We have seen in Section 6.4 that the equations for radiative transfer in a nonscatter-ing "gray" atmosphere are relatively simple. However, these equations are not applicable to the analysis of sunlight reflected by clouds in planetary atmospheres and are hence only of use in modeling the thermal-IR spectra of the planets. Even in the thermal-infrared, however, neglecting the scattering effects of atmospheric aerosols can sometimes lead to errors, especially if cloud particles are of a size approximately equal to or greater than the wavelength. The scattering effect of aerosols greatly complicates the equations of radiative transfer, as we shall see in Section 6.6. However, before we can investigate the effects of scattering, we must first introduce the basic definitions of scattering parameters and how the scattering properties of individual particles may be calculated.

Consider a single photon of wavelength A incident on a particle that is then scattered forward at an angle 9 to the original direction (Figure 6.12). This angle is defined as the scattering angle and in an experiment where light is incident upon

Figure 6.12.

Scattering angle definition.

Figure 6.12.

such a particle, the numbers of photons scattered into different directions will, for spherically symmetric particles, be a function of this scattering angle only. [NB: For nonspherical particles, discussed further in Section 6.5.3, the scattering efficiency will in general also depend on the orientation of the particle and the rotational angle of the vector of the scattered photon about its initial direction.] The function that gives the probability that a photon will be scattered into an element of solid angle dQ and scattering angle 0 is known as the phase function p{9) and by convention is normalized such that p(0) dQ:

Note that in some schemes this integral is normalized to unity, which can lead to confusion! The phase function itself is a function of wavelength, particle composition, mean particle radius, and particle shape.

The probability that a photon will actually be absorbed or scattered by a particle, regardless of direction, depends upon the absorption and scattering cross-sections cabs and ^sca, respectively. These cross-sections also depend on the wavelength, particle composition, and mean particle radius and shape. The extinction cross-section is defined as o-ext = a\,ca + o-abs, and the single-scattering albedo w is defined as the ratio w = ^sca . (6.64)

The absorption, scattering, and extinction scattering efficiencies 2abs, 6sca, and gext are defined as the ratios of the respective cross-sections to the geometric cross-sectional area of the particles.

When the wavelength is very much larger than the particle size, scattering particles tend to behave as simple dipoles and we have the condition for Rayleigh scattering that we referred to in Chapter 4. The phase function for Rayleigh scattering may be shown from standard electromagnetic theory to be (e.g., Goody and Yung, 1989)

and thus the probability that photons will be scattered at a scattering angle between 0 and 0 + d0 is

Such dipole scatterers are purely scattering and thus have zero absorption cross-sections. Their scattering cross-section varies with wavelength as 1 /A4 as described in

Chapter 4 (Equation 4.34) and the most familiar example of such scattering is in the Earth's atmosphere, where the molecules of N2 and O2 scatter a fraction of incident sunlight in all directions. Clearly, from Equation (4.34), blue light is scattered more effectively than red light leading to the familiar blue sky seen from the surface of the Earth (in the absence of cloud!).

Quantum mechanically, an atom or molecule Rayleigh-scatters a photon by first absorbing it and becoming excited to an intermediate or virtual state, whereupon it immediately relaxes back to its initial state, releasing a photon with the same wavelength in a direction governed by Equation (6.65). However, it is also possible that the atom or molecule relaxes back to a different state, thus releasing a photon of either longer or shorter wavelength than the original photon. Stokes Transitions lead to scattered photons with longer wavelengths than the incident light, while ^nti-^tokes Transitions lead to scattered photons with shorter wavelengths. The phenomenon is called Raman scattering and is usually rather weak compared with Rayleigh scattering and thus may usually be neglected. An exception to this is in the case for the giant planets (occurring in the UV spectra) of Uranus and Neptune where distinct solar spectrum features appear shifted to longer wavelengths in the observed albedos of these planets by Raman scattering associated mainly with the rotational S(Q) transition of hydrogen molecules (giving a wavenumber shift of 354 cm-1) and to a much lesser extent the S(1) and Q1(1) transitions (shifted by 587 cm-1 and 4,161cm-1, respectively). Raman scattering in the outer planet atmospheres is described in detail by Karkoschka (1994).

Should scattered photons have considerably less energy than incident photons, and thus significantly longer wavelengths, but be released quickly (within roughly 1Q-7 s), then the effect is sometimes also known as ^Morescence. It is observed that many household materials glow, or fluoresce, under UV illumination (Hecht and Zajac, 1974). If there is an appreciable delay in the release of lower energy photons, sometimes several hours, then the effect is known as ^«os^«orescence.

For particles that have a non-negligible size compared with the wavelength, Rayleigh scattering no longer applies, and calculation of the phase function and extinction cross-section becomes more complicated. However, provided the aerosol particles are spherical (and are thus liquid), and provided that the complex refractive index (nr + in;) as a function of wavelength is known, Maxwell's equations may be solved analytically via a method known as Mie theory to calculate the scattering properties. This method deals with the classical case of a dielectric sphere interacting with a plane electromagnetic wave, and is too complex to be covered in detail here. The reader is referred to a number of more detailed references for further information: Goody and Yung (1989); Hanel et a/. (2QQ3); and Hansen and Travis (1974). Using Mie theory, gext, to, and ^(0) may all be calculated as a function of wavelength. The typical

Figure 6.13. Mie scattering calculation of gext as a function of wavelength for particles of refractive index (1.4 + 0/). The solid line shows the properties of particles with a single radius of 1 ^m while the dotted line shows the properties if there is a small distribution of particle size about 1 ^m. The spectra are plotted as a function of both wavelength and size parameter, defined as 2wr/A, where r is the radius and A the wavelength.

Figure 6.13. Mie scattering calculation of gext as a function of wavelength for particles of refractive index (1.4 + 0/). The solid line shows the properties of particles with a single radius of 1 ^m while the dotted line shows the properties if there is a small distribution of particle size about 1 ^m. The spectra are plotted as a function of both wavelength and size parameter, defined as 2wr/A, where r is the radius and A the wavelength.

variation of gext with wavelength is shown in Figure 6.13. From this figure, together with the calculated phase functions, the bulk scattering properties of spherical liquid aerosols are found to be:

(1) For particles that are small compared with the wavelength, the extinction cross-section tends to 1/A4, and phase functions tend toward the Rayleigh or dipole scattering case discussed earlier.

(2) A particle is most efficient at scattering light with wavelength approximately equal to its own radius.

(3) For particles that are large compared with the wavelength, it is found that the amount of light diffracted is equal to the amount striking the particle (independent of particle shape and refractive index) and thus gext = 2 (i.e., its extinction cross-section is twice its geometric cross-section). It is also found that the phase function becomes more and more forward scattering and in the limit where the particle is very much greater than the wavelength of the incident light, p(9) = 8(9), where 8(9) is the Dirac delta function.

A number of analytical/numerical methods exist for calculating the scattering behaviour of nonspherical scatterers (i.e., ice crystals) with electromagnetic radiation. A particular example is for rotationally symmetric nonspherical scatterers in fixed or random orientations, which can be modeled using T-matrix theory (Mishchenko et al., 1996, 2006). These methods add a further layer of complexity. Fortunately, the nonspherical nature of real ice crystals may often be ignored since a set of randomly orientated crystals is, to a first approximation, indistinguishable from a set of spheres with the same mean radius. However, if crystals become aligned in some way through dynamics or other effects, the difference can become significant. The general case of nonspherical particles is outlined by Goody and Yung (1989).

The phase functions calculated for both spherical particles (using Mie theory) and nonspherical particles do not have a simple analytical expression. However, it is sometimes useful to have a more simple parameterized form of the phase function to use in radiative transfer models that can reasonably well approximate real phase functions of particles. One such representation is the double Henyey-Greenstein representation p(')={ f (1 + *2 - 2^ cos 0)3/2 + (1 - f )(1 + ,2 - 2<icos 0)3/2} (6-67)

where g1 is the asymmetry of the forward-scattering lobe (varying between 0 and 1); g2 is the asymmetry of the back-scattering lobe (varying between 0 and -1); and f is the fractional contribution (between 0 and 1) of the forward-scattering part. Calculated Henyey-Greenstein phase functions for a range of parameters are shown in Figure 6.14. An example of the use of Henyey-Greenstein phase functions is that scattering radiative transfer models may be optimized to use them. Hence, by approximating real phase functions by Henyey-Greenstein functions, great reductions in computation times may be achieved.

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