Dynamical Effects of Mountains on the General Circulation of the Atmosphere: I. Development of Finite-Difference Schemes Suitable for Incorporating Mountains

Abstract

Reexamination of schemes for the vertical finite-difference and for the pressure gradient force in the σ-coordinate system is made following the discussions of Arakawa (1972), Corby et al. (1972) and Phillips (1974) in order to find a scheme suitable for incorporating steep mountains in general circulation models and numerical forecasting models. It is possible to classify the schemes proposed so far by using integral constraints such as the conservation of the total energy, the conservation of the potential temperature (θ) and its square and the surface torque condition. Furthermore, a new scheme is obtained from the classification.
Truncation errors of the hydrostatic relation, the pressure gradient force and the advection terms are compared among the schemes obtained by the classification for the case of the atmosphere at rest. By revising schemes errors are reduced to be considerably small over a steep mountain. For example, the errors of the pressure gradient force and the advection terms of the momentum equation are about 1m/s in geostrophic wind over a slope with 4km/300km inclination if the grid size is 300km.
We also examined truncation errors in a more realistic case. A steep mountain like the Tibetan Plateau is placed in the westerly flow. It is assumed that the atmosphere has a stable stratification in the vertical and that the westerly flow is forced so as to be restored to a barotropic equilibrium wind. Computational domain is a half-hemisphere.
The results obtained by the runs having different grid resolutions and different schemes are compared with that of a high resolution run which has 12 layers and the 2.5° horizontal grid size. The result of the low horizontal resolution run with the 5° grid size is a little different from that of the high resolution run. The difference in wind is about 10m/s over the mountain but is negligibly small in the other region. Therefore, it seems that the errors in the low horizontal resolution run are tolerable for the numerical simulation of the largescale flow of the atmosphere generated by the steep mountain. Difference between the 6-layer run and the 12-layer run is negligibly small, probably because in this situation (baro-tropic equilibrium state) the effect of vertical resolution is not so important as that of horizontal resolution. Difference among the various schemes is also very small compared with that due to horizontal grid resolution. As a whole truncation errors over a steep mountain are considerably smaller than that has been expected so far by some workers.