Convection and Dynamo Action in Massive Starsby
K. C. Augustson
B.S., Oregon State University, 2006
M.S., University of Colorado Boulder, 2009

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
2013Augustson, K. C. (Ph.D., Astrophysical and Planetary Sciences)
Convection and Dynamo Action in Massive Stars
Thesis directed by Professor Juri Toomre

Contact between numerical simulations and observations of stellar magnetism is sought,with an emphasis on those stars that are the most readily observed and those that may havemagnetic activity cycles: the Sun, F-type, and B-type stars. Two approaches are taken instudying stellar dynamos and dynamics, utilizing three-dimensional MHD simulations run
on massively parallel supercomputers with the full spherical geometry and employing a newcompressible code in the spherical wedge geometry. A 3D MHD simulation of the solar
dynamo that utilizes the Anelastic Spherical Harmonic (ASH) code is presented. This simulation self-consistently exhibits four prominent aspects of solar magnetism: activity cycles,polarity cycles, the equatorward field migration, and grand minima. The ASH framework andthis simulation’s ability to capture many aspects of the solar dynamo represent a first step
toward a more complete model of the Sun’s global-scale magnetic activity and its cycles. The
dynamics and dynamos of F-type stars are studied through global-scale ASH simulations,with significant contact made between the observed differential rotation and magnetic cycle
periods of these stars and those achieved in the simulations. Separately, ASH simulationsof core convection in the massive B-type stars show that generation of superequipartition
magnetic fields with peak strengths above 1 MG is possible within their cores, which has
implications for the evolution of these stars as well as for the properties of their remnants.
The internal waves excited by overshooting convection and rotation in these stars radiative
exteriors are assessed for their asteroseismic signatures. The results of 3D compressive MHD
simulations of the solar near-surface shear layer with the Compressible Spherical Segment(CSS) code are shown, with such layers arising in the coupled dynamics of ASH and CSS as
well as in a more rapidly rotating, thin convective envelope of an F-type star.

Dedication

To the RK4:
time cruelly marches on.
To RK45:
error tolerance, smooths
over the rough spots.

v

Acknowledgements

I would foremost like to thank Brad Hindman, Mark Miesch, Mark Rast, Juri Toomre,
and Jeff Weiss for agreeing to serve on my thesis committee. Hopefully, it was not too
daunting a task and the wide range of topics covered not too disparate.
Through his kindness and steady support, Juri Toomre has provided phenomenal opportunities throughout my tenure as a graduate student: from the influential months at the
Les Houches summer school on dynamo theory to appreciating the idea-crystallizing opportunities that proposal writing can create. He has particularly been a kind guide and a patient
friend during difficult times, allowing me the time to rekindle my passion for mathematicalphysics and turbulence theory. I am deeply appreciative the many lunches Juri and I haveshared while discussing a plethora of topics we each enjoy. Moreover, the frequent dinners
and after-dinner conversations he graciously invited me to share in were always delightful.
To that end, Linda Toomre also deserves a hearty thank you.
Sacha Brun has been a mentor from afar, kindly offering several opportunities to workin France and to gain further exposure to the astrophysics community of Europe. During
his extended stays in Boulder, I have always enjoyed his company, energy, and enthusiasm,
whether it be the long hours spent slaving away in LCD or fine dining at Closerie de Lilas.
Even though it has not yet happened, someday we will have our jam session!
I also have enjoyed conversations with Mark Rast, whose interest in the depths ofturbulence theory has spurred my interest in alternative approaches to mean-field dynamics. Furthermore, his exuberance, openness, and philosophical approach to theoretical fluid
mechanics encouraged my curiousity about the mathematical side of turbulence theory.

vi
Similarly, Ben Brown’s contagious enthusiasm for many aspects of fluid dynamics andapplied mathematics kept me engaged in my first few years of graduate school, and has
continued to yield interesting new avenues of research. For the untold hours of amusing
conversation when we shared an office and for trying to coax me out of LCD during thefour months of thesis writing, Nick Featherstone merits a thank you. Regner Trampedachhas been a fount of knowledge and patient in answering my sometimes esoteric questions,but he also read portions of the thesis and his comments helped clarify them for which
I am grateful. Gwen Dickinson certainly also deserves my gratitude for our wide-ranging
conversations that almost always conclude with something depressingly political, and forbeing the wizard behind the curtain that keeps the group organized.
Finally, I wish to acknowledge my parents. Despite the anxiety and chaos of an earlylife in poverty, they formed a united front and insulated my siblings and me from the worst ofit, and emerged from it through dedication and education. Though the hardest of those times
have long since passed, they have left their lingering imprints. They nevertheless managed
to foster in us a curiosity about the workings of the world and encouraged us to explore it
through our imaginations and creativity. Had it not been for my mother’s dedication to thefamily and my father’s courage to be the first in his family to attend college, all done whileworking several jobs, I certainly would not have been able to wander down this long road.

vii

Contents

Chapter
1 A Perspective on Solar Magnetism

1

1.1

Spatial Scales of Solar Magnetism . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Modern Sunspot Record . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.3

Origins of Modern Sunspot Observations . . . . . . . . . . . . . . . . . . . .

18

1.4

Beginning of the Western Sunspot Record . . . . . . . . . . . . . . . . . . .

25

1.5

Terrestrial Record of Solar Variability . . . . . . . . . . . . . . . . . . . . . .

28

1.6

Short Time Scale Solar Variability . . . . . . . . . . . . . . . . . . . . . . . .

31

2 Observations of Stellar Magnetic Activity

33

2.1

Photometric Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.2

Chromospheric Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.3

Spectropolarimetry and Zeeman-Doppler Imaging . . . . . . . . . . . . . . .

48

2.4

X-Ray Variability and Luminosity . . . . . . . . . . . . . . . . . . . . . . . .

54

2.5

Magnetic Activity-Rotation Correlation . . . . . . . . . . . . . . . . . . . . .

61

2.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3 Stellar Dynamo Theory

72

3.1

The MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

3.2

Dynamo Waves and Low-order Approximations . . . . . . . . . . . . . . . .

82

3.3

Mean-field Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

viii
3.4

Three-Dimensional Numerical Simulations of Nonlinear Dynamos . . . . . . 100

3.5

Solar Convection and Differential Rotation . . . . . . . . . . . . . . . . . . . 103

3.6

Global-Scale Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.7

Global-Scale Solar Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 110

3.8

An Extension of Balbus’ Theory . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.9

Global-Scale Solar Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . 124

4 Anelastic Formulations for Stable Stratification and Slope-Limited Diffusion

135

4.1

Sound-proofing Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.2

Constrained Lagrangian Dynamics

4.3

ASH and the Lantz-Braginsky-Roberts Equation . . . . . . . . . . . . . . . . 147

4.4

Diffusion Methods in ASH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152. . . . . . . . . . . . . . . . . . . . . . . 137

5 Achieving Regular Cycles in ASH

161

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.2

Computational Methods and Modifications . . . . . . . . . . . . . . . . . . . 164

5.3

Dynamics and Cyclical Convective Dynamo Action . . . . . . . . . . . . . . 166

5.4

Grand Minima and Long-Term Modulation . . . . . . . . . . . . . . . . . . . 173

5.5

Generating Global-Scale Magnetism . . . . . . . . . . . . . . . . . . . . . . . 175

5.6

What Might Be Setting Cycle Periods? . . . . . . . . . . . . . . . . . . . . . 181

5.7

Equatorward Dynamo Wave Propagation . . . . . . . . . . . . . . . . . . . . 188

5.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6 Global-scale Hydrodynamic Simulations of F-type Stars
6.1

6.2

193

Introduction: Stellar Convection and Rotation . . . . . . . . . . . . . . . . . 194
6.1.1

Observations of Differential Rotation . . . . . . . . . . . . . . . . . . 195

6.1.2

Global Models of F-type Star Convection . . . . . . . . . . . . . . . . 197

Formulating the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

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6.3

6.4

6.2.1

Anelastic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.2.2

Numerics and Experimental Configuration . . . . . . . . . . . . . . . 202

6.2.3

Diffusion and Sub-grid-scale Models . . . . . . . . . . . . . . . . . . . 204

6.2.4

Scaling Diffusion with Rotation . . . . . . . . . . . . . . . . . . . . . 209

Convective Patterns and Mean Structure . . . . . . . . . . . . . . . . . . . . 210
6.3.1

Case A: 1.2 M⊙ Simulations. . . . . . . . . . . . . . . . . . . . . . . 211

6.3.2

Case B: 1.3 M⊙ Simulations . . . . . . . . . . . . . . . . . . . . . . . . 215

6.3.3

Mean Zonal and Meridional Flows . . . . . . . . . . . . . . . . . . . . 216

6.3.4

Mean Temperature Profiles. . . . . . . . . . . . . . . . . . . . . . . 219

Modulated Convection and Shear Instabilities . . . . . . . . . . . . . . . . . 220

7 Examining the Convection Zone Dynamics of F-type Stars
7.1

226

Examining Dynamics within Case A10 . . . . . . . . . . . . . . . . . . . . . 226
7.1.1

Typical Flows and Thermal Properties . . . . . . . . . . . . . . . . . 227

7.1.2

Energetics of the Convection . . . . . . . . . . . . . . . . . . . . . . . 230

7.2

Convective Overshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

7.3

Thermal Structuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.4

7.5

7.6

7.3.1

Principal Contributions to Temperature . . . . . . . . . . . . . . . . 236

7.3.2

Thermal Signature of Differential Rotation . . . . . . . . . . . . . . . 239

Maintaining Mean Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.4.1

Decrypting the Axial Torques . . . . . . . . . . . . . . . . . . . . . . 246

7.4.2

Meridional Flows Induced by Axial Torques . . . . . . . . . . . . . . 250

Scaling with Rotation and Mass . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.5.1

Differential Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7.5.2

Latitudinal Temperature Gradient . . . . . . . . . . . . . . . . . . . . 256

7.5.3

Meridional Circulation Kinetic Energy . . . . . . . . . . . . . . . . . 257

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

x
8 Global-scale Simulations of Dynamo Action in F-type Stars
8.1

8.2

262

Introduction: Stellar Magnetic Fields and Rotation . . . . . . . . . . . . . . 263
8.1.1

Global Simulations of Convection . . . . . . . . . . . . . . . . . . . . 267

8.1.2

Stellar Dynamo Modeling . . . . . . . . . . . . . . . . . . . . . . . . 268

Formulating the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
8.2.1

Anelastic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

8.2.2

Experimental Configuration . . . . . . . . . . . . . . . . . . . . . . . 274

8.3

Overview of Primary Findings . . . . . . . . . . . . . . . . . . . . . . . . . . 276

8.4

Convective Patterns and Mean Structures . . . . . . . . . . . . . . . . . . . 277

8.5

Dynamics within Case A10m . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

8.6

Generating Global-Scale Magnetism . . . . . . . . . . . . . . . . . . . . . . . 283

8.7

8.6.1

Toroidal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

8.6.2

Poloidal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Maintaining the Mean Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
8.7.1

Feedback of Magnetism on Mean Flows . . . . . . . . . . . . . . . . . 292

8.7.2

Thermal Wind Balances . . . . . . . . . . . . . . . . . . . . . . . . . 295

9 Cyclic Dynamo Action in a Rapidly Rotating F-type Star

297

9.1

Exploring Case A20m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

9.2

Morphology of a Reversal

9.3

Wreaths of Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

9.4

Mean Toroidal Field Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 302

9.5

Successive Toroidal Field Samplings During a Reversal . . . . . . . . . . . . 305

9.6

Polar-Slip Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.7

Evolution of Poloidal Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

9.8

Assessing the Production of Magnetic Energy . . . . . . . . . . . . . . . . . 311

9.9

Dynamo Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315. . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

xi
9.10 Characterizing the Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.11 A Second Fitting Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
9.12 Reversal Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
9.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
10 Convection and Dynamo Action within the Cores of B-type Stars

332

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
10.2 Recent Observational Studies of B-type Stars . . . . . . . . . . . . . . . . . . 336
10.2.1 Aspects of Convective Cores and Their Observed Properties . . . . . 336
10.2.2 Properties of Rotation in B-type Stars . . . . . . . . . . . . . . . . . 338
10.2.3 Magnetism Observed on Massive Stars . . . . . . . . . . . . . . . . . 339
10.3 Simulating Massive Stars with ASH . . . . . . . . . . . . . . . . . . . . . . . 341
10.4 Formulating ASH Simulations for B-type Stars. . . . . . . . . . . . . . . . 342

10.4.1 Experimental Configuration . . . . . . . . . . . . . . . . . . . . . . . 346
10.5 Hydrodynamics of the Core and Radiative Envelope . . . . . . . . . . . . . . 348
10.5.1 Convective Patterns and Mean Structures . . . . . . . . . . . . . . . 349
10.6 Dynamo Action Achieved in a B-type Star Core . . . . . . . . . . . . . . . . 354
10.7 Internal Gravito-Inertial Waves . . . . . . . . . . . . . . . . . . . . . . . . . 357
10.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
11 Modeling the Solar Near-Surface Shear Layer

363

11.1 Equations Solved in CSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
11.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
11.2.1 Radial Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
11.2.2 Latitudinal Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 370
11.2.3 Open Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 372
11.3 Diffusion Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
11.3.1 Newtonian Eddy Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 375

xii
11.3.2 Slope-Limited Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 376
11.4 Implementation and Evolution of CSS. . . . . . . . . . . . . . . . . . . . . 377

11.4.1 Program Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
11.4.2 Time Evolution and Error Analysis . . . . . . . . . . . . . . . . . . . 378
11.4.3 Modified Compact Finite Differences . . . . . . . . . . . . . . . . . . 380
11.4.4 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
11.5 Euler Equation in 1D with SLD . . . . . . . . . . . . . . . . . . . . . . . . . 386
11.6 Numerical Tests Through Linear Waves . . . . . . . . . . . . . . . . . . . . . 389
11.6.1 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
11.6.2 Alfvén Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
11.7 Effects of Diffusion Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
11.7.1 Formulating the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 394
11.7.2 Comparing Two Diffusion Schemes . . . . . . . . . . . . . . . . . . . 396
11.8 Entropy Rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
11.8.1 Formulating the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 398
11.9 Modeling Granular Convection . . . . . . . . . . . . . . . . . . . . . . . . . . 399
11.9.1 Nature of the Convection . . . . . . . . . . . . . . . . . . . . . . . . . 400
11.10Incorporating External Flows into CSS . . . . . . . . . . . . . . . . . . . . . 403
11.11Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
12 Perspectives and Ponderings

411

Bibliography

414

Appendix
A Derivation of the Multi-Fluid Electrodynamically-Coupled Field Equations

448

xiii
B An Exposition on Meridional Flows and Gyroscopic Pumping

456

B.1 Navier-Stokes Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . 456
B.2 Meridional Flows and Zonal Torques . . . . . . . . . . . . . . . . . . . . . . 458
B.3 Meridional Flows Due to Zonal Torques in ASH . . . . . . . . . . . . . . . . 460
B.4 Thermal Wind, Geostrophy, and Gyroscopic Pumping . . . . . . . . . . . . . 460
C Derivation of the Shear Instability Equations

466

C.1 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
C.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
C.3 Assumptions and Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . 468
C.4 Streamfunction Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
C.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
C.6 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
D Entropy Diffusion

474

E Reduced Sound Speed

479

E.1 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
E.2 Additional Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
E.3 Modified Sound Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . 482

xiv

TablesTable
6.1

Properties of F-type star simulations . . . . . . . . . . . . . . . . . . . . . . 196

6.2

Diagnostic flow parameters for the hydrodynamic F-type stars . . . . . . . . 208

7.1

Global properties of the evolved convection . . . . . . . . . . . . . . . . . . . 248

8.1

Diagnostic flow parameters for global-scale F-type stars dynamos . . . . . . 271

8.2

Global properties of evolved dynamos . . . . . . . . . . . . . . . . . . . . . . 293

10.1 Diagnostic flow parameters in the B-type star simulations . . . . . . . . . . . 345
10.2 Global properties of the evolved convection in B-type stars . . . . . . . . . . 355
11.1 Entropy rain properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

xv

FiguresFigure1.1

Solar granulation observations . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Supergranulation with HMI . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Solar transition region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4

Solar coronal observations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.5

The interplanetary magnetic field . . . . . . . . . . . . . . . . . . . . . . . .

10

1.6

Diagram of the heliotail . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.7

A Maunder diagram of modern sunspot observations . . . . . . . . . . . . .

14

1.8

Large-scale patterns of the solar cycle . . . . . . . . . . . . . . . . . . . . . .

17

1.9

A Maunder diagram of S. Schwabe’s sunspot observations (1825-1868) . . . .

21

1.10 A Maunder diagram of J. Staudacher’s sunspot observations (1670-1796) . .

23

1.11 Early sunspot drawings of G. Galilei and C. Scheiner . . . . . . . . . . . . .

27

1.12 Long-term sunspot variability . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.1

Photometric observations of V833 Tau . . . . . . . . . . . . . . . . . . . . .

37

2.2

Photometric observations of CoRoT-2a with modeled spots . . . . . . . . . .

38

2.3

Kepler observations of stellar differential rotation . . . . . . . . . . . . . . .

40

2.4

Structure of the chromosphere . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.5

Three-dimensional simulations of the Chromosphere . . . . . . . . . . . . . .

44

2.6

Photometric brightness and Ca II H and K variations . . . . . . . . . . . . .

45

xvi
2.7

Correlations of photometric amplitude and choromspheric activity . . . . . .

46

2.8

Scaling of magnetic activity cycles with Rossby number . . . . . . . . . . . .

49

2.9

Scaling of stellar magnetic field morphologies . . . . . . . . . . . . . . . . . .

51

2.10 Magnetic polarity reversals imaged on τ Böotis . . . . . . . . . . . . . . . . .

52

2.11 Magnetic field morphology of a massive star . . . . . . . . . . . . . . . . . .

54

2.12 Opacity bumps and convection zones in massive stars . . . . . . . . . . . . .

57

2.13 X-ray luminosity observations across the HR diagram . . . . . . . . . . . . .

59

2.14 Evolution of rotation period of stellar populations . . . . . . . . . . . . . . .

63

2.15 The magnetic activity-rotation correlation . . . . . . . . . . . . . . . . . . .

64

2.16 Activity-rotation correlation across spectral types . . . . . . . . . . . . . . .

68

2.17 The properties of magnetic activity in M-type stars . . . . . . . . . . . . . .

70

3.1

Sketch of the fast and slow dynamos . . . . . . . . . . . . . . . . . . . . . .

76

3.2

A menagerie of mean-field models . . . . . . . . . . . . . . . . . . . . . . . .

86

3.3

Babcock-Leighton flux transport dynamo with dual meridional cells . . . . .

88

3.4

Energy spectra for low and high-P rm dynamos . . . . . . . . . . . . . . . . . 102

3.5

Rotation within the solar interior . . . . . . . . . . . . . . . . . . . . . . . . 104

3.6

High-resolution global-scale simulation of solar convection . . . . . . . . . . . 111

3.7

A diagram of the action of the Coriolis force on convection . . . . . . . . . . 112

3.8

Differential rotation profiles in theory and observation . . . . . . . . . . . . . 114

3.9

The solar differential rotation arising from Balbus’s theory . . . . . . . . . . 123

3.10 A menagerie of global-scale dynamo simulations . . . . . . . . . . . . . . . . 127
3.11 Dynamo regimes old and new . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.12 A sketch of buoyant magnetic structures . . . . . . . . . . . . . . . . . . . . 131
3.13 A composite of time-latitude diagrams of toroidal field . . . . . . . . . . . . 133
4.1

The second-order TVD region and example limiting functions . . . . . . . . 153

4.2

Elements of slope-limited diffusion . . . . . . . . . . . . . . . . . . . . . . . . 155

xvii
5.1

Radial velocities with depth in a slope-limited ASH simulation . . . . . . . . 165

5.2

Properties of the differential rotation in D3S . . . . . . . . . . . . . . . . . . 168

5.3

Time variability of Bθ and the volume-averaged energy densities in Case D3S 170

5.4

Nature of the toroidal magnetic field in D3S . . . . . . . . . . . . . . . . . . 172

5.5Grand minima in a global-scale convective dynamo simulation . . . . . . . . 174

5.6

Generation of mean toroidal magnetic energy (TME ) in D3S . . . . . . . . . 177

5.7

Generation of mean poloidal magnetic energy (PME ) in D3S . . . . . . . . . 180

5.8

Auto and cross-correlation of Lorentz-force and toroidal field production bymean-shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.9

Comparison of poloidal and toroidal magnetic energy generation in an average
cycle of D3S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5.10 Comparison of poloidal field production by fluctuating EMF and toroidal field
production by mean-shear in an average cycle of D3S . . . . . . . . . . . . . 186
5.11 Dynamo-wave interpretation of the equatorward propagation in D3S . . . . . 188
5.12 Cuts of constant radius in the mean-shear production and fluctuating EMF . 190
6.1

A comparison of theoretical and observational differential rotation in F-typestars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.2

Evolved stratification in hydrodynamic F-type star simulations . . . . . . . . 205

6.3

Convective patterns in global-scale simulations of 1.2 M⊙ F-type stars . . . . 212

6.4

Convective patterns in global-scale simulations of 1.3 M⊙ F-type stars . . . . 217

6.5

Large-scale shear instability in Case A20 . . . . . . . . . . . . . . . . . . . . 220

6.6

Large-scale shear instability in Case B20 . . . . . . . . . . . . . . . . . . . . 222

6.7

Rayleigh instability criterion and differential rotation in Case B20 . . . . . . 224

7.1

Radial velocity and temperature patterns in Case A10 . . . . . . . . . . . . . 228

7.2

Radial energy fluxes for Case A10 and Case B10 . . . . . . . . . . . . . . . . 232

7.3

Extent of convective overshoot in global-scale simulations of F-type stars . . 234

xviii
7.4

Axisymmetric thermal fluctuations in Case A10 . . . . . . . . . . . . . . . . 237

7.5

Zonal vorticity balance in Case A10 . . . . . . . . . . . . . . . . . . . . . . . 241

7.6

A gyroscopic interpretation of angular momentum flux . . . . . . . . . . . . 245

7.7

Scaling relationships with mass and rotation rate . . . . . . . . . . . . . . . 254

8.1

Observed latitudinal differential rotation of τ Böotis . . . . . . . . . . . . . . 264

8.2

A comparison of theoretical and observational differential rotation in F-typestars with updated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

8.3

Comparison of convective patterns in Case A10 and Case A10m . . . . . . . 275

8.4

Comparison of convective patterns in Case A20 and Case A20m . . . . . . . 279

8.5

Snapshot of the magnetic field in Case A10m . . . . . . . . . . . . . . . . . . 281

8.6

Generation of mean toroidal magnetic energy (TME ) in Case A10m . . . . . 284

8.7

Generation of mean poloidal magnetic energy (PME ) in Case A10m . . . . . 288

8.8

The balance of axial torques in Case A10m . . . . . . . . . . . . . . . . . . . 290

9.1Time variation of volume-integrated energy densities for Case A20m . . . . . 299

9.2

Field line tracings of banded magnetic wreaths in Case A20m . . . . . . . . 301

9.3

Global magnetic field morphology evolving during a cycle in Case A20m

9.4

Evolving azimuthally-averaged poloidal field . . . . . . . . . . . . . . . . . . 310

9.5

Evolution of volume-integrated toroidal magnetic energy generation . . . . . 312

9.6

Evolution of volume-integrated poloidal magnetic energy generation . . . . . 314

9.7

Temporal cross-correlations of volume-averaged magnetic energy generation. . 303

rates for Case A20m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
9.8Reconstructed components of the mean-field α-effect for Case A20m . . . . . 317

9.9

A second reconstruction of α for Case A20m . . . . . . . . . . . . . . . . . . 322

9.10 A sketch of the basic processes occurring during a reversal in Case A20m . . 327
10.1 Magnetic field structure and evolution in a B-type star . . . . . . . . . . . . 334

xix
10.2 Mean stratification established in Case M4 . . . . . . . . . . . . . . . . . . . 347
10.3 Horizontally and time-averaged energy flux balance in Case M4 . . . . . . . 349
10.4 A hydrodynamic and MHD comparison of core convection . . . . . . . . . . 350
10.5 Convection in global-scale simulations of B-type stars . . . . . . . . . . . . . 351
10.6 Angular momentum balance in Cases H4 and M4 . . . . . . . . . . . . . . . 353
10.7 Magnetic and velocity fields in case M16 . . . . . . . . . . . . . . . . . . . . 354
10.8 Gravity waves excited in the radiative exterior of M4 . . . . . . . . . . . . . 358
10.9 Averaged rms velocities in B-type star simulations . . . . . . . . . . . . . . . 359
10.10Wave instability due to entropy diffusion and diffusion comparison . . . . . . 361
11.1 MHD simulations with CSS . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
11.2 Runge-Kutta stability regions . . . . . . . . . . . . . . . . . . . . . . . . . . 379
11.3 Compact finite difference dispersion errors . . . . . . . . . . . . . . . . . . . 381
11.4 Bandwidth increases using an optimized MPI process layout . . . . . . . . . 384
11.5 ASH and CSS scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
11.6 Sod shock tube problem with CSS . . . . . . . . . . . . . . . . . . . . . . . . 387
11.7 Comparison of slope-limited diffusion and Newtonian eddy diffusion . . . . . 395
11.8 Velocity PDF and energy flux in a SLD and NED . . . . . . . . . . . . . . . 396
11.9 Early simulations with entropy rain . . . . . . . . . . . . . . . . . . . . . . . 400
11.10More recent CSS simulations with plumes . . . . . . . . . . . . . . . . . . . 402
11.11Colliding flows in CSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
11.12Angular velocity profiles yielded in CSS simulations of solar dynamics . . . . 406
11.13Rotationally constrained convection in CSS . . . . . . . . . . . . . . . . . . . 407
11.14Angular velocity profiles with shear layers . . . . . . . . . . . . . . . . . . . 408
B.1 Comparison of meridional flow stream functions for 1.2 M⊙ F-type stars . . . 461
B.2 Comparison of meridional flow stream functions for 1.3 M⊙ F-type stars . . . 462

Chapter 1
A Perspective on Solar Magnetism“Stars... they are like living entities in a way, quite unpredictable.” – Timicin
Magnetic fields and phenomena associated with them pervade the observable universe.
The study of these phenomena has taken on a broad and necessary role in modern astrophysics. Indeed, magnetism impacts a wide variety of processes across the known progressionof spatial and temporal scales of the universe: from its microscopic imprint in synchrotron
emission emerging from supernova remnants to its macroscopic manifestations in collimating jets from supermassive blackholes in galactic centers. It also impacts processes fromsmall-scale phenomenon such as the pair production of electron and positrons in the hypermagnetized atmospheres of young neutron stars to large-scale processes such as Faraday
rotation of light traversing the vast gulfs between galaxies.
Magnetism may have even had a role in our universe’s cosmogenesis, either through battery effects or through phase changes in the quantum fields dominant during early inflation(e.g., Grasso & Rubinstein, 2001; Brandenburg & Subramanian, 2005). Thus magnetic fields
likely existed prior to the formation of the first stars. Those stars could amplify the primordial field through concerted dynamo action during their formation and evolution. Throughtheir often explosive demise, they may have quickly injected strong magnetic fields into thenascent universe. It was through numerous recyclings of this first field that permitted the
Sun to form from strongly magnetized plasma.
Yet the discovery of magnetic fields of stars is a relatively recent event, with the first2
observations of solar magnetism by George Hale in 1908. This discovery inspired a search formagnetism on other stars, giving rise to the field of the solar-stellar connection and to stellar
dynamo theory. It has turned out that magnetic fields impact processes in nearly all stars
at some point during their lifetimes. Indeed, magnetic fields likely are a crucial ingredientin the accretion processes of star formation as they can easily modify pressure balances andangular momentum transport. They also may lead to the acceleration and collimation ofpolar jets observed in young stellar objects.
Most lower mass stars on the main sequence, namely those possessing a convective outerenvelope like the Sun, are affected by magnetic processes. These stars exhibit a rich variety
of magnetic behavior that influence the properties of stars and their environment, including
any planets and circumstellar disks that they may host. Higher mass main-sequence stars
too possess magnetic fields that lead to both enhanced angular momentum and mass loss,
partly arising from dynamos in their convective cores. These fields are likely to have profound
impacts on the lifetimes of such massive stars and on the resulting magnetism of any stellar
remnants such as black holes, neutron stars, and white dwarfs.
Magnetic activity is now believed to be the primary driver for many of the propertiesof these stellar remnants. These include the X-ray spectrum and dramatic variability ofaccreting black holes and neutron stars, and the powerful relativistic jets that they produce.
Magnetic fields are also invoked as the main ingredient that focuses the most energetic
explosions in the universe: cosmic gamma-ray bursts. Yet despite the disparate regimes inwhich these stellar magnetic phenomenon exist, much of the physics that is applied in theseareas of astrophysics is based directly on our understanding of the Sun’s magnetic field.
To link these fields of study, a brief historical sketch provides an overview of some ofthe progress over the last century in the field of solar and stellar magnetism. This is thenfollowed by a summary of current observational and theoretical efforts to understand the
dynamics of magnetic fields in stars. Given that this field has largely observational origins
and that these observations provide key constraints on stellar dynamo theory, they will be3expounded upon first.
1.1

Spatial Scales of Solar Magnetism
The dynamics proceeding within the Sun and its sphere of influence exhibit a vast

range of spatial and temporal scales. A wide swath of our current understanding of physics
must be simultaneously invoked to understand these dynamics, from the detailed atomic andmolecular physics occurring at small-scales, which are linked across astrophysical scales to the
dynamics captured by kinetic theory and radiation-magnetohydrodynamics. The influenceof the Sun and its magnetism extends from the planets it hosts to the heliopause, where
it melds with the interstellar medium. Some of this range of scales is apparent in Figures
1.1 to 1.6: from some of the smaller scales of magnetic bright points in the roughly 100 km
wide intergranular lanes and to sunspots with 20 Mm scales (Figure 1.1), to supergranular
scales (about 30 Mm) and networks of field on the solar disk (Figure 1.2), to the activity
over its surface and in the transition region which occurs on scales of about 50 Mm as in
Figure 1.3, to the broad behavior of the Sun’s corona that exhibits scales of motion on theorder of several solar radii (Figure 1.4), to the interplanetary magnetic field with scales ofastronomical units (AU) (Figure 1.5), and finally to the outer boundary of the Sun’s influence
shown in Figure 1.6.
A small patch of the Sun’s surface is shown in Figure 1.1a, from a modern ground-based
telescope with adaptive optics. This exceptional image of the solar photosphere shows ingreat detail what was but “corrugations” to Herschel (William 1801). The patterns seen in
the image are representative of the ubiquitous granular pattern of convection that covers thesolar surface. Solar granulation occurs on a scale of about 1 Mm, with individual granules
lasting about 10 minutes. In Figure 1.1a, the granules are imaged in photometric intensity,reflecting the temperature characteristics of the flow. Namely, a granule consists of a warm
broad upflow rising from the deeper interior and cool narrow downflows. Rapid radiative
cooling brakes these buoyantly driven upflows as photons can escape from the photosphere.

4

Figure 1.1: (a) High resolution solar granulation in wide-band 3964 Å Ca II emission,
illustrating granulation with its 1 Mm scale, mesogranulation, bright points, and intergranular magnetism (23 May 2010, Swedish Solar Telescope). (b) Photospheric emission near
7060 Å passed through a TiO filter with a 10 Å bandpass, showing a sunspot with strong
magnetic fields in its dark umbra and filamentary penumbra (active region NOAA 1084) andthe surrounding granulation (2 July 2010, Big Bear Solar Observatory/NJIT).

The nearly sonic horizontal flows of these cells whisk this now cool fluid into the narrow
downflow lanes that define a granule’s boundary, where additional cooling further accelerates
them inward. These flows also sweep fibrils of magnetic field into the intergranular lanes.
These field fibrils form the bright points and networks seen in Figure 1.1a and represent
the smallest currently resolvable magnetic feature on the Sun. Magnetic structures likely
exist below this scale. Many of these smaller scales may be influenced by small-scale local
dynamo action occurring both in the regions of strong shear between upflows and downflows
and within the helical turbulence of the downflow itself (Vögler & Schüssler, 2007; Stein,2012).
Magnetism has a substantial influence on the processes occurring below the solar surface. Figure 1.1b shows the omnipresent pattern of granulation, but punctuated with thedark umbra and penumbra of a single sunspot and lightened by emission from magnetic

5
structures in the intergranular lanes. Sunspots are regions where a fluted loop of strong
magnetic field perforates the photosphere and can persist for just a few hours or up to several months. The magnetic field associated with the spot is typically several kG in strength,
which largely suppresses convection in the region. In this image it appears dark since theblackbody emission from the region has a much lower temperature, about 1000 K cooler,than the surrounding photosphere. In this region, the magnetic pressure is a large fractionof the gas pressure. Since the spots exist very much longer than the sound crossing time
across it. The sunspot is roughly in pressure equilibrium with the surrounding plasma, which
requires that the gas pressure (and correspondingly the temperature) be lower in the spot.
This leads to less emission in the observed bandpass, and to diminished emission at manyother wavelengths as well. Recently, there have been major advances in understanding not
only the surface dynamics but also the deeper dynamics within the upper 20 Mm of the Sun
in regions around a sunspot, arising from both helioseismology and numerical simulations(Rempel & Schlichenmaier, 2011).
Supergranulation is a distinct convective scale seen at the solar surface, particularly
in Doppler images, which exhibit spatial scales of about 30 Mm and persist for about a day(Leighton et al., 1962; Simon & Leighton, 1964; Rieutord & Rincon, 2010). This larger convective pattern is shown in Figure 1.2a in a Dopplergram from the Helioseismic and Magnetic
Imager instrument (HMI) on the Solar Dynamics Observatory (SDO). Supergranulation is
one of three distinct scales of convective patterns seen on the solar surface, along with granulation and the less well defined mesogranulation (which in turn has scales of about 5 Mm
and lifetimes of a few hours, November et al. (1981); Rast (2003)). The mesogranular scale issomewhat evident as the smaller scale corrugations within the dominant supergranular signal. As with granulation, the supergranular signal is complicated as the line-of-sight (LOS)
velocity changes across the disk: near disk center the LOS velocity of the plasma is predominantly radial, while near the limb it is mostly horizontal. So what is imaged across mostof the disk are the fast 500 m s−1 diverging horizontal flows, giving rise to the alternating

6

Figure 1.2: (a) The supergranular scale becomes apparent in time-averaged line-of-sight
Dopplergrams of the surface convection from 21 July 2013. Blue tones indicate material
moving toward the observer, and red tones the opposite. (b) An HMI magnetogram also
from 21 July 2013, showing the line-of-sight magnetic field. The magnetogram is derived
from spectropolarimetric measurements of the solar surface and serves to illustrate the supergranular magnetic networks in less active regions and the strong magnetic field linking
sunspots in active regions. The field shown in red (blue) is directed toward (away from) the
observer.

pattern of red and blue shifts seen in Figure 1.2a, with the front of a supergranule moving
toward the observer and the back away. Figure 1.2b shows that the small-scale magnetic
field horizontally advected by granulation and swept into the intergranular lanes organizes
into larger-scale networks. This serves to demarcate the supergranules as magnetic fieldcollects in their narrow downflow lanes. This figure also displays active regions that bristlewith strong magnetic field that link sunspots of opposite polarity in arcades that soar above
the photosphere as seen in Figure 1.3a.
Figure 1.3 shows emission in the Fe IX resonance line at 171 Å, which samples temperatures of about 0.6 MK. These temperatures occur in the transition region between thechromosphere and the corona. A single snapshot of this region is seen in Figure 1.3a, with a7

Figure 1.3: (a) The solar transition region imaged in the 171 Å channel with AIA on SDOfrom 20 July 2013, dark tones indicate areas of high emission and light tones low emission. (b) A composite of 25 images from the period between 16 March 2012 to 15 March2013 in the same Fe IX resonance line, illustrating latitudinal bands of magnetic activity(NASA/SDO/AIA & S. Wiessinger).

prominent Ω loop of magnetic field linking two sunspot groups visible near the prime meridian in the northern hemisphere. Indeed, the magnetic field associated with sunspots canrise high into the solar atmosphere leading to inhomogeneity and local structural changes
in the chromosphere and transition regions due to the loss of hydrostatic balance. This isevident in the composite image in Figure 1.3b, where the presence of active regions andstrong magnetic fields at low heliographic latitudes are associated with a more extensive
chromosphere. This is partially due to the low plasma β (ratio of gas pressure to magnetic
pressure) in these regions, which causes the dynamics do be dominated by changes in themagnetic field. It is also a result of hot plasma being guided along the field lines and lifted
higher into the atmosphere. Thus, over this year-long sampling, the base of the corona isfurther from the solar photosphere at low latitudes than at the poles, with the transition to
the corona occurring at a height two to three times greater. More to the point, however,8

Figure 1.4: Solar corona imaged during the 22 July 2009 total eclipse, showing the structureof the coronal magnetic field to about 5 solar radii (M. Druckmüller). A STEREO Helium
304 Å image from the time of the eclipse is lain over what would otherwise be the moon
illuminated by earthshine (NASA/STEREO).

the emergence of active regions in latitudinal bands is a property of the global nature of themagnetic field and hints at the deeper mechanisms behind solar magnetic activity and its
variability.
In the corona, plasma is largely frozen into the magnetic field and accelerated awayfrom the Sun in its wind, with its density dropping quickly with increasing distance from thephotosphere whereas the temperature remains nearly constant to nearly 10 R⊙ . These fields
are normally difficult to image without something to mask the intense barrage of photons
coming from the solar disk, so eclipses present an excellent opportunity for both enthusiasts
and professionals to capture great images such as the one in Figure 1.4. This image shows

9
the structure of the fields lines in visible light near a minimum in the solar activity cycle. The
field lines can be seen out to roughly 5 solar radii, with nearly open field lines stretching fromthe polar regions. Such open magnetic fields occur in coronal holes, which allows plasma
to be rapidly evacuated along the field lines and so form the fast solar wind. On the other
hand, the field is much more structured at low latitudes with some closed loops, often withthe cusped field lines called helmet streamers. It is more difficult for the plasma to escapethese regions, where the wind must do work against magnetic tension in order to escape andconsequently loses some of its kinetic energy to the magnetic field. Hence, it is thought thatthese lower-latitude structures are responsible for the slow component of the solar wind. Yet
once this plasma has escaped the Sun’s surface, it can impact the planets and speed toward
the boundary of the Sun’s magnetic influence, where it eventually merges with magnetic
fields and flows of interstellar space.
As the Sun rotates, the magnetic fields emanating radially from its surface are slowly
wound, approximately producing an Archimedean spiral, which has been dubbed the Parker
spiral after Parker (1958). Examples of this field can be seen in Figure 1.5, where three fieldlines are traced from the inner solar system to the heliopause. This figure also shows how
the magnetic field becomes more tightly wound outside the termination shock. This shockforms when the pressure of the solar wind, which decreases as the square of the distance fromthe Sun, becomes equal to the ambient pressure in the interstellar medium (ISM). It forces
the solar wind to slow to become submagnetosonic, heating and compressing the plasma andcausing the streamlines of the solar plasma to slowly bend toward streamlines of the plasma
flow in the ISM. This means that it is a dynamical, asymmetric surface that is modulated by
the solar cycle, which has associated increases in particle fluxes, magnetic field strength, andwind speeds (Stone et al., 2008). The termination shock is believed to vary between 75 to
90 AU from the Sun, as measured by the two Voyager spacecraft (Stone et al., 2005). This
shock also causes the magnetic field to be compressed, which sets the stage for its connection
to the external magnetic field of the ISM.

10

Figure 1.5: A sketch of the Sun’s magnetic field, showing magnetic field lines spiraling
outward from the photosphere, crossing the termination shock, and merging with the ISM
at the heliopause. Three field lines are shown, one from the solar equator, and two fromincreasingly higher heliographic latitudes (S. Suess/NASA).

In order to probe the outer limits of the Sun’s magnetic influence, two recent advances
must be considered. The first is that the long-running Voyager mission has revealed a muchstronger magnetic field outside of the termination shock than previously expected. It is on
the order of 5 µG oriented about 30◦ above the galactic plane (Opher et al., 2009), whereas
previous estimates had placed this value closer to 2 µG and at higher inclinations. Thus the
local ISM is dominated by magnetic pressure with β ∼ 1/2. This has several implications
for the heliosheath, which is the region where the solar wind and the Sun’s magnetism
merge with that of the local ISM. It has recently been postulated that Voyager has alreadycrossed the boundary into interstellar space and that the transition was just not as abrupt
as expected (Swisdak et al., 2013). The second observational advancement comes from thesurprising observations of the Interstellar Boundary Explorer (IBEX).

11
North

(a)

3.0

Fast Solar Wind
Slow
Solar
Wind

Slow Solar Wind

30
Spectral Index

Vism

60

(b)

2.5

2.0

150

Starboard

120

90

60

30

0

Port

Downwind

1.5

Fast Solar Wind

Bism

-30

1.0

Downﬁeld
-60

South
-90

Figure 1.6: Schematic diagram showing (a) a cut parallel to the ISM velocity (vism ) throughthe heliotail. (b) A Mollweide projection of the downwind hemisphere of the all-sky integrated IBEX spectral index observations, with the downwind direction at the origin showing
the full latitude but only the longitudes between −15◦ and 175◦. This shows the twistingof the solar wind plasma sheet in the direction of the external magnetic field (Bism ). Fast
solar wind regions in (a) are indicated in blue and slow solar wind regions in green. The tailstructure in (a) fades with distance as charge exchange losses deplete the heliospheric ions,and is tapered as external magnetic tension flattens it and the ions thermalize with the ISM(Adapted from McComas et al. (2013)).

Figure 1.6a presents a diagram of a potential heliosheath configuration, showing its
cylindrical structure parallel to the direction of the mean flow of the interstellar medium(ISM). In Figure 1.6b the heliosheath is viewed as if looking down the tail, where observations
from IBEX are displayed. IBEX detects back scattered solar wind ions that have been
converted to energized neutral atoms through charge exchange interactions with the ISM
plasma in the inner heliosheath. The intrinsically very low brightness of the heliosheathmeans that it cannot currently be observed by other means. Some of these scattered atoms
are aimed at Earth, and since they are virtually unaffected by magnetic fields they travel instraight lines.
Moreover, these observations show that the kinetic energy spectra of weakly energizedneutral atoms have slopes that vary spatially, being dominated by its quadrupolar component. Figure 1.6b shows that the slow-wind structure is tilted by roughly 10◦ , which islikely due to an interaction with the external ISM magnetic field. The direction of this field,

12
projected onto the observing plane, is indicated in the lower right of Figure 1.6b. The slow
wind coming from the upwind side of the Sun is initially headed in the upwind direction. So
it fills a thin layer just inside of the heliopause; the latter is shown with a thin black line.
Given that there is a spectrum of wind velocities, the charge exchange interaction produces
a heliotail with an extinction distance that roughly varies with the angular distance from thedownwind flow direction. Homogenization through the exchange interaction also means thatthe tail is effectively evaporated within about 1000 AU. The heliosheath and heliotail should
be squeezed along the ISM flow direction as the external magnetic and particle pressuresincreasingly push in from the sides. These processes are sketched in Figure 1.6a, where theheliospheric tail fades out and is tapered. It is here that the Sun’s influence finally wanes,
defining the largest spatial scale of its magnetism.
With the spatial extents of the Sun’s observable magnetic field mapped out, the nextquestion to ask is how does it vary with time? Thus the following sections detail the timescales exhibited by the Sun’s observed magnetic fields. This ranges from indirect measurements utilizing our current knowledge of the solar-terrestrial connection and paleogeologic
records to the most reliable and complete record of the Sun’s magnetic activity: direct observations of sunspots. The modern era of sunspot observations provides a reasonable point
to launch this retrospective look at the Sun’s temporal variability.
1.2

Modern Sunspot Record
The modern record of sunspots begins in May 1874, with subsequent observations

taking place across the world. The powerhouse behind the meticulous collection of sunspot
areas and positions was the Royal Greenwich Observatory (RGO). Between this observatory
and its sister sites in Cape Town, Kodaikanal, and Mauritius, photographic plates were used
to collect sunspot data continuously between 1 May 1874 and 31 December 1976. These
data included both overall sunspot area and just the umbral area, which were corrected forforeshortening effects and were given in micro-Hemispheres (µHem or 2π×10−6 steradians) as

13in Figure 1.7b. In 1977 these measurements became the domain of the U. S. Air Force (USAF)
and the National Oceanic and Atmospheric Administration (NOAA), with data being taken
at five USAF Solar Observing Optical Network (SOON) sites. The techniques between
the two observing campaigns differ in that sunspot areas are measured differently and thatumbral areas are not measured. Other observatories have contributed to the overall effort
with sunspot data available from Catania (1978-1999), Debrecen (1986-1998), Kodaikanal(1907-1987), Mt. Wilson (1917-1985), Rome (1958-2000), and Yunnan (1981-1992). This isuseful when constructing a contiguous record, filling in any gaps in either the NOAA or RGO
records (Hathaway, 2010).
The Wolf sunspot number R has enjoyed long-term use since its inception in 1848 by
Rudolf Wolf (e.g. Wolf, 1850, 1861). It is defined as a linear function of the number of sunspot
groups g and the total number of distinct spots s, with R = KO (10g+s), where KO is constantthat is intended to compensate for different observational methods. The later is particularly
relevant when attempting to extend the modern record back to the first Western observationsin 1610. An alternative measure is the Group sunspot number, which was introduced byP
Hoyt & Schatten (1998). The Group number is defined as RG = 12.08/N N
i ki Gi , with

N the number of observers, ki the i-th observer’s correction factor, and Gi the number ofsunspot groups observed by observer i. The factor of 12.08 normalizes RG to R.
Though they have a shorter record than either the Wolf or Group sunspot numbers,
sunspot areas are thought to be a more physical measure of solar activity since their area
correlates with the flux of the magnetic field. Furthermore, as opposed to earlier observations, the modern record also provides accurate positional information. This allows the timedependent joint distribution of sunspot area and latitude (see Figure 1.7a), and/or longitude, to be easily examined without having to infer the coordinate system from dates anddrawings as in earlier observations. Such figures are called Maunder or “butterfly” diagrams,
following Edward Maunder’s work at the turn of the last century (Maunder, 1904).
Such diagrams show some of the fundamental temporal and spatial features of the

14

Figure 1.7: (a) A Maunder diagram that shows sunspot areal coverage in heliographic latitude
and time, illustrating the equatorward propagation of spot emergence over the course of themany 11 year Schwabe cycles. Colors indicate sunspots with areas in the lower (gray), middle(red), and upper (blue) tertiles of their distribution. (b) Average monthly sunspot area inµHem, displaying the variability and longer-term modulation in sunspot area over the modern
sequence of Schwabe cycles. (c) Average monthly sunspot area in each hemisphere (units as
in panel b), indicating the phase differences between the northern (blue) and southern (red)hemispheres. (Here constructed from data provided by D. Hathaway/NASA)15
solar dynamo, with the foremost such time scale being the roughly 11 year Schwabe cycle insunspot area (Schwabe, 1844). During these magnetic activity cycles the total sunspot areaand emergence frequency increases as the cycle progresses toward maximum and declines in
its tail end. Each of these cycles is quite self-similar, with the migration of the magnetically
active latitudes from mid-latitudes at the beginning of a cycle to low latitudes near its end.
The quantification of this behavior is one of Gustav Spörer’s contributions from the latenineteenth century, and has been dubbed Spörer’s law of zones (Maunder, 1903). Thereis no mathematical rigor to this law, and it was first hypothesized by Richard Carrington
in 1863 (Carrington, 1863). Yet the law of zones is qualitatively accurate for most known
Schwabe cycles (11 year magnetic activity cycle) and was formulated and demonstrated bySpörer (1889): “Un peu avant le minimum, il n’y a de taches que près de l’equateur solaire,
entre +5◦ et −5◦ . A partir du minimum, les taches, qui avaient depuis longtemps déserté
les hautes latitudes, s’y montrent brusquement vers +30◦ . Puis elles se multiplient, un peu
partout, à peu prés entre ces limites, jusqu’au maximum, mais leur latitude moyenne diminue
constamment jusqu’à l’époque du nouveau minimum.”As Spörer describes, this migratory pattern of sunspot emergence from mid-heliographic
latitudes near the minima in magnetic activity to low-heliographic latitudes near maxima is
consistent between cycles. In particular, for the records available since the mid-eighteenth
century, even as the Sun entered the Dalton Minimum (1798-1825, see Figure 1.12), thecycles in magnetic activity are remarkably self-similar (Figure 1.7a). However, the precise
origins of this particular behavior remain somewhat mysterious. Though, as will be shownin Chapter 3, there are theoretical clues that may lead to a better understanding of the solar
cycle.
As evident in Figure 1.7a, it is notable that the cycles overlap by about a year withspots of the former cycle still emerging near the equator while spots of the succeeding cycle
begin to appear at higher latitudes. The area of sunspots also builds towards a maximum(Figure 1.7b), which indicates that the mechanism producing them may be rising closer to the16
surface or is increasing in intensity. The latitudinal extent of the activity bands widen during
the rise to the maximum and narrow during the decline. Furthermore, the stronger cycles,
namely those with greater monthly area, tend to have sunspots emerging at higher latitudes
at the beginning of the cycle than the weaker cycles. The cycles are highly symmetric about
the equator which is likely reflecting the strong dipole linking the hemispheres and a possible
underlying toroidal magnetic field structure in the deeper interior.
If the cycles are considered independently, it is also clear that the Schwabe cycle is only
approximate as cycles can vary in length by several years, although, the standard deviation of
the cycle length is only 14 months. Appealing to Figure 1.7b, it is clear that cycles exhibit apositive skew, with a rapid rise toward maximum and a longer decay toward minimum. Thereis additionally a great deal of shorter time scale variability within a given cycle, particularlyin the daily sunspot data, though, it is also evident in the monthly data displayed here. The
multi-peaked nature of some of the cycles occurs when the two hemispheres become slightly
out of phase so that they peak at slightly different times, as can be seen in Figure 1.7c. This
phasing difference is most apparent as the cycle winds down with one hemisphere tending
to have a greater number of spots.
Sunspots tend to erupt successively in groups that are more extended in longitude thanin latitude. These groups often have one magnetic polarity associated with the westerly
spots and the opposite polarity associated with the trailing spots (Figure 1.8b), reflecting
the symmetry arising from the Sun’s rotation. This feature of the Sun’s magnetism was firstdescribed by Hale et al. (1919): “We already know that the preceding and following spotsof binary groups, with few exceptions, are of opposite polarity, and that the corresponding
spots of such groups in the Northern and Southern hemispheres are also of opposite sign.
Furthermore, the spots of the present cycle are opposite in polarity to those of the last cycle.”This is now codified as Hale’s law.
The spots can form large groups, with the leading spots positioned at lower latitudes
than the trailing spots, as in Figure 1.8a, and is called Joy’s law after Hale et al. (1919): “It17

Figure 1.8: (a) A sunspot group in the southern hemisphere exhibiting Joy’s law, with thesmaller trailing spot (lower left) tilted away from the E-W line. (b) Magnetograms fromsunspot cycles 22 (left), 23 (middle), and the current cycle 24 (right) displaying the lineof-sight magnetic field, with negative field in blue and positive field in yellow. This shows
Hale’s polarity law. (c) A magnetic Maunder diagram built from longitudinally averaged
Carrington maps of the line-of-sight magnetic field, which were obtained from Kitt Peak and
SOHO. This serves to illustrate the polar field reversals, and the poleward advection or slipof magnetic field (July 2013, D. Hathaway/NASA).

18
is probably a significant fact that the axis of the group usually makes only a small angle withthe equator. A study by Joy [...] shows that there is little change in this angle during thelife of the group, but that in the mean the angle bears a definite relation to the latitude of
the group. [...] The following spot of the pair tends to appear farther from the equator thanthe preceding spot, and the higher the latitude, the greater is the inclination of the axis tothe equator.” This is thought to represent the twisting of the underlying magnetic field by
convection as the flux tube of the sunspot rises toward the Sun’s surface layers (Fan, 2009).
This hypothesis has recently been challenged in an extensive and systematic statistical studyof bipolar regions from the Michelson Doppler Imager (MDI) instrument on the Solar andHeliospheric Observatory (SoHO) (Stenflo & Kosovichev, 2012). They find that about 4% of
all large bipolar magnetic regions violate both Joy’s and Hale’s laws, with a larger fraction
for the smaller-scale bipoles.
Figure 1.8c displays the actual magnetic properties of sunspots during a Schwabe cycle.
Primarily, the magnetic polarity of the sunspots reverse between cycles, with the reversal
beginning as the cycle nears and enters a minimum in activity. This simple fact was a tremendous surprise when first uncovered during the solar minimum of 1912 (Hale, 1915), with agreat deal of effort going into confirming this unexpected behavior. The polar magnetic
fields, on the other hand, reverse polarity earlier at about the time of maximum activity, aconsequence of a large amount of magnetic flux having been transported to the pole as firstdescribed in Babcock (1959).
1.3

Origins of Modern Sunspot Observations
The modern methods of solar observations were largely pioneered in the nineteenth

century, being a product of those vigorous times early in the industrial revolution. There
were several important discoveries that would bear fruit more immediately, and a few thatwould have to wait nearly a century for our theoretical understanding to catch up to theobservations.

19
Though it did not immediately lead to multiple breakthroughs in astronomy, given
Joseph von Fraunhofer’s predilection for optical physics, his invention of the spectrographin 1819 and improvements in the production of quality of optical glass were critical to later
observations. It did, however, lead to his rediscovery of the “dark lines” in the solar spectrum, which had been noticed 15 years earlier (Wollaston, 1802), that piqued the interestof many astronomers. Yet Fraunhofer pursued these observations primarily as a means to
produce standards by which the quality of optical glasses could be quantitatively compared(Fraunhofer, 1817). Yet with the spectroscope in the hands of the famous duo of a chemist
Robert Bunsen and a physicist Gustav Kirchoff, it was realized that elements could be identified through their spectral lines by passing white light through heated gases (Kirchhoff &Bunsen, 1860). It was quickly understood then that the Fraunhofer lines could be used toinfer properties of the solar atmosphere (Kirchhoff, 1860). This was a major breakthrough inastrophysics leading to the discovery of many new objects and providing new ways of understanding and classifying stars, and eventually to Hale’s discovery of the magnetic characterof sunspots.
While the direct study of the Sun’s magnetism was not yet available, several dedicatedobservers throughout the nineteenth century provided nearly continuous, reliable, and independent observations of the Sun between November 1825 and the beginning of the modern
record in 1874. There were four significant contributors to solar astronomy during this period: Richard Carrington, Samuel Schwabe, Gustav Spörer (mentioned above), and Rudolf
Wolf. They each became engaged in work regarding the primary solar research focus of thetime, which was the nature of sunspots and the solar atmosphere.
By virtue of birth, Samuel Schwabe was the first of these three to begin observing. He
fastidiously recorded the solar disk nearly every day from 5 November 1825 to 29 December
1867, producing 8,486 drawings thereof. Initially, this apothecary turned astronomer was
hunting for a planet inside of Mercury’s orbit and not for sunspots. Yet his meticulous
observations amounted to a total of 135,000 sunspot measurements. Overall his drawings

20
are accurate to around 5% of the solar radius or to about 3◦ in heliographic coordinates.
Most of these drawings provide a coordinate grid aligned with the celestial coordinate system,
though a subset of 1168 drawings have no indication of their orientation. Despite not finding
the hypothesized planet Vulcan after 17 years of dedicated observation, Schwabe described
the fundamental frequency exhibited by the Sun in Schwabe (1844): “Vergleicht man nun
die Zahl der Gruppen und der fleckenfreien Tage mit einander, so findet man, dass die
Sonnenflecken eine Periode von ungefähr zehn Jahren hatten und daß dieselben fünf Jahre
hindurch so häufig erschienen, daß in dieser Zeit wenig oder keine fleckenfreie Tage stattfanden.”Given its import, it bears repeating that by “comparing the number of days withsunspot groups and those without, we find that the sunspots have a period of about ten
years and that within this period, five years are so heavily spotted that hardly any days
were spot-free.” This hint at a periodicity in solar maculation prompted a great deal ofinterest, though only after being popularized by Alexander Humboldt in his famous treatise“Kosmos” (Humboldt & Bromme, 1850). This is not to say that Schwabe slackened his
observational ardor after publishing 1844, rather he continued apace until 1868.
The entire data set collected by Schwabe has recently been digitized and collected ina data base (Arlt et al., 2013), and are shown in Figure 1.9. The properties of the solar
cycles here are quite similar to those of the modern record, with the equatorial propagationof sunspot emergence locations as the cycle progresses (Figure 1.9a). Yet there does appear
to be a greater difference in phase between the hemispheres than in later cycles, which ismore clearly illustrated in 1.9c. This may be related to a recovery in magnetic activity
following the Dalton Minimum that began in April 1798 (Usoskin et al., 2009), thoughdetailed observations from 1796 to 1825 seem to be lacking. Figure 1.9b shows the Groupsunspot number (RG ) as the orange filled curve with the data provided by D. Hathaway. The
overlain blue curve shows the sunspot area as derived from Schwabe’s data. Overall there isgood agreement between these two activity measures, though, there is a curious discrepancy

21

Figure 1.9: (a) A Maunder diagram showing sunspot areal coverage in latitude and time
between 1825 and 1868. Sunspot area is broken into 12 bins determined from Schwabe’s
drawings (Arlt et al., 2013). Colors indicate sunspots with areas in the bottom four ofthese classes (gray), medium sized spots of the middle four classes (red), and larger spots
and groups in the upper four classes (orange). (b) Group sunspot number multiplied by 20
approximates the area and is the filled orange curve, while the area integrated over all theclasses is shown in blue, displaying the monthly variability in sunspot area over these four
Schwabe cycles. (Constructed from data provided by NOAA). (c) Average monthly sunspot
area in each hemisphere (units as in panel b), indicating the phase differences between
the northern (blue) and southern (red) hemispheres (Realized with data provided by R.
Arlt/AIP).

22
between them in the cycle beginning in roughly 1834.
The period between November 1853 and April 1861 is one that can be cross-correlatedwith Richard Carrington’s observations (Carrington, 1863). Largely inspired by Schwabe’s
observations of possible solar cycles and also by his distaste for the lack of systematic observations, Carrington became one of Schwabe’s contemporaries, embarking on a crusade tosystematically observe an entire solar cycle. Ultimately, after seven years of devising new
observing techniques and proper coordinate systems, he divested himself of these pursuits
after he was excluded from the directorship of the Cambridge observatory by the university’s
syndicate. Nonetheless he continued his solar observations for two additional years and in yet
two more years time published his tome on sunspots “Observations of the Spots on the Sunfrom 9 November 1853 to 24 March 1861, Made at Redhill” in 1863. This work contained
major contributions to solar physics: the accurate quantification of the solar differential rotation (though his was not its initial discovery), the co-discovery of equatorward migrationof sunspot emergence with Gustav Spörer, the hitherto most precise measurement of thesolar rotation axis relative to the ecliptic, and a happenstance observation of a white lightsolar flare. The last discovery was a furtherance of the growing evidence for the linking of
geomagnetic and solar magnetic activity. This connection between the 11-year solar activity cycle and terrestrial magnetic phenomenon was a notion devised independently by fourscientists in 1852, though Edward Sabine was the first (Sabine, 1852), and will be discussed
further in §1.5.
Rudolf Wolf was one of these four scientists, and another inspired by Schwabe’s findings.
He too undertook a multi-decade observing campaign beginning in 1848 and continued until
shortly before his death in 1893. To better understand the Sun’s cycles, Wolf aimed toconstruct as complete a record of observations as possible, assembling a large trove of verbal
and pictorial sunspot observations from the previous two centuries. Having begun thispursuit in 1850 (Wolf, 1850), he had largely finished reconstructing this record by 1866
(Wolf, 1866). He was able, in addition to his own observations, to produce a fairly complete

23

Figure 1.10: (a) A Maunder diagram showing sunspot areal coverage in latitude and time
between 1670 and 1796, as determined from multiple sources in Ribes & Nesme-Ribes (1993)
and from Staudacher’s drawings (Arlt, 2009). Colors indicate sunspots with areas in thesmaller areas (gray), medium-sized spots (red), and larger spots and groups (orange). (b)Group sunspot number multiplied by 20 as it approximates the area (filled orange curve),
constructed from data provided by NOAA.

observational record dating back to 1610. Yet he suffered from the problem of calibrationthat often appears when comparing disparate data. To accommodate the varying data sets
Wolf established a consistent measure now called the Wolf, Zürich, or international sunspot
number, which is still in prominent use today. In particular, it provides a significant portionof the information used to construct the Group sunspot number (Hathaway, 2010). This
Group number is what has been shown here and in Figures 1.7, 1.9, and 1.10.
The observations of Galileo and others in the seventeenth century provided the impetus
for subsequent observations throughout much of the Maunder Minimum (a long period with

24
very few sunspots between 1645-1715, see Figure 1.12) and into the eighteenth century,
including those of a contingent of French observers at the Observatoire de Paris such as Jean
Picard (Ribes & Nesme-Ribes, 1993). The set of daily measurements from the Observatoire
accumulated to about 8,000 between 1671 and 1719. These drawings cover the later halfof the Maunder Minimum, showing sunspot positions and area which are displayed as a
Maunder diagram in Figure 1.10. These observations had been previously known, but not
fully quantified as they were in Ribes & Nesme-Ribes (1993). Indeed, in two papers shortly
before his death, Spörer described the period later dubbed the Maunder Minimum. He,
and subsequently Edward Maunder, presented evidence that for much of this period there
were no sunspots, and that for about half its 70 year duration spots were only sporadicallyobserved in the southern hemisphere (e.g., Spörer, 1890; Eddy, 1976). These states withsingle hemispheres of dynamo action have also been seen in some simulations and models(e.g., Tobias, 1997; Gallet & Pétrélis, 2009; Augustson et al., 2011). What is also of someinterest, following from the work of Eddy and Ribes, is that these early sunspot observations
indicated that the Sun’s differential rotation was markedly different during the Maunder
Minimum. Sunspot tracking suggests that the solar differential rotation was actually stronger
and had a much steeper latitudinal decline rate between the equator and poles, supporting
the notion of an equatorially confined jet (Eddy et al., 1976).
Also shown in Figure 1.10 are another large set of sunspot drawings that encompass
several cycles, though it was not until 1749 that these regular measurements began. This
was largely the work of Johann Staudacher, who spent nearly 47 years drawing observationsof the solar disk. His drawings over this period, unlike Samuel Schwabe’s, did not possessan accurate coordinate system and often lacked the orientation information for the solar
disk. Nonetheless digitized Staudacher’s drawings have been used to determine sunspot
positions for the interval 1749 to 1796. Several techniques have been employed to reconstruct
the sunspot positions, the choice of which depends on the quality and accuracy of a given
drawing. These methodologies permitted the determination of the heliographic latitudes and25longitudes for 6,285 sunspots over 999 days of this 47 year interval (Arlt, 2009).
The results of this reconstruction are shown in Figure 1.10, where the resulting Maunder diagram exhibits a dense population at or near the equator during the first two cycles(cycles 0 and 1 in the current parlance). They seem to occur even near the beginning of thecycle, where they would normally be at much higher latitudes. Cycle 2 shows an intermediate behavior with two prominent bands at higher latitudes, but still with a great deal ofequatorial activity. However, cycles 3 and 4 show a typical butterfly shape. This could be theresult of Staudacher’s improving skill, but that does not seem particularly likely given that
he had already been observing for many years when cycle 1 began. A tentative explanation
may be the transient dominance of a quadrupolar magnetic field during the first two cycles,
as can occur in the chaos of deterministic systems (e.g., Weiss & Tobias, 2000; Brandenburg& Spiegel, 2008) and as can be seen in some dynamo models (Dikpati & Gilman, 2001).
1.4

Beginning of the Western Sunspot Record
In this retrospective look at solar magnetism, the clock has ticked back 350 years to

the midpoint of the Maunder Minimum. How far back does this record go and is it still
reliable and sufficiently accurate to characterize the longer term variability? At this point,
the direct documentation of solar activity becomes increasingly sparse. However, there isevidence that solar observations were pursued in China some 2,900 years ago, with the firstsunspots being recorded during the Xi Han dynasty about 165 BCE. Looking across multiple
cultures and through 1,500 years of historical records, there at least 235 drawings and verbal
accounts of naked eye sunspot sightings spanning the interval between 165 BCE and 1684
CE (Wittmann & Xu, 1987). Yet the world had to wait until the seventeenth century forthe advent of optics and telescopes before they could be more routinely observed.
More systematic study and observation of sunspots began following Galileo Galilei andThomas Harriot’s solar observations in 1610, with examples of such early observations shownin Figure 1.11. However, the first published work was that of Johannes (Goldsmid) Fabricius

26
in 1611 entitled “De Maculis in Sole observatis, et apparente earum cum Sole conversione
narratio,” or “Of spots observed in the Sun, and a narrative on their apparent rotation with
the Sun.” When translating the title, it is easy to overlook the latter clause. Yet it is equally
profound that in the short few months between Galileo’s observations, published in his poetry
inspiring “Sidereus Nuncius” and his subsequent formal letters to Markus Wesler from the
Accademia dei Lincei, Fabricius found that the spots also indicated the Sun’s rotation.
However due to Galileo’s prominence, his telescopic and heliocentric revelations about theuniverse, such as the phases of Venus, Jupiter’s moons, and of course sunspots, greatly
disturbed many in the Ptolemaic school of thought. This included Christoph Scheiner who,
given that he was a Jesuit priest, was deeply entrenched in Ptolemaic dogma. The academic
rivalry and personal animosity that grew between these two scholars drove the initiallyskeptical Scheiner to several major discoveries. Particularly, Scheiner’s stubborn adherence
to the perfection of the Sun drew Galileo’s wrath. Though it probably did not help that
Scheiner was a Jesuit given the near constant threat of inquisition raised by the Catholic
church. Moreover, despite many years of observations, Scheiner continued to believe for sometime that the spots were satellites in a very close orbit. Galileo did eventually disabuse himof this notion after several vibrant exchanges with one of Scheiner’s benefactors.
Scheiner’s greatest contributions were the determination of both the inclination angleof the solar rotation axis relative to the ecliptic and the magnitude of the Sun’s surface
differential rotation. These discoveries were both detailed in Scheiner’s blandiloquently titled
four-volume magnum opus published in 1630: “Rosa Ursina, sive sol ex admirando facularum
et macularum suarum phoenomeno varius: necnon circa centrum suum et axem fixum ab
occasu in ortum annua, circaque alium axem mobilem ab ortu in occasu conversione quasi
menstrua, super polos proprios, libris quatuor mobilis ostensus.”
This translates to “The Bear’s Rose, that is the Sun and on the admiration of flares andsunspots and their various phenomenon: as well as on its center of rotation and its annually
fixed rotation axis, and on another rotation axis leading to a separate monthly rotation of

27

Figure 1.11: Drawings of sunspots from (a) G. Galilei in 1612 and from (b) the 1625 editionof the Rosa Ursina by C. Scheiner.

the poles, in a four volume set.” In deference to his patron, “Ursina” is a play upon thefamily name of the Duke of Bracciano Paolo Giordano Orsini II. Thus, an ursine theme isubiquitous in the book’s many illustrations. Despite this excessive flattery and Scheiner’sdifficulty in selling the book, it became the seminal text in the field of solar astronomy fornearly a century. Though, as Jack Eddy remarks in his 1976 paper, perhaps both of thesecircumstances may be due to its excessive and pedantic detail (Eddy, 1976).
This rapid pace of innovation spurred on by multiple scientists in the early part of theseventeenth century quickly came to a halt as the number of sunspots dwindled during theMaunder Minimum between 1645 and 1715. However, there is one nagging question and thatis why were sunspots discovered only late in the Renaissance, especially given that manyother observations had occurred centuries earlier. Part of the answer to this is that there had
been a prolonged solar grand minimum for much of the preceding 300 years, a period that
had followed the so called Medieval Maximum that ended around 1350. Despite this paucityof spots during the birth of modern science, sunspots have enjoyed more or less continuous

28
observation in the succeeding 400 years, as is illustrated in Figure 1.12, with even sufficientfidelity to be able to reconstruct the Maunder diagram for a substantial portion of this time.
Here the extent of the post-Renaissance sunspot number record is evident in Figure 1.12b,with yearly data between 1610 and 1749, monthly data between 1749 and 1874, and daily
sunspot counts afterward (Hoyt & Schatten, 1998). The three most prominent features are
the Maunder Minimum (1645-1715), the Dalton Minimum (1798-1835), and the Modern
Maximum in the 1960s.
1.5

Terrestrial Record of Solar Variability
As with most of the longer time scales of the Sun, including the Schwabe cycle, theirregularity and even existence is difficult to ascertain. For instance if we wished to ask how
long the 11-year cycle has been active, we can only directly assess this for the last 400
years. Yet the Sun and its magnetism have existed for billions of years. What might its
cycles, if it had them at all, have been like 1,000, 10,000, or millions of years ago? Dolarger scale ISM or galactic phenomenon have any impact on these longest of time scales?
Furthermore, do any internal structural or rotational changes that the Sun has undergone
over the eons influence the timing of the dynamo? Indeed, deciphering its longer-term
variability is an uncertain business, but there are a few techniques that allow solar activity
on these millennial or even longer time scales to be measured. But before moving on to thesolar-terrestrial connection, it should be noted that there are a few longer term periodicities
visible in the direct observations. As evident in the full data set shown in Figure 1.12, there
is a modulation of about 88 years known as the Gleissberg cycle (Gleissberg, 1939). Its
cousin, the 209 year Suess cycle may also be present in this data (Suess, 1980), but it would
be seen at most twice. Since we have but snapshot of the Sun’s magnetism on geologicaltime scales, this latter cycle’s behavior may be muddied by even longer time scale processes.
As mentioned in §1.4, the link between geomagnetic activity and solar activity was
made as early as 1852. Once the theoretical connection was made, it was inevitable that

29

Figure 1.12: (a) A Maunder diagram covering the period between 1670 to 2013. Colors
indicate proxies for sunspot areas, or in the case of the modern data the actual areas, in thelower (gray), middle (red), and upper (blue) tertiles of the respective distributions of thedata. (b) Average monthly Group sunspot number, displaying the variability and longerterm modulation in sunspot area over the many Schwabe cycles. (c) Solar cycle variability
over the last 12,000 years as derived from 14 C sunspot number proxies (Usoskin, 2008).
Grand minima in activity fall below the blue line and grand maxima above the red. The
Maunder Minimum is indicated for reference, so modern telescopic observations occupy only
the last 3% of panel (c).

30someone would go looking for it. Indeed, the two cycles just mentioned, and perhaps even
the Schwabe cycle, are seen in many proxies for solar activity throughout a large portion ofEarth’s geological history. Both of these periodicities, and many others, seen in magnetic
activity indicators could simply be harmonics of the 11-year Schwabe cycle. However, these
longer term cycles led to my inclusion of Figure 1.12c, wherein a robust proxy for solar
paleomaculations is shown. This sunspot proxy is a normalized concentration of radioactive
14

C in dendrochronological samples or polar ice cores (e.g., Usoskin, 2013). These data

elucidate the multi-periodic and chaotic nature of the solar dynamo, though not without
caveats. Those involve two major influences on the radiocarbon proxy method that mustbe accounted for: seasonal weather changes that heavily influence short term variation (lessthan decadal time scales) and the secular changes in the Earth’s magnetic field that impact
millennial time scales. These shortcomings can be partly ameliorated by smoothing anddetrending the data, after which several periodicities and extrema become apparent.
An analysis of the extrema seen in Figure 1.12c indicate that the distributions ofboth grand minima and maxima deviate from exponential characteristics, and so are likely
not purely random. Moreover, the minima show a bimodal distribution with most minima
being like the Maunder Minimum, lasting an average 60 years, and a secondary peak atlonger Spörer-like minima that last about 150 years (Usoskin et al., 2007). The periodicities
present in this radiocarbon data are the centennial Gleissberg, duocentennial Suess, andduomillenial Hallstadtzeit cycles. There have been explanations of these time scales in this14

C proxy record that invoke a mechanical forcing due to planetary orbits the most recent

being Abreu et al. (2012), with the idea originating with the sunspot pioneer Rudolf Wolf
in the mid-nineteenth century (Charbonneau, 2013).
This sunspot narrative does not end lost in a thicket of tree rings or frozen below the ice.
In fact, there has been evidence of the Schwabe, Hale (the 22-year polarity cycle), Gleissberg,
Suess, and longer potentially solar related periodicities in the fossilized sediments of Jurassic,
upper-Permian and upper pre-Cambrian lake beds. The sedimentary varve records in the

31
170 Myr old Jurassic Todilto formation in New Mexico form an unbroken 900 yr record ofclimatic variability associated with solar forcing (Anderson & Kirkland, 1966). The 240 Myr
old Castile formation, also in New Mexico, has a 250,000 year record and high laminar
resolution. This means that intermediate time-scales can be identified between the 22-year
Hale cycle and the decamillenial Milankovitch cycles, which are periodicities associated withorbital forcings such as changes in eccentricity, axial title, and precession (Anderson, 1982;
Anderson & Dean, 1995; Kirkland, 2003).
Finally, the most compelling geological record can be found in the 650 Myr old preCambrian sandstone formations of the Southern Australian Elatina formation (Williams &Sonett, 1985; Bracewell, 1988). This formation has the Schwabe, Hale, Gleissberg, and Suesscycles prominently etched in a record untouched by biological processes and unharmed byoxidation due to the anoxic atmospheric chemistry during its sedimentation. This is a mostfascinating record. If this result is robust, it shows that the temporal characteristics ofthe solar cycle have remained essentially unchanged since the late Proterozoic eon. Whatis compelling is that the solar magnetic cycle may have continued unabated even as theunderlying solar structure evolved, which is a given as this period represents about 14% of
the Sun’s current age, during which the solar luminosity has increased by 2%.
1.6

Short Time Scale Solar Variability
The most obvious short term variability arises from the rotation of the Sun, occurring

on a 27 day time scale, which gives more information about the spatial organization of thesolar activity than its temporal organization. However, there exist mid-range time scales ofvariability between 27 days and 11 years. The most prominent of these mid-range time scales
is the 153 day Rieger oscillation, which primarily occurs near solar maximum. This period
actually varies between 140 and 170 days, and is detected in a variety of observations. It
was first seen in gamma-ray flares (Rieger et al., 1984), but has been found in active region
parameters and in the 10.7 cm solar radio signal (Lean & Brueckner, 1989), as well as in32variations of the interplanetary magnetic field (Cane et al., 1998). Additionally, there aretime scales of 51, 128, and 257 days that are slightly less statistically robust, with false
alarm probabilities of about 10% (Bai, 2003). These periodicities are again associated withsolar maxima. The origins of these quasi-periodic time scales have been variously attributed
to magnetic Rossby waves operating either in the tachocline (Zaqarashvili et al., 2010) or
near the solar surface (Lou, 2000), among several other mechanisms. Furthermore, there
is also evidence for oscillations of the solar neutrino flux, with a period of about 38.7 days
(Sturrock et al., 2013). This would be quite interesting as it would imply that the nuclear
energy generation rate in the core is varying with time.
For completeness, the briefest time scales observed on the Sun occur on the order ofmilliseconds, such as the reconnection events that lead to flares, CMEs, and the waves associated with them. There is of course the granular convective overturning time scale of about
10 minutes, which sets the pace for many of the time scales in the chromosphere as well as
the photosphere. One such impact is on the magnetic fields responsible for photosphericbright points, which dance about each other in the intergranular lanes. This dance continues as individual magnetic flux bundles repel or annihilate depending upon their relative
helicity, with this process taking about 20 minutes. There are indeed a very large number ofprocesses taking place in the solar atmosphere that possess a very diverse set of short timescales. However, these processes are largely beyond the scope of the dynamo and convection
simulations that make up the bulk of this thesis and will not be covered in depth.
The multitude of spatial scales of the Sun (§1.1) are thus accompanied by an equally
diverse set of time scales that stretch across billions of years as the Sun evolves to the
10 minute time-scales of granulation and to the even shorter time-scales of reconnection thatlaunch flares and CMEs. Yet explaining and understanding how all these processes occurringin the Sun give rise to a magnetic activity cycle that seems to persist for nearly a billion
years is largely a mystery. Indeed, one of the oldest problems in physics lies cloaked below
the tumult of the solar atmosphere: what drives this rhythmic heartbeat of the Sun?

Chapter 2
Observations of Stellar Magnetic Activity“Thousands of stars, in the same stage of evolution as the Sun, doubtless exhibit similar
phenomena, which are hidden from us by distance... The simplest and most obvious stepwould be to make just such a study of the physical phenomena of selected stars,
representing different phases of evolution, as we are now making of the Sun.” – George
Ellory Hale (1915)
One path to attempt to solve the mystery of the persistent magnetic activity cycles of
the Sun is to appeal to observations of other stars. This, as most things in stellar astronomy
and astrophysics, has a substantial history behind it. Indeed stellar observations had beenongoing for at least two centuries before it was realized that other stars might behave likethe Sun, with many displaying magnetic activity and a distinguished few that also haveobserved activity cycles. There were several pieces that had to fall into place before evensolar magnetism was considered plausible. The first being the invention of the spectroscope,
which spurred many astronomers to take spectra of ever fainter objects and stars throughout
the nineteenth century. It took the additional development of electromagnetic theory andatomic physics before the connections between electromagnetic phenomena and solar activity
were made in the early twentieth century.
During the spring of 1896, three years after his doctoral thesis work on the Kerr
effect, Pieter Zeeman rebelliously repurposed his supervisor’s laboratory equipment in order
to measure the splitting of spectral lines by magnetic fields. This pioneering work in early

34quantum phenomenology led to his 1902 Nobel prize and to many other profound discoveriesin physics, such as Lorentz’s theoretical considerations of the electron. Its implications
were just as great for stellar astrophysics. Indeed, a short decade later and enabled by thediscovery of the Zeeman effect, George Hale ushered in a century of astronomical study ofmagnetic phenomenon with his observations of the magnetic field strength in sunspots (Hale,1908). At this point it was not a huge leap to ask if other stars might be like the Sun.
After such detection of magnetic fields on the Sun, the field of stellar magnetism began
to unfold. Indeed, in a “minor contribution” to the Astrophysical Journal (Eberhard &Schwarzschild, 1913), the first detection of variable Ca H and K emission from another star
was published just five years after the discovery of magnetic field in sunspots. In this paper
it was stated that the “reversal of lines in spectra are not rare [...],” and also presciently
speculated that “[... though] it remains to be shown whether the emission lines of the star
have a possible variation in intensity analogous to the sun-spot period.” The H and K linesof singly ionized calcium is a robust indicator of chromospheric structure. Indeed, theirutility in constraining the spatial character and temporal variability of structures in the highsolar atmosphere has been recognized since Hale’s initial investigations. A stream of papers
followed this early work based on the premise that other stars may also exhibit variability
in the Ca H and K lines. This literature typically presented individual stars with either afairly short or undetectable time variability in the calcium lines over a broad range of stellar
spectral types. These early searches for stellar magnetic variability were cataloged in Joy &Wilson (1949) and Bidelman (1954), which provided the basis for Olin Wilson’s later work.
In the early 1950s, Wilson began laying the groundwork for the Calcium H & K Projectat the serendipitously named Mount Wilson Observatory (MWO) (Wilson, 1978; Hall et al.,2007). This long-term monitoring project now covers 99 stars including the Sun. Of these
99 stars, 72 show variability, 14 are not variable but have activity, and the remaining 13
show no activity within the measurement uncertainty. The full MWO archive holds justover 6000 stars that have been less systematically observed in these Ca H and K lines and35possess calibrated measures of their magnetic activity (Mittag et al., 2013).
Despite there having long been this indirect evidence and theoretical motivation for
the existence of magnetism in lower mass stars, nearly forty years passed between the firstrecognition of magnetic fields on the Sun and the direct detection of the magnetic fieldon another star. In 1947, Horace Babcock uncovered the presence of 1500 G magnetic
fields on the chemically peculiar A-type star 78 Virginis (Babcock, 1947). There were many
subsequent detections of magnetic fields among this class of higher-mass, chemically-peculiar
stars, which was primarily due to their relatively strong mean magnetic fields. This leads
to widely Zeeman-split spectral lines and thus to easier detection with the limited spectral
resolution of the instruments of the time.
The next set of stars with directly detected magnetic fields were compact objects. In
1970, following his theoretical and laboratory work on gray-body magneto-emission, James
Kemp pursued an astronomical detection of this phenomenon. To first order, this spectrum ofemission arises from a thermal source in a magnetic field and is characterized by a spectrally
diffuse circular polarization directed along the line-of-sight component of the magnetic field.
The circularly polarized visible light signature of this kind of emission was indeed found ona white dwarf (Kemp et al., 1970), utilizing the relatively modest telescopes at the Pine
Mountain Observatory in Oregon. This gray-body magneto-emission betrayed the presenceof a very strong magnetic field of about 106 G, which is roughly 1,000 times greater thanthat of the Sun.
Using a phased-array radio telescope, composed of simple dipoles in a wire frame,
Jocelyn Bell and Antony Hewish serendipitously discovered one of the most magnetizedobjects in the known universe: the pulsar. These stellar remnants have magnetic fieldstrengths at least 10 billion times that of the roughly 1 kG fields on stars like the Sun. The
direct detection of the enormous magnetic fields of these stars came a decade later withthe measurement of cyclotron absorption lines seen in the X-ray spectra of Hercules X-1(Trümper et al., 1977).

36
After the polarimetric detection of magnetic fields on compact objects, it took justa few more years of innovation in both observing technology and data analysis techniques,particularly in spectropolarimetry, before it was possible to directly detect the magnetic fields
on distant lower mass stars similar to the Sun. If we may draw upon the magnetic phenomena
observed on the Sun as proxies for other such stars, the difficulty in observing fields on thesestars largely arises from the local cancellation of bipolar magnetic flux structures during the
averaging processes undertaken when observing a point source in polarization measurements(e.g., Schrijver & Zwaan, 2000). However, a new method was invented that can partiallyovercome this difficulty. The essence of it was to perform a Fourier deconvolution of a nonmagnetically sensitive line profile from a magnetically sensitive one. It goes a bit further inthat there is a very rudimentary model of the stellar surface magnetism accompanying theline deconvolution. Indeed, there is a least-squares minimization of the deconvolution where
the model function parameterizes the supposition that some fraction of the stellar surface iscovered in magnetic field of some strength in the Zeeman splitting parameter ∆ (Robinson,1979). Thus it was finally possible to assess these averaged signals to produce a single
measurement of the magnetic field strength, though it still lacked directional and spatial
information about the field. This technique was first successfully applied in measurementsof the surface magnetic field of two lower mass stars, namely ξ Bootis (a G8V-type star) and
70 Ophiuchi A (a K0V-type star) (Robinson et al., 1980).
Before delving into the plethora of recent observations of stellar magnetism, it is usefulto provide an outline of the techniques that are currently being employed to make assessments of the long-term behavior and morphology of stellar magnetic fields. To understandthe various aspects of the photospheric, chromospheric, and coronal magnetic fields and theirvariability, ongoing observations utilize primarily four techniques: photospheric photometric variability and spot modeling, long-term chromospheric Ca H and K line monitoring,
spectropolarimetry and Zeemann-Doppler imaging, and coronal X-ray observations.

37

Figure 2.1: (a) Slightly more than 100 years (see upper abscissa) of photometric data fromphotographic and photoelectric measurements of the K7V star V833 Tau, showing a longterm Gleissberg-like modulation. (b) Short-term variability of V833 Tau, showing its shorter
2.4-year periodicity and a bright white-light flare occurring during the peak of one of thesecycles (Adapted from Oláh et al. (2001)).

2.1

Photometric Variability
The simplest of observational technique is to measure the relative photometric bright-

ness variations of a star. However, beneath this simplicity lies a great deal of complexity
and information as the signal is the integrated intensity over some bandpass of the star’s
spectrum, usually in the visible. This means that anything in the line of sight, say a planet,or phenomenon on the stellar surface such as starspots can cause intensity variations. Furthermore, pulsational modes, either acoustic or buoyancy driven gravitational waves, cause
minute deviations in the brightness of the star. These qualities are those that motivatedrecent planet-finding and photometric-monitoring missions such as MOST, CoRoT, and Kepler.
One interpretation of these photometric variations is that a portion of the signal arises
from a changing number of starspots on the stellar surface that is caused either by the star’s
rotation or through the shorter lifetime of the spots. With the Sun as a reference point, thisis taken to mean that the star may be undergoing magnetic cycles or at least possess some

38

Figure 2.2: (a) Photometry from the CoRoT satellite showing the variability of the mainsequence wide binary star system CoRoT-2a, with the primary component being a G7V typestar and its companion being a K9V type star, with 140 days of data shown. (b) Star-spotmodeling of the primary component of this binary system, illustrating active longitudes andspot migration (Adapted from Lanza et al. (2009)).

magnetic variability (e.g., Strassmeier, 2005). This is the case in Figure 2.1, where 100 yearsof photometric data are shown for a main sequence K-type star. This star, much like theSun, shows multiple periodicities with an approximately 60-year cycle long-term cycle thatappears to be like the Gleissberg cycle of the Sun and two short-term periodicities of 6 and2.4 years. The latter period is taken to be this star’s Schwabe cycle, while the 6 year period
is likely its Hale cycle (Oláh et al., 2001).
Given that photometry simply counts photons to measure this kind of variability, one
can measure such activity on a very large sample of stars. This provides detailed statistics about possible magnetic phenomenon across the Hertzsprung-Russell diagram, enablingnearly anyone with a telescope and a method to accurately monitor photometric variability
to make measurements of potential stellar magnetic activity. However, there are a few drawbacks. For instance, the brightness variations of main-sequence stars do not always correlatewith other measures of their activity such as the chromospheric Ca H and K observations ormeasurements of X-ray luminosities (e.g., Pallavicini et al., 1981; Radick et al., 1998). These39
correlations also depend upon the nature of the target being observed, as binary or multiple
star systems can behave quite differently from single stars (e.g., Berdyugina, 2005). Thus itis essential to follow up with spectroscopic measurements to better characterize the target.
The second approach utilizing photometric data attempts to extract maps of coolstarspots from the stellar photometric variations (e.g., Berdyugina, 2005; Lanza et al., 2007,2009). One of these methods is direct light-curve modeling. This is a trial-and-error method
wherein a number of spots with a predetermined shape are varied in area, latitudinal andlongitudinal position, and in spot temperature relative to the effective temperature. Thesespots are rotationally modulated in an attempt to reconstruct the light curve. These models
obviously have a large number of free parameters and the solutions are not unique, andthey are best reserved for peculiar higher mass stars that tend to have only one or two large
spots. The second technique fits for a fractional spot coverage with a two temperature model
against which the observed light curve can be inverted to give information about the timedependent longitudinal distribution of spots. Both of these starspot modeling techniques
can be improved to yield information about the latitudinal dependence by incorporating
temporal information from Doppler-shifted spectral lines (e.g., Strassmeier, 2009).
Differential rotation is certainly one of the primary components of a polarity-reversing
dynamo. Although internal differential rotation is crucial, it is often difficult to observe.
However, photometric measurements of spotted stars can provide access to information about
the surface differential rotation. This is typically possible in stars with a sufficiently large
differential rotation that it can be distinguished as a secondary rotational signal in a LombScargle periodogram (effectively a Fourier transform for irregularly sampled data). The
Kepler mission has allowed the long-term photometric observation of a large sample of stars,
for the main-sequence stars in this sample a recent effort has been made to identify therates of rotation and differential rotation (Reinhold et al., 2013). The author has amassed
differential rotation measurements for roughly 20,000 Kepler stars and has attempted tocorrelate them with stellar effective temperatures and with a photometric proxy for magnetic

40

Figure 2.3: Differential rotation scaling with Rossby number and stellar mass in 20,000 Kepler stars. (a) Joint PDF of absolute differential rotation ∆Ω and Rossby number, with color
indicating the number of stars in each pixel. The three dashed lines indicate possible Rossby
number scalings in the normal regime, the fast rotator branch and in a super-saturated
branch. The blue circled area indicates those stars that may be running super-saturated
dynamos (see text). (b) Joint PDF of ∆Ω and stellar mass, showing little variation outsideof a fairly narrow band. Colors as in (a). (c) Marginal distributions of Rossby number,
integrating over ∆Ω for five classes of activity levels, with five being the highest activity as
measured by a photometric proxy. The most active stars represent those that may possess
super-saturated dynamos. (Constructed with data from Reinhold et al. (2013)).

activity. Indeed, they find a correlation between rotation period and relative differential
rotation (α, shearing rate over rotation rate) that decreases roughly as α ∝ P −1 . They alsofind that α varies little with effective temperature (Tef f ) for stars with Tef f < 6000 K, witha fairly rapid increase in α above this temperature as also found in Barnes et al. (2005);
Reiners (2006), and Ammler-von Eiff & Reiners (2012).
Some additional insight could be gained if Reinhold and coauthors also use previously
defined empirical conversions between Tef f and Rossby number as in Noyes et al. (1984).
The Rossby number Ro is a measure of the relative strengths of the nonlinear convective
force to the Coriolis force, which here is defined as the ratio of the rotation period to an
empirically determined convective overturning time (e.g., Kiraga & Stepien, 2007). Such an
approach helps to lift the mass dependency as the definition of rapid rotation becomes oneof low Rossby number rather than absolute rotation rate. This approach is attempted here,41
using the data from Reinhold et al. (2013) to construct the Rossby number correlations of∆Ω. The Rossby number is derived from an empirical fit to Tef f as in Wright et al. (2011).
Figure 2.3a shows that there may be several regimes of Rossby number dependence, whichhave been fitted as follows. The bulk of the stars, where the colors are darker tones, seem tohave ∆Ω ∝ Ro−0.3 as indicated by the middle dashed line. The upper branch of stars have∆Ω ∝ Ro−0.7 . One might suspect that these stars would be more massive, but this turns out
not to be the case as there are stars of every mass within the sample in this branch. Stars in
the lower branch, with ∆Ω ∝ Ro−0.15 , have an anomalously low differential rotation for the
given Rossby number. These stars are also the most active and have a markedly different
marginal distribution in Rossby number from the other stars in the sample (Figure 2.3c). It
could be suggested that they are pulsating stars, however they tend to be lower mass stars ofspectral type G and K and thus this scenario is less likely. Another possibility is that they arerunning saturated dynamos with strong Lorentz-force feedbacks that modulate the amplitudeof the differential rotation. Nonetheless, it is curious that there appear to be two classesof rapidly rotating stars, which causes one to ponder what might be structurally different
between stars of the same mass and Rossby number that allows such a state. Finally, as
mentioned in the caption of Figure 2.3b, it is evident that ∆Ω is nearly independent of mass,
with only a small sample of stars having more differential rotation than about 0.2 rad d−1 .
This does not contradict the findings in Reinhold et al. (2013), as it is being shown as a
distribution, which tends to hide the scatter in ∆Ω at higher masses due to there being fewerof such stars. This is in contrast to plotting all the data points as is done in their paper for
Tef f .
2.2

Chromospheric Activity
It is helpful to first define the chromosphere, which literally means “colored sphere”

and refers to its red hue in visual observations during eclipses. This is a rather difficult task
in itself as the chromospheric region is heterogeneous due in part to magnetic structures and

(a)

(b)

42

Figure 2.4: (a) The mean solar chromosphere, showing the temperature structure with heightin kilometers above the photosphere as well as with column density ‘m’ (the second set of
abscissa). The temperature reaches a minimum about 500 km above the photosphere andhas a very steep rise into the transition region. The heights of line formation regions fora wide variety of lines are indicated, such as the six regions of Mg II and Ca II H and K
emission (From Vernazza et al. (1981)). (b) Temperature as a function of pressure (heightin km with the upper abscissa) above several classical solar features: Model C (quiet-Sun),
Model P (faculae), Model H (plage), Model R (penumbrae), Model S (umbrae), with image
from Fontenla et al. (2006).

a low plasma β and also in part to non-LTE radiative processes. A working definition is
given in Hall (2008), in which the chromosphere is taken to be “[...] the regions of a stellar
atmosphere (a) where we observe emission in excess of that expected in radiative equilibriumand (b) where cooling occurs mainly by radiation in strong resonance lines [...] of abundantspecies such as Mg II and Ca II H and K.”
Figure 2.4 illustrates the mean chromospheric temperature structure of the Sun as wellas the range of heights over which certain emission and absorption features arise. The Ca
II H and K lines form throughout much of the chromosphere between 500 km and 2000 km
above the photosphere, which is due to the relative insensitivity of these line to the plasma
density in this region. This behavior is manifest given that the region is nearly isothermal
whereas the density drops exponentially. The structure of the solar chromosphere may betaken as an approximate homomorphism for the chromospheres of other lower mass stars.

43
However, it is a Goldilocks homology, wherein the structure of the atmospheres of other stars
must not be too cool so that a dusty plasma forms and not so hot that the atmosphere isseverely out of local thermodynamic equilibrium (LTE). Regardless, Figure 2.4 illustrates thehomological structure of the atmospheres above other lower mass stars with respect to theSun. This picture has recently been bolstered with the detection of a temperature minimum
and chromospheric temperature variations in the atmosphere of our nearest stellar neighbor
α Centaurus A, mimicking the Sun’s chromospheric structure (Liseau et al., 2013). Thus forthe majority of lower mass stars it is likely that, if they have surface magnetic features andemission in the Ca H and K lines, this kind of emission can be used as a reasonable proxy
for their magnetic activity.
Although the corona and transition region are coupled to the photosphere throughthe chromosphere by magnetic fields and waves, the properties of the plasma in these tworegions differ significantly. Chromospheric plasma is much less dense than the photosphere,
which has a number of implications for observations and also for modeling it. Primarily,
the low density of the plasma results in the plasma β becoming less than unity and a muchsmaller optical depth, with line emission and non-LTE effects, such as scattering, becoming more important than continuum emission. The need to account for non-LTE effects
complicates the interpretation of most of these diagnostics. Yet recent advances in groundand space-borne instrumentation, such as the Sunrise balloon mission and the IRIS mission,
have provided a route to observe the small spatial scales of the dynamic and intermittent
structures seen in current simulations of the chromosphere (e.g., Wedemeyer-Böhm et al.,2009; Leenaarts et al., 2013). The coexistence and interaction of magnetic fields and large
amplitude magnetosonic waves makes modeling the chromosphere a challenging task. Although observations have exhibited the inhomogeneous structure of the chromosphere formany years, only recently has modeling been able to show that its structure is not as simple
as in 1-D models (Figures 2.4b). Instead, both the magnetic and thermodynamic fields canvary horizontally and radially in the presence of magnetism as demonstrated in Figure 2.5

44

Figure 2.5: (a) A vertical cut through a 3D simulation of the solar chromosphere, showing
logarithmic magnetic field strength between 1 G (deep blue) and 3 kG (dark red) and its
horizontal expansion with height. The photosphere is indicated with the corrugated black line
near zero height or optical depth with τ = 1. A pore is evident with field lines expanding into
the chromosphere. (b) Temperature in the same slice ranging between 2,500 and 12,000 K.
The yellow line indicates the height at which the plasma β = 1 (Adapted from Freytag et al.
(2012)).

as well as in Freytag et al. (2012). This variability in temperature and density gives rise
to greater emission from regions where magnetic field threads through the chromosphere, as
seen in the synthetic Stokes inversions of de la Cruz Rodrı́guez et al. (2012). Indeed, thetemperature sensitivities of Ca H and K line formation causes the emergent intensity variations. The intensity variations can be inverted to produce maps of density and temperature,and of the line-of-sight magnetic field much like those in Figure 1.2b.
Having assessed the photometric measures of stellar variability, it is now useful to compare it against other methods of measuring magnetic activity. It is also worth building a

45

Figure 2.6: (a) Comparison over 30 years of total solar irradiance (TSI) changes (Fröhlich,2013) and Mt. Wilson Ca H and K emission variations (Livingston et al., 2007), illustrating
the Schwabe cycle and the close correlation between these two measurements of variability.
The red dots are Ca measurements and the blue line is an 81 day smoothing of the TSI.
(b) Photometric brightness and Ca H and K measurements from Mt. Wilson for the K1V
star HD 10476, showing its equivalent Schwabe cycles as well as its brightness and S-index(chromospheric variation) correlations. Brightness variations in blue dots with a blue line
showing a polynomial fit and red dots showing the longer-term Ca H and K observations.
(c) Photometric brightness and Ca H and K measurements from Mt. Wilson for the spotdominated G8.5V star HD 152391, showing its equivalent Schwabe cycles as well as its
brightness and S-index anti-correlations [b and c constructed from data in Radick et al.
(1998)].

framework to understand how chromospheric activity is described. Indeed, one method ofcharacterizing the observed chromospheric activity called the “S” activity index was developed during the long running MWO Calcium H & K Project. This index was introduced byVaughan et al. (1978) as a dimensionless proxy for chromospheric activity in the Ca H andK lines as measured with MWO spectrometers. More generally, the flux is measured in two
narrow bands centered on the Ca H and K and is normalized by the flux in two wide-band
continuum filters, referred to as Johnson visible (V ) and red (R). Thus the S-index values
can be constructed as
S=α

H +K
.
R+V

(2.1)

where H, K, R, and V are the values for the flux measured in the various bandpasses andα is an instrumental calibration factor. This calibration is necessary to adjust the measured

46
(b)

(a)

Faculae

Sun

Sun

Spots

Figure 2.7: (a) Correlation between average photometric amplitude and and average chro′mospheric activity (under the log RHK
measure) for 32 stars in the Mt. Wilson catalog,with the Sun labeled for reference. (b) Correlation between the photometric variation and
S-index as a function of chromospheric activity, indicating that some stars show a correlationin Ca H and K emission and in brightness variations whereas others show an anti-correlation(Figures adapted from Lockwood et al. (2007)).

activity indices from the two instruments at Mount Wilson. Values for α range from 1.3(Cincunegui et al., 2007) to 5 (Gray et al., 2003). However, the S index is a measurement of
the total Ca H and K flux from a star and not the amount due only to the chromosphere. It
takes a bit more work to isolate the chromospheric contribution, which is a better measureof magnetic activity.
The first step is to introduce the B −V color (the difference between the flux in the blue
B and visible bandpasses) and the second is to remove the photospheric component. As suchthere is an empirical affine map between the S-index and the chromospheric activity measure′′
RHK(Middelkoop, 1982). This mapping is defined as RHK= 1.34 × 10−4Ccf S − Rphot , whereCcf is the conversion factor for main-sequence stars with 0.3 ≤ B − V ≤ 1.6 and Rphot is
the photospheric contribution. Both of these factors have been empirically determined bycorrelating the S-index with other means of gathering magnetic activity and are found froma fit with a cubic polynomial in B − V Noyes et al. (1984).
With this definition in hand, some of the results from the multi-decadal stellar activity

47
can now be explored. Figure 2.6 shows a comparison between the brightness variations andthe variations in the Ca H and K lines as measured by the S-index. These temporal changes
in the Sun’s output are characterized by the 11-year Schwabe cycle and a close correlation
between Ca H and K emission and total irradiance (Figure 2.6a). This is also true of HD
10476, whose brightness and Ca emission are shown in Figure 2.6b. Such cycles are evident
across the range of low mass stars that populate the lower end of the HR diagram, and there
is now enough data collected to begin to build statistics about the nature of cycle periods
and their relation to rotation rate and stellar mass (Baliunas et al., 1995). Yet, like the Sun,
these stars exhibit a multitude of timescales, with Maunder Minimum-like grand minima as
well as many other short and long-term periodicities.
In contrast to the Sun and HD 10476, the much younger BY Draconis-type variable star
HD 152391 shows an anti-correlation between these two measures of activity induced variability (Figure 2.6c). This divergence is but one example of one of the broader behaviors of stellar
activity found using measurements of variations in the Ca H and K lines. Indeed, examining the entire Mount Wilson sample reveals a division in the brightness and chromosphericvariation between young and old stars (e.g., Radick et al., 1998; Lockwood et al., 2007). If
one assumes that old stars have activity cycles like the Sun, their photometric behavior isthen dominated by faculae and so have a positive correlation with chromospheric variations(Figure 2.7a). Conversely, it appears that young stars are spot-dominated and exhibit anegative correlation between photometric and chromospheric variations. This is illustrated′
in Figure 2.7b, where the dividing line between such behavior is near log RHK= −4.7. The′
degree of correlation between the log RHK
measure of stellar activity and the amplitude ofphotometric variation for a set of stars with long-term measurements is shown in Figure 2.7a.
′
The agreement between the rms value of the photometric variations and RHK
is reasonably

robust over about a decade in activity. This correlation does not change between the spot
and faculae dominated regimes until systematic errors arising from a lack of photons creep
in at low levels of activity.

48
The Sun has afforded us the opportunity to study a magnetic cycle in great detail.
Yet to begin to parameterize how a dynamo evolves and varies, we must turn to other solarlike stars. The Ca H & K project target stars are effectively the only source of long-term
observations of active stars. Indeed, this project provided the first results regarding longterm magnetic activity (e.g., Baliunas et al., 1985, 1996; Saar & Brandenburg, 1999), withmany of the stars in their sample appearing to have magnetic activity cycles akin to thesolar cycle in length and character. However, the results were a bit puzzling. There waslittle, if any, relationship found between cycle periods and other properties thought relevant
to dynamos such as rotation rate. Later work did find various, but often contradictory,
correlations with rotation (e.g., Saar & Brandenburg, 2001; Saar, 2009). As it stands, there
is still a great deal of uncertainty regarding the correlation between cycle period and therate of rotation due to a paucity of data. Recent work has attempted to reanalyze the Ca H
& K data with a more careful sampling of the data. When such a sampling is carried out,
as in Saar (2009), several branches in the behavior of cycle frequency ωcyc seem to emerge,
where each branch can be correlated with the inverse Rossby number (Ro−1 = τC Ω, with
τC is an empirical convective overturning time and Ω the bulk rotation rate). Figure 2.8
shows two possible interpretations of the Ca H & K data, with either a fairly linear scaling
with Coriolis number (the ratio between the overturning time and the rotation period) or a
fit that is nearly independent of Coriolis number. The latter is favored because of its lower
uncertainty in the fit. Such a result, if robust, is quite surprising as it would indicate that it
is not the rotation rate itself that sets the cycle period. Rather, another processes must be
at work that, given the regularity of the cycles, must be fairly linear.
2.3

Spectropolarimetry and Zeeman-Doppler Imaging
The direct detection of magnetic fields relies upon the lifting of the degeneracy ofquantum states in ions, atoms, and molecules when they are in the presence of a magnetic
field. Moreover, it is the interactions of photons with this split set of states that imprints

(a)

(b)

49

Figure 2.8: Relationship between cycle frequency and inverse Rossby number. (a) Primary
cycle frequency normalized by the bulk rotation rate (ωcyc /Ω) versus inverse Rossby number
Ro−1 = τC Ω, with fits to two branches: ωcyc /Ω ∝ (τC Ω)−0.98 or (τC Ω)−0.90 . (b) A different
interpretation of the same data with three potential branches, utilizing the secondary cycle
periods for the data shown as squares. The secondary cycle periods are the lower bound of
the vertical dashed lines, with a change in behavior as surface differential rotation begins todisappear (labeled as SDR breakpoint). This figure is adapted from Saar (2009).

the signature of the magnetic field on the polarization of the photons that are eventuallydetected here on Earth. The description of this effect is attributed to Zeeman (§2). The
Zeeman effect splits spectral lines and so allows many types of magnetic fields to be detected,
from the µG fields in the ISM and molecular clouds to the very strong MG fields on whitedwarfs. A large fraction of the measurements of magnetic fields reported in the literature
have been obtained through Zeeman spectroscopy, with few other options available outsideof the Hanle effect or Faraday rotation.
The well-known result from early quantum mechanics is that in the presence of a magnetic field each degenerate energy level splits into 2J + 1 sublevels j = −J . . . J, with J
the total angular momentum quantum. This in turn is due to the force that the external
magnetic field exerts on the different magnetic moments associated with each angular momentum state of the bound system of charged particles. This picture remains valid provided
the energy difference between sublevels is below the spin-orbit coupling energy, after which
the angular and spin momenta decouple due to the stronger interaction of the magnetic field50
with a particle’s spin. The energy difference δE between each of these angular momentum
sublevels is linear in the magnetic field B and is given by
δE = µ0 µ̂J · BgJ j,(2.2)

where µ0 is the vacuum permeability, µ̂J is the unit vector in the direction of the magnetic
moment of the base state J, and gJ is the dimensionless Landé factor with values of orderunity that depends on the total angular momentum of the unsplit state and upon relativistic
quantum electrodynamical effects. The relevant energy difference ∆E is between the unsplit
energy of the base level J and the split level, which is∆E = µ0 µ̂ji · B (jf ∆g + gji ∆j) ,(2.3)

where ∆g = gjf − gji and ∆j = jf − ji , with i being the initial state and f the final state.
Dipolar transitions between the levels obey the selection rule ∆j = −1, 0 or +1 and theresulting spectral lines form three corresponding groups. The lines due to transitions with∆j = 0, though with possible changes in orbital or spin states that subsequently change g
but not j, are the π components. Such components are distributed symmetrically aroundthe unsplit line, with no signal at the line core. The two groups of lines formed by transitions
with ∆j = ±1 are the σ components and are shifted symmetrically about the unsplit line,with transitions of ∆j = 1 on one side and transitions of ∆j = 1 on the other side. Though,they still have a signal at the line core.
However, the Zeeman effect is not without its limitations. For weaker fields less thanabout 1 kG, other effects tend to erase the splitting signature as the lines are broadened
due to thermal effects, rotation, and turbulent velocity fields. This problem can largely becircumvented when one uses polarizing filters to select out portions of the signal, whereinmuch weaker fields can be detected by taking differences between components with different
polarizations. This process is called spectropolarimetry and can be utilized to detect fields
with simple topologies that are as weak as a few Gauss. The photon polarization can

51
HD 141943

HD 75332

HD 76151

ξ Boo A

EKDra

1.0

HD 78366
Sun

GJ 182

HD 171488
ABDor

61 Cyg A

Stellar Mass (M )

DSLeo
DTVir

GJ 49

OT Ser

0.5
GJ 490 B

EQ Peg A

V374 Peg

Ro = 1EV Lac

Ro = 0.01

0.1

CE BooYZ CMi

EQ Peg B

0.2

ADLeo

Ro = 0.1

WXUMa
DXCnc

10

1
Rotation Period (days)

Figure 2.9: Global-scale morphology of spectropolarimetrically reconstructed magnetic fields
on low-mass stars, as a function of stellar mass and rotation rate. Symbol size indicatesrms magnetic field strength and symbol color illustrates field configurations, with blue forpurely toroidal fields and red for purely poloidal fields. The symbol geometry is meant to
depict the degree of axisymmetry of the poloidal field component, where the fully convex
decagon indicates purely axisymmetric fields and the most concave star represents purely
non-axisymmetric poloidal fields. The curves trace contours of constant Rossby number(Ro = P/τc ), with solid at Ro = 1, dashed at Ro = 0.1, and dash-dot at Ro = 0.01
(using convective turnover times τc from Kiraga & Stepien (2007)). The smallest and largest
symbols correspond to mean large-scale field strengths of 3 G and 1.5 kG on a logarithmic
scale. (Adapted from Donati & Landstreet (2009)).

probe these weaker fields because there are differences in the polarization states of emitted
photons that depend upon the energy of the photon. Namely, this energy selects one of theπ or σ states. Both states can emit linearly polarized photons due to angular momentum
transitions, but only the σ states emit circularly polarized photons due to spin transitions.
Thus passing a magnetically sensitive emission line through a polarizing filter allows the split
lines to be detected in the line core even in the presence of significant broadening. Moreover,given the variable alignment of the magnetic moment of the electronic state of the bound52

Figure 2.10: Radial magnetic field on the surface of τ Boötis reconstructed using the tomographic techniques of Zeeman-Doppler imaging. The view is aligned with the rotation axis.
Red denotes positive field pointing away from the star, blue the opposite, as indicated withthe color bar. Each sub-panel shows the reconstruction from multiple observations during
the month and year given in the upper right, whereas the source for the image is given in thelower right. The arrow indicates the direction of rotation and the small tick marks denote
the rotational phase of the star during observing nights (Images from Catala et al. (2007);
Fares et al. (2009, 2013)).

system with vector components of the magnetic field, this effect can be used to determine
the orientation of the magnetic field as the circularly polarized states gives the magnitudeof the line-of-sight magnetic field and the linear polarization gives information about themagnitude of the transverse field.
This is only the magnetic portion of the Zeeman-Doppler mapping technique, and on
its own is a robust way to detect magnetic fields. Yet it can also be taken a step further
by including multiple observations of many magnetically sensitive lines over many distinct
rotational phases of the star and Doppler information from the circularly polarized subset ofthe lines. This is essential as there is a degeneracy between the line-of-sight velocity Doppler
shifts and magnetic field Zeeman shifts, as well as an alignment or anti-alignment ambiguity.
However, this degeneracy can be partially lifted by incorporating information from the linear
polarization filters and thus the transverse field components. Finally, maximum entropy

53
modeling of the differential rotation of the star allows the morphology of the large-scale field
to be reconstructed through tomographic mapping.
Such reconstructions have been performed on a growing number of stars, slowly building a comprehensive picture of stellar magnetic field morphologies as function of mass androtation rate (Figure 2.9). Stars must be observed during multiple phases of their rotational
periods in order to accurately reconstruct the star’s rotation profile and the surface magnetic
fields, so measuring this for many stars is a significant observing campaign. Nonetheless,the basic stellar properties that have bearing on the structure of the surface magnetismof stars is being mapped out, where the total surface magnetic energy density seem to belargely determined by mass and rotation rate. The ratio of the toroidal and poloidal fieldmagnitudes also seems to depend upon those values. Indeed, Figure 2.9 suggests that stars
with masses above about 0.5 M⊙ and Rossby numbers less than unity build larger toroidal
fields and non-axisymmetric poloidal fields. In contrast, those stars with masses below about
0.5 M⊙ predominantly build strong axisymmetric poloidal fields. Furthermore, there seems
to be a close relationship between magnetic field strength and rotation rate. This of course
hints at the underlying role of a convection and rotation driven dynamo in building suchfields.
With a rudimentary picture of how spectropolarimetric observations operate, some of
its recent results can be examined. One relevant to this thesis is the detection of polarity
reversals on the planet-hosting F-type star τ Boötis. Figure 2.10, shows the morphologyof the magnetic fields on this star as extracted with the Zeeman-Doppler imaging (ZDI)
technique. The exciting realization is that it seems to have a polarity cycle, with the signof the the radial magnetic field changing at the visible pole on a fairly regular interval of
about a year (e.g., Donati et al., 2008b; Fares et al., 2013). The resolved field is quite weak,
being at most about 10 G. However, if τ Boötis is like the Sun, the measured surface-averaged
field is likely greatly reduced due to cancellation of small-scale structure surface fields. The
radial field dominates the signal, but there are contributions of about 3 G from azimuthal and

54

(a)

(b)

(c)

Figure 2.11: Magnetic field line extrapolations for the 15 M⊙ main-sequence B0V star τ
Scorpii, showing (a) the closed field line solutions at two points during the star’s 41 day
rotation and (b) the open field line solution extrapolated to twice the star’s radius. The
rotation axis is aligned with the page height (Images from Donati et al. (2006b)).

longitudinal components as well. The magnetic signal is entangled with Doppler information.
This information is dominated by the substantial bulk rotation of 2 days, with the differential
rotation between the fast equator and the slow poles being 0.4 days.
The atmospheres of massive stars present their own set of challenges such as a large
thermal broadening and often a strong rotational broadening of their spectral lines. For someof these stars, particularly those with a lower rotation rate, the fields can be reconstructed asshown for τ Scorpii in Figure 2.11. This 5 Myr old 15 M⊙ star exhibits a large-scale magnetic
field topology, with a strength of about 1 kG (Donati et al., 2006b). Surprisingly, the closed
field lines indicate that the magnetic equator is nearly aligned with the rotation axis. The
open field lines would direct the hot winds streaming from the photosphere in a roughly
dipolar fan.
2.4

X-Ray Variability and Luminosity
Observations of X-ray spectra and total bolometric luminosities offer a measure ofstellar activity similar to that of the Ca H and K lines, though it is much more sensitive to55
energetic events such as flares. However, such observations probe regions with much higher
temperatures of millions of Kelvin, as in the solar corona rather than just a few thousand
Kelvin as in the solar chromosphere. After more than 30 years of X-ray observations, which
are only possible from space, X-ray emission has been found to be ubiquitous across the
Hertzsprung-Russell (HR) diagram. The bulk of the this X-ray emission may have origins
that are independent of the mass of the star. Appealing to the Sun, it is well known that thesolar corona emits X-rays from this region of high temperature, rarefied, and magneticallyconfined plasma. Nearly all classes of low-mass stars show variable X-ray emission, which
has likewise been attributed to stellar coronal plasmas (e.g., Vaiana et al., 1981; Schmitt &Liefke, 2004). While higher mass stars do not have the same degree of variability, it is still
likely that they too possess a corona. Age also seems to impact X-ray luminosity. Young
low-mass stars tend to be among the most X-ray luminous, where LX /Lbol ∼ 10−3 is common(e.g., Telleschi et al., 2007). Older stars typically have lower X-ray luminosities, dropping
several orders of magnitude relative to young stars at the same mass. This profound decreasein stellar X-ray luminosity between the zero-age main sequence (ZAMS) and the terminal age
main sequence (TAMS) is widely supported among many stellar populations in our galaxy(e.g., Güdel et al., 1997; Wright et al., 2010).
The space-borne X-ray missions of the 1970s and 1990s, such as the High Energy
Astronomy Observatory (HEAO) series and later the Röntgensatellit (ROSAT), found that
X-ray luminosity scales with the bolometric luminosity as LX /Lbol ∼ 10−7 for young massive
O and B stars (e.g., Harnden et al., 1979; Pallavicini et al., 1981; Berghöfer et al., 1997; Nazé
et al., 2011). For some time the primary X-ray emission mechanism for these massive stars
was considered to be quite different from that occurring in lower mass stars, potentially being
produced in shocks that arise from instabilities in their radiatively driven stellar winds (e.g.,
Lucy & White, 1980; Owocki & Cohen, 1999). Since the stars generally show little X-ray
variability, these shocks could be occurring more frequently than the cadence of observation
which would imply that they are numerous and fairly homogeneous. About a decade ago,

56
spectroscopic observations from Chandra and XMM-Newton confirmed this basic picturewith X-ray emission in the extended wind giving rise to broadened lines at different depths(temperatures) within the wind (Schulz et al., 2000; Kahn et al., 2001; Waldron & Cassinelli,2001).
Yet evidence is growing that magnetically confined hot plasma either near the photosphere or in the wind plays a significant role (Babel & Montmerle, 1997; Waldron &Cassinelli, 2001; Oskinova et al., 2011). Higher resolution instruments could provide thesmoking gun for the hypothesis that these stars also have a corona and that the lack of variability largely arises from the nature of the bulk magnetic field. The fields visible at theirsurfaces are the observable portion of the fossil field remaining from their birth. However,
the impacts of such a field depend sensitively upon the ratio of kinetic energy to magnetic
energy as η = B 2 R2 /Ṁv∞ (e.g., Güdel & Nazé, 2009), with B the rms magnetic field, R thestellar radius, and the product Ṁ v∞ the terminal wind momentum. What may be crucial
is the rotation rate, since the stars that appear to have coronal activity rotate much more
slowly than the average rotation rate for O and B type stars. Raising the complexity ofthis is the observed trend that slow rotators also tend to have strong magnetic fields, which
may be of some significance as it would either mean that there was (or is) some dynamo
action building the field or that the fossil field has substantially braked the rotation of thestar. In Petit et al. (2013), a large catalog of massive stars has been compiled by the MiMeS
project. The character of the outer atmospheres of these stars has been categorized by the
behavior of their magnetospheres as either magnetodynamically or centrifugally dominated.
The magnetodynamical class have an Alfvén radius smaller than the Keplerian corotational
radius, whereas the centrifugally dominated atmospheres possess the converse. The magnetodynamical atmospheres have large meridional flows guided by the magnetic field that more
uniformly distribute high energy emission from UV and X-rays. The centrifugally dominatedwinds have equator migration of flows and field that forms a thin disk at the magnetic equator, which significantly modifies the character of the emission. These distinctions likely have57(b) Acoustic and gravity waves(a)Convection Zones

Buoyant magnetic f ux tubes

MicroturbulenceκR (cm2 g-1)

ClumpsStellar surface

Radiative Layer

Convective Zone

Radiative Layer

logT (K)

Figure 2.12: An illustration of the formation of convection zones due to distinct opacity
bumps from helium and iron. (a) Metallicity and temperature dependence of the Rosselandmean opacity. (b) A sketch of the outer iron opacity driven convection zone in a massive
star (Figures from Cantiello et al. (2009)).

some bearing on the plateau and scatter in the x-ray emission of intermediate mass stars as
in Figure 2.13.
While there is not a large degree of X-ray variability in many of these massive stars,
there is variability in both the visible photometric observations and in observations using
other bandpasses (Petit et al., 2013). This has been attributed to possible dynamo actionin the near-surface Fe-opacity bump convection zone present in stars with masses greaterthan about 5 or 6 M⊙ (Cantiello & Braithwaite, 2011), which is highly metallicity dependent
given that it arises from the peak in opacity from iron-group elements as seen in Figure 2.12a(Stothers & Chin, 1993; Cantiello et al., 2009). The character of this near-surface convection
zone is illustrated in Figure 2.12b, with the Fe-bump convection zone sitting atop the deep
radiative envelope that is sandwiched between the convective core and this layer. Convectionin this layer can excite acoustic and gravity modes above it, which may be detected using
asteroseismology. Moreover, applying a repurposed Cartesianism “Je convectent, donc jesuis une dynamo” (I convect, therefore I am a dynamo), it may be running an interface-type
dynamo like the Sun with buoyant flux tubes rising through to the surface. This opacity
bump is due to millions of spectral lines many of arising from relativistic spin-orbit coupling in

58
the iron-group elements. The iron-group alone accounts for about 50% of the total Rosselandmean opacity in the iron bump (Iglesias et al., 1992). Additionally, there is a zone of weak
convection in the more prevalent region of the He-opacity. It too could run a dynamo as it ismuch more rotationally constrained than the Fe-opacity bump convection. Depending upon
the magnetic Prandtl number there, it may build large-scale magnetic fields.

Figure 2.13: X-ray luminosity observations are displayed, showing such observations over the mass range of the main-sequence.
(a) Average X-ray luminosity, LX versus bolometric luminosity LBol , both normalized by their respective solar values. Spectral
type is indicated along the top abscissa and demarked with thin dotted lines. The black line is meant to illustrate the mean slopeof the three distinct ranges of behavior. The sources for the data are indicated in (a). (b) LX to mass ratio plotted versus mass.
This serves to highlight the region where LX depends linearly on mass and is likely wind driven. (c) The ratio LX /LBol versus
mass, highlighting the mass range where LX is linearly related to the LBol . The location of the Sun is plotted with its symbol ⊙,showing its relatively low X-ray luminosity.
5960
Some of the behavior exhibited by low, intermediate, and high mass stars is displayedin Figure 2.13. The data for the low mass stars is from Wright et al. (2011), which presentsa catalog of 824 X-ray sources with masses between 0.1 and 2.1 M⊙ and are shown as green
points in Figure 2.13. The black data points are from the ROSAT all-sky-survey (RASS)
catalog of intermediate and high mass stars (Berghöfer et al., 1997), whereas the orange data
points are from an Einstein survey of F-type and A-type stars (Schmitt et al., 1985). The
other two sets of data are from XMM-Newton observations of NGC 6231 (Sana et al., 2006,
blue points) and Carina OB1 (Antokhin et al., 2008, red points). There appear to be three
distinct mass (or luminosity) ranges where there are changes in the power law relationshipbetween X-ray luminosity (LX ) and bolometric luminosity (Figure 2.13a).
In the standard picture of stellar dynamo theory, an interaction of convection withrotation produces a magnetic dynamo within a star’s convection zone. Stars less massive
than about 1.6 M⊙ spectral type A possess an outer convective envelope above a radiative
zone, whereas very low mass stars are fully convective. The generation of magnetic field is
responsible for a plethora of magnetic phenomena in or above the stellar photosphere such asstarspots, a heterogeneous chromosphere, a magnetically constrained coronal plasma, flares,
and coronal mass ejections. The study of a nearby example of a cool star, the Sun, hasprovided a robust framework with which to interpret the X-ray emissions of other cool stars(Peres et al., 2000). Indeed, most cool stars except cool giant stars have been found to
exhibit characteristics grossly similar to what we know from the Sun. Most of these stars
have variable X-ray emission at temperatures of at least 1-2 MK and occasional flaring. Thus
the scatter in the X-ray data for lower mass stars seen in Figure 2.13 is primarily due to theinteractions between rotation, convection and magnetism.
The LX -Mass ratio in Figure 2.13b shows that the intermediate mass range between
the low-mass A-type stars and the higher mass B-type stars has LX ∝ M, which is likely due
to the dominance of shocks in their winds. When scaled by LBol , as in Figure 2.13c, two otherregimes become evident, with one for the lower mass stars and another for the massive O-type

61stars where LX ∝ LBol . The lower mass stars have a relatively high level of X-ray emission,
whereas the relative X-ray emission declines significantly with higher mass and becomes
nearly constant beyond about 16 M⊙. The saturation of LX at low masses is potentially aresult of the strong magnetic confinement of the wind, which in turn is likely due to therotation of the star and dynamo action in its exterior convective envelope. The saturation
at higher mass could be due to two mechanisms: one could be dynamo action in the Feand He-opacity bump convection zones; the other may be due to changing properties of thewinds and mass loss for the O-type stars. It would be most interesting if the mass required
for these near-surface convective layers to arise corresponds to the transition in the X-ray
luminosity behavior of the O-type stars, which may be ascertained through asteroseismic
measurements of this class of stars looking for a change in the balance of power between
g-modes and p-modes.
2.5

Magnetic Activity-Rotation Correlation
In his article “On the Nature and Construction of the Sun and the Fixed Stars,”

William Herschel appeals to the rotation of the Sun and the planets in his proposition that
other stars must also rotate (Herschel, 1795). He later clarifies his logic in a paper regarding
the periodic brightness changes in α Herculis, wherein it is stated that other stars may havespots and that the periodic light curves he observed were the result of them rotating in andout of the line-of-sight (Herschel, 1796). Nearly 80 years after his musings on the possible
axial rotation of other stars, a method for measuring stellar rotation using spectral lines
was suggested by William Abney (Abney, 1877a). Shortly thereafter Abney applied thistechnique in photographing the line broadening due to the Sun’s rotation (Abney, 1877b).
Half a century after these Doppler measurements of the solar rotation, studies of stellar
rotation shed light on two regimes of stellar rotation along the main-sequence that depends
upon mass (e.g., Shajn & Struve, 1929; Struve, 1930; Huang, 1953; Abt & Hunter, 1962;
Wilson, 1966). In particular, they found that stars more massive than about 1.6 M⊙ rotate

62
fairly rapidly at an average speed of around 140 km s−1 , with a slow increase with mass
and reaching a plateau above about 10 M⊙. Stars less massive than 1.6 M⊙, on the other
hand, show a steep decline in v sin i with decreasing mass. There is a distribution of theirprojected equatorial rotation speed (v sin i) about this mean, as the statistics in the abovereferences show. The fact that most massive main-sequence stars are rapid rotators is partly
due to their short lifespans, since only the lower mass stars have the time necessary to berotationally braked by their winds.
Spectroscopic surveys of nearby field stars and clusters revealed a potential reason forthis rapid decline in rotation. In a 1963 paper, Olin Wilson hypothesized, based on Ca
H and K observations of young clusters and old field stars, that magnetic activity declines
with age (Wilson, 1963). He further describes the link between rotation, convection, andmagnetic activity in Wilson (1966), attempting to ascertain why stars less massive thanabout 1.5 M⊙ (spectral type F5V) rotate so slowly compared to their more massive brethren.
He suggests that the reason is the loss of the outer convection zone and thus the loss of theability to generate a strong, large-scale magnetic field that can efficiently brake the rotationof a star (Wilson, 1966). Though, this would not hold for the most massive stars as they
have outer convection zones. This argument was put on a more quantitative footing witha survey of rotation in young stellar clusters that correlated with some of Wilson’s Ca H
and K measurements (Kraft, 1967). Figure 3 of that paper illustrates that F-type stars near
the canonical v sin i transition that also show Ca H and K emission have larger rotational
velocities at a given mass than stars without such emission. Shortly after which Kraft states
that: “for stars less massive than 1.25 M⊙, the largest rotational velocities are associatedwith the stars having active chromospheres.” In a concise letter, Andy Skumanich put thisin perspective and formulated what is now widely accepted as the “Skumanich t−1/2 law”
for the decay of rotation and chromospheric magnetic activity (Skumanich, 1972). Thus thelink between rotation, convection, magnetic activity, and age is made for low mass stars.
Building upon this early chromospheric activity work, it was found that there may be two

63Vaughan-Preston
Gap

Figure 2.14: Rotation period distributions of stellar populations of differing ages and mass,
showing the initially uniform distribution of a very young 1 Myr old cluster that evolves
toward slower rotation with greater age. The panel in the upper right illustrates the VaughanPreston Gap, with a population of slow rotators, of fast rotators, and very few stars between
them. The red line in the lower right two panels denotes the transition at 0.35 M⊙ fromstars with a radiative interior to those that are fully convective (Adapted from Stassun et al.
(2011)).

regimes of dynamo behavior along the main sequence. As the star spins down, there is atransition between two regimes of chromospheric activity, with a sparsely populated gap
between the two dubbed the “Vaughan-Preston Gap” after Vaughan & Preston (1980) andwhich can be seen most easily in Figure 2.14. It is also somewhat evident in the Ca H and K
data plotted in Figure 2.15, with a small gap between the most active and a second branchof less active stars. The underlying cause of this is the spin down of stars which is why it is64

Figure 2.15: X-ray and Ca H and K observations of individual stars are shown, which covera range of rotation rates and masses for lower mass stars. (a) Ratios of X-ray luminosity LX
and chromospheric Ca H and K luminosity (L′HK ) to the bolometric value LBol (normalized
so that these ratios are unity for the Sun; LX /LBol ≈ 10−6 and L′HK /LBol ≈ 10−5), displayedwith rotation period in days as abscissa. The color indicates a star’s mass, with M-type in
red, K-type in yellow, G-type in green, and F-type in blue. The dotted line demarks theboundary between X-ray and Ca H and K measurements. The solid line shows an estimatedfit for the rotation [or Rossby number in (b)] scaling of LX . (b) LX and L′HK plotted versus
an empirical Rossby number. The location of the Sun is shown with its symbol (⊙), again
indicating its relatively low X-ray and chromospheric Ca H and K luminosity [Constructedwith Ca H and K data from Soderblom et al. (1993a); Wright et al. (2004) and X-ray datafrom Wright et al. (2011)].

prominently labeled in Figure 2.14. This picture has been extended to provide a new way ofmeasuring stellar ages, based on a relation between the spin down of the star and its waning
activity, called gyrochronology (Barnes, 2007).
Beyond the connection between age and activity, the evidence began to mount thatthe mechanisms behind the activity were coupled to dynamo action that depends sensitively
upon the rotation rate of the star (Vaughan et al., 1981; Noyes et al., 1984). Indeed, therelationship between stellar rotation and proxies of magnetic activity remains an importantdiagnostic of the dynamo processes likely responsible for stellar magnetism in low mass stars.
Chromospheric variability measured in Ca H and K, and other chromospheric lines, as wellas the X-ray observations that probe stellar coronae, indicate that there is a monotonic

65
increase in activity with rotation rate. This increase is halted as the rotation rate reaches acritical point, after which either these indicators of magnetic activity or these star’s dynamos
become saturated. Indeed, both coronal and chromospheric activity measures show such asaturation (Figure 2.15), though it is more prominent in the X-ray observations. The firsttentative connection between X-ray luminosity and rotation was established by Pallavicini
et al. (1981), reporting that X-ray luminosity scaled as LX ∝ (v sin i)1.9 for low mass stars andthat the X-ray luminosities of higher mass stars depends only on the bolometric luminosity.
The scaling with rotation for low mass stars suggested that, like the Sun, X-ray emitting
coronal magnetic activity is linked to the magnetic dynamo running in the outer convection
zones of these stars, whereas stars of ever greater mass (thereby lacking convective envelopes)
produce X-rays primarily through their increasingly powerful winds.
Yet with more data the nearly quadratic proportionality between LX and v sin i wasfound to have a limited domain of applicability, since the ratio of X-ray to bolometric luminosity reaches a peak saturation of about 10−3 that is effectively independent of spectral type(e.g., Vilhu, 1984; Micela et al., 1985; Vilhu & Walter, 1987). The rotational period at which
this saturation level is reached, however, does depend upon spectral type, with lower mass
stars reaching this state at longer periods than higher mass stars (Figure 2.15a). This canbe attributed to the decrease in the vigor of the convection as one moves down the range of
masses (Pizzolato et al., 2003). It is unclear whether the constancy of the X-ray luminosity
is derived of a fundamental limit on the dynamo itself (e.g. Vilhu, 1984), a quenching arisingfrom the pervasion of strong magnetic structures across much of the stellar surface (Vilhu,
1984), or a centrifugal stripping of the corona caused by the high rotation rates (Jardine& Unruh, 1999). There could also be changes in the X-ray production mechanisms as wellas their X-ray spectral content. If these rapidly rotating stars support a Parker-like spiralmagnetic field driven away from the star by strong winds, another possible explanation is,given the limiting behavior of the toroidal magnetic field is Bφ /Br ∼ (Ωλ)/vwind (where Ωis the bulk rotation rate of the star, λ is the cylindrical radius, and vwind is the wind speed),

66
that the radius at which the field becomes tightly wound in azimuth occurs within the hotcorona itself. This would cause a transition close to the star where the wind is no longer
strong enough to advect the field, due to the plasma β becoming unity, and secondary plasma
effects cause a destabilization of the flow and magnetic field, creating a boundary layer that
may be an X-ray absorber. However, a wind could still escape along a polar route, creating astrong polar jet. Yet once saturation occurs, the X-ray emission becomes a function of only
the bolometric luminosity (Pizzolato et al., 2003), or effectively the mass, color, or radius of
the main-sequence star.
Such characteristics also hold for chromospheric measures of magnetic activity as canbe seen in Figure 2.15. This figure shows the ratio of chromospheric Ca H and K luminosity
L′HK to bolometric luminosity for 1328 main-sequence stars ranging in mass between 0.33 and1.60 M⊙, with data from Soderblom et al. (1993a) and Wright et al. (2004). The ratio of X-ray
luminosity and bolometric luminosity, also plotted with data from Wright et al. (2011), hasa much greater dynamic range, spanning six orders of magnitude compared to the roughly
1.5 orders of magnitude for the Ca H and K observations. Yet both show an increase ofactivity with decreasing rotation period and an eventual saturation. It is of note that thechromospheric activity seems to saturate at longer periods and that there are few stars thatare rotating rapidly that show Ca H and K emission, at least in this sample. As seen in many
previous papers regarding the empirical nature of dynamo action, these proxies for magnetic
activity seem to collapse into a more coherent structure when the stellar Rossby number isconsidered. This behavior follows from considering simple scaling arguments of the MHD
equations and assuming that the dynamo mechanism draws its energy from the differential
rotation. The work of Noyes et al. (1984) showed that such a relationship tightens therotation-activity correlation. An empirical Rossby number was used there that relies uponstellar models to provide the convective overturning time for each star in the sample, given
its observed log g and Teff . The resulting convective overturning times for the whole sample
are then fit with a cubic function, from which the Rossby number may be estimated given

67
the rotation rate of a star and its B-V value. The behavior of the rotation-activity curve
can be seen in Figure 2.15b, with the X-ray observations being more strongly affected. The
rather meager impact on the Ca H and K data is a result of ill-determined rotational periods,
which have been reconstructed using the cubic fits from Noyes et al. (1984), rather than aspectroscopically determined value. Moreover, there is a distinct paucity of Ca H and K
data for more rapidly rotating stars, though it is unclear whether this is an artifact due tobiased observational sampling or due to some intrinsic lack of chromospheric emission in the
more rapidly rotating stars.
If we now disassemble the X-ray observations, breaking them out by spectral type,individual trends and scaling become more obvious as in Figure 2.16. Speaking broadly,
the low mass stars of K, G, and F-type share similar characteristics in their X-ray emission
involving increasing emission with more rapid rotation and a regime of saturated emission.
What differs between them is the rapidity of the onset of this state of constant emission
with rotation, the rotation period at which it saturates, and the maximum luminosity. The
maximum luminosity seems to be determined by the bolometric luminosity, as surmised
from Figure 2.15 where the data collapse onto the same fit. The rotation period at which it
saturates may be related to the maximum achievable dynamo action, wherein the dynamo canno longer extract energy from the star’s differential rotation due to strong Lorentz feedbacks.
The rate of increase of X-ray emission with descreasing rotation period suggests a direct linkto the Rossby number of the flows within these stars in that it measures the efficiency of
the interaction of rotation and convection in building both the differential rotation and themagnetic field. In particular, there appears to be a change in the slope of the decrease of
X-ray emission with rotation period, from 1.7 to 2.5 between the F- and K-type stars (Figure
2.16a-c). This may indicate that the K-type stars have the most efficient conversion of kinetic
to magnetic energy. An interesting feature of stars about the mass of the Sun, namely the
G and F type stars, is that there seems to be a regime of super-saturation with the X-ray
luminosity decreasing at the rates of greatest rotation, which is a topic of ongoing research.

68

Figure 2.16: X-ray observations are shown for individual stars that are binned by theirspectral type, which cover a range of rotation periods and masses. X-ray luminosity (LX )normalized by the solar value and displayed with rotation period in days as abscissa. The
color indicates a star’s mass within the spectral type. (a) Spectral type F stars. (b) Spectral
type G stars, with the Sun’s position indicated. (c) Spectral type K stars. (d) Spectral typeM stars, with stars above 0.35 M⊙ as starred symbols and those below this mass as diamonds.
In (a-c), the dotted lines indicate fits to the data for the scaling with period, the saturation
value, and the rotation period corresponding to the beginning of the saturation (Constructedwith data from Wright et al. (2011)).

69
What is also evident from all of these observations is that the Sun, while thought of as amagnetically active and robustly influential object, is in fact hundreds of times less active
in both chromospheric and coronal proxies than some of its more rapidly rotating brethren,which serves to accentuate the overwhelming activity that must be present at the surfacesof these other stars.
What is possibly more remarkable in the X-ray data is the transition between stars thatpossess a radiative core and those that are fully convective (Figure 2.16d). Indeed, most of
the stars in our galaxy are less massive than our Sun, with more than half of all stars being
M-dwarfs. These are stars that are less than half the mass of the Sun and thus between tens
and thousands of times less luminous than it. These low-mass stars had been believed toharbor magnetic dynamos quite distinct from those in Sun-like stars. It seems appropriatethat there should be a difference between stars that have an interface-type dynamo, with aregion of penetration and convective overshoot, and those that are fully convective.
In the Sun, the tachocline is a site of strong radial shear revealed by helioseismology ,which is a matching layer between the differentially rotating outer convection zone and aninterior rotating as a solid body (see §3.5). Other stars with a radiative interior may also havea tachocline. In an interface-type dynamo, the tachocline is thought to play a major role in
the overall dynamo action. Such a layer below the convection zone can store and, through theradial differential rotation, comb small-scale magnetic field into a larger toroidal structure.
When the magnetic field strength is sufficiently large, this field can become buoyant and thus
could provide the seed magnetic field for the convective dynamo in the exterior convection
zone (Spiegel & Weiss, 1980; Parker, 1993).
However, stars less massive than about 0.35 M⊙ are entirely convective (Chabrier &Baraffe, 1997), and so do not have such an internal boundary layer. Thus it was expectedthat these stars would possess magnetic behavior different from more massive stars, with
only small-scale field being generated (Durney et al., 1993). Recent observations have instead
shown that such stars can instead have both small- and large-scale fields, which means that

(a)

70

(b) 10
1.0
Activity Lifetime (Gyr)8
Fraction Active

0.8
0.6
0.4
0.2
0.0

6
4
2
0

M0 M1 M2 M3 M4 M5 M6 M7 M8 M9 L0
Spectral Type

M0

M1

M2

M3 M4 M5
Spectral Type

M6

M7

Figure 2.17: Magnetic activity properties of spectral type M stars from an Hα catalog of
38,000 such stars (West et al., 2008). (a) Fraction of active stars as a function of spectral
type for the West et al. (2008) spectroscopic sample (diamonds). Older survey data areshown as triangles. (b) Activity lifetime with spectral type as abscissa. This plot serves toillustrate that M-type stars with masses greater than about 0.35 M⊙ spin down and becomeless active, whereas their lower mass counterparts remain active for much longer periods oftime. A theoretical age-activity relationship from Hawley et al. (2000) is shown as the dottedline (Adapted from West et al. (2008)).

these low mass stars can build magnetic fields with global-scale structure. Indeed, one of
the great surprises emerging from studies of stellar magnetism has been that even low-massstars can build large-scale magnetic fields. Zeeman-Doppler imaging has clearly establishedthat rapidly rotating fully-convective stars can build large-scale magnetic fields (see Figure2.9), while lacking any signs of latitudinal differential rotation (Donati et al., 2006a). Thisbehavior is in stark contrast to the expectation that such large-scale fields are only possible
with a measurable differential rotation. Yet some aspects of magnetism in low-mass stars does
depend upon mass, as is manifest in Figures 2.16d and 2.17. There is a distinct and smoothtransition between M-type stars with a radiative core and those that are fully convective
occurring at 0.35 M⊙ (spectral type M5). This is borne out in multiple observations: fromthe Hα and Ca H and K chromospheric activity proxies to X-ray emission (e.g., West et al.,2008; Morin et al., 2008; Browning et al., 2010; Morin et al., 2010). Figure 2.17a indicatesthat only about 8% of the more massive M-type stars are currently active, whereas the lower

71
mass M-type stars are nearly all active. This likely occurs because the higher mass M-typestars are able to spin down in only a few hundred million years, which eventually reduces
their activity due to decreased dynamo action. The fully convective stars, on the other hand,seem to remain active for at least ten times as long (Figure 2.17b). One potential explanationis that, in direct contradiction to interface dynamo theory, the majority of surface magnetic
features emerge from the near-surface regions (Brandenburg & Subramanian, 2005). Thus
these stars provide an interesting theoretical and observational playground for testing crucial
elements of stellar dynamo theory. For instance, one can ask what role a tachocline might
play? Since these stars are birthed with a fairly uniform distribution of rotation periods,
as visible in Figure 2.14, why does it seem that stars with a possible tachocline are able toefficiently spin down whereas those without do not?
2.6

Conclusion
In summary, there is a rich observational history for both the Sun and other stars thatspans nearly four centuries and to which modern astrophysics owes its current success. There
are a multitude of observable time and spatial scales of solar magnetism, which provides the
keystone in our understanding of stellar magnetism as well as the properties of magnetism inmany other astrophysical contexts. Yet despite the rapid advances in observing techniques
and technology over the last century reviewed in this chapter, we have only begun to shedlight on the mysteries of stellar magnetism. For instance, it is still unclear what establishes
the magnetic cycles seen in activity measures of many stars, including the Sun, or the orderly
large-scale fields present at their surfaces as determined by spectropolarimetry. Indeed,
how long-term periodicity and large-scale, organized structures emerge from the chaos andturbulence present in stars is the central theme of dynamo theory. The following chapter
will attempt to build the theoretical framework necessary to understand the processes thatlead to the observed behaviors of stellar magnetism.

Chapter 3
Stellar Dynamo Theory“TO LEARN: Non-linear Classical Hydro” – Feynman’s Last Blackboards
Dynamo theory has its roots in geophysics where an explanation was sought for the
migration of the Earth’s magnetic pole, which was well known from four centuries of compass
measurements and calibrations. The first such theory of the Earth’s magnetism, in which the
Earth is considered a permanent magnet, was put forward in William Gilbert’s 1600 treatiseon “The Magnet and Magnetic Bodies, and on That Great Magnet the Earth” (“De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure”). Carl Gauss established amuch more complete theory, which even incorporated the idea of flowing currents. Gaussconsidered the field as potential and likely due to galvanic currents contained within the
Earth’s volume (Gauss, 1838). The debate on the origins of the magnetic field that followed
would, however, not be settled for about a century. Prior to the certainty of dynamo processes being at work within the Earth, the tentative theories for the properties of the Earth’s
magnetism were laid out in 1911 by Arthur Schuster. This included the idea that eventually
emerged as dynamo theory, which he dismissed as impossible (Schuster, 1911). Schuster andmany others of the era often presumed that, due to the coincidence of the Earth’s magnetic moment and its rotation axis, the terrestrial magnetic field was directly related to
its rotation. This theory posited that the Earth’s rotation directly influenced atomic andelectronic motions, thus creating polarization. Another hypothesis was that independently
rotating magnetized iron spheres within the Earth’s interior produced the observed geomag-

73
netism. Some also thought that the fundamental equations of electromagnetism required
some obscure relativistic modifications.
Although such theories were refuted by many scientists, Joseph Larmor is largely credited with planting the seeds from which modern dynamo theory grew. His idea that mean
flows and fields could be self-inducting, though later proven incorrect, provided a basis fordisestablishing earlier theories. These ideas were succinctly presented in a two page letter,following the 87th meeting of the British Association for the Advancement of Science (Larmor, 1919). It bears repeating his words here as it summarizes parts of the theory nicely:“In the case of the sun, surface phenomena point to the existence of a residual internal circulation mainly in meridian planes. Such internal motion induces an electric field acting on
the moving matter: and if any conducting path around the solar axis happens to be open,
an electric current will flow round it, which may in turn increase the inducing magnetic
field. In this way it is possible for the internal cyclic motion to act after the manner of thecycle of a self-exciting dynamo, and maintain a permanent magnetic field from insignificant
beginnings, at the expense of some of the energy of the internal circulation.” The penultimate sentence of Larmor’s paper states that, while the mechanism he proposed for the Sun
is reasonable there, if it were to operate within the Earth it would require fluidity and aresidual circulation in deep-seated regions. This was prescient since seven years later seismic
measurements of the Earth’s interior definitively showed it is indeed a liquid (Jeffreys, 1926),
though it had been suggested earlier (Oldham, 1906). In summary, Larmor describes two
dynamos operating in the Sun: one mechanism for the large-scale field is like a flux transport
dynamo, while the other mechanism is akin to a local dynamo primarily acting in sunspots.
Following Larmor’s insight, there were rapid developments in dynamo theory. Two
prominent physicists, Thomas Cowling and Walter Elsasser, provided much of the progressin dynamo theory throughout the 1930s and 1940s. Indeed, Thomas Cowling’s interpretationof Larmor’s sunspot dynamo (Cowling, 1933) was strongly influential in the field, though it
was written in a way that emphasized the perceived flaws in kinematic dynamos and ele-

74
vated a more obscure theory in its place. In that paper, Cowling proved by example thatthe currents established by an axisymmetric magnetic field of a sunspot cannot be a selfsustaining dynamo. More generally, he showed that a field which resembles an axisymmetricfield cannot be maintained without an additional EMF. After early beginnings in quantum
theory, Walter Elsasser began a long study of the Earth’s modes of dynamo action. In a
1939 Physical Review article, Elsasser details the role of the Coriolis force in inducing statistical correlations between temperature and velocity at all turbulent scales in the Earth’s
liquid mantle, which in turn drive an electromotive force (EMF) and thermoelectric currents.
Although Elsasser found thermoelectric currents to be the dominant factor in the Earth’s
dynamo, he states at the end of this article that the fields in sunspots must instead be
maintained by a strong EMF. This, while not perhaps true of sunspots, is certainly true inthe current understanding of global-scale dynamos. After being involved in the second world
war with the US Signal Corps, Elsasser published a second work showing that toroidal flows
alone cannot sustain a field (Elsasser, 1946). Continuing their collaborative efforts, Cowling
permitted Elsasser to publish a more rigorous and general proof, elucidating why axisymmetric flows cannot sustain an axisymmetric magnetic field. This is the commonly cited“anti-dynamo theorem” (Elsasser, 1947). It does not, however, preclude the maintenance of
non-axisymmetric fields by an axisymmetric flow.
At this point, most of the work in dynamo theory had been aimed at showing how adynamo could not exist under various conditions of symmetry. Though in the 1947 article,
Elsasser builds upon his earlier ideas in describing a mechanism whereby a mean toroidal
magnetic field can be built through the inductive coupling of bulk fluid motions to the meanpoloidal field. This operates by the stretch-and-fold mechanism, with the toroidal motions of
the fluid acting to stretch poloidal field lines repeatedly over many rotation periods. These
now toroidal field lines begin to accumulate in regions with a gradient in the bulk fluid
motions. More importantly and in the same article, Elsasser states that the coupling of
turbulent convection with rotation provides a statistical means by which the poloidal field

75
can be regenerated from a mean toroidal field, which is in contrast to the generation of thetoroidal field. It is also pointed out that, while this turbulent field generation mechanism
may be a second order effect, it is crucial to the overall maintenance of the dynamo process.
These arguments are mostly qualitative and a more rigorous framework was still elusive.
Such a mechanism was confirmed to be possible with the calculations of Bullard &Gellman (1954), which evolved a kinematic dynamo in a 3D sphere with 10 radial grid points
and four spherical harmonic degrees using an early supercomputer called ACE at the National
Physics Laboratory in the UK. Their calculations showed that, if the flow field was that ofelongated eddies aligned with the rotation axis, these structures can maintain and build
poloidal and toroidal magnetic fields. They also illustrated, in painful detail, how energy
was transferred between the various spherical harmonic modes. In 1955, Eugene Parker
carried this further, describing more explicitly how a poloidal field can be regenerated froma toroidal field if there are cyclonic fluid motions present. Parker considered the interaction of
cyclonic motions with a mean toroidal field, where the motions build small loops of magnetic
field. The longitudinal field of these small-scale loops largely cancels, rather in aggregate
they act to establish a poloidal field aligned with the rotation axis (Parker, 1955).
The efforts to establish a degree of mathematical precision were still frustrated even in
1955, where Elsasser states: “We have yet to come to grips with the one problem which, whilerather arduous and frustrating from the purely mathematical viewpoint, is clearly basic forhydromagnetism, namely the question how the observed cosmic magnetic fields are actuallycreated by amplificatory processes. This is the hydromagnetic dynamo problem.” With a
bit more patience, however, rigorous proofs of self-consistent dynamo action emerged forvery idealized systems. For instance, a type of spherical dynamo was illustrated in Backus(1958). The primary result is the explicit construction of a large class of continuously differentiable velocity fields capable of indefinitely maintaining or amplifying the dipole momentof an external magnetic field even in the presence of resistive dissipation. Several importantearlier results are also confirmed in this paper, namely that if the fluid motions have no

(a)

76

(b)

Twist

Merge

StretchFold

Figure 3.1: Examples of slow and fast dynamos. (a) The “slow” Roberts kinematic dynamo.
(b) The “fast” stretch-twist-fold-merge dynamo.

radial component, the poloidal magnetic field decays as rapidly as if the fluid were at rest.
Furthermore, if no poloidal field is initially present, the toroidal field decays as if the fluid
were at rest. There it was also shown that dynamo maintenance is impossible if the local
strain-rate of the fluid is always and everywhere less than the decay rate of the axisymmetric poloidal field. One important result is that fluid motions are bounded linear operators
for the magnetic field on its Hilbert space equipped with a finite L2 (energy) norm for themagnetic field.
Other authors independently built similar solutions, though they were often a bit moreprosaic than Backus’s work (e.g., Herzenberg, 1958). Herzenberg describes a solid, uniformly
conducting medium with two or more conducting rotors in electrical contact with the otherrotors. In this configuration, dynamo action is provably possible when the rotors spin fasterthan a critical value (Gailitis, 1973; Moffatt, 1978; Brandenburg et al., 1998). The Herzenberg
dynamo belongs to the class of “slow” dynamos, with the other class being “fast” dynamos.
Slow dynamos reach their largest growth rates for moderate values of conductivity, fallingto zero in the limit of perfect conductivity. In contrast, fast dynamos, such as the Zeldovich
stretch-twist-fold dynamo (see Figure 3.1b and Brandenburg & Subramanian (2005)), have

77
a finite and positive growth rate in the same limit. More to the point, the existence of suchprovably self-sustaining dynamos was essential to the development of dynamo theory, but
the problem of greater interest became (and still is) how best to model the dynamo processesof the Earth, the Sun, as well as in other astrophysical bodies. It is prudent, at this stage, tointroduce the mathematical and physical framework within which much of the more recent
work has been set. This short mathematical diversion is contained in the next section.
3.1

The MHD Equations
The state of hydrodynamic turbulence, let alone magneto-hydrodynamic turbulence,is still best summed up by an apocryphal quote oft attributed to Werner Heisenberg: “I aman old man now, and when I die there are two matters on which I hope for enlightenment.
One is quantum electrodynamics, and the other is the turbulent motion of fluids. And
about the former I am rather optimistic.” The underlying issue being that the equations of
motion form a nonlinear, dissipative system. This inherently means that they can exhibit
quasi-periodicity, multi-stability, and other chaotic behaviors, resulting in multiple basins of
attraction in the phase space of the physical system.
There are many places from which one can begin to derive the single fluid MHD
equations, say from geometrical arguments, from a Lagrangian standpoint, or from kinetic
theory. The path that utilizes kinetic theory is tightly coupled to the classical theory offields embodied by the Lagrangian approach. Since a Lagrangian framework is used in later
sections and since kinetic theory provides the most general framework, the derivation will
start there. Under Boltzmann’s kinetic theory, statistical ensembles of large numbers of suchparticles map the otherwise 6-Np dimensional phase space into a 6-S dimensional symplectic
manifold, where S is the total number of particle species in the plasma. Appealing todifferential geometry, we know that one-forms on this manifold are isomorphic to Hamiltonian
vector fields. Moreover, such symplectomorphisms preserve volume on phase space and thus
there must be an integral invariant on this space (Flanders, 1963; Bishop, 1968). This is

78
Liouville’s theorem, which states generally that if a region of the N-dimensional phase space
RR
D0 at time t0 moves to a region D1 at time t1 then D0 µ = D1 µ, where µ is the phase spacedensity. In more practical terms and given the phase space density fs (x, v, t) of species s,

this density is constant along any trajectory in phase space and can be written as
Fs
∂fs
∂fs
,
+ v · ∇x fs +· ∇v fs =∂t
ms∂t col(3.1)

where F is the total force on species s and the collisional forces are included in the termon the right hand side. Relativistic effects and radiative transfer have been neglected asthey are irrelevant to the dynamics considered here. The goal is to remove the velocity
dependence by integrating over moments in velocity space and supplementing the resulting
equations with a closure. The derivation of the full set of equations for a multi-species plasma
coupled to an electromagnetic field that result from this approach is contained in Appendix
A. These resulting equations need to be further reduced to the more manageable single fluid
MHD equations. The MHD approximation assumes a single ion species, with a nearly equal
number of electrons. The magnetic fields are considered weak enough that the total pressure
tensor is isotropic and the time scales sufficiently long that the time variation of the electricfield is negligible. These assumptions reduce the Boltzmann equation based framework ofAppendix A to the followingn = ni ≈ ne ,
̺ = e(ni − ne ) ≈ 0,

ρe ue + ρi ui≈ ui ,
ρe + ρie
J = (ni ui − ne ue ) ≈ Jw ,
c

e Pe
Pi
e
Π=
+
I≈
Pe I,
c me micmeu=Q̇e = −ρe νei (we − wi ) ,

(3.2)

79
with c the speed of light, e the elementary charge, mq the ion or electron mass, nq the ionor electron number density, ρq the mass density of ions or electrons, Pq the partial pressuresof the ions or electrons, and I the identity tensor. A linear conservative approximation to
the momentum collision integrals Q̇q is used where νei is the electron-ion collision rate andρe νei = ρi νie , yielding Q̇e = −Q̇i . Thus the full collisional term in Equation A.41 is
X qs
̺e
Q̇s =
(we − wi )(1 + me /mi ) ≈ −νei J.
ms
cnuei
s

(3.3)

Utilizing the above assumptions and ws = us − u, the forces that yield changes in this
scenario are
X qs
s

cms

X q2n 

 qn
us
s
×B +
∇Φ,
cms
c
c
s
 e2 n ueuie2 n 
E+
E+
×B +
×B ,
=
cme
c
cmi
c
h
i
2
u
e
en
E+ ×B −
J×B
≈
cme
c
cme

F=

s

E+

(3.4)

Finally, letting the conductivity be σ = e2 n/cme νei and upon substituting the above
approximations into Equation A.41, Ohm’s law for the simplified single-ion plasma becomes


h
i
1 ∂J
σ
σ
u
+ ∇ · [J ⊗ u + u ⊗ J] + ∇Pe + J × B = σ E + × B − J,(3.5)
νei ∂tneenc
If we neglect the collisional timescale phenomenon, the electron pressure, and the Hall term
this further simplifies to the often used form
cE = c/σJ − u × B.

(3.6)

A further simplification is that of neglecting ∂t E, which can be done if the timescales ofinterest are long in comparison to 1/cσ ∼ 10−15 s for solar interior conditions. Therefore,
Maxwell’s equations become
∇·E=0

(3.7)

∇·B=0

(3.8)

∇ × B ≈ 4πJ,

(3.9)

∂B
≈ −c∇ × E = ∇ × [u × B − η∇ × B],∂t(3.10)

80
with the magnetic diffusion η = c/4πσ. Thus, after combining a few terms and explicitly
including the diffusive terms, the equation set under the standard MHD approximation are∂ρ+ ∇ · ρu = 0,
(3.11)
∂t




1
B2
∂ρu
I−
+ ∇ · ρu ⊗ u + P +B ⊗ B − D = ρ∇Φ,
(3.12)
∂t
8π
4π



∂EM
1
η 1
+ ∇ · u(EM + P ) + q − u · D −u · BB +∇B2 − B · ∇B
= 0, (3.13)
∂t
4π
4π 2
∂B
= ∇ × [u × B − η∇ × B],(3.14)∂t1
B2
EM = ρu2 ++ ǫ + ρΦ,
(3.15)
2
8π
The viscous diffusion tensor is Dij = µ(eij − 2/3∇ · uδij ), with eij being the linearized stresstensor and which assumes that the bulk viscosity is one third of the shear viscosity. The
derivation of the molecular viscous diffusion is not carried out here, as one can establish thisterm through infinitesimal deformations and ascribe a force to it through a thermodynamically dependent coupling coefficient or compute the collisional integrals in kinetic theory asin Braginskii (1965). For LES, perhaps, a more general choice in the case of large velocities
or strong gradients would be the Eulerian-Almansi finite strain tensor"#
1 ∂ui ∂uj X ∂uk ∂uk
.
+
+
Eij =
2 ∂xj
∂xi
∂xi ∂xj(3.16)k

It might be considered as an alternative to more sophisticated schemes as it nonlinearlyenhances dissipation in regions with strong gradients. More generally, the Israel-Stewart
equations (Israel & Stewart, 1979) could be invoked and parameterized with finite timecorrelations as in (Pratt, 2008).
There is now a basis for comparison between the fairly general microscopic, species anddistribution dependent magnetohydrodynamic flows and the macroscopic behavior of largerscale MHD systems. Small spatial and short temporal scales are relevant in the former,
wherein the coupled dynamics of the radiation field, magnetic and electric fields, as wellas multiple ion and chemical species play dynamical roles. This is certainly critical in low

81
density and highly variable environments such as stellar chromospheres, coronae, and winds.
In the MHD system, there is but a single ion and the free electrons that accompany it.
This greatly simplifies matters, yet the system still has many facets such as two-component
collisional dynamics, separate electron and ion pressures, and the Hall term electrons, andthus can be complex in its own regard. Even with these terms discarded, the remaining
parameter regimes for turbulent magnetohydrodynamic flows are largely unexplored. Hence,
despite being a much simpler system, the MHD regime permits the study of larger spatial andlonger temporal scales, which are most applicable to the stellar dynamo problem. However,
the transition between regimes is not at all clear, nor is it prudent to neglect the influence of
one on another. This is especially true when considering the outer envelopes of stars, where
there is a rapid change from the relatively dense low β plasma of stellar interiors and thelow density high β plasma of their chromospheres, coronae, and winds.
This effectively brings us to the current era of dynamo theory. After the 1960s, theliterature becomes intractably large to succinctly trace the roots of the various facets ofthe theory. There are currently three classes of methods in dynamo theory. The first classconsists of 1D stochastic or continuous systems of equations that model certain aspects ofdynamos. The second class utilizes the framework of mean-field electrodynamics, wherein
the effects and feedback of turbulent motions upon the mean flows and mean magnetic fields
are parameterized. The final class of methods attempt to directly simulate dynamo processes
with fully 3D models of conducting fluids in various geometries and under a wide variety ofconditions. All three of these methods seek to understand how the nonlinear interaction of
turbulent flows and magnetic fields maintain magnetic fields against resistive decay. They
also seek to comprehend what mechanisms determine the spatial and temporal scales for
local and global fields. The following subsections will provide an overview of the systems
that have relevance to stellar dynamo theory.

82
3.2

Dynamo Waves and Low-order Approximations
There are a few fundamental theories that reduce the magnetohydrodynamic (MHD)

equations to more easily studied, low-order and yet still nonlinear systems of ordinary differential equations. These methods have been developed over the last 50 years. The firstmention of dynamo waves arises from the last section of Parker’s oxyblepsic 1955 paper inwhich there is a discussion of “migratory dynamos”. In this model, a flow is prescribedand its impact upon the toroidal vector potential is parameterized as ∂t A = ΓB − η∇2 A,
with B = ∇ × Aφ̂ + B φ̂ and with Γ measuring the “violence” of Parker’s cyclones (Parker,
1955). The spatially varying functions of Γ, η, A, and B can be approximated by selecting
particular modes, which reduces the set of equations to simple ODEs, and allowing theircouplings to be examined. For instance, A could be chosen to be a pure dipole, and the particular modes that arise in B could then be found. Conversely, a toroidal field mode could
be considered and its influence upon the poloidal field examined. More complicated fields
can be constructed as well. The most rudimentary solutions of these equations eliminate one
direction of motion. In doing so, these equations admit damped traveling wave solutions, if
one considers motion only along the mean field direction. With other dimensions included,
for instance in polar coordinates, the behaviors are more complex, but still can admit traveling wave solutions (Charbonneau, 2010). Some 3D simulations also show some aspectsof this behavior, despite their turbulent conditions (e.g., Brown et al., 2010, 2011b; Käpylä
et al., 2012, 2013). Some authors have extended such simple models and considered thenonlinear versions of this equation as well. These slightly more complex approximations to
the induction equation have some bearing on the solar problem as they reveal quasi-periodic
solutions, illustrating behavior similar to the deterministic chaos of the solar cycle (Weisset al., 1984; Weiss & Tobias, 2000).
The Babcock-Leighton dynamo is one such nonlinear 1D model, in this case it is inlatitude and time (Babcock, 1961; Leighton, 1969). These have quite a few ingredients and83parameters, but overall are still quite simple. The Babcock-Leighton mechanism considers
spatially-averaged magnetic fields that vary in latitude and time. It is also assumed that thedifferential rotation is a function of latitude and depth, is constant in time, and acts only
upon the poloidal magnetic field to modify the toroidal field, with the fluid assumed to be
perfectly conducting and the flow laminar. If the toroidal field strength exceeds a criticalthreshold, the field erupts and a sunspot occurs. This eruption of the toroidal field induces
radial fields with a net meridional magnetic moment due to the Joy’s law and simultaneously
weakens the toroidal field. The emerged radial field is then dispersed by a random walk.
This model is well motivated by observations and produces solutions that mimic Hale’s law
quite well. It has also played a very prominent role in 2D mean-field modeling, which is the
subject of the following subsection. Such a Babcock-Leighton mechanism has recently been
incorporated into 3D models in order to assess the role of sunspots in the deeper dynamo
more self-consistently (Miesch & Brown, 2012).
3.3

Mean-field Electrodynamics
There are many examples of “slow” dynamos, such as the Herzenberg dynamo or theRoberts flow, the latter of which consists of a steady 2D pattern in all three directional
components of the velocity. The Roberts dynamo provides an important benchmark for alarge-scale dynamo (see Figure 3.1a and Roberts (1970)), where magnetic fields are inducedon scales larger than those of the velocity through the “α-effect.” Returning to Parker’s
dynamo wave solutions discussed above, a close inspection of the evolution equation for
the toroidal vector potential shows the first example of the scalar α-effect, which is equivalent to the First-Order Smoothing Approximation (FOSA) to turbulence (Brandenburg &Subramanian, 2005).
In work that paralleled that of Parker and many others in the West, though hidden
behind the Iron Curtain, Steenbeck and colleagues were creating an effective field theory forthe calculation of the mean Lorentz and electromotive forces for an electrically conductive,

84
rotationally influenced, and turbulent medium (Steenbeck et al., 1966; Moffatt, 1970). This
work extended Parker’s and showed that in general if the turbulent motions possess a mean
helicity, then Ohm’s law contains additional features that are aligned or anti-aligned withthe mean field. The occurrence of this kind of electromotive force (EMF) has been dubbed
the α-effect. If the time-averaged small-scale convective turbulent motions lacks reflectional
symmetry such as when it has helicity, Ohm’s law for the mean fields contains an EMF witha component directly proportional to the mean magnetic field.
The α-effect leads, as in Parker’s model, to the generation of a poloidal magnetic fieldfrom a toroidal field and thus partially resolves the dynamo problem. Primarily, a broken
symmetry leads to the possibility of overcoming resistive decay as well as producing an inverse
cascade of magnetic energy and helicity that can organize into large-scale fields. Indeed, amechanism to overcome the decay and build such fields is crucial for all dynamo theory. The
early versions of α-only dynamos have since been extended to many geometries (e.g., Rädler,1968), and have been modified so that the back-reaction of the magnetic field on the flow
can be parameterized as in the Malkus-Proctor effect (e.g., Malkus & Proctor, 1975). These
effects can be elucidated by considering first the azimuthally averaged induction equation in
spherical coordinates (obtained by applying the Reynolds averaging rules to Equation 3.14),
wherein, following from its solenoidal character, the magnetic field is expanded into poloidal
and toroidal components asmagnetic diffusion}|

meridional advection

generation}|{ poloidal
 { zz }| {1∂A
1= η ∇2 − 2 A − um · ∇ (λA) + φ̂ · u′ × B′
∂t
λ
λ


1 ∂η ∂λB∂B
1B= η ∇2 − 2 B +− λum · ∇∂tλ|λ ∂r{z ∂r }| {z λ}{z}|zmagnetic diffusion

diffusive transport

stretching(3.18)

meridional advection− B∇ · um + λ∇Ω · ∇ × Aφ̂ + φ̂ · ∇ × u′ × B′
| {z } |
{z
} |
{z
}
compression(3.17)(3.19)

toroidal generation

with B = ∇ × Aφ̂ + B φ̂ and where λ = r sin θ is the distance from the rotation axis,
um = um (r, θ) is the meridional circulation, and Ω = Ω(r, θ) is the rotation rate. Note that85
the axisymmetric velocity considered here is independent of time. With these equations,the features discussed in the first section of this chapter become quite evident. Namely,
if the turbulent electromotive terms E = u′ × B′ ≈ αB are omitted, the field simply decays away as there is nothing to regenerate A, since the equation for A is then a nothing
more than an advection-diffusion equation. The α tensor is the parameterization mentioned
above, with the correlations between the fluctuating velocity and fluctuating magnetic fieldcomponents being expressed as a spatially varying function that provide an instantaneous
temporal correlation with the mean field. The mean-field evolution equations illustrate two
new features that are intrinsic ingredients in the simulations presented in this thesis and tostellar dynamos in general. Chiefly, the stretching of the poloidal field by the differential
rotation leads to the generation of toroidal field (the Ω-effect) and the turbulent generationof poloidal field that prevents the ohmic decay of the entire field. However, it is also ofnote that differential rotation is not necessary for a dynamo to run, only that there be someinherent symmetry breaking in the system that leads to a turbulent EMF and thus an inverse cascade. To put this in the nomenclature of mean-field theory (MFT), there are three
primary types of dynamos. These are the αΩ, α2 , and α2 Ω, where the α2 indicates that the
α-effect is responsible for a substantial fraction, if not all, of the generation of both poloidal
and toroidal field.
Solar models utilizing MFT reflect many of the observed magnetic phenomena (e.g.,
Petrovay, 2000; Charbonneau, 2010). The distribution of sunspots with respect to latitude
and time can be reproduced under some assumptions for the somewhat arbitrary α-effect. To
some degree, the solar angular velocity profile could be assessed through MFT in the deeper
layers of the convection zone, which until about 1995 and the advances in helioseismology,
were largely inaccessible to observations. Although the assumptions going into the theory
initially were in contradiction with helioseismology, the coefficients of the α tensor could
be retuned to reproduce some solar phenomenon. Beyond solar dynamo models, MFT has
also been successfully applied to the terrestrial dynamo including being in good agreement

86

Figure 3.2: A diagrammatic illustration of the four primary mean-field dynamo modelingtechniques. The sites of α and Ω effects as well as the magnetic flux transport mechanisms(and their respective directions) are shown for the northern hemisphere in a meridional
plane. The four methods are: (OL) overshoot layer dynamos; (IF) interface dynamos; (CP)cospatial pumping models; and (BL) Babcock-Leighton-type models. The mechanisms at
work in each model follow from the legend on the lower right, with some clarification givenby the table in the upper right. [Adapted from Petrovay (2000)].

with 3D direct numerical simulations (Schrinner et al., 2007). While there is a bewildering
variety of MFT solar models, they all can be effectively reduced to the mean-field equations asabove with the exception of the approximation and construction of the turbulent EMF. The
menagerie of MFT models are shown in Figure 3.2 for a visual reference of the mechanisms at
work in each of them. The various models include: the usual α-type prescriptions describedabove (e.g. Ruediger & Brandenburg, 1995; Rüdiger & Arlt, 2003); flux-transport models thatspecify the meridional circulation apriori (e.g., Rempel, 2005, 2006b,a; Pipin & Kosovichev,
2013); interface dynamos, which circumvent the issues of α-quenching raised in simple αΩ87
models by placing the region of toroidal field generation away from regions where the α-effectoperates instead relying upon the radial shear of a tachocline (e.g., Parker, 1993; Dikpati &Gilman, 2001); and finally there is also the phenomenological Babcock-Leighton mechanismthat parameterizes the formation and rise of buoyant flux tubes (e.g., Wang & Sheeley, 1991;
Dikpati & Charbonneau, 1999; Charbonneau et al., 2005; Guerrero & de Gouveia Dal Pino,2008).
The ultimate goal of solar dynamo theory is to account for the 11-year sunspot cycle(22-yr magnetic activity cycle) of the Sun, within the context of other observed phenomena
such as the ubiquitous and rapidly-evolving small-scale flux that threads the solar photosphere. This striking manifestation of order amid chaos is a hallmark of solar magnetic
activity and its explanation ultimately lies in detailed studies of convective dynamics thatare inherently 3-D and nonlinear in nature. The computational challenges associated withsuch an undertaking have spurred the development of more tractable 2-D models of the solar
dynamo, such as the popular flux transport models. Typically kinematic and mean-field innature, flux transport models capture the effects of differential rotation and meridional circulation on the evolution of the axisymmetric magnetic fields. Most flux transport dynamo
models are also Babcock-Leighton models in the sense that the destabilization, emergence,and subsequent dispersal of toroidal magnetic flux play an important role in generating themean component of the poloidal magnetic field.
The coupling of mean transport effects to the emergence of magnetic structures captured by such coupled Babcock-Leighton flux transport models continues to play a prominentrole in shaping the view of the solar cycle. In particular, the transport of toroidal magnetic
field by meridional circulation in these models, and the subsequent destabilization and emergence of that flux is commonly thought to set the 11-year cycle period, accounting for theobserved equatorward migration of sunspots over the solar cycle. Until recently this circulation could be treated as a free parameter, as helioseismology had yet to constrain it.
Indeed, the widely adopted mono-cellular flow within each hemisphere now appears to be8890
Latitude (deg)

(a)

60
30
0
90

Latitude (deg)

(b)

60
30
0
90

Latitude (deg)

(c)

60
30
0
0

10

20

30

40

50

Time (yrs)

Figure 3.3: Babcock-Leighton flux transport dynamo with dual meridional cells dynamo,
showing the impact of increasing meridional flow speed. The surface radial magnetic field
is illustrated in a time-latitude diagram and in color, as in the colorbar on the right, forseveral meridional flow speeds (a) u = 0 m s−1 , (b) u = 10 m s−1 , and (c) u = 20 m s−1 .
The contours represent the toroidal field at 0.92 R⊙, with steps of 25 G between ±100 G.
[Adapted from Pipin & Kosovichev (2013)].

at odds with recent results from helioseismology that suggest the existence of multiple cells
within a hemisphere (Zhao et al., 2013). Such a multi-cellular profile may severely disrupt
the operation of a flux transport dynamo (e.g., Jouve & Brun, 2007; Pipin & Kosovichev,
2013) as is visible in Figure 3.3. Moreover, it is impossible to gauge the self-consistency ofthe parameterized 3-D effects of flux emergence and turbulent transport with the prescribedmean flows in 2-D models.
One interesting result using MFT, beyond reproducing many aspects of the solar andterrestrial dynamos, some of these models can be reduced to a simple iterative map, whichallows one to find possible basins of attraction for periodic solutions amidst those that are

89chaotic. In particular, a large sample of MFT solar simulations have been assessed for solutions with periodic behavior as the dynamo number (effectively the magnetic Reynolds
number) was increased and it shows a period-doubling bifurcation in magnetic energy thatis very similar to the amplitude bifurcations of its corresponding in iterative map (Charbonneau, 2010). Despite its reputation of being terra incognita, such a mapping of parameter
space may also be possible with global-scale 3D fully nonlinear simulations of rotating convective turbulence and has been undertaken, to a degree, by Gilman (1983), and more recently
by Brown et al. (2011a); Gastine et al. (2012); and Yadav et al. (2013).
Mean-field theory does not end with the α tensor, rather it can be extended to arbitrary
derivative orders in the mean magnetic field provided that it is smooth enough. To that end,
one of the other commonly used tensors is the rank three β tensor that links the componentsof the turbulent correlations that produce an effective magnetic diffusion to the gradient of
the mean magnetic field. Given the ever larger degrees of freedom in choosing any of thecoefficients for the α, β, or higher rank tensors, there are obvious drawbacks to the theory,
with one of them being the required scale separation between the regions of poloidal andtoroidal regeneration, whether they be spatial or temporal. As the current theory stands,
this requirement is a product of the assumption of the instantaneous correlation between
the induction on the mean field and the mean field itself. These underlying requirement
can be relaxed by including the temporal correlations in the mean field, akin to the moregeneral theories constructed by Kazantsev (1968), Orszag (1970), Kleeorin et al. (2002), andBudaev et al. (2011). This could be done through a two time-scale separation, which doubles
the number of equations. Such an approach is, in some ways, actually suggested in the
FOSA, minimum-τ , and (Eddy-Damped Quasi-Normal Markovian) EDQNM theories (e.g.,
Pouquet et al., 1976). Indeed, recent work has attempted to incorporate results from local
turbulence models and helicity cascades into the MFT framework (Kleeorin & Rogachevskii,
1999; Brandenburg & Subramanian, 2005; Brandenburg et al., 2012; Müller & Malapaka,2013). Basically, this approach could account for the time scale for the inverse cascade from90
small-scales where turbulent magnetic helicity and magnetic energy are generated to thelarge-scale mean fields.
Yet the theory is built upon shifting sands, with many underlying assumptions asmentioned above. For instance, consider the case of a moderately rotating system akin toa typical main sequence G-type star, which has a convective overturning time of 32 days
but a rotation period of 12.6 days. The Coriolis number of this system is then 32. Under
Kolmogorov scaling, the eddy correlation time is τ (k) = τ0 (ℓ0 k/2π)−2/3 , so it is possible thata large range of scales below the large-scale ℓ0 are subject to the influence of the Coriolis
force. Certainly, all the large scales of interest are under the influence of rotation. We canask, under such conditions what is the mean-field model? Under the usual assumption ofscale separation, the correlation tensors of fluctuating quantities u and b vary slowly withthe large scale R. Consider delta-correlated ensemble averages of products as f (x1 , t)g(x2 , t).
In such a case, we havef (x1 , t)g(x2, t) =Z Zfˆ(k1 , t)ĝ(k2 , t)ei(k1 ·x1 +k2 ·x2 ) d3 k1 d3 k2 ,(3.20)

where fˆ and ĝ are the Fourier transforms of f and g, respectively, and where such a notation
will be used in general. A crucial and obvious step is to transform coordinates in physicaland spectral space such that r = x1 − x2 and R = x1 + x2 , which sets the stage for the
separation of the rapid variation on the small scales r and slow variation on the large scales
R. The above correlation between f and g can then be expressed as
Z Z

fˆ(k + K, t)ĝ(−k + K, t)ei(k·r+K·R)d3 kd3 K,Z ˆ= Φ f , ĝ, k, R ei(k·r) d3 k,

f (x1 , t)g(x2 , t) =(3.21)(3.22)

with k = k1 − k2 and K = k1 + k2 . The velocity and magnetic correlations, as well as theturbulent electromotive force (E; EMF), are given by

91Zvij (k, R) = Φ (ûi , ûj , k, R) ,vij (k, R) d3 k,Z2mij (k, R) = Φ b̂i , b̂j , k, R , b = δij mij (k, R) d3 k,Zχjk (k, R) = Φ ûj , b̂k , k, R , E = ǫijk χjk (k, R) d3 k.
u2 = δij(3.23)(3.24)(3.25)

with implied sums over repeated indices. Consider the time derivative of the cross-correlation
tensor




∂χjk
˙
= Φ û˙ j , b̂k , k, R + Φ ûj , b̂k , k, R ,∂t(3.26)

which follows since the Fourier coefficients are assumed to only depend upon time. Consider
the following small-scale field equations:∂b= ∇ × U × b + u × B + u × b − u × b − ηj ,∂t∂u
= −∇Pef f + B · ∇b + b · ∇B − 2Ω × u + N + f + ν∇2 u,∂t∇ · u = 0,

(3.27)(3.28)(3.29)

where these small scale motions are assumed incompressible, the effective pressure Pef fcontains the gas and magnetic pressures, f is the external force, and the nonlinear terms
are collected in N = −u · ∇u + u · ∇u + b · ∇b − b · ∇b. It is natural to make thefurther assumption that U = 0, which is roughly valid for rapidly rotating systems. Since
the system is incompressible, the pressure can be found by a Fourier solenoidal projection
Pij = δij − k̂i k̂j , with the unit vector k̂ = k/|k|. Thus
h
i


∂ ûi
\
\
2
ˆ
= Pij B · ∇b j + b · ∇B j − 2ǫjkl Ωk ûl + fj + N̂j − νk ûj .
∂t(3.30)

where the nonlinear Lorentz force terms are convolutions in k-space. With the above definitions, we can largely proceed toward analyzing the structure of the mean-field equations.
For some brevity, the reader is pointed to Brandenburg & Subramanian (2005) §10.3 for92
the derivation of the time derivative of the cross-correlation tensor χjk (see Equation 3.25).
Briefly, the underlying assumption behind this derivation is that the evaluation of correlation integrals in Fourier space is undertaken by expanding the velocity correlation tensor ina Taylor series. Only zeroth and first-order derivatives in k′ are retained, in accord with therenormalization group analysis discussion above. This follows from the assumption that themean field B varies only on the large scales and hence small k. The result is the following
approximate equation for χjk , which is valid to first-order in large-scale spatial derivatives
∂χjk
∂χlk
+ ûj Ĝk + Ĥj b̂k − (η + ν) k 2 χjk − i (η − ν) k · ∇χjk ,= Ijk − Ajl χlk − Bjlm∂t∂Rm
(3.31)
Ajl = −2ǫjlt k̂t k̂s Ωs ,


kt Ωm
i
− 2k̂t k̂m ,Bjlm = k̂s Ωs ǫjlt δmt +
k
ks Ωs
1
Ijk = −ik · B (vjk − mjk ) + B · ∇ (vjk + mjk ) + B j,l mlk − B k,l vjl
2∂vjk ∂mjk
1+− 2k̂j k̂s B s,l mlk .
− B m,s km
2
∂ks
∂ks

(3.32)(3.33)(3.34)

where ∇ = ∂/∂R is the large-scale gradient operator in physical space and subscript indiceswith commas indicate a spatial derivative, for instance B j,l = ∂B/∂Rl . The first-ordersmoothing approximation, which says that averages of triple products of fluctuating or smallscale quantities are zero, could be applied here. However, this assumption has been provento be highly inaccurate for nonself-adjoint systems such as the incompressible Navier-Stokes
equation being considered here (Lerche, 1974). Instead, the minimum-τ approximation will
be used, which says that the such triple products will be approximated by double products
divided by a k-dependent correlation timescale τ (k). Thus, the triple product terms inEquation 3.34 are approximated as ûj Ĝk + Ĥj b̂k = −χjk /τ (k). Neglecting the microscopic
viscosity and conductivity, the evolution of the cross-correlation tensor can be written as
∂χjk
1
∂χlk
= Ijk − Djl χlk − Bjlm
,.
∂t
τ
∂Rm

(3.35)

93
where Djl = δjl + τ Ajl . If we average Equation 3.35 over a sufficiently long time, then thetime-derivative should vanish, leaving
Djl χlk + τ Bjlm

∂χlk
− τ Ijk = 0.
∂Rm

(3.36)In the weakly inhomogeneous turbulence regime, this equation can be solved perturbatively
in large-scale derivatives with the zeroth and first-order equations being
(0)

(0)

−1
χjk = Djl
τ Ilk ,
(1)

(1)

−1
−1
χjk = Djl
τ Ilk − τ 2 Djp
Bplm(3.37)∂∂Rm
−1 (0)
Dls Isk(3.38)

with the latter equation arising from substituting the zeroth-order equation into Equation(0)

3.36 and noting that Ilk = −ik · B(vlk − mlk ). The matrix inverse of D is
−1
Djl
=

h
i
1
2
δ
+
Co(k)ǫ
k̂
+
Co
(k)
k̂
k̂
jl
jlm m
j l ,
1 + Co2 (k)(3.39)

where Co(k) ≡ 2k̂s Ωs τ (k) is the scale dependent Coriolis number. As was instigated in the
introduction of this derivation, we assume the Coriolis number is large over a significant
range of scales, which is likely in rapidly rotating systems given that τ (k) ≡ τ0 (ℓ0 k/2π)−2/3
for Kolmogorov turbulence. Thus Co(k) = Co0 (ℓ0 k/2π)−2/3 , with Co0 the large-scale Coriolis number. In the limit that Co0 is moderate, say around 32 as in the introductory example,−1
Djl≈ k̂j k̂l and so is actually independent of the spectrum of the motions. Before we cancompute the mean-field expectation for the electromotive force E, a form for the velocity
and magnetic correlation tensors must be found. A form for these tensors in weakly inhomogeneous turbulence is given in either (Roberts & Soward, 1975) or Rädler et al. (2003).
Let the kinetic energy and helicity spectra be independent of the large scale R, then these
correlation tensors simplify toik̂qǫlkq F (k)kik̂qǫlkq N (k)mlk = Plk (k) M (k) −kvlk = Plk (k) E (k) −(3.40)(3.41)

94
with the kinetic and magnetic energy spectra E(k) and M(k) as well as F (k) and N(k)the kinetic and current helicity spectra. Under this assumption the mean EMF becomesR−1
E = ǫijk d3 kτ (k)Djl
Ilk + O(∇R ), or expanding using Equations 3.34, 3.40, and 3.41:
#
#
(
"
"
Z
i
k̂
i
k̂
q
q
ǫlkq (F − N) + B l,p Ppk M −ǫpkq NE i = ǫijk d3 k τ k̂j k̂l −iks B s Plk (E − M) −
k
k
"
"
#
#)


ik̂q
∂vlk ∂mlk
ik̂q
1−B k,p Plp E −− 2k̂l k̂s B s,p Ppk M −ǫlpq F − B m,s km+ǫpkq N
.
k
2
∂ks
∂ks
k
(3.42)The most simple observation is that the EMF must be real so only one of the remaining
terms depends upon the helicities. Furthermore, the spectral derivatives of vlk and mlk withrespect to ks , recalling the definition of Plk , becomes the product of (2k̂l k̂k k̂s − k̂k δls − k̂l δks )/kand E or M, respectively. Thus the above integral reduces to
Z

n
h
i
h
i
d3 k τ k̂j k̂l −k̂q k̂s ǫlkq (F − N) B s + B l,p δpk − k̂p k̂k M − B k,p δlp − k̂l k̂p E
h
i
h
i o
−1/2B m,s k̂m 2k̂l k̂k k̂s − k̂k δls − k̂l δks (E + M) − 2k̂l k̂s B s,p δpk − k̂p k̂k M + O(∇R ).

E i = ǫijk(3.43)

Simplifying further and grouping the powers of k̂ gives
E i = ǫijk

Z

h

i
d k τ k̂j k̂l k̂q k̂s ǫlkq (N − F ) B s + 1/2 (E − M) B l,k − k̂k k̂p B l,p + O(∇R ).
3(3.44)

Using the identity that ǫijk ǫlkq = δiq δjl − δil δjq , the first term in this integral vanishes. Thisis quite dramatic as it means there is no α-effect!
Indeed, the previous equation further reduces to
E i = 1/2 ǫijkZ

d3 k τ k̂j k̂l (E − M) B l,k − k̂k k̂p B l,p + O(∇R ).

(3.45)The next observation is that B is independent of k and that the spectra and eddy correlation
time τ are isotropic and depend only on k. Noting that

95

Z

Z

3

d k k̂i k̂j F (k) = 1/3 δijZd3 k F (k) ,(3.46)

3d k k̂i k̂j k̂k k̂l F (k) = 1/15 (δij δkl + δik δjl + δil δjk )Z

d3 k F (k) .

(3.47)

These identities allow the integrals to be evaluated, with the following result

where Q̃ =R

E i = 1/6 [ǫilk + 1/5 (ǫilp + ǫipl )] Ẽ − M̃ ,(3.48)

d3 k τ (k)Q(k) and Q = E, M. The latter term in this integral vanishes asǫilp = −ǫipl , thus the only remaining term in the mean-field EMF is

E i = 1/6 ǫilk Ẽ − M̃ B l,k + O(∇R ),
E = 1/6 M̃ − Ẽ ∇ × B + O(∇R ).

so(3.49)

Since we have assumed Kolmogorov scaling for the turbulent spectra, this coefficient
can be evaluated with the further assumption that M = B02 /8π(kℓ0 /2π)−α for some α ≥ 0.
The mean EMF under this approximation is
E = −ηt J + O(∇R ),
"

−4/3
4E0 τ0k ν ℓ0
3αB02
ηt =
1−
−
9ℓ0
2π
32E0

1−



k η ℓ0
2π

1−α

!#

(3.50),(3.51)

with J = ∇ × B being the mean current density. Since the global-scale ℓ0 and both diffusive
scales kν and kη are large values, these terms are negligible. The remaining term is likely still
positive provided that the magnetic fields are at most in equipartition with the kinetic energy,
or more precisely that ME < 4KE/3πα ≈ 0.1KE (with ME = B02 /8π and KE = E0 ),which tends to be the case within most global-scale convective dynamo simulations in thepresence of strong rotational constraints. Perhaps, this is the physical mechanism underlying
the saturation of the maximum magnetic energy in such simulations. Furthermore, thedependence only on J is quite remarkable as it is not the kinetic helicity of the standard

96
MFT framework that provides the mechanism of poloidal field regeneration. Rather, it is theaction of small-scale turbulence on J that can lead to a constructive rather than destructive
impact on the generation of mean magnetic fields.
It could then be asked whether or not the higher order terms proportional to ∇R better
corroborate the standard field-theoretic picture at slow rotation. The short answer is no.
The longer answer requires the calculation ofZ∂ h −1 (0) i−1
Dls Isk ,d3 k τ 2 Djp
Bplm
∂Rm
Z
∂B q
d3 k τ 2 k̂n k̂j k̂p k̂l k̂s k̂q ǫijk Cplm (vsk − msk )
= −Ωn
∂Rm
i
h
Cplm = ǫplt δtm + kt /km − 2k̂t k̂mB
E i = −ǫijk(3.52)(3.53)(3.54)

with Bplm given in Equation 3.33. Since there are no imaginary terms outside of the helicity
contributions within vsk and msk , these helicity contributions must be zero. This condition
consequently leaves the kinetic and magnetic energy spectra as the significant contributors.
However, the projection operator associated with them Psk = δsk − k̂s k̂k correlates with thek̂s term already present in the above Equation 3.53. As a result, that entire expression iszero. Hence there are no contributions to within second order in large-spatial derivatives,
leaving the sole term above, or more appropriately
E = −ηt J + O

∂2B∂Ri ∂Rj.

(3.55)

This relation is only approximate and relevant to only those scales that are under the
influence of the Coriolis force. Below the threshold where the Coriolis number falls to aroundunity, the usual picture of an α-effect will again be relevant. Nonetheless, in dynamos thathave significant rotational constraint, there could be a sizable range of scales over which
the turbulent interactions between the flows and the mean current dominate. This problem
has been looked at to some degree, though under a very different set of assumptions, inan another mean-field theory framework, and in the context of the Rädler Ω × J effect97(Moffatt & Proctor, 1982; Moffatt, 1983). Those works suggest three further explorations torefine this picture: including inhomogeneities in the turbulent spectra, including anisotropic
spectra, and perhaps employing a multi-scale expansion for such rotationally constrained
dynamos. There may also be a less perturbative means to deriving such relations thatinstead rely on the robust framework of renormalization group analyses and symmetrical
reductions constructed for quantum field theory.
Freeman Dyson was the first to formulate renormalization theory, creating a precise
computational framework that seemed to magically cure the ills of the ultraviolet divergencesin quantum electrodynamics (Dyson, 1949). Shortly thereafter, the renormalization groupconcept was introduced by Gell-Mann & Low (1954), where Dyson’s renormalization theory
was made more general and applicable beyond quantum field theory. Though, there was not
yet a systematic way of using Gell-Mann’s method and there were no clear links between
effective field theories and the renormalization group. In both of these approaches, theprevailing practice had been to artificially introduce a short-distance (high-energy) cutoff
in a field theory and then to renormalize the coupling constants of the theory, effectively
wrapping the high-energy (ultraviolet) physics into the constant.
The insight of Kenneth Wilson was to, instead of waving one’s hands and arbitrarilychoosing a cutoff, take it seriously and regard it as part of the physical formulation of the
field theory. Wilson imagined letting the cutoff shrink while simultaneously adjusting thetheory to preserve its large-scale predictions, allowing the coupling constants to run apace
with the changing cutoff (Wilson & Kogut, 1974). In this paper, he also clarified the renormalization group method, systematizing the process of integrating out short-distance degrees
of freedom. While this procedure may seem complicated, Wilson’s reasoning actually reveals
something wonderful: the equations describing the processes occurring at large-scales become
remarkably simple and completely characterized by just a few renormalized parameters. For
instance using Wilson’s approach, one recovers Dysons results in quantum electrodynamics and much more, while also establishing a robust and visually appealing picture of the

98physics.
Thanks to Wilson’s work, effective field theories are now first assessed through therenormalization group. Although this method alone is not sufficient to build an effective field
theory, a renormalization group analysis serves to illustrate the domain of applicability ofsuch theories. Indeed, such an analysis lends credence to the primary method of constructing
effective field theories, which is using symmetries and Noether’s theorem to restrict the
manifold in which the broader theory rests, reducing its degrees of freedom. In particular,
if there is a single energy scale with wavenumber k in the microscopic theory, such as thewavenumber above which a turbulent field has something akin to the Kolmogorov scaling(Kolmogorov, 1941), the effective field theory can be seen as an expansion in k.
Building an effective field theory accurate to some power of k requires an increasingly
large set of free parameters at each order of the expansion that parameterize the degrees of
freedom that have been integrated out. This technique is useful when the smallest scale has
wave number K that satisfies the condition k/K ≪ 1. This caveat holds true for mean-field
electrodynamics if one considers K to be proportional to the inverse of the molecular dissipation scale. However, the effective field theory is not necessarily renormalizable as it is not
valid at small length scales. Indeed, the increasingly large number of parameters admitted ateach order in k of an effective field theory means that they are generally not renormalizable
in the same sense as quantum electrodynamics, which requires the renormalization of only
two parameters: the bare electron mass and charge. It is likely that the MHD equations are
nonrenormalizable, but this will require some careful study.
Renormalization group analysis in wavenumber-space is usually carried out on a perturbation expansion. The standard derivation of mean-field electrodynamics does just this,being carried out as a perturbative expansion in the scale-separation limit. Yet the validityof such an expansion relies upon the hope that the underlying physics of the system is close
to that of the non-interacting, free-field system. This is most likely untrue in general formean-field electrodynamics except perhaps in the kinematic regime. In the case that it is99
close to the free field system, collecting the leading order terms in the expansion permits
calculations of quantities such as the mean-field. While the perturbative approach has beenquite successful for many theories such as those of particle physics, it usually fails for systems
whose physics is very far from any free system, for example systems with strong correlations.
MHD systems fall into this latter category, given the nonlinear field interactions present
in the MHD equations which allows for strong correlations between all scales that are unknown apriori. This fact renders the standard methods of mean-field electrodynamics rather
dubious when applied to systems in the nonlinear regime, even when such correlations areapproximated with the minimum-τ or second-order correlation approximations.
Yet there may be ways to skirt the current perturbative approach to the strongly coupled dynamics of turbulent MHD. Indeed, there have been several fairly recent attempts
to look at aspects of mean-field electrodynamics utilizing the framework built for quantum
chromodynamics, even to the extent of introducing a Feynman calculus (e.g., Adzhemyan
et al., 1999; Vasil’ev, 2004; Jurčišinová et al., 2013). There may be a route to improving
upon these direct applications of field theoretic methods to MHD, extending the resultsof Jurčišinová et al. (2013) for instance. Indeed, it may be possible to compute the allorder loop interactions (representing field self-interaction) and all-order tree interactions(representing interaction of the magnetic and velocity fields), utilizing a modified form of
the amplituhedronic approach to quantum field theory (Arkani-Hamed et al., 2012). Thislatter work makes connections between algebraic geometry on the positive Grassmannian
and the super-symmetric N = 4 Yang-Mills quantum field theory that allow the the otherwise increasingly intractable Feynman calculus of higher order interactions to be reduced
to calculations of the volume of simplices on the triangulated positive Grassmannian. A
similar geometry might be constructed for the interacting fields of the Navier-Stokes MHD
equations, allowing for a more accurate effective field theory of electrodynamics to be built.

100
3.4

Three-Dimensional Numerical Simulations of Nonlinear Dynamos
The greatest tool currently available to modern dynamo theorists are 3D numericalsimulations, as there have been few breakthroughs in turbulence theory that would allow
analytical or semi-analytical solutions to the fully nonlinear compressible MHD system. In
direct numerical simulations (DNS), all the turbulent processes are fully captured, from thedriving scale to the diffusion scales. Although these simulations are typically far removed
from the length scales important to astrophysical processes, they can shed light on the smallscale dynamics that strongly influence the transport properties of the larger-scale system.
The aim of these simulations is usually to try to make contact with theoretical predictions of
turbulent spectra, such as the Kolmogorov scaling. However, as with large-eddy simulations(LES) that parameterize the small-scale physics, they are limited by computational resourcesin obtaining ever larger resolution and are constrained in their ability to run large suites ofsimulations that accurately probe large swaths of parameter space. Some of the largest DNS
simulations to date have reached 1 trillion grid cells or roughly 10000 cells in each dimension,
such as those of P. Maninni and A. Pouquet for Taylor-Green flow and those of P. Woodward
for the National Ignition Facility.
There are many approaches to these problems: some that are true DNS in the sensethat a particular physical system with its empirically determined diffusion coefficients are
simulated, some that utilize other diffusive operators, and still others that employ only an
implicit numerical diffusion. The first method assumes that the number of atoms withina grid cell is still large enough that the diffusion approximation to the collisional interactions is valid, whereas the last assumes effectively the same thing but with a numericallydefined diffusion coefficient. This numerical diffusion could be considered a running coupling
constant under the nomenclature of renormalization theory. The same can be said for LES
schemes that explicitly and artificially increase the level of the Laplacian diffusion in thesystem. Thus there is some connection between these two schemes, though it may not be101simple. Indeed, it would be interesting to see if these two schemes act equivalently undersome properly defined problem.
For those simulations that attempt to retain the diffusion approximation to collisional
interactions, there are two dynamo regimes one of low magnetic Prandtl number (P rm) andanother of large P rm. Theoretically, one should expect that at large P rm the magnetic
Reynolds number is much greater than that of the fluid Reynolds number, which leads tothe length scales of the magnetic field being smaller than that of the velocity field. At low
P rm , the expectation should be the opposite that the generated magnetic fields roughly tend
to be of larger scale than the velocity field. This does not preclude smaller scale intermittent
fields for the low-P rm regime or vice versa for the converse, rather it is more a statement
about which components of the magnetic field contain the most energy: the large-scales
for low-P rm dynamos and the small-scales for high-P rm dynamos. Using the Braginskii
definitions of the viscous and magnetic diffusivities and stellar models one, can determine
the ranges of magnetic Prandtl numbers within a star. For dynamos in stars with convective
exteriors, the low-P rm regime is of greatest interest. Using the Sun as an example, themolecular P rm ranges between about 10−1 in the core to roughly 10−6 at the photosphere,
decreasing exponentially toward it. Other low mass stars are roughly similar, with a low
P rm throughout the bulk of their domains. High mass stars, in contrast, have relatively
large molecular values of P rm ranging from about 10 throughout the convective core andremaining nearly unity until the near-surface region is reached at which point it drops toabout 10−2 . So, the convective cores of massive stars and the convective envelopes of low
mass stars represent very different dynamo regimes with respect to the scales possessing thegreatest magnetic energy, regardless of their myriad other differences.
At small scales, the P rm parameter regimes have been studied extensively with numerical simulations. The primary results of these simulations are that, at large magnetic
Prandtl number, the magnetic field generation is of the stretch-twist-fold variety where its
structures are of a smaller scale than the viscous dissipation scale (e.g., Schekochihin et al.,

(a)

102

(b)
Magnetic energy saturates

-5/3

-5/3

k

k

EK,M

EK,M

Magnetic energy grows:
small-scale dynamo

forcing

forcing

k0

kν

kη

k0

k

(c)

kν

k

kν

k

(d)

-5/3

-5/3

k

EK,M

k

EK,M

kη

forcing
Magnetic energy grows:
large-scale dynamo

forcing

Magnetic energy saturates

k0

kη

kν

k

k0

kη

Figure 3.4: A sketch of energy spectra for low and high-P rm dynamos with wavenumber k.
(a) Kinematic phase and (b) saturation phase of the high-P rm dynamo, with kinetic energy
in blue and magnetic energy in red. (c) Kinematic phase and (d) saturation phase of thelow-P rm dynamo. Here k0 is the driving scale, kν the viscous dissipation scale, and kη themagnetic energy dissipation scale.

2004). On the other side of the Prandtl number coin, the low P rm regime has the stretchtwist-reconnect mechanism, where the flows act similar to the large P rm regime, but insteadof the flows dissipating and the magnetic field folding on itself, the magnetic field dissipatesleading to reconnection (e.g., Schekochihin et al., 2007; Brandenburg, 2009). These two dynamo regimes are sketched in Figure 3.4, showing their kinematic and saturation phases.
While such simulations are instructive for creating subgrid-scale models and understanding
the small-scale physics, they lack a direct connection to the global-scale phenomena that are
more directly observable for astrophysical objects.
Four large-scale dynamos have already been considered, or at least mentioned, those103
of Parker, Backus, Herzenberg, and Roberts. These dynamos generate magnetic fields thathave spatial scales greater than the velocity scales producing them. The Sun and the Earth,among many other astrophysical objects, seem to possess global-scale fields and as such theymust be large-scale dynamos with a cascade of energy moving from small-scale turbulence tolarge-scale magnetic fields. For the helio- and geo-dynamos, the mechanisms building largescale field is directly linked to their rotation. In these global-scale dynamos, one manner inwhich the rotation is crucial is that the Coriolis force induces a net kinetic helicity in thesmall-scale turbulent flows. Correlations between the fluctuating velocity components of the
turbulence consequently yield Reynolds stresses that act to maintain the bulk differential
rotation through an inverse cascade of kinetic helicity and energy. This inverse cascade isthus the indirect source of the differential rotation that fuels the generation of toroidal field
as in Equation 3.19, whereas direct action of the convection is largely to transform thistoroidal field into poloidal field as in an α-effect. Thus the structure of the convection andthe differential rotation that it supports is key to the overall nature of the global-scale stellar
dynamo.
3.5

Solar Convection and Differential Rotation
It is clear that both turbulent correlations in velocity and magnetic fields as well as

rotation are necessary ingredients in the brew of the solar dynamo. The plasma in the interiorof the Sun achieves its turbulence through nearly inviscid thermal convection driven by theenormous heat flux coming from fusion in its core and by the rapid cooling at its photosphere.
Despite this extreme turbulence, the Sun manages to maintain both global-scale magnetic
fields and the bulk flows of differential rotation and meridional circulation. These globalscale structures cannot emerge from only the interaction of intermittent turbulent vortices
that produce a local spontaneous symmetry breaking. The coherency time for such vortices
is simply too short in the solar context, being at most a few tens of eddy turnover times fora prominent outlier. Such spontaneous symmetry breaking consequently cannot account for

104

Eq
450
400
350

Pole

300

Figure 3.5: Helioseismically determined angular velocity profile Ω of the solar interior, with
fast rotation (red) at the equator and slow (blue) at the poles. Important regions areindicated (Adapted from Howe (2009)).

the amount of upscale transfer of energy and helicity that is required to sustain the observed
large-scale structures. Rather, the correlations induced by rotation are the only way thatthese flows and fields are possible.
It has long been known that the Sun exhibits turbulent phenomenon at its surface,
yet the role that turbulent motions have in its interior remained fairly mysterious until theadvent of helioseismology (e.g., Kahn, 1961; Leighton et al., 1962; Noyes & Leighton, 1963;
Vandakurov, 1967; Deubner & Gough, 1984; Gough & Toomre, 1991; Thompson et al., 1996).
Probing the interior of the Sun through seismology has established the surprising existenceof two zones of strong rotational shear (e.g., Deubner et al., 1979; Brown et al., 1989; Schouet al., 1998; Christensen-Dalsgaard, 2002; Thompson et al., 2003; Howe, 2009). Moreover,the isorotational contours deduced from global inversions of the rotational are conical, as
in Figure 3.5, which is in stark contrast to the cylindrical profiles one might expect from

105
the Proudman-Taylor theorem or from early global-scale simulations (Glatzmaier & Gilman,
1982). One of these zones of shear is the sharp transition between the radiative interior andthe convection zone called the tachocline. In this region, there is a sharp change from thedifferential rotation in the convection zone to the uniform rotation of the deep interior, as
was first adumbrated in Spiegel (1972) and later detected helioseismically in Brown et al.
(1989). Three mechanisms have been proposed to explain this xyresic layer: highly turbulent
horizontal motions just above the tachocline that prevent inward diffusion, as in Spiegel &Zahn (1992); magnetic confinement due to a relic magnetic field in the radiative interior(Gough & McIntyre, 1998); and that the skin-depth effect of a cycling magnetic field in a
conductive medium prevents the spread of the tachocline (Petrovay, 2003). These notions
are currently being put to the test in 3D simulations and in further theoretical work (e.g.,
Browning et al., 2006; Brun & Zahn, 2006; Zahn et al., 2007; Garaud & Garaud, 2008;
Brummell et al., 2010; Strugarek et al., 2011; Wood et al., 2011)
A natural expectation for the profile of the angular velocity is one consistent with constant angular momentum, where the convection efficiently mixes angular momentum in thevertical direction and which yields Ω ∝ r −2 (Wasiutynski, 1946; Kippenhahn, 1963). Evidence for an increasing angular velocity with increasing depth came from the proper motionof sunspots (e.g., Gilman & Foukal, 1979) and from the super-rotation of supergranulation(e.g., Gizon et al., 2003). So it was quite surprising to find that the second zone of strong
shear occupied only the outer 5% of the Sun by radius. This region has thus been dubbed
the near-surface shear layer. Yet how this layer arises naturally and blends continuouslywith the nearly conical isorotational contours of the deeper interior is still a bit of a mystery.
There are hints from simulations that the interaction of multi-scale flows and increasing rotational influences with depth regulate this behavior (Brandenburg, 2007; Augustson
et al., 2011; Hurlburt et al., 2012; Guerrero et al., 2013). Indeed, photospheric convection
simulations carried out over the last two decades have led to a picture of near-surface convection wherein cold, rapidly-descending, and small-scale plumes coalesce into a hierarchical106
tree of convective downdrafts at greater depths (Nordlund et al., 2009). More precisely,
the downflowing plumes originating at the photosphere interact and merge as they traverse
many density scale-heights, which sets the spatial scale at which weak meandering upflowsorganize in response to the emergent downdrafts (Rast & Toomre, 1993). Such an effect has
been suggested to persist deeper in the solar convection zone, with thin downdrafts resulting
from the merger of plumes effectively determining the structure of giant cells (e.g., Spruit,
1997). Collectively, the above interpretation of the dynamics may be characterized as ‘topdown’ driving. This picture of solar dynamics, with a large asymmetry between upflows
and downflows, has been posited as an explanation for the discrepancy between helioseismic
estimates of convective velocity and those realized in global-scale convection simulations, inwhich these near-surface effects are necessarily omitted. There is also a sound theoreticalreason for the structure of the near-surface shear layer, where there may be a transition froma baroclinic vorticity balance in the deep interior to a Reynolds stress balance in the nearsurface layers (Miesch & Hindman, 2011). In other words, such a shear layer would arise
from the smooth transition between the near-constant angular momentum one might expect
near the photosphere to the rotationally constrained dynamics of the deeper solar interior
where the time scales of the motions become commensurate with the rotation period.
Despite the large differential rotation and conical radial profiles, there have been timedistance analyses of solar Doppler velocity data that imply that the turbulent flow speeds
are quite low (Hanasoge et al., 2010, 2012), being nearly 100 times less than those seen in 3D
solar simulations. This helioseismic result, if confirmed, would have important implications
for the maintenance of mean flows in the Sun. However, this result is eloquently argued
against in Miesch et al. (2012). In this latter paper, lower bounds on convective velocities
are derived using the MHD equations under the constraint that the turbulent flows must
produce bulk flows consistent with those ascertained from global helioseismic inversions. Thisbound is not based on mixing-length theory nor on numerical simulations, though they are
consistent with both. It is found that the amplitude of the turbulent flows in the upper solar

107
convection zone (r ∼ 0.95 R⊙) must be at least 30 m s−1 in order to be capable of sustaining
the mean flows inferred from helioseismology. Similar estimates for the lower convection
zone (r ∼ 0.75 R⊙) imply that motions there must be at least 8 m s−1 . There are two possible
reconciliations between the lower limits in Miesch et al. (2012) with the upper limits foundby Hanasoge et al. (2012). One possibility is that there is an overestimate of the large-scaleconvective velocity amplitudes, with smaller scale turbulent flows being responsible for moreof the angular momentum transport through inverse cascades and self-organization thanpreviously realized. The second aspect, which is not considered in Miesch et al. (2012),
is the impact that magnetic fields might have on the resulting convective amplitudes. In
any case, the seeming inability of large-scale models to reproduce what should be a basic
aspect of solar convection is a definite curiosity. For if the flows are indeed so slow, what isabsorbing the kinetic energy or reducing the buoyancy of these flows as they descend into
the depths of the Sun? If such measurements could be made for other stars, would theyshow a similarly low amplitude of convective motions?
3.6

Global-Scale Simulations
There is no doubt the transfer of kinetic and magnetic energy, momentum, and helicity

from small to large scales is critical to the dynamo process operating within stars, with aninverse cascade arising from the breaking of certain symmetries with respect to isotropic
homogeneous turbulence. Such symmetry breaking in stars primarily occurs due to thepreferred directions introduced by rotation and stratification. Gravity and the resulting
stratification that it creates break the global isotropic translational symmetry, being now
only invariant in the directions perpendicular to the direction of the gravitational acceleration
vector. The stratification makes the convection multi-scaled, with the convective scales at agiven depth being roughly proportional to the density scale height. Naturally, the scales arethen large at greater depths and smaller nearer the surface where the density scale heightdecreases.

108
Next one can ask are there spatial scales that are affected by rotation? The first observation is that, in a spherically-stratified rotating star, only the longitudinal direction remains
translationally invariant as the radial direction is governed by gravity and the latitudinal
direction by rotation. The above argument regarding stratification defines a large-scale symmetry breaking, but not a locally broken symmetry. So we may ask is the reflective symmetry
breaking induced by global-scale rotation scale invariant? The answer lies within a simple
argument: the MHD equations are rotationally symmetric when rotation is not present; in arotating frame, however, the Ω×u term breaks this symmetry at all spatial scales as it transforms as a pseudo-vector with det(R)RΩ × u. The other term that might cause symmetry
breaking is the Lorentz force, but since it involves two cross products it transforms as a polar
vector and does not break the symmetry. However, this is only an argument for the spatial
scales and says nothing of time scales. Indeed, there is a minimum time scale below whichrotation has little influence which can roughly be characterized by τ < 4πRoτc , with τc theconvective overturning time. There is some additional physics that also sets a length scale
below which rotation matters increasingly less. Imagine a scenario in which the nominallyintermittent temporal scales of local vorticity can be such that they overwhelm the global
rotation. If Kolmogorov turbulence is again considered, one should expect ℓ < ℓ0 (4πRo0 )3/2 ,with ℓ0 the maximum scale for which the scaling is valid. The breaking of translationaland rotational symmetries are largely what set the minimum requirements for the time andspatial scales that must be resolved in global-scale MHD simulations of the solar interior.
In an ideal sense, a subset of these the large-scale dynamos are those that encompass
large portions or effectively all of the scales relevant to the dynamics in most astrophysicalsystems. Clearly, this is a formidable task. The aim, therefore, is to self-consistently capture
as many of the scales thought to be dynamically relevant to the global-scales as possible. For
the Sun and other rotating bodies, this is a crucial issue due to the fact that the large-scale
dynamics is strongly influenced by inverse cascades of energy and helicity. Hence there isno easy way to define the lower bound of relevancy, until the search reaches the dissipative

109
scales of the physical system. This question is quite apropos in light of the helioseismic results
described above. So, one could ask what hope is there of ever being able to capture all the
relevant scales? Moreover, how should the intermittent processes that play a significant
dynamical role over many time and spatial scales be incorporated?
The answer is, as Neil deGrasse Tyson oxymoronically defines the pursuit of scientificresearch, the informed ignorance of subgrid-scale (SGS) models. Since the individual characteristics of a range of scales can be separately simulated, or at times theoretically surmised,models can be constructed that describe the SGS transport and diffusion of energy, momentum, and magnetic fields. Such methods prescribe a treatment for how processes occurringat the unresolved scales act upon the larger scales. The most commonly used method isthat of eddy diffusion, wherein the natural diffusivities of a system are massively increased.
For instance, in many stellar simulations the viscosity is some eleven orders of magnitudelarger than its atomic value. However, this method does have a firm basis in renormalization
analysis. The eddy diffusion scheme in these models implies that there is an enhanced Laplacian operator, or even multiple hyper-elliptic operators, acting in the system. The latter ofthese two is potentially more palatable than numerical and implicit diffusion, given thatstandard flow parameters such as a Prandtl or Reynolds number can be defined. However,
the relevance of such numbers to the physical system is likely remote given the vast range ofunresolved and thus ignored dynamical scales. Though, it should be kept in mind that thescales of interest are the largest possible.
To aid in resolving the smaller scale processes, the action of unresolved motions scales
might be considered as hypo-elliptic, acting more hyperbolically than elliptically (Moffatt,
1983; Brandenburg et al., 2004). Indeed, one approach that has yet to be attempted is tocoevolve a separate SGS eddy transport model [such as the test-field method (e.g., Kraichnan,
1972; Schrinner et al., 2007)] with a large-eddy simulation (LES). This SGS model would
take the evolved fields at a grid point to be the mean test field, then a separate set of
equations would evolve a local field using an eddy closure model that is appropriate to the110
circumstances such as those found in Kazantsev (1968); Pouquet et al. (1976); Kleeorin et al.
(2002); Brandenburg et al. (2012). Rather than acting only as a diffusion, it might allow
local inverse cascades due to intermittent processes to be captured more accurately than asimple diffusive operator can. Alternatively, one could apply spectral boundary conditions
that maintain a particular spectral slope and a flux of energy and helicity consistent with thesystem being studied. The class of Smagorinski-like diffusion schemes is somewhat like thisapproach where an extrapolation is made in spectral space from some presumed resolved scalein the inertial range to the grid scale (e.g., Germano et al., 1991; Meneveau & Katz, 2000;
Nelson et al., 2013a). However, because the subgrid-scales are parameterized, global-scale
simulations are necessarily sensitive to the treatment of diffusion, whether it is a numerical
diffusion or a renormalized Laplacian diffusion. Nonetheless, global-scale simulations have
had many successes in establishing how the solar differential rotation is sustained as discussed
below in §3.7, in how magnetic fields may behave in the convection zone (§3.9), and how
these fields can have long-term regular cycles as in Chapter 5.
3.7

Global-Scale Solar Hydrodynamics
Although cyclic solutions in 3D nonlinear global-scale dynamo simulations were foundthirty years ago in the pioneering convection simulations of Gilman (1983) and Glatzmaier(1985), they were limited to laminar parameter regimes and made little contact with helioseismic observations of the solar differential rotation or with observations of the solar cycle,
exhibiting short cycle periods and poleward-propagating dynamo waves. The picture began
to change with the development of another anelastic code, which, as with Glatzmaier’s code,
captures the full spherical shell geometry. The first hydrodynamic models using this newcode found that rotationally constrained convection produces correlations in the velocity fieldthat transport momentum and drive mean flows such as differential rotation (e.g., Miesch,1998), corroborating earlier studies and physical intuition. However, given the rapid growthof computing power since the first global-scale stellar dynamo models, the first models using

111
(a)

(b)

(c)

(d)

Figure 3.6: A high resolution simulation of solar convection. The changing convective patterns with depth are illustrated through orthographic projections of the radial velocity (vr ),which show giant cells (a) near the upper boundary at 0.98 R⊙ and (b) in the upper convection zone at 0.92 R⊙, and more plume-like flows at (c) mid convection zone (0.85 R⊙), and(d) near the base of the convection zone (0.71 R⊙). Adapted from (Miesch et al., 2008).

this new code had 500 times as many grid points and could take 50 times as many time steps
for an overall increase of 25,000 in simulation fidelity. Over the last 15 years, there has beenan additional increase leading to simulations with again 500 as many grid points comparedto the those of Miesch (1998), though the number of time steps taken has only increased bya factor of four.
Under the somewhat more turbulent conditions of the much higher resolution simulations of (Miesch et al., 2000), relative to the Gilman and Glatzmaier simulations of the
1980s, the character of the transport mechanisms becomes increasingly complex, with largescale coherent vortical structures and mean flows being influenced by small-scale turbulence
through inverse cascades. Though, the dynamics of the highly turbulent solar convection
zone is likely very different from these and the earlier global-scale numerical models, which
are still limited to fairly laminar flows. In some of the first studies with this new code (Miesch et al., 2000), a region of overshoot was admitted by explicitly including a portion of
the radiative interior below the convection zone. The convective structures built in the moreturbulent simulation reported on in this paper had latitude and depth dependent structurewith the mid to low-latitude regions of the simulation being tilted Busse columns or cresentoid cells, which arise in flows with strong rotational constraints. Connected with these

112
N

(a)

E

W

(b)

Ω

r

S

convection
zone base

Figure 3.7: Sketches depicting the influence of Coriolis force. (a) Diagram of Coriolis influence on the horizontally converging flow associated with a downflow lane (vertical blackline), with westward flows deflected poleward and eastward flows deflected equatorward. (b)Coriolis force acting on the vortical flows of a downflow plume, with cyclonic, converging
flows in red and anti-cyclonic, diverging flows in blue [Adapted from Miesch (2005)].

more columnar flows are an intricate and rapidly evolving downflow network in the upper
convection zone and plume-like structures at depth and in the overshooting layer. Such structure persists at even higher levels of turbulence and resolution as in Miesch et al. (2008),
which details the dynamics of a simulation of solar convection that exhibits the patternsof giant-cells (large-scale eddies with sizes of about 200 Mm) that has a resolution of 256
points in radius, 1024 points in latitude, and 2048 points in longitude. Figure 3.6 displays
those convective patterns, with downflow lanes aligned with the rotation axis at low latitudes
and more isotropic convection at higher latitudes. The fluctuating kinetic helicity of theseconvective plumes tend to change sign at mid-convection zone, with the convergent flows
in the upper convection zone being cyclonic and those in the lower convection zone beinganti-cyclonic (Figure 3.7). Such behavior is evidence of further symmetry breaking, being in
addition to the breaking of reflective symmetry by rotation, that is in turn due to density
stratification and boundary conditions.
Relative to the local differential rotation established by the convection, the convective
patterns themselves tend to propagate prograde at low latitudes and retrograde at high113
latitudes. The profiles of the differential rotation established in the simulations of Miesch
et al. (2000) and Elliott et al. (2000) show some similarity with helioseismic determinationsof the solar rotation, yet, as with the early simulations of Gilman and Glatzmaier, they
still exhibited significant cylindrical alignment. Nonetheless, there are significant differences
between the dynamics of laminar flow and those in more turbulent regimes largely due topresence of strong, intermittent, vortical downflow lanes and plumes. In particular, thetransport of momentum and energy are prominently modified as is the structure of theovershooting layer just below the convection zone (Miesch, 1998; Miesch et al., 2000; Brun
& Toomre, 2002). Though these simulations are still quite sensitive to diffusion and thetreatment of boundary conditions as described in the thermal Prandtl number studies of
Elliott et al. (2000).
One of the goals of simulating the global-scale dynamics of the Sun is to accurately
reproduce its differential rotation profile in latitude and radius. Prior to helioseismology, thesolar differential rotation had many presumed forms and modeling efforts were geared towards
predicting its potential profile. The presumed form of the differential rotation was often
cylindrical after the Proudman-Taylor theorem and Gilman and Glatzmaier’s simulations.
So there was a bit of a theoretical scramble to try attempt to understand why the solar
differential rotation is roughly constant on radial lines and has two boundary layers. There
have been many attempts over the last 15 years to reproduce these basic features through 3D
LES or implicit large eddy simulations (ILES), 2D mean-field hydrodynamics simulations,and analytical models.
Some of the more successful simulations and analytical models are shown comparedto the solar profile in Figure 3.8. As mentioned above, the differential rotation achieved inearly models and in those without a tachocline prove to have overly cylindrical isorotational
contours, such as those in Figure 3.8d. This is known to be due to a lack of a latitudinal
entropy gradient large enough to break the Proudman-Taylor constraint (e.g., Weiss, 1964;
Thompson et al., 2003; Balbus et al., 2009). This was clearly shown in Rempel (2005), where

114
(a)

(b)

(d)

(e)

(c)

(f)

(g)

480

460

300

360

Figure 3.8: Solar differential rotation in observations and theory. (a) Isorotational contours
from GONG data, solid blue line denotes the photosphere and the red dashed line the depthof the tachocline (courtesy R. Howe). (b) Isorotational contours from an analytical model,
green dashed line indicates the tachocline and the blue curve the photosphere [Adapted from
(Balbus et al., 2012)]. (c) Mean-field hydrodynamics simulation of solar differential rotation,
dotted line indicates the tachocline and the long-dashed lines conical contours. Differential
rotation established self-consistently in 3D simulations using the anelastic spherical harmonic(ASH) code as in (d) case AB from Brun & Toomre (2002), (e) case AB3 after Miesch et al.
(2006), and (f) an AB-like case with its tachocline demarked with the black dashed line
[Adapted from Browning et al. (2006)]. The colorbar of panel d is shared with panels (e-f).
(g) The differential rotation established in an implicit large eddy solar simulation using the
EULAG code. The tachocline is marked with a dashed white line [Adapted from Racine
et al. (2011)].

2D mean-field models were applied to the solar convection zone and a portion of the subadiabatic radiative zone. The result of these calculations is shown in Figure 3.8c. Rempel foundthat the profile of the differential rotation is primarily influenced by the morphology of theentropy perturbation that thermally diffuses into the convection zone from the subadiabatic

115
region resulting. The weakness of the model is that it requires that the lower half of theconvection zone to be weakly subadiabatic for good agreement with helioseismology. In thecase of a weakly subadiabatic lower convection zone, the solar differential rotation can be
explained through entropy perturbations arising from the nonadiabatic stratification. It is
more likely, however, that the convection zone is nearly everywhere weakly superadiabatic, or
adiabatic, due to the efficiency of convection. If the subadiabatic region has the small extent
expected from the narrowness of the tachocline, additional effects such as the anisotropic
heat transport advocated by Spiegel & Zahn (1992) are then required. Another importantresult from Rempel (2005) is that the meridional flow showed very little penetration beneaththe base of the convection zone. This can be understood by invoking the concept of gyroscopic pumping, which that states that an angular momentum gradient is directly linked to avergence in angular momentum fluxes (see Appendix B for details). It is unlikely that there
is a significant Reynolds stress in the radiative zone, thus it is likewise unlikely that there isa meridional flow there, except perhaps one driven by the Eddington-Sweet mechanism.
The global-scale simulations presented in Miesch et al. (2006) provide a clear example of
how sensitive the solar differential rotation may be to the thermal structure of the tachocline
and overshoot region, though without explicitly resolving the penetrative or overshooting
convection. Rather, this latitudinal thermal structure is imposed as a boundary condition,
which provides information about the impact that a thermal wind balance (the zeroth-orderazimuthal vorticity equation) within the tachocline can have on the resulting differential
rotation and energy transport. The amplitude of the imposed entropy variations relative tothe background entropy is of order 10−5 , corresponding to a latitudinal temperature variationof about 10 K. A second caveat is that these simulations also skirt the issue of how the
sharpness of the tachoclinic transition impacts the overall character of the differential rotationand thermal structure. This does, however, have the benefit being that there is only a single
tunable parameter. Regardless of its approximations, these simulations indisputably confirm
that a baroclinic forcing, arising from tachocline-induced entropy variations, can break the116
Proudman-Taylor theorem as one might suspect from the conservation of potential vorticity.
Indeed, as the amplitude of the imposed variation is increased, initially cylindrical rotation
profiles become increasingly conical, with nearly radial angular velocity contours at midlatitudes (compare Figures 3.8d and 3.8e, for instance). In particular, Miesch et al. (2006)
found that the conical rotation profiles are maintained primarily by the resolved convective
heat flux, namely that diffusion was dominated by the transport of enthalpic variations fromthe lower boundary into the convective envelope. This sustains the baroclinicity of the fluid
and its solar-like isorotational contours, which is also pointed out in Balbus et al. (2009).
The concomitant simulations of Browning et al. (2006) [and later ones by Brown (2009),
Ghizaru et al. (2010), and Strugarek et al. (2011)] included the dynamics of a stable layer
below the solar convection zone, which were attempted despite the formidable challengethat the conditions there pose to global convection simulations. To be specific, a highspatial and temporal resolution is required in and near the tachocline to reliably capture therelatively small-scale dynamics associated with convective overshoot, tachocline instabilities,
internal waves, and other phenomena such as the magnetic boundary layers associated withsome tachocline confinement models (e.g., Rüdiger & Kitchatinov, 1997; Gough & McIntyre,
1998; Wood et al., 2011). The global-scale 3D anelastic simulation of Browning et al. (2006)
illustrated that most of the downflowing plumes overshoot into the underlying radiative
region, where they are buoyantly braked. Due to the relatively high diffusivities of thesesimulations a strong differential rotation is established by the convection, which tends tohide the difficulties with the bistability of the latitudinal entropy gradients as the simulation
attempts to establish a solar-like differential rotation in the presence of a stable region(private communication with N. Featherstone & B. Brown). Moreover, the tachocline wasmechanically forced by artificially damping the angular velocity below the tachocline, which
serves to prevent the diffusive spread of the layer.
Figure 3.8f depicts the mean angular velocity Ω with radius and latitude establishedby the convection of Browning et al. (2006). Within the bulk of the convection zone, there

117
is the expected increase of Ω from the pole to the equator. In comparison to Miesch et al.
(2006), which had no penetrative region, the angular velocity contrast between the pole andthe equator in Browning et al. (2006) is reduced and the isorotational contours are morecylindrical. Below the base of the convection zone (at ∼ 0.71 R⊙), where the stratification
is highly subadiabatic, Ω adjusts rapidly to uniform rotation due to the mechanical forcing.
Nonetheless, the simulations of Miesch (1998) and Browning et al. (2006) represent the firstattempts at including a stable layer below the convection zone in a high resolution global
model, and they do a reasonable job at establishing a solar-like differential rotation. The
recent simulations of a model solar convection zone using the ILES EULAG code also include
a stiffly stratified stable region and due to the low dissipation of the numerical model thetachocline is prevented from spreading (Ghizaru et al., 2010; Racine et al., 2011). Figure
3.8g shows the differential rotation established in this MHD simulation, which like manyof the anelastic models shows fairly cylindrical isorotational contours at low latitudes. The
difference however is the conical alignment of the contours above about ±45◦ which moreclosely match the solar profile and the profile established in case AB3 from Miesch et al.
(2006). However, there are issues with energy conservation within the anelastic approximation in regions with a stiff stable stratification as seen in Brown et al. (2012), Vasil et al.
(2013), and Chapter 10.
A significant recent result is the construction of an analytical model describing the solar
rotation profile (and perhaps the profiles of other main-sequence stars) that obeys a simple
set of equations. The basic result of this model, as shown in Figure 3.8b, is that it appears
much of the bulk dynamics of the solar convection zone may be governed by a reduced formof the vorticity equation, known as the thermal wind equation (e.g., Weiss, 1964; Thompson
et al., 2003; Balbus et al., 2009). Though there are many more terms in this equation, onethat may be the most important in the solar convection zone involves a balance between
the cylindrical gradient of the square of the angular velocity and accelerations due to themisalignment of isopycnal and isobaric surfaces, the latter arising from latitudinal entropy

118
gradients, or mathematically λ∂z Ω2 = g/γr∂θ S, where λ is the lever arm from the rotation
axis, g is gravity, and γ the adiabatic exponent.
Obviously, if both the angular velocity and the entropy are unknown, the thermal windbalance is underdetermined. However, the Balbus conjecture is that the entropy and angularvelocity are nonlinearly and apriori related, which sidesteps the underdeterminedness of thethermal wind equation. For instance, it is noted in Balbus (2009) that, if the isentropic andisorotational surfaces are aligned or anti-aligned, a subset of magnetized baroclinic modes are
marginally stable, those whose Alfvén speeds are slower than the fluid motions. However, theexact matching of the entropy and angular velocity surfaces in that case would quench any
convection. The alternative is to take the hydrodynamic approach, which is less restrictive
due to the gauge independence of the entropy thermal wind equation, where an arbitrary
function of radius can be subtracted from the total entropy (Balbus et al., 2009). This allows
the residual entropy to be separated from the mean entropy, whose gradient is responsible formuch of the energy transport. The physical basis for this argument rests on the assumptionthat convective flows transport energy and momentum along surfaces of constant residual
entropy, which then renders the total entropy profile effectively fixed in time, which thesimulations of Miesch et al. (2006) corroborate. Hence there is no restriction on the radial
gradient on the total entropy, which means convection can be quite vigorous as long as it is
not so vigorous as to break the underlying assumption of the alignment of residual entropy
and rotational surfaces. The last statement points out one weakness of the theory, withanother being the exclusion of dynamo action and magnetic feed backs. Of greater import,
one can ask how does the alignment of the residual entropy and rotational isosurfaces arise
to being with, and how is it maintained. The results are necessarily dependent upon theboundary condition, which is the helioseismic data and thus the thermal wind equation isclearly not a unified theory of the solar differential rotation.
A recent paper set out to explore how such a relationship between the convective transport and the isentropic and isorotational surfaces is maintained (Balbus et al., 2012; Balbus

119& Schaan, 2012). A surprising finding in this paper is that by using a fairly general calculation using linearized Lagrangian equations there is to leading order no poleward deflection ofwarm (relative to the background) fluid elements in the presence of a bulk rotation. Rather,it seems, such deflections occur only when a cylindrical gradient in angular velocity is already present. Though, this simply means the nonlinear transport terms must be acting
when there is not yet a substantial angular velocity gradient, such as when a star is young,rotating rapidly, and thus possessing cylindrical isorotational surfaces. However, the resultsof Balbus & Schaan (2012) point out that once such a gradient is in place it is self-reinforcing
and is to leading order the dominant dynamical process. The second point made in the paper has bearing on the order of magnitude of various processes, namely that the latitudinal
entropy perturbations and the centrifugally induced baroclinicity in the tachocline are of thesame order and as such the latter should not be neglected. They cite this latter processes
as the origin of the vorticity necessary to promote the alignment of the residual entropy
surfaces and the isorotational surfaces, though it could be suggested instead that a slightmisalignment of these surfaces could produce sufficient nonlinear poleward enthalpy transport to sustain their near alignment. The latter processes is what occurs in the global-scale
3D numerical simulations and is certainly one of their salient features. Thus the equilibrium
entropy and rotational profiles will depend sensitively upon the details of the convection, andchiefly upon how rotationally constrained it is. Yet once cylindrical gradients in the angularvelocity form, according to Balbus & Schaan (2012), the differential rotation becomes almost
self-sustaining, requiring only a weak source baroclinicity or vorticity. It will necessarily also
evolve slowly with time as angular momentum is lost through stellar winds and magnetic
breaking.
3.8

An Extension of Balbus’ Theory
One potential extension of Balbus’ theory, beyond what was expressed in Balbus & Lat-ter (2010), is to include the latitudinal pressure gradient contribution to the thermal wind, in120
order to better capture the tachocline and the stable region. The starting point for this is to
assume the fluid is hydrostatic over a long time average, with bulk flows of meridional circulation and differential rotation that vary only in meridional planes (no azimuthal variation).
If the meridional circulation is considered a first-order effect, the zeroth-order momentum
equation, which includes the differential rotation, and the longitudinal component of its curlare∇P = −ρg + λΩ2 λ̂,




∂Ω2
1
∂P ∂S ∂P ∂S
1 ∂P ∂ρ ∂P ∂ρ
λ
=
,
=− 2
−
−
∂z
rρ ∂θ ∂r∂r ∂θrρcP ∂θ ∂r∂r ∂θ

(3.56)(3.57)

where the latter equation assumes an ideal gas with a radially varying pressure specific heatcP , Ω(r, θ) is the angular velocity g is assumed spherically symmetric and in the radial
direction, λ = r sin θ, and z = r cos θ. A more thorough derivation of the averaged zonal
vorticity equation can be found in Appendix B.3. The next step is to simply insert Equation
3.56 into Equation 3.57, yielding the simplified thermal wind equation



∂Ω2
1
1
2 ∂S
2 ∂S
λ cos θΩ,g − λ sin θΩ
λ
=
+
∂z
cP
∂r
r
∂θ
∂Ω2
∂S g ∂S
cP λ
= λΩ2
+
.
∂z
∂z
r ∂θ

(3.58)

The concept upon which this fairly simple theory rests is the Balbus conjecture. This hypothesis states that S(r, θ) = f (Ω2 ) + S(r), namely isentropes and isorotational surfaces
are aligned to within an additive function. In Balbus’ theory, this amounts to a gauge freedom. Here, however, the radial entropy gradient has been retained and this symmetry is
lost, though the gauge freedom in Ω still remains at this stage. Thus Equation 3.58 can be
simplified to  ∂Ω2
gf ′ (Ω2 ) ∂Ω2
λ cP − f ′ Ω2 Ω2= λ cos θΩ2 σ +
,
∂z
r
∂θ

(3.59)

with f ′ (Ω2 ) = df /dΩ2 (so transforming the entropy gradients into derivatives in Ω) andσ(r) = dS/dr. Note that here the full nonlinearity due to the centripetal acceleration isretained. For analysis, it is prudent to further reduce the notation to

121

λ (cP − q ′ (u))

ge−u q ′ (u) ∂u∂u= λ cos θΩ2 σ +
,
∂z
r
∂θ

(3.60)

where u = ln Ω2 and q ′ (u) = df (eu )/du. The final simplification is to convert to sphericalcoordinates and minimize the number of terms, wherein the previous equation becomes
∂u
−
∂r



tan θ
ge−u q ′+ 2
rr sin θ cos θ (cP − q ′ )



∂u
σ
−
= 0.
∂θ cP − q ′(3.61)

This is a highly nonlinear equation for the logarithm of the angular velocity u, and in general
could be difficult to solve. The trick is to find a characteristic solution, which reduces thispartial differential equation into an ordinary differential equation (ODE) along a particularpath. The broader theory of implicit functions and the Lagrange-Charpit equations could be
appealed to, but that is unnecessary in this case. It is sufficient to note that we can define a, ∂u ) = 0 that encapsulates Equation 3.61. Since we are only interested
function F (r, θ, u, ∂u∂r ∂θin the morphology of the characteristics, the solution along such paths that are equippedwith a distance metric s reduces to
dr
= 1,
ds
du
σ
=
,
ds
cP − q ′dθtan θ
ge−u q ′
=−
− 2
.
ds
r
r sin θ cos θ (cP − q ′ )(3.62)(3.63)(3.64)

This system of characteristic ODEs exposes an issue, namely that the characteristics are nolonger coincident with the isotachs, which arises because du/ds 6= 0. It is worth noting,however, this is not a problem in regions that are adiabatic. Such regions are the domain of
Balbus’ solutions, though here it is still extended to the nonlinear case and with a nonconstant cP . As such, let us first find the equations for an adiabatic convection zone. First, notethe equivalency between dr and ds and that du/ds = 0 when σ = 0 and for sensible valuesof cP − q ′ . Thus, following a multiplication by 2r 2 sin θ cos θ, the above system simplifies tocharacteristics along isotachs with

122

dλ2
2ge−u q ′=−.
dr(cP − q ′ )(3.65)

Since du/ds = 0, all terms involving u are constant along these curves and can be incorporated into a single constant β(r0 , θ0 , u(r0 , θ0 )) that depends only upon the initial position(r0 , θ0 ) of a given characteristic. Hence, the general solution for the curve is
λ20 − λ2 = −β

Z r0
r

dr

g (r)
.
(cP (r) − q ′ (u0 ))(3.66)

This can be rendered independent of the choice of q(u), which arises from the nonlinearityof the centripetal terms, in the limit that q ′ ≪ cP . This constraint is almost certainly true
everywhere in the solar convection zone since the perturbations associated with the thermal
wind are about one part in 105 . Thus the above equation simplifies toλ20 − λ2 = −β

Z r0
r

h
r0 i
g
.
≈B 1−
dr
cP
r

(3.67)

under the assumption that g ∝ r −2 and that cP = const. for the latter equation. Indeed,
Balbus’ solutions can be fully recovered after a division by r02 and subtracting the equationfrom unity, as cos2 θ0 (r, θ; r0 ) = 1−r 2 /r02 sin2 θ−B 1 − rr0 . This equation is then substituted

into an appropriate fit to a rotation profile such as the common form for the Sun (e.g.,Ω = a0 + a1 cos2 θ0 + a2 cos4 θ0 ), an example of which can be seen in Figure 3.9c. It is clear
from Equation 3.67 that, if cP varies significantly across the domain as it does in the solar
convection zone due to the latent heats of ionization, there may be modifications in the
rapidity with which the term involving β becomes negligible as the near-surface shear layer
and the photosphere are approached from below. This is indeed what is found, though it
has a minimal impact on the resulting profiles of the differential rotation. It is, however, afurther indication that a thermal wind is restricted in validity to regions where turbulent
stresses are small, which is not the case in the upper convection zone (see Appendix B,
Miesch & Hindman (2011), and Balbus et al. (2012)).

123

Figure 3.9: Solar differential rotation from helioseismic observations and theoretical isotachs.
(a) Isorotational contours from a regularized least squares (RLS) inversion of the first 144
days of MDI data (publicly available at http://quake.stanford.edu/m̃jt/rls-mdi/; M. Thompson and R. Howe, 1997), solid black line denotes the photosphere and the dashed line thedepth of the tachocline. (b) Isorotational contours from the analytical model that includes
the effect of entropy gradients on the thermal wind (Equation 3.72). (c) The differential
rotation resulting from Balbus’ original thermal wind model as in Equation 3.67.

The intent now is to explore what happens when there is a significant mean radial
entropy gradient, as in the radiative interior of the Sun. In this case, as seen in Equation
3.63, the characteristic solution does not precisely correspond to an isorotational contour.
Rather, such curves are constant along isentropes modified by the pressure specific heat andby the form of the functional relationship between the angular velocity and the entropy.
However, much can be improved by simply including the lower-order terms arising from themean entropy gradient. If the centripetal terms are neglected, save for the mean entropy
gradient term, the equations become
λcP∂u gf ′ ∂u−
= λ cos θσ.
∂zr ∂θ(3.68)

Thus the characteristic equations are then
σ
du
= ,
dr
cP
tan θ
gf ′
dθ
=−
− 2
.
dr
r
r cP sin θ cos θ(3.69)(3.70)

124
Following the procedure above, this can be simplified to
2gdf /dr
dλ2
=−
.
dr
σ

(3.71)

Moreover, if f ′ (u) is expanded in powers of the ratio of the mean entropy gradient over thePpressure specific heat as f ′ = σ/cP k bk (u0 )(σ/cP )k , contact can be made with Equation
3.67, with

λ20 − λ2 = −

∞
X
k=1

bk

Z r0
r

dr

gσ k−1.
ckP(3.72)

The first term in the series corresponds to Equation 3.67, whereas the higher-order terms
account for the entropy stratification. In fact, if only the second and third-order terms are
retained, the fit to the differential rotation improves in the tachocline as one would expect
given the rapid increase in the entropy gradient there (Figure 3.9b). This improves upon
the ad hoc construction of tachoclinic isorotational contours in Balbus et al. (2012), though
here omitting the near-surface approximation made in that paper.
3.9

Global-Scale Solar Magnetohydrodynamics
Having examined some of the basic principles of how differential rotation is built andmaintained and examined how accurately global-scale models can capture such hydrodynamics, the topic of global-scale dynamo simulations in a solar setting must be addressed. To
date, several different numerical models of solar convective dynamos have achieved cyclicalpolarity reversals, whether or not a tachocline is present. In all cases of such cycling globalscale dynamos, the interaction of convection, rotation, and magnetism are crucial ingredients.
Though, precisely how the mechanisms are arranged so that a solar-like differential rotation
can be maintained, and simultaneously yield a large-scale, periodic dynamo, has yet to beascertained. Indeed, how the multiply-scaled convection of a thermaly driven system, likethe Sun, sempiternally maintains a global-scale kinetic helicity and sustains the massively
disparate scales of the Sun’s magnetism is seemingly a mystery.

125
The earliest attempts to address these issues with global-scale solar dynamo simulations
were those of Gilman and Glatzmaier, who used an anelastic and spherical harmonic based
spectral code to simulate the dynamics (Glatzmaier, 1984) rather than the finite difference
Boussinesq scheme of Gilman & Glatzmaier (1981). Gilman laid out a series of simulations
in a 1983 Astrophysical Journal Supplement paper that aimed at assessing how varying themagnetic Prandtl number P rm and the rotation rate modified the character of the resulting
dynamo. In general, it was found that magnetic cycles are a common feature, provided that
there is sufficient energy in the bulk toroidal fields and that the Lorentz force feedback on theestablished differential rotation was not too large. Moreover, Gilman also noted that there
is little preference for symmetry or anti-symmetry about the equator and that the toroidal
bands of magnetic field tended to propagate toward the poles. It was also found that, fora fixed Rayleigh number (ratio of thermal driving to diffusion), Taylor number (ratio of the
Coriolis force to viscous diffusion), and thermal Prandtl number (ratio of viscous to thermal
diffusion), there is a peak dynamo efficiency with P rm ∼ 1. For P rm below unity, theefficiency reached a plateau and for those cases with P rm above unity it decreased.
Yet the mismatch between those early simulations and helioseismic inferences effectively
ended that line of investigation for about 15 years. A similar approach to simulating globalscale dynamos was, however, restarted roughly around 2004 with the inclusion of magnetic
fields into the anelastic spherical harmonic (ASH) code (see Chapter 4 and Brun et al.
(2004); hereafter B4). These 21st century global-scale convective solar dynamo simulations
were carried out at moderate P rm , ranging between 2 and 4. Hence it was much moredifficult for a large-scale dynamo to take hold. In keeping with that notion, the fluctuating
fields dominated the magnetic energy, accounting for most of the back-reaction on the flow
via Lorentz forces, whereas the mean fields were relatively weak. There were also no hints ofperiodic reversals or dynamo wave behavior. Again as expected with a large P rm dynamo,
the magnetic fields exhibit a complex and small-scale spatial structure and highly-variable
time evolution. The radial magnetic field is swept into the downflow lanes, being stretched126
and folded there before reaching the resistive scale and reconnecting. The toroidal fields
are organized into somewhat larger scales near the equator appearing as twisted ribbons,
whereas they form smaller scales at higher latitudes (Figure 3.10a). In B4, it was also foundthat the convection maintained a solar-like angular velocity profile despite the influence of
Maxwell stresses, which tend to extract energy from the differential rotation and thus reduce
its contrast throughout the convection zone.

The next logical step was to include a stable region and to emulate a tachocline inorder to assess its impact upon the global-scale dynamo (Browning et al., 2006). This
simulation also employed the ASH code and its hydrodynamics are discussed above in §3.7.
Both this simulation and that of B4 were fairly laminar with respect to the scales present
in the velocity field. The dynamo action achieved in the convection zone of this simulationis also quite similar to the earlier calculation of B4, though with an even larger P rm . Thus,
as one might expect the scales of magnetism become much smaller than the scales of theconvection (Figure 3.10b), with the bulk of the magnetic energy appearing in the fluctuating
magnetic fields and very little in the mean fields within the convection zone. The majordifference between the dynamo action present in this simulation, as compared to that B4,
is that the stable region provided a reservoir for storing magnetic energy and the tachocline
may have provided a means of generation of large-scale magnetic field. Indeed, within thestable region, large-scale fields were able to persist, with the majority of the energy being in
the mean magnetic fields. However, there was as yet no large-scale structure that might beable to appear at the surface (i.e. no magnetically buoyant structures) nor were there any
cyclical polarity reversals.
Rapid progress over the last four years have confirmed the role that convection and rotation likely play in stellar dynamo action through global-scale dynamo simulations as well assuggested additional mechanisms that may be present in the solar dynamo. Several research

(a)

Prm=4

(b)

Ω=1Ωʘ
(d)

Prm~1

Prm=1

Ω=8Ωʘ

(c)

Ω=1Ωʘ
(e)

Prm=1/2

(h)

Prm~1

Ω=3Ωʘ

Prm=1/2

Ω=3Ωʘ
(f)

Ω=5Ωʘ

Ω=1Ωʘ
(g)

Prm=8

Prm=1/2

Ω=3Ωʘ
(i)

Prm~1/10

Ω=3Ωʘ

127

Figure 3.10: Modern 3-D global-scale convective dynamo simulations, showing snapshots of their toroidal magnetic fields in
Mollweide projection. The effective magnetic Prandtl number (P rm ) and rotation rate Ω of each simulation is indicated. (a)First dynamo simulation using ASH (Brun et al., 2004). (b) ASH simulation with a tachocline (Browning et al., 2006). (c) Largescale, persistent toroidal structures dubbed wreaths maintained amidst moderately turbulent convection at lower P rm (Brown
et al., 2010). (d) EULAG solution with regular decadal-time-scale magnetic reversals (Racine et al., 2011). (e) ASH simulation
with quasi-periodic reversals (Brown et al., 2011b). (f) Recent continuation of (c) with lower diffusion and higher resolution.
The greater levels of turbulence permits quasi-periodic cycles to emerge (Nelson et al., 2013a). (g) Cyclic dynamo solution with
the Pencil code, showing wreath formation in a spherical segment (Käpylä et al., 2013). (h) EULAG solution rotating at 3 Ω⊙,for comparison with (c), (f), and (i) at the same rotation rate (Charbonneau, 2013). (i) Recent simulation using slope-limited
diffusion in ASH exhibiting very regular cycles (see Chapter 5). Typical toroidal magnetic field strengths are between 1 and
10 kG in these simulations, with sense of fields suggested by color.

128
teams have concurrently made inroads into solar dynamo theory utilizing three distinct numerical techniques, but nonetheless realizing similar dynamo action. The Anelastic Spherical
Harmonic (ASH) code has provided a framework for constructing and understanding the dynamos that produce and sustain coherent toroidal fields exhibiting quasi-cyclic behavior. The
Eulerian-Lagrangian (EULAG) code has shown that solutions with very regular cycles canbe achieved within its implicit large eddy simulation (ILES) framework. Third, the Pencil
code that simulates compressive convection has also found cyclical solutions with large-scaletoroidal field structures within its spherical wedge geometry.
The first 3D nonlinear convective dynamo simulation, beyond those of Gilman and
Glatzmaier in the 1980s, that indicated that sustained large-scale toroidal magnetic fieldstructures could be built within the solar convection zone was presented in Brown et al.
(2010). A snapshot of this solution’s toroidal magnetic field is shown in Figure 3.10c andits evolution is shown in Figure 3.13a. The persistence of such structures had previously
been thought impossible, as one might expect that the highly turbulent convection should
rapidly eviscerate it, transporting the magnetic flux to the boundaries of the domain. What
separated this simulation from those undertaken a few years earlier was its higher rotation
rate (or lower Rossby number) and its lower P rm , being about 16 times smaller. The salient
point that this simulation brings forth was that the rate of generation of magnetic field
through the mean shear of the differential rotation could overpower the rate of its destruction
by convection and dissipation. Indeed, the convection instead acts to sustain the differential
rotation through Reynolds stresses, which indirectly aids in the generation of the toroidal
fields. The convection then in turn converts the toroidal field into poloidal field, creating
a positive feedback loop that sustains the wreaths against resistive decay. Although thissimulation hinted that perhaps fields could be sustained within the solar convection zone, it
still did not have cyclical behavior. Furthermore, it was still quite laminar, so the ability tosustain such large-scale magnetic field structures at greater levels of turbulence and increased
complexity of the velocity field was still in question.

129
That question was partly answered in concurrent dynamo calculations utilizing the
EULAG code. The ILES implementation of this code allows it to minimize the impacts ofdiffusion and achieve more complex velocity and magnetic fields at a given resolution than
ASH can achieve. The solar dynamo modeled with this computational framework exhibited
many multi-decadal cycles of roughly 40 years that were quite regular and persisted forroughly 500 years of evolution despite the fairly complex flow fields permitted by the low
numerical diffusion (Ghizaru et al., 2010; Racine et al., 2011). A snapshot of the toroidal
magnetic field from this simulation is shown Figure 3.10d and its character in time andlatitude is illustrated in Figure 3.13e. The turbulent EMF in these simulations is moreamenable to a mean-field theory description than the earlier ASH simulations, which may inpart be due to the greater complexity of the velocity field. Part of the reason for the greateragreement between these dynamos and mean-field theory may lie in the simple argument thatas the number of convective cells becomes increasingly large the average poloidal magnetic
field generation may approach a mean value. Such a characteristic might be expected if
the averaging operator obeys the central limit theorem, or if the arguments given in Parker(1955) hold.
Further exploration of the rotation rate and P rm parameter space using the ASH code
continued apace, with a quasi-periodic cycling dynamo solution published in Brown et al.
(2011b) and a multitude of solutions explored in (Brown et al., 2011a). The toroidal fieldsof the cycling solution are shown in Figure 3.10, exhibiting the high degree of longitudinalconnectivity that can be achieved within a low-P rm global-scale dynamo and also lending
credence to their moniker of wreaths. These fields form at or near the regions of the greatest
latitudinal shear in the differential rotation, which is why they tend to be generated closer
to the equator. Such characteristics can be seen in several of the dynamo solutions from
Brown et al. (2011a) as shown in Figures 3.13a–d. The dynamo behaviors exhibited within
the menagerie of simulations in this latter paper seem to roughly correspond to the regimes
described in Gilman (1983), as is illustrated in Figure 3.11. The Gilman picture suggests

130
(a)

(b)

Figure 3.11: (a) A sketch of possible dynamo regimes with rotation rate and electricalconductivity, illustrating the cycling and persistent dynamo regimes as well as those without
sustained dynamo action [adapted from Gilman (1983)]. (b) A set of numerical dynamo
experiments carried out with varying Prandtl numbers and rotation rates, exhibiting manyof the behaviors predicted in Gilman (1983) and some that were not, particularly at low
rotation rate [adapted from Brown et al. (2011a)].

that, as the magnetic diffusivity is reduced, dynamo action becomes increasingly likely andthat cycles become more likely over a small region. Eventually, however, Lorentz-force feedback quenches both the convection and differential rotation. Similarly, as the rotation rate isincreased, cycles become more likely to a point and then past this threshold they become lesslikely due to the rotational suppression of convection. Yet another set of simulations thatsought to determine the influence of the density scale height and convective driving has recently been carried out that finds distinct transitions between dipolarity, multi-polarity, andcyclic dynamos (Gastine et al., 2012), though in less of a solar context. Indeed, they foundthat their anelastic dynamo models produced a broad range of magnetic field geometries,
with dipolar configurations for the least stratified and most rapidly rotating and a gradualtransition to multipolar configurations if either the stratification or the rotation rate wasincreased. As the density stratification is increased the convective columns in their simulations become increasingly concentrated in a thin boundary layer near the upper boundary
and at low latitudes. Such patterns of convection tended to be non-axisymmetric solutions.

(a)

131

(b)
Pinning
Downﬂowϕ

Pinning
Downﬂow

0.92

0.82

r
Upﬂow
and
Buoyancyr

Diamagnetic induction
2 kG

30 kG
0.72

ϕ

Figure 3.12: (a) A sketch of the processes leading to magnetic loop formation in ASH
simulations with dynamic Smagorinski: downflows pin the foot points of the loops, shear
between the upflow and downflow builds a large magnetic diffusivity gradient at the edge ofthe loop leading to diamagnetic induction that builds radial field of the correct sign, whileadvection and buoyancy lift the inner section of the toroidal field. (b) Two magnetic loops
achieved in an ASH simulation utilizing a dynamic Smagorinski scheme shown in a field line
rendering [adapted from Nelson et al. (2011)].

Furthermore, in multipolar solutions, zonal flows became more significant and led to the
production of toroidal field through the Ω-effect. In those cases, dynamo waves seemed tothen play an important role, leading to polarity reversing dynamos. The cycle frequency of
those dynamos was

α durms
ω≈
D dr

1/2

,

(3.73)

where α is roughly related to the kinetic helicity, and D is the depth of the domain.
Continuing the rapid pace in solar dynamo modeling and under the assumptions ofdynamic Smagorinski, mildly buoyant magnetic structures have been realized in a cycling
dynamo simulation (Figure 3.12; Nelson et al., 2011, 2013a). In this fortuitous simulation, thedynamic Smagorinski diffusion permits a greater degree of turbulence than can be achievedin the eddy viscosity model. It helps to establish a pattern of quasi-buoyant magnetic
structures that form within the convection zone, which is in contrast to the common wisdomthat they could only form in the tachocline as suggested in Parker (1993). Regardless of how

132these structures come about, they tend to have statistics strikingly similar to the statistics
of solar observations regarding Joy’s and Hale’s laws (Nelson et al., 2013b). These magnetic
structures arise from the structure of the flow field, natural buoyancy, and partly fromdiamagnetic processes. The latter process has not yet been fully considered, so the logic of
it will be expanded upon a bit here. The downflows help to pin down the foot points of theloop-like magnetic fields, whereas the upflow between two downflows advects a portion of
the magnetic field toward the upper boundary of the simulation. The natural magnetic andthermal buoyancy of the structure further boost the forces compelling it to rise. Typically,
there are downflows at the edges of the loop and upflow in its interior, so there is a strong
velocity shear present at the boundaries of the loop. The dynamic Smagorinski diffusion
coefficient is proportional to this shear and the magnetic diffusivity is simply the magnetic
Prandtl number multiplied by this coefficient. Since the coefficient is proportional to the
local shear, it is highly variable in space, increasing by at least an order of magnitude near
the shearing interface of the upflows and downflows. At low latitudes where the convection
is columnar and aligned with the rotation axis, the variation in the coefficient is largely
radial and longitudinal. Two features consequently promote the formation of loops at low
latitudes; one is the induction due to the mean shear acting on poloidal fields that builds
strong toroidal fields there; the second is the strong longitudinal variation of the magnetic
diffusion coefficient. The diamagnetic term is the cross product of the electric current withthe gradient of the magnetic diffusivity. Under the circumstances arising in these simulations,the morphology of the electric current and the diffusivity gradient are arranged such thatthe diamagnetic induction reinforces a net positive growth of the radial magnetic field along
the interfaces of upflows and downflows.
Other recent convective dynamo simulations carried out in spherical segments with
the Pencil code have also manifested unprecedented magnetic self-organization, with strong
mean field generation in nonlinear dynamo regimes, and cyclic activity on decadal time scales(Figure 3.10g; Käpylä et al., 2012). There have been more recent simulations that also ex-

133
(a)
Latitude (deg)

(b)

(c)
Latitude (deg)(d)

Time (days)(e) 90Time (days)(f)Latitude (deg)

60

30
0

-30
-60

-90
0

50

100

150
200
Time (years)

250

300

1000

1500

2000
2500
Time (t urms kf )3000

Figure 3.13: Time and latitude evolution of toroidal magnetic fields achieved in several
global-scale convective dynamo solutions. (a-d) Dynamo calculations carried out with ASH
showing four dynamo regimes arising from varying the rotation rate and magnetic diffusivity[adapted from Brown et al. (2011a)]. (e) Toroidal field evolution in a solar dynamo simulations utilizing EULAG, showing regular polarity reversals as in Racine et al. (2011). (f) A
cycling case exhibiting polarward and equatorward branches of propagating dynamo waves
in a Pencil code simulation described in Käpylä et al. (2013).

hibit two branches of propagating nonlinear dynamo waves one that is equatorward at low
latitudes and another that is poleward at higher latitudes (Figure 3.13f; Käpylä et al., 2013).
As was also seen in Gastine et al. (2012), they find that the density stratification of thesimulation has a substantial impact on the overall character of the dynamo. In particular,
one of their solutions exhibits both equatorward and poleward propagating dynamo waves(Figure 3.13f), though the reason for this is unclear. The propagation characteristics of the

134
dynamo are attributed to the stratification. Yet the number of density scale heights that are
simulated in these calculations is only about 3 to 4.5 density scale heights, which is similar
to those of previous ASH simulations. It may be that the differential rotation has a regionof near-surface shear, which can promote equatorward propagation provided that the signof the kinetic helicity does not reverse at the same time. Yet another set of Pencil code
simulations have begun to investigate how the upper boundary can influence the operationof the dynamo, such as through the release of magnetic helicity (e.g., Brandenburg & Subramanian, 2005). The generation of large-scale fields in dynamos is often associated with an
upscale transfer of magnetic helicity. In order to sustain this transfer, small-scale helicity
must be either dissipated or removed from the system by passing through the boundaries.
A possible manifestation of this on the Sun is that the magnetic helicity released in coronal mass ejections (Blackman & Brandenburg, 2003). Indeed, these recent simulations havedemonstrated that convective dynamos can eject helicity through CME-like eruptions andseparately form bipolar magnetic features akin to sunspots (Warnecke et al., 2012, 2013a,b),
yet it is still unclear how such events effect the operation of the dynamo.
Though promising, all of these recent convective dynamo simulations are still in theearly stages of discovery. They are also still quite sensitive to the treatment of diffusion
where slight changes in the form and magnitudes of the diffusivities can lead to quite different behavior. Moreover, the physical mechanisms responsible for determining the cycle
properties established within these various convection models remains unknown. What isclear is that these cyclic dynamos operate very differently than typical mean field dynamo
models in that the meridional flow plays a minimal role in regulating the cycle period andflux emergence plays little or no role in poloidal field generation.

Chapter 4
Anelastic Formulations for Stable Stratification and Slope-Limited Diffusion“Minima cura si maxima vis.” – Il motto dell’Accademia dei Lincei
4.1

Sound-proofing Approximations
In studying compressible flows, particularly thermal convection in a rotating, electri-

cally conductive, and compressible medium, many kinds of waves are encountered ranging
from sound waves to gravity waves to Rossby and Alfvén waves. When pursuing numerical solutions of compressible convection, one must simultaneously resolve all the temporalscales of motion including the high frequency acoustic waves. Thus, it can be prohibitively
expensive to simulate a numerical system for a long enough period to also capture the lower
frequency convective modes.
One solution to this problem that manages to capture the dynamics in a stratified
medium without prohibitively small time steps is to attempt to filter out the sound waves.
This approach is not unreasonable given that the acoustic waves transport only a tiny fractionof the total energy in most convective systems at low Mach number. The energy contained inthese waves, at least in linear theory, involve the fluctuating density components and are what
we therefore a higher-order effect compared to the magnitude of the convective motions. The
Boussinesq approximation (Boussinesq, 1903) suppresses sound wave propagation, but theeffects of stratification are lost. Thus a more complete theory is needed if an understandingof the flows that span multiple density scale heights is to be constructed.

136
Meteorologists realized in the early 1950s that, in order to numerically model theweather, they would have to remove sound waves. The model that was used to do this is called
the anelastic approximation. For meteorologists, it was crucial to have vertical variations in
the background density stratification. The basic principle behind the anelastic approximation
is to filter out sound waves by removing the time dependence of the background density
stratification, leaving a solenoidal condition on the momentum density. There are, however,significant differences between geophysical fluid dynamics and stellar convection modeling,
particularly in the processes involved in heating the compressible medium. So a few questions
are raised: what is the model that must be used for stellar interiors, how is the systemlinearized about a background state, and how is this state chosen?
The early meteorological work on anelastic theory (Batchelor, 1953; Ogura & Charney,
1962; Ogura & Phillips, 1962) all linearize the hydrodynamic equations about local values
in a stratified atmosphere, admitting only adiabatic motions. These models are restricted
to layers with a very limited range of potential temperature variation. In a seminal work
(Gough, 1969), an anelastic formulation is derived in which the hydrodynamic equations
are linearized about a mean state obtained in a fully convective atmosphere, with the meanstate defined by horizontal averages of the thermodynamic variables. This formulation wasused in some of the first 3-D convection simulations, wherein the second convection zone
in A-type stars is simulated (Toomre et al., 1976) using modal expansions of the anelastic
equations in a plane parallel layer (Latour et al., 1976).
In a series of papers (Gilman, 1977) leading up to the creation of a 3D sphericalanelastic model, some of the effects of stratification and rotation were examined in Boussinesq
models of the Sun. A few years later, the anelastic equations were derived for convectionof a compressible fluid in a deep rotating spherical shell with the solar convection zone
in mind (Gilman & Glatzmaier, 1981). In this work, the anelastic equations in sphericalcoordinates were linearized, and the numerical method used to solve them is explained. In
this numerical recipe, the equations of motion are expanded in spherical harmonics and137
a radial basis function and then discretized in time. This model was used to extensively
study the patterns of global-scale stellar convection and dynamos (Gilman, 1983; Glatzmaier,
1984, 1985). The numerical method and processes described in those early papers have been
adapted for massively parallel computation and for dimensional quantities. Indeed, thegoverning equations and solution method are very similar to those developed and applied
to stellar convection zones by Glatzmaier. This modern massively parallel numerical codeis called the Anelastic Spherical Harmonic (ASH) code (Miesch, 1998; Clune et al., 1999;
Brun et al., 2004). The latest versions of the ASH code utilize a slightly different set of
equations that conserve energy and momentum in regions where the background stratification
is not adiabatic. In ASH, this is the Lantz-Braginksy-Roberts (LBR) formulation that solves
for a kinematic pressure rather than a thermodynamic pressure (Lantz, 1992; Braginsky &Roberts, 1995; Brown et al., 2012; Vasil et al., 2013). The latter paper provides a general
Lagrangian framework in an astrophysical context that allows for constraints to be placed
upon the ideal MHD equations in ways that are assured to conserve energy and that go
beyond the earlier formulations.
4.2

Constrained Lagrangian Dynamics
It is worth a small digression to assess the derivation of the anelastic equations from a

Lagrangian point of view, as it is implicitly energy conserving in non-adiabatic stratifications.
Indeed, energy and momentum conservation were violated in earlier perturbative derivationsof the equations in non-adiabatic stratifications. As is well known from classical mechanics,
one can best construct and understand phase space attractors and the dynamics of a system
as whole from a Lagrangian framework. Moreover, taking the classical approach to fields
allows other constraints on the system to be more easily imposed, such as imposing ananelastic-like pressure or density constraint. Having derived the MHD equations in Chapter
3 and Appendix A, other paths to their construction can be expounded upon and theirresults compared. However, one outstanding problem remains: how does one establish the

138
existence and robustness of a suitable Lagrangian for the Navier-Stokes equations, among
other dissipative systems. This has been a pernicious problem for decades. Many approaches
have been taken while building the Hamiltonian for various simplified Navier-Stokes systems,
including the local potential method, while others have relied on either entropy minimizing or
maximizing principles to build an energy functional. In contrast, it is fairly straight forward
to construct a Lagrangian system for the ideal equations, which may eventually provide aplatform for extending this approach to the dissipative regime.
First consider an ideal system wherein phase space maps possess a preserved total volume, with compensating increases in either spatial or velocimetric volumes. One of the most
simple dynamical system to consider is one of compressible and adiabatic magnetohydrodynamics, wherein
Ds
= 0,
Dt
Dρ
= −ρ∇ · u,
Dt  
D B
B= · ∇u.
Dt ρ
ρ

(4.1)
(4.2)
(4.3)

Indeed, the flow of a fluid can be considered a time dependent map of an infinitesimal
fluid element from one position to the next over some time interval t. A fluid element canbe mapped from some set of initial coordinates that label a fluid element to some otherset of coordinates, being a map from Lagrangian label space to Eulerian path space. Thus,
the simple symplectic structure of the Lagrangian allows the equations to be integrated
exactly between one time and another. Specifically, a diffeomorphic map exists between
the labeling Lagrangian coordinates ℓ and the Eulerian coordinates x(ℓ, t). Let this tensor
be Eij = ∂xi /∂ℓj , which has an inverse Λij = ∂ℓi /∂xj , and a determinant E = det(Eij ).
The Lagrangian solution for the entropy is trivial s(ℓ, t) = s0 (ℓ), as it is only an advectedscalar and so retains its value for all time in Lagrangian coordinates. The solution for thedensity (or the specific volume) requires a bit more work. First, note that one can define the139determinant of some matrix M as an expansion of matrix cofactors: det(M) =for an expansion along the jth column. Thus one can definePX∂E
δik Ckj = Cij .
=
∂Eij
k

k Mkj Ckj

(4.4)

Using the identity that C = E(Eij−1 )T = EΛji, where C is the matrix of cofactors and T
denotes the matrix transpose, it is easily seen that∂E
= EΛji.
∂Eij

(4.5)

Another useful quantity is the Lagrangian derivative of the deformation tensor Eij
D ∂xi (ℓ, t)∂ 2 xi (ℓ, t)
∂ui
DEij
=
=
,
=
Dt
Dt ∂ℓj
∂t∂ℓj
∂ℓj

(4.6)

which uses the fact that ui (ℓ, t) = ∂xi /∂t. Now consider the logarithmic Lagrangian derivative of the determinant of the deformation tensor
D ln E∂ℓj ∂uiDE∂E DEij∂ui==∇·u= E −1= E −1= Λji
Dt
Dt∂Eij Dt∂ℓj∂xi ∂ℓj(4.7)

With Equation 4.2 and the above quantities, the Lagrangian solution for the density follows
fromD ln ρ
D ln ED ln E −1= −∇ · u = −
=
.
Dt
Dt
Dt

(4.8)

Integrate both sides to see that ln ρ/ρ0 = ln E −1 , which of course implies that ρ(ℓ, t) =E −1 ρ0 (ℓ), with ρ0 the initial density configuration. This also elucidates the geometricalconnection between the density and the volume element, namely that the volume form isinvariant ρ d3 x = ρ0 d3 ℓ.
The evolution of the magnetic field can also be found analytically. The first thing tonote is that the pushforward mapping of the density normalized magnetic field vector, given
the deformation tensor Eij , isBj0 (ℓ)
Bi (ℓ, t)= Eij.
ρ (ℓ, t)ρ0 (ℓ)
Using the earlier solution for the density, the above equation implies that

(4.9)

140

Bi (ℓ, t) = E −1 Eij Bj0 (ℓ) .

(4.10)The density normalized vorticity has exactly the same solution. The underlying geometricalconnection is that both quantities transform like a line element.
The next step is to find perturbative solutions for a 3-space displacement vector. Consider two flows one with an Eulerian map x(ℓ, t) and another with the map y(ℓ, t) for which
the initial values of s0 , ρ0 , and B0 are identical. The diffeomorphism that maps x to y
is Jij = ∂yi /∂xj , hence if Xij is the deformation tensor on x and Yij the same for y thenYij = Jik Xkj . If the Eulerian deformation tensor E used above is equated with X, thecomposition of maps then takes the X fields to those fields with the deformation tensor YassY (ℓ, t) = sX (ℓ, t),
ρY (ℓ, t) = J −1 ρX (ℓ, t) ,(4.11)BiY (ℓ, t) = J −1 Jij BjX (ℓ, t) .
The Lagrangian displacement of the two flows is ξ = y − x. Therefore,
∂ξi
,
∂xj

∂ξi
+ O ξ2 ,det (Jij ) = 1 +
∂xj


∂ξi
+ O ξ2 .
det Jij−1 = 1 −∂xj
Jij = δij +(4.12)(4.13)(4.14)The Lagrangian perturbations ∆Z = ZY (ℓ, t) − ZX (ℓ, t) in a quantity Z are then easilyfound from Equations 4.11, with

141

∆s = 0,(4.15)∂ξi+ O ξ2 ,∆ρ = ρY − ρX = Y −1 − X −1 ρ0 = J −1 − 1 X −1 ρ0 = −ρX∂xi∆Bi = BiY − BiX = Y −1 Yij − X −1 Xij Bj0 = J −1 Jij − 1 BjX
 


 X

∂ξi
∂ξi
∂ξi
∂ξk
2
= δij 1 −+−1+O ξ
Bj = BjX− BiX+ O ξ2 ,
∂xi
∂xj
∂xj
∂xk
Dξi
Dyi Dxi
−
=
.
∆ui =
Dt
Dt
Dt

(4.16)(4.17)(4.18)The Eulerian perturbations δZ = ZY (x, t) − ZX (x, t) (note that the perturbations are at
a fixed coordinate) for some quantity Z crucially link the perturbative Lagrangian andEulerian frameworks. They can be connected through a Taylor expansion of the Lagrangian
perturbations in powers of ξ. In particular, let Z be some scalar, vector, or tensor quantity
Lthat has a Lagrangian value ZYL (ℓ, t) for the Y flow and ZX(ℓ, t) for the X flow. The
E
corresponding Eulerian values are ZYE (y, t) for the Y flow and ZX(x, t) for the X flow.

Therefore, the Taylor expansion about x isLE
E
∆Z = ZYL − ZX
= ZYE (x + ξ, t) − ZX
(x, t) = ZYE (x, t) − ZX
(x, t) + ξ · ∇ZYE (x, t) + O ξ 2




E
= δZ + ξ · ∇ZYE (x, t) + O ξ 2 = δZ + ξ · ∇ ZX
+ δZ + O ξ 2 = δZ + ξ · ∇Z + O ξ 2 ,(4.19)

where in the last equation we take the X flow to be the reference. To second order in ξ, theEulerian perturbations are thus δZ = ∆Z − ξ · ∇Z, which for individual quantities becomeδs = −ξ · ∇s + O ξ 2 ,δρ = −ρ∇ · ξ + O ξ 2 ,δB = B · ∇ξ − ξ · ∇B − B∇ · ξ + O ξ 2 ,

Dξ
δu =
− ξ · ∇u + O ξ 2 .
Dt(4.20)(4.21)(4.22)(4.23)

Note that the Eulerian perturbation for the magnetic field can be simplified through thevector identity ∇ × (A × B) = B · ∇A − A · ∇B + A∇ · B − B∇ · A, and also since∇ · B = 0, δB = ∇ × (ξ × B). In index notation this is142δBi = ǫijk ǫklm∂ξl Bm∂ξl Bm= (δil δjm − δim δjl )
,
∂xj
∂xj

(4.24)

where the last expression arises from the properties of the Levi-Civita symbol ǫ.
The general Lagrangian formulation for an ideal compressible conducting fluid can
now be obtained. Let the Lagrangian density of the system be L = L(Φ), where Φ =(B, u, ρ, s), then Hamilton’s principle states that the Eulerian path of motion is the one that
RR
minimizes the action functional A[Φ] =dt d3 x L. Namely, the functional derivative of A

is minimized by a particular path arising from the solution of the resulting Euler-Lagrange
equations. In the ideal compressive MHD case, this principle yields


Z Z
∂L
∂L
∂L
∂L
3
· δB +· δu +δρ +δs = 0.
δA =dt d x∂B∂u∂ρ∂s(4.25)The Eulerian perturbations δΦ are constrained above in Equations 4.1-4.3, and are parameterized to second order with Equations 4.20-4.23. Substituting these expressions into thevariation of the action and using the convention that repeated indices are summed over, it
is found that




Z Z
∂ξl Bm ∂L∂L ∂ξi∂ui∂L ∂ρξj
∂L ∂s
∂ξi
3
+
− ξj
−
− ξj
= 0.
+ uj
δA =
dt d x ǫijk ǫklm∂xj ∂Bi ∂ui ∂t∂xj∂xj∂ρ ∂xj∂s ∂xj(4.26)
To isolate the displacement vector integrate by parts and swap a few indices to see that
 





Z Z
∂ ∂L
∂
∂L
∂uj ∂L
∂L
∂
∂s ∂L
3
+
+
uj
+
−ρ
δA = B.T. −dt d x ξi∂xi ∂s∂xi ∂ρ∂t ∂ui∂xj∂ui∂xi ∂uj




∂
∂L
∂
∂L
+Bj
− Bj
= 0.
∂xj ∂Bi∂xi ∂Bj(4.27)The boundary terms (B.T.) can have several interpretations depending upon the problem
at hand. These terms must vanish if the system under consideration is infinite in extent
or fully periodic. If instead the system has a finite domain, these terms should be treatedvery carefully, especially when they are porous, permeable, or otherwise deformable. For
completeness, the boundary terms resulting from the integration by parts are143

B.T. =

XZ Z
i

dt

Y
k6=i

dxk



∂L
∂L
∂L
u i ξj− ξi ρ
− ǫijk ǫklm ξl Bm
∂uj
∂ρ
∂Bj

x1i

xi =x0i

+

Z

t

∂L 1
d x ξi∂ui t03(4.28)The beauty of this is that the impact of boundary conditions on the underlying system canbe systematically studied, particularly when placing greater constraints on the system. One
interesting feature is that an implicit assumption in the typical construction of Hamilton’s
principle used here becomes apparent with the temporal boundary condition seen in theequation for the boundary terms. While Hamilton’s principle can be used to derive equations
that function for initial conditions, it really is a boundary value problem in time. A recent
paper has reformulated Hamilton’s principle to be an initial value problem in time rather
than a boundary value problem, which provides one potential method for creating suitable
Hamiltonians and Lagrangians for dissipative systems (Galley, 2013).
For the time being the system will be considered to be fully periodic and to exist for
an infinite amount of time. In this case, the boundary terms vanish and the following EulerLagrange equations are recovered for the velocity field under the constraints of an entropy
and mass conserving system with the induction equation as
 




∂s ∂L
∂L
∂ ∂L
∂
∂uj ∂L
∂L
∂
+
+
uj
+
−ρ
∂xi ∂s∂xi ∂ρ∂t ∂ui∂xj∂ui∂xi ∂uj





∂
∂L
∂L
∂
−
= 0,
+Bj
∂xj ∂Bi∂xi ∂Bj(4.29)or in vector notation with p = ∂L/∂u (momentum) and H = −∂L/∂B (magnetic field)
X
∂L ∂p
∂L
∇s − ρ∇++ ∇ · (u ⊗ p) +(pj ∇vj ) − (∇ × H) × B = 0.
∂s
∂ρ
∂t
j

(4.30)

This so far is fairly general with no Lagrangian yet specified. We know from the derivationof the MHD equations in Chapter 3 that the total energy density, and thus the Hamiltonian
density H, is1
B2∂L
H = ρu2 ++ ǫ + ρΦ = u ·
− L.
2
8π
∂u
Since the only first term involves u, the Lagrangian density must be(4.31)

144


B2
1 2
,
L = ρ u − e (ρ, s) − Φ (x) −28π
(4.32)

with e the specific energy. Using this Lagrangian, Equation 4.30 becomes∂L∂e= −ρT,
= −ρ
∂s
∂s ρ

(4.33)1∂L1∂ep (ρ, s)= u2 − e − Φ −= u2 − e − Φ − ρ,(4.34)∂ρ2∂ρ s 2ρ∂L= ρu,(4.35)∂u∂L
B=,(4.36)
∂B
4π 

1 2
∂ρu
1
1
−ρT ∇s − ρ∇u −h−Φ ++ ∇ · (ρu ⊗ u) + ρ∇u2 −(∇ × B) × B = 0, or
2
∂t
2
4π
(4.37)∂ρu1+ ∇ · (ρu ⊗ u) + ∇p + ρ∇Φ −(∇ × B) × B = 0,∂t
4π(4.38)

where the specific enthalpy is h = e+p/ρ and with the thermodynamic identities ∂e/∂s|ρ = T
and ∂e/∂ρ|s = P/ρ2 . The last line uses the second law of thermodynamics to simplify it(ρT ∇S − ρ∇e + P/ρ2 ∇ρ = 0). It can be reduced further with the use of the continuity
equation to the usual form
11Du= − ∇p − ∇Φ +(∇ × B) × B.
Dt
ρ
4πρ

(4.39)

There are several techniques that allow dissipative processes to be included in a Lagrangian framework, but they typically require perturbations about a set of fixed and doubled
variables that represent an equilibrium flow such as with the methods of Mobbs (1982). As
a consequence, such techniques cannot be valid for large excursions from equilibrium. A
more recent insight establishes a covariant action principle in multi-fluid general relativity
to non-perturbatively find dissipative Lagrangians (Andersson & Comer, 2013). The mainidea in this approach is to treat the entropy field as a separate fluid from the matter field,

145
with the interactions between them being parameterized as interaction coefficients (diffusion
coefficients). Heat conduction is then an effective thermodynamic force within the entropy
fluid, transporting entropy from one place in the entropy coordinate space to another. The
viscous diffusion terms arise from variations in the metric leading to a generalized pressure
tensor that incorporates the shear-strain tensor. This latter variation suggests a new wayof varying the Lagrangian in the non-relativistic limit. Namely, the Eulerian coordinates
should be allowed to vary, equivalently that the volume measure should not be fixed. Thisshould not be surprising given the direct relationship between the Lagrangian and Eulerianframes through the volume element, since dissipative processes change the internal energy
and thus both the density and entropy, the volume element should be varied to accommodate
the dissipative processes. This may possibly be sketched as the following action principle
δA =

Z Z

3

dt̃ d a




∂L
∂L
∂L
∂L
∂L
∂L
δt +
· δx +· δB +· δu +δρ +δs = 0.
∂t
∂x
∂B
∂u
∂ρ
∂s

(4.40)

with L = L(t, x, B, u, ρ, s) and where t̃ is a generalized “time” coordinate that should bethought more of as a thermodynamic time that has a link to the perceived time t = t(a, t̃),the space coordinates a correspond to the thermodynamic displacements of the fields, andthe variable Eulerian coordinates are x = x(a, t̃). The way to think of this integral, inorder to make connection to recent work in mathematics, is to consider the thermodynamic
displacements more as a phase space path density, with the density of traversable paths
equivalent to the entropy. If the number density of such physical paths is governed by thesecond law of thermodynamics, the metric on the thermodynamic space must be convex with
ds2 = gµν daµ daν ≥ 0. This implies that a system in equilibrium would have ds2 = 0, withthe entropy of the system being maximum. It is also the only time-like curve available to thesystem. In this condition, the physical time and space merge wherein no further evolution ispossible and the system is totally isotropic and homogeneous. If the metric is positive, thephysical time and space instead emerge from the microscopic physics that are parameterized
in the thermodynamic space and through the dynamics that it drives.

146
A very refreshing work regarding anti-self-dual systems (ASDS; such as the NavierStokes equations) has yielded a greater mathematical formalism for variational principles
within such systems (Ghoussoub & Moameni, 2005; Ghoussoub, 2008; Ghoussoub & Moameni, 2009; Stuke, 2011). The basic premise is that the underlying manifolds of the ASDS
are monotone and convex, and that time-dependent Lagrangians must be lifted to path space
as noted above. The Lagrangian is then constructed by identifying the proper dual spaceand forming a functional on the product of the dual and physical spaces, one might notethe similarities among this approach, the earlier fixed point variational methods, and thecreation of dual matter thermodynamic spaces above. Indeed, these notions point out thesubtle problems with the Euler-Lagrange functional minimization approach. For instance,
one must show the regularity of that functional minimum, the existence of which is often
obtainable only under a weak set of assumptions. This implies that the space of admissiblefunctions must be chosen very carefully, which is quite evident in the “choice” of a Lagrangian
for the Lantz-Braginsky-Roberts formulation (see the following section) or in the choice ofan equilibrium state as in Mobbs (1982). Furthermore, not all stationary points are minima of the action in the sense that there could be multiple and nearly equal extrema. This
weakens the overarching framework as the existence of stationary, non-minimizing points isquite difficult if not impossible to show in general.
Moreover, as described in Olver (1993), the vast majority of partial differential equations fall outside the purview of the Euler-Lagrange formalism. In this paper, Bäcklund’s
theorem [which states that the most general contact (canonical) transformations are extended point (coordinate) transformations whenever both the number of independent anddependent variables exceed unity (Edelen, 1984)] is generalized to formally characterize theinternal versus external symmetries of canonical transformations relatable to the exceptional
simple Lie group G2 . This group is particularly apropos to the general internal symmetriesof the Navier-Stokes MHD equations that have eight continuous fields, with two vector fields
and two thermodynamic quantities. The solenoidal restriction of the the magnetic field re-147
duces this to seven total fields, which is the lowest order representation of the G2 group. In
this sense, perhaps some insight into the broader symmetries of the Navier-Stokes equations
could be found in the construction an octonionic Hamiltonian as an alternative framework
for analyzing the properties of conducting fluids, considering that the fundamental representation of G2 is the set of automorphisms of the octonionic algebra O. Such considerations
may be linked to the earlier equilibrium and extremally entropic variational principles of Onsager (1931), Mobbs (1982), and Glansdorff & Prigogine (1954) through the consideration of
doubled fields or dual spaces, which would fall into the 14-dimensional representation of G2 .
More generally, it could be that all the possible approaches to asking fundamental questions
about the nature of the structure of the dissipative Hamiltonian are branches of a deeper
formalism. Indeed, an abstraction of the variational action principle may be in accord with
the initial value formulation of Hamilton’s principle as suggested in Galley (2013), though
the systems considered in that work were not continuous fields.
4.3

ASH and the Lantz-Braginsky-Roberts Equation
With all the lovely Lagrangian machinery for ideal MHD at one’s fingertips, the steps

to deriving an energy conserving anelastic formulation are straight forward. To derive theenergy conserving LBR equation two pieces are needed. The first piece is to linearize thethermodynamics, and the second is to constrain the compressive Lagrangian with a Lagrange
multiplier λ that enforces the constraint ρ/ρ0 − 1 = 0, which is the underlying assumptionof the anelastic formulation. Thus the Lagrangian can be written down as
B2
1− λ (ρ/ρ0 − 1) ,L = ρu2 − e (ρ0 , s0 ) − P0 − ρT0 s − ρ0 Φ −2
8π(4.41)

with the internal energy expanded to first order in ρ and s. The terms retaining the fulldensity cancel after substituting this Lagrangian into Equation 4.30. So, the following is
found
Du
1
λ
(∇ × B) × B.
= −∇ − s∇T0 +
Dt
ρ0
4πρ0

(4.42)

148
The Lagrangian multiplier λ/ρ0 can thus be identified as the kinematic pressure p′ /ρ0 or
̟. Secondly, in a purely adiabatic atmosphere the hydrostatic background state has thefollowing equations P0 = Kργ0 , P0 = Rρ0 T0 , and ∇P0 = −ρ0 ∇Φ, which together imply that
∇T0 = −R−1 K 1/γ(γ−1) ∇Φ = −Cg. Therefore the connection is made between the morefamiliar form with gravity rather than temperature gradients in nearly adiabatic atmospheresDu1≈ −∇̟ − Csg +(∇ × B) × B.
Dt
4πρ0(4.43)

This is accompanied by the continuity equation ∇ · ρ0 u = 0 and the induction equation as
well as a modified entropy advection equation Ds/Dt = −u · ∇s0 . These are roughly theequations solved in ASH, but what sets them apart is that the above equations lack diffusive
processes.
Appealing to the derivation of the LBR equation of motion above and to the earlier
derivation of the MHD equations in Chapter 3, the equations evolved in ASH with diffusion
included and moving to the rotating frame are



∂u
1
1
(∇ × B) × B + ∇ · 2ρ0 ν e − ∇ · uI ,= −u · ∇u − ∇̟ − Csg + 2u × Ω +
∂t
4πρ0
3




(4.44)

4πη 2 2ν1
1∂s
∇ · (ρ0 T0 κs ∇s) +e : e − (∇ · u)2 + ǫ,
= −u · ∇ (s0 + s) +
J +
2
∂t
ρ0 T0
ρ0 T0 cT0
3
(4.45)∂B= ∇ × [u × B − η∇ × B] ,∂t

(4.46)∇P0 = −ρ0 g,(4.47)∇ · ρ0 u = 0,

(4.48)∇ · B = 0,

(4.49)

Here Ω is the angular velocity of the rotating frame. The diffusion coefficients are theviscous diffusion ν, the entropy diffusion κs , and the magnetic diffusion η. The viscous
diffusion acts linearly to reduce velocity gradients through its action on the stress tensor

149
e = 1/2(∇u + (∇u)T ) and on compressive motions through the velocity divergence. The
symbol I is the unit tensor and J is the electric current. The last term in the entropy equation
ǫ incorporates all spherically symmetric heating and cooling terms such as the volumetricheating arising from the diffusion of radiation down the mean temperature gradient. Both
ν and η have direct links to the microscopic fluid equations through their multiplicative
renormalization.
The role of entropy diffusion more subtle. Its primary motivation is to parameterize
the impact of unresolved flows on the energy transport in the system. The first assumption
is that the net action of the small-scale flows is to transport energy outward in a sphericallysymmetric fashion, with a diffusion carrying the energy that is not transported by the larger
scales along the mean entropy gradient. For the spherically symmetric component, this issimply the mixing length argument where the energy carried by the convection is parameterized as a diffusive flux Fconv = −ρT Kds0 /dr (e.g., Zahn, 1999). However, how much energy
the larger scales should carry is an open question, and one that is somewhat independentof the question of how much momentum they should carry. So, the level of diffusion is very
arbitrary ranging from the “let the large scales try to do everything” approach to the casewhere the resolved convection carries only a fraction of the total energy flux through thesystem.
This entropy diffusion also acts at all other resolved scales, becoming increasingly
important at smaller scales where entropy gradients tend to be steeper and due to the k 2
wavenumber dependence of the diffuse operator. This can be considered as an increased
entropy mixing at moderate scales due to the larger influence of the unresolved motions onthem. The increased influence of the small scales is in turn due to the the relative magnitudesof the integrated kinetic energy of the small scales and the kinetic energy at moderate scales.
However, its role is physically only valid in regions that are convectively unstable. Thus it
must be carefully applied. In regions where the stratification is stable and stiff, this diffusion
can violate the second law of thermodynamics. Physically, one should not expect the mixing

150
length argument to be valid in stable regions as there nominally is no convection and nothing
as direct to mix entropy. Instead, photons and waves take over the role of diffusion. The
action of the photons is to diffuse temperature gradients, whereas waves transport energy
and momentum in a highly nonlocal fashion that is difficult to parameterize and is thus
neglected.
ASH solves these 3-D anelastic MHD equations of motion in a rotating spherical shellusing a pseudo-spectral method. The numerical methods and parallelization were first documented after its construction in (Miesch, 1998) and shortly thereafter in (Clune et al., 1999).
The first version of the code solved the anelastic equations as defined in (Gough, 1969), which
are not fully energy conservative particularly away from adiabatic stratifications. This formulation was still in use when magnetism was included as in (Brun et al., 2004). Morerecently, the energy conserving formulation has been implemented, allowing greater freedom
in the choice of background states and more accurately capturing the dynamics of stableregions.
As described in Brun et al. (2004), the velocity and magnetic fields are expanded
into poloidal and toroidal potentials. This formulation ensures that both the mass flux
and the magnetic fields remain divergence free to within machine precision. It also reduces
the number of fields that must be kept in memory, but increases the order of the spatial
derivatives in the equations, which for spectral methods is usually not a problem. Thus the
kinematic pressure, entropy, and the four potentials are expanded in spherical harmonics
Yℓm (θ, φ) to resolve their horizontal structures and in either Chebyshev polynomials Tn (r)or with finite difference methods to resolve their radial structures. This requires a spatial
collocation mesh with nr × nθ × nφ points, or in spectral space this is equivalent to maximum
spherical harmonic degree of ℓmax = 2(nθ + 1)/3 with an equivalent maximum in azimuthal
wavenumber m. The spherical harmonic transform therefore relies on a triangular truncation
for which the maximum m is equal to the maximum ℓ, as opposed to other schemes such as
the rhomboidal truncation.

151
Temporal discretization is accomplished using a semi-implicit Crank-Nicolson timestepping scheme for the linear terms and a second-order explicit Adams-Bashforth method
for nonlinear terms such as the convective derivative and the dissipative terms in the entropy
equation. The Adams-Bashforth method is simple, being1
yn+1 = yn + ∆t 3fN L (tn , yn ) − fN L tn−1 , yn−1 ,2(4.50)

with y is the vector of values at each point on the spatial grid for each of the six evolved fields
and the nonlinear terms and at each grid point are contained in fN L . This scheme is stable,
but restricted by the CFL on any of the flow, Coriolis forces, Lorentz force, or the dissipative
processes. In contrast, the linear terms involving the diffusion, pressure gradients, buoyancy,and volumetric heating terms are advanced with the implicit Crank-Nicholson scheme. This
scheme for the linear portions of the time derivatives are1
yn+1 = yn + ∆t fL tn+1 , yn+1 − fL (tn , yn ) ,2(4.51)

This equation is advanced in spectral space where small nr ×nr matrices are solved for each ℓand m. This requires a careful load balancing for optimal communication and computation.
The Crank-Nicholson step implicitly determines the kinematic pressure. This character ofthe pressure has the benefit of moderating any instabilities that might arise from the lessthan ideal explicit time stepping scheme for the nonlinear terms, which have provided an
estimate for the nonlinear portions of the solution at time tn+1 . It also ensures that thesolenoidality constraint on the mass flux and the anelastic constraint on the density are
maintained. The basic layout of the ASH time evolution algorithm is that there is a stored
time step to be used for the Adams-Bashforth method, the nonlinear terms are computed
and the current Adams-Bashforth as well as the explicit portion of the Crank-Nicholson
operators are updated, then the implicit portion of the time step is computed.

152
4.4

Diffusion Methods in ASH
There are currently three methods for stabilizing the numerics in ASH, which is in-herently unstable without a sufficient level of explicit diffusion. The first is that describedabove with a renormalization of the diffusion coefficients, the second is the spectral extrapolation method as in the dynamic Smagorinski method, and most recently the numericallystabilizing slope-limited diffusion (SLD) scheme. SLD has many possible formulations, withthe one employed in ASH is similar to that found in Rempel et al. (2009) and Fan et al.
(2013). At its heart, the SLD method attempts to push the solution toward a state thatis monotonically smooth. This does not, however, guarantee the same smoothness of thefirst order derivatives. The smoothed solution is obtained through the following steps: aslope-limited piecewise linear reconstruction of the solution is made; the difference between
the left and right reconstructed values at a cell edge is computed; this difference is thenmultiplied by a diffusion rate speed to create a diffusive flux at the cell edge; finally, the divergence of this flux is computed and added to the solution. Since this diffusion is nonlinear
and a finite-volume scheme, it is computed in physical space and so must be transformed
into spectral space before being added to the other nonlinear terms of the Adams-Bashforthtime-stepping scheme described above. This incurs non-trivial overhead costs in memoryand communication.
Effectively, SLD acts to minimize the cell-edge discontinuities arising in the piecewise
linear reconstruction of the solution between adjacent cells so that the numerical method
remains stable. Thus, in a sense, the SLD operator acts more as a spatial filter than asa diffusion operator. While this model of diffusion is not physically motivated like theeddy diffusion model or the dynamic Smagorinski scheme, it holds many computational
advantages. Indeed, with SLD in ASH, one achieves a much higher level in the complexityof a solution for a given resolution than is possible in other methods. There is also no needto evaluate second-order derivatives or perform multiple transforms between spectral and153

Figure 4.1: The space of second order total variation diminishing schemes is shaded in light
blue, being bounded above by the superbee limiter (red) and from below by the minmod
limiter (dark blue). The van Leer limiter is plotted in orange, and the van Albada limiterin gray. The abscissa r is the value of the ratio of adjacent derivatives and the ordinate φ isthe value of the limiter.

physical space. Furthermore, under certain choices of slope-limiting functions and diffusion
coefficients, the SLD scheme reduces to a fourth-order accurate approximation to a Laplacian
operator as is used in the eddy diffusion model.
Before delving into the details of SLD, it is probably worth an aside to describe slope
limiters and total variation diminishing schemes. Flux or slope limiters have been employedfor hyperbolic conservation laws for over 50 years, starting with the pioneering work of
Godunov (1954, 1959) and that of van Leer (1973, 1979). Those early methods are unified in
the work of Sweby (1984), which provides a clean mathematical framework for understanding
and linking all of the previous flux limiting schemes. Such formulations of hyperbolic partial
differential equations are also known as total variation diminishing, which amounts to threeconstraints on the physical properties of a solution: there must not be any artificial entropy
decreases in expansion shocks, monotonicity must be preserved, and the total variation of

154
any physically admissible solution must not grow in time. Generally, such terms were used
to create a bounded anti-diffusive flux to counteract the excessively diffusive nature of thefirst order approximations to the hyperbolic equations, which is particularly important nearregions of sharp gradients. For the SLD method presented here, this notion is inverted as
the goal is a bounded diffusion rather than an anti-diffusion due to the use of a high-ordernumerical scheme.
The definition of total variation diminishing is that at a time step n + 1 the total
P nvariation is less than the previous time step with T V (tn+1 ) ≤ T V (tn ) = i |qi+1− qin | forsome quantity q at cell center i (Lax, 1973). The limiters that yield this property do soin the context of second-order high-resolution flux-conservative numerical methods, such as
those considered in Sweby (1984). Indeed, Sweby showed that if a second-order TVD scheme
is desired then the values that the limiter can assume are fairly restricted and in generalthey must also possess a symmetry property so that forward and backward gradients aretreated equally. The region over which this property is satisfied is shown in Figure 4.1. Also
illustrated in this figure are four examples of slope limiters: the minmod limiter that is thelower bound of the region, the superbee limiter that is its upper bound, the van Leer limiter
that smoothly bisects it, and the van Albada limiter that is commonly used in higher-ordermethods. Each of these limiters can change the character of the solution, with the superbee
limiter being more compressive (leading to more cusped structures) and the minmod limiter
being the most diffusive.
A piecewise reconstruction of values at the cell edge that are concurrent with the cell
center value takes the general form of a Taylor series as
qi+1/2 = qi +X
k=1

xi+1/2 − xi

g
kq
k ∆
,
∆xk

(4.52)iwhere each order of the numerical derivative is limited in some fashion to maintain monotonicity as is indicated with the tilde. There are a few techniques for computing thesehigher-order derivatives and their limited values, but they become increasingly complex

155

Figure 4.2: An illustration of the piecewise linear reconstruction used in slope-limited diffusion.

with each increasing order of polynomial approximation (e.g., Jiang, 1996; Suresh & Huynh,
1997; Borges et al., 2008; Čada & Torrilhon, 2009; Mignone et al., 2010). The most wellstudied schemes are linear reconstructions, which would be a poor approximation if the full
flux were treated treated with it. However, for creating a diffusive operator they will suffice.
The piecewise linear reconstruction at the cell edges requires three derivative values(calculated using the spatial derivatives internal to a given numerical method) and threequantity values at adjacent cell centers of a given quantity q in a particular direction. Eachdirection is treated equally, so the following will focus only on one of them; call it coordinate
x with index i. First define the cell-edge derivative approximation as


1 ∂q
∂q
∂q
=
+
∂x i+1/2 2 ∂x i+1 ∂x i
Next define the the average cell-centered derivative with
#
"


2
∂q
∂q
∂q
=
+ wi−1,wi+1∂x i
2 + wi+1 + wi−1
∂x i+1/2∂x i−1/2





∂q
∂q
∂q
∂q
1
+
+ wi−1+.
wi+1=
2 + wi+1 + wi−1
∂x i+1 ∂x i
∂x i ∂x i−1
where the weights w are(4.53)(4.54)(4.55)

156

wi−1 = minmod [qi , qi−1 ] ,(4.56)

wi+1 = minmod [qi , qi+1 ] ,ih min 1, x for xy > 0y.
minmod [x, y] = 0otherwise(4.57)(4.58)Such a weighting minimizes the overshoot and undershoot of the solution near a discontinuity,
allowing greater monotonicity. Let the slope-limiting function be defined so that
"

−1 #
∂q
∂q
φi = φ∂x i∂x i(4.59)

and bounded so that 0 ≤ φi ≤ 1. The limited slope at cell center is then constructed as
f
∂q
∂x

i

= (1 − φi )



∂q
∂x i



+ φi

∂q
∂x i

(4.60)

Therefore, the slope-limited and linearly reconstructed the values of q on the left and rightsides of a cell edge aref∂u1Lqi+1/2= qi + ∆xi,
2∂x(4.61)if∂u1R
qi+1/2= qi+1 − ∆xi+1
2∂xwith ∆xi = xi − xi−1 . Thus the cell-edge difference is
"
f
∂u
1
∆xi+1
δqi+1/2 = ∆qi+1/2 −
2
∂x

i+1

.

(4.62)

i+1

f
∂u
+ ∆xi
∂x

i

#

,

(4.63)

where ∆qi+1/2 = qi+1 − qi . In this case, the diffusive flux is then controlled with a coefficientthat sets the rate at which the solution achieves a state that is linearly smooth and monotonic.
x
The diffusive flux in direction x at the cell edge Fi+1/2
is thenx
Fi+1/2= ci+1/2 δqi+1/2 ,with the diffusion speed at the cell edge defined as(4.64)

157

ci+1/2 = β δqi+1/2 , ∆qi+1/2 ; αq 2′2vmin + v′2 + vA.

(4.65)

Here vmin is a minimum speed that sets the base level of diffusion and the fluctuating velocities
and magnetic fields (through fluctuating Alfvén velocities) are′2v = vr
′2
vA
=

′2

1
+ vθ′2 + vϕ′2 =

1
B′2
=
4πρ
2

"

h

2

2

(v − hvi) i+1 + (v − hvi)
2#
(B − hBi)2(B − hBi)2
+
.
4πρ
4πρ
i+1

i

i

,

(4.66)(4.67)

i

The function β further isolates the diffusion to regions of the largest discontinuities andprevents anti-diffusion. It is defined asαβ δqi+1/2 , ∆qi+1/2 ; α = minmod δqi+1/2 , ∆qi+1/2,(4.68)

with α some positive value. If the value is too great, the solution may not be stable. Let
ri be the cell center in radius of a grid cell, θj the center in colatitude, and ϕk the centerin longitude. Following the definition of finite volumes, the rate of change induced in thequantity q through the diffusive flux is∂q1=∂t Diff
VZ

1 X∇ · Fq dV =V mZ

Fq · dSm(4.69)The divergence of the diffusive flux for a scalar quantity q, which is the same for vector
components arising from tensor quantities, isZ
1 X

Vi,j,km

Fq · dSm =1 Vi,j,k

Ai+1/2,j,k ci+1/2,j,k δqi+1/2,j,k − Ai−1/2,j,k ci−1/2,j,k δqi−1/2,j,k+ Ai,j+1/2,k ci,j+1/2,k δqi,j+1/2,k − Ai,j−1/2,k ci,j−1/2,k δqi,j−1/2,k+Ai,j,k+1/2ci,j,k+1/2δqi,j,k+1/2 − Ai,j,k−1/2ci,j,k−1/2δqi,j,k−1/2 ,(4.70)The volume of a given cell is Vi,j,k and the surface area of a given face is A. In sphericalcoordinates this becomes

158

2
r
2
r
2
2
θ
θ
ri+1/2
Fi+1/2− ri−1/2
Fi−1/2∂q
3 ri+1/2 − ri−1/2 sin θj+1/2 Fj+1/2 − sin θj−1/2 Fj−1/2
=3
+ 3
3
3
3
∂t Diff
ri+1/2− ri−1/2
2 ri+1/2 − ri−1/2
cos θj−1/2 − cos θj+1/2
ϕϕ2
2
Fk+1/2− Fk−1/2
θj+1/2 − θj−1/2
3 ri+1/2 − ri−1/2,+ 3
3
2 ri+1/2 − ri−1/2
cos θj−1/2 − cos θj+1/2 ϕk+1/2 − ϕk−1/2(4.71)

where unchanging indices have been suppressed.
Since this diffusion scheme is derived from finite volumes, it is conservative to within
machine precision, with the adage “one cell’s loss is another’s gain” holding here. It isuseful to show that this diffusion reduces to a Laplacian diffusion under certain assumptions.
Assume first that the diffusion speed is set to the constant c = vmin . Second, let the function
β = 1 and further let there be no discontinuities in the domain (so that the weights w are
all unity). For simplicity, also assume a uniform mesh in one dimension, thus the divergenceof the diffusive flux in Equation 4.64 is1cδqi+1/2 − cδqi−1/2 δ ,
∆x
"
f
c
1
∂u
=
∆qi+1/2 − ∆x
∆x
2
∂x

∇·F=

= c∆x

+
i+1

qi+1 − 2qi + qi−1
1
−
∆x2
2∆x

"

f
∂u
∂x
f
∂u
∂x

i

#

"

f
f
∂u
1
∂u
− ∆qi−1/2 + ∆x
+
2
∂x
∂x

i+1

−

f
∂u
∂x

i−1

#!

i

(4.72)#!
,

i−1(4.73).

(4.74)

Notice that the first term is just the second-order approximation to the second derivative,
whereas the second is the slope-limited approximation to the second derivative. So, this may
be expressed as"#
2k+2 q∆x2k
1 ∂ 2k+2 q∂^∇ · F = c∆x−,(2k + 1)! k + 1 ∂x2k+2 ∂x2k+2
k=0
i
#
"
2
g
2

∂ q
∂ q
− 2 + O ∆x2
.
= c∆x
2
∂x
∂x
N
X

(4.75)(4.76)iwhere N is the exponent of the truncation error in the computation of the derivatives within
the numerical scheme minus two and then divided by two. For instance in ASH, N = 1 as

159
the finite differences in the radial direction have truncation error of order four. Equation 4.75
elucidates the behavior of the slope-limited diffusion operator. The difference between theunfiltered and the filtered Laplacian operators reduces the overall diffusion by subtracting
off the smoothed average Laplacian, which is by construction always less than the local
Laplacian. This leaves only the local fluctuations to be diffused away with the coefficientc∆x. In the ASH simulation described below, this value is of the order of 1012 cm2 s−1 ,which is quite typical of the diffusivities used in the turbulent eddy viscosity model. If thehigher order terms are retained, it is clear that the unfiltered hyperelliptic operators act
hyper-diffusively. In contrast, the higher-order filtered operators now act anti-diffusively
and with increasing influence at ever higher orders given the 1/(k + 1) coefficient multiplying
the unfiltered operators. This might be problematic if the derivatives were not limited, but,in addition to the slope limiting, the function β eliminates the possibility of anti-diffusion.
Some additional aspects of the SLD operator will be further explored in later Chapters.
In order to conserve energy in the domain, the diffusive processes must be accounted for.
Since the entropy diffusion is automatically energy conserving, only the dissipative exchangeof energy from the magnetic and velocity fields to the entropy field need be examined. One
caveat is that the SLD operator does not preserve solenoidality of a vector field on its own.
The projection operator from physical space into spectral space preserves this property forthe vector fields, but not for the fluxes available for calculating the dissipation. The way
around this is to project forward into spectral space, and then project back into physical space
and use the transformed fields for computing the dissipation. Such forward and backward
projections would be very costly as it would need to be done for each component of the
SLD viscous and magnetic diffusion tensors. For now, it is assumed that the departuresfrom solenoidality from time step to time step are very small given that the vector fields
themselves are solenoidal upon every entry into the SLD routine. With this in mind, theviscous component of the entropy production due to SLD is

160

ν

ρ0 T0


∂s
= ∇ · ρ0 u · τ T − ρ0 u · ∇ · τ ,∂t sld

(4.77)

where τ is the SLD tensor that is inherently non-symmetric and where ρ0 and T0 are thebackground density and temperature in an ASH simulation. Writing this in terms of theantisymmetric component of τ , the lack of symmetry can be emphasized
ν

∂s
ρ0 T0
= τ T : ∇ρ0 u − 2ρ0 u · ∇ · τ A ,∂t sld

(4.78)

with τ A the antisymmetric part and the colon denoting the full contraction of τ and thegradient of the momentum. The tensor for magnetic diffusion can be treated the same way,so the resulting total entropy gain from dissipation arising from SLD is
ν

ρ0 T0

1
∂s
1
= τ T : ∇ρ0 u + µT : ∇B − 2ρ0 u · ∇ · τ A −B · ∇ · µA .
∂t sld8π
4π(4.79)

Here µ is the magnetic SLD tensor. For the latter two terms, the tensor values must be
interpolated to the cell center for the products to be valid in the context of the ASH solution.
One further note on the implementation of SLD in ASH is that, because of the converging
longitudinal grid near the poles, the diffusion is filtered in longitude. This is accomplishedthrough two processes. First the full grid-wise SLD is computed; a second SLD is computedwith a continuously increasing cell size as the poles are approached, which incorporates agreater number of points in a given cell. The cell size increases as csc θ, with the requirementthat there be no fewer than eight large cells at high latitudes. The slope is reconstructed
over the larger cell such that the individual points within the cell see the same slope. The
two diffusive fields are weighted with sin2 θ for the first field and by cos2 θ for the second
field. This allows more uniform resolution of the SLD operator over the nonuniform gridingof the sphere. The application of this numerical diffusion technique as implemented in ASH
will be explored in Chapter 5.

Chapter 5
Achieving Regular Cycles in ASH“[Rediscovering the Maunder Minimum] was one more defeat in our long and losing battle
to keep the Sun perfect, or, if not perfect, constant, and if inconstant, regular. Why wethink the Sun should be any of these when other stars are not is more a question for social
than for physical science.” – Jack Eddy
In this Chapter, the global-scale dynamo action achieved in a simulation of a Sun-likestar rotating at thrice the solar rate is assessed. The 3-D MHD Anelastic Spherical Harmonic
code, modified with a viscosity minimization scheme, is employed to capture convection anddynamo processes in such a young Sun. The simulations are carried out in spherical shellsthat encompass 3.8 density scale heights of the solar convection zone. We find that dynamo
action with a high degree of time variation occurs, with many periodic polarity reversals
occurring roughly 6.2 years. The magnetic energy also rises and falls with a regular period,
with two magnetic energy cycles required to complete a polarity cycle. These magnetic energy
cycles arise from a Lorentz-force feedback on the differential rotation, whereas the polarity
reversals are present due to the spatial separation of the equatorial and polar dynamos.
Moreover, an equatorial migration of toroidal field is found, which is linked to the changing
differential rotation and to a nonlinear dynamo wave. This simulation also enters a grandminimum lasting roughly 20 years, after which the dynamo recovers its regular polarity
cycles. This chapter is based on work that is soon to be published in Augustson et al.

162(2013a)1 and will shortly be submitted to Astrophysical Journal Letters in a much morecondensed form. I implemented slope-limited diffusion into ASH, conducted the testing of
those routines, performed the simulation presented here, and wrote the paper. Sacha Brunprovided careful comments to this paper and suggested the dynamo-wave analysis. Mark
Miesch advised looking into the nonlinear and nonlocal effects as possible suspects for theequatorward propagation. Juri helped with revising some of the magniloquent wording andproviding guidance on how to reformulate sections for greater clarity.
5.1

Introduction
The Sun exhibits many time scales from the ten minute lifetimes of granules to multi-

millennial magnetic activity modulations. One of the most prominent of these scales is
the 11-year sunspot cycle, during which the number of magnetically active regions waxes
and wanes. The Sun also possesses longer-term variability of its magnetic activity such as
the 88-year Gleissberg cycle (Gleissberg, 1939) and less frequent phenomenon commonlydescribed as grand extrema (Usoskin, 2013). Other main-sequence stars also exhibit cyclicalmagnetic phenomenon in Ca II, photometric, spectropolarimetric, and X-ray observations(e.g., Baliunas et al., 1996; Hempelmann et al., 1996; Favata et al., 2008; Metcalfe et al., 2010;
Fares et al., 2013; Mathur et al., 2013). These observations include solar-mass stars younger
than the Sun that also possess magnetic activity cycles, yet they rotate more rapidly thanthe Sun as a consequence of the low rate of angular momentum loss in such stars (Barnes,2007). Furthermore, there are hints from both observations and from theory that a star’s
magnetic cycle period is closely linked to its rotation rate (e.g., Saar, 2009; Jouve et al.,2010; Morgenthaler et al., 2011). This may imply that the dynamo regime achieved in our
simulation of a young sun, which rotates three times faster than the Sun and has a nearly
constant magnetic polarity cycle of 6.2 years, can scale up to the solar rotation rate with a1Augustson, K. C., Brun, A. S., Miesch, M. S., & Toomre, J., 2013b, “Cycling Dynamo in a Young Sun:Grand Minima and Equatorward Propagation”, ApJL, to be submitted

163
polarity cycle period closer to the 22 year cycle of the Sun.
In addition to its large range of time scales, observations of the magnetic field at thesolar surface reveal complex, hierarchical structures existing on a vast range of spatial scales.
Despite these chaotic complexities, large-scale organized spatial patterns such as Maunder’sbutterfly diagram, Joy’s law, and Hale’s polarity laws suggest the existence of a structured
large-scale magnetic field within the solar convection zone. On the Sun’s surface active
regions initially emerge at mid-latitudes and appear at increasingly lower latitudes as thecycle progresses, thus exhibiting equatorward migration. As the low-latitude field propagates
toward the equator, the diffuse field that is comprised of small-scale bipolar regions migrates
toward the pole, with the global-scale reversal of the polar magnetic field occurring nearsolar maximum (e.g., Hathaway, 2010; Stenflo & Kosovichev, 2012).
Consequently, the large-scale field must vary with the solar cycle, likely being sustained
through dynamo action deep in the solar interior. It has been suspected for at least 60 years
that the crucial ingredients for the solar dynamo are the shear of the differential rotation andthe helical nature of the small-scale convective flows present in the solar convection zone (e.g.,Parker, 1955; Steenbeck & Krause, 1969; Parker, 1977). Yet even with the advancement tofully nonlinear global-scale 3-D MHD simulations (e.g., Gilman, 1983; Glatzmaier, 1985; Brunet al., 2004; Browning et al., 2006), achieving dynamo action that exhibits the basic propertiesof Sun’s magnetism has been quite challenging. Nonetheless, recent global-scale simulationsof convective dynamos have begun to make substantial contact with some of the propertiesof the solar dynamo through a wide variety of numerical methods (e.g., Brown et al., 2011b;
Racine et al., 2011; Käpylä et al., 2012; Nelson et al., 2013a). It is within this vein ofmodern global-scale modeling that we report on a global-scale 3D MHD convective dynamo
simulation utilizing the ASH code that possesses some features akin to those observed duringsolar cycles.

164
5.2

Computational Methods and Modifications
The simulation of convection and dynamo action presented here uses the ASH code to

evolve the anelastic equations for a conductive calorically perfect gas in a rotating sphericalshell. ASH solves these equations with a spherical harmonic decomposition of the entropy,
magnetic field, pressure, and mass flux in the horizontal directions. Either a Chebyshev
polynomial representation (Clune et al., 1999; Miesch et al., 2000), or a fourth order nonuniform finite difference in the radial direction, resolve radial derivatives. The radial finite
difference derivative scheme is used here. The solenoidality of the mass flux and magnetic
vector fields is maintained through the use of a stream function formalism (Brun et al., 2004).
The Crank-Nicholson implicit time-stepping method advances the linear terms in the MHD
equations including eddy diffusion, pressure gradients, and buoyancy. Nonlinear terms suchas the convective derivatives, Lorentz forces, and dissipative processes are handled with asecond-order Adams-Bashforth time step. The boundary conditions used are impenetrable
on radial boundaries, with a constant entropy gradient there as well. The magnetic boundary
conditions are perfectly conducting at the lower boundary and extrapolated as a potential
field at the upper boundary. Furthermore, we employ the latest version of the ASH code that
overhauls the necessary MPI communication pathways and memory layout (Featherstone,
private communication).
The authors have implemented a slope-limited diffusion (SLD) mechanism similar tothe scheme presented in Rempel et al. (2009) and Fan et al. (2013) into the reformulated
ASH code. The SLD acts locally to achieve a spatially monotonic solution by limiting thelinearly reconstructed slope in each coordinate direction. The scheme minimizes the steepest
gradients through the action of the minmod limiter as described in 4.58, while the rate ofdiffusion is regulated by the local velocity. The rate of diffusion is further reduced througha function φ of the ratio of the cell-edge difference δqi and the cell-center difference ∆qi
in a given direction i for the quantity q. Similar to Rempel et al. (2009), this function

165

Figure 5.1: Snapshots of the horizontal convective patterns arising in the radial velocity
vr from the ASH simulation presented here that uses slope-limited diffusion are shown in
Mollweide projection and are sampled at four depths: (a) 0.95 R⊙, (b) 0.87 R⊙, (c) 0.80 R⊙,(d) 0.75 R⊙. The patterns possess larger-scale columnar convection at low latitudes and thesmaller-scales at higher latitudes, with downflows dark and upflows in lighter tones. The
velocity range is the same for each panel, as indicated with the color bar.

is defined as φi = (minmod[δqi , ∆qi ])8 . This function limits the action of the diffusion toregions with large differences in the reconstructed solutions at cell-edges. Since SLD is anonlinear operator, it is computed in physical space, incurring the cost of smaller time steps
due to the convergence of the grid at the poles. The resulting diffusion fields are projectedback into spectral space and added to the solution through the Adams-Bashforth scheme forthe other nonlinear terms.
The simulation encompasses the bulk of the solar convection zone and the full sphericalgeometry, though the rotation rate of the frame is Ω0 = 3Ω⊙ . The domain stretches fromthe base of the convection zone at 0.715 R⊙ to 0.965 R⊙, omitting the tachocline and the deep
radiative interior as well as the complex physics of the near-surface layers and approximating
their action with an impenetrable boundary. The SLD has been restricted to act only on

166
the velocity field in this simulation. This mimics a lower thermal and magnetic Prandtl
number (Prt , Prm ) than otherwise obtainable through an elliptic diffusion operator. Yet theentropy and magnetic fields remain under the influence of an anisotropic and inhomogeneouseddy diffusion, with both a radially dependent entropy diffusion coefficient κS and magnetic
diffusivity η. The magnitude and form of these two diffusion coefficients are similar to thoseof case D3 from Brown et al. (2010), with κS , η ∝ ρ −1/2 , with ρ the spherically symmetricdensity. However, the stratification in this case has about twice the density contrast acrossthe domain, being 45 rather than 26, and has a resolution of Nr ×Nθ ×Nφ = 200×256×512.
In keeping with the ASH nomenclature for the simulations explored in Brown et al. (2010),
Brown et al. (2011b), and Nelson et al. (2013a), this dynamo solution has been called D3S.
5.3

Dynamics and Cyclical Convective Dynamo Action
Global-scale convective dynamo simulations in rotating spherical shells have recently

achieved the long-sought goal of cyclical magnetic polarity reversals with a multi-decadal
period. Moreover, some of these simulations have illustrated that large-scale dynamo action
is possible within the bulk of the convection zone, even in the absence of a tachocline (Brown
et al., 2010). Global-scale MHD simulations of a more rapidly rotating Sun with the pseudospectral Anelastic Spherical Harmonic (ASH) code have produced polarity reversing dynamo
action that possesses strong toroidal wreaths of magnetism that propagate poleward as acycle progresses (Brown et al., 2011b; Nelson et al., 2013a), much like the earlier work ofGilman (1983) and Glatzmaier (1985). These fields are seated deep within the convection
zone, with the bulk of the magnetic energy near the base of the convection zone. The perfectly
conducting lower boundary condition used here and in those simulations requires the field tobe horizontal there, which tends to promote the formation of longitudinal structure in thepresence of a differential rotation.
A recent simulation with ASH employs a dynamic Smagorinski diffusion scheme, wherefore a greater level of turbulent complexity is achieved than in those simulations employing

167
Newtonian diffusion as seen in the comparison of Figures 3.10c and 3.10f. Those simulations
show that the large-scale toroidal wreaths persist despite the pummeling they endure fromthe more complex and vigorous convection (Nelson et al., 2013a). Not only do the toroidsof field persevere, but portions of them can be so amplified that the combination of upward
advection and magnetic buoyancy create giant arcs of magnetic field that balloon upwards
and span hundreds of mega-meters (Nelson et al., 2013b). This lends credence to the classical picture of a Babcock-Leighton or Parker interface dynamo (Leighton, 1969; Parker,1993), with semi-buoyant flux structures that rise toward the solar surface, leading to active
regions and helicity ejection. There is the caveat that the magnetic fields in the simulation
are instead built in the convection zone.
Contemporaneously, implicit large-eddy simulations (ILES) have paved the road toward more orderly long-term cycles in a setting that mimics the solar interior. Indeed,
simulations utilizing the Eulerian-Lagrangian (EULAG) code produce regular polarity cycles occurring roughly every 80 years in the presence of a tachocline and with the bulk of themagnetic field existing at higher latitudes (Ghizaru et al., 2010). This simulation showedradial propagation of structures but little latitudinal variation during a cycle. More recent
simulations of a Sun-like star rotating at 3 Ω⊙ also produce low-latitude poleward propagating
solutions (Charbonneau, 2013). In both cases, such dynamo action is accomplished throughtwo mechanisms: first by reducing the enthalpy transport of the largest scales through a deliberately simple sub-grid-scale (SGS) model that apriori diminishes thermal perturbations
over a roughly 1.5 year time scale, serving to moderate the global Rossby number, and second through the ILES formulation of the semi-implicit EULAG MHD code that maximizes
the complexity of the flows and magnetic fields for a given Eulerian grid resolution.
Inspired by these recent ASH and EULAG results, we have attempted to make contact
with both numerical methods through the incorporation of SLD into ASH with the express
goal of achieving a low effective Prt and Prm dynamo. Thus an attempt is made to better
mimic the low Prandtl number solar setting, while keeping the eddy-diffusive approximation

Figure 5.2: The properties of the differential rotation in D3S. (a) Cuts at constant latitude through the time-averaged, azimuthallyaveraged, and normalized angular velocity hhΩii/Ω0 (double brackets indicating dual averages and Ω0 the frame rotation rateof 3Ω⊙ ), showing a fast equator and slow poles. (b) hhΩii/Ω0 shown in the meridional plane, illustrating the nearly cylindricalrotation profile. (c) A time-latitude diagram of azimuthally-averaged h∆Ωi = hΩi − hhΩii in cylindrical projection, elucidating
the propagation of equatorial and polar branches of a torsional oscillation arising from strong Lorentz-force feedback. The color
indicates enhanced differential rotation in red and periods of slower rotation in blue, with variations of up to ±10% of the bulk
rotation rate.

168

169
for entropy mixing and treating the reconnection of small-scale magnetic field as diffusive.
This effort minimizes the effects of viscosity, and so extends the inertial range as far as possible for a given resolution. Thus SLD permits more scales to be captured before entering
the dissipation range, allowing more scale separation between the larger magnetic and andsmaller kinetic scales participating in the low Prm dynamo (Ponty et al., 2005; Schekochihin
et al., 2007; Brandenburg, 2009). Subsequently, the kinetic helicity is also greater at small
scales than otherwise would be achieved, which has been shown to have a large influenceon the dynamo efficiency (Malyshkin & Boldyrev, 2010). Indeed, with this newly implemented diffusion minimization scheme, we have happened upon a solution that possesses
four fundamental features of the solar dynamo: a regular magnetic energy cycle period, andan orderly magnetic polarity cycle with a period of τC = 6.2 years, equatorward propagation of magnetic features, and poleward migration of oppositely signed flux. Furthermore
this equilibrium is punctuated by an interval of relative quiescence, after which the cycle isrecovered.
Some aspects of the nature of the convective dynamics achieved within D3S are visible
in Figure 5.1. In the upper convection zone, the downflows are roughly twice as fast as the
upflows, with the rms upflow being about 100 m s−1 and the downflow roughly −200 m s−1 .
This decreases with depth, with the asymmetry between the up and downflows nearly vanishing and with the typical velocity being about 10 m s−1 . The convection is very columnarreflecting the strength of the Coriolis force acting upon them, which tends to tilt the convective structures toward the rotation axis. In particular, the convective structures are elongatedand north-south aligned flows at low latitudes and smaller scales at higher latitudes. In aggregate, the spatial structure and flow directions along these cells produce strong Reynolds
stresses that act to accelerate the equator and slow the poles. The Reynolds stresses arisingfrom these turbulent structures are the dominant mechanism that maintains the differential
rotation, as the viscous stresses are quite small in this simulation. However, the Maxwell
stresses are certainly important during the latter phases of a magnetic energy cycle where

170

Figure 5.3: Volume-averaged energy densities and latitudinal magnetic field in Case D3S.
(a) Time traces of volume-averaged energy densities associated with the differential rotation(DRE), turbulent convective kinetic energy (CKE), meridional circulation (MCE), toroidal
magnetic field (TME), fluctuating magnetic field (FME), and the poloidal magnetic field(PME). The magnetic energy cycle is evident. (b) The horizonally-averaged and normalized
latitudinal magnetic field shown at mid-convection zone, showing the magnetic polarity cycle.

they act to quench both the convection and the differential rotation, which is apparent in
the volume averaged energy traces of Figure 5.3. Indeed, a substantial differential rotation
is established and maintained here, as can be seen in Figure 5.2. It is also apparent at
higher latitudes that this simulation has a negative radial gradient in the angular velocity Ω,akin to the near-surface shear layer that is well-known from helioseismology. The reason for
its formation is likely a numerical boundary layer effect rather than capturing the physical171
processes present in the solar near-surface shear layer. The collective action of these cells
also leads to a significant latitudinal enthalpy flux that heats the poles and sustains thethermal wind balance between the differential rotation and the latitudinal entropy gradient.
The thermal wind in concert with the Reynolds stresses serves to rebuild and maintain the
differential rotation during each cycle.
The variable nature of the convective patterns over a cycle is an important piece of thestory of the dynamo of D3S. Indeed, the magnetic fields disrupt the alignment and correlations of these cells through Lorentz forces. Particularly, as the field gathers strength during acycle, the strong azimuthally-connected toroidal fields tend to create a thermal shadow thatweakens the thermal driving of the equatorial cells. Thus their angular momentum transport
is also diminished, which explains why the differential rotation seen in Figure 5.2(b) cannot
be fully maintained during the cycle. This is captured in the modulation of the kinetic energy contained in the fluctuating velocity field, which here varies by about 50% throughout
the cycle as is visible in Figure 5.3(a). Such a mechanism is in keeping with the impacts ofstrong toroidal fields in the convection zone suggested by Parker (1987). Moreover, strong
nonlinear Lorentz force feedbacks have been seen in other convective dynamo simulations
as well (Brown et al., 2011b), and they have been theoretically realized for quite some time
(Malkus & Proctor, 1975).
Figure 5.4 illustrates the morphology of the toroidal fields in space and time. The
presence of large-scale and azimuthally-connected structures is evident in Figures 5.4(a, d).
Such toroidal structures have been dubbed wreaths (Brown et al., 2010). In D3S, there are
two counter-polarized, lower-latitude wreaths that form near the point where the tangent
cylinder intersects the spherical surface at a given radius. This point is also where thepeak in the latitudinal gradient of the differential rotation exists for much of a magnetic
energy cycle. There are also polar caps of magnetism of the opposite sense of those atlower latitudes. These caps serve to moderate the polar differential rotation, which would
otherwise tend to accelerate and hence establish fast polar vortices. The average structureFigure 5.4: Nature of the toroidal magnetic field Bφ . (a) Snapshot of the horizontal structure of Bφ at 0.95 R⊙ shown in Mollweide
projection, at the time corresponding to the vertical dashed line in (c). This illustrates the azimuthal connectivity of the magnetic
wreaths, with the polarity of the field such that red (blue) tones indicate positive (negative) toroidal field. (b) Azimuthallyaveraged hBφ i also time-averaged over a single energy cycle, depicting the structure of the toroidal field in the meridional plane.
(c) Time-latitude diagram of hBφ i at 0.95 R⊙ in cylindrical projection, exhibiting the equatorward migration of the wreaths fromthe tangent cylinder at roughly 38◦ and the poleward propagation of the higher latitude field. The color is as in (a). (d) A
rendering of magnetic field lines in the domain colored by the magnitude and sign of Bφ , with strong positively oriented field inred, and the strong oppositely directed field in blue.

172

173
of the wreaths and caps is apparent in Figure 5.4(b), which is averaged over a single energy
cycle or 3.1 years. The wreaths appear rooted at the base of the convection zone, whereas
the caps have the bulk of their energy in the lower convection zone above its base. Thisis somewhat deceptive as the wreaths are initially generated higher in the convection zone,
while the wreath generation mechanism (primarily the Ω-effect) migrates equatorward andtoward the base of the convection zone over the course of the cycle. The wreaths obtain
their greatest amplitude at the base of the convection zone and thus appear seated there.
5.4

Grand Minima and Long-Term Modulation
As with some other dynamo simulations (e.g., Brown et al., 2011b; Augustson et al.,2013b), there is also a degree of long-term modulation of the magnetic cycles in case D3S.
Figure 5.5 shows an interval of about 20 years where the polarity cycles are lost, though themagnetic energy cycles resulting from the nonlinear interaction of the differential rotationand the Lorentz force remain. During this interval, the magnetic energy in the domain is
about 25% lower, whereas the energy in the volume encompassed by the lower-latitudes
is decreased by 60%. However, both the spatial and temporal coherency of the cycles arerecovered after this interval and persist for the last 40 years of the 100 year-long simulation.
Prior to entering this quiescent period, there was an atypical cycle with only the northernhemisphere exhibiting equatorward propagation. This cycle also exhibits a prominent lossof the equatorial anti-symmetry in its magnetic polarity. The subsequent four energy cycles
do not reverse their polarity, which is especially evident in the polar regions, whereas thelower latitudes do seem to attempt such reversals.
Similar quasi-periodic dynamo action has been realized in quite a few instances, though
only in mean-field models and in reduced dynamo models not in fully nonlinear 3D MHD
simulations. Some of the most well-studied quasi-periodic systems utilize reduced dynamo
models that couple only a few modes of the overall system, such as those models of Weisset al. (1984). Weiss uses a simple αΩ dynamo model. This is accomplished through systems

174

Figure 5.5: An interval of magnetic quiescence. (a) Time-latitude diagram of hBφ i at 0.95 R⊙
in cylindrical projection, picturing the loss and reappearance of cyclical polarity reversals
as well as the lower amplitude of the wreaths. Strong positive toroidal field is shown as
red, negative in blue. (b) Normalized magnetic dipole moment (blue) and quadrupole moment normalized by the dipole absolute maximum (red). There is a significant rise in thequadrupole moment near reversals, suggesting a brief coupling between families of dipolarand quadrupolar dynamo modes.

of coupled nonlinear ordinary differential equations, where the number of included modes isvaried. This results in models that range from fourth to sixth-order in their time derivative.
These models admit simple nonlinear oscillatory solutions. The highest-order model with thelargest number of dynamic modes exhibits transitions from periodic oscillations to chaotic
behavior with longer-term modulation akin to the grand minima in the solar activity cycle.
Later work explored alternative dynamo feedback mechanisms in these low order sys-

175
tems (Tobias, 1997). Tobias explored the impacts that modulations in the toroidal magnetic
field energy and that Lorentz-force feedback on the large-scale flows have on the characterand evolution of nonlinear αΩ dynamo solutions. The former mechanism is shown to affectthe parity of the dynamo, whereas the latter can reduce the efficiency of both the α andΩ effects in the limit of low magnetic Prandtl numbers. With an appropriate choice of parameters, some solutions exhibit a long-term evolution that produces grand extrema, while
retaining the basic cycle periods, as the Sun is observed to do. In particular, it is shown thatthe magnetic field can be substantially modulated and effectively remain dipolar. During
grand minima the magnetic field is weak and is no longer a simple dipole. Such behavior isalso seen in the simulation here, where its grand minimum is associated with a much stronger
quadrupole moment, as is evident in Figure 5.5b. It also worth noting that the quadrupole
component tends to briefly spike during the polarity reversal, which may be a signal of thereversal mechanism. Indeed, recent work has explored solar observations over the past threesolar cycles to assess the magnetic energy contained in a large range of spherical harmonicmodes (DeRosa et al., 2012). It is seen there that the coupling between the dipolar andquadrupolar families of modes correlates well with the large-scale polarity reversals of the
Sun.
5.5

Generating Global-Scale Magnetism
The toroidal field shown in Figure 5.4 is initially generated and then maintained bysimilar processes. During the growth phase of the magnetic field, the shear of the differential
rotation acts to fold and wind the initial poloidal field into toroidal structures. During thiskinematic phase, the mean shear and meridional flows are largely unaffected and can beconsidered stationary relative to the time scales of the growing field. However, once themagnetic fields are strong enough they begin to quench the convective flows that cross them.
The magnetic field strength becomes saturated as the back reaction of the Lorentz forces
increases the alignment of the velocity field and the magnetic field, which reduces both its

176
generation and at times leads to its partial destruction.
This configuration of fields establishes a new balance between the various mechanismsof angular momentum transport needed to sustain the differential rotation when compared
to a hydrodynamic simulation. Here magnetic fields both transport angular momentum aswell as block formerly open channels of transport. For instance, the strong toroidal field
serves to reduce the latitudinal transport of angular momentum by Reynolds stresses, which
modifies both the differential rotation and the meridional circulations within the simulation
relative to what would be achieved in a hydrodynamic simulation.
Since the dynamo running within D3S waxes and wanes as time marches forward
through many polarity reversals, the balance between magnetic field generation mechanisms
is not instantaneous nor is it the same at any given time within the cycle. When averaged
in time over the full span of the simulation, an analysis of the terms contributing to thegeneration and destruction of magnetic energy can illustrate those mechanisms that have thegreatest overall influence on the dynamo. The evolving magnetic fields arise through many
competing processes that both produce and destroy magnetic field through the shearing andadvection of field, through compressive motions, and dissipation through resistive processes.
These production and dissipation terms are most easily understood through an evolution
equation for the magnetic energy contained in the mean magnetic fields. This equation canbe broken into its poloidal and toroidal components PME and TME as
T

T

FS
FA
TMS
TMA
z
z
zD
}|
{ z
}|
E
E{
D
}|
{
}|
{
hBφ i
∂TME
′
′
′
′
=
− (hvi · ∇) hBi|φ − (v · ∇) B
(hBi · ∇) hvi|φ + (B · ∇) v
∂t
8π
φ
φ

TCC

}|
{
TRD
}|
{
∂ ln ρ z− ∇ × (η∇ × hBi)|φ ,+ hBφ vr i
∂r
z

P

(5.1)
P

FS
FA
PMS
PMA
z
zD
z
}|
{ z
}|
E
E{
D
}|
{
}|
{
hBP i
∂PME
′
′
′
′
=
· (hBi · ∇) hvi + (B · ∇) v − (hvi · ∇) hBi − (v · ∇) B
∂t
8π

PCC

}|
{
PRD
}|
{
∂ ln ρ z− ∇ × (η∇ × hBi) ,+ hBP vr i
∂r
z

(5.2)

Figure 5.6: Generation of mean toroidal magnetic energy (TME ) in D3S. The view is from ±60◦ to emphasize the equatorial
regions. Here only the dominant production terms are shown, namely (a) mean shear (TMS ), (b) fluctuating advection (TFA ),with a cylinder tangent to the bottom of the domain (T. C.) is indicated (dashed line), (c) compressive correlations (TCC ), (d)the sum TMS + TFS + TMA + TFA + TCC , which largely act in concert to balance (e) the resistive diffusion of field (TRD ). All
panels have identical scaling. The mean generation terms shown here contribute to the TME when positive (red), and destroy it
if they are negative (blue). (f) The relative contribution of each term, with terms in red toned regions adding to the volume andtime-averaged toroidal energy generation and those in blue dissipating energy.

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178
with the hi denoting an average in longitude, v′ = v − hvi the fluctuating velocity, B′ =B − hBi the fluctuating magnetic field, and hvi and hBi the axisymmetric velocity andmagnetic field respectively. A detailed derivation of the mean-field production terms in
spherical coordinates is provided in Appendix A of Brown et al. (2010). Here T indicatesa toroidal magnetic energy generation term and P a poloidal energy generation term. The
subscript of these monikers denote production in turn by mean shear (MS), fluctuating shear(FS), mean advection (MA), fluctuating advection (FA), compressional correlations (CC),and resistive diffusion (RD).
Over a long time average, the time variability of the production of TME is removed,
leaving a balance between terms that produce and terms that destroy field. Figure 5.6
presents such a time average of these mean generation terms, involving the entire evolutionof case D3S over an 80 year interval (3300 rotations). In this statistically steady state,
the maintenance of the toroidal wreaths of magnetic field is largely governed by a balance
between the production of field by both the mean shear and the dissipation of field throughresistive processes. By comparing Figures 5.6(a) and 5.6(d), it is clear that the mean shear
TMS of Equation (5.1) is primarily responsible for maintaining the strength of the wreaths.
The production due to the compressibility of the downflows, which can be seen from theform of TCC , plays a weak but supporting role. In keeping with the dominance of the Ωeffect (TMS above), the toroidal magnetic energy generation is greatest at latitudes where
the latitudinal gradients in the differential rotation are at their largest. This maximum
occurs at latitudes outside the tangent cylinder in a roughly 15◦ swath, though there is also
generation of toroidal field at higher latitudes just inside the tangent cylinder. The resistive
dissipation and fluctuating advection of field act in tandem to destroy toroidal field. Given
the spatial distribution of TFA in Figure 5.6b, the action of the fluctuating advection could beconsidered as part of an α-effect with the small-scale flows tearing at the large-scale wreaths,
transferring energy from the toroidal to the poloidal magnetic fields.
The relative contributions of each term have been measured by integrating it over time

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and space hTi i =R

Ti dV , where i is any of the terms in Equation (5.1). This allows the

net action of a given term to be assessed, indicating whether it on average is productive
or destructive of toroidal magnetic energy. As indicated in Figure 5.6(f), the bulk of the
production is accomplished by TMS , being about 92% of the total. The remaining 8% of
the mean production is due mostly to the compressive motions of the downflows (TCC ).
In contrast, the averaged TRD , TFA , TMA , and TFS processes act in concert to dissipate
magnetic energy, with TRD responsible for 68% of the dissipation and TFA 24%.
The mechanisms generating the poloidal fields and its associated magnetic energy PME
are given in Equation 5.2. In Figure 5.7, the temporal and longitudinal averages of the generation terms with the largest average contributions are shown. The average is again carriedout over 80 years. Figure 5.7(g) shows that in the spatially integrated sense and unlike in thetoroidal energy generation, the fluctuating shear PFS is the dominant contributor to the production of poloidal field, generating on average 79% of the poloidal energy, while the resistive
diffusion PRD again is the largest dissipator, destroying 75% of it. However, other processes
contribute to the generation of poloidal field as well. The production of poloidal field arises
through the joint action of the fluctuating shear PFS and mean advection PMA , whereas thedissipation of poloidal magnetic energy arises from the combination of resistive diffusion PRD
and mean shear PMS . The bulk of the production occurs near the lower boundary and inside
the tangent cylinder, being associated with the fluctuating velocities and magnetic fields ofthe more isotropic convective cells at higher latitudes. The generation of poloidal field has
two primary balances: the first is between production through the fluctuating shear andresistive decay; the second is between the mean shear and the mean advection. The overallbalances of magnetic energy generation are subtle given the spatial separation between the
primary regions of toroidal and poloidal field generation, with poloidal field being built near
the base of the convection zone and at higher latitudes and the toroidal field being produced
at low latitudes in the upper convection zone.

Figure 5.7: Generation of mean poloidal magnetic energy (PME ) in D3S. The dominant production/destruction terms are shown,namely (a) fluctuating shear (PFS ), (b) mean shear (PMS ), (c) fluctuating advection (PFA ), (d) mean advection (PMA ) and (e) thesum PMS + PFS + PMA + PFA + PCC , with (f) the resistive diffusion of field (PRD ). The mean generation terms shown contributeto the PME when positive (red), and destroy it if negative (blue). (g) The relative contribution of each term, with those in red
adding to the volume- and time-averaged poloidal energy generation and those in blue dissipating energy.

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5.6

What Might Be Setting Cycle Periods?
There are a large set of possible and often interlinked time scales that could be relevantto the processes setting the pace of the cyclical dynamo established in D3S. For instance,
there are resistive time scales that depend upon the length scale chosen. One such time scaleis the resistive decay of the poloidal field at the upper boundary as it propagates from thetangent cylinder to the equator, which would imply that the length scale is ℓ = r2 ∆θ andso τη = ℓ2 /η2 ≈ 6.7 years and is close to the polarity cycle period, where the subscript two
denotes the value of a quantity at the outer boundary of the simulation. However, this is
likely not dynamically dominant as the the polarity reversal occurs in half that time. The
same is true of the diffusion time across the convection zone, being 4.6 years. Since the cycle
is likely not resistively controlled it must be the interplay of dynamical processes. Anothermechanism to consider is the cycle time related to flux transport by the meridional flow, withthe transit time of a magnetic element along its circuits possibly being relevant. In D3S, themean meridional flow is anti-symmetric about the equator and has two cells, with a polar
branch and a lower latitude cell that are split by the tangent cylinder. The circulation timeof the polar branch is about 0.7 years, whereas that of the equatorial cell is about a year.
So it is also unlikely that the meridional flow is setting the cycle period.
The dynamical coupling of azimuthally-averaged magnetic fields hBi and the mean
angular velocity hΩi (Figure 5.2(b)) plays a crucial role in regulating the cycle, though it
alone cannot be the sole actor as is well known from Cowling’s anti-dynamo theorem. The
significant anti-correlation of hBφ i and angular velocity variations h∆Ωi during reversals
becomes apparent when comparing Figures 5.4(c) and 5.2(c), revealing the strong nonlinear
coupling of the magnetic field and the large-scale flows. The dynamics that couples these two
fields is the toroidal field generation through the mean shear (S = λhBP i · ∇hΩi, with hBP i
the mean poloidal field) and the mean azimuthal Lorentz-force (Lφ = φ̂ · hJi × hBi), whichacts to decrease hΩi. The auto-correlation of each of these components of the MHD system

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Figure 5.8: (a) Volume-averaged temporal auto-correlation of toroidal magnetic energy generation by mean shear (S = λhBP i · ∇hΩi, blue curve) and the same for the mean Lorentz
force impacting the mean angular velocity (Lφ , red curve) plotted against temporal lags ∆tnormalized by the polarity cycle period τC = 6.2 years. Confidence intervals are shownas shaded gray regions, with the 67% interval in darker gray and 95% in lighter gray. (b)Cross-correlation of the mean poloidal energy production P = BP ·∇×Eφ′ through the fluctuating EMF and the toroidal magnetic energy production due to the mean shear T = hBφ iS,
showing the nonlinear dynamo wave character of the solution.

reveals that Lφ varies with a period corresponding to the magnetic energy cycle, whereas S
varies on the polarity cycle period (Figure 5.8). It also shows the high degree of temporal selfsimilarity between cycles, with the auto-correlation of both quantities remaining significant
with 95% confidence for a single polarity cycle and with 67% confidence for three such cycles.
Appealing to Figure 5.4(c), it is evident that B exhibits a high degree of spatial
and temporal self-similarity, though with reversing polarity. Thus the period apparent
in the auto-correlation for Lφ might be expected. Furthermore, if we simply let hBi ≈B0 (r, θ) exp(iωC t), the Lorentz forces could be characterized very roughly as Lφ ∝ Lφ ,0 exp(iωL t) ∼

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B0 · B0 /ℓ exp(2iωC t), with cycle frequency ωC = 2π/τC and some length scale ℓ. Hence, themagnetic energy or Lorentz cycle frequency ωL = 2π/τL implies that 2τL = τC . What is
potentially more curious is that S varies on the cycle period. While Figure 5.2(c) mightsuggest a reversal in the solar-like character of the differential rotation, this in fact does
not occur. Rather, the shear is significantly weakened but maintains the positive latitudinal
gradient that sustains the toroidal magnetic field, which renders the sign of ∇Ω independentof time. Therefore, the polarity reversals in hBP i require that S vary with the polarity cycle
period τC .
Therefore, the mechanism that sets the time scale for the reversal of the poloidal fieldis responsible for the reversal period. This mechanism is more difficult to assess as it requires
information about the collective action of the turbulent convection upon existing magnetic
structures. It is inherently nonlocal in space as magnetic energy from the local and small-scaleaction of helical motions upon a large-scale toroidal magnetic structure leads to a large-scalepoloidal field, and thus is also nonlocal in time as the large-scale structures evolve on longer
time scale than the convection. However, several components of the turbulent production ofpoloidal magnetic field can be illustrated. In particular, the axisymmetric collective action of
the turbulent velocity on the magnetic field that produces poloidal field Eϕ′ and its evolution
can be seen in Figures 5.9 and 5.10. In these figures and those that follow, the average cycle isshown, which has been formed by identifying the common structures in each cycle, obtaining
the times of the beginning and end of each cycle as defined through these structures, andthen stretching each cycle to be the same length in time and co-adding them. This is greatly
aided by the extreme regularity of the cycle, which varies by only a few weeks from cycle tocycle.
In Figures 5.7, 5.9, and 5.10, two primary regions of production of poloidal magnetic
field are visible: one at low latitudes and another at high latitudes. The low latitude regions of poloidal energy production are associated with convective cells acting upon theequatorially migrating toroidal wreaths. While the mechanism is somewhat similar for the

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Figure 5.9: A comparison of the temporal evolution of the poloidal magnetic energy production through the fluctuating EMF as P = BP · ∇ × Eφ′ and toroidal magnetic energy
production by mean shear with T = Bφ PM S in D3S. All cycles have been stretched to thethe same time span and then averaged to allow the progression of an average cycle to beascertained. (a) Poloidal magnetic energy generation P in the upper convection zone shownwith latitude and time in cylindrical projection, illustrating the strong field generation above
about ±30◦ and the weak generation that accompanies the equatorward propagation of thelarge toroidal wreaths. (b) P plotted with depth r/ R⊙ at a latitude of 25◦ . The taperingwith depth arises from the strong radial dependence of BP , with a downward direction of
propagation. (c) P averaged over the entire cycle rendered in the meridional plane. (d)Toroidal magnetic energy generation T exhibited over the average cycle and with latitude
in cylindrical projection, showing the equatorward migration of field generation. (e) T in
depth and time at a latitude of 25◦ , with two zones of migration evident. (f) T averaged
over the entire cycle, where the equatorial and polar regions represent spatially separated
zones of generation.

high-latitude poloidal field, it primarily originates in the helical action of convection on thehigh-latitude wreaths, which themselves are built from the differential rotation brought tothe poles through torsional oscillations. In turn, these oscillations arise from the transportof angular momentum to the poles through Maxwell stresses and from the increased vigorof the convection there at the beginning of a magnetic energy cycle leading to Reynolds

185
stresses that spin up the pole. The spatial and temporal separation of poloidal and toroidal
field generation is particularly evident when comparing Figures 5.9a and 5.9d. The greatest
generation of poloidal field is concentrated in the polar regions and near the tangent cylinder,
with the low latitudes playing much less of a role. In contrast, the generation of toroidal
magnetic field is greatest at low latitudes throughout the bulk of the cycle. In Figures 5.9b
and 5.9e, the reversal in the sign of the kinetic helicity of the convection is largely responsible
for the thin region of destruction of magnetic field in the lower convection zone, where there
is an anti-correlation between the fluctuating velocity field and the fluctuating magnetic
field.
At the beginning of the cycle, when the magnetic fields are weakest (Figure 5.10a)
and after the differential rotation has recovered (Figure 5.2c), toroidal field begins to growat roughly ±30◦ . The turbulent action of the convection on this newly generated wreath
sustains the EMF on the polar edge of the wreath, which is near the tangent cylinder (Figure5.9a). In combination with the polar EMF that emerges from the action of convection on thetoroidal fields there, this serves the purpose of sustaining the poloidal field that in turn allows
the wreaths to be maintained through the action of the differential rotation. However, oncethe polar differential rotation has been quenched the EMF that had been quite prominent at
the tangent cylinder vanishes. The EMF that remains moves equatorward, advancing withthe migration of the wreaths; yet it is still on the polar edge of the wreath. At this point,
the strong toroidal magnetic field at low latitudes has begun to significantly feedback on theequatorial differential rotation, modifying the structure of the convection and diminishingthe differential rotation. Furthermore, in D3S, as a cycle progresses the centroid for thegreatest dynamo action propagates both equatorward as evident in the time-latitude diagram
Figures 5.9e and downward in radius as is suggested in the time-radius diagrams of Figure
5.9f. Hence, the equatorial migration begun at the surface makes its way deeper into thedomain as the cycle progresses.
The wreaths eventually lose their azimuthal coherence through the lack of sufficient186

Figure 5.10: A comparison of the time and latitudinal evolution of the poloidal field production by fluctuating EMF Eφ′ and the toroidal field production by mean-shear PMS overan average cycle of D3S. (a) Azimuthally-averaged toroidal field Bφ near the upper boundary at 0.95 R⊙, with the contours of PMS overplotted with negative values as solid contours
and dash-dotted contours as positive values of PMS. (b) Toroidal magnetic field production
through mean shear PMS at 0.95 R⊙. The turbulent EMF Eφ′ is shown (c) at 0.95 R⊙ and(d) 0.72 R⊙. The contours are a visual aid to help assess the dynamo wave character of thesolution.

187
differential rotation to sustain them, the destructive influence of the convection, and also due
to cross-equatorial flux cancellation. Together these processes lead to a rapid disseminationof the remaining flux by the convection. This is evident in Figure 5.4(c), where at the end ofeach cycle the wreaths converge on the equator and their resulting destruction leads to thepoleward advection of field. This advected field is of the opposite sense of the previous cycle’s
polar cap and, being of greater amplitude compared to the remaining polar field, establishes
the sense of the subsequent cycle’s polar field. The sense of the subsequent poloidal field
seems however to be determined by the weak EMF generated at the equator, as was also
seen in Nelson et al. (2013a). This EMF begins to be generated as the toroidal fields start
their equatorward migration and is sustained throughout the rest of the cycle (Figures 5.10c,d), leading to enough field generation there for these freshly minted and oppositely signed
poloidal fields to overcome the dissipative collapse that the fields elsewhere in the domain
succumb to. This EMF originates in the action of convection on the equatorward edge of
the wreaths, which leads to a weak but influential cross-equatorial poloidal linking of thewreaths as is marginally visible in Figure 5.4.
All of these processes suggest that the reversal time scale is set by three time scales:
the rate of production of differential rotation kinetic energy, the effective rate of conversion ofdifferential rotation kinetic energy into toroidal magnetic energy, and the rate of conversionof toroidal magnetic energy into poloidal magnetic energy that in turn sustains the toroidal
field. The latter rate is related to the dynamo efficiency of the convection, which Figure
5.6f suggests to be about 24% and is close to value found in Augustson et al. (2013b). Thislevel of efficiency is what one might expect given the factor of 5 between the rate of toroidal
energy generation and poloidal energy generation seen in Figures 5.9c, f. This is more orless constant throughout the cycle, though it requires toroidal field to be present. However,
the other two processes are more difficult to pin down given that they vary throughout thecycle due to the nonlinear feedback of the Lorentz force.

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Figure 5.11: A mean-field interpretation of the D3S dynamo over the average cycle. The
isotropic α-effect arising from the kinetic helicity is shown (a) with latitude and time, (b)with radius and time, and (c) averaged over the cycle. The radial velocity Sr and latitudinalvelocity Sθ of the dynamo wave propagation within the context of the Parker-Yoshimuramechanism are exhibited in (d-i). (d) A radial cut at 0.90 R⊙ through the radial propagation Sr of the dynamo wave, outward propagation is in red and downward in blue. (e) A
latitudinal cut through Sr , colors as in (d) and the dotted line indicates the depth of theradial cut in (d). (f) Sr averaged over the cycle and shown in the meridional plane. Figures(g-i) are as in (d-f), but for Sθ , with the latitudinal propagation is equatorward at latitudes
above 30◦ and largely poleward at lower latitudes (blue in the northern hemisphere and redin the south).

5.7

Equatorward Dynamo Wave Propagation
As with ASH and EULAG, simulations in spherical segments that employ the Pencil

code have also obtained regular cyclical magnetic behavior. Some of these polarity reversing

189
solutions exhibited equatorward propagating magnetic features (Käpylä et al., 2012, 2013),
magnetic flux ejection (Warnecke et al., 2012), and 33-year magnetic polarity cycles (Warnecke, 2013). Currently, however, the mechanism for the equatorward propagation of themagnetic structures in those simulations remains unclear. Perhaps the mechanism is similar
to that seen here.
The equatorward propagation of magnetic features observed in this case, as in Figures5.4(c) and 5.5(a), arises through two mechanisms. The first process is the nonlinear feedbackof the Lorentz force that acts to quench the differential rotation, disrupting the convective
patterns and the shear-sustaining Reynolds stresses they possess. Since the latitudinal shear
serves to build and maintain the magnetic wreaths, the latitude of peak magnetic energy
corresponds to that of the greatest shear. So the region with the largest magnetic potential
energy in the form of latitudinal shear moves progressively closer to the equator as theLorentz forces of the wreaths locally weaken the shear. Such a mechanism explains theperiodic modifications of the differential rotation seen in Figure 5.2(c). However, it does
not explain how this propagation is initiated and sustained, as one might instead expect anequilibrium to be established with the magnetic energy generation balancing the productionof shear and which is further moderated by cross-equatorially magnetic flux cancellation as
the distance between the wreaths declines as seen in earlier simulations (Brown et al., 2011b;
Nelson et al., 2013a).
There are two possibilities for the second mechanism that promotes the equatorward
propagation of toroidal magnetic field structures. If we may consider the dynamo action inthis case as a dynamo wave, the velocity of the dynamo wave propagation is sensitive to thegradients in the angular velocity and the kinetic helicity in the context of an αΩ dynamo (e.g.,Parker, 1955; Yoshimura, 1975). This dynamo-wave velocity is given as S = −λαφ̂ × ∇Ω,
where λ = r sin θ. The basic α-effect has α = τo hu · ωi/3, and so depends on the convective
overturning time τo and the kinetic helicity. These quantities are shown in Figure 5.11
with time-latitude, time-radius, and time-averaged plots over the average cycle. This simple

190

Figure 5.12: A comparison of the latitudinal regions of generation by the two primary toroidal
and poloidal field production mechanisms at intervals of a quarter of the polarity cycle period
in the average cycle. This serves to show the relative displacement of the two mechanisms,
which may assist the wreaths in their equatorward propagation. The production of toroidal
field is shown in blue, and the fluctuating azimuthal EMF is in red. The time series begins
at t0 and increments by a quarter of the polarity cycle period as one proceeds to the right.
This is repeated for four depths: (a) 0.95 R⊙, (b) 0.90 R⊙, (c) 0.81 R⊙, (d) 75 R⊙.

analysis indicates that near and poleward of the edge of the low-latitude wreaths the sign of
the Parker-Yoshimura mechanism is correct to push the dynamo wave toward the equator,
but the effect is marginal elsewhere. The second possibility is that the spatial and temporal
offsets between the fluctuating EMF and the mean-shear production of toroidal field leads
to a nonlinear inducement to move equatorward. This mechanism relies on the concurrent

191
movement of the turbulent production of the poloidal field that continues to destroy gradientsin angular velocity through the production of toroidal magnetic through the action of thedifferential rotation on the renewed low-latitude poloidal field as is visible in Figures 5.9 and
5.12.
5.8

Conclusions
The simulation presented here is the first to self-consistently exhibit four prominent

aspects of solar magnetism: regular magnetic energy cycles during which the magnetic polarity reverses, akin to the sunspot cycle; magnetic polarity cycles with a period of 6.2 years,
where the orientation of the dipole moment returns to that of the initial condition, theequatorward migration of toroidal field structures during these cycles, and an interval ofmagnetic quiescence after which the previous polarity cycle is recovered. Furthermore, thissimulation may capture some aspects of the influence of a layer of near-surface shear, with aweak negative gradient in hΩi within the upper 10% of the computation domain (3% by solar
radius). The magnetic energy cycles with the time scale τC /2 arise through the nonlinear
interaction of the differential rotation and the Lorentz force. We find that the nonlinear
feedback of the Lorentz force on the differential rotation disestablishes its primus inter paresrole in the generation of toroidal magnetic energy. The magnetic fields further quench thedifferential rotation by impacting the convective angular momentum transport during thereversal. Furthermore, despite the nonlinearity of the case, there is an eligible influence ofa dynamo wave in the fluctuating production of poloidal magnetic field linked to the shearproduced toroidal field. The possible mechanisms producing the equatorward propagationof the toroidal fields have been elucidated, with the nonlinear dynamo wave character of thesolution and the nonlocal interaction of the turbulent EMF and the mean-shear productionof toroidal field all playing a role. However, the dominant mechanism behind the equatorward propagation is the nonlinear feedback of the Lorentz force on the differential rotation.
This simulation has also exhibited grand minima, similar to a Maunder minimum. Indeed,

192
there is an interval covering 20% of the cycles during which the polarity does not reverse
and the magnetic energy is substantially reduced. Hence, despite rotating three times fasterthan the Sun and parameterizing large portions of its vast range of spatio-temporal scales,
some of the features of the dynamo that may be active within the Sun’s interior have been
seen in this global-scale ASH simulation.

Chapter 6
Global-scale Hydrodynamic Simulations of F-type Stars“No man ever steps in the same river twice, for it’s not the same river and he’s not thesame man.” – Heraclitus
Differential rotation is a common feature of main-sequence spectral F-type stars. In
seeking to make contact with observations and to provide a self-consistent picture of how
differential rotation is achieved in the interiors of these stars, we use the 3-D Anelastic
Spherical Harmonic (ASH) code to simulate global-scale turbulent flows in 1.2 and 1.3 M⊙
F-type stars at varying rotation rates. The simulations are carried out in spherical shellsthat encompass most of the convection zone and a portion of the stably stratified radiativezone below it, allowing us to explore the effects of overshooting convection. We first explore
the general properties of the hydrodynamics within these thin convection zones here. Additionally, three of the simulations exhibit a global-scale shear instability within their stableregions that persists for the duration of the simulations. The flow structures associated with
the instabilities have a direct coupling to and impact upon the flows within the convection
zone.
In Chapter 7, we examine a case in detail as well as the scaling of the mean flows
and thermal state with rotation rate and mass, and link these scalings to fundamental
parameters of the simulations. These two chapters are based on work published in Augustson
et al. (2012)1 and are largely restatements of that paper, though with some new material
1Augustson, K. C., Brown, B. P., Brun, A. S., Miesch, M. S., & Toomre, J., 2012, “Convection and
Differential Rotation in F-type Stars”, ApJ, 756, 169

194
included (see §6.4). As the primary author of the paper, I conducted all of the simulations,performed all of the analysis, and wrote the paper. The coauthors provided advice andguidance throughout this process, with Sacha Brun encouraging the scaling analysis during
a delightful trip to Paris and Ben Brown kindly providing his data for the G-type stars. This
first peek into the possible scaling of hydrodynamic parameters such as differential rotationand thermal gradients with both rotation and mass has since been carried over to a larger
mass range in Olivier Do Cao’s thesis work.
6.1

Introduction: Stellar Convection and Rotation
Star-forming molecular clouds are generally endowed with angular momentum andmagnetic fields. As a star forms, gases from the cloud fall into a proto-stellar region. Since
the initial position of this gas is far from the stellar core, the initial angular momentum of
the gas is amplified many orders of magnitude before it is incorporated into the star. The
fact that most young stars are observed to have large rotation rates implies that, regardlessof the complicated processes that initiate and eventually lead to the formation of a mainsequence star, some of this angular momentum must be transferred to the star. During theearly evolution of these stars, it is thought that they lose their initially high angular velocity.
Stellar winds and magnetic braking acting throughout the lifetime of the star may lead tosuch angular momentum loss (Bouvier et al., 1997). However, even after several billion
years of angular momentum loss, stars such as the Sun still possess a substantial angularvelocity. Indeed, stars with convective envelopes exterior to their radiative envelopes are
believed to behave in qualitatively similar ways, where the complex interaction between theconvection and the star’s rotation likely yields non-uniform rotation. Hence, most stars may
be differentially rotating, with the angular velocity varying with depth and latitude. The
Sun is one such star, where the equator rotates 30% faster than the poles.
Main-sequence stars of all masses appear to have convection zones in some portionsof their interiors. What makes F-type stars of particular interest is that they bridge a

195
gap between stars that have external convection zones, like the Sun, and those that have
convective cores, such as A-type stars. They actually possess two regions of convective
instability, one exterior to their radiative envelope and a convective core. Being slightly
more massive than the Sun, F-type stars span the mass range between 1.1 and 1.6 M⊙. With
increasing mass, the convection zones of these stars become increasingly narrow and containless of the star’s total mass, ranging from 25% to 2% of the radius of the star compared to
the Sun’s 30%. The luminosity of the F-type stars increases from about 1.4 L⊙ at 1.1 M⊙ to
6.0 L⊙ at 1.6 M⊙, while the stellar radius only increases by about 27% over the same rangeof masses. Thus, the amount of energy flux that must be transported through ever thinnerconvection zones increases by a factor of 2.4. To accommodate the increasing flux, theconvection must become more vigorous in order to efficiently transport the heat from thebottom of the convection zone to the photosphere. Such properties place F-type stars in
an interesting position to address questions of how convection interacts with rotation andconvection-zone geometry to produce differential rotation.
6.1.1

Observations of Differential Rotation

Detecting differential rotation in stars is very difficult, as stars other than the Sun
generally cannot be spatially resolved. However, recent observations using spectropolarimetry permits some information about both the differential rotation and the magnetic field to
be mapped out on the surface of the star (e.g., Donati et al., 1997; Marsden et al., 2005).
Several additional techniques exist to extract information about the rotational behavior of astar from spectra (e.g., Reiners & Schmitt, 2003b; Reiners, 2006), photometry (e.g., Henryet al., 1995; Collier Cameron et al., 2002; Silva-Valio & Lanza, 2011), and asteroseismology(e.g., Thompson et al., 1996; Schou et al., 1998; Verner et al., 2011). For the F-type stars, aFourier analysis technique using stellar spectra has produced the largest number of detectionsof differential rotation to date (Reiners, 2007). In this method, many spectral lines from atarget star are averaged to produce a mean line-profile which is then Fourier transformed.

CasesA
B

Table 6.1
Properties of Cases A and B
M
L
R
Teff log10 g r1 /R
1.20 2.00 1.19 6300 4.54
0.80
1.30 2.93 1.33 6540 4.48
0.86

196
r2 /R
0.98
0.98

Table 6.1: The mass of the star being simulated (M) is given in units of M⊙ (1.98 × 1033 g);luminosity (L) in units of L⊙ (3.86 × 1033 ); stellar radius (R) in units of R⊙ (6.96 × 1010 cm);the stellar effective temperature (Teff ) in Kelvin; the logarithm of gravity in cm s−2 . The
lower and upper radial boundaries of the simulation domains are in turn r1 and r2 .

The ratio of the first two minima, which are denoted as q1 and q2 , of the resulting power
spectrum of this line is used to determine whether or not the star is differentially rotating,and if so by how much.
In Figure 6.1 the absolute differential rotation of F-type stars is shown to increase withrotation rate, reach a maximum, and then to decrease slowly with ever faster rotation; atrend that is more clear with the use of more rapidly rotating stars in Reiners (2006). For
Figure 6.1 we have only shown stars with masses between 1.15 and 1.45 M⊙ that have botha measurement of the differential rotation and its uncertainty. It is likely, however, thatrather more of the stars cataloged in Reiners (2006) rotate differentially as it is difficult
to detect when a star rotates too rapidly or too slowly. Indeed, stars with a projectedrotational velocity (v sin i) greater than 50 km s−1 have a relative differential rotation that
often falls below the detection threshold. Similarly, when using the method of Reiners& Schmitt (2002c), stars with v sin i ≤ 10 km s−1 rotate too slowly to accurately measure
the bulk rotation rate further restricting the range of detection. Those stars identified as
differentially rotating exhibit a wide range of relative differential rotation. There is eventhe suggestion that several stars may possess anti-solar differential rotation, where the polerotates more rapidly than the equator, and which may also be interpreted as cool polar spots.
There appears to be a trend in the observations of Reiners (2006) in which the numberof stars that rotate differentially seems to decrease with decreasing B-V color. However, this197
could be a two-fold selection effect. The first effect is from the non-uniform color (effective
temperature) sampling of stars. The distribution is roughly Gaussian between B-V values of
0.2 and 0.6, or spectral types A7 and G1 respectively, with a peak at a B-V of 0.4 (spectral
type F4). The second effect arises from the fact that higher mass stars tend to rotate more
rapidly and thus their q2 − q1 values fall below the differential rotation detection threshold.
Of the stars that do rotate differentially, the results for the F-type stars with the largest
differential rotation in Reiners (2006) are consistent with the trend of increasing absolutedifferential rotation found in Barnes et al. (2005) where they find that absolute differential
rotation increases strongly with effective temperature or increasing mass, as seems to be thecase in our simulations as well (Figure 6.1 and §7.5).
6.1.2

Global Models of F-type Star Convection

How the angular momentum in a star is redistributed to produce and maintain astable angular velocity profile appears to involve subtle dynamical balances. The highly
turbulent nature of stellar convection coupled with rotation involves a vast range of scales,making it quite challenging to directly model such dynamic processes. Yet these issues
must be explored if the behavior of stellar rotation is to be understood. Rapid advancesin supercomputing have enabled global-scale 3-D simulations of convection coupled withrotation that are shedding light on the dynamics of the flow achieved within stars (Bessolaz& Brun, 2011; Brown et al., 2008; Miesch et al., 2008). In a similar spirit, we are reporting on
the dynamics of turbulent global-scale convection in a realistically stratified computational
domain in two main-sequence F-type stars that are each studied over a range of rotation
rates. The two stars investigated here are a 1.2 M⊙ star (Case A simulations) and a 1.3 M⊙
star (Case B simulations) that possess the same central hydrogen abundance resulting in
main-sequence ages of 1.15 Gyr and 1.00 Gyr respectively (see Table 6.1). As mentioned
above, convection zone depth and luminosity change rapidly with mass in F-type stars.
Indeed, the tenth of a solar mass difference between the two stars simulated here leads to

198

1.0

∆Ω (rad d-1)∆Ω ( µrad s-1)

10

0.1

1

5

10

Ω sin i/Ω

20

40

Figure 6.1: Absolute differential rotation contrast (∆Ω) seen in simulations and observed on
F-type stars. The observational uncertainties of the contrast and rotation rate are shownas gray lines. The values of the ∆Ω realized in our simulations are shown in red symbols,
while those observed on F-type stars with masses between 1.15-1.25 M⊙ shown as diamonds,
1.25-1.35 M⊙ as triangles, and 1.35-1.45 M⊙ as squares. Observational data from Table B.1
of Reiners (2006).

a 50% increase in luminosity and a 50% decrease in convection zone depth, allowing us toalso probe how the vigor of the convection and convection zone geometry impact the mean
properties of the flows established in these stars.
In our analysis of these simulations, we first provide details about the anelastic equations solved, the formulation of the numerics, and the setup the numerical experiments in§6.2. The general properties of the convective patterns and mean flows of selected cases are
discussed in §6.3. We look at the dynamics in detail for a particular case in §7.1. The effectsof overshooting convection are addressed in §7.2. The link between the thermal propertiesof the simulations and the differential rotation is probed in §7.3. The connection between

199
the two mean flows, the differential rotation and the meridional circulation, is examined forthese two cases in §7.4. Finally, in §7.5, the scaling of the magnitudes of the mean flows andthe thermal signatures of the flows are presented.
6.2

Formulating the Problem
The convection zones of most stars are extremely turbulent with very small values ofviscosity and thermal diffusivity. A direct numerical simulation that effectively captures
all of the relevant scales of motion in a stellar convection zones are currently, and for theforeseeable future, impossible. Therefore, a model that eliminates the fast time-scales ofsound waves and parametrizes the dynamics at small spatial scales is necessary in order tocapture the large number of rotational periods and the full spherical geometry. Such features
are thought to be crucial in establishing an equilibrium state that maintains a differential
rotation. To this end, we use the 3-D anelastic spherical harmonic (ASH) simulation code.
ASH is a mature modeling tool which solves the anelastic MHD equations of motion in arotating spherical shell using a pseudo-spectral method. ASH simulations capture the entire
spherical shell geometry and allow for zonal jets, large scale vortices, and structures withconnectivity such as convective plumes. The simulated turbulence is still many orders ofmagnitude removed from the intensely turbulent conditions present in stellar convection
zones. Despite this discrepancy, simulations of the solar convection zone using ASH havebeen successful in matching helioseismic constraints upon differential rotation within thesolar convection zone (Miesch et al., 2000; Brun & Toomre, 2002; Miesch et al., 2006, 2008;
Brun et al., 2011).
6.2.1

Anelastic Equations

In the anelastic approximation, the time derivative of the density in the continuity
equation is eliminated. This approximation is employed to capture the effects of density
stratification without having to resolve the rapidly varying sound waves. In this approxima-

200
tion, the Courant-Friedrichs-Lewy condition on the time step will then be governed by the
subsonic flow velocity rather than the sound speed. The anelastic equations evolved within
ASH are fully nonlinear in the velocity variables, but the thermodynamic variables are linearized about the spherically symmetric and evolving mean stratification comprised of thequantities ρ, S, P , and T for the density, entropy, pressure, and temperature respectively.
The fluctuations of thermodynamic variables are taken about the mean stratification andare denoted ρ, S, P , and T . These fluctuations are generally of the order ǫ = (γ − 1) M 2
where M is the Mach number of the flow at a given depth, which in the simulations here
gives ǫ ≈ 10−6 . The resulting equations in physical units, in spherical coordinates (r, θ, φ),and with time t are (Glatzmaier, 1984; Clune et al., 1999)∂u= −ρu · ∇u − ∇P − ρg + 2ρu × Ω0 + ∇ · D − Λr̂,∂t∂S= −ρT u · ∇ S + S − ∇ · q + Φ,ρT∂t∂P
= −ρgr + Λ,∂rρ

∇ · ρu = 0,

(6.1)
(6.2)
(6.3)
(6.4)

where u = ur r̂ + uθ θ̂ + uφ φ̂ is the velocity vector, Ω0 = Ω0 ẑ is the angular velocity of therotating frame, ẑ is the axial unit vector, and cP is the specific heat at constant pressure.
The gravitational acceleration is taken to be g = −g(r)r̂. As the simulation evolves, aturbulent pressure pushes the system slightly away from the initial hydrostatic equilibrium,Λ in Equation (6.3) is the radial gradient of the spherically symmetric component of thisturbulent pressure. The components of the viscous stress tensor D are Newtonian and Φ isthe viscous heating, which are given by1
Dij = 2ρν eij − ∇ · uδij ,
3


1
2
Φ = 2ρν eij eij − (∇ · u) ,
3

(6.5)
(6.6)

where eij is the stress tensor and ν is the effective kinematic viscosity. The energy flux q is
composed of two diffusive components

201


∂S
r̂,
q = κr ρcP ∇ T + T + κρT ∇S + κ0 ρT
∂r

(6.7)

where the first component is a radiation diffusion flux with the molecular radiation diffusion
coefficient κr , The second component is an anisotropic entropy diffusion flux, with κ acting
on the non-axisymmetric entropy fluctuations and another κ0 acting only on the sphericallysymmetric component of the entropy.
Ideally, the system would adjust to a new equilibrium by modifying the Rosselandmean opacity and the thermal stratification. The overshooting convection realized in our
simulations modifies the mean stratification in the overshooting region. However, to fully
adjust the mean thermal state, and thus the radiative flux, would require evolving thesimulation for a thermal relaxation time. This timescale is about Ei /L ≈ 2400 yr forthe 1.2 M⊙ F-type stars and about 160 years for the 1.3 M⊙ F-type stars, where Ei is thetotal internal energy of the plasma in the stable region and L is the luminosity. Since
achieving these timescales is currently beyond our computational resources, we accelerate
this process by first simulating the stars with the Rosseland mean opacity (κRos ) and the
1-D thermodynamic state from the stellar model, and then updating this opacity using the
evolved thermodynamic state of the simulation. This is accomplished by Taylor expanding
the opacity using the partial derivatives of the opacity with respect to the density andtemperature extracted from the stellar model, and then updating the radiative diffusion
coefficient with the mean thermal state as
κ′Ros = κ0Ros +


∂κRos
∂κRos
(ρ − ρ0 ) +T − T0 ,
∂ρ 0
∂T 0

3

4acT
κr =
,
3cP ρ2 κ′Ros

where the subscript 0 indicates that these quantities are taken from the stellar model. This
scheme is typically applied only once, but can be used as frequently as necessary to achieve
an equilibrium. Updating the diffusion coefficient in this fashion ensures that the radiative

202
energy flux and the enthalpy flux are properly balanced in the stable region, providing for anearly constant total flux throughout the domain and also avoiding the long time evolutionthat would otherwise be required (see §7.1.2). Finally, a linearized equation of state closes
the set of equations for the fluctuations, while the ideal gas law closes the equations for themean stateρ/ρ = P/P − T /T = P/γP − S/cP ,P = (γ − 1)cP ρT /γ,

(6.8)
(6.9)

where γ is the adiabatic exponent. A stellar evolution code, MESA (Paxton et al., 2011),
is employed to evolve a realistic 1-D stellar model at the solar metallicity and helium massfraction from the pre-main sequence to a central hydrogen mass fraction of 0.54. This places
these models at about 20% of their main-sequence lifetime. This model is then used toestablish the initial mean stratification for an ASH simulation. For the numerical experiments
conducted here, we neglect the effects of stellar evolution. A single initial stellar structure
model is used for each of the two simulated F-type stars shown in Table 6.1.
6.2.2

Numerics and Experimental Configuration

In the ASH code, the mass-flux remains divergence-free through a poloidal-toroidal
streamfunction decomposition. These streamfunctions and the variations in the entropy
and pressure are expanded in spherical harmonics Yℓm (θ, φ), with spherical harmonic degreeℓ and azimuthal order m, to resolve their horizontal structures and in Chebyshev polynomials Tn (r) to resolve their radial structures. Temporal discretization is accomplishedusing a semi-implicit Crank-Nicolson time-stepping scheme for linear terms and an explicit
Adams-Bashforth scheme for nonlinear terms. ASH is designed with modern programming
constructs to yield efficient performance and scaling on massively parallel supercomputers(e.g., Clune et al., 1999). Several codes numerically solving for anelastic magnetohydrodynamics, including ASH, have recently been thoroughly tested on the same suite of nonlinear

203
problems, in which the accuracy of several numerical methods has been probed extensively
and shown to be robust (Jones et al., 2011).
The studies here explore a range of rotation rates for each mass, from 5 Ω⊙ to 20 Ω⊙ forthe 1.2 M⊙ case Case A simulations and from 8 Ω⊙ to 30 Ω⊙ for the 1.3 M⊙ Case B simulations.
All cases at a given mass use the same initial stellar structure and are initialized with random
perturbations in the fluctuating entropy. These ranges of rotation rates are chosen so that
all the cases exhibit solar-like differential rotation, where the poles rotate more slowly thanthe equator, as is discussed in §7.5. Note that the nomenclature for the cases is: Case A
or Case B depending upon the mass followed by the bulk rotation rate quoted in integer
multiples of the solar rotation rate. So, Case A10 is a 1.2 M⊙ F-type star rotating at 10 times
the solar rate. Cases that omit the stable layer, simulating only the convectively unstableregion, are denoted with an i after the case’s name.
While most parameters and the initial stratification are taken from the stellar model,
a perfect gas is assumed (Equation (6.9)). Therefore, the He and H ionization zones that
occur in the outer 1% by radius of these stars cannot be properly captured. Additionally,
ionization coupled with radiative losses drive very intense convection on small scales. These
small scale flows are nearly sonic, so sound waves play an important role in the dynamics
and cannot be neglected. Thus, the radius of the upper boundary (r2 ) in our simulations istaken to be below this region, where the Mach number is generally very small, so that theassumptions made in the formulation of the anelastic approximation remain valid.
The simulations being reported on here include a portion of the stable radiative zone
below the convection zone as seen in Figure 6.2, in a manner similar to the simulations havebeen carried out in a solar setting in Browning et al. (2006); Brun et al. (2011). Such alayer has been shown to play a large role in determining the radial and latitudinal structureof the differential rotation (Rempel, 2005). Indeed, the dependence of the morphology ofthe differential rotation upon the amplitude of a weak latitudinal entropy gradient imposedat an impenetrable lower boundary has been explored within ASH simulations of the Sun.

204
The simulations in Miesch et al. (2006) use this latitudinal entropy gradient to emulate atachocline that is in a thermal wind balance. It was found that this gradient spurred anadjustment of the contours of constant angular velocity from an alignment with the rotation
axis to a radial alignment. The impetus behind these various efforts is to understand why the
helioseismically determined angular velocity of the Sun has nearly conical contours (Schouet al., 1998), as is also found in the angular velocity contours of the simulations explored
here (see §6.3.1 and §6.3.2).
The position of the radius of the lower boundary (r1 ) is determined by how deep theovershooting motions occurring within the simulations penetrate into the stable zone andby how much resolution is necessary to fully resolve the steep entropy gradient that arises
below the convection zone. The radial extent of the stable region in these simulations is set
to be roughly twice the penetration depth of the overshooting motions. This choice reduces
the need for a high radial resolution and allows these simulations to capture the effects thatconvective overshooting and a stable layer have upon the mean flows and thermodynamic
properties of the convection. However, it does restrict the ability of the simulations tocapture the full spectrum of gravity waves excited by the convection as some of the modes
are trapped within this cavity (see Brun et al., 2011).
6.2.3

Diffusion and Sub-grid-scale Models

The anelastic system of hydrodynamic equations requires eight boundary conditions inorder to be well posed. One of the primary goals of this work is to assess the redistributionof angular momentum and energy in our simulations. Thus, we have opted for the following
torque-free and flux transmitting boundary conditions∂  uθ ∂  uφ  ∂S
= 0,
ur =
=
=
∂r r
∂r r
∂r r2 , r1

(6.10)

where the constant flux boundary condition on the entropy fluctuations implies that themean entropy gradient remains unchanged on the boundaries throughout the simulation.

205(a) 100

104 ∂S/∂r

10
1
1

A10

B10

0
-1
-2
-3
20

12
10

103 ρ

15

8

T
10

6

5
0
0.80

0.83

ρ

4

0.86

2
0
0.98

0.89
r/R

0.92

0.95

10-5 T(b)

Figure 6.2: Time-averaged mean stratification established in Case A10 (black lines) and Case
B10 (red lines). (a) The entropy gradient (∂S/∂r in units of 10−4 erg cm−4 K−1 ); the breakin the y-axis denotes the change from linear to logarithmic scaling. (b) The density (ρ inunits of 10−3 g cm−3 , dash-dot) and temperature (T in units of 105 K, solid).

The stress-free boundary conditions used in the simulations here ensure that the volumeintegrated transport terms nearly vanish. Indeed, the total angular momentum is conserved
to within the global accuracy of the radial derivatives at each time step. Over the course of
the entire simulation where the mean stratification is allowed to change, it is conserved towithin the order ǫ, roughly the order to which the anelastic approximation is valid.
Current and foreseeable computing resources, render simulations that directly capture
all scales of stellar convection from global to molecular dissipation scales unattainable. The

206
simulations reported on here resolve nonlinear interactions among a large range of scales bothin the convective and radiative zones of two F-type stars. Motions and waves within theactual star exist on scales smaller than our grid resolution, and in this sense our models should
be regarded as large-eddy simulations (LES). These sub-grid scale motions are parameterized
as effective viscous and thermal diffusivities (ν and κ respectively), which are meant toemulate the mixing properties of the unresolved turbulent eddies. These diffusivities areallowed to vary in radius only and evolve slowly in time as they are dependent upon the
profile of the mean density ρ. The radial profile of κ and κ0 are given by
−1/2κtop ρ/ρtopκ (r) =+ κmin ,1 + exp [α (rκ − r) / (r2 − r1 )]κtop
exp [β (r2 − r) / (r2 − r1 )]+ κminκ0 (r) = 0
0 ,1 + exp [δ (rκ0 − r) / (r2 − r1 )](6.11)(6.12)

where κtop is the thermal diffusivity at the upper boundary, κmin is the floor value of this
diffusivity, and α controls the steepness of the quenching of the diffusion below a particularradius rν . This allows the diffusion to be greatly reduced in stable regions, where motions
have a small amplitude and the expected SGS motions are subsequently smaller. The Prandtl
number is Pr = ν/κ = 1/4 and is fixed throughout the domain, so ν = Prκ. In both theCase A and Case B simulations, α = 300 and the minimum levels of diffusion νmin =and κmin are set to be 1000 times smaller than νtop and κtop . The radius below which the
diffusion drops rapidly is rκ = 6.7 × 1010 cm for Case A simulations and rκ = 8 × 1010 cm
for Case B simulations. This allows the convective motions to carry the majority of the flux
throughout the convection zone and avoids an inward diffusive energy flux in sub-adiabatic
regions. The diffusion acting on the mean entropy has the coefficient κ0 , which is set sothat this unresolved eddy flux carries the stellar luminosity through the upper boundary
but is exponentially tapered with depth below r2 . It is further quenched below the radiusrκ , with the rapidity of the decrease controlled by δ. In the Case A simulations, β = 60,
δ = 200, and rκ0 = 6.95 × 1010 cm, while in the Case B simulations β = 100, δ = 200, andrκ0 = 8.22 × 1010 cm.

207
The unresolved eddy diffusion associated with the spherically symmetric diffusion component acts to conduct the full stellar flux through the upper boundary. Thus, almost by definition it has a substantial influence on the spherically symmetric component of the entropy.
The monotonically decreasing initial entropy gradient from the stellar model is significantly
steepened near the upper boundary by the cooling induced by the unresolved eddy flux, as
is clearly seen as a large dip near the upper boundary in Figure 6.2. This has the effectof enhancing the buoyancy driving in the upper convection zone relative to what might be
expected if the upper boundary were open and the surface convection allowed to influence
the deeper flows. Recent 3-D surface convection simulations have shown a sizable change inthe entropy gradient in the surface layers of various stars when compared to the gradientexpected from standard mixing length theory (Trampedach & Stein, 2011). While the total
integrated entropy deficit across the convection zone remains the same, as the luminosity is
the same, its radial distribution is somewhat different. The gradient of the entropy becomes
more superadiabatic near the surface and more adiabatic in the bulk of the convection zone.
Therefore, the total entropy contrast across the domain in these simulations may be too highrelative to the actual stratification, especially in simulations at lower rotation rates where
the steepest superadiabatic gradients arise. This results in flows that begin to feel the effects of rotation at depths greater than otherwise might occur. The transition between flowsthat strongly feel the bulk rotation and those that do not takes place where the fluctuating
Rossby number of the flows drops below one. Thus, we attempt to temper the influence of
the upper boundary on the dynamics of the interior by ensuring that the fluctuating Rossby
number throughout the convection zone is less than one. This requires that the rotation rate
be high enough to attain Ro′ < 1 as is the case for the simulations here as indicated in Table
6.2.

Mass

1.2 M⊙

1.3 M⊙

Case
A5
A7
A10
A15
A20
A5i
A20i
B8
B10
B15
B20
B30
B20i

Ω0 / Ω⊙
5
7
10
15
20
5
20
8
10
15
20
30
20

Nr × Nθ × Nφ
257 × 512 × 1024
257 × 512 × 1024
257 × 512 × 1024
257 × 512 × 1024
257 × 512 × 1024
97 × 512 × 1024
97 × 512 × 1024
193 × 512 × 1024
193 × 512 × 1024
193 × 512 × 1024
193 × 512 × 1024
193 × 512 × 1024
97 × 512 × 1024

Table 6.2
Diagnostic Flow Parameters
Ra
Ta
Re Re′
1.75 × 105 2.40 × 105 84 37
2.62 × 105 7.36 × 105 140 44
4.22 × 105 2.41 × 106 240 52
7.64 × 105 9.32 × 106 451 75
1.60 × 106 2.42 × 107 644 102
1.22 × 105 2.29 × 105 82 38
1.04 × 106 2.30 × 107 410 76
4.32 × 104 9.21 × 104 62 41
6.45 × 104 1.90 × 105 108 44
1.19 × 105 7.34 × 105 207 57
2.34 × 105 1.91 × 106 363 76
4.25 × 105 4.31 × 106 324 73
9.49 × 104 1.43 × 106 248 58

Ro
0.69
0.52
0.39
0.28
0.25
0.63
0.23
0.99
0.84
0.56
0.46
0.28
0.51

Ro′ RoDR
0.48 0.29
0.35 0.24
0.25 0.21
0.18 0.19
0.15 0.16
0.44 0.23
0.14 0.14
0.73 0.07
0.61 0.11
0.40 0.12
0.31 0.13
0.19 0.09
0.30 0.21

Romc ν/1013
0.017
0.87
0.014
0.69
0.008
0.55
0.004
0.42
0.004
0.35
0.016
0.81
0.002
0.32
0.021
1.29
0.012
1.12
0.009
0.85
0.009
0.71
0.004
0.71
0.001
0.72

τc
2.8
3.0
3.3
3.7
3.9
3.2
4.2
1.1
1.2
1.3
1.5
1.7
1.5

τν
101
126
159
209
252
96
241
39
44
59
71
71
60

Table 6.2: The depth of the convection zone (d = r2 − rb , where rb is the radius of the bottom of the convection zone) is therelevant length scale in the following parameters and is 1.3×1010 cm for Case A simulations and 9.2×109 cm for Case B simulations.
We estimate and quote the following diagnostic parameters at mid-convection zone: the Rayleigh number Ra = ∆Sgd3 /cP νκ,′
Taylor number Ta = 4Ω20 d4 /ν 2 , Reynolds number Re = vrms d/ν, fluctuating Reynolds number Re′ = vrms
d/ν, Rossby number′′
Ro = |ω|/2Ω0 , fluctuating Rossby number Ro = |ω |/2Ω0 , where the hi denotes a horizontal average and ω is the vorticity
vector. The differential rotation and meridional circulation Rossby numbers at mid-convection are RoDR = |λ∇ ln Ω| andRomc = |ωφ |/2Ω0 , respectively. The bar denotes the horizontal average. Kinematic viscosity (ν) values are given in units of1013 cm2 s−1 . The mid-convection zone local overturning time τc and the viscous diffusion time τν across the convection zone arequoted in days.
208

209
6.2.4

Scaling Diffusion with Rotation

In the Boussinesq approximation, convection becomes possible above a critical Rayleigh
4/3

number, which scales with rotation rate as Rac ∝ Ω0(e.g., Chandrasekhar, 1961; Dormyet al., 2004). Anelastic systems have a similar constraint upon the minimum Rayleigh numberthat is necessary for the flows to be convective (Glatzmaier & Gilman, 1981; Jones et al.,2009). We seek here to explore the effects of rotation on stellar convection in the global
spherical geometry at a representative point of time within the main-sequence lifetime ofthese two F-type stars. This goal means that the simulations must be highly supercriticalso that the level of turbulence dominates diffusion. With unlimited computer resources,
we ideally would maintain a constant level of supercriticality, but this requires scaling theeffective diffusivities ν and κ as Ω−2
0 . However, lower diffusivities lead to both longer viscousand thermal diffusion timescales and to flows possessing finer spatial scales. Achieving
equilibrated states in these systems requires high-resolution simulations carried out overextended periods, which would be prohibitively expensive if we followed this path. We haveattempted to balance the level of supercriticality and the resolution requirements necessary
to resolve the flow in our path through parameter space. Thus, our trajectory through theparameter space of Ω0 , ν, and κ attempts to maintain strongly nonlinear dynamics without
having the increased rotation rate laminarize the convection. In our simulations, we haveconstrained this trajectory to be 1-D by requiring that the Prandtl number be 1/4 and thatthe value νtop be set relative to the most rapidly rotating case, and subsequently scaled−2/3

according to νtop ∝ Ω0

, as seen in Table 6.2.

The Rayleigh numbers at mid-convection zone are about 50 times the critical Rayleighnumber (Jones et al., 2009) for the Case A simulations about 25 for the Case B simulations.
These levels of supercriticality are equivalent to ASH simulations of lower mass stars (e.g.,
Brown et al., 2008; Matt et al., 2011). The lower level of supercriticality in the 1.3 M⊙ simulations is primarily due the stronger driving of the convection and larger superadiabatic

210
gradient, both of which are in turn due to the higher luminosity and narrower convection
zones of the higher mass F-type stars. This requires that the diffusion in the Case B simulations be about two times larger than the lower mass Case A simulation at the same rotation
rate. This is done to ensure numerical stability at a chosen resolution as the downflows
tend to be faster with larger temperature deficits in Case B simulations due to the higher
luminosity of the star. The radial extent of the convection zone is taken to be the representative scale for determining the flow parameters seen in Table 6.2. This in combination with alarger diffusion results in lower Reynolds and Rayleigh numbers for Case B simulations. As amatter of definition, an evolved case is one in which the volume averaged differential rotationand kinetic energies vary by less than 0.1% relative to an initial value over a span of about1000 days of simulation time. The differential rotation and other quantities are measured inthese evolved cases to ensure that the time averages taken are essentially stationary.
6.3

Convective Patterns and Mean Structure
The variation of convective patterns of selected Case A simulations with increasing

rotation rate Ω0 is illustrated in the first column of Figure 6.3 (Panels (a), (e), and (i)),
while the same is shown for three Case B simulations in Figure 6.4. These snapshots of
the radial velocity near the top of the domain (0.96 R) are shown in Mollweide projection
for three cases in Figure 6.3: Case A5, Case A10, and Case A20. The radial velocities
exhibit the complex, evolving, and global-scale nature of the convection established in thesesimulations. The convective cells that comprise these complex patterns are outlined by thedark downflows. A distinctive transition in the morphology of the convective cells occurs
between the columnar equatorial cells, which are essentially viewed from the side given
the spherical geometry, and the polar columns where they are viewed from the top down.
Equatorial convective columns march around the equator in a prograde sense, constantly
jockeying for position as they collide and overtake one another. Cells of convection at higher
latitudes continually form and reform as they interact with one another and are shuffled

211along by the mean flows in a retrograde sense. The mean zonal flow makes its presence felt
at mid-latitudes where there is an obvious shear between the prograde equatorial flows andthe retrograde polar flows. These flows exhibit somewhat different time scales over which
they maintain their coherence. The convective cells at the poles can last a few days, andthose at the equator last even longer.
6.3.1

Case A: 1.2 M⊙ Simulations

In Figure 6.3, there is a juxtaposition between high and low latitudes in both the scaleand structure of the convection especially for more rapidly rotating cases. The latitudinalvariation of convection patterns can be in part understood by considering a cylinder tangent to the base of the convection zone whose axis of symmetry is aligned with the rotation
axis. Within the geometry of the Case A simulations, the top of such a cylinder intersects
upper boundary at latitudes of ±32◦ . Outside the tangent cylinder, downflows are deflected
equatorward by Coriolis forces and can connect across the equator before they are strongly
braked in the stable region. For downflows inside the tangent cylinder, they generally encounter the stable region before they can be paired with a flow from the opposite hemisphere.
Such constraints on the convection become more severe as the rotation rate of the frame isincreased.
The flows outside the tangent cylinder organize into large-scale sheared cylindrical rollsaligned with the rotation axis. Individual convective cells remain coherent for roughly the
global overturning time which is between 15-30 days for these 1.2 M⊙ simulations. These
convective structures are very similar to the linearly unstable columnar modes arising in thepresence of a significant density gradient and a convex boundary (Busse, 2002; Jones et al.,2009). The convective columns form a thermal Rossby wave that has a phase velocity larger
than the local rotation rate, leading to downflows that propagate prograde faster than thelocal differential rotation. In a linear regime, the effects of compressibility counteract thetendency of the convection to occur along the tangent cylinder where it instead increasingly

212
(b)

(a)

-290 / 290 m s-1

(c)

(d)

Ω

T

1740 / 2320
nHz

-60 / 170
K

A5

(e)

(f)

(g)

4580

(h)

810

Ω

CW / CCW

-260

A10
(j)

(i)

-340 / 340

ψ

T

3270

-340 / 340

ψCW / CCW

(k)

(l)

Ω

T

ψ

6990 / 9010

-960 / 2700

CW / CCW

A20

Figure 6.3: Overview of convective patterns, mean flows, and mean temperature fluctuations
in selected Cases A. The first column of figures (a, e, i) display convective patterns in radial
velocities near the upper boundary in turn for Case A5, Case A10, and Case A20. (b,f, j) the second column presents the time- and azimuthally-averaged angular velocity withradius and latitude, with the fast prograde equator in red and slower poles in blue. (c, g,k) azimuthally- and time-averaged temperature fluctuations (where the ℓ = 0 component
has been removed) with the warm poles in red tones and the cool equator in blue tones.
(d, h, l) the streamfunction of the meridional flow (ψ), clockwise (CW) flows in red andcounter-clockwise (CCW) flows in blue. The dashed line in the second, third and fourthcolumns delineates the beginning of the stable region.

occupies the outermost portions of the domain (Jones et al., 2009). In our simulations,213
however, these effects often lead to two sets of columnar structures, one being somewhat
more confined to larger radii and another that occurs deeper along the tangent cylinder.
The primary difference in the simulations here is that they are highly supercritical with arealistic stratification, yet these structures largely persist despite their nonlinear interaction
with turbulent flows.
The downflows sheaths surrounding the upflows are elongated in azimuth forming thinsheets that spiral outward from the base of the convection zone in a prograde direction. Suchstructures are largely due to vortex stretching near the upper boundary that is enhancedby the effects of compressibility, both of which are in turn due to our choice of a Prandtl
number that is less than unity (Zhang, 1992; Jones et al., 2009). This prograde spiraling
leads to strong Reynolds stresses that act in concert with the thermal state to help build thestrong zonal flow and thus the differential rotation seen in these simulations (see §7.3.2 and§7.4). The radial differential rotation, as seen in panels (b), (f), and (j) of Figure 6.3, which
increases with larger rotation rates also must play a role in stretching out these columns. The
latitudinal differential rotation tends to shear the columns, creating the so called “banana
cells.” There also tends to be less latitudinal connectivity in the downflow networks thanin more slowly rotating cases. This enhanced connectivity at lower angular velocities canprevent the formation of the columnar structures so prevalent in cases with a larger angularvelocity.
The number of such convective modes that can fit within the circumference of the starincreases with rotation rate, and thus the Taylor number (Gilman & Glatzmaier, 1981; Dormyet al., 2004; Jones et al., 2009). This means that at a given radius these modes will have lesslongitudinal extent as exhibited in panels (a), (e), and (i) of Figure 6.3. Analysis carried out
using the modal equations of rotating Boussinesq convection within tangent cylinders (Dormyet al., 2004) and numerical computations of linear modes in compressible anelastic convection(Gilman & Glatzmaier, 1981; Jones et al., 2009) reveal that the azimuthal wavenumber ofthe most unstable mode increases as m = mc Ta1/6 , where mc is the critical wavenumber.

214
−2/3

For our cases, the viscosity scales as Ω0, thus the most unstable mode is only a functionof the rotation rate, where m = mc (Ω0 /Ω⊙ )5/9 . The critical mode is determined by using avalue estimated from those given in Jones et al. (2009) as m̂c and the values of convection
zone depth, viscosity in Case A5 and its rotation rate in the Taylor number, which yields
−1/3

1/3

mc ≈ 21/3 5−2/9 m̂c d2/3 νA5 Ω⊙ = 9. The resulting wavenumbers agree well with the modes
that are most visible at the equator in cases Case A5, Case A10, and Case A20 have azimuthal
order of m = 22, 32, and 54, where as the brief analysis here yields m = 22, 32, and 47.
There is likely a discrepancy between the order of the two modes for the fastest rotators
because an instability arises in these cases that may change this scaling (see §6.4).
At higher latitudes inside of the tangent cylinder, the convection is more isotropic andthe downflow networks organize on scales smaller than in the equatorial regions, as is seen inpanels (a), (e), and (i) of Figure 6.3. These convective cells are intricate and dynamic, withcells constantly evolving as they interact with one another and the bulk motions. Whenviewed on a spherical shell, the downflows of the high latitude convective cells take on a
crescent-like shape and precede the upflows as both are carried along by the differential
rotation. As described by the Taylor-Proudman theorem, the tendency for these flows toalign with the rotation axis becomes more pronounced at higher rotation rates. Indeed, as
the angular velocity of the frame increases the high latitude downflows form into cylindricalsheaths nearly aligned with the rotation axis. Such convective structures have a strong
vertical vorticity. The sense of this vorticity in the downflow plumes is cyclonic above themiddle of the convection zone: counterclockwise in the northern hemisphere and clockwise in
the southern. As the plumes descend deeper into the convection zone, their vorticity changes
sign and the downflows become anti-cyclonic Miesch et al. (2000, 2008). The upflows, on
the other hand, are anti-cyclonic at all depths outside the tangent cylinder and at latitudes
above about ±60◦ . However, at mid-latitudes the upflows are anti-cyclonic in the upper
convection zone but are cyclonic in the lower portion.
In Table 6.2, flow diagnostics for the last two 1.2 M⊙ F-type star simulations (Case215
A5i and Case A20i) are shown. The computational domain of these two simulations is
restricted in radius to be only in the convectively unstable portion of the domains of theother 1.2 M⊙ F-type star simulations. Placing the lower boundary of the domain at thebottom of the convection zone is tantamount to having an infinitely stiff entropy gradient
there, eliminating penetrative motions. While the convective patterns are largely unchangedin these simulations, the rate of deceleration of downflow plumes generally increases as they
approach the lower boundary. This alters the Reynolds stress correlations and turbulent
enthalpy flux, which in turn changes the differential rotation, meridional circulation, andtemperature structures established in these simulations and will be discussed further in §7.2.
6.3.2

Case B: 1.3 M⊙ Simulations

Three 1.3 M⊙ F-type star simulations are shown in Figure 6.4, Case B10, Case B20,
and Case B30. The two Case B simulations carried out at 10 and 20 times the solar rate
allow for an easy comparison to the two Case A simulations at the same rotation rate. The
third simulation (Case B30) exhibits modulated convection that is not seen in the othercases, and therefore is shown for its inherent interest. A striking similarity is seen between
the convective patterns established in the Case B simulations and the Case A simulations(Figures 6.4(a), (e), and (i)). There is still a transition from columnar convection outside thetangent cylinder, to more isotropic and smaller scale convective cells inside of it. For thesesimulations, the edges of the tangent cylinder intersect the upper boundary at latitudes of± 26◦ , six degrees closer to the equator than in the 1.2 M⊙ F-type stars.
Careful scrutiny reveals that the columnar equatorial convective modes, as seen in
the Mollweide projections of the radial velocity (Figures 6.4(a), (e), and (i)), have less ofa latitudinal extent when compared to the 1.2 M⊙ simulations. The radial velocities areabout 15% larger on average than in the Case A series of simulations, but the tails of theradial velocity distribution extend out even further to several km s−1 in the downflows. The
difference in temperature fluctuations between the center of an upflow and the downflows at

216
its edges is about 120 K, 50% larger than in the 1.2 M⊙ F-type stars. These somewhat larger
velocities and temperature fluctuations are expected given that both the lower backgrounddensity is lower and that the luminosity is 50% higher in the 1.3 M⊙ simulations. As seenin Table 6.2, there is a single case that captures only the convectively unstable region, Case
B20i. This case is meant to be a point of comparison for assessing the role of overshooting
in maintaining a strong differential rotation.
The mean angular velocities and temperature variations for Case B10, Case B20, andCase B30 are shown in the second and third column of Figure 6.4. The nine panels in thelast three columns of Figure 6.4 all demonstrate the convection zone geometry of the Case B
simulations. The convection zone is much narrower than in both the Sun and the 1.2 M⊙ Ftype stars. This narrowing of the convectively unstable region and the increased luminosityof the higher mass stars play a role in the behavior of the differential rotation established inthese simulations.
6.3.3

Mean Zonal and Meridional Flows

Shown in the second column of Figure 6.3 are the mean radial and latitudinal profilesof angular velocity (Ω = Ω (r, θ)) achieved in Case A5, Case A10, and Case A20. These
profiles are averaged in longitude and in time, over 150 days. All of the 1.2 M⊙ F-type star
simulations exhibit solar-like differential rotation profiles, with prograde (fast) equators andretrograde (slow) poles. Contours of constant angular velocity, black lines in Figures 6.3(b),(f), and (j), are largely radial but become increasingly aligned with the rotation axis withmore rapid rotation.
More specifically, the average rotation profile in depth and latitude established in Case
A10 is shown in Figure 6.3(f). The red and white tones indicate a fast prograde equator,
while the black and blue tones indicate a pole rotating more slowly than the frame rate;
this is considered solar-like differential rotation. The color associated with the frame rateof 4240 nHz is yellow. The rotational shear averaged over the convection zone between the

217
(b)

(a)

-350 / 350 m s-1

(c)

(d)

Ω

T

3530 / 4600
nHz

-310 / 770
K

B10

(e)

(f)

(g)

9160

(h)

4300

Ω

CW / CCW

-1390

B20
(j)

(i)

-360 / 360

ψ

T

6790

-400 / 400

ψCW / CCW

(k)

(l)

Ω

T

ψ

10830 / 13370-1960 / 6340

CW / CCW

B30

Figure 6.4: Overview of convective patterns, mean flows, and mean temperature fluctuations
in selected Cases B. The first column of figures (a, e, i) display convective patterns in radial
velocities near the upper boundary in turn for Case B10, Case B20, and Case B30. (b,f, j) the second column presents the time- and azimuthally-averaged angular velocity withradius and latitude, with the fast prograde equator in red and slower poles in blue. (c, g,k) azimuthally- and time-averaged temperature fluctuations (where the ℓ = 0 component
has been removed) with the warm poles in red tones and the cool equator in blue tones.
(d, h, l) the streamfunction of the meridional flow (ψ), clockwise (CW) flows in red andcounter-clockwise (CCW) flows in blue. The dashed line in the second, third and fourthcolumns delineates the beginning of the stable region.

equator and the pole is ∆Ω = 1290 nHz, corresponding to a relative latitudinal shear of218∆Ω/Ω0 = 34%. Thus, this simulation has a latitudinal shear that is quite similar to thatobserved at the solar surface. Similarly, Case A5 and Case A20 possess a equator to poleabsolute differential rotation of ∆Ω = 560 nHz or ∆Ω/Ω0 = 26% and ∆Ω = 2330 nHz or∆Ω/Ω0 = 27%. Though we have attempted to drop the diffusion within the stable region,
we are restricted both in the depth of the layer and the magnitude of the motions that stillrequire some diffusion. This leads to a slow spread of the differential rotation into the stableregion, which can be seen in Figures 6.3(b), (f), and (j).
A trend is readily discernible; the absolute contrast of angular velocity grows in latitude
and radius with more rapid rotation. These profiles show some weak asymmetry between thenorthern and southern hemispheres, but only at lower rotation rates and at high latitudes.
This behavior would likely disappear if very long time averages were taken. Variation between
the hemispheres should be expected as the patterns of convection, and thus the mean zonal
flows, are not symmetric about the equator.
The streamlines of the time-averaged meridional mass flux (ψ) are shown in the lastcolumn of Figure 6.3. The streamfunction ψ is defined such that hρum i = ∇ × ψ φ̂. The
sense of the flow is counterclockwise (CCW) for the blue contours and clockwise (CW) for
the red. There is typically a CCW cell of meridional flow in the northern hemisphere, anda CW cell in the southern hemisphere, that stretches from the inner edge of the tangent
cylinder to the pole, indicating a poleward meridional flow in the upper convection zone
and an equatorward flow near the base of the convection zone. These high latitude cells arevery large-scale flows and are thus constrained by the global spherical geometry. These flows
reside mostly on contours of constant radius and possess weaker meridional flows than the
low latitude cells. Outside the tangent cylinder the mean meridional flow is dominated by
the columnar structures seen in the Mollweide projections of the radial velocity in Figures
6.3(a), (e), and (i).
The meridional flows associated with these structures have streamlines that are quite
complicated, having multiple cells in radius and latitude. The trend is that the number

219
of cells increases with rotation rate, primarily with more cells in radius. These cells arealigned with the rotation axis and extend from the upper radial boundary to the equator
where they close, indicating that there is very little mean cross-equatorial flow or flow acrossthe tangent cylinder. The net effect of these low latitude cells is to largely isolate theequatorial region from higher latitudes inside the tangent cylinder. This behavior can beunderstood by considering the latitudinal connectivity of the downflow networks establishedin the convection, at low rotation rates there is a tendency for a large degree of latitudinalconnectivity that is reduced at higher rotation rates due to the influence of the Coriolis
forces. The meridional circulations established within the Case B simulations are similar inmorphology to those in the Case A simulations. They are multi-cellular at low latitudes andsingle celled at high latitudes, as seen in the streamlines of the meridional flow (ψ) shown in
Figures 6.4(d), (h), and (l).
6.3.4

Mean Temperature Profiles

The azimuthally- and time-averaged profiles of temperature fluctuations about thespherical mean are shown for cases Case A5, Case A10, Case A20 (Figures 6.3(c), (g), and(k)). A clear pattern arises in these simulations, where the poles are warm and the equatoris cool, while mid-latitudes are cooler yet, relative to the background temperature. Thesefluctuations are nearly constant on cylinders, especially at high latitudes. Such a pattern
has also been found in simulations of global-scale convection in G-type stars (Brown et al.,2008). The origins of these fluctuations are discussed in detail in §7.3.
These fluctuations about the spherical mean are generally still small, but become moresignificant for more rapid rotation rates. In Case A20 for instance, the temperature fluctuations near the top of the simulation are about 3% of the background value or about 3000 K.
All of the cases presented here have similar latitudinal profiles in temperature. The latitudinal temperature contrast between the equator and 60◦ latitude increases markedly at higher
rotation rates as shown in Table 7.1. Indeed, the primary difference between the Case B

220
(b)

(a)

-0.34 / 0.32
-0.14 / 0.14 Ω0

(c)

(d)

-0.50 / 0.40

-2.0 / 1.7

Figure 6.5: Large-scale shear instability existing within the stable region prints through theconvection zone as evident in Case A20. Mollweide projections of the radial vorticity (ωr )are shown relative to the vorticity of the reference frame (Ω0 ) and at four depths with them = 0 component removed (a) 0.81 R, (b) 0.86 R, (c) 0.91 R, (d) 0.96 R. Dark tones indicate
negative radial vorticity.

simulations and those shown for the Case A simulations are that the absolute latitudinal
contrasts of differential rotation and temperature fluctuations are larger (see §7.5), but thestructure is largely the same with slow, warm poles and a cool, fast equator (Figure 6.4(c),(g), and (k)).
6.4

Modulated Convection and Shear Instabilities
Spatially modulated convection, such as that discussed in Brown et al. (2008), becomesreadily apparent in Case B30 (Figure 6.4(i)). The convection in the equatorial region is lessvigorous at certain longitudes and more so at others. In this simulation, there is a roughly
m = 1 modulation in the equatorial convection. This modulation is increasingly pronouncedwith depth and is most evident in the stable region where the overshooting convection andgravity waves are confined to a horizontal region covering roughly a quarter of the sphere221
at the equator and converging at the poles. This localized overshooting convection feeds
back on to the upflows initiated at the base of the convection giving them a substantiallylarger temperature perturbation than elsewhere on the sphere generating faster flows. Thiscase is also slightly off the path in parameter space followed in the other cases in that the
diffusivity is the same as in Case B20 and thus is about 25% higher than expected given its
bulk rotation rate. This is to avoid the high cost of increasing the resolution of the simulationand the subsequently longer evolution time.
The presence of a sufficiently large differential rotation within a convectively stableregion can give rise to a global, shear instability similar to Rossby-Haurwitz or planetary
waves (Haurwitz, 1940; Longuet-Higgins, 1964; Kitchatinov, 2010). These waves can efficiently transport energy and angular momentum between the equatorial region and thepoles. Such an instability has arisen in the radiative zone of Case A15, Case A20, and Case
B20, where several low degree spherical harmonic modes are unstable as are visible for Case
A20 in Figure 6.5 and for Case B20 in Figure 6.6. These motions have a pressure and vorticity
signature that print through the convection zone, surviving despite the vigorous convection
present there as in Figure 6.6(b). The instability of these modes is sensitive to the magnitudeof the differential rotation and to its latitudinal profile (Watson, 1981; Kitchatinov, 2010).
Furthermore, it has significant impact upon the overshooting convection, allowing mixing togreater depths (Figure 7.3(c)).
To assess the character and the stability of the saturated state of the instability established within our simulations, we consider the simplified problem of a two-dimensional(horizontal) inviscid shear instability of a differentially rotating fluid as in Watson (1981);
Kitchatinov (2010). In such an approximation, the radial fluid motions are assumed to besmall compared to the horizontal flows and the effects of sound waves are ignored (Watson,
1981, hereafter W81). The neglect of radial fluid motions is justified given that the ratio of
the standard deviations of the radial velocity and the latitudinal velocity is less than 0.02
throughout the stable region in cases Case A20 and Case B20, which agrees well with the

222
(a)

-0.18 / 0.17 Ω0(b)-0.52 / 0.39

Figure 6.6: Large-scale shear instability existing within the stable region prints through theconvection zone as evident in Case B20. Mollweide projections of the radial vorticity (ωr )are shown relative to the vorticity of the reference frame (Ω0 ) and at four depths with them = 0 component removed (a) 0.86 R and (b) 0.92 R. Dark tones indicate negative radial
vorticity.

assumption in W81 that this ratio is proportional to Ω2 /N 2 , where N is the Brunt-Väisälä
frequency and Ω is the angular velocity as a function of latitude. Therefore, the dominantsource of energy for any instability is the kinetic energy of the shear rather than the baroclinic potential energy. These approximations leave us with three equations: the continuity
equation and the θ and φ momentum equations. These equations can be reduced to one
by the introduction of a toroidal streamfunction ξ, such that ρ~u = ∇ × ξr̂. If we further
consider only normal modes, with a time and longitudinal dependence of exp [im (φ − σt)],
we are left with a Sturm-Liouville problem that solves for the latitudinal dependence of thestream function ξ:

223




 ′ ′
1 − µ ξ + K (µ) −
2

′′

m2c
1 − µ2[Ω (µ) (1 − µ2 )],K (µ) = −
Ω − Ω1ξ = 0,(6.13)′′

where Ω1 is the value of Ω for which [Ω (µ) (1 − µ2 )] = 0. From Equation (6.13) it isclear that a version of Rayleigh’s necessary criterion plays a role for this instability. This′′

criterion is that [Ω (1 − µ2 )] must change sign somewhere, where µ = cos θ and primes
denote derivatives with respect to µ. This alone, however, is not enough to ensure instability.
Indeed, the magnitude of the differential rotation plays a key role in determining the stabilityof a given mode [W81]. The full solution of this problem for an arbitrary differential rotation
is given in Appendix C. As a further restriction upon which modes may be unstable, there
exists a critical longitudinal wave number mc above which the modes are stable. Thus, only
the longest waves may be unstable. This critical value is shown in W81 to be the largest
eigenvalue of the Sturm-Liouville problem of Equation (6.13).
Figure 6.7(a) shows the differential rotation profile for Case B20. There is only asmall latitudinal region where the mean angular velocity satisfies the Rayleigh criterion thattriggers this instability (Figure 6.7(b)). The most unstable modes are of azimuthal orderm = 5, 6, and 7, as can be ascertained from Figure 6.7(c). This agrees well with the sphericalharmonic modes of the radial vorticity that have the most power within the stable region.
A program that solves these linearized equations for the eigenvalues and eigenfrequencies
produces very similar results (see Appendix C), with the magnitude of the imaginary part ofthe mode set peaking at m = 2 for Case B20. A similar analysis for Case A20 shows peaks
at m = 3 and m = 5, with the latter being the maximum.
Of further interest is that not only do these modes arise, but they are sustained by theconvection. These unstable modes will grow until the instability is saturated, at which point
they become efficient at transporting angular momentum in latitude between the equator andthe poles. This mechanism quenches further growth of the modes, and without a mechanism

224(Ω-Ω0)/Ω0(a) 0.1
0.0
-0.1

-0.2
1
0
-1
-2
-3
-4
-5
(c) 100(Ω-Ω1)K(θ)/Ω0

(b)

K(θ)

80
60
40
20
0
-90

-60

-30

0
30
Latitude (deg)6090Figure 6.7: (a) The radially and time-averaged differential rotation profile (Ω − Ω0 ) /Ω0
within the stable region of Case B20. (b) The latitudes at which the Rayleigh criterion issatisfied can be seen where the function (Ω − Ω1 ) K(µ) crosses the horizontal line, whichindicates zero. (c) The function K(µ) associated with this differential rotation profile isplotted with the four most likely excited modes of m = 5, 6, 7, and 8 shown as dashed, red,green, and blue lines respectively.

to sustain the profile of the angular velocity that satisfies the Rayleigh criterion the amplitudeof the flows associated with the instability would decay. However, the turbulent overshooting
convection continually resupplies angular momentum to the stable region, thus maintaining a
differential rotation with a shear unstable profile. This leads to the shear instability persisting
throughout the statistically steady portion of the time evolution of these simulations.
Since the shear unstable modes are saturated and persist, they impact the convection
by coupling into buoyantly unstable upflows near the lower boundary. The upflows rise
carrying some of the vorticity imparted upon them toward the surface, as can be clearly seenin the radial vorticity shown at several radial levels starting in the stable region in Figure

225
6.5(a), and then proceeding into the convection zone and toward the upper boundary in
Figures 6.5(b-d).

Chapter 7
Examining the Convection Zone Dynamics of F-type Stars

After the broad overview of results in Chapter 6, the focus is narrowed to a single
case (Case A10) in order to illustrate the detailed dynamics occurring within each of thecases. The chapter then concludes with scaling results with mass and rotation rate. Indeed,
it is found that the differential rotation becomes much stronger with more rapid rotationand larger mass, scaling as ∆Ω ∝ M3.9 Ω0.6
0 . Accompanying the growing differential rotation
is a significant latitudinal temperature contrast, with amplitudes of 1000 K or higher inthe most rapidly rotating cases. This contrast in turn scales with mass and rotation rateas ∆T ∝ M6.4 Ω1.6
0 . On the other hand, the meridional circulations become much weakerwith more rapid rotation and with higher mass, with their kinetic energy decreasing as
KEMC ∝ M−1.2 Ω−0.8.
07.1

Examining Dynamics within Case A10
The convective patterns, like those discussed in §6.3.1, are complex and time dependent

with asymmetries between upflows and downflows due to mass conservation in a stratified
medium. The downflows are cold, fast, narrow and border their more leisurely ascending
counterparts. The convective cells are outlined by the dark downflows visible in the radial
velocities shown in the top four panels of Figure 7.1.

227
7.1.1

Typical Flows and Thermal Properties

The archetypal convective cell is defined by a central broad upflow bordered by narrow
downflows. Indeed, the color tones in Figures 7.1(a)-(d) are meant to construe both thesign of the temperature fluctuations (Figures 7.1(e)-(h)) associated with these structures aswell as the direction of the flow. Namely, the upflows are warm and the downflows are cool
relative to the mean temperature. Radial velocities that are within one σ of the mean at
mid-convection zone for the flows in Case A10 are between −230 m s−1 for downflows and
190 m s−1 for upflows, where the rms radial velocities are 150 m s−1. Large and small scale
thermal structures are visible in Figures 7.1(e)-(h), where the large m = 0 temperature
perturbations arising from the thermal wind balance have been removed. The average difference of temperature fluctuations between the central upflow and the bordering downflows
defining the convective cell are about 45 K at mid-convection zone and increases with height.
A large scale structure corresponding to a m = 1 spherical harmonic mode is also visible.
This global scale pattern persists for long periods of time and is most evident in the more
rapidly rotating cases.
The effects of the stratification are visible in the patterns of the convection in Figures
7.1(a)-(c). The convective cells near the surface in panel (a), show the greatest asymmetry
in areal coverage and rms radial velocity between the upflows and the downflows. Thisasymmetry is reduced at greater depths as in panel (c) as the upflows and downflows become
increasingly symmetrized both in the rms velocities achieved and in areal extent due to thelarger local density scale height. The temperature field appears slightly more diffuse than
the radial velocity due to the thermal diffusion being four times greater, as reflected in the
Prandtl number being 1/4 at all depths. However, it shows the changes in the morphology of
the convection that are similar to those of the radial velocity, where the longitudinal extent
and magnitude of the temperature fluctuations of the convective cells decreases with depth.
The areal filling factors of the upflow and downflows are roughly 2/3 and 1/3, re-

(a)

(b)

(c)

-150 / 110-250 / 230-100 / 100

(e)

(f)

(g)

-37 / 39

-28 / 26

-22 / 23(d)-0.08 / 0.08 m s-1(h)-43 / 42 K

Figure 7.1: Radial velocities (ur ) and temperature fluctuations (T ′ ) at several depths in Case A10. The temperature fluctuations
are taken about the m = 0 component, removing the largest-scale variations. The quantities ur and T ′ are shown in orthographic
projection at several radial levels with r/R equal to (a, e) 0.97, (b, f) 0.92, (c, g) 0.85, (d, h) 0.80. Light tones are warm upflows,
while dark tones are cool downflows. Panels (d) and (h) show flows within the radiative zone.
228

229
spectively, when averaged over the convection zone. There is a further asymmetry in themagnitudes of the radial velocity within these flows, with the downflows being faster thanthe upflows. Indeed, the distribution of the radial velocities has a large negative skewness
within the convection zone. The distribution of radial velocities in the downflows has a tailthat is about twice as long as the distribution of velocities in the upflows. These properties
are similar to the flows established in solar simulations detailed in Miesch et al. (2008). Further, the mean negative radial velocity is about twice as large as the mean positive radial
velocity. This asymmetry in the radial velocities reflects the density stratification and massconservation, with the caveat that the degree of asymmetry is dependent upon the level of
turbulence within a simulation where higher Reynolds numbers lead to more asymmetry.
The downflows, visible as dark lanes in (Figure 7.1(a)-(c)), are initiated by cooling near
the surface and sustained through buoyancy driving within the bulk of the convection zone.
The cooler and more dense downflows entrain fluid from the surrounding upflows throughmass conservation and diffusion. These processes act to erode the density contrast between
the downflow and the surrounding fluid. Thus, the downflows experience less buoyancy
driving as they fall deeper into the convection zone. Even without a significant buoyancy toprovide a downward acceleration, they have enough momentum to continue under their own
inertia. Once the downflows reach the bottom of the convection zone, they become buoyantly
braked as they enter the stable region. However, they continue into the uppermost reaches of
the radiative zone through their own inertia, as in Figures 7.1(d) and 7.1(h). Eventually they
are fully braked, but not before they have penetrated a significant distance into the stableregion. Such overshooting motions can excite gravito-inertial waves that propagate within
the stable region. These waves are eventually dissipated through diffusive processes, but canplay a role in the redistribution angular momentum and energy (Rogers & Glatzmaier, 2006;
Brun et al., 2011).

230
7.1.2

Energetics of the Convection

These flows give rise to the transport of energy and momentum throughout the domain.
The energy transport by resolved convective motions will not in general be equivalent to themixing-length prescriptions assumed in the 1D solar structure model, so the simulation must
adjust accordingly. In a steady state, an equilibrium is reached that balances the different
components of the mean radial transport of energy.
The large outward enthalpy flux Len seen in Figure 7.2(a) exceeds the luminosity ofthe star at its peak and is balanced primarily by the unresolved eddy flux Led near the topof the domain and by the radiative flux Lrd below mid-convection zone to produce a nearly
constant stellar luminosity throughout the domain. While the upflows contribute about 10%of the enthalpy flux, the dominant factor contributing to the enthalpy flux are the downflows.
These flows are cool relative to the mean value as seen in the temperature fluctuations of
Figure 7.1; this correlation between the negative fluctuating temperature and the negative
radial velocities leads to a radially outward enthalpy flux. There is significant latitudinalvariation in the enthalpy flux, with the largest positive flux in the polar regions and withfluxes that are about 30% less outside of the tangent cylinder. Near the base of the convection
zone, indicated by the dashed line in Figure 7.2, the enthalpy flux becomes negative due tothe overshooting convection as will discussed in the following section. The overshooting
enthalpy flux attains its most negative value near the poles, though when integrated overthe stable and overshooting regions it peaks at the tangent cylinder. This variation in theenthalpy flux reflects the latitudinal variation in the depth of the overshooting.
The asymmetrically larger velocities in the downflows give rise to an inward kinetic
energy Lke (inside the tangent cylinder) that is due to the cubic dependence on the velocity.
Outside the tangent cylinder, the columnar equatorial flows produce a net outward kinetic
energy flux, dominated by the large zonal velocity component, that are on average slightly
larger than the higher latitude components and lead to a small net outward flux for Case231
A10 (Figure 7.2(a)).
7.2

Convective Overshooting
The Péclet number (Pe = Re′ Pr) of the radial flows is of essential importance indetermining the properties of the overshooting convection (e.g., Brummell et al., 2002; Brunet al., 2011). Small values of Pe give rise to convective overshoot, in the sense that thestratification remains sub-adiabatic. Large Pe values, those much greater than one, promote
efficient entropy mixing such that the stratification becomes nearly adiabatic, and the baseof the convection zone can spread downward in time if nothing acts to oppose the spreading.
In the stable region, the Peclet number is about 1/40 at the lower boundary, around 1 in theovershooting region and approximately 10 in the bulk of the convection zone. Therefore, it
is expected that the motions at the base of the convection zone are overshooting rather thanpenetrative. This is indeed the case; as demonstrated in Figure 6.2(a), where the evolved
mean entropy gradient is shown for cases Case A10 and Case B10. However, all the cases
reported on here possess a strongly sub-adiabatic region below the convection zone that isessentially fixed in time. It extends from the lower boundary to the radius at which theentropy gradient changes sign, or the bottom of the convection zone. The mixing there isinefficient and the stratification remains largely unchanged as indicated by the Peclet number
there (Zahn, 1991; Spiegel & Zahn, 1992; Brummell et al., 2002). The width of this layer
is likely overestimated because this width scales as the square root of the filling factor ofthe downflowing plumes, which are likely larger in these simulations than in the actual stars(Zahn, 1991; Rempel, 2004).
In the region of overshooting, on the other hand, the entropy gradient is steepened
significantly compared to the stellar model. This is a consequence of the downflows within
the convection dragging cool fluid to the bottom of the domain, where they are then quickly
heated due to buoyancy braking in the region of overshooting as will be seen in the following
section. While there is not a physical boundary at the bottom of the convection zone,

232

(a) 3

1.2 M
Lrd

2
L/L

Len
1

Led
Lke

0
-1
0.80

Lvd
0.83

0.86

0.89
r/R

0.92

0.95

(b) 4

0.98

1.3 M

Lrd
3

Led

L/L

Len
2
1
0
-1
0.86

Lke

Lvd
0.88

0.90

0.92
r/R

0.94

0.96

0.98

Figure 7.2: Time- and horizontally-averaged radial energy fluxes for (a) Case A10 and (b)Case B10. Fluxes are shown as luminosities (L = 4πr 2 F ) and taken relative to the solar
luminosity with the total (Ltot ) shown as a solid black line. The components of the total flux
are the kinetic energy flux (Lke , dashed blue), enthalpy flux (Len , solid blue), radiative flux
(Lrd , solid red), unresolved eddy flux (Led , solid orange), viscous diffusive flux (Lvd , dashedred). The vertical dashed line indicates the radius at which the mean entropy gradientchanges sign.

the rapid increase in the entropy gradient acts very much like a wall that rapidly slows
downflows. However, these flows overshoot into the stable region where the work done bybuoyancy rapidly brakes and heats them. This allows the fluid to thermally equilibratewith the surrounding fluid. The stronger downflows, those that are cooler and faster thanthe majority in the convection zone, retain enough inertia to flow past the bottom of the233
convection zone and continue to be buoyantly heated. They quickly become warmer than thesurrounding fluid but are still traveling into the stable region. Thus, correlations between
the fluctuating temperature and the radial velocity in the overshooting convection gives rise
to a substantial negative enthalpy flux as seen in Figure 7.2 where it reaches around half ofa solar luminosity. This is in contrast to 1-D stellar structure models, where there is eithera small, negative overshooting enthalpy flux or none depending upon how the overshooting
is parameterized.
In Figure 7.3, we quantify the beginning of the overshooting region by the radius rc
where the enthalpy flux becomes negative, the radius at which the entropy gradient equals
zero (rbcz ) is demarked by the dashed line, and the bottom of the overshooting region occurs
at a radius where the negative enthalpy flux of the overshooting convection equals one percentof the most negative value (ro ). It is clear from Figure 7.3(a) that the convection zone in Case
A10 is most shallow around ±20◦ (at about 3% of the stellar radius R), and is deepest at theequator (0.0075 R), with an average width of 0.015 R. The overshooting region in Case B10
is 1.5% of the stellar radius at its deepest around ±20◦ , 0.009 R at its most shallow at theequator, and 0.0125 R on average. The depth of overshooting at the equator is reduced as thecentroid of the region of the overshooting occurs at a greater depth due to the penetration of
the convection, resulting in convection that feels a much more stiff stratification. At higher
latitudes, the centroid of the overshooting region occurs at a larger radius, which results ina larger region of overshooting as the convection feels a less stiff stratification.
The bottom of the overshooting region is deepest at low latitudes and most shallow at
the poles in Case A10, which implies that the radiative zone is slightly prolate. In Case B10,
the bottom of the overshooting region is essentially spherically symmetric. This is also true ofthe radius of the base of the convection zone (rbcz ), which has almost no latitudinal variationin both cases. The radius at which the enthalpy flux becomes negative (rc ) is everywherelarger rbcz . This property arises from some of the downflow plumes having a change in thesign of their fluctuating temperature (from cool to warm) that precedes their arrival at rbcz .

234

Radius (r/R)(a) 0.85

rc

0.84
0.83
0.82

ro

0.81
0.80
(b) 0.90
0.89
Radius (r/R)

Convection Zone

Radiative Zone

rc

Convection Zone

0.88

ro

0.87
0.86
0.85

(c)

Radiative Zone
0

15

0

5

30

45
60
Latitude (deg)

75

90

30

35

0.4

do/HP

0.3
0.2
0.1
0.0
10

15

Ω/Ω

20

25

Figure 7.3: Changes in the radial extent of overshooting motions with latitude. Shown asa shaded region below rbcz for case (a) Case A10 and (b) Case B10. The radii where theenthalpy flux is negative encompasses the full shaded region. The radius at which there is atransition to the convection zone is denoted by rc . The lower boundary of the overshooting
motions is denoted by ro . The scaling with rotation rate of the depth of the overshooting
region relative to the pressure scale height at rbcz (do /HP ) for cases A and B.

The downflows are diffusively heated by the warm upflows that surround them, which occurs
because the time for the downflow to cross the distance between rc and rbcz is longer thanthe horizontal diffusion time across the downflow. This results in a negative enthalpy fluxthat begins before the bottom of the convection zone and builds to its most negative values
as the downflows continue into the stable region where they are buoyantly heated.

235
We have conducted simulations with the same initial stellar model, but without theregion of penetration, these are Case A5i, Case A20i, and Case B20i. It is clear from Table
7.1 that these simulations have much weaker latitudinal entropy and temperature contrasts
than their counterparts with a stable region. However, the equator to 60◦ differential rotation
contrast is only slightly lower. Hence, the inclusion of a stable region alters several aspects ofthe mean thermal state and flows, where it primarily leads to a tilting of the contours of theangular velocity toward being conical. It also leads to a stronger scaling of the latitudinaltemperature gradient with rotation rate. The primary driver behind this is that there isan enhanced equatorward latitudinal enthalpy flux relative to the cases that have a stableregion. The meridional circulation kinetic energy, on the other hand, is lower in cases with aregion of overshooting than those without. This behavior arises due to a change in the axial
torque near the lower boundary. The impenetrable lower boundary imposes a much stronger
constraint upon the flows, indeed all flows must halt at this point. This means that thedivergence of both the Reynolds and the viscous stresses must be larger in simulations with
an impenetrable boundary when compared to simulations with a region of overshooting, as
is indeed the situation in Case A5i, Case A20i, and Case B20i. The equation of gyroscopic
pumping (Equation (7.6)), implies that this leads to a larger meridional circulation kinetic
energy as is evident in the energies given in Table 7.1.
7.3

Thermal Structuring
As an example of the mean thermal structure established in these simulated stars,

consider the time- and azimuthally-averaged temperature fluctuations for Case A10 (Figure
7.4(a)). It is clear that the perturbations do not have a zero mean, as expected from thebroken spherical symmetry arising from the rotation of the system. The temperatures range
from the hot poles at 840 K to the cool equator at −280 K with yet cooler mid-latitudes.
This equator to pole temperature contrast is slightly more than two orders of magnitude
greater than the observational limits on the latitudinal temperature profile of the Sun (Rast236
et al., 2008). However, these variations are taken relative to the mean temperature thatvaries between 1.3 × 106 K at the bottom of the domain and 105 K at the top, so they arestill relatively small being of order one percent. These averaged temperature fluctuations arenearly constant on cylinders at latitudes above ±45◦ , while it has a positive radial gradientat lower latitudes. This distribution of temperature fluctuations impacts the local energy
flux balance, and is intimately linked to the differential rotation and meridional flows, suchas those seen for Case A10 in Figures 6.3(f) and 6.3(h).
7.3.1

Principal Contributions to Temperature

The time- and azimuthally-averaged temperature fluctuations are decomposed into thefluctuating pressure and entropy components (T = P/ρcP + T S/cp ), from which it becomesapparent that the pressure is the largest contributor (Figure 7.4(b) and 7.4(c)). This isespecially true at latitudes above 45◦ , where the pressure perturbation has nearly radial
contours and contains about 80% of the amplitude of the temperature fluctuations. At thesehigher latitudes, the entropy contribution tilts the contours of the temperature fluctuations
to be nearly cylindrical. At low latitudes, the two components play more equal roles, at least
within the deep convection zone. Figure 7.4(d) exhibits the latitudinal structure of the mean
temperature fluctuations at several depths: near the top of the domain, at mid-convection
zone, and in the stable region. There is a substantial change in the temperature structurewith depth outside the tangent cylinder where it swings from −280 K in the stable region
to almost 0 K near the top of the domain at the equator. The entropy and pressure act intandem to depress its value at the base of the convection zone and the pressure drives its
value above zero near the surface where the entropy contribution wanes. Figure 7.4(e) shows
the mean entropy fluctuations at the same depths as in panel (d). The entropy fluctuations
increase nearly monotonically in latitude and have little radial variation.
In Case A10, the background temperature decreases by about a factor of 10 across theconvection zone, and since it weights the entropy fluctuations it causes an equivalent decrease

237

(a)

(b)

(c)

810 K
P
ρ cP
-160 / 750

T

TS
cP
-150 / 150-260 K(d) 800
600
T (K)

400
200
0
-200

(e)

15

105 S/cP

10
5
0
-5
-10
-90

-60

-30

0
Latitude (deg)

30

60

90

Figure 7.4: Fluctuating thermal structures found within Case A10. Azimuthally-averaged
variations are shown for the temperature (a) and the scaled contributions from the fluctuations of (b) pressure [P/ρcP ] and (c) entropy [T S/cP ]. Cuts through (d) the averaged
temperature fluctuations and (e) the averaged entropy fluctuations are displayed at fixedradii: at the top of the domain (solid line, 0.98 R), at mid-convection zone (dotted line,
0.89 R), and in the stable region (dashed line, 0.80 R). The vertical lines denote the edges of
the tangent cylinder.

238
in them (Figure 7.4(c)). The decrease is in part due to the weak radial differential rotation inCase A10 (Figure 6.3(f)). Indeed, a time steady meridional force balance leads to a linkingof the entropy to the ageostrophic component of the differential rotation (see §7.3.2), which
implies that a strong radial differential rotation could lead to a larger entropy contribution.
But as is evident in Figure 7.4(e), the entropy perturbations have little variation in radius,reflecting the weak radial differential rotation. In an azimuthal and time average of the
momentum equation (Equation (6.1)), the time and azimuthal derivatives of the mean flows
vanish, leaving the advection, viscous, Coriolis, buoyancy, and pressure gradient terms. If
we keep only the largest contributing terms, the gradient of the zonally averaged pressure
fluctuations is then primarily balanced by three terms: the buoyancy arising from the meandensity fluctuations, the Coriolis forces acting on the mean zonal flow, and the centripetal
force due to the differential rotation arising from the largest of the advection terms. These
terms combine to give∇hP i ≈ ρλ Ω2 − Ω20 λ̂ − hρigr̂,(7.1)

where Ω = Ω(r, θ) is the angular velocity and λ = r sin θ is the distance from the rotation
axis. When integrated over the domain, the relative difference between the pressure gradientand the terms on the right hand side (R) of the Equation (7.1), which is given by k∇hP i −Rk/k∇hP ik, shows that this balance is satisfied to within 2% over the entire domain. The
agreement is best at higher latitudes, but is significant everywhere. The Coriolis and theadvection terms comprise about 66% of the magnitude of the gradient, while the buoyancy
terms make up about 34%. Thus, the structure of the pressure arises primarily from thegeostrophy of the flow (Pedlosky, 1982), which is particularly important for the low latitude
peak in the pressure and temperature in the upper reaches of the convection zone. The
buoyancy arising from the mean density fluctuations, however, is also important. This term
follows from the equation of state (Equation (6.8)), and arises from the baroclinicity of theflow. Indeed, the differential rotation plays a crucial role in determining the structure of

239
these fields.
7.3.2

Thermal Signature of Differential Rotation

If the planetary vorticity of a system is large relative to the turbulent vorticity, theRossby number is small, which is the case in the simulations conducted in this paper (seeTable 2). In these systems, the dynamics that arise tend to minimize variations acrosscylinders aligned with the rotation axis. Indeed, in constant density settings, the TaylorProudman theorem states that there is no variation along these cylinders. On the other
hand, the density and pressure isosurfaces can lose alignment in stratified flows, leading
to baroclinicity and flows that can violate this constraint as is seen in the cylinder crossing
meridional circulation in Figures 6.3 and 6.4. One might suspect that, given the large equator
to pole thermal contrast established in these simulations, a strong meridional circulation
would arise to homogenize the temperature. However, as will be shown in the following
section (§7.4), the differential rotation and turbulent Reynolds stresses mediate the mean
meridional flow. Thus another mechanism must act to maintain this large thermal contrast.
In the simulations reported on here, there is a balance between latitudinal enthalpy flux andentropy diffusion that establishes the thermal contrast, which in turn evolves in conjunction
with the meridional flow and differential rotation. It is clear from the previous section thatquasi-geostrophic flow accounts for a portion of the average temperature fluctuations, where
the differential rotation leads to temperature perturbations, while another mechanism is
responsible for the ageostrophic component.
The bulk rotation of these simulations is sufficiently rapid to increase the axial alignment of the convective structures, which leads to a more cylindrically aligned transport ofenthalpy and angular momentum. The domains are highly stratified and thus baroclinicity
plays an important role in them. These tendencies lead to a mean thermodynamic state
and the two mean flows (the meridional circulation and the differential rotation) that areclosely coupled and maintained through the time-averaged properties of the turbulent trans-

240
port of energy and momentum. We here explore the connection between the mean thermal
structures and the differential rotation by examining the zonal vorticity equation, which isthe azimuthal component of the curl of the momentum equation divided by the backgrounddensity as detailed in Brun et al. (2011). We begin with the time- and azimuthally-averaged
zonal vorticity equation that has been divided by Ω20 in order to aid our assessment of how
the various terms scale with rotation rate. It is important to note that the advection terms
can be split into three contributing pieces, those due to the mean zonal velocity, the mean
meridional circulation, and the Reynolds stresses arising from the fluctuating velocities. The
advection term arising from the mean zonal velocity cancels the contribution from the Coriolis force leaving the quadratic dependence upon the angular velocity seen on the left handside (LHS) of the following equation
Baroclinicity

Ro

}|
{
z }|DR { z21
λ ∂Ω∂S ∂hP ig ∂hSi
= 2
+ 2
2
Ω0 ∂z
Ω0 rCP ∂θΩ0 rρCP ∂r ∂θ
1
1
1+ 2 hRSi + 2 hMCi + 2 hVSi,ΩΩΩ| 0 {z } | 0 {z } | 0 {z }
Ro′2

Ro2mc

(7.2)

Ek Ro

where z = r cos θ is the axial coordinate and the overbraces and underbraces indicate thescaling of a particular term with its associated Rossby number (Ro) and label the baroclinic
terms. The baroclinic terms have been simplified using Equation 6.3. The viscous stresses
hVSi scale as the Ekman number (Ek = Ta−1/2 ) times the Rossby number. The turbulent
Reynolds stresses hRSi arising from the fluctuating velocity component subsequently areproportional to the fluctuating Rossby number (Ro′), and the advection of the meridionalvorticity by the meridional circulation hMCi) vary with Romc . These stresses are given
explicitly by

(b)

(a)

(c)

0.14
LHS
LHS

(e)

(d)

(f)

0.18
1.2 M

-0.14

Baroclinic
Baroclinic-0.14 /-0.14
0.14 / 0.14

Difference
Difference-0.027-0.027/ 0.027/ 0.027

LHS
LHS

1.3 M

Baroclinic
Baroclinic-0.18 /-0.18
0.18 / 0.18

Difference
Difference-0.057-0.057/ 0.057/ 0.057-0.18

Figure 7.5: Components of the time- and azimuthally-averaged zonal vorticity balance. (a) Depicts the left hand side of Equation(7.2), (b) the baroclinic terms, and (c) their difference, which is composed of the viscous and Reynolds stresses, for Case A10.
(d, e, f) Show the same quantities for Case B10. The color tables used in each panel are scaled by the indicated values.

241

242

E
D
E D u′ ω ′ E D
φ λ− ω ′ · ∇u′φ
hRSi = u′ · ∇ωφ′ +λD u′ ω ′ E D∂ ln ρ E
λ φ− ωφ′ u′r,−λ∂r
∂ ln ρ
hωφ ihuλi− hωφ ihur i,
hMCi = humc i · ∇hωφ i −
λ
∂r


E
E
1D∂
1 ∂ ln ρ∂ D
hVSi =∇ · D · θ̂ .
∇ · D · r̂ −
−
+
ρr ∂θ
r
∂r
∂r

(7.3)
(7.4)
(7.5)

Here λ is the cylindrical radius; ωλ and uλ are the cylindrically radial directed vorticity
and velocity, respectively. Within the convection zone of these simulations, the pressure
contribution to the baroclinic terms is small, but it becomes appreciable in the stable region
and is retained. The Reynolds stresses of the fluctuating velocities (u′ ) in Equation (7.3)
possess two primary components of stretching and advection. The first two terms of thisequation are the advection of the zonal vorticity and the second two are the vortical stretchingof the zonal velocity, and the last term arises from the compressibility of the flows. The
terms associated with the meridional circulation (umc , Equation (7.4)) are similar in thatthe meridional flow advects meridional vorticity and retains the effects of compressibility,
but they lack vortical stretching with contributions from only the component due to thegeometry (hωφ ihuλ i/λ). The definition of D from Equation (6.5) is used here for the viscousstress terms given in Equation (7.5).
Consistent with the strong thermal gradients seen in Table 3 and discussed in §7.3,
Figure 7.5 demonstrates the extent to which the baroclinic terms are linked to the differential rotation through a meridional force balance. Indeed, from Equation (7.2) it is clearthat the cylindrical gradient of the square of the angular velocity (labeled LHS in Figures
7.5(a) for case Case A10 and 7.5(d) for Case B10) is almost entirely balanced by the latitudinal gradients of the entropy and pressure. These two thermal gradients arise due tothe baroclinicity of the flows and are labeled as such in Figures 7.5(b) and 7.5(e). The entropy gradient dominates the meridional force balance within the convection zone because

243
the pressure gradient is weighted by the nearly adiabatic mean entropy gradient. Within
the stable region, however, the mean entropy gradient is about four orders of magnitudelarger than in the convection zone, leading to a significant contribution form the latitudinal
gradient of the pressure. The baroclinic terms in the bulk of the convection zone change sign
at higher latitudes. This reversal in sign occurs at a lower latitude in the 1.3 M⊙ cases, atabout ±45◦ rather than ±60◦ as in the 1.2 M⊙ cases. The presence of the latitudinal pressure
gradient causes the large negative structure that occurs around the tangent cylinder in thenorthern hemisphere (positive in the south) to stretch to the poles in the stable region.
The balance, however, is not perfect as there are effects near the upper boundary thatare primarily due to the Reynolds stresses associated with the turbulent velocity field, as
has been seen in previous simulations of G-type stars (e.g., Brun et al., 2011; Brown et al.,2008; Brun & Toomre, 2002). These departures from a pure thermal wind balance are easily
seen in Figures 7.5(c) and 7.5(f), where the difference between the angular velocity gradientand the baroclinic terms are shown for Case A10 and Case B10. It is also apparent that the
turbulent Reynolds stresses play more of a role in the 1.3 M⊙ simulations than in those of
the 1.2 M⊙ F-type stars, where the rms values of the Reynolds stresses contribute to about
10% of the balance versus 5% for the lower mass 1.2 M⊙ F-type stars. This is expected
given that the fluctuating Rossby number in Case B10 is larger than in Case A10 as seenin Table 2. There are additional contributions from the viscous stresses and the advection
terms associated with the meridional circulation, but they play a much smaller role in themeridional force balance than the baroclinicity and turbulent Reynolds stresses. Indeed,
the importance of these various terms scale well with the values of the Rossby numbers
given in Table 2, with the viscous stresses being proportional to Ek Ro ≈ 10−4 and hMCi
varying as Ro2mc ≈ 10−4 . For comparison, hRSi scales as Ro′2 ≈ 10−1 . Accordingly, we may
safely neglect these small contributions to the meridional force balance due to hMCi and
hVSi. This leads to an expected proportionality of the baroclinic terms in Equation (7.2) of
RoDR + O (Ro′2 ) ≈ Ro.

244
7.4

Maintaining Mean FlowsThe convective motions in these simulations are influenced by the bulk rotation, spher-

ical geometry, and stratification of the domain. The mean flows and thermodynamic structures, such as those described above, are established and maintained by the transport ofmomentum and energy supported by these motions. Here we will consider how the transport of angular momentum leads to a co-evolution of and a direct link between the meridional
circulation and the differential rotation. To illustrate the subtle balances achieved within
this turbulent convection that lead to the maintenance of these mean flows, consider the timeand azimuthally averaged zonal component of the momentum equation (Miesch & Hindman,2011), which yields the evolution equation for the specific angular momentumhihρui · ∇L = −∇ · Fλ λ̂ + Fz ẑ ≡ T ,
∂Ω
,
∂λ
∂Ω
Fz = ρλhu′z u′φ i − ρνλ2,∂zFλ = ρλhu′λu′φ i − ρνλ2

(7.6)
(7.7)
(7.8)

where L = λ2 Ω = λ (huφ i + λΩ0 ), λ = r sin θ is the cylindrical radius, and u′ = u − hui isthe fluctuating velocity, λ̂ is the cylindrical radial unit vector, and hui is the axisymmetricvelocity. The advection of angular momentum by the meridional flow could be incorporated
into the flux, but is retained separately for later analysis. The axial torque T is the divergenceof the sum of both the radial and cylindrical angular momentum fluxes, which are due to theviscous stresses arising from the mean zonal velocity and the Reynolds stresses produced by
the turbulent alignment of the cylindrical and radial velocity components with the fluctuating
longitudinal component. These transport mechanisms act in concert during the evolutionof a case to accelerate the equator and decelerate the poles until a steady state is reached
in which the axial torques act to balance the meridional advection of angular momentum,
while also helping to maintain the differential rotation. The axial torque for Case A10, as
well as the angular momentum fluxes that contribute to it, are shown in Figure 7.6.

245

(a)

(b)

(c)

4.2 x 106TorqueTorque

RS RS

VS

VS

-4.7 x 106<Fλ>z (1037 g s-2)

(d)
5
0
-5
0.6

<Fz>λ (1036 g s-2)

(e)

0.7

0.8
λ/R

0.9

1.0

6
4
2
0
-2
-4
-6
-1.00

-0.75

-0.50

-0.25

0.00
z/R

0.25

0.50

0.75

1.00

Figure 7.6: The axial torques that balance the meridional advection of angular momentum
are shown for Case A10. (a) The net axial torque of the right hand side of Equation (7.6).
(b) The axial torque due to Reynolds stresses (RS). (c) The axial torque due to viscousstresses (VS). (a)-(c) are scaled relative to the net torque and shown in units of g cm−1 s−2 .
(d) Vertically integrated angular momentum fluxes in the cylindrical radial direction (hFλ iz )and (e) cylindrical angular momentum fluxes integrated over the cylindrical radius (hFz iλ )with Reynolds stresses as blue lines, transport by the meridional circulation in green, viscousstresses in red, and the total in black. The dashed line in (d) indicates the value of λ at thetangent cylinder at the base of the convection zone. In (e) the dashed line shows the valuesof z along the tangent cylinder until it intersects the upper radial boundary.

246
In a statistically steady state where ∂L/∂t ≈ 0, the angular momentum advected by
the meridional flows must balance the net torque T as in Equation (7.6). Although theangular velocity profile Ω of these simulations is roughly conical at most latitudes, as shownin Figures 6.3 and 6.4, the specific angular momentum L is much more cylindrical, increasingaway from the rotation axis due to the weighting by the square of the lever arm. For some
intuition, Equation (7.6) can be understood by considering the largest component of theleft hand side, hρuλ i∂L/∂λ ≈ 2Ω0 λhρuλ i. This approximate relationship implies that theaxial torque is balanced largely by the cylindrically radial component of the meridional flow.
Moreover, this can be interpreted as hρuλ i ≈ T /2Ω0 λ namely that an axial torque induces
a meridional flow either toward or away from the rotation axis depending on its sign. Thus,
Equation (7.6) implies that a net prograde torque (T > 0) will induce a steady meridional
flow away from the rotation axis, while a retrograde torque (T < 0) will generate a flow
toward the rotation axis. Thus, the steady state meridional circulation is achieved in partthrough the Coriolis force acting on the mean zonal flow, a process referred to as gyroscopic
pumping (e.g., McIntyre, 1998; Wood & McIntyre, 2007; Miesch & Hindman, 2011).
7.4.1

Decrypting the Axial Torques

Figure 7.6 depicts the balance of the axial torques expressed in Equation (7.6) thatis established in Case A10, averaged over approximately 30 rotation periods. The sum of
the torques due to the Reynolds and viscous stresses (Figures 7.6(b) and (c)) balances themeridional advection of angular momentum so closely that it is visually indistinguishable
from Figure 7.6(a), and thus the left hand side of Equation (7.6) is not included in the
figure. This indicates that the mean flows have indeed reached an equilibrium state, as
can be deduced from the very small departure from zero of the integrated total cylindricalradial and axial angular momentum fluxes shown in Figures 7.6(d) and (e). The net axial
torque shown in Figure 7.6(a) illustrates that there is a prograde equatorial torque andretrograde torque inside the tangent cylinder as one would expect given the sense of the247
differential rotation (see §6.3.1 and §6.3.2). The mean net torque is sustained over a long
period of time, as the average taken here is over several hundred rotational periods. Given
that energy is conserved within the system, this implies that energy is being transferred fromthe thermal state to the kinetic energy of the system and through dissipation it is transferredback to the thermal reservoir.
By comparing Figures 7.6(a) and (b), it is apparent that the Reynolds stresses are
the dominant mode of angular momentum transport inside of the tangent cylinder. In theregions above 45◦ latitude, the axial component of the Reynolds stress transports angularmomentum outward along cylinders and is balanced by the meridional circulation advecting
angular momentum inward, as is seen in Figures 7.6(a) and (e). At mid-latitudes between
the tangent cylinder and about 45◦ , the radial component of the Reynolds stress angularmomentum flux is positive, so angular momentum is transported toward the surface, but itthen reverses sign above 45◦ . The torque resulting from the divergence of these Reynolds
stress angular momentum fluxes is negative at mid-depth, decelerating the mid-convection
zone, and positive at the boundaries, thus accelerating the upper convection zone and theovershooting region (Figure 7.6(b)). Inside the tangent cylinder, the downflow plumes arenearly aligned with the rotation axis and tend to conserve their angular momentum. This
supports an inward angular momentum transport and results in the zonal deceleration seen
at higher latitudes and throughout the bulk of the convection zone. However, the downflows
are buoyantly braked and diverge as they approach the bottom of the convection zone andenter the overshooting region. During this process the flows begin to dilate and becomeanti-cyclonic so that they accelerate the lower convection zone and overshooting region.
This torque results in an a gyroscopically pumped meridional flow that is equatorward near
the base of the convection zone and poleward closer to the surface, where it must cross L
contours (Figures 6.3(h) and 7.6(b)).
Outside the tangent cylinder, where the columnar convective structures are so prominent, the viscous and the Reynolds stresses are of nearly equal importance. The integrated

Mass

1.2 M⊙

1.3 M⊙

Case KE/106
A5
7.46
A7
15.0
A10
29.4
A15
67.1
A20
91.9
A5i
6.92
A20i
28.5
B8
2.23
B10
6.70
B15
15.5
B20
35.0
B30
26.4
B20i
15.3

Table 7.1
Global Properties of the Evolved Convection
6
KEDR /10
KEMC /103
KEC /106∆Ωr /Ω0 ∆Ω60 /Ω06.35 (85.1%) 5.39 (0.072%) 1.10 (14.8%)
0.065
0.195
13.9 (92.9%) 4.31 (0.029%) 1.07 (7.11%)
0.063
0.212
28.4 (96.6%) 3.68 (0.013%) 1.01 (3.42%)
0.054
0.211
65.3 (97.4%) 2.44 (0.004%) 1.74 (2.60%)
0.043
0.200
90.4 (98.4%) 1.99 (0.002%) 1.51 (1.65%)
0.037
0.177
5.65 (81.7%) 6.04 (0.087%) 1.26 (18.2%)
0.057
0.199
27.5 (96.5%) 2.48 (0.009%) 0.99 (3.47%)
0.049
0.114
1.41 (63.1%) 2.93 (0.131%) 0.82 (36.8%)
0.030
0.101
5.99 (89.3%) 2.46 (0.037%) 0.71 (10.7%)
0.038
0.177
14.6 (94.4%) 1.97 (0.013%) 0.86 (5.55%)
0.041
0.204
33.8 (96.5%) 1.53 (0.004%) 1.22 (3.49%)
0.036
0.224
25.6 (97.2%) 0.88 (0.003%) 0.75 (2.83%)
0.024
0.142
14.6 (95.5%) 1.43 (0.009%) 0.69 (4.51%)
0.038
0.182

∆T60
-117
-246
-501
-1048
-1635
-101
-714
-190
-613
-1444
-2799
-3745
-1797

∆S60
do /HP-207500.088-448300.089-930100.101-184400 0.339-290300 0.332-13900—-44940—-310000.121-111670 0.122-267900 0.137-503200 0.334-812000 0.358-224100—

Table 7.1: Total (KE), differential rotation (KEDR ), meridional circulation (KEMC ), and convective (KEC ) kinetic energy densities
are both averaged in time and volume and are shown in units of erg cm−3 , and relative to the total kinetic energy in parentheses.
The spatial averages are taken over only the convection zone to allow comparison between the cases with and without overshooting.
The relative radial and latitudinal differential rotation (∆Ωr /Ω0 and ∆Ω60 /Ω0 ), the latitudinal temperature difference (∆T60 , inK) the latitudinal entropy difference (∆S60 , in erg K−1 cm−3 ) are measured between the equator and 60◦ latitude (averaged overboth hemispheres). The depth of the overshooting region (do ) is shown measured relative to the pressure scale height (HP ) at
the base of the convection zone and is horizontally averaged.

248

249
fluxes shown in Figures 7.6(d) and (e) reflect this balance. A strong columnar feature saturates the color table and is visible in the viscous and Reynolds stress torques (Figures
7.6(b) and (c)), where it nearly follows the tangent cylinder as is clear in Figure 7.6(e). The
Reynolds stresses arise from the thin, spiraling, columnar equatorial convective cells. The
properties of the torque due to Reynolds stresses associated with these structures are clear;
they act to reduce the angular momentum at depth and increase it nearer the surface, whilethe viscous stresses attempt to counteract this action but are slightly weaker. The axiallyaligned equatorial flows seen in these simulations are strongly affected by the boundary geometry. In Busse (2002), it is shown that the convex curvature of the caps of the cylindricalannular convective domain geometry leads to prograde tilted columnar cells. These tilted
columns transport prograde momentum outward and retrograde angular momentum inward,
as occurs with the equatorial convection cells here. This occurs because the thermal Rossby
wave associated with the columnar convection tends to propagate faster in the upper convection zone and slower in the lower convection zone (Busse & Hood, 1982). The prograde
torque of the Reynolds stresses above the mid-convection zone and the retrograde torquein the lower convection zone and overshooting region, visible in Figure 7.6(b), indicate thatthese equatorial convective structures do indeed play a crucial role in the angular momentum
transport. This is manifest in the radial dependence of the angular velocity (Figures 6.3 and
6.4.
There are two sets of equatorial columns in this simulation. One set of columns are
confined to the deep convection zone, terminating at the base of the convection and extending
up to the middle of the convection zone. A second set of columns exists at the upper boundaryand extend down to the middle of the convection zone. This situation arises because theradial differential rotation is large enough to shear the columns and because the density scaleheight acts to confine columns near the upper boundary, which can be understood as a resultof a large Taylor number. This configuration of columns manifests itself in the axial torqueas a region of negative torque at the base of the convection zone, a positive torque just below

250
the middle of the convection zone, a negative torque at the base of the outer columns and apositive torque near the upper boundary as in Figure 7.6(b).
In the limit that the outer boundary becomes spherical, the equatorial columns described in Busse (2002) become increasingly sheared in longitude with increasing latitude,
where the fastest prograde flows are at the equator and are slower near the latitude at which
the tangent cylinder intersects the upper boundary. This process is accentuated by the thinshell geometry of the F-type stars, where the spherical upper boundary has an increasing
slope as one approaches the equator, which in turn causes the phase speed of the thermal
Rossby waves to increase. This leads to faster equatorial flows and slower flows at higher
latitudes. The net axial torque reflects these effects, with a retrograde torque (flux divergence) at low latitudes between the tangent cylinder and about ±8◦ and a prograde torque(flux convergence) near the equator (Figure 7.6(a)).
7.4.2

Meridional Flows Induced by Axial Torques

The Reynolds stresses arising from the convective cells outside the tangent cylinder
are not perfectly cylindrical and have a convergent structure, where the latitude of the outerboundary of the cell is dependent on the spherical radius as is evident in Figures 7.6(a)and (b). On the other hand, the viscous stresses in the bulk of the convection zone arenearly cylindrical. The slight misalignment of these two torques gives rise to the net torque,as they almost cancel one another outside of the tangent cylinder (Figure 7.6(d)). Indeed,
the complicated morphology of the net torque gives rise to four reversals in the sign at theequator, resulting in four cells of low-latitude meridional flow (Figures 7.6(a) and 6.3(h)).
These cells close on the equator and end either at very low latitudes or merge and stretch
up to the tangent cylinder. At latitudes below the tangent cylinder, but above the lowest
latitudes, there are two meridional cells. In the northern hemisphere, a clockwise circulationis centered on the tangent cylinder and the counter-clockwise circulation at larger cylindricalradii is a combination of a cylindrical cell at mid-convection zone and a curved cell that

251
stretches along the outer boundary.
A similar balance of angular momentum fluxes is realized in both the Case B simulation
at 10 Ω⊙ and also at other rotation rates. The trend is that there are a larger number of
meridional circulation cells outside the tangent cylinder with faster bulk rotation rates, while
the equatorial region remains isolated from the higher latitudes by the meridional circulation
cells at the tangent cylinder. At higher latitudes inside the tangent cylinder, the Reynolds
stresses support a dual celled meridional circulation at higher rotation rates. These twocells are split in radius, one nearer the surface and one at depth. The upper cell connectswith the cell that sits on the tangent cylinder, moving fluid with lower angular momentum
from the upper layers at high latitudes to the base of the convection zone which is thenwhisked poleward by the polar circulation cell at the base of the convection zone. However,in cases with the shear instability (§6.4) there is a substantial peak in both the diffusive andReynolds stress components of the angular momentum transport at higher latitudes inside
the tangent cylinder not seen in other cases. The time dependence of the instability leads
to angular momentum fluxes that fluctuate much more than in the cases that do not havethis instability.
7.5

Scaling with Rotation and Mass
With our choices in the scaling of the diffusivities with rotation rate, the parameter

space covered by these simulations is essentially restricted to varying the rotation rate (orRossby number) and mass of the system. Thus, we explore the variation of the differential
rotation (∆Ω), meridional circulation (KEMC ), and latitudinal temperature gradient (∆T)
with these parameters. The latitudinal differential rotation is not known apriori and isfit for, however, the latitudinal temperature gradient follows directly from ∆Ω given thequasi-geostrophy and thermal wind balance of the flow. The energy associated with themeridional circulation also follows from the differential rotation, but through the mechanismof gyroscopic pumping and the influence of the Reynolds stresses.

252
The Rossby numbers of these simulations and those in Brown et al. (2008) (see Table
6.2 and Figure 7.7(a)) are fit as a function of mass (with M between 1.0 M⊙ and 1.3 M⊙) andbulk angular velocity (Ω0 ), which yields
Ro =

h|ω|i
∝ M5.7 Ω−0.77
0
2Ω0

(7.9)

This is close to the value expected from mixing length theory, where Romlt = vmlt /RΩ0 andvmlt = (L/R2 ρcz )1/3. In the mass range of stars covered here, the depth of the convection zone

decreases rapidly so we restrict our consideration to stellar masses between 1 M⊙ and 1.3 M⊙.
In this range, the mean density of the convection zone scales as ρcz ∝ M−15.5 , while the radiusand luminosity of the stars simulated here are proportional to R ∝ M and L ∝ M4 . Thisimplies that the mixing length velocity scales as vmlt ∝ M5.8 , which leads to a an estimate of
the Rossby number that scales as Romlt ∝ M4.8 Ω−1
0 . The exponents of the mass and rotation
rate for this estimate are close to the fit to the Rossby number shown above, but imply aslightly smaller dependence upon the mass and more dependence upon the rotation rate.
Numerical simulations, on the other hand, suggest that the Rossby number is inversely
proportional to a power of the rotation rate that is less than unity. This scaling is exhibited here, in G-type star simulations using ASH (Brown et al., 2008), in spherical shell
segments (Käpylä et al., 2011), and in Cartesian f-plane simulations (Käpylä et al., 2004).
The rotational scaling of the Rossby number in these simulations, with a power less thanone, is expected given the influences of rotation upon the convective patterns and the rms
velocities of the flow, effects that mixing length theory does not explicitly treat. One cansee this directly from the convective flux in the simulations, where it can be greater or lessthan the luminosity of the star depending upon the rotation rate. This leads to a mixing
length velocity that varies with rotation rate, and thus a more complicated dependence of
the Rossby number upon the rotation rate.
On our path through parameter space, a complication arises, where there is a thresholdbulk rotation rate Ωth below which the differential rotation becomes anti-solar. Such behavior

253
is seen in both simulations of spherical domains and Cartesian domains at various levels of
turbulence in both the ASH simulations (Brun & Palacios, 2009; Matt et al., 2011) and insimulations carried out in Cartesian domains (Käpylä et al., 2004) as well as in sphericalsegments (Käpylä et al., 2011). We observe that this transition between solar-like and antisolar-like differential rotation occurs within flows that have a Rossby number of nearly one.
It is also apparent that this change in the rotational behavior of the simulations is a functionof mass, as it occurs around 0.8 Ω⊙ for 1.0 M⊙ G-stars in Brown et al. (2008), near 4 Ω⊙
for the 1.2 M⊙ F-type stars, and at about 7 Ω⊙ for the 1.3 M⊙ F-type stars. We have beenworking to understand how and why there is this branch solutions possessing anti-solar-like
differential rotation, and will be subsequently reporting on it with models of F-, G-, andK-type stars. Briefly, this transition arises from over-driven convection occurring due to our
choices of boundary conditions, diffusion profiles, and initial stratification, all of which resultin a Rossby number is greater than one throughout the bulk of the convection zone and areversal in the sign of the Reynolds stresses for rotation rates below the threshold. However
due to this existence of this branch of solutions, we have chosen the bulk rotation rates ofthe simulations reported on here to be above this threshold rotation rate.
7.5.1

Differential Rotation

The relative latitudinal differential rotation (∆Ω/Ω0 ) established in these simulations
is measured between the equator and 60◦ latitude (Table 7.1). The relative radial differential rotation measured between the bottom of the domain and the top at the equator is also
shown in Table 7.1. The trend for increasing relative latitudinal differential rotation at lower
rotation rates, and decreasing ∆Ω/Ω0 at higher rotation rates is clear. There is a peak in∆Ω/Ω0 in both of the 1.2 M⊙ and the 1.3 M⊙ simulations, with the former achieving a maximum around 7 Ω⊙ and the latter at 20 Ω⊙. In Brown et al. (2008), a peak in the latitudinaldifferential rotation is reached at about 2 Ω⊙. These maxima, however, are sensitive to thelevel of turbulence in both the simulations as shown in Brown et al. (2008), which is likely

(a)

(b)

10

(c) 104

(d)
3 104

0

10

7

MCKE ( erg cm-3 )

103

3
2

∆T ( K )∆Ω ( rad d-1 )

Ro-1

5

10-1

102

101

104

3 103

103
1

10-2
1

2

5
Ω/Ω

10

20

100
1

2

3
Ro-1

5

7

10

1

2

3
Ro-1

5

7

10

1

2

3
Ro-1

5

7

10

Figure 7.7: Scaling of the inverse Rossby number with mass and rotation rate, as well as latitudinal differential rotation (∆Ω),
latitudinal thermal contrasts (∆T), and the volume averaged meridional circulation kinetic energy (KEMC ) with mass and inverse
Rossby number. (a) inverse Rossby number for F-type star Case A simulations (diamonds) and Case B simulations (triangles),and for 1.0 M⊙ G-type star simulations (crosses) [from Brown et al. (2008)]. The fit to the Rossby number is shown as solid
colored lines blue, red, and gray for the Case A, Case B, and G-type star simulations respectively. These symbols and coloredlines are employed in the following panels: (b) ∆Ω with the solid lines as in Equation (7.10). (c) ∆T with the lines given in
Equation (7.14). (d) KEMC with lines given in Equation (7.21).

254

255
also the case here but is not explicitly explored.
An intriguing feature of these simulations is that the absolute differential rotation inboth radius and in latitude increases with rotation rate (decreasing Rossby number), as
seen in Figure 7.7. It is also apparent that the mass of the star plays an important role
through its influence upon the depth of the convection zone as well as the strength of theconvective driving arising from the increased luminosity of the star. Indeed, we find herethat the absolute latitudinal differential rotation ∆Ω60 established in these simulations, andmeasured between the equator and 60◦ latitude, scales with the mass of the star and theinverse of the Rossby number as∆Ω60 = 5.3 × 10

−2

−1

rad d



M
M⊙

6.2

Ro−1.05 .

(7.10)

This scaling of the absolute latitudinal differential rotation with Rossby number andmass is shown in Figure 7.7(b). Although the differential rotation decreases consistently withincreasing Rossby number among all the cases studied here, there is some indication from thelevels of differential rotation achieved in Brown et al. (2008) that there may be a change inthe rate of increase of the absolute differential rotation at larger rates of rotation. This may
arise either because there is a maximum in absolute differential rotation or simply slower
growth at larger rotation rates. However, probing these higher rotation rates is beyond thecurrent capabilities of ASH due to the exclusion of centrifugal forces.
Mass also plays a crucial role in building increasingly strong differential rotation inhigher mass stars. The strong mass dependence of the fit in Equation (7.10) reflects themore vigorous convection and consequently larger Reynolds stresses that arise during thetransition from the Sun to the slightly more massive F-type stars. The dependence of themagnitude of the differential rotation upon the mass, or more concretely the dependence
upon effective temperature, is also clear in observations over a large range of temperatures
Barnes et al. (2005); Reiners (2006); Collier Cameron (2007). The scaling of the latitudinaldifferential rotation in Collier Cameron (2007) is ∆Ω ∝ Teff 8.6 ∝ M4.3 , where Teff ∝ M1/2 .

256
When the latitudinal differential rotation is fit for mass and bulk rotation rate, it scales as∆Ω60 = 8.3 × 10

−2

−1

rad d



3.9  0.6
M
Ω0
,
M⊙
Ω⊙

(7.11)and is therefore close to the observational mass scaling, within the bounds of the uncertaintyof the observational data. The reason for the discrepancy in the expected mass scaling
between Equation (7.10) and Equation (7.11) is that the above fit is applied to shifted data,
making the mapping between rotation rate and Rossby number more complicated. Thisis due to the fact that the rotation rate mass fit to the differential rotation of our models
crosses zero and becomes negative below a particular value of the rotation rate that depends
upon the mass of the star. We remove this variation by shifting the data so that they may
be fit consistently as in Equation (7.11).
7.5.2

Latitudinal Temperature Gradient

A striking property of the meridional force balance in Equation (7.2) is that increasingΩ0 leads to increasingly cylindrical rotation profiles, as one would expect from gyroscopic
pumping (§7.4). In the absence of baroclinicity, gyroscopic pumping will produce a meridional circulation that acts to make the contours of the angular momentum, and thereby the
angular velocity, more cylindrical (Miesch & Hindman, 2011). We find, however, that thebaroclinicity of the flows increases with more rapid rotation. The scaling of the latitudinal gradients in temperature and entropy increase at a rate that nearly keeps pace with thequadratic scaling of the rotation rate, reducing the tendency for increased alignment with therotation axis. This variation of the temperature with mass and Rossby number is shown in
Figure 7.7(b), where ∆T60 decreases substantially with increasing Rossby number (increase
with increasing rotation rate) as∆T60 = 5.3 K



16.1
6.4  1.6

Ω0
M
M
−2.3
Ro
= 5.3 K.
M⊙
M⊙Ω⊙

(7.12)

257
To understand the origin of these scalings, consider the latitudinal gradients of theentropy and pressure that scale with the ageostrophic component of the angular velocity,which in turn scales as the Rossby number. To find how the latitudinal gradient of thetemperature should vary with rotation rate, we combine the geostrophic pressure balance(Equation (7.1)) and the ageostrophic thermal wind (Equation (7.2)), which leads to thefollowing


2
∂hTi
rλT ∂ Ω2 − Ω20
.
=
∂θ
g ∂z
T

(7.13)

This allows a simple homology relationship to be developed with the assumptions thatthe effective temperature and the radius of the star depend upon only the mass of the staras mentioned above. We find that when integrated over latitude the temperature gradientbecomes∆T60 ≈ ∆T0



M
M⊙

2.5 



9.6
Ω0
M
∆Ω = ∆T0
Ro−1.3 ∆Ω.
Ω⊙M⊙(7.14)The solid curves in Figure 7.7(b) depict this homology relationship, where it overlays theactual values of the temperature contrast in the simulations.
7.5.3

Meridional Circulation Kinetic Energy

Volume-averaged energy densities for our simulations are shown in Table 7.1. At thelowest rotation rate, fluctuating convective kinetic energy (defined as 0.5ρhu′2 i, KEC ) andthe kinetic energy in the average differential rotation (0.5ρhuφ i, KEDR ) are comparable. As
the rotation rate is increased, the KEDR grows strongly and the convective energy decreases
slightly, leading the KEDR to dominate the total kinetic energy. The energy in meridional
circulations (0.5ρ(hvr i + hvθ i), KEMC ) is always small and decreases in both magnitude andpercentage of the total energy with more rapid rotation.
The volume averaged meridional circulation kinetic energy clearly decreases with rotation rate in both these simulations and those of (Brown et al., 2008). If we fit for thisscaling, as we have for the absolute differential rotation, we find that

258

−7.4
M
KEMC = 2.4 × 10 erg cm
Ro1.04 ,M⊙
−1.2  −0.8

Ω0
M
4
−3
= 2.4 × 10 erg cm
.
M⊙
Ω⊙
4

−3



(7.15)(7.16)

We can understand this scaling by returning to the equations of gyroscopic pumping
and considering the scaling of the axial torques. First, though, it is useful to extract thedependence of the streamfunction upon the gradients of the angular momentum and thetorques. Since the mean poloidal mass flux is divergence-less, the meridional flow can be
represented as a stream function ψ such that hρui = ∇ × ψ φ̂. Expressing Equation (7.6)
in terms of this streamfunction and then integrating the result clockwise from the lower
boundary along L contours we arrive
1
ψ=
λ

Z

λF
ds,
C |∇L|(7.17)

where C is the contour of integration. The denominator in this equation is dominated by
the λ component such that ∇L ≈ 2Ω0 λ. This reduces Equation (7.17) to the following
1
ψ≈
2λΩ0

Z

C

F ds ∝ M−7.4 Ro1.3 F [M, Ro] .

The meridional circulation kinetic energy is consequently proportional to2
Z ∇ × ψ φ̂1
dV ∝ M−1.3 Ro2.6 F [M, Ro]2 .
KEMC =V V
ρ

(7.18)(7.19)The proportionality of the torque F is difficult to predict apriori. One could assume that the Reynolds stresses follow the Λ effect (Rüdiger, 1989), in which case theaxial torque associated with the Reynolds stresses in our parameter regime should scale as2
ρcz vmlt
Romlt −1 ∝ M−4 Romlt−1 ∝ M9.8 Ω0 . However, this would lead to a KEMC that is con-stant with the rotation rate, which is clearly not the case here. Thus, without guidance topredict the scaling of the torque, it must be fit for. Indeed, we find that the Reynolds stressaxial torque scales as259

F ∝ M−3 Ro−0.6 .

(7.20)

Therefore, the KEMC that arises from gyroscopic pumping (Equation (7.6)) is proportional
to
KEMC ∝ M−7.3 Ro1.4 .

(7.21)

which correlates well with the KEMC found in our simulations as seen in Figure 7.7(d) andEquation (7.15). That is that the torque associated with the Reynolds stresses increases
with rotation rate, rather than decreases as expected from mean-field theory.
7.6

Conclusions
As is apparent from the observation of F-type stars, they rotate on average much more

rapidly than does the present Sun. In these stars, rotation must strongly influence the convective motions and lead to differential rotation. Such differential rotation has been observedin many F-type stars (e.g., Reiners, 2007). To assess how this differential rotation may arise,
we have explored here the effects of rotation rate, convective overshooting, and stellar mass
upon the patterns and mean properties of the global-scale convection in simulations of two
F-type stars.
We have found that the mean zonal flows of differential rotation become much stronger
3.9 −0.4
with more rapid rotation and larger mass, scaling as ∆Ω ∝ M3.9 Ω0.6
0 or as ∆Ω/Ω0 ∝ M Ω0(§7.5). This corresponds well with observational detections of differential rotation for increasing rotation rate (Reiners, 2007) and with mass (Barnes et al., 2005; Reiners, 2006; Collier
Cameron, 2007). Accompanying the growing differential rotation is a significant latitudinal temperature contrast, with amplitudes of 1000 K or higher in the most rapidly rotatingcases. This contrast in turn scales with mass and rotation rate as ∆T ∝ M6.4 Ω1.6
0 . Themaximum temperature contrast near the surface occurs between the hot poles and the cool

260
mid-latitudes at about ±30◦ , near the tangent cylinder. These temperature fluctuations
about the spherically symmetric mean temperature are decomposed into its geostrophic andageostrophic components, both of which are linked to the rotation of the star and its baroclinicity (§7.3). If this latitudinal temperature contrast prints through the vigorous convection
at the stellar surface, it may appear as an observable latitudinal variation in intensity. The
thermal contrast would presumably persist for long periods compared to stellar activity, offering a way to disentangle this intensity signature from that caused by spots of magnetismat the stellar poles.
Additionally, we have seen that the mean meridional circulation becomes much weakerwith more rapid rotation and with higher mass, consistent with the concept of gyroscopic
pumping, where the energy contained in these circulations drops approximately as KEMC ∝M−1.2 Ω−0.8. However, the scaling of these quantities are sensitive to the variation Reynolds
0stresses, and thus their variation with the level of turbulence in the simulations must be better
understood. As suggested in Brown et al. (2010, 2008), the weaker meridional circulations
in more rapidly rotating stars will have a strong impact on many theories of stellar dynamo
action, including the Babcock-Leighton flux-transport model favored for Sun-like stars as
recently investigated (Jouve & Brun, 2007; Jouve et al., 2010; Miesch & Brown, 2012).
Indeed, the lack of large single cells of meridional circulation at all latitudes in many models
of more rapidly rotating stars suggests that the most common profile used for these flows
may be oversimplified, and is actually associated with anti-solar-like differential rotation.
The simulations here include a portion of the stably stratified radiative interior below
the convection zone. It has been shown here and in previous works to have a significant
impact upon the morphology of the mean flows, and upon the magnitude of the latitudinal
thermal contrast. The largest impact is upon the thermal field, where it leads to about a
50% greater latitudinal temperature contrast but only about 10% more differential rotation.
However, its inclusion does lead to a tilting of the angular velocity contours to be moreconical than cylindrical, as occurs in cases without a stable region. The convection exhibited261
in the simulations with a stable zone yield a region of overshooting that is very slightly
prolate and occupy about 1% of the star by radius.
Three of the cases presented here have exhibited a global-scale shear instability that
operates within the stable region and persists for the duration of our simulations. Thisinstability is sensitive to the magnitude and the latitudinal profile of the differential rotation,and is present in both mass stars but only at higher rotation rates. It has a significant impact
upon the convection, causing greater depths of overshooting (§7.2) and prints through into
the convection zone (§6.4). The instability and angular momentum transport and mixing
properties associated with it will be examined in a subsequent paper (Augustson et al., 2012).
Our simulations are hydrodynamic, and it is possible that the inclusion of magnetism
may alter the scalings of differential rotation and latitudinal temperature contrast with rotation rate. Prior MHD simulations of stellar convection have demonstrated that in someparameter regimes, strong dynamo-generated magnetic fields can react back on the differential rotation, acting to lessen angular velocity contrasts as energy is transferred into themagnetic fields (e.g., Brun et al., 2005; Browning, 2008). Magnetic F-type star simulations
will be examined in the following Chapter 8. Unfortunately, the computational cost of such
simulations is too great to run simulations at each roration rate examined in Chapters 6
and 7, which would allow the assessment of whether the scaling trends identified here fordifferential rotation and latitudinal temperature contrasts as a function of Ω0 survive. Given
previous experience with G-type stars it is likely that the general functional dependencewith rotation rate is retained (Brown et al., 2010, 2011b). Our simulations stop short of the
turbulent stellar surface, and it is thus difficult to estimate how all of the properties of theconvection seen here may affect stellar observations in detail. However, we have found thatthe strong differential rotation and large thermal contrasts are a robust feature over a rangeof parameters, and it is likely that they will have observable consequences.

Chapter 8
Global-scale Simulations of Dynamo Action in F-type Stars“There are, it seems, two muses: the Muse of Inspiration, who gives us inarticulate visions
and desires, and the Muse of Realization, who returns again and again to say ’It is yet
more difficult than you thought.’ This is the muse of form. It may be then that form serves
us best when it works as an obstruction, to baffle us and deflect our intended course. It
may be that when we no longer know what to do, we have come to our real work and when
we no longer know which way to go, we have begun our real journey. The mind that is notbaffled is not employed. The impeded stream is the one that sings.” – Wendell Berry
Magnetic activity and differential rotation are commonly observed features on mainsequence F-type stars. We seek to make contact with such observations and to provide aself-consistent picture of how differential rotation and magnetic fields arise in the interiors ofthese stars. The 3-D MHD anelastic spherical harmonic (ASH) code is employed to simulate
global-scale convection and dynamo processes in a 1.2 M⊙ F-type star at two rotation rates.
The simulations are carried out in spherical shells that encompass most of the convection
zone and a portion of the stably stratified radiative zone below it, allowing us to explore theeffects a stable zone has upon the morphology of the global-scale magnetic fields. In thischapter, we examine the more slowly rotating case Case A10m that rotates at 10 Ω⊙ andpossesses persistent wreaths of magnetism established and maintained by dynamo action.
Compared to their hydrodynamic progenitors seen in the previous two chapters, the dynamo
states here involve a marked reduction in the exhibited latitudinal differential rotation, which263
also vary during the course of a cycle.
In Chapter 9, a cycling dynamo in the presence of a tachocline will be explored. These
two chapters represent work recently published in Augustson et al. (2013b)1 and are thus
largely restatements of those results, though with some additional thoughts included in §9.11.
As the primary author of the paper, I conducted all of the simulations, performed all of theanalysis, and wrote the paper. The coauthors provided advice and guidance throughout thisprocess, with Sacha Brun providing detailed comments and helpful conversations, as well assuggesting some lines of analysis for the polar-slip instability and the α-effect. Juri Toomre
had to bear reading through many drafts of the paper, and both his suggestions for the text
and for ways of pinning down the reversal mechanism were quite useful.
8.1

Introduction: Stellar Magnetic Fields and Rotation
Observations of main sequence stars across the Hertzsprung-Russell diagram indicatethat most stars rotate and that many possess both substantial magnetic variability and strong
axisymmetric magnetic fields (e.g., Donati & Landstreet, 2009). In particular, main-sequencelate-type stars such as the F-type stars exhibit a positive correlation between the stellar
rotation rate and surface magnetism (e.g., Pizzolato et al., 2003; Wright et al., 2011). The
precise origins of this magnetic activity and its correlation with rotation, however, remain
unclear. Convection and differential rotation are likely crucial elements in the dynamo actionthat manifests itself as starspots, coronal emission, and variable magnetic fields seen on thesurfaces of these stars (e.g., Noyes et al., 1984; Berdyugina, 2005; Miesch & Toomre, 2009;
Reiners, 2012).
The polarity reversals observed on the Sun occur roughly every 11 years and are theclearest evidence for cyclical magnetic field regeneration by dynamo processes. However,there have recently been detections of magnetic cycles in other stars as well. Advances
1

Augustson, K. .C, Brun, A. S., & Toomre, J. 2013, Dynamo Action and Magnetic Cycles in F-type
Stars”, ApJ, 777, 153

264

Figure 8.1: Observed latitudinal differential rotation (∆Ω) of τ Boötis, and of 1.2 M⊙ and1.3 M⊙ main sequence F-star models. Observed ∆Ω and equatorial rotation rate ΩEQ from A(Catala et al., 2007), B (Donati et al., 2008b), C1 -C3 (Fares et al., 2009), and D (Ammler-vonEiff & Reiners, 2012), with uncertainties in gray. For comparison, hydrodynamic simulationsof 1.2 M⊙ (red diamonds) and 1.3 M⊙ (red triangles) stars from Chapters 6 & 7, and our Case
A10m involving the MHD simulation of a 1.2 M⊙ star rotating at 10 Ω⊙ (blue diamond).

in asteroseismology enabled by the Kepler and CoRoT missions and in Zeeman-Doppler
spectropolarimetric imaging (ZDI) techniques have enabled long-term observations of manymagnetically active stars. Such measurements have revealed that cyclic magnetic reversals
are not a unique feature of the Sun, but seem to be a fairly common feature of late-type stars.
Some of the stars with the best observational coverage are F-type stars, being that they are
fairly luminous and have a large mean rotation rate. Several F-type stars have observations
indicating that they may harbor magnetic cycles, including τ Boötis, HD 49933, HD 52265,
and HD 181420, (Fares et al., 2009; Mathur et al., 2010, 2013).
Given the crucial role that differential rotation likely plays in the dynamo action occurring within these stars, we note that spectropolarimetric observations not only permit amapping of the surface magnetism of these stars, but also reveal information about the differential rotation on the surface of the star (e.g., Donati & Landstreet, 2009; Reiners, 2012).

265
Other methods can also shed light on the nature of the differential rotation of stars. For
instance, high resolution observations of stellar absorption lines can be processed to produce
an average line profile that is then Fourier transformed. The ratio of the values of the firsttwo zero crossings of this transform give a measurement of the differential rotation (e.g.,
Reiners, 2006, 2012). Starspot modeling of photometric data can also be used to infer the
inclination angle and differential rotation profile (e.g., Berdyugina, 2005; Strassmeier, 2009).
In another technique, photometric data can be analyzed for their asteroseismic signatures,
yielding information about the rotational behavior of a star (e.g., Gizon & Solanki, 2004;
Cunha et al., 2007). For the F-type stars, the Fourier analysis technique using stellar spectra
has produced the largest number of detections of differential rotation to date (Ammler-vonEiff & Reiners, 2012). In this work, 18 F-type stars are identified as having differential
rotation and it is shown that the horizontal shear expressed as a latitudinal angular velocity
contrast ∆Ω increases both with increasing effective temperature and with increasing angular rotation rate Ω, extending and corroborating earlier work (Barnes et al., 2005; Reiners,
2006). These data are shown in Figure 8.2 along with the differential rotation achieved inour earlier hydrodynamic simulations and the two MHD simulations presented here.
It is instructive to look in detail at one specific F-type star, namely τ Boötis, a 1.3 M⊙
star rotating at about 10 Ω⊙. This star is magnetically active and appears to reverse its
magnetic polarity with a period of about 2 years (Fares et al., 2009). The first measurementof the differential rotation of this star was made using the line profile analysis technique(Reiners, 2006), and again a year later in Catala et al. (2007). Subsequent measurements
were made using spectropolarimetric techniques as in Donati et al. (2008b). Three separate
observations were made in Fares et al. (2009) over the course of several months in 2008,
showing a potential variation in the magnetic field. Subsequent observations suggest that
τ Boötis does indeed possess stable, periodic reversals (Vidotto et al., 2012; Fares et al.,2013).
In Figure 8.1 it appears that in addition to the possible time varying magnetism of266

Figure 8.2: Absolute differential rotation contrast (∆Ω) seen in simulations and observed on
F-type stars. The observational uncertainties of the contrast and rotation rate are shown asgray lines. The values of the ∆Ω realized in hydrodynamic simulations are shown in blue
symbols (diamonds are 1.2 M⊙ simulations and triangles 1.3 M⊙), MHD simulations presented
here in red, and the observed F-type stars are shown as gray squares. Observational datafrom online sources provided by Ammler-von Eiff & Reiners (2012).

τ Boötis deduced from the ZDI studies, there may also be temporal variation in the differential rotation. Such variations have been observed using ZDI methods on more rapidly
rotating stars such as AB Doradus (Donati et al., 2003). It could be that the dynamo action
likely occurring within the star is impacting the differential rotation achieved at its surface.
Indeed, global simulations of G-type stars rotating faster than the Sun (Brown et al., 2010,2011b, hereafter B10 & B11), as well as the simulations of F-type stars being reported upon
here, show similar behavior. More precisely, the differential rotation achieved in these simulations varies in accord with the waxing and waning strength of the magnetic fields built by
dynamo action in their convection zones.

267
8.1.1

Global Simulations of Convection

The hydrodynamic behavior of F-type stars has recently been explored for a range of
masses and rotation rates using the anelastic spherical harmonic (ASH) code as in Augustson
et al. (2012), hereafter Chapters 6 & 7. It was found that the both the magnitude and thescaling of the differential rotation with rotation rate achieved in the simulations of Chapters
6 & 7 have a quantitatively good agreement with those observed in the F-type stars. We
show in Figure 8.1 a comparison of the latitudinal differential rotation ∆Ω observed onτ Boötis to those achieved in our simulations from Chapters 6 & 7 and this work withrotation rates closest to τ Boötis. We here will extend the results of Chapters 6 & 7 by
explicitly admitting dynamo processes, so that we might assess the impact of magnetismon the differential rotation achieved in these stars and comment upon what time scales ofmagnetic variability arise in them. To assess the role that magnetism may have in the internal
dynamics of these stars, we have selected two of the hydrodynamic 1.2 M⊙ simulations from
Chapters 6 & 7 that have differing rotation rates and have evolved them with magnetic
dynamo processes enabled. The latitudinal differential rotation in one of these magnetic
cases is shown in Figure 8.1 for comparison to both τ Boötis and our previous models, withthe minimum and maximum of its time varying value indicated by the error bar.
Helioseismology has presented detailed observational evidence for the presence of differential rotation in both latitude and radius in the Sun, and asteroseismology is beginning toprovide estimates of differential rotation in other stars (e.g., Thompson et al., 2003; Cunha
et al., 2007; Chaplin & Miglio, 2013). The definitive evidence for substantial differential
rotation in the Sun and advances in supercomputing technology spurred a major series ofnumerical experiments using global-scale simulations. These simulations aimed to establish which mechanisms might be sustaining the Sun’s differential rotation. Utilizing the
global-scale ASH code, early work revealed the role of turbulent Reynolds stresses and themeridional circulation in transporting angular momentum to the equator to yield rotation

268
profiles in the bulk of the convection zone largely in accord with helioseismic findings (Miesch
et al., 2000; Brun & Toomre, 2002; Miesch et al., 2006). Higher resolution simulations haveenabled detailed study of the dynamics of the giant cells of convection seen in these simulations (Miesch et al., 2008). Deeper 3-D models that capture both the solar convection zone
and much of the radiative interior are feasible and exhibit the nonlinear coupling between
the overshooting convection and the gravity wave field in the Sun’s interior (Brun et al.,2011).
8.1.2

Stellar Dynamo Modeling

A major focus of the global-scale 3-D simulations has been the solar dynamo and its
accompanying 22-year reversal period. Early simulations of the global-scale solar dynamo
with the magnetohydrodynamic (MHD) version of the ASH code showed that complex magnetic field structures were built through sustained dynamo action, with irregular polarity
reversals, and that the solar differential rotation profile was maintained despite the influenceof Maxwell stresses (Brun et al., 2004). The role of a tachocline of shear at the base of theconvection zone upon the dynamo action was examined by Browning et al. (2006). Other
3-D MHD simulation codes also capture aspects of the Sun’s complex behavior, for instancecyclical magnetic polarity reversals have been found in solar convective dynamo simulations(Ghizaru et al., 2010; Käpylä et al., 2012). There is yet no definitive answer as to what sets
the cycle period and its variability nor how it links to surface magnetic activity.
Recent MHD studies of G-type stars that rotate more rapidly than the Sun, reminiscent of the younger Sun, were conducted using ASH (B10; B11). The dynamo actionin these simulations generated significant mean fields as large-scale bands of toroidal field,
dubbed wreaths, in their convection zones without the need for a tachocline. These wreathsof magnetism were robust features over a range of rotation rates and diffusivities in theseSun-like G-type stars. The dynamo action in these simulations involves a rich set of behavior depending upon rotation rate, variously exhibiting intermittency, persistence, or cyclic

269
polarity reversals. Most recently, buoyant magnetic structures have been obtained in low
dissipation simulations of such a G-type star rotating at 3 Ω⊙, where they buoyantly riseaway from strong wreaths of magnetism built in its convection zone (Nelson et al., 2011,2013a,b).
The ASH code has also been used to simulate the dynamics of turbulent global-scalehydrodynamic and MHD convection in a range of main-sequence stars, spanning the mass
range between the M dwarfs and the more massive A-type stars. At the lower end of themass range, the global MHD simulations of fully convective M dwarfs have shown that they
likely possess little latitudinal differential rotation and strong axisymmetric fields (Browning, 2008), a finding that is commensurate with observations of differentially rotating andmagnetically active M dwarfs (Donati & Landstreet, 2009; Reiners, 2012). In studies of core
convection in A-type stars, the ASH simulations showed that dynamo action can support
magnetic fields in equipartition with the convective kinetic energy, with peak field strengthsof about 100 kG (Brun et al., 2005). When this convective core dynamo was threaded with apoloidal fossil field, it can achieve a ten-fold superequipartition dynamo state (Featherstone
et al., 2009). Such prior work with global simulations using ASH provides perspectives tothe dynamo processes that we now considers in F-type stars.
In turning to our two F-type star dynamo simulations, we provide in §8.2 the MHD
equations solved in ASH and the details of the setup of our numerical experiments. The
general properties of the convective patterns and mean flows established within the MHD
simulations are discussed in §8.4, where they are also compared to their hydrodynamic
counterparts described in Chapters 6 & 7. We then examine in §8.5 the magnetic simulationof a 1.2 M⊙ F-type star rotating at ten times the solar rate (10 Ω⊙) and its resulting persistentmagnetic wreaths. We explore in §8.6 the dynamo mechanisms maintaining these wreaths,
and in §7.4 the impact such structures have upon the transport of angular momentum andupon their thermal structure. We then examine in §9.1 the dynamo action with reversing
fields achieved in the more rapidly rotating (20 Ω⊙) of our two cases. In §9.2 we consider the270magnetic field morphologies achieved during the course of a polarity reversal, in §9.8 analyze
the evolving toroidal and poloidal field generation terms and examine the many time scales
involved in the cycles. In §9.13 we reflect upon the significance of these F-star dynamo
simulations.
8.2

Formulating the Problem
The generation of the global-scale magnetism and the maintenance of the large-scale

flows observed in the Sun and other stars involve the continuous transfer of energy and momentum amongst a vast range of spatio-temporal scales. This requires that subtle balancesarise between the dynamical components present within stars that both builds and sustains
strong, organized magnetic fields and bulk flows. Capturing all the relevant time and spatial
scales existing within stars from the global spherical geometry through the true dissipation
range within a direct numerical simulation in a single simulation is currently intractable.
Yet, studying carefully selected portions of these complex systems can yield insight into theobserved behavior of stellar magnetism and rotation.
Therefore, a modeling tool is required for stellar dynamo studies that parameterizes
unresolved spatial scales and eliminates processes that occur on time scales much fasterthan the relevant dynamics, along with taking account of the spherical geometry of the star.
The 3-D ASH simulation code is one such approach and it is employed in our study ofthe two F-type star models here. ASH solves the anelastic MHD equations of motion in arotating spherical shell using a pseudo-spectral method with spherical harmonic expansions
in the horizontal and Chebyshev polynomials in radius. Simulations using ASH capture theentire spherical shell geometry, allowing for global connectivity of magnetic structures andturbulent flows. However, the simulated turbulence is still quite removed from the intensely
turbulent conditions likely present in actual stellar convection zones.

Case Ω0 / Ω⊙
A10
10
A10m
10
A20
20
A20m
20

Table 8.1
Diagnostic Flow Parameters in the Four Cases
Nr × Nθ × NφRa
Ta
Re Re′ Rem
257 × 512 × 1024 4.22 × 105 2.41 × 106 240 52–
56257 × 512 × 1024 4.11 × 10 2.41 × 10 141 52
46
257 × 512 × 1024 1.96 × 106 2.41 × 107 592 79–
67257 × 512 × 1024 1.60 × 10 2.42 × 10 236 78
78

Re′m
–
17
–
26

Ro
0.39
0.37
0.24
0.25

κtop
6.35
6.35
4.00
4.00

τc
3.3
3.2
3.8
3.4

τν
159
159
252
253

Table 8.1: The depth of the convection zone d = r2 − rb (where rb is the radius of the bottom of the convection zone) is the relevant
length scale in the following parameters and is 1.3 × 1010 cm. We estimate the following diagnostic parameters at mid-convection
zone: the Rayleigh number Ra = ∆Sgd3 /cP νκ; Taylor number Ta = 4Ω20 d4 /ν 2 ; Reynolds number Re = vrms d/ν; fluctuating′
Reynolds number Re′ = vrmsd/ν; magnetic Reynolds number Rem = Prm Re; fluctuating Reynolds number Re′ = Prm Re′ ; Rossby
number Ro = ξ/2Ω0 , where ξ is the rms enstrophy. Thermal diffusion (κtop ) values are in units of 1013 cm2 s−1 . The mid-convection
zone local overturning time τc and the viscous diffusion time τν across the convection zone are quoted in days.

271

272
8.2.1

Anelastic Equations

In the anelastic approximation, the effects of a mean density stratification and Alfvénwaves are retained, while high frequency sound waves are filtered out. Thus the CourantFriedrichs-Lewy condition on the time step will be governed by the subsonic flow velocity
and Alfvén waves rather than the sound speed. In ASH, the anelastic MHD equations solved
are fully nonlinear in the momentum and magnetic variables (Glatzmaier, 1984; Clune et al.,
1999; Brun et al., 2004). However, the density, entropy, pressure, and temperature arelinearized about their spherically symmetric and evolving means ρ, S, P , and T respectively,
which are functions of the radial coordinate only. These linearized thermodynamic variables
are denoted ρ, S, P , and T . The equations solved in ASH retain physical units, are in
spherical coordinates (r, θ, φ), and are evolved in time t. These equations arecontinuity: ∇ · ρv = 0,

(8.1)

solenoidal: ∇ · B = 0,

(8.2)

∂P
= −ρgr + Λ,
∂r


∂v
1
momentum: ρ+ FBV Z P g= −ρv · ∇v − ∇P +∂tγP1
+ 2ρv × Ω0 +(∇ × B) × B + ∇ · D − Λr̂,4π
∂S
energy: ρT= −ρT v · ∇ S + S − ∇ · q + Φ,∂t∂B
induction:= ∇ × [v × B − η∇ × B] ,∂thydrostatic:

(8.3)

(8.4)
(8.5)
(8.6)

with the velocity field being v = vr r̂ + vθ θ̂ + vφ φ̂, and the magnetic field being B =Br r̂ + Bθ θ̂ + Bφ φ̂. Ω0 = Ω0 ẑ is the angular velocity of the rotating frame, ẑ is the direction
along the rotation axis, and the gravitational acceleration is g = −g(r)r̂.
The term FBV Z = (gρ)−1 dS/dr ensures that the motions in regions where the entropy
gradient is large are energy conserving (Brown et al., 2012). The Λ (r) term in Equation (8.3)
is a radial gradient of the mean turbulent pressure that maintains hydrostatic equilibriumas the mean state evolves. The viscous diffusion tensor D and the diffusive heating Φ are

273



1
Dij = 2ρν eij − ∇ · vδij ,
3


4πη
1
2
Φ = 2ρν eij eij − (∇ · v) + 2 J2 ,
3
c

(8.7)
(8.8)

involving the stress tensor eij , the effective kinematic viscosity ν, the magnetic resistivity η,and the current density J = c/4π∇ × B. The energy flux q is comprised of a radiation flux
(in the diffusion approximation) and a two-term anisotropic turbulent entropy diffusion flux,∂Sr̂,
q = κr ρcP ∇ T + T + κρT ∇S + κ0 ρT
∂r

(8.9)

with κr the molecular radiation diffusion coefficient, and cP the specific heat at constantpressure. The entropy diffusion flux has the effective thermal diffusivity κ acting on theentropy fluctuations with spherical harmonic degree ℓ > 0 and another κ0 acting only on
the spherically-symmetric (ℓ = 0) entropy. The form and impact of these diffusive transportcoefficients are discussed in §8.2.2. A calorically perfect gas equation of state is used for themean state, about which the fluctuations are linearized as
P = (γ − 1)cP ρT /γ,(8.10)

ρ/ρ = P/P − T /T = P/γP − S/cP ,(8.11)

with γ the adiabatic exponent. The anelastic system of MHD equations requires 12 boundary
conditions in order to be well posed. One of the primary goals of this work is to assess thegeneration of magnetic field and how it impacts the organization of angular momentum andenergy in our simulations. Thus we have opted for the following impenetrable, torque-free,
and flux transmitting boundary conditions∂  vφ  ∂S
∂  vθ 
=
=
vr =
= 0,
∂r r
∂r r
∂r

on r = r1 and r2 .

(8.12)The magnetic boundary conditions are perfectly conducting at the lower radial boundary(r1 ) and matching to a potential field at the upper radial boundary (r2 ), implying that274

Br |r1 = 0 and

B|r2 = ∇Ψ ⇒ △Ψ = 0,(8.13)

with Ψ the magnetic potential. The solution of Laplace’s equation defines the three components of B at the upper boundary. Further details of the implementation and formulationof the ASH code can be found in Clune et al. (1999) and Brun et al. (2004).
8.2.2

Experimental Configuration

The studies here explore two 1.2 M⊙ stars, one rotating every 2.72 days or 10 Ω⊙ (Case
A10m) and the other every 1.36 days or 20 Ω⊙ (Case A20m), as summarized in Table 8.1.
These cases were initialized from the evolved hydrodynamic Case A10 and Case A20 from
Chapters 6 & 7, after introducing a weak 10−4 G oblique dipolar field oriented at 45◦ relative to the rotation axis. The hydrodynamic progenitor cases were originally initialized
from a realistic 1-D stellar model evolved using the stellar evolution code MESA (Paxton
et al., 2011). The convective dynamics in these simulations slightly modifies the 1-D stellar
structure, establishing a mean thermodynamic state as shown in Figure 6.2. The radialboundaries of the original hydrodynamic simulations, and thus in the simulations presented
here, are carefully chosen locations within the star. The upper boundary is located below
the surface to avoid the very intense convection driven by radiative losses and ionization in
the uppermost layers of the actual star. Our models include a realistically stratified stablezone and possess a greater luminosity and a geometrically thinner convection zone than in
the solar models with wreaths studied as in B11. Despite being stable against convective
instability, a broad spectrum of motions arise in this region due to overshooting motions andwaves.
The strongly nonlinear interactions are resolved up to spherical harmonic degree ℓ =340 in the simulations here. However, because the convective motions and waves within theactual star exist on scales smaller than our grid resolution, these models should be regarded

275

Figure 8.3: Overview of convective patterns, mean flows, and mean temperature fluctuations in the hydrodynamic case Case
A10 (above) and its magnetic counterpart Case A10m (below). This serves to illustrate the modified and reduced latitudinaldifferential rotation in the magnetic case, and accompanying changes in the temperature and meridional circulation. (a, e)
Convective patterns in radial velocities (vr ) near the upper boundary in global Mollweide projection, with downflows in darktones and amplitude ranges indicated. (b, f) Time- and azimuthally-averaged angular velocity (Ω) with radius and latitude,with the fast, prograde equator in reddish and slower poles in bluish tones. (c, g) Azimuthally- and time-averaged temperaturefluctuations (T , with the ℓ = 0 component removed), with the warm poles in red tones and the cool equator in blue. (d, h)Streamfunction of the meridional flow (ψ), with clockwise (CW) flows blue and counter-clockwise (CCW) flows red. The dashedline in the second, third and fourth columns delineates the beginning of the stable zone.

276
as large-eddy simulations (LES). We parameterize the effects of these sub-grid scale motions
as effective viscous and thermal diffusivities and magnetic resistivity (ν, κ, and η respectively). Our parameterization of these motions are meant to mimic the mixing properties ofunresolved turbulent eddies. The functional form with radius of these diffusivities are as in
1

Chapters 6 & 7, where they are proportional to ρ− 2 and are tapered in the stable zone by afactor of 1000 below the radius (0.83 R). An effective magnetic resistivity is also introduced.
The magnetic Prandtl number Prm = ν/η = 1/3 is fixed throughout the domain, henceη = Pr/Prm κ. The thermal Prandtl number is Pr = ν/κ = 1/4. These diffusive parameters
give rise to radial variations in the Rayleigh, Taylor, and Reynolds numbers. Table 8.1 provides representative values of these parameters, which are sampled at mid-convection zone
depth.
8.3

Overview of Primary Findings
The global-scale convective hydrodynamics of F-type stars of differing masses were

explored in Chapters 6 & 7, with an emphasis on the sensitivity of the simulations to thebulk rotation rate. We found that the latitudinal differential rotation ∆Ω increased bothwith mass and with rotation rate. The increasing differential rotation induced a decreasing
meridional circulation kinetic energy because of weakened Reynolds stresses and an increasing latitudinal temperature gradient due to a strong thermal wind effect. The differential
rotation achieved in those models, however, was near the upper bound of the reported observational data over the range of rotation rates examined in Chapters 6 & 7. In addition to
general interest in dynamo processes, this encouraged us to explore the effects of magnetism
upon the mean properties of F-type stars.
By admitting MHD processes in two of these simulations, we find that both models
rapidly build strong toroidal magnetic structures through dynamo action. These prominent
toroidal magnetic fields (as in Figures 8.5, 9.2, and 9.3) may be called wreaths due to theirlongitudinal connectivity and filamentary latitudinal structure. Particularly in Case A10m,

277
the wreaths are persistent in their polarity, with their toroidal field sense maintained, though
they vary in time and in magnitude. In both of our magnetic cases, the differential rotationis reduced by at least 50% compared to their hydrodynamic counterparts. This prominentchange in rotation profile means that the latitudinal temperature contrasts are also sharply
diminished. As is discussed in §8.7, the meridional flow is significantly modified, for it isdifficult for such flows to cross the toroidal bands of field due to the Lorentz force backreaction upon them.
Much as in the studies by B11 of young G-type stars, we find here that lower diffusivities
and more rapid rotation leads to field reversals in both the toroidal and poloidal polaritiesof the wreaths. The toroidal fields of Case A20m possess a dipolar character (wreaths of
opposite polarity) in the stable zone and have a more complex latitudinal structure within
the convection zone. Indeed, in the convection zone the toroidal structures are fairly longlived octopolar wreaths. These multipolar wreaths slowly slip poleward, are eviscerated by
the small scale polar convection, and then are dissipated. The linking between this polar
slip, the poloidal field, and the wreaths generated in the convection zone that together leadto field reversals is explored in detail in §§9.2-9.12.
8.4

Convective Patterns and Mean Structures
Our hydrodynamical simulations of F-type stars in Chapters 6 & 7 revealed that theconvection involved structures that depend upon the location of a convective cell relative
to the tangent cylinder of the convection zone. This tangent cylinder is centered on andextends along the rotation axis of the simulation, and touches the bottom of the convection
zone. For the 1.2 M⊙ star simulations, this cylinder extends through the upper boundary
at ±32◦ latitude. There is a distinct transition in the nature of the convection inside (athigher latitudes) and outside the tangent cylinder. Outside the tangent cylinder (at low
latitudes) the convective structures are columnar rolls that possess an elliptical cross-sectionin the equatorial plane that is tilted prograde by Coriolis forces and that are aligned with the278rotation axis. However, the convective motions associated with these equatorial columns arelargely perpendicular to the rotation axis. Inside the tangent cylinder (at higher latitudes),the convective structures are also columnar and aligned with the rotation axis, but in contrast
to their low-latitude counterparts have convective motions mainly aligned with the rotation
axis.
This change of convective structure with latitude leads to differing transport propertiesof the flows, wherein the low latitude structures have Reynolds stresses that act to accelerate
the mean zonal flow in a prograde direction, whereas the high latitude structures act toaccelerate flows in the retrograde direction. When in a statistically steady state, the mean
angular momentum transport properties of the convection maintains both a steady zonal flow
and a steady meridional flow through gyroscopic pumping as discussed in Chapters 6 & 7. In
turn, it is shown there that this steady zonal flow establishes a largely geostrophic pressurebalance and a thermal wind balance that leads to the latitudinal temperature contrast.
In Figure 8.3, the hydrodynamic progenitor Case A10 from Chapters 6 & 7 and its
magnetic counterpart Case A10m are compared in terms of their radial velocity convective
patterns and the profiles in radius and latitude of the mean angular velocity, mean temperature fluctuations, and meridional circulation streamfunction ψ. The same is shown forCase A20 and Case A20m in Figure 8.4. It is clear from the snapshots of the radial velocities shown in Figures 8.3(a, e) and 8.4(a, e) that the convective patterns of the magnetic
cases are only moderately changed relative to the hydrodynamic progenitor cases. As in
the non-magnetic simulations, the convection is largely isotropic inside the tangent cylinder(high latitudes), whereas outside of this cylinder the convective structures are prograde-tilted
columns roughly aligned with the rotation axis (low latitudes). This is particularly true forCase A10m, where only small changes in the convective patterns are evident visible at low
latitudes. This is a result of the weakened latitudinal shear, which leads to less disruptionof the poleward edges of the low-latitude columnar convective cells.
For Case A20m, the convective columns at low-latitudes are of a smaller scale comparedFigure 8.4: As in Figure 8.3, but for Case A20 and and its magnetic companion Case A20m. (a, e) Although the radial velocity
patterns are similar, Case A20m involves two-fold faster convective flows. (b, f) There are substantial differences in the latitudinaldifferential rotation (∆Ω), here reduced by a factor of 3.2 between the hydrodynamic and magnetic case. (c, g) This reductionin ∆Ω leads to a thermal contrast reduced by a factor of 10, and in (d, h) to significantly modified meridional flows.
279

280
to Case A20. The regions that had latitudinal shear strong enough to suppress convection
now have elongated convective cells with radial velocities nearly twice as large in the magnetic
case and with much greater connectivity between the equatorial regions and mid-latitudes
extending to about ±30◦ . Also evident are the bands of weakened convection that coincidewith the locations of the strong toroidal wreaths at about ±60◦ and ±20◦ in latitude. These
wreaths tend to suppress convection where they are strongest due to the Lorentz force
feedback on the rising and descending flows.
The magnetic fields also have a more indirect impact on the mean flows and the thermodynamic state of the simulations. For instance, the angular velocity is decreased inamplitude for Case A10m and is both diminished and banded in character for Case A20m
(Figures 8.3(b, f) and 8.4(b, f)). As shown in Table 8.2, the mean difference in angularvelocity between 60◦ and the equator is reduced by 50% in Case A10m and by 60% in Case
A20m relative to their hydrodynamic counterparts. This leads to a substantial reduction in
the strength of the thermal wind, with the latitudinal temperature contrast (Figures 8.3(c,g) and 8.4(c, g)) dropping from 500 K to 300 K (a factor of 1.7) in Case A10m and from1600 K to 250 K (a factor of 6.4) in Case A20m.
Moreover, due to the modified Reynolds stresses and an introduction of the Maxwell
stresses, the meridional flow also changes in structure. In Case A10m, it is changed only
slightly with the cells occupying slightly larger extents in latitude (Figure 8.3(d, h)). The
hydrodynamic Case A20 has meridional flows that are multi-celled and possess a strong polarcell that stretches between the tangent cylinder and the pole, while in contrast the magnetic
case has a very weak polar cell with only two low latitude cells, one roughly at the tangent
cylinder and another of the opposite sense at lower latitudes near the equator (Figure 8.4(d,h)).

Figure 8.5: Snapshot of the magnetic field in Case A10m, coincident with radial velocities in Figure 8.3(e). (a) Radial componentof the magnetic field (Br ) shown near the upper boundary, where multiple zonal bands are visible in the convection zone, (b)Mean poloidal field lines shown with radius and latitude, exhibiting the quadrupolar nature of the poloidal field. (c) Toroidal
magnetic field (Bφ ) in the convection zone displaying the primary wreaths occupying lower latitudes and the secondary bands offield at higher latitudes. (d) Time- and azimuthally-averaged hBφ i. The toroidal fields obtain their greatest amplitudes in thestable zone, where the peak strength of hBφ i is 11 kG. Also evident is the radial dependence of the latitudinal connectivity ofBφ , where the wreaths are connected in the stable zone but are disrupted in the convection zone by the presence of weak field of
the opposite sense at mid-latitudes, as in (c).

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8.5

Dynamics within Case A10m
We begin with a detailed analysis of Case A10m, for it has the least temporal variability

in its magnetic structures. This allows us to more readily assess the various processes thatmaintain the mean magnetic fields and the mean flows. After considering here the generalproperties of the fields, we then turn in §8.6 to the generation of the strong toroidal wreaths,
examining the role of the various production terms in the mean induction equation. Thenin §8.7 we consider the impact of the magnetic field on the balance of angular momentum
transport terms and the thermal wind that accompanies it.
The magnetic fields as shown in Figure 8.5 for Case A10m are prominent longitudinal
structures, with the Bφ field within the stable zone positively oriented in longitude in thesouthern hemisphere and oppositely directed in the northern hemisphere as in panels (c, d).
The radial fields Br show the same sense, but are more tessellated with fine structure (panels(a, b)). In the convection zone, there are two toroidal magnetic structures at low latitudes.
The strongest of these two wreaths resides at about ±20◦ and is of the same sense as the fields
in the stable zone. There is also a weaker wreath situated at mid-latitudes in the convection
zone that rests on top of the strong, latitudinally-extended wreaths that are seated in thestable zone. The toroidal fields have peak field strengths that range between about 1.2 kG at
the upper boundary and 16 kG in the stable zone. The longitudinal component of the vector
potential is traced with streamlines in Figure 8.5(b). These streamlines show the orientationof the poloidal field, namely whether it is clockwise (CW) or counter-clockwise (CCW) in themeridional plane. The magnitude of the poloidal field is generally weaker than the toroidal
field, with about 600 G peak radial field strength and 2 kG peak meridional field strength.
The magnetic fields built and maintained through dynamo action in Case A10m wax
and wane in strength by about a factor of two with a period of about 1000 days. Similarly, thelatitudinal differential rotation varies by up to 20% on the same time scale (Figure 8.1). The
variations arise primarily from the interplay of the Lorentz forces acting on the mean flow

283
and the resulting reduced magnetic field generation due to the diminished shear. However,while the magnetic fields vary in magnitude, they do not reverse their polarity during the3000 day evolution of this case. This time span covers 1100 rotation periods, 15 magnetic
diffusion times, and many hundreds of convective overturnings. Thus these toroidal wreaths
are remarkably persistent magnetic structures.
In Case A10m, the strongest bands of toroidal field are established in the relatively
quiescent stable zone and extend into the churning convection zone. Given their long-lived
nature, these fields have both mean magnetic helicity and cross-helicity, which is achievedby the enshrouding character of the poloidal fields and to the stretching and twisting offield due to differential rotation and meridional flows. Due to the perfectly conducting lower
boundary, the magnetic helicity is somewhat suppressed in the stable zone. It is insteadlargest at the base of the convection zone where the toroidal magnetic field and the toroidal
component of the vector potential (representing the poloidal fields) have the greatest overlap.
The cross-helicity on the other hand is largest in the stable zone, where the fields and mean
flows are able to align given that they are free of the convective pummeling occurring in theconvection zone.
8.6

Generating Global-Scale Magnetism
The mean toroidal field shown in Figure 8.5(d) was initially generated and then main-tained by similar processes. During the growth phase of the magnetic field, the shear of thedifferential rotation acts to fold and wind the initial poloidal field into toroidal structures.
During this kinematic phase, the mean shear and meridional flows are largely unaffected andcan be considered stationary relative to the time scales of the growing field. However, oncethe magnetic fields are strong enough they begin to quench the convective flows that crossthem. The magnetic field strength becomes saturated as the back reaction of the Lorentz
forces increases the alignment of the velocity field and the magnetic field, which reduces bothits generation and at times leads to its partial destruction.

Figure 8.6: Generation of mean toroidal magnetic energy (TME ) in Case A10m. The view is from ±60◦ to emphasize theequatorial regions. Here only the dominant production terms are shown, namely (a) mean shear (TMS ), (b) fluctuating shear(TFS ), a cylinder tangent to the bottom of the domain (T. C.) is indicated (dashed line), (c) fluctuating advection (TFA ), (d)the sum TMS + TFS + TMA + TFA + TCC , which largely act in concert to balance (e) the resistive diffusion of field (TRD ). All
panels have identical scaling. The mean generation terms shown here contribute to the TME when positive (red), and destroy it ifthey are negative (blue). (f) The relative contribution of each term, with terms in red adding to the volume- and time-averaged
toroidal energy generation and those in blue dissipating energy.

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285
This configuration of fields establishes a new balance between the various mechanismsof angular momentum transport needed to sustain the differential rotation when comparedto the hydrodynamic cases. Here magnetic fields both transport angular momentum as wellas block formerly open channels of transport. For instance, the strong toroidal field serves toreduce the latitudinal transport of angular momentum by Reynolds stresses, which modifies
both the differential rotation and the meridional circulations within the simulation.
8.6.1

Toroidal Generation

The mean magnetic field strengths in Case A10m wax and wane as time marches forward, so this balance is not instantaneous. This time evolving dependence of the mean
fields is due to many competing processes that both produce and destroy magnetic field
through the shearing and advection of field, through compressive motions, and dissipation
through resistive processes. These production and dissipation terms are most easily understood through an evolution equation for the mean magnetic energy. We break this equation
into its poloidal and toroidal components PME and TME as
T

FS
TMS

}|
E{
}|
{ zD
∂TME
hBφ i z′′=(hBi · ∇) hvi|φ + (B · ∇) v
∂t
8π
φ

T

FA
TMA
}|
E{
z
}|
{ zD
′
′
− (hvi · ∇) hBi|φ − (v · ∇) B

φ

TCC

}|
{
TRD
}|
{
∂ ln ρ z− ∇ × (η∇ × hBi)|φ ,+ hBφ vr i∂rz

(8.14)

P

FS
PMA
PMS
z
z
}|
E{ z
}|
{ D
}|
{
hBP i
∂PME
=
· (hBi · ∇) hvi + (B′ · ∇) v′ − (hvi · ∇) hBi
∂t
8π

PFA

P

CC
}|
{
z
PRD
}|
E{
z
}|
{
ρ
∂
ln
′
′
− ∇ × (η∇ × hBi) ,− (v · ∇) B + hBP vr i∂rzD

(8.15)

with the hi denoting an average in longitude, v′ = v − hvi the fluctuating velocity, B′ =B − hBi the fluctuating magnetic field, and hvi and hBi the axisymmetric velocity and286
magnetic field respectively. A detailed derivation of the mean field production terms in
spherical coordinates is provided in Appendix A of B10. Here T indicates a toroidal magnetic
energy generation term and P a poloidal energy generation term. The subscript of thesemonikers denote production in turn by mean shear (MS), fluctuating shear (FS), mean
advection (MA), fluctuating advection (FA), compressional correlations (CC), and resistive
diffusion (RD).
Over a long time average, the time variability of the production of TME is removed,
leaving a balance between terms that produce and terms that destroy field. Figure 8.6
presents such a time average of these mean generation terms, involving a 500 day interval or
180 rotations. In this statistically steady state, the maintenance of the toroidal wreaths ofmagnetic field is largely governed by a balance between the production of field by both themean shear and the dissipation of field through resistive processes. By comparing Figures
8.6(a) and 8.6(d), it is clear that the mean shear TMS of Equation (8.14) is primarily responsible for maintaining the strength of the wreaths, with the production due to fluctuating
fields TFS , TMA , and TCC playing weak but supporting roles.
The relative contributions have been measured by integrating over time and space toRgive a mean production or destruction of toroidal magnetic energy hTi i = Ti dV , where

i is any of the terms in Equation (8.14). As indicated in Figure 8.6(f), the bulk of the
production is accomplished by TMS , being about 89% of the total. The remaining 11% of
the mean production is due mostly to the TFS term. In contrast, the averaged TRD and TFA
terms act in concert to dissipate magnetic energy, with TRD being the 98% partner.
In keeping with the dominance of the Ω effect (TMS above), the spatial distribution of
toroidal magnetic energy generation is greatest at latitudes where the latitudinal gradientsin the differential rotation are at their largest. This maximum occurs in a 15◦ swath justinside the tangent cylinder in the stable zone at about ±25◦ and in a patch centered at ±15◦
in the convection zone. The toroidal sense of the wreaths shown in Figure 8.5 are maintained
throughout the simulation by these same mechanisms. Indeed the sign of the gradient of the287
latitudinal differential rotation gives rise to the sign of the wreaths, provided the poloidal
field does not change its sense during this extended simulations, which it does not.
The fluctuating shear (TFS ) plays less of a role because it both creates and destroys
field, with the field being built where the tangent cylinder intersects the bottom of theconvection zone with regions of destruction above and below it. This can be regarded as theresult of small variations in the differential rotation as it migrates northward and southwardin latitude as each of the convective cells varies somewhat in its latitudinal extent and in
the strength of its resulting Reynolds stresses. These variations can produce field when their
gradients are aligned with the fluctuating poloidal field, and destroy it when anti-aligned with
it. The local variations in the differential rotation also induce changes in the meridional flow
as seen in the fluctuating advection terms (TFA ), which can similarly build or destroy field
when the gradients of the fluctuating toroidal field are anti-aligned or not with the fluctuating
meridional flow. The resistive diffusion (TRD ) acts to maintain an energetic balance in thesaturated state whereby it effectively counteracts the sum of all the other terms in Equation(8.14).
8.6.2

Poloidal Generation

We next examine the mechanisms generating the poloidal fields and its associatedmagnetic energy PME as in Equation 8.15. Figure 8.7 shows the temporal and longitudinal
averages of the these generation terms, displaying the terms with the largest average contributions. The averaging window is here also 500 days. The primary balance of poloidal
magnetic energy generation in the stable zone is between the mean shear PMS and the mean
advection PMA (Figures 8.7(a, b)). In particular, at latitudes inside the tangent cylinder,
mean shear produces field, while it destroys it at lower latitudes. In contrast, advection
by the mean flow creates poloidal field in a small but diffuse region about where a tangent
cylinder intersects the base of the convection zone and it acts to destroy poloidal field inside
this cylinder. The physical mechanism behind this balance is that the mean Lorentz force

Figure 8.7: Generation of mean poloidal magnetic energy (PME ) in Case A10m. The dominant production terms are shown,namely (a) mean shear (PMS ), (b) mean advection (PMA ), a cylinder tangent to the bottom of the domain (T. C.) is indicated(dashed line), (c) fluctuating shear (PFS ), (d) fluctuating advection (PFA ) and (e) the sum PMS + PFS + PMA + PFA + PCC , with (f)the resistive diffusion of field (PRD ). The mean generation terms shown contribute to the PME when positive (red), and destroy it
if negative (blue). (g) The relative contribution of each term, with those in red adding to the volume- and time-averaged poloidal
energy generation and those in blue dissipating energy.

288

289
is directed such that the mean poloidal field in the northern hemisphere is nearly alignedwith the mean meridional flow in the stable zone (anti-aligned in the southern hemisphere).
Given this, the sign difference between the PMS and the PMA terms ensures their balance.
Further, this alignment of meridional velocity and poloidal field give rise to the particularquadrupolar nature of the poloidal field seen in Figure 8.5(b), with counter-clockwise fieldat higher latitudes and clockwise field at lower latitudes.
In the convection zone itself, the story is quite different. Here the predominant balanceis between fluctuating shear PFS , fluctuating advection PFA , and resistive diffusion PRD(Figures 8.7(c), (d), (f)). These terms are associated with the fluctuating velocities andmagnetic fields of the more isotropic convective cells at higher latitudes, with the bulk of
the production occurring at the interface between the convection zone and the stable zone.
The fluctuating shear produces field inside the tangent cylinder, whereas the fluctuating
advection destroys field there. Yet just outside the tangent cylinder their roles are reversed.
Figure 8.7(g) shows that in the spatially integrated sense and unlike in the toroidal
energy generation, the fluctuating shear PFS is the dominant contributor to the production ofpoloidal field, generating on average 72% of the poloidal energy, while the resistive diffusion
PRD again is the largest dissipator, destroying 87% of it. However, the balances are more
subtle given the distinction between the spatial localities of the energy generation in theconvection zone versus the stable zone.
8.7

Maintaining the Mean FlowsThe transport of momentum, energy, and magnetic field effected by the convective

motions in these simulations establish and maintain the mean flows, mean magnetic fields,
and thermodynamic structures. Here we focus on the transport of angular momentum ina statistically steady state, in which a direct link between the meridional circulation andthe differential rotation can be found. The balance of angular momentum transport fluxes
gives rise to the statistical time invariance of the angular velocity, with those fluxes arisingFigure 8.8: The balance of axial torques in Case A10m. (a) The net axial torque (T ), as in Equation (8.16), is in near pointwise
agreement with the advection of angular momentum by the meridional flow (not shown). The components of the net axial torqueare in turn (b) Reynolds stresses (RS), a cylinder tangent to the bottom of the domain (T. C.) is indicated (dashed line), (c)viscous stresses (VS), (d) mean magnetic stresses (MM), and (e) fluctuating magnetic stresses (FM). (a)-(e) are scaled relative
to the net torque with units of 5.4 × 106 g cm−1 s−2 . (f) Vertically integrated angular momentum fluxes in the cylindrical radial
direction (hFλ iz in units of 1037 g s−2 ) showing near perfect balance, with fluxes due to RS (dark blue), Coriolis forces on themeridional circulation (CF, light blue), VS (green), MM (red), FM (yellow), and the total in black. The dashed line in (f)indicates the radius of the cylinder tangent to the base of the domain.

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291
from the Reynolds, Maxwell, and viscous stresses. In a time-averaged sense, these transportmechanisms act in sum to balance the advection of angular momentum by the meridional
circulation, while also maintaining the differential rotation. To elucidate the individual
contributions of these fluxes, consider the evolution of the mean specific angular momentumasTz h }|i{∂L+ hρvi · ∇L = − ∇ · Fλ λ̂ + Fz ẑ ,
∂t
λ
λ
∂Ω
Fλ = ρλhvλ′ vφ′ i − ρνλ2−
hBλ ihBφ i −hB ′ B ′ i,∂λ 4π
4π λ φλλ∂Ω− hBz ihBφ i − hBz′ Bφ′ i,Fz = ρλhvz′ vφ′ i − ρνλ2| {z } | {z∂z} |4π {z} |4π {z }
RS

VS

MM

(8.16)(8.17)(8.18)

FM

with L = λ2 Ω = λ (hvφ i + λΩ0 ), λ = r sin θ the cylindrical radius, v′ = v−hvi the fluctuating
velocity, B′ = B − hBi the fluctuating magnetic field, λ̂ the cylindrical radial unit vector,
and hvi and hBi the axisymmetric velocity and magnetic field respectively.
The axial torque T is the divergence of the angular momentum flux F = Fλ λ̂ + Fz ẑ.
There are four components to this flux: the Reynolds stresses produced by the turbulentalignment of the fluctuating poloidal and toroidal velocity components which are the firstterms in Equations (8.17) and (8.18) (RS), the second term being the viscous stresses associated with the mean zonal velocity VS, and the mean and fluctuating Maxwell stresses (MMand FM) are the third and fourth terms in these equations. These fluxes are integrated inz and are shown in Figure 8.8(f). The divergence of each of these components of the fluxes,
resulting in axial torques, are shown in Figures 8.8(b)-(e).
In a statistically steady state, the advection of mean angular momentum by the meridional circulation, the left hand side of Equation (8.16), then acts against this torque so as
to produce no net torque. Therefore, the meridional flow may be considered an extrinsic
property of the system that depends upon the angular momentum fluxes and the overallprofile of the angular momentum that are in turn established by the convection present inthese simulations. This framework for understanding the linking of the angular momentum,

292
meridional flow, and angular momentum transport processes is called gyroscopic pumping(e.g., McIntyre, 1998; Miesch & Hindman, 2011). The basic idea of gyroscopic pumping isthat axial torques induce meridional circulations that act to balance the total torque. Suchan approach allows us to focus on the transport properties established by the convection
with the influence of the meridional flow removed. However, the assumption of statisticalsteadiness is not required for much of this understanding to apply. Indeed, one need only
retain the explicit time dependence of the angular momentum in Equation (8.16) to havegreater generality, which can be of importance when considering the time variability of themean magnetic fields due to cyclic activity for instance.
8.7.1

Feedback of Magnetism on Mean FlowsIn Figure 8.8(a), the axial torque T gives an indication as to where the zonal flow isbeing sustained in prograde and retrograde directions and by which transport mechanism.
There are two regions to consider in detail, one is the convection zone and the other is thestable zone below it. We first examine the convection zone, where, by comparing Figures
8.8(a) and 8.8(b), it is clear that the Reynolds stresses dominate the torque inside of thetangent cylinder and act to accelerate the zonal flow in a retrograde direction (negative).
Outside the tangent cylinder, the Reynolds stresses and viscous stresses act to nearly oppose
each other. The Reynolds stresses of the low latitude columnar convective structures are yetsomewhat larger, leading to prograde zonal flow in the bulk of the interior. This balance inthe convection zone is most evident in Figure 8.8(f), where the dark blue and green lines
mirror each other to the right of the dashed line that indicates the base of the convection
zone.
The mean Maxwell stresses (MM) displayed in Figure 8.8(d) contributes to the retrograde torque on the low-latitude, equatorward edge of the wreaths (as in Figure 8.5(c))
and to the positive torque at latitudes above 35◦ in the lower convection zone where themagnetic field strength is greater. While the Reynolds and viscous stresses align very closelyCase KE/106
A10
29.4
A10m
8.84
A20
80.1
A20m
6.43

KEDR /106
28.4 (97%)
7.87 (89%)
78.8 (98%)
5.60 (86%)

Table 8.2
Global Properties of the Evolved Convection
6
KEC /10
KEMC /103
ME/105
TME/105
FME/105
1.01 (3.4%) 3.68 (0.013%)–––
0.97 (11.%) 3.06 (0.035%) 3.05 (3.4%) 2.46 (80%) 0.49 (17%)
1.33 (1.7%) 1.95 (0.002%)–––
0.83 (14.%) 1.82 (0.028%) 5.41 (9.4%) 4.44 (82%) 0.71 (14%)

PME/105 ∆Ω60 /Ω0–
0.211
0.10 (3.1%)
0.107
–
0.177
0.25 (4.4%)
0.075

∆T60
-501
-285
-1635
-255

Table 8.2: Total kinetic energy density (KE) and those of differential rotation (KEDR ), meridional circulation (KEMC ), andconvective (KEC ). Total magnetic energy density (ME) and of toroidal (TME), fluctuating (FME), and poloidal (PME). Bothenergies are averaged in time and volume and are shown in units of erg cm−3 , and component percentages in parentheses, withthe exception of ME which is shown relative to KE. The relative latitudinal differential rotation is ∆Ω60 /Ω0 , and the latitudinaltemperature difference (∆T60 , in K) are both measured between the equator and 60◦ latitude (averaged over both hemispheres),with the latter providing a more stable measure in contrast to the pole.

293

294
along the cylinder tangent to the base of the convection zone, both the mean and fluctuating
Maxwell stresses (MM and FM) act together to balance the hydrodynamic stresses (RS andVS) along a cylinder that is tangent to the bottom of the domain in the stable zone. This
leads to the tilting of the net retrograde torque away from the rotation axis and to a small
inclusion of prograde torque approximately where this latter cylinder meets the base of theconvection zone.
In comparing the mean angular velocities in the convection zones of Case A10 andCase A10m (Figures 8.3(b, f)), the tilt of contours in the polar regions have transitioned
from the cylindrical and almost lenticular alignment in Case A10 to a radial alignment inCase A10m. This is in keeping with Ferraro’s theorem, which states that the isorotational
contours become aligned with the poloidal field as can be seen in Equation 8.14. Given theradial field present at high latitudes in Case A10m (Figure 8.5) and that there is very little
net production of azimuthal magnetic field (Figure 8.6), a situation arises that promotes theconical alignment of the isorotational contours.
In the stable zone, the axial torque is confined to a roughly 45◦ band in latitude
where the toroidal magnetic fields reach their greatest strength. In this region, that is
predominantly inside the tangent cylinder, the Maxwell stresses and diffusion are the largest
sources of the torque T . The mean Maxwell stresses decelerate the lowest latitudes and act
to accelerate higher latitudes. This torque is due to the Lorentz force feedback of the wreathson the differential rotation, where they are acting to minimize the latitudinal gradients of
the angular velocity in the stable zone. Yet, the magnitude of the latitudinal shear of thedifferential rotation and the higher density in the stable zone lead to a balance between
the diffusion and the magnetic stresses, and thus to a stable net torque and steady induced
meridional flow.
The result of these torques, beyond maintaining the rotation profile Ω, is to also sustain
the meridional flow through the mechanism of gyroscopic pumping. With the addition ofmagnetism in these F-type star models, a new cell of meridional flow is built and maintained

295
through the interaction of the magnetic fields and the relatively unchanged convective flow
patterns. This cell stretches from about ±30◦ to ±55◦ and occupying much of the convection
zone (Figures 8.3(d, h)). The primary action of this cell is to short circuit the transportof angular momentum by the meridional flow between higher latitudes and the equator,
isolating it somewhat from the poles. Instead high angular momentum fluid from the poles
is now most efficiently advected to this mid-latitude cell along the base of the convection
zone. So, not only do Lorentz forces keep angular momentum from being easily advectedacross the strong toroidal wreaths by deflecting the meridional flow, but they also set up a
meridional circulation that further weakens this transport.
The magnetic disruption of the formerly efficient equatorward angular momentum
transport means that more angular momentum is trapped and recirculated in the polar
regions. It is a situation markedly different from the hydrodynamic solutions, where momentum slips more easily equatorward to enhance the differential rotation. This suggeststhat the latitudinal differential rotation can be significantly reduced in the magnetic Case
A10m, as is indeed evident in the sharply reduced differential rotation kinetic energy andlatitudinal differential rotation ∆Ω60 in Table 8.2.
8.7.2

Thermal Wind Balances

The convective structures in these simulations are largely axially aligned due to the
influence of the bulk rotation, which leads to both the enthalpy and angular momentum fluxes
also being aligned on cylinders. Baroclinicity arises from the misalignment of pressure anddensity surfaces and becomes increasingly important in regions that are stratified, such as
the domains in the simulations here where we capture roughly four density scale heights. It isthese tendencies that lead to a linking of the mean flows and the mean thermodynamic statethrough the time-averaged properties of the turbulent transport of energy and momentum.
This linking becomes more explicit when considering the time- and azimuthally-averaged
zonal vorticity equation, which is the longitudinal component of the curl of Equation (8.4)

296
divided by background density ρ. Here, we show only the dominant balance in the zonal
vorticity equation, omitting the small amplitude Reynolds, magnetic, and viscous zonal
torques that are covered in greater depth in Chapters 6 & 7. This leads to the approximate
thermal wind equationg ∂hSi
1 ∂S ∂hP i
∂Ω2
=
+
.
λ
∂z
rCP ∂θ
rρCP ∂r ∂θ

(8.19)

As is evident from Equation (8.19), the latitudinal thermal perturbations depend
quadratically on the magnitude of the differential rotation. Thus, the diminished latitudinaldifferential rotation in Case A10m compared to Case A10 requires less poleward enthalpy
flux and results in a reduced thermal contrast. This strong dependence of the mean temperature perturbations upon the differential rotation is realized when comparing the latitudinaldifferential rotation ∆Ω60 and the thermal contrast ∆T60 between these two cases in Table
8.2.
Although the magnetic torques do contribute to the thermal wind, it is a very minor
effect as they constitute only about 10−3 of the total balance. Indeed as seen in Chapters 6& 7, the baroclinic terms are by far the largest contributors outside of a small region near
the upper boundary, where the Reynolds stresses play the dominant role. The two baroclinic
terms contribute in different regions, wherein the entropy gradient is the dominant term in
the convection zone, whereas the pressure term largely controls the balance in the stablezone.

Chapter 9
Cyclic Dynamo Action in a Rapidly Rotating F-type Star

Following the detailed analysis of Case A10m, this chapter focuses on Case A20m as
it is found that dynamo action with a high degree of time variation occurs in this 1.2 M⊙
star rotating more rapidly at 20 Ω⊙. Indeed, the polarity of the mean field reverses on atimescale of about 1600 days. Between reversals the magnetic energy rises and falls witha fairly regular period, with three magnetic energy cycles required to complete a reversal.
The magnetic energy cycles and polarity reversals arise due to a linking of the polar-slip
instability in the stable region and dynamo action present in the convection zone.
9.1

Exploring Case A20m
Unlike its more diffusive and more slowly rotating partner, Case A20m has not only

oscillations in the total magnetic energy, but it also exhibits global scale changes in thepolarity of the poloidal and toroidal magnetic fields. Such transitions to some form ofcyclical and polarity reversing dynamo action have been realized in explorations of the Sunlike stars at varying rotation rates (B11), though unlike here those simulations did notincorporate a stable zone. Figure 9.1(a) presents the time evolution of the volume averagedenergy densities of Case A20m, with the time-averaged values given in Table 8.2 for reference.
The simulation has been evolved for a total of 14200 days (10450 rotations), with the first8000 days being the evolution of the hydrodynamic case. We have not shown the kinematic
growth and saturation phases of the field evolution, a process that occupies about 1000 days

298
in the early evolution. Starting at about day 9000, the magnetic field cyclically waxes andwanes in strength while also undergoing occasional polarity reversals. At later times, after
about 13000 days, the system enters a more quiescent state, during which a relatively highmagnetic energy and weaker differential rotation are maintained. It may be beginning arenewed phase of cycles as was seen in case D5 of B11, but this behavior could not be
pursued further at this stage due to the high cost of the simulation.
The energy density of the convection and of the meridional flows show only modest
temporal variations, with changes on the order of at most 20%. In contrast, the differential
rotation and magnetic energy densities possess major swings, with the strong nonlinear
feedback between the differential rotation and the magnetic fields. The energy density of thedifferential rotation oscillates by about a factor of three between roughly 3×106 erg cm−3 and9×106 erg cm−3 , which is indicative of up to 70% changes in the magnitude of the differential
rotation. The minima and maxima of the differential rotation kinetic energy is about 90◦
out of phase with the magnetic energies. The magnetic energy densities can change evenmore substantially, with the poloidal energy dropping by more than an order of magnitude
between cycles and the toroidal energy decreasing by up to a factor of six. The fluctuating
magnetic energies are somewhat less variable, but still show swings of about a factor of two.
As illustrated in Figure 9.1(b), three global-scale polarity changes in the toroidal andpoloidal fields are realized during the simulation, occurring roughly every 1600 days. There
are also several failed reversals. Additionally, there is a lag of about 150 days in thesereversals between the convection zone and the stable zone. This phase difference is reflectedin the relative timing of both the local minima and maxima of the toroidal and poloidal
magnetic energy densities. However, further time evolution would be necessary to assess theconsistency and periodicity of such cycles and reversals. The interval containing the second
reversal and the times t1 through t5 indicated in Figure 9.1 will be examined in detail toelucidate the reversal mechanism and the properties of the field generation.

299

Figure 9.1: Time variation of volume-integrated energy densities for Case A20m. (a) Time
traces of kinetic and magnetic energies in units of erg cm−3 over 4500 days of evolution(3200 rotation periods), showing the differential rotation kinetic energy (DRE, black), convective kinetic energy (CKE, dark blue), meridional circulation kinetic energy (MCE, lightblue), toroidal magnetic energy (TME, red), fluctuating magnetic energy (FME, orange),and poloidal magnetic energy (PME, yellow). (b) Time variation of the sign and magnitudeof the horizontally-averaged latitudinal magnetic field Bθ , indicating poloidal polarity reversals, with separate averages at mid-convection zone (blue) and in the stable zone (red). The
labels t1 through t5 indicate the times at which we examine the solution in detail (dashedgray lines), while the solid gray lines indicate the reversal interval studied in detail.

300
9.2

Morphology of a Reversal
Case A20m clearly shows strong variability in both its magnetic fields and its mean

flows. We seek perspective on how a reversal is achieved and identify the mechanisms responsible for destroying and building field around the time of a reversal, which in the example
reversal discussed here occurs at t4 . We begin by emphasizing the global connectivity of themagnetic field in this system as shown in Figure 9.2. We then turn to a detailed analysis of
this reversal sampled at times t1 through t5 in Figure 9.3 that provides a reasonable proxyfor what may be occurring more generally in this simulation. In studying a reversal, we firstexamine the evolution of the mean toroidal field, then turn to successive samplings of fullglobal views of the toroidal fields at two depths, and then consider the evolving propertiesof the poloidal field. To help pull all of this together, we summarize in §9.12 the elements
involved in the reversal, and forward reference these enumerated elements in presenting our
results here.
9.3

Wreaths of Magnetism
The dynamo generated magnetic fields of Case A20m have global connectivity not onlyin their mean components, but also possesses long distance links between more localized
fields. Figures 9.2(a, b) show the field lines extended outside of the surface of the star using
a force-free potential field extrapolation (Schatten et al., 1969). This serves to accentuate theglobal-scale linkage of the fields built by dynamo action within this simulation of an F-type
star. The field is shown at two times, near times t1 and t5 . The coloring of the lines indicatesthe polarity of the magnetic field. Given that at the upper radius our boundary condition isa potential field, this extrapolated field is close to the exterior field that implicitly links themagnetic fields in the interior.
At time t5 , there are distinct bridges of field between individual adjacent wreaths, andthe magnetic fields share longer distance connections as well. For instance if we focus on theFigure 9.2: Field line tracings of banded magnetic wreaths in Case A20m near the top of the convection zone. (a) A snapshotof the 3-D force-free magnetic potential field line extrapolation at time t1 . Lines trace the magnetic fields and color denotes thepolarity of the radial field Br (red, positive; blue, negative). Here the view is orthographic and tilted so the northern pole is visible,
showing the global connectivity of the magnetic field. Meridians and circles of latitude are solid black lines. Field lines bridge
the wreaths at higher latitudes and connectivity is evident across much of the sphere. (b) The same is shown after a polarity
reversal at time t5 . (c) A snapshot of multiple wreaths at t5 in a volume rendering of the upper convection zone, when magnetic
fields reach maximum strength. Lines trace the magnetic fields and color denotes the amplitude and polarity of the longitudinal
field Bφ (red, positive; blue, negative), showing the global connectivity of the magnetic field and the oppositely-directed wreathat mid-latitudes with the equator marked by a dashed line. (d) A view restricted to +30◦ and −10◦ about the equator detailingthe cross-equatorial linkages and the magnetic field bridging the wreaths at higher latitudes.

301

302
the bottom right of Figure 9.2(b), there are field lines linking the low-latitude wreaths witha polar wreath. This is also the case in the bottom left of Figure 9.2(a), where there is astrong long-distance connection between the low-latitude wreath in the southern hemisphereand the pole. Given the global-scale interconnectedness and spatial complexity of the fieldbuilt in Case A20m, it is no surprise that local changes in the structure of the field and flows
can influence the global topology of the entire magnetic field.
To have a clearer impression of the 3-D structure of the magnetism and its connectivity
within its convection zone, we show a field-line tracing of the magnetic structures at time
t5 in Figure 9.2(c) where the field lines are colored by the strength and sign of the toroidal
field. The multiple bands in latitude of large-scale toroidal field are most prominent, whereat this time there is octopolar symmetry in the convection zone. The cross-linking of these
toroidal bands of field includes an equatorial bridging of the two wreaths with the greatest
strength.
Also visible are the remnant wreaths that have slipped poleward and are being recycled
into poloidal field by the vigorous small-scale convection there. Figure 9.2(d) shows thedetails of the fields linking the two strongest wreaths that straddle the equator. The imprintof the low-latitude convective columns is visible on the blue-toned wreath in the northernhemisphere, where the upflows and downflows alternately draw field upward and punch it
down resulting in the corrugation seen there. Weak poloidal field also threads throughthese columns, field that arises from the stretching of toroidal field into poloidal field by therotation-axis-aligned helical flows of the columns. Thus the convection gives the wreaths
their braided character.
9.4

Mean Toroidal Field Evolution
In Figures 9.3(a, b), the time-latitude diagrams show the evolution of the mean toroidal

magnetic field at two depths, one near the top of the simulation in the convection zone andanother in the stable zone. Comparing these two panels, it is clear that the the morphologies

303

Figure 9.3: Global magnetic field morphology evolving during a cycle in Case A20m. Timelatitude diagrams showing the time evolution with latitude of the mean toroidal magnetic
field hBφ i over a cycle at (a) in the convection zone near the upper boundary and (b) inthe stable zone near the lower boundary (polar-slip onset indicated by arrows). (c-g) Leftcolumn shows successive snapshots of the toroidal magnetic field Bφ in the convection zone
at times t1 -t5 , with arrows here indicating the “oppositely-directed” wreaths; those wreathsin the convection zone that are directed opposite of the wreaths in the stable zone. (h-l)
Middle column shows Bφ in the stable zone. (m-q) Right column presents hBφ i averaged
over 10 days (7 rotation periods) centered on times t1 through t5 respectively. Uniform color
scaling is used with scales indicated in panels (c, h, m).

304of the fields are related, but have very different latitudinal dependencies. The multipolar
arrangement of the field in the convection zone is evident, where a wreath of the opposite
sense (as in Figure 9.2(c) and indicated with arrows in Figure 9.3(c-g)) is nestled between theradially extended lobes of the more latitudinally extended fields seated in the stable zone.
These toroidal fields merge and dissipate as time evolves, with some of them propagating
poleward and others equatorward.
Within the stable zone in Figure 9.3(b), toroidal field begins as two weak wreaths of
opposite sense in each hemisphere, these wreaths gain strength as the cycle proceeds andbegin to slowly rise along cylinders due to magnetic buoyancy. The Lorentz forces pulling on
the fluid within the wreath become strong enough to break off the poleward facing portionof it [see Element 2, §9.12]. This leaves a new toroidal segment of field that, as we show in§9.6, undergoes a polar slip instability (Spruit & van Ballegooijen, 1982b; Moreno-Insertis
et al., 1992). The onset of these polar slip events are indicated by arrows. Indeed, once thispoleward branch of field breaks away, it propagates poleward with the meridional Alfvénspeed, with the polar branch of the toroidal wreaths slipping from about 30◦ to the pole inroughly 480 days. The remaining field, with its buoyancy and Lorentz forces reduced, is thenpulled equatorward by the meridional circulation and Coriolis forces. This also impacts thefields in the convection zone, as they can also be seen propagating equatorward with a timedelayed response.
The hemispheres do not change their polarity at the same time, as the poloidal field
connecting the wreaths across the equator is too weak to rapidly affect such changes. In
particular, at time t2 the northern hemisphere reaches a maximum in field strength, while
the southern hemisphere is undergoing a polarity reversal. There was a failed reversal attime t1 in the northern hemisphere, leading to the quadrupolar symmetry maintained in the
portion of the cycle between times t1 and t4 . Likewise, the southern hemisphere fails toreverse at time t4 leading again to a dipolar symmetry in the toroidal field. When the field
does reverse its polarity, the time scale for the reversal at a given latitude is very short on the305order of a few days. The reversal begins at mid-latitudes where the latitudinally extended
swaths of toroidal magnetic field seen in Figure 9.3(b) begin to break apart [Element 4],
as occurs around day 11350 in the southern hemisphere and at about day 12000 in thenorthern hemisphere. This reversal then propagates poleward and equatorward, though at
different rates for each direction with the equatorial reversal taking about 150 days and thehigher-latitude reversal about 300 days.
9.5

Successive Toroidal Field Samplings During a Reversal
The variations in the toroidal field configurations, their waxing and waning field in-tensities, and the impacts of small-scale convection and MHD instabilities are shown as
the cycle proceeds between day 11000 and 12650 at times t1 through t5 in the left columnof Figures 9.3(c-g) for the convection zone and in the middle column of Figures 9.3(h-l)
for the stable zone. For reference, the right column of Figures 9.3(m-q) shows a time- andazimuthal-average of Bφ over 10 rotation periods centered at each time ti , which displays thelatitudinal and radial dependence of the wreaths as well as their overall sense. These mean
toroidal field snapshots reveal that the fields have their greatest strength in the stable zone,
much as in Case A10m. During most of the cycle, they possess a large degree of latitudinalconnectivity as the deeper field slips toward the poles [Figures 9.3(e, k, q) and Elements 2 &4]. They also extend radially into the convection zone nearly reaching the upper boundaryof the simulation. Weak, oppositely-directed wreaths are generated at about ±25◦ where
the latitudinal shear of the differential rotation is large [Elements 3 and 6].
The left column of panels in Figure 9.3 shows Mollweide views of Bφ in the upper
convection zone. The impact of convection on the wreaths gives rise to the complex patterns
of magnetic field seen there. The high-latitude convection destroys much of the signatureof the deeper wreaths, leaving small-scale yet still persistently polarized toroidal magnetic
fields. Similarly, the downflows of the low-latitude convective columns imprint themselves
as alternating and latitudinally inclined dark and light patterns on the stronger equatorially306
seated wreaths. All of these panels share the same color scaling, thus providing an overall
impression of the increase and decrease of the global field strength over the time samples
here. The strongest fields occur at the sample times t1 , t3 , and t5 (Figures 9.3(c, e, g)),which correspond to local maxima in the total magnetic energy. The bands of magnetic fieldin the convection zone whose toroidal sense is opposite that of the fields in the stable zone
play critical roles during a reversal are indicated with arrows in these panels [Elements 3
and 6].
The middle column of panels shows a global view of the toroidal field at the baseof the stable zone, where the strongest fields in the simulation reside. These fields canat times encompass nearly the entire horizontal surface in the stable zone, as a portion of
the strong low-latitude wreaths slips poleward leaving a sheet of fairly strong toroidal fieldin its wake [Element 2]. While the field interior to the edges of this sheet is still quite
strong, it is a tenuous enmeshment of field lines held together by magnetic tension. This
field connecting the equatorial and polar-slipping wreaths is stretched over the sphere andpossesses membrane Alfvén modes arising from the interaction with the fluid motion present
in the stable zone, which leads to a natural thinning and strengthening in places (Figures9.3(h, j, l)). Therefore, there is small-scale, fibril structure contained within this latitudinallyextended toroidal field.
At times t1 and t5 (Figures 9.3(h, l)), the detachment of the polar sheet of toroidal
field from the primary low-latitude wreath can be seen. This process begins at an isolated
latitude and longitude where the latitudinal tension force is near a minimum. But is an
inherently non-local phenomenon, as the reason the field detaches is due to the strong shear
experienced by the polar wreath, which acts through the Ω effect to build stronger field on
the poleward side of the sheet. However, this shear also advects field on a time scale fasterthan the Alfvén travel time across the sheet, leading to a loss of cohesion in the toroidal
sheet. Thus, without magnetic tension to hold the sheet together, the polar and equatorial
wreaths contract in on themselves. This process leads to a reconnection of the poloidal field307
as discussed in the following section [Element 4].
There is also evidence for an intermittent global-scale shear instability occurring at
an azimuthal order m of about 14 at times t2 and t4 , though most prominently at time
t4 . This arises from the latitudinal shear satisfying the Rayleigh criterion for instability at
certain latitudes, which leads to a rapid mixing of potential vorticity at these and nearby
latitudes (Watson, 1981; Gilman & Fox, 1997). Hence, this instability can significantly reduce
the latitudinal gradient of the differential rotation, but occurs only intermittently when its
instability criteria are met.
9.6

Polar-Slip Instability
In a non-rotating frame, the equations of motion of a toroidal flux tube are coupled in aplane parallel to the tube. Motions out of this plane, which are in latitude, are independentof the other directions and in a quiescent medium depend only upon the tension in themagnetic field (Spruit & van Ballegooijen, 1982b). Thus the latitudinal motion of such atube satisfies the dispersion relationship ω 2 = vA2 (m2 −1)/2r02 , with vA the local Alfvén speed
and r0 the radius of the centroid of the tube. This relation implies that only m = 0 flux
tubes are unstable to the polar-slip instability.
This instability may still occur for such axisymmetric flux tubes in the presence ofrotation. Yet rotation does place some constraints on when the instability can occur as itcouples longitudinal and latitudinal motions. These coupled motions of the flux tube lead tothe dispersion relationship ω 2 = −vA2 /r02 + Ω20 − Ω2e , which must be negative for the polar-slip
instability to arise (Ferriz-Mas & Schüssler, 1993; Schüssler et al., 1994). In this dispersion
relationship Ω0 is the rotation rate of gas within the flux tube and Ωe the rotation rate outside
the tube. In Case A20m, the fields in the stable region satisfy the constraint vA2 > r02 (Ω20 −Ω2e )about 75% of the time. However, given the paucity of polar slip events realized in thesimulation, there must be other processes that inhibit the instability from occurring more
often. Having accounted for rotational influences above, the only other latitudinal force to308
contend with in the stable zone is advection due to the meridional circulation.
Overcoming the equatorward meridional return flow in the stable zone is much moredifficult than satisfying the above constraint. It requires that the magnetic field strengths bein energy superequipartition with this flow. More precisely, the magnetic tension forces need
to be at least as large as the force arsing from advection, which means that the field strength2must satisfy the following B 2 & 4πρ0 vM
C . In Case A20m, the minimum field strength

required to overcome the meridional advection is therefore 19 kG. The peak field strengths
achieved in this case on the other hand are 30 kG, so it is possible for these strongest fields
to slip toward the pole. Hence it is no coincidence that the fields slip poleward when the
low latitude torii of field reach their peak strengths. We find that the mean fields satisfy
this constraint only 0.2% of the time, which implies that the slip instability should occur
roughly every 500 days. This is very close to the average time between the slip events androughly the energy cycle time in Case A20m. This time should be expected given that it isequivalent to the time for a meridional Alfvén wave to propagate from the latitude of theslip to the pole in this simulation (τ ≈ πr/3vA ≈ 480 days). In Case A10m, the kinetic
energy in the meridional circulation is about twice as large as that in Case A20m. Following
the arguments above, this requires a minimum field strength of 27 kG to have the magnetic
tension overcome meridional advection. However, the peak field strength achieved in thismore slowly rotating case is only 16 kG, which may explain why the polar-slip is not seen
there.
Next we assess the buoyancy of these wreaths using the following thin flux tube instability criterion for axisymmetric fields with motions in the plane parallel to the flux tube121βδ − f 2 (1 − s) + f −
2
γ
γ



1 1
−
γ 2



> 0,

(9.1)

where β is the ratio of gas pressure to magnetic pressure, δ is a measure of the adiabaticityof the stratification, f is the ratio of the scale height to the local radius of curvature, and γ
is the adiabatic exponent as in Spruit & van Ballegooijen (1982a). In Case A20m the plasma

309
β is fairly large, being at minimum about 106 near the stable zone and at maximum 107
near the upper boundary. The stratification is quite sub-adiabatic in the stable zone withδ = ∇ − ∇ad ≈ −0.1 throughout much of it. Hence in the bulk of the stable zone, the fieldis very stable to radial motions. Yet as shown above, the field can still slip toward the polealong horizontal surfaces. In the upper portion of stable zone and in the convection zone ofCase A20m this criterion is easily satisfied where |δ| ≤ 10−6 , meaning that fields have thepossibility of radial buoyancy.
9.7

Evolution of Poloidal Field
The variability of the poloidal field over this reversal interval is presented in Figure 9.4,

showing both the field in our computational domain and a potential field extrapolation of it
into the atmosphere above. The field begins at time t1 in a quadrupolar state with clockwiseoriented field wrapped around the deep-seated wreaths in the stable zone. The asymmetricfield reversals are also visible, with the southern hemisphere reversing between times t1 andt3 , followed by the northern hemisphere between t3 and t5 . As the simulation evolves throughthe global polarity reversal between t1 and t5 , this quadrupolar field begins to transition toa dipolar configuration as the toroidal field begins to weaken. It then again migrates toward
a quadrupolar symmetry as the dipolarity of the toroidal wreaths is reasserted.
During the detachment process of the poleward propagating wreaths in the stablezone, large portions of the wreath eventually break away from the equatorial wreath. Thesetoroidal fields have enshrouding poloidal fields associated with them, as can be deduced bycomparing Figures 9.3(m-q) and the panels of Figure 9.4. Hence, as the sheet of toroidal
field is torn in two, the poloidal field reconnects. If the polar branch retains its polarity, thepoloidal field associated with the equatorial wreath usually reconnects with the oppositelysigned equatorial poloidal field and vice versa. The failure or success of a reversal hinges on
which branch reconnects with the poloidal field of the oppositely-directed wreath present at
mid-latitudes in the convection zone. This occurs at times t2 in the northern hemisphereFigure 9.4: Azimuthally-averaged poloidal field at times (a) t1 , (b) t2 , (c) t3 , (d) t4 , and (e) t5 showing the quadrupolar reversal
and a transitory dipolar state. The field lines are extended to 1.2 R using a potential field extrapolation, indicated with a dashedline, whereas the surface of the star is shown with a solid line. The base of the convection is also illustrated with a dashed line.
The magnitude of the poloidal field is given by the depth of the color, while its orientation is given by the color sense. Namely,
blue regions are oriented clockwise, whereas those in red are counter-clockwise.

310

311
(Figure 9.4(b)) and t4 in the southern hemisphere (Figure 9.4(d)). When this occurs, theTMS that had been acting to build the field now acts in concert with resistive diffusion to
rapidly dissipate field and to later build toroidal field of the opposite sense [Elements 4 and5].
The oppositely-directed wreaths have weak poloidal fields also of the opposite sense
associated with them. Yet once the poloidal field of the deeper wreaths weaken, these fields
can become dominant as in Figures 9.4(b, d). Hence, despite being usually weaker than thefields associated with the primary toroids of field in the stable zone, these poloidal fields
provide the seed field for a reversal [Element 6].
The potential field extrapolation to 1.2 R in Figure 9.4 provides some idea of what thefield structure outside of the star might look like if the dynamo action seen here is occurring inreal stars. Namely, one may detect transitions between dipolar and quadrupolar dominancein ZDI of more rapidly rotating F-type stars. In addition, as a star transitions between
dynamo states, many of the magnetic field lines become open possibly leading to a stellar
wind with oscillatory behavior.
9.8

Assessing the Production of Magnetic Energy
The toroidal magnetic energy (TME ) generation in Case A20m is shown in Figure 9.5,

with the stable zone and the convection zone analyzed individually. Much as in Case A10m,
the dominant balance is between the production of field due to the mean shear (TMS ) and
its destruction by resistive dissipation (TRD ). The former responsible on average for 90% of
the field generation and the latter for 95% of its annihilation, with such a dominant balanceachieved in both zones. However, the stable zone contributes about three times greater
energy production than the convection zone, reflecting the stronger fields present there.
The oscillation in toroidal magnetic field energy (Figure 9.1(a)) follows naturally fromthe variation in the toroidal magnetic energy generation terms seen in Figure 9.5. The
generation rises and falls depending upon the strength of the mean poloidal field and the312

Figure 9.5: Volume-integrated toroidal magnetic energy (TME in units of erg cm−3 s−1 ) generation terms traced in time for Case A20m. (a) TME generation for the stable zone (SZ),
with a dominant balance between production through TMS (red) and dissipation by TRD(dark blue) evident. The remaining terms are TMA (green), TFS (orange), and TFA (lightblue). (b) The same quantities in the convection zone (CZ). The average production in the
SZ is three times greater than the CZ due to the weaker field present there.

313
steepness of the gradients of the differential rotation. The production of mean toroidal fieldin the stable zone appears less periodic than in the convection zone. This is due to the polar
slip instability and the resulting detachment events, which occur somewhat stochasticallyand cause shifts in the magnetic energy cycle period. The timing of peak generation by TMSin the stable zone tends to precede that in the convection zone.
The poloidal magnetic energy (PME ) generation is displayed in Figure 9.6, with the
production terms shown separately for the stable and convection zones. The PCC term
(omitted) is negligible at all times. The dominant production term in the convection zone
(Figure 9.6(a)) is the fluctuating shear (PFS ), and is counteracted by resistive dissipation(PRD ). The time averaged generation of PME by PFS accounts for 82% of the total, whereas
PRD makes up 88% of its total dissipation. The time lag between this production and the
production of toroidal field arises from the advection and rise time of the field moving fromthe stable zone into the convection zone. Once this toroidal field rises into the convection
zone, it is pummeled by the convection and shredded into poloidal field. Hence there isa rapid increase in the turbulent production terms [Element 3], where this deeper-seatedtoroidal and poloidal field serves as a source of magnetic energy for the more complex fielddeveloped in the convection zone.
In contrast, in the stable zone there is a balancing act between PMS and PMA . This
arises from the alignment of the poloidal field and meridional flows in the stable zone. The
mean shear here accounts for 50% of the production in a time averaged sense. In comparing
Figures 9.5(a) and 9.6(a), it is clear that the peak in TME production by TMA and thepeak in PME production by PMS occur at identical times. It is no coincidence that thisalso accompanies a minimum in TME production, as it coincides with the detachment of thepoleward side of the toroidal field in the stable zone and thus to the reconnection of thepoloidal field.

314

Figure 9.6: Volume-integrated poloidal magnetic energy (PME in units of erg cm−3 s−1 ) generation terms traced in time for Case A20m. (a) PME generation in the stable zone (SZ),
with a phase lag between two production mechanisms: production by PMS (red) and field
destruction by PMA (green) precedes the balance between PFS (orange) and PRD (dark blue).
PFA (light blue) plays a minor role. (b) The same quantities in the convection zone (CZ).
Here the dominant contributors are PFS and PRD .

315

Figure 9.7: Temporal cross-correlations of volume-averaged magnetic energy generation rates
for Case A20m. Cross-correlation of TMS and PFS in the stable zone (red) and convection zone
(blue). Cross-correlation of TMS between the stable and convection zones (green). Relevanttime scales are indicated as average magnetic energy cycle time τC , convective overturning
time over the entire convection zone τO , radial Alfvén wave τA crossing time across the shell,
and the polar-slip time τS .

9.9

Dynamo Time Scales
To elucidate the relevant time scales for the dynamics of the global magnetic field, we

examine cross-correlations of the magnetic energy generation terms as in Figure 9.7. The
underlying magnetic energy cycle in Case A20m is somewhat obscured by other signals in
Figures 9.5 and 9.6, but is revealed in the cross-correlation of TMS and PFS in the convection
zone (blue curve). Here the average cycle period is 580 days (τC ). There is also a 60 day
shift τO at no temporal offset. This is likely related to the convective overturning time in theconvection zone that can link the deeper wreaths with poloidal field in the convection zone.
The energy cycle period is not the sole time scale governing this system. In particular,
the same cross-correlation in the stable zone (red curve) indicates three relevant time scales,with a prominent anti-correlation between TMS and PFS at 480 days (τS ). This time scalecorresponds to the time it takes the poleward branch of the magnetic field to slip from low
latitudes to the pole [Element 2]. This is an anti-correlation because TMS is often at aminimum when PMS is growing or near a maximum. This itself may be linked to the time316
required for an Alfvén wave to circumnavigate the toroidal sheet of field in the stable zone,
which also happens to be about 480 days.
Another important time scale is that which separates the energy cycle period and thepolar-slip time scale. This roughly 100 day time scale is a lagged response of the convection
zone to the destruction of the polar wreath in the stable zone through dissipation andinstabilities. To make this time scale more evident, we have shown the volume-averaged
TMS production terms cross-correlated between the convection zone and the stable zone in
Figure 9.7 (green curve). There is a peak at the time scale labeled τA in this correlation.
This is approximately the Alfvén wave crossing time from the lower boundary at the base of
the stable zone to the upper boundary of the simulation. These waves travel along these fieldlines and serve to link the convection zone in regions where the radial field threads throughboth zones (Figure 9.4).
We conclude that in Case A20m it is the underlying dynamics of the poleward slip
instability and Alfvénic waves that act together to set the magnetic energy cycle period
τC = τS + τA . This gives rise to the time scales for reversal Elements 2-4. However, τCof course is not the polarity reversal timescale. The transitions between field polarities areaccompanied by minima in the magnetic field strength, each of which are separated by anaverage period of τC . We find here that the polarity reversal time scale of 1600 days typicallyinvolves three magnetic energy cycles.
9.10

Characterizing the Dynamo
To place the prior analysis in perspective, it is useful to make a connection to themean-field theory (MFT) of electrodynamics (e.g., Moffatt, 1978; Krause & Raedler, 1980;
Brandenburg & Subramanian, 2005). It is fairly simple to understand the mechanisms building the mean toroidal field, where the presence of a poloidal field and differential rotation
readily generate it. The origin of the axisymmetric poloidal field is more subtle, with the
turbulent convection being primarily responsible for its generation. The meridional circula-Figure 9.8: Reconstructed components of the mean-field α effect and its accompanying relative uncertainties. (a-e) Components
of α that contribute to the turbulent EMF (Eφ ) are shown in the meridional plane. (f) Relative uncertainty of the contracted α
tensor, a measure of goodness of fit. (g-k) Relative uncertainty in the reconstructed EMF as measured against the EMF of thesimulation in 20 day intervals centered on each of the sample times t1 -t5 , a measure of effectiveness and reproducibility. (l) The
same but measured over the entire reversal interval of 1600 days.
317

318
tion, turbulent pumping, and diffusion on the other hand bring it into the stable zone. Thus
we examine a constant in time but spatially varying α-effect. This provides a quantitative
measure of the influence of the turbulent properties of the flow on the turbulent electromotive
force (EMF, E), which is defined as hEi = hv′ × B′ i.
As just seen in §9.8, E is largely responsible for the generation of poloidal magnetic fieldin the convection zone and also at times in the stable zone (Figure 9.6(a),(b)). Therefore wewill concentrate on the generation of poloidal field in terms of the azimuthal component of
the mean vector potential hAφ i under the Coulomb gauge, which evolves as∂hAφ i= φ̂ · (hvi × hBi + hEi) + η∇2 hAφ i.
∂t

(9.2)

We now explore the connection between MFT and the EMF achieved in our simulations by
first noting that the first-order expansion of E around the mean magnetic field is
hEi = αhBi + β∇hBi + O ∂hBi/∂t, ∇2 hBi ,(9.3)

where α is a rank two pseudo-tensor and β is a rank three tensor. We approximate the βtensor by including only its action on the anti-symmetric part of ∇hBi, which is the current
hJi = ∇ × hBi, so that
βijk

∂hBj i∂hBj i≈ −χim ǫmkj= −χim (∇ × hBi)m = −χhJi
∂xk
∂xk

(9.4)

with the sign of χ chosen to conform to the standard notation mean-field theory, ǫmkj theLevi-Civita symbol, and with an implied sum over repeated indices. In our analysis α isexpanded as αhBi = αS hBi + γ × hBi, with αS being the symmetric portion of α and γ
the anti-symmetric portion. The latter is also known as the turbulent pumping velocity. For
the diffusion tensor χ, we apply the first order smoothing approximation and take only thelowest order moments, wherein χ becomes purely isotropic as χij ≈ 1/3τ0 hv′ · v′ iδij (e.g.,Brandenburg & Subramanian, 2005). While χ becomes isotropic under these assumptions,
it may still be inhomogeneous through both τ0 and hv′ · v′ i. We define τ0 to be the local

319
convective overturning time τ0 (r) = Hρ /ur,rms, which here ranges between 40 days at thebase of the convection zone to about one day near the upper boundary of the simulation.
To reconstruct the α tensor from our data, its individual components are solved forfrom a temporal sequence of data at each radial and latitudinal grid point using a method
similar to the least-squares singular value decomposition (SVD) methodology described in
Racine et al. (2011). This local fitting technique assumes that each point may be treatedindependently, which precludes the capture of temporally and spatially displaced correlations
that can influence the dynamo action achieved in our simulations (B11, Nelson et al. (2013a)).
Yet it has the advantage that the magnetic fields and electric currents from the simulation
constrain the components of α. The higher-order tensor components neglected here can befound using test field methods (e.g., Kraichnan, 1972; Schrinner et al., 2007), or calculated
analytically as in (Brandenburg & Subramanian, 2005). The reconstruction is carried out
over the entire reversal interval, and then its applicability at a given stage of the reversal
is assessed. To this end, Figures 9.8(a-e) show the five components of α that contributeto Eφ . The magnitude of the components of the α tensor is fairly large in comparison tosolar simulations (Racine et al., 2011; Nelson et al., 2013a; Simard et al., 2013), being about
five to ten times greater. This is due to the intrinsically faster flows present in the Ftype stars, which in turn arises from their higher luminosity and the lower density in theirconvection zones. The dynamo efficiency of these flows can be estimated by finding the
average magnitude of the α-effect relative to the rms value of the fluctuating velocity field.
We find that this measure of efficiency is roughly
α

3
h
i∼
3
vrms
2 (r2 − r13 )ZZ

drdθr 2 sin θsαφ,j αφ,j≈ 0.3.
hv′ · v′ i

(9.5)

The spatial structure of αφ,r , αφ,φ, and γr visible in Figures 9.8(a, c, d) is roughly
asymmetric between the two hemispheres, with the other two components being nearly
north-south symmetric. The former attribute arises from the influence of Coriolis forceson the flows, with the cyclonic turbulence present at higher latitudes being an important

320
contributor to the sign and magnitude of these fields. All of the components shown also havestructures that know about the bottom of the convection zone, where there is a change insign as one moves from the upper boundary to the bottom of the domain. This is particularly
evident in αφ,φ . The divergence and dilution of vortical downflows and the convergence ofswirling upflows gives rise to this feature. Not surprisingly there is a latitudinal dependence
to these structures, where a distinct transition in flow morphology across the tangent cylinder
makes itself known. The amplitude of these components is fairly low in the stable zone except
near the poles, which is consistent with the fluctuating velocities being relatively small inthis region. The impact of the β-effect is largest near the upper boundary, within the viscousboundary layer, and within the stable region, where the eddy correlation time becomes large.
The diffusion coefficient is nearly independent of latitude, with small variations of the orderof 1% at a given radius. However, it increases by about three orders of magnitude fromabout 1013 in the bulk of the convection zone to 1016 near the upper boundary and in thestable zone.
Figure 9.8(f) indicates the quality of the fit over a reversal, with σα being the standardpdeviation of a scalar contraction of the α tensor αi,j αi,j as follows
σα2 =

i X
Xh
2
2
σα2 i,j /
,
αi,j
αi,j
i,j

(9.6)

i,j

with the standard deviations of each component σα2 i,j computed using the least-squares merit
function and components from the SVD decomposition. Under this measure, the accuracy of
the SVD fit to α over the interval t1 -t5 has an uncertainty that is at worst about 5 m s−1 , andis on average about 1 m s−1 . Given the small values of the fluctuating velocity and magnetic
field in the stable zone, it is no surprise that the fits are most uncertain there. It is of note
that, when the contributions of the β-effect and the current are included, the accuracy of
the fit to the α tensor is increased by about 20%.
The lower row of Figure 9.8 shows the relative uncertainty in the reconstructed EMF
averaged over 20 day intervals surrounding times t1 -t5 , with panel (l) showing the average321over an entire reversal. The agreement between the mean-field estimate of the EMF and the
EMF of the simulation is highly time dependent, with some instances showing agreement
and other times being more poorly represented by the temporally constant α and β tensors.
Yet the longer term agreement between the two is reasonably good, as in Figure 9.8(l), withthe mean uncertainty being only about 30%. If only the α-effect is taken into account, theaverage uncertainty of the mean-field estimate relative to the EMF of the simulation is larger
at about 50%. This indicates that the combination of the α and β-effects is capturing therough outlines of the reversal, though the EMF resulting from mean-field theory tends to bemuch less spatially variable than the EMF in the simulation.
Under a L2 norm, no single one of the components of α is overtly dominant in the meanfield estimate of the turbulent EMF generating poloidal field, rather they each contributealmost equally to the total EMF. The one exception is the αφ,φ component, which contributes
about 40% to the total α-effect, whereas the other quantities are nearly evenly split inimportance and sustain the other 60% of the α-effect. This balance is sensitive to whether
or not the β-effect is included, with a lower overall contribution of the αφ,φ component when
it is not included. In either case, however, this means that the turbulent production of meanpoloidal field by an α-effect achieved through the action of convection chewing away at themean toroidal field is not the sole effect. Indeed, as the above analysis indicates, the meanpoloidal field is also generated through the action of the smaller-scale vortical turbulence
at high latitudes and the columnar convection at low latitudes amplifying existing poloidal
field.
From the above analysis it is evident that the action of convection on both the poloidal
and toroidal fields effectively sustains and regenerates the mean poloidal field, whereas thetoroidal fields are primarily built through the interaction of the poloidal field and the rotational shear. In Figure 9.5, there is a weak influence of the α-like TF S and TF A terms, which
are roughly a factor of ten smaller than the Ω-effect (TM S ). Yet this is only the case for thetoroidal magnetic energy. The difference between the same terms for the generation of the322

Figure 9.9: Reconstructed components of the mean-field α and β-effects and their accompanying relative uncertainties. (a-e) Components of α that contribute to the turbulent EMF(Eφ ) are shown in the meridional plane. (f) Relative uncertainty of the contracted α tensor,
a measure of goodness of fit. (g-k) Components of χ that contribute to Eφ . (l) A measure of
fit to χ. (m-q) Relative uncertainty in the reconstructed EMF as measured against the EMF
of the simulation in 20 day intervals centered on each of the sample times t1 -t5 , a measure ofeffectiveness and reproducibility. (r) The same but measured over the entire reversal intervalof 1600 days.

323
field itself is only a factor of three. Therefore, given the presence of the differential rotation
to sustain the toroidal field and the secondary but non-negligible action of the EMF, thedynamo operating in this simulation could be characterized as an α2 -Ω dynamo.
9.11

A Second Fitting Approach
We now explore higher-order mean-field approximations, noting that the second-order

expansion of E around the mean magnetic field is hEi = αhBi+β∇hBi+O (∂hBi/∂t, ∇2 hBi),
where α is a rank two pseudo-tensor and β is a rank three tensor. This is the logical extensionof the first-order approximation used above. We approximate the β tensor by including only
its action on the anti-symmetric part of ∇hBi, which is the current hJi = ∇ × hBi, so that
βijk

∂hBj i∂hBj i≈ −χim ǫmkj= −χim (∇ × hBi)m = −χhJi
∂xk
∂xk

(9.7)

with the sign of χ chosen to conform to the standard notation mean-field theory, ǫmkj theLevi-Civita symbol, and with an implied sum over repeated indices. The notation for α
is as in §9.10. The diffusion tensor χ can also be decomposed into symmetric and antisymmetric components as χhJi = χS hJi + ξ ×hJi, which can be interpreted as an anisotropic
conductivity χS and a helicity interaction term ξ.
To reconstruct these two tensors from our data, both of their individual components
are simultaneously solved for from a temporal sequence of data at each radial and latitudinal
grid point using a method that extends the SVD technique used in §9.10. The differenceis that both α and χ are reconstructed from the simulation data, which we find improves
the overall numerical agreement between the mean-field theoretic EMF and the EMF fromthe simulation. It has the advantage that the magnetic fields and electric currents from thesimulation constrain the components of α and χ. We again carry out the reconstruction overthe entire reversal interval, assessing its applicability at a given stage of the reversal, with
Figures 9.9(a-e) showing the five components of α that contribute to Eφ and with Figures
9.9(g-k) displaying the equivalent components of χ.

324
The spatial structure of αφ,r , αφ,φ , and γr visible in Figures 9.9(a, c, d) shares roughly
the same characteristics as in §9.10. Figure 9.9(f) indicates the quality of the fit over areversal, with σα being the standard deviation defined in Equation 9.6. The same procedure
is applied to the χ tensor to measure the accuracy of the SVD fit. Under this measure, theaccuracy of the SVD fit to α over the interval t1 -t5 has an uncertainty that is at worst about
5 m s−1 , and is on average about 1 m s−1 . The fit to χ is not as accurate as is apparent
in Figure 9.9(l), with the median uncertainty being about 6% and with a relatively longtail of larger values in the PDF of the uncertainty. Given the small values of the fluctuating
velocity and magnetic field in the stable zone, it is no surprise that the fits are most uncertain
there. It is of note that, if the contributions of the β-effect and the current are neglected,
the accuracy of the fit to the α tensor is reduced by about a factor of nine.
The component χφ,φ provides the most spatially uniform contribution to the EMF(Figure 9.9(i)). Since it is negative most everywhere, this component acts as a turbulent
anti-diffusion. This is in marked contrast with the often used approximation that χ shouldbe everywhere positive and fairly isotropic, which is valid in isotropic turbulence that is
not rotationally constrained (e.g., Brandenburg & Subramanian, 2005). In Chapter 3 it was
shown that under strong rotational constraints the isotropic components of β can in fact benegative. The common form of the β tensor has χij ∝ τ hv′ ·v′ iδij , δij the identity tensor, andτ an eddy correlation time, as is employed in §9.10. In this simulation, however, the components of χS and the ξ vector capture the impact of the anisotropy of the convection due torotation and stratification. Such an underlying anisotropy in the small-scale turbulence leads
to an anisotropic diffusion as well as to the anti-diffusive nature of some of the mean-fieldinteractions. The anisotropic turbulent diffusion of the current arises from the off-diagonalcomponents of the symmetrized χ tensor χ(φ,r) and χ(φ,θ) , as is evident upon inspection of
Figures 9.9(g, h). However, the global influence of the kinetic helicity is captured by thenorth-south asymmetry in χ(φ,θ) and ξθ , with the former acting diffusively in the northernhemisphere and anti-diffusively in the southern hemisphere.

325
The lower row of Figure 9.9 shows the relative uncertainty in the reconstructed EMF
averaged over 20 day intervals surrounding times t1 -t5 as in Figure 9.8, with panel (r) showing
the average over an entire reversal. The agreement between the mean-field estimate of the
EMF and the EMF of the simulation is highly time dependent, with some instances showing
agreement and other times being more poorly represented by the temporally constant α
and β tensors. Yet the longer term agreement between the two is reasonably good, as in
Figure 9.9(r), with the mean uncertainty being only about 14%. If only the α-effect is taken
into account, the average uncertainty of the mean-field estimate relative to the EMF of thesimulation is much larger at about 40%. This indicates that the combination of the α andβ-effects is capturing the rough outlines of the reversal, though the EMF resulting frommean-field theory tends to be much less spatially variable than the EMF in the simulation.
We find here that the overall α-effect accounts for about 43% of the generation ofpoloidal field through the azimuthal component of the turbulent EMF, whereas the β-effectis the other 57%. Under a L2 norm, no single one of the components of α or χ is overtly
dominant in the mean-field estimate of the turbulent EMF generating poloidal field, rather
they each contribute almost equally to the total EMF. As in §9.10, the αφ,φ component contributes about 40% to the total α-effect, with the other quantities evenly split in importanceto sustain the other 60% of the α-effect. This balance is again sensitive to whether or not
the β-effect is included, with a lower overall contribution of the αφ,φ component when it is
not included. In either case, however, this means that the turbulent production of meanpoloidal field by an α-effect achieved through the action of convection chewing away at themean toroidal field is not the sole effect. Indeed, as the above analysis indicates, the meanpoloidal field is also generated through the action of the smaller-scale vortical turbulence
at high latitudes and the columnar convection at low latitudes amplifying existing poloidal
field. In contrast, the components of the β-effect that contribute to the azimuthal EMF
are nearly all equal, with no dominant component. However, the existence of anti-diffusive
values of the β-effect is of some significance for this dynamo. In particular, it indicates that326
nonlinear processes involving the current (such as the Lorentz force) that lead to magnetic
field production have been captured within our fitting procedure. Moreover, the dominantanti-diffusive character of the χφ,φ component is quite surprising as it is quite different fromthe expectation of how the diagonal components of the χ tensor should behave.
From these two approaches of mean-field theoretic analysis presented in §9.10 and§9.11, it is fairly evident that the dynamo processes active in Case A20m lead to an α2 Ω typedynamo, as discussed above. Yet given the importance of the β-effect, it may be more apt
to classify this MHD simulation as an αβΩ dynamo. However, the spatio-temporal variationof the generation of poloidal field in the simulations makes an exact characterization of the
dynamo in the context of mean-field theory difficult.
9.12

Reversal Elements
Having viewed the evolving properties of the toroidal and poloidal fields in a reversal,

it may be useful to enumerate the elements that appear to be essential. We have crossreferenced these elements in our prior discussions and sketched them in Figure 9.10.
Element 1. At the beginning of each 580 day magnetic cycle, strong monolithic toroidal fieldis built through the Ω effect in the stable zone over the course of about 150 days, utilizing
the remaining poloidal field from the previous cycle (Figure 9.10(a)).
Element 2. Continued dynamo action through the Ω-effect serves to strengthen the wreaths
and to broaden them in latitude. Eventually the structure reaches a latitudinal extent andstrength great enough for Lorentz forces to begin tearing it in two, with the poleward sideof the wreath breaking away (Figure 9.10(b)). This field then accelerates toward the poles
under the influence of the combined action of buoyancy, Lorentz, and Coriolis forces. Yet
it retains some of its linkage with the primary low-latitude bound wreath, forming a sheetof toroidal field in the stable zone. The entire slip between low-latitudes and the pole takes
about 480 days, which is the Alfvénic time-scale that characterizes the polar-slip instability.

Figure 9.10: A sketch of the basic processes occurring during a reversal in Case A20m, with panels (a-g) depicting Elements 1-7.
The radial extent is expanded by a factor of two for clarity and the dotted line indicate the base of the convection zone. Solid
structures represent toroidal fields and isolated curves are meant to convey the poloidal field lines. The color gives the sense of
the field with red positive and blue negative. The dotted line illustrates the bottom of the convection zone.

327

328
Element 3. Poloidal field wraps around the deeper wreaths, encasing them. The action of
convection above the stable zone is to turn this poloidal field into toroidal field throughshearing due to the differential rotation (TMS ). This builds and maintains a wreath at midlatitudes in the convection zone over the course of a magnetic energy cycle, whose toroidal
sense is opposite to that of the deeper field (Figures 9.3(c-g) and 9.10(c)).
Element 4. While the poleward branch of toroidal field in the stable zone slips further toward
the pole, the toroidal sheet of field linking it to the low-latitude is shorn apart by thedifferential rotation. This initiates a reconnection of the poloidal field enshrouding bothbranches of toroidal field. During this reconnection event, either the equatorial or the polar
branches of this field reverse their magnetic helicity (Figures 9.10(d, e)). This reconnection
process takes only a few days at a given latitude.
Element 5. The fields whose helicity does not reverse continue to grow until the differential
rotation weakens under Lorentz force feedbacks. The rate of magnetic field generation due
to the mean shear then falls below the rate of resistive dissipation, eventually destroying thefield. The field with the reversed helicity is more rapidly dissipated by the combined actionof the resistive dissipation and the mean shear, which due to the reversed poloidal field sense
now acts to destroy field. The reconnection and field dissipation migrates equatorward fromthe initial mid-latitude reconnection on a time scale of about 100 days (Figures 9.3(b) and9.10(e, f)). It is this time scale and the polar-slip that are the dominant processes setting
the 580 day energy cycle period.
Element 6. As the field decays away, the convection which is relatively unchanged during
the cycle rebuilds the differential rotation on the timescale of a few global convective overturnings. As this is occurring, the poloidal field that had been sustained around the deeper
wreaths is no longer available to build the weaker, oppositely directed toroidal field in theconvection zone. This field is pumped into the stable region over a convective overturning329
time scale (Figure 9.10(f)).
Element 7. There are now two paths to rebuilding field in the stable zone to begin an energy
cycle anew. One path is that both the toroidal and poloidal field remnants of the weaker
wreaths in the convection zone are pumped into the stable zone. This provides the seed ofthe polarity for the new cycle and gives rise to a polarity reversal, which is then built uponby the Ω effect. This appears to occur every three energy cycles, leading to the 1600 day
reversal period. Typically following the end of the first cycle there is a failed reversal inwhich the polarity of the former wreaths is retained. In this cycle, the deeper field begins tostrengthen before it had decayed sufficiently to be overcome by the remnant wreath pumped
in from the convection zone (Figure 9.10(g)) .
9.13

Conclusions
We have explored the interaction of rotation, convection, and magnetism in a 1.2 M⊙

star rotating at 10 and 20 times the solar rate with the 3-D MHD ASH code. We findthat strong dynamo action occurs both in the convection zone and in the stable zone ofthese simulations. The magnetic fields resulting from this dynamo action are organized intolarge scale structures, forming strong toroidal bands (wreaths) of field with intricate fibril
structure and with enshrouding poloidal fields that the serve to link the wreaths. The portionof the radiative zone present in the simulations here plays an active role in both storing andbuilding these global-scale magnetic fields.
These MHD simulations of F-type stars reveal spatially complex and temporally dynamic behavior, with time variation in the magnetic energy arising in both cases. In our
analysis of these two cases, the primary mechanism for building toroidal field is the Ω effect underlying TMS , and that for building the poloidal field is PFS . This corresponds wellwith prior simulations of G-type stars (B11; Nelson et al. (2013a)), where these are also thedominant sources of magnetic energy production. To put these simulations in the context of

330
mean-field theory electrodynamics, the dynamos in both Case A10m and Case A20m exhibit
some characteristics similar to those seen in a classical α2 -Ω dynamo. We also find that the
β-effect is an important factor in obtaining a reasonable agreement between the mean-field
estimate of the turbulent EMF and the time-averaged turbulent EMF of the simulation.
The limited temporal variability of Case A10m allows us to examine time averages ofthese magnetic structures in terms of their generation, maintenance, and impacts upon themean flow and thermal structure of the simulation. Unlike previous simulations of young
G-stars that yield strong, coherent wreaths of magnetism in their convection zones, the simulations here explicitly include a stable zone. In our assessment of the generation of magnetic
field in the stable zone, we find that the PMS and PMA poloidal magnetic energy generation
terms are largely equal. This arises due to the alignment of the mean poloidal magnetic
field and the meridional flow. This poloidal field is directly tied to the deep toroidal wreaths
present in the stable zone. These structures act to block channels of angular momentum
transport that had efficiently accelerated the equator in the hydrodynamic cases of Chapters
6 & 7, leading to a reduced latitudinal differential rotation in their MHD counterparts here.
In contrast to the modest variability of Case A10m, Case A20m exhibits a broad
spectrum of dynamical time scales, with large variations in the magnetic energies and theirgeneration rates. Indeed, we have found magnetic energy cycles with a roughly 580 day
period and cyclical polarity reversals occurring roughly every 1600 days, or every threeenergy cycles. A mechanism for the polarity reversals in Case A20m has been enumerated,
where the crucial elements are the coupling between the stable zone and the convection zone
and the polar-slip instability. This reversal mechanism merits further investigation to better
assess its statistical properties over many polarity reversals and to establish its potential rolein other stellar configurations, such as in the Sun or τ Boötis.
The mean flows established in these 3-D MHD simulations are quite different fromthose of the purely hydrodynamic variety. The differential rotation and its accompanying
thermal contrast of Case A10m and Case A20m are reduced relative to their hydrodynamic

331
counterparts Case A10 and Case A20 of Chapters 6 & 7. In particular, the average latitudinaldifferential rotation has dropped by a factor of 2.0 in Case A10m and by a factor of 2.4 inCase A20m. The global-scale magnetism established in both Case A10m and Case A20m
leads to variations in the latitudinal differential rotation with time. Here the differential
rotation varies by about 20% in Case A10m and by up to 70% in Case A20m on time scalescommensurate with the variation of the magnetic fields. Time varying bulk flows have been
seen in prior MHD simulations of the G-type stars (B11). Some modulation of the differential
rotation with magnetic activity appears to be also observed on τ Boötis and several otherstars (Donati et al., 2003).
The 3-D numerical simulations presented here and in Chapters 6 & 7 cover rotation
ranges that are accessible to the line profile technique, and thus are directly comparable
to recently expanded data sets of observed differential rotation in F-stars, such as those in(Ammler-von Eiff & Reiners, 2012). Indeed, the differential rotation achieved in the priorsimulations of Chapters 6 & 7 tended to match the earlier observational data (Reiners,
2006), an agreement further improved upon here. The hydrodynamic models of Chapters 6& 7 fell well within the uncertainties of the measured differential rotation, but often a given
model had a ∆Ω matching that of the stars with the greatest differential rotation at a given
rotation rate. The time-averaged latitudinal differential rotation of the two magnetic models
explored here, on the other hand, now follows the average trend line of the observed data
for the F-type stars.
Our 3-D simulations of F-type star dynamo action reveal that quite complex processes
can be at work, and that we have sampled only a small subset of them. However, tools
in computational astrophysics such as ASH that study the interior dynamics of stars willcontinue to improve with rapid advances in supercomputing. This holds out the promiseof increasing contact between global-scale simulations and observations of stellar magnetic
variability.

Chapter 10
Convection and Dynamo Action within the Cores of B-type Stars“When I had satisfied myself that no star of that kind had ever shone before, I was led intosuch perplexity by the unbelievability of the thing that I began to doubt the faith of myown eyes.” – Tycho Brahe
In a first foray into the realm of massive star convection, the following brief paper was
the first published ASH simulation of a B-type star. It is included here as an introduction(§10.1) into the possibilities of interesting dynamo states within the cores of B-type stars.
Again this is my work, with wordsmithing assistance from Juri, and is published in Augustson
et al. (2011)1 . The remaining sections discuss the broader aspects of observations andsimulations that build upon those first steps.
10.1

Introduction
Main-sequence massive stars possess convective cores that likely harbor strong dynamo

action. To assess the role of core convection in building magnetic fields within these stars, weemploy the 3-D anelastic spherical harmonic (ASH) code to model turbulent dynamics within
a 10 M⊙ main-sequence (MS) B-type star rotating at 4 Ω⊙. We find that strong (900 kG)
magnetic fields arise within the turbulence of the core and penetrate into the stably stratified
radiative zone. These fields exhibit complex, time-dependent behavior including reversals inmagnetic polarity and shifts between which hemisphere dominates the total magnetic energy.
1Augustson, K. C., Brun, A. S., & Toomre, J., 2011, “Convection and dynamo action in B stars”, Proc.
IAU Symp., 271, 361

333
Surface magnetic fields have been found on a few MS massive stars (e.g., Donati &Landstreet, 2009). To some degree both the fossil fields and dynamo generated fields in thesestars must coexist, although how they interact to produce and maintain these surface fields
is unclear. Recent work has shed some light on the interaction between a superequipartition
core dynamo and fossil magnetic fields in A-type stars (Brun et al., 2005; Featherstone et al.,2009). We extend this work to a much more luminous 10 M⊙ star with a rotation period ofseven days (4 Ω⊙) which is typical for active MS B stars.
Using the 3-D ASH code, we study convection and dynamo action realized in the coreand part of the surrounding radiative envelope of this 7200 L⊙ B star. ASH is a maturemodeling tool which solves the anelastic MHD equations of motion in a rotating sphericalshell using a pseudo-spectral method (e.g., Brun et al., 2004). The mean structure in thisZAMS star is obtained from a stellar evolution code. We capture the full spherical geometry
with a radial domain that occupies 0.6 R (covering 7 pressure scale heights), with the inner
0.2 R being convectively unstable. The innermost 0.02 R is excluded to avoid the coordinate
singularity at the origin in the ASH code. The upper and lower radial boundary conditions
are stress-free and impenetrable for the velocity field and perfect conductor (lower) andpotential field (upper) for the magnetic field.
The intricate and time-varying flows established in this simulation are largely alignedwith the rotation axis. These columnar convection cells break the spherical symmetry due toequator-crossing meridional circulations and a north-south asymmetric differential rotation.
A central columnar flow (occupying the inner 0.1 R at the equator) stretches north-south
across the entire core, rotates retrograde to the reference frame, and gently flares out to about
25◦ in latitude at the core boundary (Fig. 1a). Along the rotation axis within this column arestrong vortical flows. Outside the central column, there are typically five columnar convection
cells that rotate prograde to the reference frame. These cells transport angular momentum
between the central column and the overshooting region, where there is a weak prograde
equatorial flow. These flows maintain a mean rotation rate that increases monotonicallyFigure 10.1: (a) Velocity streamlines within the core, cut along the equator. Fast core-crossing flows and several columnar flows
are visible. Orange tones indicate fast flow speeds (|v| > 300 m s−1 , peak 1000 m s−1 ), slower speeds in blue tones. (b) An
isocontour rendering of magnetic energy showing the dominance of the northern hemisphere and equatorward tilted magnetic
structures (rotation axis vertical). (c) North to south ratio of hemispherical averages of ME shown for a decade of time evolution;(a) and (b) are rendered at 31.5 years (dashed line).

334

335
from the center of the star to become nearly uniform within the radiative envelope, with an
overall radial differential rotation of 25%.
A strong dynamo operates within the core, generating magnetic fields with peakstrengths reaching 900 kG (200 kG rms). These fields form equatorward tilted strands
that encircle the core (Fig. 1b). The fluctuating component of the magnetic field comprises
76% of the total magnetic energy (ME) in the core, while 21% remains in the mean toroidal
field and 3% in the mean poloidal field. On average the total ME is 55% of the convective
kinetic energy, but there are intervals where it approaches 86% indicating that the ME isnearing equipartition.
The time evolution of magnetic field is complex and multi-periodic. When averaged
over several decades, the ME of the northern hemisphere is 1.7 times greater than thatof the southern hemisphere. There are intervals, however, when the southern hemispherecomes to dominate the magnetic energy (Fig. 1c), but only by a factor of at most 2.5. The
northern hemisphere, on the other hand, dominates the ME for periods of up to two years bya factor as great as 6.7. These magnetic field configurations have quadrupolar and dipolar
components that are nearly equal and opposite, which have been shown to exist when there
is weak equatorial symmetry breaking (Gallet & Pétrélis, 2009).
The greatest extent of convective overshooting into the stable radiative envelope occurs
at mid-latitudes. The sustained overshooting pushes magnetic field and lower entropy fluid
into the stable layer, making the core prolate and stochastically exciting gravity waves. The
strongest magnetic fields (900 kG) and fastest flows (1 km s−1 ) typically occur along the edgeof the central column and are maximum where this column transects the core boundary. As
this field is advected into the overshooting region it is combed into a large-scale toroidal
field (∼30 kG) by the flows in the stable region. Therefore, in this region, the velocity andmagnetic fields are nearly aligned creating a force-free state.
To better understand the hemispherical dynamo state achieved within this B-starmodel, simulations at varying rotation rates and lower diffusivities must be run. Mini-

336
mal diffusion is especially important if buoyant magnetic structures that might arise fromthe strongest fields in these models are to be captured.
10.2

Recent Observational Studies of B-type Stars
The B-type stars span a range of stellar masses from about 2.1 to 16 M⊙ (e.g., Habets& Heintze, 1981). When on the main sequence the radius of these stars is about 2 to 7 R⊙.
The fusion in their cores rapidly consumes hydrogen, with energy streaming from the stellar
surface at luminosities of up to 30,000 times that of the Sun. This large energy output
drives strong winds and ultra-violet radiation from their photospheres. Indeed, once thesestars are born in a molecular cloud, they quickly erode the cocoon of nebular gas harboringthem and in the process suppress nearby star birth. These rapidly evolving stars are also
the crucibles in which the majority of the elements are forged. Their short and energetic
lives of just a few million years soon end in a supernova, redistributing hydrogen and heavier
elements throughout their environment. Thus these massive stars are the engines driving
the evolution of galaxies.
Much is unknown about the dynamics occurring within the interiors of these stars.
During their main-sequence lifetime, there is a convective core deep within a massive starand thin convective shells near the surface with a large radiative envelope between them.
Yet how these regions interact through instabilities, convective driving, and magnetic linking
remains largely unknown. A portion of this problem is addressed here by examining the coreenvelope coupling and sustained dynamo action through 3-D MHD simulations of a 10 M⊙
main-sequence B-type star.
10.2.1

Aspects of Convective Cores and Their Observed Properties

The convective core is usually spherical due to its large mass and density, except in
the most rapidly rotating stars. The core also occupies up to 50% of the star by mass and
30% by radius. The energy burning of the CNO fusion cycle in the cores of massive stars

337
drives an energy flux that must be carried to the surface in order for the star to remain inpressure balance. However, neutrino and photon radiation can only carry a portion of thetotal energy flux due to their moderate opacities and the weak temperature gradient in thecore. Thus, convective motions must carry the remaining flux through the core, which tends
to be roughly 50% of the total luminosity. The convective flows are extremely turbulent
and hence are inherently time-dependent and three-dimensional. The convective core has astrong influence on the structure and evolution of these massive stars, though little is known
about its detailed properties. Beyond modifying the structure of the core itself, this core
convection also influences the dynamics of the radiative zone above it and may influence thesurface properties of the star through three primary mechanisms: convective overshooting,rotational mixing processes, and magnetic linkages.
In standard one-dimensional stellar models, the effects of stellar convection are usuallycalculated using simple mixing-length prescriptions (e.g., Claret, 2007), but such approaches
involve considerable uncertainties. Mixing-length theory can only provide rough estimatesof the turbulent properties of the convection, which has important consequences for compositional mixing and redistribution of angular momentum within massive stars. Overshooting
from convection zones raises other problems. It has long been realized that convective motions do not halt at the boundary between the convection zone and the stable radiativezone above it (e.g., Saslaw & Schwarzschild, 1965; Maeder, 1975). Upward moving fluid
parcels will penetrate into the stable zone, decelerating and mixing with their surroundings.
Such overshooting motions may bring fresh nuclear fuel into the core, thereby prolonging thestar’s lifetime on the main sequence as well as increasing its luminosity and radius (Brottet al., 2011). Accurately estimating the character and extent of this overshooting is challenging, but 3-D simulations provide a route to understanding aspects of it such as turbulent
entrainment.
Asteroseismology, enabled by current space-born missions as well as ground based
photometric observing campaigns, has been very successful in multiple fronts. It has provided

338
precision measurements of the fundamental parameters of many massive stars such as theirmass, age, radius, and metallicity (e.g., Handler et al., 2003; Degroote et al., 2010a; Aerts
et al., 2011). For a few stars with well resolved modes, it has even been possible to detectand partially constrain the effects of overshooting at the core boundary (e.g., Aerts et al.,
2003; Briquet et al., 2011).
10.2.2

Properties of Rotation in B-type Stars

Observations have shown that the majority of main sequence B-type stars are fairly
rapid rotators (Abt et al., 2002). The average equatorial velocities of these stars are about
140 km s−1 on the main sequence, but have an equally large standard deviation. Indeed,
spectroscopic measurements of single stars in the Small and Large Magellanic Clouds aswell as those of field stars have determined a wide range of rotational velocities, between
undetectably slow rotation to about 70% of breakup or about 450 km s−1 (e.g., Keller, 2004;
Hunter et al., 2008). Younger B-type stars may rotate slightly more rapidly, but likely spin
down quickly due to strong winds and magnetic torques.
The distribution of angular momentum achieved within the interiors of massive stars
is quite uncertain. However, for a handful of B-type stars, asteroseismic inversions of theradial profile of angular velocity have shown that the core may be more rapidly rotating
than its surrounding radiative envelope (e.g., Aerts et al., 2003), though the uncertainties inthese measurements are fairly large. Yet the presence of a such a non-uniform rotation deep
within these massive stars could have major consequences for their bulk properties.
Differential rotation can give rise to shear instabilities that stir and mix material,
and may serve to build strong magnetic fields through sustained dynamo action. Stellar
evolution codes attempt to capture these instabilities and the convection in the core using
1-D advection-diffusion approximations. Such approaches yield radial profiles of angularvelocity that are flat in the core and fall off slowly toward the surface (e.g., Maeder &Meynet, 2012; Potter et al., 2012), which agrees with these few observed profiles.

339
In the solar convection zone, Reynolds stresses give rise to a prominent radial andlatitudinal differential rotation that has been studied extensively through helioseismology(e.g., Thompson et al., 2003). Explaining this differential rotation within the Sun has beena major challenge for both theory and simulation, but 3-D spherical shell simulations of
turbulent convection have made substantial contact with the helioseismic results (e.g., Miesch
et al., 2000; Brun & Toomre, 2002; Miesch et al., 2006). If it also the case that there
is a significant radial or latitudinal differential rotation either in the core or the radiativeenvelopes of the massive stars, 3-D simulations may again provide a basis for exploring how
this may arise, as has been attempted with simulations of an A-type star (Browning et al.,
2004; Brun et al., 2005; Featherstone et al., 2009).
10.2.3Magnetism Observed on Massive Stars

Recent observational efforts have directly detected magnetic fields on the surfaces ofabout 10% of B-type stars using spectropolarimetric Zeeman Doppler Imaging techniques(e.g., Donati & Landstreet, 2009; Grunhut et al., 2013). There is also indirect evidence
that a larger fraction may possess magnetic fields, but that they may be too weak to befound using spectropolarimetry. In particular, spectral Doppler measurements have shownthat for a well selected sample population that there is a bimodal distribution of projectedrotational velocity (Dufton et al., 2013). There it is clear that there are two populations ofmain sequence B-type stars, a more rapidly rotating one with an average rotational velocity
of of about 160 km s−1 and another peaking near no detectable rotational velocity. It is likelythat the stars in this latter population possess magnetic fields that act to spin them down.
What remains largely unknown is how these magnetic fields arise to begin with, nor
is it clear how the magnetic fields in massive stars influence both their evolution and theirenvironment. There are three complementary theories that attempt to address the originsof these magnetic fields: a strong core dynamo (Charbonneau & MacGregor, 2001), a sheardriven dynamo in the radiative zone (Mullan & MacDonald, 2005), or a fossil field remaining

340
from the birth of the star (Braithwaite & Nordlund, 2006).
There is observational support for each of those hypotheses. For core convection, one
could appeal to the fact that correlations between magnetic fields and rotation have beeninferred in main-sequence stars ranging from spectral type M through F (Wright et al., 2011).
These observed magnetic fields are likely the result of convective dynamo action, though
how thoroughly they permeate the interior beyond the convective envelope is unknown.
However, massive stars with convective cores are likely to admit a similar form of dynamo
action, for they too possess the necessary ingredients: intensely turbulent convection, ahighly conductive medium, and the global effects of rotation which are all thought to be
crucial for building magnetic fields.
How magnetic fields built through dynamo action in the core could be transported tothe surface across the radiative zone of these stars remains unclear. If there is no expeditious
means of such transport, the fields will take roughly the main sequence lifetime of the starto diffuse from the core to the surface. On the other hand, sufficient turbulence and shear
in the radiative zone may allow the fields to rise to the surface on a reasonable timescale. In
particular, if the Tayler-Spruit dynamo mechanism is operating in the radiative zone, it may
be possible that a coupling between the core dynamo and a shear dynamo in the radiativezone allows strong magnetic fields to be transported to the surface.
The dynamo action within the cores of massive stars may produce buoyant fields, butthey must overcome the compositional gradients and stiff stratification present at the coreboundary (MacDonald & Mullan, 2004). One mechanism that partially circumvents thisconstraint, reducing the magnetic field strength required to buoyantly rise the surface, isconvective overshooting and penetration into the stably stratified envelope above the core.
This leads to mixing in this region, which reduces the compositional and sub-adiabatic
gradients there. It also deposits magnetic field above the steepest gradients, where the field
can be strengthened through further dynamo action and further deposition of field. This
field may grow to strengths sufficient to become buoyant. A pre-existing magnetic field might

341
link the surface and the core, potentially providing a conduit for the buoyant fields to more
readily move along. If so, the core dynamo may have direct bearing on the regions of strong
magnetic field observed on the surfaces of magnetically-active massive stars.
Regardless of the detailed evolution of the star, the magnetic field present in the molecular cloud during the formation of a massive star will almost certainly be interred into its
interior, providing the seed field for any dynamo action occurring there. During their main
sequence lifetimes, dynamo action in the convective core likely amplifies these fossil magnetic
fields and is linked to any field present in their radiative envelopes. As these stars evolve off
the main sequence, they possess a convective exterior which will also serve to build magnetic
field through dynamo action. When massive stars shed these envelopes during their explosive demise, the magnetic fields built and stored in their interiors over their lifetimes will
be dispersed into the interstellar medium and a portion of it trapped in a stellar remnant.
Such remnants provide the greatest evidence of magnetic fields being present in the deep
interior of massive stars. Thus, given the many impacts that magnetism may have upon theevolution and observable properties of massive stars, a better understanding of the complex
flows and magnetism established within their interiors should be sought.
10.3

Simulating Massive Stars with ASH
There are many difficulties to overcome when modeling the interiors of these stars. The

nuclear reactions occurring in the core constantly change its isotopic composition. These leadto double diffusive instabilities and may hinder some of the penetrative convection and large
scale circulations. In the Sun, it is thought that the tachocline plays a critical role in the
global dynamo. It is possible that the differential rotation established in the radiative zone isdifferent from that built in the convective core leading to a reverse tachocline at the boundary
between the two regions further complicating the modeling of the interior.
The regions of strong magnetic field observed on the surfaces of some massive stars may
be linked to a primordial large-scale interior magnetic field that extends through the stellar

342
surface. Studies of the interaction between such a relic magnetic field and the convective
core of an A-type star have illuminated the richness of the dynamo behavior possible inmassive stars (Featherstone et al., 2009). In these studies a coupled core-fossil field system
has been constructed by placing a fossil fields with several morphologies in the radiativezone that link to the convective core of the A-type star. These simulations used the 3-D
magnetohydrodynamic (MHD) simulation with global spherical geometry using the anelastic
spherical harmonic (ASH) code.
The interaction between the existing dynamo and the fossil field has allowed the system
to generate significantly higher (roughly tenfold) magnetic energy than kinetic energy. This
remarkable super-equipartition state is characterized by an equatorial belt of magnetic fieldthat has persisted for several magnetic diffusion times, along with complex fields built within
the convective core. Peak field strengths in this torus reach as high as 500 kG. It was also
found that such states depend upon the helicity of the initial configuration of the relic field,
where states with the largest poloidal content increase the level of superequipartition. Thisbehavior is in stark contrast to simulations in the absence of a fossil field where these stars
generated magnetic energies that were only 70% of the convective kinetic energy (Brun et al.,2005). These studies give the general flavor of the structures and strengths of fields that may
be built in B-type stars. However, the convection within the cores of the more massive Bstars is much more vigorous, where the average convective velocities are ten times those seen
in a typical A-type star. Therefore, the field strengths and flow structures maintained within
the core of these stars remain uncertain given the nonlinearity of the underlying physics.
10.4

Formulating ASH Simulations for B-type Stars
Turning to the 10 M⊙ B-type star dynamo simulations, this section describes the MHD

equations solved in ASH and elaborates on the details of how these numerical experiments
are constructed. The subgrid-scale motions present in the simulation domain must be parameterized. The turbulent eddy diffusion model does so through renormalized effective

343
momentum, thermal, and magnetic diffusivities. For B-type stars, the sound speed in thecore is on the order of 1000 km s−1 , while the convective flows are about 0.5 km s−1 . Thus
the anelastic approximation is ideally suited for simulating the interior of these stars. The
Courant-Friedrichs-Lewy condition on the time step thus depends either on the local subsonicflow velocity or the speed of Alfvén waves, rather than the local speed of sound.
The turbulence achieved in these models is still quite removed from the intensely turbulent conditions likely present in actual stellar convection zones. Yet with ASH, simulations ofconvection and dynamo action in many spectral types of stars have made significant contact
with observations and led to new ways of thinking about stellar dynamos. In particular, it
has been used to simulate a wide range of stars: from fully convective M dwarfs (Browning,2008), to Sun-like stars (Brown et al., 2011b; Nelson et al., 2013a) and F-type stars (Chapters 6, 7, 8 & 9), and extended to the convective cores of an early A-type star (Brun et al.,2005; Featherstone et al., 2009).
The version of the ASH code utilized here implements the Lantz-Braginsky-Roberts
co-density formulation (Lantz, 1992; Braginsky & Roberts, 1995), so that gravity waves areproperly captured in the stably stratified, radiative envelope (Brown et al., 2012). The
anelastic MHD equations evolved in ASH are still fully nonlinear in the momentum andmagnetic variables (Brun et al., 2004). The thermodynamic variables (S, P , T , and ρ) arelinearized about a spherically-symmetric and hydrostatic background state S, P , T , and ρ,
which are functions of the radial coordinate only. The equation of state for this backgroundstate is a calorically perfect gas. The equations solved in ASH retain physical units, are in
spherical coordinates (r, θ, φ), and are evolved in time t. These equations are∂v/∂t = −v · ∇v − ∇̟ + Sc−1
P g − Λr̂ + 2v × Ω0

(10.1)+ (4πρ)−1 (∇ × B) × B + ρ −1 ∇ · D,
∂B/∂t = ∇ × [v × B − η∇ × B] ,
−1[∇ · q + Φ + ǫ] ,∂S/∂t = −v · ∇ S + S + ρT

(10.2)(10.3)

344
with v = vr r̂ + vθ θ̂ + vφ φ̂, B = Br r̂ + Bθ θ̂ + Bφ φ̂, and the co-density ̟ = P ρ −1 . The
solenoidality of the mass flux (∇ · ρv = 0) and the magnetic field (∇ · B = 0) are maintained
through the use of a streamfunction formalism. The angular velocity of the rotating frameis Ω0 = Ω0 ẑ. The gravitational acceleration is g = −g(r)r̂. In Equation (10.1), Λ (r) is theradial gradient of the mean turbulent pressure that maintains hydrostatic equilibrium as thespherically symmetric state evolves. In the induction equation, η is the magnetic resistivity.
The volumetric heating arising from the combination of the CNO and PP nuclear reaction
chains are parameterized in ǫ. The energy flux q is comprised of a spherically-symmetric,
diffusive radiation flux and a turbulent entropy diffusion flux,
q = −κr ρcP ∇T − κρT ∇S,(10.4)

where κr the radiation diffusion coefficient, and cP the specific heat at constant pressure. The
entropy diffusion flux has an effective thermal diffusivity κ acting on the entropy fluctuations.
The viscous diffusion tensor D and the diffusive heating Φ are
Dij = 2ρν [eij − 1/3 (∇ · v) δij ] ,Φ = 2ρν eij eij − 1/3 (∇ · v)2 + (4π)−1 η (∇ × B)2 ,(10.5)(10.6)

Here eij is the stress tensor, and ν is the effective kinematic viscosity. The use of this set ofanelastic MHD equations requires 12 boundary conditions in order to be well posed. Since
it is desirable to assess the generation of magnetic field and how it impacts the organizationof angular momentum and energy in these simulations, the following boundary conditions
are imposed: impenetrable, perfectly conducting, torque-free, and thermal flux transmittingboundary conditions at both the inner (r1 ) and outer (r2 ) radial boundaries of the simulation
vr = Br =∂  vθ ∂  vφ  ∂S
= 0.
=
=
∂r r
∂r r
∂r r1 , r2

(10.7)

Case Ω0 / Ω⊙
H4
4
M4
4
M8
8
M16
16

Table 10.1
Diagnostic Flow Parameters in the Four Cases
Nr × Nθ × NφRa
Ta
Re Re′ Rem Re′m
500 × 256 × 512 9.51 × 108 1.00 × 109 1850 610––
89500 × 512 × 1024 4.18 × 10
1.24 × 10
540. 520 2160 2080
99500 × 512 × 1024 1.18 × 10
7.11 × 10500. 490 2000 1960
500 × 512 × 1024 2.93 × 109 4.45 × 1010 420. 410 1680 1640

Ro
Λ
κ1
τc
τκ
0.48 – 8.0 75. 8600.
0.35 330 7.2 87. 9600.
0.16 493 6.0 114 11500
0.06 383 4.8 165 14400

Table 10.1: The depth of the convection zone d = rc − r1 (where rc is the radius of the top of the convection core) is the relevant
length scale in the following parameters and is 6 × 1010 cm. The following diagnostic parameters are estimated at mid-convection
zone: the Rayleigh number Ra = ∆Sgd3 /cP νκ; Taylor number Ta = 4Ω20 d4 /ν 2 ; Reynolds number Re = vrms d/ν; fluctuating′
Reynolds number Re′ = vrmsd/ν; magnetic Reynolds number Rem = Prm Re; fluctuating Reynolds number Re′ = Prm Re′ ; Rossby
number Ro = ξ/2Ω0 , where ξ is the rms enstrophy. Elsasser number Λ = B2rms Prm /(κ1 ρ0 Ω0 ). Thermal diffusion (κ1 ) values are inunits of 1012 cm2 s−1 . The mid-convection zone local overturning time τc and the thermal diffusion time τκ across the convective
core are quoted in days.

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10.4.1

Experimental Configuration

The inner 65% by radius of a young main-sequence B-type star of 10 M⊙ is simulated
at the three rotation rates of 4, 8, and 16 Ω⊙. These bulk angular velocities are equivalent
to surface equatorial velocities of 28, 56, and 112 km s−1 , or 0.16, 0.66, and 2.6% of the
breakup velocity. The luminosity at the surface of the star is 5700 L⊙. The model of the starconsists of a convective core that occupies the inner 23% by radius of the star, or about 30%of the computational domain, and an overlying radiative exterior that extends to the outerradius of the domain. To avoid the small time steps required when including the coordinate
singularity at origin, the innermost 1% of the core has been excluded. The stratification
is taken from a MESA stellar evolution model that had evolved a 10 M⊙ star from the premain sequence to about 1 Myr, or just past the beginning of its main-sequence life. The
MESA code employs realistic microphysics and employs the mixing-length approximation
to the convective dynamics within the core (Paxton et al., 2011). The volume-averaged
values of ∂ ln ǫ/∂ ln T and ∂ ln ǫ/∂ ln ρ are computed in the core of the 1-D stellar model andthen they are applied in ASH as a volumetric heating arising from nuclear burning, which
15.33

gives ǫ = ǫ0 ρ2 T 6

and ǫ0 = 2.31 × 10−20 erg cm3 g−2 s−1 . The initial conditions for thesesimulations are random entropy perturbations, and for magnetic cases there is also a very
weak dipolar field confined to the core with an initial peak strength of 10−4 G.
A comparison between the 1-D stellar model and the background state used in these3-D simulation of B-type stars is shown in Figure 10.2. The discrepancy between the nuclear
energy generation rate ǫ and the Brunt-Väisälä frequency N displayed in Figure 10.2(b)arises from the use of a calorically perfect gas equation of state, where both Γ1 and cP havebeen required to be constant with radius. The logarithmic density and temperature are also
required to be well fit. This leaves ǫ and N less well described, but still within about 10%of the stellar model.
The effective diffusivities ν, η, and κ vary in radius only. The diffusion is greatly

347

Figure 10.2: Mean stratification established in Case M4 as a function of proportional radiusr/R. (a) Density normalized to the central value (ρ, solid blue) and normalized temperature(T , solid red), with the stellar model density and temperature as black dash-dotted lines. (b)Normalized Brunt-Väisälä frequency (N, solid blue) and nuclear energy generation (ǫ, solidred), with stellar model values as black dash-dotted lines. The core boundary is indicatedwith a dashed vertical line. (c) Comparison of thermal (P rt , black) and magnetic Prandtl(P rm , blue) numbers and Roberts number (Rb, red) from the atomic values in Braginskii(1965) (dash-dotted lines) and from these B-star simulations (solid lines).

reduced in the stable region, where motions have a small amplitude and the expected SGS
motions are subsequently smaller. In the convective core, the Prandtl number is fixed at

348
Pr = ν/κ = 1/4, so ν = Pr κ. Similarly, the Roberts number is constant in the core with
Rb = η/κ = 1/16, leaving η = Rb κ. Outside of the core, the diffusion is tapered to profiles
that give Pr = Rb = 1. The radial profile of the thermal diffusivity κ is given by
κ (r) = (ρ/ρ1 )−1/2 κ1 (1 + exp [α (r − rκ )])−1 + κ2 ,(10.8)

where ρ1 is the background density value at the lower boundary and κ1 is the thermal diffusivity at the lower boundary. The values of κ1 are shown in Table 10.1 for each simulation,
along with many other diagnostic flow parameters from the middle of the convective core.
The floor value of the diffusivity is κ2 . The steepness of the transition between κ1 and κ2 iscontrolled by α and occurs at the radius rκ = 6.7 × 1010 cm, which is just outside the core.
The minimum levels of diffusion are equal with κ2 = ν2 = η2 = 4 × 108 cm2 s−1 . As visible
in Figure 10.2c, the magnetic Prandtl number P rm regime likely operating within the true
stellar core is numerically accessible as the atomic values of the diffusivities yield P rm ≈ 10.
Being a fairly large magnetic Prandtl number, the dynamo that may be running within
the star and within the simulations presented here should be expected to be a small-scale
dynamo.
10.5

Hydrodynamics of the Core and Radiative Envelope
A very large heat flux is driven through the convective cores of massive stars, and theone being modeled here is no exception. Near the center of the star, the radius is quite small
so the heat flux as plotted in terms of a “luminosity” in Figure 10.3 is corresponding small.
Nonetheless, the volumetric heating due to nuclear fusion deposits a massive amount of heatin the lower half of the core as visible in Figure 10.2b. Given the small temperature gradient across the core, neither radiation or conduction are sufficient to carry the increasingly
large flux throughout a large fraction of the core. Thus convection ensues that carries theequivalent of several thousand solar luminosities of heat flux through the core. This is accomplished through the turbulent alignment of fluctuating temperature and radial velocities

349

Figure 10.3: Horizontally and time-averaged energy flux balance in Case M4. The energy
fluxes are shown as luminosities (L = 4πr 2 F ) for the total flux (L, black), radiative flux
(LRD , red), enthalpy flux (LEN , blue), and the flux due to unresolved motions (LED , green).
The magnetic energy flux is minimal and not shown. The vertical dashed line indicates thecore boundary, and the horizontal dashed line shows the zero ordinate.

that act in concert to produce a net outward enthalpy flux. However, as the outer portion of
the core is reached, the temperature gradient steepens and radiation becomes more efficientthan convection for carrying heat. This process acts somewhat like the solar photosphere
where radiative losses halt the upflows, rapidly cooling them. Here that processes takes placeover a much larger radius and involves a great deal more mass. These simulations employ
a realistic stable stratification in the radiative envelope, whose stiffness helps to prevent the
diffusive spread of the differential rotation and latitudinal thermal gradient from the core
into the stable region. The small zone of overshoot is visible near the core-radiative zone
boundary in Figure 10.3 near 0.22 R.
10.5.1

Convective Patterns and Mean Structures

Figures 10.4(a, d) show the enstrophy ξ = |∇ × v| of the flows from two simulationsin their cores and their overshooting regions. Case H4 is purely hydrodynamic and theother Case M4 is an MHD simulation. The enstrophy in these cases show the tangled andFigure 10.4: Comparison of the hydrodynamic Case H4 in panels (a)-(c) and MHD Case M4 in (d)-(f). (a, d) Orthographic
volume projection of the rotation rate normalized enstrophy ξ/Ω0 in the core. (b, e) Time- and azimuthally-averaged angularvelocity Ω shown with radius and latitude. (c, f) Radial dependence of Ω at specific latitudes. The location of the projectedvolume projection is indicated with dashed lines.

350

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Figure 10.5: Radial velocity shown in an orthographic volume projection for Case M4,
with the velocities outside the core scaled to 10 cm s−1 and those in the core scaled to
100 m s−1 . Downflows are dark, upflows light. Gravity waves are visible in the radiativezone and columnar convective structures can be seen in the core. (b) Instantaneous velocity
streamlines in the convective core, cut at the equator in Case M16. Color indicates themagnitude of the velocity, with slow flows of about 10 m s−1 in red and fast flows of about
500 m s−1 in blue. (c) Instantaneous velocity streamlines of single convective column, showing
only a weak linking between adjacent columns and a helicity reversal across the equator.
Color indicates the direction of the vertical velocity vz , with red positive and blue negative.

filamentary nature of the flows within their cores and the high radial wavenumber gravity
modes that the convective overshooting excites in their radiative exteriors. It also serves
to highlight the dramatically different amplitude of the modes excited in the two cases, thehydrodynamic case shows modes with a fair amount of enstrophy, whereas the magnetic
case has much less. The reason for this becomes clear when one considers Figures 10.4(c,f). Case H4 has a profound radial differential rotation with the deepest portions of the corerotating 50% more slowly than the frame rate, and 70% more slowly than the more rapidly
rotating flows near the overshooting region at 0.2 R. Thus the flows carrying energy in thissimulation are also very effective at transporting angular momentum out of the core. The
meridional profile of the angular velocity in this case is shown in Figure 10.4b, where a
prominent Stewartson column is formed along the rotation axis. It is quite remarkable that352
this situation is completely reversed in the magnetic case. The slow central column in Case
H4 becomes a region of more rapid rotation, though it rotates only about 7% more rapidly
than the radiative exterior. The MHD Case M4 shares all parameters with H4, save for the
addition of magnetic fields and a higher horizontal resolution. Yet the radial contrast indifferential rotation for M4 is only about 10% that of the hydrodynamic case.
The flow patterns in Case M4 become more clear in Figure 10.5a, where strong swirlingcolumnar flows pervade the core and high radial order gravity modes are visible outside of it.
These columns are also present at larger rotation rates as seen in the velocity field line tracingof Figure 10.5b, where the equatorial cut shows the cross-section of the convective columns
in Case M16. There are many weaker columns that are fairly localized. The stronger flows incontrast have a much greater radial extent. These flows are the primary conduits for radial
thermal angular momentum transport. An isolated convective column is shown in profile in
Figure 10.5c. The color indicates the orientation of the vertical flows vz . The reversal of
the sign of the vertical flows mirrors the helicity reversal across the equator. Unlike earlier
simulations in A-type stars, there is less cross-linking between adjacent columns. This is
likely due to the much stiffer stable stratification in these simulations, with the entropy
gradient being at least several hundred times greater.
The angular momentum flux balance for Cases H4 and M4 are shown in Figure 10.6.
Only the fluxes in cylindrical radius λ = r sin θ are shown as the vertical transport is relatively small. The fluxes are averaged in time and azimuth and then integrated cylindrically(with z = r cos θ). The hydrodynamic Case H4 has only three possible fluxes of angularmomentum: Reynolds stress ρλhvλ′ vφ′ i, angular momentum advection by meridional circu. The primary balance seen
lations hρvM C iλhvφ i, and those due to viscous stresses ρνλ2 ∂Ω∂λin Figure 10.6a is between the Reynolds stresses and the viscous stresses as is typical of ahydrodynamic ASH simulation with moderate levels of turbulence. The Reynolds stresses
arising from the convective columns pushes angular momentum outward, whereas the viscousstresses transport it inward. The meridional flows play a passive role accounting for about

353

Figure 10.6: A comparison of angular momentum transport mechanisms in Cases H4 andM4. The cylindrically-averaged angular momentum fluxes hFλ iz are shown with proportional
cylindrical radius λ/R for (a) Case H4 and (b) Case M4, in units of 1040 g s−2 . The flux
components are the total flux (black), the Reynolds stress (RS; blue), the advection by themeridional circulation (MC; teal), viscous stresses (VS; green), mean Maxwell stresses (MM;red), and fluctuating Maxwell stresses (FM; orange).

5% of the overall transport. The flux balance is not perfect as the time scale for the viscous
spread of the differential rotation into the stable region is many thousands of years, much
longer than can be simulated.
Figure 10.6b exhibits the angular momentum fluxes in cylindrical radius for the MHD
Case M4. The balance relative to the hydrodynamic case have been radically altered with
the inclusion of magnetic fields and their associated angular momentum transport throughMaxwell stresses. The primary balance is between the angular momentum flux arising fromλthe fluctuating Maxwell stress 4πhBλ′ Bφ′ i and that of the Reynolds stress, with weaker conλtributions from the meridional circulation and the mean Maxwell stress 4πhBλ ihBφ i. The

Maxwell stresses transport angular momentum inward countering the outward transport by
the Reynolds stress that arises from the rotationally constrained convective columns. Now

354

Figure 10.7: A comparison of magnetic and velocity fields in Case M16. (a) Magnetic field
line tracing in a snapshot of Case M16, colored by field amplitude, looking down the rotation
axis at the equatorial cross-section of the magnetic field structures. The peak fields seen here
are 0.75 MG (purple, which are also the most coherent in the core) and the weakest fields
about are 1 G (red). The strongest magnetic fields are the most interconnected, threading
throughout the core. (b) Instantaneous velocity field lines for the strongest flows, and (c)with an overlay of magnetic energy for the strongest magnetic fields. This exhibits the slightspatial offset between the strongest flows and strongest magnetic fields.

comparing Cases H4 and M4, the role of the magnetic field is very similar to that of theviscous diffusion, elucidating its visco-elastic nature in a convective dynamo. The differencein transport mechanisms in these two models is partly responsible for the differences in theirdistributions of angular momentum. However, it is only part of the story as the role of themagnetism is two-fold: first it provides an alternative channel for transport and second thepresence of magnetism acts to reduce the mean shear utilizing it as an energy source forbuilding magnetic field as seen in earlier chapters. In the core convection dynamos, the magnetic field acts to suppress the central Stewartson layer that otherwise becomes the dominant
flow structure in the core as in Case H4.
10.6

Dynamo Action Achieved in a B-type Star Core
The magnetic fields built through dynamo action in the core is of an even finer scalethan the velocity field, being twisted, braided, and pulled through the core by the strong

Case
H4
M4
M8
M16

KE
73.1
3.91
2.20
1.42

KEDR
66.4 (91.%)
0.17 (4.5%)
0.14 (6.2%)
0.09 (6.5%)

KEC
6.57 (8.9%)
3.72 (95.%)
2.06 (94.%)1.32 (93.%)

Table 10.2
Global Properties of the Evolved Convection
KEMC
ME
FME
TME
PME
ME/KE Bmax
13.0 (0.1%)
–
–
–
–
–
–
2.34 (0.6%) 4.08 3.71 (91.%) 0.30 (7.4%) 0.07 (1.7%)
1.06
0.76
0.81 (0.4%) 4.52 4.19 (93.%) 0.20 (4.5%) 0.13 (2.8%)
2.07
1.06
0.54 (0.4%) 5.42 5.17 (95.%) 0.15 (2.8%) 0.10 (1.9%)
3.86
1.17

∆Ωr /Ω00.495-0.078-0.017-0.012

Table 10.2: Total kinetic energy density (KE) and those of differential rotation (KEDR ) and convective (KEC ) in units of
107 erg cm−3 . The meridional circulation (KEMC ) kinetic energy is in units of 105 erg cm−3 . Total magnetic energy density (ME)and of toroidal (TME), fluctuating (FME), and poloidal (PME). Both energies are averaged in time and volume and are shownin units of 107 erg cm−3 , and component percentages in parentheses, with the exception of ME which is shown relative to KE
which is shown as a ratio in the third to last column. The peak magnetic field strength in units of 106 G is Bmax . The relative
radial differential rotation across the core is ∆Ωr /Ω0 .

355

356
radial flows. The magnetic field is also stored in the overshooting region allowing it tobe slowly amplified, forming a magnetic cocoon around the core as is somewhat visible in
Figure 10.7a. The knotted and braided character of the field is established through successive
core-crossing events where strong downdrafts knead the field into shape. Given the tangled
filamentary field seen in Figure 10.7, it is no surprise that over 90% of the magnetic energy
is contained in the fluctuating fields. This is largely due to the large P rm of the system,
which tends to build small-scale field as seen in §3.4.
As the rotation rate is increased, the differential rotation becomes increasingly small
in the cores of the simulated B-type stars (Table 10.2). While there is still a central regionof differential rotation in each of these systems, the amount of energy contained in this mean
flow becomes ever weaker. Combined with slightly weaker convective flows, this leads to thedecreasing kinetic energy seen in the first column of Table 10.2. Accompanying this reduced
differential rotation and total kinetic energy is an increase in magnetic energy. Indeed, theease with which these core dynamos became superequipartition is remarkable, where themagnetic energy exceeds the kinetic energy by a factor of 1.06 for M4, 2.07 for M8, 3.86
for M16. The diffusivities within these simulations was decreased to maintain the levelof supercriticality, and yet the amount of total kinetic energy in the system continues todecrease. It begs the question whether there is a point at which the fields become so strongas convection is no longer efficient enough to carry the heat building up in the core, leading tothermal pulses as convection is briefly reestablished. Further, given that superequipartition
states in previous stellar dynamo simulations of A-type stars had only been achieved withthe aid of a fossil field present in the radiative zone, it would be quite interesting to extend
these simulations by admitting a model primordial field to assess whether states of evengreater superequipartition might be achieved.
The magnetic field with the greatest amplitude tends to outline the convective columns,
which are strongly rotationally constrained and aligned with the rotation axis. Similar
structures have been seen in previous core dynamo simulations (Featherstone et al., 2009),

357
where there are spatio-temporal offsets between the magnetic structures and the velocity
structures. Similarly, the convective dynamo action within the cores of these B-type stars
builds magnetic fields that minimize the Lorentz forces through these offsets. Figure 10.7bshows a snapshot of the velocity field lines associated with the flow structures with thegreatest kinetic energy. That image is meant to be compared directly with Figure 10.7c,
which has an overlay of the magnetic energy. Together these figures exhibit the spatialoffsets between the strongest flow structures and the strongest magnetic structures. Suchstructures arise in superequipartition systems, for if they did not the magnetic fields would
quench the convection and disrupting its ability to transport heat. This arrangement must
be more energetically favorable than an increased amplitude of the thermal perturbations,which would also increase the heat flux at the expense of enhanced buoyancy and thus faster
flows. These two features are the underpinnings of the magneto-Archimedean-Coriolis force
balance, wherein the magnetic, thermal, and vortical structures are arranged to minimize thesum of those forces. Here this is accomplished through the spatially offset magnetic fields,
which reduce the impact of the Lorentz forces, while the Coriolis forces are minimized througha local baroclinic balance between thermal gradients and the local vortical structures.
10.7

Internal Gravito-Inertial Waves
The overshooting convection excites gravity waves as the flows are buoyantly braked

as they enter the stable region. These waves propagate mostly in the horizontal directions,though they also have a weak radial component that allows them to carry energy through theradiative zone. The spectra of these waves are now considered at two radii in the radiativezone of Case M4. These modes nominally would propagate and be dissipated, but here
this region is not captured and the waves reflect off of the upper boundary. This leads
to trapped resonant modes and an enhanced discrete spectrum of modes at those resonant
frequencies. Indeed, by comparing Figure 10.8a with Figures 10.8(b, c), it is clear that
discrete gravity modes are being excited in Case M4 within the radiative cavity above the

(a)

(b)

(c)

Figure 10.8: Horizontal power spectra of gravity waves excited in the radiative exterior of M4. (a) A reference gravity wave
spectrum excited in an isothermal atmosphere, with the fundamental mode in black and increasingly larger radial wave numbers
from blue to red. (b) The time and m-averaged power spectrum of radial velocities just above the overshooting region on alogarithmic color scale, a bar of low frequency power is visible as are ridges of power far above the local Brunt-Väisälä frequency.
(c) The same as in (b) but mid-way through the radiative exterior. The lower black line is the rotation frequency, the black lineat ω/NV B = 1 is the local Brunt-Väisälä frequency, and the upper black line is the upper limit of frequencies of modes present
at a given radius.

358

(a)

359

(b)

Figure 10.9: Shell and time-averaged rms velocities in the B-type star simulations. (a) The
total rms velocity vrms , showing little overall change in rms velocity with rotation rate, with
M16 in black, M8 in blue, and M4 in red. (b) Rms radial velocity vrrms , showing a 100-foldrise in radial velocity in the radiative exterior, color as in (a).

convective core. Figures 10.8b, c show the time and azimuthal-wavenumber averaged radial
velocity power spectra at two depths one just above the overshooting region and anothermid-way through the radiative envelope. The influence of the convection is still present near
the overshooting region, as is visible in the low-frequency bar of power. The ridges of power
extend far beyond the Brunt-Väisälä frequency, indicating that modes reflected from higher
in the radiative envelope are propagating in the region. This seems to be more prevalentcloser to the core where perhaps a broader spectrum of waves can be excited, or waves thathave been reflected from the upper boundary and are now impinging on the core.
What is of potentially greater interest is the ability of those waves to transport energy
and angular momentum from the core to near the surface or to a critical layer where they

360
may break. Figure 10.9 shows the total rms velocity for the three magnetic cases. The change
in vrms with a factor of four increase in rotation rate is fairly small, with slightly larger rms
values in the core for slower rotation, and the converse in the radiative zone. If one examines
the rms radial velocities, there are again modest differences in the core. Yet in stark contrast
to the total rms velocities, the radial velocities climb substantially with rotation rate, being
100-fold greater for the same factor of four increase in rotation rate. This encourages afurther analysis of the wave transport properties as they scale with rotation rate. Indeed,beyond the increase in the rms radial velocity, there is also an increase in the amplitude of
the thermal perturbations. Together these perturbations establish an enthalpy or heat fluxthat increase with rotation rate
Figure 10.10a shows where log ΓS (the growth rate of waves arising from entropy diffusion as described in Appendix D) is greater than zero over a range of frequencies varying
between zero and 1.4 times the resonance frequency of the domain, which can loosely be
Rrdefined as ω0 = r12 N/rdr. The actual Brunt-Väisälä frequency divided by the resonancefrequency is the black line. The growth rate is largest in the overshooting region where κ isstill large, with a growth rate of about 6 hours. This is about on par with the the excita-tion rate, but it decays exponentially away from this region dropping to about 280 days at
mid-radiative zone. Modes propagating near the upper boundary, whose local Brunt-Väisälä
frequency is above resonance, can also grow due to entropy diffusion though very slowly.
The most worrisome place to be is obviously when ω ≈ N as the growth rate can be comevery large.
The remedy for any possible issues arising from entropy diffusion would come from using
only the thermal diffusion arising from the combination of the Braginskii conductivity andradiation under the diffusion approximation. The radiation diffusion dominates the thermal
conductivity by at least three orders of magnitude in these stars, as seen in Figure 10.10b.
The radiation is clearly doing the yeoman’s work, smoothing out temperature perturbations.
Moreover, replacing the entropy diffusion by the thermal diffusion looks very promising, as

(a)

(b)

361

Figure 10.10: (a) Regions of wave instability due to entropy diffusion as evidenced throughthe wave growth rate ΓS , indicating that waves with frequencies close to the Brunt-Väisälä
frequency can grow faster than they are dissipated. (b) A comparison of the magnitudesof the entropy diffusion and the thermal diffusion coefficients, showing that the thermal
diffusion is more than sufficient to maintain stability.

the thermal diffusion is actually quite a bit larger than the entropy diffusion. Surprisingly,
the gravity waves should be much more damped than in the current models. Thus there is no
reason one should be using entropy diffusion rather than the temperature diffusion for thesemodels: it instead should be zero in the radiative zone. It may further be possible to remove
this problem entirely by splitting the thermal diffusion into two components, one that uses
the molecular values on the mean temperature and an artificially increased effective thermal
diffusion that acts on temperature perturbations in the convection zone but is then tapered
back to its molecular value in the radiative zone.
10.8

Conclusions
These preliminary results are promising, with superequipartition fields being built

through dynamo action that reach MG strengths. It is likely that a number of different

362
superequipartition states can be realized. With recent experience with geodynamo studies
and those with core convection in A-type stars, an external magnetic field that threads thecore but that resides in a separate region can lead to strongly superequipartition states.
These states may also be sensitive to the functional forms of the SGS diffusivities that are
imposed. The ASH simulations of B-type stars presented here may set the groundwork forfinding buoyant fields within more turbulent settings, particularly those that admit fossil
fields in the radiative exterior. In keeping with previous results (Brun et al., 2005; Featherstone et al., 2009), there is a marked difference in the differential rotation between magnetic
and hydrodynamic simulations. The gravity wave spectrum seen here shows discrete modes
that may be trapped in the cavity of the truncated radiative envelope. Furthermore, tobetter assess the properties of the internal waves, there are hints about how to better handle
diffusion in regions with stable stratifications within the context of the anelastic approximation.
There is much promise in studying the coupled core and radiative envelope dynamics ofmassive O and B-type stars. In order to make contact with the observed surface magnetismof those stars and to better understand their wind-driven mass-loss, simulations that capture
the convective core and the internal waves that connect it to the surface of the star will requirea unified treatment of both of these regions. The simulations presented here represent an
initial description of such coupled dynamics, and thus it is anticipated that substantial
advances in understanding the properties of this coupling and its observational implications
will be made in the near future. These massive stars are the crucibles in which the elements
are forged and they are the mechanism by which the galaxies are seeded with those elements.
Hence it is of great interest to ascertain how they end their lives. Namely, do they explodeas supernovae and distribute their elements in a single burst, or do their strong winds andthe subsequent mass-loss provide a slow and steady means for that distribution. The relative
importance of these mechanisms likely depends on the internal dynamics of these stars, which
involves convection, rotation, magnetism, and internal gravity waves.

Chapter 11
Modeling the Solar Near-Surface Shear Layer“Truth is much too complicated to allow anything but approximations.” – John von
Neumann
This chapter presents some aspects of the Compressible Spherical Segment (CSS) code,from its numerical foundations to some of its applications. The science goal that started workon this code was a project to assess the role of thermal plumes in the solar near-surface shear
layer, which since blossomed into many other projects. The first version of the 2-D code wasdescribed in Hurlburt & Rucklidge (2000). However, the only remaining similarity between
the two codes is the acronym. The original version of the code was not scalable and requiredlarge eddy diffusivities for stability. In the intervening time, the code has been completely
rewritten (several times over) so as to be highly scalable, numerically stable at low levels ofdiffusion, and to have a wide range of features. Thus, there is no piece of the original codethat remains.
The motivation for the chapter is to understand some aspects of the solar differential
rotation. To that end, it is worthwhile to recap some of the highlights of features within thesolar interior. While the radiative interior of the Sun rotates at a nearly uniform angularvelocity, the convection zone rotates differentially with latitude and radius (e.g., Deubner
et al., 1979; Brown et al., 1989; Thompson et al., 1996; Schou et al., 1998; Howe et al.,
2000; Thompson et al., 2003). The interface between the two has come to be known as thetachocline (Spiegel & Zahn, 1992), a region important in determining both the global dy-

364
namics of the overlying convection zone (Miesch, 2005) and likely the operation of the cyclic
solar dynamo (Charbonneau, 2010). The angular velocity in the bulk of the convection zonereflects departures from a cylindrically invariant Proudman-Taylor state. Such departures
are likely due to the presence of radial shear in the stably stratified tachocline, which induces latitudinal temperature gradients there. This thermal structures is subsequently felt
throughout the convection zone, where a thermal wind or baroclinic balance is establishedbetween cylindrical gradients in the differential rotation and the thermal field (Rempel, 2005;
Miesch et al., 2006; Balbus et al., 2012). This yields an angular velocity profile in the bulk of
the convection zone with only weak radial gradients bounded by the tachocline at the baseand the near surface shear layer at the top. It is this the origin of the upper shear layer
which is largely the scientific subject of this chapter.
As in Figure 3.5, the near surface shear layer (NSSL) occupies the upper ∼ 5% of
the Sun by radius. Within this region, the angular velocity increases by about 15 nHz
from the photosphere to about 0.95 R⊙. The gradient close to the photosphere is fairly
large, being consistent with angular momentum conservation per unit mass (Corbard &Thompson, 2002; Howe et al., 2006). The gradient becomes flatter with increased depth
and with increasing latitude, with some suggestion, not shared by all analyses, that it may
reverse in the uppermost layers for latitudes above ∼ 45 − 55◦ (Corbard & Thompson, 2002;
Howe et al., 2006). These features may imply that the near-surface differential rotation canhave a substantial variation in time throughout the solar cycle. It is unknown what role the
NSSL plays in the global dynamics of the Sun or in the operation of the solar dynamo, but
torsional oscillations within it show strong cycle dependencies.
Much of the dynamics in the NSSL arises from the vigorous granular-scale convection
driven by radiative cooling and large superadiabatic gradients in the photosphere. The
collective interaction of granular-scale flows (typical size of 1 Mm, lifetime of 0.2 hr) is amajor component in the formation of mesogranular (5-10 Mm, 5 hr) and supergranular (1535 Mm, 24 hr) scales (Rast, 1995; Rieutord et al., 2000; Rast, 2003), and these in turn likely

365interact in the NSSL with larger-scale motions generated below, possibly leaving signaturesin the surface flows (Lisle et al., 2004). The shear layer depth is of the same order as thelength scale of supergranulation, which is weakly influenced by rotation. This dynamical
coupling between vigorous small-scale convection and larger-scale rotationally influenced
motions results in angular momentum transport within the layer maintaining the observed
radial shear (De Rosa et al., 2002). Finally, recent results using the latest HMI Dopplergram
data have ascertained some information about the even larger scales of giant cells, including
the Reynolds stresses that accompany them (Hathaway et al., 2013). Those results givesome hope for contact between theory and observations that may disentangle the dynamicaldifferences between the deep interior and the near-surface region.
11.1

Equations Solved in CSSWith §3.1 as a reference, consider the vector form of the resistive MHD system of

equations in a rotating spherical coordinate system∂ρ+ ∇ · ρu = 0,
(11.1)
∂t




∂ρu
1
B2
I−
+ ∇ · ρu ⊗ u + P +B ⊗ B − D = ρ [∇φeff + 2v × Ω0 ] ,(11.2)
∂t
8π
4π



1
η 1
∂EM
2
+ ∇ · u(EM + P ) + q − u · D −u · BB +∇B − B · ∇B
= 0, (11.3)
∂t
4π
4π 2
∂B
= ∇ × [u × B − η∇ × B],(11.4)∂tB2
1+ ǫ + ρφeff ,(11.5)
EM = ρu2 +28πwith Ω0 the frame rotation rate, the effective gravitational potential φeff = φ + λ2 Ω20 /2,
and λ the distance to the rotation axis. First the energy equation is reduced to an evolution
equation for the entropy. This allows greater compatibility between the ASH and CSS codes,
but one does lose some nice properties of the conservation equations. In any case, the entropy
equation follows from the application of the second law of thermodynamics and through thesubstitution of Equations 11.1, 11.2, 11.4, and 11.5 into Equation 11.3, yielding

366

1
∂s
= −u · ∇s −[∇ · q − Φ − ρǫ] + Ds ,∂t
ρT

(11.6)

where Ds represents an entropy diffusion that will be defined later, ǫ is heating or coolingterm, Φ is the combined ohmic and viscous heating, q is a heat flux. In addition to theentropy, the form of the equations evolved in CSS is slightly modified so as to keep theprimitive variables to allow easier implementation of certain boundary conditions, speed
computation, and to minimize communication, yet the conservative form is retained formost of the Eulerian terms as∂ρ= −∇ · ρu + Dρ,(11.7)∂tu ∂ρ 1B2
11∂u=−− ∇ · ρu ⊗ u +I−
B ⊗ B − ∇P + ∇φeff + 2v × Ω0 + Du ,
∂t
ρ ∂t
ρ
8π
4π
ρ
(11.8)∂A= v × B + DA ,∂t
B = ∇ × A,

(11.9)(11.10)with Dρ , Du , and DA being diffusive terms whose form depend upon the choice of diffusion
scheme. The pressure gradient is isolated to allow a variety of boundary conditions to be
imposed. The vector potential form of the induction equation is employed to ensure thatthe solenoidality constraint on B is met to within the accuracy of the numerical scheme.
Expanding these equations in spherical coordinates and reserving the diffusion related terms
for a later section, the evolved equations are

367

1 ∂
1 ∂
1 ∂
∂ρ
= − 2 r 2 [ρur ] −sin θ [ρuθ ] −[ρuφ ] + Dρ,(11.11)∂tr ∂rr sin θ ∂θr sin θ ∂φ





1
1
ur ∂ρ
1 ∂ 2
Br Bθ
∂
∂ur
2
2
2
2
Bθ + Bφ − Br −=−−r ρur +sin θ ρur uθ −∂tρ ∂t ρr 2 ∂r
8π
ρr sin θ ∂θ
4π


1
Br Bφ
1 ∂P
∂
−
ρur uφ −−− g + r sin2 θΩ20 + 2Ω0 sin θuφ + r̂ · Du ,ρr sin θ ∂φ
4π
ρ ∂r












(11.12)222
Br + Bφ − Bθ

∂uθ
Bθ Br
1
uθ ∂ρ
1 ∂
1
∂
−
=−
− 2 r 2 ρuθ ur −sin θ ρu2θ +
∂t
ρ ∂t
ρr ∂r
4π
ρr sin θ ∂θ
8π


Bθ Bφ
1 ∂P
∂
1
ρuθ uφ −−+ r sin θ cos θΩ20 + 2Ω0 cos θuφ + θ̂ · Du ,−ρr sin θ ∂φ
4π
ρr ∂θ


(11.13)

∂uφ
Bφ Br
1
uφ ∂ρ
1 ∂
Bφ Bθ
∂
−
=−
− 2 r 2 ρuφ ur −sin θ ρuφ uθ −
∂t
ρ ∂t
ρr ∂r
4π
ρr sin θ ∂θ
4π



1
1
1 ∂P
∂
−
ρu2φ +
Br2 + Bθ2 − Bφ2 −− 2Ω0 (sin θur + cos θuθ ) + φ̂ · Du ,ρr sin θ ∂φ8π
ρr sin θ ∂φ
(11.14)∂s uθ ∂s
uφ ∂s
∂s
= −ur
−
−
∂t
∂r
r ∂θ r sin θ ∂φ

1 1 ∂ 21 ∂1 ∂−r [qr ] +sin θ [qθ ] +[qφ ] − Φ − ρǫ ,
ρT r 2 ∂rr sin θ ∂θr sin θ ∂φ

(11.15)and the vector potential equations in component form are∂Ar= uθ Bφ − uφ Bθ + r̂ · DA ,∂t
∂Aφ
= uφ Br − ur Bφ + θ̂ · DA ,∂t
∂Aφ= ur Bθ − uθ Br + φ̂ · DA .
∂t

(11.16)(11.17)(11.18)The closure of the system is the equation of state given ρ and s, which arrives in the form of
P = P (ρ, s). For a calorically perfect gas this is P = (γ − 1) CV ρT = ργ eS/CV . The physicalheat flux is currently defined in the radiation diffusion approximation asq=−

4acT 3∇T (ρ, s) ,3ρκR (ρ, s)(11.19)

(a)

(b)

-2.5°

+2.5°

(c)

-2.5°

(d)

368

+2.5°

Figure 11.1: MHD simulations with CSS. Simulations of the magneto-rotational instabilityin spherical coordinates with (a) radial velocities and (b) radial magnetic field near the upperboundary of the simulation. An MHD convection simulation with (c) radial velocities and(d) radial magnetic field near its upper boundary.

where κR (ρ, s) is the Rosseland mean opacity. When not using the perfect gas equationof state, the temperature, pressure and opacity are interpolated within a tabular equationof state such as OPAL or the Milhalas-Hummer-Dappen equation of state. Finally, there
is an additional feature in CSS that allows the artificial reduction of the sound speed bymodifying the continuity equation as seen in Appendix E. Two example MHD calculations
in local domains near the equator are shown in Figure 11.1. The simulation illustrated in
Figures 11.1(a, b) had a stable stratification and admitted the magnetorotational instability,
whereas the other simulation was in the same domain with an adiabatic stratification and isconvectively unstable.
11.2

Boundary Conditions
There are several choices of boundary conditions within CSS. These choices are de-scribed in following two subsections, one regarding the radial boundaries and the other regarding the latitudinal boundaries. Currently, the only option for the longitudinal direction
is periodicity.

369
11.2.1

Radial Boundaries

The entropy or its radial gradient may be specified at the radial boundaries as s (θ, φ, r1 )= f (θ, φ)
Dirichlet(11.20)∂s(θ, φ, r1 ) = g (θ, φ)
Neumann∂r

and similarly on the upper boundary s (θ, φ, r2 )= f (θ, φ)∂s(θ, φ, r2 ) = g (θ, φ)∂r

Dirichlet(11.21)

Neumannwhere r1 and r2 are the lower and upper radial boundaries of the computational domain,and f and g are arbitrary functions of the horizontal coordinates.
The latitudinal velocity on radial boundaries may be stress-free or no-slip as follows: uθ |r1 ,r2=0
No-Slip(11.22)uθ∂uθ=Stress-Free∂r r1 ,r2
r r1 ,r2

as is also the case for the longitudinal velocity uφ |r1 ,r2
=0
uφ
∂uφ

=

∂r r1 ,r2
r r1 ,r2

No-Slip
Stress-Free(11.23)

If the boundaries are impenetrable, then ur |r1 ,r2 = 0; the case of open boundaries ishandled below. To ensure that the pressure gradient at the boundary is consistent with animpenetrable boundary to machine precision, it is set at the latitudinal and radial boundaries.
For the radial boundaries, first all terms in the radial momentum equation except for thepressure gradient are evaluated, which yields





1
1
1 ∂ 2
Br Bθ
∂
ur ∂ρ
2
2
2
2
Bθ + Bφ − Br −− 2 r ρur +sin θ ρur uθ −
δur = −
ρ ∂t
ρr ∂r
8π
ρr sin θ ∂θ
4π


Br Bφ
∂
1
ρur uφ −− g + r sin2 θΩ20 + 2Ω0 sin θuφ + r̂ · Du .
(11.24)−
ρr sin θ ∂φ
4π
The remaining terms are

370

1 ∂P
∂ur
= 0 = δur |r1 ,r2 −.
∂t r1 ,r2
ρ ∂r r1 ,r2(11.25)Thus to ensure the consistency of the velocity evolution equations, the radial gradient of thepressure is set at the upper and lower radial boundaries as∂P= ρδur |r1 ,r2 .
∂r r1 ,r2

(11.26)The magnetic boundary conditions also have two orthogonal options. They can beperfectly conducting or normal-field. The perfectly conducting boundary implies a surface
current such that the normal field vanishes at the boundary. While there are many choices
due to the gauge freedom of the vector potential, only the simplest is implemented wherein Aθ |
r1 ,r2 = 0
Perfect Conductor(11.27) Aφ |
r1 ,r2 = 0
while Ar is left free on the boundaries. This satisfies the constraint that Br |r1 ,r2 = 0, andthat the horizontal field is nearly unconstrained, yet remains divergence free. On the other
hand, for the normal-field condition, the horizontal fields must vanish implying that



=0

 Ar |r1 ,r2


 ∂Aθ
Aθ
=−
Radial Field(11.28)∂r r1 ,r2
r r1 ,r2



∂Aφ
Aφ



=−
 ∂r
r r1 ,r2
r1 ,r211.2.2

Latitudinal Boundaries

The latitudinal boundaries are set such that the system is thermally insulated, namely
that
∂S
= 0.
∂θ θ1 ,θ2

(11.29)

371
As with the radial velocity boundary condition, the latitudinal conditions may be
stress-free or no-slip, with each directional component treated individually. The boundary iscurrently only impenetrable with uθ |θ1 ,θ2 = 0. The radial and longitudinal velocity boundary
conditions are
ur |θ1 ,θ2







 uφ |θ1 ,θ2
∂ur



∂θ θ1 ,θ2




∂uφ



∂θ θ1 ,θ2

=0

No-Slip=0=0Stress-Free(11.30)= cot θuφ

Again to maintain the numerical consistency of the pressure derivatives at the latitudinal boundaries, all terms of the latitudinal component of the momentum equation arecomputed less the pressure derivative as 



uθ ∂ρ
Bθ Br
1
1 ∂ 2
1
∂
2
2
2
2
δuθ = −−Br + Bφ − Bθ
− 2 r ρuθ ur −sin θ ρuθ +
ρ ∂t
ρr ∂r
4π
ρr sin θ ∂θ
8π


Bθ Bφ
∂
1
ρuθ uφ −+ r sin θ cos θΩ20 + 2Ω0 cos θuφ + θ̂ · Du ,(11.31)−
ρr sin θ ∂φ
4πthen, with impenetrable boundaries, the remaining terms are∂uθ
1 ∂P= 0 = δuθ |θ1 ,θ2 −.
∂t θ1 ,θ2ρr ∂θ θ1 ,θ2(11.32)Thus, the radial gradient of the pressure is set at the upper and lower radial boundaries as∂P= rρδuθ |θ1 ,θ2 .
(11.33)∂θ θ1 ,θ2
 are again either normal field or perfectly conducting.
The magnetic conditions Ar |
θ1 ,θ2 = 0
Perfect Conductor(11.34) Aφ |
θ1 ,θ2 = 0

while Aθ is left free on the boundaries. This satisfies the constraint that Bθ |θ1 ,θ2 = 0, andthat the horizontal field is nearly unconstrained, yet remains divergence free. On the other

hand, for the normal
field condition, the horizontal fields must vanish implying thatAθ |θ1 ,θ2
=0




 ∂Aφ
= − cot θAφ |θ1 ,θ2
Normal Field(11.35)∂θ θ1 ,θ2



∂Ar



=0
 ∂θ
θ1 ,θ2

372
11.2.3

Open Boundary Conditions

The general framework for the open boundary conditions is largely in place, as describedin the previous subsection. There need only be a subtle shift in perspective between theimpenetrable and open boundary conditions. The impenetrable boundary requires that thetime derivative of the radial velocity vanish, and so constrains the pressure gradient there.
A numerically sensible open boundary for a convection simulation is one that conserves massand momentum within the domain. This is accomplished by requiring that the horizontallyaveraged flow be hydrostatic at the boundary. In the current state of CSS, there is no
implementation of an open boundary for the magnetic fields nor are there open conditions
on the latitudinal boundaries.
When considering a purely open boundary, one with no information provided fromoutside the domain, there is only a small change in how the horizontal velocities are evolved
at the boundary. However, if the boundary condition is no-slip, they are directly specified
to be zero. A new term is added to the evolution equations of the horizontal velocities thatcompensates for any outgoing momentum through the addition of a uniform momentum into
the inflows.
′

∂uθ
∂uθ
hpθr1 ,r2 i+ f (θ, φ)=,∂t r1 ,r2
∂t r1 ,r2
hmin
r1 ,r2 iτr1 ,r2′
hpφr1 ,r2 i∂uφ∂uφ+ f (θ, φ)=.
∂t r1 ,r2
∂t r1 ,r2
hmin
r1 ,r2 iτr1 ,r2(11.36)(11.37)

Here f is a position dependent mask selecting points that are incoming or outgoing depending
upon the sign of the velocity normal to the boundary plane, pθ,φ are the horizontally-averaged
values of the horizontal momenta within the outflows, min
r1 ,r2 is the mass of the inflows at
the boundary, and τr1 ,r2 is the drag time scale, so

373


 1

If incoming, 0
If outgoingRf (θ, φ) ρr 2 sin θdθdφin∂VRhm ir =,f (θ, φ) r 2 sin θdθdφ∂VR(1 − f (θ, φ)) ρuθ r 2 sin θdθdφ
Rhpθ i = ∂V,
2 sin θdθdφ
(1
−
f
(θ,
φ))
r
R ∂V
(1 − f (θ, φ)) ρuφ r 2 sin θdθdφR,
hpφ i = ∂V
(1 − f (θ, φ)) r 2 sin θdθdφ∂V∆rτr1 ,r2 =.
max|ur |f (θ, φ) =(11.38)(11.39)(11.40)(11.41)(11.42)

r1 ,r2The equations for the radial velocity and the entropy are modified so that they are
pulled toward the values of the nearest interior plane on the grid crossing time scale of thefastest flow directed out of the domain on this interior plane. This time scale is typically
far less than the sound crossing time given the large sound speed near the bottom of theconvective domains being simulated, so the excitation of sound waves by the boundary is
minimized. Furthermore, the pressure gradient is set to be hydrostatic, in a fashion similar
to that for the impenetrable boundaries.
The radial velocity at the boundary must also take into account the density stratification of the domain, so the values of the radial velocity at the boundary plane are pulled
toward the values of the interior plane multiplied by the ratio of the mean density of thetwo planes. Similarly, the entropy is dragged towards the interior value, but with a mean
entropy gradient enforced by dragging the mean value at the boundary toward the desired
value at the same rate as the pointwise values. To make this more concrete,

374




ρr1 +∆r
hprr1 i,ur |r1 +∆r − ur |r1 −ρr1
hmr1 i∆t
hprr2 i1 ρr2 −∆r∂ur,=
ur |r2 −∆r − ur |r2 −∂t r2 τr2
ρr2
hmr2 i∆t1∂ur=∂t r1 τr1

(11.43)(11.44)1∂s=(s|r1 +∆r − s|r1 − hsr1 i + sr1 ) ,∂t r1 τr1

(11.45)1∂s=(s|r2 −∆r − s|r2 − hsr2 i + sr2 ) ,∂t r2 τr2

(11.46)withand where A =

R

∂V

Z
1
ρr 2 sin θdθdφ,
hmi =
A ∂V
Z
1
ρur r 2 sin θdθdφ,
hpr i =
A ∂V
Z
1
hsi =
sr 2 sin θdθdφ,
A ∂V(11.47)(11.48)(11.49)r 2 sin θdθdφ the area of the boundary. The pressure gradient conditionis thus modified to enforce the constraints placed on the radial velocities as∂P∂ur= ρδur |r1 ,r2 − ρ.
∂r r1 ,r2
∂t r1 ,r2

(11.50)with the time derivative evaluated as in Equations 11.43 and 11.44.
11.3

Diffusion Schemes
To begin, let us define the components of the momentum, induction, and energy equa-tions that were omitted in §11.1. These are diffusive terms that, in the Navier-Stokes equations, are the first order approximations to the net effects of particle interactions at the
micro-scale.

375
11.3.1

Newtonian Eddy Viscosity

The Newtonian eddy viscosity scheme is one in which a dynamic viscosity, magnetic
diffusion, and a thermal conductivity are prescribed. The values of these diffusivities are
typically many orders of magnitude higher than the expected molecular values. This has theeffect of moving the dissipation scale from the molecular scale into the spectrally resolved
domain. However, the magnitude of diffusion necessary to maintain numerical stability isoften much higher than the minimum value required to displace the dissipation scale to the
Nyquist wavenumber of the simulation. In any case, the diffusive terms omitted in §11.1 canthus be defined asDρ = 0,(11.51)

Ds = ∇ · [κT ∇T (ρ, s) + κS ∇s],
 

1
1
Du = ∇ · 2µ e − I∇ · u ,ρ3(11.52)DA = η∇ × B,

(11.53)(11.54)

where the dynamic viscosity µ, magnetic diffusivity η, thermal diffusion κT , and entropy
diffusion κs are functions of radius and where e is the stress tensor and I the unit tensor.
Yet the entropy diffusion term must be used with caution as it can violate the second law ofthermodynamics. Finally, the viscous and ohmic heating is thus12
Φ = 2µ e : e − (∇ · u) + ηJ2
3(11.55)where the colon denotes a full tensor contraction on e and J = ∇ × B is the electric current.
From a computational standpoint, the downside to using the eddy diffusion model in CSS
is that it requires extra storage, communication, and computation to handle the second
derivatives and the cross-derivatives necessary for it.

376
11.3.2

Slope-Limited Diffusion

The slope-limited diffusion scheme, as discussed in Chapter 4, is implemented in CSS
as well. The difference between the implementation in ASH and here is that the diffusion
is not filtered at high latitudes and there is sometimes a need to use the density diffusion
for numerical stability. The latter does not always have to be used, as the density usuallyremains smooth if the velocity field is well resolved and no shocks are present. On the other
hand, it will be needed when the number of radial grid points is chosen to be low, whichallows larger time steps, but can under-resolve the stratification leading to two-zone gridscale oscillations. In this scheme then, let S be the discrete slope-limited diffusion operator
defined in Equation 4.71. The diffusive terms then becomeDρ = S [ρ] ,(11.56)

Ds = S [s] ,(11.57)Du = S [ur ] r̂ + S [uθ ] θ̂ + S [uφ ] φ̂,(11.58)DA = S [Ar ] r̂ + S [Aθ ] θ̂ + S [Aφ ] φ̂.

(11.59)Since Dρ is now non-zero, there is a term in the momentum equation arising from the time
derivative of ρ that compensates for any change in the momentum due to mass diffusion.
Recall from §4.4 that the cell-edge diffusion fluxes need to be interpolated to cell-center tocompute the velocity and magnetic diffusion tensors, being τ and µ respectively. Note thelatter tensor is not directly proportional to the diffusion of the vector potential. Rather
it is proportional to its curl, so this heating is only accurate to within the precision of
the numerical scheme. These tensors are almost always non-symmetric and so have antisymmetric components that contribute to the heating. Such terms vanish for the eddy
diffusion approximation as the diffusion tensor is symmetric. The viscous and ohmic heating
arising from these diffusive processes are thus
Φ = τ T : ∇ρu +1 T
1µ : ∇B − 2ρu · ∇ · τ a −B · ∇ · µa .
4π
2π(11.60)377
with the superscript a denoting the anti-symmetric portion of the tensors and T being thetranspose operator.
11.4

Implementation and Evolution of CSS
CSS has gone through many incarnations: from the original code to a hybrid OpenMP/MPI

code to a vector optimized hybrid code and so on until it reached its current implementation. It is written in Fortran 90 for optimal speed relative to C++. The current version
solves the MHD equations discretized on a nonuniform spatial grid in spherical coordinates
using one of two time-stepping algorithms. A truncated compact finite difference scheme isemployed for the spatial derivatives. The computational workload is evenly divided acrossall the requested message passing interface (MPI) communicators with a 3-D decomposition.
The computations are parallelized using only the MPI 2.2 standard, as will be described
below.
11.4.1

Program Design

The layout of the program is fairly straightforward. The primary program calls all
the initialization routines of each module. First all processors read a namelist-formatted filecontaining all the parameters of the run, including domain size, relevant physical parameters, problem size and node configuration and so on. Given these parameters, the processors
initialize MPI communications and define a Cartesian node layout that optimizes node placement for the highest bandwidth and minimum latency. Derivative coefficients and matrices
are precomputed and memory is allocated on each MPI communicator based on the layout
and problem size. The physical problem is then initialized either from a checkpoint of aprevious run or by constructing the initial primitive variable values from the physical input
parameters.
The fundamental computational loop of the program calls only three subroutines. The
first conditionally computes the step size, which is infrequently called as the time step is con-

378
strained by the Currant-Friedrichs-Lewy (CFL) condition on the fast magneto-sonic modes
and is most often very much greater than the flow speeds. The second calls the time stepping routine. The third conditionally checks whether or not the current time step number
corresponds to the desired output cadence for a particular output type. So, as one would
usually want, the vast majority of the computational effort resides in evolving the simulation.
When I/O is desired, it handled with MPI I/O routines that maximize bandwidth to thedisk, which is particularly important for checkpointing.
There are two optional features that include additional physics into the simulation.
This can be inflows specified from ASH and flows specified at the upper boundary throughstatistical methods derived from photospheric simulations.
11.4.2Time Evolution and Error Analysis

There are currently two time-stepping algorithms available within CSS. On a per time
step basis, the fastest is a total variation diminishing (TVD) third-order Runge-Kutta scheme(Gottlieb & Shu, 1998). The second is the standard fourth-order Runge-Kutta scheme(Runge, 1895). The slope-limited diffusion is calculated only once per time step and added
to the solution with a forward Euler scheme. It could be computed at each evaluation in theRunge-Kutta methods, but this has been found to be unnecessary and much more expensive.
The third-order Runge-Kutta (RK3) method has three stages that are given by
v1 = vn + ∆tR (tn , vn ) ,





1
1
1
3
v2 = vn + v1 + ∆tR tn + ∆t, v1 ,
4
4
4
4


1
2
2
2
vn+1 = vn + v2 + ∆tR tn + ∆t, v2 .
3
3
3
3

(11.61)(11.62)(11.63)where n denotes the N-th time step, ∆t ≤ ∆tCF L is the physical time step in seconds, R isthe call to the right-hand side of the discretized version of Equations 11.11-11.18, and v isthe vector containing all primitive variables at all locations within the domain of the MPI
communicator. Note that all coefficients are positive, which is ideal for a TVD method.

379

3

(a)

(b)
2
10 -110 -2-

1

ΕA, ΕPIm z

10 -30

10 -410 -5--1

10 -6-2
10 -70.0

0.5

1.0

1.5

x
-3

-3

-2

-1

0

1

Re z

Figure 11.2: (a) Stability regions of three Runge-Kutta methods, with second-order in blue,
third-order in magenta, and fourth-order in brass. The shaded region is the region of stability.
(b) Phase (blue) and amplitude (red) errors as a function of step size for the third-ordermethod.

Similarly, the fourth-order Runge-Kutta (RK4) method has four stages that are givenbyk = ∆tR (tn , vn ) ,1
vn+1 = vn + k,611
k = ∆tR tn + ∆t, vn + k ,2
2
1
vn+1 = vn+1 + k,
 3

1
1
k = ∆tR tn + ∆t, vn + k ,2
2
1
vn+1 = vn+1 + k,3
k = ∆tR (tn + ∆t, vn + k) ,1
vn+1 = vn+1 + k,6

(11.64)(11.65)(11.66)(11.67)(11.68)(11.69)(11.70)(11.71)

where k is the time derivative evaluated at stage i. There is clearly greater overhead andcomputational expense with the fourth-order method. It has a slightly larger region of380stability with respect to the linear problem y ′ = λy with λ being some complex number.
This problem reduces to an iterative map yn+1 = r(z)yn , with r(z) = 1 + z + z 2 /2 + z 3 /6 forthe TVD RK3 method and r(z) = 1 + z + z 2 /2 + z 3 /6 + z 4 /24 for the RK4 method. These
regions of stability are shown in Figure 11.2a.
The time stepping routine calls at most six procedures depending on if there are open
boundaries with specified flows or not. If flows are specified by ASH at the lower boundary,
the first call is to read and interpolate an ASH inflow. The second call is to the slope-limited
diffusion routine. Then three or four calls are made to the “right-hand-side” routine R thatcomputes time derivatives at all points within the domain of the MPI communicator. A
routine that sets boundary values is called. If the boundaries are open and specified, three
more routines are called.
The time stepping routine uses at most six storage arrays, two for the slope-limited
diffusion, two storage arrays for temporary values between evaluations of R, one for storing
the values of the fields for use in the next time step. One additional storage array is needed
if eddy diffusion is used.
11.4.3Modified Compact Finite Differences

Compact finite differences are a particular numerical derivative scheme that can attainnearly spectral resolution (Lele, 1992). They are finite differences in the sense that they
have a fixed stencil width. What sets them apart from the Padé scheme and from simple
finite differences is that they result from a double expansion of the derivative value of somefunction f at a grid point xk as
i=M
X

i=−M

j=N
′

αi f (xk+i ) =X

aj f (xk+j ) .

(11.72)j=−Nwhere 2M is the bandwidth of the matrix with coefficients αi that is required to solve for the
now implicitly defined derivative f ′ . The finite difference stencil of the right-hand side has
coefficients aj and a stencil size 2N. It is the implicit definition of the derivative that enhances

381

Figure 11.3: Dispersion errors |k − κ(k)|k −1 for a single Fourier mode as the number ofpoints Npts resolving it is increased in four different compact finite difference (CFD, red)and standard finite difference (FD, blue) schemes. The dotted blue line is a fourth-orderFD, solid blue is sixth-order FD, solid red is sixth-order CFD, and dotted red is tenth-order
CFD.

their resolution, or more appropriately it is what reduces their dispersive numerical errors.
This effectively links the derivative values of all the grid points together. Thus the compact
finite difference method approximates a spectral method, which would have M = N = K/2,
with K the total number of grid points. The version used in CSS is
′
′
′
′
βfi−2
+ αfi−1+ fi′ + αfi+1+ βfi+2=cfi+3 − fi−3
fi+2 − fi−2
fi+1 − fi−1+b+a
. (11.73)
6∆x
4∆x
2∆x

This could be generalized for a nonuniform grid and is a planned change in CSS, where thenonuniform grid is currently accomplished through a Jacobian transformation. The simple
sixth-order scheme has β = c = 0, α = 1/3, a = 14/9, and b = 1/9, which yields a truncation
error of 4/7!∆x6 f (7) . The coefficients are found through matching terms in Taylor expansions
about fi′ and fi that provide a set of equations for the coefficients, with one free parameter α.

382With the choice of α = 1/3, the scheme becomes sixth order. The implicit definition of thederivative now requires a tridiagonal solve in addition to the finite difference. Thus the total
operation count including both the necessary tridiagonal solve and finite difference is 7K − 2
multiplications and 6K − 2 additions, which should be compared to the 6K multiplications
and 6K additions for a simple centered finite difference scheme of the same order. The
dispersion modified wavenumber κ of the derivative scheme isκ(k) =

a sin k + b/2 sin 2k + c/3 sin 3k,
1 + 2α cos k + 2β cos 2k(11.74)

which is found by admitting a Fourier mode with f (x) = exp (iks). Here the scaled wavenumber k ∈ [0, 1] is k = 2w∆x/L and the scaled coordinate is s = x/∆x, with w the physicalwavenumber. Thus the exact derivative with respect to s is f ′ (s) = ikf (s), or the spectral
derivative should have no modified wave number with κ(k) = k. The dispersive error relative to a spectral scheme is shown in Figure 11.3. The relative error indicates how many
points across a peak or a trough of a given Fourier mode are necessary for a particularderivative scheme to resolve the mode to certain level of accuracy. That is it measures theresolving power of a given derivative scheme. The fourth-order standard finite difference
scheme clearly does not perform well, requiring about four times as many points to reachfloating point machine precision when compared to the sixth-order standard finite difference.
Likewise, the sixth-order compact finite difference needs only 50% of the points as the sixthorder standard finite difference does to resolve the same features. The tenth-order compact
scheme is about a factor of two better than the sixth-order scheme, for a modest increase incomputational effort.
For the second-order derivatives f ′′ , a similar brew of coefficients yields a compact
finite difference, where
′′
′′
′′
′′
βfi−2
+ αfi−1+ fi′′ + αfi+1+ βfi+2=cfi+3 − 2fi + fi−3
fi+2 − 2fi + fi−2
fi+1 − 2fi + fi−1
+b
+a
.
2
2
9∆x
4∆x
∆x2
(11.75)

383
A scheme with a sixth-order truncation error −(23/11)/7!∆x6 f (8) has coefficients β = c = 0,
α = 2/11, a = 12/11, b = 3/11, which are again found through collecting terms in the Taylor
expansion by their order in ∆x. So far, this has only dealt with interior points. The derivativevalues at the physical boundaries of the domain can be found in the same fashion as one
need only relax the assumption of using centered differences. However, another method uses
the properties of the numerical derivative for the interior points to choose the proper values
in ghost cells outside the boundary so as to satisfy a given boundary condition. This is
usually more efficient than computing the value at the boundary itself as that requires moreconditional statements and asymmetrical workloads between boundary points and interior
points. This is used in CSS as it is most compatible with how the derivatives are handled
at internal boundaries between computational sub-domains that result from the domain
decomposition. At such internal boundaries, a sixth-order standard finite difference is used
to precondition the tridiagonal solve. This results in some loss of the global dispersion
properties of the compact finite difference, but the gains in computational efficiency andparallelizability outweigh the losses. This approach gives the following implicit solve for thederivatives within a sample sub-domain


 

′
′
1 α 0 0 0 ··· 0
f−1
f−1,F
D 


 


 

α 1 α 0 0 · · · 0   f ′   r0  0  



 

0 α 1 α 0 · · · 0  f ′   r 1  

1


 

 ..





.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .  =  . 


 



 

 0 · · · 0 α 1 α 0  f ′   rn−1   n−1  



 


 ′  

0
·
·
·
0
0
α
1
α
f
r

 n  

n



 
′
′
0 ··· 0 0 0 α 1
fn+1
fn+1,F D(11.76)′′
where f−1
and fn+1
are ghost cell values one grid points outside the sub-domain. The values r

are the right-hand side computations of the compact finite difference as above. In contrast,
the boundary values are computed using a sixth-order centered standard finite difference
constructed from values passed using MPI from the nearest neighbor sub-domains.

(b) 700(a) 1.00E+001(c) 12600

1.00E-001

1.00E-002

10

500
8
400

Ratio

Bandwidth (GB/s)

Time (s)1.00E+000

300

4

200

2

100
0

1.00E-003
10

100

1000

Message Size (kB)

10000

100000

6

0
10

100

1000

Message Size (kB)

10000

100000

10

100

1000

10000

100000

Message Size (kB)

Figure 11.4: Bandwidth increases using an optimized MPI process layout versus message size in kilobytes (kB) in a 3D domain
decomposition using round-robin process assignment (blue) and an optimized nearest-neighbor assignment (red). (a) Total timespent exchanging boundary information for 1536 Westmere cores on Pleiades at NASA Ames per time step. (b) Bisection
bandwidth in gigabytes per second (GB/s). (c) The ratio of the bisection bandwidth using the two process assignment methods.

384

385
11.4.4

Parallelization

The parallelization of the code is accomplished through MPI. After initialization, when
all processes have an identifier and can talk to each other, the MPI CART CREATE command is used to reorder nodes to conform to the domain decomposition. This accounts forthe periodicity in longitude and optimally places processes with considerations for hardware
structure and communication paths to assure that nearest-neighbor domains are physicallynearby. This increases bandwidth by over a factor of ten, yielding an order of magnitudedecrease in the clock cycles spent waiting on messages as can be seen in Figure 11.4. Furthermore, the communication is overlapped as much as possible with computation using
non-blocking send and receives. This is accomplished binarily: first through splitting thecommunications by direction, where each node separately communicates with its two neighbors in radius, latitude, and longitude; and then each direction is split into two stages. Suchan organization of the communication accommodates the layout of the computation of thetime derivatives, which is broken out by direction to optimize memory usage and instruction
reuse.
For each communication direction, the first stage initiates the message exchange prior
to some heavy computational workload, where the message is packed into 1D buffer array.
The odd numbered processors post a send first, whereas the even numbered processors firstpost a receive. After the workload is completed, the second stage finalizes the exchange
through a wait all statement and unpacks the 1D buffer into the boundary value array. Thisway each MPI processes has to wait for at most two other MPI processes to finish their task,
which is highly likely given the computational load between the two calls. All of these facetsof CSS have permitted it to be highly scalable, reaching up to 50,000 cores as can be seenin Figure 11.5. The scaling there was still linear, so the wide availability of machines withmillions of cores is eagerly being awaited.

386

Figure 11.5: Strong scaling, fixed problem size and increasing core counts, for ASH-FD and
CSS are shown on Kraken and Pleiades. (a) Time steps per wall-hour achieved in ASH
simulations with 400 radial mesh points and horizontal resolution of spherical harmonic
degree ℓ = 680 on Kraken (blue, 680K), and on Sandy Bridge cores on Pleiades (red, 680S),
and likewise for ℓ = 340 and 170. (b) Time steps per wall-hour achieved in CSS simulations
varied between 2562 and 10242 horizontal grid points and 192 radial points for Kraken (blue,labeled 256K to 1024K). With larger memory nodes, CSS simulations with 576 radial grid
points and up to 20482 horizontal grid points were run using Pleiades (red, labeled 512S to2048S). CSS modeling shows strong linear scaling between 100 and 50,000 cores, reachingnearly 1 million time steps per wall clock hour (equivalently a month of evolution of ahigh-resolution solar convection simulation).

11.5

Euler Equation in 1D with SLD
A first and simple test of the numerical scheme employed in CSS is to apply it to

the Sod shock tube problem, which is a standard test of the ability of a code to capture
shocks as it has an analytical solution. This requires the solution of the Euler equations, so
slope-limited diffusion is applied to achieve numerical stability. Quickly, the equations being
evolved use are the 1-D Euler equations with a sixth-order compact finite differences for thederivatives and the TVD RK3 time stepping algorithm. These equations are387

Figure 11.6: Sod shock tube problem solved using CSS numerical algorithms, showing theconvergence to the exact solution with increasing resolution.

∂ρ
∂ρ
∂u
= −u
−ρ
+ Dρ ,
∂t
∂x
∂x
∂u
1 ∂p
∂u u
=−
−u
− D ρ + Du ,
∂t
ρ ∂x
∂x ρ
∂e
∂h
∂u
= −u
−h
+ De ,∂t∂x  ∂x
1 2
p = (γ − 1) e − ρu ,2

(11.77)(11.78)(11.79)(11.80)h = e + p,

(11.81)1e = ρǫ + ρu2 ,2(11.82)where γ is the adiabatic exponent for this perfect gas, p is the pressure, h is the enthalpy,
and e the total energy. The slope-limited diffusion operator is D. The classical Sod shocktube problem is defined over the interval x ∈ [−1, 1] and has the initial conditions

388

ρ (x) =



 1


 1/8


 1
p (x) =

 1/10x≤0,(11.83)

x>0
x≤0

,

(11.84)x>0

u (x) = 0,

p
,
γ−1
5
γ= .
3

e (x) =

(11.85)(11.86)(11.87)The solution of this problem is shown in Figure 11.6. The conservation of mass and energy isvery good, even at low resolution. For the case with Nx = 64, the relative difference between
the volume integrated mass at t1 = 0.25 and t0 = 0.0 is δm = |mt1 − mt0 |/mt0 = 6 × 10−10
and for the volume integrated energy it is δe = |et1 − et0 |/et0 = 8 × 10−10 . The other cases
converge rapidly to the machine epsilon for double precision: with δm = 1.0 × 10−14 andδe = 1.4 × 10−14 for Nx = 128; δm = 7.5 × 10−15 and δe = 9.9 × 10−15 for Nx = 256;δm = 9.8 × 10−15 and δe = 2.8 × 10−15 for Nx = 512. The solution is very close to monotone,
but at low resolution is not perfectly monotone. Indeed, this shows that the SLD operator
does not guarantee this property for non-conservative schemes, rather it attempts to maintaina solution is close to monotone. On the other hand, it is very good at conserving mass andenergy even if the phase and amplitude of the solution have modest errors, which is apparent
in the Nx = 64 solution. If the resolution is only twice that, these errors quickly disappear
given the excellent resolution properties of the compact finite differences and due to the
TVD RK3 algorithm. It is quite remarkable that one can capture such shocks so well witha fairly simple high-order finite difference scheme such as CSS.

389
11.6

Numerical Tests Through Linear Waves
The following subsections rely upon the linearized Euler and Euler-Maxwell equations.

The linearization of these equations give rise to the hydrodynamic and MHD wave equations,which in turn can be solved analytically under many circumstances. It should be noted that
if the fields (pressure, density, entropy, velocity, and magnetic fields) of these equations are
separated into the mean field and the fluctuating fields, then to first order in the fluctuating
fields the equations are linear. To make this precise, first take the expansion about the mean
fields as P = P0 + P ′ , ρ = ρ0 + ρ′ , S = S0 + S ′ , v = v0 + v, and B = B0 + B′ . An archetypal
field F0 is independent of time and longitude, but can depend upon the two other spatial
dimensions; in contrast, the fluctuations F ′ depend upon three spatial dimensions and time.
Under such an expansion, the equations of 11.1 become∂ρ′= −∇ · ρ0 v − ∇ · ρ′ v0 ,(11.88)
∂t


ρ′
1
1
∂v′
′
Ω0 R + 2v0 × Ω0 −= −v0 · ∇v − v · ∇v0 − ∇P +(∇ × B0 ) × B0 − gr̂
∂t
ρ0
ρ0
4π
1
+ 2v′ × Ω0 +(∇ × B0 × B′ + ∇ × B′ × B0 ) + Du ,(11.89)4πρ0∂B′= ∇ × (v′ × B0 ) + ∇ × (v0 × B′ ) + DB ,(11.90)∂t
1∂s′∇ · q0 + q′ + Φ(11.91)= −v′ · ∇s0 − v0 · ∇s′ −∂t
ρ0 T0
11.6.1

Sound Waves

The first and most simple test of the numerical schemes employed in CSS is to assess
their treatment of a monochromatic sound wave propagating in the periodic dimension of
φ. To this end, the equations of §11.1 are solved under the constraint that there is no
stratification (e.g. no gravity, which implies that ρ0 is constant), rotation, mean flows,transverse flows, magnetic fields, entropy variations, or diffusion (g = Ω0 = v0 = vr′ = vθ′ =B = s = D = 0). With such restrictions, the above linearized equations become390

ρ0 ∂u′φ∂ρ′=−,∂tr sin θ ∂φ
∂u′φ
1
∂P ′
=−
.
∂t
ρ0 r sin θ ∂φ

(11.92)(11.93)This is of course contingent upon the equation of state. So, assume an isothermal state,
which requires that the speed of sound is constant with P ′ = c2s ρ′ and c2s is constant. The
subsequent step is to take the second temporal derivative of equation 11.92. Upon thesubstitution of equation 11.93, this yields∂ 2 ρ′
c2s ∂ 2 ρ′=.
∂2t
r 2 sin2 θ ∂φ2(11.94)

ρ′ = ρ1 exp i (ωt − kφ),(11.95)Thus, the solution is simply

where ρ1 = ρ1 (r, θ) is the spatially dependent perturbation amplitude. This solution implies
that the dispersion relationship isω2 =c2s k 2.
r 2 sin2 θ(11.96)Since CSS is not a spectral code, the real part of equation 11.95 must be taken to give
 

cs t
′
ρ = ρ1 cos k−φ .
(11.97)r sin θThe azimuthal velocity that corresponds to this compressive density wave is found by inserting the previous expression for ρ′ into equation 11.93, which yields
 

cs t
ρ1
′
−φ .
uφ = cs cos k
ρ0r sin θ(11.98)Equations 11.97 and 11.98 provide analytical comparisons for the numerical solutions, sothat any phasing and amplitude errors introduced by the various numerical schemes in CSS
can be determined. They also provide the initial conditions for the code, wherein t is set
to zero in equations 11.97 and 11.98. The propagation of this sound wave under various
conditions is assessed below, with the three time-stepping schemes implemented in CSS, a
non-uniform radial grid, and varying the longitudinal resolution.

391
11.6.2

Alfvén Waves

Unlike the compressive, traveling sound waves assessed in the previous subsection, finding an Alfvén wave solution in a sphere is much more difficult. The most simple assumptions
that can be made are that the guiding magnetic field is force-free (∇ × B0 = 0), and obviously divergence-less. Simultaneously, the velocity field must also be solenoidal so thatthe slow and fast magneto-acoustic waves are filtered out, this is accomplished by setting
ρ′ = 0 and by taking an isothermal equation of state for the background. Thus, the linearized
Euler-MHD equations are solved under the constraint that there is no stratification, rotation,mean flows, entropy variations, diffusion, or resistivity (g = Ω0 = v0 = S = DV = DB = 0).
With such restrictions, the linearized equations become∂v′
1∇ × B′ × B0 ,
=
∂t
4πρ0
∂B′
= ∇ × (v′ × B0 ) .
∂t(11.99)(11.100)

Nominally, to find a traveling Alfvén wave the guide field B0 would be directed in
the periodic dimension φ to accommodate the phase and group velocities that are directedalong the guide field. However, no simple, mono-directional magnetic field that is curlfree and divergence-less exists. The work around is to solve for a standing radial Alfvénwave. This wave is directed along a radial guide field that is curl and divergence-free,namely B0 = Br22 /r2 r̂, where r2 is the radius of the outer boundary. Since Alfvén waves aretransverse, only the azimuthal quantities are allowed to vary such that vφ′ = vφ′ (r, θ, t) andB′φ = B′φ (r, θ, t). This assumption requires that vr′ = vθ′ = B′r = B′θ = 0 and implies that thevelocity and fluctuating magnetic field are both automatically divergence-less. Therefore,
the reduced equations are∂vφ′
1
φ̂ · [∇ × B′ × B0 ] ,
=
∂t
4πρ0
∂B′φ
= φ̂ · [∇ × (v′ × B0 )] ,∂t

(11.101)(11.102)392
First, let the vector potential be A = vφ′ B0 θ̂ with the Coulomb gauge (∇ · A = 0), which
requires that the solution have the form vφ′ = f (r, t)/ sin θ. If a second time derivative of
equation 11.102 is taken, it reduces this system to the following single equation∂ 2 vφ′B0
φ̂ · [(∇ × ∇ × A) × r̂] ,
=
∂t2
4πρ0



B0
=
φ̂ · ∇∇ · A − ∇2 A × r̂ ,
4πρ0



B0
φ̂ · ∇2 A × r̂ ,
=−
4πρ0


1
B0
2
∇ − 2 2 Aθ .
=4πρ0r sin θ(11.103)

Now substituting A back in and expanding the derivatives in equation 11.103, it is found
that


f B0
1
B0 sin θ∂2f2∇ − 2 2=2∂t
4πρ0r sin θ sin θ
 2

2
∂
2 ∂
2
B0
f,
−
+
=
4πρ0 ∂r 2 r ∂r r 2(11.104)(11.105)

where the θ derivatives give rise to a term that cancels the last term in equation 11.103
and thus vanish. The guide field magnitude B0 is pulled through the radial derivatives andis now on the left side of the brackets, which serves to modify some of the coefficients of
the radial derivatives. Since a standing wave solution is sought, f must be of the form
f = exp(iωt)u(r), where ω is the angular frequency of the wave. Therefore the previous
equation becomes


∂2
2 ∂
−
+
2
∂r
r ∂r



ω2
2
+
r 2 vA2

u = 0,(11.106)√
where vA = vA (r) = B0 / 4πρ0 is the radially dependent Alfvén speed. When the functional
form of vA is then inserted into this equation, this yields


∂2
2 ∂
−
+
2
∂r
r ∂r



2
+ a2 r 4
r2



u = 0,(11.107)

where a2 = 4πρ0 ω 2 /B 2 r24 . This equation is simply a modified Bessel equation and has thegeneral solution

393

h

i
3
3
u(x) = x c1 J− 1 x + c2 J 1 x ,3
2

6(11.108)

6

where x = 3−1/3 a1/3 r and where c1 and c2 are arbitrary constants. To make use of thiswithin CSS, the function u requires two boundary conditions: one on r1 and another at r2 .
For simplicity, let the solution have no-slip conditions on the radial boundaries. Therefore,
the full solution is
h

i


3
vφ′ (x, θ, t) = v0 csc θ cos (ωt) x 2 J 1 x31 J− 1 x3 − J− 1 x31 J 1 x3 ,
6

6

6

6

(11.109)

where v0 is a constant velocity that is chosen such that v0 ≪ vA and where x1 = 3−1/3 a1/3 r1 .
The choice to fix vφ′ = 0 on r1 provides one no-slip condition and gives rise to the above
equation. The no-slip condition on r2 requires that r2 be chosen so that it lies at a zero of
the function u(r). This also determines the dispersion relationship of the wave, which canbe seen if k is set to k = 2π/r2 thenr ω  ω iβω h
J 1 αωk 2 J− 1 β− J− 1 αωk 2 J 1 β= 0,(11.110)
6
6
6
6
k
k
k
√
where α = r13 /12π 2 vA0 , β = 2π/3vA0 , and vA0 = B/ 4πρ0 . To be implementable in CSS, avector potential is required that corresponds to both the guide field and fluctuating magnetic
field. For the guide field, the following is a viable choiceBr220
cot θ,Aφ = −
r(11.111)

The fluctuating vector potential can be found by appealing to the induction equation, with∂A′θ= vφ′ B0 .
∂t

(11.112)Integrating this equation in time givesA′θ (x, θ, t) = A0 +

h



i
3
B0 v0
csc θ sin (ωt) x 2 J 1 x31 J− 1 x3 − J− 1 x31 J 1 x3 ,
6
6
6
6
ω

(11.113)

394or A′θ = A0 + vφ′ B0 tan ωt/ω, where A0 is an integration constant. Finally, choose A0 = 0 sothat A′θ satisfies the normal perfect conductor boundary conditions shown above. Thus, whent = 0 the initial conditions are found for the standing Alfvén problem, which can provide astringent test of the numerics in the MHD version of CSS or other spherical segment MHD
codes.
11.7

Effects of Diffusion Schemes
It is currently infeasible to try to simulate the entire range of spatial and temporal scales

on the Sun. It is even fairly difficult to simultaneously model a few tens of supergranules
as well as capture the granulation, or capture supergranules and the deeper flows of thesolar interior. However, it may still be possible to characterize the influence of granulationand deep global flows on the near-surface shear layer (NSSL) by coupling CSS with global
convection models from below such as those from (Miesch et al., 2008) and with surface
convection simulations from above (e.g. Rempel et al., 2009; Nordlund et al., 2009; Rempel& Schlichenmaier, 2011). Such a coupling, whether it is statistical or direct, requires acareful treatment of diffusion and boundary conditions (see §11.2 or §11.8). For instance,
low diffusion is necessary to preserve the spatial structure and advective timescales of thedowndrafts flowing into the CSS domain from the surface convection above. Given that thegrid used here is five times coarser than that of typical surface convection simulations, thisis no easy task.
11.7.1

Formulating the Problem

Consider two CSS numerical simulations in a 20◦ square patch centered on the equatorthat encompass most of the NSSL and some of the deep interior, rotating at the solar
rate. Thus the spherical segment domains used in these simulations involve large portionsof the Sun’s inherent spherical geometry, which is necessary to properly capture the effectsof rotation on supergranular scales. To simulate the larger scales of motion that are likely

395

Figure 11.7: Radial velocities (ur ) at 0.99 R⊙ in (a) the slope-limited diffusion Case B, and (b)the Newtonian-eddy diffusion Case A, and at 0.95 R⊙ in (c) Case B and (d) Case A. Dark
tones denote downflows and light tones denote upflows. Scaling values are −1.34 km s−1 ,0.59 km s−1 for (a) and (b) and −0.73 km s−1 , 0.24 km s−1 for (c) and (d), where positive urpoints outward.

to occur in the solar convection zone, a large-eddy simulation (LES) model is employed. In
the Newtonian eddy diffusion scheme (Case A), the scales that are not explicitly computedin these simulations are parameterized and included in a sub-grid scale model of turbulent
transport and diffusion. In the slope-limited diffusion (Case B ), the primary goal is numericalstability of the code. Though, as was seen in §4.4, SLD can act as a filtered eddy diffusionthat removes diffusion above the filtered scale. This could be considered an eddy diffusion
model that limits the impact to only the largest wavenumbers. These simulations explore
the effects of Newtonian-eddy and slope-limited diffusion schemes with an open lower radialboundary and closed upper boundary, as such they are identical except for the diffusion
scheme.
In the Newtonian eddy diffusion (NED) scheme, the momentum and entropy diffusivities (µ and κS ) are calculated based upon the desired Rayleigh number at the upperboundary with the constraint that the Prandtl number and the kinematic viscosity be constant throughout the domain. There are no directly comparable diffusive parameters in the
SLD scheme as the eddy diffusion terms have been dropped. One may still control the levelof diffusion, however, with specific choices of which slope-limiter is employed and what char-

396

Figure 11.8: (a) PDFs of radial velocities at 0.99 R⊙ (solid line) and 0.95 R⊙ (dashed), forCase B (black) and Case A (red). Time-averaged radial energy flux balances for Case A (b)and Case B (c), with total flux in purple, radiative flux (blue), enthalpy flux (green), kinetic
energy flux (orange), and acoustic flux (red).

acteristic velocity is used at cell interfaces. Figure 11.7 depicts snapshots of typical flows at
two radii from the two diffusion schemes.
11.7.2Comparing Two Diffusion Schemes

The simulations encompassing a 20◦ × 20◦ domain are relatively low resolution with128 radial mesh points and 256 points in latitude and longitude. The MESA stellar evolution
code is used to establish a realistic initial stratification for the simulations. Since a perfectgas is assumed, the He and H ionization zones were not simulated. The impenetrable upperboundary is taken to be at 0.995 R⊙ in order to exclude most of the radial extent of thesezones. The permeable lower boundary is placed at 0.915 R⊙, yielding a density contrast acrossthe domain of 400. Each case has the lowest diffusivities that allow numerical stability in
the NED (Case A) and SLD (Case B ) regimes. Using a constant ν with depth in the NED
case is rather restrictive in that the diffusion is larger than necessary away from the upperboundary, but for this analysis it is sufficient to characterize the differences between the two
schemes.
Turning to the radial velocity patterns sampled in Figure 11.7, there is a striking shift397
toward smaller spatial scales in Case B with considerably less power at large scales. At
0.99 R⊙ the peak of the spatial power spectrum in Case A is around 40 Mm. In Case B,
however, there is a broad peak at scales around 20 Mm, more closely matching the supergranular scales of the Sun. As one descends deeper into the simulations (Figures 11.7c,d),the flows appear significantly more turbulent in Case B than in Case A.
Indeed, the downflowing structures of Case B become much narrower at depth withmany more isolated plumes. The greater turbulence also manifests itself in larger extrema ofeach of the physical variables. For example, the radial velocities at 0.95 R⊙ in Case B haveextrema that are twice that of Case A (Figure 11.8a). In Case A, a precipitous drop in the
upflowing radial velocity probability density distribution (PDF) outside of about 0.1 km s−1
is indicative of just how uniform the upflows are. In stark contrast, the radial velocity PDF
of Case B has twice the dynamic range, emphasizing that the upflows are indeed much
more structured than those of Case A. While the radial mass flux PDF has higher velocity
wings that contribute only one percent to the overall distribution, the bimodal nature of
the temperature perturbation PDF has the effect of significantly amplifying the importanceof these outlying values. Thus, the net effect of the larger wings in these PDFs is to shift
the mean of the enthalpy flux in Case B to a higher value. The additional velocity in boththe horizontal and radial flows greatly alter the kinetic energy distribution, as in Case B
where the mean kinetic energy has nearly doubled and correspondingly decreased the kinetic
energy flux (Figure 11.8c). A tentative next step is to have both diffusion schemes activeat the same time. This will allow a systematic study on the convergence of solutions as themagnitude of the NED coefficients is lowered and SLD takes over as the primary diffusion.
11.8

Entropy Rain
Precisely how small-scale granulation formed at the photosphere helps to drive super-

granulation and even larger scales of convection is still a bit of mystery. No single simulation
can yet follow the coalescing networks of downflows as the descend from the photosphere398
into the deep interior. One possibility is to model the action of the granulation as localized
cooling events in a large-scale simulation. The idea is to mimic the local cooling associatedwith mixing at the edge of the cool downflows associated with the hierarchy of downflows
descending from the surface. To this end, local sites with a decreased entropy sufficient to
extract a solar energy flux are introduced at the upper boundary of a 3-D simulation ofcompressible convection in a rotating spherical shell segment.
As before, in order to investigate the dynamics of the 100-1000s of supergranules necessary for adequate statistics, the small scales that are not captured must be modeled. In
this section, two methods of carrying the required solar energy flux through the top of thedomain are considered, one using a spherically symmetric (uniform) cooling function while
the other is a granular-scale stochastic cooling model. The dynamics and differential rotation profiles achieved in these simulations are subsequently assessed. It is found that witha sufficient number of stochastic cooling events compared to uniform cooling, supergranular
scales begin to be realized in addition to a differential rotation that becomes increasingly
like the NSSL.
11.8.1

Formulating the Problem

To begin to understand the dynamics of the near-surface shear layer, numerical simulations in spherical shell segments have been constructed that encompass most of the nearsurface shear layer and some of the deep interior. The impenetrable upper boundary (r2 )is taken to be 0.995 R⊙ in order to exclude these zones. The lower radial boundary (r1 ) is
at 0.915 R⊙, which corresponds to a density contrast (∆ρ = ρ1 /ρ2 = 400, where ρ1 = ρ(r1 )and ρ2 = ρ(r2 )). The momentum and entropy diffusivities (µ and κs ) are calculated based
upon the desired Rayleigh number (Ra) at r2 and so that the Prandtl number (P r) is unity
throughout the domain, and so these subgrid-scale diffusivities are functions of radius.

Table 11.1: Properties of the Computational Domain
Case 1Case 2
Case 3
Case 4
hN i
0
250
500
1000
6
6
6
hRai 5.98 × 10
7.63 × 10
7.90 × 10
7.94 × 106
hRei
133
152
151
150
hRoi
13.3
17.7
16.9
16.6
σ−
228
248
249
250
σ+
124
132
135
137

399

Table 11.1: For each case, the depth of the convection zone as modeled is 60 Mm, ∆ρ is 400,
the angular size is 20◦ × 20◦ , with a resolution of 128 × 256 × 256. hNi is the mean numberof cooling sites. All cases were run with the same diffusivities and initial background. hRai,
hRei, and hRoi, are the average Rayleigh, Reynolds, and Rossby numbers in the domain. σ+
and σ− are the standard deviations of the positive and negative radial velocity probability
distribution function in m s−1 , respectively.

11.9

Modeling Granular Convection
The strong downflows that occur at edges of a granule drag more dense, lower entropy

fluid into the interior. Hence, in this simple model it is assumed that the net mass flux iszero and the net entropy flux is negative when integrated over the volume of a granule. It isalso assumed that the collective action of all the granular-scale dynamics contained within
the simulated domain transports the required energy flux through the upper boundary.
Therefore, the granules are modeled as cooling events that are advected with the flow, last
for an average of 10 minutes, and have as small a spatial scale as the grid and diffusion willallow. The spatial structure of a cooling event is evident in Equation 11.114. The amplitudesof the cooling events are chosen at random from an exponential distribution with a mean
determined by Equation 11.115. The horizontal extent of the cooling events is normally
distributed with a mean angular size of 2∆φ (hσi) in θ and φ. The spread of the radial
exponential is fixed to σr = 2∆r to speed computation. The cooling function is thus


N (t)
2
2
2
1 −(r2 −r)/σr X
2(11.114)eQn (t)e−[(θ−θn (t)) +(φ−φn (t)) ]/2σn ,+ǫ = −∇ · (qR r̂) =r σrn=1
where Qn (t) is the amplitude of the cooling event at time t, and the cooling events are alsoθhorizontally advected with the flow: θn (t) = θn0 + ∆tu, and φn (t) = φ0n +r2∆tuφ. The average
r2

400

Figure 11.9: (a) A 20◦ × 20◦ snapshot of radial velocities (ur ) taken at 0.99 R⊙ in case 2:
downflows are black, upflows are white. (b) Horizontal scale ℓ in ur power spectra averaged
in time and latitude from cases 1 (solid), 2 (dotted), 3 (dashed), and 4 (dash-dotted). (c)Horizontally averaged rotation rate Ω(r), lines as above.

number of cooling events (hNi), with (hσi), determines the mean strength
(φ2 − φ1 ) (cos θ1 − cos θ2 )
L
∂S
∂T
hQi =
+ κS
+ κr
.
2πhNihσi2
4πr 2
∂r
∂r r2

(11.115)

Cooling events have a mean lifetime of 10 minutes, but can deviate significantly from thisas the lifetimes are normally distributed as well. When a cooling event has reached the endof its lifetime, it is deleted and at least one new cooling event is generated. An exponential
distribution is used to determine how many new events are generated, with the mean being
unity. Similarly, at each time step there is also a chance to add new events.
11.9.1Nature of the Convection

In a typical simulation, the convective cells are broad upflows bordered by strong,narrow downflows as seen in Figure 11.9a. The upflows are warm and the downflows arecool relative to the mean temperature. The typical radial velocities in Figure 11.9a are
between −1400 m s−1 for downflows and 400 m s−1 for upflows. Some basic parameters of theflows developed in these simulations can be seen in Table 11.1. The spatial scales present in
the four cases can be seen in power spectra (Figure 11.9b). The supergranule-like scales in401
Case 1 are about 40 Mm, just outside the accepted range for supergranules. A sphericallysymmetric cooling is used in Case 1 which leads to horizontal scales near the surface thatare primarily selected by the geometry of the domain. In cases with granular-scale cooling,
the size scales seen in the simulation decrease due to the granular-scale forcing as evidenced
by the leftward shift in the power of the peaks below 40 Mm in Figure 11.9b. In Cases 2-4,
supergranular scales become much more prevalent with power increasing by 50-100% in theband between 10 Mm and 35 Mm. In Case 2, a prominent peak occurs around 10 Mm dueto the relative paucity of cooling sites. Since there are only 250 cooling sites on average, eachsite necessarily induces a strong downflow, which ensures that the enthalpy flux is sufficient
to carry out the solar flux. Furthermore, there are typically 15 sites at a single longitude orlatitude so that the average distance between them is 10-15 Mm. Thus, since each coolingsite is likely to be a strong downflow, power is present on the 10 Mm scale. Such a peakdoes not occur to the same extent in Case 3 or Case 4 because the probability of inducing
a strong downflow at a cooling site decreases by factor between 2 to 4. However, a larger
hNi reduces the average distance between cooling sites and thus allows for more interactionof the flows generated at the granular scales due to the advection and coalescence of thecooling sites.
As hNi is increased, the granular scale cooling disrupts the mean horizontal flows in
the upper 20% of the domain which reduces the mean rotation rate as seen in Figure 11.9c.
The reason the mean flows are affected is that when a cooling site forms in an upflow, it
begins cooling nearby gas causing contracting horizontal flows and a downflow. So, thenet effect of this cooling site is to reduce the horizontal flows within the upflow as well as
slow the upflow’s ascent as it has been cooled. Hence, as hNi increases the average numberof cooling sites per upflow increases thus decreasing the mean horizontal flows as in Figure
11.9c. The radial boundaries used for these studies were impenetrable and stress-free leading
to overturning of upflows at the top and downflows at the bottom of the domain. This is
the primary reason the rotation rate at the bottom does not flatten out and why there is an402

Figure 11.10: Evidence of merging plumes arising in a CSS simulation carried out with ‘entropy rain’ and open boundaries. The base simulation (20◦ in latitude and 10◦ in longitude)
has been replicated once in longitude, and reflected across the equator to emphasize thelargest scales. Horizontal planes of radial velocities (ur ) cut through the simulation domainat (a) 0.985 R⊙ just below the open upper boundary, (b) 0.950 R⊙ near the middle of the domain, and (c) 0.895 R⊙ just above the open lower boundary. Downflows are dark tones. (d)Horizontally-averaged rotation rate Ω(r) with radius shown for this simulation (solid black)with the solid blue curve corresponding to the conservation of angular momentum per unitmass. Helioseismically determined Ω at the equator (dashed curve), while the frame rate is
the horizontal solid line.

unrealistic shear layer near the upper boundary.
Building upon this early work, more recent simulations have utilized open radial boundaries that have somewhat eliminated the ill-effects of impenetrable boundaries. Such simulations also incorporate a more advanced plume model that employs flow statistics from a
photospheric simulation, which will be described in greater detail in the next section. These
plumes have both an entropy deficit and a momentum associated with them as they enter
the simulation at the upper boundary. Figures 11.10(a-c) show that flows with large-scale
structures can form even in the presence of open boundaries and small-scale plume driving.
The large-scale flows deeper in the domain have some similarity to the banana cells, which
are the most unstable modes of global-scale rotating convection and often seen in the global-

403
scale ASH simulations. However, despite admitting more physical boundary treatments, thisapproach does not yet produce a reasonable approximation to the NSSL. Instead, a new issue arises where too much momentum is carried deep into the domain as is visible in Figure11.10d yielding a radial profile of differential rotation that is consistent with angular momentum conservation per unit mass. The method that currently seems to have the greater
potential is having an open upper boundary and an informed lower boundary, where flows
from deep within an ASH simulation are imposed.
11.10

Incorporating External Flows into CSS

Simulations (De Rosa et al., 2002) with ASH of supergranular and larger scales of
convection in thin shells (extending from 0.90 to 0.98 R⊙) revealed that a rotational shear
layer can be achieved through partial conservation of angular momentum (cf. Gilman &Foukal, 1979) by scales of convection comparable to supergranulation. These sense the
Coriolis forces only weakly, thereby yielding spin down as fluid is carried radially outward
and away from the rotation axis at low latitudes. Although encouraging, such modeling was
not capable of either capturing the giant cells very well, which requires much deeper shells,
nor in describing the near-surface physics of supergranulation adequately. The outstanding
issues concern how is the NSSL formed and maintained, and what determines its thickness,
which may vary with latitude.
Giant cells of convection seen in ASH simulations (see Chapters 3, 5–9) have horizontal
scales roughly comparable to the depth of the solar convection zone (200 Mm) (Miesch et al.,2008). However, giant cells have been observationally elusive, despite many to directly
identify their signatures at the solar surface (Beck et al., 1998; Lisle et al., 2004), thoughperhaps recently they have been detected (Hathaway et al., 2013). To begin to resolve suchfundamental issues, a few preliminary CSS simulations have sought to clarify how the giant
cells of convection, so evident in global convection models, might be disrupted as thosescales interact with the smaller scales dominant near the solar surface. These simulations

+60°

(a)

(b)

(c)

-60°

Figure 11.11: Near-surface convection and colliding flows studied in a CSS simulation, with large-scale ASH flows imposed frombelow and statistically defined plumes from above. Shown are orthographic projections at (a) 0.985 R⊙ below the open upperboundary, (b) 0.950 R⊙ near the middle of the domain, and at (c) 0.900 R⊙ just above the imposed flows at the lower boundary.
Downflows are dark lanes in (a), which become shear unstable and merge into larger structures as in (b). The interaction between
merging downflows structures from above and imposed large-scale flows from an ASH simulation becomes visible in (c). The fulldomains are fairly large, spanning 120◦ in latitude and 90◦ in longitude and 2240 × 1680 × 256 in resolution.
404

405
were informed at their upper boundaries with a statistically determined flow and informedat their lower boundaries by an ASH simulation. The ℓmax = 170 ASH simulation used at
the lower boundary of CSS was four-fold periodic to match the longitudinal size of the CSS
domain and extended from the base of the convection zone to 0.96 R⊙. The full domains for
the CSS simulations were fairly large, spanning 120◦ in latitude and 90◦ in longitude andwere between nθ × nφ × nr = 1024 × 768 × 256 and 2240 × 1680 × 256 in resolution. They also
encompassed the upper 80 Mm of the convection zone. The boundary conditions defined in§11.2 are only slight modified to incorporate the entropy and the flows from ASH. The flows
from ASH are first interpolated in time and space to match the grid size, the meridional flows
are truncated at the latitudinal boundaries over a thin layer. Instead of pulling the values ofthe boundary plane to those of an interior plane of the simulation, the time evolving values
from the prior ASH simulation are substituted for the interior values as1∂u=u|ASH − u|r1 ,
∂t r1
τr1
1
∂s
=
(s|ASH − s|r1 − hsr1 i + sr1 ) ,
∂t r1
τr1

(11.116)(11.117)

where the averaged terms are defined as in §11.2 and the pressure gradient is computed in
the same fashion.
The CSS simulations were informed by evolving downflow structures at the upperboundary whose statistics are consistent with those realized in the surface convection simulations with MSC, as described in Trampedach (2010). Unlike the cooling-site entropy rainmodel used in the simulations of the previous section, these flows consisted of fast, entropy
deficient downflow plumes that enter the CSS domain through its upper radial boundary.
The plumes were randomly placed on the upper boundary and were assumed to have a
Gaussian spatial structure. The widths of these Gaussian plumes are determined by the statistical distribution of the widths of downflows in the MSC surface convection simulations.
Similarly, their duration, entropy deficiency, and radial velocity are all chosen from statistical distributions and correlations of downflows derived from the MSC simulations at the

406
480

Ω (nHz)

460

440

420

Sun
Coll.
No Coll.
C. Ang.

400
0.90

0.92

0.94
r/R

0.96

0.98

Figure 11.12: Horizontally-averaged rotation rate Ω(r) attained in two CSS simulations
involving open upper boundary conditions and “low entropy rain” statistics informed by
MSC modeling (red), and alternatively with those descending flows colliding with large-scaleconvective flows entering from the lower boundary (blue). Simple conservation of angularmomentum is shown (dashed red), along with helioseismic profile (black).

depths corresponding to the CSS upper boundary. The correlations between these quantities
have important implications for the dynamics of the shear layer, and thus cannot be chosen
independently nor randomly.
The angular momentum transport across the upper and lower radial boundaries of
CSS has a direct bearing on the maintenance of the NSSL. The upper boundary here was
required to have zero net angular momentum flux, but the transport at the lower boundary
was specified by the flows from ASH. The upper boundary conditions for the entropy rain arevery similar to the ASH inflows. The entropy rain is simply treated as an exterior simulationthat evolves on the boundary, so407

Figure 11.13: Rotationally constrained convection in CSS. (a-c) A Sun-like star rotating at
10 Ω⊙ at three depths, as labeled above each panel. (e-f) A Sun-like star rotating at 3 Ω⊙ atthree depths, showing less rotational influence.

1
hpr2 i∂u=u|rain − u|r2 −
,
∂t r2
τr2
hmr2 i∆t∂s1=(s|rain − s|r2 − hsr2 i + sr2 ) ,
∂t r2
τr2

(11.118)(11.119)

These simulations have explicitly specified the evolving flows at the lower boundaryof CSS with velocities and thermodynamic perturbations from a fairly low resolution ASH
simulation. Indeed, they have begun to enable the study of the challenging near-surface
region in some detail where the intense downdrafts of the surface convection interact andmerge with the large-scale upwelling and horizontally-sweeping flows of giant cells. The

(a)

(b)

408

Figure 11.14: Shell-averaged angular velocity profiles with shear layers realized in CSS simulations. (a) A Sun-like star rotating at 10 Ω⊙, the color denotes the time evolution of theangular velocity with several hundred days represented, showing the presence of a near surface shear layer. (b) A Sun-like star rotating at 3 Ω⊙, lacking a shear layer due to the larger
Rossby number throughout the domain.

simulation whose radial velocities at several depths are shown in Figure 11.11 has yielded aradial differential rotation that possesses a region of near-surface shear that becomes more
uniform with greater depth as depicted in Figure 11.12. The transition to uniformity partly
arises through the angular momentum flux into the domain from the ASH simulation andalso through the transition to more rotationally constrained flows with the Rossby number
falling below unity. The near-surface shear appears to be a consequence of the fast, angularmomentum conserving flows falling from the upper boundary and continuously mixing withthe broad upflows as the descend through the domain. This mixes flows with locally low
Rossby number and those with locally larger Rossby numbers leading to the gradual lossof the angular momentum conserving properties of the downflows and to the change in the
profile of the radial differential rotation.
Other recent CSS simulations have shed some light on this as well. These closed

409
domain simulations mimic the solar interior but are rotated more rapidly to attain a lower
Rossby number in the domain. With a rapid enough rotation, a clear transition between
surface convection cells and deeper rotationally constrained flows becomes evident as shownin Figures 11.13 and 11.14. Thus, the suspicion that the base of the NSSL reflects a shift
in the nature of the meridional force balance, from baroclinic forcing in the deep convection
zone to Reynolds and Maxwell stresses in the NSSL, may indeed occur and be due to theeffective Rossby number of the local flows. Hence, future CSS modeling may focus on thecharacter of the resulting meridional circulation which responds to zonal stresses in both theupper and lower convection zone through the mechanism of gyroscopic pumping (Miesch &Hindman, 2011).
11.11

Conclusions

Many aspects of the code need to be finalized and its capabilities truly put to the test.
A thorough testing and understanding of all the options within the code, particularly thoserelated to coupling codes together, needs to be undertaken. However, there is also muchto be exploited in its already existing form. Indeed, the preliminary results using CSS arevery encouraging, with properties of the solar near-surface shear layer beginning to emerge
through two separate lines of investigation. One method presented here involves coupling
the dynamics of the deep interior obtained in ASH simulations and a statistical descriptionof the unresolved flows arising near the photosphere. The flows established in the firstsimulations utilizing these informed boundaries have the properties necessary to produce a
region of near-surface shear and to transition to the more uniform rotation of the deep solar
interior. Similarly, a second approach that decreases the bulk Rossby number of the flows
in simulations of the full convection zone of F-type stars also yields a near-surface shear
layer. The latter simulations are fully self-contained and self-consistent, suggesting that themethod to self-consistently achieve a region of near-surface shear is to decrease the Rossby
number of the flows. The only viable way to do this in a solar simulation is to decrease the410convective driving on the large scales, effectively controlling the resolved heat flux that is
carried through the domain.

Chapter 12
Perspectives and Ponderings

Humanity’s long road to understanding the nature of the Sun and the curiosity ofthe stars continues to be traveled. This thesis has only brushed the surface of the many
intertwined phenomena at work within stars, beginning with their observed characteristics
and then delving into the theories that are currently thought to describe their inner workings.
With some of the latest computational tools, aspects of stellar magnetism have been exploredthat may be making significant contact with the observed character of solar and stellar
magnetic activity. The hydrodynamics of stars slightly more massive than the Sun, the Ftype stars, has been examined with an eye on how properties intrinsic to stellar dynamos
scale with mass and rotation rate and how those properties change in the presence of a stableregion below their convective regions. Subsequently, how dynamos may operate within the
F-type stars has been explored, with the stable region playing a crucial role in the dynamo.
The convective and magnetic properties of the cores of very massive stars has begun to
be probed, with MG fields forming within the core. This bodes well for building a better
understanding of the properties of highly magnetized stellar remnants. Finally, a new MHD
code has been developed that with the results in hand holds good promise for untangling someof the dynamics of the solar near-surface shear layer as well as the properties of convection
zones for which compressibility may play a role.
With those paths pursued, where do they now fork and diverge? There are many
threads that have been sprinkled throughout this thesis that may provide intriguing routes

412
to understanding the deeper mathematical and physical properties of turbulence in general,
which of course also as applicable to stellar dynamo theories. For the primary work presented
here, the most immediately promising is the application of mixed subgrid-scale methods forbuilding stellar dynamos. Indeed, the simulation presented in Chapter 5 may provide thelaunching point for understanding some of the observed global-scale properties of the solar
dynamo, such as what might be setting the cycle period and what processes lead to grandminima. Subsequent analysis of simulations utilizing some of these techniques may help topin down those processes leading to the global-scale magnetic field reversals. This path is one
that is straight and narrow, with well defined parameters that can be varied. Assessing theimpacts of the slope-limited diffusion operator is paramount to ensure that it is not impacting
the solution in adverse ways. If it is not, one may embark on a parameter study of rotation,
convective driving, magnetic Reynolds number, as well as effective thermal and magnetic
Prandtl numbers through the variation of the resolution of the simulations. This still skirts
the issue of what happens as a larger number of density scale heights are admitted in thesystem. Fortuitously, slope-limited diffusion permits simulations with it to admit larger
density stratifications and yet remain numerically stable.
Exploring dynamo action in other low mass stars like the Sun may too yield broader
insight into the fundamental mechanisms leading to stellar magnetic activity. For instancethe role of a tachocline may be particularly evident for low mass stars with some possessing
one and others slightly less massive that are fully convective. It may also be intriguing tolook at the higher mass F-type stars with CSS that may also harbor a tachocline below
their thin convection zones, which have fairly high Mach number flows throughout them andfast thermal relaxation times. Additionally, such stars rotate more rapidly than the Sun on
average, and those that have observed activity cycles tend to have fairly short cycle periods.
This permits greater contact and potential for refinement between observation and theory.
The short-lived massive stars also present intriguing prospects for developing a better
understanding of how magnetism, rotation, and convection might impact their evolution and413
their remnants. Several opportunities seem to arise with the preliminary results presented in
Chapter 10. The most obvious is to include primordial fields to assess how superequipartition
the magnetic field configuration might become, given that it is already four times greaterthan the kinetic energy. Utilizing more accurate subgrid-scale models in the radiative exterior
may permit the entire gravity wave cavity to be simulated within ASH. Previous attempts at
this had been made, but failed due to large mode amplitudes near the upper boundary. The
amplification of the waves is in part due to their propagation down the density gradient,which naturally leads to amplification. Once the full-sphere implementation in ASH has
been thoroughly vetted, it would be quite useful to retain the center-most portion of thecore, rather than omitting it as has been done here. Then one could assess what impacts if
any the core cutout has on the connectivity of the flow and magnetic structures. It would
also be useful to implement the generalized pseudo-incompressible equations into the ASH
framework in order to simulate the more extreme environments of both evolved massive stars
and of the more massive main-sequence stars for whom radiation pressure plays an important
if not dominant role.
Finally, there are many prospective directions to take with CSS. The current lines
of investigation can be extended with recent modifications to the code that increase its
speed by a factor of two more than what was seen in the strong-scaling plots of Chapter
11. Furthermore, extending beyond the solar realm to regimes that are similar but lessrestrictive may provide useful insight into processes such as magnetic buoyancy and theinfluence of a tachocline. In the longer term, it will be possible to build a generalized
pseudo-incompressible framework that admits higher-order terms in the kinematic pressure
allowing for sound waves. This will allow a mathematically-sound scale separation problem
to be solved that retains acoustic modes where necessary and omits them where they are not.
Such a code would be the next generation pseudo-spectral stellar convection code, being able
to accurately and self-consistently evolve scales from the photosphere to the deep interior.

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Appendix A
Derivation of the Multi-Fluid Electrodynamically-Coupled Field Equations

A set of equations is needed from which a Lagrangian can then be constructed. There
are many places one can start from, for instance a fairly general starting point could be
P 2
PN
the classical mechanics Hamiltonian of the universe HU =n=1 Hn =n pn /2mn +Φn (t, q1 , · · · , qn , · · · qN ), where Φn embodies the interaction of particle n with all otherparticles in the universe through gravity and electromagnetic forces. This, while somewhat
flippant, underscores one of the fundamental problems in dealing with systems that involve
more than a single particle: the coupling between greater numbers of particles quickly leads
to an unsolvable system. One can instead parameterize the problem, utilizing statisticalensembles of large numbers of such particles. Under this assumption the 6-Np dimensional
phase space tracks of all the individual particles become indistinguishable, forming a 6-S
dimensional Riemannian manifold, where S is the total number of particle species in theplasma. The evolution of an infinitessimal volume of this manifold is governed by Liouville’s
theorem which states that, given the phase space density fs (x, v, t) of species s, this density is constant along any trajectory in phase space. Since the manifold is Riemannian thisevolution can be written as
Fs
∂fs
∂fs
,
+ v · ∇x fs +· ∇v fs =∂t
ms∂t col(A.1)

where F is the total force on species s, the collisional forces are included in the term on theright hand side, relativity has been neglected, and by definition the number density of species

449
s is ns =R

dvfs . Further, we have assumed that the radiation field acts instantaneously

to equilibrate a given species and have not accounted for its interaction with the plasma.
The goal is to integrate out the velocity dependence by integrating over moments in velocity
space and supplementing the resulting equations with a closure. Let χs (v) be the generatorof the velocity moments. After integrating by parts and using the fact that χs is only afunction of the velocity, one can find the generalized transport equation for the ensembleRaveraged value of χ = 1/ns dvχfs
∂ns χ
∂ns χ
Fs
,
+ ∇ · ns χv − ns
· ∇v χ =∂t
ms∂t col(A.2)

The zeroth moment is that of the mass density with χ = ms and species mass ms , whichyields∂ms ns+ ∇ · ms ns∂tZ

dvfs v = ms

Z

dv

∂fs
= Ṁs .
∂t col(A.3)

The collisional term subsumes all interactions that lead to the change of one species intoanother, such as chemical reactions, nuclear fusion or fission, or ionization for example, andas such acts as a source or sink of mass (Ṁs ). Let the total velocity be v = us + v′ , withR
us the bulk velocity and v′ fluctuations such that 1/ns dvfs v = us (so v′ = 0) and letρs = ms ns be the mass density. Then the equation for the conservation of mass is∂ρs+ ∇ · ρs us = Ṁs .
∂t

(A.4)

The first moment gives rise to the momentum equations, wherein χ = ms v. Thus
Equation A.2 becomes ZZ∂ρs us∂ρs fs v+ ∇ · ρs dvfs v ⊗ v − ns Fs == Q̇s ,
dv∂t
∂t col(A.5)

The dyadic form ρs v⊗v produces the bulk velocity dyadic but also the correlations of fluctuRating velocities as ρs v ⊗ v = ms dvfs us ⊗ us + 2us ⊗ vs′ + vs′ ⊗ vs′ = ρs us ⊗ us + ρs vs′ ⊗ vs′ .

The fluctuating velocity dyadic ρs vs′ ⊗ vs′ may be interpreted as the partial pressure ten-sor Ps . The collisional term Q̇s describes the sources and sinks of momentum. Thus themomentum conservation equation is450∂ρs us+ ∇ · [ρs us ⊗ us + Ps ] = Q̇s + ns Fs .
∂t

(A.6)

The final equation, before introducing a closure, is the conservation of energy whereχ = 1/2ms v2 . Again appealing to Equation A.2, we have1 ∂ρs v2 11
+ ∇ · ρs v3 − ns Fs · v =
2 ∂t
2
2

Z

dv

∂ρs fs v2= K̇s .
∂t
col

(A.7)

The first term on the left hand side is
Z


 ∂E
1 
∂ 1  2s
2
dv ρs u2s + 2us · vs′ + v′ s =ρs us + v′ 2s =
,
(A.8)
2
∂t 2
∂t


where the energy density is Es = 1/2ρs u2s + v′ 2s . The energy flux divergence in Equation∂
1 ∂ρs v2=2 ∂t∂t

A.7 isZ1
dv ρs v3
2
Z

1 
2
= ∇ · dv ρs u2s + 2us · vs′ + v′ s (us + vs′ )
2
Z

1 
2
3
= ∇ · dv ρs u3s + u2s vs′ + 2vs′ · us ⊗ us + 2us · vs′ ⊗ vs′ + us v′ s + v′ s (A.9)
2



1
3
2
3
′
′
′
′
= ∇ · ρs us + 2us · vs ⊗ vs + us v s + v s
2


1
3
′
= ∇ · Es us + us · Ps + ρs v s2

1∇ · ρs v3 = ∇ ·2The last term in the energy flux divergence is an internal heat flux (qs = 1/2ρs v′ 3s ), arisingfrom the triple product correlation of the fluctuating microscopic velocity field that describes
the internal redistribution of kinetic energy. The work done by forces acting on the species
is Fs · v, which reduces to Fs · us + Fs · vs′ . The fluctuating portion of the work will be
zero if there are no velocity dependent forces. Hence the species specific energy conservation
equation is∂Es+ ∇ · [Es us + us · Ps + qs ] = K̇s + ns Fs · v.
∂t

(A.10)

451
At this point, since the goal is to derive the single fluid MHD equations relevant tothis thesis, the force can be directly expressed as Fs = qs (E + v/c × B) + ms∇Φ, with qs the
charge of species s, the electric field E, magnetic field B, and Φ the external gravitational
field under the assumption of minimal self-gravity. Thus the force term in the momentum
equation is ns Fs = ̺s E + ̺s us /c × B + ρs ∇Φ and the work that these forces do isns Fs · v =Z

dvqs fs (E +v
× B) · v + fs ms v · ∇Φc= ̺s us · E + ρs us · ∇Φ

(A.11)

where the charge density is ̺s = qs ns . Therefore the full system of equations for each speciesis∂ρs+ ∇ · ρs us = Ṁs ,∂t∂ρs usus+ ∇ · [ρs us ⊗ us + Ps ] = Q̇s + ̺s E + ̺s × B + ρs ∇Φ,∂tc∂Es+ ∇ · [Es us + us · Ps + qs ] = K̇s + ̺s us · E + ρs us · ∇Φ.
∂t(A.12)(A.13)(A.14)PPP
The next step is to sum over all species, where ρ = s ρs , ρu = s ρs us , ̺ = s ̺s ,P
and J = s ̺s us /c. If there is no net mass, momentum, or energy loss or gain in the system
PPP
arising from collisions, so s Ṁs = s Q̇s = s K̇s = 0. Thus the continuity equation is
X ∂ρs
s

∂t

+ ∇ · ρs us =∂ρ+ ∇ · ρu = 0.
∂t

(A.15)

It is also useful to transform the individual species bulk velocities to the frame of thebulk motion. The differences in the bulk motion of each species provides an additional,
but constant, internal energy and partial pressure. The velocity difference ws = us − u
is a constant for each species and so has no correlation with the fluctuating motions, withPPPs Ps +ρs ws ⊗ws ,s ws = 0. Thus the pressure becomes P =s ρs (vs − u) ⊗ (vs − u) =PPsimilarly the momentum dyadic is s ρs us ⊗ us = ρu ⊗ u + s ρs ws ⊗ ws . Hence themomentum equation is452

X ∂ρs us∂ts

+ ∇ · [ρs us ⊗ us + Ps ] − Q̇s − ̺s E − ̺sus
× B − ρs ∇Φ = 0,c∂ρu+ ∇ · [ρu ⊗ u + P] − ̺E − J × B − ρ∇Φ = 0.
∂t
The total energy density is E =

P

2
2
′2
s 1/2ρs (u +ws +v s ) =where ǫ is the internal energy. The heat flux becomes
q=

X1
s

=

X
s

2

(A.16)

P

s Es +

P22
s 1/2ρs ws = 1/2ρu +ǫ,ρs (vs − u)3 ,1
1
qs + ρs v′ 2s ws + ws · Ps + ρs ws3 .
2
2(A.17)The energy fluxes within the divergence operator in the energy conservation equation are
Flux =Xs=u

Es (u + ws ) + (u + ws ) · Ps + qs

X

Es +

s

= uE +

s

X1

2

X
s

+q−

X

s

ws Es + u ·

X

Ps +

s

X
s

ws · P s + q −ρs ws u2 + 2u · ws + ws2 + v′ 2s + u ·11
ρs v′ 2s ws + ρs ws3
2
2



X 1
2

s

P−

X
s

ρs v′ 2s ws + ws · Ps +ρs ws ⊗ ws

= uE + u · P + q!

1
ρs ws3
2(A.18)

Thus the full set of fluid equations for the plasma are∂ρ+ ∇ · ρu = 0,∂t∂ρu
+ ∇ · [ρu ⊗ u + P] = ̺E + J × B + ρ∇Φ,∂t∂E
+ ∇ · [uE + u · P + q] = J · E + ρu · ∇Φ,∂t1
E = ρu2 + ǫ.
2

(A.19)(A.20)(A.21)(A.22)



453
It is worth noting that the pressure tensor, as it currently stands, has two primary components the standard isotropic pressure P = T r(P) and an anistropic part that is related toviscous diffusion tensor Dij , so that Pij = Pδij + Dij where δij is the identity tensor. So
far, the evolution of the plasma has been described without the coevolution of the electromagetic field, though without a direct coupling to the radiation field. To this end, we willneed Maxwell’s equations.
∇ · E = 4π̺

(A.23)∇·B =0(A.24)

∇×E=−

1 ∂B
c ∂t

∇ × B = 4πJ +(A.25)
1 ∂E
c ∂t

If Equation A.13 is multiplied by qs /cms , the resulting equation for J is
#
"

X F
X us
qs
qs
∂ X us̺s + ∇ ·
Ps =
̺s
+
Q̇s
̺s ⊗ us +
∂t s
c
c
cms
cms cmsss(A.26)(A.27)

Applying the conventions and definitions from above this becomes ̺X  ̺s qs 
us
qs
∂J
s
E+
Q̇s ,
+ ∇ · [J ⊗ u + u ⊗ (J − ̺u) + Π] =× B + ∇Φ +
∂t
cm
c
c
cm
s
s
s
(A.28)
where we have used the relationshipXsX  wsu
1 ′us̺s⊗ ws + vs ⊗ vs′ ,̺s ⊗ us = ̺ ⊗ u + u ⊗ Jw + Jw ⊗ u +
c
c
c
c
s


u
+ Π,
=J⊗u+u⊗ J−̺c
X  ws1 ′′Π=̺s⊗ ws + vs ⊗ vs ,
c
c
s
X ws
Jw =
̺s ,
(A.29)cs

454
and where Jw is the current arising from differences between the bulk velocity and the mean
velocity of a given species. This is a fairly general Ohm’s law. Thus the evolution of theelectromagnetic field and charged currents remain tightly coupled to the properties of theconstituents of the plasma, whereas the fluid equations are greatly simplified for the plasma
as a whole. In summary, the electrohydrodynamic equations for a single species are
Fluid Equations∂ρs+ ∇ · ρs us = Ṁs ,∂t∂ρs us+ ∇ · [ρs us ⊗ us + Ps ] = Q̇s + ̺s E + Js × B + ρs ∇Φ,∂t∂Es+ ∇ · [Es us + us · Ps + qs ] = K̇s + Js · E + ρs us · ∇Φ,∂t
1 
Es = ρs u2s + v′ 2s ,2(A.30)(A.31)(A.32)(A.33)

Maxwell′ s Equations∇ · E = 4π̺,
∇×E=−

1 ∂B
,
c ∂t

∇ · B = 0,
∇ × B = 4πJ +(A.34)
1 ∂E,
c ∂t(A.35)

Ohm′ s Law
 ̺
∂Js
us
̺s qs 
qs
s
E+
Q̇s .
+ ∇ · [Js ⊗ us ] =× B + ∇Φ +
∂t
cms
c
c
cms

(A.36)

Similarly, for the entire fluid and summed over species and noting Maxwell’s equations above,
the plasma electrohydrodynamic equations are455

Fluid Equations∂ρ+ ∇ · ρu = 0,∂t∂ρu
+ ∇ · [ρu ⊗ u + P] = ̺E + J × B + ρ∇Φ,∂t∂E
+ ∇ · [uE + u · P + q] = J · E + ρu · ∇Φ,∂t1
E = ρu2 + ǫ,
2

(A.37)(A.38)(A.39)(A.40)

Ohm′ s Lawi X ̺ q 
h

 ̺
us
u
qs
∂J
s s
s
+Π =
E+
Q̇s .
+∇· J⊗u+u⊗ J−̺
× B + ∇Φ +
∂t
c
cm
c
c
cm
s
s
s

(A.41)

Appendix B
An Exposition on Meridional Flows and Gyroscopic Pumping

B.1

Navier-Stokes Momentum Equation
We begin with the equation that expresses the conservation of momentum in a mag-netized, compressible fluid within an inertial frame.
ρ∂v
1+ ρv · ∇v = −∇P + ρg + ∇ · D +(∇ × B) × B
∂t
4π

(B.1)

A standard notation is used: v is the bulk velocity of the fluid, B the magnetic field, ρ is thedensity, P is the pressure, g is the acceleration due to gravity, and D is the viscous stresstensor, with elements1
Dij = 2ρν eij − (∇ · v) δij
3



(B.2)

where ν is the kinematic viscosity. We use spherical polar coordinates (r, θ, φ) in this derivation. In a rotating coordinate system, equation B.1 becomesρ∂u
1+ ρu · ∇u = −∇P + ρgef f − 2ρΩ0 × u − ∇ · D +(∇ × B) × B
∂t
4π

(B.3)



where Ω0 = Ω0 ẑ, gef f = ∇ Ψg + 21 λ2 Ω20 , Ψg is the gravitational potential. Multiplying thezonal component of equation B.3 by λ = r sin θ yields
ρ

∂λuφ
∂P
∂Ψg
λ
= −ρλu · ∇uφ − (2Ω0 λ + uφ ) ρuλ −+ρ
− λφ̂ · ∇ · D +(∇ × B) × B (B.4)
∂t
∂φ
∂φ
4π

The first two terms on the right hand side can be combined and the left hand side reduced= −∇ · ρu) to giveusing the continuity equation ( ∂ρ
∂t

457



λ
∂Ptot
∂Ψg
∂ρL
= −∇ · ρuL −Mφ̂ −+ρ− λφ̂ · ∇ · D
∂t
4π
∂φ
∂φ

(B.5)

2

where Ptot = Pgas + B8π is the total pressure and M is the Maxwell stress tensor. The lastterm can be written as the divergence of a diffusive flux, if we note that∇ · λD = φ̂ · [∇ · λD]


1
∂Dφφ
∂
1 ∂ 2
+ 2 sin θDrφ + 2 cos θD(B.6)
r [λ] +sin θ [λ] +=θφDrφ r 2 ∂r
Dθφ r sin θ ∂θ
∂φ
h

i 

= λ ∇ · D φ̂ + D φ̂ · λ̂
where D = D φ̂ = Drφ r̂ + Dθφ θ̂ + Dφφ φ̂ and M = Mφ̂ = Mrφ r̂ + Mθφθ̂ + Mφφφ̂. Thus,
equation B.5 becomes


∂ρL
λ
∂Ptot
∂Ψg
= −∇ · ρuL + λD −
M −
+ρ
∂t
4π
∂φ
∂φ
If we now take a temporal and azimuthal average of this equation we arrive at


λ
∂Ptot
∂Ψg
∂ρL
it,φ = −h∇ · ρuL + λD −M it,φ − h
+ρ
it,φ
h
∂t
4π
∂φ
∂φ

(B.7)

(B.8)

Now introduce mean fields that depend only on r and θ and time dependent fluctuating fields
with zero mean such thatP = P + P′
ρ = ρ + ρ′
L = L + L′ = λ2 Ω0 + λuφ + λu′φ

(B.9)

u = u + u′

B = B + B′
With such a substitution, the φ derivatives here drop out leaving


λ
∂ρL
i = −h∇ · ρuL + λD −
M i
h
∂t
4π

(B.10)

458
Since D is a linear operator on v, and after applying the average to the expansion of thesefields, we find



λ
′
′
′
′
′
′
′
′
′
′
′
′
∇ · ρuL + ρ u L + uρ L + ρu L + ρ u L + λD + λD −BP Bφ + BP Bφ ≈ 0
4π(B.11)
where the mean diffusive flux is D = 2ρνD [u] φ̂, the fluctuating mean is D′ = 2νρ′ D [u′ ]φ̂,and BP is the poloidal field. In the anelastic approximation ∇ · ρu = 0, which implies thatρ′ = 0. Therefore, the averaged compressible equation above is reduced toλ∇ · ρuL + ρu′ L′ + λD −BP Bφ + B′P Bφ′ ≈ 0
4π
B.2(B.12)

Meridional Flows and Zonal Torques
The anelastic averaged zonal momentum equation B.12, can then be considered tohave two parts where the divergence of the mean meridional mass flux falls out because it isdefined to be zero. Hence,






λ
λ
ρu·∇L ≈ −∇· ρu′ L′ + λD −BP Bφ + B′P Bφ′ = −∇· λρu′ u′φ + λD −BP Bφ + B′P Bφ′4π
4π
Further, the mean diffusive flux can be simplified to λD = ρλν∇Ω, where Ω = Ω0 + uφ /λ.

So, equation B.12 becomesρu · ∇L ≈ −∇ ·λρu′ u′φ + ρλ2 ν∇Ω −
λBP Bφ + B′P Bφ′4π

(B.13)The right hand side is defined to be the zonal torque, so that ρu · ∇L = F . Since the massflux is divergence-less, it can be represented as a stream function ψ such that ρu = ∇ × ψ φ̂.
This can be done because the φ components drop out by definition. Define a unit vectorgθgrθ̂ wherer̂ + |g|normal to contours of angular momentum per unit mass (L), which is n̂ = |g|g = ∇L. Then equation B.13 can be simplified so thath i
∇ × ψ φ̂ · ∇L = F
h i
F∇ × ψ φ̂ · n̂ =
∇L
1 ∂
F
1 ∂
(sin θψ) nr −(rψ) nθ =r sin θ ∂θ
r ∂r
|∇L|
If this last equation is multiplied by λ, then

459

∂λψ
λF
1 ∂λψ
nr −
nθ =
r ∂θ
∂r
|∇L|

(B.14)

We now define a vector t̂ that is tangent to contours of L. This requires n̂ · t̂ = 0 and a choicein its orientation. Here we choose its orientation to be clockwise such that t̂ = −nθ r̂ + nr θ̂,
where nr and nθ are the components of n̂. Then, equation B.14 can be reformed so that∇ (λψ) · t̂ =

λF
|∇L|

(B.15)

Integrate equation B.15 along portions of the contours of L that are defined by a param
eterized curve r = r(s)r̂ + θ(s)θ̂ such that L(r) = L0 where L0 ∈ min L, max L . The

endpoints of this curve are a at the lower boundary and b at the upper end point. Let s be
the arclength along this contour, then the integral isZC∇ (λψ) · t̂ds =

Z

λF
ds
C |∇L|(B.16)

However, ∇ (λψ) · t̂ is simply the derivative of λψ along the contours of L. Therefore, the left
R
hand side of equation B.16 is simply C ∂λψ
ds = λψ|ba . Let us for the moment make a few∂sassumptions, first the boundary conditions on the flow field are stress-free and impenetrable

at the upper and lower radial boundaries. This means that the contours of L are all open
and have end points on these boundaries. Since the boundaries are impenetrable, we enforce
the boundary condition that ψ = 0 on the radial boundaries. The integral in equation B.16
then proceeds clockwise from the lower boundary in the southern hemisphere to an arbitrary
point along the contour which yields
1
ψ=
λ

Z

λF
ds
C |∇L|(B.17)

460
B.3

Meridional Flows Due to Zonal Torques in ASH
To numerically compute the integral in equation B.17, we use IDL’s routine contour

which generates the paths in polar coordinates. These paths are then used to identify which
points of the right hand side of the same equation are on the path. The arclength along thepath is computed and the right hand side as defined previously is integrated along it. This
procedure has been applied to the torques found in the F-star simulations analyzed in thisthesis. The following plots show this for six cases.
B.4

Thermal Wind, Geostrophy, and Gyroscopic Pumping
We begin with equation B.3, where a time and azimuthal average are taken yielding′i = −h(ρ + ρ′ ) u · ∇ui + hρigef f − 2hρΩ0 × ui − h∇ · Di −∇hP tot + Ptot

1h∇ · Mi4π

Further, if we assume that the flows are anelastic, in hydrostatic balance, and also take thefluctuating fields from equation B.9 with the exception that u = um + uφ φ̂ + u′ , then the
averaged advection term is
u2φ
1
1
hu · ∇ui = ∇hu′ · u′ i + um · ∇um + um · ∇ (λuφ ) φ̂ − λ̂ + h~ω ′ × u′ i2λ2λ(B.18)

When the Coriolis terms are expanded as well, it is seen that′∇Ptot
= −ρhu · ∇ui + hρ′ igef f + 2Ω0 ρ
1uφ λ̂ − uλ φ̂ − h∇ · Di −h∇ · Mi
4π(B.19)

u2

For the moment we will neglect the largest term in the advection equation ( 2λφ ), and fromdimensional analysis we can neglect all other terms except the Coriolis term if the system isrotating sufficiently rapidly. For the cases mentioned above, this is largely the case. Thus,
equation B.19 becomes′∇Ptot= 2Ω0 ρuφ λ̂(B.20)

461

Figure B.1: The meridional flow stream function ψL as defined in equation B.17 for the
1.2 M⊙ F star ASH simulations at (a) 5 Ω⊙ , (b) 10 Ω⊙ , and (c) 20 Ω⊙ . The second row of
figures is the stream function as derived directly from the meridional mass flux for the samecases in panels (e-g).

Therefore, the primarily pressure balance is in cylindrical radius, namely the cylindricalderivative of P ′ is small relative to the derivative in cylindrical radius (λ). Next we take thecurl of equation B.3, that has been divided by the density and expanded into fluctuating
and mean components, and examine its azimuthal component.



 ′

∂ωφ
1
ρ
′
= φ̂ · −∇ × (u · ∇u) − ∇ ×∇Ptot + ∇ ×
gef f − 2Ω0 ∇ × (ẑ × u)
∂t
ρ
ρ




1
1
∇·D +∇×∇·M(B.21)
− ∇×
ρ
4πρ

462

Figure B.2: The meridional flow stream function ψL as defined in equation B.17 for the
1.3 M⊙ F star ASH simulations at (a) 5 Ω⊙ , (b) 10 Ω⊙ , and (c) 20 Ω⊙ . The second row of
figures is the stream function as derived directly from the meridional mass flux for the samecases in panels (e-g).

where ω~ is the vorticity vector. Using some simple vector derivative identities, it is seen that
∇ × (u · ∇u) = ∇ × 12 ∇ (u · u) + ~ω × u = u · ∇~ω − ω~ · ∇u + ~ω ∇ · u.
∂ωφu λ ωφ − u φ ωλ∂ ln ρ∂uφ
1= ~ω · ∇uφ − u · ∇ωφ ++ ωφ u r+ 2Ω0+ φ̂ · ∇ ×
J×B
∂t
λ
∂r
∂z
ρ







 ′
1
1
1
ρ
′
gef f − φ̂ · ∇ ×∇P − φ̂ · ∇ ×∇ · D − φ̂ · ∇ ×∇·D+ φ̂ · ∇ ×
ρ
ρ
ρ
ρ
1
where J = 4π
∇ × B is the current density. Simplifying the thermodynamic terms, expanding

diffusive terms, and taking a azimuthal and temporal average, we arrive at

463

∂uφ
g ∂S ′
1 ∂S ∂P ′∂hωφ i
= 2Ω0
−
−
∂t
∂z
rCP ∂θ
rρCP ∂r ∂θDu λ ωφ − u φ ωλ∂ ln ρ E+ ~ω · ∇uφ − u · ∇ωφ +− ωφ u rλ∂rE
E 1 D 1 ∂ ln ρ
1D∂∂+(∇ · D) · θ̂(∇ · D) · r̂ −
−
+
ρr ∂θ
ρ
r
∂r
∂r
D
E
Bφ Jλ − Bλ Jφ ∂ ln ρ1B · ∇Jφ − J · ∇Bφ ++(Jr Bφ − Jφ Br ) . (B.22)+
ρλ∂r
If the time average taken is long enough then the l.h.s. should vanish leaving
2Ω0g ∂S ′
1 ∂S ∂P ′
∂uφ
≈
+
∂z
rCP ∂θ
rρCP ∂r ∂θD∂ ln ρ E
u φ ωλ − u λ ωφ
+ ωφ u r+ u · ∇ωφ − ~ω · ∇uφ +
λ
∂r
D
D
E
E
1 ∂
1 ∂ ln ρ1∂−(∇ · D) · θ̂(∇ · D) · r̂ +
−
+
ρr ∂θ
ρ
r
∂r
∂r
E
1D
Bλ Jφ − Bφ Jλ ∂ ln ρ+J · ∇Bφ − B · ∇Jφ ++(Jφ Br − Jr Bφ ) . (B.23)
ρλ∂r

Within the convection zone of ASH simulations the pressure contribution to the baroclinic
terms is small. The Reynolds stresses (line two of equation B.22) and diffusive terms (linethree of the same equation) are important only near the boundaries. Summarizing the results
for geostrophy (equation B.20) and a thermal wind, we have∂P ′= 2Ω0 ρuφ∂λ2Ω0 CP r ∂uφ
∂S ′
=
∂θ
g
∂z
This provides a integrable form for uφ that depends only upon the background and thermodynamic state of an anelastic system. Therefore, we can compute the dependence of thedenominator in equation B.17 upon the thermodynamic state of this system, where"
2 
2 # 12∂uφ∂uφ|∇L| =
2Ω0 λ + uφ + λ
+ λ
∂λ
∂z
"

2 
2 # 12g∂S ′1 ∂ λ ∂P ′
(B.24)+= (2Ω0 λ) 1 +4Ω20 λ ∂λ ρ ∂λ4Ω20 CP r ∂θTherefore, equation B.17 becomes

464

ψ=

1
2Ω0 λ

Z

C

F

"

1+

1 ∂
4Ω20 λ ∂λ



′

λ ∂P
ρ ∂λ

2

+



g

∂S

′

4Ω20 CP r ∂θ

2 #− 12ds

(B.25)

However, we have found in the F star simulations that the buoyancy terms are of nearly
equal importance when compared to the Coriolis terms. This means that the azimuthally
averaged density perturbations are large enough to make this term important, thus we havethe relations∇P ′ = 2Ω0 ρuφ λ̂ − ρ′ gr̂∂S ′2Ω0 CP r ∂uφ
=
∂θ
g
∂z

If we consider the energy equation, an equation for the entropy that depends only upon
the meridional flow and background state can be found. Also taking the divergence of thepressure equation in the line above, leads to a system of equations for the entropy, pressure,and meridional flow (Too be expanded upon)...
∂ ln ρ̂T̂ κS ∂S ′ ∇ × ψ φ̂−
· ∇S ′ + ǫ
∂r
∂r
ρ̂κS
1 ∂ 2 ′ 
2Ω0 ∂
r ρg
(ρ̂λuφ ) − 2∇2 P ′ =
λZ ∂λ
r ∂r
λF
1
ψ =
ds
λ C |∇L|∇2 S ′ =(B.26)(B.27)(B.28)The thermal wind relationship between the entropy and the mean azimuthal velocity
allows us to eliminate one or the other from the above set of equations. Whether attempting
to determine the thermal structure of the star from the differential rotation or vice versa,
this set of equations can be used in conjunction either with a measured or modeled Reynolds
stress to predict the meridional flows that are established in a particular system. Also, if we
consider Miesch & Hindman (2011), this set of equations will allow us to estimate the extent
to which the large latitudinal temperature contrasts seen in our global scale models extend
into the near surface region.

465
In the near surface regions it is likely that rather than a thermal wind balance, aReynolds stress balance is achieved. This means that the denominator in the integral forthe meridional flow streamfunction will be modified as the differential rotation near thesurface likely has a different character from that of the interior. However, if we can model
the angular momentum fluxes in these layers, we can predict how the meridional flows will
respond to the changing differential rotation near the surface. Then, using the energy andpressure equations, we may ascertain how the turbulent surface layers respond to a latitudinal
gradient of temperature. Namely, we can estimate the latitudinal thermal contrast at thesurface given the thermal structure established at depth.

Appendix C
Derivation of the Shear Instability Equations

C.1

Euler Equations
We begin with the equations that expressing the conservation of mass, momentum,

and entropy, assuming adiabatic motions, in a compressible fluid within an inertial frame.
∂ρ+ ∇ · ρv = 0,∂t
∂ρv
+ v · ∇ρv = −∇P + ρg,∂t∂S+ v · ∇S = 0.
∂t

(C.1)
(C.2)
(C.3)

A standard notation is used: v is the bulk velocity of the fluid, ρ is the density, P is thepressure, g is the acceleration due to gravity. We use spherical polar coordinates (r, θ, φ)in this derivation. We make the assumption that there is a bulk azimuthal flow v0 = v0 φ̂,
where v0 = v0 (r, θ). There is also a background state with P0 = P0 (r, θ), ρ0 = ρ0 (r, θ), andS0 = S0 (r, θ). If we expand the above Euler equations about the background state, letting
the perturbation quantities be v′ , ρ′ , S ′ , and P ′ , we arrive at∂ρ′+ ∇ · ρ0 v0 + ∇ · ρ0 v′ + ∇ · ρ′ v0 = 0,
∂t
∂v′
∂ρ′
ρ0
+ v0
+ v · ∇ρv = −∇P0 − ∇P ′ + ρ0 g + ρ′ g,
∂t
∂t
∂S ′
+ v0 · ∇S0 + v0 · ∇S ′ + v′ · ∇S0 = 0,
∂t

(C.4)
(C.5)
(C.6)

where the time derivatives of the background state quantities have been omitted as they are
zero by definition. It is helpful to separately expand the advection operator due to its lengthand direction dependence as shown in the following

467

v0 + vφ′ ∂v′ ∂∂[(ρ0 + ρ′ ) vr ′ ] + θ[(ρ0 + ρ′ ) vr ′ ] +[(ρ0 + ρ′ ) vr ′ ]∂rr ∂θr sin θ ∂φh
2 i
1
− (ρ0 + ρ′ ) vθ′2 + v0 + vφ′
,
(C.7)
r
v0 + vφ′ ∂
vθ′ ∂
′ ∂
′
′
′
′
θ̂ · [v · ∇ρv] = vr[(ρ0 + ρ ) vθ ] +[(ρ0 + ρ ) vθ ] +[(ρ0 + ρ′ ) vθ′ ]∂rr ∂θr sin θ ∂φ
h
i

1
2
+ (ρ0 + ρ′ ) vr ′ vθ′ − cot θ v0 + vφ′
,
(C.8)
r
 v′ ∂ ∂ 
(ρ0 + ρ′ ) v0 + vφ′ + θ(ρ0 + ρ′ ) v0 + vφ′
φ̂ · [v · ∇ρv] = vr ′
∂r
r ∂θ
 1

v0 + vφ′ ∂ +(ρ0 + ρ′ ) v0 + vφ′ + (ρ0 + ρ′ ) v0 + vφ′ [vr ′ + cot θvθ′ ] ,r sin θ ∂φrr̂ · [v · ∇ρv] = vr ′

(C.9)

C.2

Linearization

Examining the zeroth order terms of all equations, we find that∇ · ρ0 v0 = 0,

(C.10)

v0 · ∇S0 = 0,

(C.11)∇P0 + ρ0 gr̂ + [v · ∇ρv]0 = ∇P0 + ρ0 gr̂ −cot θρ0 v02
ρ0 v02θ̂ = 0,r̂ −r
r(C.12)

this indicates that the star is indeed in hydrostatic balance with a geostrophic balance of thecentripetal acceleration, which is in turn due to the bulk rotation of the star. Next, we lookat the linear terms of the advection operator before turning to the fully linearized equations
ρ′ v02
ρ0 v0 ∂vr ′ 2ρ0 v0 vφ′−−,(C.13)r̂ · [v · ∇ρv]1 =r sin θ ∂φrr
2 cot θρ0 v0 vφ′
ρ0 v0 ∂vθ′cot θρ′ v02θ̂ · [v · ∇ρv]1 =−−,(C.14)
r sin θ ∂φrrvθ′ ∂ρ0 v0
ρ0 v0 ∂vφ′v02 ∂ρ′ ρ0 v0′ ∂ρ0 v0φ̂ · [v · ∇ρv]1 = vr++++
(vr ′ + cot θvθ′ ) .
∂rr ∂θr sin θ ∂φr sin θ ∂φr
(C.15)

468
The linearized Euler equations are then∂ρ′v0 ∂ρ′′+ ∇ · ρ0 v += 0,∂t
r sin θ ∂φ∂vr ′
ρ0 v0 ∂vr ′ 2ρ0 v0 vφ′
ρ′ v02 ∂P ′
ρ0
+
−
−
+
+ ρ′ g = 0,∂tr sin θ ∂φ
r
r
∂r
′
′
′
2
cot
θρ
v
v
ρ0 v0 ∂vθcot θρ′ v02 1 ∂P ′
∂v
0 0 φ
−
−
+
= 0,
ρ0 θ +∂tr sin θ ∂φ
r
r
r ∂θ
∂vφ′
∂ρ′
∂ρ0 v0 vθ′ ∂ρ0 v0
ρ0 v0 ∂vφ′v02 ∂ρ′ρ0+ v0+ vr ′
+
+
+
∂t
∂t
∂r
r ∂θ
r sin θ ∂φr sin θ ∂φ
′
ρ0 v0
1 ∂P
+
(vr ′ + cot θvθ′ ) += 0,rr sin θ ∂φ
∂S ′v0 ∂S ′
∂S0 vθ′ ∂S0++ vr ′+= 0,∂t
r sin θ ∂φ
∂r
r ∂θ
C.3

(C.16)(C.17)(C.18)(C.19)(C.20)

Assumptions and Simplifications

By employing the Euler equations from the start, we have implicitly assumed that diffusion
is not present. This is not the case in our models, nor in a real star where a small but
non-zero diffusion is indeed present, where it will act couple radial and horizontal motions.
This complicates the issue and renders some of the assumptions made below invalid, andis currently ignored because the diffusive timescales in both our models and in stars ismuch longer than than the rotational timescale. Moving forward with the Euler equations,
we further suppose that there are no compositional gradients are present in the region of0
and also note thatinterest, and therefore define the Brunt-Väisälä frequency as N 2 = cgP ∂S∂rv0Ω = r sin
. Since we are considering motions deep within a star, inside the radiative zone,θthe speed of sound is large and the sound crossing time is much shorter than the rotationaltime scale which is of interest in this problem. Thus, acoustic waves are eliminated fromthe solutions by setting the time derivative of the density perturbations to zero. We also
assume that the isochoric surfaces are spherical. With these assumptions equations C.16C.20 become

469

∂ρ′
= 0,
∂φ
∂vr ′
∂vr ′
∂P ′
ρ0
+ ρ0 Ω− 2 sin θρ0 Ωvφ′ − ρ′ r sin2 θΩ2 ++ ρ′ g = 0,
∂t
∂φ
∂r
∂v′
∂v′
1 ∂P ′
ρ0 θ + ρ0 Ω θ − 2 cos θρ0 Ωvφ′ − ρ′ r sin θ cos θΩ2 +
= 0,
∂t
∂φ
r ∂θ
∂vφ′
∂vφ′
∂ρ0 v0∂ρ′∂ sin θΩρ0+ vr ′+ ρ0 vθ′+ ρ0 Ω+ r sin θΩ2
∂t
∂r
∂θ
∂φ
∂φ
′
1 ∂P
= 0,
+ ρ0 Ω (sin θvr ′ + cos θvθ′ ) +r sin θ ∂φ
∂S ′∂S ′ cP N 2 vr ′ vθ′ ∂S0
+Ω
+
+
= 0,
∂t
∂φ
g
r ∂θ

∇ · ρ0 v′ + Ω(C.21)(C.22)(C.23)(C.24)(C.25)For the F-type stars N is typically much larger than Ω, with N being a few mHz and Ωa few µHz. Let us now expand the energy equation in terms of the ratio of the rotationalfrequency to the buoyancy frequency, namely
cP vr ′∂S ′vθ′ ∂S0
1 ∂S ′++Ω+= 0.
N 2 ∂t∂φg
rN 2 ∂θ(C.26)
20≈ cPg λ ∂Ωwhere λ and z
Now, if we consider the star to be in thermal wind balance 1r ∂S∂θ∂z

are the cylindrical radius and height respectively, then



 ′
vr ′
Ω2
∂S ′
∂ ln Ω2
g
∂S
+ 2 λ
= 0.
+Ω
+
vθ′
cP N 2 vθ′ ∂t∂φN∂z(C.27)

Further, since we are interested in the shear instability due to the differential rotation, thetime scale of consideration is the rotation period. Thus, consider ∂t → Ω, so that the above
equation becomes∂ ln Ω2
g S′
Ω2
vr ′
= 0.
+λ
+ 2
vθ′
N Ωvθ′ cP∂z(C.28)For the problem we are considering, both of the terms inside of the square brackets are oforder one. Thus, for the stiffly stratified medium within the stable region of the F-typestars, we have Ω2 /N 2 ≪ 1 which implies that vr ′ /vθ′ ≪ 1. Therefore, the motions we

470
expect are primarily horizontal with a very small radial component due to gravity waves.
We now consider the radial component of the momentum equation, where it has been nondimensionalized by assuming that vr ′ = RNu′r , t̃ = Nt, vφ′ = RΩ0 u′φ , P ′ = ρ1 R2 Ω20 P̃ ′ ,
ρ′ = ρ1 ρ̃′ (where ρ1 is a reference density), Ω = Ω0 Ω̃, and r = Rr̃

−1
2
2
∂u′r Ω0 ∂u′r
ρ1 Ω20 ∂ P̃ ′ ρ1 cP ρ̃′ ∂S0
′ Ω0
′
2
2 Ω0
Ω̃
+
= 0.
− 2 sin θΩ̃uφ 2 − ρ̃ r̃ sin θΩ̃ 2 +
+
N ∂φ
N
N
ρ0 N 2 ∂r
ρ0 R
∂r
∂ t̃
(C.29)
−1∂u′r ρ1 cP ρ̃′ ∂S0
+
= 0.
ρ0 R
∂r
∂ t̃

(C.30)

here Ω0 is an appropriate reference rotation rate. If only the terms of zeroth order in Ω0 /N
are kept as in C.29, we find equation C.30. Taking the result from above that vr ′ ≈ 0 tosecond order in Ω0 /N, it is clear that equation C.30 implies that ρ′ = 0. With both soundand gravity waves eliminated from the system the equations to be solved reduce to∇ · ρ0 v′ = 0,
∂m′θ
∂m′θ
1 ∂P ′+Ω− 2 cos θΩm′φ +
= 0,
∂t
∂φ
r ∂θ
∂m′φ
∂m′φ
1 ∂P ′∂ sin θΩ+ m′θ+Ω
+ cos θΩm′θ +
= 0,
∂t
∂θ
∂φ
r sin θ ∂φ
∂S ′m′ ∂S0
∂S ′
+Ω
+ θ
= 0,
∂t
∂φ
rρ0 ∂θ

(C.31)(C.32)(C.33)(C.34)

where m′ = ρ0 v is the momentum density.
C.4

Streamfunction Formulation
Since the flow is anelastic, with no radial flow to O (Ω20 /N 2 ), as shown in equation C.31,

the system of equations may be reduced to a single equation for the streamfunction Ψ, which
and m′φ = − ∂Ψ
. We haveis defined such that m′ = ∇ × (Ψr̂). This means that m′θ = sin1 θ ∂Ψ∂φ∂θomitted the radius r in the spherical curl operator has been omitted as there is no longer
radial dependence at any given spherical shell within the region of the star where Ω0 /N ≪ 1
is a valid assumption. This generally occurs within the radiative zone of the F-type stars.

471
The pressure perturbations are eliminated by taking the curl of the momentum equations,which yields a single equation for Ψ as below


∂2Ψ
1 ∂
∂ sin θΩ ∂Ψ∂2Ψ∂Ψ ∂P ′− sin θ+− sin θΩ+ cos θΩ+sin θ ∂θ
∂t∂θ
∂θ ∂φ
∂θ∂φ
∂φ
∂φ


2
2
′
1 ∂ Ψ
Ω ∂ Ψ
∂Ψ ∂P
1 ∂
= 0,
+
+ 2 cos θΩ+−
2sin θ ∂φ sin θ ∂t∂φ sin θ ∂φ∂θ∂θ

(C.35)

which reduces to


∂2Ψ
Ω ∂3Ψ
∂2Ψ
1 ∂
−∇
sin θΩ−− 2 cot θΩ−∂t∂θ∂φ sin θ ∂θ∂θ∂φsin2 θ ∂φ3




1 ∂ ∂
1 ∂
∂Ψ
∂Ψ
+
+
cos θΩ= 0.
(sin θΩ)sin θ ∂θ ∂θ
∂φsin θ ∂θ∂φ
2 ∂Ψ(C.36)

Expanding the θ derivatives we arrive at
∂Ψ
Ω ∂3Ψ
∂3Ψ
∂2Ψ
−
−
Ω
−
3
cot
θΩ
∂t
∂θ2 ∂φ∂θ∂φsin2 θ ∂φ3
2
1 ∂
∂Ψ
1 ∂
∂Ψ
+
(sin θΩ)+(cos θΩ)= 0.
2
sin θ ∂θ∂φsin θ ∂θ∂φ

− ∇2

(C.37)

Further simplifying this by combining terms into the spherical Laplacian operator and reducing the remaining terms the previous equation becomes

 2


∂ Ω
∂
∂Ω
∂Ψ
∂
2
∇ Ψ−
+Ω
+ 3 cot θ
− 2Ω
= 0.
2
∂t
∂φ
∂θ
∂θ
∂φ

(C.38)∂∂= sin1 θ ∂θ
, and recast the second term as follows
Now define µ = cos θ, which implies ∂µ  ∂Ψ
∂∂2 ∂∇2 Ψ − 2 1 − µ 2 Ω+Ω= 0.
(C.39)
∂t
∂φ
∂µ
∂φ

C.5

SolutionsLet Ψ =

P∞m
−imωm t, where n and m are integer spherical harmonicn=m an Yn (µ, φ) e2∂2orders and ωm is the frequency of the fixed m mode, and where V = ∂µ2 [(1 − µ ) Ω], thenequation C.39 becomes
∞
X

n=m

{[(Ω − ωm ) n (n + 1)] + V } an Ynm = 0.

(C.40)

472
Note the multiplication by eimωm t and division by the im that resulted from the φ and tderivatives. To extend the work of Watson (1981) and Gilman & Fox (1997), among others,P0
expand the rotation profile Ω and the function V in Legendre polynomials as Ω = ∞ℓ=0 bℓ YℓP
0
and V = ∞ℓ=0 cℓ Yℓ where we have assumed that Ω and V are symmetric about the equator.
This produces the following
∞
X

n=m

[ωm n (n + 1) an Ynm ] −

∞ X
∞
X

[n (n + 1) bℓ + cℓ ] an Yℓ0 Ynm = 0.

(C.41)

n=m ℓ=0

The coefficients of Ψ are then extracted by integrating over the spherical shell against a thirdmconjugate spherical harmonic Y q , which gives
∞
X

n=m


Z 1 Z 2π
1/2m
an ωm n (n + 1)
dφdµ
Y q Ynm 1 − µ2

−

−1

∞
X

[n (n + 1) bℓ + cℓ ]0

Z 1 Z 2π
−1

ℓ=0

0

m
Y q Ynm Yℓ0

1−µ


2 1/2

dφdµ



= 0.

(C.42)

Next we use the orthogonality relation for the spherical harmonics to give a final form to
the most general solution. Since m > 0 and n ≥ m no factorials are less than zero, and n isnonzero so we may divide the solution through by n (n + 1) in equation C.42 and redefining
the arbitrary coefficient an . This leads to the following eigenvalue problem
)
(
∞
∞
X
X
an −(−1)m ωm δq,n +Gm
q,n,ℓ [n (n + 1) bℓ + cℓ ] ,n=m(C.43)ℓ=0(C.44)
Following Cruzan (1962), the Gaunt integral coefficients (Gmq,n,ℓ , Gaunt (1929)) are



1/2
n ℓ
[(2q + 1) (2n + 1) (2ℓ + 1)] q n ℓ   q
√
Gm



q,n,ℓ =
4πn (n + 1)
−m m 00 0 0

(C.45)

 j1 j2 j3 
where the Wigner-3j symbols  have been used. We omit the general form ofm1 m2 m3
the second Wigner-3j symbol for simplicity. However, the first Wigner-3j symbol in equation
C.45 is a special case of these symbols, where

473





h
i1/2

 (−1)J/2 (J−2q)!(J−2n)!(J−2ℓ)!

q n ℓ 
=


 0
0 0 0

(J+1)!

(J/2−q)!(J/2)!
(J/2−n)!(J/2−ℓ)!

if J is even,(C.46)

if J is odd

here J = q + n + ℓ. It is clear from this last relation that there is a decoupling of even andodd spherical harmonic degrees n. In particular for a given n, only terms that are integer
multiples of two distant from n are included in the computation of the coefficients an . Thisis a consequence of the choice to expand the angular velocity Ω and V in only even order
spherical harmonics, reflecting the assumption that they are symmetric about the equator.
C.6

Eigenvalues
To find the eigenvalues, namely the frequency ωm , one in principal would compute thedeterminant of the matrix A that arises from equation C.43, as they are determined from thecondition that there be an non-trivial solution for Ψ. This matrix will be band diagonal withthe width of the band determined by the maximum degree ℓ to which the angular velocity
and V are expanded, namely it will be 2ℓ + 1. Unlike Watson (1981), there is a uniquematrix for each m. Omitting the eigenvalue identity matrix, the matrix A has the following
structure P(q+n) Gm [n (n + 1) bℓ + cℓ ]ℓ=|q−n| q,n,ℓm
An,q =
 0

for |n − ℓ| ≤ q ≤ n + ℓ(C.47)

otherwise

where the sum over ℓ has been truncated by the use of the triangle condition on ℓ withrespect to q and n. There is also of course a triangle condition on the degree q, wherein|n−ℓ| ≤ q ≤ n+ℓ. This limits the band to 2ℓ+1. To solve this system, it must be truncated.
The minimum rank of the matrix is at least 4ℓ + 2, but typically must be much larger toaccurately capture the eigenfrequency ωm . This in turn depends upon the accuracy of thetruncation of the expansion of Ω and V , which at maximum for ASH simulations can beℓmax ≈ 1000. Usually, however, the expansion is very good for ℓmax ≈ 24.

Appendix D
Entropy Diffusion

In a simple fluid system with no mass flux across boundaries, the total rate of entropychange of the system is
Z

dV dQ
Ṡ =
+
V T dtZ

Fe · dA
,
T
∂V

(D.1)

where the rate of change in internal energy is integrated over the relevant volume; the external
heat fluxes into the system are accounted for in the second term. Through the first law ofthermodynamics, the rate of change in internal energy in the presence of diffusive processes
can be written in coordinate space asdQ= −∇ · F + Φ.
dt
Using ∇ · (F/T ) = (∇ · F) T + F · ∇ (1/T ) and Gauss’s law, Equation 1 becomes
Z
Z
Φ F
1
− · ∇ ln T dV +(Fe − F) · dA.
Ṡ =
T
∂V T
V T

(D.2)

(D.3)

Assume flux boundary conditions such that the boundary integral vanishes. Then, for entropy diffusion
Ṡ =

Z

Φ
+ κS ∇S · ∇ ln T dV.
V T

(D.4)

This looks innocuous, yet it hides some pernicious behavior. In a sub-adiabatic region, where∇T is negative and ∇S is positive, this clearly can violate the second law unless the viscousheating is much larger than the entropy diffusion. If instead the viscosity is negligible,
trouble rears its head and Ṡ ≤ 0 in a sub-adiabatic region. It could be that the convection

475
zone is safe from this issue, but there can also be local sub-adiabaticity in the convection
zone. While such fluctuations about adiabaticity are weak and temporary, it is nonethelessunphysical. However, if there is a fixed flux through the system, the boundary integral does
not vanish indicating that there is an inbalance in the net flux entering and exiting thesystem. In this case, the second law would remain safe. For a thermal conductivity, thepositive definite form for the entropy evolution isZΦṠ =+ κT (∇ ln T )2 dV.
V T

(D.5)

The entropy diffusion can be recast into a usable form. For a perfect gas,ZΦ+ κS (∇ ln T )2 − (γ − 1) ∇ ln ρ · ∇ ln T dV,
Ṡ =
V T

(D.6)

In these terms entropy diffusion amounts to a thermal diffusion as well as a mass diffusion fora perfect gas (it could also be a pressure diffusion). If instead only a thermal conductivity isadmitted, it couples the energy equation and the momentum equations through the pressure.
Thus a thermal conduction can be regarded as modifying the pressure rails that guide thesystem. Effectively, in the context of Chapter 4, there should be a modified Lagrangian
and a set of modified LBR equations due to diffusion. The entropy equation with thermal
conduction is,
dS
1
=
dt
ρT





∇ · κT∂ ln T∂ ln T
∇P +
∇S
∂P S
∂S P




+Φ

(D.7)

So, another way to think about the thermal conductivity in regards to entropy productionis that it is simultaneously a sub-grid-scale model for entropy mixing through the ∇S term
and for the entropy production by unresolved motions that are wrapped into the pressure,effectively being an additional subgrid-scale viscous heating.
Next consider the effect of entropy diffusion on entropy production in gravity waves.
For wavelengths much shorter than gradient length scales in the system, the rate of entropy
production isṠwaves =κ0 ρ0∇S ′ · ∇T ′ .
T0

(D.8)

476
For an ideal gas,
P 1/γ
S
= ln
.
cP
ρ

(D.9)

T′
S′
γ − 1 P′
=
+
.
T0
cP
γ P0

(D.10)

Linearizing this equation giveswith primes denoting perturbations. Let the dissipation be weak and assume gravity waves
under the Lantz-Braginsky-Roberts equations, which has the kinematic pressure ̟ (seeChapter 4). Thus the linearized momentum equation is−ω 2 ξ = −∇̟ − S ′ ∇T0 .

(D.11)The horizontal divergence of this equation, under the anelastic condition of ∇ · ρ0 u, leavesd ln ρ0dξz
2
2= −kh2 ̟,(D.12)+ ξz∇h ̟ = −ωdzdzwhere the last step is valid for plane waves. The wave equation for the vertical displacement
ξz is thenω2 d
ω ξz − 2
kh dz
2



dξz
d ln ρ0
+ ξz
dz
dz



gS ′
=−
.
cP

(D.13)

Under the WKB approximation (e.g., large wavenumber local modes), the solution is simply
 Z
ξ0ξz = √
exp i kz dz .
(D.14)
ρ0 kz
Substituting it back into the wave equation gives the expanded resultω 2 ξz

"

2

2 #)
2
2
1
gS ′
d
ln
ρ
d
ln
ρ
1
1
1
dk
1
d
ln
ρ
d
ln
k
0
0
z
0
z
=
−
1 + 2 kz2 + i
−
+
−
.
+
kh
dz
2 dz 24dz
2 dz 24
dz
cP(D.15)(

If the changes in the background potential are indeed slow, then the derivative terms may
be omitted. Thus, the wave equation for ξz becomes the simple form


gS ′
kz2
2
.
ω ξz 1 + 2 = −khcP(D.16)

From Chapter 4, it is known that the Eulerian perturbations for the entropy are S ′ =−ξ · ∇S0 . For the situation considered here S0 has only vertical derivatives so

477

dS0
.
dz

(D.17)

g dS0
,
cP dz

(D.18)gS ′.
N 2 cP(D.19)

S ′ = −ξzNote that the Brunt-Väisälä frequency isN2 =and the vertical displacement can be writtenξz = −

With these equations in hand, the dispersion relationship can be found by substituting theminto Equation D.16 yielding
kz2 = kh2




N2
−1 ,
ω2

(D.20)

which is the large-wavenumber limit of the gravity wave dispersion relation.
With these relationships in hand, the growth rate of waves may be assessed by using theestimated wave damping rate argument of (Braginskii, 1965), but taking into account thepresence of entropy diffusion in a stable stratification. First use the linearized equation ofstate with the substitution of the kinematic pressure ̟ for P ′, which yields
S′
γ − 1 ρ0 ̟
T′
=
+
.
T0
cP
γ P0
Then use Equation D.12 to eliminate the kinematic pressure to find


T′
S′
γ − 1 ρ0 ω 2 dξz
d ln ρ0
=
+
.
+ ξz
T0
cP
γ P0 kh2 dz
dz
Recalling the WKB approximation, this becomes



S′
γ − 1 ρ0 ω 2T′
1 d ln ρ0 d ln kz
=
+
ξz .
−
ikz +
T0
cP
γ P0 kh22dz
dz
Luckily, Equation D.19 allows this to be simplified further to




γ − 1 ρ0 gω 2S′
1 d ln ρ0 d ln kz
T′
1−
,
=
−
ikz +
T0
cP
γ P0 N 2 kh2
2dz
dz(D.21)(D.22)(D.23)(D.24)

Only the real-valued terms in the equation can contribute to entropy production, as imaginary terms produce only phase offsets. The derivative of the wavenumber kz follows fromthe dispersion relationship, assuming that the horizontal wavenumber does not vary in z, as

478

−1
1 dN 2d ln kz
N 2 − ω2
.
=
dz
2 dz

(D.25)

If the stratification remains so that dρ0 /dz < 0, that term in the equation only contributes to positive entropy production. The question then remains what is the possible
wave growth rate in the presence of entropy diffusion? If the wavenumber is assumed to varymuch more rapidly than the density, the term involving its derivative will dominate that
expression. Furthermore, if one makes kh increasingly large relative to kz , the expression for
the temperature fluctuations simplifies to(γ − 1) gω 2 d ln N 2 S ′
T′
≈ 2 2
T0
4kh cs,0 (N 2 − ω 2 ) dz cP(D.26)

Returning to the expression for entropy production or destruction through its diffusionq = κS ∇S · ∇T , it is clear that if ∇S > 0 and ∇T < 0 then the second law is violated.
What about for the waves, could they be growing unphysically as well? This depends upon
the sign of dN 2 /dz, so they too can be violating the second law. The remaining question is
how fast might they be growing? Following (Braginskii, 1965), which gives an argument thatconnects the wave damping rate Γ with the rate of entropy production in the limit of weakdamping (|Γ/ω| ≪ 1). Extending this argument to wave growth and utilizing the expression
for the LBR wave energy density (E = ρS ′2 g 2 /N 2 c2P ) from (Brown et al., 2012), it is found
that
ΓS =

q
κS kB T0 N 2≈
2E
8mg 2HP (N 2 − ω 2 )(D.27)

If this growth rate ΓS is greater than the diffusion damping time, the gravity waves
can grow artificially. With the form of the denominator, this is particularly true for waves
with frequencies close to the Brunt-Väisälä frequency. This growth rate is assessed for theradiative envelopes of the B-type stars in Chapter 10. Beyond the uncontrolled growth of
linear waves, this result and the simple thermodynamic argument above point to problems
with the concept of entropy diffusion despite its original physical motivation.

Appendix E
Reduced Sound Speed

This chapter outlines the reduced sound-speed equations as implementated in the Compressive Spherical Segment (CSS) code. The following set of Navier-Stokes equations are fora fully compressible, rotating fluid in a spherical domain with standard orthonormal coordinates and unit vectors. The rotation axis is chosen to align with the z-axis such thathiΩ = Ωẑ = Ω cos θr̂ − sin θθ̂ .
(E.1)

The centripetal acceleration (ac = Ω × r × r) due to the rotating frame depends upon theaxial radius vectorhi
R = r sin θ sin θr̂ + cos θθ̂ .

(E.2)

∂ρ
∂ρuj
= −(1 − α)
,
∂t
∂xj

(E.3)

∂P
∂µDij
∂
∂ρui
[ρui uj ] −+ ρfi +
+ 2ρ (u × Ω)i + ρΩ2 Ri ,
=−
∂t
∂xj
∂xi
∂xj

(E.4)

Thus, the equations of motion are mass continuity,conservation of momentum,

and conservation of internal energy,





∂
1 2
∂qj
∂
1 2
P
ρe + ρui = −ρuj−
ui + e +
+ ρǫ.
∂t
2
∂xj
2
ρ
∂xj

(E.5)

The viscous stress tensor D is defined as1
Dij = 2 eij − ∇ · uδij ,
3

(E.6)

480
and the viscous heating term Φ is defined as


1
1
2
Φ = D : D = 2 e : e − (∇ · u) .
2
3

(E.7)

In both equations E.6 and E.7 the tensor e is the strain rate tensor and is itself a function of
u. With these equations in hand, we can move forward to reduce the system to the primitive
variables and recover the modified Navier-Stokes equations. First, the momentum equation
will be dealt with and then the energy equation. The first step to simplifying the momentum
equation is to use the continuity equation as so:
∂ρ
∂ui
∂ρuj
∂ui
+ ui
= ρ
− (1 − α)ui
,
ρ
∂t
∂t
∂t
∂xj
∂ui
∂ρuj
∂ρui ujρ= (1 − α)ui−+··· ,
∂t
∂xj
∂xj
∂ui
∂ρuj
∂ui
ρ
= −αui− ρuj+··· ,
∂t
∂xj
∂xj
∂ui
∂ρuj
∂ui
∂P
∂µDij
ρ
− ρuj
−
+ ρfi +
+ 2ρ (u × Ω)i + ρΩ2 Ri .
= −αui
∂t
∂xj
∂xj
∂xi
∂xj
Thus, the momentum equation remains essentially the same but with the addition of a term
proportional to α that modifies the convective terms in the Navier-Stokes equations. Theenergy equation is slightly more complicated in that we must deal with the kinetic energy
terms.
∂ 12 ρu2i
1 2 ∂ρ
∂ui
=
ui
+ ρui
,
∂t
2 ∂t
∂t
∂ui
∂P
∂µDij
∂ρuj
∂ρuj
1
− αu2i− ρui uj
− ui
+ ρui fi + ui
+ ρΩ2 uiRi
= − (1 − α) u2i
2
∂xj
∂xj
∂xj
∂xi
∂xj
1
∂ui
∂P
∂µDij
∂ρuj
= − (1 + α) u2i− ρui uj
− ui
+ ρui fi + ui
+ ρΩ2 ui Ri
2
∂xj
∂xj
∂xi
∂xj
Hence, the energy equation becomes
∂ρuj
∂ρe
1
∂ui
∂P
∂µDij
=
(1 + α) u2i+ ρui uj
+ ui
− ρui fi − ui− ρΩ2 uiRi
∂t
2
∂xj
∂xj
∂xi
∂xj



∂qj
1 2
P
∂
ρuj
−
ui + e +
+ ρǫ,
−
∂xj
2
ρ
∂xj
∂e
∂ρuj
1
∂ui
∂P
∂µDij
∂ρuj
ρ − (1 − α) e=
(1 + α) u2i+ ρui uj
+ ui
− ρui fi − ui− ρΩ2 uiRi
∂t
∂xj
2
∂xj
∂xj
∂xi
∂xj



∂qj
1 2
P
∂
ρuj
−
ui + e +
+ ρǫ,
−
∂xj
2
ρ
∂xj

481
now eliminate the kinetic energy u2i terms and second simplify the convective derivatives of
e
ρ

∂e
∂ρuj
α ∂ρuj∂P∂µDij− (1 − α) e
= u2i
+ ui
− ρui fi − ui− ρΩ2 uiRi
∂t
∂xj
2 ∂xj
∂xi
∂xj



P
∂qj
∂
ρuj e +
−
+ ρǫ,
(E.8)
−
∂xj
ρ
∂xj




∂e
∂e
1 2
∂P
∂µDij
∂ρuj
ρ
=α
+ uj
ui − e+ ui− ρui fi − ui− ρΩ2 ui Ri
∂t
∂xj
2
∂xj
∂xi
∂xj
∂qj
∂P uj
−
+ ρǫ.
(E.9)
−
∂xj
∂xj

Lastly, simplify the terms involving the pressure



1 2
∂e
∂e
∂ρuj
∂P
∂µDij
=α
ρ
+ uj
ui − e+ ui− ρui fi − ui− ρΩ2 uiRi
∂t
∂xj
2
∂xj
∂xi
∂xj
∂qj
∂P uj
−
+ ρǫ,
(E.10)
−
∂xj
∂xj




1 2
∂µDij
∂e
∂ρuj
∂e
=α
− ρui fi − ui− ρΩ2 ui Ri+ uj
ui − e
ρ
∂t
∂xj
2
∂xj
∂xj
∂qj
∂uj
−
+ ρǫ.
(E.11)
−P
∂xj
∂xj


E.1

Equations of State
While equations E.3-E.5 are to be solved for ρ, P , T , e, and the three components of u,

they comprise only five independent relations. Since only two of the four state variables are
independent, the system is closed by first specifying an equation of state (typically relating
the state variables ρ, P , and T ) and then deriving an equation for e. If we assume a nearperfect gas, the equation of state isP =γ−1
CP ρT.
γ(E.12)

where γ is the adiabatic index and CP is the constant pressure specific heat. Thus, theinternal energy is
e = CV T.

(E.13)

482
Another useful state variable is the specific entropy. For a perfect gas this is1
ln P − ln ρ .
s = CPγ(E.14)orP = ργ eγs/CP
E.2(E.15)

Additional Constraints
q = − κr + κsc T 3 ∇T − κs ∇S.

(E.16)

Using the equation of state above and assuming that CP is constant, an equation for theevolution of the specific entropy can be derived from the energy conservation equation. To
do so, use the second law of thermodynamics
de = T ds − P dV = T ds +

P
dρ,
ρ2

ds P dρ
de
= T
+
.
dt
dt ρ2 dt

(E.17)(E.18)The energy equation is


1 2
∂ρuj
∂µDij
∂uj
∂qj
de
= α
ui − e− ρui fi − ui− ρΩ2 ui Ri − P
−
+ ρǫ,
ρ
dt
2
∂xj
∂xj
∂xj
∂xj


ds P dρ
1 2∂ρuj−··· ,
ρT+= αui − e
dt
ρ dt
2
∂xj


P ∂ρuj
∂µDij
∂qj
1 2
ds
= α
ui − e −− ρui fi − ui− ρΩ2 ui Ri −
+ ρǫ.
ρT
dt
2
ρ ∂xj
∂xj
∂xj
where the equation dρ/dt = α∇ · ρu − ρ∇ · u has been invoked. However, e = CP T − Pρ , sothe α enhanced entropy equation becomes


∂µDij
∂qj
1 2
∂ρuj
ds
− ρui fi − ui− ρΩ2 uiRi −+ ρǫ.
=αu i − CP T
ρT
dt
2
∂xj
∂xj
∂xj
E.3

(E.19)

Modified Sound Wave Propagation
Under a simple linearization of the above equations in which we assume a local subdo-

main with the following properties483

ρ = ρ0 + ρ1 ,s = s0 ,γρ1
P = P0 1 +,ρ0where ρ0 = s0 = P0 = const.. Then to first order the above equations become∂ρ1= − (1 − α) ρ0 ∇ · u,∂t1∂u
= − ∇P∂t
ρ0
If we assume wavelike solutions of the form
ρ1 = ρ2 ei(k·r−ωt) ,u = u0 ei(k·r−ωt)then, we haveωρ2 = (1 − α) ρ0 k · u0 ,
ωu =γkP0 ρ2
ρ0 ρ00
noting that γP= c2s this becomesρ0

ω 2 u = kc2s (1 − α) k · u0 ,√
ω = kcs 1 − α.