• Dempsey, D., , and C. Davis, 1998: Error analyses 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.

  • Dudhia, J., 1995: Reply. Mon. Wea. Rev., 123, 25732576.

  • Gallus, W. A., , and J. B. Klemp, 2000: Behavior of flow over step orography. Mon. Wea. Rev., 128, 11531164.

  • Gary, J. M., 1973: Estimate of truncation error in transformed coordinate, primitive equation atmospheric models. J. Atmos. Sci., 30, 223233.

    • Search Google Scholar
    • Export Citation
  • Janjić, Z. I., 1989: On the pressure gradient force error in σ-coordinate spectral models. Mon. Wea. Rev., 117, 22852292.

  • Klemp, J. B., 2011: A terrain-following coordinate with smoothed coordinate surfaces. Mon. Wea. Rev., 139, 21632169.

  • Leuenberger, D., , M. Koller, , O. Fuhrer, , and C. Schär, 2010: A generalization of the SLEVE vertical coordinate. Mon. Wea. Rev., 138, 36833689.

    • 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, 918922.

    • Search Google Scholar
    • Export Citation
  • Mesinger, F., 1982: On the convergence and error problems of the calculation of the pressure gradient force in sigma coordinate models. Geophys. Astrophys. Fluid Dyn., 19, 105117.

    • Search Google Scholar
    • Export Citation
  • Miura, H., 2007: An upwind-biased conservative advection scheme for spherical hexagonal–pentagonal grids. Mon. Wea. Rev., 135, 40384044.

    • Search Google Scholar
    • Export Citation
  • Queney, P., 1948: The problem of airflow over mountains: A summary of theoretical studies. Bull. Amer. Meteor. Soc., 29, 1626.

  • Schär, C., , D. Leuenberger, , O. Fuhrer, , D. Lüthi, , and C. Girard, 2002: A new terrain-following vertical coordinate for atmospheric prediction models. Mon. Wea. Rev., 130, 24592480.

    • Search Google Scholar
    • Export Citation
  • Steppeler, J., , H.-W. Bitzer, , M. Minotte, , and L. Bonaventura, 2002: Nonhydrostatic atmospheric modeling using a z-coordinate representation. Mon. Wea. Rev., 130, 21432149.

    • Search Google Scholar
    • Export Citation
  • Zängl, G., 2002: An improved method for computing horizontal diffusion in a sigma-coordinate model and its application to simulations over mountainous topography. Mon. Wea. Rev., 130, 14231432.

    • Search Google Scholar
    • Export Citation
  • Zängl, G., 2003: A generalized sigma coordinate system for the MM5. Mon. Wea. Rev., 131, 28752884.

  • Zängl, G., , L. Gantner, , G. Hartjenstein, , and H. Noppel, 2004: Numerical errors above steep topography: A model intercomparison. Meteor. Z., 13, 6976.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 56 56 10
PDF Downloads 35 35 2

Extending the Numerical Stability Limit of Terrain-Following Coordinate Models over Steep Slopes

View More View Less
  • 1 Deutscher Wetterdienst, Offenbach, Germany
© Get Permissions
Restricted access

Abstract

To extend the numerical stability limit over steep slopes, a truly horizontal pressure-gradient discretization based on the ideas formulated by Mahrer in the 1980s has been developed. Conventionally, the pressure gradient is evaluated in the terrain-following coordinate system, which necessitates a metric correction term that is prone to numerical instability if the height difference between adjacent grid points is much larger than the vertical layer spacing. The alternative way pursued here is to reconstruct the pressure gradient at auxiliary points lying at the same height as the target point on which the velocity is defined. This is accomplished via a second-order Taylor-series expansion in this work, using the hydrostatic approximation to transform the second derivatives into first derivatives to facilitate second-order accurate discretization in the presence of strong vertical grid stretching. Moreover, a reformulated lower boundary condition is used that avoids the extrapolation of vertical derivatives evaluated in potentially very thin layers. A sequence of tests at varying degrees of idealization reveals that the truly horizontal pressure-gradient discretization improves numerical stability over steep slopes for a wide range of horizontal mesh sizes, ranging from a few hundreds of meters to tens of kilometers. In addition, tests initialized with an atmosphere at rest reveal that the spurious circulations developing over steep mountains are usually smaller than for the conventional discretization even in configurations for which the latter does not suffer from stability problems.

Corresponding author address: Günther Zängl, Deutscher Wetterdienst, Frankfurter Straße 135, D-63067 Offenbach, Germany. E-mail: guenther.zaengl@dwd.de

Abstract

To extend the numerical stability limit over steep slopes, a truly horizontal pressure-gradient discretization based on the ideas formulated by Mahrer in the 1980s has been developed. Conventionally, the pressure gradient is evaluated in the terrain-following coordinate system, which necessitates a metric correction term that is prone to numerical instability if the height difference between adjacent grid points is much larger than the vertical layer spacing. The alternative way pursued here is to reconstruct the pressure gradient at auxiliary points lying at the same height as the target point on which the velocity is defined. This is accomplished via a second-order Taylor-series expansion in this work, using the hydrostatic approximation to transform the second derivatives into first derivatives to facilitate second-order accurate discretization in the presence of strong vertical grid stretching. Moreover, a reformulated lower boundary condition is used that avoids the extrapolation of vertical derivatives evaluated in potentially very thin layers. A sequence of tests at varying degrees of idealization reveals that the truly horizontal pressure-gradient discretization improves numerical stability over steep slopes for a wide range of horizontal mesh sizes, ranging from a few hundreds of meters to tens of kilometers. In addition, tests initialized with an atmosphere at rest reveal that the spurious circulations developing over steep mountains are usually smaller than for the conventional discretization even in configurations for which the latter does not suffer from stability problems.

Corresponding author address: Günther Zängl, Deutscher Wetterdienst, Frankfurter Straße 135, D-63067 Offenbach, Germany. E-mail: guenther.zaengl@dwd.de
Save