First, let's apply some of the results in the preceding section to clouds. For simplicity consider a uniform plane-parallel cloud of thickness hc composed of N water droplets per unit volume, all with the same radius a, per unit volume of cloud. The total volume of water in this cloud per unit (horizontal) cross-sectional area is

where v is the volume of a single cloud droplet, f = Nv is the volume fraction of the cloud occupied by liquid water, and hw is the liquid water path of the cloud expressed as the depth of water that would result if the cloud were compressed into a continuous slab of water. A large value for hw is of order centimeters, and even this puny amount requires clouds thousands of meters thick. Thus the volume fraction f of water in clouds is quite small, of order 10~6. Despite their apparent solidity, clouds are mostly air. It follows from Fig. 5.12 that a slab of liquid water (or ice) a few centimeters thick is insufficient to markedly attenuate visible light of any wavelength. But, as we noted previously, multiple scattering can greatly amplify photon paths. To find out how much we have to estimate the magnitude of the product of the attenuation coefficient a [Eq. (5.74)] and hc. Because the single-scattering albedo for cloud droplets is very close to 1 over the visible spectrum (Fig. 5.15) we have ahc m Khc\ ^—^. (5.84)

The absorption and scattering coefficients are

where Cabs is the absorption cross section of a droplet and Csca its scattering cross section (Secs. 2.8 and 3.5). Cloud droplets are large compared with the wavelengths of visible light

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Figure 5.15: Asymmetry parameter and single-scattering albedo, from UV to IR, of a water droplet 10 |m in diameter. The reflectivity, from Eq. (5.72), is for an infinite medium composed of water droplets this size.

and weakly absorbing in that the product of droplet diameter and the absorption coefficient of water, kw, is much less than 1. Because of this the absorption cross section is approximately Kwv, where v is the droplet volume (Sec. 2.8). This expression is not exact but is good enough for our purposes. The scattering cross section is approximately 2na2. With these approximations and a bit of algebra Eq. (5.84) can be written

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