A minimal model for vertical shear instability in protoplanetary accretion disks

TL;DR

This paper introduces a simplified analytical model for vertical shear instability in protoplanetary disks, revealing its physical mechanism and conditions for instability, with implications for turbulence in disk midplanes.

Contribution

It provides a minimal, analytically tractable framework for understanding vertical shear instability in protoplanetary disks, connecting it to known atmospheric and oceanic instabilities.

Findings

Vertical shear instability resembles Earth's mid-latitude convective instability.

Instability occurs when fluid parcel slopes exceed mean absolute momentum slope.

Anelastic dynamics produce oscillatory unstable modes.

Abstract

The Vertical Shear Instability is an axisymmetric effect suggested to drive turbulence in the magnetically inactive zones of protoplanetary accretion disks. Here we examine its physical mechanism in analytically tractable ``minimal models" in three settings that include a uniform density fluid, a stratified atmosphere, and a shearing-box section of a protoplanetary disk. Each of these analyses show that the vertical shear instability's essence is similar to the slantwise convective symmetric instability in the mid-latitude Earth atmosphere, in the presence of vertical shear of the baroclinic jet stream, as well as mixing in the top layers of the Gulf Stream. We show that in order to obtain instability the fluid parcels' slope should exceed the slope of the mean absolute momentum in the disk radial-vertical plane. We provide a detailed and mutually self-consistent physical explanation from three perspectives: in terms of angular momentum conservation, as a dynamical interplay between a fluid's radial and azimuthal vorticity components, and from an energy perspective involving a generalized Solberg-H{\o}iland Rayleigh condition. Furthermore, we explain why anelastic dynamics yield oscillatory unstable modes and isolate the oscillation mechanism from the instability one.