Comparative numerical integration of simple atmospheric models on a spherical grid

Abstract

The non‐divergent barotropic model, the two‐layer quasi‐geostrophic baroclinic model, and the two‐layer balanced baroclinic model of variable static stability are integrated in parallel on a spherical grid with comparable boundary and initial conditions for the case of 00 GCT,3 7 December 1957. The integrations of the barotropic and quasigeostrophic models show the expected accuracy decay to 72 hr, characterized by excessive momentum transport into the jet and an overall failure to predict significant baroclinic development. The balanced model displays an initially similar behavior, although the vertical motions are about twice those of the quasi‐geostrophic case. Beyond 12 hr the balanced model's vertical motions become excessively cellular and of increased magnitude, and a progressively serious distortion of the solutions takes place by 24 hr. The energy transformations of the variable‐stability model indicate a disturbance evolution significantly different from that of the quasi‐geostrophic model, although beyond 12 hr these transformations are themselves seriously influenced by the computed vertical motions. A number of recommendations for further tests are made, of which the improvement of the finite‐difference schemes is probably the most important.

Numerical Integration of the Primitive Equations on a Spherical Grid

“…We made an analysis of the high-frequency oscillation, computational mode, and short waves by comparisons of the results of Runs 1, 2, 3, and 11. As described before, Run 1 was affected by all of these three undesirable elements. In some cases, high-frequency oscillation in the solution of primitive equations is a numerically caused noise.…”

“…In figure 11, the latitudinal distributions of zonal mean of u at the end of the 10th day are illustrated for Runs 1,11,12, and 14. The distribution for Run 14 differs from the others.…”

“…Then, if we see the wind a t the pole from the grid point 2) of System 1, we will observe x p as an eastward component and 7 j p as a northward component. In this case the pole is considered to be the point (1,1). While from the grid point (2, 2), 7 j p will be observed as an eastward component and…”

“…Based upon the previously used grid systems described briefly above we propose a new grid method. The purposes of this study are as follows: (1) to establish a new spherical grid system and set up the elementary rules of finite differencing on the system, (2) to test the usefulness of the new grid system and the computation schemes by integrations of primitive equations for a barotropic model with a free surface, (3) to investigate the ways to suppress a high frequency oscillation as well as a growth of short waves in the results, and (4) to analyze the effects of the different types and the different amounts of viscosity.…”

THE EFFECT OF INITIAL UNCERTAINTIES ON PREDICTIONS MADE WITH SOME BAROTROPIC AND BAROCLINIC MODELS1

“…9 ticity advection and the Laplacian of the thermal advection, the remaining three terns arise due to the divergent part of the wind and the variation of the C o n o h parameter. These terms cannot be explicitly evaluated; hence, the solution of this equation is obtahed by an iterative procedure similar to the one used by Gates and Riegel (1963). After obtahhg a convergent solution of a, the vorticity equation 8can be solved, and numencall integration can be carried out to obtain forecast heights at any future time.…”