A mathematical-physical formulation of a mesoscale limited area model in meteorology

Abstract

In this paper we give a detailed mathematical-physical derivation of a limited-area meteorological model, used in several applications, describing the motions and the behaviour of the atmosphere, in the troposphere layer, at mesoscale range, modeled as a mixture of compressible, heat conducting, non viscous components. The mixing ratios of water vapor, cloud water, rainwater, ice, …, referred to the dry air density are introduced. We work in a non inertial reference frame, solidal to the rotating Earth, where Coriolis force is neglected, being Rossby number at the considered mesoscale much bigger than 1. We derive the momentum balance equation, the conservation law for the dry air density, the balance equation for the temperature and the balance equations for the densities of the fluid mixture components in two cases: when a local rectangular Cartesian reference frame, called z-system, solidal to the rotating Earth is used and when a new generalized nondimensional η-system is introduced, by means of the hydrostatic pressure. Detailed calculations (with particular assumptions) are worked out, starting from the case where the atmosphere is modeled as a perfect fluid, the dry air.