FIGURE 14.8 (a) Meaning of equivalent width, IV; (b) Doppler and Lorentzian line-shapes for equivalent half-widths; (c) transmission curves for an absorption line for a weak and strong absorber, respectively (adapted from Lenoble, 1993).

The absorption cross section a depends on the frequency even for a single absorption line due to various line-broadening processes that impart a finite width and particular shape to the absorption line. Figure 14.8b, for example, shows a typical line-shape due to collisional broadening, the Lorentz

FIGURE 14.8 (a) Meaning of equivalent width, IV; (b) Doppler and Lorentzian line-shapes for equivalent half-widths; (c) transmission curves for an absorption line for a weak and strong absorber, respectively (adapted from Lenoble, 1993).

lineshape, and a typical shape due to Doppler broadening. Under atmospheric conditions near the earth's surface, the linewidth is determined primarily by collisions; i.e., the Lorentz half-width is much larger than the natural linewidth or that due to

14. GLOBAL TROPOSPHERIC CHEMISTRY ANL) CLIMATE CHANGE

Doppler broadening. Collisional broadening becomes less important at the lower pressures found at higher altitudes, so that the Lorentzian half-width and the Doppler half-width become comparable at altitudes of approximately 30-40 km.

The absorption cross section can be expressed as the product of two factors, an intrinsic line strength S and a shape factor, gx, which depends on the distance x = (u — u(]) from the line center:

xs to x_s around the line center v{), where the absorption is saturated, and one in the wings, where the absorption is weaker:

center winds'

In the center region from xs to where the absorption is strong, e~NISSx -> 0 and

gx is the normalized shape factor for which jyiy_gxdx = 1. The equivalent linewidth in Eq. (E) thus becomes

In the wings, a. Weak-Absorber Regime

For weak absorptions, i.e., when the combination of concentration and path length Nl is small, (1 — e-Nlsz) ~ NISg. Equation (H) becomes

where the factor of 2 takes into account symmetrical absorption in the wings both on the low- and high-frequency sides of the band center. Thus for strong absorption,

since the integral is the normalized shape factor, which is unity. The absorbed radiance Lfl = L\VW, so that the net absorption varies linearly with (Nl), i.e., with the column burden of the absorbing gas. This is what is known as the weak-absorber regime and generally applies to such greenhouse gases as the chlorofluorocarbons. However, it should be noted that even for some trace gases such as CF4, the weak-absorber regime is only obeyed up to ~0.1 ppb (current atmospheric concentrations of CF4 are -0.07 ppb; IPCC, 1996). At f ppb, significant deviations are found because CF4 has unusually sharp lines that saturate in the center and that are overlapped by the absorption bands of other atmospheric gases (Freckleton et al., 1996).

b. Strong-Absorber Regime

However, when the combination of concentration, N, and path length, /, is not small, the weak-absorber approximation is not valid, and the line-shape needs to be taken into account. The reason for this is that in the limit when absorption of light is already saturated at the center of the line, absorption due to added gas occurs only in the wings of the absorption line, which is sensitive to the lineshape. In the case of saturation at the center of the absorption line as shown in Fig. 14.8c, the absorption can be thought of as occurring in two regions, one from and the lineshape in the center of the line is not important, whereas that in the wings at frequencies beyond xs is.

Let us return to the definition of equivalent linewidth in Eq. (H). Since lines at most pressures of interest here can generally be described as Lorentzian in shape, the shape factor in Eq. (H) is given by

where yL is the Lorentzian line half-width (i.e., peak width at half-maximum) and x = (v - v{)) as before. This can be substituted into Eq. (H) and solved as described in detail elsewhere to get an expression for the equivalent width (e.g., see Liou, 1980; Goody and Yung, 1989; and Lenoble, 1993). The result for the limit of a strong absorber where absorption in the wings is important can be obtained readily for frequencies in the wings such that x » yL. In this case, the lineshape factor reduces to

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