  We consider a model of a circular lenticular vortex immersed into a deep and
vertically stratified viscous fluid in the presence of gravity and rotation.
The vortex is assumed to be baroclinic with a Gaussian profile of angular
velocity both in the radial and axial directions. Assuming the base state to be
in a cyclogeostrophic balance, we derive linearized equations of motion and
seek for their solution in a geometric optics approximation to find amplitude
transport equations that yield a comprehensive dispersion relation. Applying
algebraic Bilharz criterion to the latter, we establish that stability
conditions are reduced to three inequalities that define stability domain in
the space of parameters. The main destabilization mechanism is either
stationary or oscillatory axisymmetric instability depending on the Schmidt
number ($Sc$), vortex Rossby number and the difference between the radial and
axial density gradients as well as the difference between the epicyclic and
vertical oscillation frequencies. We discover that the boundaries of the
regions of stationary and oscillatory axisymmetric instabilities meet at a
codimension-2 point, forming a singularity of the neutral stability curve. We
give an exhaustive classification of the geometry of the stability boundary,
depending on the values of the Schmidt number. Although we demonstrate that the
centrifugally stable (unstable) Gaussian lens can be destabilized (stabilized)
by the differential diffusion of mass and momentum and that destabilization can
happen even in the limit of vanishing diffusion, we also describe explicitly a
set of parameters in which the Gaussian lens is stable for all $Sc>0$.
