NUMERICAL INTEGRATION OF THE PRIMITIVE EQUATIONS ON A SPHERICAL GRID

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Abstract
A new spherical grid system whose grid density on the globe is almost homogeneous is proposed. The elementary rules of finite differencing on the grid system are defined so that a desirable condition for numerical area integration is satisfied. The integrations of primitive equations for a barotropic atmosphere with free surface are made. The patterns of initial fields are the same as Phillips used in 1959 for a test of his map projection system and computation schemes. Ten test runs are performed for a period of 16 days. Three of these are without viscosity and integrated with different time integration schemes. Four runs include the effect of non-linear viscosity with different coefficients, and the remaining three are computed with different amounts of linear viscosity. A noticeable distortion of the flow pattern does not occur in an early period in any run. Analyses of the results suggest that the damping of high frequency oscillation of both long and short wavelengths can be achieved by an iterative time integration scheme, e.g., the modified Euler-backward iteration method, with little effect on the prediction of a trend of the meteorological wave. Either the non-linear or the linear viscosity can be used to suppress a growth of short waves of both low and high frequency modes, if the optimum amount of viscosity for that purpose does not exceed the amount representing the actual diffusion process in the atmosphere. Analyses are also made concerning the effects caused by different specifications of the parameter in the viscosity term in the equations.

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