Magnetically-Forced Axisymmetric Zonal Accelerations in Earth's Outer Core Colin More Doctor of ...

165 Pages·2017·3.49 MB·English

Magnetically-Forced Axisymmetric Zonal
Accelerations in Earth’s Outer Core
by
Colin More
A thesis submitted in partial ful(cid:12)llment of the requirements
for the degree of
Doctor of Philosophy
in
geophysics
Department of Physics
University of Alberta
(cid:13)c Colin More, 2017
Abstract
Azimuthal accelerations of cylindrical surfaces co-axial with the rotation axis have been in-
ferred to exist in Earth’s (cid:13)uid core on the basis of magnetic (cid:12)eld observations and changes
in the length-of-day. These accelerations have a typical timescale of decades. However,
the physical mechanism causing the accelerations is not well understood. Scaling argu-
ments suggest that the leading order torque averaged over cylindrical surfaces should arise
from the Lorentz force. Decadal (cid:13)uctuations in the magnetic (cid:12)eld inside the core, driven
by convective (cid:13)ows, could then force decadal changes in the Lorentz torque and generate
zonal accelerations. This hypothesis is tested by constructing a quasi-geostrophic model of
magnetoconvection, with thermally-driven (cid:13)ows perturbing a steady, imposed background
magnetic (cid:12)eld. It is found that when the Alfv(cid:19)en number is similar to that estimated for
Earth’s (cid:13)uid core, temporal (cid:13)uctuations in the torque balance are dominated by the Lorentz
torque, with the latter generating mean zonal accelerations. The model reproduces both
fast, free Alfv(cid:19)en waves and slow, forced accelerations, with ratios of relative strength and
relative timescale similar to those inferred for the Earth’s core. The temporal changes in
the magnetic (cid:12)eld which drive the time-varying Lorentz torque are produced by the under-
lying convective (cid:13)ows, which shear and advect the magnetic (cid:12)eld on timescales associated
with convective eddies. These results support the hypothesis that temporal changes in the
magnetic (cid:12)eld deep inside Earth’s (cid:13)uid core drive the observed decadal zonal accelerations
of cylindrical surfaces through the Lorentz torque.
ii
Physics is the science of what you can throw away.
- Keith Cu(cid:11)
iii
Acknowledgements
The successful completion of a PhD project is not due to the brilliance of a single person, but
to that of the medium-sized army of friends, colleagues, acquaintances, teachers, professors,
mentors, and family members which helped along the way. I have been extraordinarily lucky
to have been surrounded by one of the (cid:12)nest such armies throughout my academic career.
The successes of the project rightly belong to all of them, while its weaknesses lie wholely
with myself.
I would particularly like to thank my PhD supervisor, Mathieu Dumberry, for the remark-
able amount of patience, support, and intelligence he has displayed over the course of the
six and a half years it took to produce this thesis. I am sure better supervisors exist, but
I have trouble imagining how. I would also like to thank my committee members, Moritz
Heimpel and Bruce Sutherland, for their own commitment to the project. Bruce in particu-
lar displayed a remarkable capacity for punishment by signing up for a second tour, having
also supervised my master’s thesis!
This being a terminal degree, now seems to be an ideal time to look back on all the teachers
who have contributed to getting me here. A good teacher is, in the truest sense of the word,
priceless, and I have had many. A non-exhaustive list would include Mrs. Bullock and Ms.
Smith, Mr. Fowler and Mr. Dencer, Tim Cooper, George McGuire, Derek Harnett, Rob
Woodside, and Je(cid:11) Chizma. An exhaustive list would be signi(cid:12)cantly longer (compared to
iv
what, you ask, Tim? Why, the non-exhaustive list, of course!). Thank you all.
I have been very fortunate to have had the support of NSERC post-graduate scholarships at
several points in my graduate school career. I certainly appreciated them when I won them,
and I appreciated them even more after they ran out!
I have been even more fortunate with the o(cid:14)cemates I’ve had over the course of this degree,
and with the friends I’ve made. Life really is largely about the people you’re surrounded
with on a daily basis, and my surroundings have been wonderful. Thanks in particular to
Ben, Dan, Daniel, Drew, Kathrin, Keith, Olivia, and the Sarahs. That was a good time!
The trials of the past several years were eased tremendously by my girlfriend, Julie. While
many people truly cared for me and o(cid:11)ered their support, she did so while having to actually
live with me! It certainly wasn’t great fun for her much of the time, but she never com-
plained, and whenever I thought her patience was at an end she instead managed to give me
even more. It was also pretty great having food appear while I was in the depths of writing.
I love you.
Like the Matrix, nobody can be told what it’s like to be in graduate school. Happily, my
dearest sister, Heather, has always been a year or three ahead of me. It’s meant so much to
have someone who knows exactly what is going on, while also knowing exactly what I need.
Such as musical suggestions at strange hours of the night, or encouragement to outside for
a change. Thanks, from your favourite brother who is really really happy to be done!
Finally, my parents. You have been constant sources of encouragement, support, and love
throughout this journey and throughout my life. None of this could have happened without
you. Thank you.
v
Contents
List of Tables ix
List of Figures x
1 Introduction 1
1.1 Geomagnetic Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Geostrophy and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Length-of-Day Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Agreement of Observation and Prediction . . . . . . . . . . . . . . . . . . . . 15
1.5 Unexplained Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.7 Scienti(cid:12)c Advancements Contained in this Thesis . . . . . . . . . . . . . . . 21
2 Theory 23
2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Axial Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Axial Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Geostrophic Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 The Quasi-Geostrophic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Representation of Fields . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.5 Transformation to QG . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Summary of Quasi-Geostrophic Equations . . . . . . . . . . . . . . . . . . . 37
3 Implementation 39
3.1 Solution Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Velocity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Magnetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.3 Temperature Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Axisymmetric Nonlinear Averages . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.1 Maintaining Model Validity . . . . . . . . . . . . . . . . . . . . . . . 46
vi
3.4.2 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.3 Earth-Like Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.4 Selection Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Results 54
4.1 The Non-Axisymmetric Lorentz Torque . . . . . . . . . . . . . . . . . . . . . 54
4.2 Overview of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Model Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.2 Exploration of Parameter Space . . . . . . . . . . . . . . . . . . . . . 57
4.3 General features of Experiment #52 . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Time-averaged axisymmetric torque balance . . . . . . . . . . . . . . . . . . 67
4.5 Zonal accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Taylorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Discussion 76
5.1 Slow Zonal Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Free Alfv(cid:19)en waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Taylorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6 Conclusion 81
6.1 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Bibliography 86
Appendix A Useful Mathematical Relations 99
A.1 Gradient Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.2 Cylindrically Radial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.3 Azimuthal Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.4 Axial Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.4.1 First Radial Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.5 Azimuthal Average of Two Real-Valued Functions . . . . . . . . . . . . . . . 109
Appendix B Dimensional Governing Equations 112
B.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.2 Buoyancy Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
B.3 Magnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.4 Summary of Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . 115
Appendix C Nondimensionalization 116
C.1 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C.2 Temperature Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
C.3 Induction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C.4 Auxiliary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
C.5 Summary of Nondimensional Equations . . . . . . . . . . . . . . . . . . . . . 119
vii
Appendix D Constructing a QG Model 120
D.1 Axial Vorticity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
D.1.1 Simpli(cid:12)cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
D.1.2 Axial Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
D.2 Zonal Flow Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
D.2.1 Simpli(cid:12)cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
D.2.2 Axial Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
D.3 Temperature Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
D.3.1 Background Pro(cid:12)le . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
D.3.2 Axial Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
D.4 Induction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
D.5 Zonal Induction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Appendix E Axisymmetric Multiplication in Fourier Space 144
E.1 (cid:0) - Reynolds Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
R
E.2 (cid:0) - Maxwell Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
L2
E.3 Zonal Induction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Appendix F Conservation of Axial Angular Momentum 153
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List of Tables
2.1 Physical parameters in the outer core . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Scales for nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Nondimensional parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Summary of models run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
ix
List of Figures
1.1 Evolution of the radial magnetic (cid:12)eld at the CMB . . . . . . . . . . . . . . . 5
1.2 Spatial power spectrum of the geomagnetic (cid:12)eld . . . . . . . . . . . . . . . . 8
1.3 Core (cid:13)ow at the CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Geostrophic cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Time variability of the zonal (cid:13)ow . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Observed and predicted LOD variations . . . . . . . . . . . . . . . . . . . . . 15
1.7 Mechanism of free Alfv(cid:19)en waves . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Snapshots of unrealistic velocity and magnetic (cid:12)eld structure . . . . . . . . . 55
4.2 Zonal accelerations when including the non-axisymmetric Lorentz torque . . 56
4.3 Family tree of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Mechanical energy vs. time . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Thermal energy vs. time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6 Zonal energy vs. time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7 Snapshots of vorticity and current structures . . . . . . . . . . . . . . . . . . 65
4.8 Snapshots of velocity structures . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.9 Snapshots of magnetic perturbation structures . . . . . . . . . . . . . . . . . 66
4.10 Time-averaged axisymmetric torque balance . . . . . . . . . . . . . . . . . . 67
4.11 Time-averaged axisymmetric magnetic torque balance . . . . . . . . . . . . . 69
4.12 Time-dependent axisymmetric torque balance . . . . . . . . . . . . . . . . . 70
4.13 Characteristic length and time scales . . . . . . . . . . . . . . . . . . . . . . 71
4.14 Filtered time-dependent axisymmetric torque balance . . . . . . . . . . . . . 73
4.15 Time-dependent Taylorization factor . . . . . . . . . . . . . . . . . . . . . . 75
x