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

Global well-posedness of large scale moist atmosphere system with only horizontal viscosity in the dynamic equation