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. 1), are so depleted by molecular scattering during the and because shortwaves are scattered more efficiently oblique transit through the atmosphere that the sun than longwaves by air molecules and by most aerosol appears to an observer located on the left-hand side particles. Under typical conditions for an observer at of the diagram to be either orange or red, and even sea level, a horizontal sunbeam must pass through the scattered sunlight at a given point, initially blue about 38 times as much air as a vertical sunbeam, to an observer on the right-hand side of the diagram, and the gradient of optical thickness is so large just above the horizon that the sun and moon exhibit a distinct color gradation from yellow at the top to red at the bottom when seen at the horizon. The horizon-level sun and moon appear even redder for elevated observers (e.g., in an airplane or on a mountain peak), because the light must pass through up to twice as much air as that for FIG. I. Comparison of the path through the atmosphere and the observers at sea level. But, even when resulting apparent color of the sun for an observer at sea level for a the sun appears at the horizon, its sun at the zenith and at the horizon is shown. The thickness of the radiance is much greater than that atmosphere, indicated by blue dots, is greatly exaggerated.
AMERICAN METEOROLOGICAL SOCIETY APRIL 2008 BAF1S- I 473
Unauthenticated | Downloaded 10/09/21 01:28 AM UTC FIG. 2. Colors of (top) direct and (bottom) scattered sunlight seen by an observer located at any point in a molecular atmosphere when the sun is near the horizon. Atmospheric thickness depicted here is 30 km and is greatly exaggerated.
is orange after passing through the atmosphere FIG. 3. Model for a halo or corona produced by (top) a almost horizontally For an observer higher above the horizontal cloud layer and (bottom) a cloud element of surface, the light is less red because the air is thinner finite horizontal extent. The zenith angles of sun and and less scattering has taken place. observer are sun and 0obs, respectively. During twilight, the horizon sky is either orange or red because it is directly illuminated only at great Minnaert (1993) noted that the tops of 22° halos distance from the observer, so that scattering through are seen about 3 times as often as the bottoms, and the great optical air mass has removed almost all are almost invariably brighter, largely because the of the shortwaves from the initially blue skylight greater optical thickness of the long, oblique path beam. This leads to a general rule that in clear skies through the atmosphere for all but the thinnest cloud without volcanic aerosols, scattering that adds to a layers near the horizon dims the bottoms. Many light beam makes it bluer than the source light, while spectacular halos form so close to the observer that scattering that depletes a light beam reddens it. This brilliant spots of light from individual crystals can explains why distant dark objects, such as shaded be seen (Tape 1994; Tape and Moilanen 2006). This mountain slopes, appear blue and brighten with in- is true for three reasons. First, when the crystals are creasing distance from the observer, and why distant so close to the observer, the phenomena tend to be bright, sunlit objects appear to be red or pink. These produced in such a small cloud volume that varia- are examples of phenomena that artists call aerial tions of particle size and shape are likely to be small. perspective. Second, the closer the crystals are to the observer, the less light either derives from or is scattered by the Halos, coronas, rainbows, and glories. The brightest intervening atmosphere. Third, when the halo cloud halos and coronas are seen through thin clouds is close, it is more likely to consist of an element with because multiple scattering in thick clouds even finite horizontal extent, so that when the sun is low in blots out the sun. The clouds in the models of the the sky it does not need to go through a long optical "Radiative transfer theory for optical phenomena" path to reach the observer. and "Simulations of atmospheric optical phenomena" Thin clouds or fog banks produce the most brilliant sections are represented in Fig. 3 as either a layer em- glories (Lynch and Futterman 1984). But, because bedded in clear sky or an element of finite horizontal rainbows and glories are produced by backscattered
extent. The observer looks up at zenith angle 0obs and light (Fig. 4), they are even seen in the thickest clouds
the sun is located at zenith angle 0sun. (Gedzelman 2003). For similar reasons, the bright-
474 I BAflfr APRIL 2008
Unauthenticated | Downloaded 10/09/21 01:28 AM UTC est rainbows are produced by moderate showers (Gedzelman 1980; Gedzelman 1982). But, because rainbows and glories are produced by backscattered light, they are even seen in the thickest clouds. For example, the radiance of glories in thick clouds is up to 10% greater than that of the general background (Lacis and Hansen 1974; Spinhirne and Nakajima 1994).
LIGHT SCATTERING PROCESSES BY INDIVIDUAL PARTICLES. Atmospheric optical phenomena produced by hydrometeors can only appear brilliant when the hydrometeors scatter light in such a way that at any given wavelength their angular scattering phase functions P(ij/) of radiance have pro- nounced local maxima that preferably have half-widths that are narrower than the angular diameter of the sun (0.53°). However, the phenomena can only exhibit pro- nounced coloration when they are much wider than the sun, that is, when the angle of dispersion or angle be- tween the maxima of P(f) at A = 0.4 and at A = 0.7 ^m is much greater than the sun's angular diameter. Figure 5 shows P(f) of scattering by air molecules (Rayleigh FIG. 4. Model for a glory produced by (top) a horizontal scattering) and spherical cloud droplets with radius cloud layer and (bottom) a rainbow or glory produced r = 8 fim (calculated with Mie theory), and by reflection by a vertical rain shaft or fog bank. Here and 0 are and refraction using geometric optics for spherical the solar zenith and azimuth angles relative to a line raindrops; thick, randomly oriented right hexagonal normal to the vertical plane of the rain shaft, respec- plate crystals (aspect ratio, 0.99); and long, horizontally tively. In all model runs here 0 = 0. oriented right hexagonal columns (aspect ratio 5). The inset to Fig. 5 shows a celestial hemi- sphere view of halos produced in a cloud of thick plates. The primary rainbow can appear bright because P(f) for large spheri- cal raindrops increases abruptly by a factor of 200 from Alexander's dark band (the region between the primary and secondary bows), and its half-width at half maximum is only 0.10°. The primary bow can also have vibrant colors because the angle between the peaks of P(\j/) at A = 0.4 and A = 0.7 ^m is 2.01°, or almost 4 times the sun's angular diameter. The 22° halo can appear bright and distinct at its inner edge, because P(f) for thick hexagonal plates in- creases abruptly by a factor of about FIG. 5. Angular scattering phase function, P(y/) at A = 0.55 pm for I) an 50 times from the darker region air molecule (blue), 2) a spherical raindrop (red), 3) a thick ice crystal plate (black), 4) a cloud droplet with 8-pm radius (green), and 5) a inside the halo. However, the outside long, horizontal column crystal (purple). Monte Carlo simulations of the halo is less distinct because the assuming geometric optics and with three-point smoothing were run half-width at half maximum of P(yi) for the raindrop. Mie theory was used for the droplet. The small inset is 2.4°. Furthermore, the colors of shows the distribution of scattered light over the celestial hemisphere the 22° halo are much less saturated from thick ice crystals.
AMERICAN METEOROLOGICAL SOCIETY APRIL 2008 BAF1S- I 475
Unauthenticated | Downloaded 10/09/21 01:28 AM UTC than those of the primary bow because the peaks of simple crystal types and/or orientations that can P(f) at A = 0.4 and A = 0.7 f/m are only 0.76° apart. produce awesome halo displays. Most clouds contain Most 22° halos exhibit little more color than a faint such a large variety of crystals and crystal clusters reddish inner edge. with so many facets that the resulting halos appear All halos that are produced when light passes quite feeble (McFarquhar et al. 2002). through alternate rectangular faces, that is, those One factor that has less of a negative impact on oriented 60° from each other, have similar color the brightness and color of rainbows and halos is the limitations. Circumscribed halos are produced when small sensitivity of P(T//) to size once a raindrop or ice light passes through alternate faces of long column crystal is large enough, so that geometric optics are crystals that fall with horizontal C axes when the a good approximation to scattering. Thus, rainbows sun is near the zenith. They are often incorrectly are often bright and colorful, provided the drops identified as 22° halos—the two coincide when the are larger than r >100 ^m, but are not so large that sun is at the zenith—but the circumscribed halos are their bottoms flatten much (r < 1500 ^m) (see, e.g., much brighter and narrower. Removing one degree Pruppacher and Klett 1996, p. 396). of freedom from the possible crystal orientations The situation is different for coronas and glories. increases the peak of P(f) by a factor of about 8 and Because P(f) at any wavelength resulting from scat- shrinks its half-width at half maximum from 2.4° tering by cloud droplets (e.g., r <20 [im) varies so to only 0.15°. Even so, the circumscribed halo does rapidly with droplet radius (Bryant and Cox 1966; not have a vibrant range of colors because the angle Lock and Yang 1991), coronas and glories are only between the peaks of P(f) at A = 0.4 and A = 0.7 \im bright and colorful when the relevant droplet size is only 0.76°, the same as that for the 22° halo. Thus, spectrum is extremely narrow, as at the fringes of it grades from red at its inner edge to yellow, and has mountain wave clouds (Shaw and Neiman 2003) for a slightly bluish outside. coronas or near the tops of some clouds for glories. Halo phenomena produced by light passing Integrating the light scattered by droplets with a through one rectangular and one hexagonal crystal wide size spectrum flattens the effective scattering face (oriented at 90°) are potentially quite colorful phase function so much that contrasts of radiance because the angle between the peaks of P(f) at A = 0.4 and color of the phenomena are washed out (e.g., and A = 0.7 ^m is 2.17°. The 46° halo is not often Cowley et al. 2005). seen and has faint colors, because under optimal conditions (simple, thick plate crystals), P(f) has RADIATIVE TRANSFER THEORY FOR a maximum less than 5 times the value inside the OPTICAL PHENOMENA. Because light that halo and a large half-width at half maximum (4.4°). passes through or emerges from a cloud or rain shaft However, circumhorizontal arcs can be brilliant and may be scattered any number of times, the brightness have spectral colors because when plate crystals fall and color purity of atmospheric optical phenomena with C axes, vertical P(\j/) approaches the form of a are problems of radiative transfer. The radiance and d function (Konnen 1983). color of sunlight are accurately described by a simple Halos are much less likely than rainbows to single scattering model in which sunlight is removed approach their maximum potential radiance and col- from the beam seen by the observer as it passes or purity. Most raindrops have similar shapes (almost through the atmosphere (Vollmer and Gedzelman spherical), while ice crystals come in a wide variety 2006). All of the other atmospheric optical phenom- of shapes (habits) and orientations as they fall (Tape ena we consider here involve at least two distinct 1994; Tape and Moilanen 2006; Libbrecht 2005). scattering events. Sunlight is first scattered into the Crystals take the form of plates, pencils, needles, beam seen by the observer, but is then reduced by a and stellar dendrites. They may have pyramidal second scattering event on its way to the observer. or bulleted endings, hollow or terraced interiors, This is illustrated in Fig. 6, with the terms for each and rounded edges. Aggregation, particularly near event described below. 0°C, and riming further complicate crystal shapes The addition of multiply scattered light to the (Heymsfield et al. 2002). Thick clouds extend over beam seen by the observer is relatively small until a wide ranges of temperature and humidity, so that significant fraction of light has been scattered, pro- crystal habit varies widely (Wang and Sassen 2002). vided that P(i//) has a marked and narrow maximum. All of these factors greatly increase the possible For r»l, the color of scattered light (neglecting paths of light and render P( f) less peaked and more absorption) approaches that of the source light so that isotropic. Few clouds contain the limited range of any theory that neglects multiply scattered skylight
476 I BAflfr APRIL 2008
Unauthenticated | Downloaded 10/09/21 01:28 AM UTC overestimates both contrast and color saturation. Thus, rainbows appear muted or may even disappear in the bright white central column of spray from fountains, and extremely distant shaded mountains appear almost white. Modeling the color and radiance of atmospheric optical phenomena begins with sunlight or moon- light. We approximate sunlight radiance atop the atmosphere IQ as a Planck radiator at T = 5800 K. When a pencil of monochromatic radiation at wave- length A with radiance /(A) passes through either air or cloud, it decreases in proportion to its magnitude and to the increment of optical thickness dr traversed according to Bouguer s law,
d/(A) = -/(A) dr. (1)
Equation (1) is valid when multiple scattering is neglected. When a light beam penetrates a thin horizontal layer of atmosphere or cloud obliquely at an angle 0, an increment of optical thickness dr is FIG. 6. Model showing incremental gains and losses to related to the normal or vertical increment of optical the light beam seen by an observer, including the con- depth drN of the layer by tribution of singly scattered sunlight and subsequent loss by scattering on the way to the observer. dr = sec(0)dr , (2) lost in a second scattering from its point of origin to where 0 is constant for light beams in a plane-parallel the bottom of the layer, and then integrating over atmosphere that are not refracted. The error in Eq. (2) the layer or is small up to 0 ~ 80°, after which it increases rapidly. % •rsec(0sun) ~(%™r)sec(^obs) Thus, the impact of Earth's curvature and refraction Ibot (A) = P(y/)Ism sec(0oJrfT, (5) is negligible for all light beams that are not near the horizon. Near the horizon, we replace sec(0) in all which yields model runs with a typical value of a quantity known as the optical air mass (Vollmer and Gedzelman f P(l//-)sec(0obs)lmn(k) f -Sec( of /bot(A) occurs at rNsec(0sun) = 1. This confirms the The subscripts top and bot refer to the top and bottom reasoning that clouds with modest optical thickness of the layer, respectively. An element of sunlight of produce the brightest halos and coronas. radiance /(A) scattered into the beam seen by the For an observer looking down on a cloud or air layer, observer from a thin layer is the radiance of up welling light I (A) (Fig. 4a) is d7obs(A)=P(v)/(A)sec(0obs)drN (4) ]• (7) [sec(0i>o) + sec(0oJ] The radiance of the scattered light beam /bot(A) pro- duced within and reaching the bottom of the layer by Equation (7) also gives the radiance of rainbows or sunlight / (A) incident on the layer is obtained by glories seen in vertical rain shafts or fog banks when reducing each increment from Eq. (4) by the amount both 0~su n and 0' ob. s are measured from the normal AMERICAN METEOROLOGICAL SOCIETY APRIL 2008 BAF1S- I 477 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC to the vertical plane of the rain swath. It increases SIMULATIONS OF ATMOSPHERIC monotonically with r and approaches a maximum OPTICAL PHENOMENA. We now incorporate asymptotically. Most of the increase of singly scat- the equations derived above in models to simulate the tered light occurs for r < 1, while multiply scattered radiance and color of the sky and halos and rainbows, light continues increasing significantly for rdd > 1, and and their variation with solar zenith angle, cloud eventually dominates (Gedzelman 2003). This is why height or distance, optical depth, and light of the the brightest rainbows and glories occur in modest background sky. To produce realistic color, the equa- rain showers with dark backgrounds. tions are solved for 61 uniformly spaced wavelengths The radiance of the halo or rainbow is reduced by over the visible light spectrum. The angular scattering scattering in the clear air both above and below the phase functions are given byP(y/) = 0.5[l+cos2(i//)] for cloud. The mathematics is greatly simplified with molecular scattering in clear air, by Mie theory for little error if the cloud is assumed to be a thin sheet cloud droplets, and by equations that fit the curves at a single pressure Pdd, where only hydrometeors in Fig. 5 for raindrops and ice crystals. scatter light (Gedzelman 1980). The scattering of light by air molecules and by Sky color simulations. Simulated skylight spectra for the aerosol particles is calculated using Angstrom coef- molecular atmosphere using Eqs. (6) and (8) are shown ficients a = 4 and a = 1. The optical depth rN(A) of in Fig. 7. When the sun is at 0sun = 45°, sky color grades the molecular atmosphere from its top to pressure p from deep blue at the zenith to white at the horizon is then approximated by (optical air mass = 38) and agrees closely with obser- vations. When the sun is at = 88.5°, simulated sky K, P sun = (8) color grades from pale blue at the zenith to orange at * Po the horizon. Observed zenith sky color at sunrise and 27 4 Here, KR = 8.66 x 10~ m is the scattering coefficient sunset is a much deeper blue than indicated by the for the molecular atmosphere and p0 = 101,325 Pa is simulations, because the model does not include pref- the mean sea level pressure. Losses due to scattering erential absorption of yellow and orange light in the of sunlight above the cloud and of the halo beam Chappuis bands of 03 (Adams et al. 1974). The model below the cloud are found by substituting Eq. (8) also cannot simulate twilight sky colors, because it does into Eq (3). These factors reduce the radiance of the not include the curvature of Earth and atmosphere. single-scattered halo beam penetrating a cloud of Gedzelman (2005) constructed a model of sky color optical thickness r ld seen by an observer to that includes these and other effects. Halo simulations. _ - sec( sum of 1) the single-scattered beam /obs(A), 2) skylight in (top) a vacuum, (middle) a molecular atmosphere, that penetrates the cloud undisturbed, 3) skylight and (bottom) a hazy atmosphere with an aerosol scale that originates below the cloud (see "Simulations of height = 2 km, Angstrom coefficient a =1, and aerosol atmospheric optical phenomena"), and 4) cloud light. optical depth twice that of the molecular atmosphere. Variations of radiance at each wavelength across the In all three cases, the radiance of the bottom of halo or corona then determine how striking these the halo (0 ~ 82°) peaks when the cloud is relatively phenomena will appear. thin (r ~ 0.25), and falls to less than 1% of its peak 478 I BAflfr APRIL 2008 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC FIG. 7. Spectrum of radiance of single-scattered light value before the cloud thickens to r.el,d ~ 3. Radiance of the halo top (obs ~ 33°) peaks for somewhat thicker clouds (rld ~ 0.63) than the halo bottom, and is still more than 10% of its peak value at rdd « 3. Radiance is greater at the top than at the bottom of the halo for 12 3 all clouds that are thicker than rel, d> ~ 0.8 in a vacuum x (cloud) and r > ~ 0.2 in hazy air. Peak radiance occurs at ld 0.6 much smaller r , and decreases more rapidly as r ld |>sun = 60°, X = 0.55 pm increases at halo bottom than at the top because the «V ) slant paths through the cloud and clear air below the c z> cloud base are much longer at halo bottom (Fig. 9). xmo,=0.094 > 0,4 Similarly, the higher the cloud base, the greater the xaer=0.188 JS difference of pathlength below the cloud, and hence a> on the more pronounced the difference between halo - halo top top and bottom. halo bottom These results of the single-scattered halo beam model can be compared with simulations using a Monte Carlo dot model, modified from Gedzelman (1994), that includes multiple scattering both in the 12 3 cloud and in a molecular atmosphere outside the x (cloud) cloud. The dot model (HALODOTS) is available online (www.sci.ccny.cuny.edu/~stan). In the dot FIG. 8. Radiance of halo beam top and bottom model, the number of random sunbeams at each of due to a cloud of thick crystal plates at 61 wavelengths is weighted by the Planck function at 300 hPa as a function of RC|D for A = 0.55 JWM, when = 60° (top) in a vacuum, (middle) T= 5,800 K. Each beam is aimed at a random point on sun a molecular atmosphere, and (bottom) hazy right hexagonal ice prisms. The beams are subject to air. Diffuse sky and cloud light are not included scattering by air molecules in the clear air and to re- here. fraction and/or reflection by ice crystals in the cloud layer. Light that is scattered a second time is assumed Figure 10 shows halo simulations of the dot model to be isotropic. Each ray is plotted as a dot, with color for cloud optical thickness r ld = 0.333, 1, and 3.0 for corresponding to that of the light beam. the same conditions as in Fig. 8. At r ld = 0.33 and 1, AMERICAN METEOROLOGICAL SOCIETY APRIL 2008 BAF1S- I 49 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC the surrounding sky than the 22° halo, and also has a slightly blue outside. Thus, the simulations hint at how to distinguish the circumscribed halo from the 22° halo when the sun is high in the sky. Color in the ring model is produced by integrating over 61 wavelengths of the visible spectrum (from 0.4 to 0.7 ^m in 0.005-^m steps), assigning red-green- blue (RGB) values of each wavelength based on the sensitivity of the human eye and then applying the equation FIG. 9. Comparison of pathlength through the cloud for the top and bottom of the 22° halo when = 55°. (ii) RGB = 255 red predominates at the inner edge of the 22° halo, which is far brighter than the faint 46° halo. Both Here, JRGB(0) is the luminance of red, green, or blue halos contrast more sharply with the background light at a point in the sky, obtained from tabulated skylight at r = 0.333 than at r = 1. At r = 3, only tristimulus values for each wavelength, and I is the dd dd ld o max the top of the 22° halo remains visible. As the sun maximum luminance in each frame. The exponent e nears the horizon, the slanted paths through both is used because the relation between RGB values on cloud and air lengthen so that halos produced by the monitor and the perceived brightness of a scene cloud layers (but not necessarily cloud elements of is not linear. We set e = 0.25 for halos and rainbows finite extent) fade at smaller because this value produced values of r..eld. realistic looking halos on the Halo dot models have monitor. Unfortunately, at a discrete appearance that least one unpleasant aspect does not match well with ob- of the appearance of the served halos. More visually simulated halos and rain- convincing simulations of bows remains, namely, that circular halos using a ring they appear opaque, whereas model based on the "Radia- in nature part of the allure tive transfer theory for opti- of halos (and particularly cal phenomena" and available rainbows) derives from their on the Web as HALORING translucence. are shown in Fig. 11 for 0sun = Model runs (not shown) 0°. A ring model is only pos- indicate that the 22° and sible when 0obs = f (i.e., for 46° halos first appear in circular halos when the sun a molecular atmosphere is at the zenith). In this case, for r,el.d « 0.0003 and 0.003,' circular halos are azimuthal- respectively. Thus, visible ly symmetrical, and spherical halos can be produced by trigonometry is not needed clouds of simple, thick hex- to relate 0obs and f. Further- agonal prisms so tenuous more, when the sun is at the that the sky is deep blue and zenith, the inner edge of the appears to be clear. With circumscribed halo coincides such simple crystals, the with the 22° halo. Indeed, the radiance of the 22° halo two are compared in Fig. 12 peaks at about 8 times that for the same conditions as of the background sky when those in Fig. 11 with T = 1.0. FIG. 10. Monte Carlo simulations of halos rcld ~ 0.4, while the peak Both have distinct red inner radiance of the 46° halo in a molecular atmosphere produced by edges, but the circumscribed thick hexagonal plates at p = 300 hPa is less than twice that of halo is much narrower and the background sky. The for (bf SI 60°, with r = (top) 0.333, much brighter compared to (middle) 1.0, and (bottom) 3.0. 46° halo disappears when 480 I BAflfr APRIL 2008 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC r ld « 4.5 and the much brighter 22° halo disappears when r.el,d ~ 7.0. However, all such ideal halos, produced by clouds of simple, thick hexagonal plates, are far brighter than almost any ever seen, except under extraordinary conditions, such as sometimes occur at the South Pole. Cloud optical depth has a similar impact on the appear- ance on coronas, with a few minor differences (Gedzelman 2003). Most coronas extend only a small angular distance from the sun, so that there is little variation in appear- ance from the top to bottom. Thicker clouds vitiate halos because crystal shapes become more varied, while coronas fade because the droplet size spectra FIG. 12. Simulations for d> = 0° broaden. < sun and rc|d = 1.0 comparing the (top) halos produced by randomly ori- Rainbow simulations. Rainbows, ented thick hexagonal plates with unlike most halos, involve re- the (bottom) circumscribed halos flection and appear in the half produced by pencil crystals (aspect of the sky opposite the sun (at ratio 5.0) falling with their long or C axis horizontal. The maximum scattering angles around 138° RGB value in each frame is set to and 129°; see Fig. 5). The rain 255 and all other colors are scaled shaft is illuminated by direct by Eq. (13). sunlight, by skylight from the clear air behind the observer, sunlight and the rainshaft, 0 = 0. and by background skylight FIG. II. Halo simulations for Under these near-field condi- under the cloud base and be- (b' su n =0° with all other conditions tions, the color and radiance of yond the rain shaft. Here, Bb, as in Fig. 10. The maximum RGB the bow and of the background the ratio of the radiance of the value in each frame is set to 255 sky are independent of 0obs, and background to that of a clear and all other colors are scaled by thus are azimuthally constant sky, is chosen as a free param- Eq.(l3). (i.e., along the rainbow arc). In eter in the model because lighting conditions under actual situations the sunlit portion of the rain shaft and beyond rain showers and thunderstorms are is invariably smaller at the top of the bow than at the highly varied. horizon (Gedzelman 1982). In the rainbow model BOWRING, the spectrum Radiance of the rainbow ray increases as the of the background skylight is assumed to be that optical depth rdd of the rain shaft increases, but less of sunlight reaching the observer, and shading by background light penetrates the rain shaft. Once r the overhanging cloud is assumed to limit skylight increases beyond a certain value, multiply backscat- from behind the observer to half of what it would be tered sunlight and skylight increase more rapidly under clear-sky conditions. To simplify the model than does the singly backscattered sunlight of the and eliminate the need to use spherical trigonometry, rainbow ray. Figure 13 shows the contrast between the model is designed for an observer standing at the the radiance maximum Jmax of the geometric optics edge of the rain shaft, so that there is no intervening rainbows and minimum I . in Alexanders dark band min skylight and that the azimuth angle between the as a function of r.,, when 0 = 60° for both dark eld ' sun AMERICAN METEOROLOGICAL SOCIETY APRIL 2008 BAH* I 481 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC rainbows calculated by Mie theory (see diagrams in Lee 1998) always have a dark blue arc for all raindrops with r > 100 the faintness or absence of blue from the observed rainbows must be due to masking by multiply scattered light. A similar model can be applied to the glory. However, several factors make bright rainbows more common than bright glories (Gedzelman 2003; Lynch and Livingston 2001). First, rain shafts tend to be optically thin, while most glory-producing clouds are optically thick. Second, the large drops producing rainbows have much more sharply peaked but much less size-dependent maxima of P(Y/) than do the tiny cloud droplets that produce glories. 0.001 0.001 0.01 0.1 1 10 100 X SUMMARY AND CONCLUSIONS. We pre CLD sented a radiative transfer theory that was used to FIG. 13. Contrast between the spectrally integrated create models simulating the appearance of atmo- radiance minimum I . in Alexander's dark band. min * spheric optical phenomena, including sky color, and maximum /max of the primary (thick lines) and halos, and rainbows. These phenomena are treated secondary (thin lines) rainbow peaks as a function of as singly scattered beams of sunlight that are, in rc|d when 0sun = 30° and BB = 0.01 (solid lines) or 0.25 (dashed lines). turn, depleted by scattering as they approach the observer. In all model runs a plane-parallel atmo- (J3q = 0.01) and bright (Bb = 0.25) backgrounds. For sphere was assumed and the clouds consisted of a these conditions, the primary and secondary bows geometrically thin horizontal layer or a vertical rain first become visible at r ~ 0.001 and 0.004, respec- shaft. The angular dependence of the scattered light tively, when the background sky is dark (Bfi = 0.01). radiance was represented by that of Rayleigh scat- Maximum contrast for both bows occurs when tering by molecules for skylight and by equations fit T ld ~ 0.25 and the bows remain visible for all larger to calculated values based on the geometric optics values of r ld. Thus, rainbows are visible in quite light of hexagonal plate crystals for halos and spherical showers and appear most intense for showers whose drops for rainbows. In some model runs, simple ap- sunlit parts have only modest optical thicknesses and, proximations for multiply scattered sky and cloud of course, dark backgrounds (BB « 1). light were included. When the background sky is relatively bright (5B = 0.25), the primary and secondary bows do not Tcld = 10.0 Ji. become visible until rdd ~ 0.02 and 0.2, respectively Contrast for both bows is much lower at small r eld,' and maximum contrast occurs when -.rel,d . ~ 1.0. Radiance,' o normalized to the peak of the primary bow, is shown c / t .2 as a function of scattering angle for Bb = 0.01 for the \ <3 three cases of r.el,d = 0.01, 0.251, and 10.01, and with no oc multiply scattered light (efficient) in Fig. 14. u Color simulations of the rainbow ring model are N shown in Fig. 15 for the safne cases as in Fig. 14. Note o again that each panel displays radiance relative to z the maximum in that panel; in fact, radiance of the Efficient I A- sky increases monotonically as rdd increases. When r = 0.01, the sky is blue and the blue inner part of 30 35 40 45 50 55 dd Angle from Shadow ( ) the bow is distinct and coincides with the radiance minimum. Both bows appear most vibrant when FIG. 14. Ratio of spectrally integrated radiance I to its maximum value in the primary bow / versus angle rdd = 0.25. As rld increases, multiply scattered light max from the low sun reddens the sky and rainbow, while from the antisolar point for conditions of Fig. 13 when R, = 0.25 and 10. the blue inner part of the bow disappears. Because 482 I BAflfr APRIL 2008 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC Skylight simulations for an observer standing encompassed the range of at the edge of a rain shaft colors that extend from the oriented at right angles to rich blue of the zenith sky the vertical plane of the in a molecular atmosphere, sun and the observer and when the sun is high in not shaded by the cloud the sky, to the orange of above. Simulations of halos, the horizon at sunrise or coronas, and rainbows that sunset. However, skylight do not assume azimuthal almost invariably reduces symmetry are performed both color purity and con- by programs (HALOSKY, trast of radiance for all CORONASKY, and optical phenomena pro- BOWSKY) that are avail- duced by hydrometeors. able online (http://sci.ccny. Thus, simulated halos and cuny.edu/stan). rainbows appear most dra- Of course, real clouds matic when scattering by do not consist of simple air molecules and aerosol hexagonal crystals or uni- particles is neglected and form size spherical drops when all sunlight is scat- or droplets. As a result, tered just once by crystals the brilliant rainbows or drops. and halos that can be pro- In reality, tenuous clouds duced by simulations or have too few particles to under laboratory condi- produce bright halos or tions (Walker 1976; Bohren rainbows because most 1987; Vollmer and Tammer sunlight passes through 1998; Vollmer and Greenler the cloud without being 2003) are either seldom or scattered. Optically thick never seen in nature. clouds contain so many This makes it all the more particles that most sunlight wondrous that atmospheric is scattered many times FIG. 15. Rainbow simulations for AMERICAN METEOROLOGICAL SOCIETY APRIL 2008 BAF1S- I 483 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC , and D. R. Huffman, 1983: Absorption and Scatter- Lacis, A. A., and J. E. Hansen, 1974: A parameterization ing of Light by Small Particles. Wiley Press, 544 pp. for the absorption of solar radiation in the Earths , and A. B. Fraser, 1985: Colors of the sky. Phys. atmosphere. /. Atmos. Sci., 31, 118-133. Teach., 23, 267-272. Lee, R. L., Jr., 1998: Mie theory, airy theory and the Boyer, C. B., 1987: The Rainbow—From Myth to Math- natural rainbow. Appl. Opt., 37, 1506-1519. ematics. Princeton University Press, 376 pp. Libbrecht, K. A., 2005: The physics of snow crystals. Rep. Bryant, H. C., and A. J. Cox, 1966: Mie theory and the Prog. Phys., 68, 855-895. glory. /. Opt. Soc. Amer., 56, 1529-1536. Lock, J. A., and L. Yang, 1991: Mie theory model of the Cowley, L., P. Laven, and M. Vollmer, 2005: Rings corona. Appl. Opt., 30, 3408-3414. around sun and moon: Coronae and diffraction. Lynch, D. K., and S. N. Futterman, 1984: Ulloa's ob- Phys. Educ., 40, 51-59. servations of the glory, fogbow, and an unidentified Dubovik, O., B. N. Holben, T. F. Eck, A. Smirnov, Y. J. phenomenon. Appl. Opt., 30, 3538-3541. Kaufman, M. D. King, D. Tanre, and I. Slutsker, 2002: , and W. Livingston, 2001: Color and Light in Nature. Variability of absorption and optical properties of 2nd ed. Cambridge University Press, 277 pp. key aerosol types observed in worldwide locations. McFarquhar, G., P. Yang, A. Macke, and A. Baran, /. Atmos. Sci., 59, 590-608. 2002: A new parameterization of single scattering Fowles, G. R., 1989: Introduction to Modern Optics. solar radiative properties for torpical anvils using Dover Publications, 332 pp. observed ice crystal size and shape distributions. /. Fraser, A. B., 1983: Why can the supernumerary bows Atmos. Sci., 59, 2458-2478. be seen in a rain shower? /. Opt. Soc. Amer., 73, Meinel, A., and M. Meinel, 1983: Sunsets, Twilights, and 1626-1628. Evening Skies. Cambridge University Press, 163 pp. Gedzelman, S. D., 1975: Sky color near the horizon Menzel, W. P., T. J. Schmit, R. M. Aune, G. S. Wade, during a total solar eclipse. Appl. Opt., 14, 2831- G. P. Ellrod, and D. G. Gray, 1998: Application of 2837. GOES-8/9 soundings to weather forecasting and now- , 1980: Visibility of halos and rainbows. Appl. Opt., casting. Bull. Amer. Meteor. Soc., 79, 2059-2077. 19, 3068-3074. Meyer, R., 1929: Die Haloerscheinungen. Probleme der , 1982: Rainbow brightness. Appl. Opt., 21, Kosmischen Physik, C. Jensen and A. Schwaftmann, 3032-3037. Eds., Vol. XII, Henri Grand, 64-69. , 1994: Simulating rainbows and halos in color. Mie, G., 1908: Beitrage zur Optik triiber Medien, Appl. Opt., 33, 4607-4613, 4958. speziell kolloidaler Metallosungen. Ann. Phys. 25, , 2003: Simulating glories and cloudbows in color. 377-445. [Translated by HMSO, 1976, Royal Aircraft Appl. Opt., 42, 429-435. Establishment Library No. 1873, Contributions to , 2005: Simulating colors of clear and partly cloudy the optics of turbid media, particularly of colloidal skies. Appl. Opt., 44 5723-5736. metal solutions, 72 pp.] , and J. Lock, 2003: Simulating coronas in color. Minnaert, M. G. J., 1993: Light and Color in the Outdoors. Appl. Opt., 42, 497-504. Springer-Verlag, 451 pp. Reprint translated and re- Greenler, R., 1980: Rainbows, Halos, and Glories. vised by Len Seymour. First published 1937. Cambridge University Press, 195 pp. Pernter, J. M., and F. M. Exner, 1922: Meteorologische Heymsfield, A. J., A. Bansemer, P. R. Field, S. L. Durden, Optik. 2nd ed. Wilhelm Braumuller, 799 pp. J. L. Stith, J. E. Dye, W. Hall, and C. A. Grainger, Pruppacher, H. R., and J. D. Klett, 1996: Microphysics of 2002: Observations and parameterizations of par- Clouds and Precipitation. 2nd ed. Springer, 954 pp. ticle size distributions in deep tropical cirrus and Shaw, J. A., and P. J. Neiman, 2003: Coronas and iri- stratiform precipitating clouds: Results from in situ descence in mountain wave clouds. Appl. Opt., 42, observations in TRMM field campaigns. /. Atmos. 476-485. Sci., 59, 3457-3591. Spinhirne, J. D., and T. Nakajima, 1994: Glory of clouds Humphreys, W. J., 1940: Physics of the Air. 3rd ed. in the infrared. Appl. Opt., 33, 4652-4662. McGraw Hill, 676 pp. Tape, W., 1994: Atmospheric Halos. Antarctic Research Konnen, G. P., 1983: Polarization and intensity distri- Series, Vol. 64, Amer. Geophys. Union, 143 pp. butions of refraction halos. /. Opt. Soc. Amer., 73, , and J. Moilanen, 2006: Atmospheric Halos and the 1629-1640. Search for Angle X. Amer. Geophys. Union, 238 pp. Koziol, A. S., and J. Pudykiewicz, 1998: High-resolution Trankle, E., and R. Greenler, 1987: Multiple scattering modeling of size-resolved stratospheric aerosol. /. effects in halo phenomena. /. Opt. Soc. Amer., A4, Atmos. Sci., 55, 3127-3147. 591-599. | BAH5- APRIL 2008 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC Tricker, R. A. R., 1970: Introduction to Meteorological and moon: The role of the optical air mass. Eur. J. Optics. Mills and Boon, 285 pp. Phys. 27, 299-306. Vollmer, M., and R. Tammer, 1998: Laboratory ex- Walker, J., 1976: Multiple rainbows from single drops of periments in atmospheric optics. Appl. Opt., 37, water and other liquids. Amer. /. Phys. 44, 421-433. 1557-1568. Wang, Z., and K. Sassen, 2002: Cirrus cloud micro- , and R. Greenler, 2003: More halo and mirage physical property retrieval using lidar and radar experiments in atmospheric optics. Appl. Opt., 42, measurements. Part II: Midlatitude cirrus micro- 394-398. physical and radiative properties. /. Atmos. Sci., 59, , and S. D. Gedzelman, 2006: Colours of the sun 2291-2302. THE LIFE CYCLES OF EXTRATROPICAL CYCLONES Edited by Melvyn A. Shapiro and Sigbjorn Gronas Containing expanded versions of the invited papers presented at the International Symposium on the Life Cycles of Extratropical Cyclones, held in Bergen, Norway, 27 June-1 July 1994, this monograph will be of interest to historians of meteorology, researchers, and forecasters. The symposium coincided with the 75th anniversary of the introduction of Jack Bjerknes's frontal-cyclone model presented in his seminal article, "On the Structure of Moving Cyclones." The monograph's content ranges from a historical overview of extratropical cyclone research and forecasting from the early eighteenth century into the mid-twentieth century, to a presentations and reviews of contemporary research on the theory, observations, analysis, diagnosis, and prediction of extratropical cyclones. The material is appropriate for teaching courses in advanced undergraduate and graduate meteorology. The Life Cycles of Extratropical Cyclones is available for $75 list/$55 members. Prices include shipping and handling. Please send prepaid orders to Order Department, American Meteorological Society, 45 Beacon St., Boston, MA 02108-3693 or call (617) 227-2426 ext. 686. Visa, MasterCard, or American Express accepted. AMERICAN METEOROLOGICAL SOCIETY APRIL 2008 BAF1S- I 485 Unauthenticated | Downloaded 10/09/21 01:28 AM UTC Unauthenticated | Downloaded 10/09/21 01:28 AM UTC