Comparison is made of the numerical solutions of the Rossby-Haurwitz waves by the shallow water equations on a sphere using staggered and non-staggered grid systems. Integrations are made with a high resolution grid net with the 2.50° grid size and a low resolution grid net with the 5° grid size for the two grid systems. The results of the high resolution grid nets of the two grid systems agree well with each other. These results also agree with those of other studies using a spectral method and a very high resolution latitude-longitude grid. Therefore, we may say that these solutions are converging. The results of the low resolution grid nets are considerably different from those of the high resolution grid nets. The solutions are also different depending on the grid system and the choice of the initial conditions.
Truncation errors of the latitude-longitude grid and the skipped grid are also reexamined. As shown by Williamson and Browning (1973), in the latitude-longitude grid, the absolute value of the error of a second-order finite-difference approximation becomes of first order near the pole where cosφ(φ: latitude of a grid point) is of the same order as Δφ (the latitudinal grid size). However, it is shown that if Δφ→0 the area of the region of first order accuracy becomes smaller and hence its erroneous effect may become negligible. There is a possibility in some skipped grids that the error of finite-difference approximations is finite due to the interpolation of the variables. The error appears not only near the pole but also in lower latitudes. However, the error may become of smaller order if the flow has a very small deviation in the zonal direction (nearly zonal flow) and hence the solution may converge to that obtained with the latitude-longitude grid when Δφ→0. In the case of the cross-polar flow the error would not be smaller whatever high resolution grid net may be used.