  In order to find a better physical model to describe the large-scale
cloud-water transformation and rainfall, we consider a moist atmosphere model
consisting of the primitive equations with only horizontal viscosity in the
dynamic equation and a set of humidity equations describing water vapor, rain
water and cloud condensates. To overcome difficulties caused by the absence of
vertical viscosity in the dynamic equation, we get the local existence of $v$
in $H^{1}$ space by combining the viscous elimination method and the $z-$weak
solution method and using the generalized Bihari-Lasalle inequality. And then,
we get the global existence of $v$ under higher regularity assumption of
initial data. In turn, the existence of quasi-strong and strong solutions to
the whole system is obtained. By introducing two new unknown quantities
appropriately and utilizing the monotone operator theory to overcome
difficulties caused by the Heaviside function in the source terms, we get the
uniqueness of solutions.
