Mathematical analysis of a cloud resolving model including the ice microphysics

Abstract

We extend our study for the warm cloud model in [ 13 ] to the analysis of a more general cloud model including the ice microphysics in [ 28 ]. The moisture variables comprise water vapor, cloud condensates (cloud water, cloud ice), and cloud precipitations (rain, snow), with respective mass ratios \begin{document}$ q_v $\end{document} , \begin{document}$ q_c $\end{document} and \begin{document}$ q_p $\end{document} . A typical assumption in [ 13 ] for the calculation of condensation rate is that the warm clouds are exactly at water saturation with no supersaturation in general. When the ice microphysics are included, the situation becomes more complicated. We have to consider both the saturation mixing ratio with respect to water ( \begin{document}$ q_{vw} $\end{document} ) and the saturation with respect to ice ( \begin{document}$ q_{vi} $\end{document} ) when the temperature \begin{document}$ T $\end{document} is below the freezing point \begin{document}$ T_w $\end{document} but above the threshold \begin{document}$ T_i $\end{document} for homogeneous ice nucleation. A remedy, acceptable from the physical and mathematical viewpoints, is to define the overall saturation mixing ratio \begin{document}$ q_{vs} $\end{document} as a convex combination of \begin{document}$ q_{vw} $\end{document} and \begin{document}$ q_{vi} $\end{document} . Under this setting, supersaturation can still be avoided and we have the constraint \begin{document}$ q_v \le q_{vs} $\end{document} with \begin{document}$ q_{vs} $\end{document} depending itself on the state. Mathematically, we are led to a system of equations and inequations involving some quasi-variational inequalities for which we prove the global existence and regularity of solutions.