thesis.tex

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\usepackage[T1]{fontenc} % see http://tinyurl.com/67zdxwf
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\usepackage{mydeluxetable} % deluxetable customized to play well with ucthesis
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\usepackage{appendix} % needed for appendices in each chapter, if required.


\begin{document}
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\begin{abstract}
Rotating planets are most stable when spinning around their maximum moment of inertia, 
and will tend to reorient themselves to achieve this configuration. 
Geological activity redistributes mass in the planet, making the moment of inertia a function of time.
As the moment of inertia of the planet changes, the spin axis shifts with respect to
a mantle reference frame in order to maintain rotational stability.
This process is known as true polar wander (TPW). 
Of the processes that contribute to a planet's moment of inertia,
convection in the mantle generates the largest and longest-period
fluctuations, with corresponding shifts in the spin axis.
True polar wander has been hypothesized to explain several physiographic features on planets and moons in our solar system.
On Earth, TPW events have been invoked in some interpretations of paleomagnetic data.
Large swings in the spin axis could have enormous ramifications for paleogeography,
paleoclimate, and the history of life.

Although the existence of TPW is well-verified, it is not known whether its rate and
magnitude have been large enough for it to be an important process in Earth history.
If true polar wander has been sluggish compared to plate tectonic speeds, 
then it would be difficult to detect and its consequences would be minor.
Herein I investigate rates of true polar wander on convecting planets using scaling, numerics,
and inverse problems.

I perform a scaling analysis of TPW on a convecting planet, identifying a minimal
set of nondimensional parameters which describe the problem. The primary nondimensional numbers
that control the rate of TPW are the ratio of centrifugal to gravitational forces ($m$)
and the Rayleigh number ($\mathrm{Ra}$). The parameter $m$ sets the size of a planet's rotational bulge,
which determines the amount of work that needs to be done to move the spin axis.
The Rayleigh number controls the size, distribution, and rate of change of
moment of inertia anomalies, all of which affect the rate of TPW.
I find that the characteristic size of moment of inertia anomalies decreases with
higher $\mathrm{Ra}$, but that the characteristic response time for TPW also decreases.
These two effects approximately cancel. However, the orientation of the principal axes of
the moment of inertia becomes less stable to perturbations at high $\mathrm{Ra}$, thereby increasing the rate of TPW.
Overall, I find that a more vigorously convecting planet (one with a higher $\mathrm{Ra}$)
is more likely to experience large TPW events.
If early Earth had more vigorous convection, it may have experienced
more TPW than present-day Earth.

Flow induced by density anomalies in the mantle deflects free surfaces at the surface
and the CMB, and the mass anomalies due to these deflections contribute to the moment of inertia.
A full accounting of the moment of inertia anomalies must include these surface effects.
Numerical models of mantle convection with a free surface have suffered from numerical
sloshing instabilities. I analyze the sloshing instability by constructing a generalized
eigenvalue problem for the relaxation time spectrum. The minimum relaxation time of the spectrum sets the
maximum stable timestep. This analysis gives the first quantitative explanation for
why existing techniques for stabilizing geodynamic simulations with a free surface work. 
I also use this perspective to construct an alternative stabilization scheme based
on nonstandard finite differences. This scheme has a single parameter, 
given by an estimate of the minimum relaxation time, and allows for still larger timesteps.

Finally, I develop a new method for analyzing apparent polar wander (APW) paths
described by sequences of paleomagnetic poles. Existing techniques, such as spline fits
and running means, do not fully account for the uncertainties in the position
and timing of paleomagnetic pole paths. Furthermore, they impose regularization
on the solution, and the resulting uncertainties are difficult to interpret.
Our technique is an extension of paleomagnetic Euler pole (PEP) analysis.
I invert for finite Euler pole rotations that can reproduce APW paths
within a Bayesian Markov chain Monte Carlo (MCMC) framework. This allows us
to naturally include uncertainties in age and position, and provides error estimates
on the resulting model parameters. Regularization can be accomplished via
physically motivated choices for the parameters' prior probability distributions.

I applied the Bayesian PEP technique to the Mesoproterozoic Laurentian APW track,
which primarily comes from the Keweenawan Midcontinent Rift. I fit
the track with one and two Euler rotations. Both inversions did a good job
of reproducing the Keweenawan track, though the two Euler pole inversion
has a closer fit. I find that the implied Laurentian plate speeds exceeds
22.9 cm/yr at the 95\% confidence level. These speeds are significantly faster than
Cenozoic plate speeds, and could be explained by either faster plate speeds in
the Proterozoic or a TPW event.


\end{abstract}

\hypersetup{pageanchor=true}
\begin{frontmatter}

\begin{dedication}
\null\vfil
{\large
\begin{center}
For my family
\end{center}}
\null\vfil
\end{dedication}

\tableofcontents
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% If using code.sty, can also add:
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%% \addcontentsline{toc}{chapter}{List of Code Examples}

\begin{acknowledgements}

An undertaking as large as a Ph.D cannot be done alone, despite the occasional
use of the first person singular in this document.
I have received encouragement, help, insight, and support from many people at every step of the way,
and I would not have gotten to this point without them.

First and foremost, I thank my advisor, Bruce Buffett.
Bruce is an inspiring scientist, with a breadth and depth of knowledge that I have yet to exhaust,
and a range of research projects that spans the whole planet.
Even in the times when I have come to him with a problem which he as not previously encountered,
his instincts almost always point me in the right direction.
More than this, Bruce has given me the intellectual space
to craft my own path, to make my own mistakes, and form my own conclusions.

Thanks to Nicholas Swanson-Hysell, who arrived at Berkeley during my degree,
and jumped into the work I was doing without hesitation.
His thoughts and insights have been invaluable.
Thanks to Michael Manga, who taught me so many things about science and academia,
even when I did not initially understand them.
Thanks to Phil Marcus, a skilled fluid dynamicist who
expanded my toolbox of analytical and numerical techniques, and more importantly,
expanded my intuition for the behavior of PDEs and their solution
using computers.

Thanks furthermore to two non-Berkeley faculty, Timo Heister and Wolfgang Bangerth, who taught me how to to program.
They are both top-notch mathematicians with many demands on their time.
Despite that, they view both education and community-building as equally important as their research,
and I have benefited immensely from their efforts.

Thanks to my undergraduate advisors Dave Evans, Jun Korenaga, and Mark Brandon, who
incited my love of Earth science.

Two institutions have supported my development as a scientist in such a fundamental
way that I would certainly not have gotten here without them.
The first institution is CIDER, the deep Earth summer program held 
at the Kavli Institute for Theoretical Physics, 
which has been an amazing resource for learning and research.
The directors of the program, led by the incomparable Barbara Romanowicz,
have reshaped the field of deep Earth research and built a durable community.
The second institution is CIG, led by Louise Kellog, which has done more than any other institution
to promote the open development and use of software in Earth science.
CIG-developed (and CIG-maintained) software has touched nearly every aspect of this dissertation.

More broadly than CIG, open source software has contributed enormously to my work.
At some point in the near future, the transparent and reproducible use of scientific
software will be universally seen as a crucial component of good science, and many great
open source projects are showing the way. These projects have dozens or hundreds of
contributors, many of whom contribute for no pay.
They cannot all be thanked here, but I have particularly benefitted from
the work put into \texttt{ASPECT}, \texttt{deal.II}, \texttt{Burnman},
Numpy, Scipy, Matplotlib, and PyMC.
This dissertation was typeset using the ucastrothesis \LaTeX\ template.

My fellow graduate students have been an incredible resource, and I
have learned as much from them as any coursework. I want to thank
Max, Edwin, Amanda, Leif, Ved, Scott, Tyler, Jesse, Nick, Percy, Daniella, Patrick, Jennifer, and Seth,
for being both colleagues and friends.
Importantly, I thank Carolina, Brent, Noah, Zack, Slayer, Pam, and Sanne for being friends first when it mattered.
My non-Earth science friends, including 
Caitlin, Katherine, Andy, Anna, Emily, Chase, Lauren, Jenny, Mary, Marc, Alice, Claire, Gerstle, Jay, and Jacob
provided important encouragement and perspective.
David Mangiante, my sometimes officemate, oftentimes co-schemer, and always friend, has kept me thinking about what is most important in life.
\\
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Finally, my family has been an unending source of unconditional love and support. 
Mom, Dad, Noah and Morgan, I would not be here without you. Thank you.

\end{acknowledgements}
\end{frontmatter}

\include{intro/intro}
\include{tpw_rate/tpw_rate_chapter}
\include{free_surface/free_surface_chapter}
\include{bayesian_plate_reconstruction/bayesian_plate_reconstruction_chapter}
\include{conclusion/conclusion}

\bibliography{thesis}
\end{document}