Vertical shear instability with partially reflecting boundary conditions

ABSTRACT The vertical shear instability (VSI) is widely believed to be effective in driving turbulence in protoplanetary discs (PPDs). Prior studies on VSI exclusively exploit the reflecting boundary conditions (BCs) at the disc surfaces. VSI depends critically on the boundary behaviours of waves at the disc 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 disc surfaces naturally lead to non-symmetric modes relative to the disc mid-plane. The potential implications of our studies for further understanding planetary formation and evolution in PPDs are also briefly discussed.