Editorial Type:
Article
Article Type:
Research Article

Numerical Consistency of Metric Terms in Terrain-Following Coordinates

Joseph B. Klemp National Center for Atmospheric Research,* Boulder, Colorado

Search for other papers by Joseph B. Klemp in
Current site
Google Scholar
PubMed Close
,
William C. Skamarock National Center for Atmospheric Research,* Boulder, Colorado

Search for other papers by William C. Skamarock in
Current site
Google Scholar
PubMed Close
, and
Oliver Fuhrer Atmospheric and Climate Science, ETH, Zurich, Switzerland

Search for other papers by Oliver Fuhrer in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Jul 2003
DOI:
https://doi.org/10.1175/1520-0493(2003)131<1229:NCOMTI>2.0.CO;2
Page(s):
1229–1239
Restricted access

Abstract

In numerically integrating the equations of motion in terrain-following coordinates, care must be taken in treating the metric terms that arise due to the sloping coordinate surfaces. In particular, metric terms that appear in the advection and pressure-gradient operators should be represented in a manner such that they exactly cancel when transformed back to Cartesian coordinates. Noncancellation of these terms can lead to spurious forcing at small scales on the numerical grid. This effect is demonstrated for a mountain wave flow problem through analytic solutions to the linear finite-difference equations. Further confirmation is provided through numerical simulations with a two-dimensional prototype version of the Weather Research and Forecasting (WRF) model, and with the Canadian Mesoscale Compressible Community (MC2) model.

Corresponding author address: Dr. Joseph Klemp, NCAR, P.O. Box 3000, Boulder, CO 80307. Email: klemp@ucar.edu

Abstract

In numerically integrating the equations of motion in terrain-following coordinates, care must be taken in treating the metric terms that arise due to the sloping coordinate surfaces. In particular, metric terms that appear in the advection and pressure-gradient operators should be represented in a manner such that they exactly cancel when transformed back to Cartesian coordinates. Noncancellation of these terms can lead to spurious forcing at small scales on the numerical grid. This effect is demonstrated for a mountain wave flow problem through analytic solutions to the linear finite-difference equations. Further confirmation is provided through numerical simulations with a two-dimensional prototype version of the Weather Research and Forecasting (WRF) model, and with the Canadian Mesoscale Compressible Community (MC2) model.

Corresponding author address: Dr. Joseph Klemp, NCAR, P.O. Box 3000, Boulder, CO 80307. Email: klemp@ucar.edu

Save
Cover Monthly Weather Review
Monthly Weather Review

  • Dempsey, D., and C. Davis, 1998: Error analysis and tests of pressure-gradient force schemes in a nonhydrostatic, mesoscale model. Preprints, 12th Conf. on Numerical Weather Prediction, Phoenix, AZ, Amer. Meteor. Soc., 236–239.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer Press, 465 pp.

  • Epifanio, C. C., and D. R. Durran, 2001: Three-dimensional effects in high-drag-state flows over long ridges. J. Atmos. Sci., 58 , 1051–1065.

    • Search Google Scholar
    • Export Citation

  • Gal-Chen, T., and R. C. J. Sommerville, 1975: Numerical solutions of the Navier–Stokes equations with topography. J. Comput. Phys., 17 , 276–310.

    • Search Google Scholar
    • Export Citation

  • Janjic, Z. I., 1977: Pressure gradient force and advection scheme used for forecasting with steep and small scale topography. Beitr. Phys. Atmos., 50 , 186–189.

    • Search Google Scholar
    • Export Citation

  • Mahrer, Y., 1984: An improved numerical approximation of the horizontal gradients in a terrain-following coordinate system. Mon. Wea. Rev., 112 , 918–922.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and N. W. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19 , 173–179.

    • Search Google Scholar
    • Export Citation

  • Phillips, N. A., 1957: A coordinate system having some special advantages for numerical forecasting. J. Meteor., 14 , 184–185.

  • Schär, C., D. Leuenberger, O. Fuhrer, D. Lüthi, and C. Girard, 2002: A new terrain-following vertical coordinate formulation for atmospheric prediction models. Mon. Wea. Rev., 130 , 2459–2480.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 406 138 6
PDF Downloads 361 135 4