We have now laid out all the ingredients that go into a real gas radiation model, and are ready to begin assembling them. The ingredients are:

• A means of computing the band-averaged transmission over a specified wavenumber range

• The band-averaged integral (Eq. 4.11,4.12, or 4.13) giving the band-averaged solution to the Schwartzschild equation in terms of the preceding transmission functions and the recipe is:

• Divide the spectrum into bands of a suitable width

• Prepare in advance: Malkmus coefficients or exponential sum coefficients H(ln k) for each band, for each greenhouse gas present in significant quantities in the atmosphere

• Program up a function to compute the band averaged transmission in each band, using the coefficients prepared in the previous step.

• If there are multiple greenhouse gases, do the preceding for each individual greenhouse gas and combine the resulting transmission functions, allowing suitably for the nature of the overlap between absorption bands of the competing gases (for advanced chefs only!)

• Use the resulting transmission in a numerical approximation to the integral in Eq. 4.11,4.12 or 4.13 in each band to get the band-averaged fluxes.

• Sum up the fluxes in each band to get the total flux

• Serve up the fluxes to the rest of the climate model and enjoy

In the typical climate simulation application, one is given a list of values of temperature and greenhouse gas concentrations tabulated on a finite grid of pressure levels pj for j = 0, ...N, and one must compute the fluxes based on this information. Either of Eq. 4.11 or 4.12 provides a suitable basis for numerical evaluation when one is working from atmospheric data tabulated on a grid. In writing down the approximate expressions for the flux, we will adopt the convention that j = 0 at the top of the atmosphere and that j = N represents the ground. We shall use the superscript (k) to refer to quantities averaged or integrated over the band k, centered on frequency v(k) and having width A(k). Let's define the gridded quantities: