The expression for the downward flux follows a similar form. Bj is the blackbody emission from layer j , and the flux at a given level is a weighted sum of the emissions from each layer below (for upward flux) or above (for downward flux) layer ¿. The weighting coefficient ej characterizes the joint effects of the emissivity at layer j and the absorptivity by all layers between i and j.

Exercise 4.4.3 Write down the analogous trapezoidal-rule approximation to I_

Exercise 4.4.4 Write down analogous trapezoidal-rule approximations to 1+ and I_ based on the form of the solution given in Eq. 4.12. What would be the advantages of using this form of solution?

To implement Eq. 4.86 and its variants as a computer algorithm, one generally writes a function which computes the transmission between levels p^ and pj. The rest of the algorithm is independent of the form this function takes, and so one can easily switch from one representation to another (e.g. Malkmus to exponential sums) by simply switching functions. One can equally easily use different representations for different bands. The transmission function requires as arguments the transmission parameters for the band under consideration (e.g. R and S parameters for Malkmus, or the H distribution for exponential sums), as well as enough information to compute the equivalent path. For a well-mixed greenhouse gas, if we are ignoring temperature scaling effects the equivalent path is simply qG 1 |(p2 — p2)|/(pogcos6), and one can simply make the concentration qG and the pressures pi and p2 arguments of the transmission function. For an inhomogeneous path, arising when qG varies with height or one needs to take into account temperature scaling which also varies in height, the path is determined by an integral. In this case, it is inefficient to recompute the path from scratch each time. Since the equivalent path can be computed incrementally using ¿(pi,p2 + Sp) = i(pi,p2) + l(p2,p2 + Sp), it is better to use the equivalent path as an argument to the transmission function, and compute the path from layer i to each layer j iteratively in the same loop in Eq. 4.86 where the weighted emission is computed.

In the preceding algorithm, we have used exponential sums to represent the transmission function appearing in the integral form of the solution to the Schwartzschild Equations. However, because the equations are linear in the fluxes, and because the exponential sum method is a weighted sum of calculations for a number of different absorption coefficients, exactly the same results can be obtained by organizing the calculation in a quite different way. Namely, instead of working from the integral form of the solution, we can work directly with a set of independent Schwartschild equations (Eq. 4.8) - one for each k going into the exponential sum for a given band; as usual, the band would be chosen narrow enough that B(v,T) could be assumed independent of frequency within the band, so we wouldn't need to know anything about which set of wavenumbers each k correponded to. With a 10-term exponential sum, for example, we would solve the Schwartschild equation for each of the 10 values of k, then form a weighted sum of the 10 resulting fluxes. This alternate formulation is not available with band-averaged transmission function models such as the Malkmus model. The weighted differential-equation approach offers a number of advantages over the transmission-function approach. A single solution gives the fluxes at all levels, making optimal use of calculations for preceding levels. This is useful when computing heating rate profiles. Moreover, it is easy to use a high-order integrator to obtain high accuracy with fewer levels. These two advantages make the method computationally attractive, but there is a further advantage that is even more compelling for our purposes: it is straightforward to extend this method to incorporate scattering, whereas it is essentially impossible to do so with band-averaged transmission approaches. We will carry out this program in Chapter 5. One could well ask why the weighted differential-equation approach hasn't completely taken over the business of radiation modelling. There is some evidence that, when scattering is unimportant, pressure and temperature scaling can be done more accurately in band-averaged transmission models, but in large measure the transmission models are a holdover from an earlier day when many radiation calculations were done on paper, and when slow computers required highly tuned special approximations in order to speed up the calculation. It does seem that the exponential sum (and its close cousin the correlated-k) approach are gradually taking over. We have nonetheless chosen to introduce the transmission function approach first, because it corresponds better to what is going on in most existing radiation models, and because the form of the solution gives considerable direct insight into the factors governing the fluxes at a given level.

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