## N

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.