ATMOSPHERIC OPTICAL PHENOMENA AND RADIATIVE TRANSFER BY STANLEY DAVID GEDZELMAN AND MICHAEL VOLLMER Sky colors, rainbows, and halos are simulated using models that include light scattered as it passes through clear air and clouds of finite optical depth. ivid rainbows, ice crystal halos, coronas, iridescence, glories, mirages, sky colors, and crepuscular rays have Valways inspired awe and wonder. This makes simulating atmospheric optical phenomena both a scientific and aesthetic undertaking. Atmospheric optics has a venerable history (Pernter and Exner 1922; Minnaert 1993; Humphreys 1940; Tricker 1970; Greenler 1980; Meinel and Meinel 1983; Lynch and Livingston 2001), because the phenomena appear so simple and striking, and because scientists emphasized this branch of atmospheric science at a time when it was far more difficult to examine large-scale weather systems. Discoveries made about or involving the rainbow by Rene Descartes, Isaac Newton, and Thomas Young rank among the early triumphs of the scientific revolution (Boyer 1987). All optical phenomena are produced when air molecules, aerosol particles, or hydrometeors either scatter or absorb light as it passes through the atmosphere. Many of the observed features of the optical phenomena can be reproduced by applying a scattering theory of light to a single particle. This can be done at various levels of complexity. The most accurate and perhaps most intricate Rainbow. " University Corporation for Atmospheric theories involve • i Research, Photo by Carlye Calvin BAH5- AMERICAN METEOROLOGICAL SOCIETY Unauthenticated | DownloadedAPRIL 2008 10/09/21 01:28 AM UTC solving Maxwell's equations with appropriate bound- flattening of drops (Fraser 1983), while models of ary conditions. Thus, for example, rainbows, coronas, complex halo displays include a number of ice crystal and glories are assumed to be generated by light scat- shapes and orientations (Greenler 1980). Even so, tered by homogeneous spheres. This was the approach these theories still essentially treat light as if it were taken by Mie (1908), and is often referred to as Mie scattered once by a single "integrated" particle. theory (Bohren and Huffman 1983). In fact, all optical phenomena are produced when Much simpler approximations to scattering can light is scattered as it passes through a cloud, a rain reproduce many of the principal features of most opti- shaft, and the atmosphere. Light reaching the ob- cal phenomena. Thus, for example, many features of server may be scattered any number of times. Except rainbows and halos are explained by using the geomet- when the sun is near the horizon, most sunlight ric optics of reflection and refraction as an approxi- passes through clear air and through very tenuous mation to scattering by spherical (or near spherical) clouds without being scattered, while most sunlight raindrops and by simple hexagonal ice crystal prisms. is scattered several times before exiting thick clouds. In a similar manner, many features of coronas can be Thus, as was noted by Meyer (1929), the brightest described using the Fresnel-Kirchhoff formula for halos are produced by clouds of modest optical depth diffraction as an approximation to light scattered by r,,, because extreme tenuous clouds contain too few clcr cloud droplets (Fowles 1989). Similarly, some features ice crystals to scatter much light, while thick clouds of sky color are explained by the simple theory first de- contain so many ice crystals that light is scattered rived by Rayleigh to express light scattered by particles many times before exiting the cloud so that halos much smaller than the wavelength of light, such as air and coronas and even direct sunlight are blurred into molecules, and by Mie theory to represent scattering incoherent, almost isotropic cloud light. by aerosol particles, even though most aerosol particles Optical depth and optical thickness are two terms are neither homogeneous nor spherical. we use here to express how effectively a layer of cloud These theories for describing atmospheric optical or air scatters or absorbs light. Assuming Bouguer's phenomena amount to finding the angular distri- law [see Eq. (1)] to be valid, a medium has optical bution of radiance scattered by a single particle. depth (thickness) r when a fraction e~T of a vertical One major limitation of such an approach becomes (oblique) sunbeam penetrates without being scattered obvious when applied to sky colors, because no or absorbed. single particle that scatters shortwaves with greater The color and radiance of atmospheric optical efficiency than longwaves can produce the orange or phenomena, therefore, depend on the optical depth red color of the horizon sky at twilight. of the cloud and/or clear air and must be treated When the angular distribution of radiance var- as problems in radiative transfer. A robust theory ies with particle size, shape, and orientation, it is of atmospheric optical phenomena must include necessary to integrate over all of the illuminated the role of multiple scattering of light. This article particles. Recent models of coronas, glories, and is, therefore, designed to develop simple radiative rainbows include the drop size distribution (Lock and transfer models that show how the radiance and Yang 1991; Cowley et al. 2005) and size-dependent color of atmospheric optical phenomena depend on the optical depth of the cloud and clear air (Meyer 1929; Minnaert 1993). This approach is routinely AFFILIATIONS: GEDZELMAN—Department of Earth and taken in climate modeling (Lacis and Hansen 1974), Atmospheric Sciences, and NOAA/CREST Center, City College remote sensing (Menzel et al. 1998), and models of of New York, New York, New York; VOLLMER—Physikalische skylight and color (Adams et al. 1974; Gedzelman Technik, Brandenburg University of Applied Sciences, 1975; Bohren and Fraser 1985; Gedzelman 2005), but Brandenburg, Germany not often in modeling rainbows, halos, coronas, and CORRESPONDING AUTHOR: Stanley David Gedzelman, glories (Gedzelman 1980; Trankle and Greenler 1987; Department of Earth and Atmospheric Sciences, and NOAA/ Gedzelman 2003; Gedzelman and Lock 2003). CREST Center, City College of New York, New York, NY 10013 E-mail: [email protected] Any rigorous theory involving multiple scattering is complex and cumbersome, but a model that treats The abstract for this article can be found in this issue, following the atmospheric optical phenomena as beams of singly table of contents. scattered sunlight that are depleted by a second DON 0.1175/BAMS-89-4-47I scattering on their way to the observer provides a In final form 5 October 2007 ©2008 American Meteorological Society simple first-order approximation for two reasons. First, the brightest rainbows, halos, coronas, and 472 I BAflfr APRIL 2008 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC glories are produced when the optical thickness of of skylight except in extremely hazy air. This indi- the light path through the atmosphere and/or cloud cates that the predominant role of scattering for the is small, so that relatively little light is scattered more appearance of sunlight is depletion. than once. Second, rainbows, halos, coronas, and Aerosol particles scatter less selectively than air glories involve light scattering from particles with molecules, but most still scatter shortwaves more effi- pronounced peaks of radiance in certain directions ciently than longwaves, and thereby act to redden the and hence are much brighter than multiply scattered sun further. The wavelength dependence of scattering light, which spreads more uniformly around the sky. is often approximated by A~a, where a is the Angstrom When modeling these phenomena, we use simple, coefficient. For air molecules a ~ 4, a ranges from 0.1 approximate expressions for multiply scattered back- to 1.2 for desert dust and from 1.1 to 2.4 for urban ground light. As a result, the theory developed here aerosols and products of biomass burning (Dubovik is simple enough and the resulting simulations are et al. 2002). Large forest fires or massive volcanic convincing enough to be presented in courses that eruptions eject micrometer-sized particles that may treat atmospheric optics. scatter longwaves more efficiently than shortwaves The rest of this article is organized as follows. (Bohren 1995; Koziol and Pudykiewicz 1998), and so Observed "Features of optical phenomena related to have Angstrom coefficients a < 0. In these rare cases radiative transfer", are described and then explained they may turn the sun and moon green or blue even qualitatively. "Light scattering processes by individual when these celestial bodies are high in the sky. particles" including air molecules, cloud droplets, raindrops and ice crystals are presented. "Radiative Sky colors. Sky colors are more varied than the colors transfer theory for optical phenomena" used to con- of the sun because skylight is both produced and de- struct the models is then developed and used to create pleted by scattering. When the sun is high in the sky "Simulations of atmospheric optical phenomena." and the air is pure, the sky is blue with a maximum The final section contains the "Summary and color purity of 42% for a dominant wavelength of conclusions." 0.475 f/m at the zenith, but grades to near white at the horizon (Bohren and Fraser 1985). At that time, FEATURES OF OPTICAL PHENOMENA skylight consists mainly of singly scattered sunlight RELATED TO RADIATIVE TRANSFER. that has been depleted by a second scattering before it Colors of the sun and moon. The brightness and color reaches the observer. As the sun nears the horizon, its of sunlight and moonlight constitute the simplest optical path through the atmosphere lengthens. This applications of radiative transfer theory for optical increases the probability of scattering, particularly of phenomena. Sunlight and moonlight dim and redden shortwaves, so that both sunlight and skylight redden. as they approach the horizon because they must pass Thus, in Fig. 2, the violet and blue colors in sunlight through much more air to reach an observer (Fig.