• Bannon, P. R., 1995: Potential vorticity, conservation, hydrostatic adjustment, and the anelastic approximation. J. Atmos. Sci., 52 , 23022312.

    • Search Google Scholar
    • Export Citation
  • Bannon, P. R., 1996: On the anelastic approximation for a compressible atmosphere. J. Atmos. Sci., 53 , 36183628.

  • Bannon, P. R., , J. M. Chagnon, , and R. P. James, 2006: Mass conservation and the anelastic approximation. Mon. Wea. Rev., 134 , 29893005.

    • Search Google Scholar
    • Export Citation
  • Bolin, B., 1956: An improved barotropic model and some aspects of using the balanced equation for three-dimensional flow. Tellus, 8 , 6175.

    • Search Google Scholar
    • Export Citation
  • Bubnová, R., , G. Hello, , P. Bénard, , and J-F. Geleyn, 1995: Integration of the fully elastic equations cast in the hypostatic pressure terrain-following coordinate in the framework of the ARPEGE/Aladin NWP system. Mon. Wea. Rev., 123 , 515535.

    • Search Google Scholar
    • Export Citation
  • Côté, J., , S. Gravel, , A. Méthot, , A. Patoine, , M. Roch, , and A. Staniforth, 1998: The operational CMC–MRB Global Environmental Multiscale (GEM) model. Part I: Design considerations and formulation. Mon. Wea. Rev., 126 , 13731395.

    • Search Google Scholar
    • Export Citation
  • Cressman, G. P., 1958: Barotropic divergence and very long atmospheric waves. Mon. Wea. Rev., 86 , 293297.

  • Cullen, M., , T. Davies, , M. Mawson, , J. James, , S. Couther, , and A. Malcolm, 1997: An overview of numerical methods for the next generation UK NWP and climate models. Numerical Methods in Atmospheric and Oceanic Modeling, C. A. Lin, R. Laprise, and H. Ritchie, Eds., NRC Research Press, 425–444.

    • Search Google Scholar
    • Export Citation
  • Davies, T., , A. Staniforth, , N. Wood, , and J. Thuburn, 2003: Validity of anelastic and other equation sets as inferred from normal-mode analysis. Quart. J. Roy. Meteor. Soc., 129 , 27612775.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., 1989: Improving the anelastic approximation. J. Atmos. Sci., 46 , 14531461.

  • Durran, D. R., 2008: A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow. J. Fluid Mech., 601 , 365379.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., , and A. Arakawa, 2007: Generalizing the Boussinesq approximation to stratified compressible flow. C. R. Mec., 335 , 655664.

    • Search Google Scholar
    • Export Citation
  • Dutton, J. A., , and G. H. Fichtl, 1969: Approximate equations of motion for gases and liquids. J. Atmos. Sci., 26 , 241254.

  • Eckart, C., 1960: Hydrodynamics of Oceans and Atmospheres. Pergamon Press, 290 pp.

  • Janjic, Z. I., 2003: A nonhydrostatic model based on a new approach. Meteor. Atmos. Phys., 82 , 271285.

  • Janjic, Z. I., , J. P. Gerrity, , and S. Nickovic, 2001: An alternate approach to nonhydrostatic modeling. Mon. Wea. Rev., 129 , 11641178.

    • Search Google Scholar
    • Export Citation
  • Jung, J-H., , and A. Arakawa, 2008: A three-dimensional anelastic model based on the vorticity equation. Mon. Wea. Rev., 136 , 276294.

  • Klemp, J., , and R. Wilhemson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci., 35 , 10701096.

  • Klemp, J., , W. C. Skamarock, , and J. Dudha, 2007: Conservative split-explicit time integration methods for the compressible nonhydrostatic equations. Mon. Wea. Rev., 135 , 28972913.

    • Search Google Scholar
    • Export Citation
  • Konor, C. S., , and A. Arakawa, 2007: Multipoint Explicit Differencing (MED) for time integrations of the wave equation. Mon. Wea. Rev., 135 , 38623875.

    • Search Google Scholar
    • Export Citation
  • Laprise, R., 1992: The Euler equations of motion with hydrostatic pressure as an independent variable. Mon. Wea. Rev., 120 , 197207.

  • Lipps, F. B., , and R. S. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci., 39 , 21922210.

    • Search Google Scholar
    • Export Citation
  • Miller, M. J., 1974: On the use of pressure as vertical co-ordinate in modeling convection. Quart. J. Roy. Meteor. Soc., 100 , 155162.

    • Search Google Scholar
    • Export Citation
  • Miller, M. J., , and R. P. Pearce, 1974: A three-dimensional primitive equation model of cumulonimbus convection. Quart. J. Roy. Meteor. Soc., 100 , 133154.

    • Search Google Scholar
    • Export Citation
  • Miller, M. J., , and A. A. White, 1984: On the non-hydrostatic equations in pressure and sigma coordinates. Quart. J. Roy. Meteor. Soc., 110 , 515533.

    • Search Google Scholar
    • Export Citation
  • Nance, L. B., , and D. R. Durran, 1994: A comparison of the accuracy of three anelastic systems and the pseudo-incompressible system. J. Atmos. Sci., 51 , 35493565.

    • Search Google Scholar
    • Export Citation
  • Ogura, Y., , and J. G. Charney, 1962: A numerical model of thermal convection in the atmosphere. Proc. Int. Symp. on Numerical Weather Prediction, Tokyo, Japan, Meteorological Society of Japan, 431–451.

    • Search Google Scholar
    • Export Citation
  • Ogura, Y., , and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19 , 173179.

  • Ooyama, K., 1990: A thermodynamic foundation for modeling the moist atmosphere. J. Atmos. Sci., 47 , 25802593.

  • Phillips, N. A., 1966: Equation of motion for a shallow rotating atmosphere and the “traditional approximation.”. J. Atmos. Sci., 23 , 626628.

    • Search Google Scholar
    • Export Citation
  • Phillips, N. A., 1968: Reply. J. Atmos. Sci., 25 , 11551157.

  • Richardson, L. F., 1922: Weather Prediction by Numerical Process. Cambridge University Press, 236 pp.

  • Rossby, C-G., and Coauthors, 1939: Relations between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J. Mar. Res., 2 , 3855.

    • Search Google Scholar
    • Export Citation
  • Schlesinger, R. E., 1975: A three-dimensional numerical model of an isolated deep convective cloud: Preliminary results. J. Atmos. Sci., 32 , 934957.

    • Search Google Scholar
    • Export Citation
  • Siebert, M., 1961: Atmospheric tides. Advances in Geophysics, Vol. 7, Academic Press, 105–187.

  • Skamarock, W. C., , and J. B. Klemp, 1992: The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations. Mon. Wea. Rev., 120 , 21092127.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., , and J. B. Klemp, 1994: Efficiency and accuracy of the Klemp–Wilhemson time-splitting technique. Mon. Wea. Rev., 122 , 26232630.

    • Search Google Scholar
    • Export Citation
  • Smolarkiewicz, P. K., , and A. Dörnbrack, 2008: Conservative integrals of adiabatic Durran’s equations. Int. J. Numer. Math. Fluids, 56 , 15131519.

    • Search Google Scholar
    • Export Citation
  • Smolarkiewicz, P. K., , L. G. Margolin, , and A. A. Wyszogrodzki, 2001: A class of nonhydrostatic global models. J. Atmos. Sci., 58 , 349364.

    • Search Google Scholar
    • Export Citation
  • Steppeler, J., , R. Hess, , U. Schättler, , and L. Bonaventura, 2003: Review of numerical methods for nonhydrostatic weather prediction models. Meteor. Atmos. Phys., 82 , 287301.

    • Search Google Scholar
    • Export Citation
  • Tanguay, M., , A. Robert, , and R. Laprise, 1990: A semi-implicit, semi-Lagranjian fully compressible regional forecast model. Mon. Wea. Rev., 118 , 19701980.

    • Search Google Scholar
    • Export Citation
  • Wedi, N. P., , and P. K. Smolarkiewicz, 2004: Extending Gal–Chen and Somerville terrain following coordinate transformation on time-dependent curvilinear boundaries. J. Comput. Phys., 193 , 120.

    • Search Google Scholar
    • Export Citation
  • Wedi, N. P., , and P. K. Smolarkiewicz, 2006: Direct numerical simulation of the Plumb–McEwan laboratory analog of the QBO. J. Atmos. Sci., 63 , 32263252.

    • Search Google Scholar
    • Export Citation
  • White, A. A., 1989: An extended version of a nonhydrostatic, pressure coordinate model. Quart. J. Roy. Meteor. Soc., 115 , 12431251.

  • Wiin-Nielsen, A., 1959: On baroropic and baroclinic models, with special emphasis on ultra-long waves. Mon. Wea. Rev., 87 , 171183.

  • Wilhemson, R., , and Y. Ogura, 1972: The pressure perturbation and the numerical modeling of a cloud. J. Atmos. Sci., 29 , 12951307.

  • Wolff, P. M., 1958: The error in numerical forecasts due to retrogression of ultra-long waves. Mon. Wea. Rev., 86 , 245250.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 51 51 14
PDF Downloads 56 56 13

Unification of the Anelastic and Quasi-Hydrostatic Systems of Equations

View More View Less
  • 1 Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
  • | 2 Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
© Get Permissions
Restricted access

Abstract

A system of equations is presented that unifies the nonhydrostatic anelastic system and the quasi-hydrostatic compressible system for use in global cloud-resolving models. By using a properly defined quasi-hydrostatic density in the continuity equation, the system is fully compressible for quasi-hydrostatic motion and anelastic for purely nonhydrostatic motion. In this way, the system can cover a wide range of horizontal scales from turbulence to planetary waves while filtering vertically propagating sound waves of all scales. The continuity equation is primarily diagnostic because the time derivative of density is calculated from the thermodynamic (and surface pressure tendency) equations as a correction to the anelastic continuity equation. No reference state is used and no approximations are made in the momentum and thermodynamic equations. An equation that governs the time change of total energy is also derived. Normal-mode analysis on an f plane without the quasigeostrophic approximation and on a midlatitude β plane with the quasigeostrophic approximation is performed to compare the unified system with other systems. It is shown that the unified system reduces the westward retrogression speed of the ultra-long barotropic Rossby waves through the inclusion of horizontal divergence due to compressibility.

Corresponding author address: Dr. Celal S. Konor, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371. Email: csk@atmos.colostate.edu

Abstract

A system of equations is presented that unifies the nonhydrostatic anelastic system and the quasi-hydrostatic compressible system for use in global cloud-resolving models. By using a properly defined quasi-hydrostatic density in the continuity equation, the system is fully compressible for quasi-hydrostatic motion and anelastic for purely nonhydrostatic motion. In this way, the system can cover a wide range of horizontal scales from turbulence to planetary waves while filtering vertically propagating sound waves of all scales. The continuity equation is primarily diagnostic because the time derivative of density is calculated from the thermodynamic (and surface pressure tendency) equations as a correction to the anelastic continuity equation. No reference state is used and no approximations are made in the momentum and thermodynamic equations. An equation that governs the time change of total energy is also derived. Normal-mode analysis on an f plane without the quasigeostrophic approximation and on a midlatitude β plane with the quasigeostrophic approximation is performed to compare the unified system with other systems. It is shown that the unified system reduces the westward retrogression speed of the ultra-long barotropic Rossby waves through the inclusion of horizontal divergence due to compressibility.

Corresponding author address: Dr. Celal S. Konor, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371. Email: csk@atmos.colostate.edu

Save