  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.
