Research Note

Linear analysis of the vertical shear instability: outstanding issues and improved solutions

¹ Space Sciences Division, NASA Ames Research Center, Moffett Field, CA 94035, USA
e-mail: orkan.m.umurhan@nasa.gov
² SETI Institute, 189 Bernardo Way, Mountain View, CA 94043, USA
³ Queen Mary University of London, School of Physics and Astronomy, 327 Mile End Road London, E1 4NS, UK
⁴ Niels Bohr International Academy, The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark

Received: 8 May 2015
Accepted: 13 July 2015

Abstract

Context. The vertical shear instability is one of several known mechanisms that are potentially active in the so-called dead zones of protoplanetary accretion disks. A recent analysis of the instability mechanism 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. This trend suggests that, possibly, the model system is ill-posed.

Aims. This research note both examines the energy content of these modes and questions the legitimacy of assuming separable solutions for a problem whose linear operator is fundamentally inseparable.

Methods. 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 introduced in this paper.

Results. Reconsidering the solutions of the reduced equations by using the separable form shows that, while the low-order body modes have converged eigenvalues and eigenfunctions (for both variations in the model atmosphere’s vertical boundaries and radial numerical resolution). It is also confirmed that the corresponding high-order body modes and the surface modes indeed show unbounded growth rates. 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 to filter 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 match the growth and length scales observed in previous nonlinear studies of the instability.

Conclusions. Linear stability analysis of the vertical shear instability should be done assuming non-separable solutions, especially for settings involving boundaries in the radial direction. We also conclude that the surface modes are relatively inconsequential because of the little energy they contain, and are artifacts of imposing specific kinematic vertical boundary conditions in isothermals disk models.