Vortex Evolution in a Protoplanetary
Disk
Sanford Davis
The theory that planets
form from a thin disk of dust and gas was first proposed in the 18th
Century and is now a generally accepted fact. The process by which planets actually
emerge from this tenuous state is a subject of intense current study. Recent
research points to vortex motion as a possible intermediary where dust particles
are captured, concentrated, and finally accumulated by gravitational attraction.
These mass accumulations gradually grow to kilometer-sized objects (planetesimals)
and ultimately to full-sized planets. Assuming that the disk can support a turbulent
flow, it was shown that vortices arise naturally and persist as long as turbulent
energy is present. Other possibilities are that vortices arise from certain
instabilities in the rotating disk or from external impacts of clumpy infalling
gas. In either case, coherent vortices could lead to important and far-reaching
processes in the protoplanetary disk.
A research study is underway
to determine the effect of vortices on the wave structure in a typical disk
which may also play a role in the planet-formation process. It is well known
that discrete vortices in a sheared flow do not retain their coherence. This
coherence time depends on the local shear rate, the strength, and the size of
the vortex. During the shearing epoch, and depending on the nature of the medium,
a vortex can emit a variety of wave systems. In this study, the equations of
motion have been simulated using a high-resolution numerical method to track
Rossby and acoustic/shock waves. Rossby waves are slowly moving waves of vorticity
generated in flows with large-scale vorticity gradients. Acoustic waves are
waves of expansion and contraction that occur in all compressible media. The
protoplanetary disk is a rotating compressible gas with a radially variable
rotation rate. It can support both wave systems.
A typical result from the
simulation is shown in fig 1. It is a sequence of snapshots of the perturbed
vorticity defined as the difference between the total vorticity and the baseline
flow. This baseline flow is a Keplerian flow (rotational velocity = const x
(radius)-1/2) and the initial vortex is shown in the third quadrant
in fig 1(a). The vortex becomes elongated about its initial location at r =
4 (blue-red streaks on the white circle in fig 1(b)) and both inward and outward-bound
counterclockwise spiral vorticity waves are spawned. The outward-bound waves
evolve to an axisymmetric wave pattern ("circularization") with shock
waves (fig 1(d)). It is interesting that the vortex-induced waves can induce
a supersonic radial flow. Another wave system (Rossby waves) appears in the
region r < 10. The shock waves are axisymmetric while the Rossby waves have
a cosine angular dependence. Each wave system has a characteristic radial speed.
In follow-up work we will
augment the numerical simulations with particle and/or granular gas models to
examine the effect of these vorticity-induced waves on particle migration, accumulation,
and (possibly) planetesimal formation.
Figure 1. Perturbation vorticity
bitmaps showing density (shocks) and Rossby waves. (a)-(d): 0, 16, 32, and 48
vortex revolutions respectively.
