Magnetic Wreaths, Cycles, and Buoyant Loops inConvective Dynamos
byNicholas J. Nelson
B.S., Brigham Young University, 2007
M.S., University of Colorado, 2010

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 & Planetary Sciences
2013This thesis entitled:
Magnetic Wreaths, Cycles, and Buoyant Loops in Convective Dynamos
written by Nicholas J. Nelson
has been approved for the Department of Astrophysical & Planetary Sciences

Juri Toomre

John Bally

Keith Julien

Mark Miesch

Mark Rast

Michael Shull

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

Nelson, Nicholas J. (Ph.D., Astrophysical & Planetary Sciences)
Magnetic Wreaths, Cycles, and Buoyant Loops in Convective Dynamos
Thesis directed by Prof. Juri Toomre

Solar-like stars exhibit a rich variety of magnetic activity, which is driven by dynamo
action in the stellar interior. In the Sun, strong dynamo action creates global-scale magnetic
fields which undergo cyclic reversals as well as smaller-scale dipolar active regions which haveglobal-scale organization. Dynamo action is a highly nonlinear process which is enabled by
the interplay of turbulent convection, rotation, and stratification. Seeking to explore theconvective origins of magnetism in sun-like stars, we have used 3D MHD simulations with
the Anelastic Spherical Harmonic (ASH) code to model elements of these dynamos. Previous
simulations have demonstrated that large-scale “wreaths” of toroidal magnetic field can beachieved in the convection zone without a tachocline of shear at its base, as was thought tobe necessary, and that these wreaths can yield reversals in global magnetic polarity.
We find that cyclic reversals of global magnetic polarity in wreath-building dynamos
can be achieved by increasing the level of turbulence in solar-like simulations. By decreasing the effective diffusion we demonstrate that large-scale magnetic wreaths can persist in
simulations where explicit diffusion has been decreased to levels at which it no longer plays
a significant role in the key dynamical balances required to achieve wreath-building dynamo
action. Magnetic reversals are attained when resistive diffusion of the poloidal magnetic
fields becomes too small to prevent turbulent magnetic induction from generating opposite
polarity poloidal fields.
In order to attain even less diffusive simulations, we explore more a dynamic Smagorinsky model. Using the dynamic Smagorinsky model, we achieve a dynamo simulation capableof building buoyant magnetic loops which rise coherently through our simulated domain.
These loops ascend via a combination of magnetic buoyancy and advection by convective

iv
giant cells. These buoyant loops originate within sections of the magnetic wreaths in which
turbulent flows amplify the fields to much higher values than is possible through laminar processes. We measure statistical trends in the polarity, twist, and tilt of these loops. Loops areshown to preferentially arise in longitudinal patches somewhat reminiscent of active longitudes in the Sun, although broader in extent. We show that the strength of the axisymmetrictoroidal field is not a good predictor of the production rate for buoyant loops or the amountof magnetic flux in the loops that are produced.
Finally, we explore the effects of a new upper boundary condition on ASH simulations.
Previous simulations have employed an impenetrable upper boundary condition, which imposed an unphysical viscous boundary layer in the upper layers of the convection zone. We
have implemented and tested an alternative boundary condition which imposes small-scaleconvective plumes on the upper boundary, mimicking the small-scale convective motions
from the near-surface layers. We find that for suitable choices of plume parameters we canlargely remove the viscous boundary layer and significantly decrease the convective velocities at mid-convection zone, thus increasing the level of rotational constraint and helping
the simulations to achieve more solar-like behavior.

Dedication

To my adviser, Juri Toomre, for his infectious enthusiasm for stars. To my parents,
Neil and Laurie Nelson, for preparing me to follow this path. To my endlessly supportive
wife, Rachel Nelson, who reminds me that sub-grid scale representations of turbulence are
not the most important things in life. To God, for whom I have quite a few questions about
dynamo action and magnetic buoyancy in stars.

vi

Acknowledgements

In February 2007 I met Juri Toomre for the first time and his singular enthusiasm forhis research convinced me to come to the University of Colorado. I have never regretted
that decision. Over the past six years, Juri has become a trusted mentor and a valued
friend. Mark Miesch has been my “adviser in the trenches”, so to speak, who has guided
me through all sorts of nuanced and detailed physical, mathematical, and computational
arguments. Sacha Brun has been my adviser from afar, providing insight and support withhis regular visits to Boulder. In many ways advising me has been a group collaboration
between Juri, Mark, and Sacha. I would not be where I am today without their combined
efforts.
I learned much of my craft from the graduate students who came before me, especially
Ben Brown, Geoff Vassil, Nicholas Featherstone, and Kyle Augustson, and from the seniorresearchers in our group Brad Hindman and Regner Trumpadarch. Special thanks are due
to Ben Brown, François Hibbert and Christopher Chronopoulos, who endured sharing an
office with me, and thus engaging in many hours of conversations related to various detours
and roadblocks over the years. It takes a lot of patience to act interested in my ramblings
for extended periods of time, something they all did very well. I thank them for putting up
with me.
Finally, I would be remiss if I did not acknowledge Gwen Dickinson for keeping ourresearch group alive and functional. I never had enough time to chat with Gwen as muchas I would have liked. One of my greatest achievements in graduate school has been staying
off her “problem child” list.

Contents

1 Solar and Stellar Magnetic Activity
1.1

1Magnetism in Sun-like Stars . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Ingredients for Solar Dynamo Action . . . . . . . . . . . . . . . . . .

5

Guidance and Challenges from Observations . . . . . . . . . . . . . . . . . .

9

1.2.1

The Magnetism of Our Local Star . . . . . . . . . . . . . . . . . . . .

10

1.2.2

The Challenge and Promise of Observing Stellar Magnetism . . . . .

11

1.3

Theoretical Frameworks for Convective Dynamos . . . . . . . . . . . . . . .

13

1.4

Thesis Overview: Wreaths, Cycles, and Buoyant Loops in 3D Dynamo Models 20

1.2

2 Modeling Convection and Dynamo Action with ASH

22

2.1

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

24

2.2

Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.3

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.3.1

Thermal Boundary Conditions . . . . . . . . . . . . . . . . . . . . . .

32

2.3.2

Momentum Boundary Conditions . . . . . . . . . . . . . . . . . . . .

34

2.3.3

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

36

Overview of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.4.1

41

2.4

Comparison with Other Dynamo Simulations . . . . . . . . . . . . .

3 Enhanced Subgrid-Scale Models for Stellar Convection
3.1

The Large-Eddy Simulation Framework . . . . . . . . . . . . . . . . . . . . .

45
45

viii
3.2

SGS Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.3

The Dynamic Smagorinsky Model . . . . . . . . . . . . . . . . . . . . . . . .

50

3.3.1

The Dynamic Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.3.2

Implementation of the DS Model in ASH . . . . . . . . . . . . . . . .

53

3.3.3

Non-dimensional Parameters with the DS Model . . . . . . . . . . . .

56

3.4

Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.5

Direct Comparisons of SGS Models . . . . . . . . . . . . . . . . . . . . . . .

60

3.5.1

Global Properties of Diffusion in SGS Models . . . . . . . . . . . . .

64

3.5.2

Local Examples of Diffusion in SGS Models . . . . . . . . . . . . . .

66

3.5.3

Comparison with Evolved Diffusion Fields . . . . . . . . . . . . . . .

68

Evolved Comparisons of SGS Models . . . . . . . . . . . . . . . . . . . . . .

69

3.6.1

Radial Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.6.2

Azimuthal Flows and Differential Rotation . . . . . . . . . . . . . . .

74

3.6.3

Convective Transport . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

3.6

3.7

4 The Discovery of Buoyant Magnetic Loops

84

4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.2

Convection, Rotation and Magnetism . . . . . . . . . . . . . . . . . . . . . .

85

4.3

Simulation Parameters and Properties . . . . . . . . . . . . . . . . . . . . .

86

4.4

Buoyant Magnetic Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.5

Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5 Magnetic Wreaths

96

5.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.2

Dynamos at 3Ω

97

5.3

Magnetic wreaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Magnetic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

ix
5.3.2

Non-axisymmetric Fields . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4

Maintaining Rotational Shear . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5

Generation of Toroidal Magnetic Energy . . . . . . . . . . . . . . . . . . . . 111

6 Cycles of Magnetic Activity
6.1

117

Cyclic Reversals Achieved by Reducing Diffusion . . . . . . . . . . . . . . . . 117
6.1.1

Reversals in Global Magnetic Polarity. . . . . . . . . . . . . . . . . 118

6.1.2

Variability at Higher Magnetic Prandtl Number . . . . . . . . . . . . 120

6.1.3

Collapse of Resistive Balance Leading to Reversals

6.1.4

Exploring An α-Like Effect. . . . . . . . . . 122. . . . . . . . . . . . . . . . . . . . . . . 124

6.2

Turbulence-Regulated Flux Emergence . . . . . . . . . . . . . . . . . . . . . 127

6.3

Richness of stellar dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7 Buoyant Magnetic Loops Over Magnetic Activity Cycles

146

7.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.2

Flux Emergence and Convective Dynamos . . . . . . . . . . . . . . . . . . . 147

7.3

Nature of the Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.4

Identifying Magnetic Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.5

Properties of Rising Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.5.1

Dynamics and Timing of Loop Ascents . . . . . . . . . . . . . . . . . 155

8 Collective Properties of Emergent Magnetic Structures

161

8.1

Statistical Distribution of Twist and Tilt

8.2

Magnetic Cycles with Buoyant Loops . . . . . . . . . . . . . . . . . . . . . . 164

8.3. . . . . . . . . . . . . . . . . . . 161

8.2.1

Relation of Loop Emergence and Mean Field Strength . . . . . . . . 165

8.2.2

Preferential Longitudes for Loop Creation . . . . . . . . . . . . . . . 167

Summary and Reflections: Buoyant Loops in Convective Dynamos . . . . . . 168

x
9 Plume Boundary Conditions in ASH

171

9.1

Challenges at the Upper Boundary . . . . . . . . . . . . . . . . . . . . . . . 171

9.2

The Consequences of Impenetrable Boundaries . . . . . . . . . . . . . . . . . 174

9.3

Formulation of the Plume Boundary Condition . . . . . . . . . . . . . . . . . 175
9.3.1

Plume Boundary Conditions on vr and S . . . . . . . . . . . . . . . . 177

9.3.2

Plume Boundary Conditions on P . . . . . . . . . . . . . . . . . . . . 183

9.3.3

Plume Boundary Conditions on Horizontal Velocities . . . . . . . . . 185

9.3.4

Applying the Plume Boundary Model . . . . . . . . . . . . . . . . . . 190

9.4

Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.5

Understanding the Effects of the Plume Boundary Condition in ASH . . . . 193

9.6

9.5.1

Replacing Diffusive Boundary Layers with a Boundary Interface . . . 194

9.5.2

Changes in the Bulk of the Domain . . . . . . . . . . . . . . . . . . . 196

9.5.3

Strengthening Solar-like Differential Rotation . . . . . . . . . . . . . 197

Discussion of the Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . 198

10 Ongoing Puzzles and Future Challenges

206

A Generation of Differential Rotation Kinetic Energy

222

Tables

2.1

Computational parameters for the ASH simulations . . . . . . . . . . . . . .

38

2.2

Dimensional parameters for ASH simulations . . . . . . . . . . . . . . . . . .

39

2.3

Dimensionless parameters for ASH simulations . . . . . . . . . . . . . . . . .

40

2.4

Dimensional parameters for selected models using the Pencil code . . . . . .

42

3.1

Selected parameters for cases Na, Nb, and S . . . . . . . . . . . . . . . . . .

59

5.1

Overview of dynamo simulations . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.2

Volume-averaged energy densities and differential rotation rates . . . . . . . 103

5.3

Production and dissipation of differential rotation kinetic energy . . . . . . . 109

9.1

Parameters for plume boundary condition used in case P . . . . . . . . . . . 181

9.2

Selected parameters for cases Ia, Ib, Ic, and P . . . . . . . . . . . . . . . . . 191

Figures1.1

Sample sunspots as seen by SDO . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

The Solar butterfly diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Solar differential rotation and schematic model of the solar dynamo . . . . .

6

1.4

Sample shell-slices of radial velocity for Pencil, EULAG-MHD, and ASH simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5

16

Magnetic wreaths shown in 3D field line tracing and isosurface rendering of
longitudinal magnetic field in case D3 . . . . . . . . . . . . . . . . . . . . . .

18

2.1

3D rendering of selected spherical harmonics . . . . . . . . . . . . . . . . . .

28

2.2

Schematic overview of ASH’s numerical method . . . . . . . . . . . . . . . .

29

3.1

Schematic diagram of velocity correlations on the test, grid, and true scales .

52

3.2

Radial profile of νS in case S . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.3

Spatial and probability distributions of νS in case S . . . . . . . . . . . . . .

56

3.4

Radial viscous force for the EE and DS SGS models with an identical flow field 63

3.5

Local radial viscous force for a convective plume in the EE and DS SGS models 65

3.6

Radial diffusion terms for cases Na, Nb, and S in equilibrated states . . . . .

3.7

Radial velocities, rotation profiles, and meridional circulations for cases Na,

67

Nb, and S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.8

Probability distribution function of radial velocity in cases Na, Nb, and S . .

72

3.9

Radial velocity power spectra for cases Na, Nb, and S . . . . . . . . . . . . .

73

xiii
3.10 Luminosity transport in cases Na, Nb, and S . . . . . . . . . . . . . . . . . .

78

3.11 Comparison of RMS enthalpy flux and temperature perturbations . . . . . .

79

3.12 Signed enthalpy flux spectra . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.1

Snapshots of flows and fields in case S3 when a buoyant loop begins to rise .

87

4.2

Field reversals with time in cases D3b and S3 . . . . . . . . . . . . . . . . .

89

4.3

Analyzing a rising magnetic loop . . . . . . . . . . . . . . . . . . . . . . . .

91

5.1

Magnetic wreaths at 3Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2

Probability distribution functions for unsigned Bφ . . . . . . . . . . . . . . . 105

5.3

Differential rotation and angular momentum transport in case D3b . . . . . 108

5.4

Generation of differential rotation kinetic energy . . . . . . . . . . . . . . . . 110

5.5

Temporal variability of differential rotation in case D3b . . . . . . . . . . . . 111

5.6

Production terms of magnetic energy in the mean toroidal fields . . . . . . . 114

6.1

Time-latitude plots of hBφ i for cases D3, D3a, and D3b . . . . . . . . . . . . 119

6.2

Snapshots of Bφ over a magnetic polarity reversal . . . . . . . . . . . . . . . 136

6.3

Temporal variability of Bφ in case D3-pm1 . . . . . . . . . . . . . . . . . . . 137

6.4

Temporal variability of Bφ in case D3-pm2 . . . . . . . . . . . . . . . . . . . 138

6.5

Time-evolution of hAφ i for case D3 . . . . . . . . . . . . . . . . . . . . . . . 139

6.6

Time-evolution of hAφ i for case D3b . . . . . . . . . . . . . . . . . . . . . . . 139

6.7

Best-fit α-tensor components related to production of 0φ . . . . . . . . . . . 140

6.8

Magnitude of cross-correlation in time of {0φ } and {Br }, {Bθ }, and {Bφ } . . 141

6.9

Time-evolution of longitudinally-averaged odd ` spherical harmonic coefficients and mean Bφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.10 Schematic description of the reversal mechanism for cyclic convective dynamos 143
6.11 Isosurfaces of magnetic field amplitude in case S3 . . . . . . . . . . . . . . . 144

xiv
6.12 Buoyant magnetic loops evolving from small-scale wreath sections amplified
by turbulent intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1

Radial velocity, toroidal magnetic field, differential rotation, and mean toroidal
magnetic field in case S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2

Evolution in time of the longitudinally-averaged toroidal magnetic field . . . 150

7.3

Field-line renderings of magnetic wreaths and two buoyant loops . . . . . . . 153

7.4

Radial position of three rising buoyant loops in time . . . . . . . . . . . . . . 155

7.5

Volume renderings of magnetic field lines which comprise three buoyant loops
during their ascent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.6

Three viewpoints the same volume-rendering of magnetic field lines in Loop 3 159

7.7

Unsigned magnetic flux in five sample loops as they rise through the convective
layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.1

Histograms of latitudinal tilt and magnetic twist in buoyant loops . . . . . . 162

8.2

Time-latitude plot of mean toroidal magnetic field with buoyant loops overplotted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.3

Relation between mean toroidal magnetic fields and the generation of buoyant
magnetic loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.4

Time-averaged Bφ over cycle 1 with initial latitude and longitudes of buoyant
loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9.1

Shapes of imposed plumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

9.2

Total volumetric torque as a function of latitude in three models using acorrective torque to conserve angular momentum. . . . . . . . . . . . . . . 189

9.3

Sample plume field applied as an upper boundary condition . . . . . . . . . 200

9.4

3D volume rendering of vr for cases Ia, Ib, Ic, and P

9.5

Radial energy transport for cases Ib and S . . . . . . . . . . . . . . . . . . . 202. . . . . . . . . . . . . 201

xv
9.6

Variation of vr over the plume mixing layer in case P . . . . . . . . . . . . . 203

9.7vr at three depths in cases Ia, Ib, Ic, and P

9.8

Differential rotation profiles for case Ia, Ib, Ic, and P . . . . . . . . . . . . . 205. . . . . . . . . . . . . . . . . . 204

Chapter 1
Solar and Stellar Magnetic Activity

1.1

Magnetism in Sun-like Stars
Stellar variability due to magnetic activity is one of the most enduring topics of astro-

nomical research. Scientific observations of sunspots began with Galileo’s regular descriptions
and drawings of sunspots over 400 years ago (see Drake, 1957). Shortly after the turn of the20th Century, Hale (1908) showed that sunspots contain strong, coherent magnetic fields.
We now know that sunspots appear as dark patches on the Sun’s surface due to the inhibition of convective heat transport by strong, organized magnetic fields. Solar magnetism isof course not limited to sunspots. The Sun’s magnetic fields exist on a wide range of scales
from the magnetized solar wind which fills the heliosphere to the smallest resolvable scales
on the photosphere. Magnetic processes in these widely disparate regimes drive much of thetemporal variability of the Sun’s surface and extended atmosphere.
Figure 1.1 illustrates the magnetic nature of sunspots. In visible light they appear as
small, dark patches with temperatures typically lower than the surrounding solar surface by
about 1000K. These lower temperatures are the result of strong magnetic fields inhibiting theconvective mixing present over the solar surface, which constantly replenishes the energy lostto radiation. Thus material in sunspots is initially of the same temperature and brightness
as the rest of the solar surface, however radiative losses quickly lead to a strong deficit intemperature in the spot. The dark spots shown here in modern space-based observations
from the Solar Dynamics Observatory are the most easily observed consequences of solar

2

Figure 1.1: Sample views of magnetic features on the solar surface taken by the Solar
Dynamics Observatory (SDO) on October 25, 2012. (a) Sunspots are visible as dark spots inthe visible continuum centered at 4500Å using the Atmospheric Imaging Array (AIA). Herethree major sunspot groups can be seen, two in the southern hemisphere and one in the north.
(b) Companion magnetogram from the Heliospheric and Magnetic Imager (HMI) where color
gives line-of-sight magnetic field strength (blue negative, red positive, peak field strengths
are about 10 kG). Here the strongest fields are clearly concentrated in the sunspot groups,
though considerable magnetic fields are also visible in plage regions surrounding sunspot
groups and the magnetic network which extends over most of the solar surface. (c) Zoom-in
view of the region around one sunspot pair indicated in (b), showing the complex nature of
the fields which emerge through the solar surface.

magnetism (see reviews Borrero & Ichimoto, 2011; Rempel & Schlichenmaier, 2011).
When narrow-band imaging is used to detect Zeeman splitting, line-of-sight magnetic
field can be measured and the magnetic nature of these spots becomes easily apparent. The
magnetogram in Figure 1.1(b) reveals the strong magnetic fields which make up the sunspot
pairs (see zoomed-in region in Figure 1.1(c)), but also makes visible the rich topology of themagnetic network. Sunspots typically have field strengths on the order of 10 kG and areas on
the order of 100 Mm2 . Also visible are regions of weaker, less coherent magnetic field known
as plage which surround active regions with magnetic field strengths on the order of 1 kG
(Guglielmino et al., 2012). At even smaller scales and field strengths is the magnetic network,
which roughly traces the boundaries of supergranules with typical field strengths on the orderof 100 G (Rieutord & Rincon, 2010). On granular scales high resolution observations reveal

3

Figure 1.2: Azimuthally-averaged magnetic fields on the sun as a function of latitude attime from 1975 to 2013 (color gives polarity). Clear indications of spatial and temporalorganization can be seen in the equatorward propagation of active latitudes, the 11-year
cycles of magnetic activity, and the 22-year cycles of magnetic polarity. Courtesy David
Hathaway, NASA-Marshall. For more information see Hathaway (2010).

magnetic bipoles as small as 100 km in size which emerge at a rate of roughly four per minuteper granule (Danilovic et al., 2010). It is likely that magnetic fields continue at even smallerscales, however there are currently no telescopes capable of resolving such small scales (seereview Mackay & Yeates, 2012).
Both sunspots and the magnetic network are most easily observed at the surface.
However they continue up into the solar atmosphere and down into the solar interior. In thesolar atmosphere, magnetism can be monitored using UV and X-ray observations (Mackay& Yeates, 2012). In the interior it becomes extremely difficult to detect magnetic fields
directly, however considerable effort has been made to map the flows and thermal fields
around sunspots using helioseismology (Hindman et al., 2009; Borrero & Ichimoto, 2011).
Understanding these flows and thermal structures provides clues as to the topology andmagnitude of sub-surface fields.
While sunspots are relatively localized structures whose individual lifetimes, ampli-

4tudes, and locations of appearance are at least somewhat random, collectively these spots
exhibit global-scale organization both spatially and temporally. Figure 1.2 shows one versionof the famous solar butterfly diagram. Sunspots undergo magnetic activity cycle of roughly
11 years in duration. At the start of each cycle sunspots appear at mid-latitudes. Eachsunspot group persists for anywhere from a few weeks to a few months. Over the courseof each cycle new spots appear on average at progressively lower latitudes until they reachthe equator. New spots then begin appearing at mid-latitudes and the cycle begins again.
Upon further inspection, it becomes clear that additional global-scale organization exists
in the form of preferential polarities on the poleward and equatorward sides of each active
band in each hemisphere. For example, in the previous activity cycle from roughly 1997 to2008 (cycle 23), sunspot pairs preferentially appeared with the negative polarity spot slightly
poleward from the positive polarity spot in the northern hemisphere. In the current cycle
this preference has reversed. Thus the 11-year activity cycle is really part of the longer 22year magnetic polarity cycle (see review Hathaway, 2010).
Indian, Chinese, Korean, and possibly Native American sources have recorded naked
eye observations of sunspots since at least 165 BC (Whittmann & Xu, 1987). Telescopic
sunspot observations began with Galileo in 1611 (Sakurai, 1980). The 11-year activity cycle
appears in the observational record from at least 1611 until 1645, and then again after 1715 to
the present. Since 1715 there have been 24 11-year activity cycles. From about 1645 to 1715,
the sunspot cycle apparently ceased in what is known as the Maunder Minimum (Hathaway,2010). Longer-term records through surrogate methods indicate that this pattern of regular11-year activity cycles with occasional interruptions by Grand Minima, such as the Maunder
Minimum, have been occurring for at least the last 10,000 years (Usoskin, 2008). Usoskin
et al. (2007) calculate that since 9,500 BC the Sun has spent roughly one-sixth of its existencein Grand Minima.
The Sun is not alone in its ability to generate magnetic spots and cycles of magnetic
activity. Solar-type stars generate magnetism almost without exception, particularly at ro-5tation rates greater than that of the current Sun (see review Reiners, 2012). Proxies forstellar magnetism such as Calcium H and K emission and X-ray flux have long indicatedthat many sun-like stars are magnetically active (Hall, 2008). More recent developments
in high-precision photometry and spectropolarimetry are providing new and independentmeasurements of stellar magnetism (Donati & Landstreet, 2009). These advances in observational techniques are, for the first time, permitting us to understand the Sun in context
rather than as a single example of stellar dynamo action.
As observations of solar and stellar magnetism improve, there is a need for corresponding advances in theoretical models of convective dynamo action. In this thesis we will explore
how 3D computational models inspired by solar and stellar observations can help us understand the physical mechanisms which drive dynamo action in sun-like stars.

1.1.1

Ingredients for Solar Dynamo Action

The interior of a sun-like star may be divided into three distinct regions. The centermost portion of the solar interior is the nuclear-burning core, which takes up roughly 25%,or about 173 Mm (1.73 × 108 m), of the solar interior by radius. Moving outward, the next
region is the radiative zone which extends up to 71% of the solar radius or 490 Mm. In
this layer energy is transported by the slow outward diffusion of photons. In the radiativezone the stratification is stable to overturning motions. The outermost of the three regions
is the convective zone which begins at 71% of the solar interior by radius (a depth of 200Mm) and extends to the solar surface. In this region the plasma becomes too opaque forphotons to transport the solar luminosity outward and thus energy must be transported by
fluid motions. This convective transport continues up to the solar photosphere where theplasma becomes largely transparent to visible light, allowing photons to stream freely andefficiently carry the solar luminosity again. This interior structure has been confirmed byhelioseismic measurements (Christensen-Dalsgaard, 2002). Using continuous observations of6

Figure 1.3: (a) The solar differential rotation profile (in angular velocity Ω) as revealedby helioseismology using 12 years of data from the Michelson-Doppler Imager (MDI) instrument on the Solar and Heliospheric Observatory (SOHO). Areas left white represent
regions where reliable measurements are not currently possible. Adapted from Howe (2009).
(b) Schematic overview of the generally favored model for large-scale solar dynamo action.
Small-scale fields are generated by turbulent convection and pumped down into the lower
convection zone and tachocline. There they are organized into large-scale toroidal structuresby differential rotation. Once they become sufficiently strong and coherent, the large-scaletoroidal structures can rise via magnetic buoyancy effects. Some loops are destroyed by turbulent convection while others emerge through the photosphere to become sunspots. Image
courtesy Nic Brummell, UC-Santa Cruz.

resonant acoustic modes, remarkably precise measurements can be made of the Sun’s averageradial structure. Coupling these measurements with 1D structure models has produced solarmodels which match profiles derived from helioseismic measurements and constraints fromsolar neutrino observations to within a few percent (Christensen-Dalsgaard et al., 1996).
In the convection zone three general classes of flows are believed to play important rolesin the dynamo mechanism (see review Miesch, 2005). First is the Sun’s differential rotation
as measured by the angular velocity Ω as a function of radius and latitude. While it haslong been known that the solar surface displays faster rotation at the equator than at highlatitudes, global helioseismology has yielded detailed measurements of the solar differential

7
rotation as shown in Figure 1.3(a). The magnitude of the shear is quite striking with acontrast of roughly 30% between equator and pole. The radiative interior rotates as a solid
body (Howe, 2009). The boundary between the radiative and convective regions is hometo the tachocline, which is marked by a sharp transition to differential rotation with a fast
equator and slow poles. How such sharp gradients are maintained is a subject of considerable
interest as the timescale for the spread of differential rotation into the radiative zone ismuch shorter than the age of the Sun (e.g., Spiegel & Zahn, 1992; Rudiger & Kitchatinov,
1997; Gough & McIntyre, 1998; Strugarek et al., 2011). However a definitive answer to thisquestion remains elusive. In the bulk of the convective zone the surfaces of constant rotation
are roughly conical (tilted about 25◦ from the rotation axis), particularly at mid-latitudes.
Finally the outer 5% by radius (35 Mm) of the Sun forms a near-surface shear layer where
the radial gradient of rotation rate becomes negative (see review Miesch & Toomre, 2009).
The total rotational velocity peaks at about 2 km s−1 . Subtracting the bulk rotation rate of
the Sun gives differential rotation velocities on the order of 100 m s−1 .
The second class of flow is the meridional circulation. Direct observations of the solar
surface show a roughly 10 m s−1 poleward flow in each hemisphere (Hathaway, 1996). Helioseismic measurements demonstrate that this flow continues through at least the near-surface
shear layer (e.g., Haber et al., 2002; Ulrich, 2010). Below this layer, however, reliable measurements of the meridional circulation become extremely challenging. The deep meridional
flow plays a key role in many flux transport dynamo models which rely on a single circulation
cell to transport magnetic flux to high latitudes and then down to the base of the convection
zone (Charbonneau, 2010). Recently, Hathaway (2012) has reported measurements of equatorward meridional flow at depths of about 50 Mm. The structure of the meridional flow asa function of depth and latitude in the bulk of the solar convection zone remains a subjectof active debate (e.g., Sivaraman et al., 2010; Dikpati, 2011).
The third class of flows involves the highly turbulent convective motions responsible
for carrying the solar luminosity. These convective motions are inherently three-dimensional,

8highly variable, and chaotic in time. To add difficulty, the turbulence is neither homogeneous
nor isotropic since rotation, stratification, and magnetism play important roles on a wide
range of spatial and temporal scales. The dominant scale of convection changes at different
depths in the solar interior. At the surface convection is dominated by granulation of theorder of 1 Mm in size with typical velocities on the order of 5 km s−1 (Nordlund et al., 2009).
In the near-surface layers supergranulation on scales on the order of 20 Mm becomes thepreferred convective scale with typical velocities of 300 m s−1 (Rieutord & Rincon, 2010).
In the deep interior it is believed that flows on still larger scales exist. Termed giant cells,
these convective structures are realized in theoretical and numerical treatments of solar
convection, where they have typical length scales between 100 and 200 Mm (see review
Miesch, 2005). Local helioseismology has provided detailed descriptions of the near-surface
layers of the solar interior (see review Gizon & Birch, 2005). Detection of the granular andsupergranular convective motions has been achieved, however the deep giant cell convection
remains elusive (Miesch & Toomre, 2009), though their amplitude can be constrained fromtheoretical arguments to be at least 30 m s−1 (Miesch et al., 2012).
Given the Sun’s differential rotation, meridional circulation, and turbulent convection,
schematic models of the solar dynamo can be constructed (see review Charbonneau, 2010).
The details vary but many nonlinear dynamo models follow the general outline shown in
Figure 1.3(b). Turbulent motions generate small-scale, incoherent fields that pervade thebulk of the convection zone. The convective upflows and downflows are continuously providing shearing motions which can amplify small-scale magnetic fields to magnetic energy
densities in roughly equipartition with the kinetic energy of the convection itself. In thesolar convection zone that suggests disorganized fields on the order of a kilogauss should becommon in the deep interior. Some of these small-scale fields are pumped downward into
regions of strong shear at the base of the convection zone and the tachocline. Figure 1.3(a)reveals that strong latitudinal and radial gradients in rotation rate can be found in and above
the tachocline. The differential rotation tends to comb the small-scale fields into coherent

9large-scale magnetic structures which are predominantly longitudinal, as radial or latitudinal
fields will tend to be sheared into longitudinal fields by the differential rotation. Portionsof these toroidal structures eventually become sufficiently strong and coherent that theirmagnetic pressure begins to replace the gas pressure required to keep them in equilibriumwith their surroundings. This leads to the gas density dropping in regions of very strong
field, which makes them buoyant. Magnetic buoyancy will tend to make regions of strong
field rise through the convection zone. Once these regions of strong field have entered theconvective layer there are also convective upflows which can assist in the flux emergenceprocess. Some of the loops will be disrupted by unfavorable interactions with the turbulent
convective flows, while others will successfully reach the photosphere where they emerge assunspot pairs.

1.2

Guidance and Challenges from Observations
With this general schematic in hand, we can now examine what is known observa-

tionally about dynamo action in sun-like stars. Solar observations can provide exquisitely
detailed data about a single star. Stellar observations are far more limited for any given
star, though the large numbers of stars studied by a wide range of techniques is helping
unlock additional understanding of dynamo processes and their dependence on parameterslike rotation rate and stellar type. Observations of stellar magnetism can be neatly divided
into two groups: our nearby star which has been and continues to be studied in unmatcheddetail, and all other stars for which observations are far more difficult but the potential
targets are far more numerous.

10
1.2.1

The Magnetism of Our Local Star

Proxies for solar magnetic activity can reliably measure solar magnetic activity much
farther into the past than recorded observations of sunspots. Perhaps the best proxy forsolar activity involves measuring cosmogenic radionuclides such as 10 Be and 14 C in ice cores,
tree rings, and marine sediments. These radioactive isotopes are primarily generated byspallation when high-energy cosmic rays strike the upper atmosphere. Stronger and morevariable heliospheric magnetic fields increase the probability that high energy cosmic rays will
be deflected out of the heliosephere or scattered to lower energies. Both mechanisms reduce
the total energy flux of cosmic rays hitting the Earth’s upper atmosphere (Potgieter, 2013).
This drives a strong inverse correlation between solar magnetic activity and the generationof cosmogenic radionuclides. The concentration of these isotopes provides reliable estimatesof the Sun’s magnetic activity for the last 12,000 years (see review Usoskin, 2008). As usefulas such long-term data are, they are highly limited both in their coarse temporal resolution
and due to their nature as an integrated quantity over essentially the entire heliosphere.
Since the advent of telescopic observations of sunspots in 1611, data have been available
as to the size and position of sunspot groups on the solar surface. Tracking these features
permitted the first measurements of the solar activity cycle as well as the bulk rotation rateand surface differential rotation (see review Hathaway, 2010). These data have allowed someof the important large-scale signatures of the solar dynamo to be observed. These include theequatorward migrations of active latitudes resulting in the “butterfly diagram” of Figure 1.2;
Hale’s polarity law describing the hemispheric preference as to the eastward and westward
polarities of sunspot pairs in each activity cycle; and Joy’s law for the average latitudinaltilt of sunspot pairs. These patterns were first deduced by Hale et al. (1919) and have been
confirmed many times since (e.g., Babcock & Babcock, 1955; Wang & Sheeley, 1989). Most
recently Stenflo & Kosovichev (2012) have used 16 years of space-based observations with theMichelson Doppler Imager (MDI) on SOHO to catalogue these patterns in 160,079 pairs of

11bipolar magnetic spots, revealing some previous unknown features of these laws. These data
demonstrate the continuation of Joy’s law to magnetic bipoles much smaller than sunspots,
and show that roughly 5% of sunspot pairs violate Hale’s polarity law.
Helioseismic observations have been used extensively to study the near-surface structure of sunspots (see review Rempel & Schlichenmaier, 2011). One recent developmentof particular interest is the detection of subsurface active regions prior to their emergencethrough the photosphere by Ilonidis et al. (2011). Detailed studies of the statistical properties of pre-emergence active regions are providing some clues as to the physical source of
these detections (Leka et al., 2013; Birch et al., 2013). As these observations improve they
may provide important observational guidance for the process of flux emergence, which iscurrently only constrained observationally by surface measurements and near-surface helioseismic constraints on flows and thermal structures (Gizon & Birch, 2005).
Advances in spectropolarimetry have permitted measurements of full vector magnetic
fields. These vector magnetograms have revealed a hemispheric helicity law. Helicity represents the degree of twist in magnetic field lines and plays a key role in many theoreticalmodels of dynamo action and flux emergence. Sunspot pairs show statistical preferences
for the handedness of their twist in each hemisphere with left-handed twist preferred in thenorthern hemisphere and right-handed twist preferred in the south (Pevtsov et al., 1995).
This preference has been confirmed to be constant over the past three magnetic activity
cycles (Pevtsov et al., 2008).

1.2.2

The Challenge and Promise of Observing Stellar Magnetism

The Sun is not alone in its magnetic variability. Solar-type stars generate magnetism
almost without exception, particularly at rotation rates greater than that of the current
Sun. Young, rapidly rotating suns appear to have much stronger magnetic fields at theirsurfaces than our Sun (Hall, 2008). Observations reveal a clear correlation between rotation

12and magnetic activity, as inferred from proxies such as coronal X-ray and chromospheric
EUV emission (Saar & Brandenburg, 1999; Pizzolato et al., 2003; Wright et al., 2011).
These measurements, however, are generally only available at a single epoch and thus cannot
comment on the variability of stellar magnetic activity due to cycles. There have been
a number of attempts to monitor the magnetic activity cycles of other stars using solarcalibrated proxies for magnetic activity such as the Calcium H and K lines which are strongly
sensitive to magnetic activity in the Sun (e.g., Baliunas et al., 1995; Hempelmann et al., 1996;
Oláh et al., 2009; Metcalfe et al., 2013). These programs require long, sustained periods ofconsistent observations, and are therefore rare. To date, the largest such project is theMount Wilson HK survey, which measured chromospheric Ca lines as a proxy for magnetic
activity for 111 solar-like stars over a 25-year period ending in 1991. In that study almost
half of the stars showed cyclic behavior, including 21 with regular periods between 7 and25 years (Baliunas et al., 1995). Recently long-term X-ray observations have provided thefirst evidence for a companion coronal activity cycle to a previously measured chromosphericactivity cycle on a young solar-like star (Sanz-Forcada et al., 2013). The existence of sunlike stars both with and without clear cycles of magnetic activity suggest that there may
be families of dynamo models that lie very close together in parameter space but exhibit
markedly different degrees of temporal variability in their large-scale magnetic features.
All of the methods described above are disk-integrated measures of proxies for stellar magnetism. Ongoing developments in new observational techniques are permitting observations of the magnetic topology of magnetism on other stars (Donati & Landstreet,2009). Perhaps of equal importance, data are becoming available on stellar bulk and differential rotation, which play a critical role in stellar dynamo mechanisms (Aerts et al., 2010).
Spot-tracking from Kepler photometry is beginning to permit measurements of the numbers,
latitudes, and migration rate of starspots (Meibom et al., 2011; Llama et al., 2012). ZeemanDoppler imaging (ZDI) combines spectropolarimetry with Doppler shifts due to stellar rotation to create surface maps of stellar magnetism (Petit et al., 2008; Gaulme et al., 2010;13
Morgenthaler et al., 2012). This ZDI technique is yielding estimates of the size, frequency,
and magnetic flux of starspots and the topology and spatial variability of photospheric magnetic fields. The development of large optical interferometers such as the CHARA array
are allowing resolved imaging of main-sequence stars. To date this capability has primarily
been used in conjunction with spectral and asteroseismic observations to constrain stellar
parameters such as radius, mass, and age with errors on the order of 1% (Huber et al.,2012; White et al., 2013). The forthcoming optical interferometer at the Magdalena Ridge
Observatory should achieve angular resolution of a few milliarcseconds, allowing it to image
solar-like stars out to distances of about 10 parsecs (Buscher et al., 2013).

1.3

Theoretical Frameworks for Convective Dynamos
Cyclic dynamos are fundamentally three-dimensional, nonlinear, and chaotic. No single

3D numerical model has yet been able to capture all of the physical mechanisms required toreproduce solar dynamo action (see review by Charbonneau, 2010). This has led dynamo
theorists to explore reduced treatments. Much of the groundwork for modern dynamo theory
has been laid in analytic mean-field models (e.g., Parker, 1955; Moffatt, 1978; Krause &Raedler, 1980). The generation of toroidal field as differential rotation acts on a poloidal
field, for example, can be well described using these models. The so-called Ω-effect relies on
shear from differential rotation in the convection zone or the tachocline at its base to stretch
poloidal field into bands of toroidal field. The regeneration of poloidal field or the generationof opposite polarity poloidal field is parameterized in mean-field theory through the α-effectwhich is intended to represents small-scale turbulent correlations between the velocity andmagnetic field.
Mean-field models attempt to reduce the dynamo problem to a 2D formulation. As
Cowling (1933) famously showed, the magnetic induction equation does not permit a 2D
dynamo solution. The root of this prohibition is that the axisymmetric induction equation

14
for poloidal magnetic field does not contain a source term. Mean-field models circumvent thisproblem by approximating the effects of 3D turbulent induction using a Taylor expansion of
the 2D axisymmetric fields. Taking this expansion to first order yields two possible source
terms for the poloidal magnetic field. By far the most favored of these parameters is arank-two symmetric tensor αij which seeks to represent turbulent correlations between smallscale velocities and magnetic fields (Krause & Raedler, 1980). Current mean-field models
favor non-local generation of poloidal field by the unresolved rise and twist of toroidal field(Babcock, 1961; Leighton, 1964). The Babcock-Leighton mechanism has been employed in avariety of mean-field models which can be used to obtain cyclic dynamo solutions which canreproduce the 22-year magnetic cycle, the sunspot butterfly diagram, and Hale’s polarity law(e.g., Dikpati & Gilman, 2006; Choudhuri et al., 2007). These models are not particularly
reliable when predicting the amplitude of future solar cycles such as the current cycle 24, but
are the only dynamo models to date which can offer any predictive capability (see reviews
Hathaway, 2009; Petrovay, 2010).
To confront the complex nature of solar-like dynamo action, 3D numerical models havebeen developed to explore aspects of various dynamo processes. As no single numerical model
can include or resolve all relevant physical processes, modelers are forced to choose between
global and local models. Global models capture the largest-scales of interest and the correct
global geometry, but must approximate the effects of unresolved small-scale dynamics. Local
models can offer much higher resolution over a small volume, which removes much of thedependence on the unresolved scales, but these simulations cannot then model the large
scales of solar magnetism which are generally of the most interest.
Local simulations have been effectively used to model magnetic flux emergence. Flux
emergence models track the rise of buoyant magnetic structures that have been inserted
into stratified domains and then allowed to rise. One class of models uses the thin fluxtube approximation where there is no back-reaction of the magnetic structures on the fluid
via the Lorentz force (e.g., Caligari et al., 1995; Moreno-Insertis & Emonet, 1996; Fan &15Fisher, 1996; Weber et al., 2011, 2012). Thin flux-tube models have the advantage that they
are far less computationally demanding than MHD models and so a much broader rangeof parameter space can be explored. MHD models, while computationally expensive, morefaithfully represent the dynamics of rising magnetic structures. By inserting a magnetic
structure into the base of the simulated convective layer, these models can track the buoyant
rise of these structures and assess processes which may disrupt or weaken them as theyrise (e.g., Fan & Abbett, 2003; Fan, 2008; Jouve & Brun, 2009). A third type of flux
emergence model uses a forced shear layer to generate magnetic structures which then become
buoyant, thereby assuring that the magnetic structure can be generated through dynamo
action (e.g., Cline et al., 2003; Vasil & Brummell, 2009). In a variation on this method,
Guerrero & Käpylä (2011) have shown that dynamo action in a domain with convection
and a forced shear layer can produce strong dynamo action and yield buoyant magnetic
structures. Finally, near-surface models which include radiative transfer have succeeded in
reproducing many essential features of sunspots by introducing bundles of magnetic field into
their high-resolution simulations as a time-dependent lower boundary condition (Rempel
et al., 2009; Cheung et al., 2010). These “sunspot in a box” simulations are of sufficientquality that it can be difficult to distinguish synthetic observations of the model from real
observations of sunspots.
Pioneering work into global dynamo models by Gilman (1983) and Glatzmaier (1985)
produced the first 3D MHD simulations of cyclic dynamo action in a rotating sphericalshell. Major efforts have been made towards building convective dynamo models capableof addressing the generation of large-scale magnetic fields. In these models correct global
geometry plays a key role. Here we will highlight three major codes which are being used toaddress convective dynamo action in stratified, rotating, spherical shells, namely the Pencil,
EULAG-MHD, and ASH codes. Figure 1.4 shows samples of convective patterns from eachof these codes. While the general pattern of rotationally-aligned convection near the equatorand more isotropic convection at mid- to high-latitudes occurs in all three simulations, the16

Figure 1.4: Radial velocity patterns in three convective dynamo simulations from (a) Warnecke et al. (2012) using the Pencil code, (b) Ghizaru et al. (2010) using EULAG-MHD, and(c) Nelson et al. (2013a) using ASH. Shown are (a) an orthographic projection for case A5 at
0.89R (scale not given in original), (b) a Mollweide projection at 0.945R (scale indicated),and (c) a Mollweide projection for case D3b at 0.95R (scale indicated). For all images darktones indicate downflows while light tones indicate upflows. All three simulations shown
produce cyclic dynamo action.

details of the flows realized are quite different due to a myriad of factors including thecomputational method, the numerical resolution of the simulation, the choices of diffusion
models, the background stratification, the convective driving, and many more. Here we will
not attempt to catalogue all possible sources of these variations, but rather will attempt togive a brief description of each code in order to give some idea about the major differences inthese three numerical tools. We will return to the ASH code in greater detail in Chapter 2.
The Pencil code uses a high-order finite difference scheme to solve the compressible

17
MHD equations in 3D. Numerical cost limits the Pencil code’s ability to reach high latitudes, thus the polar regions are generally excluded from the simulation domain. Using thePencil code to simulate spherical wedges, Käpylä et al. (2011) explored the generation andmaintenance of strong, solar-like differential rotation. When magnetic fields are added, thesemodels achieved cyclic dynamo action with reversals of global magnetic polarity (Käpylä
et al., 2012b). Warnecke et al. (2012) further explored the ability of these models to resolve
the rise and emergence of magnetic flux structures.
The EULAG-MHD code has been used to simulate solar-like dynamo action. Numerically EULAG-MHD uses an implicit large-eddy simulation (ILES) framework in which
there is no explicit physical diffusion and numerical operators are chosen to be as stable aspossible, leading to significant numerical diffusion. It is also capable of simulating entire
spherical surfaces and thus does not require the poles to be excluded from the simulated
domain. Using EULAG-MHD, Ghizaru et al. (2010) have shown large-scale organizationof the toroidal field as well as magnetic activity cycles in a solar-like simulation. Regularreversals of global magnetic polarity with a roughly 60 year period for a complete cycle wereachieved. Racine et al. (2011) has interpreted these results in terms of mean-field dynamo
theory, while Simard et al. (2013) have used the resulting mean-field dynamo parameterssuch as the α-tensor to generate companion mean-field models which display similar cyclic
behavior.
The third code is the Anelastic Spherical Harmonic (ASH) code. All of the simulations
presented here use the ASH code. ASH is a large-eddy simulation (LES) code which requires
an explicit sub-grid scale (SGS) model to parameterize the effects of unresolved scales on
the resolved dynamics. The use of pseudospectral methods allows ASH to treat dynamics
over full spherical shells (Clune et al., 1999). Specifically, ASH uses spherical harmonictransforms in the longitudinal and latitudinal directions, and can either use a projection
onto Chebyshev polynomials or a finite difference scheme in the radial direction. Miesch
et al. (2006, 2008) have explored the interplay of rotation, stratification, and moderately

18

Figure 1.5: Magnetic wreaths from a convective dynamo simulation of a young Sun rotating
at three times the solar rate (case D3) shown in two 3D volume renderings taken at the same
instant. (a) Magnetic fields lines colored by longitudinal magnetic field Bφ which highlightthe strong connectivity between the two wreaths and between each wreath and the restof the domain. (b) Isosurfaces of Bφ at 20 kG (brown), 10 kG (red), 5 kG (yellow), -5
kG (green), -10 kG (blue), and -20 kG (purple) between ±30◦ of latitude which highlightthe non-axisymmetric components of the of the wreaths which are strongly modulated byconvective flows. Perspective is looking from approximately 45◦ N through the center of thestar. These magnetic wreaths are able to persist in the bulk of the convection zone for as
long as the simulation was run (here 61.6 years). Case D3 is discussed in detail in Brown
et al. (2010).

turbulent convection to produce strong differential rotation in a hydrodynamic setting. Whenmagnetism is admitted the resulting dynamo produces reversals of global polarity but isdominated by non-axisymmetric fields with little global organization (Brun et al., 2004).
By adding an overshooting region of strong shear which mimics the solar tachocline, globalscale organization of the toroidal field was achieved but without reversals in global magnetic
polarity over about 30 simulated years (Browning et al., 2006). Miesch & Brown (2012) have
explored 3D convective dynamo action with a Babock-Leighton term in order to include flux
transport by means of a parameterization of magnetic buoyancy.

19
ASH has also been used to simulate the dynamics within the deep convective envelopesof young stars rotating faster than our current Sun. Brown et al. (2008) began with hydrodynamic simulations involving a range of rotation rates up to ten times the solar rotation rate
(10Ω ), finding that strong differential rotation is realized, and that the columnar convection
at low latitudes can exhibit significant modulation in amplitude with longitude, even appearing as nearly isolated active nests. Brown et al. (2010) examined dynamo action achieved ina MHD simulation carried out at 3Ω , finding that the convection can build ordered globalscale magnetic fields that appear as two wreaths of strong toroidal field positioned aboveand below the equator. These striking structures shown in Figure 1.5 can persist for long intervals despite being embedded within a turbulent convective layer. These wreaths stronglyinteract with the convective flows, causing strong non-axisymmetric components which canbe seen in both Figures 1.5(a) and 1.5(b). However the convection does not destroy thewreaths. It also does not fully pin them to the base of the simulation, as might be expectedfrom arguments about magnetic pumping, nor does it quickly eject the magnetic fields out
the top of the domain, as might be expected from considerations of magnetic buoyancy inconvective layers (Thompson & Weiss, 2009).
Turning to dynamo action proceeding at a faster rotation rate of 5Ω , Brown et al.
(2011) showed that self-consistently generated magnetic wreaths at low latitudes can undergo reversals in global magnetic polarity and even quasi-cycles of magnetic activity. The
complex steps involved in the magnetic field reversals are accompanied by variations in the
differential rotation, including bands of relatively fast and slow fluid propagating toward thepoles. In these models, ASH simulations have made contact with some of the key features
believed to be present in the solar dynamo, specifically the strong toroidal structures which
can yield reversals of global magnetic polarity. The work of Brown et al. (2010, 2011) provides much of the inspiration for this thesis, which seeks to understand how these wreaths
behave in more turbulent environments and how magnetic reversals are achieved.

20
1.4

Thesis Overview: Wreaths, Cycles, and Buoyant Loops in 3D Dynamo
ModelsEncouraged by the results of previous 3D ASH simulations, we have conducted a newseries of related simulations with ASH designed to explore three fundamental science questions:(1) How might sun-like stars build large-scale magnetic fields?
(2) How can those fields undergo cycles of magnetic activity and reversals in global
magnetic polarity?
(3) How can magnetic structures rise from their generation sites toward the stellar surface?
In order to address these questions, we will begin with a discussion of the ASH code which
has enabled simulations of convective dynamo action. Chapter 2 will discuss the numerical
implementation of ASH code and how our simulations with ASH compare to other simulations of solar-like convective dynamo action. The implementation and validation of a new
sub-grid scale (SGS) model in ASH will be covered in Chapter 3. Our initial discovery ofbuoyant magnetic loops in a convective dynamo simulation will be presented in Chapter 4.
These buoyant magnetic loops are enabled by the use of a dynamic Smagorsinsky SGS model
which greatly reduces the effects of diffusion in our simulations.
In Chapter 5 the effects of reduced explicit diffusion on magnetic wreaths are investigated. Here we show that strong magnetic wreaths can be maintained as simulations are
made more turbulent and that the fundamental balances which are required to maintainwreath-building dynamo action can persist as explicit diffusion is reduced. Next we will explore the cyclic magnetic activity which occurs in our more turbulent models in Chapter 6.
Special attention will be focused on the nature of the magnetic reversals achieved. We theninvestigate in Chapter 7 the most turbulent dynamo simulation yet achieved with the ASH

21code, which generates large numbers of buoyant magnetic loops. Chapter 8 discusses thecollective properties of the generation of buoyant loops over a magnetic activity cycle. Theseloops mimic some observed properties of solar active regions, specifically Hale’s polarity law,
Joy’s law for latitudinal tilts, and the hemispheric helicity rule.
As ASH is not capable of resolving the small-scales of motion which dominate dynamics near the photosphere, all previous ASH simulations have used an impenetrable upperboundary condition placed around 0.97 or 0.98R . Chapter 9 investigates a new boundary
condition which allows plumes modeled on those achieved in local near-surface simulations
to pass through the upper boundary and into the computational domain. Finally, we willconclude in Chapter 10 with a discussion of the over-arching themes of this work and thepotential for continued advances.

Chapter 2
Modeling Convection and Dynamo Action with ASH

We seek to investigate large-scale convective dynamo action in stellar interiors likethose of our Sun. To do this we need a numerical tool which can include physical processes
such as spherical geometry, rotation, turbulent convection, and magnetic induction on scales
comparable to the size of the solar convection zone. Spherical geometry is required to capture
the convective motions driven in a spherical shell between the base of the solar convection
zone at approximately 0.72R

and the photosphere. As differential rotation plays a keyrole in essentially all theoretical models for large-scale dynamo action, our numerical model
must be capable of generating bulk zonal flows, preferably due to redistribution of angularmomentum by the resolved convective flows. Dynamo action is defined as the self-sustaining
generation of magnetic fields without an influx of magnetic energy through either the lower
or upper boundary. Beyond simply permitting dynamo action, if one wishes to study possible
mechanism for the generation of magnetic loops the numerical tools must allow sufficiently
fine resolution and avoid diffusing these magnetic structures on short time scales.
The final and perhaps most challenging element needed is the ability to achieve turbulent flows. The level turbulence as measured by the Reynolds number in our simulations ismany orders of magnitude smaller than the level of turbulence in real stars. As computing
resources continue their exponential growth convective dynamo models have shown a remarkable ability to keep pace. Gilman (1983) published the first convective dynamo simulation
which reported on models with 16 radial, 36 latitudinal, and 72 longitudinal grid points with

23
roughly 8000 time steps. The largest computational simulation reported in this thesis (caseD3b) uses 145 radial, 512 latitudinal, and 1024 longitudinal grid points over 9.8 million time
steps (see Nelson et al., 2013b). Thus over 20 years the size of convective dynamo models has
increased by a factor of 221 . By comparison, Intel Corporation’s state-of-the-art commercial
chip increased in performance by a factor of 214.6 between 1983 and 2013 (Hellemans, 2013).
The largest reason for the improvement of convective dynamo models is that the number oftime steps which can be computed has increase by a factor of 103 while the spatial resolution
has only increased by factors of roughly 10 in each direction. This is in part a product of
the increased availability of large competitively available supercomputing resources fundedby agencies such as the National Science Foundation and NASA.
With the continued growth of computing power, one can speculate on when computational resources might permit a global simulation resolve the Kolmogorov scale, which
has been estimated based on atomic values to be on the order of 10 to 100 m (Braginskii,1965). If Moore’s law continues to hold and convective dynamo models continues to mimic
its growth, the answer is roughly 200 years. It is likely, however, that most if not all of
the global-scale behaviors are not dynamically linked to scales over a million times smaller.
Thus we proceed with these models, hopeful that by resolving the largest scales of motion
and achieving the most turbulent states possible with current computational resources, wemay be able to capture some or perhaps all of the physical mechanisms which are essential
to realize large-scale dynamo action as it occurs in sun-like stars.
To conduct these simulations we use the equations of magnetohydrodynamics (MHD).
Specifically for the solar interior where the flows are orders of magnitude slower than thesound speed, we use the anelastic approximation for subsonic convective flows developed by
Gough (1969) and further formulated for stellar convection by Latour et al. (1976). Our
computational tool for solving these equations is the anelastic spherical harmonic (ASH)code. The hydrodynamic portion of ASH is described in detail in Clune et al. (1999), while
the inclusion of magnetism is covered in Brun et al. (2004). ASH solves the 3D anelastic

24
MHD equations in rotating spherical shells. ASH uses 1D stellar structure models to provide
the background stratification or reference state. The 3D equations of mass conservation,
momentum conservation, energy conservation, and magnetic induction are solved using a
pseudo-spectral or mixed pseudo-spectral and finite difference method for spatial derivatives.
Time evolution is computed using a semi-implicit Crank-Nichelson scheme for linear terms
and an Adams-Bashforth explicit scheme for nonlinear terms.
What follows is not intended to provide a complete overview of the ASH code but
rather a description of its key elements. Particular attention will be paid to the components
which are of particular importance to this thesis. Specifically, we will provide an overview
of the anelastic MHD equations and how ASH solves them, while providing increased focus
on the sub-grid scale models used in ASH which are crucial for our discussion of buoyant
magnetic loops in Chapters 4, 7, and 8, as well as ASH’s boundary conditions which will
inform our discussion of plume boundary conditions in Chapter 9.

2.1

The Anelastic MHD Equations
The anelastic MHD equations are an approximate version of the fully compressible

MHD equations which have been specifically designed for regions such as stellar convective
layers. The anelastic approximation allows for strong density stratification without requiring
the computational model to track sound waves which can propagate at tens of kilometers per
second in the deep interior while convective motions are orders of magnitude slower. Thus
the anelastic approximation permits much larger time steps than fully compressible codes.
The anelastic formulation also formally decomposes the thermal state into a backgroundwith small perturbations around it. In this way we can avoid the very long timescale forchanging the mean thermal stratification. ASH is thus able to avoid both the very shorttime scales of sound propagation and the very long time scales of stellar evolution in orderto focus on the time scales of convective overturning and dynamo action. In the Sun these25
are on the order of weeks to tens of years. For comparison the sound-crossing time in the
Sun is about two hours and the thermal relaxation time scale for the solar convective layer(or its Kelvin-Helmholtz time) is about 105 years.
The anelastic approximation splits all variables into a spherically-symmetric reference
state and 3D fluctuations about that reference state. The reference state is generally held
constant, but can be updated by periodically removing the mean value at each radial levelfrom the fluctuations and adding it to the reference state. The reference state for thermodynamic variables such as density ρ̄, temperature T̄ , entropy S̄, and pressure P̄ are assigned
using the results of 1D stellar structure models (where over-bars denote the reference state),
while it is assumed that the reference state for velocity and magnetic field is zero. Fluctu~ where ~u and B~ areations about the reference state are denoted as ρ, T , S, P , ~u, and B,
respectively the velocity and magnetic field. The decomposition into reference and fluctuating components are written asρtot (r, θ, φ, t) = ρ̄(r) + ρ(r, θ, φ, t)(2.1)

Ptot (r, θ, φ, t) = P̄ (r) + P (r, θ, φ, t)(2.2)

Ttot (r, θ, φ, t) = T̄ (r) + T (r, θ, φ, t)(2.3)

Stot (r, θ, φ, t) = S̄(r) + S(r, θ, φ, t)(2.4)~utot (r, θ, φ, t) = ~u(r, θ, φ, t)(2.5)~ tot (r, θ, φ, t) = B(r,~ θ, φ, t).
B

(2.6)

The anelastic MHD equations consist of two equations of constraint, seven evolution
equations, and an equation of state for both the reference state and the thermodynamic
fluctuations. The constraint equations are the anelastic conservation of mass equation andthe solenoidal constraint on the magnetic field, given by∇ · (ρ̄~u) = 0

(2.7)

26
~ = 0.
∇·B(2.8)

The evolution equations express conservation of momentum and energy as well as the magnetic induction equation. They are solved in a reference frame which is rotating at angularvelocity Ω0 . This gives two vector equations and one scalar equation written as
ρ̄


∂~u
1 
~ ×B
~
= −ρ̄ (~u · ∇) ~u − 2ρ̄Ω0 × ~u − ∇P + ρ~g − ∇ · D −
∇×B
∂t
4π
ρ̄T̄




∂S
= − ρ̄T̄ ~u · ∇ S + S̄ + ∇ · κr ρ̄CP ∇T̄ + κρ̄T̄ ∇ S + S̄
∂t


2
η 
1
2
~
∇ × B + 2ρ̄ν eij eij − (∇ · ~u)
+
4π
3




~
∂B
~ − ∇ × η∇ × B~ ,= ∇ × ~v × B
∂t

(2.9)

(2.10)(2.11)

where ~g is gravitational acceleration, D is the viscous stress tensor, eij is the strain-rate
tensor, and CP is the specific heat at constant pressure. Diffusion coefficients for momentum,
entropy, radiative diffusion, and magnetic fields are given by ν, κ, κr , and η, respectively.
ASH follows the large-eddy simulation (LES) paradigm where those scales of motion
which are smaller than the computational resolution are represented by a sub-grid scale(SGS) model. The viscous diffusion tensor is dependent upon the SGS model chosen, but
for Newtonian diffusion it can be written as1
D = Dij = −2ρ̄ν eij − (∇ · ~u) δij .
3(2.12)

We will reserve further discussion of SGS models in ASH for Chapter 3.
The equation of state used in ASH is that of a perfect gas. For the reference state thatis writtenP̄ = Rρ̄T̄ ,(2.13)

where R is the gas constant. For our ideal gas it is R = 2CP /5. The equation of state forthe fluctuations is linearized in the anelastic approximation, which leads to
ρ
P
T
P
S
= − =
−
.
ρ̄
CP
P̄
T̄
γ P̄

(2.14)

27
The linearized equation of state prevents ASH from investigating domains where perturbations become significant compared to mean values.

2.2

Numerical Methods
ASH incorporates the equations of constraint on the mass flux and the magnetic fieldby solving for streamfunctions. Toroidal streamfunctions Z and A describe, respectively,
streamlines and magnetic field lines which close on a spherical surface, while the poloidal
streamfunctions W and C describe streamlines and magnetic fields which do not. Formallythe decomposition is given byρ̄~u = ∇ × (Z r̂) + ∇ × [∇ × (W r̂)](2.15)~ = ∇ × (Ar̂) + ∇ × [∇ × (C r̂)] .
B(2.16)

Thus the evolution variables used in ASH are the mass flux streamfunctions W and Z,
the specific entropy S, the pressure P , and the magnetic streamfunctions C and A. Equations 2.9, 2.10 and 2.11 lead to evolution equations for W , Z, S, C, and A. The resulting
equation for P is not an evolution equation but rather is elliptic and must be satisfied at each
instant over the entire domain. Physically, this indicates that information about pressure
perturbations in the anelastic approximation is instantly propagated everywhere.
As suggested by its name, ASH uses spherical harmonic transforms to compute derivatives in the latitudinal and longitudinal directions. Spherical harmonics are formally the
angular portion of solutions to Laplace’s equation on a sphere, and as such are the natural basis functions when working in spherical coordinates. Figure 2.1 shows 3D renderingsof three representative spherical harmonic functions. Following convention, we write thesefunctions as Ym` (θ, φ), where m is the azimuthal wavenumber in units of the circumferenceand ` represents the latitudinal order of the function. Spherical harmonics are the product of sine and cosine functions of φ and associated Legendre polynomials of cos θ. Thus

28

Figure 2.1: 3D volume rendering of the real parts of three selected spherical harmonic
functions. Here the radial position gives the amplitude of the function for each value of θand φ. Note that the number of nodes in longitude is given by m and the number of nodes
in latitude in given by ` − m.

ASH accomplishes the spherical harmonic transform with a combination of a Fast Fourier
Transform (FFT) and a Legendre transform. Unfortunately there does not yet exist a robust
fast Legendre transform, so that aspect of our numerical algorithm scales as the maximum
spherical harmonic degree `max squared, whereas the FFT scales as 2`max ln (2`max ).
These equations are solved using one of two numerical methods for the radial spatial
derivatives. The first uses Chebyshev polynomials in the radial direction. This version of
ASH was developed in the late 1990’s. The use of spherical harmonics allows for uniform
resolution over the spherical surface, provides spectral accuracy, and removes the coordinate
singularity at the poles. The use of Chebyshev polynomials in the radial direction also
has many positive properties. Chebyshev polynomials place large numbers of grid-points inthe boundary layers at the top and bottom of the layer where convective boundary layers
require high resolution. This version of ASH also achieves spectral convergence properties
in the spatial derivatives, although the overall accuracy of the scheme is limited by thetime-stepping used. When using Chebyshev polynomials ASH also employs a Chebyshev
colocation scheme in radius, which reduces the Chebyshev transform to a discrete cosine
transform, which scales as Nr ln Nr , where Nr is the number of radial grid points. This

29

Figure 2.2: Schematic overview of the numerical method used to solve the anelastic MHD
equations in ASH. Three data configurations are represented, as well as the steps needed tomove between each. (Courtesy Nicholas Featherstone, University of Colorado).

colocation scheme concentrates resolution near the boundaries of the domain which canhave negative implications for the size of the time step if significant radial flows exist in
those regions.
The second version of ASH uses a hybrid pseudo-spectral/finite difference scheme forspatial derivatives. It retains the use of spherical harmonics in the θ and φ directions but
uses a variable-order centered finite difference scheme in the radial direction. All simulations
here use a 5th order scheme. This permits the use of arbitrary grids in the radial direction.
It also reduces the computational cost of the radial derivative scheme, which scales as Nr .
This decrease in cost is accompanied by a decrease in accuracy. The Chebyshev transformis spectrally accurate while the finite difference scheme is not. This version of ASH is often
referred to as ASH-FD. Both versions have been incorporated into a single ASH framework,
allowing direct comparison between the different radial derivative schemes.
ASH is designed to efficiently operate on modern massively parallel supercomputers.
Thus special attention has been paid to the parallelization of its numerical scheme. The
existence of an elliptic equation for the pressure field requires at least one global communication at each time-step. In ASH, we chose a numerical technique which incorporates global
communication as required by the elliptic pressure equation and also allows other desirable

30
features such as spectrally accurate derivatives and semi-implicit time stepping of linear
terms. This is accomplished through a highly efficient series of data transposes. Figure 2.2
gives a visual representation of the numerical method employed by ASH. Radial derivatives
and implicit time-stepping occurs in our ‘Implicit Solve Configuration’ with spherical harmonic modes distributed and all radial points at a given ` and m in processor. A global
transpose (achieved using the Message Passing Interface (MPI) command ‘AlltoAllv’) moves
data to our ‘Spectral Space Configuration’ so that radial and m grid points are distributedand ` in processor. In this configuration linear terms are computed and angular derivatives
are taken. Through a FFT, another global transpose, and a Legendre transform data is
moved to physical space with radial and latitudinal grid points distributed and longitudinal
grid points in processor. This is our ‘Physical Space Configuration’. In physical space thenonlinear terms can be calculated and then transformed and transposed back to the Spectral
Space Configuration where the next time-step can begin.
When simulating physical systems with ranges of physical and temporal scales beyond
the capacity of computational resources, modeling the behavior of the largest scales requires
some approximation to be made about the behavior of scales below some cutoff so that thenumerical requirements match the computational resources. ASH follows the philosophy ofa Large-Eddy Simulation (LES). The LES paradigm relies on sub-grid scale (SGS) models torepresent the effects of unresolved scales. Explicit SGS models can range from substituting
so-called turbulent diffusivities for the much smaller atomic diffusivities to much more complex schemes which attempt to model the turbulent cascade of energy from large to smallerscales and use physical arguments such as scale-invariance or self-similarity to extrapolate
the behavior of unresolved scales based on resolved ones. ASH is an explicit LES code, as
are the Pencil (Brandenburg, 2003) and MagIC (Gastine et al., 2012) codes which have beenused extensively for convective dynamo simulations.
Alternatively, implicit SGS models have gained wide-spread popularity in a numberof settings. Outside of the field of convective dynamos, Wachtor et al. (2013) have used31
the radiative combustion code RAGE to simulate inertial confinement fusion and Patnaik
et al. (2012) have used the atmospheric chemical mixing code FAST3D-CT to simulate thetransport of pollution in cities. Both RAGE and FAST3D-CT follow the ILES philosophy
which uses highly stable spatial and temporal derivative schemes in the place of explicit
diffusion terms. These derivative operators include significant numerical diffusion, but only
when needed for numerical stability. ILES models can generally achieve much lower levelsof diffusion than explicit LES codes. However, they have the disadvantage that their diffusion is motivated purely by numerical considerations rather than physical ones, and thatthey generally cannot measure their diffusion as it is not explicitly computed. Implicit LES
codes which have been used to model stellar dynamo action include EULAG-MHD (Prusa
et al., 2008). A number of convective dynamo codes have been designed using implicit LES
numerical methods and can also include explicit SGS models, making them either implicit
or hybrid implicit/explict LES codes. These hybrid codes including MuRAM (Cheung et al.,2010), FSAM (Fan et al., 2013), and Stagger (Beeck et al., 2012).

2.3

Boundary Conditions
As with any system of partial differential equations, the solutions to the anelastic MHD

equations depends strongly on the choice of boundary conditions. In the solar convection
zone the inner and outer boundaries are at the radiative interior and in the solar atmosphere.
While it is possible to simulate a radiative interior (e.g., Browning et al., 2006; Brun et al.,2011), the simulations presented here chose to omit that region. Whereas simulations which
include a tachocline are extremely interesting, they are also extremely challenging, bothconceptually and computationally. In the past several years there has been significant work
done to address the long-standing question of the maintenance of the tachocline using 3D
simulations (Strugarek et al., 2011; Brun et al., 2011). There has also been significant work
to address the validity of the anelastic approximation in stably stratified layers (Brown et al.,32
2012; Vasil et al., 2013). The effects of a convective over-shooting region with strong shear
on convective dynamo simulations in ASH is an active area of research that is beyond thescope of this thesis. We will discuss the possibilities for future research in this direction in
Chapter 10.
The upper boundary in ASH is perhaps even more challenging than the lower boundary. Sun-like stars go through extremely large changes in their stratification near theirsurfaces. Radiative transport and non-ideal thermodynamic processes such as ionization
play important roles. The preferred scales of convection drop dramatically until at thesurface the granulation is on the order of 1 Mm or at spherical harmonic degree of about`max ≈ 4500. The largest ASH simulations to date achieve a maximum spherical harmonic
degree of `max = 1360. ASH does not have the resolution or the physics needed to simulate
surface convection. Thus we must place an upper boundary somewhere in the interior of the
Sun.

2.3.1

Thermal Boundary Conditions

The convective driving of our simulations is the result of the transport of thermalenergy through our spherical shell. Heat is input at the base of the domain through aradiative diffusion term which is taken from a 1D stellar model. This acts as a source term
which carries the entire luminosity at the bottom of the domain and decreases to essentiallyzero over the bottom 15% of the domain. Likewise we must transport heat out of the top of
the domain. We generally chose to do this through a diffusive flux, although in Chapter 9
we will discuss an alternative method which uses an imposed pattern of plumes to transport
the energy flux through the upper boundary via convection.
For our entropy equation we can set one of two boundary conditions. Either we can set
the value of S or the value of ∂S/∂r. As we are simulating a convective layer, we generally
chose to set ∂S/∂r = 0 at the lower boundary. By doing so we define the bottom of our33
domain as the base of the convection zone. For the upper boundary we have groups ofsimulations which use either condition. Earlier cases were generally run with ∂S/∂r set
such that the diffusive flux out of the domain was exactly one solar luminosity. Our more
recent cases have set S on the upper boundary, thereby allowing the convection to build
a boundary layer self-consistently in order to achieve the same goal. The first method iscomputationally cheaper as the thermal equilibration time is quite short, however the second
removes a potentially troubling issue where the total entropy of the domain is unconstrained
when both the upper and lower boundary conditions are of the second type. As the lower
boundary is designed to represent the base of the convection zone where ∂S/∂r = 0 by
definition, we find it conceptually preferable to set the gradient of S to zero at the base of
the domain and the value of S at the top of the domain.
Coupled with the choice of entropy boundary conditions is our choice of the thermal
diffusivity κ. As was mentioned, a radiative flux is responsible for carrying the energy flux
at the lower boundary in both our simulations and real stars. There is no such radiative flux
at the top of our computational domain. In stars the flux would be carried by convection all
the way to the photosphere, though with impenetrable boundary conditions placed below
the photosphere this is not possible. Thus we employ diffusion on the spherically symmetric
entropy profile to carry the required luminosity. This means that our choice of the thermal
diffusion coefficient κ plays an important role in specifying the thermal state of our upperboundary.
We have several parameters with which to specify κ, but generally the parameter which
has the greatest impact on the upper boundary layer is our specification of the thermal diffusion on the spherically-symmetric entropy, or κ00 . We can either strongly taper κ00 so that
it is large at the upper boundary and small though the rest of the domain, or set κ00 = κ
so that thermal diffusion acts uniformly on all modes. In this work, all simulations which
employed a ∂S/∂r boundary condition at the upper boundary also used a strongly tapered
profile for κ00 , while all those which set S at the boundaries set κ00 = κ.

34

2.3.2

Momentum Boundary Conditions

Previous ASH simulations have employed an impenetrable upper boundary condition
such that the radial velocity
vr = 0.

(2.17)In Chapter 9 we will present a new boundary condition which allows specified flows to crossthrough the upper boundary and thus leave or enter the computational domain.
On both the upper and lower boundaries, ASH simulations are generally run withstress-free boundary conditions. These can be written as∂  vφ ∂  vθ 
=
= 0.
∂r r
∂r r

(2.18)

The combination of impenetrable and stress-free boundary conditions prevents any hydrodynamic torques on the domain from the boundaries, thus conserving angular momentum
to the numerical accuracy of our simulations. As many of our simulations can continue formillions of time-steps, the total angular momentum of the system can and in practice doeschange as a random walk away from the initial values. In order to more precisely conserve
angular momentum over long time evolutions, we can also employ what we term angularmomentum cleaning. This takes advantage of our streamfunction formalism as follows.
The total angular momentum in our computational domain about the axis of rotation
can be written as
Z ro Z π Z 2π
Z ro Z π Z 2π2Lz =(~r × ~u) · ẑ r sin θ dr dθ dφ =ρ̄uφ r3 sin2 θ dr dθ dφ,
ri

0

0

ri

0

(2.19)

0where ~r is the radial vector from the origin and ẑ is the unit vector along the axis of rotation.
Note that we have not considered the rotation rate of the frame, so initially Lz = 0. Using
the streamfunction formalism
ρ̄uφ =

1 ∂ 2W1 ∂Z−.
r sin θ ∂r∂φ r ∂θ

(2.20)

35
Carrying out the φ integral removes the W term which contains a φ derivative, leaving
Z ro Z π∂hZi 2 2
r sin θ dr dθ,∂θ

Lz = −2πri0(2.21)

where ∂hZi is the longitudinally-averaged portion of Z. Thus we can already see that Lz∞
P
only depends on the spherical harmonic components of Z with m = 0. Thus hZi =Z̃`0 Y`0`=0

Now let us consider the latitudinal integral such thatZ ro
Lz = −2π

T r2 dr,(2.22)

ri

where
Z π∂

∞
P

T =

Z̃`0 Y`0`=0sin2 θ dθ.

∂θ

0(2.23)

If we now take the derivative through the sum and use a derivative identity for sphericalharmonics (Dennery & Krzywicki, 1996), we find that∂Y`0
= CY`0 ,∂θ

(2.24)

where C is simply a constant which depends on the normalization used. Thus we can write

Z π
∞ 
X
0
0
Z̃`
Y` sin θ sin θ dθ .
(2.25)T =C
0`=0

Recognizing the integral to be an inner product between two spherical harmonics,
T =D

∞ 
X


Z̃`0 δ`1 = DZ̃10 ,(2.26)`=0where again D is merely a constant dependent upon the normalization chosen.
Returning to Equation 2.22, we can now write the total angular momentum about theaxis of rotation as
Z ro
Lz = −2πDZ̃10 r2 dr.

(2.27)ri

We can therefore assure that angular momentum is conserved by keeping
Z rori

Z̃10 r2 dr = 0(2.28)

36
at each time step. It can be shown that the other two components of total angular momentum can similarly be expressed as radial integrals of Z̃11 and Z̃1−1 . By choosing the boundary
values of these three components of Z such that Equation 2.28 is satisfied, we can enforce
conservation of angular momentum to machine precision at each time step.

2.3.3

Magnetic Boundary Conditions

While there are clear physical arguments for our standard hydrodynamic boundary
conditions, the choices of magnetic boundary conditions are somewhat more varied. In ASH
we have three choices, which are as follows.
~ is• The boundary is a perfect conductor, such that the electric field E
Eθ = Eφ = 0.

(2.29)

Combined with the impenetrable condition described above, this leads to a condition~ given by
on B
Br =

∂Bφ Bφ∂Bθ Bθ
+
=
+
=0
∂r
r
∂r
r

(2.30)• The boundary matches to an external potential field with~ = ∇Φ
B(2.31)∇2 Φ = 0.

(2.32)

which means that

• The magnetic field is required to be purely radial at the boundaries, such as would
happen at the boundary to a highly permeable external medium. This is expressed~ = Br r̂.
at B
For all of the magnetic simulations presented here we have chosen to employ a perfect conductor on the inner boundary and a potential field on the outer boundary. Brown (2009)

37
contains additional discussion of the rationale for various magnetic boundary conditions andtheir impact on dynamo simulations.

2.4

Overview of Simulations
Using ASH we have conducted three major sets of numerical experiments to explore

questions related to convective dynamo action in ASH. The first set of simulations, containingcases Na, Nb, and S, were designed to test the dynamic Smagorisnky SGS model in ASH
and compare it to the enhanced eddy viscosity model. This comparison will be discussed
primarily in Chapter 3. They were run in 2009 on the Sun Constellation Linux Cluster
named Ranger at the Texas Advanced Computing Center.
The second set of simulations are convective dynamo models designed to probe physicalmechanisms which may be present in solar-like dynamos. Case D3 was originally begun by
Benjamin Brown and was discussed in detail in Brown et al. (2010). Cases D3-pm1 and D3pm2 were also run primarily by Bejamin Brown as part of his thesis work at the Universityof Colorado. Working with Benjamin, I ran cases D3a, D3b, and S3 in order to explore theeffects of decreased diffusion on the dynamo-generated magnetic wreaths attained in case
D3. Cases D3a and D3b were run from 2007 through 2010 on a number of systems including
the IBM Blue Gene Cluster at the San Diego Supercomputing Center, the Ranger system,and the Cray XT5 system Kraken at the National Institute for Computational Sciences.
These models will be discussed in detail in Chapters 4 through 8.

Table 2.1: Computational parameters for ASH simulations. Shown are the number of radial, latitudinal, and longitudinal grid
points. The SGS model employed for viscous, thermal, and magnetic diffusion (if applicable) is shown as either the enhanced
eddy (EE) model or the dynamic Smagorisnky (DS) model. Thermal diffusion on the spherically-symmetric entropy profile iseither equal to thermal diffusion on all other modes or strongly tapered near the upper boundary. The number of time-steps
taken is given by Nt . Cases are either magnetic or purely hydrodynamic. Boundary conditions are either closed (C) or imposed
plumes (P) (see Chapter 9). All simulations here use stress-free (SF) boundaries. For closed boundaries one can also select ifthey impose values of the entropy (S) or the entropy gradient (DS). Magnetic boundary conditions are either perfect conductors(PC) or match to an external potential field (PF). The method of taking radial derivatives ∂/∂r is either using a projection onto
Chebyshev polynomials (Cheby) or finite differences (FD).
Computational Parameters for ASH Simulations
Case
Na
Nb
S

Nr , Nθ , Nφ
SGS Model
192, 512, 1024
EE
257, 1024, 2048
EE
192, 512, 1024DS

κ00 = κ?
Nt
No
1.2 × 106No
4.3 × 105
No
7.5 × 105

D3
D3-pm1D3-pm2D3a
D3b
S3

97, 256, 512
145, 256, 512
145, 512, 1024
97, 256, 512
145, 512, 1024
145, 512, 1024

EE
EE
EE
EE
EE
DS

No
No
No
No
No
No

Ia
Ib
Ic
P

300, 512, 256
500, 1024, 512
500, 1024, 512
500, 1024, 512

EE
EE
EE
EE

Yes
Yes
Yes
Yes

Magnetic?
No
No
No

Lower B. C.’s
C, SF, DS
C, SF, DS
C, SF, DS

Upper B. C.’s
C, SF, DS
C, SF, DS
C, SF, DS

∂/∂r
Cheby
Cheby
Cheby

5.8 × 106
5.7 × 106
4.8 × 106
1.2 × 107
9.8 × 106
3.4 × 106

Yes
Yes
Yes
Yes
Yes
Yes

C, SF, DS, PC
C, SF, DS, PC
C, SF, DS, PC
C, SF, DS, PC
C, SF, DS, PC
C, SF, DS, PC

C, SF, DS, PF
C, SF, DS, PF
C, SF, DS, PF
C, SF, DS, PF
C, SF, DS, PF
C, SF, DS, PF

Cheby
Cheby
Cheby
Cheby
Cheby
Cheby

6.5 × 105
9.3 × 105
8.6 × 105
2.7 × 105

No
No
No
No

C, SF, DS
C, SF, DS
C, SF, DS
C, SF, DS

C, SF, S
C, SF, S
C, SF, S
P

FD
FD
FD
FD

38

Table 2.2: Dimensional parameters for the ASH simulations. Shown are the location of the inner Ri and outer Ro radialboundaries in units of the solar radius R = 6.95 × 1010 cm, the fractional density contrast over the domain ρi /ρo , the RMS
velocity vRMS in m s−1 and RMS magnetic field strength BRMS in kG at mid-convection zone, the bulk rotation rate Ω0 in unitsof the mean solar rotation rate Ω = 2.7 × 10−6 s−1 (thus the period P = 2π/Ω = 26.9 days), the viscous ν, thermal κ, andmagnetic η diffusion coefficients in units of cm2 s−1 , the mean time step ∆t in seconds, and the total evolution time TE in days foreach simulation. For cases S and S3 using the dynamic Smagorinsky SGS model, the values quoted are based on the time-averaged
rms viscosity, conductivity, and resistivity at mid-depth, noting that these diffusion coefficients have near hundred-fold spatial
variations.
Dimensional Parameters and Diagnostics for ASH Simulations
Case
Units
Na
Nb
S

Ri /R
–
0.719
0.719
0.719

Ro /R
–
0.967
0.967
0.967

ρi /ρo
–
24.3
24.3
24.3

vRMS
m s−1
54.4
57.2
55.2

BRMS
kG
–
–
–

Ω0 /Ω
–
3.0
3.0
3.0

ν
cm s−25.15 × 1011
3.43 × 1011
1.49 × 1010

κ
cm s−22.06 × 1011
1.37 × 1011
5.98 × 1010

η
cm s−2
–
–
–

¯
∆t
s
340
142
319

TE
days
4820
1120
2760

D3
D3-pm1D3-pm2
D3a
D3b
S3

0.719
0.719
0.719
0.719
0.719
0.719

0.967
0.967
0.967
0.967
0.967
0.967

24.3
24.3
24.3
24.3
24.3
24.3

51.6
50.1
50.5
53.8
55.9
56.4

3.24
5.20
9.39
6.13
5.79
5.31

3.0
3.0
3.0
3.0
3.0
3.0

1.44 × 1012
1.44 × 1012
1.44 × 1012
9.40 × 1011
5.67 × 1011
2.18 × 1010

5.77 × 1012
5.77 × 1012
5.77 × 1012
3.76 × 1012
2.27 × 1012
8.71 × 1010

2.89 × 1012
1.44 × 1012
7.22 × 1011
1.88 × 1012
1.13 × 1012
4.35 × 1010

337
104
89.9
175
54.5
40.0

22500
6860
4980
24500
6180
1520

Ia
Ib
Ic
P

0.719
0.719
0.719
0.719

0.983
0.983
0.983
0.983

149
149
149
149

41.8
70.9
74.5
101

–
–
–
–

1.0
1.0
1.0
1.0

2.41 × 1012
1.20 × 1012
6.02 × 1011
6.02 × 1011

9.64 × 1012
4.81 × 1012
2.41 × 1012
2.41 × 1012

–
–
–
–

998
628
280
171

7510
6800
1960
830

39

Table 2.3: Dimensionless parameters for the ASH simulations. Evaluated at mid-depth are the rms Reynolds number Re =0vrms L/ν and fluctuating Reynolds number Re0 = vrms
L/ν, the Rayleigh number Ra = (−∂ρ/∂S)(dS̄/dr)gL4 /ρνκ, the Taylor
number Ta = 4Ω20 L4 /ν 2 , the Rossby number Ro = ω/2Ω0 , and the convective Rossby number Roc = (Ra/Ta Pr)1/2 , and the
Prandtl number Pr = ν/κ. For MHD simulations the magnetic Reynolds number Rm = vrms L/η and fluctuating magnetic
0L/η, and magnetic Prandtl number Pm = ν/η are also given. Here the fluctuating velocity v 0 has
Reynolds number Rm0 = vrms
the axisymmetric component removed: v 0 = v − hvi, with angle brackets denoting an average in longitude. For cases S and S3
using the dynamic Smagorinsky SGS model, the values quoted are based on the time-averaged rms values of Re, Re0 , Ra, Ta,
Rm and Rm0 at mid-depth, noting that these parameters depend on diffusion coefficients which have near hundred-fold spatial
variations (see Table 2.2).
Dimensionless Parameters for ASH Simulations
Case
Na
Nb
S

Re
Re0
1080 337
1650 517
9210 2870

Ra
2.13 × 106
4.02 × 106
2.76 × 108

Ta
8.03 × 107
1.81 × 108
1.28 × 1011

Ro
Roc
Pr
0.725 0.307 0.25
0.859 0.287 0.25
0.735 0.0293 0.25

Rm
–
–
–

Rm0
–
–
–

Pm
–
–
–

D3
D3-pm1D3-pm2
D3a
D3b
S3

173
149
145
244
343
8050

104
102
101
154
273
5750

3.28 × 105
2.98 × 105
3.08 × 105
5.84 × 105
1.11 × 106
7.68 × 108

1.22 × 107
1.22 × 107
1.22 × 107
2.41 × 107
6.08 × 107
4.46 × 1010

0.374
0.372
0.370
0.447
0.566
0.581

0.315
0.300
0.306
0.295
0.257
0.262

0.25 860.25 149
0.25 291
0.25 122
0.25 1710.25 4030

52
102
202
77
136
2880

0.5
1.0
2.0
0.5
0.5
0.5

Ia
Ib
Ic
P

94
215
458
328

52
177
418
253

6.20 × 104
1.50 × 105
4.97 × 105
1.60 × 105

5.66 × 105
2.27 × 106
9.07 × 106
2.27 × 106

1.34
1.88
2.09
2.07

0.462
0.515
0.568
0.532

0.25
0.25
0.25
0.25

–
–
–
–

–
–
–

–
–
–
–

40

41
The final set of simulations are four hydrodynamic models designed to explore theeffects of an alternate upper boundary condition in ASH which is designed to more faithfully represent the dynamics present in the upper portion of the solar interior. This plumeboundary condition is a novel addition to the ASH code which is described in Chapter 9.
Cases Ia, Ib, and Ic use the standard impenetrable upper boundary condition, but extend to
0.983R to include as much of the convective layer as possible. Case Ia has moderately highvalues of ν and κ and has a solar-like differential rotation profile. It is modeled after caseAB2 from Brun & Toomre (2002). Both diffusion coefficients are reduced by a factor of two
in case Ib, which has a weak solar-like differential rotation profile. Case Ic has diffusivities
that are again reduced by a factor of two and produces an anti-solar differential rotation
profile. Case P uses our plume upper boundary condition with ν and κ equal to those ofcase Ib and retains a strong solar-like differential rotation profile. These simulations wererun on the Dell PowerEdge Cluster named Stampede at the Texas Advanced Computing
Center and the SGI ICE Cluster named Pleiades at NASA’s Ames Research Center.
Tables 2.1, 2.2, and 2.3 present, respectively, the computational, dimensional, and dimensionless parameters for all the ASH simulations which will be referenced in this thesis.
Though many of these parameters will be restated later in our analysis of the various simulations, this presentation is intended to provide an easy and comprehensive reference to thereader as well as an overview.

2.4.1

Comparison with Other Dynamo Simulations

As was previously discussed, there are several other codes which are used to conduct
similar simulations to those that we discuss here. It is generally quite challenging to makedirect comparisons between results from ASH simulations and those using other codes. Manyother codes use implicit LES formulations, which generally means that they do not haveaccess to parameters related to viscous, thermal, or magnetic diffusion. Additionally no two42
Table 2.4: Dimensional parameters from selected simulations conducted with the Pencil
code. These values are adapted from the non-dimensional parameters given in Käpylä et al.
(2011), Warnecke et al. (2012), and Käpylä et al. (2013) (abbreviated K11, W12, and K13).
Dimensional parameters include the RMS velocity in units of m/s, the viscosity in units of
cm2 s−1 , the magnetic diffusivity in units of cm2 s−1 , the rotation rate, and the luminosity,
both in solar units.
Dimensional Parameters for Selected Pencil Code Simulations
Reference
K11
K11
K11
K11
K11
K11
K11
K11
K11
K11
K11
K11
K11
K11

Run vRMS (m s−1 )
A0
10100
A1
9610
A2
9610
A3
9610
A4
12700
A5
9610
A6
7870
B0
8740
B1
8740
B2
7870
B3
6990
C1
3500
D1
5240
D2
5240

ν (cm s−2 )8.78 × 1013
8.87 × 1013
8.87 × 1013
8.63 × 1013
8.77 × 1013
8.87 × 1013
8.71 × 1013
5.37 × 1013
5.09 × 1013
5.22 × 1013
5.28 × 1013
9.67 × 1013
1.93 × 1013
1.96 × 1013

κ (cm s−2 )
–
–
–
–
–
–
–
–
–
–
–
–
–
–

Ω0 /Ω
0.0
1.1
2.1
4.3
10.6
21.9
42.5
0.0
10.4
21.4
43.0
23.5
23.6
35.8

L/L
2.9 × 106
2.9 × 106
2.9 × 106
2.9 × 106
2.9 × 106
2.9 × 106
2.9 × 106
2.9 × 106
2.9 × 106
2.9 × 106
2.9 × 106
1.56 × 106
6.46 × 105
6.46 × 105

W12
W12
W12

A5
A5a
Ar1

3150
4590
1750

3.17 × 1014
1.52 × 1013
1.53 × 1013

3.17 × 1013
1.52 × 1013
1.53 × 1013

19.6
18.3
77.6

1.81 × 107
1.81 × 107
1.81 × 107

K13
K13
K13
K13
K13
K13
K13
K13
K13
K13

A1
A2
B1
B2
C1
C2
D1
D2
E3
E4

404
373
584
531
929
823
473
572
929
897

5.21 × 1013
5.21 × 1013
8.83 × 1013
8.83 × 1013
8.83 × 1013
8.83 × 1013
1.42 × 1014
7.59 × 1013
8.82 × 1013
1.06 × 1014

5.21 × 1013
5.21 × 1013
8.83 × 1013
8.83 × 1013
8.83 × 1013
8.83 × 1013
4.73 × 1013
3.80 × 1013
8.82 × 1013
1.06 × 1014

3.6
3.3
5.2
4.7
8.2
7.3
4.2
5.1
8.2
8.0

7.92 × 105
7.92 × 105
7.92 × 105
7.92 × 105
7.92 × 105
7.92 × 105
1.31 × 105
1.31 × 105
7.92 × 105
7.92 × 105

codes solve exactly the same numerical evolution equations. One notable example can befound when comparing the energy equations solved by ASH and EULAG-MHD. EULAG-43
MHD includes a thermal drag term which keeps the thermal state from moving away fromthe specified background stratification (Prusa et al., 2008; Ghizaru et al., 2010). This term
is small compared to other terms in the EULAG-MHD energy equations. However upon
careful examination this term plays a crucial role in allowing EULAG-MHD simulations at
the solar rotation rate to retain a solar-like differential rotation (fast equator, slower poles)
and achieve cycles of magnetic activity, whereas ASH simulations at the solar rotation rate
which are sufficiently turbulent to achieve magnetic cycles have anti-solar differential rotation
profiles. This is one example of the variations between codes that make direct comparison
difficult.
In contrast, Jones et al. (2011) conducted a detailed comparison of ASH with several
codes designed for planetary dynamo simulations. Included in this analysis was the MaGIC
code, which has also been used in low-mass stellar models (Gastine et al., 2012). They
found that when carefully formulated, benchmark problems can be used to test differencesin numerical methods and computational implementation. Unfortunately, similar work has
not been done to compare ASH, EULAG-MHD, and the Pencil code, in part because thesethree codes emily fundamentally different computational approaches.
We have undertaken to compare ASH results with those from the Pencil code, specifically the simulations presented by Käpylä et al. (2011), Warnecke et al. (2012), and Käpylä
et al. (2013). The Pencil code is especially favorable for comparing with ASH as both codes
generally use explicit diffusion which is only a function of depth. The Pencil code is nondimensional while ASH uses dimensional units. Thus to compare the two codes we havecomputed dimensional units for the Pencil code simulations with the help of Jörn Warnecke,
which are given in Table 2.4. Simulations using the Pencil code have achieved convective
dynamo simulations with strong solar-like differential rotation, the ejection of magnetic structures generated by dynamo action, and equatorward propagation of magnetic features. It
is thus very useful to understand the similarities and differences between those simulations
and others using ASH.

44
Dimensional parameters from ASH simulations are given in Table 2.2 and similar parameters for the Pencil code simulations are shown in Table 2.4. Upon comparison it becomes
immediately clear that although some parameters such as the diffusion coefficients are very
similar, the luminosities and RMS velocities are much larger in Pencil code models than thevalues in our ASH simulations. The Pencil code simulations also explore a much broader
range of rotation rates than the ASH simulations shown here. The widest range of rotation
rates explored for solar-like stars in ASH was done by Brown et al. (2008) who consideredrotation rates between one and ten times the solar value. The ability to achieve qualitatively
similar physical mechanism in such widely disparate regions of parameter space may indicatethat some aspects of convective dynamo action are robust over a broad range of physicalsettings. It may equally inspire caution as one might interpret these results to mean that
one can find regions of parameter space which produce qualitatively similar results for very
different physical reasons. We will discuss the possibility for further comparisons between
the results of ASH, Pencil, and EULAG-MHD models in Chapter 10.

Chapter 3
Enhanced Subgrid-Scale Models for Stellar Convection

As discussed in Chapter 2, the ASH code follows an explicit LES framework in which
the largest scales of the solar interior are simulated and the effects of scales below our numerical resolution must be parameterized using a SGS model. In this chapter we will present
two SGS models, which we term the Enhanced Eddy and Dynamic Smagorinsky models.
The Enhanced Eddy (EE) model has been used in ASH extensively (e.g., Brun & Toomre,
2002; Brun et al., 2004; Miesch et al., 2006; Brown et al., 2010; Brun et al., 2011), while theDynamic Smagorinsky (DS) model was only recently implemented in the ASH code. This
chapter outlines an analysis designed to asses the validity and effectiveness of the DS model.
Sections of this chapter follow the text of my Master’s Thesis, submitted to the University ofColorado in 2009, but with significant revisions and additions. I was the sole author of thatthesis. The DS SGS model was originally implemented in ASH by Mark Miesch. I performed
significant changes to this implementation to make it numerically stable. I conducted andanalyzed all of the simulations presented here.

3.1

The Large-Eddy Simulation Framework
We seek to use computational tools to explore the nature of large-scale solar and stellar

convective and dynamo processes. Turbulence likely plays a key role in both the convection
and the generation of magnetic fields. It links the scales of interest with scales that are46
below the resolution limits of current numerical simulations. In this work we seek to use
subgrid-scale (SGS) models which parameterize the effects of this turbulent cascade in orderto improve our global models of solar and stellar convection.
Convection in the solar interior is a highly nonlinear process that involves a wide rangeof scales in both length and time. As an example, the solar convection zone spans roughly 200
Mm in radius while the length scale for diffusion of momentum, or Kolmogorov scale, may
be on the order of meters – a separation in scales of a factor of 108 . A brute-force approachthat attempted to resolve all scales of motion in the solar convection zone would require far
more powerful computers than are envisioned in the foreseeable future. The problem is madetractable by the use of SGS models which parameterize the effects of small-scale motions.
Simulations can then capture large-scale features at reasonable computational costs. The
use of SGS models has allowed remarkable advances towards creating large-scale models
consistent with solar and stellar observations, such as simulating solar differential rotation(Brun & Toomre, 2002; Miesch et al., 2006) or cycles of magnetic activity and reversals ofmagnetic polarity (Brown et al., 2011). By improviFng our SGS treatment, we hope toachieve even more realistic solar and stellar models.
In all ASH cases, finite computational resources prohibit the use of molecular values
for the diffusion of momentum, heat, and magnetic fields. This requires a SGS scheme which
replaces molecular viscosity, thermal conductivity, and magnetic resistivity with turbulent
values designed to represent unresolved small-scale mixing. Previous ASH simulations haveused an Enhanced Eddy (EE) SGS model that uses a diffusion term of the same form asthat used in the Navier-Stokes equation, but which has a kinematic viscosity that is manyorders of magnitude larger than molecular values. Additionally, this viscosity is scaled by
the background density (in this work ν ∝ ρ̄−1/2 ), and thus is constant on horizontal surfaces.
This EE viscosity model is computationally convenient and retains the same mathematicalform as a direct numerical simulation (DNS) of the solar interior. It has produced important
insights into convection, differential rotation, and dynamo action in the sun and other stars.

47
However this SGS prescription becomes prohibitively expensive for very large Reynolds numbers where the simulation cost scales roughly as Re4 . This scaling assumes that doubling
the Reynolds number requires doubling the number of grid points in each dimension andsuffering a corresponding halving of the time step due to CFL constraints. More complex
SGS models hopefully represent small-scale effects more faithfully, allowing less diffusively
influenced dynamics to be achieved without increasing resolution, thus reducing computational cost. In this chapter, we will explore the effects of an additional SGS turbulence
parameterization, the Dynamic Smagorinsky model (hereafter the DS model) developed byGermano et al. (1991), and compare it with simulations using the EE diffusion model.

3.2

SGS Models
In the LES framework the resolved scales are formally expressed as a convolution of

the true velocity and thermodynamic fields with some filter function which removes theunresolved scales. In practical terms this filter is usually the discrete grid on which thenumerical equations are solved. We consider the resolved velocity field ûi to be the truevelocity field ui convolved with a grid-scale filter G, which we write as a hat for convenience.
We can write the evolution equation for ~u as


∂ ~uˆ
= − ~uˆ · ∇ ~uˆ − ∇ · (udi uj − ûi ûj ) + ...
∂t

(3.1)

where we have omitted the additional terms which include the effects of density stratification, viscosity, gravity, and magnetism for brevity. The full set of terms can be found in
Equation 2.9. Clearly, the evolution of the resolved velocity field depends on not only the
resolved scales but on correlations in the unresolved scales as well. To close the equation wemust writeud
i uj − ûi ûj ≈ T ,

(3.2)

48
where T is the unresolved stress tensor. This approximation becomes exact if one can a findthe correct form of the stress tensor on the grid scale, which is not possible in most, if notall, physical systems. In computational settings T must be calculable. Thus the components
of T can be constants or functions of position, the background thermal state, or the resolved
3D thermal and velocity fields. Choosing to make T a function of the evolutionary variables
introduces additional nonlinearities into the mathematical system, which may or may not
be undesirable for a given numerical scheme. This term is given by the divergence of asecond-rank tensor, which is the same form as the viscous diffusion term in the momentum
equation. If small-scale motions are purely dissipative, it then makes sense for the SGS model
to take the general form of a diffusive operator and set T = D. This is of course a significant
assumption and there are likely non-dissipative effects from small-scale turbulence, but theleading effect is likely to be diffusive in nature.
Previous ASH simulations using the EE viscosity model have used a stress tensor of
the form

D = Nij = −2ρ̄νEE
1eij − (∇ · ~u) δij ,
3

(3.3)

where D is the viscous stress tensor found in Equation 2.12 and νEE is a function of radius
only (for a derivation see Kundu & Cohen, 2004). Since molecular values of ν and κ are beyond current computational capabilities, much larger values must be used for computational
stability. In practice, values of ν and κ are chosen to be as small as possible for a given levelof resolution. In this work those coefficients are on the order of a few times 1012 cm2 s−1(see Table 2.2).
The EE viscosity model has several drawbacks. First, the Sun operates at very large
Reynolds numbers, possibly as high as 1014 . With the EE diffusion model the computational
cost of a simulation scales roughly as Re4 . This makes continued progress towards the
Reynolds number of the solar convection zone slow at best. Additionally many of the smallest
scales in models using the EE SGS scheme are dominated by diffusive effects rather than truly

49
resolving convective dynamics, causing a simulation’s effective resolution to be less than the
total computational resolution. In other words, one must pay for scales that one knows are
not reproducing physical behavior. More advanced SGS treatments can provide much lower
levels of diffusion at lower computational cost. The DS model uses the scale-invariance of theinertial range in a turbulent cascade to encode effects from unresolved scales. It attempts tolimit diffusion on resolved scales, increasing the effective resolution of the simulation.
A second motivation for more advanced SGS treatments is that the viscosity in theEE model is required to be large across the entire domain. If the eddy viscosity represents
small scale mixing one might imagine that it should be lower in regions with small gradientsin flows or thermal fields. The EE viscosity model uses a uniform viscosity across horizontal
surfaces that is set to be as small as computationally allowed. The viscosity is allowed tovary with radius, but the radial contrast is generally quite small (here νtop /νbot ≈ 5) andis input into the model as a parameter. Thus the viscosity over the entire domain is set by
the minimum viscosity allowable at any grid point. While some small number of grid points
demand high viscosities for computational stability, the majority of the domain could run
at much lower values of ν. The EE model sets the viscosity for the entire domain based on
its worst handful of grid points. The DS model allows for lower levels of viscosity in regionsthat computationally permit lower viscosities in a self-consistent fashion.
There exist a number of SGS models beyond the DS model referenced here. Someof the most used SGS models that will not be covered here are similarity models (Bardina
et al., 1980) and the Lagrangian-averaged Navier Stokes (LANS) alpha model (Hecht et al.,2008). We hope to extend our analysis to include these models at a later date. For a review
of various SGS models, see Meneveau & Katz (2000).
Additionally a number of numerical diffusion models which are not designed to model
sub-grid scale behavior but rather are numerical techniques to reduce diffusion or limit it toa very small range of scales exist. These models include hyperviscosity (e.g., Stein & Nordlund, 2006; Trampedach & Stein, 2011) and slope-limited diffusion models (e.g., Cheung

50
et al., 2010; Augustson et al., 2011). These numerical diffusion models proceed essentiallyalong the same philosophical direction as Implicit LES codes. Though we will not explore
them here, we do anticipate doing so in the near future.

3.3

The Dynamic Smagorinsky Model
The EE viscosity model uses a stress tensor (see Equation 3.3) of the form expected

in a DNS simulation. In a true DNS, molecular values of the diffusion coefficients would befunctions of the thermodynamic state and thus would essentially be functions of radius withvery small variations due to perturbations of the thermodynamic state caused by convection.
However the effective diffusion due to the scales of motion not resolved by our simulations
should be orders of magnitude greater than mixing due to atomic diffusion. If the eddy
viscosity represents unresolved small-scale mixing, one might expect it to depend on thecharacteristics of the resolved flow. The Smagorinksy model defines a stress tensor Sij of thesame form as Dij except replacing ν(r) with a spatially varying νs (r, θ, φ, t), resulting in
D = Sij = −2ρ̄νs
1eij − (∇ · ~u) δij .
3

(3.4)

√
In the original model put forward by Smagorinsky (1963), νs = Cs ∆2 eij eij , where ∆ isthe local grid spacing and Cs is a constant of order unity that depends on the geometry,
Reynolds number, boundary conditions, and other simulation parameters. For anisotropic
grids like those used in ASH simulations, ∆ is defined as the geometric mean of the grid
spacings in each direction ∆ = (∆r ∆θ ∆φ )1/3 . By defining νs as a function of the local shear,a strong nonlinearity is introduced in what was previously a linear term. For an extensive
discussion of this model refer to Pope (2000). In this work, we also use a dynamic thermal
diffusivity κs that is defined, using an eddy Prandtl number Pre , to be κs = νs /Pre . As in
the cases using the EE model, Pre = 1/4 and is constant over the computational domain.

51
3.3.1

The Dynamic Procedure

A major limitation to the usefulness of the Smagorinsky model is the constant Cs whichmust be hand-tuned for each simulation. Using scale-invariance arguments Germano et al.
(1991) have formulated a dynamic procedure to calculate Cs from resolved components ofthe flow. The dynamic procedure is not a SGS model but rather a way to use the resolved
flow and an assumption of scale-invariance to set any undetermined coefficients in a given
SGS model.
The resolved flow fields are spectrally filtered at some test filter scale β∆. Traditionallyand in this work β = 2, which is assumed to be in the inertial range of the turbulent spectra.
Only high resolution ASH simulations with a maximum spherical harmonic degree of 340
or higher were able to assure that β∆ (`β = `max /β & 170) was suitably within an inertial
range. Lower resolution simulations produced essentially noise-dominated values of Cs andwere computationally unstable. This requirement for high spatial resolution indicates that
this dynamic procedure would have been unfeasible as little as a decade ago.
We define the stress tensor at the grid-scale ∆ asŜij = udˆji uj − ûi u(3.5)

with hats denoting the grid-scale filtering. If we have this term exactly then our closure approximation is no longer an approximation. It encodes the difference between the unresolved
velocity correlations (which we do not simulate) filtered on the grid scale, and the resolved
velocity correlations. The stress tensor at the test filter scale β∆ is defined as
˜ = ug
˜ ˜ˆj
Ŝ
d
ij
i uj − ûi u(3.6)

with tildes denoting test-scale filtering. This encodes the difference between the stresses at
the test scale the true correlations filtered on the test scale. Finally, the resolved stresses
can be written as
Rij = ûgˆj − û˜i u˜ˆj .
iu

(3.7)

52

Figure 3.1: Schematic diagram of correlations between two components of a fictitious
velocity field on the test scale β∆, the grid scale ∆, and the true scale set by atomic
dissipation in the physical system. Circles represent discrete cells in a numerical simulation
while the velocity field in the physical system is continuous.

The resolved stresses are simply the difference between the correlations in the test scale
velocities and the correlations in the grid scale velocities filtered at the test scale. Note
˜ require an SGS
that Rij can be computed directly from the resolved flows while Ŝij and Ŝ
ij
model, such as the Smagorinsky model, to represent the unresolved correlations.
Figure 3.1 illustrates a sample velocity field on the three scales of interest – the test
scale β∆, the grid scale ∆, and the true scale of the physical system, which is of course
not a single scale but a range of scales which is limited only by atomic diffusion. In a
numerical code we must compute filtered values of the field at discrete grid points, which
are represented by the circles, which each contain values for two vector components. The
nonlinearity of the correlation procedure means that it does not commute with the filtering
procedure, hence taking the correlations at the grid scale and filtering them is not equal tofiltering the velocity fields and then taking the correlations.
By assuming scale-invariance in the inertial range of the turbulent spectra, one canstate that˜ − Ŝ .
Rij = Ŝ
ij
ij

(3.8)

This simple statement hides a profound assumption about the nature of turbulence. We

53
are using the resolved stresses which encode the difference between the test and grid scales
to determine the difference between the true correlations in the velocity field filtered at thetest and grid scales with the correlations in the filtered velocity field at the test and gridscales. Essentially, this is demanding that the behavior between the test and grid scales be
replicated when comparing the differences between the test and true scales with the grid andtrue scales. Thus Equation 3.8 is stating that the behavior between the test and grid scales
is representative of the behavior all the way down to the scales of atomic dissipation. This
constitutes the dynamic procedure. Other SGS models with undetermined coefficients canalso be put through this dynamic procedure by invoking scale-invariance.
3.3.2

Implementation of the DS Model in ASH

Now that we have defined the dynamic procedure, we must apply it to the Smagorinsky
SGS model. This constitutes an additional assumption, namely that unresolved correlations
on both the test scale and grid scale can be represented by the same parameterization. Thisagain requires that the test scale be well beyond any of the convectively-driven scales in thesimulation. Applying the Smagorinsky model and contracting with êij to obtain a scalar
equation results in
Rij êij = −2Cs β 2 ∆2 ˆẽij ˆ
ẽij êij − ∆2 |êij | êij êij .

(3.9)

Rij êij.
Cs = − 2 β 2 ∆2 ˆẽij ˆ
ẽij êij − ∆2 |êij | êij êij

(3.10)

Solving for Cs gives

To prevent the denominator from becoming zero it is generally averaged in some way. In
this work a spectral filter is applied that removes high wavenumber components in both thenumerator and denominator. Additionally a nearest-neighbor averaging is applied to Cs inphysical space to prevent grid-scale ringing in the Cs field. Note that the strain-rate tensor
and therefore νs are not subject to these averaging and smoothing procedures. Equations
3.4, 3.7, and 3.10 constitute the DS model.

54

Figure 3.2: Radial profile of νS in case S. Kinematic viscosities in case S showing thehorizontal and temporal mean value of νs as a function of radius (dashed red line) with ±1σvariations shown in the orange shaded region, along with the EE viscosity (solid blue line)
at the top and bottom of the domain for stability, and the viscosity ceiling (solid green line)
to prevent large viscosities near the top and bottom of the domain from limiting the time
step. The dynamic procedure attempts to produce the required viscosity at the boundaries,
but is not successful in stabilizing the simulation without additional EE viscosity.

One of the known problems with the DS model is its inability to mimic behaviors nearboundaries. As a turbulent flow nears a wall it becomes laminar in the viscous boundary
layer and anisotropic due to the introduction of a strongly preferred direction perpendicular
to the wall. Both of these violate the assumption of scale-invariance, which occurs in isotropic
turbulence, upon which the dynamic procedure is predicated. In our implementation of the
DS model we also find that viscous boundary layers are not correctly formed at the top
and bottom of the domain. This results in convective plumes hitting the upper and lower
boundaries at high speeds, causing excessive ringing. To resolve these problems, we include
a smoothly-varying EE viscosity that is significant over a distance of 0.02 R at the bottom
boundary and 0.01 R near the top of the simulation.

55
In the DS model the nonlinear nature of the diffusion term requires an explicit time
stepping scheme which imposes an upper limit on the size of our time-step. In order to control
the time-step, an artificial ceiling is placed on the DS viscosity. In case S, on average, theceiling is applied to 700 out of 76 million grid points at each time step. The functional formof the ceiling is given byνmax = tmax ∆2min ,(3.11)

where ∆min is the smallest local grid-spacing in any direction and tmax is the desired size
time step. In case S tmax is set to 400 seconds.
Both the near-boundary EE diffusion buffers and the DS viscosity ceiling effects areillustrated in Figure 3.2, which also clearly shows that the DS viscosity is dominant throughout the bulk of convection zone. Future simulations could add a stable radiative zone at thebottom of the simulation, which would remove the need for the EE viscosity buffer andpossibly the DS viscosity ceiling at the bottom of the domain.
In astrophysics, the non-dynamic version of the Smagorinsky model has been usedin low-resolution (Nr = 32, `max = 31) simulations of stratified, rotating solar convection(Glatzmaier, 1985), similar to those proposed here, and more recently in simulations of
the geodynamo at low to moderate resolution (Buffett, 2003). In both cases, however, thedynamic procedure would likely have failed. The DS model has been used in simulations of
turbulence in the interstellar medium (Chernyshov et al., 2008) and in convective excitationof solar p-modes (Jacoutot et al., 2008).
Figures 3.3(a) shows the values for νs over a spherical shell at 0.84 R

at a single

instant in time. The largest values of νs are concentrated at the edges of downflow lanes
where strong horizontal gradients of vertical velocity are present. Figure 3.3(b) shows thetime-averaged probability distribution function of the DS viscosity. Probability distributionfunctions (PDF) are denoted FC (a), where a is the desired variable and C represents thesimulation. All PDFs in this work are given as the amount of surface area covered on average

56(a) νS(b) FS (νS )Figure 3.3: (a) Kinematic viscosities from the DS SGS model νs at 0.84 R in Mollweide
projection. Values are in units of cm2 s−1 . Colors are on a logarithmic scale, showing smallvalues in dark tones and large values in light tones. High-viscosity regions tend to occur on
the edges of downflows while upflows tend to have very little diffusion. (b) Probability distribution function of kinematic viscosities FS (νs ) from case S averaged over roughly 1000 days.
The distribution peaks at about 8.2 × 109 cm2 s−1 and then follows a roughly exponential
decay towards higher values. Peak values of νS at mid-convection zone are about 1.5 × 1011
cm2 s−1 . For comparison Case Nb has a value of 3.4 × 1011 cm2 s−1 at the same depth.

by the chosen field over a small range of values of the field at mid-convection zone. Surface
area is computed rather than number of pixels to account for the non-uniform grid spacing
over spherical surfaces. As expected the bulk of the domain has relatively low viscosity withthe peak of the distribution at 8.26 × 109 cm2 s−1 and then follows a roughly exponential
decay to a maximum value of about 1.5 × 1011 cm2 s−1 .

3.3.3

Non-dimensional Parameters with the DS Model

The strong spatial and temporal variations in the coefficients of viscosity, thermal conductivity, and magnetic resistivity make it difficult to construct meaningful non-dimensionalparameters involving diffusive processes for comparison with simulations using other SGS
models. In Table 2.3 we used the RMS values for νS , κS , and ηS which were averaged over

57
long times, however other possible methods of estimating parameters exist.
Let us take the Reynolds number as an example. In case S we report a value for the
Reynolds number of 9210 when calculated asq(v 2 )L
Re = q,2(νS )(3.12)

where the overbars denote averages over a spherical shell, which here are taken at midconvection zone, and L is the depth of the convection zone. This includes the differential
rotation; thus we also define a fluctuation or convection Reynolds number with the axisymmetric flows removed, given byq(v − hvi)2 L
0
q
,
Re =
2
(νS )

(3.13)

where brackets denote the longitudinal average. For case S this yields a value of 2870. When
compared with case Nb, which has Re = 1650 and Re0 = 517, and, as we will show, exhibits
comparable or even greater levels of turbulence by other metrics, the values for case S seeminflated.
Alternatively, we can construct another Reynolds number, which we will term the mean
local Reynolds number or Rem , by|v − hvi|
Rem = L.
νS

(3.14)For case S the mean local Reynolds number is 8530, which still seems to over-estimate thelevel of turbulence achieved in the simulation. This is caused by correlations between highvelocities and low levels of diffusion. Those areas with strongest shear (and hence strongest
diffusion) tend to be the downflow lanes where the velocities are highest. In case S regionswith v − hvi > 100 m s−1 have an average local Reynolds number of 11600, while regionswith v − hvi < 100 m s−1 have an average value of 6450.
Finally, we can estimate the Reynolds number by using a scale-analysis on the DS

58
viscosity itself. We can approximate νS using scale arguments as√νs = Cs ∆2 eij eij ≈ Cs ∆v,(3.15)

thus yielding a Smagorinsky Reynolds number ReS as
ReS =

L
.
Cs ∆

(3.16)The Smagorinsky Reynolds number can be thought of as simply the range of scales in thesimulation scaled by the dynamic parameter, which for case S yields a value of 2040.
We have shown four possible methods for computing a Reynolds number for case S
which yield values differing by as much as a factor of four. Clearly a range of possible values
for any non-dimensional parameter involving diffusion coefficients become viable when using
the DS SGS model. This discussion is not aimed to provide a definitive conclusion as towhich method for computing these parameters is best but rather to illustrate the difficulty
in interpreting these parameters, whichever method is chosen.

3.4

Simulation Parameters
We have conducted three simulations that compare the two SGS models. Various

properties of these three simulations are listed in Table 3.1 (see also Tables 2.1, 2.2, and
2.3). Two cases labeled Na and Nb use the EE diffusion SGS model and case S uses the DSmodel. Cases Na and S have a maximum spherical harmonic degree of 341 while case Nb
has a maximum spherical harmonic degree of 682. In the EE diffusion cases the values of ν
were chose to be as small as permissible at the given levels of resolution. Although they are
not DNS, cases Na and Nb will be used as benchmarks to assess the performance of case S.
All three simulations are conducted at a rotation rate of three times the solar value.
Previous studies with ASH have shown that for high levels of turbulence and rotation rates
less that the solar rate, the simulations produce an anti-solar differential rotation profile

59
Table 3.1: Shown are the number of radial, latitudinal, and longitudinal grid points. The
SGS model employed for viscous and thermal diffusion is shown as either the enhanced eddy(EE) model or the dynamic Smagorisnky (DS) model. The total evolution time TE for eachsimulation is given in days. The viscosity ν in units of cm2 s−1 is shown at mid-convection
0L/ν
zone. For case S the RMS value is used. The fluctuating Reynolds number Re0 = vrms
gives some indication of the relative level of turbulence in each model. Again the RMS valuefor νS was used for case S.
Selected Parameters for Cases Na, Nb, and S
Case
Nr , Nθ , Nφ
SGS ModelNa
192, 512, 1024
EE
Nb
257, 1024, 2048
EE
S
192, 512, 1024DS

TE (days) ν (cm s−2 )4820
5.15 × 1011
1120
3.43 × 1011
2760
1.49 × 1010

Re0
337
517
2870

(Brown, 2009). This transition is caused by inertial forces becoming dominant comparedto Coriolis forces, removing the rotational influence which drives the equatorward transportof angular momentum needed to maintain solar-like differential rotation. By using a more
rapid rotation rate, differential rotation retains some aspects of a solar differential rotation
profile, including a fast equator and slow mid-latitudes. Thus while these simulations may
be at a dimensional rotation rate faster than that of the current Sun, their non-dimensional
Rossby number is on the order of 0.5. Without direct measurements of convective giant
cell velocities the exact value of the solar Rossby number is somewhat uncertain, but canbe constrained from observations of differential rotation and meridional circulation to be no
more than about 0.5 (Miesch et al., 2012).
All cases are qualitatively similar to case G3 in Brown et al. (2008), from which case
Na originated. Case G3 was evolved for approximately 4000 days when case Na was spawned
from it. Case Na was then run for roughly 3000 days to a statistically steady state in termsof its volume-averaged energy densities and angular momentum transport. Cases Nb and S
were then created using case Na as initial conditions. Case S was evolved to a statisticallysteady state, while case Nb was run for as long as available resources would allow.

60
SGS models are designed to more faithfully represent turbulent processes at lower
resolution, thus reducing the computational expense. In our discussion of the various SGS
models, it is also important to keep in mind the computational cost associated with eachmodel. Table 3.1 shows the total time over which each case was evolved. Cases Na and S wereable to be run as long as desired to achieve a statistically steady state. For case Na additional
evolution was computed in order to provide very long time averages for computation ofspectra and probability distribution functions. Case Nb is more than an order of magnitude
more expensive than case Na and therefore has not been evolved as long as the other cases.
In addition to considering the effectiveness of the DS SGS model, we must also consider
the additional computational resources required to compute the Smagorinsky viscosity. The
DS model is roughly four times more expensive than case Na while case Nb is roughly twelve
times more expensive than case Na. This means that to be worthwhile case S needs to only
show a third as much improvement in the desired aspect of a simulation over case Na as case
Nb shows over case Na. While it is difficult to define quantitative metrics for this improvement, one might be interested in a simulation very low levels of explicit diffusion. We willshow that case S displays more than a order of magnitude less viscous force per unit volume
than case Na or case Nb without any obvious pathologies in other aspects of the simulation(e.g., differential rotation, convective transport, etc.). As the objectives of simulations using
the DS SGS model will vary, we do not attempt to compute a single metric for simulation
quality. Rather we will describe several possible methods for comparing aspects of cases
Na, Nb, and S. Hopefully this will permit the reader to assess the usefulness of the DS SGS
model for their specific numerical experiment.

3.5

Direct Comparisons of SGS Models
There are two important ways to compare and evaluate SGS models. One can either

examine the form of the diffusive terms added to the equations in each model or use statis-

61
tical measures to compare solutions evolved with each SGS treatment. We will call these
direct and statistical comparisons. Meneveau & Katz (2000) refer these two classes of tests,
respectively, as a priori and a posteriori comparisons. Evolved comparisons involve examining simulations using various SGS models, as we have done with cases Na, Nb, and S, where
the solution has experienced feedback from the SGS model and settled into a statisticallysteady state. The ultimate objective of any SGS treatment is to reproduce the results ofobservations, so it may be argued that evolved comparisons are all that matter. As theseare highly nonlinear systems, altering the diffusive processes will also alter the flow patterns,
energy transport mechanisms, angular momentum transport, and potentially any number ofother features of our simulations. By comparing the statistically steady states into which
each case evolves, we can assess how the effects of a SGS model propagate through thesimulation.
In contrast to evolved comparisons, direct comparisons involve directly examining thechanges to the viscous term in the momentum equation. They can provide valuable insight
into the physical processes that are changed by a SGS model. In implementing a SGS
model one is modifying the underlying physics of diffusion of momentum and entropy insometimes non-intuitive ways. Once the system has begun to feedback upon itself, it becomes
extremely difficult to track the origin of changes in behavior. Thus it is useful to perform
direct comparisons because they isolate the immediate changes to the diffusive terms in ourequations in a way that permits detailed comparisons to be made between the two SGS
treatments.
In this work, we will directly compare the diffusion term in the radial component of
the momentum equation, which is given by(∇ · D)r =


1 ∂ 2
1 ∂
1 ∂Dφr Dθθ + DφφrD+(sin θDθr ) +
−
,
rr
2
r ∂r
r sin θ ∂θr sin θ ∂φr

(3.17)

where the rr, θr, φr, θθ, and φφ components of the stress tensor for both the EE and DS

62models are, respectively,∂ur ur ∂ ln ρ̄
Drr = −2ρ̄ν
+
∂r
3 ∂r


1 ∂ur ∂uθ uθDθr = −ρ̄ν
+
−
r ∂θ
∂r
r


1 ∂ur ∂uφ uφDφr = −ρ̄ν+−r sin θ ∂φ∂rr

1 ∂uθ ur ur ∂ ln ρ̄Dθθ = −2ρ̄ν
+
+
r ∂θ
r
3 ∂r


1 ∂uφ ur cot θuθ ur ∂ ln ρ̄Dφφ = −2ρ̄ν+++.
r sin θ ∂φ
r
r
3 ∂r

(3.18)(3.19)(3.20)(3.21)(3.22)The difference between the EE and DS models comes purely from the definition of ν, whichin the EE model only varies with radius, whereas in the DS model it is a strongly varying
function of all three dimensions and time.
We chose to examine the radial viscous force in order to gain some intuition into themodified physics of the DS model. We could have just as easily looked at the viscous force
in another direction, but the radial component is particularly important as it impacts theprocess of acceleration of the convective plumes and their entrainment of the surrounding
fluid. Entrainment is essential to compressible convective transport, making it an appropriate
diagnostic. To do this we have run three simulations matching the parameters of cases Na,
Nb, and S but starting all three simulations from identical initial conditions taken from case
Na in an evolved state. Since the starting point for each simulation was taken from case Na,
we would expect no ringing in the diffusion term for case Na due to the transition (or lackthereof). If the radial viscous force in case S shows large deviations from the morphology
seen for cases Na or Nb, we may reasonably conclude that it is not appropriately capturing
the diffusive dynamics around convective plumes.
When the form of the equations being solved changes abruptly, some Gibbs ringing is
expected from the transition, which is essentially a small temporal discontinuity. Each casewas evolved for exactly five time steps of exactly 100 seconds each to allow most of the Gibbs
ringing from the transition to dissipate but not enough time for any significant evolution of

63

Figure 3.4: Radial viscous force for the EE and DS SGS models with an identical flow field
Radial diffusion terms (see Equation 3.17) in units of force per unit volume (dyn cm−3 ) for
cases Na, Nb, and S shown in Mollweide projection at 0.84 R for an identical flow field.
The highlighted region in the top panel corresponds to the regions shown in Figures 3.5(a) 3.5(d). In all cases the underlying flow field is virtually identical, allowing direct comparisons
between these images.

64
the velocity fields. The velocity and thermodynamic fields were compared to ensure no significant evolution had occurred. The maximum variation at mid-convection zone between
any two simulations was less than roughly 1 part in 105 . By using identical flows, we canthen examine the direct effects of the SGS model without considering secondary effects or
feedback mechanisms.

3.5.1

Global Properties of Diffusion in SGS Models

Figure 3.4 shows the diffusion term for each of the three cases for the mentioned
identical flows over the entire spherical surface at mid-convection zone. One can see clear
similarities between the cases and identify common structures in each, however the rangesof values achieved in each case is surprisingly different. Case Na and Nb have, as expected,
almost identical morphologies for their viscous force term, since the velocity field ~u is thesame in both as is the form of the diffusion operator. The only difference is the magnitudeof the viscosity, hence we should expect that the point-by-point ratio of the two fields will be
the ratio of their viscosities. In fact if we divide each pixel in case Na by each corresponding
pixel in case Nb, after filtering case Nb in order to have equal resolution, we find that theresult is almost exactly 3/2 = νNa /νNb over the entire domain. This verifies that the theunderlying flows are nearly identical and that our procedure behaves as expected for the EE
model.
The diffusion term in Figure 3.4 for case S also looks qualitatively similar to cases Na
and Nb, however scaling has changed dramatically. If we take a pixel by pixel ratio of thedisplayed shell slices for cases Nb (spectrally filtered to the correct resolution) and S, we findthat on average the diffusion in case Nb is larger by a factor of 86.5. However, in the 2% ofregions in case Nb where the diffusion is strongest that ratio drops to 31.4. As expected, theDS model is focusing the diffusion in areas where it is needed and greatly reducing diffusion
over the majority of the domain.

65(a) vr(b) Radial Viscous Force for Case Na(c) Radial Viscous Force for Case Nb(d) Radial Viscous Force for Case S

Figure 3.5: Local radial viscous force for a convective plume in the EE and DS SGS models.
(a) Radial velocity field and (b) - (d) radial viscous force terms in units of force per unit
volume (dyn cm−3 ) for the EE viscosities used in cases Na and Nb and the DS viscosity fromcase S zoomed in on a region spanning 40 degrees in latitude and 45 degrees in longitude.
Regions correspond to the highlighted region in the top panel of Figure 3.4. All three viscous
force fields are morphologically similar. Downflow plumes are characterized by their strong
positive (upward) cores surrounded by negative (downward) entrainment regions. The DSmodel yields lower amplitudes at all points. Away from the three downflow plumes values
can be as much as a hundred-fold smaller in case S than in case Nb.

66

3.5.2

Local Examples of Diffusion in SGS Models

To gain further physical insight into these SGS models, we have highlighted one small
area from the three images in Figure 3.4 for a more detailed examination. This area, whichis indicated by the outlined region in the top panel, stretches from 20 degrees below theequator to 20 degrees above the equator and covers 45 degrees in longitude. Figures 3.5(a) 3.5(d) show the radial velocity field in this region as well as the diffusion term for all threecases. The region contains three strong downflows each interacting with a broader, slower
upflow. Two of the downflows have strong latitudinal alignment while the downflow on theleft appears to be a more circular event. All three events are also resolved in each of the
diffusion fields. In all three plumes, the diffusion is qualitatively similar for cases Na, Nb,
and S.
Let us now examine the strong downflow plume on the right of Figures 3.5(a) - 3.5(d),indicated by the green box in Figure 3.5(b). This downflow plume is plunging into a relatively
strong upflow and therefore exhibits some of the strongest gradients in the velocity field in
the domain. For cases Na, Nb, and S the viscous structure associated with this downflow
shows a surprising degree of complexity. The viscous force in the core of each structure isstrongly positive, which would lead to an upward acceleration on the core of the downflow,
slowing it down. This braking region is surrounded by a region where the viscous term
becomes negative, producing a downward acceleration on the fluid. This happens at theinterface of the downflow structure with the surrounding upflow as the downflow entrains
fluid from the upflow. In Figures 3.5(b) - 3.5(d), there is evidence for even more complex
internal structure in the plumes, indicating that there may be important dynamics on scales
smaller than the scale of the plumes.
The net effects of diffusion on the downflow plume on the right of Figures 3.5(a) 3.5(d) vary strongly between the two SGS models considered here. For case Na and Nb the

67

Figure 3.6: Radial diffusion terms in units of force per unit volume (dyn cm−3 ) for cases
Na, Nb, and S after significant time evolution, shown in Mollweide projection at 0.84 R .
The top panel is identical to the top panel of Figure 3.4. For each case the flow field is uniqueand has been taken from each case near the end of their computed evolution (see Table 2.1).

68
net forces per unit volume of fluid over the highlighted area are −1.85 × 10−2 dyn cm−3 and−1.28×10−2 dyn cm−3 respectively. This net downward acceleration indicates that while thecore of the plume is decelerating, the plume structure at this depth is entraining new fluid
at a high rate to compensate for the slowing of the core, coming to a rough force balance.
For case S, the net force per unit volume on this plume is 7.35 × 10−3 dyn cm−3 . Here theplume structure is decelerating on average, as would be expected from a fast plume passing
through a viscous medium. The plume is clearly still entraining fluid, but strong upwardviscous forces in the core of the downflow dominate the downward forces on the edges of theplume. For comparison the average buoyant force on this structure is−5.76×10−3 dyn cm−3 .
Clearly changing the amplitude of the viscous force by switching to the DS model will havesignificant impacts on the radial acceleration of this plume. Thus while the morphologicalsimilarities of the viscous force fields are encouraging, the change in the detailed balance
between the core of the plume and the surrounding entrainment region indicates that we can
expect changes to the structure and transport properties of the convection once feedbacks
are allowed.

3.5.3

Comparison with Evolved Diffusion Fields

One of the problems with a direct examination of the diffusion fields is that the flow field
to which each diffusion operator is applied is not a relaxed solution of the evolution equations.
For example, when a DS viscosity is computed using the flow field from a simulation thathas been evolved using an EE model, there are natural questions about validity of that test.
Since the flow fields are also modified as the SGS model is changed, as will be shown in
the following evolved comparison tests, it is useful to compare the diffusion fields computedin the direct comparison to those taken from evolved, relaxed simulations. This will help
provide a means to gauge the usefulness of the direct comparison.
Figure 3.6 shows the diffusion fields for each of the three cases after significant time

69
evolution. A comparison with Figure 3.4 reveals that for each case, the diffusion fields havequalitatively similar properties with cases Nb and S demonstrating more small-scale variability than case Na. The magnitude of the diffusion in each case, however, has changed fromthe direct comparison test. For case Nb the root-mean-squared (RMS) value of the diffusion
field for the direct comparison is 1.76 × 10−3 dyn cm−3 , while the RMS value for the evolved
state is 1.06 × 10−3 dyn cm−3 . This reduction by a factor of two may be explained partiallyby reduced ringing, but modifications in the flow field have probably also reduced gradientsin the flow field. This seems to indicate that for case Nb the decrease in diffusion overcase Na is more than the 50% reduction in the enhaced eddy viscosity ν, possibly reducing
diffusion by as much as a factor of 4. For case S, the RMS value of the diffusion field forthe direct comparison is 4.61 × 10−5 dyn cm−3 , while the RMS value for the evolved state is
1.82 × 10−5 dyn cm−3 . As in case Nb, this reduction by a factor of roughly 2 can be partiallyattributed to ringing, though again the values are close to those from the direct comparison.
For both cases Nb and S we can therefore be somewhat confident in the trends shown by
direct comparisons.

3.6

Evolved Comparisons of SGS Models
The objective of any SGS model is to reproduce observed features of a turbulent sys-tem. Here we will compare various aspects of the flows, thermodynamic fields, and transport
properties of the three cases. This is often referred to as a posteriori testing of a SGS model(see Meneveau & Katz, 2000). Although all cases trace back to the same initial conditions,the chaotic nature of these solutions make it impossible to compare any two cases at a single
point in time after significant time evolution. Instead, we will compare temporal averages oflarge-scale features and statistical descriptions of small-scale behaviors. In the optimal case,the changes in moving from case Na to case S would be of the same form as the changes seenin comparing cases Na and Nb, although possibly of different amplitude We will show that

70

Figure 3.7: Radial velocities, rotation profiles, and meridional circulation streamlines forthe three cases. Left: Radial velocities at 0.95 R seen in Mollweide projection with broad,warm upflows in light tones and narrow, cool downflows in dark tones. Center: Mean rotation
profiles averaged in time and longitude showing some aspects of solar-like differential rotation
with fast equators and slow mid-latitudes. All three cases display a fast south pole, which
may be a transient phenomena that would decay if significant additional time evolution were
undertaken. Right: Mean meridional circulations averaged over time and longitude, where
red shows counter-clockwise flow, blue shows clockwise flow and darker colors indicate faster
flows.

71
while case S succeeds in this metric for some features of case Nb, it does not reproduce allimportant aspects of the higher resolution EE case.

3.6.1

Radial FlowsAll three of the cases exhibit patterns of turbulent, compressible convection characterized by broad upflows and small, fast downflows. Examining the radial velocity plots in thelefthand column of Figure 3.7, all three cases exhibit large, rotationally-aligned convective
structures at the equator, referred to as banana cells, as well as more vortical, smaller-scaleconvective structures at high latitudes. Similar convective patterns have been previously
seen in solar simulations (Miesch et al., 2008) and in case G3 in (Brown et al., 2008). Thereis a qualitative increase in small-scale motions from case Na to case Nb, as expected, whichis evident in both the banana cells at low-latitudes and the smaller-scale cyclonic convection
at high latitudes. Case S looks qualitatively more similar to case Nb than to case Na withenhanced small-scales features in both low- and mid-latitudes.
The mean meridional circulation patterns for all three cases are shown in the right-handcolumn of Figure 3.7. In all three cases the circulations are multi-cellular and highly variablein time. Cases Na and Nb show similar patterns, with a large cell in both hemispheres inside
the cylindrical region tangent to the inner boundary (or tangent cylinder) which fill the
domain in radius. Case S produces very similar behavior to cases Na and Nb.
We can examine the small-scale flow patterns quantitatively using a time-averaged PDF
of the radial velocity field at mid-convection zone. Time averages are taken over about 1000days for cases Na and S and 500 days for case Nb. Figures 3.8(a) shows that case S produces
velocity distributions that generally follow those from cases Na and Nb. If we look at thefractional variation between cases Na and cases Nb and S, we can highlight the differences inthe distributions which otherwise look extremely similar. We define the fraction difference

72(a) F (vr )(b) ∆F (vr )Figure 3.8: Time-averaged probability distribution functions (PDFs) for radial and latitudinal velocities at mid-convection zone for the three cases. (a) PDFs are extremely similarin amplitude and shape. (b) By examining the fractional change relative to case Na we cansee that case Nb has become less peaked (has a larger kurtosis) with a decrease in the coreof the PDF and an increase in the wings. Case S follows the trend established by case Nb
but to a lesser degree. Averages are carried out over roughly 1000 days for cases Na and S
and over roughly 500 days for case Nb. Note that the extreme wings of the PDFs are notshown in order to highlight the core of the distributions.

as∆FC (a) =FC (a) − FNa (a).
FNa (a)(3.23)

Figure 3.8(b) shows the fractional difference for all three cases.
The PDF for case Nb’s radial flows becomes less Gaussian compared to that of caseNa with a decreased core and increased wings. Statistically this is described by the excess
kurtosis of the distribution increasing (becoming more leptokurtic) for the more turbulent
case, as expected from turbulence theory (see Pope, 2000). Excess kurtosis is defined as the
kurtosis relative to a Gaussian distribution. It is formally defined as
R∞ 0(a − ā)4 f (a0 ) da0−∞
kurt{a} = hRi2 − 3,∞2000(a − ā) f (a ) da−∞

(3.24)

where f (a) is the probability distribution function for quantity a. The time-averaged distribution of radial velocity for case Na shows a standard deviation of 54.1 m s−1 and an excess

73(a) v̆r2(b) v̆r2 for 15 < ` < 130

Figure 3.9: Power spectra for radial velocities latitudinal velocities at 0.84 R shown with(a) logarithmic scales over all spherical harmonic modes, and (b) linear scales over modes
dominated by convective driving with 15 < ` < 130. Case Nb shows a slight reduction of
power at near the peak of the spectra with more power at high-`, as would be expected froman increased level of turbulence. Case S generally follows the other two cases, however it has
more power at peak convective scales around ` ≈ 30 than either case Na or Nb. At higher
values of ` case S generally lies between cases Nb and Na, matching onto case Nb for ` > 120.
Note that the ` = 0 component of v̆r is zero to machine precision, since the net mass flux
ρvr into any radius is required by the anelastic approximation to be zero. For convenienceand ease of viewing, the ` = 0 component is not shown. For case Nb the spectra continue
beyond the plotting range out to ` = 682.

kurtosis of 1.36, while the distribution for case Nb has a standard deviation of 57.0 m s−1
and an excess kurtosis of 1.67. This is caused by increased turbulence levels, which drive
more high velocity small-scale motions and increase the probability of extreme events on thetails of these distributions. Case S does an excellent job of broadening FS (vr ) to imitate caseNb, though it does not achieve the same magnitude of fractional difference. For case S thestandard deviation in the distribution is 55.8 m s−1 and the excess kurtosis is 1.52.
Figures 3.9(a) and 3.9(b) show the time-averaged power spectra for the radial flow
field at mid-convection zone. We examine the spherical harmonic spectra at mid-convection
zone after summing over m values and averaging the spectra in time. We denote sphericalharmonic transforms with a breve and therefore define ă (r, `, m, t) as the spherical harmonic

74
transform of field a at a given radius and over a given time interval. The power spectra of
a is given by1ă (r, `) =
∆t
2

where

∗

Z te X`tsă (r, `, m, t) ă∗ (r, `, m, t) dt,(3.25)

m=0

denotes the complex conjugate and ∆t = te − ts . For all spectra in this chapter∆t > 500 days in order to reduce noise and average over many convective overturning times
and rotation periods.
In Figures 3.9(a) and 3.9(b) we can see several features. The spectra of case S showsroughly the same level of agreement with the spectra of both cases Na and Nb. The spectraof case S diverges from the spectra of case Nb for ` > 60, while the spectra from case Na
diverges from case Nb for ` > 50. Case S shows peak values in excess of either case Na or Nb
and then quickly falls, crossing the power in case Nb at about ` = 60 and finally matching
on to case Na for ` > 120. Case S diverges from case Na again at the smallest scales of thesimulation. We can compute the RMS variation in the power spectra for cases Na and S,
and for cases Nb and S at each ` asv
umax
u 1 `X2t∆v˘r (A, B) =`max i=02 v˘r 2A − v˘r 2Bv˘r 2A + v˘r 2B

 !2,(3.26)

where A and B denote the two cases being compared. In doing so we find that ∆v˘r 2 (N a, S) =0.137 and ∆v˘r 2 (N b, S) = 0.522. For comparison ∆v˘r 2 (N a, N b) = 0.619. By this metric case
S more closely replicates case Nb than case Na, though the differences are fairly small.

3.6.2

Azimuthal Flows and Differential Rotation

Previous ASH simulations have explored differential rotation in sun-like stars (Brun &Toomre, 2002; Miesch et al., 2008; Brown et al., 2008). There is little reason to expect that a
SGS model should directly influence the differential rotation of the simulations, but the nonlinear nature of these models means that any change in the system leads to often unexpected75changes elsewhere. Thus it is prudent to establish that a SGS model can achieve solar-like
differential rotation under similar conditions to where such a profile may be achieved with
the EE SGS model.
All three cases were initiated with a solar-like differential rotation profile. Cases Na
began using case G3 of Brown et al. (2008) as initial conditions while both cases Nb and S
used case Na as initial conditions. Case G3 has a solar-like differential rotation profile witha fast equator and monotonically decreasing angular velocity with increase latitude, howevercase Na developed a region of faster rotation at the southern pole before cases Nb and S werecreated from it. The center column of Figure 3.7 shows the mean differential rotation profiles
established by each case with a high-angular velocity region at each south pole. These polar
vortices regularly appear in hydrodynamic models of sun-like stars as well as those of F-typestars. We have found that by adding magnetism these fast poles are quickly spun down,
resulting in monotonic differential rotation profiles as a function of latitude.
Radial and latitudinal shear are important factors in dynamo calculations, so for simulations related to solar and stellar dynamos, the mean differential rotation profile achieved
should mimic the solar differential rotation. All three cases show comparable solar-like differential rotation at the end of their time evolution, with the exception of their south poles.
It is possible that case Nb might experience continued evolution in its rotation profile, as wewere limited by computational expense to only about 1100 total days of evolution. There isno indication that case Nb was undergoing strong changes at the time these averages werecomputed, but we have observed significant changes to the differential rotation of other simulations over timescales of thousands of days. Thus while we feel confident in our analysis,
we cannot preclude the possibility that case Nb has not reached a fully equilibrated state.
We define the mean latitudinal differential rotation contrast by averaging over bothhemispheres to get a measure of latitudinal differential rotation. All cases, irrespective of
the symmetry about the equator at their poles are roughly symmetric about the equator
for latitudes up to about 60◦ . We therefore chose to compute the latitudinal differential

76
rotation between the equator and ±60◦ latitude for a rotation profile averaged over at least
200 days at the end of each simulations’ temporal evolution. By this measure cases Na andNb have latitudinal differential rotation contrasts at the upper boundary of the domain of
28.9% and 28.6% respectively. The radial differential rotation between the top (0.96R ) andbottom (0.72R ) of the domain at the equator for cases Na and Nb are 14.6% and 15.0%
respectively. For comparison, in the sun the latitudinal differential rotation contrast at 0.96
R is roughly 18% and the radial differential rotation contrast at the equator is roughly 7%(Howe, 2009). These values are close to those in a similar ASH model, (case G3) reported
by Brown et al. (2008).
For case S the differential rotation profile has remained very similar to cases Na and
Nb. The latitudinal differential rotation contrast at the top of the domain in case S is 29.0%
and the radial contrast at the equator is 14.7%. Overall the differential rotation is case
S appears to have remained largely unchanged by the use of the DS model. One shouldbear in mind, however, that the diffusion time across the domain has increased by morethan an order of magnitude, so it is possible that if diffusive processes are still importantin modifying the differential rotation of case S, it may not have been simulated for a longenough time to capture the change. However no trend in the differential rotation of the simulation is seen in either a sequence of 50 day time-averages spanning over 1,500 days or thevolume-averaged differential rotation kinetic energy over the same period. This indicatesthat the DS SGS model does not disrupt the transport of angular momentum needed tosustain solar-like differential rotation, either directly or through feedbacks on the properties
which drive convective transport.

3.6.3

Convective Transport

In addition to looking at the flows themselves, we can also examine the balance ofenergy transport in these models. The average radial flux of energy in the anelastic approx-

77
imation can be written as
Lr = LKE + LEN + LVD + LTD + FRD ,(3.27)

where the terms represent to luminosity flux due to kinetic energy, enthalpy, viscous diffusion,
thermal diffusion, and radiative diffusion, respectively. Each term can be written asρ̄ur 2
ur + u2θ + u2φ2LEN = CP ρ̄ur T − T̄

LKE =LVD = (~u · D)r∂ S̄ + S∂r∂ T̄
LRD = −κrad ρ̄CP.
∂r

LTD = −κ00 ρ̄T̄(3.28)(3.29)(3.30)

(3.31)(3.32)The viscous diffusion flux depends on the SGS model chosen and so is left in a general form
here.
Figure 3.10 shows the transport of the solar luminosity through the domain by kinetic
energy flux, enthalpy flux, and the total flux as functions of radius. For cases Na, Nb,
and S we have chosen to use identical values for κ00 and κrad . Additionally, LVD is always
less than 3% of the solar luminosity for all three cases. We have chosen to omit the threediffusive fluxes from Figure 3.10 for visual clarity. The primary balance through the bulk of
the convection zone is between outward enthalpy and inward kinetic energy transport. All
three cases produce similar balances, though case Nb shows a slightly larger inward kinetic
energy flux and case S shows a slightly larger enthalpy flux. This illustrates that on a basic
level the DS SGS model can achieve an average convective transport which settles into an
appropriate balance.
Figures 3.11(a) and 3.11(b) show the time- and shell-averaged root-mean-squared valDqE2
ues of the temperature perturbations
hT iθ,φ and their fractional difference from thoset

in case Na. Time averages are taken over about 500 days for each case. All three cases

78

Figure 3.10: Shell- and time-averaged (over at least 500 days) luminosities as a functionof radius for each case, showing the total luminosity as well as the luminosity from theoutward enthalpy flux and inward kinetic energy flux for each case. All cases achieve anapproximate balance. In all cases, a radiative flux carries a solar luminosity into the bottomof the domain and an unresolved flux representing small-scale convective transport from
granular scales carries a solar luminosity out of the domain. All cases use identical radiative
and unresolved fluxes. Also note that the viscous luminosity flux is not shown, but is small(< 0.03L ) through the bulk of the domain for all cases.

show the same general pattern with roughly the same amplitude of thermal perturbations
at each radial level. When we examine the variations from case Na in Figure 3.11(b) we
see that case S has not changed in same way as case Nb. Case Nb shows slightly larger
perturbations in temperature near the surface and in the lower half of the convective layer
as might be expected from increased levels of turbulence, though there is an odd dip around
0.90R . Comparing with Figure 3.10 we notice that this dip corresponds to the peak in theconvective enthalpy transport. Case S shows smaller values for the entire convection zone
below about 0.93R with a small rise near the top of the domain. This inability to mimiccase Nb is somewhat troubling as it indicates that the DS SGS treatment has altered thethermal structure of the plumes in a way that differs from that of a higher resolution EE
SGS treatment. Although there is a clear morphological difference between cases Nb and S,
both changes are very small relative to the RMS values in case Na. With variations from

79

(a)

Dp

hT 2 iθ,φ

E
t

(b)

Dp
E
hT 2 iθ,φ Relative to Case Nat

Figure 3.11: (a)Time-averaged RMS temperature perturbations as a function of radius forcases Na, Nb, and S. All three cases show similar radial variation with thermal perturbations
decreasing in amplitude with increasing radial location. (b) Fractional variation of the same
temperature perturbations for cases Nb and S relative to case Na, highlighting that case S
has changed in a very different way relative to case Na than case Nb has changed relative tocase Na.

case Na on the order of 1%, these changes may not produce significant modifications to theoverall convective dynamics.
Another spectral measure of convective energy transport can be found by examining theenthalpy flux spectrum. Continuing to use breves to denote spherical harmonic transforms,
we can define the enthalpy flux spectra as#"Z`
cp ρ̄ te X
T̆ (r, `, m, t) v˘r (r, `, m, t) dt ,ε̆ (r, `) = Re∆t ts m=0(3.33)

where Re[x + iy] = x for real x and y. The enthalpy flux spectra shows the scales on which
enthalpy transport from convection is occurring. As the energy flux due to enthalpy LEN is
the product of only two terms which vary on spherical surfaces, the enthalpy flux spectra canbe compute such that it represents the scale-by-scale contribution to the total enthalpy flux
through a given spherical surface. It should also be noted that both the entropy variance andthe enthalpy flux spectra use the entropy and temperature perturbations about the mean,

80(a) ε̆

(b) ε̆C

Figure 3.12: (a) Signed enthalpy flux spectra ε̆ at 0.84R showing the mode-by-mode contribution of the temperature and radial velocity fields to the total enthalpy flux. Convective
scales (roughly 20 < ` < 100) dominate the energy transport. Case Nb shows a lower peakand slower decline than case Na, while case S shows a higher peak value and then asymptotically approaches the value case Na at high `. (b) Cumulative enthalpy flux spectra ε̆C at
the same depth, highlighting that case S does not follow the trend from cases Na to Nb but
rather moves in the opposite direction of higher peak values and overall more total enthalpy
flux.

and so the ` = 0 component of both are zero to machine precision. We can also construct
the cumulative enthalpy flux spectra ε̆C (r, `) such thatε̆C (r, `) =`Xε̆ (r, i) .

(3.34)

i=0This gives the total enthalpy flux at a given radial level for all modes with spherical harmonic
degree of ` or less.
Figure 3.12(a) shows the enthalpy flux spectra for all three cases. The spectrum forcase Nb has a lower peak and slower fall-off at high ` compared to case Na. Case S has
increased enthalpy flux for 20 < ` < 60 compared to both case Na and case Nb. At ` ≈ 65
the spectrum for case S crosses the spectrum of case Nb, and by ` ≈ 100 matches onto thespectrum of case Na. The increase in the energy carried at the peak of the spectra by case
S while case Nb showed a decrease for the same modes indicates that the DS model is not81correctly capturing more turbulent convective transport. On the other hand, the increasein enthalpy transport by case S relative to case Na for the high-` tail of the distribution
is mimicking the behavior of case Nb. The DS model is therefore simultaneously failing tobehave as a more turbulent simulation for the peak of the spectrum while succeeding to doso for the tail of the distribution.
This picture is reinforced by the cumulative enthalpy flux spectra shown in Figure 3.12(b). In this view case Nb has clearly moved to a flatter cumulative distribution
compared to case Na. The increased flux from the moderate ` modes in this view has completely overwhelmed the flattening that should occur at higher `. These are small changes
and are partially explained by the higher total enthalpy flux in case S, but they do showthat the DS model does not provide the same effect as a higher resolution case with the EE
SGS treatment, particularly at moderate scales.

3.7

Discussion
In this work we have numerically investigated the application of SGS models in global-

scale 3D simulations of the convection zone of a sun-like star rotating at three times thesolar rate. We have found that the DS model, while expensive, reproduces the form of the
diffusion term at mid-convection zone of higher resolution simulations with much lower RMS
values of the diffusion term in the radial momentum equation, even compared to the highest
resolution case using the EE SGS model. The DS model also mimics the effects of higher
resolution simulations in turbulent flows while maintaining the overall form of the differential
rotation and meridional circulations. The DS model, however, mimics only some aspects ofthe convective transport in our higher resolution simulation using the EE model. While thegeneral properties of the temperature perturbations and enthalpy flux spectra are similar to
those using the EE SGS model, using the DS model does not drive changes in these quantitiesof the same sense as are seen in higher resolution EE viscosity cases. These deviations are

82
generally small and do not seem to impact large-scale behaviors such the differential rotation
or meridional circulation.
Perhaps it should not be surprising that the DS SGS model applied to anelastic convection performs best in metric which consider the viscous force term and the turbulent
convective velocities. The DS model was designed to produce a turbulent viscosity whichmimics the turbulent diffusion of momentum by unresolved scales. This would produce themost immediate impact on the velocity fields, which seem to yield the best results. We
have additionally applied it to the thermal diffusion in our system by means of a fixed eddy
Prandtl number. It may be more appropriate to construct a separate thermal SGS model.
However if SGS diffusion primarily represents small-scale mixing by turbulent eddies it may
be perfectly acceptable to apply a constant Prandtl number of order unity.
Unfortunately, compressible convective turbulence has been far less studied than incompressible turbulence, so this remains an area of active research. Of particular note is theon-going work in the mesoscale meteorology community where thermal effects due to solar
heating of the Earth’s surface, as well as water condensation and evaporation, play large
roles (see Sullivan & Patton, 2011). We have also not considered magnetic fields here. MHD
turbulence is far less understood than hydrodynamic turbulence. Suitable SGS models for
MHD turbulence are the focus of considerable ongoing study (e.g., Chernyshov et al., 2008,2010).
The DS SGS model shows potential as an effective method for reducing diffusion in
ASH simulations without the computational expense of increased resolution required by theEE model. The DS model does not cause drastic changes to the fundamental properties of
the underlying convection. While expensive, the DS model is less expensive than comparable
simulations using the EE model. Effective resolution can be defined in many ways, but ona rough level we can gauge that the effective resolution increase in moving from case Na tocase S must be less than the increase from case Na to case Nb. Since we know that caseNb has twice the resolution of case Na, we can state that case S must have an effectively

83higher resolution than case Na but not by more than a factor of two. Judging from theradial velocity PDFs, we might estimate that the DS SGS model in ASH provides roughly
a 50% increase in effective resolution while increasing computational expense by a factorof three. This is considerably less than the factor of roughly twelve cost increase required
from simply increasing the resolution of the simulation and using the EE SGS model withsmaller coefficients of viscosity and thermal conductivity. Additionally, the DS model allows
for levels of effective diffusion which are on average smaller than a comparable simulation
by factors of 20 or more. As we will show in Chapters 4, 7, and 8, the ability to achieve
extremely low levels of diffusion plays a key role in the creation of buoyant magnetic loops
which rise through our simulated domain.

Chapter 4
The Discovery of Buoyant Magnetic Loops

We begin our discussion of convective dynamo action in ASH simulations of sun-likestars by describing our discovery of buoyant magnetic loops which are self-consistently andspontaneously generated by convective dynamo action. This chapter will largely focus oncases D3b and S3, discussed in Tables 2.1, 2.2, and 2.3. The processes in case D3b which build
and maintain the magnetic wreaths and which are responsible for the reversals in magnetic
polarity will be discussed further in Chapters 5 and 6. The buoyant magnetic loops fromcase S3 will be explored in further detail in Chapters 7 and 8. This chapter is based on
work published in Nelson et al. (2011b)1 and is largely a restatement of that paper. I was
the primary author of that paper and conducted the simulations presented here. I was also
chiefly responsible for all of the analysis in this chapter. My co-authors provided invaluableadvice and guidance in recommending and developing the parameters of these simulations
and the analysis techniques used on them.
4.1

Overview
The current dynamo paradigm for the Sun and sun-like stars places the generationsite for strong toroidal magnetic structures deep in the solar interior. Sunspots and starspots on sun-like stars are believed to arise when sections of these magnetic structures
become buoyantly unstable and rise from the deep interior to the photosphere. Here we
1

Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S., & Toomre, J. 2011b, Buoyant Magnetic Loops
in a Global Dynamo Simulation of a Young Sun, ApJ, 739, L38

85
present the first 3-D global magnetohydrodynamic (MHD) simulation in which turbulent
convection, stratification, and rotation combine to yield a dynamo that self-consistently
generates buoyant magnetic loops. We simulate stellar convection and dynamo action ina spherical shell with solar stratification, but rotating three times faster than the current
solar rate. Strong wreaths of toroidal magnetic field are realized by dynamo action in theconvection zone. By turning to a dynamic Smagorinsky model for subgrid-scale turbulence,
we here attain considerably reduced diffusion in our simulation. This permits the regions ofstrongest magnetic field in these wreaths to rise toward the top of the convection zone via a
combination of magnetic buoyancy instabilities and advection by convective giant cells. Sucha global simulation yielding buoyant loops represents a significant step forward in combining
numerical models of dynamo action and flux emergence.
4.2

Convection, Rotation and Magnetism
The clearest signature of the global solar dynamo is the emergence of sunspots at thephotosphere. Creating these coherent magnetic structures likely requires several dynamical processes operating at various locations in the solar interior. A single 3-D numericalsimulation of solar magnetism that extends from the deep interior through the Sun’s upper
atmosphere, while resolving all relevant scales, is intractable with current computational
resources. This leads to three main classes of simulations that address elements of solar-likedynamo processes (see reviews Fan, 2009; Charbonneau, 2010). One approach to study how
loops may emerge is to insert a compact magnetic field structure into a spherical domain
and track its buoyant rise (e.g., Caligari et al., 1995; Fan, 2008; Jouve & Brun, 2009; Weberet al., 2011). Another approach uses local planar models with mechanical forcing to generatelarge-scale shear that drives dynamo action and creates buoyant magnetic loops (e.g., Clineet al., 2003; Vasil & Brummell, 2009; Guerrero & Käpylä, 2011). Planar models have also
been used to study 3-D buoyancy instabilities in a magnetized layer that can lead to risingelements (e.g., Kersalé et al., 2007). The third approach uses global convective MHD models.

86
These incorporate the rotating spherical-shell geometry needed to self-consistently generate
differential rotation and meridional circulation through Reynolds stresses (see review Miesch,2005). Such models have captured the formation of magnetic structures and cycles in solar(Ghizaru et al., 2010; Racine et al., 2011) and rapidly-rotating sun-like stellar models (Brown
et al., 2010, 2011), yielding differential rotation, dynamo action, and large-scale magnetic
fields, but not buoyant magnetic loops that rise toward the top of the convective layer.
Here we report on a global convective dynamo simulation of a sun-like star rotating at
three times the mean solar angular velocity (3Ω ), such as our Sun did when it was younger
and as do many solar analogues (Petit et al., 2008). This simulation (i) attains a differential rotation profile created by the interplay of convection, rotation and stratification (e.g.,
Brun & Toomre, 2002; Miesch & Toomre, 2009), (ii) forms global-scale toroidal magnetic
structures that undergo cycles of magnetic activity and reversals of global polarity, and (iii)achieves buoyant magnetic loops from the strongest portions of the toroidal structures whichrise from the base of the convection zone. This work extends the work of Brown et al. (2010,2011) in which simulations of rapidly-rotating suns with moderate levels of diffusion wereable to accomplish (i) and (ii). The formation of buoyant loops is facilitated in our current
work by adopting a dynamic Smagorinsky subgrid-scale model (Germano et al., 1991), which
serves to minimize the diffusion of well-resolved structures.

4.3

Simulation Parameters and Properties
We have conducted 3-D MHD simulations of turbulent convection and dynamo action

in a spherical shell spanning the bulk of the convection zone from 0.72R to 0.97R involving
a density contrast of 25, and rotating at 3Ω (1240 nHz, once every 9.3 days). We use theanelastic spherical harmonic (ASH) code (e.g., Brun et al., 2004). The anelastic treatment
lets us follow the subsonic flows in the deep convection zone. Within this nearly adiabaticallystratified region, we expect that magnetic buoyancy instabilities captured by our anelastic

87

Figure 4.1: Snapshots of flows and fields in case S3 when a buoyant loop begins to riseat time tb . (a) Radial velocity vr in global Mollweide projection (equator is dashed) near
the top of the computational domain, showing fast, narrow downflows (dark tones) andbroad, slower upflows (light tones). (b) Companion view of toroidal magnetic field Bφ at
mid-convection zone. Effects of turbulent convection contribute to the ragged nature of thewreaths. Several buoyant magnetic loops (see Figure 4.3) are generated in the negativepolarity wreath segment just above the equator and right of image center. (c) Time andzonal average of rotation profile hΩi, possessing an equatorial region with fast rotation andslower rotation at higher latitudes. (d) Longitudinally-averaged toroidal magnetic field hBφ irevealing a prominent axisymmetric field component.

treatment differ from fully compressible treatments by no more than a few percent in either
growth rate or scale (Berkoff et al., 2010). ASH is a large-eddy simulation (LES) code that
resolves the largest scales of motion and uses a subgrid-scale (SGS) model to parameterize
the effects of unresolved, small-scale turbulence. The dynamo simulations of Brown et al.
(2010) and Brown et al. (2011) used a SGS model where the turbulent magnetic diffusivity
ηt was constant on spherical shells and in time, and varied only slowly with depth as theinverse square-root of the background density. Brown et al. (2010) examined a simulation(case D3, at 3Ω ) which exhibited persistent toroidal magnetic structures, whereas Brown
et al. (2011) studied a simulation that achieved cycles of magnetic activity and global polarity
reversals (case D5, at 5Ω ). These simulations had ηt = 2.64 × 1012 and 1.88 × 1012 cm2 s−1
respectively at mid-convection zone.
Here we consider a new ASH simulation, case S3, which achieves much lower levels ofdiffusion through the use of a dynamic Smagorinksy (DSMAG) SGS model. This assumes
self-similar behavior in the resolved portion of the inertial range of scales in a turbulent flow

88in order to extrapolate the effects of unresolved small-scale motions on the resolved scales.
The resulting viscosity νS is determined by the properties of the grid and the flows, andvaries by orders of magnitude in all three spatial dimensions and in time. To determine
the thermal and magnetic diffusion coefficients we assume constant thermal and magnetic
Prandtl numbers. In cases D3, D5 and S3 these are set to 0.25 and 0.5 respectively. We
reserve further discussion of the properties of ASH simulations using the DSMAG SGS model
for Chapters 7 and 8. In case S3 the DSMAG SGS model allows a simulation (with 1024
longitudinal, 512 latitudinal, and 193 radial grid points) to achieve a mean magnetic diffusion
coefficient at mid-convection zone of η̄t = 4.8 × 1010 cm2 s−1 . This reduction in diffusion bya factor of about 40 from case D3 is critical for the formation and coherent rise of buoyant
magnetic loops.
Case S3 exhibits turbulent convective patterns shown in Figure 4.1(a) which are largely
vortical at high latitudes and aligned with the rotation axis near the equator. The convection builds and maintains a strong differential rotation that is prograde at the equator andretrograde at mid to high latitudes (Figure 4.1(c)). This organized shear drives the creationof toroidal magnetic structures at low latitudes in each hemisphere, as demonstrated in B10.
Here the increased level of turbulence enhances the power in smaller-scale components of thetoroidal field Bφ (Figure 4.1(b)) while still retaining a substantial zonally-averaged toroidal
field hBφ i (Figure 4.1(d)).
In addition to creating strong magnetic structures near the base of the convective
region, case S3 also undergoes cycles of magnetic activity and reversals of global magnetic
polarity similar to those described in case D5 in B11. This is consistent with results fromparameter surveys with ASH simulations, which indicate that decreasing both ν and Ω canyield cyclic behavior seen at 5Ω

at lower rotation rates (Brown et al., 2011). Becauseof the large computational cost of the DSMAG SGS model, case S3 was started using aless diffusive descendant of case D3 in B10, which we label case D3b, as initial conditions(Nelson et al., 2011a). Figure 4.2(a) shows the temporal evolution of the hemispherical

89

Figure 4.2: Field reversals with time. (a) Hemispherical volume-averaged toroidal magnetic
field [Bφ ] of progenitor case D3b over nearly 6000 days, displaying irregular magnetic activity
cycles. Case S3 branched from case D3b at time ts (dotted line). (b) [Bφ ] for case S3 overabout 800 days. Case S3 continues the cyclic behavior of D3b, but additionally produces
buoyant loops. The creation of loops which pass 0.90R are indicated by tick marks in thelower panel. Detailed information on the buoyant loop at time tb = ts + 683 days (dottedline) is shown in Figures 4.1 and 4.3.

volume-average of toroidal magnetic field, [Bφ ], in the progenitor case over approximately
5000 days, demonstrating the irregular cycles this model yields. Case S3 continues thisbehavior over about 1300 simulated days starting from time ts . The temporal evolution of[Bφ ] in case S3 is shown in Figure 4.2(b), revealing two reversals of global magnetic polarity.
Some caution should be used in interpreting any LES dynamo simulation, given thepotential sensitivity of dynamo action to magnetic dissipation and the nonlinear, nonlocal
nature of turbulent magnetic induction, which makes reliable SGS modeling difficult. How-90
ever, we believe the essential large-scale dynamics exhibited in this simulation are robust andare largely insensitive to the SGS model. Indeed convective dynamo simulations with differing prescriptions for SGS diffusion exhibit similar large-scale magnetic structures (Brown
et al., 2010, 2011; Ghizaru et al., 2010; Racine et al., 2011).
Here we will discuss buoyant magnetic structures which coherently rise above 0.90R
while remaining connected to the large-scale toroidal wreaths. Using these criteria, we haveidentified nine buoyant magnetic loops, indicated by hash marks in Figure 4.2(b). Eightloops are seen in the northern hemisphere and one in the southern hemisphere. We expectthat the apparent asymmetry is simply the result of having studied only two magnetic cycles.

4.4

Buoyant Magnetic Loops
Buoyant magnetic loops arise from the cores of toroidal magnetic wreaths near the baseof the simulated domain. These wreaths have significant hBφ i components that peak around5 kG while also having strong non-axisymmetric fields. Figure 4.1(b) shows a typical Bφconfiguration involving a negative polarity wreath in the northern hemisphere spanning 95◦
in longitude and a positive polarity wreath in the southern hemisphere extending over 270◦in longitude. As demonstrated in cases D3 (B10) and D5 (B11), these magnetic wreaths
are highly nonuniform and display significant internal variation as well as a high degreeof connectivity with the rest of the domain. In case S3 portions of the wreaths can havecoherent cores in which Bφ can regularly exceed 25 kG and have peak values as high as
54 kG. In these cores, bundles of magnetic field lines show very little local connectivity withthe rest of the domain or even the other portions of the wreath. A single wreath of a given
polarity may not form a coherent core at all or may have more than one core, and a single
core may produce multiple buoyant loops. Of the nine buoyant loops investigated here to
rise past 0.90R , one coherent core produces four buoyant loops, another produces three,
and two more cores each yield a single buoyant loop.

91

Figure 4.3: Analyzing a rising loop. (a) 2-D cuts in longitude at successive times (tracking
in longitude at the local rotation rate of the loop) showing toroidal magnetic field over radiusand latitude. The rising magnetic loop A is seen in cross-section starting at 0.81R at t = tb
and rising to 0.91R after roughly 15 days. Proto-loop B is also seen rising starting at 8.6
days, but the top of loop B never rises above 0.88R . (b) 3-D visualization of magnetic fieldlines in the core of a wreath which produces four loops (two shown here, one of which isloop A) at tb + 14.6 days. Perspective is looking down along the rotation axis toward theequatorial plane. Coloring indicates field magnitude. Dashed lines indicate radial position.
Dotted line shows the cutting plane used in the left-most panel above. (c) Radial location of
the top of a buoyant loop as a function of time since tb , along with movement attributableto magnetic buoyancy (red lines) or to advection by convective upflows (blue lines).

Some of the coherent wreath cores can become buoyant magnetic loop progenitors
or proto-loops. In these proto-loops the strong Lorentz forces result in highly suppressed
convective motions. If we examine extended regions in the cores of wreaths with a local ratioof magnetic to kinetic energy above a fiducial value of 100, we identify at least 35 proto-loops
at the times where the nine buoyant loops arise. Thus the large majority of proto-loops do not

92
evolve into mature buoyant loops, generally due to unfavorable interactions with convective
flows. When magnetic field strengths exceed 35 kG the proto-loops become significantly
underdense as magnetic pressure displaces fluid, causing buoyant acceleration. With somerise a proto-loop can enter a region of less suppressed giant cell convection. These flows will
advect portions of the proto-loop downward at cell edges and upward in the core of the giant
cells. The rise of the top of a magnetic loop is shown in cross-section by sampling Bφ roughly
every 2 days in Figure 4.3(a). Not all proto-loops become buoyant loops by our criteria. For
example, loop B in Figures 4.3(a,c) begins to rise but is prevented from passing 0.88 R
when the top of the loop encounters a strong downflow.
One way to track these buoyant loops is to use 3-D tracings of magnetic field lines using
the VAPOR software package (Clyne et al., 2007). In our simulations with finite resistivity,
individual field lines do not maintain their identity in time. However, one can achieve somemeasure of consistency as the structure moves and evolves by tracking field line ensembles.
We track the very strong fields at the bottom of the loops near the base of the domain
and we randomly seed large numbers of field lines (here 1000) in those highly magnetizedfootpoints at each time step. Figure 4.3(b) shows a 3-D rendering of magnetic fields lines fortwo sample loops near the peak of their rise. Similar field line tracings have been studied atvarious times during the rise of these loops.
At maximum rise, the sample magnetic loop A extends from 0.73R to 0.93R . The
magnetic fields exceed 40 kG at the base of the loop but become much weaker near the topof the loop, with field strengths as low as 2 kG. Such loops are embedded in the muchlarger wreaths which have an average cross-sectional area of 13800 Mm2 . The cross-sectional
area of loop A is 120 Mm2 at 0.795R and 520 Mm2 at its peak radial position of 0.923R .
Accounting for the continued expansion that would likely occur if this loop were able torise further, the cross-sectional area is reasonable compared with the typical area of a large
sunspot at the solar surface, which is roughly 2500 Mm2 Zwaan (1987). If the loop wererising adiabatically over the same interval, the cross-sectional area should change in inverse

93
proportion to the change in background pressure, which decreases here by a factor of 17.1,
rather than the observed expansion by a factor of 4.3. The top of the loop must then havea net outflow of heat or material in order to avoid expanding adiabatically. The loops show
a measurable deficit in density and thermodynamic pressure relative to their surroundings,
but they do not possess any detectable signature in temperature or entropy. This indicatesthat they are thermally “leaky” and able to equilibrate quickly compared to the timescale forradial motion. A simple estimate of the thermal diffusion time across one of these structuresat mid-convection zone is on the order of 50 days, implying that there is likely also a divergent
flow at the top of the loop, moving fluid along field lines. We see some evidence for suchflows with roughly 1 m s−1 speeds.
Once a loop has begun to rise, its radial motion is dominated by advection and magnetic buoyancy. Figure 4.3(c) illustrates the motion of loop A which begins to rise buoyantly
at tb , while also indicating the components of the motion due to advection and magnetic
buoyancy. To compare motion due to magnetic buoyancy, we define a magnetic buoyancy
velocity vmb at the times sampled in Figure 4.3(c). Magnetic buoyancy acceleration is here
the fractional density deficit in the loop compared to the average density of the surrounding
fluid times the local gravitational acceleration. For a magnetic structure in local thermal
equilibrium, this reduces to the ratio of magnetic pressure inside the loop to thermodynamic pressure in the surrounding fluid times gravitational acceleration. To compute vmb
we integrate the magnetic buoyancy acceleration over the intervals between times plotted in
Figure 4.3(c) (roughly 2 days), which likely provides a lower bound on this velocity. The
advective velocity vad is the volume-averaged velocity of the surrounding fluid. The pressureand velocity of the surrounding fluid are calculated by taking averages over the convective
updraft while excluding regions with field magnitude greater than 4 kG. Initially the sample
proto-loop experiences an upward vmb = 46.1 m s−1 . After 3 days of movement dominatedby magnetic buoyancy, the loop gets caught in a convective updraft and vad becomes greaterthan vmb . Even though advective motions dominate, magnetic buoyancy continues to drive

94
an average upward motion at 32.3 m s−1 relative to the surrounding fluid. Continued buoyant
acceleration of the loop as the magnetic pressure weakens is achieved because its density perturbation decreases at roughly the same rate as does the background density stratification.
Once the top of the loop has entered the main convective upflow it experiences advection at
an average velocity of 53.1 m s−1 . The presence of magnetic buoyancy forces allow this loop
to rise in 14.6 days while the average upflow traverses the same distance in 21.7 days andmagnetic buoyancy alone would require 30.6 days.
Additional accelerations are present but not shown, including thermal buoyancy, which
is significant early in the rise of the loop, and magnetic tension, which is of the same orderof magnitude as the advective motion near maximum radial extent at 14.6 days and helps
tether the loop to that height. Thermal buoyancy is distinguished from magnetic buoyancy
by averaging over the convective updraft but excluding regions with magnetic fields above
4 kG. An additional apparent motion at early times is produced as toroidal magnetic fieldused to track the loop is converted to radial magnetic field in the sides of the loop. Because
advection plays a crucial role in the transport of these magnetic loops, their size scale is set
by the size of the convective giant cells. The nine loops studied here have an average extent
of 15.4◦ in longitude when measuring across the bottom of the loop, whereas the average
distance between convective downflows in the equatorial region is 16.4◦ in longitude.

4.5

Reflections
In this chapter we have presented a 3-D MHD simulation that combines turbulent

convection, rotation, and stratification to produce solar-like differential rotation and wreathsof large-scale toroidal magnetic field at the base of the convection zone. These undergo cycles
of magnetic activity and reversals of global magnetic polarity. Most notably the wreaths also
exhibit buoyant magnetic loops capable of coherently traversing much of the convective layer.
Such loops can only be realized when the field amplitude in a portion of a wreath exceeds

95
35 kG, the diffusion timescale across the proto-loop (here 50 days) is much longer than thetimescale for rise due to magnetic buoyancy, and the interactions between rising loops andconvective flows are favorable. These buoyant loops which appear at cycle maximum canhave toroidal field strengths of 45 kG at their base and 5 kG at their top. Their size scales
are set by the size of the convective giant cells and they have cross-sectional areas at 0.90R
that are reasonable compared to the area of a large sunspot.
We must be cautious in suggesting that these rising magnetic loops can make it throughto the surface of the star. Our global simulations here only extend to 0.97R and currently
place an impenetrable boundary there, for we cannot cope with the intense small scales of
convection seen as supergranulation and granulation near the surface. The presence of thedomain boundary deflects all flows, leading to some uncertainty about the fate of the risingloops that could only be resolved by linking flows and magnetism in the upper reaches of
ASH to another high-resolution compressible domain closer to the surface. This is a task weare now pursuing in parallel with global modeling.
It is noteworthy that within this simulation convection generates differential rotation
which in turn generates toroidal flux which then buoyantly destabilizes and rises. Each linkin this chain is physically well established. Our primary accomplishment here is to capture
all these processes self-consistently within a single simulation. This represents an essential
step toward unifying numerical models of global-scale convective dynamos and surface flux
emergence.

Chapter 5
Magnetic Wreaths

Having discussed the discovery of buoyant magnetic loops in a convective dynamo
simulation at 3Ω , we now turn to a more systematic survey of dynamo action at the samerotation rate with varying levels of explicit diffusion. We study the large-scale magnetic
fields which can be achieved at various levels of turbulence. In Chapter 6 we will further
examine the temporal variability of these large-scale magnetic fields and the possibility ofcyclic reversals in global magnetic polarity.
This chapter is the first of two based on the work published in Nelson et al. (2013b) 1
and is largely quoted from that publication. I was the primary author of that paper. I ran
most of the simulations presented here and conducted all of the analysis. Notably Benjamin
Brown ran cases D3, D3-pm1, and D3-pm2. My co-authors provided essential contributions
in formulating the simulations and the analysis, and in providing significant guidance in thewriting process.
5.1

Overview
Solar-type stars exhibit a rich variety of magnetic activity. Seeking to explore theconvective origins of this activity, we have carried out a series of global 3D magnetohydrodynamic (MHD) simulations with the anelastic spherical harmonic (ASH) code. Here we
1

Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S., & Toomre, J. 2013a, Magnetic Wreaths andCycles in Convective Dynamos, ApJ, 762, 73

97report on the dynamo mechanisms achieved as the effects of artificial diffusion are systematically decreased. The simulations are carried out at a nominal rotation rate of three times
the solar value (3Ω ), but similar dynamics may also apply to the Sun. Our previous simulations demonstrated that convective dynamos can build persistent toroidal flux structures(magnetic wreaths) in the midst of a turbulent convection zone and that high rotation rates
promote the cyclic reversal of these wreaths. Here we demonstrate that magnetic cycles
can also be achieved by reducing the diffusion, thus increasing the Reynolds and magnetic
Reynolds numbers. In these more turbulent models, diffusive processes no longer play asignificant role in the key dynamical balances that establish and maintain the differential
rotation and magnetic wreaths. Magnetic reversals are attributed to an imbalance in thepoloidal magnetic induction by convective motions that is stabilized at higher diffusion levels. Additionally, the enhanced levels of turbulence lead to greater intermittency in thetoroidal magnetic wreaths, promoting the generation of buoyant magnetic loops that risefrom the deep interior to the upper regions of our simulated domain. The implications ofsuch turbulence-induced magnetic buoyancy for solar and stellar flux emergence are also
discussed.
5.2

Dynamos at 3Ω
We study convection and dynamo action in the deep interior of solar-like stars using

the anelastic spherical harmonic (ASH) code (Clune et al., 1999; Brun et al., 2004). Our simulation approach is briefly described here, but is more fully explained in Brown et al. (2010).
ASH solves the anelastic MHD equations in a rotating spherical shell with a backgroundstratification taken from a 1D model of solar structure. We focus on simulating the bulk of
the solar convection zone from 0.72R to 0.97R (R is solar radius) with a density contrast
of about 25. We do not model the near-surface layers of the sun, for we are limited by theanelastic approximation to subsonic flows. Additionally we cannot resolve the small-scalesof motion needed to simulate granular and supergranular scales. We also do not include the98stably-stratified radiative zone or the tachocline in these simulations, although simulations
including those components are an active area of research (see Brun et al., 2011). We havedone some preliminary work in adding a tachocline to these simulations and found that it
does not drastically change the dynamo action in the bulk of the convective layer. The effects
of a tachocline will will explored further in a future paper. Our results tend to support the
recent studies with mean-field dynamo models, which suggests that the differential rotationof the convection zone may play a greater role in the generation of toroidal magnetic field
than the tachocline (e.g., Dikpati & Gilman, 2006; Muñoz Jaramillo et al., 2009). We use
impenetrable and stress-free boundary conditions on both the top and bottom of the domain.
We impose the entropy gradient at the top and bottom of the domain for the thermal boundary conditions. For the magnetic fields we use a perfect conductor condition on the bottom
boundary and match to an external potential field on the top boundary. These conditions
and our evolution equations are described in detail in Brown et al. (2010).
ASH is a large-eddy simulation which employs a subgrid-scale model to account for
the effects of unresolved scales of motion. The standard subgrid-scale (SGS) model in ASH
simulations uses enhanced values of viscosity, thermal diffusivity, and magnetic resistivity
relative to those expected from atomic values in order to represent additional mixing due
to unresolved turbulent motions. In this enhanced eddy SGS model, viscosity ν, thermal
diffusivity κ, and magnetic resistivity η all scale as ρ̄−1/2 , where ρ̄ is the spherically-symmetricbackground density of the simulation. This prescription, along with constant Prandtl andmagnetic Prandtl numbers throughout the domain, follows that of Brown et al. (2010, 2011).
All cases presented in this paper use Pr = ν/κ = 0.25, but variable Pm (see Table 5.1).

Table 5.1: Dynamo simulations at three times the solar rotation rate. 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. Evaluatedat mid-depth are the Rayleigh number Ra = (−∂ρ/∂S)(dS̄/dr)gL4 /ρνκ, the Taylor number Ta = 4Ω20 L4 /ν 2 , the rms Reynolds
0L/ν, the magnetic Reynolds number Rm = vrms L/η andnumber Re = vrms L/ν and fluctuating Reynolds number Re0 = vrms
00fluctuating 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 0 has the axisymmetric component removed: v 0 = v − hvi, with angle
brackets denoting an average in longitude. For all simulations, the Prandtl number Pr = ν/κ is 0.25 and the magnetic Prandtl
number Pm = ν/η ranges between 0.5 and 4. The viscous and magnetic diffusivity, ν and η, are quoted at mid-depth (in unitsof 1011 cm2 s−1 ). The total evolution time TE for each simulation is given in years. The values for case S3 with the dynamic
Smagorinsky SGS model utilize the mean viscosity at mid-convection zone averaged on horizontal surfaces as well as in time. For
case S3 using the dynamic Smagorinsky SGS model, the values quoted are based on the time-averaged rms viscosity, conductivity,
and resistivity at mid-depth, noting that these diffusion coefficients have near hundred-fold spatial variations.
Case
D3

Nr , Nθ , Nφ
97 × 256 × 512

Ra
3.28×105Ta
1.22×107

Re
173

Re0
104

Rm
86

Rm0
52

Ro
0.374

Roc
0.315

ν
13.2

η
26.4

Pm TE
0.5 61.6

D3a
D3b

97 × 256 × 512
145 × 512 × 1024

5.84×105
1.11×106

2.41×107
6.08×107

244
343

154
273

122
171

77
136

0.447 0.295
0.566 0.257

9.40
5.92

18.8
11.8

0.5
0.5

67.1
16.9

D3-pm1D3-pm2145 × 256 × 512
145 × 512 × 1024

2.98×105
3.08×105

1.22×107
1.22×107

149
145

102
101

149
291

102
202

0.372 0.300
0.370 0.306

13.2
13.2

13.2
6.60

1
2

18.8
13.6

S3

145 × 512 × 1024

7.68 × 108

4.46 × 1010

8050

5750

4030

2880

0.581

0.218

0.435

0.5

4.01

0.262

99

100
In addition, we have also implemented a more complex SGS treatment, the dynamic
Smagorinsky model developed by Germano et al. (1991). By using the dynamic Smagorinsky
model in ASH simulations we are able to reduce the mean diffusion at mid-convection zone
by a factor of 50 without an increase in resolution. Our implementation of the dynamic
Smagorinsky model is summarized in Appendix A. This SGS treatment is only used in caseS3, which was first presented in Nelson et al. (2011b).
Table 5.1 presents the computational resolution, relevant non-dimensional parameters,
diffusion coefficients, and total evolution time for each of the six cases we will discuss here.
We have explored two main branches in parameter space. The first branch includes cases
D3, D3a, and D3b, where viscosity ν, thermal diffusivity κ, and magnetic resistivity η haveall been dropped together, thus keeping a constant magnetic Prandtl number. The second
branch includes cases D3, D3-pm1, and D3-pm2, where ν and κ are held constant and only
η is decreased, resulting in increasing magnetic Prandtl numbers. We will refer to the two
branches as the constant Pm and increasing Pm branches, respectively. The constant Pm
branch was found to be more compelling, as cases D3a and D3b generally produced strong
magnetic wreaths that were anti-symmetric about the equator, whereas the high Pm branch
produced a wider variety of symmetric and anti-symmetric toroidal field states and wastherefore less amenable to study. Such behavior is not unexpected as dynamos with higher
magnetic Prandtl number tend to promote small-scale dynamo action. We will generally
focus on the constant Pm branch of simulations while referencing the increasing Pm branch
to provide additional insight.
Case D3 was initiated from a well developed hydrodynamical simulation that wasseeded with a small random magnetic field. Each subsequent case along both branches wasstarted from the preceding case. Thus both cases D3a and D3-pm1 were started using caseD3 as initial conditions, case D3b was started using case D3a, and so on. We have restarted case D3a from a random seed field to verify that it settles into a similar region ofsolution space as the version continued from case D3 and found no strong differences in the

101

Figure 5.1: Magnetic wreaths. (a)-(c) Shown in global Mollweide view (equator at middle,
poles at top and bottom) is the radial velocity of the convection at 0.95R in cases D3,
D3a, and D3b, respectively. (d)-(f) Also in Mollweide few, longitudinal magnetic field Bφ at
mid-convection zone at times t1 indicated in Figure 6.1. (g)-(i) Shown at the same times foreach case is a 3D field line rendering of the magnetic wreaths near the equator. In both typesof display for the magnetic field, color gives the polarity and amplitude of the longitudinal
field (red positive, blue negative). Times shown correspond to t1 for each case in Figure 6.1.

time-averaged behavior over several thousand days.
5.3

Magnetic wreaths
The dominant magnetic structures built by each of these simulations are the low lat-itude bands of predominately toroidal field, which we term wreaths. These wreaths are
generally anti-symmetric about the equator, though symmetric states are observed alongwith states where one hemisphere displays a wreath while the other does not. These irregular states are most common along the increasing Pm branch of our simulations. The
wreaths in case D3 are discussed extensively by Brown et al. (2010) and additional wreaths

102
are analyzed at somewhat faster rotation rate (5Ω ) by Brown et al. (2011).

5.3.1

Magnetic Topology

Figure 5.1 shows snapshots of the turbulent convection and the wreaths for cases D3,
D3a, and D3b at mid-convection zone in global Mollweide view as well as at low latitudes
in a 3D volume rendering of magnetic field lines colored by Bφ . In all three cases strong
longitude-averaged fields are present at low latitudes, however the nature of the wreathschange from case D3 where axisymmetric fields dominate to case D3b where a significant
axisymmetric field component is present but not dominant. In case D3b the morphology haschanged such that the wreaths are confined in longitudinal extent. Figure 5.1 shows a typicalfield configuration, but the wreaths are observed at various times to extend over as little at 45
degrees and as much as 270 degrees in longitude. All three cases show extensive connectivity
between the wreaths and the surrounding domain where magnetic fields are moderate instrength but far less coherent. The wreaths are strongly modulated by the convective flows,
producing a ragged appearance that is particularly noticeable in case D3b but present in all
three cases. In the more turbulent cases there are also significant small-scale magnetic fields
at moderate to high latitudes, and occasional locally-generated wreath-like structures near
the poles which persist for less than about 100 days at a time.

Table 5.2: Volume-averaged magnetic and kinetic energies for dynamo simulations at three times the solar rotation rate, as wellas the magnitude of the differential rotation contrast in radius at the equator ∆Ωr and the average contrast at the top of thesimulated domain between the equator and ±60◦ latitude ∆Ωθ . Shown in units of 106 erg cm−3 are the total magnetic energy(Total ME), axisymmetric toroidal magnetic energy (TME), axisymmetric poloidal magnetic energy (PME), fluctuating magnetic
energy (FME), total kinetic energy (Total KE), differential rotation kinetic energy (DRKE), meridional circulation kinetic energy(MCKE), and fluctuating kinetic energy (FKE). The percentage of the total energy is shown for total magnetic energy (Total
ME) and total kinetic energy (Total KE). The percentage of the total magnetic or kinetic energy for each component is shown inparentheses. Values for differential rotation rates are in units of nHz (3Ω = 1240 nHz). Values are averaged in time over longintervals.
Case
D3

Total ME
0.68 (9%)

TME
0.29 (43%)

PME
0.029 (4%)

FME
0.36 (53%)

Total KE
DRKE
6.67 (91%) 4.35 (65%)

MCKE
0.010 (0.1%)

FKE
2.31(35%)

∆Ωr
112

∆Ωθ
192

D3a
D3b

0.88 (12%) 0.32 (36%)
0.82 (13%) 0.10 (12%)

0.030 (3%)
0.011 (1%)

0.52 (59%) 6.41 (88%) 3.71 (58%) 0.011 (0.2%) 2.68 (42%)
0.70 (85%) 5.42 (87%) 2.45 (45%) 0.012 (0.2%) 2.96 (55%)

101
95

163
131

D3-pm1D3-pm2

1.04 (18%)
1.17 (21%)

0.26 (25%)
0.15 (13%)

0.033 (3%)
0.028 (2%)

0.75 (72%)
0.99 (85%)

4.87 (82%)
4.34 (75%)

2.63 (54%)
2.29 (53%)

0.010 (0.2%)
0.009 (0.2%)

2.23 (46%)
2.04 (47%)

87
74

139
121

S3

0.83 (13%)

0.072 (9%)

0.0060 (0.7%)

0.75 (91%)

5.50 (87%)

2.32 (42%)

0.013 (0.2%)

3.17 (58%)

95

133

103

104
The shift from structures dominated by axisymmetric fields in case D3 to the patchy
wreaths in case D3b is illustrated by the changes in the time- and volume-averaged energy
densities shown in Table 5.2. Between cases D3 and D3b there is a roughly 30% increasein the total magnetic energy of the simulation, though the energy in both the axisymmetrictoroidal (TME) and poloidal (PME) fields decreases by roughly a factor of three. The
doubling of the energy in the non-axisymmetric magnetic fields more than compensates forthe decrease in mean fields. When compared with the kinetic energy densities, the changes in
the magnetic energies becomes even more striking. Viscous, thermal, and magnetic diffusion
in case D3b are all reduced by the same factor relative to case D3. However the total kinetic
energy in case D3b dropped by 19%. The non-axisymmetric kinetic energy (FKE) rose
only moderately compared to the decrease in differential rotation kinetic energy (DRKE).
The high magnetic Prandtl number cases also show a tendency towards larger total andfluctuating magnetic energies, as well as reduced axisymmetric toroidal magnetic energy as
the magnetic diffusion is reduced.
It is illustrative to compare cases D3b and D3-pm1, as they have roughly equal levels ofmagnetic diffusion, with case D3b having comparatively lower levels of viscosity and thermal
diffusion. The largest differences are in the axisymmetric magnetic energies which are bothabout three times greater in case D3-pm1 than in case D3b. This may be due to the morelaminar flow in case D3-pm1, which would tend to create fewer sharp gradients in the largescale magnetic structures and thus lower the effective dissipation in case D3-pm1 comparedto case D3b, even though the diffusion coefficients in the induction equation are nearly thesame. Case D3-pm1 also show significantly less differential rotation contrast both in radiusand latitude compared to case D3b, pointing to the key role of magnetic torques in decreasing
differential rotation, which will be discussed further in §5.

105

Figure 5.2: Probability distribution functions for unsigned Bφ at mid-convection zone
for cases D3 (purple), D3a (green), D3b (red), and S3 (blue) showing the surface area
covered by fields of a given magnitude. Distributions are averaged over about 300 days whenfields are strong and as steady as possible. Dashed vertical lines show the field-strength
at which equipartition is achieved with the maximum fluctuating kinetic energy (FKE) at
mid-convection zone for each case. Case D3b shows a deficit of field in the 10 kG range, but
an excess of surface area covered by extremely strong fields above 25 kG range, as well as
higher peak field strengths. Case S3 shows significantly greater regions of fields in excess of
20 kG than all other cases.

5.3.2

Non-axisymmetric Fields

Our discussion of the magnetic wreaths to this point has focused on the axisymmetricfields, which are progressively weaker in moving from case D3 to case D3b. While the axisymmetric fields weaken with increased turbulence, very strong fields become more common
when measured by the fraction of the domain they occupy. Figure 5.2 shows the probability
distribution function for Bφ at mid-convection zone in cases D3, D3a, D3b, and S3. Whilecase D3b has a deficit of fields around 10 kG compared to case D3a, there is a clear excess offields above 20 kG. Interestingly the distribution for case D3b is greater than that for caseD3 for all but the smallest bin, indicating that while case D3 may have stronger axisymmetric fields in the low latitude wreaths, case D3b compensates by having higher amplitude

106
fluctuating fields throughout the domain. The peak field strength at mid-convection zone is
32 kG in case D3, 36 kG in case D3a, and 38 kG in case D3b. Near the base of the convection
zone case D3b exhibits even stronger fields of up to 44 kG. Case S3 posses magnetic fields
of up to 45 kG at mid-convection zone and 52 kG near the base of the convective layer.
For all four cases fields are seen well in excess of equipartition energies with the maximum
fluctuating kinetic energy of the flows. This is a clear indication of turbulent intermittency
in the magnetic fields.
A statistical measure of turbulent intermittency is the time-averaged excess kurtosis
given by
 0
4
0
0
−
B̄
f
B
B
φ
φ dBφ
φ
kurt{Bφ } = hR −∞ 0 i2 − 3,2∞00Bφ − B̄φ f Bφ dBφ
−∞
R∞

(5.1)

where f (Bφ0 ) is the probability distribution function (see Pope, 2000). For reference a Gaussian distribution would have an excess kurtosis of 0. The level of turbulent intermittency ismeasured by how leptokurtic the distribution is found to be, with large values corresponding
to increased intermittency. For case D3 kurt{Bφ } = 9.6, while for case D3a kurt{Bφ } = 10.5,
and for case D3b kurt{Bφ } = 12.1. Leptokurtic distributions are likely to experience strong
coherent structures, such as the strong regions of coherent toroidal field in these simulations.
At even lower levels of diffusion than can be realized with the enhanced eddy SGS model,
the strong-field regions become sufficiently buoyant and coherent so as to form buoyant magnetic loops as realized in case S3 (Nelson et al., 2011b), for which kurt{Bφ } = 15.6. Highly
leptokurtic distributions like these indicate that extreme events are enhanced relative to a
Gaussian distribution, and the trend towards increasing kurtosis as simulations become moreturbulent points to turbulent amplification of magnetic fields. As we will discuss further in§7, this provides an alternate pathway to produce regions within the larger wreaths which
can be amplified through turbulent intermittency to produce coherent regions of strong magnetic field, which can then become buoyant. We term this the turbulence-enabled magnetic
buoyancy paradigm.

107
5.4

Maintaining Rotational Shear
A crucial component in the construction of magnetic wreaths is the strong latitudinal

and radial shear from the differential rotation. The Ω-effect has previously been shown tobe the primary production mechanism for the magnetic wreaths in cases D3 and D5 (Brown
et al., 2010, 2011), and it plays a key role in these simulations as well. Thus the angularmomentum transport required to maintain the differential rotation is an important physicalprocess in these dynamo models. In the hydrodynamic models explored by Brown et al.
(2008), angular momentum transport in simulations at 3Ω

was shown to be a balance

between Reynolds stress supporting solar-like differential rotation with the meridional circulation and viscous diffusion tending to dissipate gradients in the rotation profile. With
the addition of magnetic fields, Maxwell stress and mean magnetic torques can also transport angular momentum, changing the balance supporting the strong differential rotationachieved in the hydrodynamic cases. Even in cases without magnetic cycles such as case D3,
Brown et al. (2010) showed that there are significant feedbacks on the differential rotation
profile due to variations in the strength of the magnetic fields over time. It is thus useful to
examine not only the steady state balance of angular momentum transport over long time
averages covering many magnetic cycles, but also to look at the temporal variability of thosebalances.
In order to explore the transport of angular momentum, let us examine the physicalmechanisms which come into play. The balance of specific angular momentum along therotation axis is determined by taking the product of the cylindrical radius λ = r sin θ andthe longitudinal component of the longitude-averaged momentum equation, which can be
expressed as∂Lz~= ∇ · F.
∂t

(5.2)

~ into radial and latitudinal
We decompose the flux vector of mean angular momentum F
components following prior convention (Elliott et al., 2000; Brun et al., 2004; Brown et al.,

108

Figure 5.3: Differential rotation and the terms contributing to the accompanying redistribution of angular momentum in case D3b averaged over many magnetic cycles. (a) Angularvelocity Ω profile with radius and latitude, accompanied in turn by profiles of specific angular momentum flux in cylindrical radius (λ) by Reynolds stress (RS), meridional circulation(MC), viscous diffusion (VD), Maxwell stress (MS), and mean magnetic torques (MT), respectively. Terms are defined in detail in Equations A.1 and A.2. They are here averaged
in time and longitude, and given in units of 1015 g s−2 . (b) Scalar plot of z-integrated
fluxes of angular momentum with cylindrical radius λ in units of 1038 g cm2 s−2 . Reynolds
stress (RS, red) balance Maxwell stress (MS, blue), with viscous diffusion (VD, green) andmagnetic torques (MT, purple) playing less of a role. Contribution from the meridional
circulation (MC, brown) are small. The sum of all five terms are also plotted (black dashedline).

2011). We also decompose the flux vector into cylindrical coordinates along cylindrical radius(λ) and along the rotation axis (z), which in many ways is advantageous for displaying
these quantities. A detailed description of this decomposition is given in Appendix B. The
cylindrical flux of angular momentum is shown for case D3b over a long time average in
Figure 5.3. The differential rotation is again clearly maintained by the Reynolds stress(RS), however here the terms opposing the differential rotation have changed comparedto similar hydrodynamic cases. In case D3b the Maxwell stress (MS) is the largest term
opposing the Reynolds stress with viscous diffusion (VD) and the mean magnetic torques(MT) each playing a small role, while contribution of the meridional circulation (MC) isalmost insignificant.

109
Table 5.3: Total production and dissipation of kinetic energy in the axisymmetric differential rotation profile over the entire simulated volume and averaged in time. Values forenergy production rates are in units of 1032 erg s−1 . Production and dissipation terms are
split following Equation (5.3) into contributions from viscous dissipation, Reynolds stress,
meridional circulations, Maxwell stress, and mean magnetic torques, respectively. Production terms are defined in Appendix A.
Production and Dissipation of Differential Rotation Kinetic Energy
Case
D3

LRS
4.26

LMC
-0.020D3a
D3b

3.18 -0.032 -1.11 -1.36 -0.70
2.59 -0.003 -0.64 -1.94 -0.19

D3-pm1D3-pm23.64
3.32-0.014-0.024

LVD
-2.45

-1.68
-1.61

LMS
-1.36

-1.87
-1.95

LMT
-0.68

-0.35
-0.31

We can write the evolution of the total energy of the differential rotation EDR as∂EDR= LVD + LRS + LMC + LMS + LMT ,
∂t

(5.3)

where the terms on the right-hand side represent the sources and sinks of kinetic energy inthe differential rotation due to, respectively, viscous diffusion, Reynolds stress, meridional
circulations, Maxwell stress, and mean magnetic torques. Appendix A provides a derivationof Equation (5.3) and an expansion of the sources and sinks.
Using this decomposition, we can examine the balance of production and dissipationof EDR averaged over long time intervals in each simulation. The balances are represented
in Table 5.3 and Figure 5.4. For the increasing Pm branch the Reynolds stress change only
slightly while the mean magnetic torques and viscous diffusion are systematically replacedby the Maxwell stress. Similar trends are observed in the constant Pm branch of cases,
though here the magnitude of the Reynolds stress and viscous diffusion terms decrease more
dramatically. This shift from unresolved dissipation in the form of SGS viscosity to resolved,
small-scale torques from the Maxwell stress indicates that the balances which maintain strong

110

Figure 5.4: Companion to Table 5.3, showing the balance of time-averaged generation
terms for the kinetic energy in the differential rotation profiles for each case indicated. In
all cases the differential rotation is maintained by a balance between the Reynolds stressand a combination of viscous diffusion and fluctuating and mean magnetic torques. The
contribution from meridional circulations are not shown due to their small magnitude.

differential rotation can persist in less diffusive regimes, assuming that magnetic energies
remain significantly smaller than kinetic energies.
Turning to the temporal variability in these balances, we find that for case D3b the
departures from the values presented in Table 5.3 and Figure 5.4 are about 10% for LMS and
LMT when averaged over about 10 days, whereas those in LRS , LMC and LVD are about 1%.
This leads to decreases in the differential rotation when magnetic fields are strong, such asnear the peak of the magnetic activity cycles. Conversely, we observe modest increases inthe differential rotation when magnetic fields are weak, such as during reversals of magnetic
polarity.
Figure 5.5 shows the differential rotation in case D3b over several magnetic reversals.
The differential rotation profile at mid-convection zone in Figure 5.5(a) is persistent, though
there are small systematic variations in Ω revealed in Figure 5.5(b) during each magnetic

111

Figure 5.5: Temporal variability of differential rotation in case D3b over the same interval
as in Figure 6.9. (a) Longitudinally-averaged rotation rate at mid-convection zone as afunction of time and latitude. (b) Temporal variations are accentuated by subtracting thetime-averaged Ω at each latitude. Bands of faster rotating fluid move poleward on about
the cycle period. (c) Rotation contrasts in radius at the equator ∆r (red, solid) and inlatitude between the equator and ±60◦ in the upper convection zone ∆θ (red, dashed). The
volume-averaged toroidal field strength is also shown (blue, solid), with a phase-lag between
peaks in magnetic field strength and decreases in differential rotation. Dotted lines indicate
times t2 through t6 from Figure 6.2.

cycle. Figure 5.5(c) shows the differential rotation contrast in both radius and latitude overtime as well as the volume-averaged toroidal magnetic field strength, indicating that themodest variations in the differential rotation are related to those in the magnetic field.
5.5

Generation of Toroidal Magnetic Energy
The transition from persistent wreaths in case D3 to cyclic wreaths and global polarity

reversals in case D3b indicates that by reducing the levels of diffusion in these simulations
we have fundamentally altered the balance of terms in the magnetic induction equation.
The details of the reversal mechanism are likely to be very subtle in these highly nonlinear

112
systems. In order to better understand the reversal mechanism, we explore the nature of
the balances in the production and dissipation of toroidal and poloidal magnetic fields andprovide indications of where and why changes in those balances are occurring, as well assome hints as to the nature of the reversal mechanism.
In Brown et al. (2010) a detailed analysis of the balance of toroidal component of theaxisymmetric induction equation was presented. We write the toroidal component of theinduction equation as

i


∂Bφ h
~
= ∇ × ~v × B− ∇ × η∇ × (Bφ φ̂) .
∂t
φ

(5.4)

Using vector identities, the first term on the right-hand side can be written as the sum ofshearing terms, advection terms, and a compression term; additionally all of these terms canbe decomposed into mean and fluctuating components (for a full derivation, see Appendix
A in Brown et al., 2010). That work also showed that the wreaths in case D3 are primarily
generated by the Ω-effect and dissipated by a combination of small-scale advection, shear,and diffusion.
Here we perform a similar analysis, but instead of examining the generation of hBφ i,
we choose to examine the generation of the volume-integrated energy of the axisymmetrictoroidal fields over the entire computational domain V, given by
hBφ i2dV.
8π
VZ
ETME =(5.5)

We can construct an evolution equation for ETME by multiplying Equation (5.4) by hBφ i.
The result can be written as∂ETME= GMS + GFS + GMA + GFA + GAC + GRD ,
∂t

(5.6)

where the six terms on the right-hand side represent, from left to right, the shearing of
axisymmetric magnetic fields by mean flows associated with the Ω-effect (GMS ), the averageof fluctuating flows shearing fluctuation fields (GFS ), the advection of mean fields by mean

113
flows (GMA ), the average of fluctuating flows advecting fluctuating fields (GFA ), the anelastic
compression of fields (GAC ), and the resistive diffusion of mean fields (GRD ). Unlike inprevious analyses which looked at the generation of magnetic field vectors, here we areconcerned with scalar quantities.
The terms on the right-hand side of Equation (5.6) are computed as
Z
hBφ i

GMS =

h

 i
~
hBi · ∇ h~v i dV,

Z

hD 
 Ei
0
~
hBφ i
B · ∇ ~v 0
dV,

GFS =

(5.7)

φ

V

(5.8)

φ

V

Z

h
i
~
hBφ i (h~v i · ∇) hBidV,

GMA = −Z
GFA = −

hBφ i

(5.9)

φ

V

hD

~0
(~v · ∇) B
0

Ei

V

dV,

(5.10)

φ

Z
GAC =

D
∂ ln ρ̄ E
hBφ i vr Bφ
dV,
∂r
V

(5.11)hBφ i∇ × (η∇ × hBφ i) dV.

(5.12)Z
GRD = −V

For consistency, angle brackets denote longitude averages.
Figure 5.6 shows the temporal evolution of the integrated energy of the axisymmetrictoroidal field for cases D3 and D3b and the behavior of the production terms governing thevariation of ETME . We have chosen to combine the contributions of the mean shear andadvection terms and the fluctuating shear and advection terms for ease of viewing. The
mean advection term GMA is generally positive and always much smaller than the mean
shear term GMS . The fluctuating shear and advection terms are both generally negative, ofapproximately the same magnitude, and tend to vary in phase with each other.
Let us first look at the average levels of each term plotted in Figure 5.6 to get asense for the basic balance of terms. The production of ETME is dominated by the mean
shear term which is large and always positive in both case D3 and D3b. The compression
term in both cases is roughly an order of magnitude smaller but is again always positive
due to the asymmetry in upflows and downflows in compressible convection, which gives

114

Figure 5.6: Volume-integrated production terms of magnetic energy in the mean toroidal
fields from Equation (5.6) for (a) case D3 and (b) case D3b. We have combined the mean
shear and mean advection terms (blue line) and the fluctuating shear and fluctuating advection (purple line). In both simulations, energy is produced primarily by the shearing ofmean fields by mean flows. Also in both cases compression of fields (green line) plays a verysmall role. In case D3 diffusion (red line) and the advection and shear of fluctuating flows on
fluctuating fields destroy energy, with diffusion generally a factor of 2.5 larger. In case D3b,
however, the dissipation of energy by fluctuating advection and shear is 2.2 times greater on
average than diffusion. Thus in case D3b resolved turbulence is the primary mechanism fordissipating the magnetic wreaths.

preference to downward pumping of magnetic field causing an increase in magnetic energy
due to compression. The production of magnetic energy is opposed by the resistive diffusion,

115
fluctuating advection, and fluctuating shear terms. In case D3 resistive diffusion is roughly
three times larger than the sum of the two fluctuating terms, while in case D3b the roles
are reversed and resolved turbulent dissipation does most of the destruction of ETME while
the unresolved turbulent dissipation represented by our explicit resistivity is relegated toa less prominent role. Supporting this transition from unresolved to resolved dissipative
processes, in case D3 the sum of the fluctuating terms does not show noticeable changes inbehavior when the magnetic energy is high versus when it is low. Instead the response isseen primarily in the resistive dissipation term. In case D3b, however, the fluctuating terms
show strong variations in response to changes in the magnetic energy.
This transition from wreath-building dynamos that rely on our SGS diffusion to wreathbuilding dynamos that are sufficiently turbulent to be dominated by resolved turbulent dissipation answers one important question relative to the extension of this dynamo mechanism
to even more turbulent states. It had long been postulated that global-scale magnetic structures could not exist in the convection zone as they would be quickly destroyed by the intense
turbulence. While it is clearly possible that our wreaths may not be able to survive if wewere able to simulate far more turbulent conditions, case D3b marks an important milestonealong the path towards the possibility of magnetic wreaths coexisting with highly turbulent
convection.
Returning to Figure 5.6, let us now look at the time-variation in the production terms.
Both cases show variability of ETME , but for case D3b we have chosen to show a time periodthat includes states before, during, and after a reversal in global magnetic polarity. For bothcases, careful examination shows that changes in ETME are initiated primarily by changes in
GMS , not by changes to the terms dissipating energy. The terms representing both resolved
and unresolved diffusion respond to changes in ETME rather than drive them. In case D3 thisis supported by the cross-correlation of GMS and GRD peaking at a 39 day lag, while there isno significant cross-correlation between GMS and either GFS or GFA for any shift in time. Incase D3b both the cross-correlation of GMS with GFS , and GMS with GFA both peak at a lag

116
of 11 days. Resistive diffusion responds faster in case D3b with a peak in cross-correlation
for a lag of only 5.6 days. This demonstrates that the variability in the toroidal fields isdriven by changes in the generation of field by the Ω-effect.
If we more closely examine the structure of GMS from Equation (5.7), we can expand
it to
Z 
GMS =

∂hvφ i hBφ ihBθ i ∂hvφ i
+
∂r
r
∂θ
V

2
2
hBφ i hvr i hBφ i hvθ i
+
+
dV.
r
r tan θhBφ ihBr i(5.13)The third and fourth terms are geometric terms from the spherical coordinate system which
are generally small. In order to produce a change in GMS , the dynamo can either change
the axisymmetric poloidal field or modify the differential rotation of the domain. We haveexamined both the amplitude and geometry of the mean shear due to differential rotationand find only very small changes in any of the cases presented here. Additionally, reversals
in the polarity of the wreaths such as those seen in cases D3a and D3b require a change insign for the generation term (obtained by dividing by hBφ i) and there is never a change ofsign in the shear profile of the differential rotation observed in any of these cases. Thus weare left with the conclusion that reversals in the polarity of the axisymmetric toroidal fields
must be initiated by changes in the axisymmetric poloidal fields.

Chapter 6
Cycles of Magnetic Activity

We now turn to a discussion of the temporal variability of the magnetic wreaths discussed previously. Specifically, we will examine the cycles of magnetic activity and reversals
of magnetic polarity which can be achieved by reducing the explicit diffusion in our simulations. This chapter is the second of two based on the work published in Nelson et al.
(2013b)1 and is largely quoted from that publication. I was the primary author of that
paper. I ran most of the simulations presented here and conducted all of the analysis. Notably Benjamin Brown ran cases D3, D3-pm1, and D3-pm2. I wrote most of the text, withsignificant contributions from Mark Miesch in what is now Section 6.2.

6.1

Cyclic Reversals Achieved by Reducing Diffusion
In addition to building strong magnetic wreaths, cases D3a and D3b exhibit cyclic

reversals of global magnetic polarity. As is believed to occur in the Sun, the general patternof the cycles is that the toroidal fields peak at roughly the time when the poloidal fieldis reversing sign, and the poloidal fields peak in amplitude when the toroidal fields are reversing sign. There are also a number of variations on this pattern, where one hemispheremay develop considerably stronger fields than the other or where both hemispheres have thesame sense of toroidal field, pointing to large contributions at these times from quadripolar
1

Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S., & Toomre, J. 2013a, Magnetic Wreaths andCycles in Convective Dynamos, ApJ, 762, 73

118
poloidal fields. Cases D3-pm1 and D3-pm2 also display strong variations in the strength andtopology of their axisymmetric fields. However the irregularities are more pronounced forthese cases over the time simulated.

6.1.1

Reversals in Global Magnetic Polarity

Figure 6.1 shows the temporal evolution of the longitude-averaged toroidal field hBφ i
at mid-convection zone over the history of cases D3 (Figure 6.1(a)), D3a (Figure 6.1(b)),
and D3b (Figure 6.1(c)). In case D3 we see persistent wreaths centered at about 20◦ aboveand below the equator. These wreaths persist for about 68 years or as long as we haverun the simulation. The polarity of the wreaths is constant in time, though variations onroughly 6 year time scales can be seen in both the amplitude of the low latitude wreaths aswell as the propagation of field to higher latitudes. The behavior of this case is discussed indetail in Brown et al. (2010). Figure 6.1(b) shows case D3a over a comparable length of time
as in the first panel. Case D3a undergoes reversals in global magnetic polarity as well as
three significant irregular states. Additionally there are modulations in the amplitude of thewreaths and poleward movements of field on roughly 3 year time scales. These variations
are not always synchronized between the two hemispheres, and neither are the reversals,
indicating that the poloidal field can have a complicated structure.
Figure 6.1(c) shows the temporal evolution of hBφ i at mid-convection zone for case D3bover about 13 years, with indications of cycles of magnetic activity and reversals of global
polarity. We have simulated 10 reversals as measured by the time-smoothed antisymmetriccomponent of hBφ i changing sign. The time between reversals ranges from 0.6 to 1.9 years
and, as in case D3a, the two hemispheres are not always synchronized. There are several
times when one hemisphere shows significantly stronger fields than the other or when bothhemispheres have the same sense of fields. This is partly due to the averaging procedure used
to create these figures and the fact that we are only looking at a single depth. Analysis of the119

Figure 6.1: Time-latitude plots of longitude averaged toroidal magnetic field hBφ i at 0.79R
for (a) case D3 over about 56 years, (b) case D3a over the same amount of time, and (c) caseD3b over about 13 years. Dotted lines show times referenced in Figures 5.1, 6.2, 6.5, 6.6,
and 6.9. Dashed lines on (b) indicate the time period covered by (c). Case D3b was started
from case D3a at t2 (dotted line). The evolution of case D3b is limited by the increased
computational cost of the higher resolution required for computational stability.

120
full 3D data shows that there is almost always a wreath-like structure in each hemisphere.
Figure 6.2 shows a sequence of snapshots of Bφ at mid-convection zone in case D3bover a full spherical shell and of hBφ i over the domain before, during, and after a reversal in
the polarity of the wreaths. Each snapshot is roughly 120 days after the previous snapshot.
The wreaths start appearing as strong mean-field structures in the longitudinal average,
but the non-averaged cut at mid-convection zone shows that there is significant longitudinalvariation in the wreaths, with the northern wreath covering roughly 120◦ and the southern
wreath covering 180◦ in longitude. There is also substantial evidence for interactions between the wreaths near the center of the image in Figure 6.2(a). As time progresses, theaxisymmetric fields weaken as the non-axisymmetric components begin to dominate. Small
patches of strong field persist, but they are largely washed out in the longitudinal averages.
After about 480 days (Figure 6.2d) strong patches of opposite polarity field begin to appear
and by the final frame (Figure 6.2e) the strong mean fields have been reestablished in the
opposite hemispheres from the initial configuration.

6.1.2

Variability at Higher Magnetic Prandtl Number

The simulations on the increasing Pm branch also show increased temporal variability
relative to case D3. There is also evidence for a change in the nature of the dynamo action inthese simulations. Figure 6.3 shows the evolution of hBφ i at 0.79 R over the history of caseD3-pm1, along with snapshots of the toroidal magnetic field at mid-convection zone at three
representative times. This case has selected a configuration of toroidal field that is largely
symmetric about the equator and of essentially the same polarity at most times. Some
periods of positive polarity field are seen, though the dominant field in both hemispheres isclearly of negative polarity. Unlike cases D3a and D3b, case D3-pm1 does not undergo a true
global reversal of magnetic polarity. It does, however, exhibit strong temporal variability inthe wreaths seen in both hemispheres to an extent not seen in cases D3 or D3a.

121
Figure 6.4 shows a similar view of case D3-pm2 over its temporal evolution. Again
it tends to avoid the anti-symmetric states characteristic of cases D3, D3a, and, to a lesser
extent, D3b. This case, however, does exhibit clear reversals of global magnetic polarity.
Interestingly, these reversals do not appear to occur at regular intervals, and often one
hemisphere can reverse without a noticeable change in the other hemisphere. As an example,
the southern hemisphere maintains a positive polarity wreath between about t = 5.5 years
and t = 10.5 years while the northern hemisphere exhibits four reversals in that same timeinterval.
The preference for irregular polarity states in Bφ along the increasing Pm branch isclearly related to the decreased level of magnetic diffusion, though it may also be indicativeof a shift in behavior due to the transition from small to large magnetic Prandtl number.
In cases D3, D3a, and D3b magnetic diffusion occurs on scales larger than those related tothe diffusion of momentum. This tends to promote the concentration of magnetic energy
at large scales. For high Pm dynamos, the resistive scale is smaller than the viscous scale,
which tends to promote the growth of magnetic energy at small scales (e.g., Schekochihin
et al., 2004). There is still considerable large-scale organization of magnetic field by thedifferential rotation, but the increasing Pm branch exhibits less ordered behavior than theconstant Pm branch of simulations.
When examining the relative importance of decreased magnetic diffusion and increased
levels of turbulence, it is perhaps most instructive to compare cases D3b and D3-pm2. Table 5.2 shows that the division of magnetic energies between the axisymmetric toroidal,
axisymmetric poloidal, and fluctuating magnetic fields are roughly equivalent in the twocases, although case D3-pm2 has more magnetic energy overall. The kinetic energies in caseD3-pm2 are, however, more similar to case D3 than case D3b with the exception of decreased
differential rotation kinetic energy due to enhanced Lorentz force feedbacks. This suggeststhat the onset of reversals is driven primarily by decreasing magnetic diffusion rather than bysome subtle shift in the velocity fields or correlations between magnetic fields and velocities

122
on small scales.

6.1.3

Collapse of Resistive Balance Leading to Reversals

The key to understanding the reversals seen in cases D3a and D3b lies in the generationof poloidal field. When the poloidal field reverses sign the Ω-effect can then build wreathsof the opposite polarity and reverse the sign of the axisymmetric toroidal field. It is difficult
to identify a simple model for the generation of poloidal field in these cases, particularly incase D3b. We can, however, identify the change in the generation mechanism that occurred
between cases D3 and D3b.
Following the work of Brown et al. (2010, 2011), we choose to examine the evolution~ This is convenient as hAφ iof the φ component of the mean vector magnetic potential hAi.
completely specifies the components of the axisymmetric poloidal magnetic field by∇ × hAφ iφ̂ = hBr ir̂ + hBθ iθ̂.

(6.1)

The temporal variations in the magnetic wreaths are driven by changes in the shear ofmean poloidal magnetic fields by mean differential rotation and that only the axisymmetric
poloidal fields can change sign, hence changes in the polarity of the wreaths can be traced
back to the evolution of hAφ i. Further, the key region of the domain in which we should
monitor hAφ i is near the equator where the gradients in differential rotation are largest andwhere the wreaths are primarily generated.
The evolution of hAφ i is governed by∂hAφ i ~= h~v i × hBi+ (hv 0 × B 0 i)φ − ηhJφ i.
∂t
φ

(6.2)

We have ignored a gauge term in Equation (6.2) which is permissible for any longitudinallyperiodic gauge. We take a time-integral of this equation to look at the changes in hAφ i over

123
about 500 days in cases D3 and D3b, and define the time-integral of each term as
Z t2 
~h~v i × hBi
dt(∆Aφ )ME =Z t2(∆Aφ )FE =

(6.3)

φ

t1

(hv 0 × B 0 i)φ dt

(6.4)

t1

Z t2
(∆Aφ )RD = −ηhJφ idt.

(6.5)

t1

Thus the change in hAφ i can be written as∆hAφ i = (∆Aφ )FE + (∆Aφ )ME + (∆Aφ )RD .

(6.6)

Figures 6.5 and 6.6 show the evolution of hAφ i in cases D3 and D3b, respectively, as
well as the time-integrated production of terms shown above and the net change over thetime interval. For case D3b we chose a time period spanning a reversal in global magnetic
polarity. In both cases (∆Aφ )ME is small and the evolution is primarily governed by thebalance between fluctuating EMF and resistive diffusion. The primary difference between
cases D3 and D3b is the collapse of the resistive balance. Both cases show similar patternsin (∆Aφ )FE , namely that the fluctuating EMF in both cases is seeking to create a regionof opposite polarity poloidal field near the equator while reinforcing the current sense ofpoloidal field at mid-latitudes. Thus in both cases D3 and D3b the turbulent correlations
between the existing field and the convective turbulence tends to build poloidal field near
the equator of the opposite sense than the field that built the current wreaths through theΩ-effect. The difference between cases D3 and D3b is that in case D3 the diffusion term issufficiently large to prevent the reversal by diffusing away the opposite polarity poloidal fieldat the equator before it can accumulate sufficiently to cause a reversal.
What causes the fluctuating EMF to display this propensity towards reversing thepolarity of hAφ i near the equator? It would seem that there should be some link back to thestrong toroidal wreaths, however when we expand the fluctuating EMF, we find that(hv 0 × B 0 i)φ = hvr0 Bθ0 − vθ0 Br0 i.

(6.7)

124
Clearly, neither the axisymmetric nor fluctuating components of Bφ come into play here,
indicating that to complete a reversal we need to connect the large-scale toroidal fields tocorrelations between small-scale poloidal fields and poloidal flows. As shown in Figure 5.1(i),the wreaths are not purely toroidal structures, thus the small-scale fields needed in Equation 6.7 may be supplied by the wreaths themselves. However, we have not been able todefinitively link the poloidal components of the wreaths to the reversal process. While thesubtle nature of this process remains difficult to pin down, we do have some hints at its origin.

6.1.4

Exploring An α-Like Effect

The final step in the reversal process is what is often described in the parlance ofmean-field dynamo theory as the α-effect (see Charbonneau, 2010). Generally, the α-effectis the source of the axisymmetric component of the turbulent EMF, defined as~ 0 i.
h~0 i = hv~0 × B(6.8)

Specifically, we are interested in the zonal component which generates the mean poloidal
field and its connection to the axisymmetric toroidal field, which might be expressed ash0φ i = αφr hBr i + αφθ hBθ i + αφφ hBφ i.

(6.9)

In its simplest formulation, the components of αij in Equation (6.9) are constants, however
more complex formulations exist.
For case D3b, we have computed the value of the three components of the α tensor
in Equation (6.9) using a singular value decomposition following the work of Racine et al.
(2011). We compute values for αij at each radial and latitudinal location, assuming thatαij is constant in time. The results of Figure 6.7 demonstrate that the αφφ hBφ i is the mostimportant term in the generation of the fluctuating toroidal EMF. Thus, it is particularly
intriguing to focus on the connection
h0φ i = αφφ hBφ i.

(6.10)

125
Brown et al. (2010) showed that for one formulation of an α-effect in case D3, αφφ was
spatially nonlocal, which would not be picked up in our fitting procedure.
The exact mechanism for connecting mean fields and the fluctuating EMF is subtle,
but we find that in case D3b an α-like effect emerges, which is nonlocal in time, acting
on the same time scale as convective overturning. If we consider correlations between thevolume-averaged magnetic field components and similarity the fluctuating toroidal EMF,
we find evidence that the α-like effect in case D3b is not instantaneous but rather acts ona time scale (47 days) which is commensurate with the convective overturning time. The
volume averages, denoted by curly braces, are computed separately for each hemisphere overall depths and longitudes, and between the equator and ±30◦ in latitude. Combining thedata for both hemispheres, the cross-correlation is computed and shown in Figure 6.8 as afunction of the temporal interval ∆τ by which {0φ } is offset relative to {Br }, {Bθ }, and {Bφ }in turn. The peaks in the cross-correlation which exceed 2σ in significance occur when {0φ }leads {Bφ } by 312 days and when {Bφ } leads {0φ } by 47 days.
Analysis of the autocorrelation of both {Bφ } and{0φ } indicates that the two peaks are
not due to periodicities in either of the two time series individually. Further, the widths ofthese peaks largely originates from the coherence time for hBφ i of about 100 days. The firstpeak at 312 days represents the time scale for the Ω-effect and agrees well with the estimate
from mean-field theory τΩ given byτΩ = ProtΩ0 hBφ i,∆Ω hBpol i(6.11)

where Prot is the rotation period, Ω0 is the frame rotation rate, ∆Ω is the differential rotation
rate, and hBpol i is the strength of the poloidal field. For case D3b, this yields a value of 324
days. The second peak in the cross-correlation between the two fields occurs when τ∆ = −47days. This peak suggests that the correlations which generate the turbulent zonal EMF
are related in some way to the axisymmetric toroidal fields, and that whatever mechanism
establishes this correlation, it has a time scale of about 50 days. This temporally and126
spatially nonlocal α-effect clearly points to convection as a key player, as other mechanisms
like the meridional circulation are at least an order of magnitude slower.
In addition to a timescale for an α-like effect which is commensurate with the convective
overturning time, we also find evidence for an upscale transfer of magnetic energy related
to magnetic reversals. Figure 6.9 shows the temporal evolution of hBφ i at mid-convection
zone over both latitude and spherical harmonic degree `. Several reversals of global magnetic
polarity are evident in Figure 6.9(b), including the reversal shown in detail in Figure 6.2.
Figure 6.9(a) shows the coefficients of the spherical harmonic expansion of the axisymmetrictoroidal field for the anti-symmetric (odd `) modes with 1 ≤ ` ≤ 29 over roughly threemagnetic activity cycles. In both physical space and spectral space, it is clear that eachcycle has opposite polarity from the preceding cycle.
There is a preference for antisymmetric modes with odd values of `, as would be
expected from the Ω-effect acting on a poloidal field that is preferentially symmetric about
the equator (even `). The upscale cascade involving odd modes is expected from boththeoretical and observational studies families of dynamo modes (see Nishikawa & Kusano,
2008; DeRosa et al., 2012). As a reversal occurs we see power showing up first at moderate` and then cascading upscale to smaller ` values until it peaks at ` = 3 or 1 depending on
the cycle. The reversal starts at 25 . ` . 29 and then each successive mode reverses. Thereis considerable overlap between cycles, in some cases reversals are seen in the high-` modes
in as little as a hundred days after the previous reversal is completed at low-`. We notethat convective power peaks at spherical harmonic degrees between about 25 and 40 in thesesimulations. This suggests that the reversals are caused by turbulent processes interactingwith the wreaths, yielding an upscale energy transfer which organizes the large-scale fields.
Combined with our cross-correlation analysis, this upscale transfer indicates the key role ofconvection in connecting mean toroidal magnetic fields with the fluctuating toroidal EMF.
As illustrated schematically in Figure 6.10, the reversal mechanism involves three mainprocesses. First, axisymmetric wreaths of toroidal magnetic field (Figure 6.10(a)) lead to127
correlations in the non-axisymmetric poloidal velocity and magnetic fields which drive an
axisymmetric turbulent EMF through an α-like effect. The upscale transfer of magnetic
energy and the fact that the correlation between the magnetic energy of the wreaths and theturbulent EMF peaks on roughly a convective overturning time would seem to point towards
the convective motions as a key player in this α-like process. In the second step, the turbulent EMF reinforces the dominant poloidal field at mid-latitudes but is the opposite signnear the equator (Figure 6.10(b)), creating an octopolar configuration, with strong radial
field concentrations at low latitudes (Figure 6.10(c)). As the reversal progresses, the regionof new poloidal field shown in red in Figure 6.10(c) will expand and eventually replace theold sense of field shown in blue. The third step involves axisymmetric poloidal magnetic field
being sheared by differential rotation. Here the differential rotation is largely cylindrical,
thus radial poloidal field is primarily converted into toroidal magnetic field through the Ωeffect, which results in axisymmetric toroidal fields of the opposite polarity (Figure 6.10(d)).
The process then repeats with the opposite polarity.

6.2

Turbulence-Regulated Flux Emergence
Photospheric active regions are thought to arise from the buoyant destabilization, rise,and emergence of coherent, subsurface toroidal flux structures. It is often argued that thesesubsurface flux structures originate below the convection zone, where the strong shear of
the tachocline promotes toroidal flux generation and the subadiabatic stratification of theovershoot region promotes flux storage by inhibiting the buoyancy instability (Galloway &Weiss, 1981; van Ballegooijen, 1982). In this section we offer an alternative viewpoint thatis inspired and supported by the numerical models presented here. Namely, we argue thatbuoyant flux structures may be produced in the Sun and stars not only in the tachocline
but also in the lower convection zone through the combined action of rotational shear andturbulent intermittency.

128
In previous papers we have demonstrated that organized systems of toroidal flux can
persist within a turbulent convection zone despite the inhibiting influence of turbulent dispersal (Brown et al., 2010, 2011). Here we have demonstrated that this continues to holdas we decrease the diffusion, crossing a threshold beyond which resolved motions replace
artificial dissipation in the dynamical balances that sustain mean flows and fields. Furthermore, as the diffusion is decreased, intense, localized wreath cores form where the magnetic
energy density exceeds the surrounding kinetic energy density (Figure 5.2). This trend ishighlighted most dramatically by case S3, where the much lower diffusion promotes coherent wreath cores strong enough to become buoyant, as first demonstrated by Nelson et al.
(2011b).
Here we explore in more general terms the link between magnetic wreaths and flux
emergence, addressing in particular on how it might operate in real stars where the dissipation is many orders of magnitude less than in simulations. We begin by noting thatthe Ω-effect does not just operate on axisymmetric fields; poloidal fields of all longitudinalwavenumbers (m) in the convection zone are converted to toroidal fields (of the same m) andamplified by rotational shear, blurring the distinction between mean and fluctuating fields.
Turbulent intermittency in the surrounding convection can further amplify shear-generated
flux structures, promoting the generation of fibril magnetic fields and coherent, localized
wreath cores (Figure 5.2).
The low Mach number of stellar convection zones ensures that the gas pressure adjusts
rapidly to any imbalance of mechanical and magnetic stresses. Thus, the formation offibril, intermittent flux concentrations (wreath cores) will induce a pressure perturbationδP ∼ Pt − Pm , where Pt is the turbulent (kinetic plus magnetic) pressure of the surrounding
medium and Pm is the magnetic pressure associated with the coherent flux that defines
the wreath core. We have neglected the turbulent pressure within the wreath core which
may be suppressed by magnetic tension, providing a positive feedback mechanism that canfurther promote the formation of coherent, superequiparition wreath cores and buoyant loops

129(Kleeorin et al., 1989; Rogachevskii & Kleeorin, 2007; Käpylä et al., 2012a).
Weak magnetic flux concentrations, Pm < Pt , are not susceptible to buoyancy instabilities because their magnetic pressure is insufficient to balance the surrounding turbulent
pressure, resulting in δP > 0. It is only the strongest wreath cores that develop a pressure
deficit δP < 0, in particular only those cores in which the magnetic pressure Pm exceeds thestabilizing influence of the surrounding convective motions. This implies that a necessary
but not sufficient condition for the wreath cores to become buoyant is that they must be superequipartition relative to the surrounding convection. The surrounding flows may in turnenhance or retard the tendency for such structures to rise. As demonstrated in Figure 5.2,
this is indeed achieved in our simulations and it becomes more pronounced as the artificial
diffusion is reduced, eventually inducing buoyant rise.
If these superequipartition wreath cores form adiabatically, this pressure deficit will beaccompanied by a density deficit  = δρ/ρ ∼ δP/(γP ), established by diverging flows along
the axis of the wreath core. Radiative heating can further warm and rarify the wreath cores,
enhancing the the density deficit to  ∼ δP/P = (Pm − Pt )/P on a time scale ofτr−1 =∂
2r ρT Cp ∂t



∂T
2
r ρT CP κr∂r(6.12)

where κr is the radiative diffusivity (Fan & Fisher, 1996). Inserting values from Model S(Christensen-Dalsgaard et al., 1996) for  ∼ 10−6 yields τr < 100 days through most of thesolar convection zone. This value of  corresponds to an emergence time τe ∼ (2D/g), of
about 10-15 days, where D is the depth of the convection zone, and a magnetic field strengthof B ∼ (8πP )1/2 ∼ 20-40 kG over and above the equipartition value.
Convection can also promote the buoyant rise of a wreath segment by introducing afinite-amplitude undular displacement, resulting in a draining of fluid from from the apexof the loop (Jouve & Brun, 2009; Nelson et al., 2011b; Weber et al., 2011). This couldin principle operate for any field strength but in practice weak fields will by shredded andreprocessed by convection before they emerge (e.g., Fan, 2009).

130
The dynamics discussed here are indeed exhibited by our most turbulent simulation,case S3. Relative to more diffusive simulations, this case generates more regions of strong,
superequipartition fields, as demonstrated in Figure 5.2, and these regions are located incoherent, intermittent wreath cores, as illustrated in Figure 6.11. Figure 6.12 highlights two
examples in which such wreath cores become buoyant and rise. As discussed in Nelson et al.
(2011b) and Nelson et al. (2013a), the loops rise through the convection zone through the
combined influence of magnetic buoyancy and advection, reaching as high as 0.94R before
they are dissipated by diffusion. The wreath which formed these two loops (and two others
not shown) is not axisymmetric; rather, it spans about 180◦ in longitude, reaching peakfield strengths of 45 kG. We expect the process to be even more efficient in stars where theintermittency is presumably much more extreme.
In summary, this paradigm of turbulence-induced flux emergence postulates that the
combined action of turbulent intermittency and rotational shear generates a broad distribution of toroidal magnetic structures and it is only the most extreme events, in the high-B
tail of the pdf, that become buoyant. It is analogous to the theory of turbulence-regulated
star formation, whereby supersonic turbulence in interstellar molecular clouds generates aspectrum of density fluctuations but only the extreme events on the tail of the pdf are denseenough to trigger the Jeans instability and condense to form protostars (Krumholz & McKee,2005). It is also closely related to the negative magnetic pressure instability described by
Kleeorin et al. (1989) (see also Rogachevskii & Kleeorin, 2007; Käpylä et al., 2012a; Kemelet al., 2012), although it does not necessarily rely on the assumptions that underlie that
instability analysis, namely scale separation, the invariance of the small-scale turbulent energy, and the proportionality between variations in the mean and turbulent magnetic energy(attributed to kinematic shredding).
The radial location of the flux bundles that ultimately form active regions depends on
the kinetic energy density in the convection (FKE) relative to that in the differential rotation(DRKE), as well as the efficiency of magnetic pumping. In the simulations presented here,131
DRKE/FKE & 1, suggesting that the generation of the wreaths is efficient enough that they
can persist in the convection zone despite magnetic pumping. If this ratio falls much below
unity, as might be expected for lower rotation rates, the wreaths may get pushed toward thebase of the convection zone. Likewise, if the simulations are over or under-estimating theefficiency of magnetic pumping, this will influence the location of flux generation and thethreshold to trigger the magnetic buoyancy instability. However, the basic paradigm shouldstill be valid.
The scenario outlined here may resolve several current observational and theoretical
puzzles. In particular, the non-axisymmetric nature of turbulence-induced flux emergence isconsistent with the results of Stenflo & Kosovichev (2012) who find that many large bipolar
active regions on the Sun violate Hales’s polarity rules, and furthermore, that the anti-Haleregions often occur at the same latitude as bipoles that obey Hale’s rules. The fractionof anti-Hale magnetic regions increases from about 4% for the largest active regions (fluxΦ & 1023 Mx) to more than 25% for smaller bipoles with Φ ∼ 1020 Mx. The result that
more than 70% of intermediate-sized bipoles (Φ ∼ 1020 Mx) obey Hale’s laws suggests thepresence of organized toroidal flux systems throughout the convection zone since all of theseregions are unlikely to be anchored in the tachocline. Meanwhile, the diminishing of magnetic
activity patterns with decreasing flux, including an increasing fraction of anti-Hale bipoles aswell as an increased scatter in tilt angles and emergence latitudes, is often attributed to the
influence of convection on rising flux tubes (Jouve & Brun, 2009; Weber et al., 2011, 2012;
Jouve et al., 2013; Weber et al., 2013). We propose that this intimate coupling between flux
tubes and convection exists not only in their rise, but also in their very formation. Finally,
the non-axisymmetric nature of turbulence-induced flux emergence may also account for thephenomenon of active longitudes (Nelson et al., 2013a).
The observed tilt angles and emergence latitudes of bipolar magnetic regions on the
Sun is best reproduced by models of rising flux tubes with initial field strengths of 20-100kG(e.g., Fan, 2009; Jouve & Brun, 2009; Weber et al., 2011; Pinto et al., 2011). However,

132
generating such super-equipartition fields is not a trivial matter and in fact represents aformidable, unresolved problem in solar dynamo theory (e.g., Rempel & Schüssler, 2001).
Laminar amplification of toroidal fields by rotational shear, the Ω-effect, tends to saturate atfield strengths well below equipartition due to the back-reaction of the Lorentz force (Vasil &Brummell, 2009; Guerrero & Käpylä, 2011). Turbulent intermittency can help by tapping theenergy in the convection that is ultimately provided by the solar luminosity. It is clear from
Figure 5.2 that the coupled action of turbulence and shear can generate superequipartition
fields of the required amplitude.
The paradigm proposed here may also help address other difficulties with tachoclinebased dynamos discussed by Brandenburg (2005). For example, toroidal flux generation does
not rely on the radial shear of the tachocline, which is maximum near the poles. Instead,the expected location of flux generation is where |∇Ω| is maximum in the convection zone.
This corresponds to the latitudinal shear at mid-latitudes, precisely where active regions firstemerge at the beginning of a cycle, as emphasized by Spruit (2011). Note that the potential
role flux emergence plays in establishing the solar cycle is a separate question that we do
not address here.

6.3

Richness of stellar dynamos
In this chapter we have explored the complex behavior of a class of numerical simula-tions of convective dynamo action in rapidly rotating solar-like stars. More broadly, however,
we have also touched upon the rich landscape of convective dynamo simulations by discussing
both persistent wreath-building dynamos such as cases D3 and D3-pm1, and cyclic wreathbuilding dynamos including cases D3a, D3b, D3-pm2, and S3. Although the simulations
considered here are ostensibly rotating three times faster than the Sun (3Ω ), the Sun may
actually be in a similar Rossby number regime, as noted in §1. Thus the results presented
here may have some bearing on the solar dynamo as well as the dynamos of younger, more

133
rapidly-rotating solar analogues.
We have focused on two open questions that arose out of our previous work on wreathbuilding dynamos. The first is “Can magnetic wreaths persist in the highly turbulent conditions of a stellar convection zone” and the second is “What physical mechanisms establish
and regulate the magnetic cycles we see in our simulations?” We have also touched upon athird question that all solar and stellar dynamo models must eventually face, and that is“How are sunspots and bipolar active regions produced from dynamo-generated magnetic
fields?”.
The principal issue with regard to the first question is whether magnetic wreaths canpersist in stellar convection zones where the magnetic, viscous, and thermal diffusion coefficients are many orders of magnitude lower than in our simulations. We have investigatedthis question by systematically decreasing the diffusion in our simulations along two complementary paths, one in which only the magnetic diffusion coefficient, η, was reduced, and one
in which the magnetic and viscous diffusivitiy, η and ν, were reduced together, keeping themagnetic Prandtl number constant (at a value of 0.5). In both cases magnetic wreaths withquasi-cyclic polarity reversals were attained, although the constant Pm branch exhibited
more regular spatial and temporal behavior and thus became the focus of our analysis (seeFigure 6.1).
Although no simulation can approach the extreme parameter regimes of stellar interiors, we have demonstrated a shift in the dynamical balances that bodes well for the possible
persistence of magnetic wreaths at much higher Reynolds and magnetic Reynolds numbers.
In short, our simulations suggest that the answer to the first question may be “Yes, magnetic
wreaths may indeed occur in actual stars”. We have investigated in particular the balanceof angular momentum transport which maintains the differential rotation in our simulations
and the balance of processes responsible for creating and destroying the magnetic energy of
the wreaths. In both cases, as we move from case D3 to case D3b we find that resolved
turbulent dissipation has taken the place of SGS dissipation (see Figures 5.4 and 5.6). This

134
is an important milestone towards demonstrating that wreaths can exist in highly turbulent
settings and that they are not reliant on the explicit diffusion in previous simulations.
We have found that magnetic wreaths persist in our higher-resolution, lower-dissipation,
more turbulent simulations, yet their nature is altered in a fundamental and significant way.
Most notably, they are no longer axisymmetric. In our more turbulent simulations suchas case D3b, the nearly axisymmetric wreaths of case D3 are replaced by coherent wreath
segments, typically spanning between 45◦ and 270◦ in longitude. This is associated with ashift in the magnetic power spectrum from longitudinal wavenumber m = 0 to moderate mvalues. It also has important implications for flux emergence, as discussed with regard toquestion 3 below.
The first clues as to the origin of magnetic cycles in our simulations (question 2) were
uncovered by Brown et al. (2010, 2011), showing that one can move from a persistent wreathbuilding dynamo state to a cyclic one by increasing the rotation rate. Here we have shownthat a similar transition from persistent to cyclic wreaths can be achieved by decreasing
the effective magnetic diffusion, and thereby increasing the magnetic Reynolds number at
a fixed rotation rate. As mentioned above, the constant-Pm branch of solutions exhibited
more regular cyclic behavior despite the higher degree of turbulence.
We have not obtained a definitive exposition of the physical mechanisms that give riseto and regulate magnetic reversals. However, we have traced their operation to the zonal
component of the turbulent electromotive force (EMF) near the equator. In case D3 diffusion
prevented reversals in the polarity of the axisymmetric poloidal field by locally offsetting thecreation of mean poloidal field by turbulent fluctuations. In the lower-dissipation case D3b,
this balance breaks down, leaving a residual turbulent EMF near the equator that creates
poloidal field with a polarity that is opposite to that of the pre-existing field, as shown in
Figure 6.6. Once magnetic reversals are thus initiated, the overall reversal process follows
the schematic description found in Figure 6.10.
Our simulations cannot directly address the third question regarding how solar and135
stellar dynamos produce sunspots and bipolar active regions. The detailed dynamics of flux
emergence are too intricate to reliably capture in any current global dynamo simulation.
However, the change in the nature of the wreaths as the dynamical balances shift suggeststhat they may play an important role in generating buoyant magnetic loops in actual stars.
As discussed in §7, these simulations suggest that strong, coherent magnetic structures ofmoderate angular extent can be created in the cores of the magnetic wreaths. If this trend
were to continue to the extremely low diffusion regimes of actual stellar convection zones,
one would expect these flux bundles to become buoyantly unstable and rise. Indeed, thisexpectation is confirmed by our simulation Case S3 that employs a less diffusive SGS modeland that exhibits the self-consistent generation of buoyant toroidal flux tubes in a global
convective dynamo simulation. This picture of flux emergence as a fundamentally turbulent
process contrasts strongly with more idealized scenarios where the principal role of convection
is simply to produce a differential rotation. One might expect this revised paradigm tohave observable consequences in such active region characteristics as distribution, tilt angle,
and helicity. Furthermore,it may call into question our traditional reliance on sunspots
as a straightforward proxy for the axisymmetric toroidal field at or below the base of theconvection zone.
The rich behavior of these systems provides important insight into the dynamo models
for the Sun and solar-type stars. The trend towards non-axisymmetric fields with enhanced
turbulence, while still maintaining global-scale organization, pushes at the boundaries of ourunderstanding of dynamo theory in solar-like settings. That these mechanisms are accessible
with current computational resources clearly invites further intensive study of these topics.

136

Figure 6.2: Reversal in magnetic polarity of the toroidal wreaths in case D3b shown in Bφ
in Mollweide projection at mid-convection zone on left, and in hBφ i in longitudinal averageover latitude and radius on right. Color indicates strength of toroidal magnetic field withthe color table saturating at ±7 kG for the Mollweide images and ±3 kG for the longitudinal
averages. Five snapshots corresponding to t2 through t6 from Figure 6.1(c) are shown eachseparated by roughly 120 days.

137

Figure 6.3: (a) Time evolution in case D3-pm1 of the axisymmetric toroidal magnetic field
at 0.79 R over roughly 15 years of simulated time. Strong variability of the mean fields
is seen in both hemispheres. (b, c) Companion snapshots of Bφ at 0.84 R showing the
spatial variability and non-axisymmetric nature of the wreaths. Successive snapshot times
are indicated by dashed lines in (a).

138

Figure 6.4: (a) Time evolution in case D3-pm2 of the axisymmetric toroidal magnetic
field at 0.79 R over roughly 13 years of simulated time. Strong variability of the mean
fields is seen in both hemispheres, along with irregular reversals in polarity, at times in only
one hemisphere and at other times globally. (b, c) Companion snapshots of Bφ at 0.84
R showing the spatial variability and non-axisymmetric nature of the wreaths. Successive
snapshot times are indicated by dashed lines in (a).

139

Figure 6.5: Time-evolution of hAφ i between ±45◦ latitude for case D3 over about 500 days.
Times t1 and t2 for case D3 correspond to times indicated in Figure 6.1. Shown are hAφ i
at (a) the beginning and (b) end of the time interval, (c) the net change between thosetimes ∆hAφ i, the changes in hAφ i due to (d) the fluctuating EMF (∆Aφ )FE , (e) the mean
EMF (∆Aφ )ME , and (f) resistive diffusion (∆Aφ )RD . Of particular importance is the regionof positive production in (∆Aφ )FE , which if left unimpeded by diffusion would lead to areversal in global magnetic polarity. The color table has been chosen with a sharp transition
from light blue to yellow around zero, thus low-amplitude signals, such as seen in (c) and(e), are highlighted.

Figure 6.6: Same as Figure 6.5, but for case D3b. Times t2 and t6 for case D3b correspondto times indicated in Figures 6.1 and 6.9. The turbulent EMF induces field of the opposite
sense to that which was present at t2 and is opposed by the resistive diffusion. Note that(∆Aφ )FE and (∆Aφ )RD for both cases are topologically similar, but that (∆Aφ )RD is smaller
in case D3b, rendering it unable to prevent the reversal of hAφ i by the fluctuating EMF
which begins with the positive region near the equator.

140

Figure 6.7: (a-c) Values for the three components of α tensor relevant to the generationof h0φ i as a function of radius and latitude. Values are computed using a singular value
decomposition over approximately 3000 days of simulation time with the assumption thatthese components of α are spatially local and do not vary in time. The αφr component
is very small, whereas the αφθ and αφφ components show significant spatial variability andcomparable amplitude. (d-f) Values for components of αφj hBj i, showing the effect of eachcomponent on h0φ i. Magnetic fields have been averaged over the same interval as in Figure 6.6(about 480 days). Here the contribution of αφφ hBφ i is dominant, with a smaller but still
significant contribution by αφθ hBθ i. This figure has been corrected from the version included
in Nelson et al. (2013b). A minor error was discovered after publication that changed thedetails of this figure, but did not substantially alter the conclusions drawn from this figure.

141

Figure 6.8: Magnitude of cross-correlation in time of {0φ } and {Br } (green), {Bθ } (red),and {Bφ } (blue) for case D3b. Cross-correlation is computed as a function of the temporal
offset τ∆ ,/ with negative offsets indicating magnetic fields precede the toroidal EMF. Also
shown are the 2σ confidence levels (dashed), computed using a Markov chain Monte Carlo
method (Wall & Jenkins, 2003). The only statistically significant peaks are those relating{0φ } and {Bφ }.

142

Figure 6.9: Companion plots of the time evolution in case D3b of (a) the spherical harmoniccoefficients for antisymmetric modes with 1 ≤ ` ≤ 29 and m = 0 for Bφ , and (b) hBφ i in
physical space as a function of latitude, both at mid-convection zone. Dashed lines show
times referenced in Figures 6.1, 6.2, 6.5, and 6.6. A factor of (−1)(`−1)/2 is applied tothe spherical harmonic coefficients to remove the effect of the wreaths confinement to low
latitudes. There is a clear progressive spectral transfer of magnetic energy from high ` modes
to low ` modes as each cycle progresses. Reversals begin at moderate scales (high `) andthen progress to large scales (low `).

143

Figure 6.10: Schematic description of the reversal mechanism for cyclic convective dynamos
in four steps. (a) Two toroidal wreaths at low latitude which generates a turbulent EMF
via a nonlocal “α”-effect, either through nonlinear interactions across the equator or via
helical convection. The sign of the EMF changes at roughly the location of the wreaths. (b)Correlations in turbulent poloidal velocities and fluctuating magnetic field drive an inductionof mean poloidal field which is roughly octopolar. (c) Mean poloidal field near the equator issheared by differential rotation to generate mean toroidal field through the Ω-effect. In thesesimulations, the largest component is the shearing of radial field lines by radial gradients inthe differential rotation. (d) Toroidal wreaths of opposite polarity are generated.

144

Figure 6.11: Three-dimensional volume renderings of isosurfaces of magnetic field amplitude in case S3. Blue surfaces have amplitudes of 10 kG, green surfaces represent 25 kG,
and red surfaces indicate 40 kG fields. Grid lines indicate latitude and longitude at 0.72 R
as they would appear from the vantage point of the viewer. Small portions of the cores ofthese wreaths have been amplified to field strengths in excess of 40 kG while the majority of
the wreaths exhibit fields of about 10 kG or roughly in equipartition with the mean kinetic
energy density (see Figure 5.2).

145

Figure 6.12: Buoyant magnetic loops evolving from small-scale wreath sections amplified
by turbulent intermittency. (a) Field line rendering of magnetic wreaths at low latitudes incase S3. Field lines are colored by Bφ (negative in blue, positive in red) to highlight the two
wreaths present. (b) Zoom-in on region indicated in (a) showing field line tracings of the~ (weakcore of the buoyant magnetic loops at the same instant colored by magnitude of B
fields in purple, intense fields in yellow). Volume rendering shows Bφ using the same color
scheme as in (a). (c) The same region 4 days later, showing the continued rise of the loops
through the stratified domain and their expansion.

Chapter 7
Buoyant Magnetic Loops Over Magnetic Activity Cycles

The dynamo models discussed in the previous three chapters offer a new paradigm influx emergence. Previous models had sought laminar generation mechanisms for the strong
toroidal field at the base of the convective layer. In contrast we have provided an example
of a dynamo simulation where turbulent intermittency can create relatively highly coherent,
small-scale magnetic fields with energies well in excess of equipartition with the convective
kinetic energy which surrounds them.
Now we turn to a detailed analysis of the data set for case S3, which yields a large
number of buoyant magnetic loops. In this chapter we will explore the properties of thesewreaths and the variety of buoyant magnetic structures this simulation can achieve. In thefollowing chapter we will examine the statistical properties of these loops over a magnetic
activity cycle. This is the first of two chapters which are largely quoted from Nelson et al.
(2013a)1 . I was the primary author of this paper, and as such I carried out all of thesimulation and analysis work. I primarily wrote the text of the paper. My co-authors
provided essential contributions in formulating the simulations and the analysis, and inproviding significant guidance in the writing process.
1

Nelson, N. J., Brown, B. P., Brun, Sacha A., Miesch, M. S., & Toomre, J. 2013b, Buoyant Magnetic
Loops Generated by Global Convective Dynamo Action, Sol. Phys. Online First, DOI:10.1007/s11207-0120221-4

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7.1

Overview
Our global 3D simulations of convection and dynamo action in a Sun-like star revealthat persistent wreaths of strong magnetism can be built within the bulk of the convention
zone. Here we examine the characteristics of buoyant magnetic structures that are selfconsistently created by dynamo action and turbulent convective motions in a simulation
with solar stratification but rotating at three times the current solar rate. These buoyant
loops originate within sections of the magnetic wreaths in which turbulent flows amplify
the fields to much larger values than is possible through laminar processes. These amplified
portions can rise through the convective layer by a combination of magnetic buoyancy andadvection by convective giant cells, forming buoyant loops. We measure statistical trends
in the polarity, twist, and tilt of these loops. Loops are shown to preferentially arise inlongitudinal patches somewhat reminiscent of active longitudes in the Sun, although broader
in extent. We show that the strength of the axisymmetric toroidal field is not a good predictorof the production rate for buoyant loops or the amount of magnetic flux in the loops thatare produced.
7.2

Flux Emergence and Convective Dynamos
Convective dynamo action in the interior of the Sun is source of the magnetism which

creates sunspots and drives space weather. Such magnetism is not limited to the Sun, as
magnetic activity is observed to be ubiquitous among sun-like stars. To understand theorigin of sunspots and starspots, the processes which generate magnetic structures and thentransport them through the convection zone to the surface must be explored. Here we
present the results of a global numerical simulation, called case S3, which self-consistently
generates wreaths of strong magnetic field by dynamo action within the convective zone.
Case S3 models the convection zone of a sun-like start nominally rotating at three times thecurrent solar rate, or 3Ω . The wreaths reverse polarity in a cyclic fashion, yielding cycles

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of magnetic activity. Portions of these wreaths form buoyant magnetic structures, or loops,
which rise through our convective envelope. Initial results on the behavior of a small numberof these loops were reported in Nelson et al. (2011b).
Here we discuss the properties of a much larger number of loops in order to get a statistical description of their properties. We find coherent magnetic structures with a variety
of topologies, latitudinal tilts, twists, and total fluxes. Additionally, we observe only a weakcorrelation between the unsigned magnetic flux in a buoyant loop and the axisymmetrictoroidal magnetic field at that latitude and time, indicating that the generation mechanism
for these loops relies on local, coherent toroidal field structures amplified by turbulent intermittency rather than large-scale instabilities of axisymmetric fields. We also find evidence
for longitudinal intervals which preferentially produce buoyant loops, hinting at a possible
origin for active longitudes for sunspots (Henney & Harvey, 2002), though out intervals arequite broad.
7.3

Nature of the Simulation
We use the 3-D anelastic spherical harmonic (ASH) code to model large-scale convective

dynamo action in the solar convective envelope. ASH solves the anelastic MHD equations
in rotating spherical shells (Clune et al., 1999; Brun et al., 2004). ASH is limited to thedeep interior due to the anelastic approximation, which limits us to low Mach number flows.
Additionally we stay away from the near-surface layers because we cannot resolve the small
scales of granulation and super granulation realized near the photosphere. Our simulation
extends from 0.72 R

to 0.965 R , covering a density contrast of about 25 from top tobottom. The details of the numerical scheme used in case S3 are described in Brown et al.
(2010), and the specific parameters are given in Nelson et al. (2013b). Of special note, incase S3 the Rossby number is 0.581, which is in the same rotationally influenced regime as
the giant cell convection realized in the solar interior (Miesch, 2005). Thus the dynamics
in case S3 may be broadly applicable to stars like the Sun in which rotational influences on149

Figure 7.1: (a) Snapshot of radial velocities vr at time t1 = 716 days in case S3 on a sphericalsurface at 0.95R shown in Mollweide projection (equator at center, lines of constant latitude
parallel) in proportional size (outer ellipse represents photosphere). (b) Rotation rate Ω
averaged in longitude and time. Strong differential rotation is achieved in radius and latitude
over the simulated domain. (c) Companion snapshot of toroidal magnetic field Bφ at 0.84R ,with a strong coherent magnetic wreath in each hemisphere (blue negative, red positive,
ranges labeled), with considerable small-scale fields also present. (d) Azimuthally-averaged
toroidal magnetic field hBφ i at the same instant. Low-latitude wreaths are evident in bothhemispheres .

convective motions are significant.
To achieve very low levels of diffusion, we employ a dynamic Smagorinsky subgridscale (SGS) model which uses the self-similar behavior in the inertial range of the resolved
turbulent cascade to extrapolate the diffusive effects of unresolved scales. In this model
the viscosity at each point in the domain is proportional to the magnitude of the strain

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Figure 7.2: Evolution in time of the longitudinally-averaged toroidal magnetic field hBφ i
at mid-convection zone shown in a time-latitude plot. Three magnetic reversals are realized,
each with a period of about 280 days (reversals indicated by hash marks, cycles labeled 1-3
for convenience). Considerable asymmetry is seen between hemispheres in both the phaseand amplitude of the reversals. The time t1 at which snapshots in Figures 7.1 and 7.3 aresampled is indicated by the dotted line at 716 days.

rate tensor and the constant of proportionality is determined using the resolved flow and anassumption of self-similiar behavior. A detailed description of the dynamic Smagorinsky SGS
model is provided in Chapter 3. Here we employ constant SGS Prandtl and magnetic Prandtl
numbers of 0.25 and 0.5, respectively. In practice this permits a reduction in the average
diffusion by about a factor of 50 compared to a simulation with identical resolution and a lesscomplex SGS model, such as in Brown et al. (2011). This reduction in diffusion is critical not
only in enhancing the turbulent intermittency of the magnetic field, but also in permitting
the buoyant loops to rise through the convective layer without diffusive reconnection altering
their magnetic topology.
Figure 7.1(a) shows a snapshot of the convective radial velocities vr in case S3 at a single
instant. The convection near the equator is dominated by convective rolls aligned with therotation axis, while the higher latitudes have more vortical motions. The rotational influenceon the convective motions is key to achieving a pronounced differential rotation (Miesch et al.,1512006). Case S3 maintains strong gradients in angular velocity Ω (Figure 7.1(b)), which are
key to generating the large-scale magnetic wreaths through the Ω-effect. Figures 7.1(c-d)
show snapshots of the wreaths, both on a spherical surface at mid-convection zone and intheir axisymmetric component. The wreaths are dominated by non-axisymmetric fields andthus have a limited longitudinal extent, while clearly still retaining global coherence.
Remarkably, the wreaths are generated and maintained in the bulk of the convective layer without a tachocline of shear. It had been reasonably postulated that coherent,
large-scale fields in the convection zone would be shredded by the intense turbulence ofthe convective motions. However, the convective turbulence evidently does not destroy thewreaths. In fact, Nelson et al. (2013b) showed that while the axisymmetric fields show somedecrease in amplitude with increased turbulence, regions of extremely strong fields actually
become more common due to increased turbulent intermittency. In regions of particularly
strong magnetic fields, the convective motions are diminished by the Lorentz force, resulting
in even less convective disruption of the wreaths.
The dynamic Smagorinsky procedure requires additional computational expense, limiting the time evolution of our simulations. Case S3 presented here was run for 3.4 million
time steps, with an average of 40 seconds of simulated time per step. In total, case S3 covers about 4 years of simulated time, compared to the rotational period of 9.3 days and theconvective over-turning time of about 50 days. Figure 7.2 shows the time evolution of theaxisymmetric toroidal magnetic field hBφ i in case S3 over about 1100 days. In this interval
there are three reversals of global magnetic polarity. While the true polarity cycle involves
two reversals, we term the interval between each reversal an activity cycle in the same way
the Sun’s 11-year activity cycles are just about half of the true 22-year polarity cycle. These
three activity cycles have durations of about 280 days, although the reversals are not generally synchronized between the two hemispheres. This nonuniform behavior hints at theimportant role of asymmetries in the flows between the two hemispheres (DeRosa et al.,2012).

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7.4

Identifying Magnetic Loops
In order to provide a consistent treatment, we define a magnetic loop as a coherent

segment of magnetic field which extends from below 0.80R to above 0.90R and back down
again (Nelson et al., 2011b). Additionally, we require that the buoyant loops have peakmagnetic field strengths greater than 5 kG above 0.90R at selected samples in time. To findmagnetic loops fitting that description, we have developed a pattern-recognition algorithm
which searches the 3-D volume of our simulation. The most direct method of finding loops
is to look for magnetic field lines which pass through a region where |Bφ | > 20 kG below
0.80R , then pass through a region above 0.90R with |Bφ | > 5 kG, and then again througha region where |Bφ | > 20 kG below 0.80R over less than 50◦ in longitude. In practice, thiscan be done much more efficiently by recognizing that the loops start as primarily toroidal
magnetic field structures, but that as they rise into a region of faster rotation the loops aretilted in longitude so that one side of the loop retains a strong component of Bφ while theother becomes almost totally radial. Thus we initially identify loop candidates by looking
for this pattern of Bφ and Br . The loop candidates are then verified using field line tracings.
Case S3 uses 1024 grid points in longitude, 512 in latitude, and 192 in radius for 8~ entropy S, and pressure P ), thus each
evolution variables (velocity ~v , magnetic field B,
snapshot in time requires over 3 GB of data. We are therefore limited in the number of time
steps we can analyze. For the 278 days of cycle 1 we have run our loop finding procedureon snapshots of the simulation spaced roughly every 4 days. In doing so we have identified
131 buoyant loops. Additionally we have sampled cycle 2 for 20 days and cycle 3 for 40 days
with the same 4 day cadence and found 27 additional loops. We anticipate that we wouldfind many more loops if we carried out a more complete search through cycles 2 and 3.
For a subset of the 158 loops found in case S3, we have carried out detailed analysesof the dynamics of the rise of 22 of the loops (11 from cycle 1 and 11 more from cycle 3).
To do this we have used data with a time resolution of about 10 hours, which is sufficient to153

Figure 7.3: (a) Volume-renderings of magnetic field lines at low latitudes colored by toroidal
field Bφ (red positive, blue negative, amplitudes labeled). Strong magnetic wreaths exist
in each hemisphere with considerable modulation in longitude. The location of two sample
buoyant loops (labeled loops 93 and 98) are indicated. In this view it is difficult to distinguishthe loops from the surrounding magnetic fields. (b-c) Close-up views of loops 93 and 98 at
the same instant with only field lines comprising the buoyant loops rendered for visual clarity.
Color shows magnitude of magnetic field strength (yellow weak, purple strong). Loop 93 ispart of the negative polarity wreath in the northern hemisphere, while loop 98 is part ofthe positive polarity wreath in the southern hemisphere. Time shown corresponds to thesnapshots in Figure 7.1 and t1 in Figure 7.2.

track loops backward in time from their peak radial position to their origins in the magnetic
wreaths. We find that while the specific evolution of each of these 22 loops varies due to thechaotic nature of the turbulent convection, all 22 loops have significant acceleration due tomagnetic buoyancy and are embedded in convective upflows which aid their rise. This agrees
with the dynamics of the sample loop studied in detail in Nelson et al. (2011b). While wecannot with certainty say that magnetic buoyancy was a significant factor in the rise of all
158 magnetic loops, we find that for all 22 of the loops studied at high time resolution the
average ascent speed due to magnetic buoyancy alone is at least 28% of the total average

154
ascent speed. Thus we assume that magnetic buoyancy is at least an important factor in therise of these loops.
Figure 7.3 displays the complex nature of the magnetic fields in case S3 with a volume
rendering of magnetic field lines in the convection zone at low latitudes, forming two prominent magnetic wreaths of opposite polarity. We also indicate the location of two buoyant
magnetic structures, labeled loops 93 and 98. The simulation continuously exhibits magnetic
fields throughout the convection zone, including strong, small-scale magnetic fields, coherent buoyant loops, and large-scale wreaths with global scale organization. Prior studies ofmagnetic buoyancy typically involved specified buoyant magnetic structures whose rise wasstudied in a largely unmagnetized domain. In contrast, our convection zone has on average
77% of our simulated volume containing magnetic fields in excess of 1.5 kG, and 21% posses
field amplitudes in excess of 5 kG. This makes identification of the buoyant loops difficult.
Figures 7.3(b-c) show close up renderings of only the field lines comprising buoyant loops 93
and 98. We have omitted rendering other field lines in those regions for visual clarity. Magnetic fields in the loops can be quite strong even near the top of our domain, with portionsof loop 93 exceeding 25 kG at 0.92R .
7.5

Properties of Rising Loops
Unlike many previous models of buoyant magnetic transport in which convective turbu-

lence is presumed to play a purely disruptive role, the buoyant loops in our models fall under
the turbulence-enhanced magnetic buoyancy paradigm discussed in Nelson et al. (2013b).
In this model turbulent intermittency plays a key role in the formation of strong, coherent
structures which are magnetically buoyant and can be advected by convective upflows. As
was shown in Nelson et al. (2011b), these loops rise through a combination of magnetic
buoyancy and advection by giant cell convection. Thus convection plays a key role both in
the dynamo which generates the buoyant magnetic fields, and also in the transport of themagnetic loops. Due to the cooperation between convective motions and magnetic buoy-

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Figure 7.4: Location of three buoyant loops (labeled loops 1, 15, and 18) as a function oftime as they rise from the core of the toroidal wreaths in the lower convection zone throughthe simulated domain to their peak radial positions between 0.91 and 0.95 R . Times are
given relative to the launch of the loops with offsets for clarity. Also plotted are the mean
motions of the loops at each time interval due to magnetic buoyancy (red lines) and advection
by the surrounding convective upflows (blue line). Additional motions due to forces such as
thermal buoyancy, viscous drag, and magnetic tension are not plotted, and account for what
may appear to be missing in this display.

ancy, the loops are able to rise from below 0.80R to above 0.90R in as little as 12 days,
as suggested by Figure 7.4.

7.5.1

Dynamics and Timing of Loop Ascents

Buoyant loops are born from the much larger and less coherent magnetic wreaths
shown in Figures 7.1 and 7.3. The wreaths in case S3 are not axisymmetric structures andare typically coherent over spans of between 90◦ and 270◦ in longitude. Wreaths exhibit ahigh degree of magnetic connectivity with the rest of the convection zone, with field lines
threading in and out, suggesting rather leaky overall structures. Wreaths in case S3 generally
have average field strengths of between 10 and 15 kG and are confined in the lower half ofthe convection zone by magnetic pumping. In the core of the wreaths convective motions
can be limited by Lorentz forces to as little as a 1 m s−1 .
Portions of these wreaths can be amplified by intermittency in convective turbulence.

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Figure 7.5: Sequence of volume renderings of magnetic field lines which comprise threebuoyant loops (labeled loops 1, 2 and 3) as they rise through the convective layer with threedays between each frame (times indicated, progressing downward). The expansion of eachloop is here evident as they rise. Magnetic buoyancy and advection by convective upflows
allow the loops to traverse the radial interval shown here in roughly 10 days. Loop 1 is also
shown in Figures 7.4 and 7.7. Loop 3 is also shown in Figure 7.6.

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Turbulence has been shown to generate strong, coherent structures in a variety of settings(Pope, 2000). In case S3 localized portions of the wreaths are regularly observed to attainfield strengths of 40 kG and be highly coherent over as much as 50◦ in longitude. Thesemagnetic structures with strong fields are able to rise into regions where vigorous convective
motions are present. Many structures are seen to emerge from the core of the wreaths only tobe pummeled by a convective downflow, disrupted by a region of unusually strong turbulence,or limited by the development of a particularly unfavorable magnetic configuration. Whether
any given magnetic structure becomes a buoyant magnetic loop is therefore not due to thepassing of some threshold, but largely a conspiracy of favorable events.
Figure 7.4 shows the radial location of the top of three different loops as they rise fromroughly 0.77R to above 0.90R . Also plotted are the contributions to the radially outward
motion due to magnetic buoyancy and advection by convective upflows. The buoyant acceleration due to magnetism is deduced by comparing the density in the region within theloop and the density of the surrounding convective plume. We do this to separate magnetic
and thermal buoyancy effects. Each of these three loops starts in a region where convective
motions are largely suppressed by Lorentz forces due to the very strong magnetic fields in the
cores of the magnetic wreaths. As they begin to rise, the magnetic energy at the core of thewreath exceeds the kinetic energy of the flows locally by a factor of 10 to 100. As the loops
rise, they enter regions of strong upflows and are advected upwards by the convective giant
cells. Averaged over their entire ascent, magnetic buoyancy drives an average upward speedof about 50 m s−1 for these three loops, in addition to the surrounding upflows which moveat an average of about 80 m s−1 . At their maximum radial extent, the loops are prevented
from rising further by our impenetrable upper boundary condition.
Figure 7.5 shows three sample loops (labeled loops 1, 2, and 3) as they rise over 10
days. The loops remain coherently connected as they rise. Here again all three loops are
aided by convective upflows while convective downflows pin the ends of the loops downward.
The direction of motion is largely radial with a deflection of as much as 10◦ in latitude toward158
higher latitudes. This deflection is largely due to the roughly cylindrical differential rotation
contours realized in this simulation.
Loops expand as they rise through the stratified domain, but less than would be expected for a purely adiabatic rise. Without any diffusion or draining of material along thefield lines, the cross-sectional area of the loops should be inversely proportional to the background pressure, leading to expansion by roughly a factor of 20. Instead loops are seen
to expand by a factor of 5. This is consistent with previous studies of buoyant magnetic
structures in which expansion of magnetic structures is seen to be inversely proportional to
the square root of the change in pressure (Fan, 2001; Cheung et al., 2010) .
The expansion of the loops is slowed by draining flows of higher entropy fluid along
magnetic field lines, which serves to cool the material at the top of the loop. These divergent
flows are too small to be measured in individual loops due to the turbulent background, butwhen averaged over 158 loops, a mean divergent flow of 47 cm s−1 is obtained along the top of
the loops. This compares well with estimates from a simple model (neglecting viscosity andthermal diffusion) which assumes that the draining flows are constant in time and uniform
perpendicular to the axis of the loop.
Axial flows along loops are also seen as the loops rise through regions of faster rotation. When averaging over many loops, a net axial flow of 5.1 m s−1 is detectable in the
retrograde direction, consistent with the fluid inside the loop tending to conserve its specificangular momentum as the loop moves radially outward. Loops often become distorted as
this retrograde motion interacts with the surround prograde differential rotation as the loop
rise across rotational contours (see Figure 7.1(b)).
The geometry of each loops we have examined is uniqued in its details, but Figure 7.6
shows three different perspectives on a single 3D volume-rendering of a typical loop. Loop3, which is also shown in Figure 7.5, is in the northern hemisphere and its top is roughly
centered at 76◦ N latitude and 12◦ W longitude. Its parent wreath-segment runs slightly
north-west to south-east at this location and time, causing the western foot-point to be159

Figure 7.6: Three viewpoints the same volume-rendering of magnetic field lines in Loop
3 at t = 683 days (same as in Figure 7.5(d)). Color indicates amplitude of magnetic field(purple strong, yellow weak). Views are looking (a) south along the rotation axis with gridlines in radius (in units of the solar radius) and longitude, (b) radially inward with grid lines
in longitude and latitude, (c) westward along the axis of the magnetic wreath.

centered further north than the eastern foot-point. The deflection away from the equator isevident in Figures 7.6(b and c) as the top of the loop is roughly 10◦ further north than thefoot-points. The roughly five-fold expansion of the loop’s cross-sectional area can be seen,
particularly in Figure 7.6(c). This loop also shows an asymmetric top due to a downflow
plume impacting the eastern side of the top of the loop, causing the western wide to extend
further in radius.
Loops start with a wide variety of field strengths and sizes and at a variety of initialradial positions. Most loops start between 0.75 and 0.78 R , although loops starting as low
as 0.73 R are evident. When loops are traced backward in time to their starting location inorder to identify the flux which will become buoyant and rise, we find that most progenitorsof loops begin with about 1025 Mx of flux. The structures loose roughly 90% of their flux
as they rise to their peak radial positions between 0.90 and 0.96R . Much of the flux islost as convection in a stratified fluid requires a large fraction of the fluid to overturn prior
to reaching the top of the domain. Figure 7.7 shows the magnetic flux as a function of the160

Figure 7.7: Unsigned magnetic flux in five sample loops as they rise through the convective
layer, including loop 1 (see Figures 7.4 and 7.5). Loops continuously loose magnetic flux
through both diffusion and leakage of fluid. Their ascent is faster than the convective upflows
in which they are embedded, leading to the loss of fluid and flux due to drag-like effects.
Here and in the 22 loops for which detailed tracking is possible, the initial and final magnetic
flux are not correlated.

radial position of the top of the loop for five sample loops. Initial flux and initial radial
location do not appear to be good predictors of either final radial location or final magneticflux.
In the specific case of loop 1, 92% of the magnetic flux it started with is lost over thecourse of its ascent while 69% of the mass flux at 0.78R turns over below 0.91R . The
overturning mass flux carries away 61% of the magnetic flux, as regions of lower field strength
preferentially are lost. The next largest contributor is resistive diffusion, which dissipates
19% of the initial flux. The remaining 12% of the flux is lost through a combination of smallscale turbulent advection and shear. Eventually diffusive reconnection realigns the fields sothat the loops are no longer distinct from the surrounding MHD turbulence.

Chapter 8
Collective Properties of Emergent Magnetic Structures

Having discussed the dynamics of our buoyant magnetic loops, we now turn to a
discussion of their collective properties. This is the second of two chapters which are largely
quoted from Nelson et al. (2013a) 1 . I developed and applied the analysis presented here. I
was also the primary author for the text of that paper.
8.1

Statistical Distribution of Twist and Tilt
Previous MHD simulations of flux emergence have emphasized that magnetic structures

must be twisted to remain coherent as they rise (see review Fan, 2009). Twist in this context
can be defined by a parameter qA , which for a uniformly twisted flux tube is defined as
Bk = a± qA λ ∇ × Ak ,(8.1)

where Bk and Ak are, respectively, the magnetic field and magnetic vector potential along theaxis of the flux tube, a± is 1 in the northern hemisphere and −1 in the southern hemisphere,
and λ is the distance from the axis of the flux tube. For the tube to remain coherent as
it rises, previous numerical simulations have suggested that twist must exceed some criticalvalue QA (Moreno-Insertis & Emonet, 1996). Fan (2008) used 3-D simulations of buoyant
magnetic structures rising through a quiescent, stratified layer and found a critical level oftwist QA ≈ −3 × 10−10 cm−1 .
1Nelson, N. J., Brown, B. P., Brun, Sacha A., Miesch, M. S., & Toomre, J. 2013b, Buoyant Magnetic
Loops Generated by Global Convective Dynamo Action, Sol. Phys. Online First, DOI:10.1007/s11207-0120221-4

162

Figure 8.1: (a) Histogram of twist rate parameter qJ values for the 131 loops observed incycle 1 along with the best-fit Gaussian distribution of those values. The distribution showsa slight preference for negative twist rates, though the mean twist rate is (−1.8 ± 2.4) ×
10−11 cm−1 . (b) Histogram of latitudinal tilt ∆θ values for the same 131 loops. Positive tilts
indicate that the leading edge of the loop is closer to the equator than the trailing edge, as
used with Joy’s law. Tilts have been calculated so that all values fall between ±90◦ for thisanalysis. Positive tilts are preferred and the mean latitudinal tilt is 7.3◦ ± 12.6◦ in latitude.

For our simulation, the loops are clearly not uniformly twisted flux tubes, so we calculate another measure of twist following the procedure used in observational studies (e.g.,
Pevtsov et al., 1995, 2003; Tiwari et al., 2009). Sunspots often show large variations in thelevel and even sign of twist, so a weighted average of the twist parameter is employed, which
we call qJ . We compute the twist parameter as Jφ,q J = a±Bφ

(8.2)

where braces denote an average over radius and latitude for a longitudinal cut taken throughthe loop and a± is 1 in the northern hemisphere and −1 in the southern hemisphere. We
restrict our averages to contiguous regions with the correct polarity and where fields arestronger than 2.5 kG. Figure 8.1(a) shows a histogram of values for the twist parameter qJ
for the 131 loops identified in cycle 1, as well as the best-fit Gaussian to that distribution
which peaks at q¯J = −1.8 × 10−11 cm−1 . For comparison,Tiwari et al. (2009) report anaverage twist parameter of q¯J = −6.12 × 10−11 cm−1 for a sample of 43 sunspots.

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It is difficult to make a direct comparison between the two measures of twist mentioned
here. In practice our loops are poorly represented by uniformly twisted tubes. It is possible
to compute the value of qA at each point in the loop and create an average value, but wefind that those averages are highly sensitive to the weighting of the points and the region
over which the average is taken. Alternatively, we have computed the value of qJ for theformulation employed in Fan (2008) and find that the value varies with the location and sizeof the magnetic structure in radius and latitude. For most reasonable parameter choices,the qJ /qA is between 1 and 2. When comparing with photospheric measurements, we mustalso remember that considerable changes may take place as magnetic flux passes throughthe upper 5% of the solar convection zone. The dynamics of twisted buoyant loops in that
region is beginning to be studied in local domains (Cheung et al., 2010).
Of the 131 loops in cycle 1, only 13 had current-derived twist parameters qJ within
an order of magnitude of the critical value QA . One explanation may be that convective
upflows assisting the rise of these loops reduces the drag that they experience, thus making
them less susceptible to disruption as they rise and therefore less dependent on twist forcoherence. Whatever the cause, we do not see a critical value of twist beyond which loops
are unable to traverse our domain.
Additionally, we can look at the latitudinal tilts of the buoyant loops. We calculatethese tilts by computing the center of each loop at all longitudes where the center is within
0.02R of its peak position and then fitting a linear trend to latitudinal locations of the loop
center. We define positive tilts to be those with the eastern side of the loop closer to theequator than the western side, as used in Joy’s law. Here we do not consider the polarity of
the loops, so values are restricted to the interval [−90◦ , 90◦ ]. The distribution of tilts seenin the 131 loops found in cycle 1 is shown in Figure 8.1(b), along with the best-fit Gaussian
to that distribution, which peaks at 7.3◦ but is quite broad. This is similar to observationsof tilts in sunspots where the trend towards Joy’s law is part of broad distribution in tilt
angles (Li & Ulrich, 2012).

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Figure 8.2: Time-latitude display of toroidal magnetic field averaged in longitude andradius during the peak of cycle 1. Over-plotted symbols indicate the time and latitude of
131 buoyant loops in the style of a synoptic map, with positive polarity loops shown as pinksquares and negative polarity loops as green diamonds. Some loops may be present from theprevious cycle, particularly prior to day 550 in the southern hemisphere. Time t1 at which
the snapshots in Figures 7.1 and 7.3 are taken is indicated by the dotted line.

8.2

Magnetic Cycles with Buoyant Loops
Case S3 achieves three magnetic activity cycles with reversals in global magnetic polar-

ity. If we define the cycle period as the time between changes in the sign of the antisymmetriccomponents of the toroidal field at low latitudes, as in Brown et al. (2011), then cycles 1
and 2 have periods of 278 and 269 days, respectively. Cycle 3 had not ended at the present
end of the simulation, but has been simulated for 228 days. The coexistence of cyclic magnetic activity and buoyant loops provides an opportunity to probe the relationship between
axisymmetric fields which are commonly used in 2-D dynamo models (see review by Charbonneau, 2010) and the buoyant transport of magnetic flux.
We have chosen to conduct our analysis primarily using cycle 1 since the process offinding and characterizing buoyant loops is too data intensive to be carried out conveniently
for all three cycles. Figure 8.2 shows a time-latitude plot of the mean toroidal field (averaged
in longitude and in radius over the lower convection zone from 0.72 to 0.84 R ), as well as
the location in time and latitude of the 131 buoyant loops detected in cycle 1. It is evident

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Figure 8.3: (a) Relative probability that a region with given mean Bφ will produce a
buoyant loop compared to the production rate of buoyant loops in cycle 1 averaged over all
events. For cycle 1 the average occurrence rate for loops was 7.6 × 10−3 day−1 degree−1 .
We normalize all probabilities by this rate, thus the dashed line represents the average loop
production rate. Note that nearly 60% of the times and latitudes considered have mean fieldstrengths of less than 1.5 kG. (b) Total magnetic flux as a function of axisymmetric toroidal
magnetic field averaged in radius at the latitude and time of each of the 131 buoyant loops in
Figure 8.2. While most loops are associated with mean magnetic fields of the correct sense,
there are 15 loops in quadrants II and IV. These loops arise from wreath segments which
are canceled in the longitudinal averaging procedure by large or stronger wreath segments
at the same time and latitude of the opposite polarity.

from this representation that the loops do not arise uniformly in time. Although loops tend
to appear at times and latitudes when the mean toroidal fields are strong, they can also
appear at times and latitudes with relatively weak mean fields. There are even examples
in which loops have the opposite polarity to the longitudinally-averaged mean fields at thattime and latitude. This is consistent with the non-axisymmetric nature of the wreaths shownin Figures 7.1(c-d) where smaller-scale segments of intense toroidal field can be masked in
the longitudinal average by larger segments of the opposite polarity.

8.2.1

Relation of Loop Emergence and Mean Field Strength

In many mean-field models it is assumed that buoyant magnetic flux (which can beused as a proxy for the sunspot number) at a given latitude and time is proportional to theaxisymmetric toroidal field strength at that location and time at the generation depth. In

166
particular, the Babcock-Leighton model postulates that the buoyant transport of magnetic
flux occurs whenever the axisymmetric magnetic field exceeds some threshold value (e.g.,
Durney, 1995; Chatterjee et al., 2004). Here we can test this assumption by looking at theprobability that a region with a given axisymmetric field strength will produce a buoyant
loop. Figure 8.3 shows the relative probability that a region with a given mean field strength
will produce a buoyant loop. Over cycle 1, the average production rate of buoyant loops isroughly one loop every two days within 30◦ of the equator. Regions with hBφ i ≤ 1.5 kG
cover about 60% of the time-latitude domain and produce loops at or below the averagerate. The generation probability per unit time and latitude rises to five times the mean
rate for regions with hBφ i ≈ 3.9 kG. Interestingly, the generation probability then falls forthe regions of the strongest hBφ i. Indeed, the strongest regions of axisymmetric field are
only about three times more likely to produce buoyant loops than the average production
rate. The relatively small sample size invites further study on this topic, as only 5% of
the domain is covered by fields above 4.2 kG, which fall in the last four bins. However the
implication that axisymmetric toroidal fields above some threshold value are less likely to
produce buoyant loops may have significant implications for mean-field models of the solar
dynamo.
To further explore the correlation between the axisymmetric field strength and theamount of buoyant magnetic flux, we can look for correlations between the amount of flux
in a given buoyant loop and the axisymmetric fields at the time and latitude of its launch.
Figure 8.3(b) shows the magnetic flux in each of the 131 buoyant loops from cycle 1 as afunction of the average value of axisymmetric toroidal field in the lower convection zone at
the time of launch. Out of 131 loops, 15 were launched when the axisymmetric Bφ was ofthe opposite sense. Interestingly, Stenflo & Kosovichev (2012) report that roughly 5% ofmoderate to large active regions violate Hale’s polarity law.

167

Figure 8.4: Time-averaged toroidal magnetic field on a spherical surface between ±45◦ oflatitude at 0.79R during cycle 1. Symbols indicate the rotational phase in longitude of the
131 buoyant loops from Figure 8.2 at the time they were launched. Squares indicate positive
polarity loops while diamonds indicate negative polarity loops. Both the wreaths and loops
are confined in longitude. Loops are particularly concentrated in the strong positive wreath
segment in the southern hemisphere.

8.2.2

Preferential Longitudes for Loop Creation

The longitudinal concentration of sunspots into so-called active longitudes has been
observed for the past several solar cycles (Henney & Harvey, 2002). These active longitudes
provide observational evidence that the creation of buoyant magnetic structures is not apurely axisymmetric process. Magnetic wreaths in case S3 tend to be confined in longitude,
as was shown in Figure 7.1(c). These wreath segments are generally between 90◦ and 270◦in longitude. Loops tend to be generated in these wreath segments, and thus more likely
to appear in those longitudinal patches than other longitudes. Figure 8.4 shows the timeaveraged value of Bφ at 0.80R over cycle 1 with the longitudinal position of the 131 buoyant
loops over-plotted. Loops are much more likely to appear over a roughly 180◦ patch inlongitude in the southern hemisphere. Whereas we have some longitudinal modulation, it isstill far from the 10◦ to 20◦ confinement seen in active longitudes on the Sun.
The existence of longitudinal patches of both magnetic polarities in case S3 also pro-168
vides a potential explanation for the small fraction of active regions of the “wrong” polarity
seen in Figure 8.3(b). This provides a possible mechanism for the analogous phenomena inwhich a small fraction of solar active regions violate Hale’s polarity law. While active longitudes in the Sun are more confined than those seen here, the longitudinal confinement of thewreaths in case S3 may provide a possible pathway toward understanding active longitudes.
8.3

Summary and Reflections: Buoyant Loops in Convective Dynamos
Chapters 7 and 8 have explored in detail the first global convective dynamo simulationto achieve buoyant magnetic loops which transport coherent magnetic structures throughthe convection zone. These buoyant structures arise from large-scale magnetic wreaths,
which have been previously described in both persistent (Brown et al., 2010) and cyclicstates (Brown et al., 2011; Nelson et al., 2013b). In this work we have focused on case S3,
which possesses large-scale magnetic wreaths which undergo cycles of magnetic activity andproduce many buoyant magnetic loops. Case S3 was able to achieve buoyant loops due tothe use of a dynamic Smagorinsky SGS model which greatly reduced diffusive processes inthe simulation.
Although case S3 has a rotation rate greater than the current Sun, the dynamics
achieved may be applicable to solar dynamo action. The most salient non-dimensionalparameter for the creation of toroidal wreaths is the Rossby number, which considers the
local vorticity ω and rotation rate Ω as Ro = ω/2Ω. In case S3 the Rossby number at
mid-convection zone is 0.581, indicating that the convection is rotationally constrained as isalso expected in the bulk of the solar convection zone.
Much of the work on buoyant magnetic flux has generally regarded convection as apurely disruptive process. In our dynamo studies here, convection plays a key role in boththe creation of the strong, coherent magnetic fields and the advection of magnetic flux radially outward. Turbulent intermittency provides an effective mechanism for the amplificationof magnetic fields to energy densities well above equipartition with the resolved flows (Nel-169
son et al., 2013b). Convection also assists in the transport process by the upflows helping
to advect the loops. Without convection, buoyant transport of magnetic flux is generally
regarded as a low-wavenumber instability on axisymmetric fields. With convection, buoyant
loops are formed on convective length scales as the result of non-axisymmetric processes.
The loops realized in case S3 are not large-scale instabilities of axisymmetric flux tubes,
but rather they result from turbulently amplified coherent structures becoming buoyant andbeing advected by convective upflows. Similar upward advection of magnetic structures by
convection has been seen when considering the impact of convection on flux tubes (Weberet al., 2011, 2012, 2013) or specified magnetic structures (Jouve & Brun, 2009).
When we consider moderate numbers of buoyant loops over an activity cycle, we finda number of trends in their collective behavior. In all of these trends, it is important to notethat our statistical sample of 158 loops is significant but still relatively small. First, loops inour simulation clearly show a hemispheric polarity preference analogous to Hale’s polarity
law for solar active regions, although case S3 shows a slightly higher rate of violations tothis trend compared to the Sun. Second, the buoyant loops tend to show latitudinal tilts
similar to Joy’s law for solar active regions. As in the Sun, a wide variety of tilt angles areobserved, though the average tilt angle places the leading edge of the buoyant loop closer
to the equator than the trailing edge. Third, the buoyant loops tend to show a degree oftwist similar to the twist inferred from photospheric measurements of vector magnetic fields.
Again a wide variety of twist parameters are measured centered about a relatively small,
negative mean value. Finally there are ranges in longitude which demonstrate repeatedemergence of magnetic flux. This longitudinal modulation in the creation of magnetic loops
is reminiscent of active longitudes observed in the Sun, but on larger longitudinal ranges
than active longitudes in the Sun.
Buoyant transport of magnetic fields is a key ingredient in many models of the solar
dynamo. Mean-field models often use parameterizations to represent this buoyant transport.
We have considered connections between the axisymmetric toroidal fields in case S3 and the170magnetic flux in the buoyant loops. We find that total flux in a given buoyant loop is
only weakly dependent on the strength of the mean field from which that buoyant loop was
generated. Additionally, we find that the probability that a buoyant loop will be generated
in regions of relatively weak mean fields is significant, and that the strongest mean fields
may be less likely to generate buoyant loops than regions of moderate axisymmetric fields.
This simulation is a first step towards connecting convective dynamo models and flux
emergence in the Sun and sun-like stars. As we consider the role of turbulent convection, we
find clear indications that it plays important roles in the dynamo which generates buoyant
magnetic loops and the transport of those loops. This simulation invites continued effort
towards linking convective dynamo models and simulations of flux emergence.

Chapter 9
Plume Boundary Conditions in ASH

After discussing convective dynamo models which have achieved very low levels ofdiffusion both through very high resolution and the use of the Dynamic Smagorinsky SGS
model, we now turn to another way in which we can improve our ASH models. In thischapter we will discuss the implementation of and preliminary results from a new boundary
condition in ASH which we term a plume boundary condition. This work began in 2011
and is on-going. While I have taken a lead role in this effort, I have benefitted greatly fromcollaborations with Kyle Augustson, Regner Trampedach, Mark Miesch, Brad Hindman, andJuri Toomre who have provided invaluable assistance. I wrote, debugged, and analyzed thisa new module in the ASH code which applies this boundary condition. This work will thesubject of a forthcoming journal article.
9.1

Challenges at the Upper Boundary
In the previous chapters we have explored the dynamo processes which yield magnetic

wreaths, cycles of magnetic activity, and buoyant magnetic loops in solar-like simulations.
With all of these models we have faced two significant challenges. First, all of the models
to this point have used an impenetrable upper boundary condition at about 0.97R . Thisrequires us to use a diffusive process to transport the solar luminosity through the upperboundary of the simulation. This boundary layer has no physical analogue in the real Sun. In
the real Sun convection on progressively smaller scales should transport the solar luminosity

172
all the way to the photosphere. As we outlined in Chapter 2, a global simulation of convective
dynamo action that extends to the photosphere is beyond the capabilities of the ASH codefor both computational and physical reasons. The scales of motion at the photosphere would
require an order of magnitude increase in the maximum spherical harmonic degree of ourmodels. The physical assumptions built into ASH such as low Mach number flows, small
variations about the background thermal state, the ideal gas equation of state, and diffusive
radiative transfer all become invalid near the photosphere. With these limitations we must
place the upper boundary of ASH simulations in the solar interior. If this boundary isimpenetrable, we will also induce a significant boundary layer in the solar interior as well.
This has serious implications for the behavior of our simulations, particularly in models
where buoyant magnetic loops enter this diffusive boundary layer.
The second major challenge in the models of convective dynamo action is maintaining
solar-like differential rotation. Differential rotation is crucial to the operation of our dynamo
models. As we seek to study solar-like dynamo mechanisms, we require solar-like differential
rotation. Fast rotation at the equator requires a sustained transport of angular momentum equatorward by rotationally aligned convective cells in the bulk of the convection zone
(Miesch, 2005). Rotational influence relative to inertial forces is measured by the Rossby
number. Rotationally-constrained systems such as the Earth’s atmosphere or liquid core
have low Rossby numbers. Solar surface convection of granulation and supergranulation,
which operates on much smaller scales and at higher velocities than the deeper giant cellconvection, has a high Rossby number and shows almost no rotational alignment. While
there is on-going debate as to the amplitude of the flow speeds in giant cells, by almost all
estimates the Sun’s giant cell convective flows sit below a Rossby number of unity (Hanasoge
et al., 2010; Miesch et al., 2012). ASH simulations at the solar rotation rate yield solar-like
differential rotation for moderate levels of turbulence (e.g., Brun & Toomre, 2002; Miesch
et al., 2006), but when these models are made less diffusive we see an increase in theirconvective velocities which pushes them into a higher Rossby number regime with a slowly

173
rotating equator and fast poles. We term this anti-solar differential rotation. Similar behavior is reported by Gastine et al. (2013) in simulations of giant planet interiors. As muchof the convective driving in these simulations occurs near the upper boundary, we believethat the diffusive boundary layer imposed there is leading to excessive convective driving.
These over-driven flows are then too fast, allowing their inertial forces to overwhelm theirrotational constraints. When we move to slightly faster rotation rates, such as 3Ω , we findthat we can maintain solar-like differential rotation, as shown in Chapters 3 and 5.
Both of these challenges are at least partially related to our choice of boundary conditions. By choosing an “open” boundary condition that mimics the convective dynamics
occurring in the Sun’s near surface layers, we can achieve a simulation without a diffusive
boundary layer at its top and with more realistic convective driving. This chapter will focus on the implementation of such a boundary condition. We will also present preliminary
results from ASH simulations using a plume boundary condition which has solar-like differential rotation for a moderately turbulent simulation at the solar rotation rate. We willcompare this simulation, which we label case P, with three cases using impenetrable upperboundary conditions labeled cases Ia, Ib, and Ic that sample decreasing diffusive processes.
Although we here look at the implication of imposing small-scale plumes entering ASH
simulations through the upper boundary, in the longer term these inflows will be largely
specified by the properties of smaller-scale solar convection studied with local simulations of
the near-surface layers. Advances in local near-surface models have paralleled the progress
made in global simulations, reaching high spatial and temporal resolution, and capturingsolar-like behaviors which can be compared with observations (e.g., Stein & Nordlund, 2006;
Rempel et al., 2009; Augustson et al., 2011). We expect that these models will provide
statistical descriptions of the downflowing plume structures and their associated thermal
structures which could thence be imposed through our plume boundary condition. These
plumes from near-surface convection models would then replace our simplified plume model,
described here. We have taken care to define this plume boundary condition to conform174
to some of the known properties of surface convection models, though the implementation
here relies on somewhat simplified descriptions. Imposing downflows from a near-surface
model, even in a statistical sense, is likely to be a challenging undertaking. This chapter
will focus on the design and implementation of our plume boundary condition and report on
preliminary results from its application to global solar convection models.

9.2

The Consequences of Impenetrable Boundaries
ASH simulations generally use an impenetrable, stress-free condition on their upperboundaries in whichvr =∂  vφ ∂  vθ 
=
= 0.
∂r r
∂r r

(9.1)

This formulation ensures that mass and angular momentum are conserved within our domain
for hydrodynamic simulations. In addition the domain is heated by a radiative flux at thebase of the simulation and energy leaves through the top of the domain via thermal diffusion. All simulations discussed in previous chapters used this formulation. In this manner
our hydrodynamic simulations resemble a spherical version of the classic Rayleigh-Benard
convection problem which has been studied through theory, simulations, and experiments for
close to 90 years (see Chandrasekhar, 1961; Getling, 1998). The major difference, however,is that in Rayleigh-Benard convection the overall heat flux through the horizontal layer isdetermined by how the convective flows respond to the boundary layers formed at the upperand lower walls, which are typically maintained at fixed temperatures. Thus in RayleighBernad convection the convection itself determines the heat transport. In ASH simulations
we demand that a solar luminosity of energy enter the system at the lower boundary andthat one solar luminosity exit the system through the upper boundary. This has importantconsequences for the resulting flows.
The primary dimensionless parameter that measures convective driving is the Rayleigh175
number given by
Ra = −dρdS∂S gL4.
P ∂r ρνκ

(9.2)

In this formulation Ra is proportional to the entropy gradient −∂S/∂r and inversely proportional to each of the diffusion coefficients ν and κ. If we want to drive a more turbulent
system by lowering ν and κ, we should get a correspondingly higher value of the Rayleighnumber. However by lowering κ we also find there is a nonlinear feedback that yields anarrower upper boundary layer. We demand that thermal diffusion carry a solar luminosity
out of the top of our domain. This provides an additional constraint on ∂S/∂r at the upperboundary such that
L
∂S
,
=−
∂r
4πRo2 κρ̄T̄

(9.3)

and thus ∂S/∂r ∝ 1/κ. A real physical system such as the solar convection zone would
not experience this additional increase in Rayleigh number as it does not rely on a diffusive
boundary layer to transport energy.
In practice we have found that when we decrease the viscous and thermal dissipationin our models at the solar rotation rate, we see the differential rotation in these models
decrease and eventually switch sign to yield a slow equator and fast poles. This state is inmany ways conceptually very simple: convection is trying to mix the angular momentumof the layer leading to fast rotation where the lever arm is short and slow rotation where islong. This trend is likely the result of a number of mechanisms which are all interrelated,but the influence of the impenetrable boundary layer is likely to play a significant role.

9.3

Formulation of the Plume Boundary Condition
Conceptually, open boundaries present a number of challenges. On a very basic levelan open boundary is an attempt to admit dynamics into a simulation which are not being
explicitly treated. Here we make a distinction between truly open, semi-open, and permeable

176
boundary conditions. Truly open boundary conditions allow the resolved interior dynamics ofthe simulation to control the behavior of the boundaries. They are particularly challenging
because they can permit net fluxes of quantities like mass or momentum, destroying the
global conservation properties of a simulation. We are not aware of any solar convectivemodels which use a truly open boundary condition.
Semi-open boundary conditions are those which are designed to be as open as possible
while using minimally invasive techniques to preserve global conservation properties. Thismay be accomplished using an open boundary condition on the velocity fields while imposingpressure gradients at the boundary to regulate the balance between upflows and downflows,or by applying a volumetric forcing near the boundary to counteract the net fluxes throughthe boundary. Semi-open boundaries have been used in several codes designed for nearsurface solar convection, including MuRAM (Rempel et al., 2009; Cheung et al., 2010),
Stagger (Trampedach & Stein, 2011), and CSS (Augustson et al., 2011). In these codes thebottom boundary condition is open rather than the upper boundary as we are considering for
ASH. As the strongest driving in solar and stellar convection is generally believed to occur
at the photosphere, opening the lower boundary may be less problematic than opening theupper boundary. Additionally these are all finite difference codes. Stagger, for example,
applies open boundary conditions on outflows but imposes flows and thermodynamic fields
on inflows in order to preserve the global conservation of momentum and mass, as well as
the solar energy flux.
The final category of open boundary conditions is what we term permeable boundary
conditions. These models impose flows through their boundaries in a specified way. They
may be constant in time or they may vary. A permeable boundary permits flows to enter orexit the domain, but only in a specified manner. Permeable boundaries can easily control
the fluxes of conserved quantities since the fluxes are specified with the boundary condition.
Our plume boundary condition in ASH is composed of three permeable boundary
conditions and one semi-open condition. In ASH we impose small-scale plumes of radial

177
velocity and entropy as a time-dependent boundary condition. We also impose a conditionon the opening angle of the plumes in order to mitigate pressure perturbations. Finally
we impose a semi-open boundary condition designed to permit the internal dynamics of thesimulation to set the differential rotation profile on the boundary and use a volumetric torqueon the near-boundary layers to enforce conservation of angular momentum.
Semi-open boundaries are perhaps more challenging in ASH than in some of the othercodes mentioned due to ASH’s pseudospectral nature. ASH applies boundary conditions
through its semi-implicit solve (see Figure 2.2). This is done in spherical harmonic spacerather than physical space, thus boundary conditions are applied to spherical harmonicmodes rather than physical locations. This makes it essentially impossible to apply different
boundary conditions to different structures in our current computational framework. For
example, we cannot allow radial outflows to continue unperturbed through a radial gradientboundary condition while at the same time imposing inflows elsewhere.

9.3.1

Plume Boundary Conditions on vr and S

Perhaps the most straight-forward of our boundary conditions in our plume boundary
scheme are those we apply to the radial momentum and entropy equations. Formally, theboundary condition for radial momentum is
vr (Ro , θ, φ) = R (θ, φ, t)(9.4)

and the boundary condition for the entropy equation is likewise
S (Ro , θ, φ) = S (θ, φ, t) ,(9.5)

where Ro is the radius of the outer boundary and R (θ, φ, t) and S (θ, φ, t) are arbitrary
functions of θ and φ and time t. We then choose R (θ, φ, t) and S (θ, φ, t) to be composed ofNp small-scale plume structures, each with some velocity amplitude V and entropy amplitude

178
E. Thus we can write these fields asR (θ, φ, t) =Np
X

Vi Pi (θ, φ) F (t)(9.6)

Ei Pi (θ, φ) F (t),

(9.7)

i=1

S (θ, φ, t) =

Np
X
i=1

where the time dependence is encoded in F (t). The plume profiles are designed such thatthey have compact support (locally specified generating functions). This greatly reduces thecomputational cost of applying the plumes. Additionally, we choose R (θ, φ, t) and S (θ, φ, t)such thatZ

Z
R (θ, φ, t) sin θ dθ dφ =S

S (θ, φ, t) sin θ dθ dφ = 0,

(9.8)

S

where the integrals are taken over the spherical surface. Thus our plumes carry no net massflux and set the mean entropy on the boundary to zero, as was done in previous ASH models(see Table 2.1). We have investigated several possible plume profiles to date. Here we will
focus on a zero-flux plume profile P which is designed so that each plume individually has
no net mass-flux or entropy becauseZ
P sin θ dθ dφ = 0.

(9.9)

S

We have also considered a scheme which imposes plumes that are only downflows, and thenshifts the mean of the radial velocity field to zero by adding a constant term to all points.
Locally mass-conserving plumes are truly local structures, so we choose to investigate themhere.
We choose our locally mass-conserving plumes to follow the profile
−1 + 20ψ 2 − 50ψ 3 + 45ψ 4 − 14ψ 5 if φ ≤ 1
P (ψ) =

0if φ > 1(9.10)

where ψ is the angular distance from the center of the plume ∆σ divided by the width ofthe plume δ. This polynomial representation is continuous in its value, and first and second

179

Figure 9.1: (a) Profile of the shape of the imposed plumes as a function of angular distancefrom the plume center σ divided by the width of the plume δ. (b) 3D surface rendering of
the plume shape. The plumes consist of the strong negative core of downflowing low-entropy
material surrounded by a weaker positive ring of higher entropy upflows. Surface integralsof this shape are identically zero by design.

derivatives at all points. Figure 9.1(a) shows the polynomial, and Figure 9.1(b) shows a 3D
surface rendering of the plume profile. The profile has a large negative core of low-entropy
downflow surrounded by a low-amplitude positive ring of higher entropy upflow. This mimics
the shape of plume profiles seen in near-surface models (Rast, 2003).
We assign these plumes initially to uniform random locations on the outer boundary,accounting for the spherical geometry. Each plume is assigned an angular width δ. We mustthen determine the angular distance from each grid point to each plume center in order toapply the piecewise function P (∆σ/δ). This is a surprisingly difficult task. The distance
between two points over the surface of a sphere is given by∆σ (~γ , ~γp ) = cos−1 (sin θ sin θp + cos θ cos θp ∆φ) ,(9.11)

where ~γ = [θ, φ], no subscripts are used for coordinate locations, and subscript p represents
the position of the plume. Unfortunately, this representation suffers from large round-off
error for small angular separations, such as those we seek here for small plumes. A number

180of different versions of this formula have been constructed to improve the computational
accuracy of this formula. For an extended discussion of this topic at the intersection ofancient geometry and modern numerical methods, see Gade (2010). We here use a numerical
implementation designed to be accurate for small angular separations given by


 

∆φ
∆θ
2
−1
2
sin
∆σ (~γ , ~γp ) = 2 sin+ cos φ cos φp sin.
22(9.12)

This implementation suffers errors for separations larger than 90◦ , but is highly accurate andcomputationally inexpensive for small angles.
We also choose to have our plume field vary in time. Each plume is assigned a lifetime
τ . The plume’s amplitude is then modified by T (t, t0 , τ ), which is defined as a function of
the time since the plume was initiated t − t0 by



t−t0


0.1τ



T (t, t0 , τ ) = 1






1 − t−t0 −0.9τ
0.1τ

0
if 0 ≤ t−t
< 0.1
τ
0
< 0.9 .
if 0.0 ≤ t−tτ

(9.13)

0
if 0.9 ≤ t−t< 1.0
τThis provides linear ramp-up and cool-down phases for the plumes, minimizing the spuriouspressure perturbations which can plague these simulations. A plume expires when t − t0 = τ .
It is then randomly restarted with new parameters. We have applied an additional constraintthat new plumes must be at least their width from any existing plumes to avoid extremely
fast downflows which limit the size of our time step.
Having decided on the shape of our plumes and their temporal dependence, we are now
left to apply plumes to our boundary fields R and S. Each plume is assigned an angular
position ~γ , an amplitude in both radial velocity Vi and entropy Ei , an angular width δi , anda lifetime τi . With all of these parameters, the boundary fields are simply given byNpX∆σ (γ, γi )T (t, t0i , τi )R (θ, φ, t) =
Vi P
δi
i=1
S (θ, φ, t) =

Np
X
i=1


Ei P

∆σ (γ, γi )δi(9.14)T (t, t0i , τi ) .

(9.15)

181
Table 9.1: Parameters for plume boundary condition used in case P
Plume Boundary Parameters
Parameter
Range
V[240, 360] m s−1
E[8.8, 13.3] 104 erg K−1 g−1
δ[0.056, 0.084] rad.
τ
[2.0, 4.0] daysθp[0, π/2] rad.
φp[0, π/2] rad.

Correlation with V
–
0.5
0.5
0.2
0.0
0.0

Here we have chosen to use the same widths for both the momentum and entropy profiles.
This is done for simplicity, but one might expect that plume structures should have different
sizes in entropy and radial velocity. For that matter one might expect that the profiles used
for the two fields may be of different shapes. We anticipate addressing these questions infuture work.
Clearly there are a large number of free parameters in this model. Each plume requires
six parameters, for a total of 6Np parameters to choose. Correlations between these parameters for a given plume could be argued. For example, one might expect faster plumes to belarger in size, or smaller plumes to have shorter lifetimes. We chose to correlate the velocity
amplitude, entropy amplitude, plume width, and plume lifetime. The number of plumes
has a significant impact on the dynamics as well. We have conducted some preliminary
explorations of possible parameter choices guided by preliminary results from near-surface
models run with the Compressible Spherical Shell code Augustson et al. (2011), and theconstraint that we would like a solar luminosity of enthalpy flux to be transported by theplumes through the boundary on average. None of these sources of guidance are in any way
definitive and we plan to conduct extensive additional exploration of these parameters. At
this stage we have found that using Np = 1000 with four-fold longitudinal periodicity in
ASH (thus 4000 plumes for a full spherical shell), and the parameters shown in Table 9.1,

182
yields an instructive simulation in the form of case P, along with a series of test simulations
that preceded it.
The angular widths of the plumes is chosen to be as small as possible for a given spatial
resolution. We find that under-resolved plumes led to significant ringing in our sphericalharmonic transforms. For ASH simulations with `max = 682 we find that plumes must be atleast resolved by placing 12 grid points across their radius. As our grid-spacing is largest at
the equator, we have an angular spacing between points there of 0.0031 radians, giving us aminimum plume width of 0.037 radians. We elect to make our minimum plume width 0.056
radians to avoid any potential problems. We have done some tests with plumes widths as
small as 0.040 radians which appear stable over time evolutions of roughly 100 days.
Once we have determined the size of the plumes we then choose the number of plumes
such that, on average, three-quarters of the boundary is covered by a plume. Simulations
with more of the surface covered make it difficult to avoid placing new plumes on top of old
plumes, which can produce extremely fast downflows and thus constrain our time step to besmaller than desired. The plumes are placed at uniformly random locations over the sphere.
We avoid putting the center of a plume within another plume, thus plumes can initially be
no closer to the center of another plume than their width.
Our choice of the peak downflow velocity is motivated partly by near surface models
which show radial velocities on the order of a few hundred meters per second. Additionally werequire the plumes to carry a solar luminosity of enthalpy flux through the upper boundary.
Energy transport due to enthalpy flux is defined asρ̄ur
LEN = CP ρ̄ur T − T̄ =T̄CP P (γ − 1)
S − CP +P̄ γ
,(9.16)

where CP is the specific energy at constant pressure and γ is the adiabatic index. As longas S/CP > P/P̄ in our plumes, our entropy perturbations will drive enthalpy transport
across the boundary. This places a constraint on the product of the velocity and entropy
perturbations needed to maintain a given enthalpy flux. Thus if we want faster plumes

183
we must make them less entropy deficient and vice versa. We have experimented with R
ranging from 50 to 1000 m s−1 . For high speed plumes we see rapid deceleration and large
amounts of viscous heating as the plumes slam into the higher density maternal below them.
For very slow plumes we find that they must be so cold that they experience rapid buoyant
acceleration away from the boundary. In extreme cases they can coherently penetrate all
the way through the domain, reaching speeds of as much as 2 km s−1 at mid-convection
zone despite starting with speeds of 50 m s−1 We find that when the radial velocity plume
amplitude V is V ≈ 300 m s−1 the plumes are fairly well behaved.
We have also included the ability for horizontal flows to advect the plumes around theouter boundary. This is done by solving a simple advection equation for the plumes as if
they were Lagrangian test particles in the horizontal flow. This advection procedure uses
a velocity field that is smoothed somewhat over the area of the plume to provide smoothertrajectories. Advection using the unsmoothed horizontal velocities has been found to generate some ringing in the pressure field on the boundary. The advection procedure adds
measurable computational cost as it forces the processors responsible for the boundary tosolve an additional equation that is not solved by the rest of the computational domain. This
leads to load imbalance. It is anticipated that ASH could be rebalanced by assigning fewer
grid points to processors responsible for the boundary. We have tried test cases both withand without plume advection and find no systematic differences between them over roughly
100 days. Due the additional computational cost, the test cases discussed in this chapter
and case P were run without plume advection.

9.3.2

Plume Boundary Conditions on P

As discussed in Chapter 2, the anelastic approximation leads to an elliptic equation forpressure. Rather than directly solving this equation in ASH, we implicitly solve for pressure
by solving evolution equations for both the radial momentum and its horizontal divergence.

184
ASH uses a streamfunction formalism whereρ̄~u = ∇ × (Z r̂) + ∇ × ∇ × (W r̂) .

(9.17)

Solving for each component of the mass flux, this equation yields
ρ̄ur = −∇2⊥ W(9.18)ρ̄uθ =

1 ∂ 2W1 ∂Z+r ∂θ∂r r sin θ ∂φ

(9.19)

ρ̄uφ =1 ∂Z
1 ∂ 2W−,r sin θ ∂φ∂r r ∂θ

(9.20)

where ∇2⊥ is the horizontal Laplacian operator. The evolution equations for ASH in sphericalharmonic space are equations for the evolution of W , ∂W/∂r, and Z. To specify the radial
velocity field we must specify W over the outer boundary. This means that our choice for∂W/∂r has two immediate consequences. First, it will implicitly set the boundary conditionon the pressure field; second, if ∂W/∂r is not set to zero our radial velocity boundary
condition will partial specify vθ and vφ .
Initially, we chose to simply set ∂W/∂r = 0 on the boundary. However, if we look at
the evolution equation for ∂W/∂r we find that this choice leads to problems elsewhere. The
evolution equation for ∂W/∂r is given by
∂
∇2⊥

∂t



∂W
∂r





 3
d ln ρ̄ d ln ν ∂ 2 W
∂ W
+
−
∂r3
dr
dr
∂r2


2 d ln ρ̄ d2 ln ν 2 d ln ν d ln ρ̄ d ln ν ∂W
2+ ∇⊥ −
−
−
−
r dr
dr2
r dr
dr dr
∂r



d ln ρ̄ 2 d ln ν 2
−
+
+
∇2⊥ W
dr
3 dr
r

= − ∇2⊥ P + ν∇2⊥

+ QCOR + QADV ,(9.21)

where we have condensed the Coriolis and advective terms into QCOR and QADV , respectively.
If we calculate these terms explicitly, we find that the Coriolis and advective terms are
generally much smaller than the viscous term and the pressure term. If we examine the185balance of these terms we find that if we choose a boundary condition such that ∂W/∂r = 0,then we implicitly force the existence of large horizontal variations in the pressure field one
grid point in from the boundary. In fact if we run in this manner we find that pressure
fluctuations on the order to of P̄ build up on timescales of about 500 days. These pressure
perturbations soon overwhelm the dynamics of the simulation.
If instead we look at Equation 9.21 in one of these models, we find that the two largest
terms of the four viscous terms are those which concern ∂W/∂r and W . If we simply choose
a boundary condition for ∂W/∂r such that
∂W
=
∂r



d ln ρ̄ 2 d ln ν 2
+
+
dr
3 dr
r


W = CG W,(9.22)

where we have collected the terms due to geometry, stratification, and viscosity variation into
CG , we find that pressure perturbations due to the plumes are several orders of magnitude
smaller. In effect, this boundary condition is setting an opening angle for the plumes as they
enter the domain that is constant with the spherical geometry and the variations in densityand viscosity. If we substitute Equation 9.22 into Equations 9.18, 9.19, and 9.20 and takethe horizontal divergence of the flow, we find that this boundary condition leads to∇⊥ · (ρ̄~u) ∝

ρ̄ur
,
r

(9.23)

indicating that inward plumes with negative ur should be slightly converging as they enter
the domain, as one might expect from the spherical geometry.

9.3.3

Plume Boundary Conditions on Horizontal Velocities

The last remaining boundary condition in our plume boundary formulation concerns
the behavior of horizontal flows on the boundary. Ideally we would like these flows to be
as unconstrained as possible so that their impact on the differential rotation profile of asimulation is minimal. This could be done, for example, by simply setting radial gradients

186of the horizontal components of momentum to zero with∂ (ρ̄uθ )∂ (ρ̄uφ )
=
= 0.
∂r
∂r

(9.24)

If we examine Equations 9.19 and 9.20 we can immediately see that to do so will require aboundary condition on ∂W/∂r as well as boundary condition on Z. Unfortunately we havealready used the ∂W/∂r boundary condition to minimize pressure perturbations. Thus thebest we can do is to derive a condition on the toroidal streamfunction Z which will make theradial gradient of the contribution from Z to the horizontal mass fluxes vanish at the upperboundary. The can be done by imposing∂Z Z− = 0.
∂rr(9.25)

With these boundary conditions, the horizontal velocities are partially constrained such thatthey must avoid large pressure perturbations and partially free to be determined by the
behavior of the interior of the system.
This boundary condition combined with our choice of imposing non-zero radial velocities on the boundaries leads to the possibility of a net flux of angular momentum acrossthe boundary. This would led to a net accumulation or deficit of angular momentum in thesystem, effectively spinning up or down the domain. Test cases have shown that for mostchoices of plume parameters there tends to be a net outward flux of angular momentum
through the boundary, causing the domain to spin down. For a case at the solar rotation
rate, this spin down can be as large as 1% of the mean solar rotation rate roughly every 300
days. Clearly this is not desirable.
To avoid the net change of angular momentum in our simulations caused by theseboundary conditions, we have two options. The first employs the angular momentum cleaning discussed in Chapter 2 and the second uses a corrective volumetric torque. Both methods
present difficulties. The angular momentum cleaning is achieved via
Z roZ̃1m r2 dr = 0,ri

(9.26)

187
where Z1m are coefficients for the spherical harmonic modes of the toroidal mass-flux stream
function Z with ` = 1 and m = −1, 0, and 1. This is a non-local condition designed toprevent the accumulation of numerical noise in the net angular momentum of the system.
When the net change in the angular momentum of the system is on the order of the accuracyof our numerical scheme this condition works quite well. When we apply it here we find that
although it does conserve the total angular momentum of the system, it also introduces
infrequent but intense radial pressure perturbations which can drive strong ringing in oursystem. This ringing is not confined to the near-boundary layers but is seen at all depths
in our domain. Simulations using this condition are generally numerically stable, but theaccuracy of our numerical algorithm becomes less certain under these conditions. While thisis an area that deserves further study, we have chosen not to use this method to enforce
global angular momentum conservation in case P.
The second mechanism for enforcing global angular momentum conservation is to measure the net flux of angular momentum across the outer boundary and then use a volumetrictorque to add or subtract the correct amount of angular momentum from a thin layer near
the top of the domain. The net torque from the boundary is on the order of 10−6 of theaverage angular momentum of the bulk rotation rate. Thus a torque on the same order applied to a thin layer near the boundary can prevent the accumulation of angular momentum
in the system. The question then is what latitudinal and radial profile should be used forthis corrective torque. We have chosen a radial profile given by a Gaussian generated on theouter boundary with an adjustable width. In all tests discussed here we have used a widthor 0.2∆R, where ∆R is the depth of the convective layer.
We have tried four latitudinal profiles for the volumetric torque: a uniform weighting,a uniform plus sin θ weighting, a uniform plus sin2 θ weighting, and a latitude-by-latitude
replacement scheme. The latitude-by-latitude scheme attempted to measure the net flux ofangular momentum at each latitude in the simulation and then applied a corrective torque atthat latitude. This model was found to exhibit numerical instabilities in that it was possible

188
for run-away latitudes to emerge where both the boundary flux of angular momentum andthe local torque grew to very large values. The other three models are all numerically stable
and so we can assess their usefulness. This can best be done by examining the latitudinal
profiles of the average total torque on the volume Ŷtot (θ). We define
R
~ dV
∇·F
Ŷtot (θ) = Ŷcor (θ) + V= Ŷcor (θ) − Ŷp (θ) ,V(9.27)~ is the angular momentum
where Ŷcor is the latitudinal profile of the corrective torque, F
flux vector (see Equation A.1 and Appendix A for more details), and hats represent averages
in radius and longitude. Using the divergence theorem we can convert the volume integral
into a surface integral where we take all but the θ integral to preserve the latitudinal profile,which we label Ŷp . By definition if we average Ŷtot in latitude the result will be zero. Our
three corrective torque profiles have the functional formsuŶcor(θ) = Y0

(9.28)sŶcor(θ) = Y0 + Y1 sin θ(9.29)s2Ŷcor(θ) = Y0 + Y2 sin2 θ.

(9.30)

Similarly the torques due to angular momentum flux across our upper boundary in thesethree test cases will be Ŷpu (θ), Ŷps (θ), and Ŷps2 (θ).
Figure 9.2 shows the torques due to cross-boundary angular momentum flux and corrective procedure for all three latitudinal profiles. Each profile was averaged in time overroughly 100 days. In all three cases there is a large negative torque being applied by thecross-boundary flux at the equator. At high latitudes that is reversed although at much
lower amplitudes. The three profiles lead to an effective poleward transport of angular momentum, which promotes anti-solar differential rotation. If we take the average value ofŶp (θ) − Ŷcor (θ) between ±30◦ latitude for each case we find that the uniform case on averagetransports nearly twice as much angular momentum towards the poles as the sine weighted
case, and nearly four times more than the sine-squared weighted case. Interestingly all three

189

Figure 9.2: Total volumetric torque as a function of latitude in three models using acorrective torque to conserve angular momentum. For three test cases the average volumetrictorque due to cross-boundary flows Ŷp (θ) and the corrective torque applied to conserve global
angular momentum Ŷcor (θ) are plotted. The case using a uniform corrective torque (purple
lines) shows a net removal of angular momentum at the equator and a net gain at the poles.
The case using a sine weighting in latitude (blue lines) again shows a latitude-by-latitude
imbalance but one that is much less severe. The case using a sine-squared weighting (redlines) shows moderately low levels of local imbalance between the corrective torque and thecross-boundary flux torque, although there is still a net transport of angular momentum
towards the poles.

profiles shown here under-correct for the angular momentum flux across the boundary at low
latitudes and over-correct at high latitudes. In the future we will investigate the use of otherlatitudinal profiles which may yield even better results. Unfortunately, we find that eachtorque profile produces feedback on the latitudinal profile of the angular momentum flux
through the boundary, so we cannot simply measure the profile of the angular momentum
transport for one simulation and then assume that it will remain constant if we change theprofile of the volumetric torque. This can be seen, for example, by comparing the highlatitude regions of the three cases in Figure 9.2.

190
9.3.4

Applying the Plume Boundary Model

Even with a complete set of boundary conditions specified, the plume boundary model
is sufficiently complex that a wide range of morphologies and amplitudes in our velocity andthermal fields can be imposed, either explicitly or implicitly, on the boundary. The conditions for vr and S include time-dependent plumes whose parameters are selected randomly.
The condition on P is applied implicitly via a condition on the opening angle of the plumes.
The conditions on the horizontal velocities are perhaps most opaque as they are only partially specified. Figure 9.3 shows, for case P, the radial velocity and entropy fields applied
as boundary conditions, as well as the pressure fields which is implicitly applied, and thehorizontal velocity fields which are both partially imposed and partially determined by the
behavior of the interior of the simulation. With large numbers of plumes, it is common forplumes to at least partially overlap, creating larger plume complexes. This becomes evenmore likely for the largest plumes. Due to the imposed correlations between plume size andamplitude, these plumes tend to create strong downflow regions.
The implicit pressure boundary condition produces well-behaved pressure signatures
for the plumes. These are generally low pressure regions, signaling the horizontal convergence of the plumes as well as their low temperatures. With low entropy and low pressure
signatures in the downflows, their enthalpy transport is radially outward, producing thedesired energy transport properties. The horizontal flows show evidence of the converging
flows indicated by the pressure field, but also contain significant flows which are unrelated tosingle plumes. Interestingly, complexes of plumes can be seen producing coherent horizontal
flows on scales larger than individual plumes.

191
Table 9.2: Shown are the number of radial, latitudinal, and longitudinal grid points. The
total evolution time TE for each simulation is given in days. The viscosity ν in units of cm2
0
L/ν gives
s−1 is quoted at mid-convection zone. The fluctuating Reynolds number Re0 = vrms
some indication of the relative level of turbulence in each model. The convective Rossby
number Roc = (Ra/Ta Pr)1/2 demonstrates the level of rotational constraint. See Tables 2.1,
2.2, and 2.3 for additional parameters.
Selected Parameters for Cases Ia, Ib, Ic, and P
Case
Ia
Ib
Ic
P

9.4

Nr , Nθ , NφTE (days)
300, 512, 256
7510
500, 1024, 512
6800
500, 1024, 512
1960
500, 1024, 512
830

ν (cm s−2 )2.41 × 1012
1.20 × 1012
6.02 × 1011
1.20 × 1012

Re0
48
177
418
253

Roc
0.462
0.515
0.568
0.503

Simulation Parameters
As we now have defined our plume boundary conditions, we can begin a preliminary

investigation into the behavior of a simulation using this choice of boundary condition. To
do this we have selected our most promising test case and evolved in for roughly 800 days todate. We label this simulation case P. For comparison we have also run three cases using thestandard impenetrable boundary condition, which we term cases Ia, Ib, and Ic. Table 9.2
gives selected parameters for these models. Further details can be found in Tables 2.1, 2.2,
and 2.3. Case Ia is designed to be analogous to case AB2 from Brun & Toomre (2002), whichshowed a strong solar-like differential rotation profile at the solar rotation rate. Case Ib is likecase Ia, but with a decrease in ν and κ by a factor of two. Case Ic reduces ν and κ again bya factor of two. Case P has the diffusivities of case Ib. In order to reduce the computational
expense of these models, we have run them in domains which cover only 90◦ in longitude
and are thus four-fold periodic. This allows us to have high spatial resolution at moderate
expense, which is especially important for resolving the plumes in case P. This four-foldperiodicity precludes flows from crossing over the poles, which can cause the generation ofpolar vortex structures of very fast rotation. These simulations all rotate at the solar rate.

192
They extend from the base of the convection zone at 0.72R to 0.983R . The additionalradial extent compared to simulations in previous chapters provides more density contrast,thus requiring the plumes to mix and entrain fluid more rapidly as they encounter steeper
density gradients.
Case Ia was initiated from a solar structure model and run for nearly 7000 days before
case Ib was created from it, thus using case Ia as its initial conditions though interpolating
to a higher spatial resolution. Case Ia was then run for roughly an additional 500 days. CaseIb was run for about 3000 days, at which stage case Ic was created from it, using it as initialconditions. Case Ib was then run for almost another 4000 days. Case Ic was only run forabout 1900 days due to limited computational resources. It appears from examinations of
the time variation in the volume-averaged differential rotation kinetic energy that case Ic is
still experiencing significant evolution in its differential rotation. From past experience, we
anticipate that this will eventually led to strong anti-solar differential rotation.
Case P was initiated from case Ib at the end of its time evolution. Initially all plumes
we given the same t0 and allowed to turn on gradually according to the time dependence
shown in Equation 9.13. Each time a plume expires, it is given new randomly generated
parameters and reinitiated. Thus initially there was some synchronization in the plumes
turning on and off, however we would expect each plume to move away from that collective
behavior as a random walk with a step size of about 1 day compared to a mean plume
lifetime of 3 days. Thus we would expect almost no correlation after about 9 plume lifetimes
or roughly 30 days. Indeed when we examine the start times of plumes after about a hundreddays, we find that they form a uniform random distribution.
When evolving case P, we find that there is a very slow timescale for the equilibrations of flux transport in the model. Cases with impenetrable boundary conditions muststore considerable energy in their mean entropy stratification in order to build the necessary
boundary layer to achieve a sufficient diffusive flux at the top of the simulation. Case Ib has
1.2 × 1041 ergs stored in this entropy gradient, or roughly enough energy to power the solar

193
luminosity for a year. Initially both thermal diffusion and enthalpy flux from the plumeboundary condition transport a solar luminosity of flux out of the domain which begins toremove the excess energy stored in the entropy gradient, but this quickly reduces the gradient in the upper layers of the domain. Within 100 days of the start of case P the totalluminosity through the outer boundary was down to 1.11L . By 400 days the luminosity
was still at 1.07L . This process asymptotically approaches a state of flux balance, but therate of convergence is slow. At this stage case P still shows a total luminosity through thebulk of the convective layer and out the upper boundary that is 1.04L .

9.5

Understanding the Effects of the Plume Boundary Condition in ASH
We have conducted a preliminary investigation of the impact of this plume boundary

condition compared to similar cases with our standard impenetrable boundary. In doing so,
we have tried to answer three major questions:(1) How does a plume boundary condition change the portion of ASH models whichwere previously dominated by the thermal and viscous boundary layers?
(2) How does the plume boundary condition modify the flows seen at mid-convection
zone?
(3) Can a plume boundary condition promote solar-like differential rotation for moderately turbulent simulations?
While our investigations into these topics are on-going, we can offer some insights into thesequestions for one possible realization of the plume boundary condition.
Figure 9.3 shows a 3D visualization of a snapshot of radial velocity patterns in case Ia,
Ib, Ic, and P. These snapshots are most useful as qualitative tools to give us an impressionof the flows in each case. We can see that case P clearly shows higher velocities near the194outer boundary than any of the other cases. It also appears to have convective patterns
most similar to those of case Ic even though the diffusion parameters in case P are equal
to those in case Ib. Also apparent is case P are the plume trajectories that begin as small
downflows on the boundary and then rapidly coalesce into progressively larger downflows as
they descend. In contrast, case Ic shows velocity structures which form lanes of downflows.
These lanes remain coherent over most of the convective layer, though they must decelerate
as they approach the upper boundary.

9.5.1

Replacing Diffusive Boundary Layers with a Boundary Interface

The plume boundary condition was designed to remove the diffusive boundary layers
imposed in ASH simulation by our impenetrable boundary condition. The plume boundary
condition achieves this goal, however it replaces the diffusive boundary layers with what weterm an interface layer where the flows and thermal fields imposed on the boundary interact
with the flows in the bulk of the domain. As we have not solved a physical equation to
generate our plumes on the boundary there is little reason to expect that they will match
solutions to the hydrodynamic equations solved in the interior of the domain. Our plumes
have a small range in scales and are all of the same shape, whereas the convection in thebulk of the domain exhibits a range of scales and shapes. Convection tends to form cellularpatterns whereas our plume field is simply a uniform random distribution. For this reason
we can expect that there will be some layer over which we observe a collision between thegiant cells in the interior of our domain and the convective plumes applied on our outerboundary.
Figure 9.5 presents the radial profile of energy transport in case Ib and case P whichhave identical coefficients of thermal and viscous diffusion. Both cases show very similar
fluxes due to entropy diffusion LTD ≈ 0.2L

through the bulk of the convective layer,

indicating that the efficiency of convection is roughly equal in both cases through the bulk of

195
the domain. The profiles of LTD diverge at about 0.93R , climbing for case Ib and flatteningout at around 0.5L

for case P. We may expect that this will decrease somewhat withfurther thermal equilibration. The fluxes of enthalpy and kinetic energy are morphologicallysimilar for the two simulations up until near the upper boundary, though case P shows higher
magnitudes in both components.
Near the upper boundary, case P shows a sharp dip in the enthalpy flux before recovering to carry the solar luminosity specified by the boundary condition. This dip is a
result of the Prandtl number of our simulation interacting with our choice of relative shapes
for the velocity and entropy components of our plumes. Our Prandtl number is 0.25, andthe plumes imposed on the boundary have identical shapes in both vr and S. The thermal
structure of the plumes diffuses faster than the momentum signature, causing the plumes
to decelerate as they lose their initial thermal driving. The effective Prandtl number of theplumes could be thought of as the size scale of the velocity structures divided by the scaleof the entropy structure. The location of this peak tells us that the bulk of the enthalpy
transport is coming from convective structures consistent with the Prandtl number in thebulk of the domain within about the first 0.01R

from the upper boundary. As we will

see this does not mean that the plumes penetrate only that far, but rather that significant
modification to the plumes occurs very quickly as they enter the domain. This is further
demonstrated by strong viscous heating within 0.01R of the upper boundary, which points
to rapid reorganization of the plume structures as they enter the domain. Thus in somesense we have replaced a spherically-symmetric viscous boundary layer with one that acts
on much smaller scales but where diffusive processes still play a key role.
We can get a feel for the changes the plumes experience in case P by looking at Figure 9.6 which shows radial velocity snapshots over a 60◦ by 60◦ region at nine successive
depths. Plumes rapidly merge into downflow structures on larger scales and with different
morphologies. While patterns can be seen imprinting downward from one level to the next
there is little correlation with the surface at or below 0.884R . As the plumes applied on the196boundaries coalesce they appear to seed the strong downflow lanes seen at mid-convection
zone. Very little impact if any is seen on the upflows. This seems to support our choiceof plume parameters, for they have not created a pattern which simply imprints throughthe domain, but rather seems to be influencing the upper portion of the domain without
dominating it. It is possible that a better choice of parameters may be able to create evenless of a mismatch between the plumes applied on the boundary and those in interior of thesimulation, creating a gentler interface layer.

9.5.2

Changes in the Bulk of the Domain

Having examined the near-boundary layers for case P, let us now turn to the bulk of
the convective layer. It has long been known that the dynamics of convective flows canbe largely determined by their boundary layers. By removing the diffusive thermal andviscous boundary layers and replacing them with an interface layer, we can then expect thatchanges will not just be limited to regions near the boundary. Figure 9.7 shows snapshots ofradial velocity patterns for all four cases at 0.95, 0.85, and 0.79 R . While there is a clear
difference between the impenetrable and plume boundary condition cases near the top of thedomain, there is surprisingly little morphological difference between cases Ib or Ic and P at
mid-convection zone or deeper. The largest difference may be in the amplitude of the flows,which are somewhat larger in case P. This is at least partially due to case P not being fully
thermally relaxed, but there is likely also some additional effect from the increase in bothenthalpy and kinetic energy transport shown in Figure 9.5. It does appear that case P may
achieve smaller-scale convective structures than case Ib, perhaps more like those of case Ic.
Additional tests are needed to assess the differences in the flows in the bulk of the domain, but a preliminary analysis seems to support the conclusion that the plume boundary
condition does not dramatically alter the convective patterns achieved in the bulk of thedomain either in morphology or amplitude.

197

9.5.3

Strengthening Solar-like Differential Rotation

Figure 9.8 shows the time- and longitudinally-averaged differential rotation as a function of latitude and radius for all four cases (the helioseismically measured profile is shownin Figure 1.3). Case Ia shows strong solar-like differential rotation similar to case AB2 ofBrun & Toomre (2002). Moving to higher levels of turbulence, case Ib shows weak solarlike differential rotation while case Ic shows very little differential rotation. From previous
experience we believe that case Ic will produce anti-solar differential rotation if allowed to
evolve further in time. In contrast case P shows strong solar-like differential rotation similar
to case Ia.
The processes which control solar differential rotation are complex and closely connected with other topics such as the thermal structure of the convection zone (see reviews
Miesch, 2005; Miesch & Toomre, 2009). As we have shown, our plume boundary condition
includes a volumetric torque designed to correct the non-zero flux of angular momentum
allowed by our plume boundary condition. This adds another layer of complexity to analyses of angular momentum transport carried out for previous ASH simulations (e.g., Brun &Toomre, 2002; Miesch et al., 2006; Brown et al., 2008; Nelson et al., 2013b). The solar-like
differential rotation in case P is fascinating since it possesses somewhat higher radial velocities at mid-convection zone than even case Ic, though smaller values of both the Rossby
and convective Rossby numbers. A full analysis of this case will require the development ofadditional analysis tools, which are beyond the scope of this thesis at this stage.
Closer examination of the differential rotation profile reveals some additional interesting features. The polar vortices present in the cases with impenetrable boundary conditions
are stronger in case P, though there is a thin layer of slower rotation fluid within 0.02R of
the upper boundary. Case P’s differential rotation profile as a function of latitude at 0.96R
is essentially flat between 75◦ and 45◦ , then there is a shaper increase between 45◦ and 30◦

198
followed by a much more gradual increase in Ω from 30◦ to the equator. In contrast both case
Ia and the solar profile show a much more steady increase in rotation rate at a fixed depth
when moving toward the poles. Contours of constant rotation in case P are less cylindricalat mid-latitudes that in case Ia, though not as conical as the solar profile. Finally near midlatitudes the radial gradient of Ω becomes slightly negative above 0.96R , perhaps providing
a hint of the near-surface shear layer seen above 0.95R in the Sun. Studies of supergranular
scale convection have postulated that an interface between rotationally-constrained giant cellconvection and more isotropic convective motions on supergranular scales may be required toachieve the near-surface shear layer (Augustson et al., 2011). This type of model may permit
investigations of that idea. It bears repeating that all of these findings are preliminary andthat we have yet to understand the mechanisms driving these behaviors, but the potential
is tantalizing.

9.6

Discussion of the Preliminary Analysis
In this chapter we have formulated a boundary condition that applies a specified set

to small-scale plumes to the upper boundary in ASH in an attempt to explore an alternative
to our standard boundary conditions which impose an unphysical diffusive boundary layer
in a region where convective transport should dominate. We then presented some initial
analysis of a simulation, labeled case P, which utilizes this plume boundary condition. Case
P was compared with cases at a variety of turbulence levels with impenetrable boundary
conditions. This has been conducted essentially as a proof-of-concept study rather than adefinitive exploration of this novel boundary condition. We reserve that worthy undertaking
for future efforts.
Case P demonstrates that a plume boundary condition in ASH can be implemented
and applied in such a way that it greatly reduces the diffusive boundary layers near the upperboundary of an ASH simulation without producing dramatic changes in the morphology or199
amplitude of the convective flows through the bulk of the layer. This condition can also
drive flows that yield a solar-like differential rotation profile at the solar rotation rate. Thisdifferential rotation gives hints of some aspects of the solar differential rotation profile suchas conical surfaces of isorotation and the near-surface shear layer.
In addition to the extensive analyses which will be conducted on case P moving forward, this also lays the groundwork for more complex plume models based on statisticaldescriptions of convective patterns from near surface models. The plumes used in case P
follow a simple analytical perception. Real convective downflows at this layer of the solar
interior would be expected to be far more complex. By developing improved plume models
which can incorporate additional information from near-surface models, we hope to achieve
better models of giant-cell convection in the bulk of the convective layer. This would also
provide a first step towards coupling ASH with a near-surface code, enabling further development of a coupled simulation capable of treating dynamics over perhaps the entire convective
interior of sun-like stars.

200

Figure 9.3: Sample snapshot of the upper boundary condition for case P in Mollweide projection. (a) Radial velocity field applied at the outer boundary. (b) Pressure field implicitly
applied by a boundary condition on the plume opening angles. The pressure is generally
negative in the cold downflows. (c) Entropy field for the same plumes with low entropy inthe downflows and high entropy in the upflows. (d) Latitudinal velocity field which is both
applied by the implicit boundary condition on pressure and determined by the interior ofthe domain by a zero radial gradient condition on the toroidal mass flux streamfunction
Z. (e) Similarly for longitudinal velocity. We have removed the longitudinally symmetriccomponent to highlight the converging flows around strong plume systems.

201

Figure 9.4: 3D volume rendering of vr at all depths in the northern hemisphere in cases Ia,
Ib, Ic, and P. The three impenetrable boundary cases show a wide variety in their convective
patterns. Generally more turbulent cases show smaller scale features and posses larger
ranges in values. Case P seems qualitatively most similar to case Ic although its diffusion
parameters are identical to case Ib. Case P also clearly shows plumes with strong negative
radial velocities on the boundary, while cases Ia, Ib, and Ic are required to have small radial
velocities near the upper boundary.

202

Figure 9.5: Mean radial energy transport by advection of enthalpy (LEN , dashed lines),
advection of kinetic energy (LKE , dash-dot lines), and thermal diffusion (LTD , solid lines) for
case Ib (purple) and case P (green). Not shown are the radiative diffusion fluxes which carry
the solar luminosity through the bottom of the domain and is identical for both models,
and the viscous heating flux, which is never greater than 0.02L for case Ib. For case P theviscous heating flux is only significant near the upper boundary, reaching −0.09L there.
In case P the plume boundary condition imposes an outward enthalpy flux of 1.0L and an
inward kinetic energy flux of 0.03L ; these quantities are required to vanish at the upperboundary in case Ib.

203

Figure 9.6: Simultaneous snapshots in case P of vr over a 60◦ by 60◦ patch centered on
the equator at nine depths between the upper boundary at 0.983R and mid-convection
zone at 0.851R . Individual plumes rapidly merge into larger structures as they enter thedomain. Upflow structures disappear almost immediately. By 0.934R downflow patterns
are larger and clearly non-circular, although the influence of groups of plumes are still seen.
By 0.884R it is difficult to identify any unambiguous correlation between structures at thatlevel and those at the surface.

204

Figure 9.7: Snapshots of vr at three depths in cases Ia, Ib, Ic, and P. Cases Ia, Ib, and Ic
show clearly cellular patterns near their upper boundary, while case P shows much less coherent structures as plumes coalesce into larger structures. At mid-convection zone (0.85R )cellular patterns are less evident, particularly for cases Ib and Ic, while the fairly laminarconvection of case Ia retains its cellular pattern. All four cases show some rotational alignment near the equator. Case P shows a particularly strong upflow/downflow pair which do
not appear to be caused by the plume boundary condition.

205

Figure 9.8: Differential rotation profiles in angular velocity Ω (r, θ) for case Ia, Ib, Ic, and P
averaged over about 100 days each. All cases display polar vortices which may be a result of
the four-fold longitudinal periodicity of these models. Cases Ia and P shows strong solar-like
differential rotation. Case Ib shows weaker but still solar-like differential rotation. Case Ic
possess very little differential rotation. We believe case Ic will produce an anti-solar profile
if allowed to continue evolving.

Chapter 10
Ongoing Puzzles and Future Challenges

In the preceding chapters we have sought to explore the fundamental physical processes
that enable stars like our Sun to generate magnetic fields which undergo cycles of magnetic
activity and emerge from the deep interior toward the surface. The models presented here
have made progress on some of the important current questions in convective dynamos.
Specifically, we have shown that:(1) The dynamic Smagorinsky sub-grid scale model can be reliably used to significantly
reduce diffusion in ASH simulations while preserving essential elements of the convective flows.
(2) Magnetic wreaths can exist in simulations where explicit diffusion has been reduced
and replaced in fundamental balances by resolved turbulent dissipation.
(3) Turbulence plays a key role in the initiation of global reversals in magnetic polarity.
(4) Buoyant magnetic loops can be spontaneously generated in ASH models using the
dynamics Smagorinsky SGS model. These loops can rise through the convection
zone via a combination of magnetic buoyancy and advection by convective giant
cells. Their typical size is set by the size of the giant cells.
(5) The buoyant magnetic loops in our ASH models mimic the statistical distributionof solar active regions in properties such as their tilt and twist. Further these loops

207
preferentially appear in confined longitudinal patches reminiscent of active longitudes
in the Sun.
(6) The standard impenetrable upper boundary condition in ASH can be replaced with aplume boundary condition which imposes large numbers of small-scale plumes on theupper boundary. This plume boundary condition has yielded promising preliminary
results for a proof-of-concept model.
As is often the case, these discoveries have led to a series of new questions and directions
for future research, as well as reminded us of the existence of continuing puzzles. We will
not attempt to outline all of these here, but instead discuss three which we find particularly
noteworthy. First, our work with reversals in convective dynamo models has reminded usthat we still fundamentally do not understand the mechanism whereby toroidal magnetic
field regenerates the poloidal magnetic field. The dynamo community has long sought toidentify the physical mechanism which provides the α-effect, namely the connection between
the large-scale toroidal magnetic fields and the induction of correlations between small-scale
magnetic fields and flows. We have come tantalizingly close in identifying aspects of thisconnection with case D3b, going so far as to identify a timescale for this process and thatit results in an upscale progression of polarity reversals. It is likely that with additionalmodels showing cycles of magnetic activity we will be able to further identify trends in thiselusive mechanism. With current advances in the availability of computational resources andthe scalability of ASH, all that remains is for an intrepid soul to carry out the simulations,perform the analyses, and discover the final clues that we have overlooked.
Second, the plume boundary condition work has provided a tool for further study ofthe nature of convective driving in the bulk of the convection zone, possibly providing a way
to explore how the Sun’s differential rotation is maintained, as well as a means to further
explore the possible modes of giant cell convection in that may exist in the Sun. Giant cells
are essential components of any realistic model of convective energy transport in the deep

208
solar interior and are required to provide the angular momentum transport which maintains
the solar differential rotation. The giant cells are on solid theoretical footing, but they havenot been reliably detected despite considerable observational effort. Advances in helioseismic
techniques may be able to finally achieve detection of these structures if they have goodmodels of what they should be looking for. Convection in many settings has a multitude ofpreferred scales, as can be seen in our models with impenetrable boundaries, however theconvection resulting from our plume boundary condition did not show clear cellular patterns
with a preferred scale. It may be that giant cells remain undetected because they exist as
coalescing smaller-scale plumes forming tree-like structures of downflows rather than cellularpatterns. The plume boundary condition in ASH may provide a means to test these ideas
which simply was not possible with impenetrable boundaries.
Finally, the plume boundary condition in ASH is designed to remove the diffusive
boundary layer at the top of the domain. This boundary layer effectively limits the riseof the buoyant magnetic loops generated in case S3. A number of loops rose coherently
all the way to the impenetrable boundary where they were diffused away by the boundary
layer. Creating a simulation which combines case S3 with our plume boundary condition
may provide a way to allow further rise of the buoyant magnetic loops through and possibly
out the top of our simulations. This would be an exceptionality demanding simulation as
the number of parameters would be immense. Additionally, feedbacks between the dynamic
Smagorinsky model and the plume boundary condition may be very challenging to manage.
The current implementation may not be sufficiently open to permit the buoyant loops toemerge through our upper boundary. However the possibility for a model which can selfconsistently generate buoyant magnetic loops and monitor their rise from the base of theconvection zone to within a few percent of the Sun’s radius below the photosphere is enticing.
These three projects have provided only a small sampling of possible future direction
this work could inspire. On a basic level they all attempt to provide better models of components of dynamo action in stars like our Sun. With the accelerating rate of advance in this209
field, we are enthusiastic about the prospects for answering long standing questions about
the origin and variability of solar and stellar magnetism.

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Appendix A
Generation of Differential Rotation Kinetic Energy

As shown in Equation (5.2), the time evolution of angular momentum in our domain can~ The radial component
be written in conservative form as the divergence of a flux vector F.
is given by∂  vφ  d+ vφ0 vr0 + v̂r v̂φ + v̂r Ω0 λ
Fr = ρ̄λ −νr
∂r r

1 [
1
0
0
−
B̂φ B̂r ,B B −4π ρ̄ φ r 4π ρ̄(A.1)

where the terms are from left to right due to viscous diffusion, fluctuating Reynolds stress,
mean Reynolds stress from the meridional circulation, the Coriolis force with Ω0 representing
the frame rotation rate, the Maxwell stress, and mean magnetic torques. The latitudinalcomponent is given byν sin θ ∂  vφ  d+ vφ0 vθ0 + v̂θ v̂φ + v̂θ Ω0 λr ∂θ sin θ

1 \
1
0
0
−
B B −
B̂φ B̂θ
4π ρ̄ φ θ 4π ρ̄Fθ = ρ̄λ −(A.2)

where the terms have the same ordering and identities as in the radial component. We
ignore the flux due to the Coriolis force because while it can be large locally, it cannot do
any net work on the system when averaged over the full domain. We can also write thefluxes in cylindrical coordinates in terms of the cylindrical radius λ and the distance fromthe equatorial plane z. The flux in cylindrical radius is given byFλ λ̂ = Fr sin θr̂ + Fθ cos θθ̂,

(A.3)

223
while the flux in z is given byFz ẑ = Fr cos θr̂ − Fθ sin θθ̂,

(A.4)

If we multiply equation (5.2) by the longitude-averaged rotation profile Ω̂, we are left
with an equation for the time evolution of the kinetic energy density in the mean differential
rotation profile hEDR i,∂hEDR i~= Ω̂ ∇ · F .
∂t

(A.5)

We take a volume integral over the entire domain in order to calculate the total rate ofchange in the kinetic energy of differential rotation and rewrite the right-hand side as∂hEDR i
dV =
∂t
V

Z

Z h


i
~ · ∇Ω̂ − ∇ · Ω̂F
~ dV.
F

(A.6)

V

The second term in the integral can be rewritten using the divergence theorem as a surface
integral, leaving us with∂hEDR i
dV =
∂t
V

Z

Z

~ · ∇Ω̂dV −F

VZΩ̂Fr dS.

(A.7)

S

Our choice of impenetrable and stress-free boundaries causes all of the hydrodynamic terms
terms in the surface integral to vanish on both the inner and outer boundaries. Likewise
our choice of a perfect conductor boundary condition on the lower surface causes both thefluctuating and mean magnetic torques to vanish there. The choice of a potential fieldboundary condition on the upper surface forces the mean magnetic torques to be exactly
zero, however it does in principle allow the Maxwell stress to be non-zero. This reduces thesurface integral toZ π Z 2πZ−

Ω̂Fr dS =
S

0

0

Ω̂ \
Bφ0 Bθ0 Ro3 sin2 θ dθ dφ.
4π

(A.8)

We have calculated this term to be about five orders of magnitude smaller than the volume
integral term in Equation (A.7) when averaged over long periods in cases D3, D3a, and D3b.
We chose to ignore this surface term in our analysis of time-averaged quantities.

224
The generation and dissipation of differential rotation kinetic energy can be written
as the sum of five terms, as was done in Equation (5.3). Those terms, which represent viscous diffusion, Reynolds stress, meridional circulations, Maxwell stress, and mean magnetic
torques, are given in turn by"Z
LVD = −

ρ̄νr sin θ rV∂  vφ  ∂ Ω̂∂r r ∂r#sin θ ∂  vφ  ∂ Ω̂dV,
+ 2r ∂θ sin θ ∂θ
"

#
∂
Ω̂
1
∂
Ω̂
0 0
0 0
ρ̄r sin θ vd
LRS =
+ vd
dV,
φ vr
φ vθ
∂r
r
∂θ
V
"
#
Z
∂ Ω̂ v̂θ ∂ Ω̂ρ̄r sin θv̂φ v̂r
LMC =
+
dV,
∂r
r ∂θ
V
"
#
Z
0
0
\
B
B
r sin θ [
∂
Ω̂
∂
Ω̂
φ θ
LMS = −
Bφ0 Br0
+
dV,
4π
∂r
r ∂θ
V
"
#
Z
∂ Ω̂ B̂θ ∂ Ω̂r sin θB̂φ
B̂r
+
dV.
LMT = −
4π
∂r
r ∂θ
V
Z

(A.9)

The time-averaged values of these terms are reported in Table 5.3 and Figure 5.4.

(A.10)(A.11)(A.12)(A.13)