Abstract
Planets form in protoplanetary disks around young stars from the coagulation of dust grains. While it is understood that dust grains can be held
together via Van Der Waals forces at sub-centimetre scales, while large bodies can be held by self-gravitation at kilometre scales, it is unclear which
process operates between these scales: in this range, the grains are moving
too fast to stick together via Van Der Waals forces (instead they fragment
or bounce) but are too small to hold themselves together via gravity. One
mechanism that could allow dust grains to bypass this range and coagulate
into kilometre-sized planetesimals is known as the Streaming Instability
(SI). The SI is a linear instability that rapidly concentrates dust grains into
filaments in the mid-plane of a protoplanetary disk. While simulations,
starting with Johansen et al. (2007), provide tentative support for the idea
that the SI can increase the dust density to the point that they become
gravitationally bound, a variety of recent works have shown that there is
a minimum dust to gas density ratio ยต needed for this to occur, and that
the threshold is non-linearly dependent on the size (becoming more severe
for smaller grains). These issues call into question the effectiveness of the
SI at clumping grains and show that further work is needed. Further, Li
and Youdin (2021) show that non-linear clumping results do not bear an
obvious relation to the SI linear growth rates, suggesting our understanding
of the linear physics of such systems may be incomplete.
In this thesis, I studied the linear stability of protoplanetary disks in a
more general setting, including using more general "non-modal" methods to
understand finite-timescale growth, and understanding non-axisymmetric
perturbations. The purpose of doing this is to understand whether this more
general analysis can help explain simulation results showing a threshold
for dust to clump, as well as to determine whether there exists regimes
where non-axisymmetric perturbations should dominate over the standard
(axisymmetric) SI. To do this, I developed the tools and equations necessary
to apply non-modal stability methods to describe the linear dynamics of
the coupled dust-gas fluid. After an exploration of the spatial form of the
fastest growing perturbations (pseudomodes), most of our results focus on
using a shearing wave ansatz to convert the dust-gas equations into ODEs,
applying non-modal methods to these. To compare the non-axisymmetric
pseudomodes to the axisymmetric pseudomodes of the ODEs, I examined
the effective growth rates at different points in time, as well as the maximum
possible growth of perturbations over shorter time frames. I explored a large
range of parameters while doing this to better understand general features of
the system. I found that the growth of the non-axisymmetric perturbations
does not dominate over that of axisymmetric perturbations in most cases.
While it does occur in some regimes, the range of parameters is rather small
so it is unlikely that it is important. I also found that strong non-modal
growth (compared to eigenmode predictions) occurs at very small scales,
suggesting that this is unlikely of direct interest to planetesimal formation
because a small-scale perturbation implies clumping of only small quantities
of dust. Overall, while non-modal and non-axisymmetric effects show a
number of interesting behaviours, in most regimes they seem unlikely to be
of obvious relevance to explaining non-linear results.