Although the absorption spectrum has very complex behavior, the band-averaged transmission function averages out most of the complexity. The definition of the transmission guarantees that it decays monotonically as |p1 — p21 increases and the path increases, but in addition the decay is invariably found to be smooth, proceeding without erratic jumps, kinks or other complex behavior. This smoothness is what makes computationally economical radiative transfer solutions possible, and the various schemes for carrying out the calculation of fluxes amount to different ways of exploiting the smoothness of the band-averaged transmission function.

By way of example, the band-averaged transmission function for CO2 is shown for three different bands in Fig. 4.13. The calculation of Tv(p1,p2) was carried out using a straightforward - and very time consuming - integration of the transmission over frequency; at each frequency in the integrand, one must do an integral of KCO2(v, p) over pressure, and each of those k must be evaluated as a sum over the contributions of up to several hundred lines. Temperature was held constant at 296K and a constant mass-specific concentration of .0005 (330ppmv) of CO2 mixed with air was assumed. The pressure p1 was held fixed at 100mb, while p2 was varied from 100mb to 1000mb. This plot thus gives an indication of the upward flux transmitted from each layer of the atmosphere, as seen looking down from the Earth's tropical tropopause. The results are plotted as a function of the pressure-weighted strong-line path, which for constant q and T is q • (p2 — p1)/(2gpo costheta), where the reference pressure po is taken to be 105Pa. Plotting the results this way makes it easier to compare them with theoretical expectations, and also makes it easier to generalize the results to transmission between different pairs of pressure levels, which will have different amounts of pressure broadening. The rationale for using the strong-line path is that the lines are narrow enough that almost all parts of the spectrum are far from the line centers in comparison to the width, and in such cases the collision-broadened absorption coefficient increases linearly with pressure almost everywhere. This behavior is incorrect near the line centers, but the error in the transmission introduced by this shortcoming is minimal, since the absorption is so strong there the contribution to the transmission is essentially zero anyway. This reasoning - based directly on what we have learned from the strong-line limit - is at the basis of most representations of pressure-broadening effects in radiative calculations. Here, we only are using it as a graphical device, since the transmission itself is computed without approximation. Note that the strong line path becomes proportional to the (weak-line) mass path q • (p2 — p1)/(g cos 6>) when p2 ^ p1, with proportionality constant p1/po. In the present calculation, when p2 is at its limit of 1000mb, the path is about 5kg/m2, which is about half the unweighted mass path over the layer. This reflects the fact that the lower pressure over most of the layer weakens the absorption relative to the reference value at p = po.

Apart from noticing that the transmission function is indeed smooth, we immediately remark that the transmission first declines sharply, as portions of the spectrum with the highest absorption coefficient are absorbed. At larger paths, the spectrum becomes progressively more depleted in easily-absorbed wavenumbers, and the decay becomes slower. For the two strongly absorbing bands in the left panel, the transmission curve becomes nearly vertical at small paths, as suggested by the square-root behavior of the strong line limit. There is guaranteed to be a weak-line region at sufficiently small paths, where the slope becomes finite, but in these bands the region is so tiny it is invisible. In fact, the strong line transmission function in Eq. 4.69 fits the calculated transmission in the 575-600 cm-1 band almost exactly throughout the range of paths displayed, when used with the random-overlap modification in Eq. 4.70. For the more strongly absorbing 600-625 cm-1 band the fit is very good out to paths of 1.5 kg/m2, but thereafter the actual transmission decays considerably more rapidly than the strong-line form. This mismatch occurs because the derivation

Figure 4.13: The band averaged transmission as a function of path, for the three different bands, as indicated. In each case, the transmission is computed between a fixed pressure pi = 100m6 and a higher pressure p2 ranging from 100mb to 1000mb. Calculations were carried out assuming the temperature to be constant at 296K, with a constant CO2 specific concentration of q = .0005, and assuming a mean propagation angle cos theta = 1. Results are plotted as a function of the pressure-weighted path for strong lines, q • (p2 — p2)/(2gp0 costheta), where po = 1000mb. In the left panel, the best fit to the strong-line transmission function is shown as a dashed curve; the fit is essentially exact for the 575 — 600cm-1 band, so the fitted curve isn't visible. For the weaker absorption band in the right panel, fits are shown both for the strong line and the Malkmus transmission function, but the Malkmus fit is essentially exact and can't be distinguished.

of the strong-line transmission function assumes that the absorption coefficients within the band approach zero arbitrarily closely: as more and more radiation is absorbed, there is always some region where the absorption coefficient is arbitrarily close to zero, which leads to ever-slower decay. In reality, overlap between the skirts of the lines leads to finite-depth valleys between the peaks (see the inset of Fig. 4.7), and the absorption is bounded below by a finite positive value. The decay of the transmission at large paths is determined by the local minima in the valleys, and will tend toward exponential decay, rather than the slower decay predicted by the strong line approximation.

For the weakly absorbing band shown in the right panel of Fig. 4.13, a hint of weak-line behavior can be seen at small values of the path, with the result that the behavior diverges noticeably from the best strong-line fit. The representation of the transmission can be improved by adopting a two-parameter fit tailored to give the right answer in both the weak and strong limits. The Malkmus model is a handy and widely-used example of this approach. It is defined by

^ S p0 y R2 pi where R and S are the parameters of the fit 5. The parameters can be identified with characteristics of the absorption spectrum in the band by looking at the weak line (small 4) and strong line (large limits. For small the sum of the equivalent widths is S • (po/p1 = , so by comparing with Eq. 4.64 we identify S as the sum of the line intensities. For large the sum is 2v/R2Z whence on comparison with Eq. 4.69 we identify R2 as the sum of 7i(po)Si for all the lines in the band.

5The factor pi/po deals with the difference between the strong line and weak line paths, and is necessary so that the limits work out properly for small and large path. There is some flexibility in defining this factor. It is common to use 2(p 1 + P2)/po to make things look more symmetric in piandp2. This slightly changes the way the function interpolates between the weak and strong limits, without changing the endpoint behavior

The parameters R and S can thus be determined directly from the database of line intensities and widths, though in some circumstances it can be advantageous to do a direct fit to the results of a line-by-line calculation like that in 4.13 instead. One uses the Malkmus equivalent-width formula with the random-overlap transformation given in Eq. 4.70, so as to retain validity at large paths. With the Malkmus model, the transmission function in the weakly absorbing 550-575 cm-1 band can be fit almost exactly. Since the Malkmus model reduces to the strong line form at large paths, it fits the transmission functions in the left panel of Fig. 4.13 at least as well as the strong line curve did. However, it does nothing to improve the fit of the strongly absorbing case at large paths, since that mismatch arises from a failure of the strong-line assumption itself.

The Malkmus model is a good basic tool to have in one's radiation modelling toolkit. It works especially well for CO2, and does quite well for a range of other gases as well. There are other fits which have been optimized to the characteristics of different greenhouse gases (e.g. Fels-Goody for water vapor), and fits with additional parameters. Most of the curve-fit families have troubles getting the decay of the transmission right when very large paths are involved, though if the trouble only appears after the transmission has decayed to tiny values, the errors are inconsequential.

Empirical fits to the transmission function are a time-honored and effective means of dealing with infrared radiative transfer. This approach has a number of limitations, however. We have already seen some inadequacies in the Malkmus model when the path gets large; patching up these problems leads to fits with more parameters, and finding fits that are well-tailored to the characteristics of some new greenhouse gas one wants to investigate can be quite involved. It also complicates the implementation of the algorithm to have to use different classes of fits for different gases, and maybe even according to the band being considered. A more systematic and general approach is called for. The one we shall pursue now, known as exponential sums, has the additional advantage that it can be easily generalized to allow for the effects of scattering, which is not possible with band-averaged fits like the Malkmus model. As a gentle introduction to the subject, let's consider the behavior of the integral where kg is the absorption coefficient for a greenhouse gas G and i is a mass path. This would in fact be the exact expression for the band-averaged transmission for a simplified greenhouse gas whose absorption coefficient is independent of pressure and temperature. In this case, the path i between pressure pi and p2 is simply the unweighted mass path | J" qdp|/(g cos 0), which reduces to q|pi — p2|/(gcos 0) if the concentration q is constant.

The problem we are faced with is the evaluation of the integral of a function f (x) which is very rapidly varying as a function of x. The ordinary way to approximate the integral is as a Riemann-Stieltjes sum, dividing the interval up into N sub-intervals [xj, xj+i] and summing the areas of the rectangles, i.e.

The problem with this approach is that a great many rectangles are needed to represent the complex area under the curve f (x). Instead, we may define the function H (a), which is the sum of the lengths of the intervals for which f (x) < a, as illustrated in Fig. 4.14. Now, the integral can be approximated instead by the sum

Figure 4.14: Evaluation of the area under a curve by Lebesgue integration

Figure 4.14: Evaluation of the area under a curve by Lebesgue integration where we have divided the range of the function f (i.e. [fi, /bj) into M partitions. This representation can be very advantageous if H(f) is a much more smoothly varying function than f (x). To mathematicians, this form of the approximation of an integral by a sum is the first step in the magnificent apparatus of Lebesgue integration, leading onwards to what is known as measure theory, which forms the basis of rigorous real analysis.

The idea is to apply the Lebesgue integration technique to the transmission function defined in Eq 4.72, with the absorption coefficient kg playing the role of f and the frequency v playing the role of x. Thus, if H(a) is the sum of the lengths of the frequency intervals in the band for which kg < a, then H(a) = 0 when a is less than the minimum of kg and H(a) approaches the bandwidth A when a approaches the maximum of kg. The transmission function can then be written approximately as f Kmax M

<r(^)=/ e-KG£dH(Kg) e-(kj+1+kjK/2(H(«¿+i) - H(«)) (4.75)

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