Convection and Dynamo Action in Rapidly
Rotating Suns
by
B. P. Brown
B.S., Harvey Mudd College, 2003

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
2009This thesis entitled:
Convection and Dynamo Action in Rapidly Rotating Suns
written by B. P. Brown
has been approved for the Department of Astrophysical and Planetary Sciences

Juri Toomre

John Bally

Allan Sacha Brun

Keith Julien

Mark Miesch

Mark Rast

Date
The final copy of this thesis has been examined by the signatories, and we find thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

Brown, B. P. (Ph.D., Astrophysical and Planetary Sciences)
Convection and Dynamo Action in Rapidly Rotating Suns
Thesis directed by Prof. Juri Toomre

When our Sun was young it rotated much more rapidly than it does now. Observations of young, rapidly rotating stars indicate that many possess substantial magnetic
activity and strong axisymmetric magnetic fields. There is furthermore an observed correlation between the stellar rotation rate and surface magnetism. Yet the origins of themagnetic activity or the correlation with rotation remain unclear. We conduct simulations of dynamo action in rapidly rotating suns with the 3-D MHD anelastic sphericalharmonic (ASH) code to explore the complex coupling between rotation, convection
and magnetism. Here we study global-scale flows of differential rotation and meridional
circulation as well as dynamo action realized in the bulk of the convection zone for stars
rotating from one to fifteen times the current solar rate.
We find that more rapidly rotating stars generally have stronger flows of differential rotation but weaker meridional circulations that break into multiple cells in bothradius and latitude. Surprising localized states arise in the rapidly rotating simulations,with convection modulated in longitude. In the most rapid rotators convection can be
entirely confined to narrow active nests which persist for thousands of days and propagate through the shearing flow of differential rotation at their own distinct velocity.
We find that substantial organized global-scale magnetic fields are achieved bydynamo action in these rapidly rotating suns. Striking wreaths of magnetism are built
in the midst of the convection zone, coexisting with the turbulent convection. Thisis a great surprise, for many solar dynamo theories have suggested that a tachoclineof penetration and shear at the base of the convection zone is a crucial ingredientfor organized dynamo action, whereas these simulations generally do not include such

ivtachoclines. Some dynamos achieved in these rapidly rotating states build persistentglobal-scale fields which maintain amplitude and polarity for thousands of days. Other
dynamos can undergo cycles of activity, with fields varying in strength and even changing
in global-scale polarity. As the magnetic fields wax and wane in strength, the primary
response in the convective flows involves the axisymmetric differential rotation, which
begins to vary on similar time scales. Bands of relatively fast and slow fluid propagate
toward the poles on time scales of roughly 500 days. In the Sun, similar patterns areobserved in the poleward branch of the torsional oscillations, and these may represent
a response to poleward propagating magnetic field deep below the solar surface.
In one simulation, rotating at three times the solar rate, we explore how thewreaths of magnetism are built and maintained by the differential rotation and theturbulent correlations. We further explore whether a simple mean-field theory can
reproduce our 3-D results and find several discrepancies. We generally find that wreathbuilding dynamos are present in every region of parameter space we have sampled,
including simulations of the solar dynamo. We find that previous simulations hadbottom boundary conditions which make wreath formation difficult if not impossible,
but that new simulations of the solar dynamo can produce strong magnetic wreaths.
Lastly, we show that wreaths of magnetism survive in the presence of a tachoclineof penetration and shear at the base of the convection zone. These wreaths fill theconvection zone and undergo quasi-regular reversals of global-scale polarity.

Dedication

To my mother and father who have encouraged every step along this path of
discovery, and to Mindy who has made the journey one of joy.

Acknowledgements

This journey has only been possible with the help of many friends. First andforemost, I thank my advisor Juri Toomre for his support, friendship, generosity andwise advice. Juri has provided me with opportunities beyond compare. I cherish themany evenings I have spent with Juri and his wife Linda, and look forward to the many
wonderful meals and celebrations to be shared together in the years to come.
My collaborators and co-authors have become close friends over the years. Sacha
Brun has provided constant and good advice, and reflections on scientific careers and life,
even as he has undertaken his own journeys on these paths. His European perspectiveis always refreshing. I have always looked forward with pleasure to my visits with Mark
Miesch at HAO. His positive outlook and deep insights have contributed greatly to myown growth, and his excitement about the discoveries contained herein has kept my own
alive. His thesis has been at my side for many years, and it became particularly helpful
in the past months; I hope to one day write as well he does regularly. And Mark, you’reright: the high mountains do soothe the soul. My thanks also go to Matt Browning,
who walked these steps before me, and is now helping me to find my own way. Over theyears Mark Rast has been a sage source of advice, helping me remember what reallymatters in the pursuit of scientific discovery, and reminding me that the standard pathto success is not the only one, and that it is perhaps not always the the best. Gwen
Dickinson has been a friend and a constant eye of calm in the whirling storm that canbe our research. She has remained unflappably calm in the face of looming deadlines,
and has saved us all from disaster (myself in particular) many, many times.

vii
My office mates over the years, Jason Lisle, Geoff Vasil and Nick Nelson, have eachhelped shape this continuing adventure. Jason reminded me early on of the importanceof balancing life and science, and the value of time. Geoff has become a dear friend,and many evenings have been spent together thinking on science and life, in our tower
office or over good food and good beer. We both venture forth now to postdocs in colder
climes. Nick has helped me realize that I really have finally learned a few things of use.
Good luck sir, on your own path to discoveries! Kyle Augustson and Nick Featherstone
are part of our cohort as well; I would not be here now with out their aid. I lookforward to seeing where each of us ventures in the future. Along my path, I’ve had
the pleasure of meeting and working with John Clyne and Alan Norton at NCAR, andCarter Emmart at the American Museum of Natural History. These three have helpedme realize how beautiful our science can be, and how enthusiastically it can be received
when shown in the right light. With them, I look forward to many fruitful years ahead.
My close friends and housemates, Brennan Gantner and Than Putzig, have joined
me in many wonderful adventures during our years in Boulder. To them and Licia Ray,I owe my sanity. Ben Mates and Dave Schneider held the final watch at the Greenbriar
estate, through the craziness of moving, writing and planning a wedding; during thesetimes, they reminded me that this too would pass.
These adventures in science began at a young age when my father told me “there
are four fundamental forces of nature. You should learn about them, how they work,
what they are.” This fueled a love for physics and then astronomy. My mother andfather have supported me with love on every step of this adventure. In recent days mypath has been made clearer by the presence of my beloved, Mindy Wilkinson, who has
been my companion and guide. During dark days she reminds me that what we do
is fundamentally really, really neat. That it matters. That people care. And to me,that has value beyond words. On the journeys ahead, we will tread our paths together,which brings me incomparable joy.

viii
I thank Juri Toomre, Mark Rast, Mark Miesch, Keith Julien, Sacha Brun, andJohn Bally for serving on my thesis committee. Over the years, they each have acted
to kindle my scientific enthusiasm, to provide perspective on life, and to help me find
my way. It was an honor to have them present at the close of this chapter of life.

This research has been supported by NASA through Heliophysics Theory Program
grants NNG05G124G and NNX08AI57G. I have additionally been supported throughthe NASA GSRP program by award number NNG05GN08H. The simulations in thisthesis were carried out with NSF PACI support of SDSC, PSC, TACC and NICS. In
particular, these simulations ran variously on Blue Horizon, Datastar and Bluegene at
SDSC, on Lemieux and Bigben at PSC, on Ranger at TACC and on Kraken at NICS.
Volume renderings used in the analysis and the field line tracings shown were produced
using VAPOR (Clyne et al. 2007). None of this analysis would have been possible
without the resources at Laboratory for Computational Dynamics (LCD) here at the
University of Colorado.

ix

Contents

Chapter
1 Convection, Rotation and Magnetism1

1.1

Operation of the Solar Dynamo . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Theoretical Treatments of Stellar Convection . . . . . . . . . . . . . . .

6

1.3

Exploring Magnetism in Other Stars . . . . . . . . . . . . . . . . . . . .

8

1.4

Global Models of Stellar Dynamos . . . . . . . . . . . . . . . . . . . . .

13

1.5

Convection and Dynamo Action in Rapidly Rotating Suns . . . . . . . .

16

2 Elements in Modeling 3-D Stellar Convection and Dynamo Action

19

2.1

Anelastic MHD Formulation . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.2

Boundary Conditions and Their Impacts . . . . . . . . . . . . . . . . . .

22

2.3

Magnetic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . .

24

2.3.1

Potential Field Boundaries . . . . . . . . . . . . . . . . . . . . .

25

2.3.2

Perfect Conductor Boundaries. . . . . . . . . . . . . . . . . . .

28

2.3.3

Radial Field Boundaries . . . . . . . . . . . . . . . . . . . . . . .

29

2.3.4

Magnetic Boundary Conditions and Effects on CFL Limits . . .

29

2.4

Approach to Hydrodynamic Simulations . . . . . . . . . . . . . . . . . .

31

2.5

Hydrodynamic Progenitors to Dynamo Simulations . . . . . . . . . . . .

37

2.6

Studies of Dynamo Action in Rapidly Rotating Suns . . . . . . . . . . .

40

x
3 Convection in Rapidly Rotating Younger Suns

46

3.1

Early Results of Modulated Convection . . . . . . . . . . . . . . . . . .

46

3.2

Convective Patterns and Evolution with Rotation . . . . . . . . . . . . .

48

3.2.1

Radial Connectivity of Convection . . . . . . . . . . . . . . . . .

53

3.2.2

Thermal Structuring . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.3

Thermal Wind Balance. . . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.4

Angular Momentum Redistribution . . . . . . . . . . . . . . . . . . . . .

60

3.5

Differential Rotation and Scaling with Rotation . . . . . . . . . . . . . .

64

3.6

Meridional Circulations and Scaling with Rotation . . . . . . . . . . . .

67

3.7

Energy Balances and Flux Transport . . . . . . . . . . . . . . . . . . . .

70

3.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4 Dynamics Within Confined Nests of Convection

76

4.1

Spatially Localized Convection in Other Settings . . . . . . . . . . . . .

76

4.2

Properties of the Active Nests . . . . . . . . . . . . . . . . . . . . . . . .

79

4.3

Detailed Structure of an Active Nest of Convection . . . . . . . . . . . .

85

4.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5 Global Dynamo that Builds Persistent Wreaths of Magnetism

92

5.1

Patterns of Convection in Case D3 . . . . . . . . . . . . . . . . . . . . .

93

5.2

Kinetic and Magnetic Energies . . . . . . . . . . . . . . . . . . . . . . .

98

5.3

Wreaths of Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4

Wreaths Persist for Long Epochs . . . . . . . . . . . . . . . . . . . . . . 104

5.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Cyclic Dynamo Action Achieved at 5 Ω⊙

108

6.1

Patterns of Convection in Case D5 . . . . . . . . . . . . . . . . . . . . . 108

6.2

Oscillations in Energies and Changes of Polarity . . . . . . . . . . . . . 111

xi
6.3

Global-Scale Magnetic Reversals . . . . . . . . . . . . . . . . . . . . . . 113

6.4

Temporal Changes in Differential Rotation . . . . . . . . . . . . . . . . . 116

6.5

Sampling Many Magnetic Cycles in Case D5 . . . . . . . . . . . . . . . . 120

6.6Strange States and Wreaths of a Single Polarity . . . . . . . . . . . . . . 123

6.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7 Analysis of Production terms in Dynamos with Sustained Wreaths

127

7.1

Production of Axisymmetric Toroidal Field . . . . . . . . . . . . . . . . 127

7.2

Maintaining Wreaths of Toroidal Field . . . . . . . . . . . . . . . . . . . 130

7.3

Production of Axisymmetric Poloidal Field . . . . . . . . . . . . . . . . 132

7.4

Maintaining the Poloidal Field . . . . . . . . . . . . . . . . . . . . . . . 134

7.5

Exploring Mean-Field Interpretations . . . . . . . . . . . . . . . . . . . . 136

7.6

Production of Fluctuating (Non-Axisymmetric) Field . . . . . . . . . . . 140

7.7Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8 Menagerie of Wreath-Building Dynamos

142

8.1

Higher Levels of Turbulence at 3 Ω⊙ and Pm = 0.5 . . . . . . . . . . . . 148

8.2

High Pm Dynamos at 3 Ω⊙ . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3

Resulting Differential Rotation . . . . . . . . . . . . . . . . . . . . . . . 163

8.4

Extreme Rotators: 10 and 15 Ω⊙ dynamos . . . . . . . . . . . . . . . . . 165

8.5

Spinning Down to the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.6

The Mystery of Case M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.7

When the Sun is Old: Slowly Spinning Suns . . . . . . . . . . . . . . . . 183

8.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

9 Future Explorations and Wreath-building Dynamos with Tachoclines

190

9.1

Capturing a Model Tachocline within ASH . . . . . . . . . . . . . . . . 191

9.2

Magnetic Wreathes with a Tachocline . . . . . . . . . . . . . . . . . . . 194

xii
9.3

Perspective on Rapidly Rotating Suns . . . . . . . . . . . . . . . . . . . 200

9.4

The Road Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.5

Final Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Bibliography

211

xiii

TablesTable
2.1

Properties of Magnetic Boundary Conditions . . . . . . . . . . . . . . .

30

2.2

Parameters for Primary Hydrodynamic Simulations. . . . . . . . . . .

32

2.3

Parameters for Primary Dynamo Simulations . . . . . . . . . . . . . . .

43

3.1

Flux Balances and Energies . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.1

Angular Velocities of Various Structures . . . . . . . . . . . . . . . . . .

83

5.1

Near-surface ∆Ω in Cases at 3 and 5 Ω⊙

. . . . . . . . . . . . . . . . .

97

5.2

Energies in Cases at 3 and 5 Ω⊙

. . . . . . . . . . . . . . . . . . . . . .

99

8.1

Overview of Dynamos at Pm=0.5

8.2

Dynamos at Higher Pm . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.3

Dynamos That Failed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.4

Mean ∆Ω in Dynamos at 3 Ω⊙ . . . . . . . . . . . . . . . . . . . . . . . 165

8.5

Mean ∆Ω in Dynamos at 10 and 15 Ω⊙

9.1

Parameters of case T3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191. . . . . . . . . . . . . . . . . . . . . 144. . . . . . . . . . . . . . . . . . 170

xiv

FiguresFigure1.1

Solar magnetism at many scales . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Temporal variation of global-scale solar magnetism . . . . . . . . . . . .

3

1.3

Differential rotation and the solar dynamo . . . . . . . . . . . . . . . . .

5

1.4

Observational rotation-activity correlations in main-sequence stars . . .

9

1.5

Snapshot of 3-D ASH solar dynamo simulation which includes a tachocline 14

2.1

Radial variation of mean stellar structure in the ASH models . . . . . .

34

2.2

Evolution of kinetic energies in hydrodynamic case G5 . . . . . . . . . .

36

2.3

Radial structure of dynamo progenitor simulations . . . . . . . . . . . .

39

2.4

Patterns of convection in cases G5 and H5 . . . . . . . . . . . . . . . . .

41

2.5

Evolution of kinetic and magnetic energies in dynamo case D3. . . . .

44

3.1

Convective patterns in mildly turbulent simulations . . . . . . . . . . . .

47

3.2

Convective patterns in primary hydrodynamic cases . . . . . . . . . . .

49

3.3

Profile of hvφ i in case G5. . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.4

Connectivity of radial velocity with depth in case G5 . . . . . . . . . . .

55

3.5

Temperature structures within case G5 . . . . . . . . . . . . . . . . . . .

57

3.6

Thermal wind balance achieved in case G5 . . . . . . . . . . . . . . . . .

59

3.7

Scaling of ∆Sb and maximal latitudinal temperature contrast with Ω0 . .

60

3.8

Angular momentum fluxes in radius and latitude for cases G1 and G5 .

62

xv
3.9

Scaling of ∆Ω and ∆Ω/Ωeq with Ω0 . . . . . . . . . . . . . . . . . . . .

66

3.10 Changes in structure of meridional circulations with faster rotation . . .

68

3.11 Scaling of kinetic energy of meridional circulations (MCKE) with Ω0 . .

69

3.12 Average radial energy fluxes in cases G1 and G5 . . . . . . . . . . . . .

72

4.1

Nests of convection in G5 shown in time-longitude maps at three depths

80

4.2

Angular velocities of various structures with radius in case G5 . . . . . .

82

4.3

Time-longitude map of single nest of convection in case G10 . . . . . . .

84

4.4

Detailed structure of convective nest in case G10 . . . . . . . . . . . . .

86

4.5

Mean circulations associated with the nest in case G10 . . . . . . . . . .

88

5.1

Convective structures and mean flows in cases D3 and H3 . . . . . . . .

94

5.2

Magnetic and convective structures in case D3 at two depths . . . . . . 102

5.3

Field line visualization of magnetic wreaths in case D3 . . . . . . . . . . 103

5.4

Persistent wreaths of magnetism in case D3 . . . . . . . . . . . . . . . . 105

5.5

Temporal variations of differential rotation in case D3 . . . . . . . . . . 106

6.1

Patterns of convection in case D5 . . . . . . . . . . . . . . . . . . . . . . 109

6.2

Complex time evolution in case D5 with flips in polarity of magnetic
wreaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3

Evolution of Bφ during a polarity reversal in case D5 . . . . . . . . . . . 115

6.4

Time-varying differential rotation in case D5

6.5

Extended history of varying differential rotation in case D5 . . . . . . . 121

6.6

Time-latitude plots of magnetic fields in case D5 . . . . . . . . . . . . . 122

6.7

Strange single-polarity states in case D5 . . . . . . . . . . . . . . . . . . 124

7.1

Generation of hBφ i in case D3 . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2

Production of mean poloidal vector potential hAφ i in case D3 . . . . . . 135

7.3

Estimating the mean-field α-effect in case D3 . . . . . . . . . . . . . . . 138. . . . . . . . . . . . . . . 118

xvi
7.4

Comparison of emfs in case D3 . . . . . . . . . . . . . . . . . . . . . . . 139

8.1

Map of full dynamo η − Ω0 parameter space . . . . . . . . . . . . . . . . 147

8.2

Time-dependent behavior in the oscillating case D3a . . . . . . . . . . . 150

8.3

Patterns of convection in case D3a . . . . . . . . . . . . . . . . . . . . . 152

8.4

Time-dependent behavior in the cyclic case D3b

8.5

Patterns of convection in case D3b . . . . . . . . . . . . . . . . . . . . . 155

8.6

Bφ achieving a more non-axisymmetric state in case D3b

8.7

Time-dependent behavior in cyclic case D3-pm1 . . . . . . . . . . . . . . 158

8.8

Time-dependent behavior in cyclic case D3-pm2 . . . . . . . . . . . . . . 160

8.9

Patterns of convection in case D3-pm2 . . . . . . . . . . . . . . . . . . . 161. . . . . . . . . . . . . 154. . . . . . . . 156

8.10 Patterns of convection in case D3-pm4 . . . . . . . . . . . . . . . . . . . 164
8.11 Time-dependent behavior in case D10L

. . . . . . . . . . . . . . . . . . 167

8.12 Differential rotation and magnetism in case D10L . . . . . . . . . . . . . 168
8.13 Convective patterns in case D10L with nests of convection . . . . . . . . 169
8.14 Time-dependent behavior in case D15 . . . . . . . . . . . . . . . . . . . 172
8.15 Differential rotation and magnetism in case D15. . . . . . . . . . . . . 174

8.16 Convective patterns in case D15 . . . . . . . . . . . . . . . . . . . . . . . 175
8.17 Differential rotation and magnetism in case D1.5a. . . . . . . . . . . . 176

8.18 Energy traces for cases M3 and M3-pcpf . . . . . . . . . . . . . . . . . . 178
8.19 Wreaths of magnetism in solar simulations . . . . . . . . . . . . . . . . . 179
8.20 Patterns of convection in case M3-pcpf . . . . . . . . . . . . . . . . . . . 181
8.21 Patterns of convection in case M3 . . . . . . . . . . . . . . . . . . . . . . 182
8.22 Patterns of convection in case D0.5a . . . . . . . . . . . . . . . . . . . . 184
8.23 Magnetic wreaths in case D0.5a . . . . . . . . . . . . . . . . . . . . . . . 186
9.1

Profiles of mean flows and temperature in dynamo case T3 . . . . . . . 193

9.2

Time-dependent behavior in case T3 with a model tachocline . . . . . . 195

xvii
9.3

Profiles of hBφ i and hAφ i in dynamo case T3 . . . . . . . . . . . . . . . 196

9.4

Stellar structure of case T3 . . . . . . . . . . . . . . . . . . . . . . . . . 198

Chapter 1
Convection, Rotation and Magnetism

We live near a magnetic star and our Earth is embedded within its magnetized
wind. The surface of the Sun is covered in magnetism on all scales, with global-scale
structures including prominences that reach high into the solar atmosphere (Fig. 1.1a)
as well as smaller concentrated magnetic structures like sunspots, where the magnetism
becomes strong enough to largely halt the turbulent convection leaving dark regions at
the surface (Fig. 1.1b). On the smallest scales visible at the surface, magnetic fields
are swept aside by vigorous granulation which are convective cells that overturn every
10-15 minutes and have flows that are near the speed of sound. The magnetism collects
in the surrounding network of downflow lanes, where it is swept out by the slower
flows of supergranulation, whose characteristic timescales are closer to a day. There themagnetism traces out the supergranular network, and expands outwards into the solar
chromosphere and corona.
Solar magnetism is far from being static, and it evolves on many timescales, witha prominent and regular eleven-year cycle of sunspots and global-scale polarity reversals.
We stand at the start of a new cycle now (called cycle 24); few sunspots are present
on the solar disk and eruptive events and space storms are rare. Over the next elevenyears we will likely see spots emerging at the surface, initially at higher latitudes (near±30◦ ) and then at lower and lower latitudes until, near the end of the cycle, they appear
almost at the solar equator. Time-latitude maps of spot emergence form characteristic

2

Figure 1.1 — Solar magnetism at many scales. (a) On global-scales, a large prominence
lifting off the limb of the Sun is revealed in this SOHO EIT image. A scaled-image of the
Earth (inside blue circle) is provided for comparison. (b) On finer scales, as seen in thisimage from the SSVT, sunspots at the surface are regions of strong magnetism. Tickmarks indicate 1Mm spacings. Individual sunspots are roughly the size of the Earth.

“butterfly” patterns and these diagrams illustrate the migration of the active latitudes
as the cycle progresses (Fig.1.2a). When sunspots emerge at the solar surface, theygenerally have characteristic polarities and orientations. Spots can be large or small,with most as large as the Earth or larger.
Within the sunspots, the magnetism can evolve on very short timescales. In some
active regions the strong magnetic fields reconnect, leading to explosive releases of energy
in the form of high-energy photon flares and eruptive plasma coronal mass ejections. As
these photons and magnetized plasma storms strike the Earth’s magnetosphere, they can
profoundly affect our modern society, imperiling astronauts and satellites in space andscrambling communications and navigation systems on the ground. The largest solar
storms can threaten the very infrastructure of national power grids and can disrupt ordestroy computational systems.
As the cycle progresses, the sunspots will become more numerous, peaking in
number roughly midway through (Fig. 1.2b). Generally, the number of sunspots present
on the solar disk increases rapidly at the start of a new solar cycle and then declines
more gradually. It is during the later half of the cycle that the largest storms and3

Figure 1.2 — Temporal variation of global-scale solar magnetism.

(a) Butterfly diagram, or time-latitude map, showing emergence of sunspots at the surface. When a
new cycle begins, sunspots emerge first at high latitudes. As time passes, they become
more numerous and appear at lower latitudes. When their number is near maximum,
the global-scale polarities reverse. As their number declines and the sunspots approachthe equator, a cycle begins anew with opposite polarity sunspots. (b) Percentage of thevisible solar disk that is covered by sunspots. These figures adapted from Hathaway(2009).
eruptive events tend to occur, though major flares and eruptions can occur at any time.
Near the middle of the roughly eleven-year solar cycle, when the number of sunspots
is near maximum, the magnetic poles of the Sun flip in polarity. As the number ofsunspots declines and the active latitudes approach the equator, a cycle begins anew
with sunspots of opposite polarity emerging at high latitudes. Though the timing of thesolar cycle is fairly regular, the magnetic activity shows modulation on longer timescales
as well. Some cycles are strong, with many spots, while some cycles are weak. At
times,as during the Maunder Minimum, the surface of the Sun has remained barren ofsunspots for decades.

4

1.1

Operation of the Solar Dynamo
Solar magnetism and the cycles of magnetic activity must arise from organizeddynamo action in the Sun’s interior. This dynamo action is achieved by turbulent
plasma motions in the solar convection zone, which spans the outer 29% of the Sun inradius. Here vigorous convective motions and rotation couple to drive the differential
rotation and meridional circulation. These flows are important ingredients in stellar
dynamo theory, and in many theories the differential rotation plays an important rolein building and organizing the global-scale fields. The meridional circulations may
be important for returning flux to the base of the convection zone and advecting it
equatorward, enabling cycles of magnetic activity.
The manner in which the Sun achieves global-scale dynamo action is gradually
being sorted out. The seat of this dynamo is generally thought to be located in thetachocline, an interface of shear between the differentially rotating convection zone andthe radiative interior which is in solid body rotation (e.g., Parker 1993; Charbonneau& MacGregor 1997; Ossendrijver 2003). Helioseismology, which uses acoustic oscillations to probe the radial structure of the Sun as well as convective flows beneath thesurface, has revealed that the solar differential rotation profile observed at the surface
prints throughout the bulk of the convection zone with two important regions of shear(Fig. 1.3a). The near-surface shear layer occupies the outer 5% of the Sun while thetachocline is at the base of the convection zone and separates the strong differential
rotation of that region from the uniform rotation of the deeper radiative interior (e.g.,Thompson et al. 2003; Thompson 2009).
The stably stratified tachocline may also provide a region for storing and amplifying coherent tubes of magnetic field which may eventually rise to the surface of the
Sun as sunspots. It has generally been believed that magnetic buoyancy instabilities
may prevent fields from being strongly amplified within the bulk of the convection zone

5

Figure 1.3 — Differential rotation and the solar dynamo. (a) Helioseismically determined differential rotation profile for the Sun, showing the best determination to dateof the internal angular velocity Ω profile (Thompson 2009), with fast (red) equator andslow (blue) poles. Two layers of radial shear are prominently visible, one near the surface and one at the interface between the convection zone and the radiative interior.
This deeper layer, the tachocline, may be the seat of the global solar dynamo. (b) Sketchof processes likely at work in the solar dynamo. In interface dynamo models, magnetic
fields in the convection zone are wound up by helical convective flows (1) and pumped
downwards into the tachocline by the compressible convection (2). There the field isorganized and stretched into global-scale structures (3) which may become buoyantly
unstable and rise to emerge at the surface as sunspots (4). As they transit the convection
zone, the magnetic tubes are influenced by rotation and some of the field is shreddedby convection, closing the loop back to (1). Meridional circulations may additionallycontribute to the transport of field down to the tachocline.

itself (Parker 1975). In the now prevalent “interface dynamo” model, solar magnetic
fields are partly generated in the convection zone by helical convection, then transported downward into the tachocline where they are organized and amplified by theshear. Ultimately the fields may become unstable and rise to the surface. This model
is illustrated in Figure 1.3b.
In the interface dynamo model, magnetic fields are amplified by turbulent flows
throughout the convection zone. This “magnetic chaff” is pumped downwards into thetachocline by asymmetries in the compressible convection, which generally has fast andnarrow downflows and slower and broader upflows. In the tachocline, the fluctuating
magnetic fields are stored and gradually stretched out by the differential rotation into
global-scale coherent sheets of magnetism. These magnetic sheets become strong enough6
to undergo magnetic buoyancy instabilities, and this may break up the sheet into risingmagnetic tubes of predominantly toroidal magnetic field. If these tubes survive theirtransit of the convection zone, they emerge at the surface as sunspots. Coriolis forces
cause the rising tubes to twist, which leads to the observed tilt angle of emerging
sunspots and may contribute to the global-scale poloidal field.
Though some tubes of magnetism may survive, many more are likely to be shredded by the intense turbulence in the convection zone. Generally, to emerge at thesurface, the average magnetic energy density of the tube must exceed the kinetic energy
density of the strongest downflows that the tube encounters during its rise (e.g., Cline
2003; Fan et al. 2003; Abbett et al. 2004; Jouve & Brun 2009). This strong criteria may
be difficult to satisfy in the solar tachocline (Vasil & Brummell 2008, 2009). As the magnetic structures are shredded, helical convection twists the toroidal field into poloidal
field. This poloidal field is amplified in the convection zone before being caught in thevigorous downflows and pumped into the tachocline closing the loop on the dynamo
circuit, though some may also be transported by the global-scale meridional circulations. This dynamo model and the cartoon sketch shown in Figure 1.3b summarize thefundamental features found in most modern solar dynamo theories (e.g., Charbonneau
2005; Miesch 2005).

1.2

Theoretical Treatments of Stellar Convection
The elements of the solar dynamo are often interpreted in a mean-field framework,where the complex turbulent correlations in the induction equation are linearized interms of the global-scale magnetic fields. In these approaches, the magnetism andplasma flows of the interior are separated into mean components and fluctuations about
those means. These means are usually taken over large spatial scales and long temporalepochs, and are often assumed to be much larger in amplitude than the fluctuating
magnetic and velocity fields.

7
The language of mean-field theory has become the language of dynamo theory.
In this terminology, the production of mean poloidal field through the winding up of
toroidal field by helical convection is called an α-effect (e.g., Moffatt 1978; Steenbeck
et al. 1966). The stretching of mean poloidal field into mean toroidal field by theshearing flows of differential rotation is called an Ω-effect.
In the solar dynamo, the tachocline of shear at the base of the convection zone isthought to play a key role by providing a region where the Ω-effect can operate without
turbulent convection shredding and disrupting the global-scale toroidal fields. The fields
there are thought to grow in strength until magnetic buoyancy carries them upwards
into the convection zone. There, some of this toroidal field is shredded and turned into
poloidal field by the α-effect from the convection and by Coriolis forces on the rising tube.
Some flux survives to erupt at the solar surface, creating sunspots, active regions and,eventually, explosive solar magnetic activity. Some of the poloidal field generated bythis process is pumped downward by convection into the tachocline where it is amplified
into toroidal field, thus completing the dynamo cycle. Conceptual models similar to thisare called “α − Ω” dynamos, with the α-effect dominating the production of poloidal
field and the Ω-effect largely responsible for the production of the toroidal fields. Thesemodels and variants incorporating the global-scale flow of meridional circulations haveshaped our current views on solar and stellar magnetism. Other variants including α2and α2 Ω dynamos are also proposed for the Sun and particularly for other stars (e.g.,Küker & Rüdiger 1999, 2005b; Chabrier & Küker 2006).
Though instructive, these theories fall short in describing the fully nonlinear dynamo processes occurring in real stars. A fundamental assumption of mean-field theories
is that the large-scale fields can be separated from the small-scale fluctuations, and inthese theories the fluctuations are often assumed to scale in strength with the mean
fields. In the turbulent environment of stellar convection, fields of all scales interact
and are of similar magnitude, with strong local fluctuations that are not dependent on

8
the weaker large-scale fields. Major 3-D simulations (e.g., Brun et al. 2004; Browning
et al. 2006) and observations suggest that, at least in the bulk of the convection zone,
magnetism occurs over a broad range of scales with little separation between “mean”
scales and fluctuations, and that the mean fields are not the dominant players in theoverall magnetism. The impact of this lack of scale separation in both the magnetic
and velocity fields is unclear and a subject of active research in both the dynamo andturbulence communities. As an additional concern, mean-field theory is fundamentallya linear theory, and while it can describe the initial growth of magnetic fields it cannot
address their ultimate saturation as the fields strengthen and react back on the flowsthat create them. Some attempts have been made to include such nonlinear effects,
with “α-quenching” and “Ω-quenching” terms added to the mean-field equations, butsuch treatments are fairly ad-hoc and the subject of substantial debate.

1.3

Exploring Magnetism in Other Stars
Our Sun is not the only magnetic star. Indeed, magnetism appears to be aubiquitous feature in stars across the Hertzsprung-Russell (H-R) diagram. When our
Sun was younger, it must have rotated much more rapidly, as is suggested both by thesolar wind which continually removes angular momentum from the Sun and by manyobservations of rapidly rotating solar-like stars. Some of these young suns are observed
to rotate as much as 50 times faster than the current solar rate. In more rapidly rotatingsuns the coupling between rotation and convection is strong and must continue to drive
global scales of flow. These young solar-type stars, which rotate much more rapidly
than the sun’s current rate, possess much stronger magnetic activity.
Rotation appears to be inextricably linked to stellar magnetic activity. Observations indicate that in stars with extensive convective envelopes, chromospheric andcoronal activity – which partly trace magnetic heating of stellar atmospheres – firstrise with increasing rotation rate, then eventually level off at a constant value for ro-

9

Figure 1.4 — Observational rotation-activity correlations in main-sequence stars. (a) Xray activity measured by its luminosity Lx as a function of rotation period, showing
growth and saturation phases of the relationship across broad populations of stars.
(b) Different stellar types (approximately indicated) saturate at various levels of activity
Lx and reach saturation at rotation rates dependent on spectral type. The growth ofactivity with more rapid rotation is similar in all G-, K- and M-type stars (adaptedfrom Pizzolato et al. 2003). (c) Despite their lack of tachoclines, most fully-convective
stars (types M3.5-M9) show signs of magnetic activity, here measured by Hα emission.
Indeed, the fraction of active stars increases with decreasing mass (adapted from Westet al. 2008).

tation rates above a mass-dependent threshold velocity (e.g., Noyes et al. 1984; Patten& Simon 1996; Delfosse et al. 1998; Pizzolato et al. 2003). Activity may even decline
somewhat in the most rapid rotators (e.g., James et al. 2000). This rotation-activity
correlation is shown in Figure 1.4a, b for a variety of solar-like stars. Similar correspondence is observed between rotation rate and estimates of the unsigned surface magnetic
flux (Saar 1996, 2001; Reiners et al. 2009).
Global-scale magnetic activity has also been observed in the lower-mass mainsequence (dwarf) K- and M-type stars. Their convection zones occupy an increasingly
large fraction of the interior and their tachoclines must play a diminishing role in theoverall dynamics, being located in ever deeper regions of the star. Even fully convective
stars, such as M-dwarfs with masses below 0.35 solar masses, show strong surface magnetism through observations of Hα emission (e.g., Hawley et al. 1996; Mohanty & Basri2003; West et al. 2004, 2006, 2008) and magnetically sensitive FeH line ratios (Reiners &Basri 2007, 2008). Though these stars possess no tachoclines and should have distinctly
different stellar dynamos than the sun, there is no observed break in magnetic activ-

10
ity with stellar type. Indeed, the fraction of magnetically active stars increases as thestellar mass decreases (Fig. 1.4c). These stars also show rotation-activity correlations.
Recent observations of magnetic fields in a fully convective M-dwarf by Donati
et al. (2006) are raising further questions for stellar dynamo theory. There, Zeeman
Doppler imaging is used to map the surface of the rapidly rotating and fully convective
star v374 Peg. These observations indicate strong, organized axisymmetric fields thathave strengths of a few kilo-Gauss but no surface differential rotation. The globalscale fields appear to be stable on one-year timescales (Morin et al. 2008). The stableaxisymmetric nature of these fields is in striking contrast to theoretical predictions forfully convective stars (e.g., Küker & Rüdiger 1999, 2005b; Chabrier & Küker 2006).
Observations of stellar magnetism are indirectly raising serious questions about thecurrent paradigm of solar dynamo theory, where the tachocline is thought to play a
crucial role in amplifying and organizing the global-scale fields and the radial velocity
shear of differential rotation is a necessary ingredient for dynamo action.
The rotation-activity relationship is tightened across the full range of solar-likestars when stellar rotation is given in terms of the Rossby number Ro ∼ P/τc , with P therotation period and τc an estimate of the convective overturning time (e.g., Noyes et al.
1984). Expressed in this fashion, a common rotation-activity correlation appears tospan spectral types ranging from late F to late M (e.g., Patten & Simon 1996; Mohanty& Basri 2003; Pizzolato et al. 2003; Reiners & Basri 2007). These stellar magnetic fields
must be generated by dynamo action in the stellar convection zones. At present, therotational dependence of this dynamo action is unknown. In mean-field theory, bothgeneration terms are sensitive to rotation – the α-effect because it is proportional to
the kinetic helicity of the convective flows, which sense the overall rotation rate, andthe Ω-effect because more rapidly rotating stars are generally expected to have stronger
differential rotation. But the detailed nature of these effects in the solar dynamo andthe appropriate scaling with rotation has been very difficult to elucidate.

11
Rotation and stellar magnetism are inextricably linked, as the stronger magnetic
fields in rapidly rotating stars are thought to produce a larger outward transport ofangular momentum by the magnetized stellar winds, which slowly spin down the stars.
(e.g., Weber & Davis 1967; Skumanich 1972; MacGregor & Brenner 1991; Matt & Pudritz 2008). The time needed for significant spindown appears to be a strong function ofstellar mass (e.g., Barnes 2003; West et al. 2004). Solar-mass stars slow less rapidly thansomewhat less massive G and K-type stars, but still appear to lose much of their angularmomentum by the age of the Hyades (about 1 Gyr). They spin more slowly yet when
they are as old as the Sun. Present day observations of the solar wind likewise indicatethat the current angular momentum flux from the Sun is a few times 1030 dyne cm (e.g.,
Pizzo et al. 1983), suggesting a time-scale for substantial angular momentum loss of a
few billion years. Analyses of stellar spindown as a function of age and mass have thus
provided further constraints on stellar magnetism and its connections to rotation.
In addition, recent observations of solar-type stars suggest that the topology of the
global-scale fields changes with rotation rate, with the rapid rotators having substantial
global-scale toroidal magnetic fields at their surfaces (Petit et al. 2008). The overall
picture that emerges from these observations is that rapid rotation, as realized in theyounger Sun and in a host of other stars, can aid in the generation of strong magnetic
fields, and that young stars tend to be rapidly rotating and magnetically active, whereas
older ones are slower and less active (e.g., Barnes 2003; West et al. 2004, 2008).
A full theoretical understanding of the rotation-activity relationship, and likewiseof stellar spindown, has remained elusive. Some aspects of these phenomena probably
depend upon the details of magnetic flux emergence, chromospheric and coronal heating,and mass loss mechanisms – but the basic existence of a rotation-activity relationship is
widely thought to reflect some underlying rotational dependence of the dynamo processitself (e.g., Knobloch et al. 1981; Noyes et al. 1984; Baliunas et al. 1996).

12
Research into the stellar magnetism of solar-like stars is timely. These lowermass stars are the primary targets of the NASA Kepler mission and the ESA CoRoT
satellite, which both seek to find the signatures of Earth-like planets orbiting otherstars and to perform asteroseismology on some of these stars. These solar-like stars
have large habitable zones and are the closest analogues to our own Sun and solar
system. In the search for their possible life-bearing worlds, we will learn much more
about their host stars. Kepler will observe about 105 stars over the course of the mission,
searching for other Earths by the transit method. While searching for the slight dipsin stellar light from these eclipses, the satellite will make detailed measurements ofsurface magnetic activity and differential rotation through high-precision photometry.
For several hundred low-mass stars, Kepler will also carry out asteroseismology, which
will yield new insights into the masses, sizes and ages of these stars, possibly along withmeasurements of their convection zone depths and internal rotation. Such observations
are likely to lead to a renaissance in stellar physics, challenging our views on stellar
structure, aging processes and stellar magnetism. A much better understanding ofstellar magnetism and activity may well shed new light on the solar dynamo and may
be crucial for reliably detecting distant Earths.
Probing the nature of these dynamos and the impact of faster rotation on theinternal stellar dynamics requires both accurate observations and detailed dynamical
models of the stellar interiors. The faster flows of differential rotation are much easier
to detect than the relatively slow motions associated with meridional circulations; observations across the HR diagram indicate that differential rotation is a common feature
in many stars. Asteroseismic observations with the Kepler and CoRoT missions may
soon begin to constrain the internal rotation structure. At present only measurementsof surface differential rotation are available, as assessed with a variety of techniques
including photometric variability (Donahue et al. 1996; Walker et al. 2007), Doppler
imaging (Donati et al. 2003) and Fourier transform methods (Reiners & Schmitt 2003).

13
At our current state of knowledge, stellar magnetism raises three fundamental
questions, which missions like Kepler and CoRoT will begin to address in great observational detail(1) Where are global-scale magnetic fields built and organized within the interiorsof stars like our sun?
(2) Why is there a correlation between rotation rate and magnetic activity, andwhy is this behavior similar in stars with very different convection zone depths?
(3) What role do tachoclines play in stellar dynamos?
To probe these questions more deeply, we can now turn to simulations of magnetohydrodynamic (MHD) convection and dynamo action. These large computations must be
carried out on supercomputers, and these theoretical tools are helping to sort out the
mysteries of stellar magnetism.

1.4

Global Models of Stellar Dynamos
Advances in supercomputing have enabled three-dimensional (3-D) simulations

that are beginning to capture many of the dynamical elements of the solar convection
zone. Early global-scale simulations of solar convection by Gilman (1975, 1977, 1979)
under the Boussinesq approximation were extended by the pioneering work of Gilman& Glatzmaier (1981). Such global-scale simulations of solar convection conducted infull spherical shells sought to capture the largest scales of convective flows and began to
study how they can establish differential rotation and meridional circulations. However,
the range of spatial and temporal scales present in solar convection are vast and thus
the computational resources required by the modeling are daunting.
Through recent advances in massively parallel computer architectures, 3-D solar
convection simulations are now beginning to make detailed contact with the observational constraints provided by helioseismology (e.g., Brun & Toomre 2002; Miesch et al.

14

Figure 1.5 — Snapshot of 3-D ASH solar dynamo simulation which includes a tachocline.
This layer of shear and penetration is located at the base of the convection zone. (a) Radial velocity in orthographic projection at 0.88 R⊙ , showing the north pole and equatorial regions with upflows in light tones and downflows in dark tones (scale in m/s).
(b) Its mean profile of angular velocity Ω(r, θ), with a solar-like differential rotation
profile in the bulk of the convection zone and near uniform rotation in the deeper interior. (c) Radial cuts of Ω at selected latitudes. These simulations are beginning tocapture in a self-consistent fashion the key ingredients of latitudinal and radial shear.
(d) Profile of temporally- and azimuthally-averaged longitudinal magnetic field hBφ i,with substantial mean field present in the tachocline and little in the convection zone
above. (e) Accompanying mean poloidal field lines, with polarity indicated by color.
These snapshots are from a model based on the simulation of Browning et al. (2006).

2006, 2008). Other efforts have focused on the vigorous turbulence and the dynamo
action achieved in the bulk of the solar convection zone (Brun et al. 2004), with recent
studies beginning to include the tachocline as a region of penetrative overshoot, shear,and magnetic field amplification (Browning et al. 2006). Facilitated by these computational advances, models of convection and dynamo action within the cores of A-typestars have also begun to be investigated (Browning et al. 2004; Brun et al. 2005; Featherstone et al. 2007, 2009), as have models of fully-convective M-dwarf stars (Browning2008) and red giant branch stars (Palacios & Brun 2007; Brun & Palacios 2009).
To date, most models of stellar differential rotation and dynamo action in stars
like our Sun that rotate more rapidly have been carried out in 2-D under the simplifying
assumptions of mean-field theory (e.g., Rüdiger et al. 1998; Küker & Stix 2001; Küker& Rüdiger 2005a,b). The time is ripe to pursue the question with fully 3-D simulationsof global-scale stellar convection.

15
Simulations of the global-scale solar dynamo have generally affirmed the viewthat the tachocline may play a central role in building the globally-ordered magnetismin the Sun. Early 3-D simulations of solar convection without a tachocline at the baseof the convection zone achieved dynamo action and produced magnetic fields whichwere strongly dominated by fluctuating components with little global-scale order (Brunet al. 2004). When a tachocline of penetration and shear was included, remarkableglobal-scale structures were realized in the tachocline region, while the convection zone
remained dominated by fluctuating fields (Browning et al. 2006). These results areillustrated in Figure 1.5.
These simulations are making good progress toward clarifying the elements atwork in the operation of the solar global-scale dynamo, but for other stars many questions remain. In particular, observations of large-scale magnetism in fully convective
M-stars (Donati et al. 2006), along with the persistence of a rotation-activity correlation in such low-mass stars, hint that perhaps tachoclines may not be essential for thegeneration of global-scale magnetic fields. This view is partly borne out by 3-D simulations of M-dwarfs under strong rotational constraints (Browning 2008), where strong
longitudinal mean fields were realized despite the lack of either substantial differential
rotation or a stable interior and thus no classical tachocline. Major puzzles remain in
the quest to understand stellar magnetism and its scaling with stellar rotation.
With growing computational resources available, the time has come to extend
these solar simulations to broader classes of stars. Such studies will enhance our understanding of stellar dynamo action, by making detailed contact with the evolving
observations of stellar magnetism in stars like our sun. Studying dynamo action inother stars is likely to shine new light on the uncertain physical processes at workwithin the solar dynamo.

16

1.5

Convection and Dynamo Action in Rapidly Rotating Suns
Encouraged by the success of the solar simulations, we have begun exploring

convection and dynamo action in younger and more rapidly rotating suns in this thesis.
We have found that younger suns likely possess a much stronger differential rotationand that the flows of meridional circulation become weaker with more rapid rotation.
At high rotation rates, the convection becomes strongly modulated in strength withlongitude. Striking localized patterns of convection emerge at the equator, and these
active nests of convection dominate the transport of heat and angular momentum inthat region. At the highest rotation rates, the convection can be entirely confined tonarrow intervals (or active nests) in longitude.
Modulated convection has persisted under more turbulent conditions and theactive nests of convection appear in some dynamo simulations as well. When present,
these nests of localized convection persist for long intervals of time and despite theirsmall filling factor maintain a strong differential rotation. The emergence of spatiallylocalized convective states has been observed in other systems, particularly in theoreticalstudies of doubly-diffusive systems such as thermosolutal convection (e.g., Spina et al.
1998; Batiste et al. 2006), in laboratory studies of convection in binary fluids (e.g.,
Surko et al. 1991), and in simulations of magnetoconvection where isolated “convectons”
have been observed (Blanchflower 1999). In shells of rapidly rotating fluid, temporallyintermittent patches of localized convection emerged in Boussinesq simulations of thegeodynamo (Grote & Busse 2000) and in anelastic simulations of convection in young
suns with much deeper convection zones (Ballot et al. 2006, 2007). In many of thesesystems, spatial modulation occurs in the weakly nonlinear regime close to the onset of
convection. In contrast, our simulations of stellar convection in younger suns are in a
regime of fully developed turbulent convection.

17
We have begun preliminary simulations of the dynamo action possible in thesestars and have found several surprises. These MHD simulations span the convection zone
alone, as the nature of tachoclines in more rapidly rotating suns is at present unclear.
We find that a variety of dynamos can be excited, including steady and oscillating states,
and that dynamo action is substantially easier to achieve at these faster rotation rates
than in the solar simulations. Some of the oscillating simulations undergo quasi-regularglobal-scale polarity reversals, with the mean toroidal and poloidal fields exchanging
polarities.
In these rapidly rotating solar-type stars, substantial global-scale organizationof magnetic fields can occur in the middle of the convection zone. These wreaths ofmagnetism fill the convection zone and appear to be a general feature of our dynamos,
appearing now in our solar dynamos as well. Generally, we find that the dynamos
operating in the rapidly rotating suns may not require the presence of a tachocline,
being able to instead organize global-scale fields in the bulk of their convection zones.
We describe briefly in Chapter 2 the 3-D magnetohydrodynamic anelastic spherical harmonic simulation code called ASH and the parameter space explored by our
simulations that are carried out in spherical shells. In Chapter 3, we discuss the natureof convection realized in more rapidly rotating stars and the emergence of spatiallylocalized patterns of convection. Here we also examine the global-scale flows realizedin our simulations, including differential rotation and meridional circulation, and theirscaling with more rapid rotation. A more detailed exploration of the active nests of
convection is presented in Chapter 4.
We then turn to dynamo simulations, examining in Chapter 5 the persistent
wreaths of magnetism achieved in a rapidly rotating dynamo rotating at three times
the current solar rate. In Chapter 6 we examine time-dependent behavior and organizedglobal-scale polarity reversals in a dynamo rotating five times faster than the Sun. We
return in Chapter 7 to the three solar dynamo with persistent magnetic wreaths and18examine how those global-scale structures are created and maintained. In Chapter 8 we
explore the broad parameter space sampled by rapidly rotating dynamos spinning at up
to fifteen times the solar rate. We find that magnetic wreaths are a nearly ubiquitousfeature in all of these simulations. Here we also return to the Sun itself and find wreathsof magnetism as well. We further briefly consider convective and dynamo processes atwork in older, more slowly spinning suns. In Chapter 9 we show preliminary results forwreath-building dynamos that include tachoclines of shear and penetration. We findthat magnetic wreaths continue to fill the convection zone and undergo cycles of polarity
reversal. Here we summarize our explorations of rapidly rotating dynamos and look toprojects of the future.

Chapter 2
Elements in Modeling 3-D Stellar Convection and Dynamo Action

To study the coupling between rotation, magnetism and the large-scale flows
achieved in stellar convection zones, we must employ a global model which simultaneously captures the spherical shell geometry and admits the possibility of zonal jets andlarge eddy vortices, and of convective plumes that may span the depth of the convection
zone. The solar convection zone is intensely turbulent and microscopic values of viscosity and magnetic and thermal diffusivities in the Sun are estimated to be very small.
Numerical simulations cannot hope to resolve all scales of motion present in real stellar
convection and must instead strike a compromise between resolving dynamics on small
scales and capturing the connectivity and geometry of the global scales. Here we focus
on the latter by studying a full spherical shell of convection.

2.1

Anelastic MHD Formulation
Our tool for exploring MHD stellar convection is the anelastic spherical harmonic(ASH) code, which is described in detail in Clune et al. (1999). The implementationof magnetism is discussed in Brun et al. (2004). ASH solves the 3-D MHD anelastic
equations of motion in a rotating spherical shell using the pseudo-spectral method andruns efficiently on massively parallel architectures. We use the anelastic approximation
to capture the effects of density stratification without having to resolve sound waves
which have short periods (about 5 minutes) relative to the dynamical time scales of the20
global scale convection (weeks to months) or possible cycles of stellar activity (years
to decades). This criteria effectively filters out the fast magneto-acoustic modes while
retaining the slow modes and Alfvén waves. Under the anelastic approximation thethermodynamic fluctuating variables are linearized about their spherically symmetricand evolving mean state, with radially varying density ρ̄, pressure P̄ , temperature T̄
and specific entropy S̄. The fluctuations about this mean state are denoted as ρ, P ,T and S. In the reference frame of the star, rotating at average rotation rate Ω0 , theresulting MHD equations are:

∇ · (ρ̄v) = 0 ,(2.1)∇ · B = 0,

(2.2)


∂v
ρ̄
+ (v · ∇)v + 2Ω0 × v =∂t

1−∇(P̄ + P ) + (ρ̄ + ρ)g +(∇ × B) × B − ∇ · D,
4π
∂B
= ∇ × (v × B) − ∇ × (η∇ × B),∂t∂S+ v · ∇(S̄ + S) =
ρ̄T̄
∂t


∇ · κr ρ̄cp ∇(T̄ + T ) + κ0 ρ̄T̄ ∇S̄ + κρ̄T̄ ∇S

(2.3)

(2.4)



(2.5)



1
4πη 2
2
+ 2 j + 2ρ̄ν eij eij − (∇ · v) ,c
3

where v = (vr , vθ , vφ ) is the local velocity in the stellar reference frame, B = (Br , Bθ , Bφ )is the magnetic field, j is the vector current density, g is the gravitational acceleration,
cp is the specific heat at constant pressure, κr is the radiative diffusivity and D is theviscous stress tensor, given by1
Dij = −2ρ̄ν eij − (∇ · v)δij ,
3


(2.6)

21
where eij is the strain rate tensor. Here ν, κ and η are the diffusivities for vorticity,
entropy and magnetic field. We assume an ideal gas law
P̄ = Rρ̄T̄ ,(2.7)

where R is the gas constant, and close this set of equations using the linearized relations
for the thermodynamic fluctuations of
ρ
P
S
T
P
=
− .
− =
ρ̄
cp
γ P̄
P̄
T̄

(2.8)

The mean state thermodynamic variables that vary with radius are evolved with thefluctuations, thus allowing the convection to modify the entropy gradients which driveit.
The mass flux and the magnetic field are represented with a toroidal-poloidal
decomposition asρ̄v = ∇ × ∇ × (W r̂) + ∇ × (Z r̂),

(2.9)

B=

(2.10)∇ × ∇ × (βr̂) + ∇ × (ζ r̂),

with streamfunctions W and Z and magnetic potentials β and ζ. This approach ensuresthat both quantities remain divergence-free to machine precision throughout the simulation. The velocity, magnetic and thermodynamic variables are all expanded in sphericalharmonics for their horizontal structure and in Chebyshev polynomials for their radial
structure. The solution is time evolved with a second-order Adams-Bashforth/CrankNicolson technique.
ASH is a large-eddy simulation (LES) code, with subgrid-scale (SGS) treatments
for scales of motion which fall below the spatial resolution in our simulations. We treat
these scales with effective eddy diffusivities, ν, κ and η, which represent the transportof momentum, entropy and magnetic field by unresolved motions in the simulations. In
these simulations ν, κ and η are taken for simplicity as functions of radius alone and are22
proportional to ρ̄−1/2 . This adopted SGS variation, as in Brun et al. (2004), Browning
et al. (2006) and Ballot et al. (2007) yields lower diffusivities near the bottom of thelayer and thus higher Reynolds numbers. In their stellar structure, our simulations here
are similar to case AB as reported in Brun & Toomre (2002) though with a different SGS
functional form (there ν, κ ∝ ρ̄−1 ), and here we shall consider the effects of faster Ω0(there Ω0 = Ω⊙ ). Acting on the mean entropy gradient is the eddy thermal diffusion κ0
which is treated separately and occupies a narrow region in the upper convection zone.
Its purpose is to transport entropy through the outer surface where radial convective
motions vanish.
Our simulations are still separated by many orders of magnitude from the intensely turbulent conditions present within the solar convection zone. They are likely
to capture many aspects of the dynamics of solar convection, and we are encouragedby the success that similar simulations (e.g., Miesch et al. 2000; Brun & Toomre 2002;
Miesch et al. 2006, 2008) have had in beginning to match the detailed observational
constraints for differential rotation within the solar convection zone provided by helioseismology (c.f. Thompson et al. 2003).

2.2

Boundary Conditions and Their Impacts
In this thesis we will explore how patterns of convection and dynamo-generated

magnetism change in more rapidly rotating suns. In models of the solar dynamo, themagnetic fields observed at the surface are thought to be in the convection zone, withthe tachocline of penetration and shear at the base of the convection zone possibly
allowing field to be stored and organized on global-scales. The tachocline is a complex
internal boundary layer. Below the tachocline lies the radiative zone, a region of stablestratification. Above it is the intensely turbulent and highly magnetized convection zone.
Plunging downflows originating there splash into the tachocline, carrying magnetic field
downwards and likely driving gravity waves and large-scale circulations. In the Sun, the23
origin and maintenance of the tachocline remains unclear, but it seems likely that slow
meridional circulations in that layer are crucial to its long term evolution. These flows
likely have turnover times of millions of years while the fast downflows in the convection
zone evolve on timescales of a few days.
In nearly all of the simulations contained in this thesis, we will explore dynamics
within the convection zone only. Simulations which include model tachoclines are now
being conducted in ASH, with the first simulations of the solar dynamo coupled to atachocline reported on in Browning et al. (2006). Explorations that include a tachocline
will be extended to the rapidly rotating suns in the future, and preliminary results forsuch a system will be shown in Chapter 9. However, the majority of our simulations
here consist of spherical shells of convectively unstable fluid which capture the bulk of
the solar convection zone in radius. We adopt boundary conditions appropriate to thisregion, and for now we neglect the region of penetration, shear and stable stratification
at the base of the convection zone.
One area of focus in this thesis is the differential rotation which is naturally established within a shell of fluid that experiences no external torques. In these systems,
convection and magnetism built by dynamo action act to redistribute angular momentum within the convection zone and establish gradients of angular velocity in both radiusand latitude. Our velocity boundary conditions that are implemented in ASH are fairly
straight forward and are chosen to eliminate external torques. The velocity boundary
conditions imposed at the top and bottom of the convectively unstable shell are:(1) Impenetrable top and bottom: vr = 0 ,(2) Stress-free top and bottom:(∂/∂r)(vθ /r) = (∂/∂r)(vφ /r) = 0 ,(2.11)In recent solar simulations (Miesch et al. 2006, 2008) a latitudinal gradient ofentropy has been imposed to mimic the balance likely achieved within the sub-adiabatic

24
tachocline. This modified boundary condition (with S|r=rbot = F (θ) constant in time)slightly modifies the thermal wind balance achieved throughout the convection zone andcan modify the profiles of differential rotation. The overall angular velocity contrast inlatitude remains similar. Ballot et al. (2007) explored the consequences of such a boundary condition in one of their young, rapidly rotating suns with deep convection zones
and found that the results were similar to those of Miesch et al. (2006). In these simulations of rapidly rotating suns however, we do not employ such a treatment. Instead, alatitudinal contrast in entropy is naturally established throughout the convection zone
and at the bottom boundary by the convection itself. We make this choice becauseof our uncertainties about the structure of stellar tachoclines. In the case of the Sun,bounds can be placed on this thermal gradient from helioseismic observations of thetachocline.
In other stars we have no such asteroseismic observations of their tachoclines. In
the more rapidly rotating suns, we thus do not yet know how the tachoclines scale withrotation rate, magnetic activity or stellar age. As such, the possible thermal structureof those tachoclines is poorly constrained. With these observational uncertainties inmind we choose constant flux thermal boundary conditions and allow the convection
to establish its own gradients of entropy in latitude within the convection zone. The
thermal boundary conditions at the top and bottom of the shell are thus
(3) Constant entropy gradient at top and bottom:∂(S + S̄)/∂r = const .

2.3(2.12)

Magnetic Boundary Conditions
The most natural choice for the magnetic boundary conditions is somewhat lessclear, and thus deserves some detailed discussion. Typically in ASH we employ one ofthree treatments for magnetism at the boundaries. Those conditions are:

25(4) Match to external potential field at top:B = ∇Φ

and ∇2 Φ = 0 r=rtop ,(5) Perfect conductor at bottom:
Eθ = Eφ = 0
Br = (∂/∂r)(rBθ ) = (∂/∂r)(rBφ ) = 0 ,(2.13)(6) Radial field only at the boundary:
B = Br .
In our later discussions, it will be helpful to understand the impact of these different boundary conditions on the transport of angular momentum and energy throughthe boundary. We examine those properties for each boundary condition in turn.

2.3.1

Potential Field Boundaries

A potential field boundary condition is most appropriate when the region outside
the boundary mimics a vacuum or other extremely good insulator. Under the potential
field approach, no currents can cross the boundary, nor can currents be supported in theexternal volume. Magnetic fields can however extend out of the simulation, in a fashion
the preserves ∇ · B = 0. This boundary condition is most appropriate near the surfaceof the star, though it entirely neglects the dynamics present in the photosphere, as wellas the complex balances achieved in the chromosphere of the star and the likely forcefree corona. The dynamo simulations in this thesis use this as their upper boundary
condition.
Our boundaries are impenetrable to the fluid motions, and thus the transport ofenergy by magnetism is from the Poynting flux. This flux is
FPoynting = S =cc (−v × B + η∇ × B) × B ,(E × B) =4π
4π(2.14)

26
where we have used Ohm’s law to replace the electric field. We are most interested in
the flux of energy entering or leaving the volume, which is given by the radial flux atthe boundary. This is


 

c
2
Sr = −Br (B · v) + vr B + η [∇ × B] × B,
4πr(2.15)

where we use a vector identity to expand the first term in the electromotive force. Our
impenetrable boundary conditions ensure that vr = 0. Using this, and expanding the
diffusion term we obtainc
Br (B · v) + η [∇ × B]θ Bφ − [∇ × B]φ Bθ
.
Sr = −
4π(2.16)

With a potential field boundary condition, B = ∇Φ and ∇ × B = 0, and thus
the diffusive terms vanish. There is however radial field that can cross the boundary,and thus an overall energy flux of
Sr = −c
Br (B · v) .
4π

(2.17)In these simulations, both the mean and fluctuating magnetic fields contribute to a
Poynting flux through the boundary. The radial fields enter only through the Br factor,
as vr = 0 at the upper surface. With stress-free boundary conditions, vθ and vφ are
non-zero. The horizontal magnetic fields Bθ and Bφ can also be non-zero, though theazimuthally averaged longitudinal field is zero. This can be seen fromhBφ i =∂1hΦi = 0 ,r sin θ ∂φ

(2.18)

where angle brackets denote an average in longitude. This quantity is zero because
there are no longitudinal gradients in the azimuthally averaged quantities by definitionof the average.
There is clearly a fluctuating Poynting flux through a potential field boundary.
Indeed, the horizontal fields can contribute to a mean Poynting flux through their correlation with the horizontal flows, and the colatitudinal field Bθ can contribute on both

27
mean and fluctuating scales, while the longitudinal field Bφ contributes only throughcorrelations between the fluctuating fields and fluctuating velocities. This mean flux isgiven by




c
′ ′
′ ′
′ ′ ′
′ ′ ′
hSr i = −hBr i hBθ ihvθ i + hBθ vθ i + hBφ vφ i + hBr Bθ vθ i + hBr Bφ vφ i ,4π(2.19)

with fluctuating fields B ′ = B − hBi and flows v ′ = v − hvi, with again hB ′ i = hv ′ i = 0
by definition. Generally, the total Poynting flux across a shell is very small, as local
regions of inward and outward directed flux tend to cancel one another when an average
is taken across both the northern and southern hemispheres.
The transport of angular momentum in the hydrodynamic simulations will be
discussed later in detail in §3.4. In the dynamo simulations, the magnetic contribution
to the radial angular momentum flux through a boundary is given by
FrM S

= −

r sin θ ′ ′hBr Bφ i ,4π(2.20)FrM T= −r sin θhBr ihBφ i ,4π(2.21)

with F M S the angular momentum transport from fluctuating Maxwell stresses and F M T
the transport by large-scale magnetic torques (Brun et al. 2004, 2005). At a potential
field boundary, hBφ i = 0, thus F M T vanishes. In principle there could be a remaining
flux from the fluctuating terms; in practice there is not. Locally there is a flux, butwhen integrated over a full shell FrM S is near zero. This is likely related to the natureof the external potential field. With no currents and no external forces, the external
region is entirely connected to the simulation and there is no ability to remove angularmomentum entirely from the coupled system.

28
2.3.2

Perfect Conductor Boundaries

A perfect conductor boundary condition is more appropriate when the external
volume is filled with a plasma that is very highly conductive. This boundary condition
prevents magnetic field from crossing the boundary, though currents can flow through.
This boundary condition seems most appropriate at the bottom of the convection zone,
and it appears that such a boundary condition is more in keeping with the internalboundary layer that forms when we capture a mock tachocline within our simulations(see §9.2). The dynamo simulations generally have perfectly conducting bottom boundaries.
Perfect conductor boundaries have no radial Poynting flux. To show this, werecall that the horizontal electric fields Eθ and Eφ vanish at the boundary, as does theradial magnetic field. Thus the radial Poynting flux is
Sr =


c 
Eθ Bφ − Eφ Bθ = 0 .
4π(2.22)

We can see how this feeds into our magnetic boundary conditions by examining equation (2.16). At a perfect conducting boundary, that equation becomes
Sr




 
c
∂
1 ∂
c η
∂
∂
= − Br (B · v) +Br −(rBφ ) Bφ −(rBθ ) −
Br Bθ
4π
4π r
sin θ ∂φ
∂r
∂r
∂θ


 

∂
∂
c η
(rBφ ) Bφ −(rBθ ) − 0 Bθ = 0 ,(2.23)
0−= − 0+
4π r∂r∂r

and thus our requirement that ∂/∂r(rBθ ) = ∂/∂r(rBφ ) = 0 at a perfect conductor.
Perfect conducting boundaries also do not support a radial flux of angular momentum. This is guaranteed by the radial field vanishing at the boundary, which causes
both FrM S and FrM T to vanish as well.

29
2.3.3

Radial Field Boundaries

The last boundary condition, that of a radial field only, is not used in these dynamo simulations, but has often been important in magnetoconvection studies where an
external field is imposed. This boundary condition is numerically simple to implement
and corresponds to a system where the external volume is a region of high magnetic
permeability (e.g., Jackson 1998, pg. 194).
Radial field boundaries have no Poynting flux. The impenetrable condition for
velocities (vr = 0) eliminates the contribution from Br vr , and the lack of horizontal
magnetic fields eliminates the radial component of (∇ × B) × B. Likewise, there is noangular momentum flux across this type of boundary. This results again from hBφ i = 0
and Bφ′ = 0 at the boundary.
2.3.4

Magnetic Boundary Conditions and Effects on CFL Limits

The properties of our three possible boundary conditions are summarized in Table 2.1. To summarize, the perfectly conducting and radial field only boundary conditions do not permit a flux of energy or angular momentum across the boundary and thus
in or out of the system. Potential field boundary conditions do permit a flux of energy
and angular momentum, though the azimuthally-averaged magnetic fields do not contribute to the transport of angular momentum through the boundary. Generally, we findthat the total flux of energy or angular momentum through a potential field boundary issmall, as the contributions from the northern and southern hemispheres largely cancel
each other. Horizontal magnetic fields are permitted near potential field boundaries andperfectly conducting boundaries, but are not permitted at a radial field only boundary.
Likewise, radial magnetic fields can thread through the potential field boundaries andradial field only boundaries, but are not present at a perfectly conducting boundary.

30
Table 2.1.

Properties of Magnetic Boundary Conditions
EnergyAngular Momentum

Radial Alfvén CFL limited

Sr

FrM S

FrM T

va = √B4πr ρ̄c− 4πBr (B · v)

θ− r sinhBr′ Bφ′ i
4π

0

yes

Perfect Conductor

0

0

0

no

Radial Field

0

0

0

yes

Type
Potential Field

Note. — Boundary conditions affect the flux of energy and angular momentum in and out of
the system. With some boundary choices the simulation may be CFL limited by Alfvén waves
near the boundary.

The choice of magnetic boundary conditions can also have direct effect on the
typical timesteps achieved by a simulation. Our simulations will become numericallyunstable and diverge if we exceed the Courant-Friedrichs-Lewy (CFL) criteria by taking
timesteps which are too large. Thus our our timesteps must be shorter than the thefastest flow of information between individual grid points. In dynamo simulations,this becomes an issue with boundary conditions that permit a radial magnetic field.
Here, Alfvén waves traveling radially encounter the fine grid-spacing inherent to our
Chebyshev expansion. The CFL timestep limit in the dynamo modeling is dominatedby these relatively rapid waves and becomesτ = α0 τCFL,B = α0

v

A,r

∆r



= α0



1
B
√ r
4π ρ̄ ∆r,

(2.24)

with α0 a safety factor which is slightly less than unity.
The limitation arises because we are able in these dynamo simulations to admit
waves which propagate through the narrow grid near the boundaries. In contrast,
our hydrodynamic simulations of stellar convection zones do not encounter a similar
limitation, as vr smoothly tends to zero as the boundary is approached, and typicallydecreases sufficiently quickly that the narrow grid spacing is not felt. Hydrodynamic
simulations that include penetration into a stable layer may of course encounter similar

31
limitations when stacked Chebyshev domains are used to resolve the internal interface
and where the radial motions are non-zero. In our dynamo simulations, the Alfvén wave
CFL timestep, set by waves at the upper potential field boundary, can be one or twoorders of magnitude more stringent than the limits set by either the convection or theAlfvén waves propagating in the rest of the domain. In a practical sense, this stringent
CFL limit increases the computational cost of dynamo simulations with radial field or
potential field boundaries by well over an order of magnitude compared to comparablehydrodynamic cases or dynamo cases with perfect conducting boundaries at top andbottom.

2.4

Approach to Hydrodynamic Simulations
Our numerical model is a relatively simple description of the solar convection

zone that captures the essential spherical geometry and global connectivity of thatdomain. Solar values are taken for heat flux, mass and radius and a perfect gas is
assumed. Near the solar surface the H and He ionization zones, coupled with radiative
losses, drive intense convective motions on very small scales which appear at the surface
as granulation. Capturing granulation in a global simulation would require sphericalharmonic degrees of order 4000 and this is currently too demanding. We therefore
position the top of our domain slightly below these ionization layers. In these simulations
our lower boundary is positioned near the base of the convection zone, thus omitting thestably stratified radiative interior and the shear layer at the base of the convection zone
known as the tachocline. We make this choice both because the nature of tachoclines
in rapidly rotating stars is quite uncertain at present and because simulations that
include penetration into a stable region are quite challenging in terms of computational
resources.

Table 2.2.

Parameters for Primary Hydrodynamic Simulations

Case

Nr , Nθ , Nφ

Ra

Ta

Re

Re′

Ro

Roc

ν

κ

Ω0 /Ω⊙

G1
G2
G3
G4
G5
G7
G10

96 × 256 × 512
96 × 256 × 512
96 × 256 × 512
96 × 256 × 512
96 × 256 × 512
192 × 512 × 1024
192 × 512 × 1024

3.22 × 104
1.75 × 105
4.22 × 105
7.89 × 105
1.29 × 106
2.63 × 106
5.58 × 106

3.14 × 105
3.21 × 106
1.22 × 107
3.18 × 107
6.70 × 107
2.06 × 108
6.74 × 108

84
205
326
433
543
763
1051

63
85
103
119
133
154
188

0.92
0.55
0.41
0.33
0.28
0.22
0.17

0.61
0.45
0.36
0.31
0.27
0.22
0.18

2.75
1.72
1.32
1.09
0.94
0.75
0.59

11.0
6.87
5.28
4.36
3.76
3.01
2.37

1
2
3
4
5
7
10

G3a
G3b
G5b

96 × 256 × 512
192 × 256 × 512
192 × 512 × 1024

7.83 × 105
2.26 × 106
4.03 × 106

2.41 × 107
8.02 × 107
2.23 × 108

528
1121
1347

158
324
274

0.50
0.70
0.41

0.34
0.32
0.26

0.94
0.52
0.52

3.76
2.06
2.06

3
3
5

Note. — All simulations have inner radius rbot = 5.0 × 1010 cm and outer radius of rtop = 6.72 × 1010 cm,
with L = (rtop − rbot ) = 1.72 × 1010 cm the thickness of the spherical shell. Evaluated at mid-depth
are the Rayleigh number Ra = (−∂ρ/∂S)(dS̄/dr)gL4 /ρνκ, the Taylor number Ta = 4Ω20 L4 /ν 2 , the rms′
Reynolds number Re = vrms L/ν and fluctuating Reynolds number Re′ = vrms
L/ν, the Rossby number
1/2
Ro = ω/2Ω0 , and the convective Rossby number Roc = (Ra/TaPr) . Here the fluctuating velocity v ′has the differential rotation removed: v ′ = v − hvi, with angle brackets denoting an average in longitude.
The Prandtl number Pr = ν/κ is 0.25 for all simulations. The viscous and thermal diffusivity, ν and κ, arequoted at mid-depth (in units of 1012 cm2 s−1 ). The rotation rate of each reference frame Ω0 is in multiplesof Ω⊙ = 2.6 × 10−6 rad s−1 or 414 nHz. The viscous time scale at mid-depth τν = L2 /ν is 1250 days forcase G1 and is 3640 days for case G5. Additional cases considered other rotation rates at 1.25, 1.5, 1.75
and 6 Ω⊙ .

32

33
We focus here on the bulk of the convection zone, with our computational domain
extending from 0.72R⊙ to 0.965R⊙ , thus spanning 172 Mm in radius. The total density
contrast across the shell is about 25. The reference or mean state of our thermodynamic variables is derived from a one-dimensional solar structure model (Brun et al.
2002) and is continuously updated with the spherically symmetric components of thethermodynamic fluctuations as the simulations proceed. These values are illustrated in
Figure 2.1 after convection has readjusted the stratification.
Our hydrodynamic studies here explore a variety of solar-like stars rotating from
1 to 10 Ω⊙ (cases G1-G10). All cases use the same initial stellar structure. We seek
to explore the general effects of rotation on stellar convection rather than the evolutionof a particular star, which would require modifications to the stellar structure as thestar aged. In surveying the effects of more rapid rotation on global-scale convection,
we seek to achieve reasonably high levels of turbulence in the resulting flows. Thus our
trajectory through the parameter space of Ω0 , ν, and κ attempts to maintain strong
nonlinearity without having the increasing Ω0 serve to laminarize the convection. As
we increase the rotation rate, we simultaneously decrease the effective eddy diffusivitiesν and κ to maintain the supercriticality of the simulated turbulent convection. We notethat the critical Rayleigh number for the onset of convection scales with rotation as
4/3

Rac ∝ Ta2/3 ∝ Ω0 ν −4/3 for Boussinesq convection (e.g., Chandrasekhar 1961; Dormyet al. 2004). Lower diffusivities lead to both longer viscous and thermal diffusion timescales and to flows possessing finer spatial scales. Achieving equilibrated states in thesesystems requires high resolution simulations carried out over extended periods. We havetaken a middle ground between attempting to maintain constant supercriticality (which
may require scaling ν, κ ∝ Ω−20 ) and keeping resolution requirements reasonable by−2/3

scaling our diffusivities as ν, κ ∝ Ω0

. All of our cases studied are highly supercritical,

noting that the critical Rayleigh number for these simulations at 1 Ω⊙ is Rac ∼ 1000
(Gilman & Glatzmaier 1981; Miesch 1998). We have also maintained a constant Prandtl

34

Figure 2.1 — Radial variation of mean stellar structure in the ASH models. (a) Entropy
gradient (dS̄/dr) for cases G1, G5 and G10 (as labeled). At higher rotation rates theentropy gradient becomes steeper throughout the convection zone, even for our most
turbulent cases (case G5b, long dashes). (b) Temperature and density (latter ranging
from 0.203 to 0.008 g cm−3 in the region simulated) for case G1. (c) Pressure scaleheight HP (in Mm and fractional solar radii), for case G1, with cases G2-G10 similarin their mean stratification.

35
number Pr = ν/κ = 0.25 in all of our simulations. The parameters of our models aredetailed in Table 2.2. Our choice of scalings for the eddy diffusivities with rotation rate
may have some influence on the nature of the convective patterns and mean flows weachieve. To assess some of the sensitivity the choice of our path through parameter
space, we have also sampled a limited range of more turbulent simulations at a few
rotation rates (cases G3a, G3b and G5b).
Some of our simulations are initialized by perturbing a quiescent state in solid
body rotation. The growth of convection leads to velocity correlations that serve toredistribute angular momentum within the shell, building a differential rotation andmeridional circulation. We evolve the simulation for long periods compared variously
to convective overturning times, rotation periods or typical diffusive times. Other simulations were started from these evolved states and then run for long intervals after
all adjustments have been made to the frame rotation rate Ω0 and to the viscous andthermal diffusivities.
All of the hydrodynamic simulations discussed in this thesis are at approximately
the same level of maturity in their evolution. Case G1 was the progenitor case at 1 Ω⊙
and was evolved for some 3000 days after branching away from case AB from Brun &Toomre (2002), which itself has seen about 10000 days of total simulated life. Startingwith this case, each subsequent simulation was spun up from the next fastest case (i.e.,
G3 was spun up from G2) and evolved for over 4000 days, or many hundreds of rotation
periods.
The time evolution of case G5, which was started from an evolved state, is shownin Figure 2.2. At day zero the rotation rate Ω0 and eddy diffusivities ν and κ are set
to appropriate values for case G5. The convection responds and equilibrates to thesechanges on a timescale of a few hundred days but the mean flows of differential rotationand meridional circulation take significantly longer to fully equilibrate. As shown in
Figure 2.2b the differential rotation can take nearly 4000 days to equilibrate. During

36

Figure 2.2 — Evolution of kinetic energies in hydrodynamic case G5.

(a) Volumeaveraged kinetic energy densities of differential rotation (DRKE), convection (CKE)and meridional circulations (MCKE) shown in logarithmic plot for the first 6500 daysof the simulation. Convective kinetic energies adjust very quickly, but the energy of themean circulations take longer to equilibrate. After approximately 4000 days all threeenergies have reached a stationary state. (b) Linear plot of evolving DRKE and CKE.

37
this interval the convection and meridional circulations are redistributing angular momentum and heat, building a latitudinal contrast of angular velocity and of temperature
with hot poles, cool mid-latitudes and a warm equator (see Chapter 3). After 4000 days
the profile of differential rotation remains nearly steady and we are able to explore thetime evolution of the patterns of convection.
After similar intervals of evolution, all cases appear to be statistically stationary
in terms of the angular momentum fluxes and the kinetic energies. We believe that thedifferential rotation profiles presented are effectively stationary, though there are small
fluctuations as determined from short averages over a few rotation periods. Certain
cases (including G5) were evolved for much longer intervals (over 10000 days and morethan 2000 rotation periods) to explore the long-term behavior of convective patterns
in these rapidly rotating systems. To test that our results are not unduly subject tohysteresis in the system, we explored a branch of cases which were successively spun
down from 5 Ω⊙ to 1 Ω⊙ . No significant hysteresis was found.
We shall discuss the properties of cases G1-G10 in Chapters 3 and 4, and there
explore the nature of the convective patterns realized, as well as the differential rotationand meridional circulation that are achieved for a range of rotation rates.

2.5

Hydrodynamic Progenitors to Dynamo Simulations
The hydrodynamic progenitors used in our dynamo solutions are slightly differ-ent from the solutions presented as cases G1-G10. In conducting those hydrodynamic
studies, we learned that at the highest rotation rates the unresolved flux carried out
the top of the domain by κ0 was beginning to imprint deeper into the convection zone.
This occurs because dS̄/dr generally increases in amplitude in the more rapidly rotating
simulations.

38
In most of the hydrodynamic simulations, the κ0 diffusion term has the form
κ0 = κ0,top



ρ̄top
ρ̄

α

+ κ0,C(2.25)

with κ0,top the diffusivity at the top boundary, and κ0,C a small constant (of order1010 cm2 s−1 ) added to smooth out potential spikes in dS̄/dr near the bottom of theconvection zone, and α controlling the tapering of the unresolved flux in radius. The
value of κ0,top is chosen to transport a solar luminosity through the upper boundary,with
Lu r=rtop = 4πr 2 κ0 ρ̄T̄ ∇S̄

r=rtop= L⊙ .

(2.26)In all of these rapidly rotating simulations κ0,top = 2.979 × 1014 , based on the selfconsistently determined profiles of entropy, temperature and density within ASH.
In cases G1-G10, we chose α = 4. Larger values of α lead to stronger confinementof the unresolved flux to the top of the convection zone, but also lead to heating within
the entropy equation and modification of the entropy gradients. Thus, in cases H3 andH5, we chose α = 7 to better confine the unresolved flux in a narrow upper boundary
layer. As a result, this flux is largely confined within the upper 10% of the convection
zone and shows only small variations with rotation rate.
In these cases the profile of dS̄/dr is somewhat steeper near the top of the convection zone. This effect is visible in Figure 2.3, where the stellar structure of cases H3 andH5 are compared with that in cases G3 and G5. The profiles of dS̄/dr come into good
agreement at depths below 0.87R⊙ for both branches of simulations, but are deeper forcases H3 and H5 near the top of the convection zone. This leads to somewhat stronger
driving of the convection as the Rayleigh numbers are also higher near the top of thedomain. This partially explains why cases H3 and H5 have higher Reynolds numbers
and higher Rossby numbers (Table 2.3) than cases G3 and G10 (Table 2.2). The effects of this are subtle, resulting primarily in slightly stronger latitudinal gradients ofdifferential rotation and temperature in the uppermost regions of the shell.

39

Figure 2.3 — Radial structure of dynamo progenitor simulations. Shown is the entropy

gradient (dS̄/dr) for dynamo progenitor cases H3 and H5 (solid lines) compared withhydrodynamical cases G3 and G5 (dashed lines). The entropy gradient in the dynamo
progenitor cases is somewhat steeper throughout the convection zone, and the minimum
is deeper. This leads to higher Rayleigh numbers in these simulations and stronger
driving. Though the convection is driven more strongly, the temperature, density andpressure profiles are nearly identical to those shown in Figure 2.1. Overlain with a blue,
dashed line is a 1-D solar structure model computed with the CESAM code by Brunet al. (2002).

The entropy gradient from a 1-D, helioseismically-constrained solar structure
model is also shown in Figure 2.3. The equilibrated profiles of dS̄/dr in ASH tend
to be somewhat steeper than the solar model at depths below 0.95 R⊙ and significantly
shallower in the upper convection zone. The large change in dS̄/dr between about
0.93 R⊙ and the surface appears to result from heating in that layer by our adopted unresolved flux. New simulations are now being explored in ASH, with different treatmentsof the unresolved flux, and these models appear able to better match the structure ofdS̄/dr from the solar model. Here however, all of our simulations have dS̄/dr profiles
akin to those illustrated in Figure 2.3.

40
Convective structures in these hydrodynamic progenitor cases are similar to thoserealized in cases G1-G10, though the patterns tend to be slightly more complex near the
top of the shell. The patterns of convection for cases G5 and H5 are shown in Figure 2.4.
Shown in global Mollweide projection are radial velocities at a radius of 0.95 R⊙ andnear the upper boundary. These cases were well evolved and possess intricate convective
patterns and solar-like differential rotation profiles, with fast zonal flow at the equatorand slower flows at the poles. The prominent modulation visible in case G5 (Fig. 2.4a)
is less visible in case H5. Here in case H5 fine-scaled convection has filled in the regions
between the patches and obscures the modulation. These modulated convective states,
called active nests of convection, will be explored in detail in Chapter 4 for case G5 andG10. Time-longitude maps indicate that these structures are still present in case H5
and remain comparable in amplitude to those seen in case G5.
Ultimately, these changes in stratification are modest compared to the overall
structure of the star. As such, the profiles of density, temperature and pressure arenearly unaffected by the changes in the entropy gradient.

2.6

Studies of Dynamo Action in Rapidly Rotating Suns
We have explored a number of dynamo scenarios in our rapidly rotating suns.

Two solutions, one rotating at three times the current solar rate (3 Ω⊙ ) and one rotating
five times the solar rate (5 Ω⊙ ), will be the primary focus of our discussion of dynamo
action in Chapters 5, 6 and 7. We begin here by discussing the basic formulation of the
dynamo studies. The parameter space explored by our broader family of dynamos will
be discussed in turn in Chapter 8.
The dynamo simulations were initiated from mature hydrodynamic progenitor
cases which had been evolved for at least 5000 days at each rotation rate and were wellequilibrated. Our simulations at three and five times the solar rate Ω⊙ lie again on a path
−2/3

where the SGS diffusivities ν, κ and η decrease as Ω0, in order to maintain vigorous

41

Figure 2.4 — Patterns of convection in cases G5 and H5. (a) Snapshot of radial velocity
vr in global Mollweide projection for case G5 near the upper boundary (0.95 R⊙ ) withscale indicated. Downflows are dark and narrow, while upflows are broad and indicatedby light tones. A prominent modulation in longitude is visible near the equator. These
active nests of convection persist for thousands of days. (b) Radial velocities in case H5
at same depth, showing somewhat more vigorous convection. Here the active nests are
obscured by the finer-scale convection but are still visible over long intervals of time.

42
convection as rotation attempts to constrain and quench the motions. The fundamental
characteristics of our primary dynamo simulations and their hydrodynamic progenitors
are summarized in Table 2.3; other cases will be defined in Chapter 8.
To initiate our dynamo cases, a small seed dipole magnetic field was introduced
and evolved via the induction equation. The time evolution of magnetic and kinetic
energies is presented for case D3 in Figure 2.5. Here and throughout the thesis, day 0
refers to the last adjustments made to the simulation (generally to the rotation rate Ω0and eddy diffusivities ν, κ and η). This time-trace shows about 7000 days of simulated
time throughout the history of the dynamo. As shown, the energy in the magnetic fields
is initially many orders of magnitude smaller than the energy contained in the convective
motions. These fields are amplified by shear and grow to become comparable in energy
to the convective motions. Generally the dynamos spend about 2000 days reaching fully
equilibrated states.
These dynamo simulations are computationally intensive, requiring both high spatial resolution to correctly represent the velocity fields and long time evolution to capture
the equilibrated dynamo behavior, which may include cyclic variations on time scales ofhundreds to thousands of days. The strong magnetic fields can produce rapidly moving
Alfvén waves which seriously restrict the Courant-Friedrichs-Lewy (CFL) timestep limits in the upper portions of the convection zone. Case D3, rotating three times fasterthan the current Sun, has been evolved for over 7000 days (or over 2 million timesteps
with typical timesteps of 300 seconds), and case D5, rotating five times faster than theSun, has seen more than 17000 days of evolution (representing more than 10 million
timesteps, with typically 140 seconds of evolution per timestep).
These two cases were conducted at magnetic Prandtl number Pm = ν/η = 0.5, avalue significantly lower than employed in our previous solar simulations. In particular,
Brun et al. (2004) explored Pm = 2, 2.5 and 4, and Browning et al. (2006) studied
Pm = 8. The high magnetic Prandtl numbers were required in the solar simulations toTable 2.3.

Parameters for Primary Dynamo Simulations

Case

Nr , Nθ , Nφ

Ra

Ta

Re

Re′

Rm

Rm′

Ro

Roc

ν

η

Ω0 /Ω⊙

D3
D5
H3
H5

96 × 256 × 512
96 × 256 × 512
96 × 256 × 512
96 × 256 × 512

3.22×105
1.05×106
4.10×105
1.27×106

1.22×107
6.70×107
1.22×107
6.70×107

173
273
335
576

105
133
105
141

86
136
—
—

52
66
—
—

0.378
0.273
0.427
0.303

0.311
0.241
0.353
0.268

1.32
0.940
1.32
0.940

2.64
1.88
—
—

3
5
3
5

Note. — Dynamo simulations at three and five times the solar rotation rate are cases D3 and D5, and theirhydrodynamic (non-magnetic) companions are H3 and H5. All simulations have inner radius rbot = 5.0 × 1010 cm
and outer radius of rtop = 6.72 × 1010 cm, with L = (rtop − rbot ) = 1.72 × 1010 cm the thickness of the spherical shell.
Evaluated at mid-depth are the Rayleigh number Ra = (−∂ρ/∂S)(dS̄/dr)gL4 /ρνκ, the Taylor number Ta = 4Ω20 L4 /ν 2 ,′the rms Reynolds number Re = vrms L/ν and fluctuating Reynolds number Re′ = vrms
L/ν, the magnetic Reynolds′′
number Rm = vrms L/η and fluctuating magnetic Reynolds number Rm = vrms L/η, the Rossby number Ro = ω/2Ω0, and the convective Rossby number Roc = (Ra/Ta Pr)1/2 . Here the fluctuating velocity v ′ has the axisymmetriccomponent removed: v ′ = v − hvi, with angle brackets denoting an average in longitude. For all simulations, thePrandtl number Pr = ν/κ is 0.25 and the magnetic Prandtl number Pm = ν/η is 0.5. The viscous and magnetic
diffusivity, ν and η, are quoted at mid-depth (in units of 1012 cm2 s−1 ). The rotation rate Ω0 of each reference frame
is in multiples of the solar rate Ω⊙ = 2.6 × 10−6 rad s−1 or 414 nHz. The viscous time scale at mid-depth τν = L2 /ν is3640 days for case D5 and the resistive time scale is 1820 days. Rotation periods at three and five times the solar rate
are in turn 9.3 days and 5.6 days.

43

44

Figure 2.5 — Evolution of kinetic and magnetic energies in dynamo case D3.

(a)Volume-averaged kinetic energy densities of differential rotation (DRKE), convection(CKE) and meridional circulations (MCKE) shown in logarithmic plot. Magnetic energy densities in fluctuating magnetic fields (FME), mean toroidal fields (TME) andmean poloidal fields (PME) are shown growing from an initial seed field. The dynamo
saturates after roughly 1700 days. After saturation, DRKE has changed substantially
but CKE and MCKE are largely unaffected. (b) Zoom in logarithmic plot of evolving
kinetic and magnetic energies, emphasizing their behavior after saturation. Time iscounted from day 0, when the last adjustments were made to parameters controllingthe simulation (i.e. Ω0 , η, etc.).

45
reach sufficiently high magnetic Reynolds numbers to drive sustained dynamo action. In
the simulations of Brun et al. (2004) only the simulations with Pm > 2.5 and Rm′ & 300
achieved sustained dynamo action, where Rm′ is the fluctuating magnetic Reynolds
number. We are here able to use a lower magnetic Prandtl number for three reasons.
Firstly, more rapid rotation tends to stabilize convection and lower values of ν andη are required to drive the convection. Once convective motions begin however they
become quite vigorous and the fluctuating velocities saturate at values comparable toour solar cases. Thus the Reynolds numbers achieved are fairly large and we can achievemodestly high magnetic Reynolds numbers even at low Pm. Secondly, the differential
rotation becomes substantially stronger with both more rapid rotation Ω0 and with lower
diffusivities ν and η. This global-scale flow is an important ingredient and reservoir ofenergy for these dynamos, and the increase in its amplitude means that low Pm dynamos
can still achieve large magnetic Reynolds numbers based on this zonal flow. Thirdly, thecritical magnetic Reynolds number for dynamo action likely decreases with increasing
kinetic helicity (e.g., Leorat et al. 1981). Helicity generally increases with rotation rate
(e.g., Käpylä et al. 2009), so the rapidly rotating flows considered here achieve dynamo
action at somewhat lower Rm than the models of Brun et al. (2004), which rotated at
the solar rate.
Case D3, which builds persistent wreaths of magnetism, is presented in Chapter 5.
Case D5 at five times the solar rate has time-dependent wreaths that flip global-scalepolarity and is presented in Chapter 6. An analysis of terms contributing to building
and destroying the persistent magnetic fields is carried out for case D3 in Chapter 7. A
variety of other dynamo cases, some at higher turbulence levels and rotation rates, will
be in turn discussed in Chapter 8.

Chapter 3
Convection in Rapidly Rotating Younger Suns

We begin our exploration of convection and dynamo action in rapidly rotatingsuns with a series of hydrodynamic simulations which sample rotation rates between
1 − 10 Ω⊙ . These correspond to cases G1-G10 in Table 2.2. In these systems, we explore
how the patterns of convection are modified by more rapid rotation. We study the
resulting differential rotation and meridional circulation and their scaling with rotation
rate. In Chapter 4 we examine how novel modulated patterns of convection arise in themost rapidly rotating simulations.
These two chapters are based on work published in Brown et al. (2008)1 and arelargely a restatement of that paper. As the primary author of this paper, I conducted thesimulations presented here, performed the analysis and wrote the text. My co-authors
provided advice and guidance throughout the process, helping frame the questions which
form the core of the study. Preliminary versions of these results have been presented
in Brown et al. (2004), Brown et al. (2006), Brown et al. (2007a), and Brown et al.
(2007d).

3.1

Early Results of Modulated Convection
In our early simulations of rapidly rotating suns we found that strongly localizedstates of convection emerged with more rapid rotation (Brown et al. 2004). A selection
1
Brown, B. P., Browning, M. K., Brun, A. S., Miesch, M. S., & Toomre, J., 2008, “Rapidly Rotating
Suns and Active Nests of Convection”, ApJ, 689, 1354–1372.

47

Figure 3.1 — Convective patterns in mildly turbulent simulations. Cases shown arerotating at (a) one, (b) three and (c) five times the solar rate. Shown as snapshots areradial velocities near top of layer in global Mollweide projection, with upflows light anddownflows dark (scaling indicated by accompanying colorbars). Poles are at top andbottom, and the equatorial region appears at middle, with equator indicated by bolddashed line. Thin dashed lines denote circles of constant latitude or longitude, and thethin surrounding line indicates the location of the stellar surface at R⊙ . A striking
pattern of convection localized into nests near the equator emerges as the rotation rateincreases.

48of these simulations in Figure 3.1 present snapshots of radial velocity near the top of thedomain in global Mollweide projection, showing the entire spherical surface with minimal
distortion. With more rapid rotation, a prominent longitudinal modulation appears
in the patterns of equatorial convection. At the higher rotation rates the equatorialconvection is confined to one or two nests, with streaming zonal flow filling the regions
outside these nests of convection. These nests can persist for intervals spanning many
hundreds of rotation periods, often with little change. Two nest states sometimes evolve
into single nest states as one nest overtakes another.
The simulations shown in Figure 3.1 are less turbulent than the cases presented
in the rest of this chapter, each possessing Reynolds numbers that are about three-foldsmaller near the surface than in our new simulations. In these early models, a large
portion of the energy transport in the upper convection zone was carried by unresolved
scales of motion. This parametrized SGS flux, represented by κ0 , dominated transport
in the upper 30% of the convection zone, leading to weaker enthalpy transport andweaker resolved convection there. This parametrized flux acts as a volume cooling termthat removes flux from the regions where it is dominant; the dynamics were influenced
by the presence of this cooling layer. The cases presented in detail in this chapter havea narrower unresolved flux layer, confined now to the upper 10% of the convection zone,
and consequently much more vigorous convection is realized throughout the domain. In
these more turbulent cases, the phenomena of localized nests of convection is realized
at somewhat higher rotation rates.

3.2

Convective Patterns and Evolution with Rotation
The variation of convective patterns with increasing rotation rate Ω0 in our more

rapidly rotating suns is illustrated in Figure 3.2. Snapshots of the radial velocity near
the top of the domain (0.95R⊙ ) are shown in Mollweide projection for four cases: G1,
G3, G5 and G10. The convection patterns are complex and time dependent, with

49

Figure 3.2 — Convective patterns in primary hydrodynamic cases.

Shown are radial
velocity patterns in Mollweide projection at 0.95R⊙ (left) and differential rotation profiles (middle, right) with increasing rotation rate in (a, e) for case G1, (b, f ) for G3, (c, g)
for G5, and (d, h) for G10. At higher rotation rates the horizontal scale of convective
cells shrinks at all latitudes and cells are more strongly aligned with the rotation axis. A
striking pattern of modulated convection emerges at low latitudes with faster rotation,
consisting of spatially modulated or patchy convection. These active nests of convection
are propagating structures which persist for long periods of time. At middle are profilesof mean angular velocity Ω with radius and latitude. These differential rotation profiles
all involve fast equators (prograde relative to the frame rate Ω0 , indicated by tickmarkon scale) and a monotonic decrease of Ω as the poles are approached. At right are radial
cuts of the angular velocity at selected latitudes, as labeled. The dark dashed contour
denotes the constant propagation rate of the nests where discernible.

50
asymmetries between the upflows and downflows owing to the density stratification.
Thus narrow, fast downflow lanes are surrounded by broad, relatively weak upflows.
There is a clear difference in both the scale and structure of convection at highand low latitudes. In the equatorial regions (roughly ±30◦ in latitude), the downflowsorganize into large structures (loosely called banana cells) aligned with the rotation axis,
thus extending in the north-south direction. At high rotation rates this tendency foralignment becomes pronounced, largely in the spirit of the Taylor-Proudman theorem,
and the downflow network exhibits little of the east-west branching visible in case G1.
These downflow lanes propagate in a prograde sense relative to the bulk rotation rateand do so more rapidly than the differential rotation which they themselves establish.
The nests of convection, when they appear at the higher rotation rates, propagate at anintermediate rate as denoted by the heavy dashed contours in Figure 3.2f − h. We defer
discussion of the nature of these nests of convection to Chapter 4. Individual convective
cells persist for about 10 to 30 days.
In the higher latitude regions, the convection cells are more isotropic and thedownflow network organizes on smaller scales. Convection in these regions is vigorousand complex, with upflows and their downflow networks in a constant dance. The
convective cells have a cusped appearance, with downflows leading upflows as bothpropagate in a retrograde fashion (most apparent in Figs 3.2b, c). Strong vortical plumes
form in the interstices of the downflow network at both high and low latitudes. In thepolar regions above the middle of the convection zone, the sense of vorticity in these
downflow plumes is generally cyclonic: counterclockwise in the northern hemisphere andclockwise in the southern. As they descend through the mid-layer their vorticity changes
and they become largely anti-cyclonic. In contrast, the polar upflows are anti-cyclonicat all depths.

51
The latitudinal variation of convection patterns can be in part understood byconsidering a cylinder tangent to the base of the convection zone and aligned with therotation axis. Within our geometry, this tangent cylinder intersects the outer boundaryat about ±42◦ of latitude. It is well known that in a rotating convective shell the flow
dynamics are different inside and outside of the tangent cylinder, owing to differences in
the connectivity of the flows, the influence of the Coriolis forces and distance from therotation axis (e.g., Busse 1970). These differences become more evident as the rotation
rate, and hence the rotational constraints on the convection, increases. With more rapid
rotation the longitudinal extent of the convective cells becomes progressively smaller.
Linear theory, in the Boussinesq approximation, predicts that the wavenumber of the
1/3

most unstable mode scales with rotation as m ∝ Ta1/6 ∝ Ω0 ν −1/3 (e.g., Chandrasekhar
1961; Dormy et al. 2004) for polar and equatorial convection. This effect is found inanelastic systems as well (Glatzmaier & Gilman 1981) and in our simulations is evident
at both high and low latitudes.
Shown at right in Figures 3.2e−h are the profiles of differential rotation (as angular velocity Ω) realized in these simulations. These Ω(r, θ) profiles are azimuthally andtemporally averaged over a period of roughly 200 days. All of our more rapidly rotatingstars exhibit solar-like differential rotation profiles, with prograde (fast) equators andretrograde (slow) poles. Contours of constant angular velocity are aligned nearly on
cylinders, influenced by the Taylor-Proudman theorem, though recent simulations ofsolar convection suggest that this is sensitive to the treatment of the bottom thermalboundary condition (Miesch et al. 2006). As first shown in mean-field models by Rempel(2005) and then in global-scale convection models by Miesch et al. (2006), introducing
a weak latitudinal gradient of entropy at the base of the convection zone, consistent
with a thermal wind balance in a tachocline of shear, can serve to rotate the Ω contours
toward the more radial alignment deduced from helioseismology without significantly
changing either the overall Ω contrast with latitude or the convective patterns. We

52
expect similar behavior here, as briefly explored by Ballot et al. (2007) in younger sunswith deeper convection zones, but have not explored this issue in detail at the higher
rotation rates. More rapidly rotating suns may very well also possess tachoclines, but
at this stage there is no observational evidence of this. Thus we have simplified thesesimulations by imposing a constant entropy at the bottom boundary. Our contours ofΩ in Figure 3.2 show some differences between the northern and southern hemispheres,particularly at higher latitudes, and these differences decrease with more rapid rotation.
The patterns of convection are not simply symmetric about the equator, and thus theaccompanying mean zonal flows can be expected to show some variations between thetwo hemispheres. Also shown are radial cuts of Ω at six fixed latitudes that make evident
the angular velocity contrasts with radius and latitude achieved in these simulations.
The absolute contrast in latitude and radius clearly grows with rotation rate, and will
be discussed in §3.5.
A most striking result of our simulations is the emergence of persistent, spatiallymodulated convection in the equatorial regions at high rotation rates. At these low
latitudes, convection becomes modulated in longitude and forms distinct active nests
where the convective vigor is enhanced compared to the regions outside. The amplitudeof the convective motions and enthalpy transport is larger within these nests, and indeed
at the highest rotation rates, convection in the equatorial band is confined entirely tothe nests. These nests of convection propagate at a velocity distinct from either thezonal flow of differential rotation or that of the individual cells of convection and persist
for very long periods of time (more than 5000 days in case G5). The nature of these
active nests of spatially localized convection will be explored in detail in Chapter 4.
Weak modulation in longitude is already evident at low rotation rates. Whenlong time series are considered, we have positively identified nests of convection in all
simulations rotating at Ω0 & 3 Ω⊙ . As the rotation rate increases, the modulation level
gradually increases; at the highest rotation rates (& 7 Ω⊙ ) the equatorial convection is53
almost solely confined to the nests. The convection realized in case G10 (Fig. 3.2d) ismarked by this extreme modulation, with strong upflows and downflows inside the nest
and very little convection in the surrounding regions. These most rapidly rotating cases
maintain a strong differential rotation profile, even though the equatorial convection
occupies only a narrow interval in longitude. The regions outside the nest are filledwith fast streaming zonal flows consistent with the differential rotation.

3.2.1

Radial Connectivity of Convection

The nests of enhanced convection span the convection zone and propagate everywhere at a constant prograde angular velocity relative to the bulk rotation rate of thestar. A contour corresponding to this characteristic propagation rate is overplotted on
the differential rotation profiles in Figures 3.2f − h for cases G3, G5 and G10. As isevident from these profiles, the angular velocity associated with the differential rotation
exceeds that of the propagation rate of the nests near the surface and is slower thanthat near the base of the convection zone. The nests of convection therefore live within
an environment of substantial zonal shear with radius, as is quantified for case G5 in
Figure 3.3. Here the shearing zonal velocity of differential rotation is plotted in latitude
at six radial depths. At all depths there is substantial zonal flow through the nests of
convection.
Other studies of solar convection with ASH show that strong downflow lanes extend throughout the entire depth of the domain (Miesch et al. 2000; Brun & Toomre
2002; Miesch et al. 2008). In our more rapidly rotating stars, this connectivity withdepth changes markedly, as is illustrated for case G5 in Figure 3.4 showing radial velocities throughout the convection zone. In these rapidly rotating suns, the strong variationof mean zonal flow with radius in the equatorial regions prevents all but the strongest
downflows from spanning the convection zone. Within the nest of enhanced convection
the plumes are able to traverse the convection zone. Yet in the quieter regions outside,54

Figure 3.3 — Profile of hvφ i in case G5. The variation of mean zonal velocity hvφ i withlatitude for case G5 is sampled at six radial cuts as labeled and shown here relative tothe uniform propagation rate of the nests of convection. The nests experience a strong
prograde zonal flow (positive) near the top of layer and a prominent retrograde flow
within the lower half.

55

Figure 3.4 — Connectivity of radial velocity with depth in case G5. Shown at thesame instant in Mollweide view are vr (a) near top of domain (0.95R⊙ ), (b) at mid
depth (0.85R⊙ ), and in (c) for an equatorial cut in longitude over full depth range.
Strong plumes span the convection zone in the equatorial regions only within the nestof enhanced convection. The weaker cellular flows outside the nests are confined byshear to the upper reaches.

56
the weaker downflow plumes are truncated by shear before reaching the middle of theconvection zone. It is evident (Fig 3.4c) that the amplitude of convective motions is
pronounced at all depths within the nest of active convection.
The downflowing plumes are influenced by the strong radial shear and some breakinto multiple cells with radius even before the full blown nests of localized convection
emerge, as is evident already in our simulation rotating at twice the solar rate (case G2).
When the downflow networks only span a portion of the convection zone and experiencea limited range of the full density stratification, the importance of compressible effects
decreases. This has important consequences for the energetics of the simulations, particularly the radial kinetic energy flux, as will be addressed in §3.7. In contrast, thedownflowing plumes in the polar regions experience much less shear from either thedifferential rotation or the relatively weak meridional circulations and continue to spanthe entire convection zone depth.

3.2.2

Thermal Structuring

In these rapidly rotating suns, the turbulent alignment of convection with therotation axis leads to a net latitudinal transport of enthalpy, yielding a prominent latitudinal gradient of temperature. The resulting thermal structuring in case G5 is shownin Figure 3.5, presenting both the mean temperature profile and representative temperature fluctuations in a snapshot near the surface. In the latter, individual convection
cells are associated with small fluctuations with amplitudes of a few K. Downflows are
generally cool while upflows are relatively warmer. The enhanced enthalpy transport
within the active nests of convection appears as positive temperature fluctuations in theequatorial region.
Evident at high latitudes (Fig. 3.5c) are broad spatial structures (in addition
to small-scale convection) which appear in the temperature fluctuations and are notreadily visible in the maps of radial velocity (see Fig. 3.4 at same instant). These

57

Figure 3.5 — Temperature structures within case G5. Mean latitudinal variations intemperature are shown relative to their spherical average T̄ in (a) as contours with radiusand latitude and (b) as cuts at fixed radii at the top (solid, 0.96R⊙ ), middle (dashed,
0.84R⊙ ) and bottom (dotted, 0.72R⊙ ) of the domain. (c) Temperature fluctuations ina snapshot near top of domain (0.95R⊙ ) relative to the mean structure in (a).

structures are long lived and appear to be a separate phenomena from the nests of
convection. The polar patterns propagate in a retrograde sense more rapidly than the
differential rotation in which they are embedded, and though streaming wakes from theactive nests print weakly into the polar regions, the polar patterns and nests appear
to be distinct phenomena. The large-scale polar patterns are not evident in the slowly
rotating cases (G1 and G2); in the most rapidly rotating cases this modulation attains
a more complicated form than the two-lobed structure shown here.
The zonally-averaged thermal structure (Fig. 3.5a, b) is quite smooth and is characterized by warm poles and a cool equator, with yet cooler mid-latitudes. In contrast,
the mean entropy increases monotonically from equator to pole, due to effects of pressure. All of the more rapidly rotating cases have similar latitudinal thermal profiles,
though the temperature difference between equator and pole increases with more rapid
rotation, as will be discussed further in §3.3. In case G5, the latitudinal pole to equatortemperature contrast is approximately 100 K throughout the convection zone. These
latitudinal variations remain small at all rotation rates in comparison to the sphericallysymmetric background T̄ , which ranges from 2.7 × 105 K near the surface to 2.2 × 106 K
near the bottom of the convection zone (as shown in Fig. 2.1).

58

3.3

Thermal Wind Balance
Rapidly rotating systems are constrained by the Taylor-Proudman theorem tohave minimal variations in flow dynamics along the direction of the rotation axis. In
stratified flows, gradients in density and pressure contribute to baroclinic terms in thevorticity equations (Pedlosky 1982; Zahn 1992) which maintain flows that can break the
Taylor-Proudman constraint. In our rapidly rotating suns, convective plumes tilt toward
the rotation axis as rotation effects increase. This results in latitudinal as well as radialtransport of enthalpy and builds a latitudinal gradient of temperature and entropy. Such
gradients arise naturally in a rotating convective system even with uniform thermalboundary conditions. For a nearly adiabatic stratified, rotating, non-magnetized fluid
it can be shown that in the limit of small Rossby number and negligible viscous effects
the zonal component of the vorticity equations reduces to the well known thermal windbalance (e.g., Brun & Toomre 2002; Miesch et al. 2006):
∂ vbφ
∂ Sb
g
=
,
∂z
2CP rΩ0 ∂θ

(3.1)

where z is directed along the rotation axis and a hat denotes an average in longitude
and time. We have further assumed that the turbulent pressure is negligible.
From equation (3.1) it is clear that departures from rotation constant on cylinders(as observed in the solar interior by helioseismology) can be maintained by a latitudinal
gradient of entropy. The left and right hand sides of equation (3.1) are shown forcase G5 in Figure 3.6. In the bulk of the convection zone, the differential rotation
profiles realized in these more rapidly rotating suns are substantially in thermal windbalance. Significant departures arise near the inner and outer boundaries (Fig. 3.6c)
where Reynolds stresses and boundary conditions play a dominant role, as was foundin earlier simulations of solar convection (Brun & Toomre 2002).

59

Figure 3.6 — Thermal wind balance achieved in case G5. (a) Gradients of vbφ along therotation axis, ∂ vbφ /∂z, (b) the scaled latitudinal entropy gradient from the right-handside of eq. (3.1), and (c) their difference, with contours in the latter rescaled to show
the departures near the boundaries. The bulk of the convection zone is in thermal windbalance, but substantial departures arise near the top and bottom of the domain where
Reynolds stresses dominate.

Another striking property of the thermal wind balance is that increasing Ω0 leadsbto more cylindrical profiles of vbφ unless ∂ S/∂θ
also adjusts with the rotation rate. In

our more rapidly rotating suns we find that the latitudinal gradients of temperature andentropy increase with more rapid rotation. The growth of ∆Sb (difference between thesurface value of Sb at say 60◦ and the equator) with increasing rotation rate Ω0 is shown in
Figure 3.7. The latitudinal structure of entropy is always monotonic in these simulations,with lower entropy at the equator and higher entropy at the poles. Convection in thesemore rapidly rotating systems establishes stronger latitudinal gradients of entropy, but
not enough in these simulations to maintain the Ω profiles unchanged.
Accompanying the growth of ∆Sb is a growth in the latitudinal temperature con-trast, as shown by the maximum temperature contrast in latitude near the stellar surface
in Figure 3.7. Typically, the maximal contrast occurs between the poles and latitudesof ±40◦ , as seen in Figure 3.5 for case G5 with a contrast of about 100 K. In the

60b and maximal latitudinal temperature contrast with Ω0 .
Figure 3.7 — Scaling of ∆S

The latitudinal contrast of entropy ∆Sb (plotted as diamonds) is measured between
equator and high latitudes at 0.96R⊙ . It increases with more rapid rotation. The moreturbulent cases (G3a, G3b and G5b as labeled) have larger entropy contrasts, in keepingwith their generally stronger differential rotation. Blue triangles indicate the maximum
temperature contrast in latitude at the upper boundary in each simulation.

rapidly rotating simulations, the primary flux balance in latitude is between thermal
eddy diffusion κρ̄T̄ h∂S/∂θi and convective enthalpy transport Cp ρ̄hvθ′ T ′ i. Here convective transport moves warm material to the poles as the downflows align more strongly
with the rotation axis while eddy diffusion works to erode the gradient. The meridional
circulations appear to play a relatively minor role in maintaining the overall latitudinal
entropy contrast.

3.4

Angular Momentum Redistribution
In these simulations of stellar convection, complex couplings between rotation andconvection build the profiles of differential rotation and meridional circulation. Withstress-free boundary conditions at the top and bottom of the shell there are no net
torques and thus the total angular momentum is conserved. Couplings between rotationand convection lead to a global-scale redistribution of angular momentum, resulting in61
the sustained flows of both differential rotation and meridional circulation. To assess thetransport of angular momentum in these systems we follow the approach of Miesch et al.
(2008), examining the average radial and latitudinal angular momentum transport as
detailed in their equations (10)-(12) (see also Brun & Toomre 2002; Miesch 2005). The
angular momentum fluxes from Reynolds stresses (RS), meridional circulations (MS)and viscous diffusion (VD) are
F RS = ρ̄r sin θ hvr′ vφ′ ir̂ + hvθ′ vφ′ iθ̂ ,F MC = ρ̄L hvr ir̂ + hvθ iθ̂ ,F VD = −ρ̄νr 2 sin2 θ∇Ω,

(3.2)
(3.3)
(3.4)

where
L = r sin θ (Ω0 r sin θ + hvφ i)(3.5)

is the specific angular momentum.
The total radial and latitudinal fluxes of angular momentum are shown for case G1and G5 in Figure 3.8. Here we have integrated in co-latitude and radius respectively
to deduce the net fluxes through shells at various radii and through cones at various
latitudes (c.f., Miesch 2005). The three major contributions arise from Reynolds stresses,
meridional circulations and viscous terms. Velocity correlations lead to net angularmomentum transport by Reynolds stresses as convective structures develop organizedtilts and align partially with the axis of rotation (e.g., Brummell et al. 1998; Brun &Toomre 2002; Miesch et al. 2008). This alignment is particularly prominent in the fast
downflow lanes, and becomes stronger as rotation increases.
Turning first to our solar case (G1, Fig. 3.8a, b), we see that in radius the meridional circulations and Reynolds stresses play similar and nearly equal roles in transporting angular momentum outward. The viscous flux meanwhile is negative and transports
angular momentum inward, in keeping with the positive radial gradient of the differential rotation profile (eq. 3.4), and the total flux in radius is nearly zero. The transport

62

Figure 3.8 — Angular momentum fluxes in radius and latitude for cases G1 and G5.
Shown are time averages of the integrated radial (Fr ) and latitudinal (Fθ ) angularmomentum flux for case G1 (a, b) and case G5 (c, d). Contributions arise from Reynolds
stresses (RS), meridional circulations (MC) and viscous diffusion (VD). Their total isalso shown (Sum). Transport by viscous diffusion remains comparable in all cases, while
the transport by Reynolds stresses and meridional circulations changes markedly withmore rapid rotation.

63
in latitude is somewhat different. Here meridional circulations combine with viscous
fluxes to transport angular momentum away from the equator and toward the poles(i.e., positive in the southern hemisphere and negative in the northern hemisphere).
This tendency is opposed by Reynolds stresses, which continuously accelerate the equatorial regions and dynamically maintain the angular velocity contrast ∆Ω in latitude.
The transport of angular momentum in our more rapidly rotating cases are all
similar in form and are well represented by case G5 (Fig. 3.8c, d). In these more rapidly
rotating stars, the radial balance is dominantly between the Reynolds stresses transporting angular momentum outward and viscous terms transporting it inward. The viscoustransport is similar in magnitude to that of case G1, though the radial boundary layers
are now much narrower. The transport by Reynolds stresses is nearly twice as large, andthis likely arises from the strong alignment of convective structures in both polar andequatorial regions. In these stars the weaker meridional circulations become relatively
minor and disorganized players in the radial flux balance, moving angular momentum
outward in some regions of the shell and inward in others. This opposing behavior
between the upper and lower convection zone arises from the meridional circulations
breaking into multiple cells in radius.
The balances achieved in the latitudinal transport in case G5 (Fig. 3.8d) are more
complex. As the rotation rate has increased, the total viscous transport has remained
nearly constant, with the significantly stronger gradients of angular velocity in the
differential rotation profiles offset by the lower turbulent diffusivities dictated by ourpath through parameter space. That these two opposing actions should conspire to
produce a nearly constant profile of viscous angular momentum transport is striking
and not intuitive. This is particularly apparent when we examine the two other terms
in the flux balance. The meridional circulations have reversed their role from our solarlike case G1 and now work with the Reynolds stresses to accelerate the equator andspin down the polar regions. The reduced contribution of the meridional circulations

64
to the total balance arises as the flows become both weaker and multi-celled in radiusand latitude. The smaller transport by Reynolds stresses appears to result from thedestruction by radial shear of some of the downflow plumes.

3.5

Differential Rotation and Scaling with Rotation
In analyzing our simulation results, it is the differential rotation established by

the convection that may yield the most direct contact with observations. Stellar observations across the HR diagram indicate that differential rotation is a common feature
in many stars, particularly stars of spectral class F and later. In the sun, differential
rotation has been measured throughout the bulk of the convection zone (as reviewed byThompson et al. 2003), but at present for more distant stars only the surface differential rotation can be inferred. A variety of observational techniques have been employed,
ranging from photometric variability studies (e.g., Donahue et al. 1996; Walker et al.
2007), Doppler imaging techniques (e.g., Donati et al. 2003) and Fourier transform
methods (e.g., Reiners & Schmitt 2003). Typically, these observations seek to measure
the amount of angular velocity contrast at the stellar surface, denoted as ∆Ω∗ , though
what is being measured may be somewhat uncertain. Variations in ∆Ω∗ have beenfound with both rotation rate and spectral type, however these quantities are correlated
and in observations to date it is difficult to disentangle their possible separate effects(Reiners 2006).
Different techniques measure fundamentally different tracers of surface differential rotation, either following variability of Ca emission (photometric), darkening frominferred starspot presence (photometric and Doppler imaging) or rotational broadeningof absorption lines of unspotted stars (Fourier transform methods). Each techniquealso is most applicable in only a limited region of stellar parameter space. As such,overlapping surveys are in short supply. Generally, most observations indicate that therelative shear, ∆Ω∗ /Ω∗ , depends on the stellar rotation rate Ω∗ as a power law, though65
different surveys find different scalings for the differential rotation, expressed as
∆Ω∗
∝ Ωn∗
Ω∗

(3.6)

(e.g., n = −0.3 ± 0.1 in Donahue et al. 1996 and n = −0.34 ± 0.26 in Reiners & Schmitt
2003, but n = −0.85 ± 0.10 in Barnes et al. 2005). Whereas some global models of
convection in more rapidly rotating stars have been conducted (e.g., Rüdiger et al.
1998; Küker & Stix 2001; Küker & Rüdiger 2005a,b), these have been largely carriedout in 2-D under the simplifying assumptions of mean-field theory.
The amount of latitudinal shear observed at the surface ∆Ω is an important
quantity both for interpreting stellar observations and for many dynamo theories. Here
we define ∆Ω more specifically as the difference in angular velocity between the equatorand say at 60◦ latitude, namely∆Ω = Ωeq − Ω60 ∝ Ωm
0 .

(3.7)

Going to higher latitudes yields comparable behavior. As shown in Figure 3.9, we findthat ∆Ω increases with rotation rate in our simulations, with m = 0.3 in the most
rapidly rotating simulations. The radial shear also increases with more rapid rotation,
and at the equator the difference between the mean angular velocity at top and bottomof the shell scales as m = 0.4 for the rapid rotators. Because m < 1, the relative
shear ∆Ω/Ωeq decreases with rotation rate Ω0 for the rapid rotators, in nearly a powerlaw fashion for the path through parameter space explored here. The scaling exponent
from Equation (3.6) exhibited by these cases is n = −0.6, but this may be influenced
by our choice in the scaling of diffusivities with rotation. We are encouraged that ourmore turbulent cases G3b and G5b, with the same diffusivities at different rotation
rates, exhibit similar behavior. Our choice of low Prandtl number also has an effecton this scaling (see Ballot et al. 2007). Additionally, different treatments of the SGS
unresolved flux, which has the most effect in the upper 10% of the convection zone, can

66

Figure 3.9 — Scaling of ∆Ω and ∆Ω/Ωeq with Ω0 . (a) Angular velocity contrast ∆Ω

in latitude between equator and 60◦ (diamonds) and in radius across the shell at theequator (blue triangles). The more rapidly rotating cases appear to follow a power law,which for the latitudinal contrast is m = 0.3 and for the radial contrast is m = 0.4 (asin eq. 5.1). (b) Relative latitudinal angular velocity contrast ∆Ω/Ωeq , with the shownpower law having n = −0.6 (as in eq. 3.6). The scaling may vary with the path inparameter space, as suggested by cases G3a, G3b and G5b.

alter the particular scaling law. The early simulations presented in Brown et al. (2004)
and shown in Figure 3.1, which had a much thicker unresolved flux layer, had a scalingof n = −0.8 in the rapid rotation limit. We have found that normalizing ∆Ω by Ω0
rather than Ωeq leads to a systematic offset for n of about -0.05 in the inferred scaling
law.

67

3.6

Meridional Circulations and Scaling with Rotation
The meridional circulations realized within our simulations are of significancesince they can variously transport heat, angular momentum and even magnetic fields
between the equator and the poles, though the latter are not included in the present
simulations. Our time and longitude-averaged meridional circulation patterns are shownin Figure 3.10 for cases G1, G5 and G10, depicted as streamlines of mass flux Ψ,r sin θhρ̄vr i = −

1 ∂Ψ
r ∂θ

and

r sin θhρ̄vθ i =

∂Ψ
∂r

(3.8)

and averaged here over a period of at least 150 days.
In our more rapidly rotating cases the meridional circulations have broken into
several cells strongly aligned with the rotation axis (Fig. 3.10b, c), particularly in theequatorial regions. Weak connections between the equatorial and polar regions persist
at the highest rotation rates studied, with organized flows along the tangent cylinder.
These internal flows weaken with more rapid rotation. The meridional circulations arecomplex and time dependent, with large fluctuations around the statistically-steadystates shown here, involving variations comparable to or larger than the mean values
themselves. The circulations are driven by small imbalances between relatively large
forces and their nature is subtle. The variation of the meridional flows near the surface(0.96R⊙ ) with rotation rate is shown in Figure 3.10d. The amplitude of the flows
decreases substantially with more rapid rotation. Peak velocities drop from 22 m s−1 incase G1 to 14 m s−1 in G5 and about 7 m s−1 in G10.
The total energy contained in these meridional circulations decreases quickly withmore rapid rotation, as shown in Figure 3.11. This drop in energy is independent of
the detailed structure of the convection, showing no change in behavior at the transition to spatially modulated convection. In contrast to the energy contained in convection (CKE) and differential rotation (DRKE), the energy in the meridional circulations(MCKE) is much less sensitive to the level of turbulence in any particular simulation, as

68Figure 3.10 — Changes in structure of meridional circulations with faster rotation.

Shown are profiles of time and azimuthally averaged meridional circulations with latitude and radius for (a) case G1, (b) case G5 and (c) case G10 with streamlines of massflux Ψ overlaid. Colors indicate the sense (red counter-clockwise, blue clockwise) andmagnitude of the meridional velocity hvmi = hvr ir̂ + hvθ iθ̂. With more rapid rotation the meridional circulation cells align strongly with the rotation axis and weakenin amplitude. (d) Amplitude of the mean latitudinal component vθ at the top of thesimulation for case G1 (blue), G5 (black), and G10 (red), with regions of poleward andequatorward flow denoted.

69

Figure 3.11 — Scaling of kinetic energy of meridional circulations (MCKE) with Ω0 . The
MCKE is normalized by that energy in case G1 at the solar rate (2.5 × 104 ergs cm−3 ).
The kinetic energy of these circulations decreases with rotation rate; a power law scalingof Ω−0.9
is shown for reference.
0

indicated by cases G3, G3a and G3b (detailed in Table 3.1). The meridional circulations
remain important to the global-scale dynamics as their gradual redistribution of angularmomentum contributes to the large angular velocity gradients in latitude. Yet they are
inefficient at transporting heat out of the star and at redistributing thermal material
to maintain the latitudinal gradients of temperature and entropy (which correspond to
the thermal-wind component of the achieved differential rotation).
This finding is in striking contrast to the assumptions of many Babcock-Leighton
dynamos, which often take the meridional velocity to scale as vm ∝ Ω or vm ∝ log Ω(e.g., Charbonneau & Saar 2001; Dikpati et al. 2001). In these models faster meridional
circulations lead to shorter dynamo cycles as surface flux is returned more rapidly tothe tachocline. Currently, the observational data does not appear good enough to distinguish between the competing models and we will have to await better measurementsof the scaling between cycle period and rotation rate and possible observations of themeridional circulations themselves (Rempel 2008). Existing 2-D mean-field models of

70
rapidly rotating suns (Küker & Stix 2001; Rüdiger et al. 1998; Küker & Rüdiger 2005a)
also predict an increase of meridional circulation velocities with more rapid rotation.
This is in contrast to their decrease in our simulations.

3.7

Energy Balances and Flux Transport
Convection is responsible for transporting the stellar flux emerging from the deep

interior through the convection zone. In these simulations, the total luminosity L(r)and its components are
Fe + Fk + Fr + Fu + Fν =L(r)= Ft ,
4πr 2

(3.9)

with
Fe = cp ρ̄vr T ′ ,Fk = 21 ρ̄vr v 2 ,Fr = −κr cp ρ̄ ddrT̄ ,Fu = −κ0 ρ̄T̄

dS̄
,
dr

(3.10)
Fν = −v · D r ,(3.11)

where Fe is the enthalpy transport by convective motions, Fk is the kinetic energy flux,
Fr is the transport by radiation, Fu is the unresolved SGS heat flux for parametrizedtransport by scales of motion below the resolution of our simulation and Fν is the
SGS viscous flux. Figure 3.12 shows the flux balance with radius achieved in cases G1
and G5, averaged over horizontal surfaces and converted to relative luminosities. In
the deepest layers the radiative flux becomes significant as the radiative conductivity
steadily increases with depth. By construction this flux suffices to carry all of theimposed flux through the lower boundary, where the radial velocities and convective
flux vanish. A similar role is played near the top of the convection zone by the sub-gridscale transport which yields Fu . The main role of Fu is to transport energy outward
through the impenetrable upper boundary where the convective fluxes vanish and theremaining fluxes are small, thereby avoiding the building of strong superadiabatic radial
gradients there.

71
The functional form of κ0 (r) is chosen so that the entire stellar luminosity will betransported at the surface of the simulation by Fu . A subtlety of this treatment of the
SGS flux lies in dS̄/dr. In the more rapidly rotating simulations, we find that convectionis less able to establish an adiabatic profile throughout the convection zone. Instead,much of the convection zone remains slightly superadiabatic (dS̄/dr < 0). This property
is also realized in local-domain simulations of rapidly rotating convection, where more
rapid rotation leads to enhanced horizontal mixing through vortex interactions and a
resulting decrease in enthalpy transport as vertical velocities and thermal perturbations
become more decorrelated (Julien et al. 1996; Brummell et al. 1996). The change ofdS̄/dr with rotation rate is shown in Figure 2.1a for four of our simulations. Fixing
the amplitude and structure of κ0 (r) across simulations leads Fu to influence a greater
portion of the convection zone at more rapid rotation, as indicated by the slight growthof Fu in case G5. Though this effect becomes stronger as our simulations rotate more
rapidly, at no point in these simulations does Fu transport more than 10% of the totalluminosity at mid convection zone.
In all these simulations, the strong correlations between radial velocities and temperature fluctuations yield the enthalpy flux Fe , which dominates the energy transport
at mid convection zone. Both warm upflows and cool downflows serve to transport flux
out of the star, and the two carry comparable amounts of flux to one another in the
rapidly rotating simulations. In going to the more rapidly rotating simulations, we findthat the average convective enthalpy flux through the polar regions is greater than thatthrough the equator. This is shown in Table 3.1. In case G1 (at the solar rate) slightly
more flux is transported through the equator than the poles, but as the rotation rateincreases significantly more flux is transported through the poles than the equator. This
latitudinal variation becomes somewhat weaker in our more turbulent cases (G3a, G3b
and G5b), but remains present. Interestingly, cases G7 and G10 follow similar trends intheir equatorial enthalpy transport, despite the emergence of strong nests of convection

72

Figure 3.12 — Average radial energy fluxes in cases G1 and G5.

Energy fluxes areshown with radius with case G1 (dashed) and G5 (solid) overlain for comparison. Shownare the radiative luminosity (Fr ), convective enthalpy transport (Fe ), unresolved flux
(Fu ), kinetic energy transport (Fk ), and the total flux through the simulation (Ft ),all normalized by L⊙ /(4πr 2 ). Case G1 has a large positive convective enthalpy flux
throughout the domain, with the excess in luminosity largely balanced by an inward
flux of kinetic energy. At higher rotation rates (as in G5), the kinetic energy flux isnearly zero throughout the domain. In both cases Fν is negligible and not shown.

at the equator and the suppression of convection in the rest of that region.
In case G1, convection transports slightly more enthalpy than the solar luminosity.
This over luminosity is balanced by an inward transport of kinetic energy, which is primarily due to compressible effects and the transport of vr2 within strong downflows thatspan the convection zone and feel the full density stratification (Hurlburt et al. 1986).
In the more rapidly rotating cases, the prominent differential rotation shears apart theconvection cells. The downflows lose much of their coherence and only the strongest
downflows within the nests of localized convection survive to span the full convection
zone. Individual downflows thus feel less density stratification, and compressible effects
become less important, leading to a balance in the transport of vr2 between the upflows
and downflows. Instead, as shown in Table 3.1 the sense of Fk reverses in the equatorial
regions, becoming dominated by the outward transport of vφ2 . The polar regions remain
largely unchanged and the total Fk across spherical surfaces is nearly zero.

73

Table 3.1.

Flux Balances and Energies

CaseFe,pole aFe,eq aFk,pole a

Fk,eq aCKEb DRKEb MCKEb ∆Tmax c

G1
G2
G3
G4
G5
G7
G10

1.014
1.300
1.349
1.327
1.329
1.298
1.236

1.140
0.684
0.628
0.631
0.625
0.581
0.623

-0.118-0.088-0.077-0.065-0.079-0.097-0.093-0.190
0.046
0.071
0.073
0.078
0.080
0.090

3.28
2.64
2.40
2.21
2.11
1.69
1.51

2.26
13.2
20.5
25.5
30.1
38.7
47.1

0.025
0.015
0.011
0.009
0.007
0.005
0.003

5.50
28.0
53.5
78.5
107
171
271

G3a
G3b
G5b

1.268
1.101
1.172

0.655
0.780
0.668

-0.071-0.065-0.047

0.068
0.043
0.081

2.73
3.34
2.44

27.7
37.9
57.4

0.012
0.013
0.008

58.7
62.4
134

a Average convective enthalpy (F ) and kinetic energy (F ) fluxes at mid-layerekscaled by the solar flux, shown for polar (latitudes above ±60◦ ) and equatorial(from ±30◦ ) regions.

b Kinetic energy density relative to the rotating coordinate system, for convec-

tion (CKE), differential rotation (DRKE) and meridional circulations (MCKE),
averaged over the full shell and over ∼ 300 days; units are 106 erg cm−3 .
c Maximum temperature contrast at 0.96R

pole and ±40◦

⊙ in K, typically occurring between

74
Volume-averaged energy densities for a selection of our simulations are shown′in Table 3.1. At the solar rotation rate, convective kinetic energy (with kernel 21 ρ̄v 2
and labeled CKE) and the kinetic energy in the average differential rotation ( 21 ρ̄hvφ i2 ,DRKE) are comparable. As the rotation rate is increased, DRKE grows strongly andconvective energy decreases slightly, leading DRKE to dominate the total energy budget.
This is true even in our significantly more turbulent solutions. The energy in meridional
circulations ( 12 ρ̄(hvr i2 +hvθ i2 ), MCKE) is always small, and decreases in both magnitude
and percentage of the total energy with more rapid rotation.

3.8

Conclusions
When stars like our sun are young they rotate much more rapidly than the present

sun. In these stars rotation must strongly influence the convective motions and maylead to stronger global-scale dynamo action. In this chapter, we have explored theeffects of more rapid rotation on global-scale convection in simulations of stars like oursun. The mean zonal flows of differential rotation become much stronger with more
−0.6
. In striking contrast, therapid rotation, scaling as ∆Ω ∝ Ω0.3
0 or as ∆Ω/Ωeq ∝ Ω0

meridional circulations become much weaker with more rapid rotation, and the energy
contained in them drops approximately as Ω−0.9. Accompanying the growing differential
0
rotation is a significant latitudinal temperature contrast, with amplitudes of 100 K or
higher in the most rapidly rotating cases. The maximum temperature contrast near thesurface occurs between the hot poles and the cool mid latitudes at about ±40◦ . If thislatitudinal 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.

75
These simulations are entirely hydrodynamic and this provides a major caveat to
our findings on the scaling of differential rotation and latitudinal temperature contrast
with rotation rate Ω0 . Prior MHD simulations of stellar convection have demonstrated
that in some parameter regimes strong dynamo-generated magnetic fields can react backstrongly on the differential rotation, acting to lessen angular velocity contrasts or largely
eliminate them (e.g., Brun et al. 2005; Featherstone et al. 2007; Browning 2008). It is
unclear whether the scaling trends identified here for differential rotation as a functionof Ω0 will survive in the presence of dynamo action and magnetic fields. Likewise,
magnetic fields may lessen the strong temperature contrasts realized here, doing sothrough their feedback on the convective flows and energy transport. We expect that the
weaker meridional circulations may be less affected by magnetic feedbacks, and thus theprediction that meridional circulations lessen in energy and amplitude with more rapid
rotation may be of greater significance though harder to confirm observationally. 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 solar-type stars. To answer such questions, we will turn in due course
in Chapters 5-8 to MHD simulations that capture dynamo action in the bulk of theconvection zone.

Chapter 4
Dynamics Within Confined Nests of Convection

The emergence of localized nests of convection at higher rotation rates is a striking
feature that calls out for an explanation. In many ways, it is quite surprising thatconvection chooses to be confined to narrow intervals in longitude, but such states havealso been realized in a number of other dynamical systems. Generally the appearanceof nests is a challenge to explain in detail, yet the onset of spatially modulated states
which are their precursor is better understood.

4.1

Spatially Localized Convection in Other Settings
The phenomena of spatially localized convection has a rich history, variously ap-

pearing in laboratory experiments and numerical simulations. Much interest in confinedstates of convection was motivated by the discovery of such states in binary fluid convection (e.g., Anderson & Behringer 1990; Kolodner & Glazier 1990; Niemela et al. 1990;
Surko et al. 1991), where traveling waves of convection appear via subcritical Hopf
bifurcations and near onset are seen to evolve into traveling patches of convection separated by regions of nearly quiescent fluid. From a theoretical perspective, these confinedstates near onset are accessible to weakly nonlinear theory and considerable progress has
been made in understanding their nature (e.g., Riecke 1992; Barten et al. 1995; Batiste& Knobloch 2005; Batiste et al. 2006; Burke & Knobloch 2007). The confined statesfound in binary convection differ from our active nests of convection in several respects.

77
The most important is that in binary convection there is no net vertical transport ofsolute. The confined states instead pump solute horizontally and create regions of stablevertical stratification in the quiescent regions. A possibly related phenomena is that oflocalized states in magnetoconvection, as studied by Blanchflower (1999) and in 3-D byBlanchflower & Weiss (2002). Here single convective cells (called “convectons”) formed
in a region of initially uniform strong vertical magnetic field by a process of flux expulsion. Convection within the localized states was strong and was entirely suppressed in
the surrounding medium. These convectons could contain several convective cells andwere generally stationary, though some solutions exhibited time dependent behavior.
Recently, progress has been made addressing these systems in approximate 2-D models(Dawes 2007).
Confined states are also realized in other doubly diffusive systems, as in theoreticalstudies of thermosolutal convection (Knobloch et al. 1986; Deane et al. 1987, 1988; Spinaet al. 1998). In the latter studies a variety of traveling convective patches were found,and in these the convective transport of heat and solute was enhanced compared to thatin the interpatch regions. In all cases the patches propagated in the same direction as
the individual convective cells, though more slowly. Such behavior persisted for long
periods of time. These localized states occurred well above the onset of convection,and convection continued in the interpatch regions. There may also be analogues inconvection within the Earth’s atmosphere, where deep convection in the tropics tends tobe organized on global scales into regions of locally enhanced convection which propagate
in a prograde sense. These organized convective structures are called the Madden-Julien
Oscillation and appear to have their origin in the coupling of convective motions withequatorially trapped waves (perhaps Rossby or Kelvin waves; see review by Zhang 2005).
The spatially modulated states in our simulations of stellar convection exist atRayleigh numbers far above the onset of convection. Spatially modulated states inthis parameter regime have also been observed in geophysically motivated 3-D Boussi-

78
nesq simulations of convection within a thick, rotating spherical shell (Grote & Busse
2000; Busse 2002; Busse & Simitev 2005). In these studies, spatially localized states
emerged at moderate Rayleigh numbers, involving an equilibrium between the shearing
flow of differential rotation destroying convective eddies and the Reynolds stresses in
the convection driving the differential rotation. These effects led to localized states
where convection occupied a limited portion of the domain and the region outside theconvective patch was filled with quiescent streaming flow and almost no radial motions.
In the quiescent regions the thermal gradients become increasingly unstable until they
are advected back into the patch where they help sustain the convective eddies. The
patches in these geophysical simulations moved slowly retrograde and persisted up toRayleigh numbers of about 106 . Beyond this point the differential rotation became sostrong that no sustained convection was possible. Instead the system began to behave
as a relaxation oscillator, with short bursts of convection temporarily building a strong
differential rotation which then sheared out the convective eddies. Convective transportremained suppressed until the shear of differential rotation decayed viscously, whereupon a new burst of convection would begin the process anew. In all cases with localized
convection, significant oscillations were seen in the kinetic energies of both differential
rotation and convection (Grote & Busse 2000).
Similar states have also been realized in anelastic simulations of stellar convectionin spherical shells for a rotating younger sun with a much thicker convection zone (Ballot
et al. 2006, 2007). The spatially modulated states found there appear in the equatorialconvection for models with low Prandtl number (Pr = 0.25, as in the models of thispaper). Localized states turn into bursty, vacillating convection at large Taylor numbers(Ta & 109 ), much like those in Grote & Busse (2000). Localized states observed inthick convective shells (with typical aspect ratios χ = rbot /rtop = 0.58) differ in many
important respects from the states we find in our relatively thin shells of convection(χ = 0.76), most notably in their temporal stability, which we will next discuss.

79

4.2

Properties of the Active Nests
Our active nests of convection appear first as regions of mildly enhanced con-vective amplitude. As the rotation rate increases, convection in the equatorial regions
gradually becomes more localized (as in Fig. 3.2) and eventually is present only within
the active nest. In some of our systems we observe two nests or patches in longitude(such as case G5) and in some only a single nest (as in case G10). Generally, convection
at the highest rotation rates possesses only a single nest, and for moderate rotation rates
the system can alternate between two-nest states and single-nest states, suggesting thatseveral attractors exist within the phase space.
To study the temporal evolution of our convective patterns in more detail, we
employ time-longitude maps as shown in Figure 4.1. Here radial velocities are sampled
at the equator for all longitudes, considering case G5 at three depths (near the top,
middle and bottom of the convection zone) and doing so over an interval of 260 days
(or about 45 rotation periods). In these maps, structures propagating in a prograde
fashion relative to the frame of reference are tilted to the upper-right and patterns
propagating in a retrograde sense tilt to the upper-left. To construct these maps wehave chosen a frame of reference propagating in a prograde sense relative to the bulk
rotation frame (Ω0 = 1.3 × 10−5 rad s−1 or 2070 nHz), with constant relative angularvelocity 6.75 × 10−7 rad s−1 (107 nHz, for a total rotation rate of 1.052 Ω0 ). This
corresponds to the propagation rate of the modulated convection pattern, and thus
the nests appear stationary in this frame. In the Ω0 reference frame the nest takes
about 108 days to complete one lap around the equator, thus for the time interval
shown in Figure 4.1 the nests travel about 870◦ in longitude and completes about 2.4
circuits around the equator. The active nests of convection can persist for extremely
long periods of time. The two nests visible in Figure 4.1 remain coherent for over 5000
days of simulated time.

80

Figure 4.1 — Nests of convection in G5 shown in time-longitude maps at three depths.
Radial velocity vr is sampled (a) near top (0.95 R⊙ ), (b) middle (0.85 R⊙ ) and (c) bottom(0.73 R⊙ ) of layer. These samples are extracted at the equator, using a reference frame
tracking the nests and starting from a mature state in the simulation. Two nests arevisible, and individual convective cells appear as paired upflows and downflows, which
propagate slightly faster than the mean zonal flow that they establish and thus passthrough the nests. (d, e, f ) Time averages with longitude of entropy fluctuations Se and
e in these samples.
convective kinetic energy density K

81
Individual convective upflows and downflows appear as streaked red and blue regions respectively. In the upper convection zone (0.95R⊙ ), the convective cells propagate
more rapidly than the nests of convection. Here they overtake the nests from behind(from smaller longitudes) and then slowly swim through at a reduced speed. Within
the nests convective cells collide and interact, and the strongest survive to emerge fromthe front of the nest, where they speed up as they then propagate through the morequiescent region. When this occurs, typically a small wave train comprised of 2 to 3
upflow/downflow pairs escapes, and as they enter the quiescent regions these convective cells speed up to once again outpace the zonal flow. In the lower convection zone
(0.73R⊙ ), the nests propagate more rapidly than the convective cells, and the individual
upflows and downflows appear as strong retrograde-directed streaks. Also visible in thelower convection zone are low-amplitude velocity structures of rapid retrograde propagation. They appear to be the weak equatorward extension of the large-scale (retrograde
rotating) polar patterns evident in Figure 3.5. At mid-convection zone (0.85R⊙ ), nearly
all vertical flow is occurring within the nests of active convection, though the strongest
cells outside the nest in the upper convection zone manage to weakly print down to thisintermediate depth.
The typical angular velocities of the differential rotation, the individual convective
cells and the active nests of convection are shown in Figure 4.2 for case G5 and detailed
for several cases in Table 4.1. The nests of convection propagate at a constant angularvelocity at all depths in the convection zone and over the entire range of latitudes (±30◦ )where they are present. In contrast, the angular velocity Ω of the differential rotation
varies substantially with radius and latitude. At all depths, the individual convective
cells propagate more rapidly than the zonal flow of differential rotation which they
drive. Because the nests of convection propagate at an intermediate prograde rate, theyexperience a head-wind from the differential rotation in the deep convection zone and a
tail-wind near the surface. Despite this strong radial shear, the nests remain coherent

82

Figure 4.2 — Angular velocities of various structures with radius in case G5. The mean
shearing zonal flow of differential rotation Ω is denoted for all depths at the equator(solid curve) and at ±15◦ latitude (dotted). The propagation rate of the active nests(diamonds) is constant across the entire latitude and radial range where they appear(±30◦ ). The average velocity of individual convective cells at the equator (asterisks) ismore rapid than the mean zonal flow they establish. Their propagation is faster than
the nests near the top and slower near the bottom. The global rotation rate Ω0 is 2070
nHz (marked).

across the entire convection zone for long periods compared with either the lifetimeof individual convective cells (10-30 days), the rotation period of the star (5 days) or
typical thermal and viscous diffusion times (τκ = 910 days, τν = 3640 days, both at
mid-depth). The time for the differential rotation to lap the nests at the equator isabout 112 days near the surface and 143 days at the bottom of the shell.
At the higher rotation rates, single stable nests of convection dominate the equatorial region. Time-longitude maps are shown in Figure 4.3 for the equatorial radial
velocity at two depths in case G10, our most rapidly rotating simulation. Within thenest, convection remains vigorous, with the strongest downflow networks still spanningthe entire depth of the convection zone. This nest again propagates in a prograde sense
relative to Ω0 , with constant relative angular velocity 8.3 × 10−7 rad s−1 (132 nHz, for83

Table 4.1.

Angular Velocities of Various Structures

Case

Ω0

Ωnest

Ωeq,topΩeq,botΩc,topΩc,bot

G1
G3
G5
G10

2.60
7.80
13.00
26.00

—
0.511
0.675
0.830

0.341
1.094
1.319
1.497

0.060
0.220
0.197
0.140

0.535
1.176
1.381
1.623

0.180
0.330
0.288
0.280

G3a
G3b
G5b

7.80
13.00
13.00

0.690
0.670
0.750

1.295
1.555
1.778

0.265
0.250
0.246

1.421
1.590
1.800

0.373
0.413
0.430

Note. — All angular velocities in µrad s−1 and, except for
the frame rate Ω0 , are given relative to Ω0 . The differential rotation at the equator Ωeq is measured at 0.95R⊙ (top)and 0.73R⊙ (bot). The mean propagation rate of individual convective structures Ωc is measured at the same depths
in time-longitude maps of vr and has a typical variance of±0.04 µrad s−1 .

84Figure 4.3 — Time-longitude map of single nest of convection in case G10.

As in
Fig. 4.1, radial velocity vr is shown (a) near top and (b) near bottom of layer. Herea single nest is realized, with individual convective cells almost entirely confined toethe nest, though they continue to propagate within it. (c, d) Time averages Se and K
constructed in the same reference frame.

85
a total rotation rate of 1.032 Ω0 ). Over the interval shown the nest completes almost 3
circuits of the equator relative to the Ω0 reference frame. Individual convective cells continue to swim through the nest, moving more rapidly near the surface and more slowly
in the deep convection zone. The very strong radial shear prevents all but the strongestof downflows from spanning the full convection zone. In the region outside the nest,
convection is strongly suppressed and the main features are the weak flows associatedwith the retrograde propagating polar pattern. In the upper convection zone, occasional
weak convective disturbances appear upstream of the nest. As these fluctuations enter
the nest they grow in amplitude.
Our nests of active convection may owe their existence to a competition between
the shearing action of differential rotation acting on the individual convective eddies
and Reynolds stresses within the convection helping to maintain the strong zonal flows.
Unlike the systems studied by Grote & Busse (2000) and Ballot et al. (2007), our patchy
convection is not accompanied by relaxation oscillation behavior or large exchanges
between the kinetic energy in convection and in the differential rotation. Rather, ournests are not bursty in time and instead persist for long periods in quasi-steady states.
This is true even for our most rapidly rotating case considered here (G10) and atmuch higher turbulent driving (G5b), though these simulations exist at Taylor numbers
below the threshold suggested by Ballot et al. (2007) (Ta & 109 ). Coupling between
equatorially trapped waves and convection may also have a role, but the contributionof this coupling to spatial localization has been difficult to elucidate.

4.3

Detailed Structure of an Active Nest of Convection
We focus here on the structure of the nests so evident in Figures 4.1 and 4.3 forcases G5 and G10, but devote particular attention to the single nest realized in thelatter as a representative case. The active nests extend throughout the depth of theconvection zone. This is illustrated most clearly in Figure 4.4a, showing the vertical

86

Figure 4.4 — Detailed structure of convective nest in case G10. This snapshot of nest

structure is shown at relative day 221 of Fig. 4.3. (a) Radial velocities in an equatorial
cut shown for all longitudes and radii. (b) Companion entropy fluctuations S about
their spherical means S̄ (in cgs units), with high entropy fluid in red tones. (c) Radial
velocity in the equatorial region shown near the top of layer.

87
profile of radial velocities with longitude in a cut at the equator. Convection is broken
into multiple cells, one set above 0.9R⊙ and another below 0.8R⊙ . Only within the nest
do strong downflows span the convection zone. The nest is embedded in a region ofstrong latitudinal and radial zonal shear (as shown by the tilted nature of its structure),
yet the pattern propagates at a constant angular velocity at all depths and latitudes.
At the surface and near the equator the nest propagates more slowly than the zonal
wind, whereas at the base of the convection zone it propagates more rapidly. Individual
convective structures swim still more rapidly and both enter and exit the region ofenhanced convection. As such, the nest experiences a strong retrograde flow at the baseand a strong prograde flow near the surface.
The thermal structure of the nests is revealed by their entropy fluctuations, as
shown in Figure 4.4b. In the upper convection zone, the mean zonal flow advects
low entropy fluid into the nests from the left side. This fluid is then swept away byintermittent downflows and replaced by higher entropy fluid from below. In the lower
convection zone, higher entropy fluid is swept into the nest from the right and lower
entropy exits to the left. At mid convection zone, regions outside the nest remain
convectively unstable, but only weak radial motions are driven here.
The mean longitudinal structure of the nests can be assessed by forming temporal
averages of various properties in a frame co-rotating with the nests of convection (these
averages denoted by a tilde). This is done at the equator for entropy Se and convectivee (with same form as CKE) for case G5 in Figure 4.1d − f andkinetic energy density K

for case G10 in Figure 4.3c, d. There are distinctive differences between the leading (tothe right) and trailing (to the left) portions of the profiles, with Se showing a gradual riseand steeper drop in going to decreasing longitudes (from right to left). The envelope of
Se is largely similar in form at the top and bottom of the convection zone. In contrast

e which traces the fluctuating velocities of convection, changes form with depth, being
K,

skewed in the direction of the mean zonal flow. At the top of the convection zone this88

Figure 4.5 — Mean circulations associated with the nest in case G10. (a) Time averaged
radial (colors) and horizontal flows (arrows) near the top and (b) near the bottom of
the shell, in a reference frame tracking the nest. The strong zonal flow of differential
rotation has been subtracted at each latitude and is shown by the solid curves (bothscaled by 125 m s−1 arrow length, with zero velocity relative to the nest indicated by
the vertical line). Within the nest strong eddy currents partially decelerate the flow,
while outside the streaming mean zonal flow dominates.

89
sense of skew is toward the right, and at the bottom it is toward the left.
These nests must have gradual circulations associated with them, though these
are a challenge to discern as they are much weaker than the strong mean zonal flows.
We can get some sense of them by averaging the vertical and horizontal flows in thesurroundings of a nest while tracking it over long time intervals. Shown in Figure 4.5 are
the slow circulations associated with the nest in case G10 near the top and bottom of theconvection zone. These flows have been averaged over a period of 615 days starting at
the beginning of the interval shown in Figure 4.3. The flows suggest a systematic zonal
circulation from fore to aft in the upper convection zone and toward higher latitudes
and diverging from the nest at depth, though to achieve this we have subtracted thefast mean zonal flow which varies with latitude as shown. The mean upflows anddownflows in the nest are quite weak, with amplitudes of a few m s−1 , as compared withthe convection which can have peak velocities of a few hundred m s−1 . Test particles
released in the flow would not simply circulate according to these mean circulations
and would be instead swept along by the strong mean zonal flow and of course by thevigorous convection cells that propagate through the nest. Some test particles would
encounter strong downflows and would be swept down through the nest to the bottomof the convection zone while others would likely be carried horizontally out the nest andremain at a similar depth. Yet Figure 4.5 indicates that a weak large-scale circulationis realized, and this may serve to slowly pump fluid through the system. These mean
circulatory flows may also serve to inform analytic models of such nests of convection.

90

4.4

Conclusions
A striking feature of these simulations is the emergence of a pattern of strongly

modulated convection in the equatorial regions. These nests of active convection are
regions of enhanced convective vigor and transport which propagate at rates distinct
from either the mean zonal flows of differential rotation or the individual convective
cells. In the most rapidly rotating systems, such as case G10, convection at the equatoris entirely dominated by motions inside the nest with only very weak radial motions
present in the regions outside the nest. Though their impact on the convection is mostobvious in the rapidly rotating limit, we find some evidence for weak modulation evenin our more slowly rotating cases.
All of our simulations stop short of the turbulent stellar surface, and it is thus
difficult to estimate how these nests of active convection may affect stellar observations
in detail. The convective velocities associated with the nests are small compared to thenearly supersonic flows in stellar granulation, and in the Sun such global-scale convective structures have evaded direct detection despite intensive searches throughout thenear-surface layers by helioseismology. The extremely localized nests found in our most
rapidly rotating cases may however influence the thermal stratification and thus convective vigor in the near surface regions, as most of the flux at the equator is transported
through a narrow range of longitudes. These nests may act as traveling hot spots withenhanced convection even in the surface layers where the higher emerging flux escapes
the system.
These spatially localized states of convection may also have some bearing on theactive longitudes of magnetic activity observed in the Sun, if the enhanced pummelingof the tachocline by the convection within the nest preferentially destabilizes magnetic
structures within the tachocline that then rise to the surface. If they do survive in amagnetic environment, then their strongest signature is likely to emerge in magnetic

91
stars, where magnetic fields threading the bulk of the convection zone may be concentrated in the nests and mimic giant, propagating starspots which survive for very longepochs in time. If the nests lead to active longitudes of enhanced magnetic activity inrapidly rotating stars, we might expect these long-lived magnetic structures to propagate at a rate different from the stellar rotation rate as measured at the surface or fromthe stellar differential rotation.
We recognize that our simulations remain separated by many orders of magnitude
from the parameter space of real stellar convection. As such, we must be cautious with
our interpretations of the overall dynamics. However, we have found that these nestsof convection are a robust feature over a range of parameters and that they are able
to persist as entities for as long as we could pursue their modelling. Thus one should
be prepared to consider the possibility of their presence also in real stellar convection
zones, where they may appear as long-lived propagating features.
Our current family of dynamo simulations indicate that nests of convection cancoexist with magnetism in portions of parameter space. We will explore this possibility
in Chapter 8, but first we turn to a more general exploration of dynamo activity in these
rapidly rotating suns.

Chapter 5
Global Dynamo that Builds Persistent Wreaths of Magnetism

Having explored the coupling of convection and rotation in our hydrodynamic
simulations, it is now time to turn to the question of possible dynamo action in rapidly
rotating suns. The hydrodynamical simulations showed us that the shearing flows ofdifferential rotation generally grow in amplitude with more rapid rotation, possessing
rapid equators and slower poles, while the meridional circulations weaken and break up
into multiple cells in radius and latitude. More rapid rotation can also substantiallymodify the patterns of convection in a surprising fashion. With more rapid rotation,
localized states begin to appear in which the convection at low latitudes is modulated
in its strength with longitude. At the highest rotation rates, the convection can become
confined to active nests which propagate at distinct rates and persist for long epochs.
Motivated by these discoveries, we turn here to explorations of the possible dynamo action achieved in solar-type stars rotating at three and five times the current
solar rate. Magnetism leads to strong feedbacks on the flows, particularly modifying
the differential rotation and its scaling with overall rotation rate Ω0 . The magnetic
fields which form in these dynamos have prominent global-scale organization within theconvection zone, in contrast to previous solar dynamo simulations (Brun et al. 2004;
Browning et al. 2006). Quite strikingly, we find that coherent global magnetic structures arise naturally in the midst of the turbulent convection zones. These wreath-likestructures are regions of strong longitudinal field Bφ organized loosely into tubes, with

93
fields wandering in and out of the surrounding convection. These wreaths of magnetism
differ substantially from the idealized flux tubes supposed in many dynamo theories,
though they may be related to coherent structures achieved in local simulations of dynamo action in shear flows (Cline et al. 2003; Vasil & Brummell 2008, 2009). Here we
explore the nature of magnetic wreaths realized in our global simulations, and discuss
their temporal behavior.
The contents of Chapters 5, 6 and 7 are based on work submitted for publicationas Brown et al. (2009)1 and are mainly a restatement of that paper under review. As theprimary author of this paper, I conducted the simulations presented here, performed theanalysis and wrote the text. My co-authors provided advice and guidance throughout
the process, helping frame the questions which form the core of the study. Preliminary
versions of these results have also been presented in Brown et al. (2007b), Brown et al.
(2007c), Brown (2009a), and Brown (2009b).
We begin by turning to case D3 whose formulation is shown in Figure 2.5. This
dynamo yields fairly persistent wreaths of magnetism in its two hemispheres, thoughthese structures did wax and wane somewhat in strength once established. Examining
the properties of this dynamo solution helps to provide a perspective for the greater
variations realized in our more rapidly rotating case D5.

5.1

Patterns of Convection in Case D3
The complex and evolving convective structures in our dynamo cases are sub-stantially similar to the patterns of convection found in our hydrodynamic simulations.
Our dynamo solution rotating at three times the solar rate, case D3, is presented in Figure 5.1, along with its hydrodynamic progenitor, case H3. The radial velocities shownnear the top of the simulated domain (Figs. 5.1a, e) have broad upflows and narrow
1

Brown, B. P., Browning, M. K., Brun, A. S., Miesch, M. S., & Toomre, J., 2009, Persistent wreathes
of magnetism in a rapidly rotating sun, ApJ, under review.

94

Figure 5.1 — Convective structures and mean flows in cases D3 and H3. (a) Radial
velocity vr in dynamo case D3, shown in global Mollweide projection at 0.95R⊙ , with
upflows light and downflows dark. Poles are at top and bottom and the equator is the
thick dashed line. The stellar surface at R⊙ is indicated by the thin surrounding line.
(b) Profiles of mean angular velocity Ω(r, θ), accompanied in (c) by radial cuts of Ωat selected latitudes. A strong differential rotation is established by the convection.
(d) Profiles of meridional circulation, with sense of circulation indicated by color (redcounter-clockwise, blue clockwise) and streamlines of mass flux overlaid. (e − h) Companion presentation of fields for hydrodynamic progenitor case H3. The patterns ofradial velocity are very similar in both cases. The differential rotation is much stronger
in the hydrodynamic case and the meridional circulations there are somewhat weaker,
though their structure remains similar.

downflows as a consequence of the compressible motions. Near the equator the convection is aligned largely in the north-south direction, and these broad fronts sweep throughthe domain in a prograde fashion. The strongest downflows penetrate to the bottomof the convection zone; the weaker flows are partially truncated by the strong zonal
flows of differential rotation. In the polar regions the convection is more isotropic andcyclonic. There the networks of downflow lanes surround upflows and both propagate
in a retrograde fashion.

95
The convection establishes a prominent differential rotation profile by redistributing angular momentum and entropy, building gradients in latitude of angular velocity
and temperature. Figures 5.1b, f show the mean angular velocity Ω(r, θ) for cases D3
and H3, revealing a solar-like structure with a prograde (fast) equator and retrograde(slow) pole. This property is also realized for cases D5 and H5 with faster rotation. Figures 5.1c, g present in turn radial cuts of Ω at selected latitudes, which are useful as we
consider the angular velocity patterns realized here with faster rotation. These Ω(r, θ)
profiles are averaged in azimuth (longitude) and time over a period of roughly 200 days.
Contours of constant angular velocity are aligned nearly on cylinders, influenced by the
Taylor-Proudman theorem.
As discussed in Chapter 1, helioseismology has revealed that in the Sun the contours of angular velocity are aligned almost on radial lines rather than on cylinders. The
tilt of Ω contours in the Sun may be due in part to the thermal structure of the solar
tachocline, as first found in the mean-field models of Rempel (2005) and then in 3-D
simulations of global-scale convection by Miesch et al. (2006). In those computations,
it was realized that introducing a weak latitudinal gradient of entropy at the base of
the convection zone, consistent with a thermal wind balance in a tachocline of shear,can serve to tilt the Ω contours toward a more radial alignment without significantly
changing either the overall Ω contrast with latitude or the convective patterns. We
expect similar behavior here, but at present, observations of rapidly rotating stars only
measure differential rotation at the surface and do not offer constraints on either the
existence of tachoclines in young suns or the nature of their internal differential rotation
profiles. As such, we have neglected the possible tachoclines of penetration and shear
entirely in these models and instead adopt the simplification of imposing a constantradial entropy gradient at the bottom of the convection zone.
The differential rotation achieved is stronger in our hydrodynamic case H3 thanin our dynamo case D3. This can be quantified by measurements of the latitudinal

96
angular velocity shear ∆Ωlat . Here, as in Chapter 3 we define ∆Ωlat as the shear near
the surface between the equator and a high latitude, say ±60◦∆Ωlat = Ωeq − Ω60 ,(5.1)

and the radial shear ∆Ωr as the angular velocity shear between the surface and bottomof the convection zone near the equator∆Ωr = Ω0.97R⊙ − Ω0.72R⊙ .

(5.2)

We further define the relative shear as ∆Ωlat /Ωeq . In both definitions, we average themeasurements of ∆Ω in the northern and southern hemispheres, as the rotation profile
is often slightly asymmetric about the equator. Case H3 achieves an absolute contrast∆Ωlat of 2.22 µ rad s−1 (352 nHz) and a relative contrast of 0.247. The strong globalscale magnetic fields realized in the dynamo case D3 serve to diminish the differential
rotation. As such, this case achieves an absolute contrast ∆Ωlat of only 1.17 µ rad s−1(186 nHz) and a relative contrast of 0.136. This results from both a slowing of theequatorial rotation rate and an increase in the rotation rate in the polar regions. These
results are quoted in Table 5.1, along with related measurements for our five solar
dynamo case D5 and hydrodynamic cases H3 and H5. It is interesting to note thatin both dynamo cases the amount of latitudinal and radial shear is almost the same,
whereas in the hydrodynamic simulations the more rapidly rotating case H5 has larger
angular velocity contrasts than the slower case H3.
The meridional circulations realized in the dynamo case D3 are very similar to
those found in its hydrodynamic progenitor (case H3). As illustrated in Figures 1d, h,the circulations are multi-celled in radius and latitude. The cells are strongly alignedwith the rotation axis, though some flows along the inner and outer boundaries crossthe tangent cylinder and serve to couple the polar regions to the equatorial convection.
Flows of meridional circulation are slightly stronger in the dynamo cases than in thepurely hydrodynamic cases, and these flows weaken with more rapid rotation.

97

Table 5.1.

Near-surface ∆Ω in Cases at 3 and 5 Ω⊙

Case

∆Ωlat

∆Ωr

∆Ωlat /Ωeq

Epoch

D3
D5avg
D5min
D5max
H3
H5

1.17
1.14
0.91
1.43
2.22
2.77

0.71
0.71
0.39
0.98
0.94
1.31

0.136
0.083
0.067
0.102
0.246
0.192

6460-6920
3500-5700
3702
4060
-

Note. — Angular velocity shear in units ofµrad s−1 , with ∆Ωlat and ∆Ωlat /Ωeq measured near
the surface (0.97R⊙ ) and ∆Ωr measured across thefull shell at the equator. For the dynamo cases, thesemeasurements are taken over the indicated range ofdays. In oscillating case D5, these measurements areaveraged over a long epoch (avg), and are also taken
at two instants in time when the differential rotation is particularly strong (max) and when magnetic
fields have suppressed this flow (min). The hydrodynamic cases are each averaged for roughly 300 days.
Case D3 also shows slow variations in ∆Ωlat overperiods of about 2000 days.

98

5.2

Kinetic and Magnetic Energies
Convection in these rapidly rotating dynamos is responsible for building the dif-ferential rotation and the magnetic fields. In a volume averaged sense, the energy
contained in the magnetic fields in case D3 is about 10% of the kinetic energy. About
35% of this kinetic energy is contained in the fluctuating convection (CKE) and about
65% in the differential rotation (DRKE), whereas the weaker meridional circulations
contain only a small portion (MCKE). The magnetic energy is split between the contributions from fluctuating fields (FME), involving roughly 53% of the total magnetic
energy, and the energy of the mean toroidal fields (TME) that are 43% of the total. Theenergy contained in the mean poloidal fields (PME) is only 4% of the total magnetic
energy. These energies are quoted in Table 5.2 and are defined as
CKE =DRKE =
MCKE =
FME =

TME =
PME =

1 h
ρ̄ (vr − hvr i)2 + (vθ − hvθ i)2 +2i(vφ − hvφ i)2 ,1ρ̄hvφ i2 ,21 ρ̄ hvr i2 + hvθ i2 ,2
1 h
(Br − hBr i)2 + (Bθ − hBθ i)2 +8πi(Bφ − hBφ i)2 ,1hBφ i2 ,8π

1 
hBr i2 + hBθ i2 .
8π

(5.3)
(5.4)
(5.5)

(5.6)
(5.7)
(5.8)

where angle brackets denote an average in longitude.
These results are in contrast to our previous simulations of the solar dynamo,
where the mean fields contained only about 2% of the magnetic energy and the fluctuating fields contained nearly 98% (Brun et al. 2004). In simulations of the solar dynamothat included a stable tachocline at the base of the convection zone (Browning et al.
2006), the energy of the mean fields in the tachocline can exceed the energy of the99

Table 5.2.

Energies in Cases at 3 and 5 Ω⊙

Case

CKE

DRKE

MCKE

FME

TME

PME

D3
D5avg
D5min
D5max
H3
H5

2.31
1.85
1.70
1.85
2.56
2.27

4.35
4.46
2.85
7.52
22.2
34.3

0.010
0.006
0.005
0.007
0.012
0.008

0.36
0.55
0.50
0.39
-

0.29
0.43
0.25
0.65
-

0.029
0.048
0.062
0.042
-

Note. — Volume-averaged energy densities relative to therotating coordinate system. Kinetic energies are shown forconvection (CKE), differential rotation (DRKE) and meridional circulations (MCKE). Magnetic energies are shown forfluctuating magnetic fields (FME), mean toroidal fields (TME)and mean poloidal fields (PME). All energy densities are reported in units of 106 erg cm−3 and are averaged over 1000 day
periods except for time-varying case D5, where intervals are
as in Table 5.1.

100
fluctuating fields there by about a factor of three, though the fluctuating fields still
dominate the magnetic energy budget within the convection zone itself. Simulationsof dynamo activity in the convecting cores of A-type stars (Brun et al. 2005) achieved
similar results. There in the stable radiative zone the energies of the mean fields wereable to exceed the energy contained in the fluctuating fields, but in the convecting core
the fluctuating fields contained roughly 95% of the magnetic energy. Simulations ofdynamo action in fully-convective M-stars do however show high levels of magnetic energy in the mean fields (Browning 2008). In those simulations the fluctuating fields stillcontain much of the magnetic energy, but the mean toroidal fields possess about 18% of
the total throughout most of the stellar interior. In our rapidly rotating suns, the mean
fields comprise a significant portion of the magnetic energy in the convection zone andare as important as the fluctuating fields.
Convection is similarly strong in all four rapidly rotating cases, and CKE is similar
in magnitude. The differential rotation in the dynamo cases is much weaker than in
the hydrodynamic progenitors and DRKE has decreased by about a factor of five. The
magnetic fields and differential rotation in case D5 change in time, but the averageenergy contained in DRKE is nearly the same in both dynamo cases despite their very
different rotation rates. This is in striking contrast to the behavior of the hydrodynamic
cases, where DRKE is much larger in the more rapidly rotating case H5 than in case H3.
Meridional circulations are comparably weak in all cases.
The amount of energy contained in the magnetic fields is different in these two
dynamo cases, with energies generally stronger in case D5 than in case D3. In an
average sense, all three magnetic energies are about 1.5 times greater in case D5 thanin case D3. Case D5 shows substantial time variation, and at periods either FME or
TME can become quite similar to those values realized in case D3. Meanwhile, PME isalways stronger in D5 than in D3.

101

5.3

Wreaths of Magnetism
These dynamos produce striking magnetic structures in the midst of their turbu-

lent convection zones. The magnetic field is organized into large banded, wreath-likestructures positioned near the equator and spanning the depth of the convection zone.
These wreaths are shown at two depths in the convection zone in Figure 5.2. The dominant component of the magnetic wreaths is the strong longitudinal field Bφ , with eachwreath possessing its own polarity. The average strength of the longitudinal field at
mid-convection zone is ±7 kG and peak field strengths there reach roughly ±26 kG.
Threaded throughout the wreaths are weaker radial and latitudinal magnetic fields,
which connect the two structures across the equator and also to the high-latitude regions.
These wreaths of magnetism survive despite being embedded in vigorous convective upflows and downflows. The convective flows leave their imprint on the magnetic
structures, with individual downflow lanes entraining the magnetic field, advecting it
away, and stretching it into Br while leaving regions of locally reduced Bφ . The slower
upflows carry stronger Bφ up from the depths. Where the magnetism is particularly
strong the convective flows are disrupted. Meanwhile, where the convective flows arestrongest, the longitudinal magnetic field is weakened and appears to vanish. In reality,
the magnetic wreaths here are diving deeper below the mid-convection zone, apparently
pumped down by the pummeling action of the strong downflows.
The deep structure of these wreaths is revealed by field line tracings throughout
the volume, shown in Figure 5.3 for the same instant in time. The wreaths are topologically leaky structures, with magnetic field lines threading in and out of the surrounding
convection. The wreaths are connected to the high-latitude (polar) convection, and on
the poleward edges they show substantial winding from the highly vortical convection
found there. This occurs in both the northern and southern hemispheres, as shown in102

Figure 5.2 — Magnetic wreaths and convective flows sampled at the same instant incase D3. (a) Longitudinal magnetic field Bφ near the top of the shell (0.95R⊙ ) and(b) at mid-depth (0.85R⊙ ). Strong flux structures with opposite polarity lie above andbelow the equator and span the convection zone. (c, d) Weaker radial magnetic field Br
permeates and encircles each wreath. (e, f ) Strong convective upflows and downflows
shown by Vr pass through and around the wreaths. The regions of strong magnetism
tend to disrupt the convective flows while the strongest downflows serve to pump thewreaths to greater depths.

103

Figure 5.3 — Field line visualization of magnetic wreaths in case D3.

(a) Snapshotof two wreaths in full volume at same instant as in Fig. 5.2. Lines trace the magnetic
fields, color denoting the amplitude and polarity of the longitudinal field Bφ (red, positive; blue, negative). Magnetic field threads in and out of the wreaths, connecting thetwo opposite polarity structures across the equator (i.e., region A) and to the polar
regions where the magnetic field is wound up by the cyclonic convection. (b) Samesnapshot showing south polar region. (c) Zoom in on region A showing the complex
interconnections across the equator between the two wreaths and to high latitudes.
Convective flows create the distinctive waviness visible in all three images.

104
two views at the same instant (north, Fig. 5.3a and south, Fig. 5.3b). It is here that the
global-scale poloidal field is being regenerated by the coupling of fluctuating velocities
and fluctuating fields. Magnetic fields cross the equator, tying the two wreaths together
at many locations (Fig. 5.3c). The strongest convective downflows leave their imprint on
the wreaths as regions where the field lines are dragged down deeper into the convection
zone, yielding a wavy appearance to the wreaths as a whole.

5.4

Wreaths Persist for Long Epochs
The wreaths of magnetism built in case D3 persist for long periods of time, withlittle change in strength and no reversals in global-scale polarity for as long as wehave pursued these calculations. The long-term stability of the wreaths realized by the
dynamo of case D3 is shown in Figure 5.4. Here the azimuthally-averaged longitudinal
field hBφ i and colatitudinal field hBθ i are shown at mid-convection zone at a point after
the dynamo has equilibrated and for a period of roughly 5000 days (i.e., several ohmic
diffusion times). During this interval there is little change in either the amplitude or
structure of the mean fields. This is despite the short overturn times of the convection(10-30 days) or the rotation period of the star (∼ 9 days). The ohmic diffusion time at
mid-convection zone is approximately 1300 days.
Though the mean (global-scale) fields are roughly steady in nature (Figs. 5.4a, b),
the magnetic field interacts strongly with the convection on smaller scales. Several
samples of longitudinal field Bφ are shown in full Mollweide projection at mid-convection
zone (Fig. 5.4c). The magnetic fields are clearly reacting on short time scales to theconvection but the wreaths maintain their coherence.
There are also some small but repeated variations in the global-scale magnetic
fields. Visible in Figure 5.4b are events where propagating structures of hBθ i reachtoward higher latitudes over periods of about 1000 days (i.e., from day 3700 to day
4500 and from day 5600 to day 6400). These are accompanied by slight variations

105

Figure 5.4 — Persistent wreaths of magnetism in case D3. (a) Time-latitude plotsof azimuthally-averaged longitudinal field hBφ i at mid-convection zone (0.85R⊙ ) in aview spanning latitudes from ±70◦ , with scaling values indicated. The two wreathsof opposite polarity persist for more than 4000 days. (b) Mean colatitudinal magnetic
field hBθ i at mid-convection zone over same interval. (c) Snapshots of Bφ in Mollweide
projection at mid-convection zone, shown for three times indicated in a, b. The wreaths
maintain constant polarity over long time intervals, but still show variation as theyinteract with the convection. Time t2 corresponds to the snapshot in Fig. 5.2b.

in the volume-averaged magnetic energy densities and the comparable kinetic energyof the differential rotation. These variations are also visible in the differential rotation
itself, as shown in Figure 5.5. The differential rotation is fairly stable, though some time
variation is visible at high latitudes. This is better revealed (Fig. 5.5b) by subtracting thetime-averaged profile of Ω at each latitude, revealing the temporal variations about thismean. In the polar regions above ±40◦ latitude, speedup features move poleward over
500 day periods. These features track similar structures visible in the mean magnetic
fields (Fig. 5.4b).
These evolving structures of magnetism and faster and slower differential rotation
appear to be the first indications of behavior where the mean fields themselves begin to
wax and wane substantially in strength. As the magnetic Reynolds number is increased,
this time varying behavior becomes more prominent and can even result in organizedchanges in the global-scale polarity. Such behavior is evident in our case D5.

106Figure 5.5 — Temporal variations of differential rotation in case D3. (a) Angularvelocity Ω at mid-convection zone (0.85R⊙ ), with ranges in both nHz and µrad s−1 .
The equator is fast while the poles rotate more slowly. (b) Temporal variations areemphasized by subtracting the time-averaged profile of Ω(r, θ), revealing speedup structures at high latitudes and pulses of fast and slow motion near the equator. Thesebands have average amplitudes of ±20 m s−1 and peak amplitudes of about ±60 m s−1 .
(c) Angular velocity shear ∆Ωlat (eq. 5.1) near the surface (upper curve, green) and at
mid-convection zone (lower, red).

107

5.5

Conclusions
The ability for a dynamo to build wreaths of strong magnetic fields in the bulkof the convection zone has largely been a surprise, for it had generally been supposedthat turbulent convection would disrupt such magnetic structures. To avoid these difficulties, many solar and stellar dynamo theories shift the burden of magnetic storage,
amplification and organization to a tachocline of shear and penetration at the base of
the convection zone where motions are more quiescent. In contrast, our simulations ofrapidly rotating stars are able to achieve sustained global-scale dynamo action within
the convection zone itself, with the magnetic structures both being built and able tosurvive while embedded deep within the turbulence. These dynamos are able to circumvent the Parker instability by means of turbulent Reynolds and Maxwell stresses
that contribute to the mechanical force balance and prevent the wreaths from buoyantly
escaping the convection zone. This striking behavior may be enabled by the stars rotating three to five times faster than the current Sun, which yields a strong differential
rotation that is a key element in the dynamo behavior. It is quite interesting that in our
dynamo cases the angular velocity contrast in latitude and radius is almost constant at
differing rotation rates, whereas our hydrodynamic cases tend to have increasing ∆Ωwith more rapid rotation Ω0 .
As we have seen in this chapter, we have achieved some dynamo states that are
persistent. Others flip the sense of their magnetic fields. In our case D3 the global-scale
fields have small vacillations in their amplitudes, but the magnetic wreaths retain theiridentities for many thousands of days. This represents hundreds of rotation periods andseveral magnetic diffusion times, indicating that the dynamo has achieved a persistentequilibrium. As we will see next in Chapter 6, increasing the rotation rate yields more
complex time dependence.

Chapter 6
Cyclic Dynamo Action Achieved at 5 Ω⊙

Our more rapidly rotating simulation, case D5 at five times the current solar rate,
also builds strong wreaths of magnetism that span the convection zone. However, in
this simulation the dynamo is not steady in time and instead goes through cycles ofactivity. During a cycle, the global-scale magnetic fields wax and wane in strength andat the lowest field strengths they can flip their polarity. We shall begin by looking at
the general properties of the convective flows and their associated differential rotationand meridional circulation, and then turn to examining the nature of the magnetic fields
and their time-varying behavior.

6.1

Patterns of Convection in Case D5
Figure 6.1a shows a snapshot of the patterns of convection realized in case D5

near the top of the domain. Much as in the radial velocity patterns shown for case D3(Figs. 5.1a,5.2e), here with faster rotation we continue to have prominent north-southaligned cells in the lower latitudes and more isotropic patterns near the poles. There issome modulation with longitude in the equatorial roll amplitudes. The downflows arefast and narrow, while the upflows are broader and slower. The convection establishes
a prominent differential rotation, with a fast equator and slow poles (Fig. 6.1c).
As in case D3, here too the convective downflow structures propagate more rapidly
than the differential rotation in which they are embedded. In the equatorial band, these109

Figure 6.1 — Patterns of convection in case D5. This simulation is rotating at five times
the solar rotation rate. (a) Radial velocity vr in Mollweide projection at 0.95R⊙ forcase D5. This snapshot samples day 3880 (time t1) when the magnetic fields are strong.
(b) Companion hydrodynamic case H5, whose stronger differential rotation shears out
convective structures in the mid-latitudes. (c) Profile of mean angular velocity Ω(r, θ)for case D5, with (d) radial cuts of Ω at selected latitudes. (e) Meridional circulations
for case D5, with magnitude and sense of circulation indicated by color (red counterclockwise, blue clockwise) and streamlines of mass flux overlaid.

110
structures move in a prograde fashion and at high latitudes in the retrograde sense. In
the polar regions, the radial velocity patterns have a somewhat cuspy appearance, withthe strongest downflows appearing to favor the westward and lower-latitude side of eachconvective cell. This may be a consequence of the strong retrograde differential rotationin those regions.
The convective structures are quite similar to those realized in the hydrodynamic
case H5 (Fig. 6.1b), though there are some noticeable differences, particularly at the
mid latitudes (around ±30◦ ). In the hydrodynamic case there is little radial flow inthese regions, as the strong differential rotation shears out the convective cells. Thisregion is equatorward of the tangent cylinder, an imaginary boundary tangent to thebase of the convection zone and aligned with the rotation axis. For rotating convective
shells, it has generally been found that the dynamics are different inside and outside thetangent cylinder, due to differences in connectivity and rotational constraint in these
two regions (e.g., Busse 1970). The tangent cylinder in our geometry intersects withthe stellar surface at roughly ±42◦ of latitude. In our compressible simulations, we
generally find that the convective patterns in the equatorial regions are bounded by aconic surface rather than the tangent cylinder (Brown et al. 2008). In case H5 the strong
differential rotation serves to disrupt the convection at the mid-latitudes. In contrast,in the dynamo case D5 the differential rotation is substantially weaker in both radial
and latitudinal angular velocity contrasts (Table 5.1). As is evident in Figure 6.1a, theconvective cells fill in this region quite completely.
The time-averaged angular velocity profile Ω(r, θ) is shown for case D5 in Figures 6.1c, d. The latitudinal angular velocity contrast ∆Ωlat and radial contrast ∆Ωr inthis case is remarkably similar in amplitude to that realized in case D3 (Table 5.1), eventhough the basic rotation rate Ω0 is substantially faster. This is in marked contrast to
our hydrodynamic companion cases where faster rotation leads to greater angular velocity contrasts. The accompanying meridional circulation patterns for case D5 (Fig. 6.1e)

111
appear to have three major cells of circulation in each hemisphere. These flows are
weaker than in case D3. They are very similar to the circulations found in case H5.

6.2

Oscillations in Energies and Changes of Polarity
The dynamo action realized by the convection in case D5 exhibits significant

changes in time. This time-varying behavior is readily visible as oscillations of thevolume-averaged kinetic and magnetic energy densities, as shown in Figure 6.2a at atime long after the dynamo has saturated and reached equilibration. Here the kinetic
energy of differential rotation (DRKE) undergoes factor of five changes on periods of
500-1000 days. As DRKE decreases the magnetic energies increase. Moving in concert
are the mean toroidal (TME) and mean poloidal (PME) magnetic energies. The meanpoloidal fields appear to lag slightly behind the mean toroidal fields as they both change
in strength. The fluctuating magnetic energies (FME) track the largest rises in themean fields but decouple during many of the deepest dips. In contrast, the variations inconvective kinetic energies (CKE) show little organized behavior in time, and appear tochange substantially only when the differential rotation is highly suppressed during theperiod from day 7500 to day 8300. The energy contained in the meridional circulations(MCKE) is weaker and not shown. Though it varies somewhat in time, there is not aclear relation to the changes in magnetic energies.
Magnetic energies in case D5 can rise to be a substantial fraction of the kinetic
energies. Averaged over the nearly 16000 days shown here, the magnetic energies areabout 17% of the kinetic energies. During individual oscillations the magnetic energies
can range from a few percent of the kinetic energies to levels as high as 50%. The kinetic
energy is largely in the fluctuating convection and differential rotation, with CKE fairly
constant and ranging from 15-60% of the total kinetic energy as DRKE grows andsubsides, itself contributing between 40 to 85% of the kinetic energy. The magnetic
energies are largely split between the mean toroidal fields and the fluctuating fields, with

112

Figure 6.2 — Complex time evolution in case D5 with flips in polarity of magnetic

wreaths. (a) Volume-averaged energy densities for kinetic energy of convection (CKE),
differential rotation (DRKE) and for magnetic energy in fluctuating fields (FME), inmean toroidal fields (TME) and in mean poloidal fields (PME) as labeled. Oscillations
on roughly 500-1000 day periods are visible in the magnetic energies and in DRKE,
though CKE stays nearly constant. (b) Mean toroidal field hBφ i averaged over entire
northern and southern hemispheres (labeled) at mid-convection zone (0.85R⊙ ). Early inthe simulation, opposite polarities dominate each hemisphere. Several reversals occur,along with several extreme excursions which do not flip the polarity of the globalscale field. During the interval from roughly day 7700 to 10200, the dynamo falls
into peculiar single polarity states, with one polarity dominating both hemispheres.
Bracketed interval from day 3500 to 5700 spans one full polarity reversal; (c) shows
volume-averaged energy densities during this period, and (d) the mean toroidal fieldwith same vertical axis scales as in (a, b). Thick labeled tick marks above a, c indicate
time samples used in later images.

113
TME containing about 35% of the magnetic energy on average, FME containing about
61% and PME containing 4%. The roles of these energy reservoirs change somewhat
through each oscillation. At any one time, between 10 and 60% of the magnetic energy
is in TME while FME contains between 30 and 85% of the total. Meanwhile, PME cancomprise as little as 1% or as much as 10% of the total. Generally, PME is about 12%
of TME, but because PME lags the changes in TME slightly, there are periods of time
when PME is almost 40% of TME.
The global-scale magnetic fields can reverse their polarities during some of theoscillations in magnetic energies. This is evident in Figure 6.2b showing averages at midconvection zone of the longitudinal magnetic field hBφ i over the northern and southernhemispheres. Reversals in field polarity occur periodically, with typical time scalesof roughly 1500 days. These reversals appear to happen shortly after peak magnetic
energies are achieved, but do not occur every time magnetic energies undergo a full
oscillation. Rather, it appears that several failed reversals occur where the magnetic
energies drop and the average fields decline in strength, only to return with the samepolarity a few hundred days later, for each successful polarity reversal.
We focus in the following discussion on one such reversal, shown in closeup in Figures 6.2c, d and spanning the interval of time between days 3500 and 5700. Two reversals
occur during this interval, with the global polarities flipping into a new state at roughly
day 4100 and then changing back again at about day 5500. Detailed measurements ofkinetic and magnetic energies during this interval are shown in Table 5.2.

6.3

Global-Scale Magnetic Reversals
The nature of the global-scale magnetic fields during the reversal spanning days

3500-5700 are presented in detail in Figure 6.3. Several samples of longitudinal magnetic
field Bφ are shown at mid-convection zone spanning this time period. The timing ofthese samples is indicated in Figure 6.2 by numeric labels and likewise in Figure 6.3a

114
which shows azimuthally-averaged hBφ i in a time-latitude map that spans the reversal.
Before a reversal, the magnetic wreaths of case D5 are very similar in appearance
to the wreaths realized in case D3. They are dominated by the azimuthally-averaged
component of Bφ , while also showing small-scale variations where convective plumes
distort the fields (Fig. 6.3b). At mid-convection zone, typical longitudinal field strengths
are of order ±13 kG, while peak field strengths there can reach ±40 kG. Meanwhile hBφ i
is fairly anti-symmetric between the northern and southern hemispheres (Fig. 6.3g).
Shortly before a reversal, the magnetic wreaths strengthen in amplitude and become
more anti-symmetric about the equator.
They reach their peak values just before the polarity change at roughly day
4000 but then quickly begin to unravel, gaining significant structure on smaller scales(Fig. 6.3c). At the same time, prominent magnetic structures detach from the higherlatitude edges and begin migrating toward the polar regions. Meanwhile, hBφ i loses
its anti-symmetry between the two hemispheres, with hBφ i in one hemisphere typicallyremaining stronger and more concentrated than in the other (Fig. 6.3h). The stronger
hemisphere (here the northern) retains its polarity for about 100 days as the fields inthe other hemisphere (here southern) weaken and reverse in polarity. At this point the
new wreaths of the next cycle, with opposite polarity, are already faintly visible at theequator.
Within 100 days these new wreaths grow in strength and become comparablewith the structures they replace, which are still visible at higher latitudes (Figs. 6.3d, i).
The mean hBφ i begins to contribute significantly to the overall structure of the new
wreaths, and soon the polarity reversal is completed. In the interval immediately after
the reversal, small-scale fluctuations still contribute significantly to the overall structureof the wreaths, and Bφ has complicated structure at mid-convection zone. At thistime, the peak magnetic field strengths are somewhat lower, at about ±20 kG. As
hBφ i becomes stronger, the wreaths return to an anti-symmetric state, with similar

115

Figure 6.3 — Evolution of Bφ during a polarity reversal in case D5. (a) Time-latitude
plot of hBφ i at mid-convection zone. (b − f ) Snapshots of Bφ in Mollweide projection at
mid-convection zone (0.85R⊙ ) at times indicated by numbers in a. Between reversals thefield is dominated by the mean component, but during reversals substantial fluctuations
develop. (g−k) Accompanying samples of azimuthally-averaged hBφ i, showing structureof mean fields with radius and latitude at same instants in time.

116
amplitudes and structure in both the northern and southern hemispheres (Figs. 6.3e, j).
They look much as they did before the reversal, though now with opposite polarities.
The wreaths from the previous cycle appear to move through the lower convection
zone and toward higher latitudes. This can be seen variously in the time-latitude map
at mid-convection zone (Fig. 6.3a), in the Mollweide snapshots (Figs. 6.3c − e), as wellas in the samples of hBφ i (Figs. 6.3h − j). This poleward migration is likely due to hoop
stresses within the magnetic wreaths and an associated poleward-slip instability (e.g.,
Spruit & van Ballegooijen 1982; Moreno-Insertis et al. 1992). Even at late times somesignatures of the previous wreaths persist in the polar regions, and are still visible in
Figures 6.3e, j at day 4390. They are much weaker in amplitude than the wreaths at
the equator, but they persist until the wreaths from the next cycle move poleward andreplace them. As they approach the polar regions, the old wreaths dissipate on bothlarge and small scales, for the vortical polar convection shreds them apart and ohmic
diffusion reconnects them with the relic wreaths of the previous cycle.
Though reversals occur on average once every 1500 days, substantial variations
can occur on shorter time scales. Here at mid-cycle the mean longitudinal field hBφ ibecomes quite weak as the wreaths become concentrated in smaller longitudinal intervalsof the equatorial region (as in Figs. 6.3f, k at day 4780). At other times the mean
longitudinal fields become quite asymmetric, with one hemisphere strong and one weak(i.e., during days 4900-5200) before regaining their anti-symmetric nature shortly prior
to the next reversal.

6.4

Temporal Changes in Differential Rotation
The strong magnetic fields in case D5 suppress the global-scale flow of differential

rotation. As the fields themselves vary in strength, the differential rotation responds
in turn, becoming stronger as the fields weaken and then diminishing as the fields areamplified. These cycles of faster and slower differential rotation are visible in the traces

117
of DRKE shown in Figure 6.2a. We revisit here the interval explored in closer detailboth in Figure 6.2c and in Figure 6.3, spanning days 3500 to 5700 of the simulation and
one full polarity reversal.
The angular velocity Ω at mid-convection zone is shown for this period as a timelatitude map in Figure 6.4a. Here again the timing marks t1–t5 refer to the snapshots ofBφ shown in Figures 6.3b− k. In the equatorial regions, the differential rotation remains
fast and prograde, but with some modulation in time. Prominent structures of speedup
are visible propagating toward the poles at the high latitudes. These structures aremuch more evident when we subtract the time-averaged profile of Ω for this period ateach latitude (Fig. 6.4b). They appear as strong, tilted fast (red) structures extending
from roughly ±30◦ latitude poleward. In the northern hemisphere, three such structures
are launched over this interval. In contrast, in the south only two such structures areevident. One is perhaps launched around day 4500 but does not survive or propagate.
Comparing these features with the propagation of magnetic fields shown in Figure 6.3aover the same interval, we find that velocity speedup features are well correlated withthe poleward migration of mean longitudinal magnetic field. The velocity features bearsome resemblance to the poleward branch of torsional oscillations observed in the solar
convection zone over the course of a solar magnetic activity cycle, though on a much
shorter time scale here as befits the correspondingly shorter time between magnetic
polarity reversals in these dynamo simulations.
These velocity features propagate toward the poles relatively slowly. In a periodof roughly 500 days they travel about 40◦ in latitude, or a distance of about 410 Mm.
Their propagation velocity is about 0.8 Mm day−1 or about 9 m s−1 . This is considerably slower than the fluctuating latitudinal flows associated with the convection whichat this depth have peak speeds of ±200 m s−1 during this time period. The meridional
circulations have amplitudes of about ±6 m s−1 here but do not have a latitudinal structure at all similar to the pattern propagation. The propagation speed of the speedup

118

Figure 6.4 — Time-varying differential rotation in case D5. (a) Time-latitude mapof angular velocity Ω at mid-convection zone (0.85R⊙ ). There are substantial temporal variations at both the equator and high latitudes. (b) These are accentuated bysubtracting the time-averaged profile of Ω(r, θ) at each latitude. Visible are poleward
propagating speedup structures at high latitudes and more uniform modulations near
the equator. (c) Corresponding variations in ∆Ωlat near the surface (upper curve, green)
and at mid-convection zone (lower, red).

119patterns is closer to the Alfvén velocity of the mean latitudinal magnetic fields, namelyhBθ ivA,θ = √
.
4π ρ̄

(6.1)

At mid-convection zone the mean density is about 0.065 g cm−3 and hBθ i in the poleward propagating plumes ranges between roughly ±1.5 kG, yielding Alfvén velocities
of about ±17 m s−1 . This poleward migration may also be due to a poleward-slip instability arising from the strong toroidal fields. In this scenario, if we neglect rotationand turbulent pumping in latitude (e.g., Moreno-Insertis et al. 1992; Jouve & Brun2009), the propagation speed should be approximately the Alfvén velocity of the mean
toroidal magnetic fields, or about ±45 m s−1 based on a mid-convection zone Bφ ofapproximately ±4 kG in the propagating features. If the poleward-slip instability is
occurring, the velocity speedup features may result from conservation of angular momentum in the plasma that travels poleward with the wreaths. Rotation is likely topartially stabilize wreaths against poleward-slip (Moreno-Insertis et al. 1992), and thismay help explain their slower poleward propagation. The leaky topology of the wreaths
will allow plasma to escape these structures, and this may modify the rate of their poleward propagation. The weak propagating features seen in case D3 (Fig. 5.5) required
nearly 700 days to propagate a similar distance in latitude. This difference may be dueto the somewhat lower magnetic field strengths achieved in that dynamo.
With the expanded sensitivity of Figure 6.4b, we can see that the equatorial
modulation appears as fast and slow pulses which span the latitude range of ±20◦ .
These variations are fairly uniform across this equatorial region. The velocity variations
at the equator do not correspond with the equatorial propagating branch of torsional
oscillations seen in the Sun (Thompson et al. 2003). In the Sun, the equatorial branch
may arise from enhanced cooling in the magnetically active regions (e.g., Spruit 2003;
Rempel 2006, 2007).

120
The temporal variations of the angular velocity contrast in latitude ∆Ωlat is shownfor this period in Figure 6.4c. At mid-convection zone (sampled by red line) the variations in ∆Ωlat are modest, varying by roughly 15%. Near the surface (green line) ∆Ωlatshows more substantial variations, with large contrasts when the fields are strong in themagnetic cycle (prior to t1) and smaller contrasts when the fields are in the process ofreversing (t2, t3). These near-surface values of ∆Ωlat are reported in Table 5.1, averaged
over this entire period (avg) and at points in time when the contrast is large (max, atday 3702) and small (min, at day 4060).

6.5

Sampling Many Magnetic Cycles in Case D5
The variations of angular velocities over considerably longer intervals of time

for case D5 are shown in Figures 6.5a, b. Here too we see the equatorial modulation
over many magnetic cycles and the poleward propagating speedup bands. Asymmetries
between the northern and southern hemisphere are evident at many times in different
cycles. The latitudinal angular velocity contrasts shown in Figure 6.5c exhibit large
variations. Successive magnetic cycles can have distinctly different angular velocity
contrasts, and there are additional long-term modulations that span many magnetic
cycles.
A sampling of the associated magnetic field behavior is shown in the time-latitude
maps of hBφ i and hBθ i in Figures 6.6a, b. From day 1500 to 7300, four cycles occur inwhich wreaths of opposite polarity are achieved in each hemisphere. After this period,
the dynamo explores unusual single-polarity states. Here either a single wreath is built(t7), or two wreaths of the same polarity (t6) occupy the two hemispheres. Afterday 10700 the dynamo emerges from this state and returns to building two wreaths of
opposite polarity which flip in their sense an additional three times as the simulation
continues.

121

Figure 6.5 — Extended history of varying differential rotation in case D5. (a) Variationsof Ω(r, θ) at mid-convection zone. (b) Temporal variations are emphasized by subtracting the time-averaged profile of Ω. Poleward propagating speedup structures are visible
in each magnetic oscillation. (c) Variations in ∆Ωlat near the surface (upper curve,
green) and at mid-convection zone (lower, red).

122

Figure 6.6 — Time-latitude plots of magnetic fields in case D5. Shown at midconvection zone are (a) mean longitudinal field hBφ i, and (b) mean colatitudinal fieldhBθ i. Cycles of activity are visible, with fields changing polarity in the equatorial region. Also prominently visible are plumes of field reaching toward the polar regions ina manner recalling Fig. 5.4. The time samples used in Figs. 6.3 and 6.7 are indicated.

123
The poleward propagating magnetic features shown previously in Figures 6.3a areevident throughout this longer time sampling, now appearing as nearly vertical streaks
in both hBφ i and hBθ i. These features continue to correlate well with the velocity
speedup features evident in Figure 6.5b.

6.6Strange States and Wreaths of a Single Polarity
These oscillating dynamos occasionally wander into distinctly different states,

and this occurs for case D5 around day 7300. Instead of the two nearly anti-symmetricwreaths of opposite polarity above and below the equator, the dynamo enters a state
where the polarity in each hemisphere is the same, as shown in Figures 6.7a, b at day
8903. Here two wreaths of same polarity occupy the two hemispheres and persist for aninterval of more than 500 days. The positive-polarity Bφ reaches average amplitudes of
18 kG while the negative polarity structures have average amplitudes around 3 kG. The
azimuthally-averaged profiles of hBφ i emphasize that these wreaths span the convection
zone and have the same polarity everywhere. During this interval of time, the meanpoloidal field has changed from an odd-parity state, with strong contributions fromthe odd-ℓ components, to an even-parity state where the even-ℓ components are more
prominent.
The dynamo can also achieve states where only a single wreath is built in theequatorial regions, as in Figures 6.7c, d at day 9590. Here a single strong wreath of
positive polarity fills the northern hemisphere, with hBφ i reaching a peak amplitudeof +18 kG. This unique structure persists for about 800 days before the dynamo flips
polarity and builds a strong wreath of negative polarity. The predecessor of this new
wreath can be seen in profiles of hBφ i where a much weaker structure of negative polarity
is visible in the lower convection zone.
The strong magnetic fields realized in the single wreath states react back on theconvective flows. This is evident in the accompanying snapshot of radial velocities

124

Figure 6.7 — Strange single-polarity states in case D5.

(a) Snapshot of Bφ at midconvection zone, showing two strong wreaths of the same polarity. (b) Instantaneous
profile of hBφ i at same time. (c) Snapshot of Bφ at mid-convection zone at a time
when a single wreath is formed. (d) Weaker negative polarity structures are visible inprofile of hBφ i at same instant. (e) Accompanying snapshot of vr at mid-convection
zone, showing flows strongly affected by magnetism. (f ) The instantaneous differential
rotation, shown here as profile of Ω(r, θ), is unaffected by the strong wreath.

125
at mid-convection zone (Fig. 6.7e). In a narrow band spanning 0 − 20◦ latitude andcoinciding with the strong tube, the upflows and downflows have been virtually erased.
Fluctuations in vφ and vθ are also very small in this region, and the flow is dominatedby the streaming flows of differential rotation. Within the wreath the total magnetic
energy (ME) at mid-convection zone is locally about 10 to 100 times larger than the
total kinetic energy (KE), while outside the wreath KE exceeds ME by factors of roughly
10 to 104 at this depth. We see similar restriction of the convective flows whenever themagnetic fields become this strong.
The differential rotation itself (Fig. 6.7f ) is largely unaffected by the presence ofthe strong magnetic wreath. There is no clear signature of faster flow down the middle of
the wreath. Likewise, there is no sign of the structure in profiles of the thermodynamic
variables P, T, S, or ρ, with the mean profile instead dominated by latitudinal variations
consistent with thermal wind balance.

6.7

Conclusions
In our case D5 the oscillations can become large, and may result in the global-

scale fields repeatedly flipping their polarity. At times this dynamo appears to be cyclic
but in other intervals it behaves more chaotically.
The transition to richer time dependence with increasing rotation rate in case D5
appears to arise from subtle changes and phasing relationships between the toroidal andpoloidal magnetic production terms. These are difficult to assess, since the production
terms themselves are complex in space and now vary in time as well. It is evident thatthe mean poloidal fields lag the changes in the mean toroidal fields, and there is clear
migration of magnetic field to the higher latitudes during a reversal. This latitudinal migration may result from a polarward-slip instability, triggered by the stronger magnetic
fields that are generated by this dynamo, and this migration may lead to the oscillations in field strength and polarity. The analysis of magnetic field production that we

126
have carried out for case D3 (Chapter 7) requires significant averaging of the turbulent
correlations, accomplished there by time averaging, to obtain a coherent picture of thebalances achieved. This has not yet been tractable for the oscillating solutions, as thegeneration terms change on shorter time scales than needed to obtain stable averagesof the turbulent processes. It appears that the imbalances in the production and destruction of mean magnetic fields during cyclic behavior are modest, and currently wecannot pinpoint just which terms out of the large medley serve to drive the oscillations.
These dynamo oscillations are not special to case D5. Indeed, we have explored abroader class of oscillating dynamo solutions, which will be detailed in Chapter 8. Someof these solutions are realized by taking our more slowly rotating case D3 to higher levelsof turbulence by reducing the eddy diffusivities, while others are achieved at even higher
rotation rates. We find such global-scale oscillations and polarity reversals fascinating,since these appear to be the first self-consistent 3-D stellar dynamo simulations which
achieve such temporally organized behavior in the bulk of the convection zone.
Accompanying the oscillations in global-scale magnetic field are changes in the
differential rotation, and these signatures are the most likely to be found through stellar
observations. The angular velocity contrast ∆Ωlat between the equator and high latitudes can vary by 20% over periods of hundreds or thousands of days. The patterns ofspeedup which we find propagating toward the polar regions of these dynamos may havecorrespondence with the polar branch of the torsional oscillations which are observedin the Sun. The speedup features may arise from conservation of angular momentum in
fluid which is partially trapped within the wreaths as they slip toward the poles. The
leaky nature of our wreaths will modify this somewhat and that may explain why thewreaths are not stabilized by rotation (e.g., Moreno-Insertis et al. 1992).
In the next chapter, we return to our steady case D3 to examine how the magnetic
fields are produced and maintained by the turbulent convection and global-scale flows
against resistive dissipation.

Chapter 7
Analysis of Production terms in Dynamos with Sustained Wreaths

The magnetic wreaths formed in case D3 are dominated by strong mean longitudinal field components and show little variation in time. To understand the physicalprocesses responsible for maintaining these magnetic wreaths, we examine the terms
arising in the time- and azimuth-averaged induction equation for case D3. We derive
diagnostic tools to evaluate the generation and transport of magnetic field in a magnetized and rotating turbulent convection zone. This derivation is in spherical coordinates,
and is under the anelastic approximation.

7.1

Production of Axisymmetric Toroidal Field
We begin our analysis by exploring the maintenance of the mean toroidal fieldhBφ i. Here it is helpful to break the induction term from equation (2.4) into contributions from shear, advection and compression. In the induction equation (2.4), the firstterm on the right hand side represents production of magnetic field while the second
term represents its diffusion. We rewrite the production term to make the contributionsof shear, advection and compressible effects more explicit as∇ × (v × B) = (B · ∇)v − (v · ∇)B − B(∇· v).

(7.1)

128
Under the anelastic approximation the divergence of v can be expressed in terms of thelogarithmic derivative of the mean density because∇· (ρ̄v) = 0 = ρ̄(∇· v) + (v · ∇)ρ̄,and therefore∇· v = −vr∂ln ρ̄.
∂r(7.2)

The induction equation thus becomes∂∂B= (B · ∇)v − (v · ∇)B + vr B ln ρ̄ − ∇ × (η∇ × B)
{z
}
{z
}
{z
}
|
|
∂t
{z } || ∂r
shearing advection compression
diffusion(7.3)

As labeled, the first term represents shearing of B, the second term advection of B, thethird one compressible amplification of B, and the last term ohmic diffusion.
To identify the processes contributing to the production of mean (axisymmetric)
field, we separate our velocities and magnetic fields into mean and fluctuating components v = hvi + v ′ and B = hBi + B ′ where angle brackets denote an average inlongitude. Thus hv ′ i = hB ′ i = 0 by definition. Expanding the production term ofequation (7.3) we obtain the mean shearing termh(B · ∇)vi = (hBi · ∇) hvi + h(B ′ · ∇)v ′ i,

(7.4)

the mean advection term−h(v · ∇)Bi = − (hvi · ∇) hBi − h(v ′ · ∇)B ′ i,

(7.5)

and the mean compressibility term
hvr B

 ∂
∂
ln ρ̄i = hvr ihBi + hvr′ B ′ iln ρ̄.
∂r
∂r

(7.6)

In a similar fashion, the mean diffusion term becomes−h∇ × (η∇ × B)i = −∇ × (η∇ × hBi).

(7.7)

129
The axisymmetric component of the induction equation is written symbolically
as:

∂hBi
= PMS + PFS + PMA + PFA + PMC + PFC + PMD
∂t

(7.8)

With PMS representing production of field by mean shear, PFS production by fluctuating shear, PMA advection by mean flows, PFA advection by fluctuating flows, PMC
amplification arising from the compressibility of mean flows, PFC amplification arisingfrom fluctuating compressible motions, and PMD ohmic diffusion of the mean fields. Inturn, these terms are
PMS = (hBi · ∇) hvi ,PFS =h(B ′ · ∇)v ′ i ,PMA = − (hvi · ∇) hBi ,PFA = − h(v ′ · ∇)B ′ i ,PMC = (hvr ihBi)∂
ln ρ̄ ,∂r

PMD = − ∇ × (η∇ × hBi) .

(7.9)

 ∂
PFC = hvr′ B ′ iln ρ̄ ,∂r

We now expand each of these terms into their full representation in spherical coordinates.

∂hBφ i
∂t
PMS
PFS
PMA
PFA
PMC
PMD

= PMS + PFS + PMA + PFA + PMC + PFC + PMD
hBφ ihvr i + cot θhBφ ihvθ i∂
hBθ i ∂=
hBr i+
(7.10)
hvφ i +
∂r
r ∂θ
r
# 
"
Bφ′ ∂
hBφ′ vr′ i + cot θhBφ′ vθ′ i
Bθ′ ∂
′ ∂
vφ′ +
+
+
(7.11)Br=∂rr ∂θ r sin θ ∂φrhvφ ihBr i + cot θhvφ ihBθ i
hvθ i ∂∂+
(7.12)hBφ i −
= − hvr i
∂r
r ∂θ
r
#
"


hvφ′ Br′ i + cot θhvφ′ Bθ′ i
vφ′
vθ′ ∂
∂
′
′ ∂
Bφ −
+
+
(7.13)= − vr∂r
r ∂θ r sin θ ∂φ
r
 ∂
∂
=
(hvr ihBφ i)ln ρ̄ ,PFC = hvr′ Bφ′ iln ρ̄(7.14)∂r∂r
ηhBφ idη 1 ∂(rhBφ i)(7.15)= η∇2 hBφ i − 2 2 +∂rr sin θ dr r

130

7.2

Maintaining Wreaths of Toroidal Field
The evolution of the mean longitudinal (toroidal) field hBφ i is described sym-

bolically in equation (7.8), with individual terms defined in equation (7.9). When weanalyze these terms in case D3, we find that hBφ i is produced by the shear of differential
rotation and is dissipated by a combination of turbulent induction and ohmic diffusion.
This balance can be restated as∂hBφ i≈ PMS + (PFS + PFA + PMD ) ≈ 0 ,∂t(7.16)

with PMS representing production by the mean shearing flow of differential rotation, PFS
by fluctuating shear, PFA by fluctuating advection, and PMD by mean ohmic diffusion.
Those terms are in turn
PMS =(hBi · ∇) hvi φ ,(7.17),(7.18),(7.19)PMD = −∇ × η∇ × hBi φ ,(7.20)PFS =PFA = −

B′ · ∇ v′

v′ · ∇ B′

φφ

where brackets again indicate an azimuthal average and primes indicate fluctuating
terms: v ′ = v − hvi. The detailed implementation of these terms is presented for ourspherical geometry in equations (7.10-7.15). These terms are illustrated in Figure 7.1
for case D3, averaged over a 450 day interval from day 6450 to 6900.
The structure of hBφ i is shown in Figure 7.1a. The shearing flows of differential
rotation PMS (Fig. 7.1b) act almost everywhere to reinforce the mean toroidal field. Thus
the polarity of this production term generally matches that of hBφ i. This production isbalanced by destruction of mean field arising from both turbulent induction and ohmic
diffusion (sum shown in Fig. 7.1c). The individual profiles of PFS , PFA and PMD arepresented in turn in Figures 7.1d, e, f . The terms from turbulent induction (PFS and
PFA ) contribute to roughly half of the total balance, with the remainder carried by ohmic

131

Figure 7.1 — Generation of hBφ i in case D3. The view is from ±45◦ latitude to empha-

size the equatorial regions. (a) Mean toroidal field hBφ i with wreaths strongly evident.
(b) Production by PMS serves to build hBφ i. This rate term generally matches the senseof hBφ i, thus being negative (blue in colorbar, with ranges indicated) in the core of thenorthern wreath and positive (red) in that of the southern wreath. (c) Destruction ofmean toroidal field is achieved by the sum of the two fluctuating (turbulent) induction
terms and the ohmic diffusion (PFS + PFA + PMD ). This sum clearly has opposite sense
and similar magnitude to PMS . We break out these three destruction terms in the following panels. (d) Fluctuating (turbulent) shear PFS is strongest near the high-latitudeside of each wreath, and (e) fluctuating (turbulent) advection PFA is strongest in the
cores of the wreaths. The sum of these terms (PFS + PFA ) is responsible for about half
the destructive balance, with the remainder coming from (f ) the mean ohmic diffusion
PMD . Some differences arise in the boundary layers at top and bottom.

diffusion of the mean fields (PMD ). In the core of the wreaths, removal of mean toroidal
field is largely accomplished by fluctuating advection PFA (Fig. 7.1e) and mean ohmic
diffusion PMD (Fig. 7.1f ), with the latter also important near the upper boundary.
Turbulent shear becomes strongest near the bottom of the convection zone and in theregions near the high-latitude side of each wreath. Thus PFS (Fig. 7.1d) becomes thedominant member of the triad of terms seeking to diminish the mean toroidal field there.
We find that the mean poloidal field is regenerated in roughly the same region.
In the analysis presented in Figure 7.1 we have neglected the advection of hBφ i by
the meridional circulations (denoted by PMA ), which we find plays a very small role in
the overall balance. We have also neglected the amplification of hBφ i by compressibility

132
effects (namely, PMC and PFC ), though it does contribute slightly to reinforcing theunderlying mean fields within the wreaths.
To summarize, the mean toroidal fields are built through an Ω-effect, where production by the mean shearing flow of differential rotation (PMS ) builds the underlying
hBφ i. In the statistically steady state achieved, this production is balanced by a combination of turbulent induction (PFS + PFA ) and ohmic diffusion of the mean fields(PMD ).

7.3

Production of Axisymmetric Poloidal Field
When analyzing the production of the mean poloidal field, two contributions arise.

One is for the mean radial field and one for the mean colatitudinal field. Evaluating
the production of poloidal field in terms of the radial and colatitudinal field is a bitcomplex, and in practice the production of the mean poloidal vector potential is a more
useful quantity. For clarity however, it is useful to first exhibit the production terms
for radial and colatitudinal in a fashion similar to those evaluated for the mean toroidal
field. With the same analysis approach, the induction equation for those two fields arein turn∂hBr i
∂t
PMS
PFS
PMA
PFA
PMC
PMD

= PMS + PFS + PMA + PFA + PMC + PFC + PMD ,hBθ ihvθ i + hBφ ihvφ i
hBθ i ∂∂+,
hvr i −
=
hBr i
∂r
r ∂θ
r
# 
"
′
′
hBθ′ vθ′ i + hBφ′ vφ′ i
B
B
∂
∂
∂
φ
Br′
=
vr′ −
+ θ
+
,
∂r
r ∂θ r sin θ ∂φrhvθ ihBθ i + hvφ ihBφ i
hvθ i ∂∂+,
hBr i +
= − hvr i
∂r
r ∂θ
r
# 
"
vφ′
hvθ′ Bθ′ i + hvφ′ Bφ′ i
vθ′ ∂
∂
′
′ ∂
+
+
,
Br +
= − vr∂r
r ∂θ r sin θ ∂φ
r
 ∂
∂
=
(hvr ihBr i)ln ρ̄ ,PFC = hvr′ Br′ iln ρ̄ ,
∂r∂r
hBr i 2η ∂hBθ i 2η cot θhBθ i−,= η∇2 hBr i − 2η 2 − 2
r
r ∂θ
r2

(7.21)(7.22)(7.23)(7.24)(7.25)(7.26)

133
and∂hBθ i
∂t
PMS
PFS
PMA
PFA
PMC

= PMS + PFS + PMA + PFA + PMC + PFC + PMD ,hBθ ihvr i − cot θhBφ ihvφ i∂
hBθ i ∂=
hBr i+,
(7.27)
hvθ i +
∂r
r ∂θ
r
# 
"
′
′
hBθ′ vr′ i − cot θhBφ′ vφ′ i
B
B
∂
∂
∂
φ
′
′
θ
v +
+
+
, (7.28)Br=∂r
r ∂θ r sin θ ∂φ θrhvθ ihBr i − cot θhvφ ihBφ i
hvθ i ∂∂+,
(7.29)hBθ i −
= − hvr i
∂r
r ∂θ
r
# 
"
vφ′
hvθ′ Br′ i − cot θhvφ′ Bφ′ i
vθ′ ∂
∂
′ ∂
vr
= −
Bθ′ −
+
+
, (7.30)∂rr ∂θ r sin θ ∂φr
 ∂∂ln ρ̄ ,PFC = hvr′ Bθ′ iln ρ̄ , (7.31)=(hvr ihBθ i)
∂r
∂r

PMD =

η∇2 hBθ i +2η ∂hBr iηhBθ i− 2 2
r 2 ∂θ
r sin θdη 1 ∂(rhBθ i) 1 ∂hBr i
−
.
+
dr r
∂r
r ∂θ

(7.32)

In practice we use the above equations to diagnose and understand the local productionof poloidal field, but will not show these terms explicitly.
Instead, we find that when analyzing the production of mean poloidal magnetic
field, the balances achieved are somewhat clearer if we consider its vector potential
rather than the two separate fields themselves. The mean poloidal field hB pol i has acorresponding vector potential hAφ i, wherehB pol i = hBr ir̂ + hBθ iθ̂ = ∇ × hA φ i∂
1 ∂1= r sinθ ∂θ hAφ sin θi r̂ − r ∂r hrAφ i θ̂

(7.33)

DE= ∇ × Aφ φ̂ .

The other components of the poloidal vector potential disappear, as terms involving ∂/∂φ vanish in the azimuthally-averaged equations. Likewise, the φ-component of
the possible gauge term ∇λ is zero by virtue of axisymmetry. We recast the induction
equation (eq. 2.4) in terms of the poloidal vector potential by uncurling the equation

134
once and obtain∂hAφi= v × B φ − η∇ × B φ .
∂t(7.34)

This can then be decomposed into mean and fluctuating contributions, and represented
symbolically as∂hAφi= EMI + EFI + EMD ,∂t(7.35)

with EMI representing the electromotive forces (emf) arising from mean flows and mean
fields, and related to their mean induction. Likewise, EFI is the emf from fluctuatingflows and fields and EMD is the emf arising from mean diffusion. These are in turnEMI= hvi × hBi φ =EFIhvr′ Bθ′ i − hvθ′ Br′ i,
1 ∂∂hBr i= −η∇ × hBi φ = −η(rhBθ i) −
.
r ∂r
∂θ

EMD

7.4

hvr ihBθ i − hvθ ihBr i,= hv ′ × B ′ i φ =(7.36)(7.37)(7.38)

Maintaining the Poloidal Field
The production of mean poloidal field is achieved through a slightly different

balance, with turbulent induction producing poloidal field and ohmic diffusion acting
to dissipate it. The mean flows play only a modest role in the overall balance.
In case D3 we find that the mean poloidal vector potential hAφ i is produced by
the fluctuating (turbulent) emf and is dissipated by ohmic diffusion∂hAφ i≈ EFI + EMD ≈ 0 ,∂t(7.39)

with EFI the emf arising from fluctuating flows and fluctuating fields, and contributing
to the mean induction. The EMD is the emf arising from mean ohmic diffusion. These
terms are
EFI = hv ′ × B ′i φ = hvr′ Bθ′ i − hvθ′ Br′ i,(7.40)

EMD = −η∇ × hBi φ .

(7.41)

135

Figure 7.2 — Production of mean poloidal vector potential hAφ i in case D3. The viewis restricted to ±45◦ latitude to emphasize the regions of production. (a) Mean poloidal
vector potential hAφ i, with sense denoted by color (red, clockwise; blue, counterclockwise). (b) The fluctuating (turbulent) emf EFI acts to build the vector potential.
This term is strongest near the bottom of the convection zone and the poleward side of
the wreaths. (c) Mean ohmic diffusion EMD acts everywhere in opposition to EFI . The
cores of the wreaths are positioned at roughly ±15◦ latitude (Fig. 7.1a).

The contribution arising from the omitted term EMI (see eq. 7.36), related to the emfof mean flows and mean fields, is smaller than these first two by more than an orderof magnitude. Additionally, EMI has a complicated spatial structure which does notappear to act in a coherent fashion within the wreaths to either build or destroy meanpoloidal field.
The mean vector potential hAφ i is shown in Figure 7.2a, with poloidal field lines
represented by the overlying contours. The mean radial magnetic field hBr i is about±1 kG in the cores of the wreaths, whereas the mean colatitudinal field hBθ i has an amplitude of roughly −2 kG (thus directed northward in both hemispheres), concentrated
near the bottom of the convection zone.

136
The production of hAφ i by the fluctuating (turbulent) emf EFI is shown in Figure 7.2b. Here too we average over the same 450 day interval. This term generally acts
to reinforce the existing poloidal field, having the same sense as the underlying vector
potential in most regions. It is strongest near the bottom of the convection zone and isconcentrated at the poleward side of each wreath. This is similar, though not identical,
to the structure of destruction of mean toroidal field by fluctuating shear PFS (Fig. 7.1d).
It suggests that mean toroidal field is here being converted into mean poloidal field by
the fluctuating flows.
There are two terms that contribute to EFI , as shown in equation (7.40). Muchof that fluctuating emf arises from correlations between fluctuating latitudinal flows
and radial fields h−vθ′ Br′ i, which follows the structure of EFI (Fig. 7.2b) closely. The
contribution from fluctuating radial flows and colatitudinal fields hvr′ Bθ′ i is more complex
in structure. Near ±20◦ latitude, this term reinforces h−vθ′ Br′ i, but acts against it at
higher latitudes and thus diminishes the overall amplitude of EFI . The mean ohmic
diffusion EMD (Fig. 7.2c), almost entirely balances the production of hAφ i by EFI .
This shows that our mean poloidal magnetic field is maintained by the fluctuating(turbulent) emf and is destroyed by ohmic diffusion. In mean-field dynamo theory, thisis often parametrized by an “α-effect.” Now we turn to interpretations within thatframework.

7.5

Exploring Mean-Field Interpretations
Many mean-field theories assert that the production of mean poloidal field is likely

to arise from the fluctuating emf. This process is often approximated with an α-effect,
where it is proposed that the sense and amplitude of the emf scales with the mean
toroidal field
hv ′ × B ′ i = αhBi,(7.42)

137
where α can be either a simple scalar or may be related to the kinetic and magnetic(current) helicities. In isotropic (but not reflectionally symmetric), homogeneous, incompressible MHD turbulenceτ(αk + αm ) ,3= −v ′ · ∇ × v ′ ,α =(7.43)

αk(7.44)αm =
1B′ · ∇ × B′ ,4πρ(7.45)as discussed in Pouquet et al. (1976) and Brandenburg & Subramanian (2005). Here
τ is the lifetime or correlation time of a typical turbulent eddy. In mean-field theory, these fluctuating helicities are typically not solved directly and are instead solved
through auxiliary equations for the total magnetic helicity or are prescribed. Here wecan directly measure our fluctuating helicities and examine whether they approximate
our fluctuating emf.
To assess the possible role of an α-effect in our simulation, we show in Figures 7.3a, b the fluctuating kinetic and current helicities αk and αm realized in ourcase D3, averaged over the same 450 day analysis interval. To make an estimate of the
α-effect, we approximate the correlation time τ by defining
τ=

HP
,
v′

(7.46)

where HP is the local pressure scale height and v ′ is the local fluctuating rms velocity,which are functions of radius only. Estimated by this method, the turnover time τ hasa smooth radial profile and is roughly 10 days near the bottom of the convection zone,
3 days at mid-convection zone, and slightly less near the upper boundary. If we use thefast peak upflow or downflow velocities instead of the rms velocities, our estimate of τ
is about a factor of 4 smaller. Our mean-field α (eq. 7.43) is shown in Figure 7.3c. In
the upper convection zone, this is dominated by the fluctuating kinetic helicity while
the fluctuating magnetic (current) helicity becomes important at depth.

138

Figure 7.3 — Estimating the mean-field α-effect in case D3. (a) Fluctuating kinetic helicity αk . (b) Fluctuating magnetic (current) helicity αm . (c) Mean-field α, constructedby combining αk and αm with a turbulent correlation time τ .

We form a mean-field emf (right-hand side of eq. 7.42) by multiplying our derived
α (Fig. 7.3c) with our hBφ i (Fig. 7.1a), and show this in Figure 7.4a. The turbulent emf
EFI , which is the left-hand side of equation (7.42), can be measured in our simulations
and is shown again in Figure 7.4b. Although there is some correspondence in the twopatterns, there are significant differences. In particular, the mean-field emf has peak
amplitudes in the cores of the wreaths (at ±15◦ latitude) and is negative there. In
contrast, the actual fluctuating emf given by EFI is positive and has its highest amplitude
at the poleward side of the wreaths. Thus the mean-field emf predicts an incorrectbalance in the generation terms and would yield a distinctly different mean poloidal
magnetic field. To assess whether better agreement may be achieved with a latitudeaveraged emf, we average the mean-field emf and EFI separately over the northern andsouthern hemispheres and plot these quantities in Figure 7.4c. Though both have asimilar positive sense near the base of the convection zone, the hemisphere-averaged
EFI becomes small above 0.8R⊙ whereas the averaged mean-field emf αhBφ i is large
and negative there. Thus even the averaged emfs are not in accord.

139

Figure 7.4 — Comparison of emfs in case D3. (a) Profile of proposed mean-field emf

given by αhBφ i. (b) Actual turbulent emf EFI measured in the dynamo. (c) Variationof hemisphere-averaged emfs with fractional radius. The mean-field approximated emf
is shown in blue, and EFI in red. The average over the northern hemisphere is shown
solid, the southern is dashed.

In summary, it is evident that a simple scalar α-effect will predict the wrong
sign for the fluctuating emf in the two hemispheres, as hBφ i is anti-symmetric acrossthe equator while hAφ i is symmetric. An α-effect based on the kinetic helicity andmagnetic helicity may capture some sense of the fluctuating emf, as those quantities arethemselves anti-symmetric across the equator. Yet Figure 7.4 suggests that there are
significant discrepancies between this particular approximation and our turbulent emf.
In particular, this mean-field α-effect misses the offset between the generation regions
for mean toroidal and mean poloidal field. This offset in latitude of the generation
regions may be important for avoiding the α-quenching problems encountered in manymean-field theories. A more complex mean-field model, which takes spatial gradientsof hBφ i into account, may do better. In particular, the Ω × J-effect (e.g., Moffatt &Proctor 1982; Rogachevskii & Kleeorin 2003) may be at work in these systems, andpreliminary explorations indicate that this term matches the spatial structure of our140
EFI better than the above α-effect. A tensor representation of the α-effect may domuch better, and test-field techniques could be employed to measure this quantity (e.g.,
Schrinner et al. 2005, and recently reviewed in Brandenburg 2009).

7.6

Production of Fluctuating (Non-Axisymmetric) Field
Left out of this analysis is the fluctuating component of the induction equation,

which produces the small-scale but strong fluctuating magnetic fields. In case D3 thesefields do not appear to be a dominant feature, but as will be seen in Chapter 8 thefluctuating fields become prominent in our more turbulent dynamo simulations. For
completeness, we include their induction equation here. This can be derived by subtracting the mean induction equation (7.8) from the full induction equation, yielding
the following equation for the fluctuating fields
∂B ′
∂t

=

(hBi · ∇)v ′ + (B ′ · ∇)hvi + E−(hvi · ∇)B ′ − (v ′ · ∇)hBi − F+(hvr iB ′ + vr′ hBi)∂ln ρ̄ + G∂r−∇ × (η∇ × hB ′i) ,(7.47)

where the quantities E = (B ′ · ∇)v ′ − h(B ′ · ∇)v ′i, F = (v ′ · ∇)B ′ − h(v ′ · ∇)B ′ i, and∂ln ρ̄, represent the difference between mixed stresses from which
G = (vr′ B ′ − hvr′ B ′i) ∂r

we subtract their axisymmetric mean. In the standard mean-field derivation, thesequantities are siblings of the G-current involving the mean electromotive force hv × Bi
and its 3-D equivalent v × B (i.e., the so called “pain in the neck” term, Moffatt 1978).

7.7

Conclusions
In our persistent case D3 we are able to analyze the generation and transportof mean magnetic field. We find that our dynamo action is of an α − Ω nature, withthe mean toroidal fields being generated by an Ω-effect from the mean shearing flow of

141
differential rotation. This generation is balanced by a combination of turbulent induction and ohmic diffusion. The mean poloidal fields are generated by an α-effect arisingfrom couplings between the fluctuating flows and fluctuating fields, with this production
largely balanced by the ohmic diffusion. This is unlike the toroidal balance, for here themean flows play almost no role and the turbulent correlations are constructive rather
than destructive. In assessing what a mean-field model might predict for the magnetic
structures realized in case D3, we find that the isotropic, homogeneous α-effect basedon kinetic and magnetic (current) helicities fails to capture the sense of our turbulent
emf. In general, our EFI is poorly represented by an αhBφ i that is so determined.
The realization of global-scale magnetic structures in our simulations, and theirgreat strength relative to the fluctuating fields, may in part be a consequence of therelatively modest degree of turbulence attained here. Whether such structures can begenerated and sustained amidst the far more complex flows in actual stellar interiors
is not yet clear. If such structures are indeed realized in stars, they may or may notsurvive to print through the highly turbulent convection occurring just below the stellar
photosphere. If they do appear at the surface, some global-scale magnetic features may
propagate toward the poles along with the bands of angular velocity speedup. There aresome indications in stellar observations that global-scale toroidal magnetic fields may
indeed become strong in rapidly rotating stars (Donati et al. 2006; Petit et al. 2008),
though small-scale fields may still account for much of the magnetic energy near thesurface (Reiners & Basri 2009). The global-scale poloidal fields may be more successful
in surviving the passage through the turbulent surface convection. If they do, thestellar magnetic field will likely have significant non-dipole components. Thus the meanpoloidal fields observed at the surface may give clues to the presence of large wreathsof magnetism that occupy the bulk of the convection zone.

Chapter 8
Menagerie of Wreath-Building Dynamos

In the preceding chapters we have extensively explored the characteristics of two
dynamo solutions. Those simulations, cases D3 and D5 at three and five times the solar
rotation rate, are part of a much larger family of simulations that we have conductedwhich explore the interplay between convection, rotation and magnetism in younger
suns. In this chapter we present a sampling of simulations rotating from half the current
solar rate to fifteen times the solar rate. This spans a range of ages for a star like thesun, from just a few million years after formation when its rotating quite rapidly to a few
billion years older than the current age of the Sun, when our star will rotate more slowly.
Rotation and age are inextricably linked, and there are likely to be significant changes
in the stellar structure over these interval of several billion years, as the luminosity ofthe star gradually increases, helium settles into the core and the depth of the convection
zone itself shifts. Here we have neglected those changes by retaining a stellar structure
appropriate for the current Sun. We have instead explored only the effects of faster andslower rotation, to begin disentangling the varied physical effects present in real stellar
observations. Future work should clearly include an exploration of the dual variationof rotation and structure with age, as in Ballot et al. (2007), but at present observed
stellar ages remain highly imprecise which complicates such efforts.

143
The parameter space explored by the simulations is summarized in Tables 8.1-8.3.
Table 8.1 details our successful dynamos at varying levels of turbulence conducted at a
fixed magnetic Prandtl number Pm = 0.5 and fluid Prandtl number Pr = 0.25. One setof simulations (D3-D15) are conducted on the familiar parameter path with turbulent
diffusivities scaling as
ν, κ, η ∝ (Ω0 )−2/3 .

(8.1)

Other simulations (D3a, D3b, etc.) sample varying levels of turbulence at each rotation
rate. At the lower rotation rates (Ω0 ≤ 1.5 Ω⊙ ), dynamo action was not realizable
at Pm = 0.5 along this main branch of solutions, but is achieved for more turbulent
simulations (D1.5a, D0.5a, etc.). Table 8.2 details an additional selection of simulations
which were carried out at higher magnetic Prandtl numbers (Pm = 1 − 4) for various
rotation rates. Lastly, a selection of failed dynamos are reported in Table 8.3, to help
constrain the boundaries in parameter space for dynamo action in these simulations.
The parameter space explored by this family of dynamo solutions is illustrated
in Figure 8.1. Here the primary input parameters of our dynamo simulations, theturbulent diffusivity η and the bulk rotation rate Ω0 , are shown for all of our convection
zone dynamos. As detailed in the tables, triangles denote dynamos with Pm = 0.5
while diamonds denote high Pm dynamos. The failed dynamos are shown with blue
crosses. As becomes immediately apparent, near the solar rotation rate there appears
to be a deep dip in this parameter space where successful dynamos can occur, and such
dynamos must be run at substantially lower diffusivities. A few dynamos are labeledwith case names in Figure 8.1 for reference with the tables.
All dynamos shown in Figure 8.1 build large-scale magnetic wreaths in the bulkof their convection zone. The character of the wreaths changes somewhat across thisparameter space. Generally, magnetic wreaths in the rapidly rotating simulations (Ω0 &3 Ω⊙ ) fill the bulk of the convection zone, with substantial structure in both radius andTable 8.1.

Overview of Dynamos at Pm=0.5
Rm′

Ro

Roc

ν

η

Dynamos on the η ∝ Ω0path (denoted in Fig. 8.1 with black triangles)
D3
97 × 256 × 512
3.28×105 1.22×107 173 1048652
D597 × 256 × 512
1.04×106 6.70×107 268 132 134
66
D10
97 × 512 × 1024 3.22×106 6.75×108 253 228 126114
D15129 × 512 × 1024 7.05×106 2.61×109 288 272 144
136

0.374
0.272
0.153
0.119

0.315
0.241
0.134
0.101

1.32
0.940
0.592
0.452

2.64
1.88
1.18
0.904

3
5
10
15

S
C
C
C

Dynamos at other turbulence levels (denoted with colored triangles)
D0.5a
145 × 256 × 512
2.52×104 2.01×105 154 127
77
D0.5b 145 × 512 × 1024 1.43×104 8.03×105 395 347 197

63
173

2.45
3.29

0.688
0.779

1.72
0.859

3.43
1.72

0.5
0.5

S
O/C?

Case

Nr , Nθ , Nφ

Ra

Ta

Re

Re′

Rm

Ω0 /Ω⊙

Time-dependence

−2/3

D1.5a
D1.5b

97 × 256 × 512
257 × 512 × 1024

2.33×105
7.24×105

5.02×106
2.01×107

215
483

178
414

107
241

89
207

0.993
1.54

0.404
0.371

1.03
0.515

2.06
1.03

1
1

S
O/C?

D3a
D3b

97 × 256 × 512
143 × 512 × 1024

5.84×105
1.11×106

2.41×107
6.08×107

244
343

154
273

122
171

77
136

0.447
0.566

0.295
0.257

0.940
0.592

1.88
1.18

3
3

O/C
C

D10 L

97 × 512 × 1024

2.15×106

2.68×108

331

110

165

55

0.131

0.177

0.940

1.88

10

C

Note. — Dynamo simulations at many different rotation rates. All simulations have inner radius rbot = 5.0 × 1010 cm and outer radius ofrtop = 6.72 × 1010 cm, with L = (rtop − rbot ) = 1.72 × 1010 cm the thickness of the spherical shell. All quantities are the same as defined inTable 2.3. Briefly recapping, evaluated at mid-depth are the Rayleigh number Ra, the Taylor number Ta, the rms Reynolds number Re andfluctuating Reynolds number Re′ , the magnetic Reynolds number Rm and fluctuating magnetic Reynolds number Rm′ , the Rossby number Ro ,and the convective Rossby number Roc. For all simulations, the Prandtl number Pr = ν/κ is 0.25 and the magnetic Prandtl number Pm = ν/ηis 0.5. The viscous and magnetic diffusivity, ν and η, are quoted at mid-depth (in units of 1012 cm2 s−1 ). The rotation rate Ω0 of each reference
frame is in multiples of the solar rate Ω⊙ = 2.6 × 10−6 rad s−1 or 414 nHz. The viscous time scale at mid-depth τν = L2 /ν is 3640 days forcase D5 and the resistive time scale is 1820 days. Rotation periods at three and five times the solar rate are in turn 9.3 days and 5.6 days. The
time-dependence of the dynamo is indicated, with dynamos that build persistent and steady wreaths labeled by S, dynamos where the energies
oscillate significantly by O, and dynamos that achieve cycles of polarity reversal by C. These cases are denoted in Figure 8.1 with triangles.

144

Table 8.2.

Dynamos at Higher Pm

Nr , Nθ , Nφ

Ra

Ta

Re

Re′

Rm

Rm′

Ro

Roc

ν

η

Pm

Ω0 /Ω⊙

M3
M3-pcpf

129 × 512 × 1024
129 × 512 × 1024

7.18×104
7.31×104

1.25×106
1.25×106

179
162

125
123

716
651

500
492

1.04
1.01

0.656
0.663

1.37
1.37

0.343
0.343

4
4

1
1

O/C?
C

D1.25a-pm2

145 × 256 × 512

1.72×105

3.48×106

238

211

476

422

1.29

0.419

1.03

0.52

2

1.25

O?

D3-pm1D3-pm2D3-pm4145 × 256 × 512
145 × 512 × 1024
145 × 512 × 1024

2.98×105
3.08×105
3.12×105

1.22×107
1.22×107
1.22×107

149
145
161

102
101
94

149
291
644

102
202
377

0.372
0.370
0.336

0.300
0.306
0.308

1.32
1.32
1.32

1.32
0.660
0.330

1
2
4

3
3
3

C
C
C?

D5-pm1

97 × 256 × 512

8.24×105

6.70×107

180

132

180

132

0.241

0.214

0.940

0.940

1

5

C

Case

Time-dependence

Note. — Parameters are defined in Table 8.1. The dynamos shown in this table are at relatively high magnetic Prandtl numbers, with Pm = 1 − 4. These
cases are denoted in Fig. 8.1 with diamonds. Case M3 was initially published in Brun et al. (2004) and has been continued here for comparison. Cases M3
and M3-pcpf are at Prandtl number Pr = 0.125, rather than Pr = 0.25 as in the other simulations of this thesis. Case D1.25a-pm2 has been run to help
constrain the boundary of sustained dynamo action and has currently only evolved for 1000 days.

145

Table 8.3.

Dynamos That Failed

Case

Nr , Nθ , Nφ

Ra

Ta

Re

Re′

Rm

Rm′

Ro

Roc

ν

η

Pm

Ω0 /Ω⊙

D1a
D1a-pm1
D1a-pm2
D1b
D1b-pm1

97 × 256 × 512
97 × 256 × 512
97 × 512 × 1024
97 × 256 × 512
97 × 256 × 512

5.67×104
5.66×104
5.23×104
1.22×105
1.23×105

8.03×105
8.03×105
8.03×105
2.23×106
2.23×106

138
140
139
220
224

118
118
121
212
212

69
140
279
110
224

59
118
243
106
212

1.22
1.22
1.23
1.65
1.64

0.494
0.493
0.475
0.444
0.447

1.72
1.72
1.72
1.03
1.03

3.43
1.72
0.859
2.06
1.03

0.5
1
2
0.5
1

1
1
1
1
1

D1.25aD1.25a-pm1

145 × 256 × 512
145 × 256 × 512

1.72×105
1.72×105

3.48×106
3.48×106

238
238

211
211

119
238

106
211

1.29
1.29

0.419
0.419

1.03
1.03

2.06
1.03

0.5
1

1.25
1.25

D1.5
D3 L

145 × 256 × 512
97 × 256 × 512

8.73×104
1.21×105

1.21×106
2.82×106

137
109

74
41

68
54

37
20

0.682
0.279

0.516
0.409

2.10
2.75

4.19
5.50

0.5
0.5

1
3

Note. — These simulations failed to attain substantial dynamo action. Some are marginal, with very low magnetic energies, but inmost cases the magnetic energies decline exponentially. These cases are denoted in Fig. 8.1 with crosses and are labeled F.

146

147

Figure 8.1 — Map of full dynamo η − Ω0 parameter space. These parameters are theeddy diffusivity η and bulk rotation rate Ω0 . Solutions at magnetic Prandtl number
0.5 are shown with triangles while cases at higher magnetic Prandtl numbers are shownwith diamonds. Cases that build steady, persistent magnetic wreaths are labeled S,
while those that undergo oscillations but rarely flip polarities are O and those that
undergo many polarity reversals are C. Cases shown with blue crosses and labeled F
are failed dynamos, where magnetic energies drop over long periods of time. Question
marks indicate where the time-dependence remains uncertain. See Tables 8.1 (triangle
symbols), 8.2 (diamonds) and 8.3 (crosses) for simulation details.

latitude, as seen previously in cases D3 and D5 (Chapters 5-7). The mean poloidal fieldis typically complex, with generally two different polarities in the polar and equatorial
regions. These wreaths can also be single structures that cross the equator and primarily
have a single polarity, as will be seen for the most rapidly rotating cases D10 and D15. In
the more slowly rotating cases (Ω0 . 1.5 Ω⊙ ), the wreaths become increasingly confinedto the bottom of the convection zone, and while they retain their latitudinal extent,
much of the prominent radial structure seen in the rapidly rotating cases disappears.
At present it is unclear whether this is a matter of more effective turbulent pumping in
those simulations, or changes in the differential rotation and the velocity shear available

148
for amplifying the magnetic fields. The mean poloidal field in these slower rotating
simulations is often of a single polarity throughout the convection zone.
Many of the dynamos in Figure 8.1 exhibit temporal variations, with either oscillations in magnetic energies or cycles of activity where the polarity of the global-scale
magnetic fields routinely changes. The labeling denotes the temporal characteristics ofthe dynamo solution, with steady solutions that do not oscillate noticeably indicatedby S, solutions that oscillate frequently but rarely change their global-scale polarity by
O/C, and solutions that routinely interchange polarities by C.
Overall, we have found three solutions that are steady in nature (cases D0.5a,
D1.5a, and D3), and these generally appear to cluster near the boundary in parameter
space between successful dynamos and failed dynamos that do not successfully regenerate their magnetic fields. Even these steady simulations evince small oscillations intheir volume-averaged magnetic energy densities, as seen previously in case D3 (Chapter 5). As the magnetic Reynolds number increases, the flows and magnetic fields tend
to become more time-dependent, with the dynamos showing either large oscillations intheir magnetic energies or undergoing repeated global-scale magnetic polarity reversals.
We begin by returning to familiar ground, exploring with more extensive sampling
the turbulent parameter space in a series of simulations rotating three times faster thanthe Sun. We first examine the onset of time-dependent behavior in simulations at threetimes the current solar rate.

8.1

Higher Levels of Turbulence at 3 Ω⊙ and Pm = 0.5
In our simulations rotating at three times the solar rate, we have examined how

our wreaths of magnetism change as we raise the overall level of turbulence. This isaccomplished along one of two paths in parameter space. On the first path, we simultaneously decrease ν, κ and η, thus maintaining a fluid Prandtl number of 0.25 andmagnetic Prandtl number of 0.5 as the diffusivities are decreased. This path corresponds

149
to dynamo cases D3, D3a and D3b. On this path the convection becomes more complex
and turbulent and in hydrodynamic simulations would drive a substantially stronger
differential rotation. Both the fluid Reynolds number and the magnetic Reynolds number increase in the successively more turbulent simulations. In these dynamo solutions
we find that wreaths of magnetism persist but begin to undergo oscillations similar to
those found in case D5 rotating at five times the solar rate.
Case D3a – The time-dependence of case D3a is shown in Figure 8.2. This
simulation is somewhat more turbulent than case D3, with typical rms and fluctuating Reynolds numbers of 244 and 154 respectively, and rms and fluctuating magnetic
Reynolds numbers of 122 and 77. Case D3a undergoes many oscillations in energy andmean field strength with a typical timescale of 500 days, as shown in the time tracesof Figure 8.2a, b. Generally, the azimuthally averaged toroidal and poloidal fields arestable, retaining the same polarity through many such oscillations. Only very rarely(twice in the 16000 days shown here) do the global-scale fields flip in polarity. Thisis evident in time-latitude plots of hBφ i at mid-convection zone, shown in Figure 8.2c.
Here we note that the large excursion in mean field strengths occurring between days
13000-14500 corresponds to a strong, single polarity state. As in case D5 during similar
excursions, we find that during this interval of time the mean poloidal field has changed
from an odd-parity state, with strong contributions from the odd-ℓ components, to an
even-parity state where the even-ℓ components are more prominent. The dynamo exits
this state at roughly day 14500 and appears to return to a more normal state, with two
opposite polarity wreaths and a predominantly odd-parity poloidal field.
Throughout the history of this case, weak magnetic structures appear to propagate from the equatorial regions to both poles. As in case D5 (Chapter 6), thesemagnetic structures are associated with local bands of slightly faster differential rotation which also propagate toward the poles (Fig. 8.2d). These structures are observedwith each oscillation, being launched from the equatorial regions roughly every 500 days.

150

Figure 8.2 — Time-dependent behavior in the oscillating case D3a. (a) Volumeaveraged kinetic and magnetic energies, showing DRKE, CKE, TME, PME and FME
as labeled. Many small oscillations occur on roughly 500 day timescales, with a large
excursion around day 9000. (b) Mean hBφ i averaged over northern and southern hemispheres at mid-convection zone. Despite the many oscillations, a global-scale polarity
reversal occurs only once (near day 9000), with a second significant excursion between
roughly days 13000-14500. (c) Time-latitude maps of hBφ i at mid-convection zone.
During roughly days 13000-14500 a strange, single-polarity state emerges. (d) Timelatitude map of Ω′ at mid-convection zone, with time average removed to emphasize thepoleward propagating velocity structures.

151
At a few intervals in the simulation, very strong magnetic features appear near the poles(i.e. near days 5500-6000 and 9000-9500). At present it is unclear whether these areformed through local amplification of hBφ i by either the turbulence or the differential
rotation, or if they represent a geometric amplification of magnetic field as particularly
strong wreaths slip to the poles and are concentrated into a smaller volume there. Thesepolar magnetic structures are associated with strong velocity structures.
The patterns of convection achieved in case D3a are shown in Figure 8.3. The
convective cells are a little more complex than in case D3 (compare with Fig. 5.2) andthe amplitude of the motions is 10% faster than in that case. The toroidal field in thismore turbulent case is somewhat higher, with typical field strengths at mid-convection
zone of almost ±10 kG and peak field strengths of roughly ±35 kG. In comparison,
typical field strengths in case D3 were ±7 kG with peak strengths of roughly ±26 kG.
This snapshot is taken at an instant when the mean toroidal fields are quite strong(TME is at a peak). As such, the magnetic wreaths are visibly dominated by the mean
toroidal field hBφ i and their structure appears quite similar to those found in case D3.
When the fields are weaker (say at day 12500 when TME and PME are at a minimum),the structure of the wreaths is very similar, though their typical field strengths at midconvection zone are then only ±5 kG, with peak field strengths of ±20 kG. Thus at thetimes when hBφ i is weak, case D3a returns to a state very similar to case D3. Whencase D3a is in a single-polarity state, between days 13000-14500, the magnetic fields
at mid-convection zone tend to be in a single wreath of negative polarity that wanders
across the equator. Surrounding this structure is weaker positive polarity field.
The differential rotation in this case is similar to that realized in the less turbulent
case D3. As shown in Figure 8.3d, the equator remains fast and prograde while the poles
are filled with retrograde flow. There is substantial radial shear near the equator. The
mean toroidal field associated with the wreaths of magnetism (Fig. 8.3e) is comparable
in strength to the fields achieved in case D3, but the cores of the wreaths are located152

Figure 8.3 — Patterns of convection in case D3a. (a) Radial velocity vr in Mollweide
projection near the top of the shell (0.95 R⊙ ). (b) Toroidal magnetic field Bφ at midconvection zone (0.85 R⊙ ) and (c) near the bottom of the convective shell (0.73 R⊙ ).
These snapshots are shown at day 12200, a time when the mean magnetic fields arestrong. Also shown are 100 day averaged profiles of (d) angular velocity Ω, with fast
equator and slow poles, (e) mean toroidal field hBφ i and (f ) mean poloidal vector
potential hAφ i with contours representing the mean poloidal field lines.

153
slightly closer to the equator than in case D3 where they straddled ±15◦ . The poloidal
vector potential (Fig. 8.3f ) is also similar in morphology, though now the equatorial
region is comparable in strength to the poleward regions. In case D3, the vector potential
near the equator was about a factor of three weaker than that found in this case above
latitude ±15◦ .
Case D3b – When the level of turbulence is increased further (case D3b, Fig. 8.4ad), the global scale fields appear to flip far more frequently. This simulation is yet moreturbulent, with typical rms and fluctuating Reynolds numbers of 343 and 273 respectively, and rms and fluctuating magnetic Reynolds numbers of 171 and 136. Though
this simulation has not achieved nearly as much time evolution, with 3300 days of total
evolution since the last adjustment of diffusivities, the global scale fields have alreadyexchanged polarity at least once. We thus suspect that these oscillations are due tothe higher magnetic Reynolds number achieved rather than being linked intrinsically tothe higher rotation rate used in D5. At times, the magnetic wreaths survive for longintervals, with one pair existing from roughly day 1400 to day 2100. At other periods,
strong wreaths are built in each hemisphere but only survive for short periods of time,
disappearing within a few hundred days. This behavior holds true at deeper locations
within the convection zone as well. After day 2700, wreath building occurs predominantly in only the southern hemisphere. As in the other simulations, prominent velocity
structures are launched toward the polar regions. In the interval of simulated time, only
a few such structures have appeared, with possibly two in the northern hemisphere andone in the southern.
The patterns of convection achieved in case D3b are shown in Figure 8.5. The
radial velocity structures are more complex than in case D3a and average amplitudes
of motion have increased a further 20% (or roughly 35% faster than the radial flows incase D3 at this depth). The toroidal fields have amplitudes very similar to those realizedin case D3a, with typical strengths of ±10 kG at mid-convection zone and peak ampli-

154

Figure 8.4 — Time-dependent behavior in the cyclic case D3b. (a) Volume-averaged
kinetic and magnetic energies, and (b) mean hBφ i averaged over northern and southernhemispheres at mid-convection zone. Individual quantities are colored as in Fig. 8.4a, b.
Though this simulation has evolved for only a short period, one polarity reversal has
already occurred. (c) Time-latitude maps of hBφ i at mid-convection zone. The large
structures visible at the poles near day 1500 appear to be part of the initial transientas the dynamo equilibrates and adjusts the profile of differential rotation. (d) Timelatitude map of Ω′ at mid-convection zone, with time average removed to emphasize thepoleward propagating velocity structures.

155

Figure 8.5 — Patterns of convection in case D3b. (a) Radial velocity vr in Mollweide

projection near the top of the shell (0.95 R⊙ ). (b) Toroidal magnetic field Bφ at midconvection zone (0.85 R⊙ ) with two wreaths of opposite polarity. (c) Wreaths near
the bottom of the convective shell (Bφ at 0.73 R⊙ ). These snapshots are shown at
day 2050, a time when the mean magnetic fields are strong. Also shown are 100 day
averaged profiles of (d) Ω, (e) hBφ i and (f ) hAφ i.

156

Figure 8.6 — Bφ achieving a more non-axisymmetric state in case D3b. (a) Midconvection zone snapshot of Bφ at day 2400 and (b) at day 2860. At these times thewreaths have become highly non-axisymmetric structures. (c) Same fields near bottomof convection zone (0.73 R⊙ ) at day 2400 and (d) at day 2860.

tudes of ±35 kG. The toroidal fields at the poles are much stronger in case D3b thanin the previous 3 Ω⊙ simulations, with fine-scale structure and typical field strengths
of ±1.5 kG. In comparison, the polar Bφ of case D3a had amplitudes of ±1 kG, while
those fields in case D3 rarely exceeded ±0.5 kG in amplitude. The profiles of differential rotation (Fig. 8.5d), mean toroidal magnetic field (Fig. 8.5e) and the mean poloidal
vector potential (Fig. 8.5f ) are all fairly similar to those found in case D3a.
The wreaths of magnetism in case D3b frequently attain strongly non-axisymmetricstates. This is shown at two different times in Figure 8.6. At around day 2400 (Fig 8.6a),a region of reversed polarity appears in the midst of the southern hemisphere. This positive polarity Bφ replaces the preexisting negative wreath, and is more equatorially located than the previous wreaths (see Fig. 8.5b). The wreath in the northern hemisphereappears to peel apart and unwind toward the north pole. At this time the dynamo falls
into a state where the southern hemisphere is generating magnetic wreaths, while thenorthern hemisphere is full of more complex structures, as near day 2900 (Fig. 8.6b).

157
Here hBφ i in the southern hemisphere is negative, but a positive polarity structureof substantial amplitude occupies roughly 25% of the domain in longitude. Near thebottom of the convection zone (Fig. 8.6c, d) the story is similar. There are distinct
wreath-like structures in each hemisphere, but individual structures occupy only about180◦ in longitude before being replaced by a structure with opposite polarity. As aresult of this cancellation, hBφ i is very small in the lower convection zone. In case D3b
the mean toroidal fields no longer dominate the structure of the wreaths, and thesestructures must be analyzed on local scales.

8.2

High Pm Dynamos at 3 Ω⊙
To further probe the nature and sensitivity of these dynamo solutions we ex-plore a second path through parameter space where the magnetic Prandtl number isincreased. Along this path, the viscous and thermal diffusivities ν and η are held constant while the magnetic diffusivity η is decreased, yielding larger magnetic Prandtl
numbers and higher magnetic Reynolds numbers. This path corresponds to dynamocases D3, D3-pm1, D3-pm2 and D3-pm4. These dynamo simulations begin occupying
a similar region in parameter space as our previous dynamo simulations conducted at
the solar rotation rate (e.g., Brun et al. 2004; Browning et al. 2006). Along this path,the magnetic Reynolds number increases while the fluid Reynolds number should remain approximately constant, only changing as the flows themselves respond to themagnetism that they generate. The convection retains a comparable level of complexity
and the underlying driving of the differential rotation should be nearly constant.
Case D3-pm1 – At higher magnetic Prandtl numbers our dynamo simulations
also begin to experience significant time dependence. This occurs already in our firstcase, case D3-pm1 with magnetic Prandtl number Pm = 1. The rms Reynolds numberin this case is lower than in case D3, with a value of 149, owing to the weaker differential
rotation realized in this simulation. The fluctuating Reynolds number is very similar

158

Figure 8.7 — Time-dependent behavior in cyclic case D3-pm1.

(a) Volume-averaged
kinetic and magnetic energies, and (b) mean hBφ i averaged over northern and southernhemispheres at mid-convection zone. (c) Time-latitude maps of hBφ i at mid-convection
zone. (d) Time-latitude map of Ω′ at mid-convection zone, with time average removed
to emphasize the poleward propagating velocity structures.

159
with a value of 102, but the rms and fluctuating magnetic Reynolds numbers are higher
than in that case, being 149 and 102 respectively.
The time history of case D3-pm1 is shown in Figure 8.7. At day 0 the values ofη in case D3 were dropped by a factor of two. After about 500 days the steady wreaths
from case D3 begin to break apart as new wreaths are generated near the equator. Thisis visible in time latitude maps of hBφ i (Fig. 8.7c) at mid-convection zone. The mean
toroidal fields hunt between states with two wreaths of opposite polarity and states
where one or two single polarity wreaths are built (i.e. days 3000-5000). When two
wreaths are present the simulation appears to undergo frequent and rapid reversals,with typical timescales of only a few hundred days. Accompanying these reversals arestrong angular velocity structures that propagate toward the polar regions (Fig. 8.7d).
Case D3-pm2 – Our next case on the high magnetic Prandtl number path iscase D3-pm2, with Pm = 2. This case was initiated from case D3-pm1, and at roughly
day 600 of that simulation the eddy diffusivity η was dropped by an additional factor of
two, with ν and κ still the same as in case D3. In this simulation the rms and fluctuating
Reynolds numbers are almost the same as in its progenitor (145 and 101 respectively),
but the magnetic Reynolds numbers have increase by a factor of two to values of 291
and 202 for the rms and fluctuating quantities.
The time history for case D3-pm2 is shown in Figure 8.8. At first the wreaths
from the previous dynamo persist, but after about 250 days they are replaced by different polarity structures and this dynamo diverges strongly from its progenitor. Afterabout 1000 days the dynamo enters a state where almost all hBφ i is located in thenorthern hemisphere. The dynamo undergoes cycles in that hemisphere with the mean
fields reversing on short timescales of a few hundred days. In the southern hemispherethe mean fields are weaker but are able to retain their average polarity and do notexperience as many reversals. Angular velocity features launch toward the north polefrequently. These structures occasionally occur in the southern hemisphere, when the

160

Figure 8.8 — Time-dependent behavior in cyclic case D3-pm2.

(a) Volume-averaged
kinetic and magnetic energies, and (b) mean hBφ i averaged over northern and southernhemispheres at mid-convection zone. (c) Time-latitude maps of hBφ i at mid-convection
zone. (d) Time-latitude map of Ω′ at mid-convection zone, with time average removed
to emphasize the poleward propagating velocity structures.

161

Figure 8.9 — Patterns of convection in case D3-pm2. (a) Radial velocity vr in Mollweide

projection near the top of the shell (0.95 R⊙ ). (b) Toroidal magnetic field Bφ at midconvection zone (0.85 R⊙ ) with one strong wreath. (c) Two wreaths of same polarity arenear the bottom of the convective shell (Bφ at 0.73 R⊙ ). These snapshots are shown at
day 3100, a time when the mean magnetic fields are strong in the northern hemisphere.
Also shown are 100 day averaged profiles of (d) Ω, (e) hBφ i and (f ) hAφ i.

162mean magnetic fields become particularly strong (i.e. days 3000-3800).
The patterns of convection in case D3-pm2 are shown in Figure 8.9. Near thesurface, the convective cells are very similar to those found in our other simulations
at three times the solar rate with velocities similar to those in case D3, rather than
the faster flows found in cases D3a and D3b. The magnetic fields in this case aremuch stronger and show marked asymmetry, with strong fields occupying narrow rangesof longitude. At mid-convection zone, often a single strong wreath appears in thenorthern hemisphere. These states are similar to those realized occasionally in case D5,
and as there, weaker opposite polarity field surrounds the single wreath. At greater
depths, two wreaths of the same polarity occupy the two hemispheres. A profile ofthe mean differential rotation (angular velocity Ω) is shown averaged over a 100 day
interval around the snapshots (Fig. 8.9d). Accompanying profiles show hBφ i and hAφ iaveraged over the same interval (Fig. 8.9e, f ). During this interval, the wreaths possessa substantially different mean poloidal field than in our previous simulations at threetimes the solar rate. Here the mean poloidal field is almost quadrupolar in nature.
Case D3-pm4 – Our highest magnetic Prandtl number case at three times
the solar rate is case D3-pm4 with Pm = 4. This simulation is quite magneticallyturbulent. The rms and fluctuating Reynolds numbers remain comparable to the othersimulations on the high-Pm branch, with values of 161 and 94 respectively. The magnetic
Reynolds numbers are much higher, with rms and fluctuating values of 644 and 377.
The fluctuating magnetic Reynolds number is among the highest achieved in any of ourrapidly rotating dynamo simulations and is comparable to those achieved in previous
simulations of the solar dynamo (Brun et al. 2004; Browning et al. 2006). Despite thishigh level of magnetic turbulence, case D3-pm4 still builds significant magnetic wreaths
within its convection zone.

163
The patterns of convection and the magnetic structures realized are shown in
Figure 8.10 at a time shortly after the dynamo has equilibrated. Though the velocity patterns remain very similar, the magnetic fields have significantly more fine-scale
structure. Though highly complex at mid-convection zone, the toroidal fields retain an
overall polarity in each hemisphere (Fig. 8.10b). Near the bottom of the convection
zone the wreaths again are marked by the strong mean fields (Fig. 8.10c, e). Indeed, themagnetic fields generated within these wreaths are among the strongest achieved in anyof our rapidly rotating dynamos, with typical amplitudes of more than ±20 kG in thefluctuating fields and peak amplitudes of ±40 kG near the base of the convection zone.
This simulation has only experienced about 1800 days of evolution and the timesteps
are heavily limited by the Alfvénic CFL with typical limits near 30 seconds or less.

8.3

Resulting Differential Rotation
In our high magnetic Prandtl number branch of simulations, we find that the mean

profile of differential rotation is somewhat different than in our Pm = 0.5 simulations.
In these higher magnetic Prandtl number cases, the equator remains prograde withmore retrograde poles, but the overall angular velocity contrast is significantly weaker.
Measurements of the angular velocity contrast in latitude and in radius are shown for
all of our dynamos at three times the solar rate in Table 8.4. These values are timeaveraged over the indicated ranges of dates and these intervals typically span several
magnetic oscillations or reversals. Case D3b is still undergoing some evolution in ∆Ω at
this time, as is case D3-pm4, which has seen approximately 1800 days of evolution andhas uncertain time-dependence. The other cases are well equilibrated. Accompanying
each measurement is an indication of the standard deviation of the angular velocity
contrast in time, with case D3 showing little variation and the more turbulent cases
showing substantially larger excursions.

164

Figure 8.10 — Patterns of convection in case D3-pm4. (a) Radial velocity vr in Moll-

weide projection near the top of the shell (0.95 R⊙ ). (b) Toroidal magnetic field Bφ at
mid-convection zone (0.85 R⊙ ) with indistinct wreaths. (c) Two strong wreaths are athigh latitudes near the bottom of the convective shell (Bφ at 0.73 R⊙ ). These snapshots
are shown at day 1830. The dynamo is still equilibrating at this time. Also shown are
20 day averaged profiles of (d) Ω, (e) hBφ i and (f ) hAφ i.

165
Table 8.4.
Case

Mean ∆Ω in Dynamos at 3 Ω⊙

0.97R⊙∆Ωlat
0.85R⊙∆Ωrequator

Epoch
days

D3
D3a
D3b

1.18 ± 0.05
1.11 ± 0.18
1.01 ± 0.17

0.79 ± 0.06
0.68 ± 0.10
0.56 ± 0.07

0.70 ± 0.02
0.71 ± 0.13
0.71 ± 0.14

2000 − 6980
2000 − 15980
1000 − 3120

D3-pm1D3-pm2D3-pm40.87 ± 0.10
0.78 ± 0.08
0.90 ± ?

0.53 ± 0.07
0.48 ± 0.05
0.74 ± ?

0.55 ± 0.06
0.48 ± 0.06
0.38 ± ?

1000 − 6850
1000 − 4730
1830 − 1850

Note. — Angular velocity shear in units of µrad s−1 , with ∆Ωlat measured at two depths and ∆Ωr measured across the full shell at the equator.

In the Pm = 0.5 branch of dynamos, the more turbulent dynamos still have astrong differential rotation. The latitudinal angular velocity contrast ∆Ωlat in the upper
convection zone decreases slightly as the diffusivities are dropped. At mid-convection
zone the decrease is stronger. In contrast, the radial shear at the equator remains
almost constant in this group of cases. The high magnetic Prandtl number dynamos
have differential rotation profiles that become substantially weaker as η is decreased. In
this family of solutions, both the latitudinal and radial shear are markedly smaller. In
comparing to the Pm = 0.5 dynamos, case D3-pm1 and case D3b, with similar values ofη throughout the convection zone, have similar latitudinal shear at mid-convection zone
but different angular velocity contrasts in the upper convection zone and in radius.

8.4

Extreme Rotators: 10 and 15 Ω⊙ dynamos
Our most rapidly rotating dynamo simulations are currently at ten and fifteentimes the current solar rotation rate. In a star like our Sun, such rapid rotation is likely
only when the star is very young, shortly after reaching the main sequence. In thesestars, we find that the dynamos can produce magnetic fields which are strong enough tolargely quench the differential rotation. Despite this, cyclic oscillations and global-scalepolarity reversals of the mean magnetic fields continue to occur.

166
In total, there are three simulations rotating in this regime. Two of these simulations are rotating at 10 Ω⊙ , one with the same diffusivities used in case D5 (case D10L)
and one with lower diffusivities in keeping with the scaling law of equation 8.1 (case D10).
Our most rapidly rotating dynamo is case D15, at 15 Ω⊙ . Here we will briefly examine
the emergence of nests of convection in case D10L, and the suppression of differential
rotation in case D15.
Case D10L – Only one of our most rapidly rotating cases retains a strong differential rotation. This is case D10L rotating at 10 Ω⊙ and with eddy diffusivities identicalto those in both case D5 and case D3a. Typical rms and fluctuating Reynolds numbers in this simulation are 331 and 110 respectively, with rms and fluctuating magnetic
Reynolds numbers of 165 and 55. The time history of case D10L is shown in Figure 8.11, covering 6000 days of the simulation after the initial seed magnetic fields are
introduced. The magnetic energies grow quickly to near equipartition with the kinetic
energies before they react back on the differential rotation and substantially suppress it.
Time-latitude maps of hBφ i and Ω reveal how this quenching occurs. Case D10L
builds strong wreaths of magnetism which frequently propagate toward the polar regions(Fig. 8.11c). These poleward-slips of hBφ i are accompanied by substantial poleward
propagating angular velocity structures, which transport quickly rotating material fromthe prograde equator to the retrograde poles. This is evident in the time-evolutionof the latitudinal and radial angular velocity contrast ∆Ω (Fig. 8.11d). During theinitial transient, the latitudinal angular velocity contrast ∆Ωlat is substantially reduced,
dropping by a factor of about two in the upper convection zone and by nearly a factorof five at mid-convection zone. Convection rebuilds the differential rotation and by
day 2300 nearly half the angular velocity contrast has been rebuilt. At this point
however, strong magnetic wreaths are formed leading to another pulse of poleward
propagating fields and flows. The radial angular velocity contrast ∆Ωr is less affectedby the magnetism. It experiences temporal variations on similar timescales, but retains

167

Figure 8.11 — Time-dependent behavior in case D10L. (a) Volume-averaged kinetic andmagnetic energies, and (b) mean hBφ i averaged over northern and southern hemispheresat mid-convection zone. (c) Time-latitude maps of hBφ i at mid-convection zone. (d) Angular velocity contrast ∆Ω, with ∆Ωlat measured near the surface (top line, green) andat mid-convection zone (bottom line, red) and with ∆Ωr measured across the shell at
the equator (middle line, purple). (e) Time-latitude map of full angular velocity Ω at
mid-convection zone, with fast equator and slow poles. (f ) Map of Ω′ with time average
removed, emphasizing temporal variations. The initial transient slows the equator andspeeds up the poles, reducing the overall differential rotation.

168

Figure 8.12 — Differential rotation and magnetism in case D10L. (a) Profile of angularvelocity Ω averaged over an interval of about 100 days when the differential rotationis strong (days 4490-4600) accompanied by (b) radial cuts. (c) Two magnetic wreaths
are present. (d) The poloidal vector potential hAφ i is complex and largely of a single
polarity.

an average value similar to the hydrodynamic progenitor (visible at day 0).
The mean toroidal magnetic fields have a much more complex structure than in
our three or five solar dynamos. Strong concentrations of hBφ i appear and disappear on
short timescales near the equator while large wreaths occupy latitudes near ±20◦ . Those
wreaths eventually slip toward the poles and are often replaced by wreaths of opposite
polarity (i.e. the reversals that occur near day 2500 and day 4900). Though initiallythe dynamo builds quite symmetric wreaths, the wreaths which form after the initialtransient are less symmetric and frequently are substantially stronger in one hemisphere
and weaker in the other.
Time-averaged profiles in radius and latitude of the differential rotation and themagnetic fields are shown in Figure 8.12 at a time when the differential rotation isstrong (days 4490-4600). The equatorial regions still possess a strong radial shear, andthe profile and radial cuts are similar to cases D3 and D5. Two magnetic wreaths areformed, with a strong positive polarity in the northern hemisphere and a weaker wreathin the southern. Here the poloidal field is quite complex and is highly asymmetric witha major enhancement of hAφ i visible in the northern hemisphere. This is in marked

169

Figure 8.13 — Convective patterns in case D10L with nests of convection. Shown assnapshots in Mollweide projection are radial velocities vr (a) near the surface (0.95 R⊙ )and (b) at mid-convection zone (0.85 R⊙ ). Two active nests of convection are clearlyvisible. (c) Near the surface, the radial magnetic field Br is concentrated in the stronger
nest. (d) At mid-convection zone the active nests make an imprint on the magnetic
wreaths. Also visible near the south pole are the remnants of the magnetic wreath fromthe previous cycle. These snapshots are of day 4490 at a period when the differential
rotation is strong.

contrast to the poloidal fields achieved in our cases at three times the solar rate, whichwere dominated by low-ℓ structures with typically either largely octopolar (i.e. cases D3,
D3a, D3b) or quadrupolar (i.e. case D3-pm2) topologies.
The patterns of convection in case D10L are unique among our rapidly rotating
dynamos. The relatively strong differential rotation, combined with the rapid overallrotation, lead to strongly localized nests of convection in the equatorial regions. These
nests are shown in Figure 8.13 at day 4490, when the differential rotation is strong.
The localized convection once again is strongly visible in the radial velocity patterns
throughout the convection zone (Fig. 8.13a, b). At this instant, two nests are visible,
separated by almost 180◦ of longitude.
The nests have substantial magnetic signatures as well, particularly in the radial
magnetic field near the surface (Fig. 8.13c). Here the stronger nest (at left) is accompanied by a significant enhancement of radial field. At mid-convection zone the nests also

170
Table 8.5.

Mean ∆Ω in Dynamos at 10 and 15 Ω⊙

Case
0.97R⊙∆Ωlat
0.85R⊙∆Ωrequator

Epoch
days

D10L
D10

1.39 ± 0.27
0.45 ± 0.08

0.83 ± 0.21
0.22 ± 0.05

0.97 ± 0.19
0.38 ± 0.101000 − 6060
2720 − 5000

D15
D15
D15

0.33 ± 0.07
0.30 ± 0.04
0.51 ± 0.01

0.12 ± 0.04
0.10 ± 0.01
0.17 ± 0.003

0.26 ± 0.08
0.22 ± 0.03
0.50 ± 0.02

1500 − 4500
2000 − 3500
3950 − 4100

Note. — Angular velocity shear in units of µrad s−1 , with ∆Ωlatmeasured at two depths and ∆Ωr measured across the full shell at theequator. These measurements are averaged over the indicated rangeof days. Dynamo case D15 suddenly amplifies its differential rotation
for a period of time between days 3700-4300.

make an imprint on the toroidal field, locally stretching and shredding the wreaths. In
the more quiescent regions between the nests, the wreaths are stronger and have lessfine-scale structure (Fig. 8.13d).
In case D10L the active nests of convection are clearly visible and persist for many
hundreds of days. The temporal variations in the differential rotation lead to periods
when the nests are strong and the convection is highly confined in longitude, as wellas other periods where the differential rotation and nests are weaker. Time-longitude
maps reveal that on thousand day timescales the convection alternates between stateswith two or three active nests, spending a few hundred days in each state.
Cases D10 – Our other dynamo at ten times the solar rate is case D10, withdiffusivities lower by slightly more than a factor of two compared to case D10L. This
simulation is more turbulent, with rms and fluctuating Reynolds numbers of 253 and 228
and rms and fluctuating magnetic Reynolds numbers of 126 and 114 respectively. The
small spread between the rms and fluctuating values is due to the differential rotation
collapsing almost entirely.
The average angular velocity contrast achieved in these most rapidly rotating
simulations is reported in Table 8.5. In case D10, the angular velocity contrast ∆Ω171
and its temporal variations have both become much smaller than in case D10L. These
quantities are measure over an interval of almost 2300 days. There is little angularvelocity contrast in either radius or latitude.
Despite the small differential rotation, substantial toroidal magnetic fields are
generated throughout the convection zone. Now however, the mean fields comprise asmaller proportion of the total magnetic energy while the fluctuating fields become more
prominent. The fluctuations exist on large scales, and with substantial contributions
from longitudinal wavenumbers m=1 or m=2. The structure of the magnetic fields
remains wreath-like, though now more complex. The wreaths appear to still undergo
cyclic variations in polarity.
Case D15 – In our most rapidly rotating dynamo, case D15, the differential
rotation is even weaker. This dynamo is one of the more turbulent Pm = 0.5 dynamos,
with a fluctuating Reynolds number of 272 and a fluctuating magnetic Reynolds numberof 136. This case is thus comparable to case D3b. The hydrodynamic progenitor tothis case had a very strong differential rotation, and its rms and fluctuating Reynolds
numbers were 1670 and 293 respectively.
The time history of case D15 is shown in Figure 8.14. Starting from weak seed
fields, the dynamo quickly builds strong magnetic wreaths whose energies on averagereach equipartition with the convective flows. These fields quench the differential rotation, driving DRKE down into sub-equipartition with either the convection (CKE) or
the fluctuating magnetism. Though the differential rotation is weak, the dynamo still
undergoes substantial temporal variations and cycles of polarity change (Fig. 8.14b).
Time-latitude maps reveal that, as in case D3-pm2, this cyclic wreath building activity is largely confined to the northern hemisphere (Fig. 8.14c). The wreaths that areachieved begin to slip toward the north pole almost immediately, and typically persist in
the equatorial region for less than 500 days before beginning their poleward migration.

172

Figure 8.14 — Time-dependent behavior in case D15. (a) Volume-averaged kinetic andmagnetic energies, and (b) mean hBφ i averaged over northern and southern hemispheresat mid-convection zone. The azimuthally averaged data exists from day 660 onwards.
(c) Time-latitude maps of hBφ i at mid-convection zone. (d) Angular velocity contrast∆Ω, with ∆Ωlat measured near the surface (top line, green) and at mid-convection zone(bottom line, red) and with ∆Ωr measured across the shell at the equator (middle line,
purple). The latitudinal differential rotation is highly suppressed after the dynamo
equilibrates, but the radial shear at the equator is largely unaffected. (e) Time-latitude
map of full angular velocity Ω at mid-convection zone. (f ) Map of Ω′ with time-average
removed, emphasizing temporal variations. The initial transient slows the equator andspeeds up the poles, reducing the overall differential rotation. In this dynamo, the poles
actually become prograde, and retain that sense for long intervals.

173
The first magnetic wreaths that form were two strong wreaths of opposite polarity
in the two hemispheres. These first magnetic wreaths slip to the poles and transportenough angular momentum from the prograde equator that above ±70◦ the polar regions
change their sense of differential rotation from retrograde to prograde. Once prograde
flows are established in the polar regions, they persist for the full 5000 simulated days
shown here. The angular velocity contrasts become much weaker after the dynamo
saturates and reduces the differential rotation. As in case D10L, the latitudinal contrast is affected more strongly by the magnetism than the radial shear, though ∆Ωralso decreases significantly (Fig. 8.14d). Dynamo case D15 has latitudinal and radial
angular velocity contrasts which range from 0.3 to 0.5 µrad s−1 depending on the phaseof the dynamo oscillations (see Table 8.5). During a brief period (days 3800 - 4500) the
differential rotation in the upper convection zone becomes substantially stronger, withboth the latitudinal and radial shear increasing by more than a factor of two. The
mid-convection zone ∆Ω shows almost no variation during this interval. Even at theirpeak levels (about day 400), the angular velocity contrasts in case D15 are dramatically weaker than the contrast achieved in the hydrodynamic progenitor, which had anear-surface ∆Ωlat of 3.9 µrad s−1 and a radial shear ∆Ωr of 2 µrad s−1 .
The time-averaged profiles in radius and latitude of the differential rotation andthe magnetic fields of case D15 are shown in Figure 8.15 at a time when the differential
rotation is unusually strong (days 4000 - 4115). During this time interval, the radial
shear in the equatorial regions remains substantial, but the mid-latitudes (30◦ -60◦ )have lost much of their contrast in radius or latitude. The polar regions above ±75◦
are spinning prograde at all depths. At other points in time, as indicated by the time
trace of ∆Ωr in Figure 8.14d, the radial shear at the equator becomes much weaker,
decreasing by more than a factor of two. In these states, the most visible features inthe differential rotation profile are the prograde poles, which retain their strong angularvelocity gradients at all times.

174

Figure 8.15 — Differential rotation and magnetism in case D15. (a) Profile of angularvelocity Ω averaged over an interval of about 100 days when the differential rotation
unusually strong (days 4000-4115) accompanied by (b) radial cuts. At other intervals,
the equatorial shear disappears entirely. (c) Several narrow wreaths occupy the equatorial region, and two strong magnetic structures are visible at the poles. (d) As incase D10L, the poloidal vector potential hAφ i is quite complex.

The mean toroidal field in this case is substantially weaker than in many othersimulations, with average amplitudes of roughly ±3 kG during this interval (Fig. 8.15c).
This decrease is due in part to the equatorial wreaths becoming substantially nonaxisymmetric in structure. Strong wreaths are visible at the polar regions. At present
it is unclear whether these wreaths are sustained by continual poleward transport of fieldby many generations of wreaths being built at the equator, or if they are locally built
and maintained by the gradients in angular velocity near the poles. As in case D10L, themean poloidal magnetic field has a complex structure, with a significant enhancement
visible during this interval in the southern hemisphere.
The patterns of convection in case D15 are shown in Figure 8.16. The radial velocities show some modulation, but the strongly confined single nest of the hydrodynamic
progenitor has disappeared as the differential rotation has collapsed. Time-longitude
maps reveal that the modulation persists for more than a thousand days, and with the
weak differential rotation, the pattern remains almost stationary relative to the rotatingreference frame of the star. The strong magnetism disrupts the convective cells in some

175

Figure 8.16 — Convective patterns in case D15. Shown in global Mollweide projection

at day 4000 are (a) radial velocities vr near the surface, with visible longitudinal modulation and (b) toroidal magnetic field Bφ at mid-convection zone, with highly asymmetricstructures present throughout and strong structures near the poles. These snapshots
are taken at day 4000, a time when the differential rotation is particularly strong.

regions, particularly at mid-convection zone. Shadows of these magnetic regions remain
visible in the near-surface flows.
In case D15, wreaths of magnetism continue to pervade the convection zone. Now
these structures have gained significant non-axisymmetric components which can spanmore than 90◦ of longitude. These structures retain a high level of organization but
no longer encircle the entire convection zone. As a result of this, hBφ i has relatively
low amplitudes while Bφ locally can still attain high values. In this snapshot, the peak
amplitudes of Bφ at mid-convection zone are ±38 kG. This is comparable to the peak
amplitudes achieved in case D3b.

8.5

Spinning Down to the Sun
We now return to the Sun, considering cases which rotate more slowly than threetimes the solar rate. In this regime, dynamo action is somewhat harder to excite. Thisis evident already in Figure 8.1 at 1.5 Ω⊙ , with sustained dynamo action only appearing
in the more turbulent cases (D1.5a, D1.5b). In these respective cases, persistent or
temporally-varying wreaths are realized in the bulk of the convection zone.

176

Figure 8.17 — Differential rotation and magnetism in case D1.5a. (a) Profile of angularvelocity Ω averaged over an interval of about 100 days with (b) radial cuts. (c) Two wide
wreaths occupy the equatorial region. (d) At these slower rotation rates, the poloidal
vector potential hAφ i is much more dipolar.

Case D1.5a – Profiles of the differential rotation and magnetic wreaths generated
in case D1.5a are shown in Figure 8.17. In this simulation, the differential rotation
remains strong, with a fast equator and slow poles. The magnetic wreaths have lost
some of the strong radial extent seen in the more rapidly rotating dynamos (e.g. case D3),and survive here near the bottom of the convection zone as structures which are highly
extended in latitude. These wreaths persist with the same polarity for more than
5000 days after the dynamo equilibrates. The mean poloidal field is more dipolar innature than those realized in the more rapidly rotating dynamos, and the equatorial
region has the same polarity and sense as the polar regions.
The eddy diffusivity η must be reduced even further as the solar rotation rate
is approached, and below 1.5 Ω⊙ we have shifted to high magnetic Prandtl numbers to
excite dynamo action. Those simulations that achieve sustained dynamo action possessvalues of η which are comparable to our most turbulent rapidly rotating dynamos (e.g.
cases M3 and D3-pm4 at similar values of η). This choice was necessary because in thisrange of rotation rates, decreasing ν and κ at fixed Prandtl number appears to lead to adecrease in the overall differential rotation that is achieved. This is in marked contrast
to those simulations rotating faster than 1.5 Ω⊙ , where decreasing ν and κ uniformly177
leads to stronger differential rotation. A strong differential rotation greatly enhances
dynamo action in these wreath-building dynamos. However, even the failed dynamos
often initially create wreaths of magnetism. In these cases, dynamo action fails because
the poloidal fields are not regenerated quickly enough. As the poloidal fields resistively
decay away, the overall magnetic energies plummet with time.

8.6

The Mystery of Case M3
We turn now to case M3 rotating at the solar rotation rate. Case M3 was initiallypublished in Brun et al. (2004). This simulation, at magnetic Prandtl number Pm = 4,
generated strong fluctuating magnetic fields but produced only weak mean toroidal andpoloidal fields, with little global-scale organization. There are no striking magnetic
wreaths present in the simulation. The lack of wreaths is a mystery, as case M3 appears
to lie well within the parameter space of wreath-building dynamos (Fig. 8.1). The
differential rotation achieved in this case is somewhat weaker than that achieved inour rapidly rotating suns, but the high magnetic Prandtl number should still make thedifferential rotation highly effective at stretching and amplifying wreaths of magnetism.
Case M3 differs from the rapidly rotating dynamos in three characteristics, andone of these appears to be the answer to the missing wreaths. The first and most obvious
difference is that case M3 rotates relatively slowly, at the solar rotation rate. If wreathbuilding is facilitated by more rapid rotation, than the slow rotation could explain thelack of wreaths in case M3. Fortunately, this does not appear to be the case. Even cases
at 1.25 Ω⊙ build magnetic wreaths, as do simulations that rotate more slowly than thesun (i.e. case D0.5a). The second difference is that case M3 is at high magnetic Prandtl
number and has very high fluctuating and rms magnetic Reynolds numbers at valuesof 716 and 500 respectively. If wreaths are shredded and destroyed in the presence ofhigh magnetic Reynolds number flows, than the lack of wreaths in case M3 would raise
serious questions for the rapidly rotating dynamos which are generally at lower magnetic

178

Figure 8.18 — Energy traces for cases M3 and M3-pcpf. Shown in overlaying volume-

averaged energy traces are case M3-pcpf (solid) and the original case M3 (dashed), overthe same interval in time. At day 0 the bottom boundary condition in case M3-pcpf
was changed from potential field to perfect conductor. Shortly afterwards the energy
contained in mean toroidal fields (TME) grows substantially.
Reynolds numbers. But magnetic wreaths appear to persist in our high Pm simulationcase D3-pm4, with a similar fluctuating magnetic Reynolds number of 437.
The third and final significant difference between case M3 lies in the treatmentof the bottom magnetic boundary condition. In case M3, the bottom boundary was
taken to match onto a potential field solution, whereas in the rapidly rotating dynamos
we have treated the bottom boundary as a perfect conductor. Perfectly conducting
boundaries trap horizontal magnetic field, while potential field boundaries allow thatfield to exit the simulation. To test whether the treatment of boundary conditions
affects the presence of magnetic wreaths, we have modified case M3 by changing thebottom boundary to a perfectly conducting medium. This new case, case M3-pcpf has aperfectly conducting bottom boundary and potential field top boundary, in the fashionof case D3 and all of our other rapidly rotating dynamos. No other changes were madebetween the two simulations. Cases M3 and M3-pcpf were continued in parallel, andtheir volume-averaged energies are shown over the same interval of time in Figure 8.18.

179

Figure 8.19 — Wreaths of magnetism in solar simulations.

Shown are time-latitude
maps of hBφ i at 0.75 R⊙ for (a) case M3-pcpf and (b) case M3 over the same 2000 day
interval of time. The bottom boundary conditions were changed in case M3-pcpf at
day 0, and the two simulations quickly diverge. Case M3-pcpf builds strong wreaths ofmagnetism while case M3 has weaker and less organized mean fields. For both cases,
the time-latitude maps are scaled from ±2 kG.

180
The energy contained in the mean toroidal magnetic fields (TME) begins to growalmost immediately in case M3-pcpf after the change of boundary conditions. The fluctuating magnetic energies (FME) and the energy of the mean poloidal fields (PME)also grow relative to those of case M3. In response, the differential rotation is slightly
reduced, the the convective kinetic energies (CKE) and the meridional circulations(MCKE) are basically unchanged.
Time-latitude maps of the mean toroidal field reveal that case M3-pcpf builds
magnetic wreaths, while case M3 does not. Shown in Figure 8.19 are maps of hBφ i forboth cases over the same 2000 day interval in time. Within 300 days, case M3-pcpf has
built substantial magnetic wreaths with typical field strengths of ±2 kG and peak mean
toroidal fields of over 4 kG. In contrast, case M3 builds weak and disorganized mean
toroidal fields, with typical amplitudes of ±1 kG and peak amplitudes of ±2 kG. Those
fields are far less organized than the wreaths present in case M3-pcpf. The magnetic
wreaths in case M3-pcpf undergo a partial polarity reversal around day 1000, with thetoroidal fields in the northern hemisphere changing sign while those in the southernhemisphere retain their negative polarity. These time-latitude maps are made at aslightly deeper depth in the convection zone (0.75 R⊙ ) than the maps shown for the
rapidly rotating cases. This is because the magnetic wreaths here are concentrated in
the lower convection zone, as also seen case D1.5a.
The patterns of convection and magnetism in cases M3-pcpf and M3 are shownin Figures 8.20 and 8.21 respectively. The radial velocity structures are very similar inboth cases, as are the profiles of mean differential rotation. The toroidal magnetic fields
have fine structure throughout the convection zone, but those in case M3-pcpf also
show organization on large scales at mid-convection zone and especially in the lower
convection zone (Figs. 8.20b, c). In contrast, the toroidal fields in case M3 are comparably disorganized at all depths (Figs. 8.21b, c). Time-averaged profiles of hBφ i and thepoloidal vector potential hAφ i show the wreaths of magnetism strongly in case M3-pcpf

181

Figure 8.20 — Patterns of convection in case M3-pcpf. (a) Radial velocity vr in
Mollweide projection near the top of the shell (0.95 R⊙ ). (b) Toroidal magnetic fieldBφ at mid-convection zone (0.85 R⊙ ) with weak wreath signatures. (c) The wreaths are
significantly more evident are near the bottom of the convective shell (Bφ at 0.73 R⊙ ).
These snapshots are shown at day 700, a time when the mean magnetic fields are strong.
Also shown are time-averaged profiles of (d) Ω, (e) hBφ i with wreaths of magnetism and(f ) hAφ i. These averages span the interval from days 700-800.

182

Figure 8.21 — Patterns of convection in case M3. (a) Radial velocity vr in Mollweide

projection near the top of the shell (0.95 R⊙ ). (b) Toroidal magnetic field Bφ at midconvection zone (0.85 R⊙ ) with no wreaths present. (c) There are also no wreaths near
the bottom of the convective shell, and the field amplitudes are comparable to thoseat mid-convection zone (Bφ at 0.73 R⊙ ). These snapshots are shown at day 700. Also
shown are time-averaged profiles of (d) Ω, (e) hBφ i with no wreaths and (f ) hAφ i withdisorganized mean poloidal fields. These averages span the interval from days 700-800.

183(Figs. 8.20e, f ). The mean poloidal field also shows more organization than those realized in case M3 and is somewhat stronger particularly near the bottom of the convection
zone where the wreaths concentrate.

8.7

When the Sun is Old: Slowly Spinning Suns
While we have explored the likely effects of rapid rotation on convection anddynamo action in solar type stars, we have said little on the effects of slower rotationin these systems. As the Sun ages, its magnetized wind continues to carry angularmomentum away from the star, slowing its rotation over billion year timescales. Theseolder stars are relatively difficult to observe, often being in the halo of the galaxy.
As such, they have received less observational attention. However, many other stars
are effectively slow rotators. This is particularly true when we look at the rotational
constraint of convection, measured by the Rossby number. Systems with very fast
flows may act as slow rotators, even when their rotation periods are as short as theSun’s or several times shorter. The Sun is believed to have a Rossby number of nearly
unity, straddling the regimes of the slow and fast rotators. To understand the solar
differential rotation, we should understand the coupling of rotation and convection in
the high Rossby number regime as well. Stars that have high luminosities and low
average convection zone densities are likely to have extremely fast convective flows thatare generally less constrained by the effects of rotation. We expect this behavior in main
sequence F-type stars (Augustson et al. 2010) and also in the highly luminous red giantstars at the end of stellar life (Palacios & Brun 2007; Brun & Palacios 2009). Somesimulations of convection in those systems are now underway.
We have conducted a limited exploration of the slow-rotation parameter spacefor solar-type stars. Our primary simulation in this regime is case D0.5a, rotating at
0.5 Ω⊙ . This dynamo builds persistent wreaths of magnetism, though a more turbulent
simulation at that rotation rate (case D0.5b) undergoes temporal oscillations and polar-

184

Figure 8.22 — Patterns of convection in case D0.5a. (a) Radial velocity vr in Mollweide
projection near the top of the shell (0.95 R⊙ ) and (b) at mid-convection zone (0.85 R⊙ ).
(c) Time-averaged profile of angular velocity Ω with (d) accompanying radial cuts. The
differential rotation has flipped in sense relative to the rapidly rotating simulations,
here with a retrograde equator and prograde poles. There is still a substantial angularvelocity contrast in latitude and radius. (d) Meridional circulations, with streamlinesof mass flux overlain and sense of circulation indicated by color.

185
ity reversals. These successful dynamos have eddy diffusivities η which are much higher
than those necessary for dynamo action in stars rotating near the solar rate. This is
likely due to the relatively weak differential rotation in the Sun, as the slow rotators
have substantial angular velocity contrasts. Now however, the differential rotation haschanged in sense, with a retrograde equator and prograde poles.
The patterns of convection in case D0.5a are shown in Figure 8.22. The Rossby
number in this case is high, with a value of 2.45, indicating that the rotation should
only weakly influence the convective patterns. Despite this, a clear dichotomy exists
between the equatorial convection, which retains a north/south alignment on average,
and the polar convection which is more isotropic (Fig. 8.22a). In contrast to the rapidly
rotating systems, where the downflows are shorn out in radius, here the downflows
connect throughout the convection zone, with sheet-like structures near the equatorand strong vortical plumes at higher latitudes (Fig. 8.22b). This leads to a substantial
inward kinetic energy flux of almost −0.5 L⊙ at mid-convection zone, which is balancedby an outward enthalpy flux of nearly 1.5 L⊙ .
The differential rotation profile has flipped in sense from the rapidly rotating
simulations. In the slowly rotating regime, anti-solar differential rotation appears to
be a common feature, with retrograde equators and prograde poles (Fig. 8.22c, d). In
case D0.5a, the latitudinal angular velocity contrast ∆Ωlat near the surface is −0.65 µrads−1
and the radial shear ∆Ωr across the shell at the equator is −0.26 µrads−1 . This is a substantial angular velocity contrast, with the latitudinal contrast being nearly 50% of theframe rotation rate. The meridional circulations in this simulation are much stronger
and more organized than those found in the rapidly rotating simulations. Here, singlecelled structures fill each hemisphere (Fig. 8.22e). At present, it is unclear where either
the strong meridional circulations or the anti-solar differential rotation arise from. Strikingly, the thermal structure of the slowly rotating simulations retains an approximate
sense of thermal wind balance, here with warm equators and cool poles.

186

Figure 8.23 — Magnetic wreaths in case D0.5a. (a) Toroidal magnetic field Bφ at midconvection zone, with wreaths visible in the high-latitude regions. (b) Time-averaged
profiles of hBφ i, with wreaths present throughout the convection zone and (c) of hAφ i,with a single polarity of poloidal field.

The magnetic wreaths attained in this dynamo are shown in Figure 8.23. Incase D0.5a, the wreaths are located in the region strong angular velocity contrast
at high latitudes.

These wreaths have complex structure at mid-convection zone,

with substantial fluctuating fields, but the hBφ i is still visibly a major component(Fig. 8.23a). Profiles of hBφ i indicate that the wreaths extend throughout the convection zone (Fig. 8.23b). These magnetic structures extend mainly in latitude, withlittle radial structure, as also found in cases D1.5a and M3-pcpf. The accompanying
mean poloidal field shown in Figure 8.23c is of a single polarity everywhere in theconvection zone.

8.8

Conclusions
Wreaths of magnetism appear to be a ubiquitous feature in these dynamo simu-

lations. These wreaths can be persistent for long intervals of time, with little change
in their volume-averaged energies and polarities, or can show dramatic temporal oscillations including quasi-regular global-scale reversals of polarity. Both the toroidal andpoloidal magnetic fields begin to show temporal variation as the level of turbulence
is increased, with both magnetic fields waxing and waning in strength over periods ofroughly 1000 days. The differential rotation grows in turn weaker and then stronger,
being slowed by the strong magnetic fields or rebuilt by the convection.

187
During many of the temporal oscillations, remnant wreaths appear to slip toward
the polar regions. These magnetic structures propagate to high latitudes over a periodof roughly 500 days and are typically associated with bands of faster and slower rotating
fluid that also propagate poleward at approximately the same rate. At present, it isunclear whether the global-scale polarity reversals are dependent on these cycles ofwreath-amplification and poleward-slip. At least one simulation, case D3a, appears toundergo poleward-slip events without associated reversals of the global-scale polarities.
However, when case D3a does flip its global polarities, it also launches substantial
magnetic structures toward the polar regions.
Wreaths remain present in even the most turbulent simulations at all rotation
rates (i.e. cases D3b, D1.5b, and D0.5b), and are also present in our high magnetic
Prandtl number simulations (i.e. cases M3-pcpf and D3-pm4). At present, it is unclear
whether the wreaths in the high magnetic Prandtl number regime differ substantiallyfrom those realized at low magnetic Prandtl numbers. When simulations with highmagnetic Reynolds numbers at fixed rotation rate but differing Pm are compared, thefluctuating and mean fields appear comparable (i.e. cases D3b and D3-pm2) and temporal variation is achieved at similarly low levels of η.
Indeed, these dynamos generally appear to be more sensitive to the strength andstructure of the differential rotation than the complexity or amplitude of the smallscale convection. The magnetic wreaths produced in dynamos at Pm = 0.5 with highlevels of turbulence and very complex convective patterns, appear quite similar to thewreaths achieved in high Pm dynamos that have more laminar convective patterns. In
the future, high Pm dynamos should be explored in the regions of parameter spacenear the onset of sustained dynamo action, to determine whether the critical magnetic
Reynolds number scales with Pm and whether the fluctuating or rms magnetic Reynolds
number is a better diagnostic for successful dynamo action. These simulations may
additionally explore whether temporally varying behavior is linked to the small-scale

188
convection or the global-scale differential rotation. Without question it is also vitalthat more turbulent simulations be continued, to determine whether wreaths survive
in substantially more turbulent regimes. The most turbulent dynamos considered here
have fluctuating magnetic Reynolds numbers of just 500, which is far less than the values
found in astrophysical regimes.
Our most rapidly rotating dynamos, at 10 and 15 Ω⊙ , generate strong magnetic
fields which significantly modify the differential rotation. The poleward propagating
velocity features seen in our dynamos rotating at three times the solar rate (cases
spanning from D3 to D3-pm4) here become strong enough to affect the underlying
differential rotation when they ultimately reach the poles. In case D15, these structuresappear to reverse the flows in the polar regions, flipping the differential rotation there
from a retrograde sense to a prograde sense.
The modulated states found in hydrodynamic simulations (Chapters 3 and 4) remain present in regions of our dynamo parameter space, being most evident in case D10L
which has rapid rotation, strong magnetic fields and a reasonably strong differential rotation. As the differential rotation weakens, the active nests become less distinct but
remain evident when long intervals of time are examined in time-longitude maps.
Magnetic wreaths are now found in solar simulations as well as in rapidly rotatingsuns. Generally, as the rotation rate decreases, magnetic wreaths are confined to thelower portion of the convection zone and do not have the radial extent seen in case D3
and the other rapid rotators. In simulations at the solar rate, the mean toroidal fields
have relatively small amplitudes, while the fluctuating fields remain strong. Despite
this, the fluctuating fields are still organized on global-scales and wreaths are clearly
evident.
These wreath-building dynamos are sensitive to the bottom magnetic boundary
condition that is employed. In simulations at the solar rate, employing a potential fieldbottom boundary leads to dynamos without wreaths (case M3), while dynamos with

189
perfect conducting bottom boundaries build wreaths almost immediately (case M3pcpf). Indeed, when we have attempted to run a version of our three solar dynamocase D3 with potential field bottom boundaries, we find that dynamo action is notsustained at the values of η employed in case D3. It appears that case M3 may berunning a small-scale or fluctuating dynamo, which primarily relies on the turbulent
correlations to generate magnetism, while case D3 runs a large-scale dynamo that relies
primarily on the Ω-effect associated with the strong differential rotation. The successof case D3 as a dynamo is clearly enhanced when the bottom boundaries both allow
horizontal magnetic fields near the boundary and prohibit their expulsion from thesimulated domain.
At present, it is unclear how the poloidal fields are regenerated in the rapidly
rotating suns. In particular, it is unknown whether this is a process arising from turbulent correlations on the smallest scales or from instabilities in the shearing flows ofdifferential rotation. Case M3-pcpf may bridge the gap between these two regimes, possessing both a small-scale α-effect dominated dynamo and a large-scale Ω-effect dynamo
operating in the same regions of the convection zone.
Wreath-building dynamos should be present in slowly rotating stars as well. Thesestars, which include both older suns and young but highly luminous main-sequencestars, are likely to possess an anti-solar differential rotation with a retrograde rotating
equator and prograde pole. Here, wreaths appear at high latitudes, in the regions ofstrong angular velocity contrast. It is unclear whether the temporally varying behavior
seen in some of the slowly spinning suns (i.e. case D0.5b) arises from the cycles ofwreath-amplification, polarward slip and angular momentum transport that appear tooperate in many of the rapidly rotating suns.

Chapter 9
Future Explorations and Wreath-building Dynamos with Tachoclines

In this thesis we have explored a variety of dynamo solutions in stars rotating
more rapidly than the sun. Magnetic wreaths arise as a near ubiquitous feature ofthese simulations, irrespective of rotation rate or level of turbulence achieved. Thesesimulations have at present included the convection zone only. In many solar dynamo
theories, it is thought that the tachocline must play a major role. Clearly, a major thrustof future research must be the inclusion of this important internal boundary layer.
In studies of compressible convection that include regions of penetration at thebottom of the convection zone, organized magnetic fields are often efficiently pumped
downwards and out of the convection zone (e.g., Tobias et al. 2001; Browning et al.
2006). Magnetic wreaths might be efficiently expelled from the convection zone by thecompressible convection there. This could remove them from the region of amplification,
as the poloidal field may not be regenerated within the tachocline by the same processes
at work within the convection zone. This would likely cause magnetic wreaths to vanish
entirely. Such a finding would cast serious doubt on the viability of magnetic wreaths
in turbulent stellar convection zones.
We have begun explorations into whether magnetic wreaths survive in the presence of a tachocline in rapidly rotating suns. We find that the wreaths do survive, andindeed continue to fill the bulk of the convection zone, even as they become rooted
within the tachocline. These wreaths still undergo temporal oscillations and cycles of191
Table 9.1.

Parameters of case T3

Nr , Nθ , Nφ

Ra

Ta

Re

Re′

Rm

Rm′

Ro

Roc

ν

η

258 × 256 × 512

9.64×105

1.39×107

184

129

92

64

0.481

0.333

1.37

2.75

Note. — Simulation parameters for case T3 rotating at 3 Ω⊙ and with a tachocline of shear.
Parameters have same definition as in Table 8.1, with a deep inner radius of rbot = 3.8 × 1010 cm
and outer radius of rtop = 6.72 × 1010 cm. The base of convection zone is at rbcz = 4.91 × 1010 cm or
0.705 R⊙ . Here we define L = (rtop − rbcz ) = 1.81 × 1010 cm or 0.26 R⊙ .

polarity reversal. Preliminary results are reported on here for one simulation rotating
at three times the solar rate, case T3, with similar input parameters to our wreathbuilding case D3.

9.1

Capturing a Model Tachocline within ASH
The primary parameters of case T3 are reported in Table 9.1 and are similar to ourcases D3 and D3a. Case T3 has a lower boundary at r = 0.5 R⊙ and an upper boundary
at 0.965 R⊙ . There is thus an overall density contrast of 48 across the convection zone
and 133 across the entire shell. This leads to somewhat lower values of ν, κ and η thanin case D3, despite the two cases sharing the same values for eddy diffusivities at the
top of the convection zone.
After thousands of days of evolution, this dynamo establishes a stable profile
of dS̄/dr, with the base of the convection zone where dS̄/dr crosses zero located at
approximately 0.705 R⊙ . The fluctuating radial velocities extend somewhat deeper,and decrease from an average amplitude of 25 m s−1 to about 1 m s−1 by a depth ofapproximately 0.66 R⊙ . The fluctuating radial flows are much smaller below this depth.
The peak downflows are moving at almost −200 m s−1 at 0.705 R⊙ but have slowed toroughly −10 m s−1 when they reach 0.66 R⊙ .

192
To prevent the diffusive spread of the differential rotation profile from the convection zone into the stable radiative zone, we taper our eddy diffusivities with a smooth
step function
ν(r) = ν0



ρ̄0
ρ̄(r)

α

Φ(r) ,

κ(r) = κ0



ρ̄0
ρ̄(r)

α

Φ(r) ,

η(r) = η0



ρ̄0
ρ̄(r)

α

Φ(r) , (9.1)

where α = −0.5 as before, ν0 , κ0 and η0 are the values at the top of the convection zone
and our taper with radius is given by
r − rtaper1+β.
Φ(r) = (1 − β) 1 + tanh2∆taper(9.2)

We recover our treatment of eddy diffusivities in the convection-zone dynamos when
Φ = 1 at all radii. In case T3 the taper function is centered at rtaper = 4.70 × 1010 cm(0.675 R⊙ ) with width ∆taper = 5 × 108 cm (0.007 R⊙ ). The variable β controls theamplitude of the tapering function, which ranges from a value of one in the convection
zone to β in the radiative interior. Here we set β = 10−2 thus reducing our eddy
diffusivities by two orders of magnitude in the stable radiative interior.
We expect that turbulent mixing and transport should generally be lower in thestable radiative zone than in the unstable convection zone, but these eddy diffusivities
are still many orders of magnitude larger than the microscopic viscosities and diffusivities present in stellar interiors. Over long intervals of time, the differential rotation will
still diffusively spread into the radiative interior. By tapering our eddy diffusivities by
two orders of magnitude, we find that the differential rotation produced by convection
remains well confined to the convection zone for more than 7000 days. It is likely that
artificially lowering our viscosities is a more physically plausible approach to preventing“tachocline-creep” than other techniques employed in the past, which included mechanical forcing in the radiative zone and entropy forcing in the region of the tachocline toretain a comparably good structure of differential rotation (e.g., Browning et al. 2006).

193

Figure 9.1 — Profiles of mean flows and temperature in dynamo case T3. These
profiles are averaged over the interval from days 4050-4150 of dynamo case T3. The
hydrodynamic progenitor ran for an additional 600 days previous to day 0. (a) Profile ofdifferential rotation Ω with (b) accompanying radial cuts at selected latitudes as labeled.
The bottom of the convection zone at rbcz = 0.705 R⊙ is marked on both with a dashedline. The equator is prograde, the poles are retrograde and the radiative interior isin nearly solid-body rotation between latitudes ±45◦ . (c) Meridional circulations withsurface structure and amplitudes similar to those in case D3 but different at depth.
(d) Profile of latitudinal temperature variations relative to the spherically symmetricaverage T̄ .

Time-averaged profiles of the differential rotation, meridional circulations and latitudinal temperature variations of case T3 are shown in Figure 9.1. After a total of morethan 4500 days of evolution in both the hydrodynamic progenitor and then in case T3,
the differential rotation remains well confined to the convection zone (Fig. 9.1a). Within
the convection zone, there is a significant angular velocity contrast in latitude and radius, with a ∆Ωlat near the surface of 1.04 µrad s−1 and a ∆Ωr across the convection
zone at the equator of 0.89 µrad s−1 . The latitudinal angular velocity contrast is similar
to that achieved in dynamo case D3b, but the radial contrast is somewhat larger. Below
the base of the convection zone, the location of the eddy diffusivity taper is visible asa slight contour in the profile of Ω (Fig. 9.1a) and as a small kink in the radial cuts ofthe same (Fig. 9.1b). In the radiative interior, a small latitudinal angular velocity contrast persists. This is especially prominent in the polar regions at latitudes above ±60◦ .
Near the equator and up to latitudes of roughly ±45◦ the radiative interior is almost in
solid-body rotation.

194
The meridional circulations achieved in case T3 are shown in Figure 9.1c. In theupper convection zone, these circulations are similar to those found in case D3 (e.g.,
Fig. 5.1d). In the lower convection zone the flows penetrate into the tachocline andradiative interior. This largely removes the narrow return flow seen at the base of case D3from the convection zone. The latitudinal temperature variations achieved in case T3
are shown in Figure 9.1c. As in the other rapidly rotating simulations, the convectionbuilds a prominent gradient in latitude, with hot poles, cool mid-latitudes and a warm
equator near the surface. The temperature gradients appear to have spread into theradiative interior, and the thermal wind associated with this likely helps maintain thesmall differential rotation that is present there. Temperature gradients will diffusively
spread more easily than gradients in angular velocity, owing to the low Prandtl numberof this simulation (as before, Pr = 0.25). Lowering the value of β in the tapering function
would likely further curtail this spreading, as would reducing ν and κ throughout theconvection zone.

9.2

Magnetic Wreathes with a Tachocline
Case T3 continues to build large wreaths of magnetism that fill the bulk of its

convection zone. The time history of this case is shown in Figure 9.2. As in many of
our other rapidly rotating dynamos, wreaths appear at latitudes of roughly ±20◦ andundergo cycles of polarity change. In the 5800 days shown here, four polarity reversals
occur in both hemispheres after the dynamo saturates at roughly day 1000. The wreaths
are strong at mid-convection zone and stronger yet near the base of the convection
zone (Figs. 9.2c, d). In both regions the wreaths have similar latitudinal structure.
In the tachocline, the wreaths occupy a broader range of latitudes and remain strong(Fig. 9.2e). Time-radius maps at a fixed latitude of 20◦ , in the core of the magnetic
wreaths, show that with each successive polarity reversal some field of the previous
cycle migrates deeper into the radiative interior (Fig. 9.2f ). At depths below 0.6 R⊙

195

Figure 9.2 — Time-dependent behavior in case T3 with a model tachocline. (a) Volume-

averaged kinetic and magnetic energies, and (b) mean hBφ i averaged over northern andsouthern hemispheres at mid-convection zone. (c) Time-latitude maps of hBφ i at midconvection zone (0.85 R⊙ ) and (d) near the base of the convection zone (0.7 R⊙ ). Multiple cycles of wreath building occur in the roughly 5000 days after dynamo saturation.
(e) The wreaths extend into the tachocline (0.65 R⊙ ), with somewhat broader structure. At greater depths single polarity wreaths persist for the entire simulated interval.
(f ) Time-radius map at a fixed latitude of 20◦ , in the core of the northern hemispherewreaths. The black contour denotes the neutral line between successive wreaths.

196

Figure 9.3 — Profiles of hBφ i and hAφ i in dynamo case T3. (a) Mean toroidal magnetic
field hBφ i and (b) poloidal vector potential hAφ i averaged over days 4050-4150 at a timeof peak mean fields. At this time, the wreaths and associated mean-poloidal field are outof phase with the magnetic field in the deep interior. (d) hBφ i and (e) hAφ i averaged overdays 5000-5100 when the fields have flipped polarity and again reached peak amplitudes.
At this time, the magnetic fields in the convection zone have the same polarity as thosein the radiative interior. The bottom of the convection zone at rbcz = 0.705 R⊙ isindicated by the dashed semi-circle.

however a single polarity of wreathes are established and persist. During the 5800 days
shown here, these deep magnetic structures slowly grow in strength. By the end of thisinterval, the mean toroidal fields have reached peak amplitudes of nearly ±9 kG near
the bottom boundary, with the strongest fields appearing near latitudes of ±55◦ . These
deep fields may arise from turbulent transport and pumping within the convection zone
and coupled down into the deep radiative interior. It seems more likely however thatthey result from stretching of the initial weak seed field by the differential rotationthat has diffusively spread into the polar regions of the radiative interior, as the peakangular velocity contrast there occurs at the higher latitudes where the wreaths arelocated. Decreasing our tapering constant β to values of 10−3 or less should strongly
decrease the diffusive spread of differential rotation and suppress the generation of these
deep-seated wreaths.
Time-averaged profiles of the mean toroidal fields and poloidal vector potential
are shown at two intervals in Figure 9.3. At both times the wreaths of magnetism are at197
a point in the cycles of polarity reversal where the fields have attained peak amplitudes.
The mean magnetic fields associated with the wreaths extend into the tachocline down
to roughly where the eddy diffusivities taper in amplitude. This depth also roughly
coincides with the depth at which the fastest downflows have their motion substantiallybraked by the stable stratification.
The mean fields produced in the convection zone undergo cycles of polarity reversal, while those deep in the radiative interior always retain the same polarity. During
the first cycle shown in Figure 9.3, spanning days 4050-4150, the mean magnetic fields
in the convection zone have opposite polarity from the fields in the radiative interior(Figs. 9.3a, b). After the fields flip in polarity and regrow in amplitude, they match
the polarity of the radiative interior fields (days 5050-5150; Figs. 9.3c, d). At present,
it is unclear whether the dynamo behaves substantially differently when the convection
either matches or opposes the fields in the radiative interior. Convective patterns andthe global-scale flows of differential rotation and meridional circulation are very similar
during both of the intervals examined here.
These simulations of dynamo action in the presence of a tachocline are very
promising. Wreaths of magnetism continue to fill the bulk of the convection zone andundergo cycles of polarity reversal. These polarity reversals occur on similar timescales
to those found in the convection-zone dynamos studied in Chapters 6 and 8. The polarity
reversals in real stars generally have slightly longer periods than those found here, andin the Sun magnetic polarity reversals occur on a time scale of roughly 11 years, orabout 4000 days.
Our dynamo with a tachocline shows promise, but the model could be improved
significantly. The background structure used within case T3 is shown in Figure 9.4. At
present, our thermodynamic variables are in decent agreement with the solar structure
model (a CESAM model based on work in Brun et al. 2002) throughout the bulk of
the convection zone. In the stably-stratified radiative interior, ASH diverges from the

198

Figure 9.4 — Stellar structure of case T3. Shown are spherically symmetric profiles ofthermodynamic variables (a) ρ̄, (b) dS̄/dr, (c) P̄ and (d) T̄ . The adopted structure in
ASH is shown with a black, solid line and the CESAM solar structure model is shownwith a blue, dashed line (from Brun et al. 2002). The ASH entropy gradient shownin (b) has been multiplied by a factor of 100 to make it visible. The ASH structureand solar model are in good agreement within the convection zone, but diverge in theradiative interior.

model. This is particularly due to the low values of dS̄/dr used in ASH within theradiative interior. There, the values of the entropy gradient are more than two orders ofmagnitude smaller than those in the solar model. Additionally, the profile of dS̄/dr usedin case T3 has developed kinks in the radiative zone (near radii 0.65R⊙ and 0.55R⊙ ) andgrows in amplitude too slowly as the radius decreases. This leads to a softer tachocline
than that likely present in real stars, which allows radial motions to penetrate too deeply
into the radiative interior. Additionally, T̄ is too large in the interior, while ρ̄ and P̄ are
too small. Lastly, the dS̄/dr profile used in ASH is generally too large in amplitude in
the upper convection zone, especially near r = 0.95 R⊙ , as was also seen in the structureof our convection-zone models (see Figure 2.3).

199
New models are in progress which are appear likely to achieve both the steep
gradient of dS̄/dr in the tachocline and the profile near the upper surface seen in 1D solar structure models. Exploring how 3-D effects adjust the stratification of thetachocline will further require methods that accelerate the thermal relaxation of thatlayer. Such methods are in active development (Featherstone et al. 2010). Without
an accelerated relaxation scheme, the thermal structure of the tachocline cannot becorrectly captured in ASH. This problem arises due to the long timescales required for
adjustment of the tachocline, which likely requires several million years. In comparison,
we have evolved case T3 for roughly 16 simulated years. A case with a realistic stableregion will likely have its timesteps strongly constrained by the high-frequency gravity
waves propagating there. With the real solar stratification, we expect timesteps ofroughly 200 seconds. This will be an issue for hydrodynamic simulations but is unlikely
to significantly affect the dynamo simulations, as the Alfvénic CFL near the surface
applies a more stringent limit.
These simulations of dynamo action in the presence of a tachocline are computationally demanding. To achieve the 5800 days of dynamo evolution in case T3 required
over 6 million iterations of evolution, with typical timesteps of 80 seconds each. Running on a special allocation on the Kraken-XT4 at NICS, this tachocline simulation
has required nearly 750000 cpu hours, running continuously on 512 processors there.
Simulations that include tachoclines must however form the core of future explorations
into wreath-building dynamos.

200

9.3

Perspective on Rapidly Rotating Suns
Our explorations of rapidly rotating suns have taken us into interesting territory

and have ultimately raised fascinating questions about the solar dynamo. In hydrodynamic simulations, we found that generally the global-scale flow of differential rotation
becomes considerably stronger as the rotation rate is increased. In the rapidly rotating
systems we have studied, achieving greater levels of turbulence at fixed rotation rate
leads to even stronger angular velocity contrasts in both radius and latitude. In our
dynamo studies we find that the magnetism becomes so strong that it substantiallysuppresses the angular velocity contrast of differential rotation. Relative to similarhydrodynamic simulations, the angular velocity contrast in latitude and radius of theequilibrated dynamo solution can be reduced by more than a factor of two (i.e. cases H5and D5). This effect becomes more pronounced as the rotation rate increases. In themost rapidly rotating dynamos (i.e. cases D10 and D15) the differential rotation can be
wiped out almost entirely in the equatorial regions. These results may be in reasonableagreement with observations of differential rotation, where some disagreement remains
between different groups using different observational techniques. Generally however,observations indicate that differential rotation either grows with faster rotation or perhaps remains nearly constant.
In contrast, the meridional circulations become substantially weaker as the rotation rate increases, their volume-averaged energies falling almost inversely with rotation
rate. These flows, which in the Sun may be comprised of single cells in each hemisphere,
break apart into multiple cells in both radius and latitude. The meridional circulations
are less dependent on the level of turbulence attained, at least in the parameter spacestudied here. Additionally, these circulations are largely unchanged in either amplitude
or spatial structure in the dynamo solutions.

201
The weakening of meridional circulations with faster rotation raises profoundquestions for stellar dynamo theory. In the case of the Sun, current theories of fluxtransport dynamos, including the Babcock-Leighton dynamo, rely on the slow meridional circulations to return flux to the tachocline. It is the circulation time of that flowwhich largely sets the timescale of cyclic dynamo variations.
In the Sun the eleven-year activity cycle is close to estimated turnover times
for the meridional circulations, based on the observed flow at the surface and massconservation arguments. In rapidly rotating stars, measurements of either cycle period
or meridional circulation amplitude are quite difficult; the former because of the longtime scales involved (decades), and the later because of the small amplitude of themeridional circulations relative to either flows associated with surface convection or the
fast differential rotation. Observations seem to indicate that there may be a weak correlation between cycle period and rotation rate, with more rapidly rotating stars having
slightly shorter periods then the Sun (Saar & Brandenburg 1999). This is in disagreement with the meridional circulations found in our rapidly rotating simulations, which
in a flux-transport framework would likely lead to significantly longer dynamo cycles.
Recent flux-transport dynamo models are beginning to explore how such variations inmeridional circulation amplitude and 2-D structure may affect the operation of the
dynamo (e.g., Jouve et al. 2009).
In many of the rapidly rotating suns, the strong differential rotation shears convective cells out in radius. This can lead to changes in the radial transport of energy. In
particular, the inwards transport of kinetic energy is strongly reduced in these simulations. In some regimes, striking patterns of convection modulated in longitude arise. As
the rotation rate increases, convection in the hydrodynamic models can become entirely
confined to active nests with more quiescent streaming flow inbetween. These active
nests persist for thousands of days and propagate at an angular velocity which is constant at all depths of the convection zone. This angular velocity typically matches that202of the differential rotation near mid-convection zone. As such, the active nests experience strong shear from the differential rotation, with a relative tailwind in the upper
convection zone and a headwind in the lower convection zone. These nests reappear innearly the same form in one of our MHD dynamo simulations (case D10L). In otherdynamo simulations they appear to be weakly present, becoming somewhat more evident when the differential rotation is stronger. In stellar observations, such structures
may lead to the presence of magnetic features at the surface which propagate at rates
distinct from either the surface differential rotation or the overall rotation of the star.
Our dynamo simulations consistently build striking global-scale magnetic structures in the bulk of the convection zone. These structures are more topologically complex
than ideal flux tubes, whose magnetic fields are confined within bounded surfaces. The
magnetic wreaths which are self-consistently built in these rapidly rotating dynamos
have more open topologies, with field wandering throughout the convection zone, reconnecting across the equator and being wound up in the vortical flows at high latitudes.
Some of these dynamos build persistent wreaths that survive relatively unchangedin the midst of the convection zone for thousands of days. As the magnetic Reynolds
number increases, the wreaths become time-dependent, undergoing large oscillations
in magnetic energy. These oscillations can lead to global-scale polarity reversals, withthe mean toroidal and mean poloidal fields changing sign quasi-regularly. Associatedwith these oscillations or reversals are poleward propagating magnetic structures. Thesemay be wreaths that become strong enough to undergo a poleward-slip instability where
magnetic tension pulls the wreaths toward smaller cylindrical radii. Accompanying thesemagnetic features are local regions of more quickly rotating fluid. These angular velocity
structures can cause changes in the overall differential rotation. They may bear someresemblance to the polar branch of torsional oscillations observed in the Sun.
Even the more slowly rotating simulations form wreaths of magnetism, including
simulations at the solar rate and slower. These wreaths tend to be more concentrated in

203the lower convection zone and less evident at mid-convection zone. Such structures may
be pumped out of the convection zone when a tachocline of shear and penetration is
included at the lower boundary. In our preliminary explorations however, we find thatwhen a convection zone dynamo is coupled with a model tachocline, magnetic wreaths
still form and fill the bulk of the convection zone. The wreaths realized in case T3
undergo similar polarity reversals to those found in other simulations at three times thesolar rate. Case T3 has a very soft stable stratification, and this likely underestimates
the amount of magnetism which is able to remain within the convection zone. Thismay change however at higher levels of turbulence if very strong downflows are able toovercome the shearing tendencies of the differential rotation and survive to efficiently
pump the wreaths downward.
Wreaths have not been found in previous solar simulations in part due to the difference in lower magnetic boundary conditions employed. The wreath-building dynamos
appear to work far more efficiently when the lower boundaries permit horizontal fieldand prevent that field from escaping the convection zone. It is reassuring that including
a more realistic lower boundary for the convection zone by including a tachocline ofpenetration and shear leads to qualitatively and quantitatively similar behavior to theconvection zone dynamos with perfectly conducting bottom boundaries. Namely, incase T3 wreaths of magnetism form, span the convection zone, reach similar amplitudes
at mid-convection zone to other 3 Ω⊙ dynamos, and undergo cyclic polarity reversals ona similar timescale (roughly 1000 days). Simulations of the solar dynamo which include
a tachocline of penetration appear to yield very similar results, though there the wreaths
are confined almost entirely to the tachocline (Browning et al. 2006). It is unclear atpresent whether the magnetic structures in that dynamo arise locally within the shearof the tachocline or are produced in the convection zone and pumped downward. Incase T3 however, the wreaths are clearly formed in the convection zone itself.

204

9.4

The Road Ahead
The rapidly rotating dynamos have opened a rich realm for further exploration.

At present, we have followed one cut through turbulent parameter space with limited
additional sampling at interesting rotation rates. The exploration of these stars is by nomeans complete. Here briefly are future projects which should be pursued. Some of these
are underway presently or will be in the near future; others remain more speculative. A
number of the projects will require significant computation, but several would involve
deeper analysis of the rich dataset already produced in this research.
• How are poloidal fields regenerated in wreath-building dynamos? The
mean poloidal magnetic fields are regenerated on the poleward edge of the magnetic wreaths. Here, field-line tracings in case D3 reveal fields that are partly
wound up in the vortical polar convection. Larger-scale variations are also
clearly evident. It is unclear at present which scale contributes to the regeneration of mean poloidal fields. This question should be answerable with thecurrent set of simulations, by comparing features common to the three dynamosthat build persistent wreaths (cases D0.5, D1.5a and D3).
• What is the origin of cyclic reversals, and what sets their timescale?
In many of our dynamos, quasi-regular temporal oscillations are observed in
the magnetic energies and the amplitudes of the mean toroidal and poloidal
field. In many cases these are linked with global-scale reversals of polarity.
It is unclear why organized time-dependent behavior occurs when the magnetic
Reynolds number increases. Global-scale polarity reversals appear to occur evenin those cases where the differential rotation is highly suppressed (i.e. case D15).
These reversals may originate in changes to the magnetic production terms, but
elucidating their variation has proven complex. The reversals that occur in these
rapidly rotating suns typically span intervals of about 1000 days. In the Sun,

205
such reversals take roughly eleven years or about 4000 days. In other solarlike stars the timescales for reversal are comparable, though perhaps slightly
decreasing in the more rapidly rotating suns. Understanding the origin of theorganized polarity reversals, and why they are consistently shorter in thesesimulations than in real stellar dynamos, must be a high priority of future
research.
• Do mean-field theories reproduce wreath-building dynamos? In our
preliminary analysis of a simple mean-field model, we find that the EMF predicted by a simple α-effect fails to reproduce the 3-D EMF measured in ourcase D3. At present, it is unclear whether a more sophisticated approach will
yield better agreement. Test-field methods and further expansions of the meanfield EMF should be carefully explored, to both understand how the 3-D dynamos are operating and to help improve stellar mean-field models. This analysis should be extended to the time-dependent cases, including to the initial
dynamo growth phases.
• Do wreath-building dynamos survive at high Reynolds numbers? Our
present simulations still have relatively low magnetic Reynolds numbers. As
Rm′ is increased, wreathes become more complex in spatial and temporal structure and are generally stronger in the lower convection zone than at midconvection zone. At present, it is unclear whether such structures survive underhighly turbulent conditions, in which a small-scale convective dynamo is active in addition to the global-scale wreath building dynamo. Case M3-pcpf andcase D3-pm4 may be beginning to explore this coupled regime, but simulations
should be carried out with fluctuating magnetic Reynolds numbers Rm′ ∼ 1000
or higher to ensure strong dynamo action everywhere in the convection zone.

206• Are high-Pm dynamos different from low-Pm dynamos? We have conducted a limited exploration on the effects of Pm on the magnetic wreaths andgenerally find that it is the value of η rather than Pm that defines the temporalvariations and spatial structure of the wreathes. However, in these rapidly rotating dynamos the differential rotation itself changes as ν and κ are varied. This
complicates exploration of the boundaries in parameter space variously for sustained dynamo action, temporally varying magnetism, cyclic polarity reversals
or quenching of differential rotation. At certain special rotation rates however,clustering around 1.5 Ω⊙ in these simulations, the differential rotation appears
to be less sensitive to variation of ν or κ. More extensive simulations should
be carried out at 1.5 Ω⊙ , exploring how varying Pm affects the wreaths whenother factors, including the angular velocity contrast in latitude and radius, areapproximately constant.
• Why do high Rossby numbers lead to anti-solar differential rotation?
As we have taken our rapidly rotating suns back to the solar rate and thenexplored more slowly rotating stars, we have found that the differential rotation
profile often flips in sense and becomes anti-solar, with fast, prograde poles anda slow, retrograde equator. This phenomena is however not confined to the older
suns alone. Indeed, at the solar rotation rate, dropping ν and κ to the levels
necessary to achieve dynamo action at Pm = 0.5 causes the differential rotation
to collapse entirely, or even flip sense and become anti-solar. The cause of this is
as yet unknown, but appears to be related to the convection shifting from the low
Rossby number regime that the rapid rotators inhabit to a high Rossby number
regime. At rotation rates above approximately 1.5 Ω⊙ , the differential rotation
is solar in sense and becomes more strongly so as the diffusivities are dropped.
At rotation rates below about 1.5 Ω⊙ , the differential rotation becomes weaker207
as ν and κ decrease. When conditions become sufficiently turbulent, the senseof differential rotation becomes anti-solar and this inverted profile of angularvelocity is further enhanced as the diffusivities drop. The strength and senseof differential rotation appears to correlate reasonably well with the Rossby
number of the convection, but the reason for this correlation is unknown. The
Rossby number of the flows may in turn be related to the stratification we haveadopted in our simulations. Promising work is underway on new models which
better match the helioseismically determined solar stratification, but further
work must be done on the sensitivity of the differential rotation profile, in the
Sun and in other stars, to relatively small changes in the eddy diffusivities.
• Do active nests of convection survive under more realistic conditions?
We still find strongly localized active nests of convection in one of our dynamo
simulations (case D10L). It is unclear at present why they largely vanish as thedifferential rotation is quenched by strong magnetism. Weak localized structures remain and persist for thousands of days, but these structures are highly
obscured by small-scale convection which emerges as the radial angular velocity
shear decreases. We do not at present have a theory for the emergence or persistence of such modulated states. These structures appear to be hydrodynamic innature, and they seem to survive under some magnetohydrodynamic conditions.
Further work should be done on the theoretical side to understand where active
nests arise from. Simulations should be undertaken that further constrain thedependence of active nests of convection on rotation, on the angular velocity
shear of differential rotation, and on the hydrodynamic parameters of the simulations. Simulations should also be pursued which include tachoclines at thebase of the convection zone, to determine whether active nests survive underconditions that are more similar to those found within stellar convection zones.

208• What is the origin of the rotation-activity relationship? These simulations have not answered the questions posed by the observed correlation between
rotation and stellar magnetic activity. It is unclear whether the observed correlation stems from dynamos that operate more efficiently with faster rotationand produce more magnetism, or from subtle changes in the emergence of magnetism at the stellar surface which are unrelated to changes in the global-scale
dynamo itself. New observations are beginning to trace the multiple correlations between magnetic activity, differential rotation and overall rotation instars (e.g., Saar 2008). These indicate that in the rising portion of the rotationactivity relationship the differential rotation is also growing, and in this regime
there is a clear relation between magnetic activity and the amplitude of surface
differential rotation. In the saturation regime however, this relationship breaks
down and now magnetic activity appears to be uncorrelated with the amplitude of differential rotation. This is striking. Our simulations are beginning to
explore multiple regimes, some where the differential rotation and magnetism
coexist and some where the magnetism persists at similar amplitudes but thedifferential rotation is almost entirely quenched.
• Do stars with deep convection zones operate wreathy dynamos? Observations of fully convective M-dwarf stars, with masses below 0.35 M⊙ , indicate that these stars still have strong magnetic activity at their surfaces. Thesestars do not have stably-stratified radiative interiors and thus also should nothave internal boundary layers such as tachoclines at the base of their convection
zones. Additionally, K-type stars have much deeper convection zones than theSun, and tachoclines in those stars are buried under more than 50% of the starin radius. It seems highly unlikely that the traditional interface model of stellar dynamos could be successfully operating in such stars. Instead, these stars

209
may run dynamos in the bulk of their convection zones, and those dynamos
may be wreath-building dynamos. Convection and dynamo action in stars withdeep convection zones should be explored. Changes in stellar luminosity at lower
masses will additionally drive generally slower convective flows with correspondingly lower Rossby numbers. Simulations of these stars may help disentangle
how convection, rotation and magnetism couple in stellar interiors.
• Are tachoclines important? These wreath-building dynamos are able toorganize substantial global-scale magnetic fields in the bulk of their convection
zones. They generally do so without resorting to tachoclines of penetrationand shear for the storage and organization of such fields. Wreaths persist in
the presence of tachoclines and, in the very limited explorations to date, arelargely unmodified by the presence of this internal boundary layer. What rolethen do tachoclines play in stellar dynamos? These complex shear layers must
contribute in some fashion to the operation of the global-scale dynamos but theirrole remains mysterious. If wreath-building dynamos continue to thrive undermore realistic conditions, it seems likely that fully-convective stars and thosewith deep convection zones can still produce substantial global-scale magnetic
fields. Simulations of these stars should be undertaken, to learn how these stellar
dynamos are similar to and differ from the operation of the solar dynamo.

210

9.5

Final Reflections
In this thesis we have explored how stellar convection and magnetism built by

dynamo action are affected by rotation. In rapidly rotating stars, the global-scale flows
of differential rotation become stronger while the meridional circulations appear to
become much weaker. The patterns of convection can achieve novel modulated states,
and in some cases the equatorial convection is entirely confined to narrow active nestsof convection with more quiescent flow in between. The rapidly rotating stars havestrong dynamos that can build global-scale, wreath-like magnetic structures in the bulkof the convection zone. These wreaths can persist for thousands of days, or can become
time-dependent with global-scale magnetic polarity reversals occurring quasi-regularly.
Wreaths of magnetism can appear in solar simulations as well, and are also present inslowly spinning stars.
These projects sample only a small portion of the rich dynamics that arise fromthe coupling of convection, rotation, and magnetism in stellar interiors. Ultimately,
they strive to better answer fundamental questions about stellar magnetism: Where
are global-scale stellar magnetic fields built and organized, why is there a correlation
between magnetic activity and stellar rotation rate, and what role does the tachocline
play in a stellar dynamo?

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