## Conservative Scattering Reflection and Transmission

Although the solutions in the previous section are illuminating (forgive the pun), they are not especially realistic. So let's consider a more realistic problem in which photons that leak out the bottom boundary never return (i.e., F-\ = 0 at r), either because there is nothing to return them or they are absorbed before doing so; the downward irradiance at the top boundary is F0. With these boundary conditions the solution to Eq. (5.49) yields for the reflectivity R and transmissivity T

As required by radiant energy conservation, R + T = 1. But wait! We've seen Eq. (5.51) before. It has the same form as Eq. (5.14), the reflectivity of a pile of plates, the only difference being that we called t = NRi = N(1 - g)/2 [seeEq. (5.21)] the optical thickness. We did so because calling this quantity one-half the scaled optical thickness might have been perplexing. But now that you are inoculated against perplexity we can rewrite history and say that in Eq. (5.14) we really meant to write r for N. Does it make sense that N, the number of plates, is analogous to optical thickness? We think it does. Although N can take on only discrete values (1, 2,3 ...) whereas r is continuous, r = 1 corresponds (roughly) to a probability of 1 for a photon to be scattered at least once, r = 2 corresponds to a probability of 1 for a photon to be scattered at least twice, and so on. But this is analogous to what happens when we add plates to a pile. Every plate added (in discrete steps) increases the probability of a photon being scattered. With 2 plates a photon is likely to be scattered (reflected) at least twice, with 3 plates, at least 3 times, and so on. With the wisdom of hindsight, and a cup of physical intuition, we could have solved the equation of transfer (without even being aware of its existence) to obtain Eqs. (5.51) and (5.52) knowing the solution to the pile-of-plates problem and understanding the physical meaning of optical thickness. Moreover, now we don't have to plotEq. (5.51): we already did so (Figs. 5.7 and 5.8).

FromEq. (5.51) it follows that reflectivity asymptotically approaches 1 as optical thickness approaches infinity. But it is not necessary for the medium to be infinite in order for it to be optically thick, by which is meant indistinguishable from an infinite medium. For example, the average contrast threshold of the human eye is 2%, which means that humans cannot detect the difference between two light sources with luminances different by less than 2%. If we adopt this criterion, a scaled optical thickness r(l — g) of around 100 corresponds to a reflectivity of 0.98, which is within 2% of the reflectivity of an infinite medium.

With the help of Eq. (5.43) we can write Eq. (5.52) in a form that is easier to interpret:

This equation tells us that attenuation of incident light is a consequence only of downward photons scattered upward. Downward photons scattered downward continue to contribute to the downward irradiance.

With the assumption that r*/2 <C 1, Eq. (5.52) can be expanded in a power series and truncated after the second term:

But this is also the first two terms in the expansion of the exponential function:

Exponential attenuation by scattering would prevail only if multiple scattering were negligible, that is, if photons scattered out of the forward direction never returned to that direction. Contrary to what the term "multiple" might lead you to think, attenuation for which multiple scattering is taken into account is always less than attenuation for which single scattering is assumed (Fig. 5.11). No equations or figures are needed to prove this. On the basis of single-scattering arguments, a photon scattered out of its original direction never returns to that direction, whereas multiple scattering gives the photon additional chances to be scattered back into that direction.

Clouds often are said to be white (strictly, the spectrum of light reflected by them is not appreciably different from that of the light illuminating them) because scattering of visible light by cloud droplets is independent of wavelength. This leads to the notion (which takes on various guises) that when a multiple-scattering medium is observed to be white, this infallibly signals the presence of "big" (compared with the wavelength) particles. Here we have a failure to distinguish between a necessary and a sufficient condition. It is indeed true that cloud particles (water droplets, ice particles) are sufficiently large compared with the wavelengths of visible light that they scatter more or less independently of wavelength (Fig. 3.12). This is a sufficient condition for the cloud to be white upon illumination by white light, but it is not necessary, which you can demonstrate for yourself. Paint the insides of two aluminum pie pans black. Fill them with water. To one add just a few drops of milk, to the other so much milk that the bottom of the pan is not visible. Observe these two pans side by side in bright sunshine. The dilute (optically thin) suspension has a bluish cast, whereas the more concentrated (optically thick) suspension is white. And yet the particles in both pans are the same. This is yet another example in which single-scattering arguments applied to a multiple-scattering medium lead to erroneous conclusions. The individual particles in milk are sufficiently small that they scatter more at the short-wavelength end of the visible spectrum

 1.0 0.9 0.8 si 0.6 is E 0.5 s 0.4 n to H 0.3 0.2 0.1 Scaled Optical Thickness Figure 5.11: Because of multiple scattering, transmission by a plane-parallel medium decreases less rapidly (solid curve) with scaled optical thickness than exponentially (dashed curve). than at the long-wavelength end. But in an optically thick medium of many such particles the cumulative effect of multiple scattering is to wash out this spectral dependence, which follows from differentiating Eq. (5.51) to obtain the wavelength dependence of reflectivity: The spectral dependence of r* is a consequence of that of the scattering coefficient ¡3 and of g (usually much weaker). Consider the two limits, optically thin and optically thick: These are markedly different spectral dependences and yet the scatterers are the same. What is different is only their amount. For the optically thin medium [Eq. (5.57)], the spectral dependence of reflectivity is essentially that of the individual scatterers; for the optically thick medium [Eq. (5.58)], reflectivity is essentially independent of wavelength. At least two caveats should accompany pronouncements about the whiteness of clouds. You may have noticed pastel colors, called iridescence, in thin clouds or at the edges of thick ones when looking toward the sun. Or you may even have seen colored rings a few degrees across, called the corona, around the sun or moon through thin, even imperceptible, clouds (see Sec. 8.4.1). The key word here is "thin". Although total scattering (in all directions) of visible light by cloud droplets is to good approximation independent of wavelength, angular scattering is not: light is scattered more in some directions than in others depending on wavelength. We can express this within the context of our simple two-stream theory by saying Wavelength (pm) Figure 5.12: Absorption length (inverse absorption coefficient) of pure ice and liquid water over the visible spectrum. The data for liquid water were taken from Querry et al. (1991), those for ice from Warren (1984). Wavelength (pm) Figure 5.12: Absorption length (inverse absorption coefficient) of pure ice and liquid water over the visible spectrum. The data for liquid water were taken from Querry et al. (1991), those for ice from Warren (1984). that g depends on wavelength. This dependence is weak but may be observable in thin clouds. Here is yet another example of how multiple scattering qualitatively changes what is observed. Although clouds seen by reflected light are usually white (again, assuming incident white light), light transmitted by very thick clouds may have a bluish cast. This is not evident from Eq. (5.52) nor can it be because it was derived under the assumption that absorption is negligible. And it often is - but not always. Pure water, including ice, has an absorption minimum in the blue-green part of the visible spectrum and rises sharply toward the red (Fig. 5.12). But absorption by water at all visible wavelengths is weak in the sense that visible light has to be transmitted many meters through water before being perceptibly attenuated by absorption (which is why we don't think of a glass of water as a glass of blue dye). Although the total liquid water paths of thick clouds are of order a centimeter or less, multiple scattering in very thick clouds can greatly magnify the effective transmission distance. We try to make this point clearer in Section 5.3 on multiple scattering in absorbing media.