Dynamically consistent formulations in meteorological and air quality models for multiscale atmospheric studies. Part I: Governing equations in a generalized coordinate system

In recent years, the popularity of the fully compressible nonhydrostatic atmospheric models has increased due to the need for simulating multiscale dynamics from convective to synoptic weather phenomena. These recent advances in meteorological modeling techniques present a new set of implementation difficulties and challenges for air quality models and others that are dependent on prognostic meteorological models because some of the scalar quantities are not conserved adequately due to the nature of numerical formulations in these models. To provide a foundation for conservative discrete formulations in atmospheric models, this paper proposes a set of governing equations for a fully compressible atmosphere in a generalized coordinate system. The idealized set of governing equations for atmospheric studies includes conservative equations for the thermodynamic variables and trace species as well as equations for wind components. For the thermodynamic parameters, a continuity equation for air density and a conservation equation for entropy are preferred to the prognostic equations for pressure and temperature. The proposed set of governing equations allows for a unified description of atmospheric dynamics, thermodynamics, and air quality problems. Alternative prognostic and diagnostic formulations of the governing equations and their characteristics are studied for different coordinate systems. The companion paper addresses the mass conservation issues in multiscale air quality applications.