The appearance of brilliant halos and coronas leadsus to question why these phenomena so often appeardull or faint. Two main factors restrict their brilli-ance. The first involves the particles that producethe halos and coronas. Brilliant halos require simplecrystal habits. But most crystals are either multi-faceted (such as dendrites), have hollow or steppedinsides, are clustered, rimed, or are so small that dif-fraction is important [1–3]. The most brilliant haloarcs and spots are also produced when the crystalsfall with a particular orientation so that the scat-tered light is “focused” on limited parts of the sky[4–7]. Some of the arcs and spots are most pro-nounced when the Sun is at or near an optimum ze-nith angle. Thus, for example, the circumhorizontalarc, which appears at least 46° below the Sun, canonly form when the solar zenith angle ϕSun ¼ 32°and has the most brilliant coloration when ϕSun ¼22:5° because then the deviation angle, ψ , is a mini-
mum and small oscillations of falling crystalsproduce the least possible change of ψ.
Droplet shape and orientation have no impact onthe brilliance of coronas, because all cloud dropletsare spherical, but the range of droplet sizes does.Cloud droplets scatter light in circularly symmetricpatterns whose radii vary almost inversely with dro-plet size. The most brilliant, multiringed coronas aretherefore produced by clouds with the most nearlyuniform size tiny droplets, such as are found atthe edges of thin laminar wave clouds [8].
The second main factor limiting the brilliance ofhalos and coronas concerns the media through whichthe light travels. Halos and coronas are produced bylight that penetrates clouds and clear air. Cloudheight or pressure, optical depth, τcld, horizontal ex-tent, as well as surface pressure, aerosol content, andthe height of the Sun in the sky all affect the appear-ance of halos and coronas [9–13]. We use opticaldepth (thickness) to refer to a light beam that passesthrough a horizontal layer vertically (obliquely). If,for example, a cloud is too tenuous it will not haveenough droplets or crystals to produce bright coronasor halos. If the cloud is optically thick and if the air is
hazy, or if the light passes obliquely through bothcloud and air, as when the Sun and/or phenomenaare near the horizon, most light will be scattered sev-eral times, and the coronas and halos will bedimmed. Once optical thickness of the light path isgreater than about 3 most of the incident light isscattered more than once, so that halos and coronasare seriously eroded. Since the mean global opticaldepths of high (cirrus and cirrostratus) and middleclouds (altocumulus and altostratus) are found bythe International Satellite Cloud Climatology Pro-ject (ISCCP) to be 2.2 and 4.8, respectively [14],the typical cloud is too thick to produce bright halosor coronas, particularly when the Sun is low inthe sky.The impact of particle shape, size, and orientation
on the appearance of halos and coronas has been in-vestigated in some detail [3]. Many of the atmo-spheric optical phenomena have been simulatedwith an increasing sense of realism, albeit usuallyin a vacuum. Starting with the simulations of Sunpillars and circumscribed halos, [15,16] almost allthe observed and a number of hypothetical haloshave been simulated with dot models using a MonteCarlo approach of aiming a large number of randomsunbeams at crystals with selected shapes and orien-tations. Most of these models assume that each lightbeam is scattered once and only once by a crystal ordroplet. Some effects of multiple scattering such asthe possible existence of parhelia of the parheliahave been modeled [17], but generally these modelsoperate in a vacuum and without regard to theoptical depth of the cloud.The effect of the medium through which the light
must travel has begun to receive more attention. Ahalo model that includes sky backgrounds with auser-selected range of brightness, color, and fill gra-dients is available on the Internet at http://www.atoptics.co.uk [18]. However, background skylightis inserted artificially as if it were an illuminatedscreen behind the halo. As the authors point out,
“HALOSIM accurately traces rays and simulates ha-los. In contrast, the coloured sky backgrounds that canbe selected for the simulations are intended only to bepleasing and they make no claim to accuracy.”
In a similar manner, visually convincing simula-tions of glories have been produced by insertingartificial background skylight in the simulations[19,20].Artificial insertion of an atmospheric screen
behind halos captures some aspects of their appear-ance. For example, because the sky is both brighterand whiter at the horizon than at the zenith whenclouds are optically thin, halos will tend to contrastmore sharply the higher in the sky they appear. Butlight scattering by air molecules, aerosol particlesand ice crystals or cloud droplets determines the col-or and light not only of halos and coronas but also ofthe background sky.
In this paper models are developed that simulatethe impact of skylight and cloud optical depth on theappearance of circular halos and coronas for a Sun atany zenith angle. The model can thus capture asym-metries in circular halos and coronas. For example,when the Sun is low in the sky, the top of the 22° halois often brighter than the bottom and can be seenthrough much thicker cloud layers [10]. A simplermodel of symmetric circular halos and coronas, re-stricted to the case when the Sun is at the zenith,was developed by Gedzelman and Vollmer [13]. Inthat case, and when in addition the atmosphere isfree of aerosols and the clouds consist of simple, hex-agonal prisms or of droplets with a narrow range ofsizes, the phenomena first become visible in clouds sotenuous that the sky still appears an unblemishedblue ðτcld ≥ 0:003Þ, are brightest and most colorfulfor cloud optical thickness in the range ð0:05 ≤
τcld ≤ 0:5Þ, and fade to little more than monochro-matic rings amidst the gray light of clouds thickerthan τcld ≥ 4. The restriction to circular halos andcoronas, retained in this article, means that the an-gular scattering phase functions, P, depend only onthe deviation angle, ψ .
The simplest models of coronas and halos assume ahorizontally uniform cloud layer. In that case, halosnear the horizon are blotted out by clouds of evenmodest optical depth. But many spectacular halosare seen near the horizon through clouds cells offinite width, such as diamond dust halos seen atground level and the notable South Pole halo dis-plays. To simulate such halos, the model includesthe impact of the Earth’s curvature and thecloud’s width.
The purpose of this paper is to generalize the simu-lations of circular halos and coronas seen through at-mosphere and cloud by including the factorsmentioned above. The phenomena are simulatedfor any solar zenith angle and clouds of any width,and the effects of the Earth’s curvature, albeit with-out refraction, are parameterized. The model atmo-sphere also contains an ozone layer because ozoneabsorbs visible light selectively in the Chappuisbands and alters sky color noticeably when theSun is near the horizon. The restriction to angularscattering phase functions that depend only on ψ ,i.e., PðψÞ, is retained for simplicity. This limits thesimulations to circular halos and coronas and thusdoes not allow simulation of phenomena such as ir-idescence, Sun dogs, or the many halo arcs.
2. The Models
The halo and corona models HALOSKY and CORO-NASKY used to produce the simulations describedhere are available on the web [21]. They displaysky panoramas centered on the Sun that includeeither the 22° and 46° halos or a corona for a rangeof parameters listed in Table 1 that must be specifiedby the user. The Sun is approximated as a Planck ra-diator at 5750K, and its radiance is calculated for 61wavelengths from 400 to 700nm. Before reaching the
clouds, sunlight penetrates the ozone layer, where ra-diance is particularly reduced by absorption in theChappuis bands at wavelengths around 600nm.The ozone layer is given by a bell-shaped curve witha peak at heightHoz0 in the stratosphere and a lower,constant value in the troposphere (Fig. 1). The ab-sorption cross section per cm depth of O3 (1000DU, or Dobson units) is shown in Fig. 2 as a functionof wavelength [22,23], and all absorption is assumedto occur above cloud level. The height of the ozonemaximum and the total amount of ozone, given inDU, are free parameters in the models.The geometry of clouds and atmosphere in the
models is shown in Fig. 3. The user can choose a hor-izontally uniform cloud layer or a cloud cell withwidth to height ratio rcld ≡ xcld=hcld. The cloud layeris sandwiched between two layers of molecular atmo-sphere with aerosols confined to the atmosphericboundary layer. This is the scenario for halos pro-duced in cirrostratus or coronas, in altostratus.The cloud cell rests on the ground with an aerosollayer above, and both are topped by a molecular at-mosphere. This is the scenario for the sometimesstunning diamond dust halos seen right beforethe eyes.The models incorporate the impact of the Earth’s
curvature on the path length of a beam at zenith an-gle, ϕ, through a layer of thickness, H, that extends
from the ground up by use of the air mass factor (theratio of optical thickness of a slanted path to opticaldepth) Matm [24],
The air mass factor, MH;ΔH , for elevated layers ofthickness ΔH and centered at height H, such as theozone layer, is then given by
MH;ΔH ¼
�H þ ΔH
2
�MHþΔH
2 ;ΔH −
�H −
ΔH2
�MH−
ΔH2
ΔH:
ð2Þ
Values for MH and MH;ΔH as a function of ϕ areshown in Fig. 4 and compared to M ¼ secðϕÞ of aplane parallel atmosphere. When ϕ ¼ 90°, MH ¼39:9 for H ¼ 8km and MH;ΔH ¼ 11:4 for H ¼ 25kmand ΔHoz ¼ 20km. MH;ΔH is smaller than MHbecause when the Sun appears at the horizon for
Table 1. User-Selected Parameters in the Models
Parameter Symbol Choice or Range
Surface pressure PSL ≥pAERSolar zenith angle ϕSun 0 → 90°Cloud optical depth τ ≥0Cloud type Layer or CellCloud layer pressure (PC) PC <PAERCloud cell width Δx ≥0Cloud cell thickness Δh ≥0Aerosol layer pressure at top PAER PC ≤ PAER ≤ PSLCrystal aspect ratio 0.2, 1.01, 5Atmospheric turbidity β ≥1Ångstrom coefficient α −1, 0, 1, 2, 3Ozone layer height H ≥12kmTotal ozone content DU ≥0
Fig. 1. Normalized vertical profile of number concentration ofozone used in the model for 1 cm ¼ 1000 DU. In this caseHoz0 ¼ 24km.
Fig. 2. Vertically integrated absorption cross section of ozone as afunction of wavelength for an ozone content of 1000DU.
Fig. 3. Geometry of the halo and corona models for (a) a horizon-tally uniform cloud layer and (b) a cloud cell of finite width.
an observer at sea level, it crosses the elevated layerat a zenith angle of about 84°.Aerosol particles alter the appearance of the sky
and optical phenomena. In the models, atmosphericturbidity, β, is a free parameter defined as the ratio ofthe hazy atmosphere’s total scattering cross sectionintegrated over the visible light spectrum to that of apure, molecular atmosphere. Effective aerosol size,another free parameter, is given in terms of the Ång-strom coefficient, α. This is allowed integral valuesfrom 3 for the finest urban aerosols to −1 for strato-spheric aerosols from large volcanic eruptions [25].All simulations shown here use α ¼ 1.Virtually all halos and coronas consist of sunlight
that has been scattered once, but much light eitherpasses straight through clean air and tenuous cloudswhen the Sun is high in the sky or is scattered morethan once when the Sun is near the horizon, the air ishazy, and the clouds are thick. In a perfectly efficientscattering model for halos and coronas, all light isscattered once as it strikes a crystal or droplet andis not scattered again or absorbed. The next orderof approximation, and the one used here, is to treathalos, coronas, and skylight as singly scattered sun-light whose radiance, IðλÞ, at each wavelength λ, isreduced by scattering and absorption according toBouguer’s law,
IðλÞ ¼ I0ðλÞe−τ; ð3Þ
before reaching the particle that produces the phe-nomenon and then is reduced again on its way tothe observer [9,10].Multiply scattered light adds to the background.
Its contribution increases with optical thickness,dominating in thick clouds, hazy air, and when theSun or the phenomena appear near the horizon.The models only include multiple scattering withinthe cloud and consider it to derive from both skylightabove the cloud and direct sunlight that reaches thecloud. Half of the skylight above the cloud is assumedto reach the cloud and enter it at the mean effective
secðϕÞ ¼ 1:66 for transmission of radiation throughplane parallel atmospheres. Since this light has al-ready been scattered, all of it that is scattered againin the cloud and passes through cloud base is in-cluded as multiply scattered light. Direct sunlightthat is scattered twice or more within the cloudand passes through the cloud base is also includedas multiply scattered light.
Each contribution to the multiply scattered lightwas calculated using a Monte Carlo scattering sub-routine that utilized smoothed angular scatteringphase functions PðψÞ (i.e., without peaks correspond-ing to halos or coronas) for clouds of crystals or dro-plets shown in Fig. 5. The curve of PðψÞ for dropletsmatched that of a Mie scattering program for sphereswith radius rm ¼ 8 μm. The curve of PðψÞ for crystalshas the same shape as that calculated for ice crystalfractals but is larger by a factor of about 2 at all an-gles except ψ ¼ 0 because it does not include lightthat passes through opposite parallel faces of thecrystals [26,27]. All multiply scattered light exitingcloud base is treated as downward isotropic and isreduced according to Bouguer’s law on its way tothe observer.
The calculated fluxes of direct, singly scattered,and multiply scattered light that penetrate a cloudof ice crystals and droplets are shown in Figs. 6(a)and 6(b), respectively as a function of τcld forϕSun ¼ 60°. For optically thin clouds ðτcld ≤ 0:3Þ directlight is the largest term, and singly scattered lightdominates multiply scattered light. The flux of singlyscattered light peaks at about τcld ≅ 0:55, and multi-ple scattering dominates for clouds of moderate andlarge optical thickness ðτcld ≥ 1:5Þ. However peak ra-diance of halos and coronas may exceed multiplyscattered light for clouds as thick as τcld ≅ 5 becausePðψÞ for ice crystal prisms and cloud droplets havepronounced, narrow maxima.
The angular scattering phase functions, PðψÞof randomly oriented ice crystal prisms used tosimulate the 22° and 46° halos are the piecewisesmooth curves of Fig. 7 based on equations that fit
Fig. 4. Optical air mass for a plane parallel atmosphere (sec(Z))and for the spherical Earth with a layer of thicknessH ¼ 8km andwith elevated layers at Hoz0 ¼ 20 and 25km, each with width athalf-maximum ΔH ¼ 20km.
Fig. 5. Smoothed angular scattering phase functions, PðψÞ usedin calculating the albedo of ice crystal and water droplet clouds.These smoothed phase functions do not include peaks that producehalos or coronas.
simulations of a Monte Carlo halo model for hexago-nal prisms with aspect ratios, 0.2 for long columns,1.01 for thick plates, and 5 for thin plates [11]. Theseaspect ratios were chosen to show the impact of crys-tal dimensions on the relative brightness of the 46°halo, which is greatest for thick plates and least forlong columns. The crystals were assumed be largeenough so that geometric optics is valid. Because dif-fraction is neglected, radiance is not accurately re-presented near the Sun, but this has little impacton the appearance of halos.The angular scattering phase functions for coronas
are computed using a Mie scattering subroutine ofCORONASKY with a binomial droplet size distribu-tion with 9 size bins. The mean rm and standard de-viation σr of droplet radius are free parameters in themodel. Curves of PðψÞ for clouds of droplets with rm ¼6000nm and σr ¼ 10nm and 1000nm are shown inFig. 8. Peaks and troughs of PðψÞ are much more pro-nounced and regular when σr ¼ 10nm, demonstrat-ing that coronas with multiple rings are only possiblewhen the droplet size spectrum is relatively narrow.
3. The Simulations
Programs HALOSKY and CORONASKY create pa-noramic views of the sky around halos and coronas.
HALOSKY uses a wide angle view that extends 57°from the Sun. Because coronas are usually muchsmaller than halos, CORONASKY creates a viewthat extends only 16° from the Sun. Color is producedby integrating over 61 wavelengths of the visiblespectrum, assigning red-green-blue (RGB) valuesof each wavelength based on the sensitivity of the hu-man eye and then applying the equation
RGB ¼ 255
�IRGBðϕÞImax
�ε: ð4Þ
Here, IRGBðϕÞ is the luminance of red, green, orblue light at a point in the sky, obtained from tabu-lated tristimulus values for each wavelength, andImax is the maximum luminance in each image.The exponent, ϵ, is used because the relation be-tween RGB values on the monitor and perceivedbrightness of a scene is not linear. We set ϵ ¼ 0:25for halos and ϵ ¼ 0:15 for coronas because thesevalues appeared to produce halos and coronas on
Fig. 6. Calculated fractions of light flux that penetrate a cloud ofcrystals (a) and droplets (b) for direct (thin curve), singly scattered(circles), and multiply scattered (squares) light as a function ofcloud optical depth, τcld, when the solar zenith angle ϕSun ¼ 60°.
Fig. 7. Angular scattering phase functions for right hexagonalprisms with aspect ratio 0.2 (long columns), 1.01 (thick plates),and 5 (thin plates). Curves for the thick plates and thin plates havebeen reduced by a factor of 10 and 100, respectively. Jagged curvesrepresent results of Monte Carlo halo simulations, while piecewisesmooth curves represent matching functions used in the models ofthis paper.
Fig. 8. Angular scattering phase functions for clouds of dropletswith mean radius rm ¼ 6000nm and standard deviations σr ¼10nm (thick line) and 1000nm (dotted line).
the monitor that matched observations most closely.The smaller value for coronas is necessary becauseradiance fades rapidly with distance from the Sun.Even though the models were not designed to si-
mulate light and color of the clear sky, convincing si-mulations of halos and coronas require that theycapture the major features of the appearance ofthe clear sky. HALOSKY was therefore first runfor clear skies with solar zenith angle ϕSun ¼ 40°.For the molecular atmosphere the sky is deep bluealoft, and neither radiance nor color change nearthe Sun, while for hazy air, the sky is brighter andwhiter particularly around the Sun and the horizonsky has a slight orange cast. Figure 9 shows the spec-tra of calculated radiance IðλÞ of skylight near the ze-nith, near the Sun, and near the horizon (i.e.,observer zenith angles, ϕobs ¼ 0:1°, 38:3°, and88:3°). Figures 10(a) and 10(b) show clear sky panor-amas for a molecular atmosphere (β ¼ 1) and a mod-erately hazy atmosphere (β ¼ 2 and α ¼ 1). Aerosolsincrease the radiance of skylight and make it appearwhiter or redder, especially near the Sun and nearthe horizon. Indeed, the aerosol laden sky just abovethe horizon is slightly orange, even though the Sun isrelatively high in the sky.HALOSKY was next run for clear skies in a mole-
cular atmosphere with the Sun at the horizon(ϕSun ¼ 90°) as the severest test of the model because
that is when the range of sky colors near the horizon,the impact of ozone on sky color, and the relative con-tribution of multiply scattered light are greatest. Thepanoramas are shown in Fig. 10(c) for 0 DU and inFig. 10(d) for 300 DU of ozone. Without absorptionthe sunset and twilight sky would be almost whitenear the zenith, so ozone is what keeps the zenithsky blue. The simulated images therefore indicatenot only what the clear sky looks like at sunsetbut also what it may have looked like early in Earth’shistory before the atmosphere had its protectiveshield of ozone. The sky just above the horizon is al-most equally orange with or without ozone. However,ozone sharpens the sky’s vertical color gradation andincreases the spectral purity of all colors from orangeat the horizon to blue aloft. But due to the neglect ofmultiple scattering, the simulated clear sunset skynear the horizon more closely resembles the realsky when the Sun is a few degrees below the horizon.
Neglect of multiple scattering in the clear air red-dens the simulated sunset sky most near the horizonbecause almost all violet and blue light reaching theobserver has been scattered more than once. This isapparent in Fig. 11, which compares calculated radi-ance spectra of molecular atmospheres at sunsetwith and without ozone to the measured sunset spec-tra on a cloudless day with high visibility. Measuredand modeled spectra including ozone have similarform, especially at ϕobs ¼ 60°, where absorption inthe Chappuis bands is captured well (Fig. 11(b)),but the measured spectrum at ϕobs ¼ 88° has muchmore violet and blue light than the modeledspectrum.
The halo simulations are mainly designed to inves-tigate the impact of cloud optical depth, τcld, on theappearance of the sky and of the 22° and 46° halos.Simulations were run for values of τcld in the range0:0001 ≤ τcld ≤ 10 with all other conditions constant.Halos were produced in a uniform cloud layer of ran-domly oriented solid right hexagonal column crystalsat pcld ¼ 300hPa for ϕSun ¼ 60°. Aerosols with β ¼1:2 and α ¼ 1 were confined between surface pres-sure, p0 ¼ 1013hPa, and paer ¼ 800hPa. The ozonelayer was centered at 24km with total content300 DU.
For these conditions, the 22° halo first appears in adeep blue, seemingly clear sky when τcld ≅ 0:0003 be-cause the 22° halo has a peak of PðψÞ for long columncrystals more than 30 times that of the sky just in-side, and humans can detect contrasts of luminanceas small as about 2%, and because τatm ≅ Oð0:1Þ. The46° halo has a peak of PðψÞ, roughly twice that of thesky just inside and so can first be seen at τcld ≅ 0:01.
The 22° halo is brightest for 0:02 ≤ τcld ≤ 1:5, whilethe 46° halo is brightest and shows color for0:1 ≤ τ ≤ 1:5. For τcld ≥ 5, only the top of the 22° haloremains faintly visible as a pale pink arc and atτcld ≥ 6, even the halo top disappears. These rangesof τcld are much reduced when the ice crystals arenot simple right hexagonal prisms.
Fig. 9. Calculated spectra of clear sky radiance for solar zenithangle ϕSun ¼ 40° in the vertical plane of the Sun with the observerlooking near the zenith (ϕobs ¼ 0:1), near the Sun (ϕobs ¼ 38:3), andnear the horizon (ϕobs ¼ 88:3°) for a molecular atmosphere (a) anda hazy atmosphere (b) with β ¼ 2 and α ¼ 1.
Sky panoramas for the above conditions are shownin Fig. 12 for τcld ¼ 0:03, 0.3, and 3.0. When τcld ¼0:03 the 22° halo is bright and colorful, with a dis-tinct red inner rim and a white outer annulus. The46° halo is faintly visible as a pale ring. The sky isdarker inside the 22° halo and deep blue everywhereexcept just outside the 22° halo and near the horizon.Directly below the 22° halo, the sky has a pronouncedyellow-orange cast, much as when an opaque cloudcovers all but the horizon, because then most shortwaves are scattered away during the long obliquepath the light takes from the distant clearing.
Because of the pronounced peak of PðψÞ, much ofthe light that penetrates an optically thin cloud ofright hexagonal ice crystal prisms does so at and justoutside the 22° halo. As a result, when the cloud baseis high, most light that appears near the horizon haspassed through a great optical thickness of clear airbelow the cloud and has been reddened by scattering.But when the cloud base is low (e.g., at Pcld ¼700hPa) or when it does not consist of such “perfect”crystals so that the peak of PðψÞ is less pronounced,or when it is optically thin, the sky near the horizondirectly below the Sun is not orange because then theeffective light source is closer to the observer.
When ϕSun ¼ 60°, peak radiance occurs at τcld ≅
0:2 for the bottom and τcld ≅ 0:65 for the top of the22° halo. At τcld ¼ 0:3, the 22° and 46° halos aretherefore near peak brilliance and color purity butthe sky aloft has a steel blue cast, with distinctly low-er spectral purity than a clear sky. The orange hue isalso deepest just below the 22° halo. This reversedvertical color gradation from halo bottom to horizon
Fig. 10. Simulated sky panoramas for solar zenith angle ϕSun ¼ 40° in a molecular atmosphere (a) and a hazy atmosphere (b) with β ¼ 2and α ¼ 1 and for ϕSun ¼ 90° in a molecular atmosphere with 300DU (c) and 0 DU (d) of ozone.
Fig. 11. Comparison of observed spectra of radiance of clear skiesfor solar zenith angle ϕSun ¼ 90° in the vertical plane of the Sunand the observer on 07 March 2008 with calculated spectra justabove the horizon (a) ϕobs ¼ 90° and near the midpoint in thesky and (b) ϕobs ¼ 60° with 0DU (hollow) and 300DU (solid) ofozone.
is caused by the rapid decrease of PðψÞ as ψ increasesabove 22°.As τcld increases above about 1 the halos fade no-
ticeably, especially at the bottom because of the long-er oblique path through the atmosphere just abovethe horizon. At τcld ¼ 3:0, the sky is gray, andalthough the entire 22° halo is still visible, onlythe top is bright and colorful. At all heights belowthe Sun, the 46° halo has disappeared.
The radiance, Itot, of a vertical slice of the skythrough the Sun is shown in Fig. 13 as a functionof observer zenith angle, ϕobs, for cloud opticaldepths, τcld ¼ 0:1, 1.0, and 3.0. When the cloud isthin, e.g., at τcld ¼ 0:1, the halo bottom is more pro-nounced than the top. For thicker clouds, e.g.,τcld ¼ 1:0, the top is more pronounced, while at τcld ¼3:0 cloud bottom is barely visible. Thus, observersshould be able to estimate cloud optical depth fromthe color of the sky and the appearance of halos.
The above findings represent optimal conditionsfor halos produced by clouds of simple, right hexago-nal ice crystal prisms. Most halos are much fainterbecause most ice crystals in thick clouds have manyfacets and irregular features. One of the commoncrystal forms in mid latitude storm cirrus and cirros-tratus, the bullet rosette, tends to be hollow at theend that would be a plane hexagonal surface [24].Such crystals lack a peak of PðψÞ at 46° and thereforecannot produce a 46° halo. Thus, crystal form is thedetermining factor for the low frequency of observingthe 46° halo.
Halos seen near the horizon that are produced byhigh cloud layers fade rapidly for τcld ≥ 1 because ofthe long slant path the light must take below cloud toreach the observer. But halos produced by narrowcloud cells near ground level (such as diamond dustdisplays) can be brilliant at or even below the hori-zon. HALOSKY therefore includes an option to selectthe width of cloud cells via the cloud width to heightratio, rcld ≡ xcld=hcld.
At first as cell width increases, so does the width ofthe sunbeam that produces the halo. This acts to in-crease radiance of the halo and is the dominant factorfor tenuous clouds, i.e., τcld ≪ cosðϕSunÞ. But sun-beams that enter the side of a cloud enter it higherup and take longer paths through it as the cloudwidens. This acts to dim halos and is the dominantfactor when τcld ≫ cosðϕSunÞ. No further changes tothe top and bottom of the halo beam take place forclouds wider than rcld > tanðϕSunÞ and rcld >
Fig. 12. Simulated sky panoramas of the 22° and 46° halos forϕSun ¼ 60°, β ¼ 1:2, pcld ¼ 300hPa, and cloud optical depths τcld ¼0:03 (top), 0.3 (center), and 3.0 (bottom).
Fig. 13. Calculated radiance, Itot, of the sky as a function of view-er zenith angle, ϕobs, for ϕSun ¼ 60°, β ¼ 1:2, pcld ¼ 300hPa, withτcld ¼ 0:1, 1.0, and 3.0. The halo is most pronounced at the topwhen τcld ¼ 1:0 and at the bottom when τcld ¼ 0:1. It is barely visi-ble at bottom when τcld ¼ 3:0.
tanðϕobsÞ, respectively. Figure 14 shows simulatedradiance of halo top and bottom as a function ofrcld for τcld ¼ 1:0 and the same conditions as in theearlier halo simulations. As the cloud widens, radi-ance at top and bottom first increases to a maximum,then decreases, and ultimately approaches aconstant value.The impact of cloud optical thickness, etc., on sym-
metric coronas has already been simulated [12]. Lessemphasis is devoted here to coronas because mostare so small they appear radially symmetric pro-vided there is no gradient of droplet sizes. But largecircular coronas (i.e., those produced by small dro-plets) can be highly asymmetric when the Sun is nearthe horizon. In that case, the nature of their asymme-try is similar to that of halos near the horizon.Figure 15 shows a sky panorama with a corona
produced in a cloud of uniform droplets withrm ¼ 6 μm, τcld ¼ 1:0 at ϕSun ¼ 80°, and pcld ¼600hPa and all other conditions as in Fig. 12. Be-cause all droplets are the same size the corona hasmany colorful rings above the Sun. However the bot-
tom half of the corona appears so near the horizonthat only the first ring is visible and all others havebeen obscured by the long oblique path through thecloud and clear air. The sky outside the corona ap-pears almost uniformly gray because scattering bydroplets dominates for ψ ≤ 16°, since PðψÞ is muchlarger for droplets than for molecules at these smallangles. It would be interesting to see a corona photo-graphed under similar conditions.
4. Summary and Conclusions
The two models presented here (HALOSKY andCORONASKY) produce sky panoramas that capturemany features of the light and color of the clear skyand of cloudy skies with circular halos and coronas.The models were described, and the simulations illu-strated how halos and coronas vary with atmo-spheric turbidity, solar zenith angle, and cloudheight, width, and optical depth.
Light in the models is absorbed by ozone, and scat-tered by air molecules, aerosol particles, and cloudparticles. The halos, coronas, and skylight consistof singly scattered sunlight that is reduced by scat-tering and absorption as it approaches the observer.Multiply scattered background light is included onlywithin clouds and is calculated in a Monte Carlo sub-routine. The atmosphere is assumed to be horizon-tally uniform and effects of the Earth’s curvatureare included in parameterized form by using airmass factors. The aerosol concentration and sizeand ozone content and height can be specified bythe user.
When the Sun is high in the sky, the color of themolecular atmosphere grades from deep blue atthe zenith to almost white at the horizon. As aerosolcontent increases, the sky whitens and brightens,particularly around the Sun and near the horizon.When the Sun appears on the horizon, sky colorranges through the spectrum from orange or red atthe horizon to blue above when ozone is included.
Halos are simulated for clouds of right hexagonalprisms. In this optimal case, the 22° (46°) halo firstappears when cloud optical depth is roughly 0.0003(0.004) and can be seen until optical depth of a cloudlayer is roughly 6 (4.5). The brightest, most colorfulhalos occur when cloud optical depth is somewhatless than the cosine of the observer’s zenith angle.In that case and when the bottom of the 22° halois near the horizon the sky is deep blue just insidethe halo and bright white just outside, except nearthe horizon where it is orange. When the Sun islow in the sky, halos produced by cloud layers arebrighter at the top than at the bottom for larger aver-age values and ranges of cloud optical depths. Whenthe solar zenith angle is 60°, the bottom of the 22°halo produced in a high cloud layer disappears whencloud optical depth is slightly greater than 3. How-ever, halos seen near the horizon through narrowcloud cells can be much brighter and more colorfuland are visible for a wider range of cloud optical
Fig. 14. Maximum and minimum calculated radiance of top andbottom of the 22° halo as a function of the width to height ratio of acloud element with τcld ¼ 1:0 for ϕSun ¼ 60°, β ¼ 1:2 and α ¼ 1.
Fig. 15. (Color online) Simulated sky panorama of a corona pro-duced by a cloud of droplets with radius, rd ¼ 6 μm, τcld ¼ 1:0 atpcld ¼ 600hPa, for ϕSun ¼ 80°, β ¼ 1:2, and α ¼ 1.
depths because then much sunlight enters the side ofthe cloud and does not penetrate its entire depth.In CORONASKY the user can choose the mean
and standard deviation of droplet size. Coronas arebrightest and most colorful and contain many ringswhen all droplets are closest to the mean size. In thatcase many of the same conclusions made for halosproduced by clouds of right hexagonal prisms holdtrue for coronas. In particular, when the Sun is with-in about 10° of the horizon, several colorful rings canbe seen at the top of the corona, while the bottom ofall outer rings disappears when the optical depth of alayer cloud is as small as 1.In summary, while simulations produced by the
simplified models presented here can never capturethe full range or beauty of sky color, halos, and cor-onas, they can provide insight into the factors thataffect these phenomena and can brighten the spiritson gloomy days or nights.
This research was supported by a NOAA CrestGrant and by a Professional Staff Congress CityUniversity of New York (PSC CUNY) grant.
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