Vertical Shear Instability with Partially Reflecting Boundary Conditions

TL;DR

This paper investigates how partially reflecting boundary conditions affect the vertical shear instability in protoplanetary disks, revealing that boundary behaviors significantly influence the growth rates and symmetry of unstable modes, with implications for planetary formation.

Contribution

The study extends previous VSI analyses by incorporating partially reflecting boundary conditions and analyzing their impact on mode growth and symmetry.

Findings

Growth rates decrease with dominant outgoing wave components.

Non-axisymmetric modes are unstable and grow at rates decreasing with azimuthal wavenumber.

Boundary conditions induce non-symmetric modes relative to the disk midplane.

Abstract

The vertical shear instability (VSI) is widely believed to be effective in driving turbulence in protoplanetary disks. Prior studies on VSI exclusively exploit the reflecting boundary conditions (BCs) at the disk surfaces. VSI depends critically on the boundary behaviors of waves at the disk surfaces. We extend earlier studies by performing a comprehensive numerical analysis of VSI with partially reflecting BCs for both the axisymmetric and non-axisymmetric unstable VSI modes. We find that the growth rates of the unstable modes diminish when the outgoing component of the flow is greater than the incoming one for high-order body modes. When the outgoing wave component dominates, the growth of VSI is notably suppressed. We find that the non-axisymmetric modes are unstable and they grow at a rate that decreases with the azimuthal wavenumber. The different BCs at the lower and upper disk surfaces naturally lead to non-symmetric modes relative to the disk midplane. The potential implications of our studies for further understanding planetary formation and evolution in protoplanetary disks (PPDs) are also briefly discussed.