Linear analysis of the vertical shear instability: outstanding issues and improved solutions (Research Note)

TL;DR

This paper revisits the linear analysis of the vertical shear instability in protoplanetary disks, introduces an improved solution form, and clarifies the dominant modes and growth rates relevant to disk dynamics.

Contribution

It introduces an inseparable solution form that filters out surface modes, providing clearer insight into the dominant low-order body modes and their growth rates.

Findings

Low-order modes have converged eigenvalues and eigenfunctions.

High-order and surface modes show unbounded growth but contain less energy.

The fastest growing mode matches previous nonlinear study observations.

Abstract

The Vertical Shear Instability is one of two known mechanisms potentially active in the so-called dead zones of protoplanetary accretion disks. A recent analysis indicates that a subset of unstable modes shows unbounded growth - both as resolution is increased and when the nominal lid of the atmosphere is extended, possibly indicating ill-posedness in previous attempts of linear analysis. The reduced equations governing the instability are revisited and the generated solutions are examined using both the previously assumed separable forms and an improved non-separable solution form that is herewith introduced. Analyzing the reduced equations using the separable form shows that, while the low-order body modes have converged eigenvalues and eigenfunctions (as both the vertical boundaries of the atmosphere are extended and with increased radial resolution), it is also confirmed that the corresponding high-order body modes and the surface modes do indeed show unbounded growth rates. However, the energy contained in both the higher-order body modes and surface modes diminishes precipitously due to the disk's Gaussian density profile. Most of the energy of the instability is contained in the low-order modes. An inseparable solution form is introduced which filters out the inconsequential surface modes leaving only body modes (both low and high-order ones). The analysis predicts a fastest growing mode with a specific radial length scale. The growth rates associated with the fundamental corrugation and breathing modes matches the growth and length scales observed in previous nonlinear studies of the instability.