# Global Well-posedness for the Primitive Equations Coupled to Nonlinear Moisture Dynamics with Phase Changes

## TL;DR

This paper proves the global well-posedness of a coupled atmospheric model involving primitive equations and nonlinear moisture dynamics with phase changes, extending previous work by including full velocity field coupling.

## Contribution

It establishes the global existence and uniqueness of solutions for the full moist atmospheric flow model with phase changes, coupling moisture dynamics to primitive equations.

## Key findings

- Proved global well-posedness of the coupled model.
- Extended previous results to include full velocity coupling.
- Demonstrated boundedness of solutions over time.

## Abstract

In this work we study the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein and Majda in \cite{KM} and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler \cite{Ke} and Grabowski and Smolarkiewicz \cite{GS}. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In \cite{HKLT} we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao and Titi \cite{CT}, who succeeded in proving the global solvability of the primitive equations.