Instabilities in protoplanetary disk

Huang, P., & Bai, X.-N. (2025). The interplay between dust dynamics and turbulence induced by the vertical shear instability. The Astrophysical Journal, 986(1), 76. https://doi.org/10.3847/1538-4357/add345

Huang, P., & Bai, X.-N. (2025). Dust clumping in outer protoplanetary disks: The interplay among four instabilities. The Astrophysical Journal Letters, 986(1), L13. https://doi.org/10.3847/2041-8213/adcebb

Evolutions

The dust is treated as a single pressureless fluid, characterized by the stopping time $T_s$, the interaction term between the gas and the dust fluid can be find in the equations:

$$\frac{\partial\rho_g\boldsymbol{v}_g}{\partial t}+\nabla\cdot(\rho_g\boldsymbol{v}_g\boldsymbol{v}_g+P)=\rho_g\nabla\Phi+\rho_d\frac{\boldsymbol{v}_d-\boldsymbol{v}_g}{T_s}\tag{2}$$

$$\frac{\partial\rho_d\boldsymbol{v}_d}{\partial t}+\nabla\cdot(\rho_d\boldsymbol{v}_d\boldsymbol{v}_d)=\rho_d\nabla\Phi+\rho_d\frac{\boldsymbol{v}_g-\boldsymbol{v}_d}{T_s}\tag{4}$$

Instabilities

• VSI (vertical shear instability): for nonbarotropic ($\nabla\rho_g\times\nabla P\neq 0$) disk, there is a weak vertical shear

$$2R\Omega_g\frac{\partial \Omega_g}{\partial z}=-\frac{(\nabla\rho_g\times\nabla P)}{\rho_g^2}\cdot\hat{\phi}$$

lead to weak zonal flows.

• SI (streaming instability): the dust streaming motion (due to radial perssure gradient) back-reaction to the gas.
• RWI (Rossby wave instability): the maximum of “entropy modified inversed vortensity”, with strong radial pressure variations.
• KHI (Kelvin-Helmholtz instability): velocity shear across an interface.