  This paper concerns the large time asymptotic behavior of solutions to the
free boundary problem of the compressible primitive equations in atmospheric
dynamics with physical vacuum. Up to second order of the perturbations of an
equilibrium, we have introduced a model of the compressible primitive equations
with a specific viscosity and shown that the physical vacuum free boundary
problem for this model system has a global-in-time solution converging to an
equilibrium exponentially, provided that the initial data is a small
perturbation of the equilibrium. More precisely, we introduce a new coordinate
system by choosing the enthalpy (the square of sound speed) as the vertical
coordinate, and thanks to the hydrostatic balance, the degenerate density at
the free boundary admits a representation with separation of variables in the
new coordinates. Such a property allows us to establish horizontal derivative
estimates without involving the singular vertical derivative of the density
profile, which plays a key role in our analysis.
