Numerical Integration of the Quasi-Geostrophic Equations for Barotropic and Simple Baroclinic Flows

An n-level generalization of the 2½-dimensional model is derived by specialization of the complete three-dimensional quasi-geostrophic equations. In the case n = 1, it reduces to the two-dimensional single-layer barometric model. In the case N = 2, it reduces to the double-layer barotropic model, or — what is shown to be mathematically equivalent —the 2½-dimensional model. Methods of numerical integration of the 2- and 2½-dimensional equations, and the machine requirements for such integrations, are discussed. The results of a series of six two-dimensional and six 2½-dimensional forecasts for 12 and 24 hours are presented. Although the 2½-dimensional forecasts are noticeably superior to the two-dimensional forecasts, it is apparent that considerable improvement will be possible with models in which there are fewer artificial constraints. A method of integration is therefore proposed for the n-level generalization of the 2½-dimensional model, and computation schemes are outlined for the general three-dimensional quasi-geostrophic equations. The semi-Lagrangian coordinate system with potential temperature as vertical coordinate is shown to exhibit favorable properties for machine integration.