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Ludwig Umscheid
and
Peter R. Bannon

Abstract

Three finite-difference global grids [the original Kurihara (OK), a modified Kurihara (MK) and latitude- longitude (LL)] are tested by comparing numerical solutions with a barotrpic free-surface model to a high-resolution control run and by comparing forecasts with a general circulation model to observations.

With the free surface model, 30-day integrations are made for three different resolutions of each grid and with three initial conditions two mathematical patterns and one 500 mb observed field. The LL grid performed well on the mathematical patterns, especially the case with zonal wavenumber 4. The numerical solutions with the high resolution MK grid also performed satisfactorily for both mathematical patterns. The OK grid did not perform as well, particularly on the case with zonal wavenumber 1. For the observed case, the LL grid in general had lower rms errors although the solutions did not depend as strongly on the three different grid types as the solutions for the mathematical patterns.

For the three-dimensional cases, the GFDL nine-level model was used for 14-day forecasts for observed conditions in March and 3-day forecasts in November. Forecast sensitivity to the different grids is low for short range. The MK grid had the lowest 500 mb rms errors for the duration of both forecasts. Both the MK and LL grids were free of the problem of anomalously high geopotential heights over the North Pole that occurred with the OK grid.

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LUDWIG UMSCHEID JR.
and
M. SANKAR-RAO

Abstract

Two smoothing techniques are tested as a practical means of allowing a larger time step in the numerical integration of a primitive equation free-surface model. The numerical integration uses a finite-difference grid and operators based on the method of Kurihara and Holloway.

A time step six times larger can be used with a corresponding six-fold decrease in computer time, by implementing the weighted averaging procedure given by Langlois and Kwok in their description of the Mintz-Arakawa general circulation model. A Fourier filtering scheme permits the use of a time step 10 times larger, and results in a five-fold improvement in computer time. After 10 days, the geopotential and wind fields obtained with these techniques still closely resemble the unsmoothed fields, the closest correspondence being found with the Fourier filtering technique.

In another set of experiments, steady-state solutions to special cases of the governing analytic equations are used as initial conditions in a test of the accuracy of the grid and operators. These steady-state solutions are preserved satisfactorily for the 10-day integration period.

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M. SANKAR RAO
and
LUDWIG UMSCHEID JR.

Abstract

Kurihara and Holloway have proposed an integration scheme that offers advantages in the problems of geophysical fluid dynamics by rigorously conserving mass and energy. We have attempted to investigate the accuracy of the Kurihara and Holloway method by numerical experiments, applying it to a problem for which an approximate analytic solution is available.

For the case in which the planetary wave number is 4, we find that with the equal-area grid and with a latitude grid spacing of 4.5°, the planetary wave is destroyed by truncation errors within 5 days. In order to achieve a solution with acceptable accuracy, in which the planetary wave character is retained for a minimum of 10 days, the grid spacing near the Pole has to be decreased by a factor of 9.

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