Note that the factor of πr2 can be factored out, giving
which gives a value of 246 to 248 kelvin - about -27 to -25 °C - as the Earth's average temperature T. This is approximately 35 degrees colder than the average surface temperature of 282 K. This is because the above equation attempts to represent the radiative temperature of the earth, and the average radiative level is well above the surface. The difference between the radiative and surface temperatures is the natural greenhouse effect.
This very simple model is quite instructive, and the only model that could fit on a page. But it produces a result we are not really interested in - the radiative temperature - rather than the more useful surface temperature. It also contains the albedo as a specified constant, with no way to "predict" it from within the model.
The radiative-convective models have advantages over the simple model: they can tell you the surface termperature, and the effects of varying greenhouse gas concentrations on the surface temperature. But they need added parameters, and still represent by one point the horizontal surface of the earth.
and parametrisations which handle other processes: these include
radiation (solar/short wave and terrestrial/infra-red/long wave)
land surface processes and hydrology
The method by which AGCMs discretise the fluid equations may be the familiar finite difference method or the somewhat harder to understand spectral method. Typical AGCM resolution is between 1 and 5 degrees in latitude or longitude: the Hadley Centre model HadAM3, for example, uses 2.5 degrees in latitude and 3.75 in longitude, giving a grid of 73 by 96 points; and has 19 levels in the vertical.
Oceanic GCMs (OGCMs) model the ocean (with fluxes from the atmosphere imposed) and may or may not contain a sea ice model.
Coupled atmosphere-ocean GCMs (AOGCMs) combine the two models. They thus have the advantage of removing the need to specify fluxes across the interface of the ocean surface. These models are the basis for sophisticated model predictions of future climate, such as are discussed by the IPCC.
AOGCMs represent the pinnacle of complexity in climate models and internalise as many processes as possible. They are the only tools that could provide detailed regional predictions of future climate change. However, they are still under development. The simpler models are generally susceptible to simple analysis and their results are generally easy to understand. AOGCMs, by contrast, are often as hard to analyse as the real climate system.
The most modern AOGCMs simulate the observed warming over the past 150 years,
when forced by observed changes in "Greenhouse" gases and aerosols
Note that global climate models, whilst very similar in structure to (and often sharing computer code with) numerical weather prediction models are nonetheless logically distinct: see weather vs climate for details.