(see Problem ??). Therefore, substituting for X, the expression for T in the strong limit becomes
For strong lines the equivalent width is W = VS(To)-y(po)ts. In this case, the width of the chunk taken out of the spectrum increases like the square root of the path because the absorption coefficient decreases like 1/v'2 with distance from the line center, implying that the width of the spectral region within which the atmosphere is optically thick scales like the square root of the path. Unlike weak lines, strong lines really do take almost all of the energy out of a limited segment of the spectrum. The multiplicative property for transmission is equivalent to an additive property for equivalent width. The nonlinearity of the square root linking path to equivalent width in the strong line case thus means that the band-averaged transmission has lost the multiplicative property. As in our earlier general discussion of this property, the loss stems from the progressive depletion of energy in parts of the spectrum near the line center.
The pressure-weighting of the strong-line path reflects the fact that, away from the line centers, the atmosphere becomes more optically thick as pressure is increased and the absorption is spread over a greater distance around each line. Note that if we choose as the reference pressure po any pressure that remains between p1 and p2, then £s ^ £w as p1 ^ p2. In this case, one can use the strong line path £s regardless of the pressure range, since the strong line path reduces to the correct weak line path for thin layers where weak line approximation becomes valid. A common choice for the reference pressure is the average (p1 + p2)/2 but one could just as well choose one of the endpoints of the interval instead. In the case of a well-mixed greenhouse gas (constant qG) for a nearly isothermal layer, the equivalent path becomes 1/cos ^times qa(v2 — V\)/po, which reduces to the actual mass path qa(p2 — p1)/g if po is taken to be the average. In this case, one gets the correct transmission by using the conventional mass path with absorption coefficients computed for the average pressure of the layer. This is known as the Curtiss-Godson approximation.
In solving radiative transfer problems related to planetary climate, one typically takes the bandwidth A large enough that the band contains a great many lines. For example, there are about 600 CO2 lines in the band between 600 and 625 cm-1. In the weak line limit the transmission is linear in the absorption coefficient, so one can simply sum the equivalent widths of all the lines in the band to obtain the total equivalent width W = J2 W. For strong lines, the situation is a bit more complicated, because of the nonlinearity of the exponential function. For the same reasons one loses the multiplicative property of transmission upon band averaging, one generally loses the additive property of equivalent widths. There is one important case in which additivity of equivalent widths is retained, however. If the lines are non-overlapping, in the sense that they are far apart compared to the width over which each one causes significant absorption, then the absorption from each line behaves almost as if the line were acting in isolation. In this case, each line essentially takes a distinct chunk out of the spectrum, and the equivalent widths can be summed up to yield the net transmission.
The additivity of strong-line equivalent widths breaks down at large paths. Since each Wj increases like the square root of the path, eventually the sum exceeds A, leading to the absurdity of a negative transmission. What is going wrong is that, as the equivalent widths become large, the absorption regions associated with each line start to overlap. One is trying to take away the same chunk of the spectrum more than once. This doesn't work for spectra any more than it works for ten hungry people trying to eat an eighth of a pizza each. One approach which has met with considerable success is to assume that the lines are randomly placed, so that the transmission functions due to each line are uncorrelated. This is Goody's Random Overlap Approximation. For uncorrelated transmission functions, the band-averaged transmissions can be multiplied, yielding
Wi W2 WN
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