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The Vertical Shear Instability is one of two known mechanisms potentiallyactive in the so-called dead zones of protoplanetary accretion disks. A recentanalysis indicates that a subset of unstable modes shows unbounded growth -both as resolution is increased and when the nominal lid of the atmosphere isextended, possibly indicating ill-posedness in previous attempts of linearanalysis. The reduced equations governing the instability are revisited and thegenerated solutions are examined using both the previously assumed separableforms and an improved non-separable solution form that is herewith introduced.Analyzing the reduced equations using the separable form shows that, while thelow-order body modes have converged eigenvalues and eigenfunctions (as both thevertical boundaries of the atmosphere are extended and with increased radialresolution), it is also confirmed that the corresponding high-order body modesand the surface modes do indeed show unbounded growth rates. However, theenergy contained in both the higher-order body modes and surface modesdiminishes precipitously due to the disk's Gaussian density profile. Most ofthe energy of the instability is contained in the low-order modes. Aninseparable solution form is introduced which filters out the inconsequentialsurface modes leaving only body modes (both low and high-order ones). Theanalysis predicts a fastest growing mode with a specific radial length scale.The growth rates associated with the fundamental corrugation and breathingmodes matches the growth and length scales observed in previous nonlinearstudies of the instability.

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