Dynamics in the Upper Solar Convection Zone
byMarc Lyle DeRosa
B.A., Johns Hopkins University, 1994
M.S., University of Colorado, 1997

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Astrophysical and Planetary Sciences
2001This thesis entitled:
Dynamics in the Upper Solar Convection Zone
written by Marc Lyle DeRosa
has been approved for the Department of Astrophysical and Planetary Sciences

Juri Toomre

Nicholas H. Brummell

Peter A. Gilman

Jeffrey B. Weiss

Ellen G. Zweibel

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

DeRosa, Marc Lyle (Ph.D., Astrophysical and Planetary Sciences)
Dynamics in the Upper Solar Convection Zone
Thesis directed by Professor Juri Toomre

The differential rotation of the sun, as deduced from helioseismology, exhibits
a prominent layer of radial shear near the top of the convection zone. This shearing
boundary layer just below the solar surface is composed of convection possessing abroad range of length and time scales, including granulation, mesogranulation, andsupergranulation. Such turbulent convection is likely to influence the dynamics of the
deep convection zone in ways that are not yet fully understood. We seek to assess
the effects of this near-surface shear layer through two complementary studies, one
observational and the other theoretical in nature. Both deal with turbulent convection
occurring on supergranular scales within the upper solar convection zone.
We characterize the horizontal outflow patterns associated with solar supergranulation by individually identifying several thousand supergranules from a 45◦ -square
field of quiet sun. This region is tracked for a duration of six days as it rotates acrossthe disk of the sun, using full-disk (200 pixels) SOI-MDI images from the SOHO spacecraft of line-of-sight Doppler velocity imaging the solar photosphere at a cadence of one
minute. This time series represents the first study of solar supergranulation at suchhigh combined temporal and spatial resolution over an extended period of time. The
outflow cells in this region are observed to have a distribution of sizes, ranging from
14–20 Mm across, while continuously evolving on time scales of several days. Suchevolution manifests itself in the form of cell merging, fragmentation, and advection,
as the supergranules and their associated network of convergence lanes respond to theturbulent convection occurring a short distance below the photosphere.
We have also conducted three-dimensional numerical simulations of turbulent

iv
compressible convection within thin spherical shells located near the top of the convection zone. Vigorous fluid motions possessing several length and time scales are driven by
imposing the solar heat flux and differential rotation at the bottom of the domain. The
convection patterns form a connected network of downflow lanes in the surface layersthat break up into more plume-like structures with depth. The regions delineated bythis downflow network enclose broad upflows that fragment into smaller structures near
the surface. We find that a negative radial gradient of angular velocity Ω is maintained
against diffusion in these simulations by the tendency for the convective motions topartially conserve their angular momentum in radial motion. This behavior suggeststhat similar dynamics may be responsible for the decrease of Ω with radius as deduced
from helioseismology within the upper shear layer of the solar convection zone.

Dedication

To my parents, Pete and Cindy, and to my sister Lisa.

Acknowledgements

The research presented in this thesis simply would not have been completed without the guidance, encouragement, and wisdom of my advisor and mentor, Juri Toomre.
Starting with the first summer following my arrival in Boulder, and extending throughout the intervening years up to the present time, Juri has stimulated my interest in therich fields of solar physics and astrophysical fluid dynamics while providing perspective,
insight, and sage advice at several critical junctures along the way. His cheerful nature,
sparkling wit, and unending generosity have made this experience a truly enjoyable one.
I would like to acknowledge several members of the solar group here at CU, including Deborah Haber, Sacha Brun, Nic Brummell, Brad Hindman, and Jason Lisle,
who have shaped my understanding of the solar interior through several helpful discussions. I am indebted in particular to Mark Miesch, Tom Clune, and Julian Elliott for
their long, arduous hours spent programming and debugging the ASH computer code.
Special mention also goes to Kelly Cline, who endured varied mood swings and othereccentric behavior as my officemate for the last three years, as well as to Anne Hammond and Gwen Dickinson for their attention to detail and the behind-the-scenes work
necessary to keep things running smoothly.
It is also necessary to mention several folks residing at outside institutions fortheir part in this research. Analysis of the numerical simulations presented in Chapter 5
progressed largely due to several long sessions with Peter Gilman of HAO, whose broad
experience with fluid flows of all flavors within spherical shells was (and continues toviibe) a huge asset. Additionally, Dick Shine of Lockheed-Martin graciously provided his
ideas and experience (and perhaps most importantly, his computer code) associated withcorrelation tracking methods. Tom Duvall, Rick Bogart, and John Beck of the SOI-MDITeam at Stanford assisted in part in the processing and subsequent analysis of the large
MDI dataset of the near-surface solar velocity field. Without their contributions, theresearch of Chapter 3 would never have advanced to its current state. Support for thisresearch was provided by NSF grants ATM–9731676, ECS–9217394, and GRT–9355046,
and NASA grants NAG 5–8133 and NAG 5–7996.
Memories of my time in Boulder will always be associated with the many friends
with whom I have spent countless hours playing softball and volleyball, watching movies,
playing poker, rooting for the Broncos, Rockies, and Avs, hiking, and eating steak. A
few deserve special mention. Marc Swisdak, whom I met within a week of arriving in
Boulder, is an associate and close friend anyone would be lucky to have, and I will
always be grateful for his kindness, gracefulness, and his amazing ability to remember
almost anything. I would also like to recognize my housemates Eli Michael and Jason
Tumlinson for making it through the last several months without getting too annoyed
living with someone who was probably a little too stressed out. Many thanks go out toAmanda Sickafoose, who has shown me the merits of having a take-charge attitude inlife, and to Matt Beasley, for always being so effervescently happy and for showing me
how to effectively dispose of disposable income.
Finally, I am truly grateful for having such a wonderful family. My sister, Lisa,
must be commended for having grown up with an older brother who was probably
somewhat overbearing at times, and for managing nevertheless to turn into a top-notch
person who will undoubted be extremely successful at any endeavor she tries. Lastly,
I am truly indebted to my parents, Pete and Cindy, who have provided a positiveenvironment filled with direction, encouragement, motivation, and love in which a rather
shy kid could develop the confidence and determination to realize his dreams.

Contents

Chapter
1 INTRODUCTION: THE DYNAMIC SOLAR CONVECTION ZONE
1.1

1.2

1.3

OBSERVATIONS OF THE SOLAR SURFACE . . . . . . . . . . . . . .

2

1.1.1

Differential Rotation of the Surface and Interior. . . . . . . . .

2

1.1.2

Dynamics in the Near-Surface Layers . . . . . . . . . . . . . . . .

4

SIMULATIONS OF TURBULENT CONVECTION . . . . . . . . . . .

7

1.2.1

Mean-Field Hydrodynamic Models of the Convection Zone . . .

7

1.2.2

Multi-Mode Hydrodynamical Simulations . . . . . . . . . . . . .

8

1.2.3

Models Including Magnetic Effects . . . . . . . . . . . . . . . . .

13

SUMMARY OF RESEARCH PRESENTED IN THIS THESIS . . . . .

16

2 SURFACE FLOW MEASUREMENTS
2.1

2.2

2.3

1

19

AN OVERVIEW OF SURFACE FLOWS . . . . . . . . . . . . . . . . .

19

2.1.1

Direct Doppler Methods . . . . . . . . . . . . . . . . . . . . . . .

19

2.1.2

Local Helioseismic Techniques . . . . . . . . . . . . . . . . . . . .

22

2.1.3

Tracer-Type Measurements . . . . . . . . . . . . . . . . . . . . .

23

OBTAINING SURFACE FLOW FIELDS FROM MDI DATA . . . . . .

24

2.2.1

The MDI Instrument on SOHO . . . . . . . . . . . . . . . . . . .

25

2.2.2

Isolating the Mesogranules . . . . . . . . . . . . . . . . . . . . . .

26

THE CORRELATION TRACKING TECHNIQUE . . . . . . . . . . . .

29

ix
2.3.1

The Correlation Tracking Algorithm . . . . . . . . . . . . . . . .

29

2.3.2

Calibrating the Algorithm . . . . . . . . . . . . . . . . . . . . . .

31

3 THE NATURE OF SOLAR SUPERGRANULATION
3.1

38

ASSESSING CORRELATION TRACKED FLOW MAPS . . . . . . . .

38

3.1.1

Properties of the Flow Maps . . . . . . . . . . . . . . . . . . . .

38

3.1.2

Comparison With Direct Doppler Images . . . . . . . . . . . . .

44

3.1.3

Comparison With Time-Distance Results . . . . . . . . . . . . .

46

3.1.4

The Anomalous Flow Directed Toward Disk Center. . . . . . .

49

3.2

A DYNAMICAL PICTURE OF SUPERGRANULATION . . . . . . . .

52

3.3

MEASURING SUPERGRANULAR SIZES AND LIFETIMES . . . . .

60

3.3.1

Correlation Sizes and Lifetimes . . . . . . . . . . . . . . . . . . .

60

3.3.2

The Direct Identification Method . . . . . . . . . . . . . . . . . .

62

3.4

FLOWS ON LARGER SCALES . . . . . . . . . . . . . . . . . . . . . .

73

3.5

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4 NUMERICAL MODELING OF SPHERICAL SHELLS OF CONVECTION

81

4.1

NUMERICAL MODELING OF ANELASTIC FLUIDS . . . . . . . . .

81

4.2

THE ASH CODE: EQUATION SUMMARY. . . . . . . . . . . . . . .

84

4.2.1

Fluid Flow in a Rotating Frame . . . . . . . . . . . . . . . . . .

84

4.2.2

The Anelastic Approximation . . . . . . . . . . . . . . . . . . . .

88

4.2.3

Scaling of the Fully Compressible Fluid Equations . . . . . . . .

91

4.2.4

Energetics of the Anelastic Equations . . . . . . . . . . . . . . .

94

4.2.5

Streamfunction Formalism . . . . . . . . . . . . . . . . . . . . . .

95

THE ASH CODE: NUMERICAL IMPLEMENTATION . . . . . . . . .

96

4.3.1

Angular Discretization . . . . . . . . . . . . . . . . . . . . . . . .

96

4.3.2

Radial Discretization . . . . . . . . . . . . . . . . . . . . . . . . .

98

4.3.3

Temporal Discretization and Parallel Implementation

99

4.3

. . . . . .

x
5 SUPERGRANULAR CONVECTION IN THIN SHELL MODELS

103

5.1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2

SUMMARY OF SIMULATION PARAMETERS . . . . . . . . . . . . . 108

5.3

5.4

5.5

5.2.1

Initialization of the Spherically Symmetric Mean State . . . . . . 108

5.2.2

Approach to Thermal Equilibrium . . . . . . . . . . . . . . . . . 112

GENERAL FLOW CHARACTERISTICS . . . . . . . . . . . . . . . . . 114
5.3.1

Multi-Scale Convection . . . . . . . . . . . . . . . . . . . . . . . 114

5.3.2

Time-Averaged Axisymmetric Flows . . . . . . . . . . . . . . . . 123

ENERGETICS OF AXISYMMETRIC FLOWS . . . . . . . . . . . . . . 127
5.4.1

Axisymmetric Differential Rotation Balance . . . . . . . . . . . . 127

5.4.2

The Radial Shear Layer Within Case S1 . . . . . . . . . . . . . . 129

5.4.3

Differential Rotation Within Cases S2, S3, and D2 . . . . . . . . 133

5.4.4

Axisymmetric Meridional Circulation. . . . . . . . . . . . . . . 137

CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6 CONCLUDING REMARKS

143

6.1

SUMMARY OF RESEARCH PRESENTED IN THIS THESIS . . . . . 143

6.2

FUTURE OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Bibliography

150

Appendix
A CORRELATION TRACKING ALGORITHM DETAILS

155

A.1 SETTING UP THE GRIDPOINT ARRAY . . . . . . . . . . . . . . . . 155
A.2 THE MERIT FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 156
A.3 MINIMIZING THE MERIT FUNCTION . . . . . . . . . . . . . . . . . 160
A.4 ITERATING WITH BICUBIC SHIFTS . . . . . . . . . . . . . . . . . . 162

xi
A.5 CORRELATION TRACKING PITFALLS . . . . . . . . . . . . . . . . . 163
B ASH CODE EQUATIONS — DETAILED DERIVATIONS

165

B.1 ANELASTIC FLUID EQUATIONS . . . . . . . . . . . . . . . . . . . . 165
B.1.1 Order in  of All Dependent Variables . . . . . . . . . . . . . . . 165
B.1.2 Derivation of the Anelastic Mass Continuity Equation . . . . . . 167
B.1.3 Derivation of the Anelastic Momentum Equations . . . . . . . . 168
B.1.4 Derivation of the Anelastic Energy Equation . . . . . . . . . . . 168
B.1.5 Derivation of the Anelastic Equations of State . . . . . . . . . . 169
B.2 ANELASTIC EQUATION ENERGETICS . . . . . . . . . . . . . . . . . 170
B.2.1 Derivation of the Kinetic Energy Conservation Equation . . . . . 170
B.2.2 Derivation of the Internal Energy Conservation Equation . . . . 172
B.2.3 Derivation of the Total Energy Conservation Equation . . . . . . 173
B.3 THE NUMERICAL EVOLUTION EQUATIONS . . . . . . . . . . . . . 174
B.3.1 Streamfunction Formalism . . . . . . . . . . . . . . . . . . . . . . 174
B.3.2 Streamfunction Identities . . . . . . . . . . . . . . . . . . . . . . 174
B.3.3 Components of the Divergence of the Viscous Stress Tensor . . . 175
B.3.4 Derivation of the W Equation . . . . . . . . . . . . . . . . . . . . 190
B.3.5 Derivation of the P Equation . . . . . . . . . . . . . . . . . . . . 191
B.3.6 Derivation of the Z Equation . . . . . . . . . . . . . . . . . . . . 193
B.3.7 Derivation of the S Equation . . . . . . . . . . . . . . . . . . . . 195
C ASH CODE TIME STEPPING

196

C.1 THE SECOND-ORDER CRANK-NICHOLSON METHOD . . . . . . . 196
C.2 THE SECOND-ORDER ADAMS-BASHFORTH METHOD . . . . . . . 198
C.3 COMBINING THE METHODS. . . . . . . . . . . . . . . . . . . . . . 199

xii

TablesTable
4.1

Definitions of symbols appearing in §4.2. . . . . . . . . . . . . . . . . . .

86

4.2

Estimates of v, , and Ma calculated from equations (4.14) and (4.20) .

91

5.1

A summary of the parameters of the thin shell convection simulations . 107

B.1 Order in  for all quantities and operators appearing in the fully compressible fluid equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.2 Strategy employed to obtain the ASH code evolution equations. . . . . . 175

xiii

FiguresFigure1.1

Rotation rates Ω/2π inferred from helioseismic analysis of SOI data . .

4

1.2

Rotation rates Ω/2π inferred from helioseismic analysis of GONG data .

5

2.1

A full-disk velocity image . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2

Data processing: remapping the full-disk images . . . . . . . . . . . . .

27

2.3

Data processing: separating the mesogranules from the supergranules . .

28

2.4

Sample images used for correlation tracking tests . . . . . . . . . . . . .

33

2.5

Results of correlation tracking applied to Fig. 2.4a . . . . . . . . . . . .

34

2.6

Systematic and random errors for the two test images of Figure 2.4 . . .

35

3.1

Average flow map overplotted on velocity image . . . . . . . . . . . . . .

39

3.2

Average flow map overplotted on divergence image . . . . . . . . . . . .

40

3.3

Supergranules near disk center . . . . . . . . . . . . . . . . . . . . . . .

42

3.4

Effects of time-averaging on low map velocity . . . . . . . . . . . . . . .

43

3.5

Comparison between correlation tracking and direct Doppler velocity
measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.6

Visual comparison between divergence and travel-time approaches . . .

47

3.7

Statistical comparison between divergence and travel-time maps . . . . .

48

3.8

Anomalous large-scale flow . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.9

Corks subject to an evolving flow field . . . . . . . . . . . . . . . . . . .

53

xiv
3.10 Corks subject to an evolving flow field . . . . . . . . . . . . . . . . . . .

54

3.11 Corks subject to an evolving flow field . . . . . . . . . . . . . . . . . . .

55

3.12 Formation of a convergence lane . . . . . . . . . . . . . . . . . . . . . .

56

3.13 Intermittent convergence lanes . . . . . . . . . . . . . . . . . . . . . . .

57

3.14 Advection of convergence lanes . . . . . . . . . . . . . . . . . . . . . . .

58

3.15 Space-time cut of horizontal divergence. . . . . . . . . . . . . . . . . .

59

3.16 Supergranular outflow cores . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.17 Corks overlaid on supergranular outflow cores . . . . . . . . . . . . . . .

65

3.18 Corks overlaid on supergranular outflow cores . . . . . . . . . . . . . . .

66

3.19 Evolution of outflow cores . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.20 Supergranule size histogram . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.21 Supergranule sizes vs. smoothing . . . . . . . . . . . . . . . . . . . . . .

69

3.22 Supergranule lifetime histogram . . . . . . . . . . . . . . . . . . . . . . .

70

3.23 Supergranule lifetimes vs. smoothing . . . . . . . . . . . . . . . . . . . .

71

3.24 Histogram of supergranular sizes vs. lifetimes . . . . . . . . . . . . . . .

72

3.25 Large-scale flows in quiet-sun . . . . . . . . . . . . . . . . . . . . . . . .

74

3.26 Longitudinally averaged zonal and meridional large-scale flows . . . . .

75

3.27 Residual large-scale flow field . . . . . . . . . . . . . . . . . . . . . . . .

76

3.28 Comparison to ring-diagram results . . . . . . . . . . . . . . . . . . . . .

77

3.29 Scatter diagram comparing correlation tracking and ring-diagram results

78

4.1

An illustration of the spherical shell domain . . . . . . . . . . . . . . . .

85

5.1

Average rotation rates Ω/2π inferred from helioseismology . . . . . . . . 104

5.2

Initial radial structure of the simulations . . . . . . . . . . . . . . . . . . 108

5.3

Approach to equilibrium for Case D2 . . . . . . . . . . . . . . . . . . . . 112

5.4

Flux balance with Cases S2 and D2 . . . . . . . . . . . . . . . . . . . . 113

5.5

Horizontal planforms of radial velocity for Case S2 . . . . . . . . . . . . 115

xv
5.6

Horizontal planforms of radial velocity for Case D2 . . . . . . . . . . . . 116

5.7

Vertical structure of radial velocity for Cases S2 and D2 . . . . . . . . . 117

5.8

Horizontal advection of velocity structures of Case S2 . . . . . . . . . . 118

5.9Space-time velocity structures of Case S2 . . . . . . . . . . . . . . . . . 119

5.10 Histogram of radial velocities for Cases S2 and D2 . . . . . . . . . . . . 120
5.11 Horizontal temperature variation for Case D2 . . . . . . . . . . . . . . . 121
5.12 Latitudinal temperature variation for Case D2. . . . . . . . . . . . . . 122

5.13 Differential rotation profiles for all cases . . . . . . . . . . . . . . . . . . 124
5.14 Axisymmetric radial velocity profiles for all cases . . . . . . . . . . . . . 125
5.15 Axisymmetric meridional velocity profiles for all cases . . . . . . . . . . 126
5.16 DRKE source terms for Case S1 . . . . . . . . . . . . . . . . . . . . . . 129
5.17 Total DRKE and change for Case S1 . . . . . . . . . . . . . . . . . . . . 130
5.18 Radial cuts in angular velocity for Cases S1, S2, and S3 . . . . . . . . . 131
5.19 Radial cuts in angular velocity for Case D2 . . . . . . . . . . . . . . . . 134
5.20 DRKE source terms for Case S2 . . . . . . . . . . . . . . . . . . . . . . 135
5.21 DRKE source terms for Case D2 . . . . . . . . . . . . . . . . . . . . . . 136
5.22 DRKE source terms for Case S3 . . . . . . . . . . . . . . . . . . . . . . 137
5.23 MCKE source terms for Case S1 . . . . . . . . . . . . . . . . . . . . . . 139
5.24 MCKE source terms for Case S2 . . . . . . . . . . . . . . . . . . . . . . 140

Chapter 1

INTRODUCTION: THE DYNAMIC SOLAR CONVECTION ZONE

Our sun is the nearest star to earth and the most massive object in the solar
system. Most of this mass is concentrated near its center (the core), where the matter
is hot and dense enough to permit the fusion of its abundant supply of hydrogen into
helium. The energy produced via these thermonuclear reactions slowly diffuses throughthe radiative interior until it enters the convection zone, a layer roughly 200 Mm (where
1 Mm equals 1000 km) thick occupying about the outer 30% of the sun by radius. In
this region, the entropy and temperature gradients are superadiabatic and therefore
unstable to convective overturning motions; thus, it is energetically favorable for fluid
to rise buoyantly by extracting energy from the ambient thermal field. Once the heatreaches the surface of the sun (the photosphere), it escapes through the tenuous solar
chromosphere and corona and into outer space.
The focus of this thesis is the dynamics within the upper layers of the solar
convection zone. The convection zone as a whole contains fluid motions that vary overwidely ranging yet distinct length and time scales, all of which interact in complex
ways. In this introductory chapter, we first examine in §1.1 what has been learned
about the convection zone from observations of the photosphere, including from resultsof helioseismic inversions. In §1.2 we describe the current state of numerical modeling
efforts which elucidate some of the dynamical processes present when compressible fluids
are subject to rotation and stratification. Finally, we develop and motivate the research

2presented in this thesis in §1.3.

1.1

OBSERVATIONS OF THE SOLAR SURFACE

1.1.1

Differential Rotation of the Surface and Interior

Observations of the solar photosphere show that the sun possesses a differential
rotation in which the equatorial regions rotate faster than the poles. Measurements ofDoppler shifts of photospheric absorption lines (Howard & Harvey 1970; Snodgrass &Ulrich 1990) indicate that the surface plasma at the equator has a rotation period of
25 days, while in the polar regions it is about 33 days. Alternatively, measuring therotation rate of sunspots and other surface features as they rotate across the disk of thesun (Ward 1966; Howard et al. 1984; Snodgrass & Ulrich 1990; Zappalà & Zuccarello1991) also show a similar differential rotation profile, except that these rotation rates
are found to be faster than the plasma rate by a few percent at each latitude. Thissystematic difference in rotation rates most likely indicates that the sub-surface layers
rotate faster than the surface, as the magnetic features reflect an average of angularvelocity as weighted over their radial extent.
The existence of approximately 107 resonant acoustic modes of oscillation makes
it possible to probe the solar interior in some detail using helioseismology (e.g. Ulrich
1970; Gough & Toomre 1991). Helioseismology involves the measurement and analysisof the oscillation frequencies of these normal modes, as measured at the surface, inorder to infer properties of the medium through which the waves have traveled. Wavesof different frequencies provide information over a range of depths and latitudes, fromwhich it is possible to construct detailed maps of structure and large-scale flows within
the interior of the sun. For example, inversions of global modes are used to infer the
dependence of density, pressure, temperature, and sound speed with both radius andlatitude, whereas their rotational splittings provide estimates of the angular velocity

3field throughout much of the sun, as shown in Figure 1.1 for example.
Over the last seven years, two major research efforts have provided nearly uninterrupted observations of the solar photosphere from which the oscillation frequencies
can be measured. The Global Oscillation Network Group (GONG) operates a series of
six identical Doppler imaging instruments distributed approximately equally in longitude around the earth. The data from the six observing sites are subsequently combined
to form one continuous dataset on which helioseismic inversions can be performed. In
a complementary fashion, the Michelson Doppler Imager (MDI), one of several instruments on board the Solar and Heliospheric Observatory (SOHO) spacecraft, also provides continuous observations of the line-of-sight surface velocity field. From its vantage
point in space, MDI does not suffer from atmospheric seeing effects and weather-related
interruptions which can affect the data from the GONG project. On the other hand,
the ground-based GONG instruments are more accessible and can be upgraded as moreadvanced instruments become available. Together, the virtually uninterrupted coverageand high temporal resolution of the data provided by both projects enables the precise
measurements of oscillation frequencies necessary for detailed helioseismic analyses.
Global helioseismic inferences of angular velocity deduced from both GONG and
MDI data indicate that the surface differential rotation pattern of fast equatorial rotation relative to the poles largely holds throughout the bulk of the convection zone,such that radial gradients of angular velocity are small (Thompson et al. 1996; Schouet al. 1998). Figure 1.2 shows that there exists a layer of radial shear, known as thetachocline (Spiegel & Zahn 1992), located at the interface between the bottom of theconvection zone and the stably stratified radiative interior underneath. Much recent
attention has focused on the tachocline region, as it is thought to be the seat of the
global solar magnetic dynamo. In the near-surface layers, helioseismology also reveals
the existence of another shear layer occupying the outer 35 Mm or 5% of the sun. This
upper shear layer contains negative radial gradients in angular velocity, lending credence

1.0

1.0

a)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2
2d RLS

0.0
0.0

0.2

0.0
0.4

0.6

0.8

1.0

0.0

b)

nHz

days

450

25.7

400

28.9

350

33.1

300

38.6

4

2d SOLA
0.2

0.4

0.6

0.8

1.0

Figure 1.1: Average rotation rates Ω/2π as a function of radius and latitude inferredfrom two different helioseismic analyses applied to MDI velocity images. (adapted from
Fig. 5 of Schou et al. 1998).

to the notion that subsurface fluid layers rotate faster near the surface, much as wasinferred by comparing the tracer and spectroscopic rotation rates.

1.1.2

Dynamics in the Near-Surface Layers

High-resolution observations of the photosphere indicate that the upper shear
layer contains several spatially coincident yet distinct modes of convection. White lightimages show a mottled pattern known as granulation (e.g. Bray et al. 1984; Roudier
et al. 1991), representing the tops of near-surface convection cells, called granules. The
granulation pattern is observed to cover the entire visible surface, with individual granules being recognizable for about 5–10 minutes and typically measuring about 1 Mm
across. In addition, images of the line-of-sight velocity field reveal larger scales of convection such as mesogranulation (measuring 5 Mm across, existing for several hours)(November et al. 1981) and supergranulation (25–30 Mm, 1 day) (Leighton et al. 1962),
which also cover the visible surface of the sun while coexisting with the granulation
pattern.

5near-surface
shear layer

480
0°

460

Ω / 2π [nHz]

15°
440
30°
420
45°

400

380

360
0.50

0.60

tachocline

60°

0.70

0.80

0.90

1.00

r/R

Figure 1.2: Average rotation rates Ω/2π inferred from the helioseismic inversion of over
4 years of GONG data using the RLS technique (adapted from Howe et al. 2000). Shearlayers (shaded), evidenced by variations of Ω with radius, are observed near the base of
the convection zone as well as near the surface. The gradients of Ω in that near-surface
shear layer at high latitudes is somewhat sensitive to the inversion method and data
sets used (e.g. see Fig. 1.1 and Schou et al. 1998).

Time-distance and ring-diagram techniques are two examples of local helioseismology, in which helioseismic analyses are applied to more localized regions of thesun. Both methods have revealed flow patterns in the near-surface layers having aspatial scale larger than that of supergranulation. By partitioning the visible surface
into smaller sections, it is possible to infer the dependence with position and depth ofquantities such as temperature and average flow velocities by measuring the oscillation
frequencies of high-degree modes. Time-distance analyses indicate that persistent poleward meridional flows of order 20 m s−1 occupy at least a 20 Mm-deep layer below
the photosphere (Giles et al. 1998). The return flow, which must be located at a level

6
deeper than to which the time-distance analysis is sensitive, has not yet been detected.
Synoptic maps of the horizontal velocity field inferred from ring-diagram analyses show
extended regions of organized flow patterns evolving on time scales on the order of one
day, while longer temporal averages show banded zonal flows which propagate toward
the equator over the course of several years (Haber et al. 2000). The ring-diagram
analyses have revealed that the meridional circulations can possess an evolving doublecelled structure with latitude in the northern hemisphere, while that in the south issingle-celled (Haber et al. 2001).
In response to these various motions, small-scale magnetic elements present in thephotosphere are observed to be systematically advected by granular and supergranular
flows (Berger et al. 1998; Hagenaar et al. 1999). As a result, individual filaments ofemergent flux interact frequently with other magnetic elements, either merging with orannihilating other elements, depending on polarity. At the same time, larger regionsof intense magnetic field, including sunspot pairs and plage regions, can persist fortime periods ranging between a few days and several months. Sunspots in particularform regular patterns which cycle every 22 years as indicated by the timing, latitude,
and polarity of emerging sunspot pairs. At the beginning of each sunspot cycle, pairs
first appear in mid-latitude regions with an approximate east-west orientation. Sunspot
pairs in the northern hemisphere generally have the opposite polarity of pairs in thesouthern hemisphere, but throughout each cycle the polarity remains the same for agiven hemisphere. As the cycle progresses, pairs emerge in increasing numbers at latitudes progressively closer to the equator. After about 11 years, the sunspots at theequator become fewer in number while the next cycle begins as sunspots reappear at
mid-latitudes. The spots of the new cycle have their polarity reversed from the spots ofthe previous cycle, making the total cycle time about 22 years.
We have described several ordered phenomena, such as differential rotation, supergranulation, and the 22-year magnetic cycle, which coexist with the quickly evolving

7
and highly turbulent fluid motions within the convection zone. A detailed picture of
how such persistent flows and magnetic fields interact with the more intermittent fluid
motions is just beginning to emerge. Dynamical effects which are not yet accessible
to observations can only be examined with the aid of analytical models and numericalsimulations of turbulent convection, which we now review briefly.

1.2

SIMULATIONS OF TURBULENT CONVECTION

1.2.1

Mean-Field Hydrodynamic Models of the Convection Zone

With the rapid advancement of computer technology, especially over the past two
decades, theoretical modeling of turbulent convection is now playing an increasingly
larger role in gaining a better understanding of convection zone dynamics. Solar observations not only provide many clues toward such understanding, but also serve as goals
to which the theoretical models strive to attain.
Due to the complexity and vigor of the turbulence within the convection zone,
not all dynamically relevant scales of motion can be explicitly resolved in the same
simulation. The most widely adopted approach is to assume a separation of scales,
whereby large-scale or long-lived phenomena of interest, such as the differential rotation,
are typically represented by globally averaged quantities. Perturbations superposed onthese mean quantities thus describe more local phenomena. Such scale separation isachieved by filtering the relevant physical equations in space or time, and then solving
only the averaged equations. Obtaining a solvable set of equations requires one tosomehow approximate how the dynamics of the unresolved scales of motion affect thosescales which are explicitly resolved. Given the usually intricate dependencies between
large- and small-scale dynamics of a turbulent system, however, it may not be clear
what treatment of the small-scale effects is appropriate. Furthermore, it may not evenbe clear that an appropriate dividing line between small- and large-scale motions exists,

8
since the very nature of turbulence is that it is characterized by a wide range of scalesof motion. Despite these limitations, such scale-separation treatments are widely used
to investigate the intricate dynamics occurring within the solar convection zone.
The broad class of mean-field models, which typically contain only at most a
few resolved large scales of motion, provided the first theoretical picture of convection
zone dynamics. Such calculations suggested that the effects of unresolved motions arelikely to make important contributions to large-scale dynamics. For example, velocity
correlations associated with turbulent eddies, commonly known as Reynolds stresses,
were found to be very effective in transporting angular momentum throughout theconvection zone and are therefore likely to play an important role in achieving the solar
differential rotation profile. By adjusting the functional forms of the Reynolds stressterms it was possible to obtain models which compare favorably to the observed solar
surface rotation, meridional circulation, and temperature profiles (Durney & Roxburgh
1971; Kitchatinov & Rüdiger 1993). However, other choices led to very different profiles,
and there is no evident basis for preferring some functional forms over others. Manyof these models also produced cylindrically symmetric angular velocity profiles in theinterior, a result now contradicted by helioseismic inferences. One notable exception
is the model of Kitchatinov & Rüdiger (1995), which was able to produce an interiordifferential rotation profile similar to that inferred from helioseismic inversions. The
main drawback of the mean-field approaches continues to be the ad hoc parameterizationof small-scale effects.

1.2.2

Multi-Mode Hydrodynamical Simulations

The continual advancement of computing technology has enabled simulations of
three-dimensional systems that explicitly resolve a diverse spectrum of size scales, rather
than only the largest few. Several recent compressible convection simulations exhibit
rich structures, such as narrow downflow plumes and concentrated vorticity fields co-9existing with large-scale circulation (e.g. Brummell et al. 1995), suggesting that thegross parameterizations of small-scale dynamical source terms typically employed inmean-field treatments are physically inappropriate. However, the disparity between thelargest and smallest size scales of motion in the convection zone continues to present
severe challenges for modeling, as local dissipative processes operate on size scales thatare at least a factor of 106 smaller than the depth of the convection zone itself. Current computer technology can explicitly resolve only 103 size scales in each of the threephysical dimensions when memory, data storage, and time-stepping limitations are considered. In terms of the Reynolds number Re (ratio of inertial to dissipative forces), thelargest current models have Re = 104 , which is much lower than the value of Re = 1012
thought to exist within the solar convection zone. Values of other nondimensionalparameters of these simulations, such as the Rayleigh number Ra (ratio of buoyancy
driving to dissipative forces), Taylor number Ta (ratio of Coriolis to viscous forces),and Prandtl number Pr (ratio of viscous to thermal diffusion) are similarly removed
from the parameter regime applicable to the actual sun. As a result, issues of scale
separation and closure still exist, although to a lesser degree than in mean-field models
since these more detailed simulations explicitly resolve a much broader range of scales.
Three-dimensional models serve as useful analogs for the real convection zone because
many physical effects relevant to the dynamics on a global scale are present to somedegree.
Solar convection zone simulations have employed several different strategies tocope with the disparity of scales. One approach seeks to study the global dynamics
of a spherical system by approximating the convection zone as a fluid confined to arotating spherical shell. The available computational degrees of freedom are used to
explicitly model the largest scales of motion, while net contributions resulting frommotions occurring on sub-grid scales are approximated.
We now discuss global simulations of convection in rotating spherical shells. Un-10like mean-field models, the largest size scales associated with convection are now explicitly resolved in these simulations. The first extensive research in this area was performedby Gilman (1977, 1978a,b), who modeled a Boussinesq fluid (where the fluid is largely
incompressible except for density perturbations coupled with gravitational effects) at
Re ≈ 100, Ta = 105 , and Pr = 1. Simulations having Rayleigh numbers between 104 and105 were computed for various combinations of temperature and velocity boundary conditions. For Ra . 5 × 104 these models exhibited solar-like surface differential rotation
profiles, with zonal velocities faster at the equator than at mid- to high latitudes. Such
differential rotation was achieved by the equatorward transport of angular momentum
by Coriolis forces and Reynolds stresses working against diffusion. For Ra & 5×104 , thetransport of angular momentum away from the equator by poleward meridional flows
became strong enough to reverse the sense of the differential rotation profile, such thatthe angular velocity of equatorial latitudes became slower than at higher latitudes.
Compressibility was later included into these spherical shell models (Glatzmaier& Gilman 1981; Glatzmaier 1984, 1985; Gilman & Miller 1986) via the anelastic approximation, first adapted from meteorology to the stellar context by Gough (1969).
These anelastic models exhibited convective structures possessing a wide range of spatial and temporal scales. The most prominent velocity structures that emerged were theso-called banana cells, visible as fluid rolls elongated in latitude oriented perpendicular
to the equator. For the more highly stratified models that were considered, a necessary
condition for surface differential rotation was that the rotational influence, characterized
by Ta , must be large enough to allow angular momentum to be deposited in equatorial
regions by Coriolis-driven Reynolds stresses. In addition, the unresolved motions needto transport heat more efficiently than momentum, suggesting a Pr less than unity.
However, the solutions with strong enough rotation to produce a surface equatorial acceleration also had the angular velocity nearly constant on cylinders aligned with theaxis of rotation, which we now know is not the case within the actual solar convection

11
zone.
The advent of high-performance, massively parallel computing platforms provides
access to more turbulent regimes, allowing simulations with much lower diffusivities (andthus higher Re and Ra ) to be carried out. The models of Miesch et al. (2000) and Elliott
et al. (2000) represent the most turbulent simulations of a global variety performed todate. In these simulations, combinations of flow parameters and boundary conditions
were found which produced a more solar-like differential rotation, breaking the tendency
for a cylindrically symmetric angular velocity profile. The increased turbulence in thesesimulations was accompanied by a breakup of the banana cell structures and a more
complex evolution of convective structures than found in the earlier, less turbulent
simulations. As the driving was increased, however, the rotation profiles tended torevert back to being aligned on cylinders. In these simulations the level of turbulenceis still substantially lower than in the actual sun, and it remains to be seen how evenmore turbulent driving affects the dynamics within these systems.
Parallel computing opportunities have also permitted local studies of the turbulent dynamics of rotating fluids. In contrast to the scale-separation approach, thisstrategy distributes the computational degrees of freedom over the smallest scales of
motion, thereby precluding the need for parameterizing unresolved effects. As a result,the computational domains are more restricted in overall size, and are usually Cartesianrather than spherical, but do not include any sub-grid scale approximations found in
the global simulations described earlier. However, any large-scale structures that doform are limited by the size of the domain and thus may not be representative of alarger system. Because the combined effects of small-scale motions at or near the scaleof dissipation have a significant effect on the global dynamics, these models are still ofgreat interest.
Turbulent, compressible fluids in a Cartesian domain have been modeled by Cattaneo et al. (1991), with effects due to rotation later included by Brummell et al. (1996,

121998). Rotational effects on an f -plane are included by inclining the axis of rotation withrespect to the thermal stratification, effectively representing different latitudinal positions on a sphere. These models are characterized by a more laminar thermal boundary
layer at the upper surface covering a deeper turbulent interior. Of interest in the solarcase is the behavior at Pr < 1, where compact plumes of downwelling fluid span the fullvertical extent of the fluid layer. These coherent downflows, however, contribute little
to the overall vertical transport of energy as their upward heat flux is largely balancedby the downward transport of kinetic energy. The primary energy transport is therefore
accomplished by the smaller-scale, more turbulent motions of the system. Simulations
with a moderate rotational influence were found to drive a mean flow (although it israther modest when compared to the kinetic energy of the convection), owing to thetilting of downflowing structures toward the rotation vector.
Another class of local simulations designed to provide insight into granular convection in the near-surface layers has been performed by Stein & Nordlund (1998, 2000).
The fully compressible fluid equations are solved, including the effects of ionization andradiative transfer which are important for the near-photospheric layers. Their domain
is a 6 Mm × 6 Mm wide and 3 Mm deep Cartesian box, which is large enough to containmany granules but only encompasses a relatively shallow layer of convection near thesurface of the sun. However, these models do produce some extremely realistic resultsin the appearance and evolution of granulation. The granulation pattern is shown tobe thermally driven, with radiatively cooled matter being transported downward in the
network of dark lanes separating individual granules. In addition, these granulation
simulations produce photospheric absorption line profiles, acoustic oscillation frequencies and excitation characteristics in close agreement with measurements of comparable
quantities determined from observations of actual granulation.

13
1.2.3

Models Including Magnetic Effects

The inclusion of magnetic effects into hydrodynamic convection models adds another level of both complexity and realism. It is believed that both the large-scale andsmall-scale photospheric magnetic field structures are merely the surface manifestationof magnetism produced by the solar dynamo (Parker 1979) thought to be operating
within the solar convection zone and possibly just below its base in the overshoot layer.
The observed 22-year pattern of sunspot emergence during each solar activity cycle, as
characterized by Hale’s polarity laws, must be directly linked to a global solar dynamo
operating within the sun. On smaller scales, filaments of magnetic flux are observed toemerge through the photosphere and are advected horizontally in response to the organized granulation and supergranulation flow patterns. Because these small-scale flux
elements possess less ordered behavior than larger magnetic structures such as sunspots
and appear to be unaffected by the 22-year global magnetic cycle, they may be theresult of a local solar dynamo that is at least partially distinct from the global dynamo(e.g. Cattaneo 1999). Any dynamical picture must account for both the organizedlarge-scale dynamo activity and the small-scale magnetic filaments.
Shear flows such as the interior differential rotation profile are extremely efficient
at stretching out poloidal magnetic field into toroidal field. For sustained dynamo action
to occur in the sun, however, there must exist a mechanism which regenerates poloidal
field from toroidal field; otherwise, the toroidal magnetic field which results from thedifferential rotation of the solar interior would eventually diffuse away. One promising
way such poloidal field may be created is known as the α-effect, and operates via thecyclonic twisting of toroidal field by turbulent convective fluid motions, as originallysuggested by Parker (1955a). The earliest models which investigated the feasibility ofthe α-effect in producing a solar-like global magnetic cycle were of the mean-field type.
Such models showed that dynamo action is indeed possible for systems containing both

14
shear and rotation-induced helical motions (e.g. Stix 1976), but have trouble reproducing the relative strength of magnetic features observed on the sun. This effect occurs
because strong magnetic fields tend to inhibit the small-scale helical motions that may
be responsible for the α-effect, a process known as α-quenching.
We now know from helioseismology that a strong region of radial shear is locatedat the base of the convection zone within the tachocline, and it is within this region
that the strongest toroidal magnetic fields are believed to be formed. The existence of
this shear layer led to the idea of interface dynamos (Parker 1993), a class of meanfield models which contain a layer of convection placed immediately above a stably
stratified overshoot layer containing shearing flows. The flows within the shear layer
produce the toroidal field, while the poloidal field necessary to complete the dynamo
cycle is regenerated within the convection zone. Because the region of storage is spatially
distinct from the region where the α-effect occurs, the quenching problem is avoided;
however, there must now exist a mechanism that transports the magnetic flux between
each region. Toroidal magnetic structures naturally drift upward into the convecting
layer due to their magnetic buoyancy (Parker 1955b), where some fraction is converted
into poloidal field via the α-effect. Transporting this poloidal field back down into theshear layer is more problematic, and is accomplished by turbulent diffusion in someof the most recent interface dynamo models (e.g. Charbonneau & MacGregor 1997).
These models are successful in reproducing waves of antisymmetric dynamo activity
reminiscent of the 22-year sunspot cycle, but suffer from the drawback that the resulting
dynamo behavior depends strongly on the details of the α-effect, which results fromunresolved turbulent motions that can only be approximated in such a mean-field model.
Another mechanism which is able to regenerate poloidal field from toroidal fieldstems from the Coriolis force acting on large-scale concentrations of toroidal flux. Such
dynamo models, called Babcock-Leighton or flux-transport dynamos, require magnetic
structures within the convection zone large enough and long-lived enough to be influ-

15
enced by the Coriolis force. To avoid the quenching problem, the Babcock-Leighton
dynamos must also rely upon a mechanism to transport the poloidal flux down into thetachocline. Recent mean-field models (e.g. Dikpati & Charbonneau 1999) show thatthe advection of poloidal flux by a solar-like meridional circulation may be a viablemechanism, as suggested by their ability to produce realistic dynamo activity.
One drawback of the Babcock-Leighton models, which also applies to the α-effectmodels described above, is that the magnetism generated by most of these models
does not feed back on the shearing flows in any way, and thus it is unclear how theseprescribed flows are affected by the presence of a magnetic field within the domain. This
problem suggests the need for dynamically consistent models which solve for both theflow velocities and the magnetic fields. Such full MHD simulations of Boussinesq fluids
have been computed by Gilman & Miller (1981) and Gilman (1983), with compressibility
added later by Glatzmaier (1984, 1985). For the relatively laminar parameter regimes
considered, however, it was found in both the Boussinesq and compressible simulations
that waves of dynamo activity tended to propagate poleward rather than toward theequator. In addition, these simulations could not reproduce the alternating polarity
observed in the solar dynamo with each new sunspot emergence cycle. More turbulent
simulations are just beginning to be attempted.
The highly organized patterns of sunspot and active region emergence associatedwith the 22-year magnetic cycle suggest that portions of the toroidal field must risecoherently through the convection zone and emerge at the surface. Simulations of thin
flux tubes rising through a convectively unstable spherical shell (e.g. Moreno-Insertis
1986; D’Silva & Choudhuri 1993; Fan et al. 1993) reproduce many of the observed characteristics of bipolar active regions, including the emergence and tilt angles of sunspot
pairs. These simulations, while encouraging, only consider thin flux tubes and thus
neglect effects associated with the tube thickness. Such effects may be important, as
three-dimensional magnetic structures within the solar convection zone may be subject

16
to several MHD instabilities which may cause their destruction before they ever reachthe photosphere. For example, vortical motions within these flux tubes may cause theirfragmentation, after which they are likely to be shredded by the surrounding vigorous
convection (Schüßler 1979; Longcope et al. 1996). Structures which manage to remain
coherent are subject to kink instabilities which also may enhance their dissipation (Linton et al. 1996; Fan et al. 1999). In addition, strong downflow plumes within theconvection may prevent some tubes from emerging at all, as such fast downwelling fluid
motions can counteract their rising and pump the magnetic flux back into the tachocline(Tobias et al. 1998).
Theoretical modeling of turbulent fluids has provided valuable insight toward ourunderstanding of convection zone dynamics. These simulations have identified likely
mechanisms for sustaining the observed solar differential rotation profile and the global
dynamo, even though severe approximations and parameterizations were used in mostcases. These results suggest that the differential rotation throughout the solar interior isalmost certainly achieved by the influence of rotation on the turbulent motions locatedwithin the solar convection zone, while the global solar dynamo is believed to occur
via the interaction of magnetic field with strong shearing motions within the tachocline
and with the turbulent fluid motions of the convection zone. While much attention
has been focused on the relevant processes occurring within the bulk of the convection
zone and the tachocline region below, the dynamics occurring in the upper shear layer
immediately below the photosphere have not been studied in as much detail. This thesis
seeks to address this area.

1.3

SUMMARY OF RESEARCH PRESENTED IN THIS THESISThis thesis is devoted to a study of the upper shear layer within the solar convec-

tion zone, which forms the transition region between the deep convection zone interior
and the photosphere. This layer is visible in helioseismic inversions as a region of radial

17
gradients in angular velocity which occupy about the outer 5% (or 35 Mm) of the sun.
The upper shear layer is of great interest because the multiple modes of convection contained therein are likely to influence both the dynamics of the deeper convection zone
as well as the photosphere in ways that are not yet well understood. Convection onsupergranular size and time scales in particular may be weakly influenced by rotation,
facilitating the transport of angular momentum. In combination with such rotational
effects, the large horizontal and radial extent of supergranular flows suggest that they
most likely play the largest role in the dynamics of this shearing boundary layer.
We first seek to characterize the supergranulation pattern using correlation tracking methods applied to observations of line-of-sight Doppler velocity to identify supergranular outflows on the surface. Chapter 2 is devoted to an overview of surface flow
measurements, including a description of the correlation tracking technique and an assessment of the systematic and random errors associated with this technique. We thenin Chapter 3 apply correlation tracking methods to one 45◦ -square region of quiet sun
to generate flow maps of horizontal velocity for a duration of six days, representing
the longest uninterrupted time series of solar supergranulation studied to date. Distributions of supergranular sizes and lifetimes are obtained after directly identifying
individual supergranules on each image in the time series. The intricate evolution of
the supergranulation pattern is evident in the numerous examples of cell emergence, disappearance, fragmentation, and merging events, as well as in the systematic advectionof intercellular lanes.
This observational study is complemented in Chapters 4 and 5 by global numerical simulations of turbulent convection within thin spherical shells. Such simulations
approximate the conditions present in the upper solar convection zone as solar-like stratification, rotation, and thermal forcing profiles are imposed. We find that convection
subject to solar-like density gradients naturally produces convective structures on multiple scales, the smallest of which is analogous in size to solar supergranulation. We

18
investigate the influence of these supergranular-like convection cells on the global differential rotation and meridional circulation contained within the thin shells, including
their contribution to the maintenance of shearing flows within the domain. Finally,
concluding remarks and future directions associated with this research are presented in
Chapter 6.

Chapter 2

SURFACE FLOW MEASUREMENTS

2.1

AN OVERVIEW OF SURFACE FLOWS
Observations of the velocity field at the solar surface play a vital role in thequest to understand how the turbulent convection within the sun transports energy
and momentum throughout its interior. The dynamics of convection are responsible
for sustaining large-scale features such as the global differential rotation and meridional
circulation profiles, as well as more localized phenomena such as granulation, mesogranulation, and supergranulation. We will here use measurements of horizontal velocities
to analyze flows on supergranular size scales. Measurements of surface flows fall into
three general categories: those determined by direct Doppler measurements of the surface fluid, those inferred from helioseismic inversions, and those obtained by following
tracers. We now examine each measurement technique in some detail.

2.1.1

Direct Doppler Methods

Flows can be detected by direct Doppler methods by observing the shifts of photospheric spectral lines. These measurements are possible since the Doppler shifts ofthese lines are proportional to the line-of-sight velocity of the emitting plasma. Bymeasuring variations of the same spectral line across the solar disk, one obtains a mapof line-of-sight velocities with position. Images of this kind are collectively referred to
as Dopplergrams or velocity images. A sample velocity image is presented in Figure 2.1.

20

N
E

-2

-1

0
km/s

1

2

Figure 2.1: An image of line-of-sight velocity of the photosphere taken by MDI (see§2.2.1) on 1999 May 7, 00h 01m UT. The solar equator is indicated by a dashed line. Approaching (negative) velocities are dark, while receding (positive) velocities are bright.

21
Some of the more groundbreaking discoveries in solar physics have been madeusing velocity images (initially by photographic means), including the first detection ofsolar supergranulation (Leighton et al. 1962) and of mesogranulation (November et al.
1981). In addition, the frequencies of acoustic oscillations, which form the basis forhelioseismology, are most easily measured using time series of velocity images. Furthermore, measurements of large-scale velocity patterns, such as the surface differential
rotation profile (Snodgrass 1984; Snodgrass & Ulrich 1990) and torsional oscillations(Howard & LaBonte 1980; LaBonte & Howard 1982), have been performed using timeseries of velocity images.
In Figure 2.1 the Doppler signal resulting from rotation is immediately apparent as
the large-scale gradient of color across the image. Superposed on the rotation signal area network of supergranular outflows, which appear as small perturbations to the overallrotation velocity. Supergranules are cellular flows in which the fluid diverges horizontallyfrom a central region, and are recognizable on the velocity image as an association ofdark and light regions. The side of each supergranule closest to disk center is dark(relative to the rotation signal) since this side of the outflow is largely approaching theobserver. Similarly, the side of the supergranule closer to the limb is light due to its
velocity of recession relative to the observer. These dark/light associations were first
identified as outflows and attributed to solar convection by Leighton et al. (1962). Also
visible in the image are two sunspots, one above and to the right of disk center, and theother at about the same latitude near the east limb.
Using direct Doppler measurements to measure horizontal flows is problematicfor several reasons. First, since only fluid motions moving toward or away from theobserver are measured, the spherical nature of the sun causes the proportion of radial
to horizontal velocities projected into the line-of-sight to vary with the angular distancefrom disk center. This geometric effect makes measurements of horizontal flow fields difficult near disk center (where only a small amount of the horizontal velocity is projected22
into the line-of-sight) as well as near the limb (where foreshortening effects come intoplay). In addition, only one of the two horizontal velocity components are available, as
motions moving parallel to the limb move transverse to the observer and are therefore
not projected into the line-of-sight Doppler velocity. Furthermore, one needs to makesome assumptions about the nature of the flow field in order to separate the radial flow
contributions from the horizontal components to the projected line-of-sight velocity.

2.1.2

Local Helioseismic Techniques

Helioseismic techniques are used to infer the global characteristics of the solar
interior after observing acoustic oscillations visible over the entire photosphere. By
observing smaller regions of the sun and analyzing waves which propagate into andout of these regions, researchers are able to determine more localized properties of thesun. As a result, such analysis techniques are termed local helioseismology. One suchmeasurement is of the bulk horizontal velocity of the fluid as averaged over the region ofinterest, as the frequencies of waves traveling through a moving medium will be Doppler
shifted by an amount proportional to the bulk flow speed. This principle is used in bothtime-distance and ring-diagram techniques.
Time-distance helioseismology (Duvall et al. 1993; D’Silva 1996; Duvall et al.
1997) uses a time series of velocity images to measure the travel times of acoustic
modes propagating along subsurface ray paths. The method involves computing thecross-correlation function between the data at two points separated by varying distances
and times. The locations in space-time where the correlation is high indicate the upper
turning points of acoustic modes after having traversed their subsurface ray paths. The
difference in travel times between two counter-propagating waves traversing the same
ray path in opposite directions indicates the presence of a bulk flow, since waves travelingagainst the flow take longer to traverse the same distance than waves traveling with theflow. These travel-time differences can be used to construct a map of fluid velocities

23
not only at the surface but also over a range of depths. This technique has been used
to measure flows in both quiet sun and under sunspots (Duvall et al. 1996), as well as
to characterize meridional flows near the surface and at depth (Giles et al. 1997; Giles
1999).
Ring-diagram analysis (Hill 1988) is based on the multi-dimensional power spectraof the normal modes of oscillation computed from time series of velocity images. Aftertracking a localized region of the solar photosphere in a frame corotating with the sun,these time series are then Fourier decomposed, thereby transforming the (x, y, t) velocity
data into (kx , ky , ω) frequency data. The nested trumpet-like structures which appear as
maxima in frequency space indicate the eigenfrequencies of the oscillations. Projectionsof these structures in the (kx , ω)-plane yield the familiar (`, ν) diagram, but projectionsin the (kx , ky )-plane yield a series of concentric rings, for which this analysis technique isnamed. Horizontal flows within the localized region both near the surface and at depth
cause displacements in the locations of the rings, which are then be used to infer themagnitude and direction of such flows (Schou & Bogart 1998; Basu et al. 1999; Haber
et al. 2000).

2.1.3

Tracer-Type Measurements

Near-surface flows can also be measured by following recognizable features embedded in a time series of images from frame to frame as they move around. Although
only motion transverse to the line-of-sight can be detected, such techniques have provenuseful for characterizing several aspects pertaining to granulation, mesogranulation, andsupergranulation (Title et al. 1995; Strous & Simon 1998).
Feature tracking and correlation tracking are the two tracer-type techniques mostcommonly used in practice. The main difference between the two methods is that feature tracking requires the identification of structures embedded in the flow. Horizontal
velocities are then determined by identifying the same structures in subsequent images

24in the time series, measuring their horizontal displacements, and then converting these
displacements to a velocity. Alternatively, correlation tracking compares the topologyof the images surrounding predetermined measurement gridpoints with the topology inthe vicinity of the same gridpoints in subsequent images. Horizontal velocities are thencalculated by determining the optimal displacement such that the topology maximallycoincides.
The main drawback of both feature tracking and correlation tracking is thatchanges in appearance may be interpreted as proper motions. For example, changes inthe shape of an object due to its evolution may affect the velocities measured by feature
tracking and correlation tracking methods. In addition, the shape of an object rotating
across the solar disk may appear to change even if it is not evolving, as projection
effects may alter how the object is viewed by an observer. In either case, a tracer-type
measurement algorithm may interpret these changes as proper motions and ascribe
spurious velocities to the region of interest. Furthermore, the underlying assumptionthat the tracers are passive floaters in the fluid may be incorrect, as the proper motionsof tracers may be somewhat different than the actual velocity of the fluid. Despite
these drawbacks, tracer-type methods are useful for measuring several different aspectsof solar surface flows.
Chapter 3 will examine the general properties of the flow field on supergranular
scales by having applied the correlation tracking method to features present in time series of velocity images. We begin here by describing in detail how the near-photosphericflow maps are obtained from time series of observational data, and then in §2.3 discuss
the sensitivities of the correlation tracking approach.

2.2

OBTAINING SURFACE FLOW FIELDS FROM MDI DATA
The flow field on supergranular scales can be deduced by applying correlation

tracking to mesogranule-sized structures evident in time series of velocity images. Such

25features have been shown to sense the supergranular flow field quite well, even neardisk center (see §3.1) where the contrast of individual mesogranules is small (DeRosa &Toomre 1998). With this method, we are able to measure both horizontal componentsof the near-surface velocity field for several supergranular lifetimes. We begin here bybriefly describing the MDI instrument, which provides the velocity time series to which
the data processing scheme is applied. Then we summarize the data processing steps
leading up to the application of the correlation tracking technique described in §2.2.2.

2.2.1

The MDI Instrument on SOHOThe Solar and Heliospheric Observatory (SOHO) spacecraft provides extended,
uninterrupted observations of the sun by viewing our star from the L1 Lagrangian point
located approximately 1.5 × 106 km sunward from earth. The suite of 12 instruments
on SOHO observe the solar interior and atmosphere as well as the solar wind from avantage point where images and measurements free of atmospheric seeing effects can beobtained. Furthermore, the Doppler velocity of that position relative to the sun onlychanges slowly in time, unlike in near-earth orbits, and this relative stability leads togreatly enhanced sensitivity in Doppler measurements.
One of the instruments on SOHO, the Michelson Doppler Imager (MDI) (Scherrer
et al. 1995), was designed so that helioseismic techniques applied to its images could
be used to probe the interior structure of the sun. Such analyses are performed using
Doppler and intensity observations of the photosphere with high spatial and temporal
resolution, providing precision measurements of the many resonant mode frequencies.
From its space-based location, MDI is able to provide data undistorted by the earth’s
atmosphere and uninterrupted by diurnal gaps, allowing helioseismologists to observe
the solar interior in greater detail than previously possible. Such rapid, undistorted
time series are also ideal for the measurement and analysis of surface flows considered
here.

26
MDI is able to provide images of continuum intensity, line-of-sight velocity, andline-of-sight magnetic field by sampling the radiation field at 75 mÅ intervals surrounding the mid-photospheric 6768 Å absorption line of Ni i. After passing through thetelescope, the light travels through a series of fixed filters and two tunable Michelsoninterferometers which are able to provide filtergrams anywhere in the vicinity of theabsorption line. Images of the sun at five equally spaced wavelengths are then linearly
combined to compute the observables listed above. The full sun can be imaged simultaneously onto a 1024 × 1024-pixel CCD detector (with 200 pixels), or a higher resolution
field (with 0.600 pixels) can be projected onto the same detector. Telemetry constraints
limit the quantity of data which gets beamed back to earth, but dedicated campaigns ineach of the last five years have provided continuous coverage of velocity and sometimes
intensity and magnetic field at a cadence of one minute for a duration of several months.
We use data from these so-called Dynamics Campaigns providing full-disk data with few
interruptions to perform our study of supergranular surface flows.

2.2.2

Isolating the Mesogranules

We use the line-of-sight velocity images observed by MDI in full-disk mode (200 pixels) during the 1999 Dynamics Campaign at a cadence of one minute. These time series
are processed so that the evolving mesogranular pattern can be used to track surface
flow on supergranular size scales. To accomplish this objective, one 45◦ -square regionof photospheric plasma is tracked and remapped onto a latitude-longitude coordinate
system as it rotates across the disk of the sun, as illustrated in Figure 2.2. The pixel
size in the remapped images is equivalent to 1.46 Mm in both latitude and longitude,
which is approximately equal to the instrument pixel scale of 200 at disk center. The
entire region is tracked rigidly at a synodic rate of 13.5◦ day−1 , which is slightly fasterthan the Carrington rate of 13.2◦ day−1 and equal to the average supergranular rateover the latitudinal extent of the region as measured by Snodgrass & Ulrich (1990).

27
(a)

(b)

45° ≈ 560 Mm

Figure 2.2: The first step in the data processing sequence. (a) The full-disk line-of-sightvelocity image observed by MDI on 1999 May 7, 00h 01m UT, scaled to ±2500 m s−1 . The
equator is indicated by the solid black line crossing the image. (b) The 45◦ -square region
enclosed in white in (a) after remapping to a latitude-longitude coordinate system, andscaled to ±800 m s−1 . The Doppler signal due to solar rotation, visible in (a) as the large
horizontal gradient of velocity across the solar disk, is removed as part of the remapping
process. In both images, approaching velocities are dark and receding bright.

The time series spans approximately six days, or about as long as a 45◦ -squareregion centered in latitude near the equator remains accessible to our analysis on thesolar disk. Although all points in the tracked region remain on the earth-facing side of
the sun for longer than six days, most of that additional time places the regions within
about 25◦ of the limb where geometric foreshortening effects blur the objects of interest.
During the remapping stage, the large Doppler signal resulting from the overallrotation of the sun is removed by subtracting an empirical fit to the observed differential
rotation profile. This empirical fit is formed by averaging together 60 consecutive fulldisk images (spanning one hour) and then fitting a planar surface to the resulting
average image. A circular mask is used so that pixels falling outside the solar disk do
not contribute to the fit. This surface fit is subtracted from each full-disk MDI velocity
image before the remapping occurs.

28
(a)

30° ≈ 375 Mm
(b)

(c)

Figure 2.3: The next step in the data processing sequence. (a) The central 30◦ portion ofFig. 2.2b after the acoustic oscillations have been attenuated using a low-pass temporal
filter. (b) The image in (a) after 7.3-Mm Gaussian smoothing, and (c) the residualof (a) and (b). Such filtering effectively separates the supergranules in (b) from themesogranules in (c). All three images are scaled to ±500 m s−1 , with approachingvelocities dark and receding velocities bright.

To separate the radial velocity signal (arising from the acoustic oscillations) fromthe horizontal velocities (resulting from supergranulation and mesogranulation), we apply a low-pass temporal filter to the time series. The filter consists of taking a 31-minuteweighted average using a tapered Gaussian filter to attenuate the high-frequency acous-

29
tic oscillations. The weighting function (derived from Libbrecht & Zirin 1986) is





∆t2
∆t2
δt2
δt2
1+ 2 − 2 ,W (δt) = exp − 2 − exp − 2
2τ
2τ
2τ
2τ

(2.1)

where δt is the time difference from the central observing time, τ = 8 minute is the halfwidth of the filter, and ∆t = 16 minute is the half-length of the filter. This weighting
function has the property that W =dW= 0 for δt = ±∆t. For these parameters,d(δt)

typical p-mode oscillation amplitudes are reduced by a factor of 500 compared to thelonger-lived mesogranular and supergranular signals.
At this stage, it is assumed that the remaining structure in the velocity images
originates primarily from the horizontal motions of the supergranular, mesogranular,
and marginally resolved granular outflows. Mesogranules appear as small-scale undulations superposed on the supergranular outflows, as can be seen in Figure 2.3a. The next
step in the data processing sequence is to separate the mesogranular component fromthe supergranular component by applying a spatial Gaussian filter of width 7.3 Mm(equivalent to 5 pixels) to each of the individual images comprising the time series.
Retaining the low frequencies produces an image of the supergranular pattern as in
Figure 2.3b, whereas the high-frequency image contains the mesogranules as in Figure 2.3c. The sum of these two components yields the original unfiltered image. The
filter width was chosen so that the locations of individual supergranules in the field
are difficult to determine by inspecting only the mesogranular component. Once time
sequences of mesogranular component are available, flow maps are obtained by applying
the correlation tracking technique.

2.3

THE CORRELATION TRACKING TECHNIQUE

2.3.1

The Correlation Tracking Algorithm

We now describe the correlation tracking algorithm and its sensitivities. Correlation tracking methods, as applied to solar images, were primarily developed by members

30of the Lockheed-Martin Solar and Astrophysics Laboratory, most notably Neal Hurlburt, Dick Shine, and Alan Title (Title et al. 1995). The algorithm presented here is
the result of several rounds of fine-tuning, both on their part (Hurlburt et al. 1995) andours.
The correlation tracking algorithm takes as input two equally sized images I1 (x, y)
and I2 (x, y), where in this notation integer values of the pixel coordinates x and y index
the pixels in the two images. We now identify N coordinates in the image field as
gridpoints at which to measure the local displacement of the features embedded in thetwo images. Note that while it is possible for the measurement gridpoints to be locatedat fractional pixel coordinates, it is computationally more sensible to have them locatedat integer pixel coordinates. In practice, the gridpoints form a regularly spaced array.
We denote each measurement gridpoint by the pixel coordinates (xn , yn ), where thevariable n = 1, 2, · · · , N indexes the measurement gridpoints.
At each gridpoint (xn , yn ), the goal is to calculate the optimal displacement(δx, δy) such that the topology of the pixels in the neighborhood of (xn , yn ) in image I1
best coincide with the topology of the pixels in the neighborhood of the corresponding
gridpoint on image I2 . Because we consider only those pixels in the neighborhood ofeach gridpoint, the optimal displacement thus calculated serves as a measurement of
the movement of the features in the immediate area of the gridpoint. This optimal
displacement is determined by shifting the pixels in the neighborhood of each gridpointby varying amounts and determining the relative displacement of the best overlap. In
this formulation, the pixels closer to the gridpoint are weighted more heavily than pixels
farther out, as characterized by the e-folding width σ of the spatial weighting function.
The mathematical details of the algorithm have been omitted here, but are presented
in Appendix A for the interested reader. The end result is an array of displacements(δx, δy) at each measurement gridpoint, which can be converted into a velocity after
dividing by the time separation between I1 and I2 . Note that the spatial resolution

31of the resulting velocity field is determined by σ, and not by the gridpoint spacing.
In practice the flow maps are spatially oversampled by a factor of two or four in eachdimension.

2.3.2

Calibrating the Algorithm

To detect supergranular outflows, the algorithm needs to be able to detect displacements as small as 0.004 pixels, equivalent to a structure having a velocity of
100 m s−1 given the spatial and temporal resolution of the MDI full-disk solar data. We
can assess the accuracy and precision of the correlation tracking algorithm by shifting
sample images by known amounts, and then applying the correlation tracking algorithm
to the original and shifted image pairs. Before describing the results of these calibration
tests, we first briefly describe the Fourier shifting algorithm.

2.3.2.1

The Fourier Shifting Scheme

Shifting the sample images is performed by the Fourier shift technique, whereby
the image is reconstructed at shifted gridpoints once the two-dimensional Fourier spectrum of the image is known. The Fourier shifting scheme was chosen since it incorporates
global information, whereas the interpolation performed as part of the correlation tracking technique is local. Given a two-dimensional 2M ×2N -pixel image I(x, y) indexed by
x = 0, 1, · · · , 2M − 1 and y = 0, 1, · · · , 2N − 1, the discrete forward and inverse Fouriertransforms are given byˆ p , ωq ) =
I(ω

2M
−1 2N
−1
X
X
1
I(x, y) e−iωp x e−iωq y
4M N x=0 y=0(forward transform)(2.2)(inverse transform)

(2.3)

and

I(x, y) =

2M
−1 2N
−1
X
X
p=0

q=0

ˆ p , ωq ) eiωp x eiωq y ,I(ω32
where the spatial frequencies ωp and ωq are defined
ωp =

πp
M

and

ωq =

πq
.
N

(2.4)

To determine I(x + δx, y + δy), we simply evaluate equation (2.3) at the shifted
points x + δx and y + δy:I(x + δx, y + δy) =

2M
−1 2N
−1
X
X
p=0

=

ˆ p , ωq ) eiωp (x+δx) eiωq (y+δy)
I(ω

q=0

−1 h
2M
−1 2N
X
X

i
ˆ p , ωq ) eiωp δx eiωq δy eiωp x eiωq y .
I(ω

p=0

(2.5)

(2.6)

q=0

ˆ p , ωq ) allows one to calculate theConsequently, obtaining the transformed image I(ωˆ p , ωq ) modulated
shifted image I(x + δx, y + δy) by taking the inverse transform of I(ωby the function eiωp δx eiωq δy .

2.3.2.2

Assessing the Accuracy and Precision of Correlation Tracking

The calibration experiments were performed on the two 384 × 384-pixel images
shown in Figure 2.4. Image A (Fig. 2.4a) is the superposition of 100,000 two-dimensional
Gaussian functions whose positions, amplitudes, and widths in each the two directions
were randomly chosen. The amplitudes were allowed to be of either sign. Image B(Fig. 2.4b) is a 45◦ -square (heliographic) region of mesogranulation centered approximately just north of disk center, which was originally observed by MDI and processedso that the mesogranules are evident, as described in §2.2.2.
To assess the accuracy and precision of correlation tracking, we shift the two
sample images by several amounts ranging from 0.001 to 0.4 pixels, and then apply thecorrelation tracking algorithm to each shifted image and its unshifted parent image.
As stated earlier, we need to detect shifts as small as 0.004 pixels, but the behaviorof the algorithm at large shifts is also of interest. For each such pair of images, thecorrelation tracking algorithm computes the optimal shift at each gridpoint in a 48 × 48
array of measurement gridpoints. These gridpoints are spaced 8 pixels apart, with the33(a) Image A(b) Image B

Figure 2.4: Sample images used for the correlation tracking calibration tests: (a) Image A, an image of 100,000 Gaussian structures whose sign, size, height are randomly
chosen to approximate the mesogranulation pattern; (b) Image B, a 45◦ -square imageof mesogranulation similar to those analyzed in the next chapter.

e-folding distance σ also chosen to be 8 pixels. Because the overlap between neighboring
subimages is small, each of the 482 = 2304 displacements measured by the correlation
tracking algorithm at each gridpoint thus serves as a (mostly) independent measurementof the actual shift in each case.
In Figure 2.5 is shown a series of scatter diagrams containing the results of shifting
Image A by several amounts in the positive x-direction. In each panel, the optimal
displacement (δx, δy) for each measurement gridpoint is plotted as a point, with theblue cross characterizing a two-dimensional Gaussian fit to the distribution of optimal
displacements. The red cross in each panel indicates the amount each image was actuallyshifted. Comparing the locations of the two crosses therefore yields the systematic error
associated with the correlation tracking algorithm, while the arm length of each blue
cross, indicating the 2σ width of the Gaussian fit in the x- and y-directions, represents
the random error.
As shown in Figure 2.5, we find that for Image A the correlation tracking algo-

34

SHIFT = 0.001 pixels

SHIFT = 0.002 pixels

SHIFT = 0.003 pixels

0.004

0.004

0.004

0.002

0.002

0.002

0.000

0.000

0.000

-0.002-0.002-0.002-0.004
0.000

0.002

0.004

0.006

0.008

-0.004
0.000

SHIFT = 0.004 pixels

0.002

0.004

0.006

0.008

-0.004
0.000

SHIFT = 0.006 pixels
0.004

0.004

0.002

0.002

0.002

0.000

0.000

0.000

-0.002-0.002-0.002

0.002

0.004

0.006

0.008

-0.0040.002

SHIFT = 0.010 pixels

0.004

0.006

0.008

0.010

-0.004
0.004

SHIFT = 0.020 pixels
0.010

0.010

0.005

0.005

0.005

0.000

0.000

0.000

-0.005-0.005-0.005

0.005

0.010

0.015

0.020

-0.010
0.010

0.015

0.020

0.025

0.006

0.008

0.006

0.008

0.010

0.012

SHIFT = 0.050 pixels

0.010-0.010
0.000

0.004

SHIFT = 0.008 pixels

0.004-0.004
0.000

0.002

0.030

-0.010
0.040

0.045

0.050

0.055

0.060

Figure 2.5: Scatter diagrams of the x- and y-displacements computed by applying thecorrelation tracking technique to Image A (Fig. 2.4a). Each panel contains data fromeach of nine shifts in the x-directions as indicated on top of each panel. The dots
represent the displacement measured by the correlation tracking algorithm for each of
the 2304 measurement gridpoints, while the known shift is plotted as a red cross. The
blue cross represents a two-dimensional Gaussian fit to the data, with the length of thearms indicating the 2σ level in the x- and y-directions. Note that the overall scales forthe images in the bottom row have been broadened.

35
10.00

10.00
(b)
RANDOM ERRORSYSTEMATIC ERROR

(a)

1.00

0.10

0.01

1.00

0.10

0.01
0.001

0.010
0.100
ACTUAL SHIFT IN X

0.001

0.010
0.100
ACTUAL SHIFT IN X

Figure 2.6: Systematic (a) and random (b) errors for the correlation tracking algorithm
applied to Images A (green) and B (purple) of Fig. 2.4.

rithm measures an average shift in the x-direction that is systematically larger than
the actual shift, ranging from about 10% for the largest shifts to a factor of 2.5 forthe case of the 0.001-pixel shift. These results for Image A are also shown in green in
Figure 2.6a, wherein the systematic errors are plotted as a function of the known shift.
The ordinate is defined bysystematic error =measured shift− 1,
actual shift(2.7)

corresponding to the percentage increase of the measured shift relative to the actual
shift.
We believe the systematic errors are caused by the difficulty of the algorithm toaccurately locate the optimal displacement. As discussed in Appendix A, for each measurement gridpoint (xn , yn ) the algorithm searches (δx, δy)-space for the displacement
which minimizes a merit function m(δx, δy). This function is equal to the squared difference between two subimages shifted by the relative shift (δx, δy). If the function m
is somewhat lumpy in the area of the global minimum, the algorithm may settle into a

36
local minimum in the vicinity of the global minimum, rather than the global minimum,
causing the merit degradation problem described in §A.5. In all nine scatter diagrams,a large fraction (over 90%) of the gridpoints were flagged for merit degradation.
Figure 2.6b shows the random error in the x-direction for each measurement,
equal to the width of the Gaussian fit in the x-direction divided by the actual shift. We
find the random errors of the correlation tracking algorithm are generally smaller than
10% for shifts of 0.005 pixels or larger. This scatter most likely results from inaccuracies
in the interpolation scheme used to perform the image shifting. When analyzing real
solar data, we find that both spatially and temporally averaging the flow maps until
the resulting flows become coherent will attenuate the random error produced by thealgorithm. We discuss such averaging further in §3.1. The random errors in the ydirection are less than those in the x-direction, as shown for Image A by the blue
crosses in Figure 2.5.
We have performed the same calibration experiments on Image B, containing thesolar mesogranulation pattern of Figure 2.4b. We find that both the systematic andrandom errors for Image B are slightly larger than those for Image A, but otherwise
shows the same general trends. These errors are plotted in purple in Figure 2.6. We
note that the region surrounding disk center of Image B, corresponding to the washedout area of the image in Figure 2.4b just below center, causes a contrast gradient in themesogranulation pattern that is not present in Image A. This contrast gradient may be
the source of an additional source of systematic error seen in the correlation tracking
flow maps, as discussed further in §3.1.4.
The results of these simple calibration studies show that correlation tracking caneffectively measure the proper motions of moving patterns and features contained insample images down to displacements of about 0.004 pixels. For shifts of 0.005 pixels
and 0.05 pixels, the systematic errors in these correlation tracking measurements arelimited to about 10%. The random errors steadily descrease from 10% for a 0.005-

37
pixel shift to less than 2% for a shift of 0.05 pixels. The high temporal cadence ofthe time series of solar data permits spatial and temporal averaging which reduces therandom noise level below the strength of the coherent signal of interest. In the next
chapter, we use the correlation tracking technique to measure the near-surface velocity
field experienced by solar mesogranulation.

Chapter 3

THE NATURE OF SOLAR SUPERGRANULATION

We use near-photospheric flow fields to study surface convection on size scales on
the order of solar supergranulation and larger. The flow fields are obtained by applying
the correlation tracking method described in Chapter 2 to the mesogranulation pattern
extracted from time series of MDI velocity images. The time series discussed in thischapter is of a 45◦ -square region centered on the equator observed from 1999 April 14–20
during the April 1999 Dynamics Campaign of MDI.
We first provide an overview of the flow fields deduced by correlation tracking,and their relation to other determinations of the surface velocity field. An anomalous
flow of unknown origin present in the flow fields is also discussed. In §3.2 we provide
a more dynamical description of solar supergranulation, before formally characterizing
the distributions of supergranular sizes and lifetimes in §3.3. We present in §3.4 theresults of our efforts to search for larger-scale flows by applying correlation tracking to
the supergranular flow pattern. Finally, we present concluding remarks in §3.5.

3.1

ASSESSING CORRELATION TRACKED FLOW MAPS

3.1.1

Properties of the Flow Maps

As described in §2.2 and §2.3, our dataset is a time series of one corotating regionof quiet sun measuring 45◦ on a side. Each image in the time series was extracted froman MDI full-disk velocity image and remapped onto a latitude-longitude grid with a

39

20 Mm

400 m s-1

toward disk center

Figure 3.1: A 4-hr averaged flow map, generated by applying correlation tracking methods to a time series of mesogranules, is superposed as a pattern of arrows on the corresponding time-averaged, line-of-sight velocity image. This 20◦ -square region is centeredheliographically on (b, `)=(7.5◦ N, 30◦ E), with the direction toward disk center from thecenter of the image as indicated. The background velocity image has been smoothed in
the same manner as Fig. 2.3b to bring out the supergranules, and is scaled to ±500 m s−1 ,with approaching velocities dark and receding velocities light. The arrow field has anrms velocity of 194 m s−1 , and is spatially oversampled by a factor of four relative tothe gridpoint neighborhood size σ.

40

20 Mm

400 m s-1

toward disk center

Figure 3.2: The same 4-hr averaged flow map of Fig. 3.1 is superposed on an image of
its horizontal divergence. Regions of negative divergence (convergence) are dark andpositive divergence light.

pixel spacing of 1.46 Mm in each direction. The temporal cadence of the time series is
one minute. The images were then filtered (using a high-pass 7.3 Mm Gaussian kernel)
to reveal the mesogranulation pattern, after which the correlation tracking algorithm
is applied to obtain a time series of surface flow maps. For this study, we set the sizeof the measurement gridpoint neighborhood at σ = 8 pixels or 11.7 Mm (see §2.3 for41
the formal definition of σ), equal to approximately two mesogranule diameters. The
measurement gridpoints are spaced 2 pixels or 2.92 Mm apart, such that the spatial
resolution of the flow maps is oversampled by a factor of four in each dimension. The
oversampling allows the resulting flow fields to be spatially averaged in order to reducesome of the random noise introduced by the correlation tracking algorithm.
We now examine these correlation tracked flow fields and make comparisons withother horizontal velocity determinations. Figure 3.1 shows a 4-hr averaged flow field,
superimposed as a pattern of arrows on the corresponding 4-hr averaged image of line-ofsight velocity that has been smoothed to show only the supergranulation pattern. The
flow map contains several prominent outflow sites of diameter 20–30 Mm which coincidewith the supergranules evident in the velocity image. Also present are diverging lanes,
where fluid motions spread out on either side of an essentially linear feature. Theselanes can result from several supergranules packed closely enough such that the spatial
resolution of the flow map is insufficient to discern the area of converging fluid in theintercellular region.
Visually studying time series of flow map arrows can be somewhat confusing, so
we typically compute the horizontal divergence D of each flow map in our time series
and instead analyze time series of the scalar field D. The horizontal divergence D is
defined as
D(x, y) =∂ux ∂uy
+
,
∂x
∂y

(3.1)

which is computed directly from the horizontal flow field (ux , uy ) using second-orderfinite differences to approximate the spatial derivatives. Individual supergranules, which
by definition are cells of horizontal outflow, are consequently identified more easily fromimages of D than from the preceding flow fields, as such outflow features now appear
as local maxima in divergence. The divergence image computed from the flow field ofFigure 3.1 is displayed as the background image in Figure 3.2.

42
(a)

(b)

(c)

40 Mm

Figure 3.3: A 40◦ × 20◦ field of mesogranulation (a) and supergranulation (b) as seenin images of Doppler velocity, accompanied by the corresponding image of horizontal
divergence (c) deduced from correlation tracking. All three images are centered on
disk center. The locations of several prominent supergranules, most clearly seen in thedivergence image, are circled in all three images.

43

urms [m s-1]

1000

supergranularregime

100
0.1

1.0
10.0
averaging window, w [hr]100.0

Figure 3.4: The rms velocity urms as a function of the temporal averaging window w.
1
Temporal averages of random velocity vectors would fall off as urms ∼ w− 2 , indicatedby the dotted lines. The flatter parts of the spectrum reflect the existence of featureswith lifetimes of order w. To see the supergranular outflows, we need to average forabout 4 hr to obtain a coherent signal.

One advantage of applying correlation tracking to mesogranules is that we canstill reliably detect outflows located near disk center where the line-of-sight velocity
signal from horizontal fluid motions fades out. Figure 3.3 shows the velocity field in
the disk center region for one day in May 1999. The top two panels are Doppler
images showing the mesogranulation and supergranulation patterns respectively. One
can see that the detection of supergranules directly from the velocity images becomesincreasingly more difficult as one approaches the center of the disk due to their lower
contrast. While the mesogranules also show lower contrast levels, they remain coherent
enough to be recognizable in correlation studies. The bottom panel in Figure 3.3 is animage of horizontal divergence calculated directly from the horizontal flow field deduced

44
from correlation tracking. The locations of several prominent supergranules, as revealed
most clearly in the divergence image, are circled in each of the three images.
The flow maps are averaged in time (typically for several hours) so that only thelonger-lived flows remain, but also to reduce the random error of the flow maps. One
source of random error originates from the algorithm itself, as discussed in §2.3. In
calibration tests there, the algorithm is found to have a random noise level of approximately 10% for flows on the order of 100 m s−1 ; averaging for several hours or more
will reduce this error below 1%. In addition to the random noise component, it is notwell known how the residual signal originating from granulation pattern manifests itself
in the flow fields. With 200 pixels, individual granules are at best marginally resolvedin our data, though larger resolved structures associated with solar granulation (suchas exploding granules) may still be present. We do not filter the data in any way thatspecifically targets the granulation pattern, though the temporal filtering designed toattenuate the acoustic oscillations also attenuates much of the granulation signal.
Time-averaging will abate the effects of both the residual granulation signal andthe algorithmic noise in the velocity images. In Figure 3.4 is plotted the rms velocity as
a function of the time-averaging window for a sample time series of flow maps. Flatterregions of this curve indicate that the flow field contains temporally coherent features at
those time scales, while the steeper portions of the curve indicate incoherence on thosetime scales. In this study, an averaging window of at least 4 hr (corresponding to 240
flow maps) is used because much of the random noise present in the data is reduced
while preserving the evolution of individual supergranules, a phenomenon we believeoccurs on time scales much longer than 4 hr.

3.1.2

Comparison With Direct Doppler Images

To understand how well the flow maps generated by correlation tracking sense
the actual horizontal velocity field, we can compare the disk-center-directed component

45

correlation tracking velocity [m s-1]

600
r = 0.738
400
200
0
-200
-400
slope of linear fit = 0.747-600-600 -400 -2000
200 400
direct Doppler velocity [m s-1]

600

Figure 3.5: A scatter diagram showing the correlation in Fig. 3.1 between the line-ofsight velocity and the disk-center-directed component of the correlation tracking flow
field. A linear fit to the data is shown, with its slope indicating that the correlation
tracking velocities underrepresent the velocities measured in the velocity image by about
25%.

of each flow vector in the flow maps with its corresponding line-of-sight velocity as
measured directly from the velocity image. Using Figure 3.1 as an example, which iscentered heliographically at about (b, `)=(7.5◦ N, 30◦ E) so that disk center is below andto the right in the figure, we can see how well the two velocity measurements agree.
Figure 3.5 reveals a strong correlation between the disk-center-directed velocities
measured by correlation tracking and those in the corresponding velocity image. As seenin Figure 3.1, the strong outflow sites present in the flow map are coincident with thesupergranules present in the Doppler image, while other regions of slower mesogranularadvection correspond to regions in the velocity images where no organized supergranular
outflows exist. The correlation coefficient (defined in Burr 1974, for example) is found to46
be r = 0.720, which quantifies the visual impression that the two velocity measurements
agree well.
The line in Figure 3.5 is a linear fit to the data showing the correspondence between the correlation tracking and direct Doppler measurements of velocity. The slopeof the line indicates that the velocities derived from the correlation tracking calculation
underrepresent the velocities indicated on the velocity image by about 25%. Systematic errors of this order of magnitude should be taken into account when interpretingcorrelation tracking flow maps.

3.1.3

Comparison With Time-Distance Results

We now describe a study (see also DeRosa, Duvall, & Toomre 2000) where outflow sites detected using our correlation tracking technique are compared to travel-timedifferences measured by time-distance analysis. This study uses a 2048-min sequenceof velocity images observed by MDI using its high-resolution (0.600 pixels) field-of-view
taken from 1997 January 16–18.
As stated in §2.1.2, time-distance analysis is performed by computing travel-timedifferences between counterpropagating waves. Such travel-time differences arise due tothe effects of bulk fluid motions on the waves, as the mode frequencies will be Doppler
shifted when the waves are propagating upstream or downstream relative to a bulk
flow. In this study, we examine the travel-time differences between f modes (gravity
waves which propagate along the surface of the sun) traveling toward or away from eachmeasurement location.
The time-distance analysis was performed by first blocking the 2048-min timeseries into four 512-min sequences. For each block of data, travel-time differences were
measured between f modes traveling toward and away from each measurement location.
The timing measurements were averaged radially over an annulus with inner and outerradii of 3.72 Mm and 8.67 Mm surrounding each measurement location, and then aver-

47

(b)

200 Mm

(a)

-0.05
0.00
0.05
divergence, D [ms]-1

-0.30
0.00
0.30
time difference, Tio [s]Figure 3.6: Comparison between the correlation tracking and time-distance helioseismology techniques: (a) the image of divergence D calculated from correlation tracking
measurements of the near-surface velocity field for one of the 512-min blocks; (b) theimage of travel time differences Tio measured using time-distance helioseismology forthe same block.

aged azimuthally in space to remove all directional dependence. Finally, the data wereaveraged in time over each 512-min block to increase the signal-to-noise ratio. The end
result is a map of travel-time differences Tio (x, y) as a function of position, where thesubscript “io” indicates that the sense of the difference is times from incoming waves
minus times from outgoing waves.
As a result of the directional averaging, extrema in Tio correspond to locations
where the flow is divergent or convergent (such as supergranular outflows or their associated intercellular lanes). For example, maxima in Tio indicate places where f modes
approaching from any direction, on average, take longer to traverse the same ray paththan the corresponding outgoing f modes, signifying a horizontal outflow. Consequently,
Tio should be proportional to the horizontal divergence D of the near-surface flow field,
which can be computed directly using equation (3.1) on correlation tracking flow maps

48

rs = 0.887

time difference, Tio [s]

0.4

0.2

0.0

-0.2

-0.4
-0.10

-0.05
0.00
0.05
divergence, D [ms]-1

0.10

Figure 3.7: Scatter diagram of the travel-time vs. the divergence data shown in Fig. 3.6.
The best-fit line, having a vertical intercept and slope of −2.01 × 10−4 and 4.48 respectively, is determined from regression.

derived from the same data.
To test this hypothesis, we compared maps of D and Tio for each of the four
512-min blocks of data. Figure 3.6 shows the results for one of the 512-min blocks, after
artificially degrading the spatial resolution of the divergence map to match that of thetime-distance results. The accompanying scatter diagram of Tio vs. D in Figure 3.7
shows the correlation between the two images. The correlation coefficient for this block
evaluates to r = 0.887, and as averaged over all four blocks becomes r̄ = 0.890. The
statistical interpretation of this result is that 79.2% (equal to r̄ 2 ) of the scatter in D
and Tio can be explained by the fact that a linear relationship exists between the twovariables.
This result indicates that the time-distance flow maps, which are sensitive to bulk
fluid motions occurring near the solar surface, are proportional to the motions of the

49

30

heliocentric latitude [deg]

20
10
0
-10
-20
-30
-40

-20
0
20
heliocentric longitude [deg]40Figure 3.8: Anomalous large-scale flow present in the mesogranulation correlation tracking results, remapped in a heliocentric frame where the disk center point is always placedat the origin. As a result, the April 1999 dataset, which is at fixed heliographic coordinates, rotates from east to west across this coordinate system. The flow map arrows
are on the scale of 200 m s−1 .

mesogranule pattern velocities, as deduced from correlation tracking. The additional
similarity between these flows and those measured directly from Doppler images imply
that we can reliably detect fluid motions associated with solar supergranulation using
the correlation tracking technique.

3.1.4

The Anomalous Flow Directed Toward Disk Center

We now discuss another source of systematic error associated with the correlation
tracking flow maps. When applying correlation tracking to the mesogranular time series
observed by the full-disk field-of-view of MDI, we find that superimposed on the supergranular outflow pattern is a larger-scale flow of order 200 m s−1 directed everywhere

50
toward disk center (from the vantage point of MDI). To characterize this anomalous
flow, we have remapped each flow field in the April 1999 time series onto a heliocentric
coordinate system and then averaged in time over the entire dataset. The heliocentric
coordinate system is defined such that the origin always coincides with disk center. The
time-averaged results are shown in Figure 3.8.
Attempts to discover the origin of this anomalous flow have suggested only that
it is a result of the changing contrast gradient of the mesogranular features with heliocentric distance. This contrast gradient is primarily a geometric effect, since thechanging viewing angle across the solar disk causes a variation in the amount of the
proportion of the horizontal velocity projected into the line-of-sight. It is thought that
random noise present in the images coupled with such a contrast gradient produces a
non-zero average when the optimal displacement vectors are computed by the correlation tracking algorithm. As a result, this asymmetry introduces a spurious displacement
at each measurement gridpoint, thereby biasing the resulting flows in the direction of
the asymmetry. Such random variations in the mesogranular data, which we suspect
are the cause of the anomalous flow, can result from instrumental noise, unresolved
granulation, or even the signal resulting from residual acoustic oscillations.
Several artificial datasets have been constructed in order to study the anomalous flow, but to date we not been able to reproduce the effect. It is also not known
whether the spurious velocities are present only using data from the MDI instrument,
which is known to be slightly astigmatic, or whether flow maps computed using datafrom other instruments show the same effect. Flow fields determined by other authors
using correlation tracking on data observed by ground-based instruments routinely had
their mean flows removed because the pointing of these instruments was not known as
precisely as with MDI. The resulting pointing jitter introduces a registration error of an
unknown amount between each consecutive image in the time series, which the correlation tracking algorithm subsequently interprets as a real displacement. This effect is not

51
a problem for MDI as its pointing accuracy is much higher than that of ground-based
instruments.
Because the magnitude of such an anomalous flow is approximately the sameorder of magnitude as the supergranular outflows, it makes analysis of the flow maps
more difficult. However, because the spatial scale of the anomalous flow is much larger
than that of the supergranulation pattern, we typically remove the empirically determined average of Figure 3.8 from our flow fields, which adequately reduces this level ofsystematic error below the supergranular outflow signal. The flow fields shown in Figures 3.1 and 3.2 have had such an average removed. The anomalous flow is also present
in flow maps calculated using the high-resolution field-of-view of MDI, which were used
to perform the comparison with the time-distance results. However, the smaller pixel
size of the high-resolution data allows us to use a smaller gridpoint neighborhood size
σ when applying the correlation tracking method, which reduce the contrast gradientacross each measurement gridpoint cell. As a result, the anomalous flow magnitudes
are proportionally smaller.
In spite of the anomalous flow, we believe that correlation tracking is an effective
method of deducing the locations of supergranular outflows from time series of lineof-sight velocity images. We are able to measure both components of the horizontal
velocity field across most of the disk, including the disk-center region where horizontal
velocities cannot be measured directly. We have also shown that our flow maps compare
favorably with direct Doppler and time-distance measurements of the near-surface horizontal velocity field. We now use the flow maps determined by correlation tracking to
examine the dynamics of the supergranular flow field in more detail, both qualitatively
in the next section and quantitatively in §3.3.

52

3.2

A DYNAMICAL PICTURE OF SUPERGRANULATION
The flow maps determined from this dataset, consisting of about six days of data,

represent the longest uninterrupted time series of solar supergranulation to date. The
4-hr temporal cadence allows us to study the evolution of supergranular flows with
unprecedented temporal resolution. Since supergranules are primarily defined by theirvelocity structure, one way to examine these outflow patterns is by placing passive
floaters (“corks”) in our horizontal flow fields and observing their motions in response.
Figure 3.9 shows a time series of horizontal divergence images, computed fromcorrelation tracked flow fields of the April 1999 dataset. Cork locations as a function oftime are overplotted in red as they are advected horizontally by the evolving flow field.
Initially, the corks are distributed uniformly across the field-of-view. However, on timescales of about 8–12 hr, the corks within each supergranule are advected away from thecell centers at speeds of about 500 m s−1 toward lanes of convergence which occur where
neighboring outflows meet. This cleaning-out process is illustrated in the top row ofimages in Figure 3.9, where most corks in the field have reached the nearest convergencelane within 12 hr. Collectively, these corks trace out a network defining supergranuleboundaries, or in some cases, the corks outline regions of the flow field containing several
supergranules. Once within a convergence lane, the corks begin to travel along the lanes
at much slower speeds toward collection points in the network. The effect of this slower
migration toward network interstices is evident in Figure 3.9, where by the end of thetime series the connected network present at the t = 12 hr mark has split into multiple
fragments each containing corks approaching their respective collection points. Notethat the corks are permitted to coincide, accounting for the (false) impression that the
total number of corks in the figure decreases with time.
While the corks are approaching the collection points, the flow field continues
to evolve. Occasionally, a new supergranular outflow emerges at a location which was

53
h

t=0

h

t=4

h

h

t=8

t = 12

t = 16ht = 20ht = 24ht = 28ht = 32ht = 36ht = 40ht = 44ht = 48ht = 52ht = 56ht = 60ht = 64ht = 68ht = 72ht = 76h15° ≈ 187 Mm

Figure 3.9: A 15◦ -square region from the April 1999 dataset showing cork locations (red)superimposed on the corresponding horizontal divergence images (gray scale). The corks
are first advected toward cell boundaries on time scales of 12 hr, after which they travel
toward boundary intersections on longer time scales. The regions enclosed in the circle
show the emergence of a long-lived supergranule which suddenly appears at one of the
network intersections. The boxed regions are enlarged in Fig. 3.12.

54
t = 0ht = 4ht = 8ht = 12ht = 16ht = 20ht = 24ht = 28ht = 32ht = 36ht = 40ht = 44ht = 48ht = 52ht = 56ht = 60ht = 64ht = 68ht = 72ht = 76h15° ≈ 187 Mm

Figure 3.10: The same time series as in Fig. 3.9, but here additional corks have been
added at random locations with time, emphasizing the newly formed convergence lanes
which have appeared after the initial 12-hr clean-out time. The time series continues
in Fig. 3.11, where the supergranule in the circle can be seen to survive for about twodays. The boxed regions are enlarged in Fig. 3.12.

t = 80ht = 84ht = 88ht = 92ht = 96ht = 100h

t = 104ht = 108h

t = 112ht = 116ht = 120h

t = 124ht = 128ht = 132h55

15° ≈ 187 Mm

Figure 3.11: A continuation of the time series of Fig. 3.10.

formerly part of the network, as in the circled regions of Figure 3.9. At t = 36 hr, thecorks in the circled region trace out a portion of the network where three lanes intersect.
Four hours later, a new supergranule appears, as indicated by the bright spot in thedivergence image. Over the following 12 hr, the corks are advected away from this new
outflow center until they reach the edge of the emergent supergranule.
New supergranules appearing via such emergence events are less common thant = 20ht = 24ht = 28ht = 20ht = 24ht = 28h56

8° ≈ 98 Mm
Figure 3.12: A comparison between the cork movies of Figs. 3.9 (top row) and 3.10
(bottom row) showing the different cork responses to the formation of a new convergencelane (black box). By adding corks continuously throughout the time series in the lower
sequence, convergence lanes which form after the initial 12-hr clean-out time become
populated with corks, whereas these lanes appear free of corks when additional corks
are not added, as in the upper sequence. The panels here correspond to the regions
enclosed in the square boxes of Figs. 3.9 and 3.10.

those that result from the fragmentation of existing supergranules into multiple cells, as
new convergence lanes are continually forming within existing supergranular outflows.
These new lanes would not be populated by corks if they were to form after the initial
12-hr cleaning-out period, as no corks remain within the adjoining outflow cells to
populate the newly formed convergence lanes. To remove this dependence on the initialtime at which the corks are placed into the flow field, we have adopted the approachof placing additional corks at random locations continually throughout the time series,

57

t = 30ht = 32ht = 34ht = 36ht = 38ht = 40h8° ≈ 98 Mm
Figure 3.13: A time series showing a large cell outlined by corks which contains several
smaller coutflow centers which evolve on time scales of several hours. These smaller
outflows are separated by weaker convergence lanes which do not exist long enough to
get populated by corks. For this time series, corks were continually introduced into theflow field.

as illustrated by Figures 3.10 and 3.11. There, we show the same time series as in
Figure 3.9, except that additional corks are introduced at a rate such that the numberof new corks introduced over each 6-hr interval is equal to the number of corks present
initially. These added corks serve to outline convergence lanes formed after the initial
12-hr cleaning-out period, as illustrated in the close-up images of Figure 3.12.
From Figures 3.9–3.12 it is evident that some of the regions surrounded by thecork boundaries contain several peaks in divergence separated by weaker convergencelanes. Both the weaker lanes and associated outflow centers typically evolve on timescales much shorter than the 12 hr it takes the flow to clean out corks in any individual

58

t = 52ht = 55ht = 58ht = 61ht = 64ht = 67h8° ≈ 98 Mm
Figure 3.14: A time series showing two convergence lanes (indicated by the arrows)
which are advected horizontally by the evolving flow field. For this time series, corks
were continually introduced into the flow field.

cell, and are often intermittent in nature. A weak convergence lane may, for example,
either become more established or disappear completely, and in the process undergo
several weakenings and strengthenings. Such evolution is evident in Figure 3.13, where
we show a large structure outlined by corks containing several smaller outflow centers.
Consequently, if one considers the network of all convergence lanes, both weak andstrong, as defining supergranular boundaries, then the typical sizes and lifetimes ofsupergranulation are much smaller and much shorter than the cork network would
suggest.
If a convergence lane becomes strong relative to its neighboring outflow cells, itwill generally persist until a stronger feature appears nearby. Figure 3.14 shows a pair of

59

20

latitude [deg]

10

0

-10

-20
0

1

2

3
time [days]

4

5

Figure 3.15: A space-time cut across 45◦ of latitude of horizontal divergence for the
April 1999 dataset. Horizontal structures represent persistent outflow cells, with theirlengths proportional to their lifetimes.

outflow lanes which are advected horizontally in response to strengthening supergranules
in the surrounding region. In the first panel, arrows indicate two existing convergencelanes which partially enclose a supergranule. As this supergranule weakens, two stronger
supergranules emerge on either side, whose outflows cause the two convergence lanes tomove laterally in response until the original cell vanishes.
Such dynamic behavior, including the emergence, fragmentation, and lane advection events described above, is prevalent in the supergranular flow fields deducedby correlation tracking. The flow field also contains many outflow cells which survive
for several days and coexist with other more quickly evolving outflows. For example,
the cell circled in Figures 3.10 and 3.11 survives for about two days. We also show in
Figure 3.15 a space-time cut of horizontal divergence extracted from the same dataset.
The bright horizontal structures represent outflow cells which persist at the same location, with the lengths of such structures indicative of their lifetimes. It is evident that60
outflow cells with lifetimes of at least 36 hr are plentiful, accompanied by other outflow
cells having much shorter lifetimes.
Throughout this section, we have illustrated several modes of evolution associated with supergranular outflows detected at the solar surface, including instances ofcell emergence, merging, and fragmentation. The associated network of convergencelanes also changes constantly, as the network continually reorganizes itself in responseto nearby evolving outflows as well as to the turbulent motions occurring beneath thesurface. In addition to the regions where the flow pattern evolves quickly, we also findfeatures which persist for much longer time periods. In the next section, we substantiate
the qualitative impressions described here by measuring the distribution of supergranular cell sizes and lifetimes after identifying individual outflows on our flow maps.

3.3

MEASURING SUPERGRANULAR SIZES AND LIFETIMES

3.3.1

Correlation Sizes and Lifetimes

The classical technique for measuring supergranular properties involves computing spatial and temporal correlation functions of photospheric and chromospheric data.
Separations between the maxima of spatial correlation functions yield an average separation of the supergranules evident in the data. Using line-of-sight velocity images,
the average separation between supergranular cell centers is 32 Mm, as originally foundby Leighton et al. (1962) and Simon & Leighton (1964) and confirmed by several later
studies (e.g. Wang & Zirin 1989).
Such correlation analyses can also be applied to quiet-sun images of surface magnetic flux, which effectively outline supergranular boundaries, since small-scale concentrations of magnetism visible at the photosphere are quickly advected horizontally bysupergranular outflows (Schrijver et al. 1997; Hagenaar et al. 1999; Lisle et al. 2000).
This network has been observed in the chromosphere using emission in Ca ii K as a61proxy for magnetic field. Correlation studies of this chromospheric network yield average cell sizes of 30–35 Mm (Simon & Leighton 1964; Singh & Bappu 1981; Hagenaar
et al. 1997) in agreement with the measurements of supergranular cell sizes derived fromvelocity images.
Spatial correlation and power spectral analyses applied to high resolution horizontal flow maps by November (1994) also give a horizontal length scale of 32 Mm forsupergranulation, but these and other more recent flow map studies (e.g. Shine et al.
2000) show that most supergranules contain several smaller divergence centers having an
average separation of about 7 Mm, consistent with mesogranular separation distances.
The correlation tracked flow maps of this study also show large outflow cells containing
smaller divergence centers, confirming the impression that broad outflows associatedwith supergranulation are comprised of several smaller divergence centers.
Cell lifetimes can also be determined by computing temporal correlation functions
among successive images in a time series. The rate at which the function declines fromits central maximum yields a measurement of the average lifetime of the supergranules
in the field of view. Simon & Leighton (1964) were able to measure the average cell
lifetime of the Ca ii K network cells to be 20 hr, equal to the length of time required for
temporal correlations of successive images to drop by a factor of e−1 from its initial value.
However, Wang & Zirin (1989) argue that proper motions and evolutionary changes inindividual supergranules in the field of view may bias the correlation lifetime toward
shorter values, due to the possibility that supergranules may alter their appearance or
location without losing their identity, and thus the actual supergranular lifetime may
be even longer than 20 hr. These effects occur on shorter time scales than typicalsupergranular lifetimes, and are also reflected in the shape of the temporal correlation
function. Indeed, we have already demonstrated the existence of several supergranules
having lifetime of at least 36 hr (e.g. in Fig. 3.15) in our data.

62
3.3.2

The Direct Identification Method

While correlation and power spectral analysis methods are useful for obtaining
a general idea of supergranular cell sizes and lifetimes, we propose that measurementsof such quantities should take into account the more complex phenomenology by which
individual supergranules form and evolve. Events similar to those described in §3.2,
where supergranular cells frequently merge and fragment, are observed to occur much
more frequently than cases where individual supergranules simply emerge, persist, anddisappear all without interacting with other outflow cells. In order to account for such
intricate evolutionary behavior, we have devised a pattern recognition algorithm that
identifies the outflow centers on each divergence image in our time series, from which
the detailed life histories of all supergranules in the dataset can be determined. This
information is then used to compile a statistical database of the sizes and lifetimes ofall supergranules present in the April 1999 dataset.
Before identifying the supergranules, we attenuate the random noise in the divergence data by applying selected spatial and temporal filters. The data are averaged
in time using boxcar filters with widths ∆t of 4 hr, 6 hr, and 8 hr, and using spatialsmoothing using Gaussian kernels with widths w of 5.8 Mm, 8.8 Mm, and 14.6 Mm.
Such filtering operations are generally consistent with the effective spatial and temporal
resolution of the data, so we do not believe the filtering causes much loss of information. We realize that such filtering will sometimes artificially blur together some of the
distinct objects we seek to identify, and as a result we assess the effects such filtering
may have on the resulting size and lifetime distributions in what follows.
On each filtered divergence image, we identify supergranular outflow cores, defined mathematically as a contiguous grouping of pixels for which the local curvature(i.e. second derivative) of divergence in all horizontal directions is negative. By this definition, the pixels comprising each outflow core contain exactly one local maximum in63divergence. The identity of individual objects is established by matching outflow cores
which fully or partially overlap on successive images. To be considered supergranules,
the cores must cover at least 50 Mm2 in area (equal to 6 pixels) and survive for at least
4 hr, thresholds which are consistent with the effective spatial and temporal resolutionof the data.
Figure 3.16 shows both the flow field and the corresponding outflow cores. The
full 45◦ -square field from which the figure originates contains approximately 400 outflow
cores, and by implication the same number of supergranules. The cores cover about 20%of the total area of the image, so that the full area of each supergranule is approximately
five times that of its corresponding outflow core, assuming the supergranular patternfills the available area. When tabulating the areas of individual supergranules below,
we multiply the core sizes by the appropriate filling factor, determined by comparing
the area covered by the cores to the total area in the field.
To further illustrate the core identification scheme, we now show in Figures 3.17
and 3.18 the evolving cork network (colored red) of Figures 3.10 and 3.11, except that thedivergence images have been replaced by colored tiles representing the outflow cores.
Each core is assigned a solid color (other than red) at random for as long as it iscontinuously recognizable on successive images. We find that the core identification
scheme performs quite well, as most regions outlined by the corks contain objects, andobjects which overlap in subsequent images have been assigned the same color. The
black object circled in Figures 3.17 and 3.18, for example, retains its identity for about
48 hr until it disappears.
In the cases where objects merge or split, the largest fragment is chosen to retain
the identity of the outflow core. Such a process is illustrated in Figure 3.19, where thetwo fragments inside the circle merge together for a short time and then split up again.
After the cells have merged, at t = 36 hr one large cell (colored purple) exists at thelocation previously occupied by two smaller cells. The large cell retains the color of the

64

20 Mm

400 m s-1

Figure 3.16: A 23◦ -square field of supergranule outflow cores (blue), with the corresponding averaged flow field (red arrows) overlaid. Each outflow core is represented bya contiguous group of pixels. Several prominent supergranules possessing well defined
outflows are circled. The arrow scale is 400 m s −1 , the tick marks are spaced 20 Mm
apart, and the field is approximately 23◦ square. The data were filtered using 2-hr
temporal boxcar and 8.8-Mm spatial Gaussian smoothing functions prior to the identification of the outflow cores. The field is centered in space at (b, `)=(12◦ N, 38◦ E) andin time at 11h 00m on 14 Apr 1999.

largest of the two fragments which merged together. After 8 hr have elapsed, the large
cell splits back into two smaller cells. Again, the largest fragment retains the identityof the original cell. The other (smaller) fragment is assigned a new identity and color.

65
h

t=0

h

t=4

h

h

t=8

t = 12

t = 16ht = 20ht = 24ht = 28ht = 32ht = 36ht = 40ht = 44ht = 48ht = 52ht = 56ht = 60ht = 64ht = 68ht = 72ht = 76h15° ≈ 187 Mm

Figure 3.17: A time series of cork positions (red) relative to the outflow cores (coloredtiles). Outflow cores recognizable on successive images are assigned the same solid color
throughout their lifetime; however, the colors are chosen randomly, making it possiblefor different objects to have the same color. The time series shown here and continued
in Fig. 3.18 is the same as in Figs. 3.10 and 3.11. The long-lived object enclosed in thecircle is the same as identified in Figs. 3.10 and 3.11.

t = 80ht = 84ht = 88ht = 92ht = 96ht = 100h

t = 104ht = 108h

t = 112ht = 116ht = 120h

t = 124ht = 128ht = 132h66

15° ≈ 187 Mm

Figure 3.18: A continuation of the time series of Fig. 3.17.

In this way, each outflow core in the six-day dataset can be assigned a unique
identity, from which its lifetime and average area can be determined. We now have theinformation necessary to compile statistics on the supergranulation pattern, as deduced
from their outflow cores. As discussed earlier, we assume that the area of an individual
supergranule is directly proportional to the area of its outflow core. Figure 3.20 shows
the distribution of average supergranular areas, defined as the average area of eacht = 32ht = 34ht = 36ht = 38ht = 40ht = 42ht = 44ht = 46ht = 48h67

12° ≈ 149 Mm
Figure 3.19: An enlarged region illustrating splitting and merging events. The objects
enclosed in the circle merge together at t = 36 hr, and subsequently dissociate at
t = 44 hr.

supergranule throughout its lifetime. The dotted line represents the best-fit Gaussianfunction, 2
z
N (A) dA = C exp −
2

where

z=

A − hAi
dA,
B

(3.2)

where the mean, width, and height, are respectively denoted by hAi, B, and C. The dis-

68SUPERGRANULE SIZE SPECTRUM
300
0.0025frequency N(A) dA, % of total [Mm-2]

frequency, N(A)250

Ntotal = 3494 objects
binsize = 30 Mm2mean = 690 Mm2

0.0020200

0.0015150

100

0.001050

0.0005

0
0

500
1000
average area, A [Mm2]0.00001500

Figure 3.20: Average size distribution N (A) for the Ntotal supergranules identified in
the April 1999 dataset. The area A is defined as the area of the entire supergranule (notjust the outflow core) as averaged over the lifetime of each object. The outflow cores
were identified after applying 4-hr temporal boxcar and 8.8-Mm spatial Gaussian filtersto the time series. The dotted line is the best-fit Gaussian function [see eq. (3.2)] with
hAi = 669 Mm2 , B = 174 Mm2 , and C = 239.

tribution was compiled using time series of divergence data from the April 1999 dataset
after smoothing by 4-hr temporal boxcar and 8.8-Mm spatial Gaussian filters. Figure 3.20 indicates that supergranules have an average area hAi of 669 ± 174 Mm2 , whichp
is approximately equivalent to a linear extent of hAi=22–29 Mm across. However,as one would expect, this result is sensitive to the degree of spatial smoothing applied
to the data before identifying the outflow cores. Figure 3.21 shows the dependence ofpthe mean sizes hAi on the spatial smoothing width w for each of the temporal andspatial smoothing combinations. One would expect average sizes to increase with moresmoothing, and this is indeed realized. Extrapolating down to zero smoothing, we find69

supergranular size, <A>1/2 [Mm]

40

30

20

10

0
0

2

4
6
8
10
12
smoothing width, w [Mm]14p
Figure 3.21: Dependence of the mean supergranular sizes hAi on the width w of theGaussian kernel used to spatially smooth the divergence images. Extrapolating down
to zero smoothing gives a supergranular size of 17 ± 3 Mm across.

that supergranules have an average size of 14–20 Mm across. The amount of temporal
filtering appears to have little effect on the characteristic supergranular cell size.
One study of this sort has already been performed by Hagenaar et al. (1997),
wherein the Ca ii K network were used to determine supergranular boundaries. Theyfind a mean cell size of 13–18 Mm, in agreement with our result and approximately
half the average cell size measured using correlation analysis techniques. Hagenaar
et al. (1997) argue that this discrepancy arises because the correlation measurement
preferentially weights the largest cells when computing the average cell size. Direct
inspection of the Ca ii K network supports this result, in that while cells of order 32 Mm(the average cell size found using classical corrleation analysis techniques) certainly
exist, many smaller cells can also be found. We believe that the same rules applicableto the chromospheric Ca ii K network also apply to the outflows found by our direct
identification method.

70SUPERGRANULE LIFETIME SPECTRUM

frequency, N(t)Ntotal = 3494 objects
binsize = 2 hr
mean = 15 hr

0.100
frequency N(t) dt, % of total [hr-1]

1000

100

0.010

10

0.001

1
0

20

40
lifetime, t [hr]6080Figure 3.22: Distribution of supergranular lifetimes N (t) for the Ntotal supergranules
identified in the April 1999 dataset. The lifetime t is equal to the number of consecutive
frames in which each object can be identified. The outflow cores were identified after
applying 4-hr temporal boxcar and 8.8-Mm spatial Gaussian filters to the time series.
The dotted line is the best-fit exponential function [see eq. (3.3)] with the intercept
C = 434 and τ = 16 hr.

Figure 3.22 shows the distribution of supergranular lifetimes for the April 1999
dataset, along with the best-fit exponential function,
N (t) dt = Ce−t/τ dt,(3.3)

where the intercept and mean lifetime are denoted by C and τ respectively. The supergranules identified by our method have a mean lifetime of τ = 16 hr, or 25% lessthan the correlation lifetime of 20 hr. However, this mean value may obscure the factthat amidst the many short-lived supergranules there exist several cells which survive
for multiple days, suggesting that the pattern as a whole cannot be adequately characterized by a single time scale. In addition, approximately 10% of the supergranules

71frequency N(t) dt, % of total [hr-1]

0.10000.01000.0010∆t = 4 hr, τ = 14 hr∆t = 6 hr, τ = 18 hr∆t = 8 hr, τ = 22 hr
0.0001020

40
lifetime, t [hr]6080Figure 3.23: Dependence of the distribution of supergranular lifetimes N (t) on thetemporal averaging window ∆t.

present in this dataset appear on either the first or last image in the time series, suchthat the actual lifetimes of these objects are longer than those values used in compilingthe lifetime histogram of Figure 3.22. Consequently, we believe the distribution in the
figure is biased toward shorter supergranular lifetimes. We included these extra cells
in the distribution because removing them would bias the results even more so toward
shorter lifetimes, for longer-lived supergranules have a greater probability of appearing
on either the first or last image in the time series than do those supergranules withshorter lifetimes.
The effect of the temporal averaging is shown in Figure 3.23, displaying the supergranular lifetime distributions for several values of ∆t for w = 8.8 Mm. With greater
temporal averaging, the slope of the distribution function becomes flatter due to themerging of short-lived cells into longer-lived cells. Note that the smallest value of ∆t
for which a lifetime distribution is plotted is ∆t=4 hr, corresponding to the minimum
amount of temporal averaging one needs to perform in order to reduce the random noise

72

AREA VS. LIFETIME
120
Ntotal = 3494 objects
100

lifetime, t [hr]

80

60

40

20
0
0

500
1000
average area, A [Mm2]

1500

Figure 3.24: The distribution N(A,t) as a function of the average supergranular area
A vs. lifetime t for the Ntotal supergranules identified in the April 1999 dataset. The
shaded contours appear at values of N =2, 5, 10, 20, 30, 40, 50, and 60. The dots
represent locations in coordinate space where N = 1.

level in the divergence signal below the supergranular outflow signal (see Fig. 3.4). We
also find that the amount of spatial smoothing has little effect on the lifetime distribution functions.
Figure 3.24 shows the distribution of supergranular cells as a function of averagearea A and lifetime t. The general trend is for larger cells to live longer, although we
note that the longest lived cells are not the largest. This effect may be caused by thepropensity for extremely large cells to fragment into smaller cells at some point during
their existence.

73

3.4

FLOWS ON LARGER SCALES
It is evident that the pattern of supergranulation possesses a broad range of scalesof motion in time and space, as shown both by the heuristic impressions provided by
the cork studies in §3.2 as well as by the statistical analysis of the previous section.
We now attempt to measure surface flows occurring on larger spatial scales than thatof supergranulation, including differential rotation, meridional circulation, and possibly
giant cell circulations. Such large-scale flows detected at the surface would likely spana significant fraction of the convection zone, and thus would provide another way toprobe the solar interior. These flows would be large enough to interact with the solar
differential rotation profile and might play a role in the 22-yr global magnetic cycle.
Several other studies using a variety of techniques have detected large-scale flows
both near the surface and at depth. Ring-diagram analyses (Haber et al. 2000) of timeseries of MDI velocity data have shown extended regions of organized flows on the orderof 25–50 m s−1 which vary on time scales of about one day. Longer temporal averages (ofat least nine days) show bands of zonal flows of order 20 m s−1 which migrate toward
the equator and appear to vary with changes in solar activity. Such long temporal
averages also contain persistent poleward-directed meridional flows of order 20 m s−1
in both hemispheres and extending to depths of at least 10 Mm below the surface.
Similar meridional flows were also measured using time-distance analyses (Giles et al.
1998) applied to two months of MDI data, with poleward flows detected as deep as
20 Mm. In addition, spectral decompositions of line-of-sight velocity images reveal
large-scale banded flow patterns at the surface which persist for several solar rotations(Beck et al. 1998; Ulrich 1998). Such flow patterns are distinct from the torsional
oscillations discovered earlier by Howard & LaBonte (1980) and are not associated with
regions of surface magnetic activity.
The supergranular flow fields discussed throughout this chapter, which were de-

74

20

latitude [deg]

10

0

-10

-20
-20

-10

0
10
longitude [deg]20

Figure 3.25: Six-day averaged flow field deduced by applying correlation tracking to 8-hr
averaged images of horizontal divergence. The flow field arrows have an rms velocity of
22 m s−1 and are scaled to 50 m s−1 .

duced from applying correlation tracking to images of mesogranulation, presumably
also include a large-scale velocity component. Unfortunately, the speed of the outflows(200–500 m s−1 ) combined with the 200 m s−1 anomalous flow pattern (see §3.1.4) obscure the contribution to the flow maps arising from large-scale flows, which we expect
to have velocities of at most 50 m s−1 . However, if the supergranular pattern itself ishorizontally advected by flows on larger scales, it may be possible to measure such largescale flows by applying the correlation tracking algorithm to the horizontal divergence
images, themselves produced from correlation tracked flow maps. We now discuss theresults of this effort.
We have shown that supergranules in quiet sun regions are on average about
20 Mm across. Using the supergranules as tracers for the correlation tracking technique

75

zonal flow [m s-1]

20

(a)

a0 = 0.060 ° day-1
a2 = -0.721 ° day-1
a4 = -6.251 ° day-1(b)slope = 0.318±0.051
intercept = -0.396±0.683

10
0
-10

meridional flow [m s-1]

-20
20
10
0
-10
-20
-20°

-10°

0°
latitude

10°

20°

Figure 3.26: Longitudinally averaged zonal and meridional flows from Fig. 3.25. The
fit parameters for the zonal flow are of the functional form Ω(λ) = a0 + a2 sin2 (λ) +a4 sin4 (λ), where λ denotes latitude. The equatorial rate a0 is relative to the tracking
rate of the dataset of 13.5◦ day−1 synodic. The fit to the meridional flow is linear withlatitude. Negative velocities indicate eastward/southward flows while positive values
correspond to westward/northward flows.

therefore requires the size of the gridpoint neighborhood σ to be at least this size. In
practice, we find that setting σ to be about twice the size of the tracers works best, so wehere set σ=47 Mm, equivalent to a heliographic distance of 3.8◦ . Figure 3.25 shows theresulting flow field obtained by applying correlation tracking to supergranular outflow
patterns present in the divergence data. The resulting flow field is averaged over thefull duration of the time series, or about six days. The flow field arrows are binned
spatially so that the distance separating neighboring arrows is 23 Mm or 1.9◦ . Bothspatial and temporal averaging are necessary to attenuate the random errors present in
the resulting flow maps.

76

20

latitude [deg]

10

0

-10

-20
-20

-10

0
10
longitude [deg]20

Figure 3.27: Residual large-scale flow field, formed by subtracting the fits to the longitudinally averaged zonal and meridional flows of Fig. 3.25 from the flow field of Fig. 3.25.
Two organized outflows are enclosed in circles. The flow map vectors are scaled to
50 m s−1 , and are spatially oversampled by a factor of two.

From the flow field of Figure 3.25, we first compute longitudinal averages of thezonal and meridional components. The zonal average, displayed in Figure 3.26, contains
the differential rotation profile of the supergranulation pattern minus the mean rate of
13.5◦ day−1 (synodic) at which the dataset was originally tracked. We have fitted thezonal flow to a function of the formΩ(λ) = a0 + a2 sin2 λ + a4 sin4 λ,(3.4)

where the differential rotation rate Ω is a function of latitude λ. After adding back in the
tracking rate of the region, the equatorial rate a0 deduced from these data corresponds
to a sidereal rate of 14.5◦ day−1 or 466 nHz, which is slower than the equatorial rate
for supergranules of 2.972 µrad s−1 or 473 nHz quoted by Snodgrass & Ulrich (1990),

77(a) Correlation Tracked Flows(b) Flows from Ring Diagrams

160

160

20

latitude [deg]

10

0

-10

-20
170
180
190
200
Carrington longitude [deg]

170
180
190
200
Carrington longitude [deg]Figure 3.28: A comparison between the large-scale flow field of Fig. 3.27 (a) with ringdiagram results (b) for the same region. The correlation tracked flow field has been
spatially averaged to match the spatial resolution of the ring-diagram analysis.

but higher than the equatorial rate deduced spectroscopically (a0 = 2.851 µrad s−1 or
454 nHz) and from magnetic tracers (a0 = 2.879 µrad s−1 or 458 nHz) in the same data.
In addition, their supergranular rotation rate also decreases much more rapidly thanours, corresponding in our units to a2 = −2.40◦ day−1 (versus the a2 = −0.721◦ day−1
for this dataset as indicated in Fig. 3.26). As a result, we conclude that our method
for measuring differential rotation only partially senses the true supergranular rotation
rate.
The lower panel of Figure 3.26 shows the meridional flow as averaged over longitude. We detect a poleward meridional flow symmetric about the equator, to within
the error of the coefficients of the linear fit. At latitudes of ±20◦ the flow has attained
speeds of approximately 5–10 m s−1 , which may be slightly slower by about a factor of
two than other measurements of the surface meridional flow, such as those performed
using time-distance (e.g. Giles et al. 1997) and ring diagram (e.g. Haber et al. 2000)
analyses. We are not able to detect the asymmetry present in the meridional flow seen

78

20

ring diagram velocikty [m s-1]

r = 0.567
10

0

-10

-20
-20

-10
0
10
correlation tracking velocity [m s-1]

20

Figure 3.29: A scatter diagram comparing the large-scale flow field deduced by correlation tracking Fig. 3.28a to the ring-diagram results Fig. 3.28b for the same region. Boththe correlation between the x-component (crosses) and the y-component (plusses) are
shown. The correlation coefficient is r = 0.567.

in both the time-distance and ring diagram analyses which causes the zero-crossing of
the meridional flow velocity to occur at about 5◦ of latitude north of the equator. We
note, however, that both studies use substantially longer datasets to make their measurements of the meridional flow speeds, and it is possible that with greater temporal
coverage the meridional flow measurement attained using this scheme may approach theresults found in these other studies.
We now examine the residual flow field, shown in Figure 3.27, formed by removing the fits of Figure 3.26 from the large-scale flow field of Figure 3.25. Several regionsof organized fluid motions on scales of 10◦ –20◦ are evident in this residual flow field,
including two regions (circled in the figure) suggestive of large-scale outflows. These di-

79vergent flows are approximately 10◦ or about 120 Mm across and have outflow velocities
on the order of 30 m s−1 .
We believe the flows of Figure 3.27 are of solar origin for two reasons. First, theflow field possesses a high amount of spatial coherence down to the resolution limit of thedata. Furthermore, our results compare favorably with ring-diagram results for the same
region, as shown in Figure 3.28. The flow vectors determined from ring-diagram analysis
are obtained by performing helioseismic inversions on 15◦ -square time series of Doppler
velocity images. These square regions are typically separated by 7.5◦ in longitude andlatitude. For direct comparison, we show in Figure 3.28a the residual flow field deducedby correlation tracking from Figure 3.27 after reducing the resolution to match thatof the corresponding ring-diagram flow vectors for the April 1999 dataset, shown in
Figure 3.28b. A scatter diagram comparing the two methods is shown in Figure 3.29,
and while there is some scatter in the data, the points appear to be correlated. The
correlation coefficient is r = 0.567 as indicated in the figure.
We find that velocity features associated with solar supergranulation are advectedby large-scale horizontal zonal and meridional flows, and as a result these flows can bemeasured applying the correlation tracking technique to the supergranular flow field.
The supergranules in the April 1999 dataset possess a fast differential rotation profile relative to spectroscopic and magnetic features, with a meridional circulation of±10 m s−1 between −20◦ and 20◦ of latitude. In addition, organized flow patterns onscales of 10◦ –20◦ are evident in the residual flow field, after the longitudinally averaged
zonal and meridional flows are removed.

3.5

CONCLUSIONS
We have presented in this chapter evidence that the near-photospheric velocity

field on supergranular and larger scales is extremely complex, containing fluid motions
possessing a wide range of length and time scales. Using the correlation tracking tech-

80
nique applied to full-disk MDI velocity images, we are able to study the horizontal flow
field associated with solar supergranulation for time periods as long as six days with a
combined spatial and temporal resolution unavailable before MDI. We find significant
evolution of the supergranular flow field on time scales as short as 6 hr as convergencelanes are observed to appear and disappear and be horizontally advected by stronger
elements of the flow field. Amongst these quickly evolving flows are persistent features
which survive for time periods of 36 hr or more, encompassing not only long-lived supergranules but also velocity patterns evident in the large-scale flow maps deduced fromtracking the supergranular pattern. A complete understanding of these surface patterns
requires knowledge of how these structures are formed and interact within the bulk of
convection zone. To this end, we now turn to detailed three-dimensional convection
models of a stratified fluid confined to rotating spherical shells which are designed tosimulate aspects the supergranular layer of the upper solar convection zone.

Chapter 4

NUMERICAL MODELING OF SPHERICAL SHELLS OFCONVECTION

4.1

NUMERICAL MODELING OF ANELASTIC FLUIDS
Observations of supergranular and larger-scale flow fields, such as those presented

in Chapter 3, together with measurements of the smaller-scale patterns of mesogranulation and granulation, provide the only direct look at the turbulent motions which
occupy the underlying convection zone. Such observations indicate this convection isextremely complex, forming coherent features such as networks of upflows and downflows possessing a broad range of length and time scales. These vigorous overturning
motions are permeated by magnetic structures of all sizes, ranging from small flux tubes
to large-scale magnetic patterns associated with active regions and sunspots observedat the surface. To investigate further the dynamics of such turbulent flows on supergranular and larger scales, we now turn to three-dimensional numerical simulations ofconvection occupying thin spherical shells located immediately below the solar surface.
The simulations described in the next chapter are carried out using the anelastic
spherical harmonic (ASH) code. The ASH code solves the anelastic equations of hydrodynamics describing a compressible fluid confined to a rotating spherical shell heatedfrom below. The complex structures and intricate behavior of the resulting convection
requires high spatial resolution, and the flows must be studied over extended periodsof time for statistical equilibration to be achieved. As a result, the ASH code is de-82
signed to run efficiently on massively parallel architectures such as the Cray T3E andSGI Origin 2000 machines (Clune et al. 1999). This multi-processor version was devisedby Tom Clune, Mark Miesch, and Julian Elliott, although the numerical approach tothis problem was first implemented by Gary Glatzmaier in the early 1980’s (Glatzmaier
1984).
The ASH code employs a pseudo-spectral approach, where all fluid velocities andstate variables are projected onto orthogonal basis functions in each of the three spatial
dimensions. The radial structure of the solution variables is represented by an expansion
based on Chebyshev polynomials, while variations in the latitudinal and longitudinal
directions are expanded over spherical harmonic basis functions Y`m , characterized by
the angular degree ` and azimuthal order m. This discretization scheme ensures that thehorizontal resolution is uniform everywhere on a sphere when all (`, m)-pairs for a given
maximum degree `max are retained in the modal expansion. Conversely, the simplest
finite-difference scheme, where computational gridpoints are distributed along lines oflatitude and longitude, suffers from the problem that the spatial resolution varies withlatitude such that the gridpoints are more closely spaced near the poles compared toequatorial regions (colloquially known as the pole problem).
As the name implies, the ASH code solves an approximate form of the NavierStokes equations known as the anelastic equations. The anelastic approximation (Gough1969) allows us to handle the effects of compressibility while filtering out acoustic perturbations which would otherwise severely limit the size of the computational time step.
This approximation is valid when the convective fluid velocities are subsonic, which inturn implies that the stratification of the fluid is only slightly superadiabatic. The filtering is achieved by insisting that the time derivative of density vanishes in the continuity
equation, or thereby that the divergence of the momentum be zero, or that the momentum vector be solenoidal. This approximation is effectively equivalent to allowingpressure disturbances to equilibrate instantaneously, forcing the system to evolve on

83
convective rather than sound-speed time scales. It is therefore implicitly assumed thatsound waves do not play a significant role in the dynamical evolution of the system,
which is in agreement with the expectation that the coupling of convection, stratification, and rotation are the major dynamical influences throughout the bulk of theconvection zone.
As with the temporal scales of motion, fully resolving all spatial scales of motion
in a numerical simulation is infeasible at this time, as the dynamically active scales inthe solar convection zone range from 102 Mm (depth of the zone) to 10−4 Mm (typical dissipation scale), thereby encompassing a factor of 106 in scale. However, current
simulations can cope with a range of only about 103 in each of three dimensions. Consequently, the ASH code adopts the common approach of parameterizing the transport
properties resulting sub-grid scale (SGS) turbulent eddies and resolving only the largest
scales of convection, thus becoming a large eddy simulation (LES).
All LES-SGS simulations require a prescription for representing the effects of SGS
convective motions not explicitly resolved in the model. Such a scheme may incorporate characteristics of the resolved flows into their functional forms (see the reviews
by Canuto 1996; Lesieur 1997; Canuto & Christensen-Dalsgaard 1998), or may simply
enhance the molecular (viscous and thermal) diffusivities. We have adopted the latterapproach for simplicity, yet recognize that this aspect requires considerable attention
in the future. The main drawback of this scheme is that the enhanced diffusion draws
energy from larger resolved scales of motion which should be unaffected by such dissipative effects. In one alternative approach, known as hyperviscosity, one allows theenhanced eddy diffusivities to act on fourth (or higher) order derivatives of the velocity
field, thereby confining the diffusive effects more toward the smaller end of the spectrum.
Another class of SGS models involves adding extra stress terms to the equations of motion. Evolution equations for these additional contributions can then be constructed
once functional forms for the correlations between second-order variables are specified

84
using some kind of a closure hypothesis. As is true of all LES-SGS studies, one hopesthat the specific form by which the SGS motions are parameterized has a relatively
small effect on the global dynamics of the system.
Time-stepping in the ASH code is performed using an implicit second-order
Crank-Nicholson procedure for the linear terms and a fully explicit second-order AdamsBashforth procedure for the nonlinear terms. Because the explicit time-stepping procedure cannot be performed in the spectral domain, this scheme necessitates conversions
between the physical and spectral representations during each time step when switching
between solving the implicit and explicit terms in the evolution equations. However, thebenefits gained by avoiding the pole problem prevalent in finite difference representations outweigh the added computational time spent in performing the transformations
between the physical and spectral domains.

4.2

THE ASH CODE: EQUATION SUMMARY

4.2.1

Fluid Flow in a Rotating Frame

We first set down the equations valid for a fully compressible, rotating fluid before introducing the anelastic approximation. We choose to operate in spherical polar
coordinates (r, θ, φ), where r is the radius at each point, and θ and φ are respectively
the polar (latitude) and azimuthal (longitude) angles. The fluid is rotating with respect
to an inertial frame at constant angular velocity Ω, such thatΩ = Ω cos θ r̂ − Ω sin θ θ̂,(4.1)

as illustrated in Figure 4.1. All symbols appearing in this section are defined in Table 4.1
for convenience, and most have their usual meanings in a fluid dynamical context. The
fluid equations express the conservation of mass (mass continuity),∂ρ+ ∇ · (ρu) = 0,
∂t

(4.2)

85
Ω

φ
r
θ

Figure 4.1: The spherical shell domain in partial cutaway to show the empty interior.

the conservation of momentum,

ρ


∂u
+ (u · ∇)u = −∇p − ρg r̂ + 2ρ(u × Ω) + ∇ · D,
∂t

(4.3)

and the conservation of internal energy,

ρT


∂s
+ (u · ∇)s = −∇ · q eff + Φ.
∂t

(4.4)

In the momentum conservation equation, it is customary for the gravitational acceleration term to contain contributions from both centrifugal acceleration and classical
Newtonian gravitational acceleration. For the solar models considered here, however,the centrifugal acceleration (of order Ω2 R ) is several orders of magnitude smaller than

86
Table 4.1: Definitions of symbols appearing in §4.2.
GaussianFirst
SymbolUnit
Appears
Definition−1−1
cp erg g Kequation (4.8) specific heat at constant pressure−1−1
cv erg g K
γ def below specific heat at constant volume
D g cm−3 s−1 equation (4.3) viscous stress tensores−1equation (4.5) strain rate tensor
−2
g
cm s
equation (4.3) Newtonian gravitational accelerationp
erg cm−3equation (4.3) pressure−2−1
q erg cm sequation (4.8) diffusive heat flux
qeff erg cm−2 s−1 equation (4.4) effective heat flux
q turb erg cm−2 s−1 equation (4.8) turbulent heat fluxrcm
§4.2.1 intro radiusr̂ dimensionless equation (4.3) radial unit vectors erg g−1 K−1 equation (4.4) specific entropyTK
equation (4.4) temperaturetsequation (4.2) time
−1
u
cm s
equation (4.2) velocityδ dimensionless equation (4.5) Kronecker delta
γ dimensionless equation (4.10) ratio cp /cv of specific heats
κr
cm 2 s−1equation (4.8) radiative diffusivity
κs
cm 2 s−1equation (4.8) turbulent thermal diffusivity
θ dimensionless §4.2.1 intro polar angle
ν
cm2 s−1
equation (4.7) kinematic viscosity
2
−1
νeff
cm s
equation (4.5) effective kinematic viscosity
νturb
cm2 s−1
equation (4.7) turbulent kinematic viscosityρg cm−3equation (4.2) density−2−1
Φ erg cm sequation (4.4) viscous heating
φ dimensionless §4.2.1 intro azimuthal angleΩs−1
equation (4.3) angular velocity of reference frameGM), where R and M are the solar radius andR2mass. In order to use simple spherical harmonic expansions, we ignore centrifugal efthe Newtonian gravity (of orderfects in these models, thereby rendering the surfaces of constant gravitational potential
spherical: g = −g r̂. Note that this assumption precludes Eddington-Sweet circulations(e.g. Tassoul 1978).

87
The viscous stress tensor D appearing in equation (4.3) is defined as1
D = 2ρνeff e − (∇ · u) δ ,3
while the viscous heating term Φ appearing in equation (4.4) can be written12
Φ = 2ρνeff e : e − (∇ · u) ,
3

(4.5)

(4.6)

where in both equations (4.5) and (4.6) the tensor e is the strain rate tensor and is itself
a function of u.
In accordance with our SGS formulation, we define the effective viscosity νeff asνeff = ν + νturb ,(4.7)

where νturb is the turbulent eddy viscosity which accounts for viscous transport byconvective motions not formally resolved in the simulation. In the above hydrodynamic
equations, the true viscosity ν has been replaced by νeff . Additionally, we also mustaccount for the transport of heat by the SGS turbulence. We therefore define
q eff = q + qturb = −κr ρcp ∇T −κs ρT ∇s,|{z} | {z }q(4.8)

q turbwhere we have explicitly taken into account the smoothing out of entropy gradients byunresolved SGS motions by having the eddy thermal diffusivity κs acting on ∇s. Ideally,
if the simulation were to resolve all relevant scales of motion, the turbulent contributions
to νeff and qeff would be superfluous. However, fully resolved global simulations of
relevance to the sun are currently unattainable.
The heat flux q appearing in equation (4.8) should contain contributions fromboth radiative diffusion and thermal conduction. For the sun, the energy flux due tothermal conduction is much smaller than the fluxes from radiative diffusion or convection (e.g. Spitzer 1962; Hansen & Kawaler 1994) and is thus ignored in these models.
Furthermore, the radiative flux is assumed to take the form of a Fick-type diffusion law,q = −κr ρcp ∇T.

(4.9)

88
Equations (4.2)–(4.4) are to be solved for the following seven quantities: the fourstate variables ρ, p, T , s, and the three components of u. However, equations (4.2)–(4.4) comprise only five independent relations. Since only two of the state variables are
independent, the system is closed by first specifying an equation of state relating thestate variables ρ, p, and T , and then deriving an equation for s.
In the simulations presented here, we assume the fluid is a perfect gas, thereby
ignoring the effects of ionization. The equation of state is thereforep=γ−1
cp ρT.
γ(4.10)

1ln p − ln ρ ,γ(4.11)The specific entropy is thens = cpvalid to within an arbitrary constant determined by specifying the value of s given reference values of p and ρ. Equations (4.2)–(4.4), (4.10), and (4.11) now form a complete
set of equations describing a compressible fluid in a spherical geometry.

4.2.2

The Anelastic Approximation

Compressibility clearly plays an important role in the dynamics of the solar convection zone, as the fluid density varies by several orders of magnitude across the layer.
The density drops by a factor of approximately 100 between 0.90R and 0.99R , which
encompasses the upper region of the convection zone considered in this thesis. We
therefore apply the anelastic approximation to the fully compressible fluid equations inorder to accommodate a solar-like density stratification in our simulations. As discussed
earlier, sound waves and other pressure disturbances which operate on time scales fasterthan the time scale of the convection are filtered out in this formulation, since including
such dynamics would otherwise limit the size of the computational time step.
The main assumption of the anelastic approximation is that such temporal filtering is valid when the convective motions are subsonic, which occurs when the radial

89
entropy gradient responsible for driving the convection departs only slightly from themarginally stable entropy gradient (that is, whends 1). We now estimate thedrdegree of superadiabaticity at a given radius within the solar convection zone using
simplified mixing-length arguments to estimate the heat transport of a convectively
unstable parcel of fluid.
We first define the dimensionless parameter  as
=−

λ ds
,
cp dr

(4.12)

where  characterizes the superadiabaticity of the fluid in terms of the entropy gradient. To make  dimensionless, we normalize the entropy gradient by choosing cp as
the entropy scale and the mixing length λ as the characteristic length scale. The negative sign in equation (4.12) ensures that  is positive, since superadiabatically stratified (convectively unstable) fluids have negative entropy gradients. We assume thatthe superadiabatic stratificationds
is prescribed throughout the convection zone, evendrthough the very motions which it supposedly drives most assuredly feed back and alter
the stratification.
Now consider an isolated fluid parcel, initially in equilibrium with its surroundings, rising through the convectively unstable medium characterized by entropy gradientds. The motion of this parcel is driven by the excess heat it possesses after rising adrradial distance λ. Equating the kinetic energy of the parcel with this heat excess, we
have
ds
ρv 2
= −λρT,2
dr(4.13)

where v is the upward radial velocity of the parcel, and T and ρ are representativevalues of the temperature and density of the fluid. Solving for v and substituting thedefinition of  from equation (4.12) into equation (4.13), we obtain
1

v = (2cp T ) 2 .

(4.14)

90
The flux of energy Fconv =1 3
ρv carried radially outward by these convective motions
2is therefore equal to3

Fconv = ρ (cp T ) 2 ,(4.15)

where factors of order unity have been omitted from our estimate. We now determine
the degree of superadiabaticity  necessary for convection to transport the entire solar
luminosity L through the convection zone by equating Ftotal = Fconv , which implies
3
L
2 ,
=
ρ
(c
T
)
p
4πr 2

(4.16)and solving for . We thus have
1
=
cp T



L
4πr 2 ρ2
3.

(4.17)

We can also estimate the typical Mach number Ma achieved by these convective
flows. By definition,
Ma =

v
,
cs

(4.18)

where the adiabatic sound speed cs is given by

cs =

γp
ρ

1
2

.

(4.19)

Substituting the expressions for v and cs from equations (4.14) and (4.19) into equation (4.18) givesMa =cp ρT
γp

1



2

=


γ −1

1
2

,

(4.20)

where the last equality follows from the ideal gas law, equation (4.10). Equation (4.20)
implies that convective motions driven by a fluid for which  is small do not approach
the speed of sound.
Table 4.2 lists values for v, , and Ma calculated using equations (4.14), (4.17),and (4.20) for several radii within the convection zone. Values for T and ρ are taken from

91
Table 4.2: Estimates of v, , and Ma using a mixing-length approach as calculated
from equations (4.14), (4.17), and (4.20) assuming L = 3.90 × 1033 erg s−1 , R =6.96 × 1010 cm, and cp = 4 × 108 erg g−1 K−1 . The temperature T and density ρ aretaken from Model S of Christensen-Dalsgaard et al. (1993). In this model, the base of
the convection zone sits at r = 0.73R .

r [R ]
0.73
0.75
0.80
0.85
0.90
0.94
0.96
0.97
0.98
0.99
0.999

T [K]

ρ [g cm−3 ] v [m s−1 ]2.1 × 106 1.6 × 10−1
1.8 × 106 1.3 × 10−1
1.4 × 106 8.8 × 10−2
9.9 × 105 5.2 × 10−2
6.2 × 105 2.6 × 10−2
3.5 × 105 1.0 × 10−2
2.2 × 105 5.3 × 10−3
1.6 × 105 3.1 × 10−3
1.0 × 105 1.4 × 10−3
4.7 × 104 3.2 × 10−4
1.3 × 104 1.2 × 10−6

130
140
150
170
200
270
330
400
510
830
5300



Ma

1 × 10−7 4 × 10−4
1 × 10−7 4 × 10−4
2 × 10−7 5 × 10−4
4 × 10−7 7 × 10−4
8 × 10−7 1 × 10−3
3 × 10−6 2 × 10−3
6 × 10−6 3 × 10−3
1 × 10−5 4 × 10−3
3 × 10−5 7 × 10−3
2 × 10−4 2 × 10−2
3 × 10−2 2 × 10−1

an advanced one-dimensional solar model used extensively in helioseismology (Model S
of Christensen-Dalsgaard et al. 1993). Table 4.2 shows that  is no greater than about
10−4 (and thus Ma . 10−2 ) throughout the bulk of the convection zone, and so theconvective motions driven at greater depths are expected to be decidedly subsonic.
Only within the last 1% by radius does  approach unity. Above 0.99R , the rapidly
decreasing density and temperature profiles lead to faster fluid motions involving smaller
spatial and temporal scales. In addition, the recombination of electrons with helium
and hydrogen nuclei release latent heat which then must be transported outward by theconvection. As a result, the stratification must become more superadiabatic to transport
the requisite amount of energy, rendering the anelastic approximation less valid.

4.2.3

Scaling of the Fully Compressible Fluid Equations

Because the convective motions are subsonic and the superadiabaticity within theconvection zone is small, we expect that the resulting departures of the thermodynamic

92
state variables from their spherically averaged means should also be small. Therefore,
we perform a formal scale analysis on the fully compressible fluid equations (4.2)–(4.4)
introduced in §4.2.1 to separate the spherically symmetric mean state from the fluctuations resulting from the convection. We express each state variable f as the sum of aspherically symmetric mean quantity fˆ(r, t) and a fluctuating quantity f 0 (r, θ, φ, t):p(r, θ, φ, t) = p̂(r, t) + p0 (r, θ, φ, t),ρ(r, θ, φ, t) = ρ̂(r, t) + ρ0 (r, θ, φ, t),(4.21)0

T (r, θ, φ, t) = T̂ (r, t) + T (r, θ, φ, t),s(r, θ, φ, t) = ŝ(r, t) + s0 (r, θ, φ, t).
The perturbations to the state variables result from convective motions driven by thesuperadiabatic stratification of the layer, which we have previously characterized by theparameter  defined in equation (4.12). We therefore assumef0
is of order  for anyfˆquantity f in equation (4.21).
For low Mach number flows the time scale on which pressure and density perturbations get smoothed out is much faster than the time scale of the convective motions,since such perturbations dissipate on sound-speed time scales. We perform such temporal filtering by assuming
∂ρ0
=0
∂t

(4.22)

in the mass continuity equation.
The resulting anelastic equations consist of a set of equations describing the meanstate, and a system of dynamic equations which governs the evolution of the fluctuating
quantities. The details of the formal scale separation, which include the derivations ofequations (4.23)–(4.33) which follow, are presented in §B.1 of Appendix B. We assume
that the diffusivities νeff , κr , and κs are functions of ρ̂ (and thus r) only, while theparameters γ and cp are assumed constant throughout the domain.

93
The mean momentum equation becomesd(p̂ + p̂turb )= −ρ̂g,dr(4.23)

where the quantity p̂turb (r, t) represents the small departure from hydrostatic equilibrium caused by the combined effect of turbulent motions in the system. The meanequations of state areγ −1
cp ρ̂T̂ ,γdŝ
1 dp̂ 1 dρ̂
= cp
−
.
dr
γ p̂ dr ρ̂ drp̂ =(4.24)(4.25)

Upon initialization, the equations (4.23)–(4.25) are solved to determine the meanstate, with the turbulent pressure p̂turb = 0 since the system is started from rest.
Because there are only three equations but four unknown mean variables, one of them istherefore undetermined. Consequently, we take the approach of prescribing the entropy
gradientdŝ
, and then solving equations (4.23)–(4.25) to determine the other mean
drvariables p̂, ρ̂, and T̂ . Note that the gravitational acceleration g can either be assumed
constant throughout the layer, or can be prescribed along with the entropy gradient if
the self-gravity of the mean state is deemed important.
The equations governing the evolution of the dynamic quantities u and s are:ρ̂∂u= 2ρ̂(u × Ω) − ρ̂(u · ∇)u − ∇p0 − ρ0 g r̂ + ∇ · D̂∂t

(4.26)∂s0= ∇ · q̂eff − ρ̂T̂ (u · ∇)(ŝ + s0 ) + Φ̂,∂t(4.27)

andρ̂T̂

where the anelastic viscous stress tensor D̂ and the anelastic viscous heating term Φ̂are defined1D̂ = 2ρ̂νeff e − (∇ · u) δ3

(4.28)

94
and


1
2
Φ̂ = 2ρ̂νeff e : e − (∇ · u) ,3

(4.29)

and where the anelastic heat flux q̂eff is defined asq̂ eff = −κr ρ̂cp ∇(T̂ + T 0 ) − κs ρ̂T̂ ∇(ŝ + s0 ).

(4.30)The evolution of the system given by equations (4.26)–(4.27) are subject to the following
three constraints:∇ · (ρ̂u) = 0,
ρ0 T 0
p0
= + ,
p̂
ρ̂
T̂
0
p0
ρ0
s
=
− .
cp
γ p̂
ρ̂
4.2.4

(4.31)(4.32)(4.33)

Energetics of the Anelastic Equations

For future reference, we now list the equations describing the kinetic and internal energy budgets of the system, while relegating the detailed derivations to §B.2
in Appendix B. The equation describing the conservation of kinetic energy density
Ek =ρ̂u · u
is
2

 



∂Ek
ρ̂u · u00
= −u · ∇p − ρ gu · r̂ + u · ∇ · D̂ + ∇ · u.
∂t
2(4.34)The terms on the right-hand side of equation (4.34) all represent sources or sinks ofkinetic energy due to work respectively done by pressure gradient forces, buoyancy
forces, shear stresses, and by inertial forces.
The equation describing the conservation of internal energy density Es = ρ̂T̂ s0 is∂Es= ∇ · q̂ eff + u · ∇p0 + ρ0 gu · r̂ − cp ∇ · (ρ̂T 0 u) + Φ̂.
∂t(4.35)

As with the kinetic energy equation, each of the terms on the right-hand side of equation (4.35) represents either a source or sink of internal energy. The first term on

95
the right-hand side represents the transport of flux via temperature/entropy gradients
throughout the domain. The next two terms denote the extraction of heat from thethermal field by pressure-gradient and buoyancy forces. The fourth and fifth terms represent the heat generated by volume expansion and viscous dissipation, while the sixth
term may be interpreted as the radial transport of enthalpy.
The sum of equations (4.34) and (4.35) yields the following equation describing
the total energy budget of the system:
 



ρ̂u · u∂(Ek + Es )= ∇ · u · D̂ + ∇ · u+ ∇ · q̂ eff − ∇ · (cp ρ̂T 0 u).
∂t
2
4.2.5

(4.36)

Streamfunction Formalism

To solve the anelastic evolution equations (4.26) and (4.27), we recognize thatthe mass flux is solenoidal, which permits the quantity ρ̂u to be written in terms ofpoloidal and toroidal streamfunctions W and Z,
ρ̂u = ∇ × (∇ × W r̂) + ∇ × Z r̂(4.37)

such that equation (4.31) is automatically satisfied. We now replace the anelastic momentum equation (4.26) with three scalar equations describing the evolution of W ,

∂W
,
∂r

and Z. They are the radial component of the momentum equation,
−

i
h
i
h
∂p0
∂
(∇2⊥ W ) = −− ρ̂g + r̂ · ∇ · D̂ + ρ̂ r̂ · 2(u × Ω) − (u · ∇)u ,∂t∂r(4.38)

the horizontal divergence of the momentum equation,


i
h
h
i
∂ ∂
2
−
(∇⊥ W ) = −∇2⊥ p0 + ∇⊥ · ∇ · D̂ + ρ̂∇⊥ · 2(u × Ω) − (u · ∇)u , (4.39)∂t ∂rand the radial component of the curl of the momentum equation,−hi∂(∇2⊥ Z) = r̂ · ∇ × (∇ · D̂) − ρ̂ r̂ · [∇ × (u · ∇)u]∂t2ρ̂Ωuθ sin θ+ 2(Ω · ∇)ρ̂ur +.
r(4.40)

96
In the above equations, the horizontal Laplacian operator is defined


∂
1
1
∂
∂2
2
sin θ
+ 2 2
∇⊥ = 2,r sin θ ∂θ∂θr sin θ ∂φ2(4.41)

while the horizontal divergence of a vector A is defined∇⊥ · A =1 ∂(sin θ Aθ )
1 ∂Aφ
+
.
r sin θ∂θ
r sin θ ∂φ

(4.42)

Equations (4.38)–(4.40), together with the anelastic energy equation (4.27), form theevolution equations solved by the ASH code. Detailed derivations of these equations
are provided in §B.3 of Appendix B.

4.3

THE ASH CODE: NUMERICAL IMPLEMENTATION
We now summarize the discretization scheme and parallel implementation of theASH code, following the more detailed presentation of Miesch (1998).

4.3.1

Angular Discretization

The first step in the pseudo-spectral method is to express all dependent variables
as a projection over orthogonal basis functions. For the angular dependence, the basis
functions are the spherical harmonics Y`m ,Y`m (θ, φ) = C`m P`m (cos θ) eimφ ,(4.43)

where the functions P`m are the associated Legendre functions, and where the constantsC`m are definedC`m = (−1)m2` + 1 (` − m)!
4π (` + m)!

1
2

.

(4.44)The spherical harmonic functions are characterized by the angular degree ` and azimuthal order m. All dependent variables f (r, θ, φ, t) in the problem are expressed aslinear combinations of the spherical harmonics,
f (r, θ, φ, t) =∞ X̀X`=0 m=−`f`m(r, t)Y`m (θ, φ),(4.45)

97
where the spectral coefficients f`m (r, t) are found by the formulaf`m (r, t) =

Z 2π
0

= C`m

Z π

dφ
Z π
0

sin θ dθ Y`m∗ (θ, φ) f (r, θ, φ, t)
Z 2πm
sin θ dθ P` (cos θ)dφ e−imφ f (r, θ, φ, t).
0(4.46)

0Equation (4.46) constitutes the continuous spherical harmonic transform formula.
Numerical simulations are of course discretized, either by having a discrete gridin physical space or by limiting the number of mode coefficients in spectral space. To
maximize the accuracy of the transforms, Gaussian quadrature techniques are utilized in
evaluating the integrals of equation (4.46), where continuous integrals are approximatedas weighted sums over predetermined collocation points (θi , φj ) known as Gaussian
abscissae. The Gaussian abscissae depend only on the number of collocation points andthe orthogonal basis used in the expansion. Using Gaussian quadrature, the sphericalharmonic transform of equation (4.46) becomesf`m (r, t) = C`mNθ
Xwi P`m (cos θi )

i=1

=

Nφ
Nθ X
X

Nφ
X

wj e−imφj f (r, θi , φj , t)

j=1

(4.47)

wi wj Y`m∗ (θi , φj )f (r, θi , φj , t),

i=1 j=1

where Nθ and Nφ are the number of collocation points used in the θ and φ directions.
In this formula, the Gaussian abscissae in φ are the Fourier collocation points φj , where
φj =

2πj
Nφ

where

j = 1, 2, · · · , Nφ ,(4.48)

having weights wj of
wj =

1
.
Nφ

(4.49)The Gaussian abscissae in θ are the Legendre collocation points θi , whereθi = zeroes of PNθ (cos θ)

with

i = 1, 2, · · · , Nθ ,(4.50)

98
having weights wi of
2

wi =

[sin

2

θi PN0 θ (cos θi )]2.

(4.51)

In equations (4.50) and (4.51), the functions P` (cos θ) are the Legendre functions of thefirst kind, with
PN0 θ =dPNθ (cos θ).
d(cos θ) θ=θi

(4.52)The values Nθ and Nφ determine the angular resolution in physical space in boththe latitudinal and longitudinal directions. To ensure that the horizontal resolution iseven everywhere on a spherical surface, we set Nφ = 2Nθ , and truncate the sphericalharmonic expansion of equation (4.45) at a maximum angular degree `max , wherein allmodes for which 0 ≤ ` ≤ `max and −`max ≤ m ≤ `max are used in the expansion. To
attenuate aliasing errors when evaluating the nonlinear terms in the evolution equations,
we also choose `max to satisfy
Nθ ≥

4.3.2

3`max + 1.
2

(4.53)

Radial Discretization

As with the angular discretization, the radial discretization is handled in a similar
fashion. The radial dependence of each field is evaluated only at discrete collocation
points, and is transformed to spectral space using Gaussian quadrature. Each coefficientf`m (r, t) is expanded over the set of Chebyshev polynomials Tn of order n, where
Tn (x) = cos(n arccos x).

(4.54)

While the Tn appear trigonometric, they are in fact polynomials in x defined on (−1, 1).
The physical radii rk are scaled to the Chebyshev gridpoints xk by
rk =

i
1h
(r1 + r2 ) + (r2 − r1 )xk ,2(4.55)

99
where r1 and r2 are respectively the radii of the lower and upper boundary of thephysical domain such that they map to −1 and 1 in the Chebyshev domain.
Each discrete function f`m (rk , t) can be expressed as a truncated seriesf`m(rk , t) =

Nr
X
2
m
n f`n(t)Tn−1 (xk ),Nr − 1 n=1(4.56)

m are found by evaluatingwhere the spectral coefficients f`n
m
=
f`n

Nr
X

wk Tn−1 (xk )f`m (xk , t).

(4.57)

k=1

The Chebyshev collocation points xk are
xk = cos(k − 1)π
Nr − 1

wk =

k π
.
Nr − 1,(4.58)

while the weights wk are(4.59)In this formula and in equation (4.56), the constant k is equal to 1, unless k = 1 or1
k = Nr , for which k = .
2
4.3.3

Temporal Discretization and Parallel Implementation

When transformed to spectral space, the ASH code evolution equations (4.27)
and (4.38)–(4.40) are of the form∂y
= L + N,∂twhere L and N respectively designate linear and nonlinear source terms.1(4.60)
Generally,

these source terms can be a function of any of the dependent variables W , Z, p, and s
as well as a function of the independent variables r, θ, φ, and t.
The time-stepping for both the linear and nonlinear source terms is performed
simultaneously using a combination of the Crank-Nicholson scheme for the linear source
Although the Coriolis terms are linear, they are grouped with the nonlinear source terms N in the
ASH code time-stepping algorithm.
1

100
terms L and the Adams-Bashforth scheme for the nonlinear source terms N . Bothmethods are accurate to second-order. By combining the two methods (see Appendix C),
we discretize equation (4.60) to obtainyi+1 − yi
=
∆ti





1 ∆ti
1 ∆ti
Ni −
Ni−1 + ΘLi+1 + (1 − Θ)Li ,1+
2 ∆ti−1
2 ∆ti−1(4.61)

where subscripts are used to denote the index of the time step. For example, yi denotes
the value of y at the most recently computed time step (occurring at time ti ) and yi+1
denotes the yet-to-be-computed value of y at the next time step. The time separation
between subsequent time steps is ∆ti = ti+1 − ti .
After rewriting, equation (4.61) becomes





1 ∆ti
1 ∆ti
yi+1 − ∆ti ΘLi+1 = yi + ∆ti 1 +Ni −Ni−1 + (1 − Θ)Li .
2 ∆ti−1
2 ∆ti−1(4.62)
Since the term Li+1 is linear, we may write Li+1 = A0 yi+1 , where A0 is a matrix operator.
Equation (4.62) now becomes the matrix equation
Ayi+1 = B,

(4.63)

A = 1 − ∆ti ΘA0(4.64)

where

and





1 ∆ti
1 ∆ti
Ni −
Ni−1 + (1 − Θ)Li .
B = yi + ∆ti 1 +2 ∆ti−1
2 ∆ti−1(4.65)The updated fields yi+1 are determined by solving the matrix equation (4.63) once A
and B are calculated.
We now summarize the main loop of the ASH code:• At the beginning of each iteration, the dependent variables yi begin in (r, `, m)space.

101• Spatial derivatives of the dependent variables are evaluated. First, each quantity
is transformed to (n, `, m)-space via a Chebyshev transform, then the derivatives
are evaluated using Chebyshev recursion relations, and then the derivatives aretransformed back to (r, `, m)-space via an inverse Chebyshev transform.
• The linear term Li is evaluated in (r, `, m)-space.
• The dependent variables yi are transformed to (r, θ, φ)-space via an inverse
Legendre transform (to go from ` to θ) and an inverse Fourier transform (to gofrom m to φ).
• The nonlinear term Ni is evaluated in (r, θ, φ)-space.
• The nonlinear term Ni is transformed back to (r, `, m)-space via a forward
Legendre transform and a forward Fourier transform.
• The quantity B of equation (4.65) is evaluated, now that we have available theterms Li , Ni , and Ni−1 , with the latter term saved over from the previous time
step.
• The quantity A of equation (4.64) is calculated.
• The matrix equation (4.63) is solved for yi+1 using LU-decomposition.
As stated at the beginning of this chapter, the ASH code runs efficiently on
massively parallel architectures such as the Cray T3E and Origin 2000 machines (Cluneet al. 1999). These machines are of the distributed-memory configuration, where eachprocessor has its own local memory bank independent of the other processors. Sucha configuration requires to the programmer to devise sensible algorithms responsible
for evenly splitting up the workload (load-balancing) and coordinating interprocessor
communication when information is needed by multiple processors.

102
The most computationally intensive steps of the flowchart above involve global
operations such as the spectral transforms and the solution of the matrix equation. The
strategy adopted by the ASH code is to perform all transforms between the physicaland spectral domains in-processor so as to avoid interprocessor communication during
the transform. However, such a scheme requires frequent global transposes in order toarrange the data such that the dimension of the transform is local.

Chapter 5

SUPERGRANULAR CONVECTION IN THIN SHELL MODELS

5.1

INTRODUCTION
Helioseismology has shown that the latitudinally varying angular velocity profile

observed at the surface is largely maintained throughout the bulk of the convection
zone (Thompson et al. 1996; Schou et al. 1998). The angular velocity is nearly constantalong radial lines, particularly at mid-latitudes, as seen in Figure 5.1. However, regionsof strong radial shear (shown shaded) exist near both the bottom and the top of theconvection zone, and these shear layers are thought to play important roles in the
dynamics of the convection zone. The tachocline region at the base of the convection
zone has commanded much recent attention as it is likely the seat of the global solar
dynamo. Portions of the strong toroidal magnetic fields thought to be generated by theshearing motions within the tachocline will likely rise buoyantly through the convection
zone in a coherent fashion until they reach the surface. Such magnetic structures are
believed to produce the observed large-scale patterns of magnetic activity visible at thephotosphere, including sunspots and bipolar active regions, that comprise the 22-year
solar magnetic activity cycle. This flux may also contribute to the production of smallscale magnetic field elements seen at the photosphere, including bright points, pores,
and the magnetic carpet, although these features are more likely the result of local
dynamo action closer to the surface.
Prior to their emergence, the small-scale filamentary magnetic structures must104
near-surface
shear layer

480
0°

460

Ω / 2π [nHz]

15°
440
30°
420
45°

400

380

360
0.50

0.60

tachocline

60°

0.70

0.80

0.90

1.00

r/R

Figure 5.1: Average rotation rates Ω/2π inferred from the helioseismic inversion of over
4 years of GONG data using RLS inversions (adapted from Howe et al. 2000). Shearlayers (shaded), evidenced by variations of Ω with radius, are observed near the base of
the convection zone as well as near the surface, with the latter region extending from
0.95–1.00 R (where R is the radius of the sun). The gradients of Ω in that near-surface
shear layer at high latitudes is somewhat sensitive to the inversion method and data
sets used (e.g. see Schou et al. 1998).

be at least partially influenced by the dynamics within the upper shear layer of thesolar convection zone. The stratification within this region serves to drive vigorous
motions possessing a wide range of spatial and temporal scales, visible at the surface as
the convective patterns of supergranulation, mesogranulation, and granulation (Spruit
et al. 1990). The outflows associated with such convective motions are observed toreadily advect emergent magnetic flux toward intercellular lanes and concentrate thesefields on scales small enough for dissipation to occur. It is not yet known to what degreelarge-scale magnetic structures such as sunspots and active regions are impacted by the

105
turbulent convection occurring near the surface. In addition to reconfiguring magnetic
flux, these convective motions facilitate the transport of angular momentum along bothradial and latitudinal velocity gradients, and are likely to influence the dynamics ofthe bulk of the convection zone in ways that are not yet understood. The relatively
large horizontal and radial extent of supergranular flows in particular suggest that theyplay a prominent role in the dynamics within this shearing boundary layer, as suchmotions may be weakly influenced by rotational effects that create the Reynolds stresses
necessary for efficient turbulent transport of angular momentum within the layer.
Velocity features larger than the spatial scale of solar supergranulation are also
evident (e.g. Giles et al. 1998; Haber et al. 2000) in the upper shear layer. Banded
zonal velocity features which propagate in latitude are visible in long-term observationsof the photosphere (LaBonte & Howard 1982; Hathaway et al. 1996; Howe et al. 2000),
suggesting a weak organization on larger spatial scales. Helioseismic observations indicate a steady, longitudinally averaged, meridional flow toward the poles having speedsof order 20 m s−1 , extending to depths of at least 20 Mm below the visible surface(Giles et al. 1998; Haber et al. 2000). The return flow necessary to satisfy mass conservation has not yet been detected. Despite the slower fluid velocities associated withsuch large-scale flow patterns, fluid motions in the meridional plane can still effectively
redistribute angular momentum and couple widely separated regions within the solar
convection zone.
To understand more clearly some of the physical processes occurring within theupper shear layer of the solar convection zone, we have constructed numerical simulations of a compressible fluid confined to a thin spherical shell located near the top of
the convection zone. These simulations constitute the first global simulations of solar
convection at a resolution high enough to begin to explicitly resolve fluid motions onsupergranular scales. Such calculations are performed using the ASH computer code
described in Chapter 4, which runs efficiently on massively parallel computer platforms

106such as the Cray T3E and Origin 2000 machines (Clune et al. 1999). The complex
structures and intricate behavior of the resulting convection require high spatial resolution, and the flows must be studied for long periods of time for statistical equilibrium
to be achieved.
In constructing these simulations, we have adopted the viewpoint that the angular velocity profile seen in helioseismic observations is maintained within the bulk of theconvection zone, somewhere below the bottom boundary of our models, where the combined effects of convection and rotation create a net equatorward transport of angularmomentum. Numerical simulations of the full convection zone suggest that such transport is achieved by Reynolds stresses and large-scale meridional circulation, all working
against diffusion (e.g. Miesch et al. 2000; Elliott et al. 2000; Brun & Toomre 2001).
From this perspective, we are essentially assuming that the global differential rotation
profile is not substantially affected by the convection within our thin shells, and thus
can be imposed via the velocity boundary condition applied to the lower boundary. We
can therefore only explore the effects of such differential rotation within a thin sphericallayer of convection, but are able to investigate some of the ingredients associated withthe formation of shearing boundary layers analogous to the near-surface shear layer ofthe sun.
After discussing the initialization of the four thin shell simulations in §5.2, we
begin in §5.3 by illustrating the velocity patterns and time evolution of the multiplescale convection driven within our simulations. We then in §5.4 examine the energybalance responsible for the resulting axisymmetric differential rotation and meridional
circulation profiles and discuss the role of the convective motions in their maintenance.
We conclude with a discussion of the solar implications of these simulations and present
ideas for future research in §5.5.

107

Table 5.1: A summary of the parameters of the thin shell convection simulations. The
differences between Case S2 and the other three simulations are set in bold.

Case S1

Case S2

Case S3

Case D2

Radial Extent

0.94–0.98 R

0.94–0.98 R

0.94–0.98 R

0.90–0.98 R

Thickness [Mm]

28

28

28

56

Velocity BC’s
at Topstress-free
impenetrable

stress-free
impenetrable

stress-free
impenetrable

stress-free
impenetrable

Velocity BC’s
at Bottom

no-slip
uniform rot.
impenetrable

no-slip
differential rot.
impenetrable

no-slip
differential rot.
impenetrable

no-slip
differential rot.
impenetrable

Entropy BC’s
at Topconstant entropy constant entropy constant entropy constant entropy

Entropy BC’s
at Bottom

constant flux

constant flux

constant flux

constant flux

Ang. velocity Ω0 [nHz]

410

410

410

410

Rot. period P [days]

28.2

28.2

28.2

28.2

νtop [cm2 s−1 ]1 × 1012

1 × 1012

κtop [cm2 s−1 ]1 × 1012

1 × 1012

 10
2  10Density contrast

7.5

7.5

7.5

18

Prandtl number Pr

1

1

1
4

1

Taylor number Ta

5.4 × 103

5.4 × 103

5.4 × 103

2.1 × 105

Reynolds number Re

1.4 × 102

1.4 × 102

1.4 × 102

2.2 × 102

Rayleigh number Ra

1.9 × 104

1.9 × 104

1.9 × 104

5.4 × 105

40

100

200

500

Averaging Interval [days]

140

140

30

36

Nr × Nθ × Nφ`max
Ang. Periodicity

64×512×1024
340
four-fold

64×512×1024
340
four-fold

128×512×1024
340
four-fold

128×512×1024
340
four-fold

Supercriticality

Ra
Ra,0

5

11

1 × 1012

12

1 × 1012

108(a) Density(b) Pressure

1012
0.010
1011

1010

0.001
50

p [dyne cm-2]

ρ [g cm-3]

0.100

1013

(c) Density Scale Height(d) Temperature

106

30
105

20

T [K]

λρ [Mm]

40

10
104

0
0.90 0.92 0.94 0.96 0.98
r/R

0.90 0.92 0.94 0.96 0.98
r/R

Figure 5.2: The radial profiles of (a) density, (b) pressure, (c) density scale height,and (d) temperature as a function of proportional radius used upon initialization of
the simulations (black), compared with the corresponding values taken from a onedimensional solar structure model (red).

5.2

SUMMARY OF SIMULATION PARAMETERS

5.2.1

Initialization of the Spherically Symmetric Mean StateWe have constructed three shallow shell simulations (Cases S1, S2, and S3) thatspan a radial extent of 0.94–0.98 R, equivalent to a shell thickness of 28 Mm. The
remaining simulation (Case D2) spans 0.90–0.98 R or 56 Mm, and is accordingly twiceas thick as the three shallow shell simulations but otherwise equivalent to Case S2.

109
We simulate the angular velocity profile established within the deep convection zone byimposing a differentially rotating no-slip lower boundary on Cases S2, S3, and D2. For
comparison purposes, the lower boundary of Case S1 is maintained at a uniform rateequal to the angular velocity Ω0 of the computational frame, with all other attributes
equivalent to Case S2. In all four cases, the upper boundary is stress-free and both thelower and upper boundaries are impenetrable. The parameters of the four thin shell
simulations are summarized in Table 5.1.
As discussed in the previous chapter, the anelastic equations of motion are advanced in time by the ASH code using a pseudo-spectral approach, wherein functions ofθ and φ are expanded over spherical harmonic functions characterized by angular degree` and azimuthal order m. Functions of radius are projected onto Chebyshev polynomials characterized by radial order n. The four simulations presented here are calculatedusing spherical harmonic functions with `max ≤ 340, so that horizontal scales as small
as about 10 Mm are explicitly resolved. The highest order Chebyshev polynomial used
is n = 64 in Cases S1 and S2, and n = 128 in Cases S3 and D2. Since we expect theresulting convection to have a limited longitudinal scale and seek computational economics, we impose a four-fold azimuthal symmetry by keeping every fourth m value inthe spherical harmonic expansion. Such an imposed symmetry is primarily noticeable
only in the higher latitude regions, where the convergence of meridional lines near thepoles limits the longitudinal scale of the convective structures to sizes smaller than arepresent at lower latitudes.
During initialization, the radial profiles of density ρ̂(r), temperature T̂ (r), andpressure p̂(r) are determined by jointly solving the equation of hydrostatic equilibrium(4.23) and the mean equations of state (4.24) and (4.25), given radial profiles of thegravitational acceleration g(r) and specific entropy gradientdŝthroughout the dodr

main. We specify the initial entropy gradient to have a slightly superadiabatic value(e.g. −dŝ= 10−7 ), while the function g(r) is taken from the one-dimensional solardr110
model of Brun et al. (1999). The initial values of ρ̂(r), T̂ (r), and p̂(r) obtained inthis way are shown in Figure 5.2 to compare favorably with the structure model, withthe slight discrepancy in density resulting in a greater density scale height λρ in our
simulations than in the structure model.
With our current angular resolution, we are able to place the upper boundaryof each simulation at 0.98 R and thus accommodate the intricate convective structures
driven by the imposed solar-like stratification. Above this radius, the even greater degreeof stratification is likely to drive modes of convection that are smaller in physical size andthus below our current resolution limit. In addition, the anelastic approximation is likely
to break down as the convection becomes less efficient and typical convective velocities
become a significant fraction of the speed of sound. Furthermore, the ionization zoneof hydrogen is located above 0.98 R in the sun, suggesting that both the ideal gas
equation of state and the diffusive treatment of radiative effects currently used in thesecalculations become inappropriate above this level. Consequently, the upper boundaryof all four simulations is located at 0.98 R.
The ASH code is a LES-SGS simulation and thus requires a prescription to account for the transport of energy and momentum by turbulent motions not explicitly
resolved. We have adopted the simplest approach of enhancing the molecular values ofthe viscous and thermal diffusivities, while recognizing that such an SGS treatment isunlikely to capture all dynamical effects of small-scale turbulence. The eddy diffusivitiesνeff (r) and κs (r) used in these simulations are chosen to vary inversely as the squareroot of the mean density profile, so that β = 12 inνeff = νtop

ρ̂
ρ̂top

−β


and

κs = κtop

ρ̂
ρ̂top

−β
.

(5.1)

This particular value of β was chosen to allow some variation of the dissipation scalewith the density scale height λρ , but at the same time to prevent numerical instabilities
near the bottom of the domain where the dissipation length scale is smallest. We note111
that as computing technology becomes more advanced, the parameterization of SGS
transport effects in simulations of highly turbulent fluids should become less of an issue
because such global simulations will be able to explicitly resolve more of the energy
transport at small dynamical scales. Nevertheless, we fully acknowledge that our SGS
prescription is somewhat inadequate and requires considerable attention in the future.
The free parameters νtop and κtop in equation (5.1) are the viscous and thermal
diffusivities at the upper boundary, and their ratio determines the Prandtl number
Pr =

νtop
νeff
=
of each simulation. Cases S1, S2, and D2 each have Pr = 1, whileκtopκsCase S3 is constructed with Pr = 14 in order to investigate a turbulent fluid which
diffuses heat more readily than momentum. The lower Prandtl number of Case S3 isachieved by increasing κtop by a factor of 2 while simultaneously decreasing νtop by afactor of 2, relative to Case S2.
The thermal driving in all cases is accomplished by setting the heat flux at thelower boundary equal to the solar value, while the upper boundary is held at constantentropy. The radiative diffusivity κr (r) throughout the domain is taken from the structure model used above; however, we note that the radiative heat flux is several ordersof magnitude smaller in our simulations than convective heat flux throughout the bulkof the domain. We have artificially increased the thermal eddy diffusivity κs near theboundaries in order to prevent the formation of thin diffusive thermal layers with aradial extent below our current radial resolution. This enhanced κs is applied only
to the spherically symmetric component of the thermal field. Steep entropy gradientswould otherwise be required to carry the imposed heat flux, since the radial velocities
are forced to vanish, and so too the convective transport of heat near the boundaries.
This enhanced κs profile turns out to play a sensitive role in determining the nature of
the convection near the top of the thin shell simulations.

112

Kinetic Energy [erg]1010

Case D2: Approach to Equilibrium

105

100

10-5
0

10

20

30

40

50

time [s]Figure 5.3: The domain-averaged kinetic energy as a function of time for Case D2.
The ramp-up time of about 30 days from seed entropy perturbations is typical of thesimulations considered here.

5.2.2

Approach to Thermal Equilibrium

Once the spherically symmetric mean state has been arranged, small seed perturbations are introduced into the fluctuating entropy field s0 , and the simulations advanced
in time using the evolution equations (4.26) and (4.27). The seed entropy perturbations
are soon reflected in the fluctuating density field ρ0 , whose variations quickly provide
the unstable density imbalance that buoyantly accelerates the fluid from rest. Afteran initial period of adjustment during which the convection kinetic energy ramps up(typically about 30 days, as shown in Fig. 5.3 for Case D2), an approximate thermal
equilibrium is reached.
In total thermal equilibrium, the outward energy transport in these simulations
must be achieved by a balance of radiative, kinetic, enthalpy, and eddy diffusive fluxes,

113
1.5
Luminosity [Lsun](a) Case S2
total energy transportenthalpy transport
eddy diffusive transport
KE transport

1.0
0.5
0.0
-0.5
0.94

0.96

0.98

0.96

0.98

1.5
Luminosity [Lsun](b) Case D2
1.0
0.5
0.0
-0.5
0.90

0.92

0.94
r/R

Figure 5.4: The time-averaged spherically symmetric energy balance within Cases S2
and D2 as a function of radius, showing the percentage of L carried by kinetic energy
Fk , enthalpy Fe , and by unresolved eddies Fu . The radiative flux Fr (not shown) is
negligible in both cases.

as represented by the following:Fk = ur

ρ̂u2
2


,

Fe = ur ρ̂cp (T 0 − T 0 ),∂(T̂ + T 0 ),∂r∂(ŝ + s0 ),Fu = −κs ρ̂T̂∂r
Fr = −κr ρ̂cp

(5.2)
(5.3)
(5.4)
(5.5)

where the kinetic, enthalpy, radiative, and unresolved eddy fluxes are denoted by Fk , Fe ,Fr , and Fu respectively. The quantity T 0 − hT 0 i appearing in the definition of Fe
is the temperature excess relative to the mean (spherically symmetric) value of thetemperature field at each radial level. In a true steady state, the total energy transport
at each radius within the domain must equal the energy influx at the lower boundary,

114
which in terms of luminosity is equivalent to the total solar luminosity L :4πr 2 (Fk + Fe + Fr + Fu ) = L .

(5.6)

Figure 5.4 shows the time-averaged energy transport within Cases S2 and D2 as afunction of radius at a late stage in the simulations. In both cases, the energy transportachieved by unresolved eddies reflects the enhanced profile of κs applied to the spherically symmetric entropy gradient. The function κs is determined during initializationand is typically about 3 orders of magnitude larger at the top than the corresponding
κtop felt by the convection. The radial energy transport within Cases S1 and S3 isqualitatively similar to Case S2.

5.3

GENERAL FLOW CHARACTERISTICS

5.3.1

Multi-Scale Convection

The convective flow patterns in these simulations are intricate, containing complex
evolving structures occurring on multiple size scales. We illustrate the velocity patternsin Figures 5.5 and 5.6 by showing the radial velocity for Cases S2 and D2 at several
depths within each domain. The horizontal structure of the radial velocity fields realizedin Cases S1 and S3 are qualitatively similar to Case S2.
Figures 5.5a and 5.6a show that the largest scale of convection visible near the
top of both Cases S2 and D2 is associated with a connected network of downflow lanes(green-blue colors) having a spatial scale of about 200 Mm. The large areas enclosedby the downflow lanes contain a number of smaller-scale upflows (orange-red tones)
each measuring about 15–30 Mm across. Although the upflow cells in the shallow shell(Case S2) tend to be slightly larger than those in the deeper shell (Case D2), this generalsurface pattern of a network of connected downflow lanes enclosing several distinct
smaller upflows appears to be a robust property of convection within our thin shells.
We also note that the size of the smaller upflows in these simulations is approximately

115

Case S2: Radial Velocityab

TOP (r = 0.978 R)

-50

-25
0
25
radial velocity [m s-1]c

MIDDLE (r = 0.960 R)

50

-200

-100
0
100
radial velocity [m s-1]

BOTTOM (r = 0.942 R)

200

-50

-25
0
25
radial velocity [m s-1]

50

Figure 5.5: Instantaneous snapshots of radial velocity for Case S2 near the (a) top,(b) middle, and (c) bottom of the domain. Positive radial velocities (orange-red colors)
denote upflows and negative radial velocities (green-blue colors) denote downflows. Each
image in the top row is an orthographic projection of the velocity field, with the northpole tilted 20◦ toward the observer and the equator indicated by a white line. Eachenlarged image in the bottom row shows a rectangular (latitude-longitude) projectionof a 45◦ -square portion of the corresponding velocity field in the top row. The four-foldazimuthal periodicity is most noticeable near the north pole.

equal to the horizontal size scale of solar supergranulation.
Figures 5.5b,c and 5.6b,c illustrate how the horizontal planforms change withdepth for Cases S2 and D2 respectively. As the downwelling fluid reaches deeper layers,these convective structures no longer form a connected network composed of downflows
having roughly constant vertical velocities along each lane. Although it is possible
near the bottom to make out much of the pattern of downflow lanes seen near the

116

Case D2: Radial Velocityab

TOP (r = 0.979 R)

-40

-20
0
20
radial velocity [m s-1]c

MIDDLE (r = 0.940 R)

40

-400

-200
0
200
radial velocity [m s-1]

BOTTOM (r = 0.901 R)

400

-40

-20
0
20
radial velocity [m s-1]

40

Figure 5.6: Same as in Fig. 5.5 except for Case D2, showing the radial velocity structuresampled near the (a) top, (b) middle, and (c) bottom of the deep shell simulation.

surface, the vertical velocities along each lane become increasingly less uniform, suchthat the downflows fragment into more isolated and compact plume-like structures withdepth. This narrowing of scale is likely related to the larger densities found near thebottom of each domain. In addition, the strongest downflow lanes in the equatorial
region of Case D2 possess a noticeable north-south orientation that is reminiscent of
the banana-cell modes evident in earlier, more laminar, spherical shell convection simulations (e.g. Miesch 1998). The columnar structures seen in Case D2 are much thinnerand have quite a bit more variation than the banana cells in the more laminar cases, asa result of the strong small-scale convection driven throughout the domain.

117(a) Case S2

0°
-15°

15°

-30°

30°

(b) Case D2
-15°

0°

15°

-30°

30°
-300

-150
0
150
radial velocity [m s-1]

300

Figure 5.7: Images of the instantaneous radial velocity for Cases S2 and D2 as a functionof latitude and radius in a cut of fixed longitude, showing the vertical structure of thepattern of upflows and downflows within each domain.

The pattern of smaller-scale upflows visible near the upper boundary of thesesimulations also changes its character with depth. The distinct upflows enclosed by
downflow lanes seen in the upper layers gradually become more uniform, forming broad
regions of upwelling fluid surrounded by an incomplete network of downflow lanes. Near
the lower boundary, these broad upflow regions have largely disappeared, except forpartial shrouds of upwelling fluid surrounding each of the stronger downflows. Figure 5.7
contains vertical cuts of the radial velocity field showing the variation with latitude andradius for Cases S2 and D2. The radial structure of the broad upflow regions is mostevident in Case D2 (Fig. 5.7b), where the broad regions of upward moving fluid fragment
into smaller-scale yet faster upflows as the top of the domain is approached.
Figure 5.8 shows the time evolution of the radial velocity field of Case S2. Suchmovie sequences illustrate the tendency for smaller flow features to be systematically
advected by larger-scale velocity patterns. One small upflow (indicated by the arrow)as well as the downflow lane immediately to its left are both advected laterally by the

118
1

2

3

4

5

10

9

8

7

6

20°

Figure 5.8: A time series of images showing a 20◦ -square region of Case S2 near theupper boundary showing small-scale features in radial velocity being laterally advectedby larger-scale horizontal motions. The arrow points to one such small-scale upflow
which is advected away from the center of the broader upflow (indicated by the darkcross). The time index of each image is indicated in the upper-left corner, and thecadence is about 1.3 days between images.

horizontal outflow motions associated with the broader cell. The center of this broad
outflow cell is indicated by a cross. Such lateral transport of velocity features by larger
scales of convection is most apparent in movie sequences showing the time evolution of
the radial velocity field.
In Figure 5.9 we present space-time diagrams of radial velocity sampled in timefor four specific latitudes (and all longitudes) at a radius of 0.978 R for Case S2, plottedwith respect to the rotation rate of the computational frame Ω0 . The advection of radial
velocity structures appear as slanted features in each panel, with the higher latitudes
rotating more slowly (retrograde) than those near the equator. The mean advection
rate of the radial velocity patterns at the surface approximately equals the differential
rotation rate of the fluid for the same depth. The latitudinal variation of angularvelocity shown in the figure results largely from the imposed differential rotation at
the lower boundary. It is interesting to note from Figure 5.9 that the pattern speed of

119time [days]

50
40
30
20
10
0
50
40
30
20
10
0
50
40
30
20
10
0
50
40
30
20
10
0

(a) latitude=60°(b) latitude=45°(c) latitude=30°(d) latitude=0°

0°

15°

30°

45°
60°
longitude

75°

90°
−10 0
−20
−40
tachometer [nHz] −80

Figure 5.9: Near-surface radial velocity structures at four specific latitudes from
Case S2, plotted as a function of time and latitude. As labeled, the four panels correspond to latitudes of 0◦ , 30◦ , 45◦ , and 60◦ . The retrograde propagation rate of thesefeatures, quantified by the tachometer, is a reflection of the no-slip differential lower
boundary.

the advected structures varies somewhat with longitude, as structures having the same
latitude but separated in longitude may exhibit different propagation rates.
The characteristic size scales of the convective structures near the top of thesethin shell simulations are sensitive to the degree of driving they experience. We find that
adjusting either the diffusivities or the superadiabaticity within the domain will alter
the appearance of the convection. The superadiabaticity depends on the functional form

120

Velocity Histogram
Frequency [% of total]

0.06

0.04

Case D2

Case S2

0.02

0.00
-400

-200

0
200
Radial velocity [m s-1]

400

Figure 5.10: The distribution of radial velocities for the shallow and deep shells (Cases S2
and D2) attained at mid-shell as shown in Figs. 5.5b and 5.6b respectively. The corresponding rms velocities are 105 m s−1 for Case S2 and 140 m s−1 for Case D2. Positive
radial velocities denote upflows and negative radial velocities denote downflows.

of the spherically symmetric κs profile, set upon the initialization of each simulation.
The functional form of this profile in essence determines how much energy needs to betransported via convection near the top of the domain, which in turn feeds back on
the entropy gradient. We have chosen a somewhat gentle κs function, as suggested by
the profile of Fu in Figure 5.4, initialized to have an e-folding depth away from eachboundary of about 0.01 R for both the shallow and deep shell cases.
The peak radial velocities attained by the convective structures at mid-shell inCases S2 and D2 are 300 m s−1 and 400 m s−1 respectively, as shown in the radial
velocity distributions of Figure 5.10. The distribution of radial velocity for Case S2 at
0.96 R is noticeably asymmetric, such that mean downward velocities are much fasterthan the mean upward velocities. The fastest downward velocities correspond to thedownflow lane network evident in the middle of the shell, as illustrated in Figure 5.5b.
The radial velocity distribution for Case D2 is more symmetric, indicating that the flow

121

Case D2: Temperatureabc

TOP (r = 0.979 R)

-6

-3
0
3
6
temperature perturbation [K]

MIDDLE (r = 0.940 R)

-8

-4
0
4
8
temperature perturbation [K]BOTTOM (r = 0.901 R)

-6

-3
0
3
6
temperature perturbation [K]Figure 5.11: Similar to Fig. 5.6, but showing the horizontal structure of the temperature
perturbation sampled near the (a) top, (b) middle, and (c) bottom of the domain, withthe mean temperature for each level removed. Orange-red colors denote warmer fluid,
green-blue colors denote cooler fluid.

patterns at 0.94 R are more homogeneous than for higher levels.
The rms velocities associated with the radial velocity distributions of Figure 5.10
are 105 m s−1 for Case S2 and 140 m s−1 for Case D2, equivalent to an overturning
time on the order of 6–10 days, depending on the shell depth. These overturning times
suggest that the large-scale convective pattern is weakly sensitive to rotational effects,
as both Cases S2 and D2 are rotating at the solar-like mean rate of one rotation per
28 days. We will discuss the rotational influence of these convective overturning motions
in more detail in §5.4.

122

Temperature Variation with Latitude (Case D2)

Temperature [K]

15
10
5
0
-5
-90°

-60°

-30°

0°
Latitude

30°

60°

90°

Figure 5.12: The zonally averaged fluctuating temperature near the top of the domain(r=0.979 R) for Case D2.

Figure 5.11 shows the fluctuating temperature field for Case D2, corresponding
to the velocity planforms of Figure 5.6. The mean temperature at each radius has
been subtracted out. We find that the locations of the large regions of warm andcool temperatures correlate well with the locations of the broad upflows and narrow
downflow lanes visible in the radial velocity field, with the primary difference beingthat the cooler regions tend to be much broader than their radial velocity counterparts.
Near the equator, the temperature field is dominated by columns of alternating warmand cold fluid associated with the weak banana-cell-like structures visible in the radial
velocity images. In both temperature and radial velocity, these structures are shearedslightly by the differential rotation within the domain, and extend up to about ±30◦ oflatitude. These large-scale columnar temperature structures are broken up by smallerscale variations on the temperature field which also tend to correlate well with some ofthe small-scale radial velocity features. For example, the localized hot and cold spots
particularly evident in the close-up views of Figure 5.11a are coincident with some of

123
the fastest fluid motions visible in the radial velocity image of Figure 5.6.
At all radii within Case D2, there exists a significant latitudinal temperature
contrast between the equator and the poles, as shown in Figure 5.12. The temperaturesin the near-polar regions are about 10–15 K warmer than near the equator, althoughover half of the total equator-to-pole difference occurs within 10◦ of the pole. As statedearlier, we believe that many characteristics of the fluid in the near-polar regions aremost likely artifacts of the four-fold azimuthal periodicity imposed in these simulations,and should be interpreted with care.

5.3.2

Time-Averaged Axisymmetric FlowsThe time-averaged axisymmetric (zonally averaged) profiles of angular velocityΩwithin all four domains are shown in Figure 5.13, with the angular velocity of the
2πΩ0
subtracted out. Note that we have chosen to have the equatorial
computation frame
2πrate imposed at the lower boundary be equal to the computational frame rate, such thatΩ
= 0 nHz there. The differentially rotating lower boundary imposed in Cases D2, S2,
2π
and S3 decreases from 0 nHz at the equator to about −120 nHz at a latitude of 75◦ , and issimilar in contrast and functional form to the latitudinal variation of the photosphericplasma rate measured by Snodgrass (1984). In contrast, the no-slip lower boundary
imposed in Case S1 is not differentially rotating.
With the exception of the poles, the angular velocity profiles within all four
simulations are retrograde with respect to the rotating coordinate system specifying theequatorial rate, with the fastest rotation rates occurring near the bottom of each shell.
Cases S1 and S2 show a largely constant negative radial gradient of angular velocity
with radius at each point (e.g. Fig. 5.18), with the overall magnitude of Ω determined by
the rotation rate imposed at the no-slip lower boundary at the corresponding latitude.
Within Cases S3 and D2, the negative radial gradients in angular velocity throughout
the bulk of each shell are smaller, with the exception of the thin viscous boundary layer

124

Angular Velocity

0

0

0

0

-30

-5

-30

-30

Case

D2

-60

-90

Case

S1

-10

-15

-120

S2

-90

-20
nHz

Case
-60

S3

-90

-120
nHz

Case
-60

-120
nHz

nHz

Figure 5.13: Angular velocity Ω/2π relative to the rotating coordinate system as afunction of latitude and radius for all four cases, averaged over longitude and time. A
no-slip differentially rotating lower boundary is imposed in Cases D2, S2, and S3. That
imposed angular velocity decreases from 0 nHz at the equator to about −120 nHz at
a latitude of 75◦ . Case S1 has a uniformly rotating no-slip lower boundary. The tickmarks are separated by 15◦ of latitude.

present near the lower boundary in each case.
The regions poleward of 75◦ of latitude are not shown in Figure 5.13 due tothe high angular velocities present there. Even though these regions exhibit reasonable (linear) zonal velocities, the short moment arm produces higher values of Ω thanseen elsewhere within the domain at lower latitudes. In addition, the polar regions are
dominated by effects related to the four-fold azimuthal periodicity imposed in these simulations, producing dynamics which may not be otherwise present in similar simulations

125

Radial Velocity

50

20

20

20

25

10

10

10

Case
0

D2

Case
0

S1

Case
0

S2

Case
0

-25

-10

-10

-10

-50
m s-1

-20
m s-1

-20
m s-1

-20
m s-1

S3

Figure 5.14: Axisymmetric radial velocity profiles for all cases. Positive values represent
upward moving fluid.

without such an imposed periodicity.
The time-averaged meridional circulation is illustrated in Figures 5.14 and 5.15,
showing the axisymmetric profiles of uθ and uφ . In all cases, the dominant mode driven
in these simulations is that of axisymmetric rolls that occupy the entire depth of thedomain. Cases S1, S2, and D2 contain several 15◦ -wide rolls located between latitudesof −75◦ and +75◦ with typical flow speeds of 50–75 m s−1 . Rolls which have a poleward
surface component seem to be preferred, as they are generally more extended horizontally and possess faster fluid velocities. In Case S3, the rolls are smaller in size but areotherwise similar to the larger rolls of the other three simulations. We will show in §5.4

126

Meridional Velocity

100

100

100

100

50

50

50

50

Case
0

Case

D2

0

Case

S1

0

S2

Case
0

-50

-50

-50

-50

-100
m s-1

-100
m s-1

-100
m s-1

-100
m s-1

S3

Figure 5.15: Axisymmetric meridional velocity profiles for all cases. Positive values
represent southward flows.

that the rolls are primarily buoyantly driven.
In order to examine the form achieved by the differential rotation and meridional
circulation in each of the four cases, we derive equations describing the kinetic energybalance within the system. Throughout the remainder of this chapter, the axisymmetricor longitudinally averaged component of a quantity A(r, θ, φ) is denoted by an overbar,
and is given by
A(r, θ) =

1
2π

Z 2π
dφ A(r, θ, φ),
0

(5.7)

127
allowing the decomposition
A = A(r, θ) + A0 (r, θ, φ)such that

A0 = 0,(5.8)

where the prime on A0 denotes the non-axisymmetric part of A. By noting that theaxisymmetric component of a product AB is equal toAB = A B + A0 B 0 ,(5.9)

we can therefore decompose the axisymmetric kinetic energy Ek into three parts:ρ̂ u2r + u2θ + u2φ(5.10)
Ek =2ρ̂ u0r 2 + u0θ 2 + u0φ 2
ρ̂ uφ 2 ρ̂ ur 2 + uθ 2=++,
(5.11)
2 } |
2
2
| {z
{z
} |
{z
}
DRKE

MCKE

NAKE

where
DRKE = differential rotation kinetic energy,(5.12)

MCKE = meridional circulation kinetic energy,(5.13)

NAKE = non-axisymmetric (convective) kinetic energy.

(5.14)The balance of DRKE, MCKE, and NAKE within the shell models is now discussed in
more detail.

5.4

ENERGETICS OF AXISYMMETRIC FLOWS

5.4.1

Axisymmetric Differential Rotation Balance

We will now show that the radial gradients of Ω realized in each of the four
simulations are supported against diffusion by Reynolds stresses associated with nonaxisymmetric convective motions. This behavior is likely caused by the tendency of
convective fluid elements to partially conserve their angular momentum per unit mass
λ = Ωr 2 cos2 θ as they move toward or away from the axis of rotation.

128
As suggested by Foukal & Jokipii (1975), possible constancy of λ along radiallines may also explain why surface magnetic tracers on the sun have a faster rotation
rate relative to the surface fluid, if one assumes that the magnetic features are anchoredat a radius slightly below the photosphere where the rotation rate is faster. Gilman &Foukal (1979) tested this notion by numerically modeling Boussinesq convection confined
to a thin shell, and found that angular momentum was roughly conserved along local
radii for such an incompressible fluid. They demonstrated that the convective motions
were able to transport angular momentum inward, thereby maintaining the negative
radial gradient of rotation rate with radius. The models presented here suggest thatcompressible convection behaves in a similar manner.
To quantify these ideas, we examine the evolution equation for the DRKE,∂ (DRKE)= RCF + LCF + RRS + LRS + ADV + VT + DIFF + CURV,∂t(5.15)

where the abbreviations signify contributions by different physical mechanisms in amanner similar to Gilman (1977). In a statistical steady state, the eight work terms inequation (5.15) above must sum to zero. These work terms are:
RCF = Coriolis forces on radial motions = −2Ω ρ̂ ur uφ sin θ
LCF = Coriolis forces on latitudinal motions = −2Ω ρ̂ uθ uφ cos θ
RRS = radial Reynolds stresses = ρ̂ ur uφ

∂ uφ
∂r

∂ uφ
LRS = latitudinal Reynolds stresses = ρ̂ uθ uφ
∂θ



1 ∂ 2
1 ∂
ADV = DRKE advection = − 2r ρ̂ uφ ur uφ +(sin θ ρ̂ uφ uθ uφ )r ∂rr sin θ ∂θVT = viscous transport

 





uφ
1 ∂
∂ uφ
sin θ ∂1 ∂2
r ν ρ̂ uφ r+
sin θ ν ρ̂ uφ
= 2
r ∂r
∂r r
r sin θ ∂θr ∂θ sin θ( 2 2 )uφsin θ ∂∂ uφ+
DIFF = viscous losses = −ν ρ̂ r
∂r r
r ∂θ sin θ
CURV = curvature effects = −uφ(ρ̂ ur uφ sin θ + ρ̂ uθ uφ cos θ) .
r sin θ

129

0.2

Case S1(a) RCF(b) LCF(c) ADV(d) RRS(e) LRS(f) VT+DIFF

0.1

0.0

(a)

(b)

(c)

(d)

(e)

(f)

-0.1

-0.2
erg cm-2 s-1

Figure 5.16: DRKE source terms for Case S1.

5.4.2

The Radial Shear Layer Within Case S1

We begin by examining the DRKE balance within Case S1, as it contains manyof the important elements of angular momentum transfer in these simulations without
the additional effects related to the differentially rotating lower boundary condition
imposed in the other three simulations. The contributions from each of the terms in
the DRKE balance equation (5.15) are illustrated in Figure 5.16 (with the exception of
CURV, which is negligible), with Figure 5.17 showing the total DRKE and∂ DRKE∂t

for Case S1 as a function of radius and latitude. With respect to the rotating frame,uφ < 0 throughout most of the domain, such that a positive contribution by any of the

130

20

0.2

16

Case S1(a) DRKE(b) d/dt DRKE

0.1

12

(a)

0.0

(b)

8
4
0
104 erg cm-2

-0.1

-0.2
erg cm-2 s-1

Figure 5.17: Total DRKE and change for Case S1. The ratio between typical values
contained in (b) to those of (a) indicates the time scale on which the DRKE energetics
within Case S1 is changing.

source terms of equation (5.15) represents a source of DRKE which in isolation would
decrease uφ over time to more negative values. Likewise, a negative contribution to
DRKE represents an increase (to less negative values) in uφ over time. Figure 5.17
shows that Case S1 is evolving on a time scale of at least 106 s or 12 days, as indicatedby the sum total of all work terms (Fig. 5.17b) divided by the DRKE throughout theshell (Fig. 5.17a).
In the absence of convection, the DRKE profile shown in Figure 5.17a would
achieve solid body rotation (that is, relax to zero) due to viscous dissipation in thesystem. Such dissipation accounts for the negative values of the VT+DIFF profile

131
40
latitude = 0°

latitude = 45°

latitude = 30°

latitude = 60°

20(Ωtot − Ω0.94) / 2π [nHz]

0
-20
-40
40
20

Case S3
GONG result

0Case S1-20

Case S2
λ = constant

-40
0.94

0.95

0.96

0.97
r/R

0.98

0.99

1.00

0.94

0.95

0.96

0.97
r/R

0.98

0.99

1.00

Figure 5.18: Angular velocity profiles for Cases S1 (red), S2 (green), and S3 (blue) as
a function of radius for latitudes 0◦ , 30◦ , 45◦ , and 60◦ as indicated. The dashed line ineach panel represents the angular velocity of a radially moving fluid parcel for which its
angular velocity per unit mass λ is conserved, while the dash-dot lines correspond to
the GONG data plotted in Fig. 5.1.

shown in Figure 5.16f . The goal is therefore to determine which forcing terms maintainthe radial shear in uφ against viscous diffusion. The convective motions present in
the domain drive an axisymmetric meridional circulation, as well as create non-zero
velocity correlations on smaller scales which produce axisymmetric Reynolds stresses.
Contributions from one or both of these sources, which collectively make up the forcing
terms in equation (5.15), must therefore be responsible for maintaining the non-zeroradial gradient of uφ .
The Coriolis force terms, RCF and LCF, represent exchanges between the DRKE
and MCKE equations in the form of fluid parcels tending to preserve their overall
angular momentum as they move toward or away from the axis of rotation. As a result,the RCF profile shown in Figure 5.16a is well correlated with the axisymmetric radial

132
velocity profile (shown in Fig. 5.14), such that local upflows (downflows) in ur tend to
reduce (increase) uφ and thus cause a local increase (decrease) of DRKE. In a similar
fashion, the LCF term shown in Figure 5.16b is correlated to uθ where now equatorward(poleward) flows produce a local increase (decrease) of DRKE.
Both the RCF and LCF contributions to the DRKE balance result from global
fluid motions conserving angular momentum as they move throughout the domain.
Such transport will spin down the portions of the domain farther away from the axisof rotation, which naturally tends to make∂ uφ
< 0 everywhere. However, the strong∂r

axisymmetric flows associated with the persistent latitudinal banding in ur and uθ also
advect DRKE quite efficiently, such that regions of global convergence or divergence
will respectively increase or decrease their DRKE content in response. This effect is
captured in the DRKE transport term, ADV, shown in Figure 5.16c. There it can beseen that such transport effectively balances out the contributions to DRKE from the
RCF and LCF terms.
We find that it is the Reynolds stress terms, RRS and LRS, that are primarily
responsible for maintaining the radial shear in uφ . These terms represent exchanges
between the DRKE and NAKE equations, where the average tilt of non-axisymmetricconvective structures (similar to the downflow plumes of Case S2 discussed earlier in§5.3.1) in the rφ- or θφ-planes in the presence of gradients of uφ can produce contributions to DRKE. Figure 5.16d shows that the RRS profile is mainly positive, suggestingthat the ur uφ velocity correlation tends to be negative throughout the domain and thus
implies a retrograde tilt among small-scale radial motions. Such a tendency is exactly
what one would expect if the non-axisymmetric convection were to partially conserve
its angular momentum per unit mass λ in radial motion.
The LRS term (Fig. 5.16e) is lesser in magnitude then RRS but also plays a role inmaintaining the DRKE profile, especially at higher latitudes. The LRS term possesses
a bimodal structure in radius which contains positive contributions to DRKE in the133upper half of the domain and negative contributions in the lower half. Such a structure
will tend to reinforce negative radial gradients in uφ in cases where uφ < 0, as is thecase here (see Fig. 5.17a).
Figure 5.18 shows the drop in total angular velocity Ωtot = Ω + Ω0 with radiusrealized in Case S1 (plotted in red) for several latitudes between 0◦ and 60◦ . The dashedlines indicate the angular velocity an isolated fluid parcel moving in a purely radial
direction would have if λ = constant (that is, Ωtot ∝ r −2 ). There is some tendency for
fluid parcels in Case S1 to conserve λ as they move radially throughout the shell, except
near the bottom of the domain wheredΩ
is larger (less negative). For comparison, thedrdash-dot lines in Figure 5.18 indicate the helioseismic inferences of Ω within the nearsurface shear layer of the sun. The tendency to conserve λ is also apparent in the sun
but to a lesser degree than in the models between 0.94–0.98 R, although the helioseismic
results suggest that this effect may play a larger role closer to the photosphere where
the convection is more vigorous.
The thin viscous boundary layer near the bottom of Case S1 is formed in responseto the no-slip lower boundary. The greater influence of viscous dissipation within thislayer flattens out the angular velocity gradient at low latitudes and even produces apositive

5.4.3

dΩ
at higher latitudes (e.g. at 60◦ in Fig. 5.18).
dr
Differential Rotation Within Cases S2, S3, and D2

Turning to Cases S2, S3, and D2, we find that the angular velocity Ω possesses
radial gradients similar to those achieved within Case S1, though there is substantiallydifferent behavior at high latitudes. The green and blue lines of Figure 5.18, corresponding to Cases S2 and S3, along with Figure 5.19, illustrate this behavior. In this section,
we show that these other three simulations are similar to Case S1, in which these radial
gradients of Ω are primarily maintained against diffusion by Reynolds stresses associated with convective motions. The primary difference is that the latitudinal structure

134
40

latitude = 0°

latitude = 45°

latitude = 30°

latitude = 60°

20(Ωtot − Ω0.90) / 2π [nHz]

0
-20
-40
40
20
0
-20
-40
0.90

0.92

0.94
0.96
r/R

0.98

1.00 0.90

0.92

0.94
0.96
r/R

0.98

1.00

Figure 5.19: Similar to Fig. 5.18, except for the deeper shell simulation, Case D2.
As before, the dashed line in each panel represents the angular velocity of a radiallymoving fluid parcel for which its angular velocity per unit mass λ is conserved, while
the dash-dot lines correspond to the GONG data plotted in Fig. 5.1.

of DRKE has now changed, in response to the differentially rotating lower boundary
applied in these three cases.
Figure 5.20 shows the DRKE source terms for Case S2, which, aside from thedifferentially rotating lower boundary, is otherwise identical to Case S1. This boundary condition forces the fluid in contact with the lower boundary to have increasingly
negative zonal velocities with latitude, thereby constituting a continual supply of angular momentum. The DRKE associated with this imposed flow is then transported inboth the radial and latitudinal directions, to give the angular velocity profile seen in
Figure 5.13. Some of this energy is converted into MCKE, driving the more vigorous
latitudinal rolls present at high latitudes (see Figs. 5.14 and 5.15), the effects of which
feed back on the DRKE balance through the RCF and LCF work terms.

135

0.8

Case S2(a) RCF(b) LCF(c) ADV(d) RRS(e) LRS(f) VT+DIFF

0.4

0.0

(a)

(b)

(c)

(d)

(e)

(f)

-0.4

-0.8
erg cm-2 s-1

Figure 5.20: DRKE source terms for Case S2.

The DRKE energy balance within Cases D2 and S3 are shown in Figures 5.21
and 5.22 respectively. In contrast to the shallow shell (Case S2), these two cases possessa more moderate latitudinal gradient of uφ . This form is achieved by a greater advectionof DRKE, especially at mid to high latitudes, and is evident in the larger role the ADV
term plays in the DRKE balance. A relatively lower portion of the DRKE input isconverted into MCKE, as shown by the profiles of RCF and LCF.
The latitudinal rolls within Case S3 are smaller in physical size as a consequenceof the lower Prandtl number in this simulation. Both the lower viscous diffusivity andhigher thermal diffusivity associated with the lower value of Pr drive more vigorous
convection that occurs on smaller scales. These smaller scales are evident in the various

136

2.0

Case D2(a) RCF(b) LCF(c) ADV(d) RRS(e) LRS(f) VT+DIFF

1.0

0.0

(a)

(b)

(c)

(d)

(e)

(f)

-1.0

-2.0
erg cm-2 s-1

Figure 5.21: DRKE source terms for Case D2.

DRKE source terms, shown in Figure 5.22, but otherwise the dynamics are similar toCase S2.
In summary, the time scales of the largest overturning motions in our simulations
suggest that they are at least weakly influenced by rotational effects, which in turn
may enable Reynolds stresses to facilitate transport angular momentum inward. This
inward angular momentum transport balances the outward diffusive transport, thereby
maintaining a negative angular velocity gradient throughout the layer. This effect may
contribute to the decrease of Ω with radius in the near-surface shear layer of the sun as
deduced from helioseismic observations. Behavior at high latitudes is somewhat more
complex due to the presence of a viscous boundary layer near the lower boundary, and

137

0.6

Case S3(a) RCF(b) LCF(c) ADV(d) RRS(e) LRS(f) VT+DIFF

0.3

0.0

(a)

(b)

(c)

(d)

(e)

(f)

-0.3

-0.6
erg cm-2 s-1

Figure 5.22: DRKE source terms for Case S3.

likewise there is some uncertainty in the helioseismic inferences about the radial gradientin Ω achieved at latitudes of 60◦ or greater.

5.4.4

Axisymmetric Meridional Circulation

We now briefly investigate the meridional circulation driven within the thin shell
simulations, as these motions play an important role in the transport of DRKE (and thus
angular momentum) within the shells. Schematically, the MCKE evolution equation is∂ (MCKE)= RCF + LCF + RRS + LRS + GRAV + PGF∂t+ ADV + VT + DIFF + CURV,(5.16)

138
where the individual contributions are symbolized by:
RCF = Coriolis forces on radial motions = 2Ω ρ̂ ur uφ sin θ
LCF = Coriolis forces on latitudinal motions = 2Ω ρ̂ uθ uφ cos θ
∂ ur
∂ uθ
+ ρ̂ ur uθ∂r∂rρ̂ u2θ ∂ uθ
∂ ur+
LRS = latitudinal Reynolds stresses = ρ̂ ur uθ∂θr ∂θRRS = radial Reynolds stresses = ρ̂ u2r

GRAV = gravity/buoyancy driving = −ρ g ur
∂ p uθ ∂ p
PGF = work done by pressure gradient forces = − ur
+
∂r
r ∂θ

h
i

1 ∂
r 2 ρ̂ ur u2r + ρ̂ uθ ur uθ
ADV = MCKE advection = − 2
r ∂r
h
i
1 ∂ 
2
sin θ ρ̂ ur ur uθ + ρ̂ uθ uθ+r sin θ ∂θVT = viscous transport

 




1 ∂ ur
2ur
∂ uθ
∂ ur
1 ∂
2
r ν ρ̂ 2 ur+−(∇ · u)+ uθ r
= 2
r ∂r
∂r
∂r r
r ∂θ
3

 
 

1 ∂ ur∂ uθ
1 ∂sin θ ν ρ̂ ur r++
r sin θ ∂θ
∂r r
r ∂θ



1 ∂ uθ
ur2uθ
+ 2 uθ
(∇ · u)
+
−
r ∂θ
r
3
(
2 
2
1
1
1 ∂ uθur∂ ur− (∇ · u) ++− (∇ · u)
DIFF = viscous losses = −2ν ρ̂
∂r
3
r ∂θ
r
3

2
ur
uθ cos θ 1++− (∇ · u)rr sin θ3

 2 )∂ uθ
1 1 ∂ ur+r+
2 r ∂θ
∂r r



ur
uθ 
ρ̂ ur uθ sin θ − ρ̂ u2φ cos θ .
CURV = curvature effects =u2r + u2φ +rr sin θWe first examine the maintenance of the meridional circulation within Case S1,
corresponding to the profiles of ur and uθ shown previously in Figures 5.14 and 5.15.
Most of the kinetic energy associated with these axisymmetric flows is contained in
the 15◦ -wide latitudinal rolls distributed across the mid latitude and equatorial regions.

139

MCKE balance
Case S1(a) GRAV+PGF(b) ADV(c) RRS+LRS(d) RCF+LCF(e) VT+DIFF

2.0

1.0

0.0

(a)

(b)

(c)

(d)

(e)

-1.0

-2.0
erg cm-2 s-1

Figure 5.23: MCKE source terms for Case S1.

Rolls having poleward velocities near the surface are preferred over cells which have theopposite rotational sense, especially in the mid latitude regions.
The main contributions to the MCKE balance within Case S1 are illustrated in
Figure 5.23. We find that the energetics within the latitudinal rolls are achieved by anear balance between the work done by buoyancy and pressure gradients (Fig. 5.23a) andby Reynolds stresses (Fig. 5.23c). In places where the body forces drive the meridionalflows (e.g. in radial upflows) the Reynolds stresses oppose it, and conversely when the
body forces oppose the motion the Reynolds stresses act in such a way as to maintainit.
The RCF and LCF source terms represent the conversion of DRKE to MCKE

140

2.0

MCKE balance
Case S2(a) GRAV+PGF(b) ADV(c) RRS+LRS(d) RCF+LCF(e) VT+DIFF

1.0

0.0

(a)

(b)

(c)

(d)

(e)

-1.0

-2.0
erg cm-2 s-1

Figure 5.24: MCKE source terms for Case S2.

through Coriolis forces acting on the differential rotation. Because uφ < 0 throughout
most of the domain, the RCF and LCF terms will tend to enhance poleward or inward
flows associated with the latitudinal rolls, and counteract any flows moving outward inradius or toward the equator. Figure 5.23d shows that the contributions by Coriolis
forces acting on the differential rotation in Case S1 are small relative to the body forces
and Reynolds stresses.
The differentially rotating lower boundary imposed in Cases S2, S3, and D2 affects the dynamics within these domains by driving latitudinal rolls with faster velocities
than are present when the lower boundary is uniformly rotating (as in Case S1). This
energy, added to the system as DRKE, is subsequently converted to MCKE through the

141
Coriolis work terms RCF and LCF. This effect is illustrated in Figure 5.24d for Case S2,
showing a more prominent driving by Coriolis forces than for Case S1 in these regions.

5.5

CONCLUSION
We have presented results of high-resolution numerical simulations of turbulence

confined to thin spherical shells, seeking to understand some of the effects of supergranular scales of motion within a thin shearing layer nominally located near the top of thesolar convection zone. These simulations represent the first global simulations of solarlike convection possessing horizontal scales on the order of supergranulation. In a seriesof four closely related simulations, we find in all cases that the convection is organized
into a connected network of fast downflow lanes separating broad regions of warm more
slowly rising fluid. With depth, we find that the downflow network loses some of its
connectivity, instead forming more plume-like structures in the deeper layers.
Near the surface, the broad regions of upwelling fluid segment into more isolated,smaller-scale upflows, each with velocities comparable to the larger-scale network of fast
downflows. These smaller upflow cells possess a spatial scale comparable to that of solar
supergranulation and only appear near the upper boundary, suggesting that their origin
stems from the more superadiabatic stratification present near the top of the shells.
That such convective structures appear in both the shallow and deep shell simulations
further suggests that it is the nature of the driving, rather than the depth of the shell,that influences the morphology of the convection near the surface.
The angular velocity within the thin shells decreases with radius throughout eachdomain, with the exception of the thin viscous boundary layer near the lower boundary
in each case. We find that such negative radial gradients of angular velocity are maintained against diffusion by correlations between small-scale velocity components. The
net effect of such Reynolds stresses suggests that fluid motions associated with both thebroad upflows and strong downflow network tend to conserve their angular momentum

142
per unit mass as they move radially throughout the shell. We believe these Reynolds
stresses are an effect of the weak influence of rotation on the coherent convective structures realized in the simulations. We further speculate that supergranular convection in
the sun behaves in a similar fashion, thereby contributing to the near-surface shear layer
occupying the upper 5% of the sun by radius as inferred from helioseismic analyses.

Chapter 6

CONCLUDING REMARKS

We have presented the results of two complementary studies, one observational
and the other theoretical, of turbulent convection of supergranular scales of motion
within the upper solar convection zone in order to better understand the dynamical roleof such convection in shearing layers within the sun. Complex convective patterns possessing a wide range of spatial scales are evident both in observations of supergranular
outflows visible at the solar surface as well as in global numerical simulations of turbulence within thin spherical shells, and are likely to influence the dynamics of the deeper
convection zone below as well as the more tenuous photosphere, chromosphere, andcorona above. We now briefly summarize our findings from both studies, and conclude
with a discussion of future directions in §6.2.

6.1

SUMMARY OF RESEARCH PRESENTED IN THIS THESIS
Chapter 3 describes an observational study entailing the characterization of cell

sizes and evolution associated with solar supergranulation, as determined from the life
histories of several thousand supergranules individually identified by their horizontal
outflow signatures. The near-surface horizontal flow fields were measured by applying
correlation tracking methods to mesogranular flow structures within a time series of
MDI full-disk (200 pixels) line-of-sight velocity images of the solar photosphere. This
time series contains over 8000 images separated by one minute, sampling a 45◦ -square144
region of quiet sun for a duration of about six days, and represents the first study ofsolar supergranulation at such high combined temporal and spatial resolution. Sites ofstrong horizontal outflow were identified as supergranules using a pattern recognition
algorithm, from which the life history of all cells in the six-day dataset was recorded.
We find that supergranular outflow cells in this quiet sun region have a broad
range of sizes, distributed in an approximate Gaussian fashion, with typical length scalesof 14–20 Mm after image smoothing effects are accounted for. The supergranular pattern
appears to be surface-filling, with individual cells separated by a connected network ofthin convergence lanes that covers the entire field of view. The complex evolution of
the supergranular pattern is embodied in the wide spectrum of cell lifetimes, ranging
from time scales as long as several days down to the temporal resolution limit of 6 hr.
Such evolution typically occurs via merging and splitting of individual supergranules,
coupled with the emergence and disappearance of the associated interstitial convergencelanes. In other instances, segments of the convergence lane network are observed to beadvected laterally as existing supergranular outflows grow or contract.
The supergranular pattern can therefore be characterized as an evolving networkof thin convergence lanes separating broad outflow cells, and is most likely the surface
manifestation of vigorous overturning motions occurring in the near-surface layers directly below the photosphere. In addition, the hierarchy of fluid motions in the upper
convection zone is not limited to supergranular scales of motion, but also encompasses
the smaller-scale patterns of mesogranulation and granulation as well as larger scales
such as banded zonal flows and weak meridional currents. To elucidate the role such
dynamics may play within the upper solar convection zone, we have constructed detailed three-dimensional numerical simulations of a compressible fluid, confined to thinrotating spherical shells positioned near the top of the solar convection zone.
The simulations presented in Chapter 5 constitute the highest resolution simulation of solar-like convection within a spherical shell computed to date. Spherical har-145
monic modes with ` ≤ 340 are explicitly resolved, allowing structures of order 10 Mm
to be realized within the simulations. In all cases, a four-fold angular periodicity wasused to reduce the computational workload. Solar-like stratification, thermal forcing,and differential rotation profiles are imposed on the fluid, in order to approximate theconditions present in the layers where solar supergranulation is thought to be driven.
The ASH computer code, operating in a massively parallel computing environment, isused to advance in time the anelastic equations of motion. These equations allow us
to include the effects of compressibility, yet filter out acoustic waves which would otherwise severely limit the size of the computational time step. The dynamical effectsof turbulent motions not explicitly resolved in these simulations are accounted for bysimple parameterizations of energy and momentum transport included in the anelastic
equations.
The simulations suggest that convection on multiple scales may be a naturalconsequence of a compressible fluid experiencing a rapidly changing stratification. Convection within both shallow and deep shell cases having identical stratification near theirrespective upper boundaries (both located at 0.98 R) show a prominent network of narrow downflow lanes having a length scale of about 200 Mm near the top of each shell.
The broad regions in between contain rising fluid that is further segmented into more
localized sites of fast upflows each measuring about 15–30 Mm across, or about equal
to the horizontal scale associated with solar supergranulation. With depth, the networkof fast downflows becomes less uniform, forming plume-like structures that extend thefull depth of each shell, while the small-scale upflows evident near in the upper layers
gradually disappear altogether to form broad regions of upwelling fluid.
In all cases studied here, the latitudinal dependence of the differential rotation
within these thin shell simulations reflects the angular velocity profile Ω imposed at
the lower boundary. Negative radial gradients of angular velocity persist throughout
each domain, and are maintained by the transport of angular momentum by the con-

146
vective motions. Reynolds stresses associated with such motions, in particular thebroad regions of upflowing fluid and the surrounding network of downflows, are foundto transport angular momentum inward, balancing the outward transport achieved bydiffusion. These dynamics may be interpreted as the tendency for radially moving fluid
parcels to partially conserve their angular momentum per unit mass as they move radially throughout the shell, and are likely induced by the influence of rotation on theconvective motions. These results suggest that the dynamical effects of rotation, while
relatively weak given the short overturning time scales associated with the convection,
are still strong enough to create the necessary velocity correlations that account for theobserved angular momentum transport.
These simulations suggest that the upper shear layer of the sun may behave similarly, with the network of narrow downflow lanes associated with the supergranulation
pattern facilitating the inward transport of angular momentum. Such transport would
than contribute to the decrease of angular velocity with radius, as seen in helioseismic
inferences of Ω. As a result, the depth of the upper shear layer may be loosely related
to the radial extent of convection associated with the supergranulation pattern visibleat the surface.

6.2

FUTURE OUTLOOKThe research presented in this thesis represents a detailed look at the intricate

convection within the upper shear layer of the solar convection zone. We have sought tounderstand what scales of convection are driven, what determines the depth of the shear
layer, and what are the dynamical effects of this convection in maintaining the shear
layer. Since both the observational project and numerical simulations can be viewed as
initial building blocks toward such understanding, we now present some possible future
directions this research might take.
The correlation tracking algorithm outlined in §2.3 used to determine the super-

147
granular flow field was shown to have systematic errors on the order of 10% for shifts
between 0.005 pixels and 0.05 pixels. The magnitude of such errors suggest that thealgorithm can be made more accurate by using an interpolation scheme of a higher
order, thereby reducing the systematic errors generated by the algorithm.
We have left unexplored the large body of work regarding pattern formation in
turbulent systems. The ever-changing near-surface velocity field appears to evolve in aremarkably ordered fashion, especially given the level of turbulence that must be presenta short distance below. Consequently, it may be possible to create an evolving field ofcells using a rule-based system that produces statistical area and lifetime distributions
similar to those of solar supergranulation, thereby isolating the important length andtime scales operating within the system.
The mean differential rotation and meridional circulation profiles deduced fromthe large-scale flow study of §3.4, wherein correlation tracking was applied to the supergranular flow field, contain a large degree of scatter. Determining these mean flows
within many additional regions available to MDI will likely reduce this scatter.
The supergranular flow field has been shown to readily advect small-scale emergent magnetic flux toward convergence lanes, thereby concentrating the flux on relatively
larger scales. This behavior is common to regions of quiet sun, such as the 45◦ -square
field presented here. A logical extension of this research is therefore to investigate theeffects of more intense magnetism on supergranular flows, such as within active regions
and near sunspots. The character of supergranulation is likely to change under the
influence of stronger magnetic fields, as the stabilizing effects of such magnetism affects
the overturning motions associated with the convectively unstable fluid.
The numerical models possess several attributes that are drastically different thanthe upper solar convection zone. Foremost, the sun does not possess an impermeable
lower boundary in the middle of its convection zone. Allowing mass to pass through thelower boundary would be more realistic, as the current simulations require the convective

148
overturning motions as well as any large-scale meridional circulations to close within theconfines of the thin shell domains. As an alternative to allowing permeable boundaries,
we are in the process of constructing a simulation of the entire convection zone witha spatial resolution adequate to deal with a surface layer containing the small-scaleconvective structures seen in the simulations presented here. It will be interesting tosee if a shearing boundary layer near the top of these domains forms naturally.
As computing technology increases, it will become possible to construct simulations of fluid with even higher spatial resolution, which in turn would allow convection
at higher Ra and Re to be studied. Whereas attaining solar values of Ra and Re within
simulations of convection is unlikely in the foreseeable future, it is of great interest to
see whether the dynamical trends presented here also operate within more turbulent
flows.
We recognize that the representation of sub-grid scale (SGS) effects in these simulations, achieved by simply enhancing the thermal and molecular diffusivities, greatly
simplifies their true effects. As more coherent small-scale structures form within moreturbulent fluid, the inaccuracies associated with using a simplified SGS treatment arelikely to grow. It has been shown that small-scale features in the flows will not only
dissipate momentum and heat, but are also able to advectively transport these quantities. More sensible treatments of the SGS terms in the ASH code would capture moreof the dynamical effects associated with fluid motions not explicitly resolved.
The advent of continuous imaging of the solar surface carried out at high spatial
and temporal resolution has led to remarkable ways to probe the dynamics of the intensely turbulent convection zone. Such efforts will be carried to new levels with theplanned Solar Dynamics Observatory mission, and likewise the upgrades to GONG will
provide uninterrupted Doppler imaging from the ground. The continuous rapid advances with massively parallel computing architectures will enable even more detailed
simulations of turbulent convective fluids analogous to the solar convection zone. We

149
thus foresee complementary paths coupling major observational and theoretical efforts
in studying the complex evolving dynamics within the solar convection zone.

Bibliography

Basu, S., Antia, H. M., & Tripathy, S. C. 1999, ApJ, 512, 458
Beck, J. T., Duvall, J. T. L., & Scherrer, P. H. 1998, Nature, 394, 653
Berger, T. E., Löfdahl, M. G., Shine, R. A., & Title, A. M. 1998, ApJ, 506, 439
Bray, R. J., Loughhead, R. E., & Durrant, C. J. 1984, The Solar Granulation (Cambridge: Cambridge University Press)
Brummell, N., Cattaneo, F., & Toomre, J. 1995, Science, 269, 1370
Brummell, N. H., Hurlburt, N. E., & Toomre, J. 1996, ApJ, 473, 494—. 1998, ApJ, 493, 955
Brun, A. S. & Toomre, J. 2001, in Helio- and Asteroseismology at the Dawn of theMillennium, Proceedings of the SOHO 10/GONG 2000 Workshop, ESA SP–464, ed.
A. Wilson, 619–624
Brun, A. S., Turck-Chièze, S., & Zahn, J. P. 1999, ApJ, 525, 1032
Burr, I. W. 1974, Applied Statistical Methods (New York: Academic Press)
Canuto, V. M. 1996, ApJ, 467, 385
Canuto, V. M. & Christensen-Dalsgaard, J. 1998, Ann. Rev. Fluid Mech., 30, 167
Cattaneo, F. 1999, ApJ, 515, L39
Cattaneo, F., Brummell, N. H., Toomre, J., Malagoli, A., & Hurlburt, N. E. 1991, ApJ,
370, 282
Charbonneau, P. & MacGregor, K. B. 1997, ApJ, 486, 502
Christensen-Dalsgaard, J., Proffitt, C. R., & Thompson, M. J. 1993, ApJ, 403, L75
Clune, T. C., Elliott, J. R., Miesch, M. S., & Toomre, J. 1999, Parallel Comp., 25, 361
DeRosa, M., Duvall, Jr., T. L., & Toomre, J. 2000, Sol. Phys., 192, 349
DeRosa, M. L. & Toomre, J. 1998, in Structure and Dynamics of the Interior of the Sun
and Sun-Like Stars, Proceedings of the SOHO 6/GONG 98 Workshop, ESA SP–418,
ed. S. Korzennik & A. Wilson, 753–758

151
Dikpati, M. & Charbonneau, P. 1999, ApJ, 518, 508
D’Silva, S. 1996, ApJ, 469, 964
D’Silva, S. & Choudhuri, A. R. 1993, A&A, 272, 621
Durney, B. R. & Roxburgh, I. W. 1971, Sol. Phys., 16, 3Duvall, Jr., T. L., D’Silva, S., Jefferies, S. M., Harvey, J. W., & Schou, J. 1996, Nature,379, 235
Duvall, Jr., T. L., Kosovichev, A. G., Scherrer, P. H., Bogart, R. S., Bush, R. I.,
De Forest, C., Hoeksema, J. T., Schou, J., Saba, J. L. R., Tarbell, T. D., Title,
A. M., Wolfson, C. J., & Milford, P. N. 1997, Sol. Phys., 170, 63
Duvall, Jr., T. L., M., J. S., Harvey, J. W., & Pomerantz, M. A. 1993, Nature, 362, 430
Elliott, J. R., Miesch, M. S., & Toomre, J. 2000, ApJ, 533, 546
Fan, Y., Fisher, G. H., & DeLuca, E. E. 1993, ApJ, 405, 390
Fan, Y., Zweibel, E. G., Linton, M. G., & Fisher, G. H. 1999, ApJ, 521, 460
Foukal, P. & Jokipii, J. R. 1975, ApJ, 199, L71
Giles, P. M. 1999, PhD thesis, Stanford University
Giles, P. M., Duvall, Jr., T. L., Scherrer, P. H., & Bogart, R. S. 1997, Nature, 390, 52
Giles, P. M., Duvell, Jr., T. L., & Scherrer, P. H. 1998, in Structure and Dynamics ofthe Interior of the Sun and Sun-Like Stars, Proceedings of the SOHO 6/GONG 98
Workshop, ESA SP–418, ed. S. Korzennik & A. Wilson, 775–780
Gilman, P. A. 1977, GAFD, 8, 93—. 1978a, GAFD, 11, 157—. 1978b, GAFD, 11, 181—. 1983, ApJS, 53, 243
Gilman, P. A. & Foukal, P. V. 1979, ApJ, 229, 1179
Gilman, P. A. & Glatzmaier, G. A. 1981, ApJS, 45, 335
Gilman, P. A. & Miller, J. 1981, ApJS, 46, 211—. 1986, ApJS, 61, 585
Glatzmaier, G. A. 1984, J. Comp. Phys., 55, 461—. 1985, ApJ, 291, 300
Glatzmaier, G. A. & Gilman, P. A. 1981, ApJS, 45, 351
Gough, D. & Toomre, J. 1991, ARA&A, 29, 627

152
Gough, D. A. 1969, J. Atmos. Sci., 26, 448
Haber, D. A., Hindman, B. W., Toomre, J., Bogart, R. S., & Hill, F. 2001, in Helioand Asteroseismology at the Dawn of the Millennium, Proceedings of the SOHO
10/GONG 2000 Workshop, ESA SP–464, ed. A. Wilson, 213–218
Haber, D. A., Hindman, B. W., Toomre, J., Bogart, R. S., Thompson, M. J., & Hill, F.
2000, Sol. Phys., 192, 335
Hagenaar, H. J., Schrijver, C. J., & Title, A. M. 1997, ApJ, 481, 988
Hagenaar, H. J., Schrijver, C. J., Title, A. M., & Shine, R. A. 1999, ApJ, 511, 932
Hansen, C. J. & Kawaler, S. D. 1994, Stellar Interiors: Physical Principles, Structure,and Evolution (New York: Springer-Verlag)
Hathaway, D. H., Gilman, P. A., Harvey, J. W., Hill, F., Howard, R. F., Jones, H. P.,
Kasher, J. C., Leibacher, J. W., Pintar, J. A., & Simon, G. W. 1996, Science, 272,
1306
Hill, F. 1988, ApJ, 333, 996
Howard, R., Gilman, P. A., & Gilman, P. I. 1984, ApJ, 283, 373
Howard, R. & Harvey, J. 1970, Sol. Phys., 12, 23
Howard, R. & LaBonte, B. J. 1980, ApJ, 239, L33
Howe, R., Christensen-Dalsgaard, J., Hill, F., Komm, R. W., Larsen, R. M., Schou, J.,
Thompson, M. J., & Toomre, J. 2000, Science, 287, 2456
Hurlburt, N. E., Schrijver, C. J., Shine, R. A., & Title, A. M. 1995, in Proceedings ofthe Fourth SOHO Workshop, ESA SP–376, 239–243
Kitchatinov, L. L. & Rüdiger, G. 1993, A&A, 276, 96—. 1995, A&A, 299, 446
LaBonte, B. J. & Howard, R. 1982, Sol. Phys., 75, 161
Leighton, R. B., Noyes, R. W., & Simon, G. W. 1962, ApJ, 135, 474
Lesieur, M. 1997, Turbulence in Fluids (Dordrecht: Kluwer Academic Publishers)
Libbrecht, K. G. & Zirin, H. 1986, ApJ, 308, 413
Linton, M. G., Longcope, D. W., & Fisher, G. H. 1996, ApJ, 469, 954
Lisle, J., DeRosa, M., & Toomre, J. 2000, Sol. Phys., 197, 21
Longcope, D. W., Fisher, G. W., & Arendt, S. 1996, ApJ, 464, 999
Miesch, M. S. 1998, PhD thesis, University of Colorado
Miesch, M. S., Elliott, J. R., Toomre, J., Clune, T. L., Glatzmaier, G. A., & Gilman,
P. A. 2000, ApJ, 532, 593

153
Moreno-Insertis, F. 1986, A&A, 166, 291
November, L. J. 1994, Sol. Phys., 154, 1
November, L. J., Toomre, J., & Gebbie, K. B. 1981, ApJ, 245, L123
Parker, E. N. 1955a, ApJ, 122, 293—. 1955b, ApJ, 121, 491—. 1979, Cosmical Magnetic Fields: Their Origin and Activity (Oxford: Clarendon
Press)—. 1993, ApJ, 408, 707
Roudier, T., Muller, R., Mein, P., Vigneau, J., Malherbe, J. M., & Espagnet, O. 1991,
A&A, 248, 245
Scherrer, P. H., Bogart, R. S., Bush, R. I., Hoeksema, J. T., Kosovichev, A. G., Schou,J., Rosenberg, W., Springer, L., Tarbell, T. D., Title, A., Wolfson, C. J., Zayer, I., &the MDI Engineering Team. 1995, Sol. Phys., 162, 129
Schou, J., Antia, H. M., Basu, S., Bogart, R. S., Bush, R. I., Chitre, S. M., ChristensenDalsgaard, J., DiMauro, M. P., Dziembowski, W. A., Eff-Darwich, A., Gough, D. O.,
Haber, D. A., Hoeksema, J. T., Howe, R., Korzennik, S. G., Kosovichev, A. G.,
Larsen, R. M., Pijpers, F. P., Scherrer, P. H., Sekii, T., Tarbell, T. D., Title, A. M.,
Thompson, M. J., & Toomre, J. 1998, ApJ, 505, 390
Schou, J. & Bogart, R. S. 1998, ApJ, 504, L131
Schrijver, C. J., Title, A. M., van Ballegooijen, A. A., Hagenaar, H. J., & Shine, R. A.
1997, ApJ, 487, 424
Schüßler, M. 1979, A&A, 71, 79
Shine, R. A., Simon, G. W., & Hurlburt, N. E. 2000, Sol. Phys., 193, 313
Simon, G. W. & Leighton, R. B. 1964, ApJ, 140, 1120
Singh, J. & Bappu, M. K. V. 1981, Sol. Phys., 71, 161
Snodgrass, H. B. 1984, Sol. Phys., 94, 13
Snodgrass, H. B. & Ulrich, R. K. 1990, ApJ, 351, 309
Spiegel, E. A. & Zahn, J.-P. 1992, A&A, 265, 106
Spitzer, L. 1962, Physics of Fully Ionized Gases (New York: Interscience Publishers)
Spruit, H. C., Nordlund, Å., & Title, A. M. 1990, ARA&A, 28, 263
Stein, R. F. & Nordlund, Å. 1998, ApJ, 499, 914—. 2000, Sol. Phys., 192, 91

154
Stix, M. 1976, A&A, 47, 243
Strous, L. H. & Simon, G. W. 1998, in Astronomical Society of the Pacific Conference
Series, Vol. 140, 161–169
Tassoul, J.-L. 1978, Theory of Rotating Stars (Princeton: Princeton University Press)
Thompson, M. J., Toomre, J., Anderson, E. R., Antia, H. M., Berthomieu, G., Burtonclay, D., Chitre, S. M., Christensen-Dalsgaard, J., Corbard, T., DeRosa, M. Genovese,
C. R., Gough, D. O., Haber, D. A., Harvey, J. W., Hill, F., Howe, R., Korzennik,
S. G., Kosovichev, A. G., Leibacher, J. W., Pijpers, F. P., Provost, J., Rhodes, Jr.,
E. J., Schou, J., Sekii, T., Stark, P. B., & Wilson, P. R. 1996, Science, 272, 1300
Title, A. M., Hurlburt, N., Schrijver, C., Shine, R., & Tarbell, T. 1995, in Proceedingsof the Fourth SOHO Workshop, ESA SP–376, 113–120
Tobias, S. M., Brummell, N. H., Clune, T. L., & Toomre, J. 1998, ApJ, 502, L177
Ulrich, R. 1970, ApJ, 162, 993
Ulrich, R. K. 1998, in Structure and Dynamics of the Interior of the Sun and Sun-LikeStars, Proceedings of the SOHO 6/GONG 98 Workshop, ESA SP–418, ed. S. Korzennik & A. Wilson, 851–855
Wang, H. & Zirin, H. 1989, Sol. Phys., 120, 1
Ward, F. 1966, ApJ, 145, 416
Zappalà, R. A. & Zuccarello, F. 1991, A&A, 242, 480

Appendix A

CORRELATION TRACKING ALGORITHM DETAILS

A.1

SETTING UP THE GRIDPOINT ARRAYIn §2.3, it is stated that the correlation tracking algorithm is applied to two

equally sized images I1 (x, y) and I2 (x, y). Our notation is such that integer values ofthe pixel coordinates x and y index the pixels in the two images, though we will alsoneed to refer to fractional pixel values in what follows. An array of N gridpoints at
which local displacements are to be detected is denoted by the coordinates (xn , yn ),where n = 1, 2, · · · , N . By the following method, local displacements are calculated sothat the pixels in the neighborhood of (xn , yn ) in image I1 optimally coincide with the
pixels in the neighborhood of (xn , yn ) in image I2 when shifted by the relative amount(δx, δy).
Before calculating the optimal displacement, we formally define the neighborhoodof each gridpoint as those pixels contained in a square subimage centered on the gridpoint in question. For example, if the subimages have 2p + 1 pixels on a side (that is, p
is the subimage half-width), the subimage centered on the gridpoint (xn , yn ) extracted
from image I1 is thus comprised of all pixels I1 (xi , yj ) with
xi = xn − p + i

andyj = yn − p + j,

(A.1)

where the indices i and j run from i, j = 0, 1, · · · , 2p. Therefore, this subimage isbounded horizontally by x = xn −p and x = xn +p and bounded vertically by y = yn −p

156
and y = yn + p inclusive. As shorthand, we define S1n as the subimage centered on themeasurement gridpoint (xn , yn ) extracted from image I1 . The conventions defined above
are equally applicable to image I2 .

A.2

THE MERIT FUNCTION
Given the two corresponding subimages S1n and S2n centered on the same gridpoint(xn , yn ) extracted from I1 and I2 , we now wish to determine the relative shift between
the two subimages such that the topology of the two subimages maximally coincides.
To calculate this optimal displacement, we determine the displacement vector (δx, δy)such that the following merit function is minimized:
( 

2p−1
X 2p−1X
1
1
W xi − xn + , yj − yn +m(δx, δy) =2
2
i=0 j=0
 

1 δx
1 δy
× I1 xi + + , yj + +
2
2
2
2
 )

1 δx
1 δy 2.
− I2 xi + − , yj + −
2
2
2
2

(A.2)

In its essence, the merit function m(δx, δy) in equation (A.2) above is a weighted sumof the squares of the differences between the pixel values of the two corresponding
subimages after shifting the two subimages with respect to each other by the relative
offset (δx, δy).
The merit function is arranged so that the fractional pixel shifts are measured
with respect to a position a half-pixel above and a half-pixel to the right of the pixels inthe subimages; that is, the fractional pixel shifts are measured with respect to the points


1
1
1
. This formulation explains the presence of the various ’s added toxi + , yj +222
the pixel indices, as well as the upper limit of 2p − 1 for each of the sums.
Additionally, the merit function is set up so that the subimages are shifted in asymmetric fashion. For given horizontal and vertical displacements δx and δy, S1n andS2n are each shifted by the same amounts

δy
δx
and
but in opposite directions. This
2
2

157
symmetric shifting stabilizes the behavior of m(δx, δy) near the point (δx, δy) = (0, 0).
Both the one-half pixel offset and the symmetric shift simplify the computational
task of evaluating I1 and I2 at interpixel values. This scheme guarantees that theinterpolating point always falls within the square framed by the four pixels (xi , yj ),(xi , yj + 1), (xi + 1, yj ), and (xi + 1, yj + 1), as long as only shifts of less than one pixel
are considered. For this condition to be true, we require|δx| ≤ 1and|δy| ≤ 1.

(A.3)

For the magnitudes of the velocities we hope to measure in this work, this condition isalways met.
The function W in equation (A.2) above is a weighting function multiplying thesquared pixel differences so that the terms in the double sum arising from points closer
to the subimage center (xn , yn ) are weighted more heavily in the sum than those farther
away. We use a Gaussian function for W , 2
x + y2,W (x, y) = exp −
σ2

(A.4)

although any tapered function will suffice. The adjustable parameter σ is the e-folding
distance of the Gaussian, which in turn determines the spatial resolution of the resulting velocity field. The parameter p introduced above, which characterizes the size ofthe subimages, should be chosen such that the subimages extend far enough out toencompass the peak and falloff of W . Choosing p too large will only increase the computational workload while the extra outlying points contribute little to the weighted
sum in equation (A.2). Conversely, choosing p too small may not encompass all of
the points surrounding the measurement gridpoint which have any significant weighting
in the double sum. In this work, p is chosen to be slightly larger than σ. Typically,p = 11 pixels for σ = 8 pixels.
As seen by inspecting equation (A.2), minimizing the merit function requires amethod for evaluating I1 and I2 at fractional pixel coordinates. In general, if we know

158
the value of a function f at the four bounding points (x1 , y1 ), (x2 , y1 ), (x1 , y2 ), and(x2 , y2 ), we can bilinearly interpolate between the abscissae to estimate the value of f
at any point interior to the bounding points:
f (x1 + ∆x, y1 + ∆y) = (1 − A)(1 − B) f (x1 , y1 ) + A(1 − B) f (x2 , y1 )+ (1 − A)B f (x1 , y2 ) + AB f (x2 , y2 ),(A.5)

where the scaled coordinates A and B are definedA≡∆xx2 − x1

and

B≡

∆y
.
y2 − y1(A.6)

For the point (x1 + ∆x, y1 + ∆y) to be interior to the bounding points, we must have
0≤A≤1

and

0 ≤ B ≤ 1,

(A.7)

otherwise we would be extrapolating rather than interpolating.
The bilinear interpolation formula (A.5) is used to evaluate I1 and I2 at fractional
pixel coordinates. Identifying the indices x1 , x2 , y1 , and y2 in equation (A.5) withthe pixel indices xi , xi+1 , yj , and yj+1, respectively, and setting ∆x =
∆y =

1
δx
+
and
2
2

δy
1
+
, we obtain the following expression for the I1 term in equation (A.2)
22

above:
 



1 δx
1 δy
1 δx
1 δy
=
−
−
I1 (xi , yj )I1 xi + + , yj + +
2
2
2
2
2
2
2
2



1 δy
1 δx
I1 (xi+1 , yj )
+
−
+
2
2
2
2



1 δy
1 δx
−
+
I1 (xi , yj+1 )
+
2
2
2
2



1 δy
1 δx
+
+
I1 (xi+1 , yj+1 ).
+
2
2
2
2

(A.8)

159
1 δy
1 δx
−
and ∆y = − , we have
2
2
2
2
 



1 δx
1 δy
1 δx
1 δy
=
+
+
I2 (xi , yj )I2 xi + − , yj + −
2
2
2
2
2
2
2
2



1 δy
1 δx
−
+
I2 (xi+1 , yj )
+
2
2
2
2



1 δy
1 δx
+
−
I2 (xi , yj+1 )
+
2
2
2
2



1 δy
1 δx
−
−
I2 (xi+1 , yj+1 ).
+
2
2
2
2

Likewise for I2 , where now ∆x =(A.9)

Note that the restrictions on A and B given above in equation (A.7) imply that |δx| ≤ 1
and |δy| ≤ 1, as already assumed by equation (A.3). We now substitute equations (A.8)
and (A.9) into equation (A.2) to obtain
(



1
1
W xi − xn + , yj − yn +m(δx, δy) =2
2
i=0 j=0



1 δx
1 δy
×
−
−
I1 (xi , yj )
2
2
2
2



1 δy
1 δx
I1 (xi+1 , yj )
+
−
+
2
2
2
2



1 δy
1 δx
−
+
I1 (xi , yj+1 )
+
2
2
2
2



1 δy
1 δx
I1 (xi+1 , yj+1 )
+
+
+
2
2
2
2



1 δy
1 δx
+
+
I2 (xi , yj )
−
2
2
2
2
 


1 δy
1 δx
−
I2 (xi+1 , yj )
−
+
−
2
2
2
2



1 δy
1 δx
+
−
I2 (xi , yj+1 )
−
2
2
2
2



2 )
1 δy
1 δx
−
−
I2 (xi+1 , yj+1 )
−
.
2
2
2
2
2p−1
X 2p−1
X(A.10)

This expression for m(δx, δy) depends only upon the unknown displacement (δx, δy)and the data values of the pixels in the two subimages extracted from I1 and I2 .

160

A.3

MINIMIZING THE MERIT FUNCTION
To find the values of δx and δy which minimize the merit function (A.10), we

will need to evaluate the derivatives

∂m
∂m
and
. This task is simplified by first
∂(δx)
∂(δy)

expanding the squared term in equation (A.10). To proceed, first define the quantities
d1 = I1 (xj , yk ) − I2 (xj+1 , yk+1),
d2 = I1 (xj , yk+1 ) − I2 (xj+1 , yk ),d3 = I1 (xj+1 , yk ) − I2 (xj , yk+1),
d4 = I1 (xj+1 , yk+1 ) − I2 (xj , yk ),and then substitute these differences into equation (A.10):m(δx, δy) =2p−1
X 2p−1X
i=0 j=0

Wij h(1 − δx)(1 − δy) d1 + (1 − δx)(1 + δy) d2
16
i2 
,
+ (1 + δx)(1 − δy) d3 + (1 + δx)(1 + δy) d4(A.11)

11
where Wij = W xi − xn + , yj − yn +. Then, after expanding the square and22regrouping we havem(δx, δy) =2p−1
X 2p−1X
i=0 j=0

Wij h
a1 + a2 δx2 + a3 δy 2 + a4 δx2 δy 2 + a5 δx
16
i
2
2
,
+ a6 δy + a7 δx δy + a8 δx δy + a9 δx δy(A.12)

161
where, after some algebra, the a-coefficients are found to be
a1 = (d1 + d2 + d3 + d4 )2 ,a2 = (d1 + d2 − d3 − d4 )2 ,a3 = (d1 − d2 + d3 − d4 )2 ,a4 = (d1 − d2 − d3 + d4 )2 ,a5 = −2(d1 + d2 + d3 + d4 )(d1 + d2 − d3 − d4 ),a6 = −2(d1 + d2 + d3 + d4 )(d1 − d2 + d3 − d4 ),a7 = 4(d21 − d22 − d23 + d24 ),a8 = −2(d1 + d2 − d3 − d4 )(d1 − d2 − d3 + d4 ),a9 = −2(d1 − d2 + d3 − d4 )(d1 − d2 − d3 + d4 ).
Since the offsets δx and δy are independent of the quantities being summed over, we
can pull them out of the sum and define the coefficients
Am ≡

2p−1
X 2p−1X
i=0 j=0

Wij
am ,
16

(A.13)

where m = 1, 2, · · · , 9. Substituting equation (A.13) into equation (A.12), we havem(δx, δy) =A1 + A2 δx2 + A3 δy 2 + A4 δx2 δy 2 + A5 δx(A.14)
2

2+ A6 δy + A7 δx δy + A8 δx δy + A9 δx δy .
Note that the Am -coefficients can be calculated as soon as the subimages have been
extracted, as they depend only upon W and the data values contained in the two
subimages S1n and S2n .
To find the point (δx, δy) which minimizes m(δx, δy), we follow the usual procedure of setting the first partial derivatives of m equal to zero and solving the resulting
system of equations:
∂m
=0
∂(δx)
∂m
=0
∂(δy)

=⇒



A5 + A7 δy + A9 δy 2 + 2δx A2 + A8 δy + A4 δy 2 = 0,

(A.15)=⇒


A6 + A7 δx + A8 δx2 + 2δy A3 + A9 δx + A4 δx2 = 0.

(A.16)

162
At this point, we note that solving for either δx or δy using equation (A.15) or (A.16)
and substituting the resulting expression into the other yields a quintic equation. Rather
than look for zeroes in such a quintic equation, the correlation tracking algorithm uses
an iterative scheme to approximate the extremal value of (δx, δy). Starting with anestimate for δy, the algorithm obtains an estimate for δx by solving equation (A.15).
Using this estimate for δx, a new estimate for δy is calculated by solving equation(A.16). This stepping scheme is iterated until either δx and δy differ from its previous
estimate by less than 10−4 .

A.4

ITERATING WITH BICUBIC SHIFTSTo attain a more accurate estimate of the optimal displacement (δx, δy), we could

use a higher-ordered interpolation scheme (such as bicubic interpolation) to evaluate I1
and I2 at interpixel values in the merit function (A.2) above. However, we find it
simpler to use the optimal shifts (δx, δy) calculated by minimizing m given by equation(A.10), which uses bilinear shifts to estimate interpixel data values, as a guess for what
the optimal displacement would be had bicubic interpolation been used in minimizing
the merit function. Then one of the subimages is bicubically shifted by the known
amount (δx, δy) and the procedure is repeated. This iteration scheme is chosen because
it is computationally faster to bicubically shift images by a known amount, as opposed
to calculating an unknown displacement assuming bicubic interpolation was used toevaluate the functions I1 and I2 at interpixel values.
The correlation tracking algorithm therefore proceeds as follows. At each measurement gridpoint (xn , yn ), an optimal displacement is calculated using bilinear shifts
in the merit function as described in §A.2. Define this initial displacement as (δx, δy)agg .
The subimage S2n is then shifted by this offset (δx, δy)agg using bicubic interpolation.
An additional optimal displacement vector (δx0 , δy 0 ) is then calculated using the merit
function containing bilinear shifts, where S1n and the (bicubically) shifted version of S2n

163
are now used as input images. The total displacement becomes the sum of the initial displacement and this additional displacement, so that the aggregate displacement
00(δx, δy)agg = (δx, δy)old
agg + (δx , δy ).

We now continue to add to the aggregate displacement by iterating until convergence. That is, subimage S2n is now bicubically shifted by the aggregate displacement(δx, δy)agg , and another additional displacement vector is calculated using S1n and the
newly shifted version of S2n . This new displacement vector is then added to the aggregate. Convergence occurs when the displacement vector produced using S1n and ashifted version of S2n has a |δx| or |δy| less than 10−4 . This aggregate shift (δx, δy)agg isthen returned as the optimal displacement for the gridpoint in question. This bicubic
iteration process provides an estimate for what the optimal displacement would havebeen had bicubic shifts (rather than bilinear shifts) been used to evaluate I1 and I2 at
interpixel values in equation (A.2).

A.5

CORRELATION TRACKING PITFALLSIn determining the optimal displacement for each measurement gridpoint, sev-

eral problems may befall the correlation algorithm described above. As stipulated byequation (A.3), if either aggregate offset δxagg or δyagg is more than one pixel in either
direction, the algorithm sets the offsets to zero and the gridpoint is flagged. As statedearlier, we do not expect the motions in the time series analyzed in this these to possess
displacements of more than one pixel.
Two problems may occur when iterating using the bicubic shifts. First, if morethan twenty iterations occur prior to convergence, the offsets are set to zero and thegridpoint is flagged. This situation usually occurs when the merit function contains noprominent minimum causing the algorithm to wander around in (δx, δy)-space, as iscaused by when the subimages contain no prominent topological features.
The second problem which may occur during the bicubic iteration process is a164degradation of the merit function. Ideally, the value of the merit function continues
to decrease as the minimum value of (δx, δy) is approached. If the value of the merit
function is found to increase between iterations, suggesting that the next aggregate
offset is actually farther away from the minimum in m(δx, δy), the current aggregate
displacements are returned and the gridpoint is flagged. This problem generally occurs
when the topology of the merit function near the minimum value of (δx, δy) is somewhat
lumpy, containing several local extrema in the neighborhood of the absolute minimum.

Appendix B

ASH CODE EQUATIONS — DETAILED DERIVATIONSThis appendix supplements Chapter 4 by providing detailed derivations of theequations presented therein. In the following section, we perform a formal scale analysis
on the fully compressible fluid equations in order to determine the anelastic momentum
and energy equations listed in §4.2.3. We then in §B.2 derive the energy conservation
equations given in §4.2.4. Lastly, in §B.3 we incorporate the streamfunction formalism presented in §4.2.5 into the anelastic evolution equations, replacing them with theevolution equations listed in §4.2.5.

B.1

ANELASTIC FLUID EQUATIONS
We now apply the scaling outlined in §4.2.3 to the fully compressible fluid equa-tions (4.2)–(4.4) to obtain the set of anelastic equations (4.23)–(4.33), where each of
the boxed equations in §B.1.2–§B.1.4 which follow correspond to one of the anelastic
equations listed in §4.2.3. The derivations presented here follow Gilman & Glatzmaier(1981).

B.1.1

Order in  of All Dependent Variables

As stated in §4.2.3, we express the four thermodynamic state variables as thesum of a spherically symmetric mean quantity and a fluctuating quantity, as in equa-166tion (4.21) of §4.2.3:
ptotal (r, θ, φ, t) = p̂(r) + p(r, θ, φ, t),ρtotal (r, θ, φ, t) = ρ̂(r) + ρ(r, θ, φ, t),(B.1)
Ttotal (r, θ, φ, t) = T̂ (r) + T (r, θ, φ, t),stotal (r, θ, φ, t) = ŝ(r) + s(r, θ, φ, t).
Note that in these equations we have altered the notation used in equation (4.21) byremoving the primes from all fluctuating quantities.
The perturbations to the state variables result from convective motions driven by
the superadiabatic stratification of the layer, which is characterized by the parameter defined in equation (4.12). We therefore assume forf
is of order  for any quantityfˆ

f in equation (B.1), with the anelastic approximation valid when   1. To filter out
sound waves, we assume
∂ρ
= 0,
∂t

(B.2)

as in equation (4.22) of §4.2.3.
Before performing the scale separation on the compressible fluid equations, wemust first determine the order in  of all variables and operators present. We examine
variables other than the thermodynamic state variables in the remainder of this section.
A summary of these results is provided in Table B.1.
1

It is important to note that the fluid velocity u is of order  2 , as suggested byequation (4.14). The main reason for this scaling is that since the kinetic energy of the
motions is extracted from the stratification, we have1ρ̂u2
∼  and therefore u ∼  2 .
2Given a representative length scale λp and velocity u, the time scale dt on which theconvection modifies the stratification is thus on the order of ∆t =time derivatives to scale as1∂∼ 2 .
∂t1
λp∼ − 2 , causing|u|

The heuristic arguments presented above also apply to the diffusivity coefficients.
Since we expect the eddy dissipation scale to be set by the scale of the convection, we167
Table B.1: Order in  for all quantities and operators appearing in the fully compressible
fluid equations.
Order in 0Quantities

Operatorsp̂, ρ̂, T̂ , ŝ, g, δ ∇, ∇·, ∇×

2u, νeff , κs , Ω, e

∂
∂t

1

p, ρ, T , s(none)

1

1

1

find that νeff ≈ λp u, and thus νeff ∼  2 since u ∼  2 . Similar scaling arguments also
hold for κr and κs .
In the following subsections, we substitute the scaling given in equation (B.1)
into the compressible fluid equations and derive the anelastic equations. As stated in§4.2.3, the diffusivities νeff , κr , and κs are assumed to be functions of ρ̂ only, while theparameters γ and cp are assumed constant throughout the domain.

B.1.2

Derivation of the Anelastic Mass Continuity Equation

By substituting equations (B.1) into the mass continuity equation (4.2), we obtain∂ρ+ ∇ · (ρ̂u) + ∇ · (ρu) = 0.
| {z } | {z }∂t|{z}vanishes

1
∼ 2

(B.3)

3
∼ 2

As indicated, the time derivative of ρ vanishes by equation (B.2). Retaining the highestordered term yields the anelastic mass continuity equation,∇ · (ρ̂u) = 0.

(B.4)

168
B.1.3

Derivation of the Anelastic Momentum Equations

By substituting equations (B.1) into the momentum equation (4.3), and using thedefinition of D, equation (4.5), we obtain∂u+ (u · ∇)u = −∇(p̂ + p) − (ρ̂ + ρ)g r̂ + 2(ρ̂ + ρ)(u × Ω)(ρ̂ + ρ)
∂t



1
+ ∇ · 2(ρ̂ + ρ)νeff e − (∇ · u) δ .
3

(B.5)

After expanding and grouping according to their order in , we obtain∂u∂u+ ρ̂(u · ∇)u + ρ+ ρ(u · ∇)u + ∇p̂ + ρ̂g r̂
ρ̂
∂t
|
{z
} | ∂t{z} | {z0 }
∼1

∼

∼2




1
= 2ρ̂(u × Ω) − ∇p − ρg r̂ + ∇ · 2ρ̂νeff e − (∇ · u) δ
3
{z
}
|

(B.6)

∼1



1
+ 2ρ(u × Ω) + ∇ · 2ρνeff e − (∇ · u) δ .
3
|
{z
}


∼2

The mean momentum equation consists of the terms scaling as 0 , of which only ther̂-component remains,∂ p̂+ ρ̂g = 0.
∂r

(B.7)

The first-order (1 ) terms give the fluctuating momentum equation,ρ̂∂u= 2ρ̂(u × Ω) − ρ̂(u · ∇)u − ∇p − ρg r̂ + ∇ · D̂,∂twhere the anelastic viscous stress tensor D̂ is defined1D̂ = 2ρ̂νeff e − (∇ · u) δ .
3
B.1.4

(B.8)

(B.9)

Derivation of the Anelastic Energy Equation

By substituting equations (B.1) into the energy equation (4.4), and using thedefinition of Φ, equation (4.6), we obtain∂(ŝ + s)+ (u · ∇)(ŝ + s)(ρ̂ + ρ)(T̂ + T )
∂t


1
2
= −∇ · q̂ eff + 2(ρ̂ + ρ) e : e − (∇ · u) ,3

(B.10)

169
where the anelastic heat flux q̂ eff is definedq̂eff = −κr ρ̂cp ∇(T̂ + T ) − κs ρ̂T̂ ∇(ŝ + s).

(B.11)In the above equation, we have explicitly included the contributions to the diffusivetransport of internal energy by the fluctuating temperature and entropy gradients. Similarly, we also keep the fluctuating entropy gradient in the advection term, so that theanelastic energy equation becomesρ̂T̂∂s= −∇ · q̂eff − ρ̂T̂ (u · ∇)(ŝ + s) + Φ̂,∂t(B.12)

where the anelastic viscous heating term Φ̂ is defined12Φ̂ = 2ρ̂νeff e : e − (∇ · u) .
3

(B.13)

Note that we have used the anelastic stress tensor D̂ as defined in equation (B.9).

B.1.5

Derivation of the Anelastic Equations of State

Substituting equations (B.1) into the equation of state (4.10) yieldsp̂ + p =γ−1
cp (ρ̂ + ρ)(T̂ + T ).
γ

(B.14)

Collecting the zeroth-order terms gives
γ−1
cp ρ̂T̂ ,γ(B.15)

γ−1 cp ρT̂ + ρ̂T ,γ(B.16)p̂ =while the first-order terms arep=or after dividing by equation (B.15) we arrive at
p
ρ T
= + .
p̂
ρ̂ T̂

(B.17)

170
Substituting equations (B.1) into the entropy equation (4.11) yields1ŝ + s = cpln(p̂ + p) − ln(ρ̂ + ρ) ,
γ

 


 
p
ρ
1
ln p̂ + ln 1 +− ln ρ̂ + ln 1 +
.
= cp
γ
p̂
ρ̂


(B.18)(B.19)pρ 1 and 1, we can use the series representation of ln(1 + x),
p̂
ρ̂

Since

x2 x3
+
− ··· ,23

ln(1 + x) = x −valid for −1 < x ≤ 1.

(B.20)

Keeping only the highest-order term in the series, we obtain
 

 
p
ρ
1
ln p̂ +− ln ρ̂ +.
ŝ + s = cpγp̂ρ̂(B.21)The zeroth-order terms areŝ = cp1ln p̂ − ln ρ̂ ,γ

(B.22)

which after taking a radial derivative become
dŝ
= cp
dr



1 dp̂ 1 dρ̂−γ p̂ dr ρ̂ dr
,(B.23)

while the first-order terms give
p
s
ρ
=
− .
cp
γ p̂ ρ̂(B.24)B.2

ANELASTIC EQUATION ENERGETICS

B.2.1

Derivation of the Kinetic Energy Conservation Equation

The equation describing the conservation of kinetic energy density Ek =

ρ̂u · u2

is formed by taking u· each term in the anelastic momentum equation (B.8),ρ̂∂u= 2ρ̂(u × Ω) − ρ̂(u · ∇)u − ∇p − ρg r̂ + ∇ · D̂.
∂t(B.25)Starting with the time-derivative term, we have
∂u
∂
ρ̂u ·
=
∂t
∂t



ρ̂u · u
2


=

∂Ek
,
∂t

(B.26)

171
where we have made use of the fact that

∂ ρ̂
= 0.
∂t

The Coriolis term,2ρ̂u · (u × Ω) = 0,

(B.27)

vanishes since it acts perpendicular to the motion and thus cannot perform any work.
The inertial term is ρ̂u · uρ̂u · [(u · ∇)u] = ∇ · u.
2(B.28)

This relation is most easily verified by using the following vector identity,∇ · (f A) = A · (∇f ) + f (∇ · A),(B.29)

 

 

ρ̂u · uρ̂u
to expand ∇ · u= ∇·u · u , and showing that it equals ρ̂u ·2
2[(u · ∇)u]:

∇·

ρ̂u
2




ρ̂u
1
u·u =
· [∇(u · u)] +(u · u)∇ · (ρ̂u)
2
|2
{z
}

(B.30)

vanishes by equation (B.4)= ρ̂u · [u × (∇ × u)] + ρ̂u · [(u · ∇)u] .
|{z}(B.31)

vanishes

Note that this derivation shows that the inertial term can be written as the divergenceof the kinetic energy flux.
The remaining terms are simplyu · ∇p + ρgu · r̂ + u · ∇ · D̂ .

(B.32)

Collecting terms, we obtain the kinetic energy conservation equation listed as
equation (4.34) of Chapter 4,
 



ρ̂u · u∂Ek
= −u · ∇p − ρgu · r̂ + u · ∇ · D̂ − ∇ · u.
∂t
2(B.33)

172
B.2.2

Derivation of the Internal Energy Conservation Equation

Since the time derivatives of ρ̂ and T̂ are zero, the equation describing the timedependence of internal energy density Es = ρ̂T̂ s is the anelastic internal energy equation (B.12),∂Es= ∇ · q̂eff − ρ̂T̂ (u · ∇)(ŝ + s) + Φ̂,∂twhere the time derivative term ρ̂T̂

(B.34)∂s∂∂Es
has been replaced by(ρ̂T̂ s) =
.
∂t
∂t
∂t

Additional insight may be gained by rewriting the inertial term as follows:− ρ̂T̂ (u · ∇)s


γ −1p
−
= −cp ρ̂T̂ (u · ∇)
γ p̂
T̂
#
"
T dT̂
γ − 1 p dp̂
∇Tγ − 1 ∇p−r̂ +r̂= −cp ρ̂T̂ u ·−γp̂
γ p̂2 drT̂T̂ 2 drTρ̂T dT̂
p dp̂
− ur
p̂ dr
T̂ dr


1 dŝ γ − 1 dp̂
p dp̂
+= −cp ∇ · (ρ̂T u) + u · ∇p + cp ur ρ̂T
− ur
cp dr
γ p̂ dr
p̂ dr


T
dŝ
p dp̂
+ ur
= −cp ∇ · (ρ̂T u) + u · ∇p + ur ρ̂T
−
dr
p̂ dr
T̂
dŝ
ρ dp̂
= −cp ∇ · (ρ̂T u) + u · ∇p − ur
+ ur ρ̂Tρ̂ dr| {z dr}= −cp ρ̂u · ∇T + u · ∇p + cp ur≈ 0, higher order= −cp ∇ · (ρ̂T u) + u · ∇p + ur ρgwhere ur = u · r̂, and where the mean equations (B.7), (B.15), and (B.23) as well as the
dynamic equations of state (B.17) and (B.24) have been used throughout. Using thisform of the inertial term, we obtain the internal energy conservation equation listed as
equation (4.35) of Chapter 4,∂Es= ∇ · q̂ eff + u · ∇p + ρgu · r̂ − ∇ · (cp ρ̂T u) + Φ̂.
∂t(B.35)

173
B.2.3

Derivation of the Total Energy Conservation Equation

Adding together the energy conservation equations (B.33) and (B.35) yields thetotal energy equation (4.36) of Chapter 4,
 



∂(Ek + Es )ρ̂u · u= ∇ · u · D̂ + ∇ · u+ ∇ · q̂ eff − ∇ · (cp ρ̂T u).
∂t
2(B.36)


In the above equation, the viscous energy source term ∇ · u · D̂ is arrived atby combining the diffusion terms in equations (B.33) and (B.35), which we now show
using indicial notation in Cartesian coordinates.




∂
∂Dij
∂ui
u · ∇ · D̂ + Φ̂ = ui
+ Dij=
(ui Dij ) = ∇ · u · D̂ ,
∂xj
∂xj
∂xj

(B.37)

where the we have used the equality


1
∂ui
∂ui
∂ui
Dij
= 2ρ̂νeff eij− (∇ · u)
δij
∂xj
∂xj
3
∂xj


1
2
= 2ρ̂νeff eij eij − (∇ · u)
3


1
2
= 2ρ̂νeff e : e − (∇ · u)3
= Φ̂.

(B.38)since∂ui
δij = ∇ · u∂xj(B.39)(B.40)(B.41)

Going from equation (B.38) to (B.39) is achieved by using the fact that the doubly
contracted tensor product of a symmetric tensor and an antisymmetric tensor is zero.


∂uj
1 ∂ui
is symmetric (as can be seen from+
Since the strain rate tensor eij =
2 ∂xj
∂xi
∂ui
its Cartesian representation), the producteij is equal to the product of eij and the∂xj∂ui, which is simply eij .
symmetric part of
∂xj

174

B.3

THE NUMERICAL EVOLUTION EQUATIONS
We now proceed to derive equations (4.38)–(4.40) of Chapter 4.

B.3.1

Streamfunction Formalism

Any vector A for which ∇ · A = 0 can be decomposed into poloidal and toroidal
streamfunctions W and Z, respectively:A = ∇ × (∇ × W r̂) + ∇ × Z r̂,(B.42)

such that ∇ · A = 0 is identically satisfied at all times. Because∇ · (ρ̂u) = 0,

(B.43)

ρ̂u = ∇ × (∇ × W r̂) + ∇ × Z r̂.

(B.44)

by equation (B.4), we may write

The scalar quantities W and Z are the poloidal and toroidal streamfunctions for themass flux, and it is equations describing the evolution of these quantities (along with pand s) that are solved by the ASH code. We outline the derivations of these equationsin Table B.2, while presenting detailed derivations in §B.3.4–§B.3.7.
B.3.2

Streamfunction Identities

In this section, we list some identities used in the upcoming sections. Expanding
the vector cross products from equation (B.42) in spherical polar coordinates, the threecomponents of ρ̂u are found to be


 2


1 ∂Z
1 ∂Z
1∂ W
1 ∂2W
2
+
−
θ̂ +
φ̂,
ρ̂u = − ∇⊥ W r̂ +r ∂r∂θ r sin θ ∂φr sin θ ∂r∂φ r ∂θ

(B.45)

while the curl of ρ̂u is

 2



∂
1 ∂2Z
1
∂
2
r̂ + −+ ∇⊥ W +θ̂∇ × ρ̂u = −r sin θ ∂φ ∂r 2
r ∂r∂θ
 2



∂
1 ∂2Z
1 ∂
2
φ̂.
+ ∇⊥ W ++r ∂θ ∂r 2
r sin θ ∂r∂φ
∇2⊥ Z



(B.46)

175
Table B.2: Strategy employed to obtain the ASH code evolution equations.
Apply this to thisto obtain thederivedasoperator equation evolution equation forinequation∂(∇2 W )
∂t ⊥


∂
2 ∂W
∇⊥
∂t
∂r

§B.3.4(B.123)§B.3.5(B.132)(B.8)∂(∇2 Z)∂t ⊥§B.3.6(B.140)(B.12)∂s∂t§B.3.7(B.144)

r̂·

(B.8)

∇⊥ ·

(B.8)

r̂ · ∇×[none]In the above two expressions, we have used the horizontal Laplacian operator,
∇2⊥ =

1
∂
2
r sin θ ∂θ



∂
1
∂2
sin θ
+ 2 2
,
∂θ
r sin θ ∂φ2(B.47)

defined such that the total Laplacian operator is
∇2 =

1 ∂
r 2 ∂r∂
r2+ ∇2⊥ .
∂r

(B.48)The horizontal Laplacian operator commutes with radial derivatives in the following
manner:



2
∂
∂ 2
2
∇
−
f (r, θ, φ),f (r, θ, φ) = ∇⊥
∂r ⊥
∂r r
 2

 2

∂
∂
4 ∂
6
2
2
f (r, θ, φ) = ∇⊥f (r, θ, φ),∇−+∂r 2 ⊥∂r 2 r ∂r r 2
 3

 3

24
6 ∂2
18 ∂
∂
∂
2
2
− 3 f (r, θ, φ).
f (r, θ, φ) = ∇⊥∇−+ 2
∂r 3 ⊥∂r 3 r ∂r 2
r ∂r
r


B.3.3

(B.49)(B.50)(B.51)

Components of the Divergence of the Viscous Stress Tensor

In this section, we evaluate the components of the anelastic viscous stress tensor,∇ · D̂, in preparation for their use in §B.3.4–§B.3.7.

176
B.3.3.1

Preliminaries

The anelastic stress tensor is1
by equation (B.9)
D̂ = 2ρ̂ν e − (∇ · u) δ3ρ̂= 2ν ρ̂e − (∇ · u)δ
3

 
1
∇ · (ρ̂u) − u · ∇ρ̂ δ= 2ν ρ̂e −3βsince ∇ · (ρ̂u) = 0 by equation (B.4)= 2ν ρ̂e + ρ̂ur δ ,3(B.52)

where β is defined below in equation (B.56) and where we have dropped the “eff”
subscript from νeff , hereafter using ν instead. To evaluate ∇ · D̂, we will need theindividual components of the tensor ρ̂e in spherical polar coordinates,∂(ρ̂ur )− β ρ̂ur∂r
1 ∂(ρ̂uθ ) 1ρ̂eθθ =+ ρ̂ur ,
r ∂θ
r
∂(ρ̂u
1
1
cos θ
φ)
+ ρ̂ur +ρ̂uθ ,
ρ̂eφφ =
r sin θ ∂φrr sin θ1 ∂(ρ̂uθ ) β
11 ∂(ρ̂ur )− ρ̂uθ −ρ̂uθ +,
ρ̂erθ =
2 ∂r
2
2r
2r ∂θ
1 ∂(ρ̂ur ) 1 ∂(ρ̂uφ ) β1ρ̂erφ =+− ρ̂uφ −ρ̂uφ ,
2r sin θ ∂φ
2 ∂r
2
2r
cos θ
1 ∂(ρ̂uφ )1 ∂(ρ̂uθ )−ρ̂uφ +,
ρ̂eθφ =
2r ∂θ
2r sin θ
2r sin θ ∂φρ̂err =

(B.53)

where we have again used the definition of β given in equation (B.56) below. It will also
prove useful to have handy the expression for ∇ · (ρ̂u) = 0,
1 ∂(sin θ ρ̂uθ )1 ∂(ρ̂uφ )1 ∂(r 2 ρ̂ur )
+
+
= 0,
2
r
∂r
r sin θ∂θ
r sin θ ∂φ

(B.54)

which vanishes by equation (B.4). Finally, we define the following functions of r,α=d ln ν
dr(B.55)

β=d ln ρ̂.
dr(B.56)

and177
B.3.3.2

Radial Component

For any tensor T , the radial component of its divergence isr̂ · ∇ · T=1 ∂(sin θ Trθ )1 ∂Trφ Tθθ + Tφφ
1 ∂(r 2 Trr )
+
+
−
.
2
θ ∂φ
r
|r {z∂r } |r sin θ {z ∂θ|} |r sin {z
{z
}
}
1
2
4
3

(B.57)In this section, we evaluate equation (B.57) for T = D̂ using the definition of D̂ given
in equation (B.52) along with the strain rate tensor components of equation (B.53).
Starting with the first term, we have



1 ∂
∂(ρ̂ur ) 2β
2
r 2ν
−
ρ̂ur
1 = 2
r ∂r
∂r
3
 




2β
2β
∂
2ν
2 ∂(ρ̂ur )2 ∂(ρ̂ur )−ρ̂ur +r−ρ̂ur= 2 αr
r
∂r
3
∂r
∂r
3




2ν ∂
4νβ ∂(r 2 ρ̂ur ) 4ν dβ∂(ρ̂ur )∂(ρ̂ur ) 2β−ρ̂ur + 2r2
ρ̂ur . (B.58)
− 2
−
= 2να
∂r
3
r ∂r
∂r
3r {z∂r } |3 dr
|
{z
}
|
{z
} |
{z
}
1c
1d
1a
1b
The second term is
1 ∂(ρ̂uθ ) β
1 ∂11 ∂(ρ̂ur )sin θ 2ν2 =− ρ̂uθ −ρ̂uθ +
r sin θ ∂θ
2 ∂r
2
2r
2r ∂θ



ν
1
1 ∂(ρ̂ur )∂∂(ρ̂uθ )sin θ− β ρ̂uθ − ρ̂uθ +=r sin θ ∂θ
∂r
r
r ∂θ




1 ∂(sin θ ρ̂uθ )∂(ρ̂ur )∂νβ ∂(sin θ ρ̂uθ )
ν
∂
sin θ
=ν
−
+ 2
,
∂r r sin θ∂θr sin θ {z ∂θr sin θ ∂θ
∂θ
|
}
|
|
{z
}
{z
}
2b
2a
2c
(B.59)
while the third term is 
1 ∂(ρ̂ur ) 1 ∂(ρ̂uφ ) β
1
1
∂
2ν
+
− ρ̂uφ −ρ̂uφ3 =r sin θ ∂φ2r sin θ ∂φ
2 ∂r
2
2r


1 ∂(ρ̂ur ) ∂(ρ̂uφ )1
ν
∂+− β ρ̂uφ − ρ̂uφ=r sin θ ∂φ r sin θ ∂φ
∂r
r


2
1 ∂(ρ̂uφ )ννβ ∂(ρ̂uφ )∂∂ (ρ̂ur )
+ 2 2
−
.
=ν
2
∂r r sin θ ∂φr sin θ ∂φr sin θ ∂φ
|
|
{z
}
{z
}
|
{z
}
3c
3b
3a

(B.60)

178
The last term is2ν 1 ∂(ρ̂uθ )cos θ
2β1 ∂(ρ̂uφ ) 2++ ρ̂ur +ρ̂uθ +
ρ̂urr r ∂θ
r sin θ ∂φrr sin θ3

1 ∂(sin θ ρ̂uθ )1 ∂(ρ̂uφ )
4ν
4νβ
2ν
+
− 2 ρ̂ur −ρ̂ur=−r r sin θ∂θr sin θ ∂φr
3r
2ν 1 ∂(r 2 ρ̂ur )
4νβ
4ν
ρ̂ur
=
by equation (B.54)− 2 ρ̂ur −
2
r r
∂r
r
3r


4ν
4νβ
2ν ∂(ρ̂ur ) 2+ ρ̂ur − 2 ρ̂ur −
ρ̂ur
=
r
∂r
r
r
3r
2ν ∂(ρ̂ur ) 4νβρ̂ur .
=−|r {z∂r } |3r{z }4 =−(B.61)

4a4bih
To evaluate r̂ · ∇ · D̂ = 1 + 2 + 3 + 4 by equation (B.57), we combine
pieces from equations (B.58)–(B.61) as indicated in the following expressions:
1 ∂(sin θ ρ̂uθ )1 ∂(ρ̂uφ )++ 1c + 4b
1c + 2b + 3c + 4b = −νβr sin θ∂θ
r sin θ ∂φ
1 ∂(r 2 ρ̂ur )+ 1c + 4b= νβ 2by equation (B.54)r∂r
νβ ∂(r 2 ρ̂ur )
+ 4b
=− 2
3r
∂r
νβ ∂(ρ̂ur ) 2νβ=−−ρ̂ur + 4b3 ∂r
3r
νβ ∂(ρ̂ur ) 2νβ−ρ̂ur ,=−(B.62)
3 ∂r
r




1 ∂(sin θ ρ̂uθ )1 ∂(ρ̂uφ )∂++ 1b
1b + 2a + 3a = ν∂r r sin θ∂θr sin θ ∂φ
∂ 1 ∂(r 2 ρ̂ur )
+ 1b
= −ν
by equation (B.54)∂r r 2∂r 2
2∂ (ρ̂ur ) 2 ∂(ρ̂ur )+ 2 ρ̂ur ,(B.63)
=ν
+
∂r 2
r ∂r
r

2c + 3b = ν∇2⊥ (ρ̂ur )by equation (B.47).

(B.64)The remaining terms are unchanged:∂(ρ̂ur ) 2β
2ν ∂(ρ̂ur )4ν dβ−ρ̂ur −
ρ̂ur +.
1a + 1d + 4a = 2να
∂r
3
3 dr
r ∂r

(B.65)

179
Combining equations (B.62)–(B.65) and grouping by derivatives of ρ̂ur , we obtain
(


i
h
∂ 2 (ρ̂ur )β 4 ∂(ρ̂ur )++2α−r̂ · ∇ · D̂ = ν
∂r 2
3
r
∂r
)
(B.66)

4αβ 4 dβ 2β22+ ∇⊥ −−−+ 2 ρ̂ur .
3
3 dr
r
r
The last step is to eliminate the radial mass flux ρ̂ur in favor of the streamfunction
W by substituting ρ̂ur = −∇2⊥ W , by equation (B.45). Using the commutation identities
of equations (B.49) and (B.50), equation (B.66) thus becomes
(


h
i
β ∂W
∂2W
2
r̂ · ∇ · D̂ = ν∇⊥
+
2α
−
∂r 2
3 ∂r
 )

4αβ 4α 4 ∂β 4β
2
−
−
−
W .
+ ∇⊥ −
3r
3 ∂r
3r
B.3.3.3(B.67)

Polar Component

For any tensor T , the polar component of its divergence is1 ∂(sin θ Tθθ )1 ∂Tθφ Trθcos θ
1 ∂(r 2 Trθ )
Tφφ .
+
+
+
−
θ̂ · ∇ · T = 2θ ∂φrr sin{zθ }|r {z∂r } |r sin θ {z ∂θ|{z}} |r sin {z
|
}
1
2
4
5
3
(B.68)In this section, we evaluate (B.68) for T = D̂ using the definition of D̂ given in equation(B.52) along with the strain rate tensor components of equation (B.53). Starting withthe first term, we have



4
∂
1 ∂  2r 2ν ρ̂erθ = ν 2α + + 2(ρ̂erθ )
1 = 2
r ∂r
r
∂r



1 ∂(ρ̂uθ ) β
4
11 ∂(ρ̂ur )∂+2− ρ̂uθ −ρ̂uθ += 2να ρ̂erθ + ν
r
∂r
2 ∂r
2
2r
2r ∂θ



 
 2
2
dβ
1 ∂
∂
∂
+
(ρ̂uθ ) +ρ̂uθ + ν(ρ̂uθ )= 2να ρ̂erθ − ν β+| {z }∂r r
dr∂r 2 r ∂r
|
{z
} |
{z
}
1a
1c
1b
 2


∂
1 ∂
ν
ν
+
(ρ̂ur ) − 2 ρ̂uθ .
+r∂r ∂θ r ∂θr{z
} | {z }|1e
1d(B.69)

180
The second term is
1 ∂(ρ̂uθ ) 1
2ν ∂
β
sin θ
2 =
+ ρ̂ur + ρ̂urr sin θ ∂θ
r ∂θ
r
3


∂(ρ̂uθ )2ν ∂(sin θ ρ̂ur )2νβ ∂(sin θ ρ̂ur )
2ν ∂
sin θ
+ 2
+
,
= 2
r sin θ ∂θ
∂θ
r
sin
θ
∂θ
3r
sin θ {z ∂θ
|
|
{z
}
}
|
{z
}
2c
2b
2a

(B.70)

while the third term isν
1 ∂(ρ̂uθ )∂ 1 ∂(ρ̂uφ )
cos θ
3 =
−
ρ̂uφ +
r sin θ ∂φ r ∂θ
r sin θr sin θ ∂φ
2cos θ ∂(ρ̂uφ )1∂ 2 (ρ̂uθ )1 ∂ (ρ̂uφ )
− 2 2
+ 2 2
=ν 2
r sin θ ∂θ ∂φr sin θ ∂φr sin θ ∂φ22
1 ∂(ρ̂uφ )1ν ∂∂ (ρ̂uθ )+ 2 2.
=r ∂θ r sin θ ∂φr sin θ ∂φ2
{z
}
|
{z
} |
3b
3a

(B.71)The last two terms are
4 =ν ∂(ρ̂uθ ) νβνν ∂(ρ̂ur )ρuˆ θ − 2 ρ̂uθ + 2
−
r
∂r
r
r
r
| {z } | {z } | {z } | {z∂θ }
4a

4b

4c

(B.72)

4d

and
5 =−

2ν cos θ ∂(ρ̂uφ ) 2ν cos θ2ν cos2 θ2νβ cos θρ̂uρ̂ur .
−−ρ̂uθ −
r
2
2
2
2
2
∂φ
r
sin
θ
r sin θ{z
} |r sin{zθ}} | 3r sin{zθ
|
{z
} |
5b
5c
5d
5a

(B.73)ih
To evaluate θ̂ · ∇ · D̂ = 1 + 2 + 3 + 4 + 5 by equation (B.68), we combine
pieces from equations (B.69)–(B.73) as indicated in the following expressions:
11 ∂(ρ̂ur )∂(ρ̂uθ )− β ρ̂uθ − ρ̂uθ +,1a = να
∂r
r
r ∂θ


∂(ρ̂uθ ) dβ
2β ∂(ρ̂ur )3β−β−ρ̂uθ −ρ̂uθ ,
1b + 2c + 4b + 5d = ν
3r ∂θ
∂r
dr
r


(B.74)(B.75)

181
2a+ 3b + 4a = ν∇2 (ρ̂uθ ),
2
i
2a
ν ∂ h
ν
+ 3a + 4d =∇ · (ρ̂u) + 2 2 ρ̂uθ ,1d +2r ∂θ | {z } r sin θ
1c +(B.76)(B.77)=02ν
1e + 4c + 5c = − 2 2 ρ̂uθ ,r sin θ2ν ∂(ρ̂ur ),2b + 5b = 2r∂θ(B.78)(B.79)

along with2ν cos θ ∂(ρ̂uφ ),5a = − 2 2r sin θ ∂φ

(B.80)

which is left unchanged. Combining equations (B.74)–(B.80) and regrouping, we obtain
(


i
h
2β 2 ∂(ρ̂ur )∂(ρ̂uθ )12α+++ (α − β)θ̂ · ∇ · D̂ = ν ∇ (ρ̂uθ ) +
r
3
r
∂θ
∂r
)


α dβ 3β1
2 cos θ ∂(ρ̂uφ )ρ̂uθ
− αβ + +
+
+ 2 2
− 2 2
r
dr
r
r sin θ ∂φr sin θ(B.81)The polar component will be used in combination with the azimuthal component in§B.3.3.5 and §B.3.3.6.

B.3.3.4

Azimuthal Component

For any tensor T , the azimuthal component of its divergence isφ̂ · ∇ · T

=1 ∂Tφφ Trφ 2 cos θ
1 ∂(r 2 Trφ ) 1 ∂Tθφ
Tθφ .
+
+
+
+
2
∂θ } r sin θ ∂φrr sin θ|r
{z∂r } |r {z{z} |{z} | {z }
|
1
2
4
5
3

(B.82)In this section, we evaluate equation (B.82) for T = D̂ using the definition of D̂ given
in equation (B.52) along with the strain rate tensor components of equation (B.53).

182
Starting with the first term, we have
41 ∂  2
∂
=
ν
2α
+
r
(ρ̂erφ )
2ν
ρ̂e
+
2
rφ
r 2 ∂r
r
∂r



∂
1
1 ∂(ρ̂ur ) 1 ∂(ρ̂uφ ) β
4+2+− ρ̂uφ −ρ̂uφ= 2να ρ̂erφ + νr∂r
2r sin θ ∂φ
2 ∂r
2
2r


 
 2

2
∂
dβ
∂
1 ∂
+
(ρ̂uφ ) +ρ̂uφ + ν(ρ̂uφ )+= 2να ρ̂erφ − ν β∂r r
dr∂r 2 r ∂r| {z }
{z
} |
{z
}
|
1a
1c
1b
(B.83)
 2

1 ∂
ν
ν
∂
+
(ρ̂ur ) − 2 ρ̂uφ .
+r sin θ ∂r ∂φ r ∂φr{z
} | {z }
|
1e
1d

1 =

The second term isν ∂ 1 ∂(ρ̂uφ )1 ∂(ρ̂uθ )
cos θ
2 =
−
ρ̂uφ +r ∂θ r ∂θ
r sin θr sin θ ∂φ

 2

 2
cos θ ∂cos θ ∂νν∂∂ν(ρ̂uφ ) + 2 2 ρ̂uφ + 2−(ρ̂uθ ),
−
= 2
r
∂θ 2
sin θ ∂θr sin{zθ } r sin θ ∂θ ∂φ sin θ ∂φ
{z
} |
{z
}
|
|
2b
2a
2c
(B.84)
while the third term is


2ν ∂
cos θ
β
1 ∂(ρ̂uφ ) 1+ ρ̂ur +ρ̂uθ + ρ̂ur
3 =r sin θ ∂φ r sin θ ∂φrr sin θ3
2
2ν ∂(ρ̂ur ) 2ν cos θ ∂(ρ̂uθ )2νβ ∂(ρ̂ur )2ν ∂ (ρ̂uφ )
+ 2
+ 2 2
+
.
= 2 2
r sin θ ∂φ3r sin θ ∂φr sin θ ∂φ2
r sin θ ∂φ
|
{z
} |
{z
} |
{z
} |
{z
}
3a
3c
3b
3d

(B.85)The last two terms are
4 =

ν

ν
∂(ρ̂ur ) ν ∂(ρ̂uφ ) νβρ̂uφ − 2 ρ̂uφ
+
−
∂φ
r ∂r
r
r
{z
} | {z } | {z } | {z }
4c
4b
4d
4a

r 2 sin θ|(B.86)

and5 =2ν cos θ ∂(ρ̂uφ ) 2ν cos2 θ2ν cos θ ∂(ρ̂uθ )− 2 2 ρ̂uφ + 2 2.
2θ ∂φ|r sin θ{z ∂θ } |r sin{zθ} |r sin {z
}
5a
5b
5c

(B.87)

183hi
To evaluate φ̂ · ∇ · D̂ = 1 + 2 + 3 + 4 + 5 by equation (B.82), we combine
pieces from equations (B.83)–(B.87) as indicated in the following expressions:
1 ∂(ρ̂ur ) ∂(ρ̂uφ )1
1a = να+− β ρ̂uφ − ρ̂uφ ,r sin θ ∂φ∂rr
∂(ρ̂uφ ) dβ
3β
2β ∂(ρ̂ur )−β−ρ̂uφ −
ρ̂uφ ,
1b + 3d + 4c = ν
3r sin θ ∂φ
∂r
dr
r


(B.88)(B.89)

3a+ 4b + 5a = ν∇2 (ρ̂uφ ),(B.90)
2i3a
ν∂ h+ 4a + 5c =∇ · (ρ̂u) = 0,
(B.91)
1d + 2c +2r sin θ ∂φhiν1e + 2b + 4d + 5b = 2 2 1 − 2 sin2 θ − 2 cos2 θ (ρ̂uφ )r sin θν= − 2 2 ρ̂uφ .
(B.92)
r sin θ1c + 2a +The remaining terms are unchanged:
2 ∂(ρ̂ur )2 cos θ ∂(ρ̂uθ ).
3b + 3c = ν 2+ 2 2
r sin θ ∂φr sin θ ∂φ


(B.93)

Combining equations (B.88)–(B.93) and regrouping, we obtain(ihφ̂ · ∇ · D̂ = ν ∇2 (ρ̂uφ ) +∂(ρ̂uφ )2β 2 ∂(ρ̂ur )1α+++ (α − β)r sin θ
3
r
∂φ
∂r
)


2 cos θ ∂(ρ̂uθ )α dβ 3β
1
ρ̂uφ
+ 2 2
− αβ + +
+
+ 2 2
r
dr
r
r sin θ ∂φr sin θ(B.94)

B.3.3.5

Perpendicular Divergencehi
We now use the results of the previous two sections to evaluate ∇⊥ · ∇ · D̂ ,which serves as the diffusion term in the P equation. By definition,∇⊥ · A =1 ∂Aφ
1 ∂(sin θ Aθ )+,r sin θ∂θ
r sin θ ∂φ

(B.95)

1 ∂(r 2 Ar )+ ∇⊥ · A.
r 2 ∂r

(B.96)

such that
∇·A=

184
Thus,
i
h
∇⊥ · ∇ · D̂ =

 


∂
1
∂ 
sin θ θ̂ · ∇ · D̂ +φ̂ · ∇ · D̂ .
r sin θ ∂θ∂φ(B.97)The polar and azimuthal components of ∇ · D̂ are given by equations (B.81) and(B.94). We rewrite them here and label the individual terms before combining them as
per equation (B.97):
(


i
h
2β 2 ∂(ρ̂ur )1∂(ρ̂uθ )2α+++ (α − β)θ̂ · ∇ · D̂ = ν ∇ (ρ̂uθ ) +| {z } r
3r∂θ{z ∂r }
|
{z
} |
1a
1c
1b
)


2 cos θ ∂(ρ̂uφ )α dβ 3β
1
ρ̂uθ
− 2 2
− αβ + +
+
+ 2 2
r
dr
r
r sin θ ∂φr sin θ
|
{z
} |
{z
}
1d
1e
(B.98)andihφ̂ · ∇ · D̂ = ν(∂(ρ̂uφ )2β 2 ∂(ρ̂ur )1α+∇ (ρ̂uφ ) ++ (α − β)+3r∂φ| {z } r sin θ{z ∂r }
{z
} |
|
2a
2c
2b
)


2 cos θ ∂(ρ̂uθ )α dβ 3β1+ 2 2− αβ + +ρ̂uφ .
+
+ 2 2
r
dr
r
r sin θ ∂φr sin θ
|
{z
} |
{z
}
2d
2e
2

(B.99)

185
Combining the terms containing 1a and 2a yields:()
i ∂ 2aν∂ hsin θ 1a +r sin θ ∂θ
∂φ




∂ 
∂  2
ν
2
=
sin θ ∇ (ρ̂uθ ) +∇ (ρ̂uφ )r sin θ ∂θ∂φ( ∂(ρ̂uφ )2 cos θ ∂ 2 (ρ̂uθ )ν
2 ∂ (sin θ ρ̂uθ )−=∇+r sin θ
∂θ
∂φ
r2
∂θ 2
)


2
2 cos θ ∂ 2 (ρ̂uθ )∂(ρ̂uθ ) 2 cos θ
4 sin θ++− 2ρ̂uθ − 2 2r2
r sin θ∂θ
r2
r sin θ ∂φ2(hi 2 cos θ ∂ 2 (ρ̂u )2 cos θ ∂ 2 (ρ̂uθ )
ν
φ
− 2 2
=
sin θ ∇2 r ∇⊥ · (ρ̂u) + 2
r sin θr sin θ ∂θ ∂φr sin θ ∂φ2)∂(ρ̂uφ )cos θ
1 ∂(ρ̂uθ )1− 2 2 ρ̂uθ − 2 2+ 2.
(B.100)r sin θ ∂θr sin θr sin θ ∂φ
Combining the terms containing 1b and 2b yields:()
i ∂ 2bν∂ h
2β 2∇2⊥ (ρ̂ur ).
sin θ 1b +=ν α++r sin θ ∂θ∂φ3r

(B.101)

Combining the terms containing 1c and 2c yields:()
i ∂ 2c∂ hν(α − β) ∂ ∂(sin θ ρ̂uθ ) ∂(ρ̂uφ )ν
sin θ 1c ++=r sin θ ∂θ∂φr sin θ ∂r
∂θ
∂φ
h
i
ν(α − β) ∂=r∇⊥ · (ρ̂u) .
(B.102)r∂r
Combining the terms containing 1d and 2d yields:()
i ∂ 2dν∂ hsin θ 1d +r sin θ ∂θ
∂φ





∂
2 cos θ ∂(ρ̂uφ )∂2 cos θ ∂(ρ̂uθ )ν−+=r sin θ∂θ r 2 sin θ ∂φ∂φ r 2 sin2 θ ∂φ∂(ρ̂uφ )2 cos θ ∂ 2 (ρ̂uφ )2
2 cos θ ∂ 2 (ρ̂uθ )
ν
− 2
+ 2 2
+ 2 2
.
=
r sin θr sin θ ∂θ ∂φr sin θ ∂φr sin θ ∂φ2(B.103)

186
Finally, combining the terms containing 1e and 2e yields:()
i ∂ 2e∂ hνsin θ 1e +−r sin θ ∂θ∂φ
α dβ 3β= −ν αβ + ++∇⊥ · (ρ̂u)
r
dr
r
 



ρ̂uφ
∂
ρ̂uθ
∂
ν
+
−
r sin θ ∂θ r 2 sin θ∂φ r 2 sin2 θ
α dβ 3β+∇⊥ · (ρ̂u)= −ν αβ + +
r
dr
r


∂(ρ̂uφ )1 ∂(ρ̂uθ )
1
cos θ
ν
− 2
.
+ 2 2 ρ̂uθ − 2 2+r sin θr sin θ ∂θr sin θr sin θ ∂φSumming together the terms in equations (B.100)–(B.104) yields
(

i
h
i 
1 2h
α dβ 3β∇⊥ · ∇ · D̂ = ν∇ r ∇⊥ · (ρ̂u) − αβ + ++∇⊥ · (ρ̂u)
r
dr
r
{z
} |
|r
{z
}
3a
3b
)

i 
α−β ∂ h2β 22r∇⊥ · (ρ̂u) + α ++∇⊥ (ρ̂ur ) .
+
3
r
} |
| r ∂r {z
{z
}
3c
3d

(B.104)(B.105)Using equation (B.96) and the fact that ∇ · (ρ̂u) = 0, we can make the substitution
1 ∂(r 2 ρ̂ur )∂(ρ̂ur ) 2=−− ρ̂ur∇⊥ · (ρ̂u) = − 2
r
∂r
∂r
r

(B.106)such that only ρ̂ur remains, and then use equation (B.45) to eliminate ρ̂ur in favor of W .
Once in terms of W , the commutation relations (B.49)–(B.51) are used to interchangethe ∇2⊥ operator with r-derivatives. We now examine each term in equation (B.105)
above, starting with the first term:i
1 h3a = ∇2 r ∇⊥ · (ρ̂u)
r
1 2 ∂(ρ̂ur )+ 2 ρ̂ur=− ∇ r
r
∂r





∂(ρ̂ur )
1 1 ∂
2 ∂
2
r
+ ∇⊥ r+ 2 ρ̂ur=−r r 2 ∂r
∂r
∂r

 3
2 2
6 ∂(ρ̂ur )∂ (ρ̂ur ) 6 ∂ 2 (ρ̂ur )2 ∂(ρ̂ur )+ ∇⊥+ ∇⊥ (ρ̂ur )
+
+ 2
=−
∂r 3
r ∂r 2
r
∂r
∂r
r

 3
∂ W
∂W
.
(B.107)
+ ∇2⊥
= ∇2⊥
3
∂r
∂r

187
The next two terms are
α dβ 3β∇⊥ · (ρ̂u)
3b = − αβ + +
+
r
dr
r



α dβ 3β ∂(ρ̂ur ) 2++ ρ̂ur= αβ + +
r
dr
r
∂r
r


α dβ 3β ∂W+= −∇2⊥ αβ + +
r
dr
r
∂r

(B.108)andi
α−β ∂ hr∇⊥ · (ρ̂u)
r ∂r 
∂ 2 (ρ̂ur ) 3 ∂(ρ̂ur )+= −(α − β)
∂r 2
r ∂r


 2
1 ∂W
∂ W
2
.
= ∇⊥ (α − β)
−
∂r 2
r ∂r

3c =

(B.109)Lastly, the fourth term is simply




2β 2
2β 2
2
2
+
∇⊥ (ρ̂ur ) = −∇⊥ α ++∇2⊥ W.
3d = α +3r
3r(B.110)Combining equations (B.107)–(B.110) and regrouping, we obtain
(
h
i
∂3W
∂2W
2
∇⊥ · ∇ · D̂ = ν∇⊥
+
(α
−
β)
∂r 3
∂r 2


2α dβ 2β
2 ∂W
+
+
− ∇⊥
− αβ +
r
dr
r
∂r
)


2β 2
+
∇2⊥ W .
− α+3r
B.3.3.6(B.111)

Radial Component of Curlhi
We now use the results of §B.3.3.3 and §B.3.3.4 to evaluate r̂ · ∇ × ∇ · D̂ ,which serves as the diffusion term in the Z equation. By definition,hir̂ · ∇ × ∇ · D̂ =

 


∂
1
∂ 
sin θ φ̂ · ∇ · D̂ −θ̂ · ∇ · D̂ .
r sin θ ∂θ∂φ

(B.112)The polar and azimuthal components of ∇ · D̂ are given by equations (B.81) and(B.94). We rewrite them here and label the individual terms before combining them as

188
per equation (B.112):
(


i
h
2β 2 ∂(ρ̂ur )1∂(ρ̂uθ )2α++θ̂ · ∇ · D̂ = ν ∇ (ρ̂uθ ) ++ (α − β)| {z } r
3r∂θ{z ∂r }
|
{z
} |
1a
1c
1b
)


1
2 cos θ ∂(ρ̂uφ )α dβ 3β
ρ̂uθ
+
+ 2 2
− 2 2
− αβ + +r
drrr sin θ ∂φr sin θ
{z
} |
|
{z
}
1d
1e
(B.113)andihφ̂ · ∇ · D̂ = ν(∂(ρ̂uφ )2β 2 ∂(ρ̂ur )1∇ (ρ̂uφ ) +α+++ (α − β)3r∂φ| {z } r sin θ{z ∂r }
|
{z
} |
2a
2c
2b
)


2 cos θ ∂(ρ̂uθ )α dβ 3β1
ρ̂uφ .
+ 2 2− αβ + +
+
+ 2 2
r
dr
r
r sin θ ∂φr sin θ
|
{z
} |
{z
}
2d
2e
2

(B.114)
Combining the terms containing 1a and 2a yields:()
i ∂ 1aν∂ hsin θ 2a −r sin θ ∂θ
∂φ




∂ 
∂  2
ν
2
=
sin θ ∇ (ρ̂uφ ) −∇ (ρ̂uθ )r sin θ ∂θ∂φ( ∂(ρ̂uθ )2 cos θ ∂ 2 (ρ̂uφ )ν
2 ∂ (sin θ ρ̂uφ )−−∇=r sin θ
∂θ
∂φ
r2
∂θ 2
)


∂(ρ̂uφ ) 2 cos θ22 cos θ ∂ 2 (ρ̂uφ )4 sin θ++− 2ρ̂uφ − 2 2r2
r sin θ∂θ
r2
r sin θ ∂φ2(hi 2 cos θ ∂ 2 (ρ̂u )2 cos θ ∂ 2 (ρ̂uφ )
ν
θ
− 2 2
=
sin θ ∇2 r r̂ · ∇ × ρ̂u − 2
r sin θr sin θ ∂θ ∂φr sin θ ∂φ2)cos θ
1 ∂(ρ̂uφ )1∂(ρ̂uθ )− 2 2 ρ̂uφ + 2 2+ 2.
(B.115)r sin θ ∂θr sin θr sin θ ∂φ
Combining the terms containing 1b and 2b yields:()
i ∂ 1bν∂ hsin θ 2b −= 0.
r sin θ ∂θ∂φ

(B.116)

189
Combining the terms containing 1c and 2c yields:()
i ∂ 1cν∂ hν(α − β) ∂ ∂(sin θ ρ̂uφ ) ∂(ρ̂uθ )sin θ 2c −=−r sin θ ∂θ∂φr sin θ ∂r
∂θ
∂φ
h
i
ν(α − β) ∂=r r̂ · ∇ × ρ̂u .
(B.117)r∂r
Combining the terms containing 1d and 2d yields:()
i ∂ 1d∂ hνsin θ 2d −r sin θ ∂θ
∂φ
 



∂
2 cos θ ∂(ρ̂uθ )∂2 cos θ ∂(ρ̂uφ )ν
+=r sin θ ∂θ r 2 sin θ ∂φ∂φ r 2 sin2 θ ∂φ
2 cos θ ∂ 2 (ρ̂uθ )2
2 cos θ ∂ 2 (ρ̂uφ )ν∂(ρ̂uθ )
− 2 2
+ 2 2
=
r sin θ r 2 sin θ ∂θ ∂φr sin θ ∂φr sin θ ∂φ2
Finally, combining the terms containing 1e and 2e yields:()
i ∂ 1eν∂ hsin θ 2e −r sin θ ∂θ∂φ
α dβ 3β= −ν αβ + ++r̂ · ∇ × ρ̂u
r
dr
r

 


ρ̂uφ
∂
ν
∂
ρ̂uθ
+
−
r sin θ ∂θ r 2 sin θ∂φ r 2 sin2 θ
α dβ 3β+= −ν αβ + +r̂ · ∇ × ρ̂u
r
dr
r


cos θ
1 ∂(ρ̂uφ )1∂(ρ̂uθ )ν− 2.
ρ̂uφ + 2 2+
r sin θ r 2 sin2 θr sin θ ∂θr sin θ ∂φ

(B.118)(B.119)Summing together the terms in equations (B.115)–(B.119) yields
(

h
i
i 
1 2h
α dβ 3βr̂ · ∇ × ∇ · D̂ = ν∇ r r̂ · ∇ × ρ̂u − αβ + ++r̂ · ∇ × ρ̂u
r
r
dr
r
)
i
α−β ∂ h+r r̂ · ∇ × ρ̂u .
r ∂r(B.120)
We now eliminate ρ̂u in favor of Z by substituting r̂ · ∇ × ρ̂u = −∇2⊥ Z by equa-190tion (B.46) to obtain
(


i
h
1 2h 2 iα dβ 3β∇ r∇⊥ Z − αβ + ++r̂ · ∇ × ∇ · D̂ = −ν
∇2⊥ Z
r
r
dr
r
)
α−β ∂ h 2 i
r∇⊥ Z
+
r ∂r
( 


h
i
1 1 ∂
2 ∂
2
2
r
+
∇
Z
r∇
= −ν
⊥
⊥
r r 2 ∂r∂r)

α−β ∂ h 2 i
α dβ 3β2Z+− ∇⊥ αβ + +
+
r∇⊥ Z
r
dr
r
r ∂r
(

4 ∂
2
∂2
2
+
+ ∇⊥ ∇2⊥ Z+= −ν∂r 2 r ∂r r 2
)




∂
α
dβ
3β
1
Z + (α − β)∇2⊥ Z.
− ∇2⊥ αβ + +
+
+
r
dr
r
∂r r
Using the commutation identities (B.49) and (B.50), we finally obtain
(
i
h
∂Z
∂2Z
r̂ · ∇ × ∇ · D̂ = −ν∇2⊥+ (α − β)
2
∂r
∂r
 )

2α dβ 2β2+++ ∇⊥ Z .
− αβ +
r
dr
r
B.3.4

(B.121)

Derivation of the W Equation

The evolution equation for W is obtained by taking r̂· each term in the anelastic
momentum equation (B.8). Applying r̂· to the time-derivative term gives
 
∂
∂W
∂
∂u
= ( r̂ · ρ̂u) = − (∇2⊥ W ) = −∇2⊥,ρ̂ r̂ ·
∂t
∂t
∂t
∂t

(B.122)where we have used equation (B.45). As a result, a schematic representation of the Wequation is−∇2⊥∂W= WPG + WGRAV + WDIFF + WCOR + WADV .
∂t(B.123)

We now compute the five terms on the right-hand side.
The pressure gradient and gravitational force terms are simply∂p
WPG = r̂ · −∇p = − ,∂r(B.124)

191
and


WGRAV = r̂ · −ρg r̂ = −ρg.

(B.125)Equation (B.67) gives the diffusion term in terms of W :ih
WDIFF = r̂ · ∇ · D̂
(


 )

2W
4α
4
4β
∂
β
4αβ
∂W
∂β
+ ∇2⊥ −−−−W ,+ 2α −= νeff ∇2⊥
∂r 2
3 ∂r
3
r
3 ∂r
3r
(B.126)
where α =d ln νeffd ln ρ̂and β =. The Coriolis term can also be expanded in terms ofdrdr

the streamfunctions:
WCOR = 2ρ̂ r̂ · (u × Ω)
= 2Ω sin θ ρ̂uφ∂Z2Ω ∂ 2 W− sin θ.
=r ∂r∂φ∂θby equation (B.45)(B.127)

Finally, the advection term can be expanded in terms of the three components of u:hi
WADV = −ρ̂ r̂ · (u · ∇)u
#
"
u2φ
u2θ
−
= −ρ̂ (u · ∇)ur −
r
r
#
"

u2φ
u2θ
uφ ∂
∂
uθ ∂
ur −
+
+
−
.
= −ρ̂ ur∂rr ∂θ r sin θ ∂φ
r
r
B.3.5

(B.128)Derivation of the P Equation

The evolution equation for P is obtained by taking ∇⊥ · each term in the anelastic
momentum equation (B.8), where the expression ∇⊥ · A equals∇⊥ · A =1 ∂(sin θ Aθ )
1 ∂Aφ
+
,
r sin θ∂θ
r sin θ ∂φ

(B.129)1 ∂(r 2 Ar )+ ∇⊥ · A.
r2∂r(B.130)such that
∇·A=

192
The time-derivative term simplifies to


i
∂h
∂u
=
∇⊥ · (ρ̂u)∇⊥ · ρ̂∂t∂t
1 ∂(r 2 ρ̂ur )∂∇ · (ρ̂u) − 2
by equation (B.130)
=
∂t
r
∂r


∂ ∂(ρ̂ur ) 2ρ̂ur=−by equation (B.4)
+
∂t
∂r
r


∂ ∂(∇2⊥ W ) 2 2=
by equation (B.45)+ ∇⊥ W
∂t
∂r
r


∂W
2 ∂
.
by equation (B.49)(B.131)= ∇⊥∂t ∂rTherefore, the P equation is represented by


∂W
2 ∂
= PPG + PDIFF + PCOR + PADV .
∇⊥
∂t ∂r

(B.132)

Note that there is no gravity term since the quantity ∇⊥ · (ρg r̂) vanishes. We now list
the remaining terms on the right-hand side in order.
The pressure gradient term simplifies to



1 ∂
∂p
r2
= −∇2⊥ p,
PPG = −∇⊥ · (∇p) = − ∇ · (∇p) − 2r ∂r∂r(B.133)by equations (B.47) and (B.130). The diffusion term is given by equation (B.111):ih
PDIFF = ∇⊥ · ∇ · D̂
(


∂2W
2α dβ 2β
∂3W
2
2 ∂W
+
+
− ∇⊥
+ (α − β) 2 − αβ += νeff ∇⊥
∂r 3
∂r
r
dr
r
∂r
)


2β 2
+
∇2⊥ W .
(B.134)− α+
3r
The Coriolis term can be simplified by using


1 ∂Z
1 ∂Z
∇ × ∇ × Z r̂ = ∇ ×θ̂ −φ̂r sin θ ∂φ
r ∂θ





∂Z
1
∂
∂
1 ∂Z
1
sin θ
+ 2 2
r̂
=− 2
r sin θ ∂θ∂θr sin θ ∂φ r sin θ ∂φ
1 ∂2Z
1 ∂2Z
+
θ̂ +
φ̂
r ∂r∂θ
r sin θ ∂r∂φ


∂2Z
∂Z
2
−
r̂
(B.135)= −(∇⊥ Z) r̂ + ∇
∂r
∂r 2

193
and




∂2W
∂W
2
−
∇ × ∇ × ∇ × W r̂ = ∇ × −(∇⊥ W ) r̂ + ∇
r̂
∂r
∂r 2




∂2W
∂2W
∂
1 ∂
1
2
2
∇⊥ W +∇⊥ W +θ̂ +φ̂
=−r sin θ ∂φ
∂r 2
r ∂θ
∂r 2
(B.136)so thathi
PCOR = ∇⊥ · 2ρ̂u × Ω

i
h
i ∂
2 h
+
r̂ · (2ρ̂u × Ω)
= 2Ω · ∇ × (ρ̂u) −∂r rhi= 2Ω(cos θ r̂ − sin θ θ̂) · ∇ × ∇ × ∇ × W r̂ + ∇ × ∇ × Z r̂


i
2 h
∂
+
2Ω sin θ ρ̂uφ
−
∂r r


 
∂ 2 ∂Wsin θ ∂ 2 Z
1
2
2∇⊥ + 2− cos θ ∇⊥ Z −
= 2Ω
r
∂r
∂φ
r ∂r∂θ

 2


2
2
2Ω ∂ W∂ W∂+− sin θ.
(B.137)
−
∂r r
r
∂r∂φ
∂r∂θ
The advection term can be expanded in terms of the three components of u:hi
PADV = −∇⊥ · ρ̂(u · ∇)u hi ∂ 
h
i
ρ̂
∂
=−
sin θ θ̂ · (u · ∇)u +φ̂ · (u · ∇)ur sin θ ∂θ
∂φ
( "

#
u2φ cos θ
ur uθ
∂
ρ̂
sin θ (u · ∇)uθ +−=−r sin θ ∂θrr sin θ
)ur uφ uθ uφ cos θ∂(u · ∇)uφ +++∂φrr sin θ( "#
u2φ cos θuφ ∂uθρ̂∂uθ∂uθ ∂uθ
ur uθ
=−sin θ ur+++−r sin θ ∂θ∂rr ∂θr sin θ ∂φrr sin θ
)∂uφ uθ ∂uφuφ ∂uφ ur uφ uθ uφ cos θ∂ur. (B.138)
+
+
+
+
+
∂φ
∂r
r ∂θ
r sin θ ∂φrr sin θB.3.6

Derivation of the Z Equation

The evolution equation for Z is obtained by taking r̂ · ∇× each term in the
anelastic momentum equation (B.8). Applying r̂ · ∇× to the time-derivative term

194
gives
r̂ · ∇ ×∂(ρ̂u)∂Z∂∂= ( r̂ · ∇ × ρ̂u) = − (∇2⊥ Z) = −∇2⊥
,
∂t
∂t
∂t
∂t

(B.139)where we have used equation (B.46). As a result, a schematic representation of the Zequation is−∇2⊥∂Z= ZDIFF + ZCOR + ZADV .
∂t(B.140)Note that there is no pressure gradient term since ∇ × ∇p = 0, and that the gravity
term vanishes since ∇ × (ρg r̂) has no r̂-component. We now compute the three termson the right-hand side in order.
The diffusion term is given by equation (B.121):ih
ZDIFF = r̂ · ∇ × ∇ · D̂( )2α dβ 2β∂Z∂2Z22− αβ ++++ ∇⊥ Z .
+ (α − β)= −νeff ∇⊥
∂r 2
∂r
r
dr
r

(B.141)The Coriolis term simplifies tohi
ZCOR = r̂ · ∇ × (2ρ̂u) × Ωhi= 2 r̂ · (Ω · ∇)(ρ̂u)
∂(ρ̂ur ) sin θ ∂(ρ̂ur ) ρ̂uθ sin θ−+= 2Ω cos θ
∂r
r
∂θ
r


∂
sin θ ∂sin θ ∂ 2 W
1 ∂Z
2
2= 2Ω − cos θ (∇⊥ W ) +(∇ W ) + 2+.
∂rr ∂θ ⊥r ∂r ∂θ r 2 ∂φ

(B.142)

195
The advection term can be expanded in terms of the three components of u:hi
ZADV = − r̂ · ∇ × ρ̂(u · ∇)u( )hi ∂ hi∂ρ̂sin θ φ̂ · (u · ∇)u −θ̂ · (u · ∇)u=−r sin θ ∂θ
∂φ
( 


ur uφ uθ uφ cos θ∂ρ̂sin θ (u · ∇)uφ ++=−r sin θ ∂θrr sin θ"#)u2φ cos θ
ur uθ
∂
−
−
(u · ∇)uθ +∂φrr sin θ( uφ ∂uφ ur uφ uθ uφ cos θ∂uφ uθ ∂uφρ̂∂sin θ ur++++=−r sin θ ∂θ∂rr ∂θr sin θ ∂φrr sin θ"#)u2φ cos θuφ ∂uθuθ ∂uθ
ur uθ
∂
∂uθ
+
+
+
−
−
ur
(B.143)∂φ∂rr ∂θr sin θ ∂φrr sin θB.3.7

Derivation of the S Equation

The evolution equation for S is simply equation (B.12),ρ̂T̂∂s= SFLUX + SADV + SDIFF .
∂t(B.144)

where the entropy and radiative fluxes are given by
SFLUX = −∇ · q̂eff= −κr ρ̂cp ∇(T̂ + T ) − κs ρ̂T̂ ∇(ŝ + s),

by equation (B.11)(B.145)the advection term is
SADV = −ρ̂T̂ (u · ∇)(ŝ + s)


uφ ∂
uθ ∂
∂
+
+
(ŝ + s),= −ρ̂T̂ ur∂rr ∂θ r sin θ ∂φ

(B.146)and the viscous heating term is
SDIFF = Φ̂12
= 2ρ̂νeff e : e − (∇ · u) .
3

by equation (B.13)(B.147)

Appendix C

ASH CODE TIME STEPPING

We discuss here in more detail the time stepping scheme employed by the ASH
code. The basic problem is how one numerically solves an initial value problem of theform∂y= f (y(t), t).
∂t

(C.1)

Since any higher-dimensional differential equation can be broken down into a systemof one-dimensional equations, solving the one-dimensional problem is sufficient. The
ASH code uses both the Crank-Nicholson and Adams-Bashforth methods to advancethe solution in time, with each described in turn.
Subscripts are used here to denote the index of the time step. For example, yi
denotes the value of y at the most recently computed time step (occurring at time ti )and yi+1 denotes the yet-to-be-computed value of y at the next time step. The time
separation between subsequent time steps is ∆ti ≡ ti+1 − ti .

C.1

THE SECOND-ORDER CRANK-NICHOLSON METHOD
As a first step in solving equation (C.1), one notices that the function f is simply

the slope of y, suggesting the formulayi+1 − yi
= fi
∆ti

or

yi+1 = yi + ∆ti fi ,

(C.2)

197
commonly referred to as the forward Euler method. This method is explicit, meaningthat yi+1 is the only unknown quantity in the formula and thus can be readily computed.
The accuracy of this method is determined by computing the local truncation
error (LTE), which is simply the residual when the true values of y and f are substituted
into equation (C.2):
LTE = y(ti + ∆ti ) − y(ti ) − ∆ti f (y(ti ), ti ).

(C.3)

By noting that the Taylor expansion of y(ti + ∆ti ) isy(ti + ∆ti ) = y(ti ) + ∆ti
and by using the fact that f =∆t2i d2 ydy++ ··· ,
dt ti
2 dt2 ti(C.4)

dy
, substituting equation (C.4) into equation (C.3) gives
dt
LTE =

∆t2i d2 y+ ··· .
2 dt2 ti

(C.5)

Since the local truncation error only contains terms of second and higher order in ∆ti ,the forward Euler method is accurate to first order.
We now consider the backward Euler method:yi+1 − yi= fi+1 .
∆ti

(C.6)

The only difference between the backward and forward Euler methods is the use of
the unknown quantity fi+1 rather than the known quantity fi on the right-hand side.
Because we have to simultaneously solve for both yi+1 and fi+1 , this method is now
implicit, and consequently requires a matrix inversion if f is linear in y. If f is nonlinear
in y, implicit schemes such as the backward Euler become even more problematic toimplement, and as a result they are generally only used to solve linear equations. As
with the forward Euler method, the backward Euler method is accurate to first order.
The Crank-Nicholson method is a weighted average of the forward and backward
Euler methods presented above:yi+1 − yi= Θfi+1 + (1 − Θ)fi ,
∆ti

(C.7)

198
where Θ is an adjustable parameter ranging from 0 (fully explicit) to 1 (fully implicit).
The Crank-Nicholson method can be shown to be accurate to second order. In addition,
this method is numerically stable, meaning that there is no limit on the time step size
one can use (though the accuracy of the solution might decrease if too large a time step
is chosen). In contrast, the forward Euler method is conditionally stable: choosing toolarge of a time step will cause numerical errors to grow, eventually overwhelming theactual solution.

C.2

THE SECOND-ORDER ADAMS-BASHFORTH METHOD
We now examine explicit multistep methods, where the results from more than

one previous time step are used to solve equations of the form (C.1). In this case, thevalues of yi , fi and fi−1 (where the last quantity is either computed from a previous
time step or stated as initial conditions) are already available. We now assume theunknown quantity yi+1 is given by the general form
yi+1 − yi= αfi + βfi−1∆tior

yi+1 = yi + ∆ti (αfi + βfi−1 ),(C.8)

where the constant coefficients α and β will be determined to minimize the local truncation error.
For this method to be second-order accurate, we require the local truncation error
to vanish through terms of order ∆t2i . As with the forward Euler method in the previous
section, the local truncation error is calculated by substituting into equation (C.8) thetrue values of y and f at the various time steps and calculating the residual. That is,
LTE = y(ti + ∆ti ) − y(ti ) − ∆ti [αf (y(ti ), ti ) + βf (y(ti − ∆ti−1 ), ti − ∆ti−1 )] . (C.9)
We first must evaluate the Taylor series expansions of yi+1 and fi−1 around ti . These
expansions are given byy(ti + ∆ti ) = yi + ∆ti∆t2i d2 y∆t3i d3 y
dy
+
+
+ ···
dt ti
2 dt2 ti
6 dt3 ti

(C.10)199
andf (y(ti − ∆ti−1 ), ti − ∆ti−1 ) = fi − ∆ti−1∆t2i−1 d2 f∆t3i−1 d3 f
df+−
+ ··· .
dt ti
2
dt2 ti6
dt3 ti(C.11)

Using these Taylor expansions, and the fact that

d2 f
dy
df
= y, 2 =, etc., we can now
dt
dt
dt

write equation (C.9) in terms of the coefficients α and β, plus the dependent variable yand its derivatives evaluated at the current time step ti . Grouping by power of ∆ti , wenow havedy
d2 y
LTE = ∆ti
(1 − α − β) + ∆t2i
dt ti
dt2 ti



1
∆ti−1
+β
2
∆ti


+··· ,(C.12)

where only terms of first and second order in ∆ti are listed explicitly. Since we require
the parenthesized expressions to vanish for the method to be second-order accurate, wemust have
α =1+

1 ∆ti
2 ∆ti−1

and

β=−

1 ∆ti
.
2 ∆ti−1(C.13)

Substituting these values of α and β into equation (C.8), we obtain the second-order
Adams-Bashforth formula:
yi+1 − yi
1 ∆ti
1 ∆ti
fi −
fi−1 .
= 1+∆ti
2 ∆ti−1
2 ∆ti−1(C.14)

Note that higher-order Adams-Bashforth formulae can be derived in a similar manner.

C.3

COMBINING THE METHODSThe ASH code evolution equations are of the form∂y
= L + N,∂t

(C.15)

where y can represent any dependent variable in the system of equations (W , Z, p, s),and L and N respectively designate all linear and nonlinear source terms. Generally,

200
these source terms can be a function of any of the dependent variables as well as afunction of the independent variables ~r and t.
Combining the two methods, equation (C.15) becomesyi+1 − yi
=
∆ti





1 ∆ti
1 ∆ti
Ni −
Ni−1 + ΘLi+1 + (1 − Θ)Li ,1+
2 ∆ti−1
2 ∆ti−1

which is equation (4.61) in Chapter 4.

(C.16)