We present a formal derivation of a simplified version of Compressible
Primitive Equations (CPEs) for atmosphere modeling. They are obtained from
$3$-D compressible Navier-Stokes equations with an \emph{anisotropic viscous
stress tensor} where viscosity depends on the density. We then study the
stability of the weak solutions of this model by using an intermediate model,
called model problem, which is more simple and practical, to achieve the main
result.