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.

[1]  Katsuyuki V. Ooyama,et al.  A Thermodynamic Foundation for Modeling the Moist Atmosphere , 1990 .

[2]  Daewon W. Byun,et al.  Development and Implementation of the EPA’s Models-3 Initial Operating Version: Community Multi-Scale Air Quality (CMAQ) Model , 1998 .

[3]  T. Black.  The new NMC mesoscale Eta Model: description and forecast examples , 1994 .

[4]  R. Bornstein,et al.  Eddy exchange coefficients in numerical models of the planetary boundary layer , 1977 .

[5]  N. A. Phillips,et al.  A COORDINATE SYSTEM HAVING SOME SPECIAL ADVANTAGES FOR NUMERICAL FORECASTING , 1957 .

[6]  Roger A. Pielke,et al.  A terrain-following coordinate system—Derivation of diagnostic relationships , 1989 .

[7]  Richard C. J. Somerville,et al.  On the use of a coordinate transformation for the solution of the Navier-Stokes equations , 1975 .

[8]  A. D. Young,et al.  An Introduction to Fluid Mechanics , 1968 .

[9]  Mark DeMaria.  Evaluation of a Hydrostatic, Height-Coordinate Formulation of the Primitive Equations for Atmospheric Modeling , 1995 .

[10]  P. Thunis,et al.  Hierarchy of Mesoscale Flow Assumptions and Equations , 1996 .

[11]  J. Dudhia.  A Nonhydrostatic Version of the Penn State–NCAR Mesoscale Model: Validation Tests and Simulation of an Atlantic Cyclone and Cold Front , 1993 .

[12]  M. Tapp,et al.  A non‐hydrostatic mesoscale model , 1976 .

[13]  William A. Gallus,et al.  A non-hydrostatic version of the NMC's regional Eta model , 1996 .

[14]  Hartmut Höller,et al.  Entropy and Potential Temperature , 1987 .

[15]  H. Juang,et al.  A spectral fully compressible nonhydrostatic mesoscale model in hydrostatic sigma coordinates: Formulation and preliminary results , 1992 .

[16]  Monique Tanguay,et al.  A Semi-implicit Send-Lagrangian Fully Compressible Regional Forecast Model , 1990 .

[17]  Akio Arakawa,et al.  An isentropic vertical coordinate model: Design and application to atmospheric frontogenesis studies , 1992 .

[18]  René Laprise,et al.  The Euler Equations of Motion with Hydrostatic Pressure as an Independent Variable , 1992 .

[19]  J. Klemp,et al.  The Simulation of Three-Dimensional Convective Storm Dynamics , 1978 .

[20]  P. Thunis,et al.  Topographic Vorticity-Mode Mesoscale-β (TVM) Model. Part I: Formulation , 1996 .

[21]  Pierre Defrise.  Tensor Calculus in Atmospheric Mechanics , 1964 .

[22]  A. Kasahara.  Various Vertical Coordinate Systems Used for Numerical Weather Prediction , 1974 .

[23]  Lewis F. Richardson,et al.  Weather Prediction by Numerical Process , 1922 .

[24]  Slobodan Nickovic,et al.  The Step-Mountain Coordinate: Model Description and Performance for Cases of Alpine Lee Cyclogenesis and for a Case of an Appalachian Redevelopment , 1988 .

[25]  John A. Dutton,et al.  The Ceaseless Wind: An Introduction to the Theory of Atmospheric Motion , 1976 .

[26]  R. Pielke.  Mesoscale Meteorological Modeling , 1984 .

[27]  Warren M. Washington,et al.  NCAR global General Circulation Model of the atmosphere , 1967 .

[28]  Roger A. Pielke,et al.  Derivation of Slope Flow Equations Using Two Different Coordinate Representations , 1985 .

[29]  Daewon W. Byun,et al.  Dynamically Consistent Formulations in Meteorological and Air Quality Models for Multiscale Atmospheric Studies. Part II: Mass Conservation Issues , 1999 .

[30]  Thomas Matejka,et al.  The accuracy of vertical air velocities from Doppler radar data , 1998 .