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 auto conversion 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 [] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [] and Grabowski and Smolarkiewicz []. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In [] 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 [], who succeeded in proving the global solvability of the primitive equations.
Highlights
Moisture and precipitation still cause major uncertainties in numerical weather prediction models and it is our aim here to develop further the rigorous analysis of atmospheric flow models
A study of a moisture model coupled to the primitive equations has already been carried out by Coti Zelati et al in [11]
The moisture model there consists of one moisture
Summary
Moisture and precipitation still cause major uncertainties in numerical weather prediction models and it is our aim here to develop further the rigorous analysis of atmospheric flow models. In the remainder of the introduction we first state the moisture model in pressure coordinates, which have the advantage that under the assumption of hydrostatic balance the continuity equation takes the form of the incompressibility condition. In the present model the difference of the gas constants for dry air and water vapor as well as the dependence of the internal energy on the moisture components via different heat capacities is neglected. These additional terms that would arise in a more precise thermodynamical setting are small in principle and often not taken into account. Throughout this paper, unless explicitly specified, we use C to denote a generic positive constant depending only on the given functions in the boundary conditions, the initial data, and the physical parameters appearing in the original system (but not on the parameter ε arising in the approximate system introduced )
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