layers, as seen in Figure 2.17(d).
In contrast to the plumes, the columnar downflow lanes in the deep layers (Figures 2.17(b–d)) mostly have positive angular-momentum flux, consistent with the tilt of Busse columns. It isalso interesting to note that in between the axially aligned downflow lanes, the upflows are split
into two regions with alternating sign of the flux—positive on the right and negative on the left.
This is simply due to the manner in which upflows diverge and recirculate (in the center of the
upwell) to accommodate the impenetrable outer boundary and maintain mass conservation. Note,however, that in the deeper layers, the positive flux in the upflows dominates the negative flux.
This explains the weakness of the positive upflow-flux between r/ro ∼ 0.96 and r/ro ∼ 1 in Figure
2.15(c). Near the outer surface, the upflow rolls diverge symmetrically in both the positive andnegative longitudinal directions and their net angular momentum transport cancels out almost
completely.
Finally, we note that near the outer surface, the east–west downflow lanes have largely the
opposite sign of flux compared to the upflows in which they are embedded—negative on the rightand positive on the left for the upflow column straddling the central meridian. This is due to thetendency of the fluid in the east–west lanes to flow sideways toward the interstices, which have an

79

Figure 2.17: Angular Momentum Transport by Downflow Plumes. Radial structure of theangular-momentum transport by downflow plumes in the 40◦ × 40◦ equatorial patch for case N5(compare to Figure 2.11). Each panel shows the instantaneous Reynolds-stress angular-momentum
flux ρr sin θvr0 v 0 φ in the patch for successively deeper layers at a common time. Here the colorbar
is saturated to the same values for all four panels. The locations of the downflow plumes identified
in Figure 2.11 are again traced in depth by small black circles, labeled in the first panel by capitalletters A–F.

80
extremely low pressure compared to their surroundings.

2.8

Discussion
In this work, we have explored the development of near-surface shear in 3-D spherical-shellmodels of solar-like convection as a natural consequence of increasing the density contrast acrossthe shell. We find that increased stratification does indeed foster more rapid flow structures with
Reynolds stresses that enhance near-surface shear. However, this proves insufficient to create asolar-like Near-Surface Shear Layer.
Our highest-contrast model (case N5) contains two types of flow structures that influence
differential rotation: rotationally constrained Busse columns and rotationally unconstrained downflow plumes. The Busse columns transport angular momentum outward at low latitudes and thus
maintain a fast equator and slow polar regions. The plumes transport angular momentum inward
in the outer half of the layer at all latitudes. The influences of Busse columns and downflow plumes
on the Reynolds stress is cleanly summarized in Figure 2.18, which shows the correlation coefficient
Ei1/2
E h
D
D
plotted in the meridional plane for the upflows and downflows
C ≡ vr0 vφ0 / (vr0 )2 (vφ0 )2
in case N5. This correlation contains the same information as the angular-momentum fluxes depicted in Figure 2.15, but the signal has been normalized and the geometric factor r sin θ has been
removed.
For the upflows, the correlation is everywhere positive at low latitudes, implying outward
angular momentum transport—and thus dominance by Busse columns—at all depths. It is negative (except near the inner boundary) at high latitudes, consistent with inward angular momentum
transport through Coriolis-deflection. The correlation for the downflows is mostly negative everywhere, except in the inner half of the shell at low latitudes (where the Busse columns dominate) andnear the inner boundary at high latitudes. The strong positive correlation near the inner boundary
for both upflows and downflows is due to the impenetrability condition maintained by the pressure
force, which we do not investigate in detail here.
Figure 2.18 shows that the dip in angular velocity at low latitudes in case N5, while similar

81

Figure 2.18: Reynolds-Stress Correlation in Upflows and Downflows for High and Low
Density Contrast. Correlation coefficient of the convective velocities vr0 and vφ0 for the upflows,
downflows, and total flow in the meridional plane for case N5. The numbers beside each panel
indicate the extremum values for the correlation coefficient. The dashed contour in panel (c)denotes the surface in the meridional plane where the total correlation coefficient vanishes.

82
in amplitude and radial extent to the solar NSSL, arises for the wrong reasons. The Busse columns
transport angular momentum outward, while the plumes transport angular momentum inward.
The radial location where the upward flux from the Busse columns cancels the downward flux
from the plumes—i.e., the spherical surface where the total correlation is zero in Figure 2.18(c)—corresponds roughly to the peak in rotation rate at low latitudes. The low-latitude NSSL in caseN5 thus does not represent a layer that the downflow plumes slow down, but rather a layer thatthe Busse columns fail to speed up. In the Sun, by contrast, the rotation curves have only a small
positive slope with radius (see Figure 2.1), implying that Busse columns, if they are present in theSun, do not transport angular momentum in the same way as they do in simulations, or that theplumes reach more deeply than in our models.
The essential role played by Busse columns in maintaining numerical models’ near-surface
shear is even more apparent at high latitudes in case N5. Here the Busse columns have little effecton angular momentum transport (the correlations in Figure 2.18 are mostly negative for both
upflows and downflows), while fast, small-scale downflow plumes conserve angular momentum inradial motion, transporting it inward. Although this Coriolis-deflection effect is, in fact, stronger athigh latitudes than at low latitudes, there is basically no high-latitude near-surface shear in case N5.
Thus, the mechanism described by Foukal & Jokipii (1975)—namely, that there is a rotationallyunconstrained fluid layer near the outer surface of the solar CZ—cannot, by itself, explain the
NSSL, at least not using current numerical models.
Miesch & Hindman (2011) make a similar point in their investigation of how meridional
circulation and differential rotation respond to a negative axial torque. In our case N5, there isa negative torque due to the Reynolds stress. At high latitudes, the torque balance is primarily
between the Reynolds stress and meridional circulation, reducing the torque balance in Equation(2.3) to −τmc = τrs . The definition of τmc in Equation (2.4) then leads to the following equilibrium
relationship between meridional circulation and differential rotation:
ρ hvpol i · ∇(Ωr2 sin2 θ) = τrs .

(2.6)

83
The preceding equation may be satisfied in multiple ways, since both the rotation rate Ωand the meridional circulation ρ hvpol i appear on the LHS. All local models that seek to explain
the solar NSSL by inward angular momentum transport—for example, the Coriolis-deflection of
downflow plumes in our case N5—prescribe a negative right-hand side, e.g., a negative τrs . This
does not, however, constrain the rotation rate until another equation—namely, that of meridional
force balance—is specified to determine the meridional circulation. Since the meridional circulation
is fundamentally a global phenomenon, it is unlikely that any local model of angular momentum
transport will explain the NSSL.
An examination of the high-latitude meridional circulation profile in the context of Equation(2.6) reveals why near-surface shear is nearly absent at high latitudes in our case N5 and in thesimulation of Hotta et al. (2015). In both simulations, there is a narrow band of poleward meridional
circulation near the outer surface (see Figure 2.5 of this work and Figure 10 of Hotta et al. 2015),
which brings high-angular-momentum fluid from equator to pole. If, on long timescales, this process
pumps angular momentum to high latitudes more efficiently than the Reynolds stress can remove
it, there will be no shear at high latitudes in equilibrium.
Figure 2.19 shows the radial profiles of Reynolds-stress torque, latitudinal mass flux androtation rate averaged over high latitudes for case N5. Clearly, case N5 satisfies Equation (2.6)
in its outer layers by having the meridional circulation near the outer surface inherit the radial
profile of negative Reynolds-stress torque, while the near-surface rotation rate is mostly flat. In
other words, although we might expect the strongly negative Reynolds-stress torque created by
downflow plumes in the outer layers of case N5 to drive near-surface shear, the particular profile ofthe near-surface meridional circulation opposes this driving mechanism almost completely.
Because the molecular viscosity in the solar interior is so low, the balance in Equation (2.6)
likely holds in the Sun, possibly with the added complication of a Maxwell torque from the magnetic
field. This work has shown explicitly that the only way to understand the solar NSSL in the context
of local angular momentum transport—i.e., the right-hand side of Equation (2.6)—is through adetailed understanding of how the meridional circulation is established in the Sun. Once this is

84

1.00

v

rs

0.75

scaled units

0.50
0.25
0.00
0.25
0.50
0.75
1.00
0.80

0.85

r/ro

0.90

0.95

1.00

Figure 2.19: The Relation Between Meridional Circulation, Rotation Rate, andReynolds Stress in the NSSL. Radial profiles of τrs , hρvθ i, and Ω, averaged over high latitudes (between ±45◦ and ±60◦ ) for case N5. We scale hρvθ i by its maximum absolute value tomake its radial profile lie in the range (−1, 1). We scale Ω and τrs to match the shape of theirprofiles in Figures 2.3 and 2.14, respectively. The shaded region indicates the near-surface layers
in which the radial profile of the meridional circulation follows that of the Reynolds-stress torque.

85
understood, the origins of the NSSL may be determined by Equation (2.6).
Hotta et al. (2015) and Featherstone & Miesch (2015) have made substantial progress inunderstanding meridional force balance, with the former being concerned specifically with thebalance required to maintain a NSSL. Hotta et al. (2015) argue that the high-latitude features ofnear-surface shear achieved in their simulation represent a significant deviation from the TaylorProudman state, which is achieved by turbulent transport of latitudinal momentum in the radial
direction—i.e., the correlation hvr0 vθ0 i. This correlation is, they claim, produced by the sign of theradial derivative of the meridional flow (∂ hvθ i /∂r) in their NSSL. Although this indeed appears
to be a promising mechanism to break the Taylor-Proudman constraint, we argue that it cannot
provide the full picture for meridional force balance in the solar NSSL. The Sun has rotation rate
constant roughly on radial lines at high latitudes and thus is not in a Taylor-Proudman state either
within the shear layer or without. Thus, breaking the Taylor-Proudman state is a necessary, but
insufficient condition to create near-surface shear.
Although the detailed dynamical maintenance of the solar meridional circulation is unclear,
helioseismic observations provide fairly rigorous constraints on the circulation profile itself, at leastin the NSSL. In particular, the there is little near-surface variation of latitudinal flow hρvθ i withradius in the Sun as compared to case N5 (e.g., Giles et al. 1997; Zhao & Kosovichev 2004; Hathaway
2012; Chen & Zhao 2017; Mandal et al. 2018). Given that both the meridional circulation hρvpol iand rotation rate Ω are observationally constrained in the NSSL, it would be worthwhile to measure
the Reynolds-stress torque in the NSSL as well, for example using high-resolution ring-diagram
analysis (e.g., Greer et al. 2014; Greer et al. 2015). Once this is done, the relationship between
meridional circulation and differential rotation in the NSSL will be elucidated. This will be an
important guide for future simulations attempting to capture near-surface shear.
In summary, we have determined that the solar NSSL is still an unsolved problem. The
argument that a rotationally unconstrained layer near a convecting spherical shell’s outer surface
efficiently mixes angular momentum (and thus, creates a NSSL) is oversimplified, since only the
downflows are sufficiently rotationally unconstrained to pump angular momentum inward. At low

86
latitudes, the downflows must continually fight the upflows, which still exist in Busse columns andtransport angular momentum outward. In our models, this battle between upflows and downflows
seems essential to create substantial shear. By contrast, at high latitudes, the inward transportof angular momentum by both upflows and downflows only produces very weak shear due to theglobal character of the meridional circulation.
In future work, it would be useful to analyze the nature of the feedback between the meridional
circulation and differential rotation achieved in spherical-shell models of convection, in particular
its behavior in highly stratified regimes and in the presence of magnetic fields. This feedback islikely relevant in a broader context than simply NSSL dynamics. Since the meridional circulation
is dynamically linked to the solar dynamo cycle both observationally and theoretically (e.g., Wanget al. 1989; Chou & Dai 2001; Ghizaru et al. 2010; Charbonneau 2014; Komm et al. 2015), dissecting
the nature of feedback between the solar NSSL and meridional circulation will provide important
theoretical constraints on the processes by which the Sun forms its magnetic field.

2.9

More Recent Modeling of the NSSLAs mentioned in Section 1.2.2 some of the magnetic tachocline simulations of Guerrero

et al. (2016a,b) achieve shear layers strikingly similar to the solar NSSL, mostly at high latitudes. Guerrero et al. claim that their NSSLs arise from rotationally unconstrained turbulence (anegative torque from the Reynolds stress, as in cases N3 and N5), but their torque balances are
not reported. (The torques are technically plotted in Figure 3 of Guerrero et al. 2016b, but the
choice of colorbar renders the balance in the NSSL invisible). These EULAG-MHD models also have no
explicit diffusion, and the transport properties associated with the unresolved scales are unknown.
Still, the NSSLs in Guerrero et al. (2016a,b) are the most convincing of any of the global models.
Isolating the role of the meridional circulation, magnetic fields, and coupling to a tachocline in suchmodels would shed much light on the solar NSSL problem.
More recently, Choudhuri (2021) developed a simple model of the NSSL without using global
simulations. His work posits that the NSSL is in thermal wind (geostrophic) balance, in addition

87
to the bulk of the CZ. As mentioned in Section 1.2.3, thermal wind balance is assumed to holdfor high rotational constraint. This is usually considered a good approximation in the deep solar
interior, but not in the near-surface layers (e.g., Miesch & Hindman 2011; Augustson et al. 2015;
Karak et al. 2014; Hotta et al. 2015). The assumption that geostrophy is broken in the NSSL is,
however, still an open question.
Choudhuri (2021) presents the thermal wind equation in the following form:
D E
D E
∂
T̂
∂
Ŝ
2
∂Ω
g(r)
g(r)
λ
=
=
.
∂z
rcp ∂θ
rT (r) ∂θ

(2.7)

The right-most equation is valid if isochores (surfaces of constant density) coincide with spheres,
D E
D E
yielding hρ̂i = 0 everywhere in the meridional plane and thus Ŝ /cp = T̂ /T (r). It should be
D E
noted that no simulation we are aware of has yielded hρ̂i = 0 (more often they yield Ŝ ≈ 0—see
Figure 3.8).
In any case, Choudhuri’s argument goes as follows. The background temperature T (r) decreases rapidly radius, while ∂ T̂ /∂θ ∼ ∆T̂ —the pole-to-equator temperature contrast—remains
the same.3 . We thus have that ∆T̂ /T should skyrocket near the outer surface. If we also require Equation (2.7) to hold, then ∂Ω2 /∂z—which is proportional to the tilt of the isorotation
contours—must become large, thereby creating the NSSL.
Jha & Choudhuri (2021) show that Equation (2.7), assuming a radially independent ∆T̂ =2.5 K (this value is taken from continuum and Ca II K intensity observations of the solar discin Rast et al. 2008; see the following Chapter 3) reproduces the NSSL very well. They take thehelioseismically inferred rotation profile from below the NSSL and then integrate Equation (2.7)
outward, yielding a rotation profile in the NSSL matching helioseismology to within ∼5%. Of
course, it is unclear what happens if the correct form of the thermal wind equation—employing thelatitudinal entropy gradient, which unconstrained, either by helioseismology or theory—is used. If
thermal wind balance is applicable in the NSSL, however, Choudhuri (2021); Jha & Choudhuri
3
Rotationally unconstrained convection is argued to homogenize ∂ T̂ /∂r in latitude so that the buoyant driving is
not influenced by rotation. The temperature contrast ∆T̂ should then be constant in the rotationally unconstrained
layer, which Choudhuri also assumes to be the depth of the NSSL.

88(2021) have effectively shown that ρ̂ = 0 near the outer surface, which would be an interestingfinding in and of itself.

Chapter 3

Revisiting the Sun’s Strong Differential Rotation along Radial Lines

Current state-of-the-art models of the solar convection zone consist of solutions to the NavierStokes equations in rotating, 3-D spherical shells. Such models are highly sensitive to the choiceof boundary conditions. In this Chapter, we present two suites of simulations differing only intheir outer thermal boundary condition. This boundary condition is either one of fixed-entropy
or fixed-entropy-gradient, as in Equation (1.7)(e). We find that the resulting differential rotation
is markedly different between the two sets. The fixed-entropy-gradient simulations have strong
differential rotation contrast and isorotation contours tilted along radial lines (in good agreementwith the Sun’s interior rotation profile revealed by helioseismology), whereas the fixed-entropy
simulations have weaker contrast and contours tilted in the opposite sense. We examine in detailthe force balances in our models and find that the poleward transport of heat by Busse columns
drives a thermal wind responsible for the different rotation profiles. We conclude that the Sun’sstrong differential rotation along radial lines may result from the solar emissivity being invariant
with latitude (which is similar to the fixed-entropy-gradient condition in our models) and thepoleward transport of heat by Busse columns. In future work on convection in the solar context,
we strongly advise modelers to use a fixed-gradient outer boundary condition.
This Chapter is primarily a restatement of the work published in Matilsky et al. (2020). As
the primary author of the paper, I conducted the simulations, performed their analyses, createdthe Figures, and wrote most of the text. My coauthor Bradley Hindman wrote some of the initial
drafts of Sections and 3.5 and 3.6. Both my coauthors Juri Toomre and Bradley Hindman provided

90
advice and guidance throughout the process, gave detailed text edits, and suggested schematic
designs of some of the figures.

3.1

Introduction: The Sun’s Tilted Isorotation Contours
Helioseismology has revealed in detail the internal rotation profile of the solar convection

zone (CZ; e.g., Thompson et al. 2003; Howe et al. 2005), as shown in Figure 3.1. The mostnotable properties of the rotation rate are that the equator rotates significantly faster than thehigh-latitude regions and that the isorotation contours are tilted significantly with respect to therotation axis, falling largely along radial lines. Furthermore, there are two shear layers at the top
and bottom of the CZ: at the top, the contours bend toward the equator in a region known as
the Near-Surface Shear Layer (NSSL; see Chapter 2), and at the bottom, the differential rotationin the CZ transitions to solid-body rotation, over a narrow boundary layer called the tachocline.
Prior to helioseismic probing, most theoreticians had assumed that the differential rotation thatis observed directly at the surface would imprint into the interior along isosurfaces parallel to therotation axis, hence satisfying the Taylor-Proudman theorem. The helioseismic observations haveclearly demonstrated that this theoretical supposition was wrong.
For the last several decades, global, 3-D supercomputer simulations of hydrodynamic convection in rotating spherical shells have succeeded in achieving rotation profiles that are fast at theequator and slow at the poles. However, simulations generally have a weaker overall differential
rotation contrast than that of the Sun. If the contrast is defined to be the difference in rotation
rate between the equator and 60◦ latitude, expressed as a percentage of the “frame” rotation rate,then for the Sun, this magnitude is ∼20% (using the solar Carrington rate Ωas the frame rate,

for example). Most simulations, on the other hand, have rotation-contrast magnitudes of ∼10%,
although there are some notable exceptions (e.g., Brun & Toomre 2002; Brown et al. 2010; Matilsky & Toomre 2020a). To date, however, there is no systematic physical explanation for why theseparticular simulations have high rotation contrast.
Simulations have also struggled to achieve rotation contours in the bulk of the CZ that are

91

Figure 3.1: The Solar Rotation Profile in the Meridional Half-Plane (Again). Contour
map in the upper meridional plane of the internal rotation profile of the Sun, in and below theCZ, averaged in longitude and time from 1995–2009. The rotation rate has been obtained usinga regularized least-squares (RLS) inversion, which is sensitive only to the equatorially symmetricpart of the rotation. The dashed lines are at a 25◦ angle to the rotation axis and align with theisorotation contours at mid-latitudes. Image credit: Howe et al. (2005), extended to include GONG
data until 2009.

92
significantly tilted from the axis, as seen in Figure 3.1. With some exceptions (e.g., Elliott et al.
2000; Miesch et al. 2006), the simulations generally have cylindrically aligned contours. In the caseof Miesch et al. (2006), the contours were tilted systematically by imposing a modest latitudinal
entropy gradient at the base of the CZ. By modifying the magnitude of this entropy gradient, Miesch
et al. (2006) controlled the magnitude of the contour tilt. Imposing this boundary condition was
inspired by thermal wind balance in the tachocline imprinting up into the CZ.
There is also good evidence for thermal wind balance operating in the deep CZ from analytical models. Balbus (2009a) and Balbus et al. (2009) assumed a simple functional relationshipbetween the entropy and angular velocity and solved explicitly for the combined isorotational andisentropic contours as characteristics of the thermal wind equation. This yielded tilted contours
in good agreement with helioseismology for the bulk of the CZ (away from the boundary layers ofshear at the top and bottom). Such analytical work—in addition to lending strong support for thepresence of a thermal wind in the solar interior—points to the need to better understand the dynamical coupling between entropy and angular velocity in global numerical simulations, something
we address in the present paper.
In this work, we explore the role that the outer thermal boundary condition plays, in conjunction with an interior thermal wind, in modifying the resulting differential rotation. We consider thetwo most commonly used options: fixed-entropy (FE), in which the entropy at the outer boundary
is fixed to a constant value, and fixed-flux (FF), in which the radial entropy gradient at the outerboundary is fixed, therefore implying an outward conductive flux that is independent of latitude.
See Hurle et al. (1967) and Edwards (1990) for descriptions of these boundary conditions in thecontext of linear theory, and Anders et al. (2020) for a detailed analysis of the boundary conditions
in nonlinear simulations of Rayleigh-Bénard convection. The FF condition is more appropriate for
the Sun, since the radiant flux from the solar photosphere does not appear to vary substantiallywith latitude. Rast et al. (2008) analyzed full-disk images from the Precision Solar Photometric Telescope at the Mauna Loa Solar Observatory and calculated non-magnetic contributions tothe solar photospheric intensity. In both continuum and Ca II K intensity distributions, only a

93∼0.1–0.2% variation was observed, corresponding to a solar pole that is at most ∼2.5 K warmer(in terms of effective temperature) than the equator. The deduced near-spherical-symmetry of thesolar emissive flux is a significant observation, since a thermal wind strong enough to drive theobserved differential rotation would require greater differences in the interior temperature between
equator and pole. We return to this point in our concluding remarks.
We do not address the dynamics of the near-surface nor tachocline shear layers in this work. In
particular, our models have an impenetrable lower boundary that does not allow for the convective
overshoot of downflow plumes into the stable region that may play a role in the origin of thetachocline. The dynamical maintenance of the NSSL is still an open question, as discussed in
Hotta et al. (2015) and Matilsky et al. (2019), and models tend to only display signs of near-surface
shear if they have high density contrast (&100) across the CZ. To avoid high computational cost,the models in this work have a smaller density contrast of ∼20.
Solar-like differential rotation (fast equator and slow poles) in spherical-shell convection isthought to be due to the outward transport of angular momentum by Busse columns (also called“Taylor columns” and “banana cells”; e.g., Busse 2002; Jones et al. 2011; Matilsky et al. 2019).
In this work, we show that Busse columns also transport heat poleward and equivalently drive asolar-like differential rotation through a thermal wind. We find that in our FF cases, the thermal
wind drives stronger differential rotation magnitudes and achieves more significant tilts of theisorotation contours than in the corresponding FE cases. Elliott et al. (2000) noted this effect for
one simulation, but did not explore the underlying mechanism.
In Section 3.2, we describe the parameter space explored by our simulation suite, as well as themathematical details of the FF and FE boundary conditions. In Section 3.3, we describe the basic
results of our experiment, focusing on the achieved differential rotation. In Section 3.4, we quantify
the force balance achieved in our models, which, for the radial and latitudinal directions, consistsof a thermal wind in spherical geometry. In Section 3.5, we examine the latitudinal transport ofenergy by Busse columns that is responsible for the thermal wind. In Section 3.6, we discuss how
the effects of the thermal wind are modified by the outer thermal boundary condition. In Section

94
Table 3.1: Common Input Parameters for All Simulations. Each shell covers three density
scale heights of the solar convection zone, as in case N3 (see Chapter 2 and Table 2.1).
ri
ro
cp
γ
ρi
L
M
R
Pr ≡ ν/κ

5.00 ×1010 cm = 0.719 R
6.59 ×1010 cm = 0.947 R
3.50 ×108 erg K−1 g−1
1.67
0.181 g cm−3
3.85 ×1033 erg s−1
1.99 ×1033 g
6.96 ×1010 cm
1.00

3.7, we discuss our simulation results in the context of the Sun.

3.2

Numerical Experiment
We consider time-dependent, 3-D simulations of a rotating, stratified spherical shell of fluid

representative of the solar CZ. As in Chapter 2, we use the open-source code Rayleigh 0.9.1(Featherstone & Hindman, 2016a; Matsui et al., 2016; Featherstone, 2018), which solves the equations of hydrodynamics in spherical geometry. All notation and equations solved are identical tothat laid out in Chapter 2. Each simulation is similar to case N3 from that Chapter. In particular,
the background thermodynamic state is one of an adiabatically stratified polytrope and the density
contrast across all our spherical shells is exp(Nρ ) = exp(3) ∼20. Each simulation differs only by
the value of ν = κ (the Prandtl number for all simulations described in this thesis is unity), therotation rate, and the outer thermal boundary condition. Input parameters common to all thesimulations explored here are shown in Table 3.1.
The main purpose of this work is to characterize the influence of the outer thermal boundary
condition on the behavior of the resulting differential rotation. Recall that in Chapter 2, both cases
considered (N3 and N5) had top thermal boundary conditions of fixed entropy. Here we consider
models with different background rotation rates Ω0 and diffusion values (ν = κ). For each model,

95
we analyze two sub-cases:Ŝ = 0 at r = ro(fixed entropy, or FE)

(3.1)

and
∂ Ŝ
L
at r = ro=−∂r
4πro2 ρT κ

(fixed flux, or FF).

(3.2)

The solar luminosity that is injected into the system via internal heating is ultimately carriedout through the outer surface via thermal conduction, which in our models arises from entropy
gradients (see Equation (1.1b)). For the fixed-entropy condition (3.1), the interior is initiallyheated (leading to Ŝ > 0 in the lower parts of the CZ) while the entropy at the outer surface is“pinned” to zero. This naturally establishes a thermal boundary layer (sharp entropy gradients∂ Ŝ/∂r < 0) just below the outer surface. The steepness of the gradient (i.e., the strength of theoutward conductive loss of energy) is allowed to vary with latitude.
For the fixed-flux condition (3.2), the outer thermal boundary layer is present from the
beginning of the simulation. The steepness of the entropy gradient (and thus the energy loss) at
the outer surface is independent of latitude by construction, and is forced to have exactly the value
needed to carry out a solar luminosity. The fixed-flux condition is thus more “solar-like,” since in
the Sun there is little observed latitudinal dependence of the emergent intensity (Rast et al., 2008),
which is equal to the energy lost via radiative cooling at the photosphere.
For both the FE and FF cases, the thermal conductive boundary layer stands in contrast
to the real solar photosphere, in which radiative cooling removes the heat from a very thin (∼100km) outer layer. The cooling drives very small temporal and spatial scales of motion compared tothe deep interior (such as granulation and supergranulation), making its direct inclusion in global
models problematic. Researchers have sought to address this difficulty in various ways. Nelson et al.
(2018) implemented stochastic driving of convection by near-surface plumes designed to mimic theeffects of supergranulation, finding that that the flow structures and transport properties weresignificantly altered in the deep CZ. Hotta et al. (2019) simulated the whole CZ with no rotation or96
magnetic field, coupling a global spherical shell that captured large-scale flows in the deep interior
to a Cartesian box that solved the equations of radiative transfer in the photosphere (they used
the RSST code mentioned in Chapter 1). They found that the near-surface motions had a weakinfluence on the deep interior. Regardless of its relevance to interior flow structures, correctly
capturing the small-scale near-surface flows in global models is currently prohibitively expensive
computationally. In order to explore a wider range of parameter space, we thus only consider the
FE and FF boundary conditions here.

3.3

Simulation Results
We label simulations with a prefix that signifies the outer boundary condition (“FE” forEquation (3.1) and “FF” for Equation (3.2)), followed by the value of the diffusion constant ν = κ(in units of 1012 cm2 s−1 ), followed by the value of the rotation rate (in units of the sidereal
Carrington value for the Sun, Ω ≡ 2.87 × 10−6 rad s−1 , or Ω /2π ≡ 456 nHz). For example, “caseFE4-3” refers to a simulation with an FE outer boundary, for which ν = κ = 4 × 1012 cm2 s−1throughout the domain, and Ω0 = 3Ω . We note that except for the slightly differing definitionsof Ω , case FE2-3 here is identical to case N3 from Chapter 2. 1
Table 3.2 in Section 3.8 contains the values of the non-dimensional parameters, as well as thegrid resolution, for each of the 18 simulations considered in this work. Table 3.2 has four groupings
according to FE or FF at two different rotation rates. Following the notation of Featherstone &Hindman (2016b), we parameterize the strength of the imposed driving in each simulation througha bulk “flux Rayleigh number” (imposed a priori),
RaF ≡g̃ F̃ H 4
cp ρ̃T̃ ν 3(3.3)

In Chapter 2, we used Ω ≡ 2.6 × 10−6 rad s−1 , which is the value of the interior rotation rate of the Sun aspredicted from the tachocline model of Spiegel & Zahn (1992). This convention likely began in Brun & Toomre (2002)
and does not agree with the results of helioseismology (e.g., Howe et al. 2000). In this and subsequent Chapters,
we instead choose Ω to be the solar Carrington value. In any case, these different conventions likely make little
difference, since we cannot achieve solar-like differential rotation for models rotating near the solar rate anyway.
1

97
and the level of turbulence through bulk Reynolds or Péclet numbers (calculated a posteriori),
Re =Peṽ 0 H
=
.
Pr
ν

(3.4)

Since the Prandtl number for all models is unity, the Péclet number Pe = ṽ 0 Hκ−1 equals the
Reynolds number.
Similarly, we parameterize the influence of rotation through an Ekman number (imposed apriori),Ek ≡ν
2Ω0 H 2

(3.5)

and a bulk Rossby number (calculated a posteriori),
Ro ≡

ṽ 0
.
2Ω0 H

(3.6)

In the preceding equations, the length scale H is taken to be the shell depth ro − ri , the tildes
refer to volume averages over the full spherical shell. For the volume averages of the thermodynamic
variables (e.g., ρ̃), we take the average of the background variable (e.g., ρ(r)). F refers to the energy
flux associated with conduction and convection in equilibrium (see Featherstone & Hindman 2016b).
The typical convective velocity amplitude ṽ 0 refers to the rms of the velocity with the longitudinallyaveraged part subtracted, the mean being taken over time and over the full volume of the shell.
Throughout this work, temporal averages are taken during the latter portion of run time for which
there is statistical equilibrium—generally ∼3/4 of the total run time listed in Table 3.2.
Before discussing our results in detail, we note that all our models have fairly high levels ofthermal and viscous diffusion. Furthermore, all our models rotate at either two or three times thesolar Carrington rate. These choices, which stand in contrast to the physics of the solar interior,
ensure that our models have low enough Rossby numbers to avoid antisolar differential rotation (fastpolar regions, slow equator). All global spherical-shell convection codes produce high velocities atlarge scales in the solar context when sufficiently turbulent. The influence of rotation on the large
scales is therefore weak, which causes less coherence in the outward angular momentum transport

98
by the convection, and ultimately less angular momentum in the outer layers than the inner layers(i.e., an antisolar differential rotation). The overall problem—that increasing the turbulence insimulations leads to antisolar states—is now called the “convective conundrum” (O’Mara et al.,2016). The antisolar states can be avoided by raising the rotation rate, raising the diffusions, or
lowering the luminosity. We choose a combination of the former two, which requires that our models
are only moderately turbulent. Nonetheless, the viscous force and heat flux are small to leading
order in the primary dynamical balances.
Returning to our simulation results, we quantify the magnitude of the overall differential
rotation contrast as the difference in the outer-surface rotation rate between the equator and 60◦ latitude, normalized by the frame rotation rate:∆ΩΩ(ro , π/2) − Ω(ro , π/6)
≡
,
Ω0
Ω0

(3.7)

where
Ω(r, θ) ≡ Ω0 +hvφ i
r sin θ(3.8)

is the rotation rate of the fluid as a function of r and θ and the angular brackets denote a combined
temporal and longitudinal average. From Table 3.2, the FF cases have differential rotation contrasts∆Ω/Ω0 that are significantly greater—on average, by ∼40%—than those of the FE cases. For
comparison, the solar value of the rotation contrast is substantially higher than in any of ourmodels: ∆Ω /Ω = 0.20 (see Howe et al. 2000, Figure 1). For the solar estimate, we have takenΩ to be the sidereal Carrington rate and ro to lie just below the Near-Surface Shear Layer.
Figure 3.2 shows how the differential rotation fraction scales with the reduced Rayleigh number, which accounts for the increase to the critical Rayleigh number for convective onset causedby rotation (Chandrasekhar, 1961). The reduced Rayleigh number thus serves as a better parameterization of the supercriticality of the system than simply the Rayleigh number. From Figure
3.2, each type of boundary condition yields a similar scaling with the reduced Rayleigh number
R. For the Ω0 = 3Ω

cases (circles in Figure 3.2), the rotation contrast increases monotonically99

Figure 3.2: Simulated Rotation Contrasts for Upper Thermal Boundary Conditions.
Rotation contrast achieved by the suite of simulations plotted as a function of the reduced Rayleighnumber R ≡ Ra Ek4/3 . Circles mark the Ω0 = 3Ω cases and triangles the Ω0 = 2Ω cases.

(but with decreasing slope) with increasing R, so that the curves connected by circles in Figure
3.2 are concave-down. For the Ω0 = 2Ωcases (triangles in Figure 3.2), the curves “overturn” sothat a peak value of the rotation contrast (at around R ∼ 27) is achieved. This behavior (concisely described in Gastine et al. 2013) is a symptom of the convective conundrum; as models growmore turbulent, the rotation contrast increases at first, but then decreases and eventually becomes
negative (i.e., antisolar).
To illustrate exactly where the “extra” rotation contrast in the FF cases is located, we plot
the rotation rate at the outer surface for three of the cases rotating at 3Ω

in Figure 3.3. Mostof the additional contrast occurs at high latitudes, where the polar regions in the FF cases rotate
significantly more slowly than in their FE counterparts. Additionally, the equator in the FF cases
rotates slightly faster than in the FE cases. For all simulation pairs, the difference in contrast
between the FE case and the FF case is greater the smaller the value of the diffusion (or the higher
the level of turbulence).
Figure 3.4 shows contour plots of rotation rate in the upper meridional plane for some of thesimulation suite. Clearly there is a striking difference between the tilts of the rotation contours in100

Figure 3.3: Simulated Latitudinal Rotation Profiles for Each Thermal Boundary Condition. Temporally and longitudinally averaged rotation rate (normalized by the frame rate Ω0 )at the outer surface for three of the cases rotating at Ω0 = 3Ω , plotted versus latitude. Dashedlines correspond to the FE cases and solid lines to the FF cases. In the legend, ν = κ is given inunits of 1012 cm2 s−1 .

Figure 3.4: Simulated Latitudinal Rotation Profiles for Each Thermal Boundary Condition. Temporally and longitudinallyaveraged rotation rate in the meridional plane for some of the simulations in the suite, with the two hemispheres averaged assuming evensymmetry about the equator. In the titles at the top, ν = κ is given of units of 1012 cm2 s−1 , and Ω0 in units of Ω .
101

102
the FE and FF simulations.
In this paper, we define all rotation-contour tilt angles (or simply tilts) with respect to therotation axis, with zero tilt corresponding to alignment of the contour with the rotation axis. We
use the sign convention for tilt angle illustrated in Figure 3.5. Under this convention, the solar
rotation contours have positive tilts at all latitudes in the bulk of the CZ (above the tachocline andbelow the NSSL, as shown in Figure 3.1). We thus define the tilt angle of a rotation contour at agiven point in the meridional plane as−1

tilt ≡ − tan




∂Ω/∂z
,
∂Ω/∂λ

(3.9)

which is consistent with the sign convention shown in Figure 3.5 for solar-like differential rotation,in which the contours further from the rotation axis correspond to a higher rotation rate.
Describing the solar rotation contours as “tilted along radial lines,” as is often done, istechnically misleading. Radial tilt implies a specific dependence of the contour tilt angle withlatitude, namely, contours that fan radially outward from the center of the Sun. In Figure 3.1, bycontrast, the bulk-CZ tilts are roughly constant at ∼25◦ for mid-latitudes, are smaller than ∼25◦for low latitudes (where radially aligned tilts would be greater), and are greater than ∼25◦ for highlatitudes (where radially aligned tilts would be smaller). To avoid confusion, we will henceforth
not use the term “radial tilt” and instead describe the rotation-contour tilt (in the Sun and in our
simulations) simply as “positive” or “negative,” using the sign convention illustrated in Figure 3.5.
In Figure 3.6, we show the values of the rotation-contour tilt angle at mid-depth for a subset
of our models and for the Sun. The positive tilt for the FF cases is obvious, with the maximum tilt
angle being about +15◦ for the highest value of the diffusion (ν = 4 × 1012 cm2 s−1 ). This is still
substantially lower than the solar value for contour tilt, which attains a maximum value of ∼25◦ inthe middle of the solar CZ. The contours in the FE cases all have positive tilt at low latitudes. At
high latitudes, however, they have negative tilt, and are tilted the most (with a tilt angle of about−10◦ ) for the lowest value of the diffusion (ν = 2 × 1012 cm2 s−1 ).

103Figure 3.5: Schematic of the Different Types of Contour Tilt. Here we show our definitionof contour tilts: (a) Positive contour tilt (all contours tilted at a constant +25◦ ), (b) zero contour
tilt, and (c) negative contour tilt (all contours tilted at a constant −25◦ ).

Figure 3.6: Latitudinal Tilt-Angle Profiles for Each Thermal Boundary Condition. Tilt
angle of Equation (3.9) shown as a function of latitude for three of the cases rotating at 3Ω andfor the Sun. The profiles are taken at the middle of the shell for our models and the middle of
the CZ for the Sun. The Northern and Southern Hemispheres have been averaged assuming odd
symmetry for tilt angle. Dashed lines correspond to the FE cases and solid lines to the FF cases.
For the solar tilt angle, we use the inversion from GONG data 1995–2004 as reported in Howe et al.
(2005) and shown in Figure 3.1.

104

3.4

Thermal wind balance
We find that to leading order, the longitudinally and temporally averaged force balance inthe meridional directions r and θ (or λ and z) is dominated by the Coriolis, pressure, and buoyancy
forces for each simulation in this work:


hP i
−∇
ρ


+

hSi
g(r)êr + 2 hvφ i êφ × Ω0 ≈ 0.
cp

(3.10)

Here, the angular brackets denote a combined temporal and longitudinal average.
In the Earth’s atmosphere, a “thermal wind” describes a situation in which geostrophic
balance (pressure balancing the Coriolis force) holds in the horizontal directions and hydrostaticbalance (pressure balancing gravity) holds in the vertical direction. Equation (3.10) thus represents
the generalization of a thermal wind to the solar geometry, in which the vertical (radial) andhorizontal extents of the flow structures are comparable (unlike in the Earth’s atmosphere wherethe vertical extent is very small). Furthermore, the flows in the solar geometry generally have a
vertical component, unlike in a classical thermal wind for which the flows are purely horizontal.
The colatitudinal component of Equation (3.10) may be rearranged to yield1∂PΩ(r, θ) ≈ Ω0 +,
Ω0 ρr2 sin 2θ ∂θ

(3.11)

which is a purely geostrophic equation, since the buoyancy force is radial. Figure 3.7 shows a
representative example of geostrophic balance for the FE2-3/FF2-3 pair. Clearly Equation (3.11)
is very well satisfied for both cases, with deviations from geostrophy being no more than 1 part in
103 in the bulk of the meridional plane and 1 part in 102 at isolated regions by the equator andpole. The same is true for all the cases considered in this work, indicating that the differential
rotation profile in our simulations is almost completely determined by the pressure profile, andvice versa. The fact that the differential rotation magnitudes are ∼40% greater in the FF cases
compared to the FE cases is thus a consequence of greater latitudinal pressure gradients. Figure
3.7 also indicates that viscosity plays a relatively insignificant role in the force balance at large
scales.

105

Figure 3.7: Thermal Wind Balance: The Pressure Perspective. Point-by-point colatitudinal
force balance in the meridional plane for representative pair (a) case FE2-3 and (b) case FF2-3.
We show the difference between the temporally and longitudinally averaged rotation rate Ω andthe right-hand side of Equation (3.11), which we denote by ΩP (rotation rate from the pressure),
normalized by the frame rotation rate Ω0 . The two hemispheres have been averaged assuming evensymmetry about the equator.

106
To assess why there are opposite signs of tilt for the rotation contours in the FF and FE
simulations, we differentiate Equation (3.11) with respect to the axial coordinate z and use theradial component of Equation (3.10) to eliminate terms (or equivalently, take the φ-component of
the curl of Equation (3.10)), yielding
∂Ω
g
≈
2
∂z
2Ω0 r sin θcp




∂S
.
∂θ

(3.12)The tilt of the rotation contours is thus determined by the entropy distribution in the final thermodynamic state.
In Figure 3.8, we show the average profiles for entropy, pressure, and temperature in themeridional half-plane for the FE2-3/FF2-3 pair. Case FF2-3 (which is representative of all theFF cases in the simulation suite) displays a monotonically increasing entropy from equator topole. Case FE2-3, on the other hand, has a non-monotonic entropy profile: except on the outerboundary, the entropy from equator to pole increases up to ∼20◦ latitude, then decreases. In bothcases, the pressure and temperature deviations (normalized by the background reference state) aresubstantially greater (by a factor of ∼30 in the case of the pressure) than the entropy deviation.
The profiles of pressure and temperature in the meridional plane thus tend to mirror one another,with high temperature regions corresponding to high pressure regions and vice versa (compare thelast two columns of Figure 3.8).
The balance described by Equation (3.12) is shown for the representative simulation pairFE2-3/FF2-3 in Figure 3.9. There is good balance in the deep layers, although significant departuresnear the outer surface, which has been noted frequently in past work (e.g., Brun et al. 2011;
Augustson et al. 2012; Hotta et al. 2015). Quantitatively, the error in Equation (3.12) (shown in
the right-hand column of Figure 3.9) is ∼10% in the lower 80% of the layer and ∼50% in the upper
20% of the layer. For solar-like differential rotation (fast equator and slow poles), positively tilted
rotation contours (the FF cases) correspond to ∂Ω/∂z < 0, which arises from h∂S/∂θi < 0 at all
latitudes, as in Figure 3.8(d). Similarly, the FE cases (which have contours tilted negatively at highlatitudes and positively at low latitudes) all have h∂S/∂θi > 0 at high latitudes and h∂S/∂θi < 0Figure 3.8: Thermodynamic Profiles for Each Thermal Boundary Condition. Temporally and longitudinally averaged entropy,pressure, and temperature deviations from the spherically symmetric mean in the meridional half-plane (averaged assuming even symmetry
about the equator) for cases FE2-3 and FF2-3, normalized by the reference state profiles. The spherical mean h· · · isph has been removed
to show the variation from equator to pole. There should be “hats” over some of the thermodynamic variables in the panel labels; inMatilsky et al. (2020), we used overbars to indicate reference quantities and the lack of overbars to indicate perturbation quantities.

107

108
at low latitudes, as in Figure 3.8(a).

3.5

Poleward energy transport from Busse columns
In light of Equations (3.11) and (3.12), a thermal wind in spherical geometry fundamen-tally consists of pressure and entropy differences in latitude. Polar regions that are high-pressureand high-entropy relative to lower latitudes (which we have shown lead to strong differential rotation with positively tilted contours) are expected to be established by the preferentially poleward
transport of energy. In our simulations, this transport arises from the action of the convective
Busse-column rolls. These rolls manifest at convective onset as an overstable, low-frequency prograde wave (e.g., Unno et al. 1989) or, as it is called in the geophysics literature, a thermal Rossby
wave. This wave consists of a series of convective rolls, or Busse columns, that gird the equator.
Each roll is rotationally aligned and the sign of the vorticity alternates from roll to roll. Furthermore, each roll is in geostrophic balance; hence, the alternating sign of the vorticity corresponds toevery other roll being a zone of high pressure, with low-pressure rolls in between. Since the ends ofthe columns (at mid-latitudes) have neutral pressure, the pressure anomalies at the equator cause
poleward axial flow in the high-pressure rolls and equatorward flow in the low-pressure rolls (e.g.,
Figure 1 in Gilman 1983). The resulting strong correlation between pressure and the direction of
axial flow leads to a net poleward enthalpy flux through pressure work.
The effect just described is easiest to illustrate for models that are barely supercritical. Here,the profiles for the velocity and thermodynamic variables are dominated by the wavenumber associated with the most unstable mode. For the range of Ekman numbers spanned by our simulation
suite, the resulting Busse columns are mostly localized in the outer half of the shell by radius
and at low latitudes (see Jones et al. 2009 for a linear stability analysis of the problem). Figures3.10(a, b) show the instantaneous convective radial velocity and convective colatitudinal energy
transport in the highly diffusive case FE10-3, which lies in the barely supercritical regime. Each
upflow and downflow (pairs of which trace one Busse column roll) has an associated colatitudinal energy transport that is, on average, negative in the Northern Hemisphere and positive in109

Figure 3.9: Curl of Thermal Wind Balance: The Entropy Perspective. Temporally andlongitudinally averaged azimuthal vorticity balance in the meridional plane for representative pairFE2-3 and FF2-3. The two hemispheres have been averaged assuming odd symmetry about theequator. The left-hand column (a, c) shows the axial derivative of the rotation rate, ∂Ω/∂z.
The right-hand column (b, d) shows the difference between ∂Ω/∂z and the right-hand side ofEquation (3.12) ((∂Ω/∂z)S , or the axial derivative of rotation rate from the entropy), normalizedby 1.5 × 10−17 s−1 cm−1 .

110
the Southern Hemisphere, implying preferentially poleward energy transport. Note that under
the spherical-coordinate convention, with θ as the colatitude, the positive-θ direction is always
oriented north–south. Thus equatorward (poleward) transport of energy corresponds to positive(negative) colatitudinal transport in the Northern Hemisphere and negative (positive) transport in
the Southern Hemisphere.
Figures 3.10(c, d) show the radial velocity and convective energy transport in the comparatively more turbulent case FF2-3. The flow structures are more intricate and fine-scale than in
the barely supercritical regime, but the imprint of the most unstable mode remains. Many Bussecolumn rolls—which can be seen at low latitudes as columnar red and blue features in Figure3.10(c)—correspond to sites of negative colatitudinal energy transport (blue in Figure 3.10(d)) in
the Northern Hemisphere and positive transport (red in Figure 3.10(d)) in the Southern Hemisphere. Overall, there are more sites of poleward energy transport (from the Busse columns) thansites of equatorward transport in each hemisphere. Under an azimuthal average, the Busse columns
in the more turbulent case FF2-3 thus yield preferentially poleward energy transport, just as in thebarely supercritical case FE10-3.
The geostrophic nature of the Busse columns is illustrated in Figure 3.11, as is the resulting
axial component of the flow. In the top row (case FE10-3), panel (a) shows that the Busse-column
rolls alternate between high and low pressure. Panel (b) shows that the high-pressure rolls areeach anticyclonic (have negative vorticity), while the low-pressure rolls are cyclonic. Finally, panel(c) shows that each high-pressure anomaly corresponds to poleward flow (vz > 0 in the Northern
Hemisphere), while each low-pressure anomaly corresponds to equatorward flow (vz < 0). In thebottom row (the more supercritical case FF4-3), the Busse columns are less regularly spaced, but
still largely alternate between anticyclonic regions of high pressure and cyclonic regions of low
pressure (panels d, e). The axial flow associated with the Busse columns in case FF4-3 (panel f)then leads to poleward energy transport through pressure work, just as in case FE10-3.
It has long been known that Busse columns transport angular momentum outward. We
have just shown that Busse columns also transport heat poleward. The Busse columns thus define

111
a purely hydrodynamic mechanism coupling entropy and angular velocity. Balbus et al. (2009)
posited the presence of such a convective mechanism in the Sun and further argued that the motions
responsible should fall along surfaces of both constant entropy and constant angular velocity. In
that picture, the isorotational and isentropic contours should thus coincide. The Busse columns in
our simulation suite do not completely behave in this way, as evidenced by none of our simulations
having good alignment of the isorotational and isentropic contours. Independent of whether theconstant-entropy and constant-angular-velocity surfaces coincide in the Sun, a key point from ourwork is that the Busse columns provide an explicit convective mechanism to couple entropy andangular velocity.

3.6

Effect of outer thermal boundary condition
Given that Busse columns direct energy poleward, equilibrium can be achieved by forming

conductive gradients that balance the poleward convective enthalpy flux. In general, such conductive transport can be achieved in two distinct ways. As the pole heats up and the equator-to-polecontrast increases, a latitudinal gradient will form that transports heat equatorward. Additionally,
the increased temperature of the pole can lead to enhancement of the radial gradients in the outerthermal boundary layer, thus causing the poles to lose heat more efficiently (i.e., become superluminous). In the FF cases, the outer thermal boundary condition precludes the second of these options
because the radial gradients are fixed. Hence, the FF models must rely solely on the developmentof a pole-to-equator conductive gradient. In the FE models, both types of gradients are possible.
Therefore, the amount that the pole must be heated before equilibrium can be achieved is less for
the FE models than it is for the FF models. The outer thermal boundary condition thus has adirect influence on the latitudinal contrast in the temperature, entropy, and pressure, with the FF
boundary condition being conducive to strong contrast in all the thermodynamic variables. In thepresence of thermal wind balance, the FF boundary condition thus leads to enhanced contrast in
the differential rotation and positively tilted contours in the rotation rate.
Mathematically, we illustrate the combined effects of the outer thermal boundary condition

Figure 3.10: Heat Transport by Busse Columns: Laminar and Turbulent. Spherical snapshots (at r/ro = 0.910) of the radial
velocity vr0 (left-hand panels) and colatitudinal convective energy flux vθ0 [T Ŝ + P̂ + (1/2)ρ(v 0 )2 ] (right-hand panels), shown in global
Mollweide projections. The three terms in the convective energy flux correspond to advection of heat, pressure work, and advection of
kinetic energy, respectively. The top row of panels is taken from the highly diffusive, barely supercritical case FE10-3 and the bottom row
of panels is taken from case FF2-3. In both cases, the Busse-column sites in the left-hand panels are associated with sites of poleward
convective energy flux in the right-hand panels—i.e., panels (b) and (d) are on average blue in the North and red in the South.

112renderings, with the view from slightly north of the equator and the scene cut by the plane z/ro = 0.2. (b, e) Closeup views of P̂ 0 (shownin color) and ρvh0 ≡ ρ(vx0 êx + vy0 êy ) (shown as a vector field) in a portion of the plane z/ro = 0.2. (c, f) Closeup views of ρvz0 and ρvh0 in
the same portion of the plane as in panels (b) and (e). In all panels, red tones indicate positive values and blue tones indicate negativevalues. There should be “hats” over the pressure P̂ in the panel labels; in Matilsky et al. (2020), we used overbars to indicate reference
quantities and the lack of overbars to indicate perturbation quantities.

Figure 3.11: Geostrophic Balance within Busse Columns: Laminar and Turbulent. This figure illustrates the balance between
pressure and Coriolis force for Busse columns in cases FE10-3 (laminar; top row) and FF4-3 (turbulent; bottom
D E row). Here, we use the0standard Cartesian coordinates x, y, and z. (a, d) Snapshots of the non-axisymmetric pressure (P̂ = P̂ − P̂ ) shown as 3-D volume

113

114
and latitudinal energy transport using the steady-state total energy equation for the fluid. Using
Equations (1.2), this equation takes the form of a balance of fluxes:∇ · F = 0,

(3.13)F ≡ F conv + F cond + F rad + F visc + F circ(3.14)

whereis the temporally and longitudinally averaged total energy flux in the meridional plane and we havedefined the averaged convective, conductive, radiative, viscous, and meridional-circulation fluxes
through D
E 1
F conv ≡ ρ cp T̂ 0 v 0 +
v2v ,
2
D
E
F cond ≡ −κρT ∇Ŝ ,!
Z
1 ro
Q(x)x2 dx êr ,F rad ≡
r2 r

andF visc ≡ − hD · vi ,D E
F circ ≡ ρcp T̂ hvi(3.15a)(3.15b)(3.15c)(3.15d)(3.15e)E
D E
Drespectively. Note that ρT Ŝ + P̂ = cp T̂ , so the terms with T̂ 0 v 0 and T̂ hvi in the convective andmeridional-circulation fluxes represent the combined effects of heat advection and pressure work.
Technically, the flux due to transport of kinetic energy (proportional to v 2 v ) has convective terms(e.g., (v 0 )2 v 0 ) and meridional-circulation terms (e.g., hvi2 hvi). For simplicity, we include all the
kinetic-energy terms in the convective flux since they are in general small. 2
We are interested in the total latitudinal transport of energy, and so we integrate the total
flux in Equation (3.14) over conical surfaces at constant latitude:Iθ (θ) ≡ 2π sin θZ roFθ (r, θ)rdr.

(3.16)

ri
2

Recall that in Chapter 2, we use a capital Roman “F ” to denote the spherically integrated radial energy flux
(units of luminosity) and a curly “F” to denote the angular momentum flux. Here in Chapter 3, we keep the notationof Matilsky et al. (2020). We use “F” to denote the non-integrated energy flux (units of luminosity per unit area)and “Ir ” to denote the spherically integrated energy flux.

115
For the FF cases, there can be no net transport of energy in latitude due to the absence ofconductive losses in the polar regions through the outer boundary. In other words, Iθ (θ) ≡ 0.
For the FE cases, by contrast, there is a net poleward energy transport because the poles areallowed to be superluminous. Thus, Iθ (θ) will in general be negative in the Northern Hemisphereand positive in the Southern Hemisphere. Recall that equatorward (poleward) transport of energy
corresponds to positive (negative) Iθ (θ) in the Northern Hemisphere and negative (positive) Iθ (θ)
in the Southern Hemisphere.
Figure 3.12(a) shows the integrated colatitudinal energy fluxes in case FF2-3 after the system
has achieved statistical equilibrium. The total flux is very close to zero at all latitudes, indicating a well-equilibrated state. The dominant transport components are the convective flux, whichtransports energy preferentially poleward due to the Busse columns, and the conductive flux, whichtransports energy equatorward. The monotonic entropy gradient of Figure 3.8(d) and (by extension) the radially tilted isorotation contours in the FF cases, is thus seen to be a result of the
response by conduction to the convective transport of energy to the poles.
Figure 3.12(b) shows the integrated colatitudinal energy transport in case FE2-3. The poles
are clearly superluminous—i.e., there is a net poleward energy transport due to the convection.
For all the FE cases explored here, the energy loss at the poles is even greater than the heating by
the convection; the conductive flux is thus forced to change sign at mid-latitudes (around ±20◦ ),transporting energy poleward in concert with the Busse columns. This results in the non-monotonic
entropy profile of, e.g., Figure 3.8(a). The tilts of the rotation contours are thus negative at highlatitudes and positive at low latitudes, as in the top row of Figure 3.4.

3.7

Discussion and conclusions
We have shown that the differential rotation achieved in global, 3-D convection simulations iswell described by a thermal wind and highly sensitive to the outer thermal boundary condition. The
FF boundary tends to yield more solar-like rotation profiles (strong contrast with positively tilted
contours), while the FE boundary yields weaker contrast and negatively tilted contours. In light of

116

Figure 3.12: Colatitudinal Energy Transport for Each Thermal Boundary Condition.
This figure shows the total colatitudinal energy transport for (a) case FF2-3 and (b) case FE2-3—
i.e., the integrals of the fluxes in Equations (3.15) over conical surfaces at constant latitude. The
integrated fluxes are plotted as functions of latitude (= π/2 − θ). Contributions from the various
fluxes are indicated in the legend, “tot” denoting the sum of all the other fluxes. Positive (negative) Iθ indicates equatorward (poleward) transport in the Northern Hemisphere and poleward(equatorward) transport in the Southern Hemisphere.

117
these results, we now discuss the likelihood that the Sun’s strong rotation contrast and positively
tilted contours arise from thermal wind balance in the deep interior coupled with the observationthat the radiative flux from the solar photosphere does not vary appreciably with latitude.
The first question is whether the force balance Equation (3.10), which should in generalbe true for low Rossby numbers, holds in the Sun. The interior solar Rossby number is currently
unknown, but recent helioseismic estimates (Hanasoge et al., 2012; Greer et al., 2015) give Ro . 0.1
in the deep interior. Thus, it is likely that Equation (3.10) (and the derivative thermal wind
Equations (3.11) and (3.12)) apply in the solar CZ, except perhaps in the layers just below thephotosphere. thermal wind balance in the Sun is also not prohibited by observational results. We
can derive the temperature in the solar CZ from the rotation rate of Figure 3.1 assuming thermal
wind balance holds, integrating Equations (3.11) and (3.12) to get P̂ and Ŝ in the meridional planeand using Equation (1.2) to get T̂ . In this calculation, we set Ω0 to the sidereal Carrington rate
and use the same polytropic reference state employed in our models for the solar profiles. The
result is shown in Figure 3.13. The equator-to-pole temperature contrast required to drive thesolar-like differential rotation is ∼10 K (approximately uniform with radius), which is well below
the detection limit of helioseismology (e.g., Brun et al. 2010).
The second question is whether the Sun’s Busse columns send energy preferentially poleward.
In general, stellar convection transitions through a series of convective regimes as the supercriticality(measured by the reduced Rayleigh number R) increases (Hindman et al., 2020). Both the least
supercritical case in our work (FE10-3, for which R ∼ 2) and the most supercritical cases (thepair FF2-2 and FE2-2, for which R ∼ 50) have a strong preference for poleward transport byBusse columns, suggesting that the poleward transport is a feature of the most unstable mode of
convection that stays present as the flows get ever more complex.
Finally, it is an open question how the Sun might transport energy equatorward to maintainequilibrium. In our simulations, the net poleward transport of energy by Busse columns is, at its
maximum, a few percent of the solar luminosity, which is counteracted almost entirely by conductionin the FF cases (Figure 3.12). In the Sun, the thermal diffusivity associated with radiative heating

118

Figure 3.13: The Sun’s “Thermal Wind” Temperature Profile. Temperature deviation fromthe spherical mean in the solar CZ, assuming thermal wind balance holds. The temperature has
been calculated from the rotation rate inferred from helioseismology (Figure 3.1), using Equations(3.11), (3.12) and(1.2).

Figure 3.14: Total Busse-Column Heat Transport for Each Thermal Boundary Condition. Net poleward transport of heat from Busse columns for all simulations in the suite (except
the barely supercritical cases FE10-3 and FF10-3) plotted versus the reduced Rayleigh number
R ≡ Ra Ek4/3 . The poleward heat flux |Iθ, conv | has been averaged over latitude. Circles markthe Ω0 = 3Ω cases and triangles the Ω = 2Ω cases.

119
is ∼107 cm2 s−1 at mid-CZ (e.g., Hindman et al. 2020), which (if a thermal wind were operatingwith a temperature contrast of ∼10 K) would correspond to a latitudinal energy flux of ∼10−7 L .
Figure 3.14 shows how the Busse-column heat flux scales with supercriticality. The trends aredifferent between the FE and FF cases (and the Ω0 = 3Ω

and Ω0 = 2Ωcases), but there isclearly a tendency for the net transport to go down for our more supercritical FF simulations (bluecurves). This indicates that the balance between conductive and convective heat flux could hold in
the Sun, just with much smaller flux magnitudes. We admit that these results are only suggestive,
since the flux in all our models is still orders of magnitude higher than the presumed solar value of
10−7 L .
On a more practical note, it is advantageous to use an FF outer boundary condition in solar
simulations for two reasons. First, maintaining a strong differential rotation is particularly relevant
for dynamo models, since the dynamo cannot be maintained without sufficient shear (e.g., Brown
et al. 2010; Guerrero et al. 2016a; Matilsky & Toomre 2020a; Bice & Toomre 2020). Second, using
an FE outer boundary condition leads to superluminous poles, which are directly at odds withsolar observations. Figure 3.15 shows the conductive flux as a function of latitude at the top of thedomain for the FE cases. For case FE2-3, the flux in the polar regions reaches a value in excess of
the “solar-luminosity flux” (L /4πR2 ) by about 20%. This is far greater than the observationallyconstrained value of < 1% for the Sun (Rast et al., 2008).
We very much view this paper as a complement to Miesch et al. (2006). In that work,a systematic tilt of the rotation contours was achieved by imposing a small latitudinal entropy
gradient at the inner boundary, thereby ensuring that the entropy increased monotonically fromequator to pole. And indeed, for all our FF cases, there is a similar monotonic equator-to-pole
entropy gradient at the inner boundary. Monotonicity is not achieved in the FE cases. In otherwords, Miesch et al. (2006) showed that it is possible to tilt the rotation contours by imposing
a preferred geostrophic balance, and here we have shown how this preferred balance is naturally
established as the result of poleward energy transport by Busse columns and the FF outer boundary
condition.

120

Figure 3.15: The “Leaky Poles” of Fixed-Entropy Simulations. Latitudinal profile of Fcond, rat the outer boundary for several of the FE cases, normalized by the flux needed to carry out thesolar luminosity equally at all latitudes (the “solar-luminosity flux,” L /4πR2 ). The flux has been
averaged over time and longitude.

121
We conclude that any successful dynamical model of the Sun’s convection zone must be
consistent with three key observations: (1) The Sun possesses strong latitudinal differential rotation;(2) The isorotation contours are tilted positively in the sense of Figure 3.5; and (3) The emergent
intensity is nearly spherically symmetric. We have demonstrated in this paper that the transportof angular momentum and heat by Busse columns can yield results consistent with observations(1) and (2). Busse columns transport angular momentum outward, away from the rotation axis,
thus speeding up the equator. They simultaneously transport heat poleward via the enthalpy flux
associated with pressure work. These two transports reach equilibrium once thermal wind balanceis achieved.
While it is satisfying to address the observations (1) and (2) with a single mechanism (i.e., theaction of Busse columns), observation (3) presents a theoretical problem. The thermal wind balance
resulting from Busse columns requires that the solar poles be at a higher temperature than theequator, with a contrast of ∼10 K throughout the CZ. The emergent intensity at the solar surface,
however, is nearly uniform in latitude, with less than ∼2.5 K variation in effective temperature(Rast et al., 2008). This suggests that the thermal wind balance must only be maintained in the
deep CZ, and the surface layers (perhaps within the Near-Surface Shear Layer) may have different
dynamics. Those dynamics must somehow screen the surface from the imposition of the latitudinaltemperature gradient from below.
In our models, the proper proxy for the emergent intensity is the radial conductive flux atthe outer surface. For the FF cases, we impose latitudinal independence of this flux as a boundary
condition. Further, we have shown that this boundary condition is conducive to achieving strong
differential rotation and properly tilted isorotation contours. However, a fully self-consistent modelof the solar interior would reproduce the latitudinal independence at the outer boundary as anatural consequence of the near-surface convection and not as an enforced boundary condition.
Since the underlying dynamics giving rise to latitudinally independent emissivity in the Sun
are not well understood, we cannot predict a priori whether that same mechnism would be sufficient
to enforce latitudinally independent flux for other stars. It has long been postulated that oblateness122
induced by rotation in high-mass stars (in which a radiation layer overlies an interior CZ) could
result in superluminous poles (e.g., von Zeipel 1924; Collins 1963). Here, we have identified Bussecolumns as a possible mechanism for producing hot poles in low-mass stars (in which the CZ overlies
the radiation layer) that does not depend on the star’s distortion. Furthermore, Busse columns
do not necessarily require rapid rotation—they simply require strong rotational constraint, or inother words, low Rossby number. Thus, small dim stars that are expected to have weak convection
may have active Busse columns and hot poles, even with only moderate rotation. Finally, since
the homogeneity (or lack thereof) of the stellar emissive flux may significantly affect the interiordifferential rotation, we should not assume that the positive tilts of the isorotation contours inferred
for the Sun are a general feature of other low-mass stars.

3.8

Diagnostic Parameters for the Simulation Suite
Table 3.2 contains various input and diagnostic parameters for the entire simulation suite as

defined in Section 3.3. Recall that for each simulation name, “FE” or “FF” refers to the type ofouter thermal boundary condition employed, the first number gives the diffusion value ν = κ inunits of 1012 cm2 s−1 , and the second number after the dash gives the frame rotation rate Ω0 inunits of Ω ≡ 2.87 × 10−6 rad s−1 , or Ω /2π = 456 nHz. Nr , Nθ , and Nφ refer to the number ofradial, colatitudinal, and longitudinal grid points, respectively. The run time is given in units ofthe thermal diffusion time, defined to be H 2 /κ, with H ≡ ro − ri the shell depth.

Name
FE2-3
FE2.3-3
FE3-3
FE4-3
FE10-3
FE2-2
FE2.3-2
FE3-2
FE4-2
FF2-3
FF2.3-3
FF3-3
FF4-3
FF10-3
FF2-2
FF2.3-2
FF3-2
FF4-2

(Nr , Nθ , Nφ )(128, 384, 768)
(128, 384, 768)
(96, 384, 768)
(128, 192, 384)
(64, 96, 192)
(128, 576, 1152)(128, 576, 1152)(96, 384, 768)
(64, 288, 576)
(128, 576, 1152)(128, 384, 768)
(96, 384, 768)
(128, 192, 384)
(64, 96, 192)
(128, 576, 1152)(128, 576, 1152)(96, 384, 768)
(128, 384, 768)

RaF
8.52 × 105
5.60 × 105
2.52 × 105
1.07 × 105
6.82 × 103
8.52 × 105
5.60 × 105
2.52 × 105
1.07 × 105
8.52 × 105
5.60 × 105
2.52 × 105
1.07 × 105
6.82 × 103
8.52 × 105
5.60 × 105
2.52 × 105
1.07 × 105

Re = Pe
63.2
51.8
35.2
21.9
2.54
75.6
62.2
42.8
28.0
64.5
52.9
35.8
22.4
2.89
81.2
65.3
44.4
28.9

Ek
4.62 × 10−4
5.31 × 10−4
6.93 × 10−4
9.23 × 10−4
2.31 × 10−3
6.93 × 10−4
7.96 × 10−4
1.04 × 10−3
1.39 × 10−3
4.62 × 10−4
5.31 × 10−4
6.93 × 10−4
9.23 × 10−4
2.31 × 10−3
6.93 × 10−4
7.96 × 10−4
1.04 × 10−3
1.39 × 10−3Ro
3.38 × 10−2
3.21 × 10−22.90 × 10−2
2.55 × 10−2
7.45 × 10−3
5.91 × 10−2
5.61 × 10−2
5.09 × 10−2
4.51 × 10−2
3.45 × 10−2
3.28 × 10−2
2.95 × 10−2
2.46 × 10−2
8.58 × 10−3
6.33 × 10−2
5.89 × 10−2
5.30 × 10−2
4.66 × 10−2

R ≡ Ra Ek4/3
30.4
24.1
15.5
9.62
2.08
52.2
41.3
26.6
16.6
30.4
24.1
15.5
9.62
2.08
52.2
41.3
26.6
16.6

∆Ω/Ω0
0.097
0.092
0.078
0.062
0.0048
0.082
0.088
0.092
0.086
0.140
0.131
0.108
0.080
0.0024
0.110
0.121
0.128
0.117

Run time
23.6
7.77
59.6
33.4
238.5
7.78
9.14
11.9
47.7
14.7
7.90
14.4
25.5
15.9
6.68
8.27
10.3
70.4

Table 3.2: Diagnostic Parameters for Each Thermal Boundary Condition. Fluid diagnostic parameters for the different simulations. The 18 simulations here are in four groupings according to FE or FF at two different rotation rates.

123Chapter 4

Exploring Bistability in the Cycles of the Solar Dynamo through Global
Simulations

The calling card of solar magnetism is the sunspot cycle, during which sunspots regularly
reverse their polarity sense every 11 years. However, a number of more complicated time-dependent
behaviors have also been identified. In particular, there are temporal modulations associated withactive longitudes and hemispheric asymmetry, when sunspots appear at certain solar longitudes
or else in one hemisphere preferentially. So far, a direct link between between this asymmetrictemporal behavior and the underlying solar dynamo has remained elusive. In this work, we present
results from global, 3-D magnetohydrodynamic (MHD) simulations, which display both behavior
reminiscent of the sunspot cycle (regular polarity reversals and equatorward migration of internal
magnetic field) and asymmetric, irregular behavior that in the simulations we interpret as active
longitudes and hemispheric asymmetry. The simulations are thus bistable, in that the turbulent
convection can stably support two distinct flavors of magnetism at different times, in superposition,or with smooth transitions from one state to the other. We discuss this new family of dynamo
models in the context of the extensive space-telescope observations of the Sun’s surface magnetic
field with the Solar and Heliospheric Observatory (SOHO) and the Solar Dynamics Observatory(SDO), as well as earlier observations of sunspot number and synoptic maps. We suggest that thesolar dynamo itself may be bistable in nature, exhibiting two types of temporal behavior in themagnetic field.
This Chapter is primarily a restatement of the work published in Matilsky & Toomre (2020a)

125(also see the related conference proceedings paper, Matilsky & Toomre 2020b). As the primaryauthor of the paper, I conducted the simulations, performed their analyses, created the Figures,and wrote the text. My coauthor Juri Toomre recognized the novelty of the simulations I had
stumbled upon and encouraged me to pursue this remarkable phenomenon of bistability further.
He additionally provided advice and guidance throughout the writing process, gave detailed text
edits, and suggested schematic designs of some of the figures.

4.1

Introduction
Since the early observations of sunspot number, the Sun’s magnetic field has been known tofollow the fairly regular 11-year sunspot cycle. Sunspots appear at mid-latitudes at the beginningof the cycle, then at latitudes a bit higher during the peak of solar activity, and finally at sites
progressively closer to the equator as the magnetic activity wanes (e.g., Hathaway 2011). Polarimetric observations from the NSO/Kitt Peak Observatory and magnetograms from the Michelson
Doppler Imager (MDI) aboard SOHO and the Helioseismic and Magnetic Imager (HMI) aboard
SDO have enabled the detailed and continuous study of sunspots for the last ∼50 years (recall
Figure 1.2). Sunspots appear in pairs of opposite polarity sense (i.e., bipolar active regions) fromone cycle to the next, making a 22-year cycle overall (e.g., Hathaway 2015). In addition, a numberof other cycles (less obvious and regular than the 22-year cycle) have been identified, from a slow
∼100-year modulation of the peak cycle amplitudes called the Gleissberg Cycle (e.g., Gleissberg
1939; Ogurtsov et al. 2002), to a rapid ∼2-year periodicity in the global magnetic field (e.g., Ulrich& Tran 2013). On timescales in between, active longitudes (longitudes at which sunspots appear
more frequently and with greater strength) persist for several decades (e.g., Henney & Harvey2002). There are also periods (the longest being ∼50 yr) associated with hemispheric asymmetry,or times at which more sunspots conglomerate in either the Northern or Southern Hemispheres(e.g., Ballester et al. 2005). Finally, there are several recorded grand minima, or times throughout
history when solar activity is substantially diminished over protracted intervals of several decades(e.g., Hoyt & Schatten 1998).

126
All of these complicated features of the solar cycle must have their roots in the global solar
dynamo—the process by which the Sun’s interior magnetic field regenerates through dynamicalinteractions between rotation and convection. Sunspots are believed to be the result of rising
toroidal loops of magnetic field. That their sites of emergence migrate equatorward with thesunspot cycle and that their polarity senses flip every 11 years suggest an interior toroidal reservoirof magnetism that also migrates equatorward and flips polarity every 11 years. The additional
cycles and behavior mentioned above could also be indicative of corresponding cycles in the interior
magnetic field.
In the past decade or so, global, 3-D, MHD simulations of the solar convection zone have madesignificant headway in reproducing aspects of the solar cycle. Using the Anelastic Spherical
Harmonic (ASH) code, Brown et al. (2010) and Brown et al. (2011) have shown that it is possible
to build strong interior “wreaths” of magnetism amidst a chaotic, turbulent flow. These wreaths
can be space-filling, with nearly 360◦ -connectivity throughout the spherical shell, and can exhibit
regular polarity reversals, during which the longitudinal direction of magnetic field in the wreath
flips every few years of simulation time. Also using ASH, Augustson et al. (2015) have achieved
wreath-building dynamos which exhibit both equatorward propagation of wreaths at low latitudes
and poleward propagation of wreaths at high latitudes—a signature feature of the SOHO/SDO
observations (e.g., Hathaway 2015).
Perhaps one of the most significant issues facing global MHD simulations of solar convectionis that the wide range of spatial and temporal scales relevant for the fluid makes a direct numericalsimulation impossible. Researchers have attempted to address this problem by using various prescriptions for sub-grid-scale (SGS) turbulent effects or by restricting the global domain. Warnecke
et al. (2014) used the Pencil code to solve the MHD equations in a spherical-wedge geometry,showing that global convection could yield αΩ-type dynamos with the direction of propagationof the interior magnetic field being set by the Parker-Yoshimura rule. Passos & Charbonneau(2014) used the EULAG-MHD code (which incorporates implicit dissipation on the smallest spatial
scales to maintain numerical stability) to achieve regular, cyclic polarity reversals in their global

127“Millennium Simulation.” The cycles persisted over very long time-scales, with statistical features
showing long-term trends reminiscent of the observed solar Gleissberg modulation. Hotta et al.
(2016) have explored a large dynamical range for solar convection using their Reduced Speed ofSound Technique (RSST) code. They explored the coupling of a near-surface layer of small-scale(< 10 Mm) convection and deep, large-scale flows, finding that coherent magnetic structures could
persist even in the presence of very small diffusivities.
Another important outstanding issue for solar MHD simulations is that most lack the eruption of interior magnetic flux and subsequent decay of active regions believed to play an essential
role in the global dynamo. In the work of Nelson et al. (2013a,b), an extension of ASH incorporatinga Dynamic Smagorinsky treatment of the sub-grid-scale (SGS) fluid motions achieved a dynamo inwhich small loops of magnetism detached from the interior wreaths and rose to the outer surface
via magnetic buoyancy instabilities. The polarity, twist, and tilt of the loops displayed statistical properties reminiscent of Joy’s law for sunspot emergence. Using a finite-difference sphericalanelastic MHD code, Fan & Fang (2014) found that the convection gave rise to super-equipartition
magnetic flux bundles that had similar characteristics to emerging active regions on the Sun. Morerecently, dynamos with wreaths that give rise to buoyant loops have been achieved with Rayleigh(the code used in this work) in modeling the intense magnetism exhibited by M-dwarf stars (Bice& Toomre, 2020). These simulations have thus shown it is possible to connect MHD simulationsof the interior solar dynamo to the emergence of magnetic flux that is actually observed at thephotosphere.
It has generally been difficult for simulations to reproduce equatorward propagation of magnetic field as seen in the solar butterfly diagram (although see Käpylä et al. 2012 and Augustson
et al. 2015 for notable exceptions). Furthermore, two other prominent features of solar magnetic
activity—active longitudes and hemispheric asymmetry—have not been systematically explored in
global simulations. In this work, we report on a new class of dynamo simulations whose magnetism exhibits polarity-reversing cycles with equatorward propagation, as well as a quasi-regular,
hemispherically asymmetric cycle, the features of which are suggestive of active longitudes and128hemispheric asymmetry. Both cycles are present simultaneously, although usually one cycle ismore dominant than the other.
In Section 4.2 we discuss our numerical methods and the parameter space covered by our
simulations, as well as the hydrodynamic (HD) progenitor common to each magnetic simulation.
In Section 4.3 we discuss the bistable nature of the magnetism achieved in our dynamo cases.
We discuss striking flux-transport-like behavior in Section 4.4. We examine each cycling mode
individually in Sections 4.5 and 4.6. In Sections 4.3–4.6 we refer only to our primary lowestmagnetic-Prandtl-number case, and we return to the higher-magnetic-Prandtl-number cases in
Section 4.7. In Section 4.8 we explore the dynamical origins of the two cycles in terms of the
production of magnetic field. In Section 4.9 we discuss our bistable dynamo simulations in thecontext of solar observations and present our conclusions in Section 4.10.

4.2

Numerical Experiment
We consider time-dependent, 3-D simulations of a rotating, convecting shell of fluid modeled

after the solar convection zone (CZ), again using Rayleigh 0.9.1 (Featherstone & Hindman, 2016a;
Matsui et al., 2016; Featherstone, 2018). We use notation identical to that of the previous two “HD”
Chapters. However, we now consider a magnetized convection zone. Rayleigh thus solves the LBR
Equations (1.1) in full MHD.1 Recall that all simulations in this thesis have impenetrable, stressfree boundaries, a fixed-flux (∂ Ŝ/∂r = 0) bottom, and here, they have a fixed-flux top as well. For
the magnetic field, we match to a potential field at both boundaries—see Equation (1.7)(d). Other
magnetic lower boundary conditions, such as a perfect conductor or a purely radial field, could beconsidered, but there is no obvious choice from a physical point of view.
1

We briefly note that the simulations presented in this Chapter used Rayleigh 0.9.1, which had a bug in its
implementation of the magnetic induction equation. As noted in Section 1.3.3, Rayleigh evolves the induction
equation (1.1c) by representing the magnetic field by flux functions: B = ∇ × [Aêr + ∇ × (C êr )]. In spectral space(and neglecting the nonlinear emf terms), the equation for the A flux function is ∂A/∂t = η(r){[`(` + 1)/r2 ][∂ 2 /∂r2 +(d ln η/dr)∂/∂r − [`(` + 1)/r2 ]}A. In Rayleigh, up through version 1.0.0 (Featherstone et al., 2021), the term in the
A equation proportional to (d ln η/dr)∂A/∂r was not computed correctly in the Crank-Nicolson solve. This effectively
resulted in that term being too small by a factor of ∼2 for all timesteps. Since η(r) did not vary rapidly with radius,
the error likely did not affect the core bistability results presented in this Chapter and in the publications, Matilsky& Toomre (2020a,b). However, for due diligence, we re-ran case D3-1 (from t = 9610 Prot to t = 10350 Prot —an
interval of about 700 Prot ) using Rayleigh 1.0.1 and the bistability behavior was largely identical.

129
Table 4.1: Common Input Parameters for the Bistability Experiment. Input parametervalues common to all four cases (H3, D3-1, D3-2, and D3-4). See Section 4.11 for the definition of
the flux Rayleigh number RaF .
Parameter
Spherical-shell inner boundary
Spherical-shell outer boundary
Specific heat at constant pressure
Ratio of specific heats
Density at the inner boundary
Luminosity of the Sun
Mass of the Sun
Frame rotation rate
Frame rotation period
Viscosity at the outer boundary
Flux Rayleigh number
Thermal Prandtl number

Symbol
ri
ro
cp
γ
ρi
L
M
Ω0
Prot = 2π/Ω0ν(ro )RaF
Pr = ν/κValue
5.00 × 1010 cm
6.59 × 1010 cm
3.50 × 108 erg K−1 g−1
5/3
0.181 g cm−3
3.85 × 1033 erg s−1
1.99 × 1033 g
8.61 × 10−6 rad s−1
8.45 days
3.00 × 1012 cm2 s−1
2.56 × 106
1

Each case rotates at three times the solar Carrington rate, 3Ω , as discussed further below.
The background thermodynamic state (adiabatically stratified polytrope with density contrast of∼20 across the shell and a solar luminosity injected via internal heating) is identical to those of
our previously discussed three-density-scale-height models (i.e., case N3 from Chapter 2 and all thecases from Chapter 3). One difference between the bistable dynamo simulations discussed here andthe simulations from the HD Chapters is that the diffusivities ν(r), κ(r), and η(r) vary with radiuslike ρ(r)−1/2 . In the HD simulations, all diffusivities were constant in space.
We discuss three dynamo simulations in the present study, which all start from the samehydrodynamic progenitor. We define Ω ≡ 2.87 × 10−6 rad s−1 as the sidereal Carrington rotation
rate, and all our models rotate at three times this rate to ensure a low Rossby number and thus asolar-like differential rotation, in which the equator rotates faster than the poles (e.g., Brown et al.
2010). The transition from solar to anti-solar differential rotation as simulations grow increasingly
more turbulent has come to be called the “convective conundrum” (O’Mara et al., 2016). The
anti-solar regime can be avoided by either rotating faster or lowering the luminosity, and we opt
for the former. We refer to the hydrodynamic progenitor as case H3 (“H” for hydrodynamic, “3”for Ω0 = 3Ω ). Some input parameters common to case H3 and the three dynamo cases are shown130in Table 4.1.
The three dynamo simulations were initialized from the well-equilibrated hydrodynamic caseH3 after about 6400 Prot , using the full MHD equations. These dynamo cases differ in the valuesof the magnetic Prandtl number Prm = ν/η, which takes on the values 1, 2, and 4. For all cases,
the kinematic viscosity profile ν(r) is fixed, effectively keeping the Rayleigh number constant. We
set the value of Prm by varying the value of the magnetic diffusivity at the top of the domain. Welabel the corresponding dynamo simulations as cases D3-1, D3-2, and D3-4, where the last number
corresponds to the value of Prm being 1, 2, and 4, respectively. To resolve both the flow field andmagnetic structures, we use a substantial resolution of at least (Nr , Nθ , Nφ ) = (96, 384, 768) for thenumber of radial, latitudinal, and longitudinal grid points, respectively (see Table 4.2).
Our dynamo cases expand the parameter space explored by Brown et al. (2010)—who examined the case Prm = 0.5 while varying the rotation rate—to higher magnetic Prandtl numbers. In
all cases discussed here, the magnetic field was initialized from a random seed field of amplitude∼1 G (“random” in terms of the amplitude of each individual spectral component).
Solutions to the nonlinear MHD equations in rotating, convecting spherical shells involve
highly time-dependent and intricate flow structures. Figure 4.1 shows a sample of the global flow
structure achieved in the hydrodynamic progenitor H3. As seen in the orthographic and Mollweide
projections of the radial velocity vr , the flows at any given time consist of Taylor columns between
about ±20◦ -latitude (also called Busse columns and banana cells in the literature) and a broken
honeycomb network of upflows and downflows at higher latitudes. The Taylor columns transport
angular momentum outward (Brun & Toomre, 2002; Busse, 2002; Matilsky et al., 2019) and drivea strong differential rotation (Ω − Ω0 ), with an associated meridional circulation (hρvm it ), as seenin Figures 4.1(c,d).22

In this Chapter, we define the “meridional part” of a vector field A through Am ≡ Ar êr + Aθ êθ . This isequivalent to the “poloidal component” of vector fields referred to elsewhere in this thesis (e.g., Bm in this Chapter
is equivalent to Bpol in the next Chapter 5.) The terminology used here coheres with the notation of Matilsky &Toomre (2020a). Another taxonomic difference in this Chapter is that the angular brackets with the “t” subscript
denote a combined temporal and longitudinal average, while angular brackets with no subscript denote a longitudinal
average at a particular time.

Figure 4.1: Turbulent Flows in a Hydrodynamic Dynamo Progenitor. Flow structures in the hydrodynamic case H3. (a) Snapshotof the convection (as traced by vr ) near the outer surface, shown in orthographic projection. (b) The same snapshot of vr , shown in
Mollweide projection. (c) Averaged fluid rotation rate (Ω(r, θ) − Ω0 ≡ hvφ it /r sin θ) in the meridional plane. (d) Averaged meridional
circulation, with the mass flux (hρvm it ) magnitude plotted in color, overlaid on circulation streamlines. In panel (d), red and blue tones
correspond to clockwise and counterclockwise circulation, respectively.

131

132

4.3

Bistable Magnetic Cycles in the Dynamo Cases
The magnetism in our dynamo cases exhibits two striking (and distinct) cycling modes.

Usually one is dominant, but they can also coexist simultaneously. The phenomenon of bistability
in dynamical systems generally refers to the coexistence of two stable equilibrium states, and we
use the term somewhat loosely to refer to the two different types of dynamo cycle. In this and thefollowing sections, we discuss case D3-1, which exhibits bistable cycling in the cleanest manner.
We return to the higher-Prm cases in Section 4.7.
Each cycle is associated with a unique morphology of the magnetic field, as seen in Figure
4.2. In roughly the lower half of the shell at an early time as in Figure 4.2(a), there is one pairof opposite-polarity wreaths in each hemisphere (a fourfold-wreath structure) confined to within∼40◦ -latitude of the equator. At this instant, the polarity sense of the wreath structure is largely
symmetric across the equator. Each wreath mostly has 360◦ -connectivity, linking the magnetic field
in a large torus. The wreaths individually have rms field strengths of ∼5 kG and longitudinallyaveraged rms field strengths ∼2.5 kG.
Later in the simulation, near the middle of the CZ as in Figure 4.2(b), there is a strong,negative-Bφ partial wreath (wrapping ∼180◦ around the sphere), as well as a slightly less prominent
positive-Bφ partial wreath on the opposite side of the sphere. The negative-Bφ partial wreath isdominant at this time, with a peak strength of ∼40 kG, compared to the positive-Bφ peak fieldstrength of ∼20 kG. In the longitudinal average, there is a residual hBφ i that is negative and hasa peak magnitude of ∼10 kG. The partial-wreath structure occupies most of the domain in radius,
but is confined between the equator and 20◦ -south in latitude.
The temporal behavior of the magnetic field as a whole roughly consists of transitions between
the two magnetic-field structures presented in Figure 4.2. Figure 4.3 shows the evolution in timelatitude space near mid-CZ for hBφ i (throughout this Chapter, angular brackets with no subscript
denote a longitudinal average at a particular time, while angular brackets with a “t” subscript
denote a combined longitudinal and temporal average), as well as the evolution of the toroidal

133

Figure 4.2: Two Magnetic-Field Structures in a Bistable Dynamo. Snapshots of the azimuthal magnetic field Bφ (in kG) for case D3-1 at two different depth/time pairs, shown in global
Mollweide view on the left and accompanied by the longitudinal average in the meridional planeon the right. (a) The field deep in the shell at an early time in the simulation. (b) The field in
the middle of the shell at a later time. In panel (b), the color maps are symmetric-logarithmic tocapture both weak-amplitude and strong-amplitude magnetic structures. The colors in the intervalaround zero map linearly to field strength and the two intervals on either side (one positive andone negative) map logarithmically. An animated versions of each panel is available in the online
version of Matilsky & Toomre (2020a).

134

Figure 4.3: Extended Evolution of Bistable Dynamo. Extended time-latitude diagram of theazimuthal magnetic field hBφ i for case D3-1 at r/R = 0.835 (mid-CZ). Panels (a) (split acrossmultiple rows) show the longitudinally averaged field hBφ i as a function of time and latitude.
The saturation values of the color map are indicated in the top row, and the map is symmetriclogarithmic, as in Figure 4.2(b). Sharing the same time axis, panels (b) show the energy density inthe azimuthal magnetic field, MEφ = Bφ2 /8π, averaged about r/R = 0.835 over 10% of the shellby radius. This caption is continued in the main text within quotation marks.

135
magnetic energy, Bφ2 /8π. To maximize the size of this important Figure 4.3, we continue its large
caption of here: “In panels (b), the y-axis is measured in the cgs units of energy density, erg cm−3 .
The arrows mark wreaths in the fourfold-wreath cycle originating at mid-latitudes that appear toseed partial wreaths in the South. Time is measured from the instant when the seed magnetic field
was introduced to the hydrodynamic progenitor (namely, 6400 Prot ). The dashed lines mark theintervals shown in closeup view in Figures 4.7 and 4.10.”
After the dynamo has established strong fields from the initial seed field (around t = 300 Prot ),the fourfold-wreath cycle dominates. Each wreath emerges at ∼40◦ –50◦ -latitude and then migrates
steadily equatorward. The newly formed wreaths alternate between positive-hBφ i and negativehBφ i being dominant, with the time between the appearance of two wreaths of the same polarity—
the fourfold-wreath cycle period—equal to ∼25 Prot , or about six months. This is the same time ittakes an individual wreath to migrate from its mid-latitude starting point to the equator. Effectively
each hemisphere operates on its own (i.e., not necessarily in phase), producing wreaths of a given
polarity at mid-latitudes that reappear once they reach the equator.
At around t = 2100 Prot , the fourfold-wreath cycles are disturbed by the partial-wreath
cycle, which starts in the South (first arrow in Figure 4.3(a)). The dominant polarity of thepartial-wreath structure reverses quasi-periodically. We can estimate the period (time between
two successive states of one dominant polarity) from a visual inspection of Figure 4.3(a): betweent = 2950 Prot and t = 3350 Prot (an interval slightly larger than the one spanned by the second setof dashed lines), there are roughly five cycles, yielding a period of ∼80 Prot . We note that visualinspections of other intervals would yield slightly different cycle periods, making the period of ∼80
Prot only approximate. Nevertheless, we thus identify a partial-wreath cycle with a period about
three times as long as that of the fourfold-wreath cycle.
The partial-wreath pair wanders into the North around t = 2250 Prot and flips from North
to South once more before the fourfold-wreath cycle becomes dominant for the second time aroundt = 4150 Prot . The rest of the simulation displays a complex seesaw behavior between the two
cycling states. The fourfold-wreath cycle never really disappears, but does significantly decrease in

136
amplitude when the partial-wreath cycle is dominant. Examining Figure 4.3(a) in detail, it seems
that whenever the partial-wreath cycle starts, its field comes originally from the fourfold-wreath
cycle. That is, a wreath of one polarity will migrate toward the equator and then significantly growin amplitude, seeding the following partial-wreath cycle with that same polarity dominant. Threeinstances of this phenomenon are marked by the arrows in Figure 4.3(a) .
Along with the time-latitude panels in Figure 4.3(a), we show the temporal behavior of theenergy in the azimuthal magnetic field in Figure 4.3(b), averaged over 10% of the shell (by radius)
at mid-depth. The fourfold-wreath cycle is seen to correspond to a low-energy state of the magnetic
field, with sporadic peaks in energy, but no discernible energy cycle associated with the polarity
reversals. When the partial-wreath cycle first begins (around t = 2100 Prot ), the field jumps into ahigh-energy state, with local peaks spaced by roughly half the partial-wreath cycle period.

4.4

Polar Caps of Magnetism
Also visible in Figure 4.3(a) is a weak azimuthal magnetic field at high latitudes on the orderof tens of Gauss (see, for example, the positive hBφ i in the North and the negative hBφ i in theSouth during the interval from 3500 to 5000 Prot ). These polar caps of magnetism fluctuate inamplitude and occasionally reverse polarity, but not with any regularity and with no discernible
relationship to the fourfold-wreath or partial-wreath cycles.
There is a significantly stronger polar cap associated with the meridional magnetic field Bm ,with magnitude ∼1 kG. Figure 4.4 shows the radial magnetic field when there are intense polarcaps of magnetism. Near the top of the domain (left-hand column), the local structure of Br
inherits the honeycomb network of downflows associated with the Taylor columns, while deeper
down (right-hand column), the field is more homogeneous. At both depths shown, however, thecomplicated polar-cap structure on average resembles a dipolar field, with magnetic-north lining
up with geometric-north. The instantaneous views of Bθ (not shown) similarly display a dipolarstructure, with Bθ positive near the poles in both hemispheres.
Figure 4.5 provides the time-latitude diagram for hBr i near the top of the domain during an

137

Figure 4.4: Polar Caps of Magnetism. Spherical cuts of the instantaneous radial magnetic field
Br , viewed from the North Pole (top row) and the South Pole (bottom row). The instant shown
is t ≈ 4200 Prot , just after the first transition from partial-wreath to fourfold-wreath cycling, when
there is intense meridional field at both the North and South Poles. Two depths are sampled: onenear the top of the domain and one near the bottom.

138

Figure 4.5: Flux-Transport of Magnetic Field. Time-latitude diagram of radial magnetic fieldhBr i for case D3-1 at r/R = 0.917 (upper CZ), and associated energy density at the same depth.
The time interval sampled corresponds to the second and third panels of the extended time trace
in Figure 4.3.

139interval in the middle of the simulation. There is clearly a strong dipolar component to the field at
the poles. After a fairly quiescent initial interval, the dipolar field strengthens (at t ≈ 3750 Prot )a few hundred rotations before the fourfold-wreath cycle becomes dominant for the second time.
The partial wreaths produce flux that is transported to the poles, seen in Figure 4.5 as intermittent
large plumes of red and blue.
Not all the partial-wreath cycles produce field that propagates all the way to the poles, andthere does not seem to be a one-to-one correspondence between polarity reversals of the partial
wreaths and reversals in the polar caps. For nearly every partial-wreath cycle, however, the radial
field propagates much further poleward than the ±20◦ latitude limits that are maintained fairly
consistently for hBφ i.
The poleward migration of hBr i seen in Figure 4.5 bears a striking resemblance to the concept
the meridional circulation acting as a “conveyor belt” of magnetism in the flux-transport (F-T) dynamo framework (e.g., Charbonneau 2014). In that picture, the meridional circulation advects themeridional component of magnetic field to the poles near the outer surface, where it accumulates
and is eventually pumped back down to the equator in the deep CZ, also by the meridional circulation. This poloidal field is then sheared into toroidal field by the differential rotation, starting thecycle anew. In the F-T framework, the meridional circulation time thus sets the cycle period.
However, case D3-1 is more complicated than the simple F-T model for two reasons. Firstly,
the meridional circulation time for case D3-1 is on the order of 400 Prot , which is substantiallylonger than either the partial-wreath or fourfold-wreath cycle periods. Secondly, although thepoles occasionally flip polarity, there is no clean correspondence between the polarity of the fieldat the poles and the polarity of the field in the equatorial regions during subsequent cycles.

4.5

Evolution of the Fourfold-Wreath Cycle
Figure 4.6 shows the evolution of the azimuthal magnetic field Bφ during one reversal in the

fourfold-wreath cycle. At any given time, each wreath has a complicated structure due to stretchingand pummeling by the Taylor columns, but generally has the same sense for Bφ all the way around140
the sphere, indicating field lines that connect in a large torus. The accompanying snapshots of hBφ iin the meridional plane indicate that the wreaths are strongest deep in the shell and are confinedin radius to the lower half of the CZ and in latitude to within ∼40◦ of the equator.
Each wreath marches steadily equatorward as the cycle progresses. As a wreath of one
polarity reaches the equator, a new wreath of that same polarity appears at a higher latitude. In
Figure 4.6(a), for example, there are two opposite-polarity wreaths in the Northern Hemisphere
below ∼30◦ -latitude (roughly equal in amplitude) and a negative-hBφ i wreath that is just beginningto form above ∼30◦ -latitude. By the time a given wreath migrates from mid-latitudes to within∼10◦ of the equator, a new wreath of that same polarity has formed at mid-latitudes, completingthe cycle. Therefore, the configuration in Figure 4.6(e) has essentially returned to that of Figure
4.6(a).
Figure 4.7 shows a close-up view (in time-latitude space) of a small time interval containing
about eight fourfold-wreath cycles (marked by the first set of dashed lines in Figure 4.3(a)), including the instants sampled in Figure 4.6. Here the field is first seen to appear (at its weakest) at
roughly 40◦ -latitude north and south and migrates steadily equatorward, attaining its maximum
strength between 20◦ –30◦ north and south, while stopping short of the equator at about 10◦ northand south.
Interestingly, the Southern Hemisphere is often out of phase with the Northern Hemisphereand it is hard to pinpoint whether the fourfold-wreath structure is symmetric about the equatoror antisymmetric. Indeed, the simulation exhibits both symmetric and antisymmetric tendencies
at different times (for example, compare the symmetric configuration in Figure 4.2(a) to the antisymmetric configurations shown in Figure 4.6). At t = 1200 Prot (horizontal center of Figure 4.7),
the wreath structure is clearly antisymmetric in the large.
The symmetry of dynamo states in mean-field dynamo theory is usually quantified by theprevalence of the dipole (` = 1) and quadrupole (` = 2). Our dynamo cases, by contrast, aredominated by higher-order modes that serve to localize each wreath in space, leaving the dipolarand quadrupolar contributions extremely weak. Figure 4.8 shows the magnetic-field power spectra,

141

Figure 4.6: Walking though aRegular Polarity Reversal. Mollweide snapshots (at r/R = 0.748) ofBφ and meridional snapshots of hBφ i
for one fourfold-wreath cycle. Snapshots are separated by about onequarter of a cycle (∼6 Prot ). The
color map is identical to that of Figure
4.2(a), with the Mollweide plots saturated at ±4.5 kG and the longitudeaverage plots saturated at ±2.5 kG.

142

Figure 4.7: Regular Fourfold-Wreath Cycles. Close-up view of the time-latitude behavior
of hBφ i near the bottom of the CZ (r/R = 0.748) during the interval marked by the verticaldashed lines in Figure 4.3(a). This interval (between 1100 and 1300 Prot ), encompasses about eight
fourfold-wreath cycles. The dashed lines indicate the times sampled by the series of Mollweide
projections in Figure 4.6.

143

Figure 4.8: Spatial Scales of the Toroidal and Poloidal Magnetic Fields. Power spectra of
the magnetic field at r/R = 0.748 during fourfold-wreath cycling, averaged between t = 0 andt = 2200 Prot . Separate curves indicate contributions from all m, m = 0, and |m| > 0. The sum of
the total power over all ` equals 1. (a) Spectrum of azimuthal magnetic field Bφ . (b) Spectrum of
meridional magnetic field Bm ≡ Br êr + Bθ êθ .

averaged over the initial interval during which the fourfold-wreath cycle dominates. The power inthe longitudinally averaged magnetic fields (the m = 0 power) peaks around ` = 10. The powerin the fluctuating magnetic fields (|m| > 0) peaks at the smaller scales, around ` = 20 for Bφ0 and
0 .
` = 25 for Bm

The peak around ` = 10 is consistent with the fourfold-wreath structure as seen in Figures
4.2(a) and 4.6: each wreath in a given hemisphere has a latitudinal extent of about 15◦ , or roughly
one-tenth of the 180◦ in latitude covered by the full sphere. We note that for the longitudinallyaveraged fields, the dipolar and quadrupolar modes (` = 1 and 2, respectively) are weaker by afactor of ∼10 compared to any of the modes near ` = 10.
In Figure 4.9(a), we show the temporal behavior of the power in the longitudinally averaged(m = 0) field hBφ i around ` = 10 deep in the shell during the fourfold-wreath cycle. Mostof the time the fourfold-wreath structure is symmetric about the equator, with the even modes` = 8, 10, 12 containing up to 60% of the total power for hBφ i. The two hemispheres are outof phase, however, and the symmetric (odd) and antisymmetric (even) modes can sometimes beroughly equal (e.g., near t = 800 Prot ). Near t = 1200 Prot , the antisymmetric modes dominate, as
we noted in discussing Figure 4.7.

144

Figure 4.9: Changing Symmetry of Magnetic Field about the Equator. (a) Temporal
evolution of the power for hBφ i at r/R = 0.748 in the modes ` = 8, 10, 12 (black curve) and ` =7, 9, 11 (red curve). These are the even/odd modes near the peak of the spectrum at ` ≈ 10, whichcontain most of the power. Power is shown during the initial interval for which the fourfold-wreath
cycle is dominant, coincident with the top panel in Figure 4.3. (b) Lomb-Scargle periodogram ofthe ` = 7–12 contributions to hBφ i during the interval shown in panel (a). Normalized LombScargle power in each frequency component is shown versus period, with a clear peak occurring at
Penergy ≡ 12.0 Prot .

145
The m = 0 power around ` = 10 is is a proxy for the total energy contained in the fourfoldwreath structure. The oscillations in Figure 4.9(a) thus show that the fourfold-wreath energy
oscillates periodically. In Figure 4.9(b), we decompose the time series for all the power contained
around ` = 10, both symmetric (even) and antisymmetric (odd) modes, into its frequency (orperiod) components. There is clearly one period that is dominant, which corresponds to an energy
cycle of Penergy = 12.0 Prot . Since the polarity alternates between energy cycles, the fourfold-wreath
cycle period isPfourfold = 2 Penergy = 24.0 Prot ,(4.1)

similar to the estimate of ∼25 Prot mentioned previously.

4.6

Evolution of the Partial-Wreath Cycle
The partial-wreath component (when present) is strongest at mid-depth. From Figure 4.3(a),the partial-wreath structure first appears around t = 2100 Prot and cycles between positive-hBφ idominant and negative-hBφ i-dominant partial-wreath pairs. Unlike the fourfold-wreath structures(which cycle regularly even when the partial-wreath structure dominates), the partial-wreath structures cycle intermittently, turning off altogether at times, but always returning. There is some variation in the time-latitude behavior of hBφ i from cycle to cycle, but in general, hBφ i stays confined
to within ∼20◦ of the equator in a given hemisphere and shows a tendency to propagate towardhigher latitudes with time.
Even when the partial-wreath cycle is dominant, the equatorward-propagating fourfoldwreath cycle still persists with its regular period of 24.0 Prot unchanged. However, the fourfoldwreath structure is of much weaker amplitude and is less coherent in the hemisphere where thepartial wreaths reside. There is thus a wide dynamical range in the amplitudes of the magnetic
field overall. The partial-wreath cycle is associated with longitudinally averaged field strengths of∼10 kG, and when the partial-wreath cycle is dominant, the fourfold-wreath cycle’s longitudinallyaveraged field strengths are reduced to several hundred G from ∼2 kG.

146

Figure 4.10: Irregular Partial-Wreath Cycles. Close-up view of the time-latitude behavior ofhBφ i during the interval marked by the first set of vertical dashed lines in Figure 4.3(a). This
interval (between 3000 and 3300 Prot ), encompasses about four partial-wreath cycles. The verticaldashed lines denote the instances sampled by the volume-renderings of Bφ in Figure 4.11.

147
To determine in detail how the partial-wreath structure modulates over the course of a cycle,
we show in Figure 4.11 a series of volume-renderings of the azimuthal field. In each snapshot, thepositive-Bφ and negative-Bφ partial wreaths are centered roughly around φ = 210◦ and φ = 30◦ inlongitude, respectively, although “islands” of magnetism associated with each wreath extend aroundthe whole sphere. In the first snapshot (panel a), the positive-Bφ partial-wreath is dominant. Its
amplitude and longitudinal extent then steadily decline over the next half-cycle while the negativeBφ partial-wreath becomes dominant (panels b,c). In the second half of the cycle (panels d,e),
the positive-Bφ wreath expands and strengthens until it is dominant again and the configurationis roughly the same it was at the beginning of the cycle. In the longitudinal average (e.g., Figure
4.3(a)), much cancellation occurs, but the sign of hBφ i still reflects which partial wreath is dominant.
In order to track the partial-wreath structure over multiple cycles, we trace Bφ with respect
to time and longitude in a frame rotating at the same rate as in Figure 4.11. Figure 4.12 shows thetime-longitude trace for Bφ at 10◦ -south latitude and at two depths, during an interval when thepartial-wreath structure is stronger than the fourfold-wreath structure and cycling in the South.
The choice of 10◦ -south latitude cuts the partial wreaths roughly in their core, where the fieldstrengths are highest.
Approximately the same propagation pattern in time-longitude space occurs both at middepth and deep in the shell, despite the field being tracked at the same rate for both depths. Thisrate is close to the equatorial fluid rotation rate at mid-depth, suggesting that the whole asymmetricwreath structure is “anchored” in the middle of the shell and moves through the fluid as a single
entity. At mid-depth, the field pattern is uniformly located at a higher longitude than the field
pattern deep down. This displacement does not change with time, suggesting that it is due to theinherent geometry of the wreath rather than the shearing action of the flow.
In the rotating frame chosen, the field structure is approximately stationary. Each partial
wreath (either positive-Bφ or negative-Bφ ) is modulated in amplitude over a period of ∼80 Prot(the same as the cycle period identified in the time-latitude trace of hBφ i), but remains basicallyat a fixed longitude. Furthermore, the positive-Bφ and negative-Bφ wreaths are out of phase with

148

Figure 4.11:
Walking
through a Partial-Wreath
Cycle. Volume-rendering ofBφ throughout one partialwreath cycle. We track thepartial wreaths in a frame
rotating at 32 nHz faster than
the frame rotation rate Ω0 ,approximately equal to theequatorial rotation rate at
mid-depth. Only strong fields(|Bφ | > 5 kG) are depicted. To
show the field everywhere in
3-D space and simultaneously
emphasize the strong fields,
transparency varies linearlywith field strength, with structures for which |Bφ | ≥ 35 kG
being completely opaque. The
first row (panel a) samples
the field when the red partial
wreath is dominant, and subsequent samples (panels b–e)
are separated by roughly onequarter of the partial-wreath
cycle period. The left-handcolumn shows the full Bφ profile and the right-hand column
shows only positive Bφ . Theview is from ∼30◦ north ofthe equator and 90◦ longitude.
The spherical-shell boundaries
in the equatorial plane and thelocation of 0◦ -longitude aremarked by the black curves.

149

Figure 4.12: Modulation of Active Longitudes during a Partial-Wreath Cycle. Timelongitude trace of Bφ for 10◦ -south latitude (a) deep in the shell and (b) at mid-depth, tracked at
the same rotation rate as in Figure 4.11. The color map saturation values are ±12 kG in panel(a) and ±25 kG in panel (b). The interval shown in close-up view in Figure 4.10 is marked by the
thick horizontal dashed lines in panel (b). The instances sampled in Figure 4.11 are marked by thethinner dashed lines in both panels.

150
another. At any given time, there are thus two partial wreaths, but in general when the positive-Bφwreath is strong, the negative-Bφ wreath is weak, and vice versa.
The partial-wreath structure is still not exactly stationary even when tracked in the chosenframe. Figure 4.12 shows a weak zigzag pattern for each the positive-Bφ and negative-Bφ partial
wreaths, implying that the propagation rate of the wreath through the fluid is not constant withtime, or that longitudinal extents of each partial wreath shrink and expand throughout the cycle.
In light of Figures 4.11 and 4.12, the “polarity reversals” in hBφ i during the partial-wreath
cycle are in fact accomplished by each partial wreath weakening and strengthening mostly in place,with the positive-Bφ partial wreath oscillating in time roughly 180◦ out of phase with negative-Bφone. This stands in contrast to the reversals of the fourfold-wreath cycles, which are achieved byfull wreaths migrating equatorward from mid-latitudes and getting replaced by new wreaths of the
opposite polarity.

4.6.1

Partial-Wreath Cycle Period and Hemispheric Asymmetry

The partial-wreath cycle period is roughly 80 Prot , as identified previously, but it is substantially irregular. We determine the period more formally by calculating the periodicities associatedwith the relative energy content in the two partial wreaths:
Apartial ≡Bφ+ − Bφ−
Bφ+ + Bφ−

,

(4.2)

the “+” and “−” superscripts referring to instantaneous rms averages of the field strength over
the regions where Bφ is positive and negative (respectively), thus measuring the energy content in
the positive and negative partial wreaths separately. The mean in the rms is taken over longitude,
low latitudes between ±25◦ , and all depths. Unlike the linear longitudinal average hBφ i, the rmsaverage Apartial is more sensitive to the strongest fields, and thus better determines the relative
amplitudes of the partial wreaths as visualized in Figure 4.11.
Figure 4.13(a) shows the relative energy content over an interval for which no period is easily
identifiable in the time-latitude trace of hBφ i in Figure 4.3(a), but the partial-wreath cycle is still151
visible in the well-defined peaks and troughs of Apartial . In Figure 4.13(b), we show the periodogramassociated with Apartial . There is a distinctive peak in power at
Ppartial ≡ 81.5 Prot ,(4.3)

which we define to be the (dominant) partial-wreath cycle period. Due to the irregularity ofthe partial-wreath cycle—caused by some cycles being longer or shorter than Ppartial , as well as
significant amplitude and phase modulation—there is a substantial spread in the periods for which
the Lomb-Scargle power is large. The dominant period identified in Equation (4.3) is thus consistent
with the previous estimate ∼80 Prot from a visual examination of Figure 4.3(a).
The partial wreaths tend to reside predominantly in one hemisphere or the other, although
they are not strictly confined and can in some instances cross the equator. Nonetheless, thisphenomenon leads to a substantial hemispheric asymmetry, which we define for our dynamo models
asAhem ≡

BφN − BφSBφN + BφS

,

(4.4)

the “N” and “S” superscripts referring to instantaneous rms averages of the field strength over the
Northern and Southern Hemispheres (respectively). The mean in the rms is taken over longitude,
low latitudes (between the equator and 25◦ -latitude for the Northern Hemisphere and comparably
for the Southern Hemisphere), and all depths.
Figure 4.14(a) shows the temporal behavior of the hemispheric asymmetry for case D3-1 during roughly the latter half of the simulation, when the partial-wreath cycle is usually substantiallystronger than the fourfold-wreath cycle. The dominant hemisphere switches between North andSouth chaotically, with no single period easily seen. We have confirmed this aperiodic nature bycomputing the periodogram associated with Ahem shown in Figure 4.14(b): there are a wide rangeof periods that are relevant, with no obvious peak in the power spectrum. What is clear from thebroad peak near 2000 Prot in the periodogram (and from a visual inspection of Figure 4.14(a)) isthat asymmetry of a single sign—corresponding to partial wreaths cycling in preferentially in asingle hemisphere—can persist for long time-scales, up to ∼1000 Prot .

Figure 4.13: Periodogram of the Fourfold-Wreath Cycle. (a) Temporal evolution of Apartial (Equation (4.2)) during the interval
between 6000 Prot and 7000 Prot in case D3-1. (b) Normalized Lomb-Scargle periodogram of Apartial for case D3-1 during the intervalfrom 5100 Prot until the end of the simulation, during which time the partial-wreath pair is consistently stronger than the fourfold-wreath
structure.

152

Figure 4.14: Temporal Evolution of Hemispheric Asymmetry. (a) Temporal evolution of Ahem (Equation (4.4)) during the interval
between 5100 Prot and the end of the simulation in case D3-1. (b) Normalized Lomb-Scargle periodogram of Ahem for case D3-1 during
the same interval as in panel (a).

153

154

4.7

Bistability Trends with Higher Magnetic Prandtl Number (Prm )Because we keep the viscous diffusivity fixed while lowering the magnetic diffusivity, themagnetic structures grow increasingly more complex the higher the magnetic Prandtl number (Prm ).
Figure 4.15 contains snapshots of the azimuthal magnetic field Bφ (in Mollweide projection) near
the beginning of each of the three dynamo cases. At these early times, the structure of Bφ consists
largely of the fourfold wreaths. As Prm is increased, the magnetic structures become more shredded
and dominated by small-scale features. The local field strength increases, but the global coherenceof the field decreases.
Although bistability (in the form of the fourfold and partial-wreath cycles) is most clearly
evident in case D3-1, similar behavior can be seen in the higher-Prm cases D3-2 and D3-4. In Figure
4.16, we show extended time-latitude diagrams for case D3-2, with the time axes and color map
scaled to be directly comparable to Figure 4.3. Case D3-2 exhibits bistability with largely the samecharacteristics as case D3-1, namely a fourfold-wreath cycle with regular polarity reversals and asingle-hemisphere partial-wreath cycle with irregular polarity reversals. The fourfold-wreath cycle
has a weaker signal, however, due to the finer-scale magnetic structures. The system destabilizes
after only ∼250 Prot to launch the partial-wreath cycle, which is less intermittent than in case D3-1,
wandering between the Northern and Southern Hemispheres without ever fully shutting down.
From Figure 4.15(c), the fourfold-wreath structure in case D3-4 is less striking than for the
lower-magnetic-Prandtl number cases. Each wreath is also significantly tilted in latitude and thus
there is very little signal left after a longitudinal average. The partial wreaths, which appear laterin the simulation, are also less coherent. Nonetheless, the fourfold-wreath and partial-wreath cycles
in case D3-4 can still be detected in sequences of Mollweide projections.
Each of the higher-Prm cases has substantial polar caps of magnetism, which occasionally
reverse polarity but are not clearly tied to either of the two cycles. In time-latitude diagrams for
hBr i and hBθ i (not shown), there is a clear tendency for the meridional field to break off from thepartial wreaths and move to the poles, just as in Figure 4.5 for case D3-1.

155

Figure 4.15: Fourfold Wreaths at Different Levels of Turbulence. Snapshots on sphericalsurfaces of the azimuthal magnetic field Bφ shown in global Mollweide projection for the three
dynamo cases D3-1, D3-2, and D3-4 with increasing Prm from (a) to (c). Each model is samplednear the bottom of the shell (r/R = 0.748) at an early time within the dynamo simulation (lessthan 100 Prot after appreciable magnetism develops).

Figure 4.16: Bistability in a More Turbulent Dynamo. Extended time-latitude diagram of azimuthal magnetic field hBφ i and energy
trace for MEφ at mid-depth (r/R = 0.835) for case D3-2 (at Prm = 2), similar to Figure 4.3. Here the fourfold-wreath and partial-wreath
cycles are seen to coexist for most of the simulation, with the partial-wreath cycle in particular being less intermittent than in case D3-1.
156

Case
H3
D3-1
D3-2
D3-4

Prm
1
2
4

η(ro )
3.00 × 1012
1.50 × 1012
0.75 × 1012

Ro
0.05990.06090.06090.0599

Re
56.4
58.3
57.0
53.0

Rem
76.5
101
131

∆Ω/Ω0
0.228
0.183
0.127
0.063

(Nr , Nθ , Nφ )(96, 384, 768)
(96, 384, 768)
(96, 384, 768)
(128, 576, 1152)

`max
255
255
255
383

Diffusion time
115 Prot
115 Prot
230 Prot
460 Prot

Run time
6390 Prot
11100 Prot
4210 Prot
2140 Prot

Run time
55.6 TDT
96.7 MDT18.3 MDT4.65 MDT

Table 4.2: Fluid Parameters for the Different Cases in the Bistability Experiment. Diagnostic parameters, grid resolution, andsimulation run times for all cases.

157

Case
H3
D3-1
D3-2
D3-4

KE/106
19.3
13.5
7.89
3.52

KEφ /10617.5 (91.1%)
11.7 (86.7%)
6.12 (77.6%)
1.88 (53.4%)

KEm /106
0.0053 (0.027%)
0.0053 (0.039%)
0.0048 (0.061%)
0.0038 (0.11%)

KEc /106
1.70 (8.83%)
1.79 (13.2%)
1.76 (22.3%)
1.64 (46.5%)

ME/105
2.15
3.79
6.15

MEφ /1050.161 (7.50%)
0.162 (4.26%)
0.051 (0.82%)

MEm /105
0.015 (0.69%)
0.029 (0.77%)
0.031 (0.50%)

MEc /105
1.98 (91.8%)
3.60 (95.0%)
6.07 (98.7%)

Table 4.3: Kinetic and Magnetic Energies in the Bistability Experiment. Volume-averaged energy budgets for cases H3, D3-1,
D3-2, and D3-4. “KE” and “ME” refer to the kinetic and magnetic energy densities, respectively. Units are cgs, with a common exponent
divided out. The subscripts φ, m, and c refer the energy contributions from the mean azimuthal field components (hvφ i and hBφ i), the mean
meridional field components (hvm i and hBm i), and the convective components (v 0 and B 0 ), respectively. The fraction of the total energy(no subscripts; magnetic and kinetic energies considered separately) from each contribution is shown as the percentage in parentheses.

158

159
The biggest global effect of increasing the magnetic Prandtl number is to lower the overall
differential rotation, while keeping the shape of the rotation profile in the meridional plane largely
fixed. Table 4.2 contains the values of some of the diagnostic non-dimensional parameters (as well as
the grid resolution and various timescales) characterizing each system. We quantify the strength ofthe differential rotation as the difference at the outer surface of the rotation rate between the equatorand 60◦ -latitude, normalized by the frame rotation rate: ∆Ω/Ω0 ≡ [Ω(ro , π/2) − Ω(ro , π/6)]/Ω0 .
For comparison, for the solar rotation rate determined through helioseismology, ∆Ω/Ω0 = 0.197, if
we take Ω0 = Ωand ro to be the radius just below near-surface shear layer (Howe et al., 2000).

The magnitude of the differential rotation in case H3 is quite strong at 0.228—even greater than thesolar value. As Prm increases, ∆Ω/Ω0 is steadily weakened by the presence of small-scale magnetic
structures. These counteract the Reynolds stress produced by the Taylor columns (which tend todrive a strong differential rotation) through feedbacks from the non-axisymmetric Maxwell stress.
From Table 4.2, modifying the magnetic Prandtl number has little effect on the resulting levelof rotational constraint (parameterized by the Rossby number) or the level of turbulence (parameterized by the Reynolds number). This agrees with the fact that the hydrodynamic structures (interms of their distribution of length-scales and amplitudes) in the dynamo cases are largely similar
to those of the hydrodynamic case H3. The main effect of modifying Prm comes from the degreeof complexity in the magnetic structures, as seen by the steady increase of Rem with Prm .
The bulk energy budget for the simulation suite is shown in Table 4.3. Case H3 has both thehighest kinetic energy from the differential rotation (KEφ ) and the highest kinetic energy overall.
For the dynamo cases, as the magnetic Prandtl number is increased, the energy in the meridional
circulation (KEm ) and convection (KEc ) stays roughly the same as in case H3, while the energy inthe differential rotation (KEφ ) sharply decreases.
The magnetic energy density (which is dominated by the fluctuating contribution MEc for
all three dynamo cases) steadily increases with increasing Prm , at the expense of the energy in the
differential rotation. The energy in the mean magnetic fields MEφ and MEm seems to vary in noclear way with Prm , although the mean energy in the azimuthal field (MEφ ) is significantly less in160case D3-4, consistent with the shredded profile of Bφ seen in Figure 4.15(c).

4.8

The Dynamical Origins of Each Cycling Mode
The real 22-year sunspot cycle must be the result of a complex interaction between turbulent

convection, rotation, and magnetism. Our simulations, although far less turbulent than the Sun,
have the advantage that the dynamical origins of each of the two cycling modes identified can be
explored in a fair amount of detail. This is because we can probe each term in the dynamical
equations with high spatio-temporal resolution and accuracy, something we are unable to do for
the Sun. We devote this section to describing the dynamics that lead to the partial-wreath andfourfold-wreath cycles in case D3-1. We focus the discussion on the dominant terms in the induction
Equation (1.1c) and their physical origins.
We note at the outset that for both cycles, there is very little temporal modulation in either
the differential rotation (DR; the main driver of the dynamo through the Ω-effect) or the meridional
circulation (MC). This stands in contrast to the dynamos of Augustson et al. (2015) and Guerrero
et al. (2016a,b), in which significant modulation of the DR (10–20%) played a significant role.
For the fourfold-wreath cycle, modulation of mean flows is undetectable. For the partial-wreath
cycle, there is a ∼5% weaker DR when compared to the fourfold-wreath cycle (due to the stronger
magnetic fields), and a ∼1% modulation of the DR from cycle to cycle. The MC develops significant
equatorial asymmetry during partial-wreath cycling, but this appears to be mostly a response to
the magnetic field and has little effect on the dynamics.

4.8.1

Dynamics of the Fourfold-Wreath Cycle

The magnetic field in our simulations is dominated by the azimuthal component Bφ . We
therefore first turn our attention to the φ-component of the longitudinally averaged Equation

161(1.1c), i.e.,∂ hBφ i≈ (hBi · ∇ hvi)φ + B 0 · ∇v 0 φ
∂t
{z
} |
|
{z
}
MSφ

FSφ

− hv · ∇Biφ − h∇ × [η(r)∇ × B]iφ ,
|
{z
}|
{z
}
TAφ

(4.5)

RDφ

where “MS,” “FS,” “TA,” and “RD” refer to mean shear, fluctuating shear, total advection, andresistive dissipation, respectively. The generation due to fluid compression, − hBφ ∇ · vi, is very
weak and we neglect it throughout.
Figure 4.17 shows profiles of the terms in Equation (4.5) averaged over a short time interval
during fourfold-wreath cycling. The time interval chosen is not unique and the profiles in Figure
4.17 are representative of the generation terms at any time the fourfold-wreath cycle is dominant.
The governing term is clearly the mean-shear, or Ω-effect (MSφ ). The fluctuating shear (FSφ ) is
the weakest term overall and has no obvious pattern in its spatial distribution. The total advection (TAφ ) is weak and also has no cellular structure (compare to Figure 4.1(d)), suggesting thattransport of Bφ by the meridional circulation is negligible. The resistive dissipation (RDφ ) plays
some role (especially near the inner boundary) and has a sign structure opposite to that of hBφ i,acting to destroy whatever field is already present.
The mean-shear production of hBφ i has a similar pattern to hBφ i itself, but with the four
wreaths displaced closer to the equator, which tends to drive the hBφ i wreath structure equatorward.
Furthermore, like the fourfold wreaths themselves, the wreaths of MSφ reappear at mid-latitudes
after they disappear near the equator. In Figure 4.17(a), for example, the negative-MSφ wreath
straddling (−15◦ )-latitude in the Southern Hemisphere is associated with a weak negative-MSφ
wreath forming at (−40◦ )-latitude. Positive Bφ thus forms at high latitudes where originally there
was negative Bφ , creating a polarity reversal.
It is unclear exactly what stops the equatorward migration—i.e., why the wreaths havesignificantly reduced amplitude in the ±10◦ -latitude band straddling the equator. In the Sun,
there is presumably only one wreath per hemisphere, each with opposite polarity (as inferred from162
the solar butterfly diagram). When the two wreaths meet at the equator, they annihilate, as inmagnetic reconnection. The same process cannot occur in the fourfold-wreath cycle because thewreaths are often equatorially symmetric.
We now investigate the origins of the displacement of the shear-production pattern relative
to the azimuthal magnetic field. The mean shear is (neglecting curvature terms)
MSφ ≈ hBm i · ∇ hvφ i .

(4.6)

The magnetic field components hBm i and hBφ i, the azimuthal velocity field hvφ i, and the shear
production MSφ are shown as a set of snapshots in Figure 4.18. The meridional magnetic field has afourfold structure and is roughly coincident spatially with the hBφ i wreaths. From Figure 4.18(a),the streamlines of Bm are roughly counterclockwise for positive hBφ i and clockwise for negativehBφ i.
The profile of hvφ i comes from a differential rotation (at least during the fourfold-wreath
cycle) nearly identical to that of the hydrodynamic progenitor, case H3 (compare to Figure 4.1(c)).
The direction field of h∇vφ i (perpendicular to the contours in Figure 4.18(b)) is thus cylindricallyoutward and slightly tilted toward the equator. The combination of hBm i and h∇vφ i creates thepattern of MSφ shown in Figure 4.18(c).
The overall geometry depicted in Figure 4.18 thus explains the displacement of the term MSφrelative to hBφ i. In the positive-hBφ i wreath in the Northern Hemisphere, for example, the field lines
wind so that hBm i points outward in the lower-latitude part of the wreath (corresponding to positiveMSφ via Equation (4.6)) and inward in the higher-latitude part of the wreath (corresponding tonegative MSφ ). With time, the positive-hBφ i wreath is thus strengthened at lower latitudes andweakened at higher latitudes, sending the whole structure south toward the equator so that thepositive-hBφ i wreath is eventually replaced with a negative-hBφ i wreath.
The nature of the Ω-effect can also be used to estimate the fourfold-wreath cycle period.
Since the mean-shear term is responsible for the cycling of hBφ i, the cycle period should be given

Figure 4.17: Inductive Generation of the Fourfold Wreaths. Generation terms in the azimuthal component of the longitudinallyaveraged induction Equation (4.5) for case D3-1, averaged over an interval of one rotation period centered at 660 Prot . The terms are(a) mean shear, (b) fluctuating shear, (c) total advection, (d) resistive dissipation, (e) sum of all terms (approximately equal to thetime-derivative of hBφ i), and (f) hBφ i itself. The units are cgs (G s−1 in panels a–e and G in panel f).

163

164

Figure 4.18: Mean-Shear Production of the Fourfold Wreaths. Instantaneous profiles (att = 660 Prot ) of (a) the longitudinally averaged magnetic field, (b) the longitudinally averaged
azimuthal velocity, and (c) the mean-shear production MSφ between the equator and 30◦ -north forcase D3-1. In panel (a), hBφ i is plotted in color (with saturation values ±2.5 kG) and hBm i isplotted as a vector field (with the longest arrows corresponding to ∼750 G). In panels (b,c), hvφ i
and MSφ are shown in color with saturation values of ±750 m s−1 and ±0.001 G s−1 , respectively.

Figure 4.19: Inductive Generation of the Poloidal Magnetic Field. Generation terms appearing in Equation (4.9), averaged from snapshots taken at the beginning of 57 successive fourfoldwreath cycles. (a) Fluctuating transport, (b) sum of all terms (equal to the time-derivative of hBr i),and (c) hBr i itself. The units are cgs (G s−1 in panels a–b and G in panel c).

165roughly by
Pfourfold ∼4 hBφ imax
4 hBφ imax∼,
(MSφ )rms
hBm irms h∇vφ irms

(4.7)

where hBφ imax is the typical azimuthal magnetic field’s maximum amplitude in the wreath’s core
and the “rms” subscripts refer to rms magnitudes taken over the whole wreath. For the wreath
pair in Figure 4.18 (taking the mean between the equator and 30◦ -north in lower half of the CZ),
we compute hBφ imax = 2.63 kG, hBm irms = 302 G, and h∇vφ irms = 2.17 × 10−6 s−1 , yielding
Pfourfold ∼ 22.0 Prot ,(4.8)

in good agreement with the actual cycle period of Equation (4.1).
To investigate the source of the meridional field Bm , we turn to the radial component of thelongitudinally averaged Equation (1.1c), which we write as∂ hBr i= [∇ × (hvi × hBi)]r + [∇ × ( v 0 × B 0 )]r
|
{z
} |
∂t
{z
}
MTr

− h∇ × [η(r)∇ × B]ir .
|
{z
}

FTr

(4.9)

RDr

Here “MT” and “FT” refer to the mean transport (magnetic field generation from the mean electromotive force—or emf—hvi × hBi) and the fluctuating transport (or generation from the fluctuating
emf hv 0 × B 0 i), respectively.
Figure 4.19 shows the dominant generation terms for hBr i alongside hBr i itself. We only
show the fluctuating transport FTr , which is an order of magnitude larger than both MTr and
RDr . Similar to the generation of hBφ i shown in Figure 4.6, the profile for ∂ hBr i /∂t is displaced
equatorward relative to hBr i, leading to the equatorward propagation of hBr i as well as hBφ i. The
generation terms for hBθ i (not shown) possess a similar structure, with FTθ dominating the otherterms by a factor of ∼5.
The fluctuating transport is similar in spirit to the α-effect from mean-field theory, which
represents the emf caused by the helical action of convection on the magnetic field. In this respect,
the fourfold-wreath cycle is characteristic of an αΩ dynamo. However, our simulations are distinct

166from most mean-field dynamo models in that the generation of both meridional field and azimuthal
field occurs in the same location in the CZ, with the meridional circulation playing essentially norole. This stands in contrast to the flux-transport paradigm, in which the α-effect creates meridional
field at high latitudes that is then pumped deep into the CZ by the meridional circulation to seedthe Ω-effect in the following cycle.

4.8.2

Dynamics of the Partial-Wreath Cycle

The physical mechanisms responsible for the system transitioning between the fourfold-wreathand partial-wreath states remain unclear. Nevertheless, we can understand the maintenance of the
dynamo during the partial-wreath state in terms of the dominant induction terms. Furthermore,even though the partial-wreath structure is inherently non-axisymmetric, it is sufficient to perform a mean-field analysis—i.e., identify the dominant longitudinally averaged induction terms ofEquations (4.5) and (4.9). Since the longitudinally averaged field hBφ i has the same polarity as
the dominant partial wreath, there is a one-to-one correspondence between cycles in hBφ i andpartial-wreath cycles.
The mean-field generation terms during the partial-wreath cycle vary rapidly in space andtime. Therefore, significant spatial and temporal averaging must be performed to extract a meaningful signal. Figure 4.20 shows the spatially-averaged contributions to the production of hBφ i incase D3-1 from the mean shear (which is clearly dominant) and the other terms, as well as hBφ iitself, during the interval of partial-wreath cycling depicted in Figure 4.10. All three profiles areroughly sinusoidal. The profile of mean-shear production slightly leads the profile of hBφ i in time,
indicating that the rise and fall of hBφ i is directly caused by the mean-shear production. The
contribution from the other induction terms (which are dominated by the resistive dissipation) is∼180◦ out-of-phase with hBφ i, meaning the magnetic diffusion attempts to destroy whatever field
is already there.
The production of hBm i (which in turn affects the production of hBφ i through mean-shear)is highly chaotic in time during the partial-wreath cycle, and so instead of plotting the production

Figure 4.20: Inductive Generation of Partial Wreaths during a Cycle. The dominant generation terms appearing in Equation(4.9), as well as hBφ i, when the partial-wreath structure is cycling in the South, during the same interval shown in Figure 4.10). Quantities
are averaged over radius and between 0◦ and 20◦ -south in latitude. The generation terms are broken up into mean-shear and “other” (i.e.,∂ hBφ i /∂t − MSφ ) and multiplied by 5 Prot to put them on the same scale as hBφ i. The contribution to “other” dominated by the resistive
dissipation.

167

168
terms in Equation (4.9) as functions of time, we simply quote their temporally-averaged magnitudes
below. Just as in the fourfold-wreath cycle, the dominant term producing hBm i is the fluctuating
emf:(FTr )rms = 9.3 (MTr )rms

(4.10)(FTr )rms = 2.6 (RDr )rms

(4.11)

at mid-depth, where the mean is taken over the time interval (3000, 3300) Prot and the latitude
band between 0◦ and 20◦ -south. Similar relationships hold for the production of hBθ i, which is also
dominated by the fluctuating emf

4.9

Bistability in the Context of Solar Observations
Continuous, full-disc observations of the photospheric magnetic field, produced by MDI/SOHOand HMI/SDO, have greatly constrained certain features of the solar dynamo and have yielded anumber of surprises. Stenflo & Kosovichev (2012) analyzed a large set of MDI magnetograms from1995–2011 and found that Joy’s law (which states that emerging sunspot pairs are tilted on averagesuch that the leading spot is closer to the equator, and that the tilt is proportional to heliographic
latitude) holds for active regions both large and small. Furthermore, there was no observed dependence of tilt angle on total active-region flux (or region area), implying that the tilts were inherent
to buoyant flux ropes in the solar interior. This stands in contrast to the argument that the tilts are
established by the Coriolis force as the flux ropes rise through the CZ (e.g., D’Silva & Choudhuri
1993; Fan et al. 1994).
The view that the interior flux ropes are associated with a preferred sense of tilt is interestingin light of the fourfold-wreath cycle. Each of the fourfold wreaths possesses a particular sense oftwist as in Figure 4.18(a). If loops were to spontaneously break off from the wreaths (this does
not occur in our simulations because the magnetic diffusivity is too high), they would be endowedwith a tilt from the twisting magnetic field lines in the wreath. Nelson et al. (2013b) achieved theseparation of magnetic loops from interior wreaths and found that the sample of loops reaching the169
surface made contact with Joy’s law. This supports the idea that a fourfold-wreath-like structurein the Sun could establish systematic tilts associated with flux ropes in the deep interior.
Stenflo & Kosovichev (2012) further reported that a statistically significant fraction of active
regions (5%–25%, depending on the region’s size) violate Hale’s polarity law, strongly suggesting that there must be multiple reservoirs of interior magnetism from which buoyant flux ropes
originate. Additionally, the “anti-Hale” active regions could appear in the same latitude band as“Hale” active regions, suggesting close proximity in latitude for the reservoirs. This result agrees
with the geometry of both the fourfold-wreath structure (whose wreaths come in opposite-polarity
pairs with wreaths in a pair separated in latitude by ∼15◦ ) and the partial-wreath structure (withopposite-polarity partial wreaths at approximately the same latitude).
Li (2018) performed another extensive analysis of sunspot groups from 1996–2018 using both
the MDI and HMI magnetograms, also reporting that a sizeable fraction (∼8%) of sunspot pairsviolate Hale’s polarity law and confirming the result obtained by Stenflo & Kosovichev (2012). In
addition, Li (2018) found that the anti-Hale sunspots had different properties than Hale sunspots,
namely, smaller bipole separation and weaker total magnetic flux. These results further support
the idea that the Hale and anti-Hale sunspots may come from different interior reservoirs.
It is also possible that the anti-Hale active regions form an essential part of the solar dynamo.
Hathaway & Upton (2016) find that modeling the flux transport of active regions observed by SDO
during the current solar cycle can be used to predict the amplitude and hemispheric asymmetryof the following cycle. Central to such predictions is the initial distribution of active-region tilts,
which would be significantly modified by the presence of anti-Hale active regions. Although it isunlikely that the real interior solar magnetic field matches the fourfold-wreath and partial-wreath
structures in detail, such configurations offer two ways in which violations of Hale’s polarity law
could occur.
We now discuss the asymmetric features of solar activity, many of which are captured by thepartial-wreath cycle. For well over a century, sunspots have been observed to emerge preferentiallyat particular solar longitudes (e.g., Maunder 1905; Svalgaard & Wilcox 1975; Bogart 1982; Henney

170& Harvey 2002). This phenomenon, referred to as active longitudes, remains poorly understood.
It is common for active longitudes to come in pairs, separated by about 180◦ in longitude(e.g., Bai 2003). The pairs are also usually found in a single hemisphere and are part of the global
hemispheric asymmetry in magnetic activity. Finally, the pairs may be associated with opposite
polarity of the azimuthal magnetic field on opposite sides of the Sun, as suggested by synoptic
maps from the Wilcox Solar Observatory (Mordvinov & Kitchatinov, 2004). Thus, it is possiblethat the pairs are associated with nests of Hale and anti-Hale active regions on opposite sides of
the Sun, although a systematic study of the MDI/HMI magnetograms would be needed to confirm
this result.
Active longitudes and their associated hemispheric asymmetry bear a striking resemblance tothe partial-wreath structure in our simulated dynamos. The partial wreaths preferentially occupy
one hemisphere at a time. They come in opposite-polarity pairs, with the strongest magnetism on
opposite sides of the sphere (see the Mollweide projection in Figure 4.2(b) and the 3-D volumerenderings in Figure 4.11). The time-longitude diagrams for the partial wreaths in case D3-1 (Figure4.12) show that each wreath is roughly stationary once an appropriate rotating frame is chosen,
similar to the quasi-rigid structure formed by active-longitude pairs in the Carrington reference
frame (Berdyugina & Usoskin, 2003).
Figures 4.11 and 4.12 would also indicate cyclic behavior, in which the strongest active
longitude flips by about 180◦ over the partial-wreath cycle period. Berdyugina & Usoskin (2003)
reported such behavior in observations of solar sunspot number, wherein the active longitudes
appeared to flip between opposite sides of the Sun every ∼3.7 years, or every ∼53 solar Carrington
rotations. Our simulation suite displays similar properties in the relative energy content of thepartial wreaths (Figure 4.13), showing that the dominant partial wreath switches with a quasiregular period of 81.5 Prot .
The hemispheric asymmetry in solar activity can be seen in a wide variety of activity indicators, such as number of flares, sunspot number, sunspot area, and photospheric magnetic flux
(e.g., Schüssler & Cameron 2018 and references therein). Many studies have been performed on

171
the periodic behavior of the asymmetry, with most agreeing on a long-term period of ∼50 years
and a short-term period of ∼10 years (see Hathaway 2015 and reference therein). Our simulations
each possess a significant hemispheric asymmetry (Figure 4.14(a)), although there is no obvious
associated cycle (Figure 4.14(b)). Nevertheless, our simulations make contact with observations inthat the hemispheric asymmetry occurs as a long-term modulation of another cycle, which in our
simulations is that of the partial wreaths.
In summary, the dynamo cases presented here show similarities to several features of solar
activity, namely, the sunspot cycle, opposite-polarity magnetic regions in close proximity, active
longitudes, and hemispheric asymmetry. We do not suggest that all these observed features of theSun’s activity are a result of fourfold-wreath and partial-wreath structures existing in the solar
CZ. Rather, our simulations provide insights as to how a solar-like convection zone can support
dynamos exhibiting a wide range of cyclic behavior.

4.10

Conclusions
We have explored three dynamo simulations of a 3-D, solar-like convection zone at varying

magnetic Prandtl number. These dynamo cases display clear signs of bistability. In particular, there
is a fourfold-wreath structure that exhibits equatorward propagation and regular polarity reversals.
There is also a single-hemisphere partial-wreath pair that is roughly stationary and whose whose
dominant polarity changes cyclically, but with an irregular period. The fourfold-wreath cycle
captures aspects of the 22-year sunspot cycle (namely, the solar butterfly diagram), whereas thepartial-wreath cycle is reminiscent of several of the observed features of active longitudes andhemispheric asymmetry.
We have examined the dynamics of the fourfold-wreath cycle in detail, and find they areapproximately described by an αΩ dynamo. The mean shear, or Ω-effect, is responsible for thegenerating the wreaths. The regular polarity reversals and equatorward propagation are causedby the inherent geometry of the wreaths (namely the meridional field winding counterclockwise forpositive Bφ and clockwise for negative Bφ ) coupled to a solar-like differential rotation. Generation

172of the meridional field occurs through the fluctuating emf, which is similar to an α-effect, but
fully nonlinear. Both effects occur in the same localized region of the CZ (with the meridional
circulation playing basically no role), unlike in the flux-transport dynamo framework. As a whole,
the production terms for the fourfold-wreath cycle are similar to those of model “K3S” of Augustson
et al. (2015), although in that work, significant modulation of the differential rotation played a largerole, whereas in our dynamo cases, the differential rotation stays mostly constant with time.
The partial-wreath cycle has longitudinally averaged dynamics similar to an αΩ dynamo as
well, but it is not fully clear how the system transitions between the regularly cycling fourfoldwreaths and the single-hemispheric partial-wreath pair. It is possible that the transition is governed by the fourfold-wreath structure changing between equatorially symmetric and antisymmetricstates, but more work needs to be done to assess in detail how such a mechanism operates. Equatorial parity also played a key role in model K3S, wherein atypical dominance of the radial magnetic
field by the even-` modes acted as a precursor to the cycling dynamo entering a “grand minimum.”
Similarly, in the global convective dynamo simulation of Raynaud & Tobias (2016), the interactionof hydromagnetic modes with different symmetries led to transitions between distinct cycling dynamo states. The magnetic Prandtl number also may be significant in determining whether thesystem is unstable to the partial-wreath state; Nelson et al. (2013a) found magnetic structures
resembling single-hemisphere partial-wreaths in their high-Prm simulations but not low-Prm .
Our work suggests that spherical-shell convection can support complex dynamos capable ofexhibiting vastly different global structures for the magnetism simultaneously. In particular, thebistable properties of our dynamo cases are consistent with a solar dynamo containing two cycles at
once: one associated with the observed butterfly diagram and one associated with active longitudes
and hemispheric asymmetry. We recognize that our simulations here are just beginning to explore
the complexities that might be at work in the highly nonlinear and turbulent dynamics of the Sun.
The range of scales that we can capture and the diffusivities we use are still far removed from thesolar regime. However, we have found that some reasonable large-scale behavior emerges from ourdynamo models, and these may broaden our insights into the physical processes occurring deep

173
within the interior of the Sun.

4.11

Definitions of Non-Dimensional Numbers and Diffusion Times
After the example of Featherstone & Hindman (2016b), we parameterize the strength of theimposed driving through a bulk “flux Rayleigh number,”
RaF ≡g̃ F̃ H 4,
cp ρ̃T̃ ν 3(4.12)

where H is the shell depth and F is the energy flux associated with the combined conduction andconvection in a statistically steady state. The tildes refer to volume-averages over the full sphericalshell.
We parameterize the degree of rotational constraint through a bulk Rossby number,
 ^

v0
Ro ≡
,
(4.13)
2Ω0 Hv
where the typical velocity v 0 (r) refers to the rms of the convective (|m| > 0) velocity over a sphereof radius r. The typical length-scale of the flow structure Hv (r) is not well-represented by either
the scale height or the shell depth. Instead, we choose Hv (r) based on the power spectrum ofP
the convective velocity. Specifically, Hv (r) ≡ 2πr/`v (r), where `v (r) ≡ |m|>0 `|v̂`m |2 and |v̂`m |2
refers to the normalized (`, m)-power associated with v at radius r. We similarly define a spectral
length-scale HB (r) associated with the convective magnetic-field structures.
We assess the level of turbulence through a volume-averaged Reynolds number,^
v 0 HvRe ≡νand a volume-averaged magnetic Reynolds number,^
v 0 HBRem ≡,η

(4.14)(4.15)We define the thermal diffusion time (which is equal to the viscous diffusion, time since Pr= 1) asTDT =(ro − ri )2= 115 Prot ,κ(ro )(4.16)

174
which is the same for all four cases.
We define the magnetic diffusion time as
MDT =(ro − ri )2= 115, 230, 460 Prot
η(ro )

for cases D3-1, D3-2, and D3-4, respectively.

(4.17)

Chapter 5

Confinement of the Solar Tachocline by Dynamo Action in the Radiative
Interior

The helioseismically inferred tachocline (why it is observed to be so thin, at ∼0.05R ) is an
outstanding fundamental mystery of solar physics. Spiegel & Zahn (1992) (hereafter SZ92) firstrecognized that the tachocline should have “radiatively spread” to a thickness of (at minimum) ∼0.4
R

by the current age of the Sun (see Section 5.1). To remain consistent with the helioseismic

deductions, there must be a torque that forces the radiative interior to remain in solid-body rotationand prevents radiative spread. Various tachocline confinement scenarios postulate the origin of thistorque (for reviews, see, e.g., Gilman 2000a; Miesch 2005; Brun & Strugarek 2019; and references
therein). The two most commonly invoked confinement mechanisms are the “fast HD scenario” of
SZ92 (the confining torque is caused by HD instabilities and operates on the fast dynamical timescale; see Section 5.2) and the “slow MHD scenario” of Gough & McIntyre (1998) (the confining
torque is caused by a primordial magnetic field and operates on the slow meridional-circulation
time-scale; see Section 5.3). As described in the following Sections, much work has been done toassess the validity of each of these confinement scenarios. Significant progress has been made in
identifying many of the waves, instabilities, and dynamo processes present in the tachocline, but
the origin of the confining torque is still an open question.
The parameter space in which radiative spread occurs is not currently accessible by global
simulations (e.g., Brun et al. 2011; Wood & Brummell 2012, 2018; and references therein; also see
Table 5.1). Because global simulations have greatly enhanced turbulent diffusions, however, the176tachocline still spreads viscously. As described in Chapter 1, most prior work employing global
simulations has focused on the influence of an imposed or transient tachocline on the dynamo,
generally by artificially lowering the viscosity below the convection zone. No global model we areaware of has been demonstrated to achieve tachocline confinement against the inevitable viscous
spread by a self-consistently generated torque. In this Chapter (Sections 5.4–5.10), we present a
global Rayleigh simulation that achieves differential rotation in the convection zone coexistingwith solid-body rotation in the radiative interior. This shear layer, which we call a “tachocline,”remains in a statistically steady state for many viscous diffusion times. The magnetic torque froma self-excited dynamo forces the radiative interior into solid-body rotation and prevents the viscous
imprinting of the convection zone’s differential rotation.
Our work supports the hypothesis that magnetism is responsible for solid-body rotation in
the solar radiative interior, but postulates a very new form and origin of this magnetism. In contrast
to the primordial magnetic-field remnant originally envisioned in the slow scenario, the confining
magnetic fields in our simulation are primarily non-axisymmetric (similar to the partial-wreath
structures just encountered in Chapter 4), and are also oscillating in time. The more fundamental
difference, however, is that the magnetism in our MHD case is maintained inductively, even below
the layer of convective overshoot. The induction is caused by strong horizontal motions, which
we argue represent inertial oscillations, and are likely excited by the pummeling of convective
downflows from above. This work thus hints that the Sun’s “stable” radiative interior may be a
dynamically active region that can amplify magnetic field and significantly affect the operation of
the global dynamo.
The MHD case presented in this Chapter is the first fully 3-D, nonlinear, global simulationthat has been shown to achieve dynamics similar to a magnetic confinement scenario. Aspectsof our MHD case—namely, strong horizontal velocities and magnetic fields below the overshootlayer—have been frequently encountered in prior simulations, but have not been recognized to be
the result of an active dynamo, and have not been shown to confine a tachocline. In Section 5.9.,We
discuss these similarities between our MHD case and prior work. In Section 5.10, we discuss how

177
our work offers a new perspective on the solar tachocline. We assess the likelihood that the solar
dynamo penetrates deep into the radiative interior and whether it might confine the tachocline
against radiative spread. We conclude by offering some future directions global simulations mighttake to fully address the solar tachocline confinement problem.
This Chapter is based on work currently being prepared for an upcoming publication, Matilsky et al. (2022) and was briefly described in the conference proceedings paper, Matilsky & Toomre(2021). As the primary author of the paper, I conducted the simulations, performed their analyses,
created most of the Figures (with the exception of the two final Figures by Catherine Blume),
and wrote the text. My coauthors Juri Toomre, Bradley Hindman, Nicholas Featherstone, andCatherine Blume provided (and likely will continue to provide) advice and guidance throughout
the process, gave detailed text edits, and suggested schematic designs of some of the figures. Special thanks is owed to Mark Miesch for his explanation of the Gough-and-McIntyre model (slow
confinement scenario) and to Nicholas Brummell and Pascale Garaud for their explanation of thephenomenon of radiative spread.

5.1

Radiative Spread of the Solar Tachocline
SZ92 imagine the solar radiative interior as a non-magnetic (HD) ball of ideal gas, with theconvection zone at the top imposing its differential rotation as a boundary condition. Initially, theradiative interior is assumed to rotate like a solid body (at rate ΩRI ), except for a thin boundary
layer of shear at the top, which represents the tachocline. The tachocline has initial thickness h0 andradial location r0 (helioseismically, h0 ≈ 0.05R and r0 ≈ 0.72R ; see Section 1.1.2). SZ92 assume
a thin tachocline, or h0 /r0  1. They also consider a laminar fluid, hydrostatic and geostrophic
thermodynamic perturbations (here we call this “thermal wind balance,” following the terminology
of Chapter 3), strong rotational constraint (as defined in this thesis, low Ekman and Rossby numbers), weak differential rotation, axisymmetry, and the Boussinesq approximation (specifically, theBoussinesq equations in the form of Spiegel & Veronis 1960). Under these assumptions, SZ92 argue
that the tachocline naturally spreads via two distinct processes: viscous diffusion and radiative

178
thermal diffusion. SZ92 note that in the Sun, the molecular thermal conductivity can be neglectedcompared to the radiative thermal conductivity. SZ92 ultimately find that radiative spread should
dominate viscous spread in the solar tachocline, and furthermore, that unless radiative spread wereprevented, the tachocline would be much thicker than is helioseismically observed.
Viscous tachocline spread is simple to understand. The convection zone above drags theradiative interior with it via viscous forces, and the convection zone’s differential rotation thus
imprints downward. Through viscous spread, the tachocline boundary layer should grow like t1/2 ,where t is the time since the tachocline was initially confined. Furthermore, viscous spread shouldtake place on the viscous diffusion time-scale.
Radiative spread is conceptually more complicated. Unlike viscosity, thermal diffusion does
not directly torque the radiative interior; instead, it induces a meridional circulation that transports
angular momentum via advection. In brief, thermal wind balance implies that the differential
rotation is always coupled to the horizontal pressure gradient (through geostrophic balance), which
in turn is coupled to the horizontal temperature gradient (through hydrostatic balance). There isthus an equator-to-pole temperature contrast in the top layers of the radiative interior, which rotate
differentially, and no such temperature contrast in the deeper layers, which rotate like a solid body.
The equator-to-pole temperature contrast must then spread inward via radiative diffusion. As thetemperature in the radiative interior changes, buoyancy forces will drive a meridional circulationthat ultimately stops the diffusive spread and establishes an advection-diffusion balance of heat.
This circulation also advects angular momentum in a manner that imprints the differential rotation
inward. Because the meridional circulation always closely balances radiative diffusion, the spreading
process is not diffusive and does not occur on the thermal diffusion time-scale. Spiegel & Zahn
(1992) instead find that radiative spread occurs on the Eddington-Sweet time-scale tES and is“hyperdiffusive:” The temperature gradient, meridional circulation, and differential rotation are
expected to burrow inward like t1/4 .
The two diffusive spreading processes can be summarized by the following angular-momentum

179
equation, paraphrased from Spiegel & Zahn (1992):

where 4 4 0  2  2 0∂Ω0r0 ∂ Ω
r0
∂ Ω
≈−
+
,
4
∂t
tES ∂rtvisc ∂r2
 2
N2
r0
r02
tES ≡
and
t
≡
visc
ν
4Ω2RI κ(5.1a)(5.1b)

are the Eddington-Sweet and viscous diffusion times across the radiative interior, respectively.
Here κ and ν denote the radiative diffusivity and molecular viscosity of the Sun, respectively. The
buoyancy frequency is defined through N 2 = (g/cp )dS/dr and Ω0 denotes the deviation of theangular velocity from the solid-body rate ΩRI . The first term on the right-hand side of Equation(5.1a) represents the radiative spread of the tachocline and the second term represents the viscous
spread. Note that the radiative diffusivity κ acts on the temperature gradient, not the entropy
gradient. Otherwise, the notation in Equations (5.1) is identical to that of the rest of this thesis(compare to Equations (1.1)).
We note that although the time-scale tES shows up in the phenomenon of radiative-spread,the meridional circulation invoked is not the classical Eddington-Sweet (or Eddington-Vogt-Sweet)
circulation (Vogt, 1924; Eddington, 1925; Sweet, 1950). The Eddington-Sweet circulation arises
in rotating stellar radiative interiors because even small oblateness causes heating of gas by photons to depend slightly on latitude (von Zeipel, 1924). Because the equatorial regions are “puffedout” in a rotating star, the radial temperature gradient and absorbed radiative flux are smaller,
and the equator is heated less rapidly than the poles. The only way the star can maintain thermodynamic equilibrium is if a weak meridional circulation transports heat from pole to equator.
Simple considerations reveal that the required circulation overturns on a time-scale of ∼tES . In
the solar tachocline, the meridional circulation is also generated by a pole-to-equator temperature
contrast, but this contrast comes from the convection zone’s thermal wind, not the oblateness. The
circulation cells are highly elongated in latitude (they occupy the thin tachocline in SZ92’s theory)and have a circulation time of ∼(h0 /r0 )4 tES . In the context of the tachocline, tES thus does not
represent a circulation time, but rather the time it takes the circulation itself to burrow all the way
to the solar core via radiative spread.

180
Clark (1973) was the first to note the competition between viscosity and meridional circulationin transporting angular momentum. He considered a cylinder of initially uniformly rotating stably
stratified fluid subject to a time-varying rotation rate applied at the outer boundary. In thelinear calculations, the changing boundary condition induced a meridional circulation in a thermalboundary layer whose thickness grew hyperdiffusively as t1/4 . Advection of angular momentum by
this circulation caused the rotation rate imposed at the outer boundary to burrow inward at thesame hyperdiffusive rate. Overall, these results were remarkably similar to SZ92’s theory of radiative
spread. Haynes et al. (1991) obtained similar results on fast (though not strictly hyperdiffusive)
downward burrowing of circulations in the Earth’s middle atmosphere. Since these initial theoretical
calculations, the burrowing of meridional circulation has been identified as a robust phenomenon insimulations of rotating, stably stratified convection (e.g., Elliott 1997; Garaud & Brummell 2008;
Garaud & Acevedo-Arreguin 2009; Wood & Brummell 2012, 2018).
We list the Sun’s diffusions at the base of the convection zone (including the magnetic diffusivity η, for completeness) in Table 5.1. We also list other relevant solar parameters (mostimportantly, the diffusive time-scales) and the corresponding parameters in the MHD case (with atachocline) that we discuss in the following sections. We note the parameters chosen for our MHD
case, although they are unique, are typical of all global simulations we are aware of. In particular,
global simulations have Prandtl numbers of order unity and diffusivities (explicit in the case of, e.g.,Rayleigh and ASH, and implicit in, e.g., the EULAG-MHD cases) many orders of magnitude higher
than in the Sun. From Table 5.1, the time-scale tvisc for viscous spread of the solar tachocline isgreater than the Eddington-Sweet time-scale tES for radiative spread. Both time-scales are much
longer than the age of the Sun (4.6 billion years), but radiative spread is still significant because
the meridional circulation burrows faster than the diffusive rate. In the MHD case, the viscoustime-scale is significantly shorter than the Eddington-Sweet time-scale. The viscous spread thus
dominates and the computations cannot be run long enough for radiative spread to be significant.
Because of the competition between viscous and radiative tachocline spread, the defining
parameter of a given star or simulation measures the ratio of the Eddington-Sweet time to the181

Table 5.1: Diffusive Time-Scales in the Radiative Interior of the Sun and Our MHD
Case. Here we list the numerical values associated with the diffusive processes (viscous, radiative,
and magnetic) relevant for the Sun and our MHD case (with a tachocline) discussed in this Chapter.
The quoted values are at the base of the convection zone (or equivalently, the top of the radiativeinterior). Note that since the MHD case covers only a spherical shell (not the whole Sun) the
depth of its radiative interior (H ≡ rov − ri , where rov is the base of the convective overshootlayer and ri is the inner boundary of the simulated shell) is smaller than the solar value, r0 . The
molecular diffusivity values for the base of the solar convection zone are quoted from Brandenburg& Subramanian (2005) (ν and η) and Hindman et al. (2020) (κ), who in turn derived them fromthe statistical physics considerations of Landau et al. (1998) and Spitzer (1962), respectively. All
quantities are considered to have two significant figures.
Parameter

Solar ValueMHD Case Value

Depth of radiative interior (H)

5.0 × 1010 cm (also r0 )1.6 × 1010 cm

430 nHz

1400 nHz

0.19 mHz

0.19 mHz

Viscosity (ν)

10 cm2 s−1

1.1 × 1012 cm2 s−1
H2
Viscous diffusion time tvisc ≡ν

7.9 × 1012 years

7.4 years

Thermal conductivity (κ)

107 cm2 s−1

1.1 × 1012 cm2 s−1

3.9 × 1011 years

3.4 × 104 years

104 cm2 s−1

2.8 × 1011 cm2 s−1

7.9 × 109 years

29 years

10−6

1

10−4

4

0.22

68




ΩRI
2π
 
N
Buoyancy frequency
2π
Rotation frequency!

N 2 H2
tES ≡
2ΩRI
κ


Eddington-Sweet time
Magnetic diffusivity (η)

Magnetic diffusion time
Prandtl number Pr ≡H2tmag ≡
ηνκMagnetic Prandtl number Prm ≡
r
σ≡

tES
N √
=
Pr
tvisc
2ΩRI

ν
η



182
viscous diffusion time. This quantity, first identified in Garaud & Brummell (2008) and Garaud &Acevedo-Arreguin (2009), is defined through
r
σ≡

N
tES
=
tvisc
2ΩRI

r

ν
N √
=
Pr.
κ
2ΩRI

(5.2)

As would be intuitively expected, Garaud & Brummell (2008) and Garaud & Acevedo-Arreguin(2009) found that radiative spread is significant only for σ < 1. Prior global simulations had
σ = O(100) (e.g., Brun & Zahn 2006; Rogers 2011; Strugarek et al. 2011a,b) and so could notassess radiative spread. These results were further supported by the 2-D global models of AcevedoArreguin et al. (2013) (including both a radiative interior and simplified convection zone) and thefully 3-D, fairly turbulent local simulations of Wood & Brummell (2012, 2018).
The low-σ regime is currently rather difficult for global simulations to achieve. The ratio
N/2ΩRI = 200 in the Sun, and so simulations could either lower the level of stable stratificationqin the radiative interior (since N ∼ dS/dr) or run at low Prandtl number, say, Pr ∼ 10−5 .
1The stable entropy gradient should not be lowered too far below the solar value, since theovershoot layer would grow unrealistically large, and the mixing properties of the overshoot would
be unrealistic (e.g., Hurlburt et al. 1986, 1994; Brummell et al. 2002; Korre et al. 2019; Korre &Featherstone 2021). Very low Prandtl number achieved by very low viscosity is difficult because of
the fine spatial resolution it would require. Finally, there is not much leeway to lower the Prandtl
number by increasing the thermal diffusion; the sub-grid-scale thermal diffusion is usually alreadyquite high, and if raised by a factor of 10 or so, it would prevent convection entirely.

5.2

Tachocline Instabilities and the Fast Confinement Scenario
SZ92 proposed a tachocline confinement scenario in the form of strongly anisotropic turbu-

lence in the radiative interior. The very stable stratification implies effectively 2-D motions, which,if turbulent, could mix fluid in the horizontal directions much more efficiently than the vertical1

The other option—greatly increase the rotation rate—is generally undesirable since it would cause unrealistic
levels of rotational constraint. It also causes numerical problems due to the small scales introduced—see, e.g.,
Featherstone & Hindman 2016b; Hindman et al. 2020.

183(Zahn, 1974, 1992). SZ92 thus proposed that anisotropic turbulence could homogenize angular velocity in latitude below the convection zone, keeping the radiative interior rotating like a solid bodyand maintaining the tachocline. The homogenization would occur on the fast dynamical time-scaleof the instabilities (hence, “fast” confinement scenario; see, e.g., Gilman 2000a for the origin of the“slow” and “fast” terminology). Anisotropic transport was nominally expected from the theoryof geostrophic turbulence (e.g., Rhines 1979) and might be produced by shear instabilities in thetachocline due to the high Reynolds number of the horizontal shear (Zahn, 1975). Other than these
speculations, however, SZ92 did not identify the instabilities that might lead to turbulence.
Since SZ92, substantial work has been done to characterize the instabilities that may occur
in the tachocline. Charbonneau et al. (1999a) analyzed the linear shear instability of an inviscid,
incompressible fluid on a 2-D spherical surface. They performed analysis following the instability
criteria for rotating stars laid out in Watson (1981) and Dziembowski & Kosovichev (1987), findingthat the tachocline’s shear was stable to any perturbation. However, the shear was very close
to the marginal value required for instability onset and it was argued that small differences in
the helioseismic inversions of the tachocline’s radial location were consistent with instability. For
example, the horizontal shear at r/R
the tachocline, r/R= 0.75 (only slightly higher than the nominal location of= 0.72) was shown to induce the instability of low-azimuthal-order modes(m = 1, 2, 3), with growth rates on the order of years.
Charbonneau et al. (1999a) also noted that there were marginally stable solutions (neither
growing nor decaying) with oscillation periods varying continuously between the equatorial andpolar rotation rates. These solutions could be interpreted as Rossby waves (waves with a Coriolis
restoring force) in critical layers.2

Critical layers are regions bounded by turning points inlatitude where the Rossby-wave frequency equals the background rotation frequency. Nonlinear
behavior of critical-layer Rossby waves has been studied extensively for shear flows in the context of
meteorology (e.g., Killworth & McIntyre 1985; Haynes 1989) and stellar astrophysics (e.g., Watson
2

These critical-layer Rossby waves have also been described as “baroclinic” waves and instabilities (e.g., Gilman
1967a,b,c,d; Braginsky & Roberts 1975; McIntyre 1970, 1998; Gilman & Dikpati 2014).

1841981). Usually it is found that the waves become unstable in the presence of maxima in potential
vorticity (potential vorticity is the dot product of vorticity and the background stratification; inour notation, it would be ωr dS/dr, where ωr is the radial vorticity (∇ × v)r ; see, e.g. Gilman& Dikpati 2014). The waves then nonlinearly evolve into strong spiral structures that eventually
break within the critical layer and locally smooth out the potential vorticity, driving the systemback toward marginal stability.
Garaud (2001) considered the weakly nonlinear regime of the tachocline’s shear instability
explored by Charbonneau et al. (1999a). She found that once again the m = 1, 2, 3 modes were
the only unstable ones, and also confirmed the interpretation of the unstable perturbations as
breaking critical-layer Rossby waves. The system ultimately reached a state of marginal instability
that depended slightly on the chosen initial rotation profiles and relative amplitudes of the initial
m = 1, 2, 3 perturbations. In general, however, the final state had an angular momentum profile
deviating by no more than ∼0.2% from the initial state. Garaud (2001) thus concluded that thetachocline was effectively in a state of marginal stability with respect to shear, and that shear
instabilities were unlikely to cause significant angular momentum transport.
Dikpati & Gilman (2001a,b) analyzed the tachocline’s shear instability using a “shallowwater” formalism (e.g., Pedlosky 1987; Gilman 2000b), in which weak vertical motions are allowed
so long as they conserve mass. They found that in the upper part of the tachocline (the overshootlayer, where the stable stratification is weakest), shear instabilities could lead to significant angularmomentum transport. Furthermore, the instabilities were associated with a net pattern of kinetic
helicity. Dikpati (2012) continued this analysis in the fully nonlinear regime, largely confirming thelinear results and showing that the lowest-order (m = 1, 2) modes were the most unstable. She also
confirmed the presence of net kinetic helicity and proposed that the upper layers of the tachocline
could generate an α-effect, thereby driving the solar dynamo entirely from within the tachocline.
Garaud (2020) and Cope et al. (2020) presented the first investigations of 3-D shear instabilities and their mixing properties in non-rotating, non-magnetic Cartesian-box simulations
representing a local region of the tachocline. They found that in the regime appropriate for the

185
Sun (low Prandtl number and high Péclet number—where the advection of heat dominates the
diffusion of heat), shear instabilities induced substantial vertical mixing in addition to horizontal
mixing. The results were thus incompatible with SZ92’s strongly anisotropic mixing. However, Garaud (2020) noted that the presence of rotation, magnetism, and compositional stratification would
probably alter the character of the identified shear instabilities, including their mixing properties.
Significant work has also been done to characterize the tachocline’s magneto-shear, or magnetorotational, instability (e.g., Gilman & Fox 1997; Dikpati & Gilman 1999; Gilman 2000b; Dikpati et al. 2003; Dikpati & Gilman 2005; Miesch 2007; Gilman & Dikpati 2014; Lawson et al.
2015; Gilman 2018a,b). These investigations suggested that the presence of toroidal fields in thetachocline can destabilize latitudinal shear that would otherwise be stable in a purely hydrodynamic
system. Long-wavelength (low m) disturbances were once again identified as the most unstable,with m = 1, 2 modes resulting from broad toroidal wreaths (wide in latitude and radius) and modes
with m as high as 7 resulting from narrow toroidal wreaths. The m = 1, 2 modes in particularwere argued by Dikpati & Gilman (2005) to be a potential origin of the observed active longitudes
discussed in Chapter 4.
Lawson et al. (2015) argued for the presence of magneto-shear instabilities occurring in theirglobal, 3-D EULAG-MHD simulations. They further proposed that these instabilities “pumped” magnetic field downward and may have influenced the global dynamo cycle. Notably, Lawson et al.
(2015) attributed strong horizontal velocities to magneto-shear instabilities, which may have supported the anisotropic turbulence required by the fast confinement scenario. Gilman (2018a,b) examined joint magnetic buoyancy and magneto-shear instability in the solar tachocline with valuesof the horizontal and vertical shear taken from helioseismology. He found that vigorous instabilities induced by weak toroidal fields in high-m modes would mostly prevent the storage of toroidal
wreaths at high latitudes. Low latitudes, by contrast, were found to not only be stable to shear,
but also be stable to the magnetic buoyancy instability until the field strengths rose to very highamplitudes (∼10 kG). He thus argued that the shear profile of the tachocline may yield a new
explanation for the appearance of sunspots at low latitudes.

186
We finally note that Zahn et al. (1997) and Kumar & Quataert (1997) proposed a fast
confinement scenario by the breaking of prograde and retrograde gravity waves. These waves would
have their frequencies, and thus dissipation rates, shifted due to differential rotation near the base of
the convection zone. There could thus be a latitudinal gradient in the wave-field’s local dissipation
rate, which could redistribute angular momentum to maintain solid-body rotation on the time-scaleof ∼107 years (still “fast” compared to the Eddington-Sweet time-scale of ∼1012 years). Talon
et al. (2002) carried this analysis further, finding that gravity-wave breaking could extract angularmomentum from the deep radiative interiors of low-mass stars and could possibly lead to a slowly
rotating core. As of now, however, the angular-momentum transport properties of gravity waves in a
3-D, nonlinear context are unknown. The only global simulations dedicated to characterizing stellar
gravity waves in detail were the ASH models of Alvan et al. (2014, 2015). They found that a rich
spectrum of gravity modes was excited by overshooting convection. They identified the driving
mechanism of the waves, computed their eigenspectra in the context of theoretical ray-tracing
formalism (e.g., Gough 1993), and examined other properties like mode lifetimes and nonlinear
propagation patterns. Although angular momentum transport was not computed explicitly, thiswork highlighted the need to study gravity waves in 3-D spherical geometry using global simulations.

5.3

The Slow MHD Tachocline Confinement Scenario
Gough & McIntyre (1998) (hereafter GM98) argued that the fast tachocline scenarios wereunlikely to hold up in the solar interior. They agreed that instabilities in the presence of strong
stable stratification may induce essentially 2-D geostrophic turbulence. However, they noted thatgeostrophic turbulence in the Earth’s atmosphere was known to drive the system away from uniform rotation, not toward it (McIntyre, 1994). In the atmospheric sciences, this effect had beencalled “negative viscosity” and was well documented (e.g., Starr 1968 and references therein). Similarly, GM98 argued that the torques from dissipating gravity waves were not known to suppress
shear (Gough, 1977) and should (like geostrophic turbulence) enhance it (Plumb & McEwan, 1978;
Andrews et al., 1987; McIntyre, 1994).

187
GM98 thus proposed a confinement scenario wherein a tachocline-confining torque could be
generated by an axisymmetric poloidal magnetic field in the radiative interior, presumed to be a
primordial remnant from the Sun’s birth. As the differential rotation burrowed inward, the poloidal
field (which was assumed to be stationary, confined from above by the meridional circulation) would
create a toroidal field. According to Ferraro’s law of isorotation (Ferraro, 1937), the associatedmagnetic-tension torque would force isorotation contours to fall on poloidal field lines (with solidbody rotation as a special case) and could stop the burrowing. The magnetic torque would react
on the long time-scale associated with radiative spread, namely, the Eddington-Sweet time (hence,“slow” confinement scenario),
The idea that a large-scale magnetic field could significantly affect the differential rotationin stellar radiative interiors was not new (e.g., Ferraro 1937; Alfvén 1943; Cowling 1957). Afterthe work of Ferraro (1937), there was much debate on whether radiative interiors might rotate
like solid bodies if magnetism were present. This debate began well before the discovery of thetachocline (e.g., Chandrasekhar 1956; Tassoul & Tassoul 1986; Mestel & Weiss 1987; Mestel et al.
1988). In presenting the slow confinement scenario, however, GM98 were unique in making thestrong (paraphrased) statement: “To reconcile the thin tachocline revealed by helioseismology withthe theory of radiative spread, there must be a primordial magnetic field in the radiative interior.”
GM98 derived the linear magnetic-boundary-layer equations for solar tachocline confinement,
including the proposed poloidal magnetic field in the angular-momentum equation as a Lorentz
torque. The toroidal field was assumed to arise from a balance between mean shear (the Ω-effectmentioned in Chapter 1) and magnetic diffusion. In the slow confinement theory, the amplitudeof the poloidal field set the thickness of the tachocline. From the helioseismic constraints, GM98
estimated a fairly weak field (∼1 G) would be enough to produce a tachocline width of ∼0.02R .
This result largely agreed with the earlier work of Mestel & Weiss (1987), who proposed thatdifferential rotation in the radiative interior was possible only if the magnetic field did not exceed∼0.03 G. Mestel & Weiss (1987)’s value of the field could be so small (and still induce solid-bodyrotation) because it was not required to confine the shear to a thin layer.

188
Since these boundary-layer calculations, much work has been done to assess the slow scenario under more realistic conditions. Garaud (2002) and Garaud & Garaud (2008) presented
axisymmetric (2-D), nonlinear models of an isolated radiative interior (with the convection zone’s
differential rotation applied as a top boundary condition) containing a primordial magnetic field.
Garaud (2002) found that the slow scenario was achieved only for a narrow range of poloidal field
amplitudes and Garaud & Garaud (2008) showed that prior assessments of the slow scenario (e.g.,
Brun & Zahn 2006) did not reproduce the dynamics of a slow confinement scenario likely because of
inappropriate boundary conditions applied at the top of the radiative interior. Acevedo-Arreguin
et al. (2013) presented the first global numerical simulation of a slow scenario in 2-D, including
the full radiative interior and convection zone with simplified nonlinear convective dynamics. They
found that the slow scenario could at least be qualitatively reproduced, but only for the correct
parameter space (σ  1, and also small absolute magnitudes of the diffusive coefficients). Other
global attempts to model the slow scenario in 3-D (Brun & Zahn, 2006; Strugarek et al., 2011a,b)
had likely yielded null results because of the sampled parameter regime (e.g., σ > 1), not the failureof the slow scenario.
Finally, Wood & Brummell (2018) presented the first 3-D, fully nonlinear, moderately turbulent, self-consistent simulation of tachocline confinement in a local Cartesian box. They ensured
the correct ordering of time-scales: even though the simulation was far from an astrophysical parameter regime, the Eddington-Sweet time was significantly shorter than the viscous diffusion time
(σ < 1) and radiative spread was clearly shown to occur. Wood & Brummell (2018) largely confirmed the results of Acevedo-Arreguin et al. (2013) in showing that a 3-D version of the slow
scenario was possible in the right parameter regime. Furthermore, they found that the meridional
flow could confine the primordial magnetic field to the stable layer, even amidst 3-D turbulence.
This was interesting in light of the 3-D global simulations of Brun & Zahn (2006) and Strugarek
et al. (2011a,b), in which the primordial field would not stay confined to the radiative interior.
We briefly remark that in addition to the slow confinement scenario, Forgács-Dajka &Petrovay (2001) proposed a “fast magnetic scenario.” Similar to GM98, the tachocline confine-

189
ment was imagined to occur via a magnetic torque. The requisite poloidal field, however, diffusedinward from the cycling dynamo in the convection zone above. This torque would thus respond tothe burrowing meridional circulation on the time-scale of the solar cycle, i.e., ∼22 years. As long asa large magnetic diffusivity was assumed (turbulently enhanced, presumably by MHD instabilities),
an oscillating poloidal field imposed at the top of the radiative interior could confine the tachocline
against radiative spread (for more details, see Forgács-Dajka & Petrovay 2001, 2002; Forgács-Dajka
2004; Barnabé et al. 2017).

5.4

Numerical Experiment
We consider 3-D, global, nonlinear Rayleigh simulations of a solar-like star that includes aconvection zone coupled to a radiative interior. We simulate both a hydrodynamic (HD) case anda magnetohydrodynamic (MHD) case using Rayleigh 1.0.1 (Featherstone & Hindman, 2016a;
Matsui et al., 2016; Featherstone et al., 2021).3

This shell captures most of the solar convection

zone (0.72R to ro ≡ 0.95R by radius) and part of the underlying radiative interior (ri ≡ 0.49R
to 0.72R

by radius). As in our other models, each case has a solar luminosity driven throughthe layer via internal heating in the convection zone only. Each model rotates at three times
the solar Carrington rate, 3Ω , where Ω= 2.87 × 10−6 rad s−1 . Each model has stress-free,impenetrable, and fixed-flux boundary conditions, and the MHD case has potential-field-matchingboundary conditions.
As in Chapter 4, the diffusivities ν(r), κ(r), and η(r) vary with radius like ρ(r)−1/2 . They do
not approach very low values in the radiative interior, as has usually been implemented in previous
work for the purpose of delaying tachocline spread. The thermal Prandtl number (Pr ≡ ν/κ) is
unity (as always), and in the MHD case, the magnetic Prandtl number (Prm ≡ ν/η) is 4. At the
top of the domain, ν(ro ) = κ(ro ) = 5 × 1012 cm2 s−1 and η(ro ) = 1.25 × 1012 cm2 s−1 . Except for
the addition of the stable layer and slightly higher diffusions, the HD and MHD cases are identical3

In this version of Rayleigh, the bug mentioned in Chapter 4 that affected simulations with radially varying η(r)was fixed.

190
to cases H3 and D3-4 (respectively) discussed in Chapter 4. Both our Prandtl numbers are of orderunity, in contrast to the very low Prandtl numbers in the solar interior (Table 5.1). We choose
Pr = 1 largely to remain consistent with our earlier work. We do note that lower thermal Prandtl
numbers (e.g., Pr = 1/4) in early ASH and recent Rayleigh models seem to produce more turbulent
and vortical convective structures (e.g., Brun & Toomre 2002; Miesch et al. 2006; Brown et al.
2008, 2010, 2011; Orvedahl et al. 2018; Bice & Toomre 2020) and more recently have been shownto yield “patchy” active nests of convection that significantly affect the global dynamo cycles andmay produce active longitudes (Bice & Toomre, 2022). Higher magnetic Prandtl number is more
conducive to dynamo action since large-scale vortical motions can induct strong magnetic fields
that take a long time to diffuse away. We explored a range of magnetic Prandtl numbers (Prm =1/2, 1, 2, 4, and 8) and found that all cases with Prm ≥ 2 led to magnetic tachocline confinement.
The confinement was most pronounced for Prm = 4.
We simulate a combined stable radiative interior and unstable convection zone by choosing theappropriate background entropy gradient. In the convection zone, this background gradient is set tozero, as in the models from all previous Chapters. The combined effects of internal heating near thebottom and conductive cooling near the top force convection, which establishes a slightly superadiabatic entropy gradient in the perturbations. In the radiative interior, the background entropy
gradient, which is constant in space, is strongly subadiabatic: dS/dr ≡ β = 0.01 erg g−1 K−1 cm−1 .
The “stiffness” β is thus roughly the same as the solar stiffness at the base of the convection zone.4
We match the entropy gradient from its convection-zone value (0) to its radiative-interior value (β)
over a transition width of ∼0.05R . A detailed description of the models, including the backgroundstate, is given in the supplementary Section 5.11. Also see Table 5.1 for the diffusive time-scales.
The HD case is initialized from a small thermal field (Ŝ ∼ 10 erg g−1 K−1 ) randomly distributed throughout the whole shell) and run to an equilibrated state, which we define as the time
4

The solar entropy gradient, of course, varies with radius (see Brun et al. 2011; his Figure 3) and is concavedown, which yields a resonant cavity for trapped gravity modes. One consequence of choosing constant dS/dr in our
simulations’ radiative interiors is that our gravity-mode cavities have the wrong lower turning point; i.e., the waves
reflect off the impenetrable inner boundary, instead of becoming evanescent when the buoyancy frequency becomes
low enough. Our gravity-wave spectrum for the MHD case is shown in Figure 5.12.

191
interval following the volume-integrated kinetic energy reaching statistically steady values in theradiative interior. We define “equilibrated” in this way because it takes longer to achieve a steadystate in the radiative interior than in the convection zone. The MHD case is initialized identicallyto the HD case, but with a small magnetic field (amplitude ∼1 G; randomly distributed throughout
the convection zone only) in addition to the thermal field. In the MHD case, the magnetic field
exponentially grows through dynamo action to amplitudes of ∼10 kG (in the radiative interior,the seed field first spreads downward diffusively, but is amplified there by dynamo action as well).
We define the MHD case’s equilibrated state as the time interval following the volume-integrated
magnetic energy reaching statistically steady values in the radiative interior. Note that the MHD
case (since it operates a dynamo) is not really in equilibrium per se. Rather, the magnetic field
fluctuates significantly (both in topology and amplitude) about a stationary mean state.
We run the HD case in its equilibrated state for 26.5 viscous diffusion times across theradiative interior (equivalently, 8200 rotations). This ensures that the viscosity has ample time
to imprint the differential rotation all the way through the shell. We run the MHD case for 10.6
magnetic diffusion times across the radiative interior (equivalently, 13100 rotations, or 42.4 viscous
diffusion times). This ensures that the tachocline achieved in the MHD case is in good torquebalance (magnetic torque fighting viscous torque) and furthermore, that the MHD case’s dynamo
becomes fully saturated. The MHD simulation covers ∼25 dynamo cycles, as discussed in Section
5.7. The diffusion times are defined and evaluated in Table 5.1.
To distinguish between different parts of the spherical shell, we define two critical radii,
which we mark in all applicable figures: the base of the convection zone (rbcz = 0.73R ; solid black
curves in the figures) and the base of the overshoot layer (rov = 0.71 R ; dashed green curves in
the figures). Following Brun et al. (2011), we define the convection zone as the region in which theconvective heat flux (enthalpy flux; Fenth ) is positive (i.e., outward). We thus define the two criticalradii such that below rbcz , Fenth becomes negative (i.e., inward) and below rov , Fenth approaches
zero. Specifically, the base of the overshoot layer is defined to be where Fenth becomes 5% of its
minimum value achieved in the overshoot layer; this also corresponds to the location where the192vertical convective velocities drop by ∼102 . We call r > rbcz the “convection zone,” r < rov the“radiative interior,” and rov < r < rbcz the “overshoot layer.” Under the same definitions, the HDcase’s overshoot layer is slightly deeper than the MHD case’s, extending from 0.70R

to 0.73R ;the vertical convective velocities drop by ∼103 in the HD case’s overshoot layer.

5.5

Solid-Body Rotation in the MHD Case’s Radiative Interior
Figures 5.1(a,b) show the rotation-frequency profiles for the HD and MHD cases (respec-tively), plotted in the (r, θ)-plane. The rotation frequency has been averaged in longitude and intime over each case’s equilibrated state. In the HD case, the differential rotation in the convection
zone has diffusively imprinted onto the entire radiative interior, as should be expected from priorwork in the high-σ regime (e.g., Miesch 2005; Garaud & Acevedo-Arreguin 2009; Brun et al. 2011;
Acevedo-Arreguin et al. 2013; Brun et al. 2017; Guerrero et al. 2019).
In the MHD case, however, the radiative interior sustains near-solid-body rotation, while theconvection zone rotates differentially. We call the shear layer between the convection zone andradiative interior in the MHD case a “tachocline.” However, we emphasize that in the convection
zone, the differential rotation in our MHD case differs from the solar differential rotation (and fromthe differential rotation in the HD case, which is quite solar-like) in several ways. First, the rotation
contrast is too small by a factor of ∼5 compared to the observed solar value. Second, most of the
MHD case’s differential rotation is at low latitudes, unlike the Sun, which has strong shear at highlatitudes as well. Third, the MHD case has a noticeably banded average rotation profile, which is
non-monotonic in latitude.5

Overall, the tachocline achieved in our MHD case is quite similar

to that of the EULAG-MHD “millennium simulation” (Passos & Charbonneau, 2014; Lawson et al.,2015) and that of the EULAG-MHD “case RC01” (Guerrero et al., 2016a, 2019). This Chapter is
5

The Sun technically has a banded differential rotation during any given sunspot cycle due to the torsional
oscillations. This effect comes in the form of a ∼3–4 nHz “fast band” in each hemisphere that migrates equatorwardwith the solar cycle (e.g., Howe 2009). Relative to the solar differential rotation contrast (∼80 nHz between equatorand 60◦ ), this is a ∼5% effect, which is actually the same order of magnitude as in our MHD case (from Figure 5.1(c),we can estimate a fast band of ∼5 nHz, which is a ∼10% effect compared to the differential rotation contrast). Our
MHD case is still different than the Sun, however, because the temporally averaged differential rotation is banded,
whereas the Sun’s is latitudinally monotonic.

193
focused on understanding the mechanism by which solid-body rotation is maintained in our MHD
case and confined to the radiative interior. We leave the full tachocline confinement problem (i.e.,
how solid-body rotation could be maintained alongside a solar-like differential rotation profile inthe convection zone) to future work.
Figure 5.1(c) shows the rotation profile in the MHD case as a function of radius for different
latitudes. To quantify the thickness of the tachocline, we define the radially varying latitudinalrotation contrast ∆Ω(r) as the difference in rotation rate between the equator and 60◦ latitude at the
fixed radius r. In the HD case, ∆Ω(r)/Ω0 ∼ 0.2 throughout the whole shell (similar in magnitude to
the solar value). By contrast, ∆Ω(r) for the MHD case is drastically lower in the radiative interior
than in the convection zone. Explicitly, the MHD case has ∆Ω(r)/Ω0 = 4 × 10−2 ≡ ∆ΩCZ /Ω0
at the top of the convection zone and ∆Ω(r)/Ω0 = 7 × 10−4 ≡ ∆ΩRI /Ω0 at the bottom of theradiative interior. In describing the MHD case, we arbitrarily define the base of the tachocline(rtach = 0.64 R ; dotted red curves in the figures) as the location where ∆Ω(r) has dropped by afactor of 20 from its value at the top of the convection zone. We define the top of the tachocline asrov , so the tachocline occupies (rtach , rov ) ≡ (0.64R , 0.71R ) by radius.
The convective-velocity and magnetic-field amplitudes for the equilibrated MHD case areshown in Figure 5.2. Below the overshoot layer, the vertical components of v 0 and B 0 are dwarfed
by the horizontal components. The plunge in vertical velocity (Figure 5.2(a)) is a consequence ofthe stable stratification. Specifically, vr0 falls by ∼2 orders of magnitude across the overshoot layer
and by ∼5 orders of magnitude across the whole radiative interior. Because of the anelastic approximation, this means that the flows are not only horizontal, but almost completely divergenceless(i.e., incompressible). The convective horizontal velocity field is thus determined almost entirelyby the convective radial vorticity ωr0 , which acts essentially like a streamfunction for the horizontal
flow. As was discussed in Sections 5.1–5.3, such purely vortical flows have been expected fromearly theoretical considerations (e.g., Charbonneau et al. 1999b; Garaud 2001; Dikpati & Gilman
2001a,b) and have likely been present in global numerical simulations as well (e.g., Brun et al. 2011;
Lawson et al. 2015; Guerrero et al. 2016a; Bice & Toomre 2020).

194

Figure 5.1: A Simulated Radiative Interior Magnetically Forced into Solid-Body Rotation. (a) Level surfaces (contours) of (Ω − Ω0 )/2π in the HD case, plotted in the meridional plane.
Negative values are normalized separately from positive values, so that there are three equally
spaced positive and negative contours each (solid contours). The zero contour Ω = Ω0 is dashed.
(b) Like (a), but for the MHD case. (c) Rotation frequency Ω/2π at different latitudes along radiallines in the MHD case. The rotation contrasts in the convection zone and radiative interior aremarked by vertical arrows; the frame rotation frequency Ω0 /2π is marked by the horizontal dashedline. In all panels, the solid black, dashed green, and dotted red curves (vertical lines in panel (c))
refer to rbcz , rov , and rtach , respectively.

195(b) magnetic field amplitudes

101
10 1
vr0
v0
v0

10 3
10 5

0.5

0.6

0.7
r/R

0.8

0.9

Magnetic field (kG)

Convective velocity (m s 1)(a) velocity amplitudes

100

10 1

Br0
B0
B0
0.5

0.6

0.7
r/R

0.8

0.9

Figure 5.2: Flow and Field Strengths in the MHD case. (a) Rms of the convective velocity(full velocity with the longitudinal mean subtracted). The mean is taken in time over the equilibrated state and over spheres of radius r. Each component (r, θ, and φ) is plotted separately, as
shown in the legend. (b) Same as (a), but for the convective magnetic field. In both panels, thelocations rbcz , rov , and rtach , are marked by vertical lines as in Figure 5.1.

196
From Figure 5.2(b), the vertical component of the magnetic field Br0 also declines in theradiative interior, but more gradually than vr0 . It falls by a factor of ∼50 from mid-convection zone(∼0.84R ) to the base of the overshoot layer. In the very deep radiative interior, though not quite
in the overshoot layer or tachocline, Br0 can be neglected and the magnetic field is well characterized
by its horizontal components. Figure 5.3 shows this horizontal field at the base of the overshootlayer at an instant during the equilibrated state (specifically, the very end of the simulation run,t = 16050 rotations). We note that both field components are largely non-axisymmetric, mainlycomposed of spherical-harmonic orders m = 1, 2. At the particular instant sampled by Figure 5.3,m = 1 dominates. After the explorations of Chapter 4, we call these structures partial wreaths,
although we note that here m = 2 structures are present in addition to m = 1.
In general, Bφ is slightly stronger than Bθ and more concentrated near the equator. Thereis clearly some correlation between the two horizontal field components within the partial wreaths.
As shown in Figure 5.3(c), the product Bθ Bφ is mostly positive in the North and negative in theSouth. This correlation (as discussed in the next Section 5.6) indicates that (1) the toroidal field Bφ
is generated primarily from the poloidal field Bθ via mean shear and (2) the resulting Lorentz force
from bent poloidal field lines sends angular momentum poleward (which shows up in Figure 5.3
as the preferred sign structure of the product Bθ Bφ ). The poleward angular-momentum transport
tends to eliminate the latitudinal shear.

5.6

Dynamical Maintenance of the MHD Case’s Tachocline
As mentioned in Section 5.4, the solid-body rotation in the MHD case’s radiative interior has

been sustained for over 10 magnetic diffusion times (40 viscous diffusion times) across the radiativeinterior. This means that the system, on average, must be in dynamical torque balance, with zeronet torque on the system. We note that the torque balance (and the inductive balance we discussnext) only applies to the system over a long temporal average that encompasses multiple dynamo
cycles. As shown in the following Section 5.7, this system is running a cycling dynamo with periodof a few hundred rotations. The magnetic fields and differential rotation thus change slowly in time;

197

Figure 5.3: Non-Axisymmetric Partial Wreaths Forcing Solid-Body Rotation. Snapshots
during the MHD case’s equilibrated state of (a) Bφ , (b) Bθ , and (c) their product, Bφ Bθ . Eachsnapshot is plotted in full Mollweide view on a spherical surface at the base of the overshoot layer,
rov /R = 0.71. The snapshot is taken at the end of the simulation run, namely, t = 16050 rotations.

198
during a particular dynamo cycle, there may be a net torque, but the average torque approaches
zero after several dynamo cycles.
In the MHD case’s tachocline, the torques are primarily viscous and magnetic:1
τv + τmag ≡ ∇ · [ρνr2 sin2 θ∇Ω] + ∇ · [r sin θhBφ Bpol i] ≈ 0,
|
{z
} |4π
{z
}
viscous torque (τv )(5.3)

magnetic torque (τmag )where Bpol ≡ Br êr + Bθ êθ is the poloidal magnetic field and the form of each torque is derived in
Miesch & Hindman (2011). Also recall Chapter 2, Equation (2.4). The angular brackets denote a
combined longitudinal and temporal average (this combined average is also implied by Ω, which isdefined as hvφ i /r sin θ + Ω0 ). In the Rayleigh code, longitudinal averages are calculated by taking
the full 3-D snapshot of the flows in the spherical shell and averaging over all longitudinal collocation
points (or equivalently, by taking the m = 0 component of the discrete Fourier transform of the fieldin φ). In general, we output these longitudinal-average snapshots every ∼1 rotation (although thesampling is non-uniform due to the variable time-step). The temporal averages are then computedby averaging together the ten thousand or so longitudinal-average snapshots. The system is run
in its “equilibrated state” for ∼25 dynamo cycles; as more and more cycles are included in the
average, the total torque approaches zero.
We note that the magnetic torque can be interpreted as the tension force associated thepoloidal magnetic field lines bent in the toroidal direction. Since we employ this result frequently
in the following discussion, we derive it here. The magnetic torque is obtained simply by multiplying
the toroidal Lorentz force (longitudinally and temporally averaged) by the moment arm r sin θ:τmag =

r sin θh[ (∇ × B) × B) ]φ i
|
{z
}
4π

(5.4a)

r sin θh(B∇B})φ i| ·{z4π(5.4b)

1hBpol · ∇(r sin θBφ )i
4π |
{z
}

(5.4c)

1∇ · [r sin θhBφ Bpol i]
|4π
{z
}

(5.4d)MHD Lorentz force=

toroidal magnetic tension force=

tension torque from bent poloidal field lines=

magnetic torque from Equation (5.3)

199
In deriving these expressions, we have used (in order) the identity (∇×B)×B = B·∇B−(1/2)∇B 2 ,the expression for B · ∇B in spherical coordinates, and the solenoidal field condition ∇ · B = 0.
Throughout these manipulations, we have dropped terms that do not survive a longitudinal average,
like (∇B 2 )φ , which is a perfect derivative with respect to φ. The magnetic torque is thus interpreted
as either the torque from the magnetic-tension force, h(B · ∇B)φ i, or equivalently (as derived
in Miesch & Hindman 2011 and presented in Equation (5.3)), as the divergence of the angularmomentum flux, (r sin θ/4π)hBφ Bpol i.
The torque balance in the MHD case is shown in Figure 5.4. See Equation (2.4) for thedefinitions of the three non-magnetic torques (in our MHD case, the meridional-circulation torqueplays very little role all throughout the shell). This balance is dominated by the torques from the
Reynolds stress (defined and discussed at length in Chapter 2), magnetic tension, and viscosity,
which we show in the meridional plane in Figures 5.4(a,b,c), respectively. In the convection zone,
the torque balance is similar to previous HD simulations (e.g., Hotta et al. 2015; Matilsky et al.
2019; see Chapters 2 and 3). The Reynolds stress maintains a solar-like differential rotation outside
the tangent cylinder (cylinder circumscribing the radiative interior) by spinning up the fluid far
from the rotation axis and slowing down the fluid close to the rotation axis. The convection
zone’s viscous and magnetic torques both work to counter the Reynolds stress. Magnetic torque insimulated convection zones quenching the differential rotation is a common occurrence and why it
happens is still an open question (e.g., Browning 2008; Yadav et al. 2015; Lawson et al. 2015; Yadav
et al. 2016; Guerrero et al. 2016a; Bice & Toomre 2020, 2022). As was discussed in Chapter 4, thedifferential-rotation quenching is most pronounced at higher magnetic Prandtl number, which may
be due to the growing dominance of small-scale magnetism over convection (Bice & Toomre, 2022).
In the tachocline (Figure 5.4(d)), viscosity works to imprint the differential rotation fromabove by spinning up the equator and slowing down the mid-latitude regions. However, the imprinting is almost completely halted by the magnetic torque. The same viscous imprinting occurs
in the HD case, but because there is no other substantial torque on the system, the differential
rotation in the convection zone successfully drags the radiative interior along with it (Figure 5.1a).

200
We have separately confirmed that the magnetic torque in the MHD case’s tachocline is primarily
due to the correlation between horizontal field components Bφ Bθ (see Figure 5.3), with the othercorrelation Bφ Br playing a lesser role.
As discussed in Section 5.3, magnetic confinement scenarios for the solar tachocline follow
arguments similar to those invoked in Ferraro’s law of isorotation. Here we illustrate how Ferraro’s
law extends to our MHD case, which is dominated by non-axisymmetric partial wreaths. We closely
follow the arguments laid out in Mestel & Weiss (1987). Differential rotation (nonzero ∇Ω) will
bend poloidal magnetic field lines to produce a toroidal field through mean shear. This is expressed
through the toroidal component of the magnetic induction equation, which, if mean shear dominates
the other induction terms, takes the form∂Bφ
≈ r sin θBpol · ∇Ω − {∇ × [η(r)∇ × B]}φ .
∂t
|
{z
} |
{z
}
mean shear(5.5)

toroidal diffusion

See Section 5.11.2, Equation (5.15a) for a full derivation. Here we define the mean-shear term(the Ω-effect) as the axisymmetric differential rotation bending the non-axisymmetric poloidal
magnetic field. Equation (5.5) yields Bφ ≈ (r sin θBpol · ∇Ω)tbend (we ignore the diffusion in thisillustration), where tbend is the time-scale over which the bending occurs. From Equation (5.3),
2 ∇the associated torque due to magnetic tension is ∇ · [r2 sin2 θhêpol Bpolpol Ωi]tbend , which tends toeliminate gradients in Ω parallel to Bpol . Here, êpol ≡ Bpol /|Bpol | is the unit vector in the directionof Bpol and ∇pol ≡ êpol · ∇ is the derivative in the direction of Bpol . When magnetic tension
is dominant, the equilibrium state will require hêpol ∇pol Ωi = 0. For an axisymmetric poloidal
magnetic field, this reduces to hBpol i · ∇Ω = 0, which is the original Ferraro’s law. However, a
non-axisymmetric field is more restrictive, since the direction êpol changes with longitude. Forêpol ∇pol Ω to average to zero, the most likely solution is ∇pol Ω = 0 everywhere—i.e., solid-bodyrotation.6
Figure 5.5(a) shows the production toroidal magnetic energy in the MHD case’s radiative
6

More specifically, the bending time-scale tbend is the time it takes an Alfvén wave—with speed vA determined by2
the poloidal magnetic field, i.e., vA= |Bpol |2 /4πρ—to traverse a distance long enough that Ω changes substantially.
Mestel & Weiss (1987) showed that when these Alfvén waves dissipate, gradients in Ω will ultimately be eliminated(in the absence of non-magnetic forces) if the field is non-axisymmetric.

Figure 5.4: Magnetic Torque Maintaining the MHD Case’s Tachocline. (a)–(c) The dominant torque balance (from left to right:
magnetic, viscous, and Reynolds stress) in the full meridional plane for the MHD case. Two separate color tables are used for the convection
zone (top colorbar) and radiative interior (bottom colorbar). The torque densities are plotted in the cgs units of pressure (dyn cm−2 ). (d)Latitudinal profiles of the torque balance (also including the meridional circulation), averaged in radius over the tachocline.

201

202
interior, i.e., Equation (5.5)—strictly, the full Equation (5.15a)—multiplied by Bφ /4π. The resulting magnetic-energy production equation is then averaged over spheres and in time. Similar
to the torque Equation (5.3), Rayleigh outputs spherical-average snapshots of the simulation by
averaging the full 3-D profiles over longitude and colatitude. These snapshots are output at thesame ∼1-rotation cadence as the longitude-average snapshots (but non-uniformly sampled due tothe varying time-step), and then averaged over the equilibrated state to form the temporal average.
Similar to the torque balance, the magnetic-energy balance can be nonzero for a given dynamo
cycle, but becomes zero when averaged over many dynamo cycles.
Figure 5.5(a) shows that, on average, the toroidal magnetic field is produced predominantly
by mean shear and is destroyed by magnetic diffusion. Because Br is so weak in the radiativeinterior (Figure 5.2), it is the horizontal shear that dominates the production of toroidal field.
This shows that Ferraro’s law (as applied to non-axisymmetric fields after Mestel & Weiss 1987) is
responsible for the torque forcing solid-body rotation. The generation of Bφ by mean shear actingon Bθ also coheres with the correlation between Bθ and Bφ pointed out in Figure 5.3.
Toroidal field produced by the Ω-effect is a common feature of most global dynamo simulations(e.g., Brown et al. 2010, 2011; Lawson et al. 2015; Matilsky & Toomre 2020a; Bice & Toomre 2020;
Brun et al. 2022) and is the central torque-producing mechanism invoked by the fast and slow
magnetic tachocline confinement scenarios. The ultimate question for our work is thus, “How doesa poloidal magnetic field permeate the MHD case’s radiative interior and maintain large amplitudes
for over 10 magnetic diffusion times?” In the slow magnetic confinement scenario (GM98), theinterior poloidal magnetic field is a primordial remnant. It is supposedly prevented from diffusingaway (the Sun has already made it through ∼1 magnetic diffusion time; see Table 5.1) by the verticalcomponent of the meridional circulation. In the fast magnetic confinement scenario (Forgács-Dajka& Petrovay, 2001), the poloidal field is cyclically pumped downward from the convection zone above
via turbulent diffusion. There is clearly vigorous dynamo action in the MHD case’s convection zone;
fields of amplitude ∼10 kG are amplified and then sustained for many magnetic diffusion times.
Additionally, our sub-grid-scale magnetic diffusivity is quite high at all radii. We thus might suspect203that the radiative interior’s magnetic field is simply diffusing down from the convection zone above,
as in the fast confinement scenario. This is not the case, however, as we now show.
Since the magnetic field is primarily horizontal, the production of poloidal magnetic energy
is dominated by the θ-component of the MHD induction equation,
∂Bθ
≈ −Bθ
∂t
|




v 
1 ∂vφ cot θθ− ∇ × [η(r)∇ × B]θ ,+vθ + r(Br êr + Bφ êφ ) · ∇{z}r sin θ ∂φ
r
r } |
{z
{z
} |
θ diffusion
θ compression(5.6)

θ shearmultiplied by Bθ /4π. See Section 5.11.2, Equation (5.15b) for a full derivation. This production,
temporally and spherically averaged, is shown in Figure 5.5(b). Clearly the poloidal magnetic field
is not pumped in via diffusion, but instead is amplified inductively (and destroyed diffusively) at
all radii, just like the toroidal field. The induction comes from two mechanisms: compression (thelongitudinal squeezing of field lines) and fluctuating shear (the tilting of radial and longitudinal
field into colatitudinal field by the convective—non-circulatory—meridional flow). Note that in
the tachocline, the compression and shear contribute roughly equally to poloidal magnetic energy
production, but below the tachocline, production is almost exclusively by compression.
The results encapsulated in Figure 5.5(b) may prove to be profound. They imply that strong
magnetism could be maintained by dynamo action, even below the convective overshoot layer.
This picture is quite different from the mechanisms usually invoked to explain how magnetism
might enter the radiative interior—e.g., magnetic “pumping,” magnetic diffusion, and primordial
remnants. Furthermore, in the interface-dynamo paradigm, it is often argued that the radiativeinterior basically acts as an “abyssal deep” (to borrow a term from oceanography)—a passive storage
reservoir for magnetic field produced in the tachocline. The “quiescent” stable layer accepts field
pumped in from above, until the magnetism grows strong enough to form buoyant loops thatthen rise through the convection zone to form sunspots (e.g., Spruit & van Ballegooijen 1982; vanBallegooijen 1982; Moreno-Insertis et al. 1992; Parker 1993; Ferriz-Mas & Schuessler 1994). If ourresults extend to the Sun, the solar radiative interior may be a site of induction that contributes
significantly to driving the global dynamo.

Magnetic energy production (erg s 1 cm 3)

204

10 1
10 2
10 3
10 4
10 5
0
10 5
10 4
10 3
10 2
10 1
100
101

(a) toroidal production(b) poloidal production

linear scale

0.50

toroidal induction
toroidal diffusion
mean shear
0.55 0.60 0.65
r/R

poloidal induction
poloidal diffusion
shear
compression
0.70 0.50

0.55

0.60
r/R

0.65

0.70

Figure 5.5: Dynamo Action in the MHD Case’s Radiative Interior. The steady-state
productions of (a) toroidal magnetic energy and (b) poloidal magnetic energy are plotted as functions of radius below the overshoot layer. The different production rates are defined explicitly
in the supplementary Section 5.11 in Equations (5.15a) and (5.15b). These equations have been
averaged in time over the equilibrated state and over spheres of radius r. Each panel shares thesame x- and y-axes. The y-axis is scaled logarithmically (unshaded region) for absolute values> 3 × 10−6 erg s−1 cm−3 and linearly (shaded region) otherwise. In both panels, the locations rov
and rtach are marked by vertical lines, as in previous Figures.

205

5.7

Tachocline Confinement by Cycling Partial Wreaths
The tachocline in our MHD case is clearly confined by the magnetic torque (Figure 5.4).

We have further shown that this torque is generated by the bending of poloidal field lines by
the mean shear (which, although very weak, is nonzero in the radiative interior). This process iswell represented by the typical partial-wreath structure shown in Figure 5.3. Here, we investigate
how the partial wreaths vary throughout the shell, and how they cycle in time. We show thatin addition to the stationary component, the cycling components of the wreaths give rise to theconfining torque.
Figure 5.6 shows the full morphology of the horizontal magnetic fields and fluid flows throughout the shell, at the end of the simulation (same instant sampled by Figure 5.3). The magnetic
field in the radiative interior (top row) is quite smooth and mostly consists of an m = 1 partial
wreath in the North and an m = 2 partial wreath in the South. The horizontal flows, represented
by the convective radial vorticity ωr0 , are at smaller scales (m ∼ 10) and have less clear symmetry
about the equator. Higher up in the shell (bottom two rows) the magnetic fields develop morefine structure, but the imprint of the large-scale partial wreaths remains. The horizontal flows
also become finer-scale in the convection zone, with no obvious imprint from the motions in theradiative interior.
Figure 5.6 suggests that the radiative interior and convection zone have essentially distinct
flow and field properties, except for the large-scale (m = 1, 2) magnetic fields. To investigate further,
we show the time-radius diagrams of the longitudinally averaged (m = 0) horizontal magnetic fields
in Figure 5.7. Although our MHD case’s magnetic field is dominated by the non-axisymmetriccomponents, we speculate that we can use the longitudinal average hBφ i (this time with the angularbrackets not implying a temporal average as well) to trace the dynamo cycle and polarity reversals,
as we did in Chapter 4. Figure 5.7 shows that the partial wreaths in the overshoot layer lead thesystem, and the fields in the deep radiative interior follow. Any pattern realized in the overshootlayer appears near the base of the radiative interior ∼250 rotations later, which also seems to be

Figure 5.6: Full Morphology of the MHD Case’s Dynamo. Full spherical Mollweide snapshot views of the MHD case’s dynamo.
The snapshots are taken at the end of the simulation run (t = 16050 rotations; same as Figure 5.3). Different fluid variables are shown ineach column. From left to right: toroidal magnetic field Bφ , poloidal magnetic field Bθ , and convective horizontal flows (convective radial
vorticity ωr0 ). Different depths are shown in each row. From top to bottom: just below the tachocline (r/R = 0.63), the base of theovershoot layer (rov /R = 0.71), and mid-convection-zone (r/R = 0.84).

206

Figure 5.7: Time-Radius Diagram of the Confining Dynamo. The longitudinally averaged magnetic field is shown as a functionof time and radius for (a) hBφ i and (b) hBθ i. The three critical radii are marked by their usual lines. The field is sampled over the last∼4000 rotations at a latitude of 10 degrees in the Northern Hemisphere.
207

208

Figure 5.8: Confining Magnetism as Partial Wreaths. The full non-axisymmetric toroidal
magnetic field Bφ is shown as “partial wreaths,” similar to the structures encountered in Chapter 4.
The time-longitude plots (longitude on the x-axis, time running vertically downward on the y-axis)
are shown in (a) the rotating-frame frequency Ω0 /2π and (b) a frame rotating slightly slower, atthe background rotation rate, ΩRI /2π ≡ Ω0 /2π − 2.3 nHz. The field is sampled over the whole
equilibrated state at r = rov at a latitude of 10 degrees in the Northern Hemisphere.

209
about the time for a polarity reversal of the partial-wreath pair at a fixed radius. As in Chapter
4, we define the partial-wreath polarity via the longitudinal average, i.e., the sign of hBφ i or hBθ i.
This yields a rough dynamo cycle period of ∼500 rotations, though we note that the reversals are
not regular. The magnetic diffusion time across the radiative interior is substantially longer, ∼1300
rotations, suggesting once again that the magnetic field does not spread inward diffusively. In theconvection zone, there is possibly some radial propagation of the fields upward from the overshootlayer, although this is less clear. The polarity reversals are more rapid in the convection zone,
occurring every ∼100 rotations, for a dynamo cycle of ∼200 rotations.
Following the methods of Section 4.6, we examine the toroidal field in time-longitude space(Figure 5.8), choosing a radial level of rov and a latitude of 10 degrees, which from Figures 5.2 and5.3 is also approximately the region of strongest magnetic-field amplitude. If we rotate with theframe (Figure 5.8(a)), the non-axisymmetric field structure is seen to travel a full 360◦ backward
in longitude over the previously identified cycle period of ∼500 rotations. This suggests thatthe field structure rotates slower than the frame rate, likely at the radiative interior’s rotation
rate ΩRI . If we co-rotate with the radiative interior at frequency ΩRI /2π ≡ Ω0 /2π − 2.3 nHz(Figure 5.8(b)), the toroidal field is approximately stationary, staying roughly localized at the samelongitudes.7

The polarity reversal is the flip-flop back and forth between the dominant partial

wreath. Note, however, that even co-rotating with the radiative interior, a given partial wreath
zigzags in longitude, sometimes moving ∼2.5 nHz faster or slower than the background rotation
frequency. This zigzag effect was also mentioned in Chapter 4 (Matilsky & Toomre, 2020a).
Partial-wreath cycles are fairly difficult to visualize, and so far we have described them largely
from a phenomenological perspective. To be more precise, we now analyze the MHD case’s m = 1, 2
magnetic-field components in frequency space. We Fourier-transform each field component andtheir derivatives in both longitude and time. We consider an equally spaced time series of length
T , yielding frequencies fn ≡ n/T equally spaced by the frequency resolution, 1/T (the number
7

Note that we define ΩRI arbitrarily to force the time-longitude behavior of the partial wreaths to appear as
stationary as possible. This is justified by the field structures largely being advected by the background flow; from
Figure 5.1(c), ΩRI chosen in this way is clearly close to the background rotation rate of the radiative interior.

210
of frequencies Nf equals the number of samples Nt , and n runs from −Nf /2 to Nf /2 + 1). In
our convention, positive frequency indicates a retrograde-traveling wave. We denote the doublytransformed quantities with tildes and normalize them so that Parseval’s theorem takes the form
(e.g., for the toroidal field Bφ ):
Bφ2 =

XX
m

fφ ∗ (m, fn )B
fφ (m, fn ),
B

(5.7)

n

where the angular brackets here denote a combined longitudinal and temporal average, and thefφ (m, fn ))asterisk denotes the complex conjugate. This makes the spectral coefficients (e.g., B
represent the “amount” of field (e.g., Bφ ) present at a given wavenumber m and frequency fn .
D E1/2.
When added in quadrature, these amounts give the total magnetic-field amplitude Bφ2
Figure 5.9 shows distribution of the horizontal-field components in latitude-frequency spacefor m = 1. Clearly the fields are localized near a central frequency of ∼2.5 nHz (consistent,
within our frequency resolution, with the (Ω0 − ΩRI )/2π = 2.3 nHz identified from the timelongitude diagram), but have substantial components in a surrounding interval ∼5 nHz wide. Thisis basically consistent with the zigzag pattern in Figure 5.8). The fields are primarily at positivefrequencies; thus, the whole structure moves primarily retrograde in the rotating frame. The
dispersion in frequency indicates that the partial wreaths are not stationary, however, but change
their longitudinal position even in their “home” frame.
Because the expressions for magnetic torque in Equation (5.4) are quadratic in the magnetic
field, each wavenumber-frequency pair is associated with a contribution to the torque. These
contributions can be defined (using, e.g., Expression (5.4c)) as:τmag (m, fn ) ≡
with

XX
m

h
i
1 g
fφ (m, fn )[Bpol (m, fn )]∗ · ∇ r sin θB
4π

(5.8)

τmag (m, fn ) = τmag .

nBecause the magnetic field is a real quantity, each “mixed” product τmag (m, fn ) is purely real.
In Figure 5.10, we plot the torque contributions as a function of latitude and frequency atthe base of the overshoot layer. As in Figure 5.4(d), the total torque at this depth (Figure 5.10(a))

211

Figure 5.9: Cycling Frequencies of the Partial Wreaths. The absolute values of the m = 1
spectral coefficients for the horizontal magnetic fields are shown at the base of the overshoot layer
fφ | and (b) |B
fθ |. The coefficient values are plotted in gray-scale with respect to latitude
for (a) |B
and frequency. For this time series, the sampling interval is T = 1880 rotations, the frequency
resolution is ∼0.7 nHz, and the Nyquist frequency is 5000 nHz. The interval starts and ends at
t = 4420 rotations and t = 6300 rotations, respectively.

Figure 5.10: Magnetic Torque from Each Cycling Frequency. The magnetic torque from thetwo wavenumbers m = 1, 2 is shown as a function of latitude at the base of the overshoot layer.
The time series is the same as in Figure 5.9. (a) The torque contributions, defined in Equation(5.8), summed over m = 1, 2 and all frequencies, along with the torque directly computed from thelongitudinal-average data (or equivalently, the mode contributions summed over all m). Compare
to Figure 5.4. (b) The m = 1, 2 torque contributions, each frequency considered separately as afunction of latitude.

212
is concentrated at low latitudes, spinning the equator down and the higher-latitude regions up. We
sum over m = 1, 2 only, and these two modes contain nearly all the torque. Broken up by frequency(Figure 5.10(b)), the components of magnetism producing the torque are seen to roughly follow
the distribution of the toroidal field (Figure 5.9(a)). It is thus the cycling partial-wreath dynamothat produces the magnetic torque confining the tachocline against viscosity. Furthermore, each
latitudinal strip at a fixed frequency in Figure 5.10(b) roughly matches the full latitudinal profile
of Figure 5.10(a). This simply emphasizes the fact that the bending of poloidal field lines by theshear (and subsequent tension force tending to eliminate the shear) is a local process. Howevercomplicated the m = 1, 2 magnetic field structures behave in space and time, each part of thestructure (which may cycle at its own distinct frequency), produces torque to counter the viscously
imposed differential rotation.
The results of this Section offer a new possibility for magnetic confinement scenarios. Gough& McIntyre (1998) imagined the confining poloidal magnetic field to be an axisymmetric andstationary primordial remnant, entirely separate from the convection zone above. Forgács-Dajka& Petrovay (2001) imagined the poloidal field to be axisymmetric and cycling, brought in from theconvection zone via diffusion (and not allowed to react back on the convection zone), and considered
only one frequency component corresponding to a period of 22 years. Here we have identified a
new confining structure, which is non-axisymmetric (composed mostly of m = 1, 2 partial wreaths),
cycles at multiple frequencies, and is fully coupled to the convection zone above.

5.8

Inertial Modes and Dynamo Action in the Radiative Interior
Figure 5.5 shows that both horizontal components of the magnetic field are produced induc-tively in the radiative interior by horizontal motions. Here, we clearly show that the frequencies ofthese motions occur primarily in the “inertial band”—i.e., the frequency interval ±2Ω0 /2π = ±2700
nHz. We thus loosely refer to the horizontal motions as inertial oscillations and describe them using the terminology presented in, e.g., Gizon et al. (2020b, 2021) and Zaqarashvili et al. (2021).
We clearly identify the higher-frequency modes as classical equatorially confined (or “equatorial”)

213
Rossby waves, recently identified at the solar surface by Löptien et al. (2018). There are also
low-frequency oscillations that we argue represent MHD Rossby waves, possibly in critical layers(bands in latitude bounded by “critical points,” where the Rossby-mode frequency equals the background rotation frequency). The horizontal power is dominated by the equatorial Rossby waves,
but the low-frequency modes, whose energy is in equipartition with the poloidal magnetic-fieldenergy, appear to cause the induction of poloidal field inferred from Figure 5.5(b).
Strong horizontal motions in the radiative interior are expected from the many investigationsof shear and magnetic instabilities (Section 5.2). They have also been encountered frequently
in prior global simulations. For example, in the middle of the MHD case’s radiative interior,the horizontal velocity amplitude (on the order of ∼1–10 m s−1 ; see Figure 5.2) is similar to theamplitudes reported in work utilizing the ASH code (e.g., Alvan et al. 2015), as well as the EULAG-MHD
code (e.g., Lawson et al. 2015). They have likely been present in other global simulations as
well, though not explicitly reported. Racine et al. (2011), Augustson et al. (2013), and Guerrero
et al. (2016a), for example, all describe sizable α-effects, with α-tensor components ∼1–10 m s−1 ,operating in their radiative interiors. Large α-effects would, in turn, require strong horizontal
motions.8
As mentioned in Section 5.2, Alvan et al. (2014, 2015) investigated in detail the rich gravitywave-field associated with the vertical motions in global simulations. The character and sourceof the horizontal motions, however, has yet to be systematically explored. Lawson et al. (2015)
claimed that the horizontal motions were partly due to magneto-shear and Tayler instabilities, but
this was not explicitly shown. The uncertainty as to where the horizontal velocities come fromhighlights the difficulty in identifying wave and instability properties of turbulent 3-D simulations.
As discussed in Section 5.5, the horizontal motions are nearly incompressible and are essentially described by the convective radial vorticity ωr0 ≡ (∇ × v 0 )r . The longitudinally averaged
8

According to classical mean-field analysis, toroidal field produces poloidal field through an emf according to∂ hBpol i /∂t = ∇ × hv 0 × B 0 i, with hv 0 × B 0 i ≈ α · hBi, where α is a second-rank tensor related to the kinetic
and/or current helicities of the flow (e.g., Brandenburg & Subramanian 2005 and references therein). The α-tensor
values reported in global simulations can thus be interpreted roughly as convective velocities, which must be purely
horizontal in the radiative interior because of the stable stratification.

214

Figure 5.11: Rossby Waves and Other Low-Frequency Inertial Oscillations. The dispersion
relation of the horizontal motions is shown deep in the radiative interior (r/R = 0.55) for them = 1 power in the inertial band. The full inertial band is ±2Ω0 /2π = ±2700 nHz, but wehere plot a smaller sub-interval of frequencies that contains most of the power. We plot |fωr (r =
2
0.55R , `, m = 1, fn )| on a log-10 scale (indicated by the colorbar) as a function of ` and fn for(a) the full inertial-mode spectrum and (b) the low-frequency region. In both panels, we denote
the results from the classical dispersion relation (5.10), f`1 = 2Ω0 /[2π`(` + 1)], by black dots. We
note that although the radiative interior rotates at a rate ΩRI , which is slightly slower than Ω0 , thisdifference would be negligible on the scale of the above plots. In this case, the sampling interval is1650 rotations, the frequency resolution is 0.8 nHz, and the Nyquist frequency is 5000 nHz. Figure
credit: Catherine Blume.

215
component hωr i is nonzero, but simply represents the differential rotation, which cannot induct
poloidal magnetic field. An equatorial Rossby mode in a sphere rigidly rotating with frequencyΩRI /2π is described by its spherical-harmonic degree `, azimuthal order m 6= 0, and radial eigenfunction A`m (r). The horizontal eigenfunction of the mode is the spherical harmonic Y`m (θ, φ).
The full mode is a global oscillation in the radial vorticity of the form:
ωr;`m (r, θ, φ, t) = A`m (r)Y`m (θ, φ) exp (−i2πf`m t),(5.9)

whose frequency follows the dispersion relation:
f`m =

2mΩRI
.
2π`(` + 1)

(5.10)

Here, f`m is the mode frequency, t is the time, and the real part of ωr;`m gives the convective radial
vorticity due to the mode. Rossby waves in the Earth’s atmosphere and oceans were describedin Rossby (1939) and Rossby (1945), and extended to rotating spheres by Haurwitz (1940) andLonguet-Higgins (1964). In stars, Rossby waves are often called r-modes (e.g., Unno et al. 1989).
Good reviews of Rossby waves in the astrophysical context (including the equatorial modes) are
given by Zaqarashvili et al. (2021) and Gizon et al. (2021).
To identify waves in our MHD case, we analyze the temporal power spectra of the sphericalharmonic components of ωr (ignoring m = 0). As in discussing the magnetic field, we denotethe spectral coefficients of the horizontal motions by ω
fr (r, `, m, fn ). Here, we have transformed incolatitude (i.e., θ → `) in addition to longitude and time. Recall that fn ≡ n/T , with T the lengthof the uniformly sampled time series.
Figure 5.11 shows the m = 1 horizontal power as a function of harmonic degree. In Figure5.11(a), the power closely follows the classical Rossby-wave dispersion relation (for a fixed m,frequency is proportional to 1/[`(` + 1)]) for low `, but gets fairly muddy for high `. We zoom in on
the low-frequency region in Figure 5.11(b). The dominant flows at low frequencies are concentrated
in the ∼[0, 5] nHz region, just like the partial-wreath dynamo cycle, and they are clearly separate
from the expected Rossby-wave frequencies. We thus refer to the horizontal motions in the band ±5

216
nHz as low-frequency inertial oscillations and motions outside this interval (but still in the inertialband) as equatorial Rossby waves.
We believe that the low-frequency oscillations produce the induction of poloidal field of Figure5.5(b) for two reasons. First, the magnetic field “lives” in the low-frequency region, according to
Figure 5.9. For the Rossby waves to induct the magnetic field, there would have to be at least
some magnetism oscillating at high frequency, shifted by a small amount from the Rossby waves.
The beating between v 0 and B 0 could then cause an emf at the right frequencies to account for
the induction. In the deep radiative interior (r/R

= 0.60), the magnetic-field amplitude at theequatorial Rossby-wave frequencies is Bθ ∼ 5 G and the horizontal power of the Rossby waves isωr0 ∼ 2×10−8 rad s−1 , so the production of poloidal magnetic is (Bθ2 /4π)ωr0 ∼ 4×10−8 erg s−1 cm−3 .
Even if all the high-frequency magnetic power were organized into ridges like the Rossby waves,
but shifted by a few nHz (which we do not observe), there would thus not be enough magnetic fieldto yield the induction shown in Figure 5.5(b).
Second, the low-frequency inertial oscillations do have enough amplitude to account for theobserved induction. For example, at the same r/R= 0.60 depth for low frequencies, we canestimate Bθ ∼ 500 G, ωr0 ∼ 5 × 10−10 rad s−1 , and thus (Bθ2 /4π)ωr0 ∼ 10−5 erg s−1 cm−3 . Furthermore, the low-frequency inertial oscillations seem to be clustered exactly around the frequencies
associated with the cycling partial-wreath dynamo. It thus seems very likely that the inductionis due to the flows within the identified ±5 nHz low-frequency band (although we have not yet
explicitly demonstrated this).
We speculate that these inertial oscillations are similar to the critical-layer Rossby waves
described in Section 5.2 (e.g., Braginsky & Roberts 1975; Watson 1981; Charbonneau et al. 1999a;
Garaud 2001). These waves have horizontal eigenfunctions with amplitude highest between two
critical points in latitude, which are located where the mode frequency equals the backgroundrotation frequency. In the MHD case, the radiative interior rotates ∼2.3 nHz (or more conveniently
here, ∼2.5 nHz) slower than the frame rate. Furthermore, the background rotation frequency varies
over the partial-wreath cycles (like a torsional oscillation) by an amplitude of ∼2.5 nHz. At any

217
given time, the background rotation rate can thus be anywhere in the ∼[0, 5] nHz range. There
could then be critical-layer Rossby waves, having a continuous spectrum of frequencies also in the∼[0, 5] nHz range. Finally, because the low-frequency inertial oscillations appear with roughly
the same energy as the poloidal magnetic field, they may well be “MHD” Rossby waves (a hybrid
between equatorial Rossby waves and Alfvén waves). We discuss the possibility of MHD Rossby
waves more in Section 5.10.
We briefly note that the power in the vertical velocity consists of a rich spectrum of gravity
waves, which we show in Figure 5.12. Vertical power is clearly concentrated in ridges that correspond to the gravity-wave dispersion relation. Furthermore, the gravity and inertial waves appear
to live in distinct frequency intervals, with the gravity-mode frequencies well outside the inertial
band. Hence, the influence of “gravito-inertial” modes (e.g., Rieutord et al. 2001; Baruteau &Rieutord 2013; Mirouh et al. 2016) is likely small in our MHD case. Figure 5.12 clearly shows thatthe horizontal motions are not due to gravity waves, as may have been suspected from the initialinvestigations of Alvan et al. (2014) and Alvan et al. (2015). The same frequency separation likely
holds in the Sun. The solar gravity modes are not anelastic and the resonant cavity is quite different
than in our simulations.9

However, the range of frequencies in the excited solar gravity modes

should still roughly fill the space between very low frequency and the buoyancy frequency (which ison par with that of our simulations). Thus, unless the very-low-frequency (very-high-radial-order)
gravity modes are for some reason preferentially excited in the Sun, most of the gravity modes will
lie well outside the inertial band.

5.9

Magnetic Tachocline Confinement in Global Simulations
We first note that most, if not all, MHD simulations including stable layers also have strong

poloidal magnetic fields in their radiative interiors (e.g., Ghizaru et al. 2010; Racine et al. 2011;
Passos & Charbonneau 2014; Lawson et al. 2015; Guerrero et al. 2016a,b, 2019; Bice & Toomre
9In our models, the gravity waves reflect off an unphysical rigid boundary at r = ri , whereas in the Sun, they
have a lower turning point where the mode frequency becomes greater than the local buoyancy frequency.

218

Figure 5.12: Gravity Waves Below the Tachocline. The gravity-wave dispersion relation in
the MHD case is shown at three different depths in the radiative interior. The full gravity-wave
band is approximately ±N/2π = ±190 µHz (see Table 5.1; note that the gravitational acceleration,and thus N , varies slightly throughout the radiative interior), but we here plot a smaller subinterval of frequencies that contains
most of the power. The temporal power spectra of the radialPvelocity summed over all m,|ve(r, `, m, fn )|2 , is plotted on a log-10 scale (indicated by thermcolorbar) as functions of spherical-harmonic degree ` and frequency fn for three different radiallevels, radius increasing from (a)–(c). In this case, the sampling interval is ∼2.75 rotations, thefrequency resolution is 0.5 µHz, and the Nyquist frequency is 500 µHz. Figure credit: Catherine
Blume.

219
2020, 2022; Brun et al. 2022). These examples all have tachoclines implemented by enforcing
very low diffusions throughout the radiative interior. There is thus ostensibly negligible diffusive
imprinting of magnetic field from above, and we can similarly rule out “downward pumping” ofmagnetic field by downflow plumes because of the stable stratification. The likely conclusion is thatthe fields come locally from induction. The possibility of induction in these low-diffusion models is
also supported by the presence of strong horizontal motions within the stable layer.
We thus speculate that many prior simulations may have contained dynamo action in theirstable layers, though it was not recognized as such. However, it is difficult to know if the magnetism
could have been confining the tachocline in the low-diffusion models. The main goal of lowering
the diffusions is to slow down the eventual tachocline spread. If this technique is successful, thedifferential rotation never enters the radiative interior during the course of the simulation, and theprocess by which Ferraro’s law enforces solid-body rotation cannot take place.
However, even substantially lowering the radiative interior’s diffusions often does not successfully confine the tachocline (e.g., all the ASH simulations of F-type stars in Augustson et al.
2012, 2013; about half the EULAG-MHD simulations in Guerrero et al. 2019; and the “M07R01m”
and “M09Sm” ASH simulations of Brun et al. 2022). It is a puzzle why this happens, especially in
the EULAG-MHD code where diffusion is implicit and reportedly too low to be measured (e.g., Lawson et al. 2015; Guerrero et al. 2016a,b). In any case, it would be hard to explicitly demonstratemagnetic confinement in a model with low diffusions. The torque balance could not be assesseduntil after the simulation had been run past the very long viscous diffusion time across the radiative interior (which is designed at the outset to be much longer than the intended run time ofthe simulation). Furthermore, in the EULAG-MHD simulations, there does not seem to be a method
available to compute the torque associated with the implicit diffusive operator in the first place.
We thus argue that keeping the diffusions relatively high everywhere in the shell has the advantage of making a thorough dynamical analysis possible. In future global simulations assessing thetachocline-confinement problem, we strongly support explicit diffusions and running the simulation
for multiple diffusion times.

220

5.10

Revisiting Magnetic Tachocline Confinement in the Sun
We have presented the first 3-D gobal simulation to evince a mechanism similar to one of

the magnetic confinement scenarios proposed for the Sun. Here, we speculate whether the rather
specific form of the confining magnetism we have identified (non-axisymmetric, cycling partial
wreaths induced locally by dynamo action all throughout the radiative interior) could be playingout in the Sun. To that end, we briefly discuss three issues that will have to clarified in future
work. (1) Could there be strong horizontal motions, in the form of vortical Rossby waves andother low-frequency inertial oscillations, in the solar radiative interior? (2) Could such horizontal
motions induce dynamo action, with non-axisymmetric fields that are consistent with the observed
22-year cycle? (3) Do the answers to the two prior questions change between the parameter space of
simulations (in which the tachocline spreads viscously) and that of the Sun (in which the tachocline
spreads radiatively)?
(1) As discussed in 5.2, horizontal motions have long been thought to occur in the solar
radiative interior via shear instabilities (e.g., Zahn 1974, 1992; Charbonneau et al. 1999a; Garaud
2020; Cope et al. 2020. More recently, there has been much work to characterize magneto-shear
instabilities in the form of MHD Rossby waves (a hybrid between equatorial Rossby waves andAlfvén waves) in the solar tachocline (Zaqarashvili et al., 2021)). Specifically, it seems likely that
there is an exchange of energy between differential rotation, unstable MHD Rossby waves, and themagnetic fields in the tachocline (e.g., Dikpati 2012; Dikpati et al. 2017, 2018a,b). These nonlinear
energy exchanges, also called Tachocline Nonlinear Oscillations (TNOs), have been postulated tocause the observed “seasons” in space weather (bursts of solar flares and coronal mass ejections
every 6–18 months; e.g., McIntosh et al. 2015). TNOs have been simulated using the shallowwater formalism (Dikpati et al., 2017, 2018a,b) and reveal tilted flow and field patterns remarkably
reminiscent of Figures 5.3 and 5.6). Finally, Zaqarashvili (2018) and Zaqarashvili & Gurgenashvili(2018) have shown that MHD Rossby waves in the solar tachocline could yield oscillation periods
on the order of ∼10 years (i.e., close to the sunspot cycle period), as well as ∼100 years (i.e., close

221
to the Gleissberg cycle period).
(2) Whether the horizontal motions could induce dynamo action is an open question. The
results of Dikpati et al. (2017, 2018b); Zaqarashvili (2018); Zaqarashvili & Gurgenashvili (2018)
are highly suggestive, showing that unstable MHD Rossby waves are likely to at least be strongly
correlated with observed dynamo variations, and may even drive them. Additionally, recall from
Section 5.2 that some of the work on magneto-shear instabilities (e.g., Dikpati & Gilman 2001a;
Dikpati 2012; etc.) postulated that Rossby waves could induce an α-effect through their net helicity.
Furthermore, MHD Rossby waves in the convection zone have been invoked as a potential driver ofthe solar dynamo (Gilman, 1969a,b). Finally, MHD Rossby waves in the stably stratified Earth’s
core (in the form of “baroclinic instabilities,” as described by Braginsky & Roberts 1975) have been
explicitly shown to be capable of sustained dynamo action. In future work, we must assess exactly
what type of wave or instability the low-frequency oscillations of Figure 5.11 represent. If they are
MHD Rossby waves, our MHD case would be good evidence that such waves in the solar tachocline
could help drive a dynamo.
One feature of the MHD case’s dynamo (which was also a feature of many instability analyses)
is the dominance of non-axisymmetric (specifically, m = 1, 2) magnetic fields. Whether such afield in the radiative interior would be consistent with Hale’s polarity law, which would seem toindicate one largely axisymmetric reservoir of toroidal field in each hemisphere, is unclear. As
discussed in Chapter 4, there are a significant number of “anti-Hale” bipolar active regions (Stenflo& Kosovichev, 2012; Li, 2018), which might be consistent with some amount of non-axisymmetry.
Also, the magnetic field’s geometry and amplitude in the deep radiative interior are almost entirely
unconstrained by helioseismology. The path forward thus seems best suited to global modeling. To
verify whether a confining dynamo would be consistent with solar-cycle observations, future global
simulations must better assess the magnetic coupling (or lack thereof) between convection zone
and radiative interior.
(3) It is a tantalizing possibility that a dynamo driven by inertial oscillations maintains thesolar tachocline. But could the process we have identified operate in the solar parameter regime?

222
Apart from the MHD case being much more laminar than the Sun, the essential difference lies in themechanism of tachocline spread: viscous versus radiative. In the Sun, the meridional circulation
should burrow inward and also play a dominant role in the torque balance (e.g., Spiegel & Zahn 1992;
Garaud & Brummell 2008; Garaud & Garaud 2008; Wood & Brummell 2012; Acevedo-Arreguin
et al. 2013; Wood & Brummell 2018). In current global simulations (and for the foreseeable future),
this burrowing does not occur. Empirically, we cannot say if circulation-burrowing (or its absence)
should strongly affect the system’s dynamics, aside from changing the form of the torque thatimprints the differential rotation. The only 3-D (local) simulations to clearly display radiative
spread (Wood & Brummell, 2012, 2018) indicate that the imprinting of differential rotation is largely
uniform (i.e., the shear profile simply grows vertically in time with little horizontal dependence).
This would seem to be quite similar to viscous spread. Wood & Brummell (2018) further found thatradiative spread could be prevented by magnetic field in a process like Ferraro’s law; however, they
considered a primordial field as in Gough & McIntyre (1998). It is thus too soon to say definitively
whether these results would extend to a global dynamo in a solar-like parameter space.
In summary, the MHD case presented here offers a strong piece of evidence that magnetic
tachocline confinement is possible in a fully 3-D, global geometry with fairly turbulent motions. It
also clearly shows the presence of classical equatorial Rossby waves, which make up most of thehorizontal motion, as well as low-frequency inertial oscillations, which may represent MHD and/or
critical-layer Rossby waves. The low-frequency modes appear to induce dynamo action even below
the convective-overshoot layer. These results suggest that the radiative interior of the Sun is not an“abyssal deep” at all, but rather is filled with strong horizontal motions, and that dynamo action
may extend beneath the convection zone. In future work, we will assess the detailed origin of the
low frequency modes and how they cohere with our current understanding of Rossby waves in the
Sun.

223

5.11

Supplement: Numerical Specifics and Equation Identities
Our HD and MHD cases are simulations of spherical shells spanning (ri , ro ) ≡ (0.49R ,0.95R ), where R ≈ 700,000 km is the solar radius. The base of the convection zone occurs near
0.72R , so our shell covers about 160,000 km (by radius) in both the convection and radiativeinterior. In the convection zone, this translates to a density contrast of ∼20, or about three density
scale heights (to borrow our notation from Chapter 2, Nρ = 3). Our grid resolution is Nθ = 384 andNφ = 768 in the horizontal directions (the maximum spherical harmonic degree after de-aliasingis `max = 255) and Nr = (64, 64, 64) in the vertical direction. We use three stacked Chebyshev
domains in radius (each with 64 grid points) with boundaries (ri , 0.67R , 0.72R , ro ). The use of
three radial domains and two internal boundaries maximizes the grid resolution in the overshootlayer (the Chebyshev collocation points cluster near the ends of each domain). We define the viscousand magnetic diffusion times across the radiative interior as [rov − ri ]2 /ν(rov ) (this translates to
295 rotations) and [rov − ri ]2 /η(rov ) (this translates to 1180 rotations), respectively. The HD casewas run in its equilibrated state for 7810 rotations (26.5 viscous diffusion times) and the MHD casefor 12500 rotations (10.6 magnetic diffusion times).

5.11.1

Background Thermodynamic StateWe treat the thermodynamic profiles of the stable-to-unstable transition in the Sun (which
occurs at r ≡ rbcz ) slightly differently than past work has. We choose a simplified entropy-gradientprofile, which is zero in the convection zone, achieves a constant positive value in the stable layer,
and has smooth matching in between. As mentioned in Chapter 1 (and now proven), the backgroundstate is fully specified by the entropy gradient and gravitational-acceleration profile. We begin by224
assuming that the solar interior is a spherically symmetric ideal gas in hydrostatic balance:

anddP= −ρg(r),dr(5.11a)P (r) = ρ(r)RT (r),(5.11b)d ln Td ln ρ
1 dS− (γ − 1)
=
.
dr
dr
cv dr

(5.11c)Here, cv (cp ) is the specific heat at constant volume (pressure), R = kB /µmp is the gas constant(with kB Boltzmann’s constant, mp the proton mass, and µ the mean molecular weight), andγ = cp /cv .
Equations (5.11) can be combined to yield an ordinary first-order differential equation for
the temperature:


dT
1 dS
g(r)
−
,
T (r) = −
dr
cp dr
cp
which has the analytic solutioneS(r)/cp
T (r) = −
cp

Z r

r
g(e
r)e−S(er)/cp der1+ T (r1 )e[S(r)−S(r1 )]/cp ,(5.12)

where r1 is an arbitrary radius from which to start the integration. We choose r0 = rtach ≡5 × 1010 cm ≈ 0.72R , ρ(rtach ) = 0.18 g cm−3 , and T (rtach ) = 2.6 × 106 K. The thermodynamic
variables at the base of the convection zone are thus pinned to the same values as in Featherstone& Hindman (2016a); Matilsky et al. (2020); Hindman et al. (2020); and the other simulations
described in this thesis. We integrate numerically on a fine grid (5,000 uniformly spaced radialpoints) and then interpolate onto the simulation grid.
Equation (5.12), combined with Equations (5.11) and the boundary conditions, gives thetemperature, density, and pressure in terms of the entropy gradient and gravitational acceleration,
which is g(r) = GM /r2 . We choose a quartic matching between dS/dr = 0 in the convection zone

225
and dS/dr = β ≡ 0.01 erg g−1 K−1 cm−1 in the radiative interior:




β
r ≤ rtach − δ







2 i 2hdS= β 1 − 1 − r−rtach
rtach − δ < r < rtach
δ

dr






0
r ≥ rtach(5.13)The quartic matching (instead of the tanh or erf profile employed in (for example) Guerrero et al.
2016a) ensures that none of the stable gradient in the radiative interior “leaks” into the convection
zone. The solar stiffness β is (for example) ∼104 –105 times higher than the entropy gradients in
EULAG-MHD needed to drive the convection (or in Rayleigh, ∼104 –105 times higher than the entropy
gradients driven by the convection). A tanh matching (e.g., dS/dr = (β/2){1−tanh [(r − rtach )/δ]})thus results in convective stability all the way up to about rtach +5δ, since [1−tanh (5)]/2 ≈ 5×10−5 .
In the two cases presented here, we choose δ/R = 0.05.
We do use a tanh profile to put the internal heating Q(r) almost completely in the convection
zone: Q(r) = α tanh [(r − rtach )/δheat ][P (r) − P (ro )], where δheat /R= 0.03. The constant α ischosen so the volume integral of Q(r) over the whole shell is the solar luminosity. In the convection
zones of our two models, the reference state is thus nearly identical the three-scale-height models in
our prior work (Featherstone & Hindman, 2016a,b; Matilsky et al., 2018, 2019; Matilsky & Toomre,
2020a; Matilsky et al., 2020; Hindman et al., 2020; Matilsky & Toomre, 2021; Korre & Featherstone,2021).

5.11.2

Induction Equation in Spherical Coordinates

We write the MHD induction equation asor∂B= ∇ × [v × B] − ∇ × [η(r)∇ × B]∂t
∂B
= −B(∇ · v) + B · ∇v − v · ∇B + D,∂t

(5.14a)(5.14b)where D ≡ −∇ × [η(r)∇ × B]. We refer to the first and second terms on the right-hand side ofEquation (5.14a) as the “induction” and “diffusion,” respectively. The three induction terms on

226
the right-hand side of Equation (5.14b) are traditionally interpreted as “compression,” “shear,”and “advection,” respectively (recall the inductive analyses of the bistable dynamos in Chapter
4). This interpretation is problematic, however, because only compression and shear perpendicular
to magnetic field lines can cause induction, and furthermore, only gradients in the local rate ofstrain (not the velocity) can cause shear. For example, [B · ∇v]φ = r sin θBpol · ∇(vφ /r sin θ) +(1/r sin θ)Bφ ∂vφ /∂φ. Only the first term can shear poloidal field (via gradients not in the velocity
vφ but in the rate of strain vφ /r sin θ) to produce toroidal field; the second term cancels with aterm in −Bφ (∇ · v). We thus rewrite Equation (5.14b) for the horizontal directions as follows:
∂Bφ
= −Bφ
∂t
|



∂vr
∂r




+
{z

1 ∂vθ
vr
+
r ∂θ
r



“perpendicular” toroidal compression

}

 v 
φ
+ r sin θBpol · ∇− v · ∇Bφ + Dφ{z r sin θ }|(5.15a)“perpendicular” toroidal shear

and

∂Bθ
= −Bθ
∂t
|



∂vr
∂r




+


1 ∂vφ cot θvθ
vr++
r sin θ ∂φ
r
r
{z
}

“perpendicular” θ compression+ r(Br êr + Bφ êφ ) · ∇
{z
|

v 

“perpendicular” θ shearθr }− v · ∇Bθ + Dθ ,(5.15b)where the third term on the right-hand side of each equation is the advection. These Equations
follow straightforwardly from Equation (5.14b) by canceling and rearranging terms. However, the“perpendicularity” of the shear and compression may be seen by considering, at a given point(r, θ, φ), planes perpendicular to the directions (êr , êθ , êφ ) and then computing ∇ · v and B · ∇while keeping the gradient operator restricted to these planes and expressing the result in sphericalcoordinates. We will present the details in future work.
When the azimuthal flow is dominated by the mean rotation rate, it makes sense to definemean shear ≡ r sin θBpol · ∇Ω.

(5.16)

Note that our mean-shear term uses the full Bpol , not the temporally and longitudinally averaged
hBpol i that is used in the original Ferraro’s law. When the advection terms can be neglected, when

227
mean shear dominates the other induction terms in Equation (5.15a), and when the velocities arelargely horizontal (all conditions well satisfied in the MHD case’s radiative interior), Equations(5.15) reduce to Equations (5.5) and (5.6).
Because our fields are non-axisymmetric, we consider the evolution of the magnetic energy
density B 2 /8π, which is obtained from taking the dot product of the induction equation with B/4π.
As in Figure 5.4, we analyze the production of toroidal and poloidal magnetic energy (Bφ2 /8π
2 /8π, respectively) separately. In the legend of Figure 5.4, we thus refer to the magnetic
and Bpolenergy production terms due to “mean shear”—(Bφ /4π) × (mean shear); “poloidal induction”
Bpol · ∇ × [v × B]; “θ compression”—(Bθ /4π) × (θ compression); and so on.

Chapter 6

Concluding Remarks

In this thesis, we have explored the dynamics of the solar interior using the advanced machinery of fully 3-D, global, nonlinear, magnetic convection simulations. We sought to understand how
convection under the influence of rotation and magnetism could explain some of the most striking
aspects of the observed solar dynamo and helioseismically inferred differential rotation.
We began by discussing the hydrodynamic (HD) Sun, assessing the dynamics of the helioseismically inferred Near-Surface Shear Layer (NSSL) and the tilts of the solar isorotation contours.
We found that the solar NSSL—which may play a prominent role in the global dynamo as a siteof strong shear in close proximity to sunspots—is still essentially an unsolved problem. This is
likely because of our inability to understand the dynamics of the Sun’s meridional circulation, andpossibly its near-surface magnetism. In addressing the tilts of the isorotation contours, we foundthat a fixed-flux outer thermal boundary condition—which was inspired by the observed invarianceof the Sun’s emissive flux with latitude—led to solar-like contour tilts. By contrast, the commonly
employed fixed-entropy (in other work, fixed-temperature) thermal boundary condition led to the
opposite sense of tilt. We identified for the first time a purely HD mechanism (namely, heat transport by Busse columns, or convective Rossby waves) by which the Sun might move thermal energy
from equator to pole. In combination with a fixed-flux boundary condition (which prohibits “leaks”
of energy out through the poles), the Busse columns could thus maintain the thermal wind thathas long been invoked to explain the helioeseismically observed contour tilts.
In future work, there is much to be done regarding the HD Sun. First, the “convective

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conundrum” is still a major problem for simulations. Restated here, the issue is that simulated
convection pushed closer to the turbulent solar parameter regime seems unable to maintain asolar-like differential rotation. Specifically, simulations must rotate more rapidly than the solarrate, carry less luminosity than the Sun, or a combination thereof. Otherwise, they risk falling into
“anti-solar” states, with polar regions rotating faster than the equator. One approach to addressing
the convective conundrum would be to remove the simulation’s luminosity via a cooling layer (withamplitude and thickness controlled a priori), in place of the traditionally employed outer thermalboundary layer. This could potentially separate the spatial scales of the thermal driving fromthose of the angular momentum transport, thereby assessing whether the large-scale motions incurrent simulations are being driven too hard. Simulated convective velocities being too high at
the large scales is the often-invoked, though still untested, reason for the convective conundrum(recall our detailed discussion of this scale-separation issue in the Introduction, Section 1.2.3). Our
work on the NSSL and outer thermal boundary condition shows that the near-surface dynamics
may significantly affect the global properties of simulations, and the most obvious step forward is
the demolition of our rather unrealistic outer thermal boundary layer.
Second, future work may address the NSSL by way of a systematic investigation of the
dynamics of meridional circulation. We recall that our models did not reproduce the NSSL because
the meridional circulation developed in a thin band to eliminate any shear (and earlier global
models of the NSSL likely suffered from the same issue). How meridional circulation develops, in
global convection simulations or in a real star, is a difficult problem. The circulation is at a higher
order than the dominant flows (convection and differential rotation) and responds to residual force
balances that usually require very long integration times to assess in the presence of high-amplitude
convective “noise.” The approach in past work has generally been to assume that changes to thedifferential rotation (i.e., torques) drive the meridional circulation. Invoking this phenomenon, socalled gyroscopic pumping (e.g., Haynes et al. 1991; McIntyre 1998, 2002; Miesch & Hindman 2011),
may cause issues in our understanding, since it does not treat the meridional force balance directly.
Global simulations are in a privileged position to address the meridional-circulation problem, since

230
the flows, while more laminar than in a star, are fully self-consistent and the meridional force
balance can be directly assessed.
Third, the tilted solar isorotation contours were a striking surprise revealed by helioseismology
and remain largely an outstanding problem. We have made substantial progress in this area byidentifying poleward heat transport by Busse columns as a natural driver of the solar thermal
wind. However, our results relied on an artificially imposed fixed-flux outer thermal boundary
condition. How the Sun actually achieves latitudinally independent emissive flux (if there is indeed
a thermal wind in the interior) is a non-trivial question. No doubt the convective flows at small
scales (granulation, which is absent in most global simulations) may play an important role, as
may the degree of turbulence, the small-scale magnetism, and the properties of the near-surface
meridional circulation. Further inroads into addressing the convective conundrum and meridional
circulation may thus also make progress in solving the problem of constant emissive flux (and therelated problem of contour tilt).
In our explorations of the magnetohydrodynamic (MHD) Sun, we found a remarkable phenomenon of bistability. Here, our global dynamo simulations exhibited two distinct cycles simultaneously. One cycle involved a set of four “magnetic wreaths,” similar to the structures encounteredin past simulations, that in our case displayed a cycle reminiscent of the magnetic butterfly diagram. The other cycle involved our newly identified “partial wreaths,” possibly similar to the
non-axisymmetric single-hemisphere toroidal structures encountered in past dynamo simulations,that in our case displayed temporal behavior similar to the Sun’s active longitudes and hemisphericasymmetry. The bistable system began with only the fourfold-wreath cycle operating, but eventually reached a state wherein both cycles operated more or less superimposed. Looked at anotherway, two basins of attraction were available to the dynamo, which could fall into one basin orboth basins in a chaotic manner. Furthermore, small changes to the initial conditions resulted invastly different states after a long time interval, also consistent with dynamical chaos. This work
highlighted the richness of the nonlinear, chaotic dynamics present in stellar interiors, and alsooffered the practical possibility that solar and stellar dynamos observed in one state may exist in231
quite different states at other points of their evolution.
We ended this thesis by discussing our extension of the bistable dynamos to include a stably
stratified radiative layer below the convection zone. In this MHD case, the partial-wreath cycle
was dominant. The magnetism extended all the way into the radiative interior and produced atorque that forced the background fluid into solid-body rotation. Meanwhile, Reynolds stresses in
the convection zone forced a weak solar-like differential rotation. The resulting shear layer bore
resemblance to the solar tachocline and was maintained against viscous spread for many diffusion
times. This was the first time a 3-D global simulation was shown capable of self-consistently
confining a tachocline magnetically. The MHD case was similar in some ways to the early slow
and fast magnetic confinement scenarios, but implied a form for the confining magnetism thatwas distinct from the axisymmetric primordial-field remnant (Gough-and-McIntyre slow scenario)or the “turbulently pumped” axisymmetric magnetism from the 22-year cycle (Forgács-Dajkaand-Petrovay fast scenario). In particular, our confining magnetic torque came from the nonaxisymmetric, irregularly cycling partial wreaths. Furthermore, we showed that the partial wreaths
were inductively produced, likely by low-frequency inertial oscillations, even below the convective
overshoot layer. These results may profoundly challenge the idea that stellar radiative interiors are
passive reservoirs or “abyssal deeps,” simply accepting whatever magnetism the convection zone
pumps in and storing it until it grows strong enough to buoyantly rise to the surface. Instead,our calculations were consistent with a dynamically active radiative interior, in which dynamo
amplification can occur locally via the strong horizontal motions associated with inertial oscillations.
In future work, the sensitivities of bistable dynamos—and the apparently related problemof magnetic tachocline confinement—must be more systematically explored. Due to inherent computational constraints, we so far have only sampled a small portion of parameter space, and therichness of magnetism achieved even in this limited regime was still quite impressive. Pinpointing
how fourfold and partial wreaths may operate in the solar interior (which, of course, rotates atthe solar rate and is highly turbulent—a parameter space that is still inaccessible to global models
because of the convective conundrum) may yield significant insights into the solar dynamo. More

232
specifically, the Sun’s striking 22-year cycle, long-period modulations, active longitudes, and hemispheric asymmetry are the results of complex nonlinear dynamics in the interior. Future global
simulations will be powerful tools to reproduce some of the Sun’s behaviors and understand themfrom a fundamental dynamical perspective.
If partial wreaths are indeed a possible magnetic phenomenon capable of confining the solar
tachocline, future global models must assess the inductive properties of the radiative interior. We
plan specifically to classify the low-frequency inertial oscillations that gave rise to dynamo actionin our MHD case. We will assess how these magnetically important oscillations may be related
to the large body of prior work on MHD Rossby waves and tachocline magneto-shear instabilities.
If inertial oscillations really can drive a dynamo, our work hints at a profoundly different view ofstellar radiative interiors. Instead of being a passive reservoir, or “abyssal deep,” the stable layer
may need to be treated as a dynamically active region fully coupled to its neighboring convection
zone. The interplay of convective overshoot, inertial oscillations, and magnetic induction may thensignificantly affect our understanding of stellar dynamos.
In conclusion, it is our view that the field of solar and stellar interiors—although one of theoldest and most well-studied branches of astrophysics—still contains many mysteries. Observationsof the solar magnetic field, as well as the internal rotation profile deduced by helioseismology, showthat stars can exhibit remarkably well-ordered behavior. Even more striking, this behavior arises
from the nonlinear dynamics of highly turbulent convection under the influence of rotation andmagnetism. In this thesis, we have only begun to scratch the surface of some of the major observed
phenomena—the NSSL, the isorotation contour tilts, the 22-year cycle, and the solar tachocline.
Our work has shown that global simulations present a unique window into the dynamics of stellarinteriors and may provide many more insights going forward. We hope that this thesis may serve tohighlight what is possible using global modeling and that it may potentially inspire future researchin solar and stellar dynamics.

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