Numerical Integration of the Primitive Equations on a Spherical Grid

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 t o suppress a growth of short waves of both low and high frequency modes, if the optimum amount of viscosity for t h a t 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.

A Stretched Icosahedral Grid by a New Grid Transformation

“…Section 4 presented an application of the proposed transformation to the icosahedral grid. The new transformation can be applied to other homogeneous grid systems such as the cubic grid (e.g., non-conformal grids such as a gnomonic grid, Rancic et al 1996, and the Kurihara reduced grid, Kurihara 1965).…”

Atmospheric Predictability

“…In those days, however, NWP models were far from perfect, and much effort had been made to improve the models. There were many important contributions by Japanese scientists to the improvements, including novel and stable finite-difference methods developed in the 1960s, such as the Kurihara (1965) grid, the Arakawa (1966) Jacobian and the Matsuno (1966) scheme, and sophisticated parameterization schemes of sub-grid scale phenomena developed in the 1970s, such as the Arakawa and Schubert (1974) cumulus parameterization and the Mellor and Yamada (1974) turbulence closure models for planetary boundary layers.…”

A Prediction Experiment with a Global Model of the Kurihara-Grid

“…The grid was first designed by Kurihara (1965) with the idea of having a homogeneous density of gridpoints on the spherical surface. In his model, "kinetic energy conservation" in the finite difference scheme was not automatically guaranteed.…”