Linearly unstable modes in a vertically-stratified, dusty disk. Left: with gas viscosity. Right: inviscid disk. Reproduced from Lin (2020).
Schematic illustration of planetesimal formation via the streaming instability. Dust first settles into a dense mid-plane, followed by the streaming instability. The SI leads to dust-clumping, which can eventually collapse into planetesimals.
The formation of km or larger-sized `planetesimals' -- the building blocks of planets -- from mm-cm pebbles in protoplanetary disks (PPDs) is a key stage in planet formation theory. The streaming instability (SI, Youdin & Goodman, 2005) is the de facto mechanism for planetesimal formation as its ingredients are natural to PPDs: dusty gas in rotation. Indeed, modern, sophisticated numerical simulations show that the SI can produce planetesimals with properties consistent with that in the solar system (Nesvorný et al., 2019).
The theory behind the SI has been developed under the `unstratified' approximation, where the disk's vertical structure is ignored, which is valid for dynamics at the disk mid-plane. Here, the relative radial motion between dust and gas drives the SI.
However, real PPDs are stratified: dust settling results in a dusty mid-plane with particle densities declining with height. I thus extended the original linear stability analysis for the SI to stratified disks. The stratified problem involves a complex set of ordinary differential equations, for which I used the Dedalus code (Burns et al., 2019) to solve for direct solutions.
I found the dominant instability in stratified disks is a form of SI that is driven by the vertical gradient in the disk's rotation velocity, or vertical shear, but also requires partial coupling between dust and gas like the SI. That is, dust and gas need to be able to `stream' past one another. These vertically-shearing streaming instabilities (VSSIs) operate on scales of about 0.1% of the gas disk scale height. This is much smaller than the classic SI, which has characteristic scales of about 1% of the gas scale height.
VSSIs can be thought of a combination of the classic SI and another well-known instability in PPDs, the Kelvin-Helmholtz instability (KHI).
Interestingly, more than a decade ago, Ishitsu et al. (2009) also found, using direct numerical simulations, an axisymmetric instability associated with the vertical shear of a dust layer. This was most likely the VSSIs identified in my linear calculations. Ishitsu et al. found that VSSIs led to turbulence that disrupted the dust layer, which may negatively impact planetesimal formation.
Many numerical simulations of stratified, dusty disks have been conducted since the original discovery by of the SI (Youdin & Goodman, 2005), e.g. Johansen et al. (2009), Bai & Stone (2010), Yang et al. (2017), etc. In light of my linear calculations and the earlier results of Ishitsu et al., it is possible that the initial turbulence observed in other stratified simulations is not due to the classic SI, nor the KHI, but a new instability somewhere in between, the VSSIs.
For further details, see the pre-print.