latitudes. However, geodesic grids have a nearly homogeneous distribution of points over the sphere.

In mathematics, a geodesic is the equivalent of a straight line, but on a curved surface. On a spherical surface, such as that of Earth, a geodesic is the shortest path between two points, specifically a segment of a great circle. A spherical geodesic grid is defined by spherical, equilateral triangles whose edges are geodesics. One way of defining this grid is to begin with an icosahedron, the geometric solid shown in Fig. 3.10a with 20 triangular faces (major triangles), 12 vertices, and 30 edges, where the vertices touch the surface of a sphere. The vertices may then be connected by geodesics on the sphere, producing spherical triangles. A grid may be created by dividing the major triangles into smaller ones (grid triangles) using a variety of approaches. For example, bisecting each edge of the icosahedron and connecting the bisection points produces four new equilateral triangles for each original one (Fig. 3.10b). The vertices of these new triangles can then be projected onto the sphere along a radial from the center (Fig. 3.10c), and then connected by geodesics to again produce spherical triangles (Fig. 3.10d). Even though the distances between adjacent points look uniform, they are not exactly so. A hint at the asymmetries from one part of the surface to the next can be seen in the fact that the “new” vertex facing the viewer in the upper-center (Fig. 3.10d) is surrounded by six adjoining triangles, while the “original” icosahedron vertex to its right is surrounded by only five. Williamson (1968) and Sadourny et al. (1968) describe another approach for dividing the major triangles into grid triangles, where the inequality in the distance between points is less than that resulting from the method just described. Figure 3.11 shows an example of the distribution of grid points over the sphere.

Some applications of spherical geodesic grids employ the above triangular cells, while others use a related grid with hexagonal cells. To obtain the latter, Voronoi cells are constructed based on the triangular grid, where such cells consist of the set of all points that are closer to a particular vertex than to any other vertex. For the twelve original vertices in the icosahedral grid (e.g., in Fig. 3.11), the Voronoi cells are pentagons. For all the rest, they are hexagons. Figure 3.12 illustrates the geometric relationship between the triangular

In the generation of a spherical geodesic grid, the major triangles of the icosahedron (a) are subdivided, where (b) shows one approach. The vertices of the new triangles are projected (c) onto the sphere that is coincident with the vertices of the icosahedron. Geodesic lines are then drawn between the new vertices to generate spherical grid triangles (d).

                P. Lynch / Journal of Computational Physics 227 (2008) 3431–3444                              3439

dynamical formulation. Many early coupled models needed a flux adjustment (additional artificial heat and moisture fluxes at the ocean surface) to produce good simulations. The higher ocean resolution of HadCM3 was a major factor in removing this requirement. The atmospheric component of HadCM3 has 19 levels and a latitude/longitude resolution of 2.5 · 3.75, with grid of 96 · 73 points covering the globe. The resolution is about 417 · 278 km at the Equator. The physical parameterization package of the model is very sophisticated. The oceanic component of HadCM3 has 20 levels with a horizontal resolution of 1.25 · 1.25 permitting important details in the oceanic current structure to be represented. HadCM3 and HADGEM have been used for a wide range of climate studies which provided crucial inputs to the Fourth Assessment Report (AR4) of the Intergovernmental Panel on Climate Change (IPCC), published in 2007. The development of comprehensive models of the atmosphere is undoubtedly one of the finest achievements of meteorology in the 20th century. Advanced models are under continuing refinement and extension, and are increasing in sophistication and comprehensiveness. They simulate not only the atmosphere and oceans but also a wide range of geophysical, chemical and biological processes and feedbacks. The models, now called Earth System Models, are applied to the eminently practical problem of weather prediction and also to the study of climate variability and mankind’s impact on it.

3. Numerical weather prediction today
It is no exaggeration to describe the advances made over the past half century as revolutionary. Thanks to this work, meteorology is now firmly established as a quantitative science, and its value and validity are demonstrated daily by the acid test of any science, its ability to predict the future. Operational forecasting today uses guidance from a wide range of models. In most centres a combination of global and local models is used. By way of illustration, we will consider the global model of the European Centre for medium-range weather forecasts.

3.1. The European centre for medium-range weather forecasts
Perhaps the most important event in European meteorology over the last half-century was the establishment of the European Centre for medium-range weather forecasts (ECMWF). The mission of ECMWF is to deliver weather forecasts of increasingly high quality and scope from a few days to a few seasons ahead. The Centre has been spectacularly successful in fulfilling its mission, and continues to develop forecasts and other products of steadily increasing accuracy and value, maintaining its position as a world leader. The first operational forecasts were made on 1 August, 1979. The Centre is currently undergoing enlargement. A new Convention has been agreed and is in the process of ratification. The ECMWF model is a spectral primitive equation model with a semi-lagrangian, semi-implicit time scheme and a comprehensive treatment of physical processes. It is coupled interactively to an ocean wave model. Its spatial resolution is 25 km and there are 91 vertical levels. Initial data for the forecasts are prepared using a four-dimensional variational assimilation scheme, which uses a large range of conventional and satellite observations over a 12-hour time window. A sustained and consolidated research effort has been devoted to exploiting quantitative data from satellites, and now these observations are crucial to forecast quality. ECMWF produces a wide range of global atmospheric and marine forecasts and disseminates them on a regular schedule to its Member States. The primary products are listed here (explanations of technical terms
will follow).
Forecasts for the atmosphere out to 10 days ahead, based on a T799 (25 km) 91-level (L91) deterministic model are disseminated twice per day.
Forecasts from the ensemble prediction system (EPS) using a T399 (50 km) L62 version of the model and an ensemble of 51 members are computed and disseminated twice per day.
Forecasts out to one month ahead, based on ensembles using a resolution of T255 (78 km) and 62 levels are distributed once per week.
Seasonal Forecasts out to six months ahead, based on ensembles with a T159 (125 km) L40 model are disseminated
once per month.