Abstract

Models are developed that simulate the light and color of the sky and of circular halos and coronas as a function of atmospheric pressure, cloud height, width, and optical depth, solar zenith angle, aerosol concentration and size, and ozone content. Halos, coronas, and skylight are treated as singly scattered sunbeams that are depleted in their passage through the atmosphere and cloud. Multiple scattering is included only for background cloud light. Halos produced by hexagonal crystal prisms and coronas produced by monodisperse droplets are visible for cloud optical depths in the range 0.0003τcld7 and are brightest and most colorful when τcld is somewhat less than the cosine of the observer’s zenith angle. When the Sun is low in the sky, halos and coronas can be bright only at smaller cloud optical depths and tend to be faint at their bottoms when produced in high cloud layers but can be bright at the horizon when produced by narrow cloud cells near ground level.

© 2008 Optical Society of America

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References

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  1. A. J. Heymsfield, A. Bansemer, S. L. Durden, R. L. Herman, and T. P. Bui, “Ice microphysics observations in Hurricane Humberto: Comparison with non-hurricane-generated ice cloud layers,” J. Atmos. Sci. 63, 288-308 (2006).
    [CrossRef]
  2. A. J. Heymsfield, “Properties of tropical and midlatitude ice cloud particle ensembles. Part I: Median mass diameters and terminal velocities,” J. Atmos. Sci. 60, 2573-2591 (2003).
    [CrossRef]
  3. W. Tape, Atmospheric Halos, Vol. 64 of Antarctic Research Series (American Geophysical Union, 1994).
    [CrossRef]
  4. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, 1954) , pp. 235-308, reprint of 1938 edition.
  5. R. A. R. Tricker, Introduction to Meteorological Optics (Mills and Boon, 1970).
  6. R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, 1980).
  7. D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge U. Press, 2001).
  8. J. A. Shaw and P. J. Neiman, “Coronas and iridescence in mountain wave clouds,” Appl. Opt. 42, 476-485 (2003).
    [CrossRef] [PubMed]
  9. R. Meyer, Die Haloerscheinungen. Probleme der Kosmischen Physik XII (Henri Grand, 1929), pp. 64-69.
  10. S. D. Gedzelman, “Visibility of halos and rainbows,” Appl. Opt. 19, 3068-3074 (1980).
    [CrossRef] [PubMed]
  11. S. D. Gedzelman, “Simulating rainbows and halos in color,” Appl. Opt. 33, 4607-4613, 4958 (1994).
    [CrossRef] [PubMed]
  12. S. D. Gedzelman and J. Lock, “Simulating coronas in color,” Appl. Opt. 42, 497-504 (2003).
    [CrossRef] [PubMed]
  13. S. D. Gedzelman and M. Vollmer, “Atmospheric optical phenomena and radiative transfer,” Bull. Am. Meteorol. Soc. 89, 471-485 (2008).
    [CrossRef]
  14. W. B. Rossow and R. A. Schiffer, “Advances in understanding clouds from ISCCP,” Bull. Am. Meteorol. Soc. 80, 2261-2286 (1999).
    [CrossRef]
  15. R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The origin of sun pillars,” Am. Sci. 60, 292-302 (1972).
  16. R. G. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128-131 (1972).
    [CrossRef] [PubMed]
  17. E. Tränkle and R. Greenler, “Multiple scattering effects in halo phenomena,” J. Opt. Soc. Am. A 4, 591-599 (1987).
    [CrossRef]
  18. L. Cowley, “Atmospheric optics,” http://www.atoptics.uk.
  19. P. Laven, “Simulation of rainbows, coronas, and glories by use of Mie theory,” Appl. Opt. 42, 436-444 (2003).
    [CrossRef] [PubMed]
  20. P. Laven, “Atmospheric glories: simulations and observations,” Appl. Opt. 44, 5667-5674 (2005).
    [CrossRef] [PubMed]
  21. S. D. Gedzelman, “Atmospheric optics programs,” http://www.sci.ccny.cuny.edu/~stan.
  22. E. C. Y. Inn and Y. Tanaka, “Absorption coefficient of ozone in the ultraviolet and visible regions,” J. Opt. Soc. Am. 43, 870-873 (1953).
    [CrossRef]
  23. A. Amoruso, M. Cacciani, A. Di Sarra, and G. Fiocco, “Absorption cross sections of ozone in the 590- to 610 nm region at T=230 K and T=299 K,” J. Geophys. Res. 95, 20565-20568(1990).
    [CrossRef]
  24. M. Vollmer and S. D. Gedzelman, “Colours of the sun and moon: the role of the optical air mass,” Eur. J. Phys. 27, 299-306 (2006).
    [CrossRef]
  25. O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
    [CrossRef]
  26. C. G. Schmitt, J. Iaquinta, and A. J. Heymsfield, “The asymmetry parameter of cirrus clouds composed of hollow bullet rosette shaped ice crystals from ray-tracing calculations,” J. Appl. Meteorol. 45, 973-981 (2006).
    [CrossRef]
  27. D. L. Mitchell, A. Macke, and Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967-2988 (1996).
    [CrossRef]

2008 (1)

S. D. Gedzelman and M. Vollmer, “Atmospheric optical phenomena and radiative transfer,” Bull. Am. Meteorol. Soc. 89, 471-485 (2008).
[CrossRef]

2006 (3)

A. J. Heymsfield, A. Bansemer, S. L. Durden, R. L. Herman, and T. P. Bui, “Ice microphysics observations in Hurricane Humberto: Comparison with non-hurricane-generated ice cloud layers,” J. Atmos. Sci. 63, 288-308 (2006).
[CrossRef]

M. Vollmer and S. D. Gedzelman, “Colours of the sun and moon: the role of the optical air mass,” Eur. J. Phys. 27, 299-306 (2006).
[CrossRef]

C. G. Schmitt, J. Iaquinta, and A. J. Heymsfield, “The asymmetry parameter of cirrus clouds composed of hollow bullet rosette shaped ice crystals from ray-tracing calculations,” J. Appl. Meteorol. 45, 973-981 (2006).
[CrossRef]

2005 (1)

2003 (4)

2002 (1)

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

1999 (1)

W. B. Rossow and R. A. Schiffer, “Advances in understanding clouds from ISCCP,” Bull. Am. Meteorol. Soc. 80, 2261-2286 (1999).
[CrossRef]

1996 (1)

D. L. Mitchell, A. Macke, and Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967-2988 (1996).
[CrossRef]

1994 (1)

1990 (1)

A. Amoruso, M. Cacciani, A. Di Sarra, and G. Fiocco, “Absorption cross sections of ozone in the 590- to 610 nm region at T=230 K and T=299 K,” J. Geophys. Res. 95, 20565-20568(1990).
[CrossRef]

1987 (1)

1980 (1)

1972 (2)

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The origin of sun pillars,” Am. Sci. 60, 292-302 (1972).

R. G. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128-131 (1972).
[CrossRef] [PubMed]

1953 (1)

Amoruso, A.

A. Amoruso, M. Cacciani, A. Di Sarra, and G. Fiocco, “Absorption cross sections of ozone in the 590- to 610 nm region at T=230 K and T=299 K,” J. Geophys. Res. 95, 20565-20568(1990).
[CrossRef]

Bansemer, A.

A. J. Heymsfield, A. Bansemer, S. L. Durden, R. L. Herman, and T. P. Bui, “Ice microphysics observations in Hurricane Humberto: Comparison with non-hurricane-generated ice cloud layers,” J. Atmos. Sci. 63, 288-308 (2006).
[CrossRef]

Blumenthal, G.

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The origin of sun pillars,” Am. Sci. 60, 292-302 (1972).

Bui, T. P.

A. J. Heymsfield, A. Bansemer, S. L. Durden, R. L. Herman, and T. P. Bui, “Ice microphysics observations in Hurricane Humberto: Comparison with non-hurricane-generated ice cloud layers,” J. Atmos. Sci. 63, 288-308 (2006).
[CrossRef]

Cacciani, M.

A. Amoruso, M. Cacciani, A. Di Sarra, and G. Fiocco, “Absorption cross sections of ozone in the 590- to 610 nm region at T=230 K and T=299 K,” J. Geophys. Res. 95, 20565-20568(1990).
[CrossRef]

Cowley, L.

L. Cowley, “Atmospheric optics,” http://www.atoptics.uk.

Di Sarra, A.

A. Amoruso, M. Cacciani, A. Di Sarra, and G. Fiocco, “Absorption cross sections of ozone in the 590- to 610 nm region at T=230 K and T=299 K,” J. Geophys. Res. 95, 20565-20568(1990).
[CrossRef]

Drinkwine, M.

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The origin of sun pillars,” Am. Sci. 60, 292-302 (1972).

Dubovik, O.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

Durden, S. L.

A. J. Heymsfield, A. Bansemer, S. L. Durden, R. L. Herman, and T. P. Bui, “Ice microphysics observations in Hurricane Humberto: Comparison with non-hurricane-generated ice cloud layers,” J. Atmos. Sci. 63, 288-308 (2006).
[CrossRef]

Eck, T. F.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

Fiocco, G.

A. Amoruso, M. Cacciani, A. Di Sarra, and G. Fiocco, “Absorption cross sections of ozone in the 590- to 610 nm region at T=230 K and T=299 K,” J. Geophys. Res. 95, 20565-20568(1990).
[CrossRef]

Gedzelman, S. D.

S. D. Gedzelman and M. Vollmer, “Atmospheric optical phenomena and radiative transfer,” Bull. Am. Meteorol. Soc. 89, 471-485 (2008).
[CrossRef]

M. Vollmer and S. D. Gedzelman, “Colours of the sun and moon: the role of the optical air mass,” Eur. J. Phys. 27, 299-306 (2006).
[CrossRef]

S. D. Gedzelman and J. Lock, “Simulating coronas in color,” Appl. Opt. 42, 497-504 (2003).
[CrossRef] [PubMed]

S. D. Gedzelman, “Simulating rainbows and halos in color,” Appl. Opt. 33, 4607-4613, 4958 (1994).
[CrossRef] [PubMed]

S. D. Gedzelman, “Visibility of halos and rainbows,” Appl. Opt. 19, 3068-3074 (1980).
[CrossRef] [PubMed]

S. D. Gedzelman, “Atmospheric optics programs,” http://www.sci.ccny.cuny.edu/~stan.

Greenler, R.

Greenler, R. G.

R. G. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128-131 (1972).
[CrossRef] [PubMed]

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The origin of sun pillars,” Am. Sci. 60, 292-302 (1972).

Herman, R. L.

A. J. Heymsfield, A. Bansemer, S. L. Durden, R. L. Herman, and T. P. Bui, “Ice microphysics observations in Hurricane Humberto: Comparison with non-hurricane-generated ice cloud layers,” J. Atmos. Sci. 63, 288-308 (2006).
[CrossRef]

Heymsfield, A. J.

A. J. Heymsfield, A. Bansemer, S. L. Durden, R. L. Herman, and T. P. Bui, “Ice microphysics observations in Hurricane Humberto: Comparison with non-hurricane-generated ice cloud layers,” J. Atmos. Sci. 63, 288-308 (2006).
[CrossRef]

C. G. Schmitt, J. Iaquinta, and A. J. Heymsfield, “The asymmetry parameter of cirrus clouds composed of hollow bullet rosette shaped ice crystals from ray-tracing calculations,” J. Appl. Meteorol. 45, 973-981 (2006).
[CrossRef]

A. J. Heymsfield, “Properties of tropical and midlatitude ice cloud particle ensembles. Part I: Median mass diameters and terminal velocities,” J. Atmos. Sci. 60, 2573-2591 (2003).
[CrossRef]

Holben, B. N.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

Iaquinta, J.

C. G. Schmitt, J. Iaquinta, and A. J. Heymsfield, “The asymmetry parameter of cirrus clouds composed of hollow bullet rosette shaped ice crystals from ray-tracing calculations,” J. Appl. Meteorol. 45, 973-981 (2006).
[CrossRef]

Inn, E. C. Y.

Kaufman, Y. J.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

King, M. D.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

Laven, P.

Liu, Y.

D. L. Mitchell, A. Macke, and Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967-2988 (1996).
[CrossRef]

Livingston, W.

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge U. Press, 2001).

Lock, J.

Lynch, D. K.

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge U. Press, 2001).

Macke, A.

D. L. Mitchell, A. Macke, and Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967-2988 (1996).
[CrossRef]

Mallmann, A. J.

R. G. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128-131 (1972).
[CrossRef] [PubMed]

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The origin of sun pillars,” Am. Sci. 60, 292-302 (1972).

Meyer, R.

R. Meyer, Die Haloerscheinungen. Probleme der Kosmischen Physik XII (Henri Grand, 1929), pp. 64-69.

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, 1954) , pp. 235-308, reprint of 1938 edition.

Mitchell, D. L.

D. L. Mitchell, A. Macke, and Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967-2988 (1996).
[CrossRef]

Neiman, P. J.

Rossow, W. B.

W. B. Rossow and R. A. Schiffer, “Advances in understanding clouds from ISCCP,” Bull. Am. Meteorol. Soc. 80, 2261-2286 (1999).
[CrossRef]

Schiffer, R. A.

W. B. Rossow and R. A. Schiffer, “Advances in understanding clouds from ISCCP,” Bull. Am. Meteorol. Soc. 80, 2261-2286 (1999).
[CrossRef]

Schmitt, C. G.

C. G. Schmitt, J. Iaquinta, and A. J. Heymsfield, “The asymmetry parameter of cirrus clouds composed of hollow bullet rosette shaped ice crystals from ray-tracing calculations,” J. Appl. Meteorol. 45, 973-981 (2006).
[CrossRef]

Shaw, J. A.

Slutsker, I.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

Smirnov, A.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

Tanaka, Y.

Tanre, D.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

Tape, W.

W. Tape, Atmospheric Halos, Vol. 64 of Antarctic Research Series (American Geophysical Union, 1994).
[CrossRef]

Tränkle, E.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (Mills and Boon, 1970).

Vollmer, M.

S. D. Gedzelman and M. Vollmer, “Atmospheric optical phenomena and radiative transfer,” Bull. Am. Meteorol. Soc. 89, 471-485 (2008).
[CrossRef]

M. Vollmer and S. D. Gedzelman, “Colours of the sun and moon: the role of the optical air mass,” Eur. J. Phys. 27, 299-306 (2006).
[CrossRef]

Am. Sci. (1)

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The origin of sun pillars,” Am. Sci. 60, 292-302 (1972).

Appl. Opt. (6)

Bull. Am. Meteorol. Soc. (2)

S. D. Gedzelman and M. Vollmer, “Atmospheric optical phenomena and radiative transfer,” Bull. Am. Meteorol. Soc. 89, 471-485 (2008).
[CrossRef]

W. B. Rossow and R. A. Schiffer, “Advances in understanding clouds from ISCCP,” Bull. Am. Meteorol. Soc. 80, 2261-2286 (1999).
[CrossRef]

Eur. J. Phys. (1)

M. Vollmer and S. D. Gedzelman, “Colours of the sun and moon: the role of the optical air mass,” Eur. J. Phys. 27, 299-306 (2006).
[CrossRef]

J. Appl. Meteorol. (1)

C. G. Schmitt, J. Iaquinta, and A. J. Heymsfield, “The asymmetry parameter of cirrus clouds composed of hollow bullet rosette shaped ice crystals from ray-tracing calculations,” J. Appl. Meteorol. 45, 973-981 (2006).
[CrossRef]

J. Atmos. Sci. (4)

D. L. Mitchell, A. Macke, and Y. Liu, “Modeling cirrus clouds. Part II: Treatment of radiative properties,” J. Atmos. Sci. 53, 2967-2988 (1996).
[CrossRef]

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590-608 (2002).
[CrossRef]

A. J. Heymsfield, A. Bansemer, S. L. Durden, R. L. Herman, and T. P. Bui, “Ice microphysics observations in Hurricane Humberto: Comparison with non-hurricane-generated ice cloud layers,” J. Atmos. Sci. 63, 288-308 (2006).
[CrossRef]

A. J. Heymsfield, “Properties of tropical and midlatitude ice cloud particle ensembles. Part I: Median mass diameters and terminal velocities,” J. Atmos. Sci. 60, 2573-2591 (2003).
[CrossRef]

J. Geophys. Res. (1)

A. Amoruso, M. Cacciani, A. Di Sarra, and G. Fiocco, “Absorption cross sections of ozone in the 590- to 610 nm region at T=230 K and T=299 K,” J. Geophys. Res. 95, 20565-20568(1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Science (1)

R. G. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128-131 (1972).
[CrossRef] [PubMed]

Other (8)

L. Cowley, “Atmospheric optics,” http://www.atoptics.uk.

S. D. Gedzelman, “Atmospheric optics programs,” http://www.sci.ccny.cuny.edu/~stan.

W. Tape, Atmospheric Halos, Vol. 64 of Antarctic Research Series (American Geophysical Union, 1994).
[CrossRef]

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, 1954) , pp. 235-308, reprint of 1938 edition.

R. A. R. Tricker, Introduction to Meteorological Optics (Mills and Boon, 1970).

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, 1980).

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge U. Press, 2001).

R. Meyer, Die Haloerscheinungen. Probleme der Kosmischen Physik XII (Henri Grand, 1929), pp. 64-69.

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Figures (15)

Fig. 1
Fig. 1

Normalized vertical profile of number concentration of ozone used in the model for 1 cm = 1000 DU. In this case H o z 0 = 24 km .

Fig. 2
Fig. 2

Vertically integrated absorption cross section of ozone as a function of wavelength for an ozone content of 1000 DU .

Fig. 3
Fig. 3

Geometry of the halo and corona models for (a) a horizontally uniform cloud layer and (b) a cloud cell of finite width.

Fig. 4
Fig. 4

Optical air mass for a plane parallel atmosphere (sec(Z)) and for the spherical Earth with a layer of thickness H = 8 km and with elevated layers at H o z 0 = 20 and 25 km , each with width at half-maximum Δ H = 20 km .

Fig. 5
Fig. 5

Smoothed angular scattering phase functions, P ( ψ ) used in calculating the albedo of ice crystal and water droplet clouds. These smoothed phase functions do not include peaks that produce halos or coronas.

Fig. 6
Fig. 6

Calculated fractions of light flux that penetrate a cloud of crystals (a) and droplets (b) for direct (thin curve), singly scattered (circles), and multiply scattered (squares) light as a function of cloud optical depth, τ cld , when the solar zenith angle ϕ Sun = 60 ° .

Fig. 7
Fig. 7

Angular scattering phase functions for right hexagonal prisms with aspect ratio 0.2 (long columns), 1.01 (thick plates), and 5 (thin plates). Curves for the thick plates and thin plates have been reduced by a factor of 10 and 100, respectively. Jagged curves represent results of Monte Carlo halo simulations, while piecewise smooth curves represent matching functions used in the models of this paper.

Fig. 8
Fig. 8

Angular scattering phase functions for clouds of droplets with mean radius r m = 6000 nm and standard deviations σ r = 10 nm (thick line) and 1000 nm (dotted line).

Fig. 9
Fig. 9

Calculated spectra of clear sky radiance for solar zenith angle ϕ Sun = 40 ° in the vertical plane of the Sun with the observer looking near the zenith ( ϕ obs = 0.1 ), near the Sun ( ϕ obs = 38.3 ), and near the horizon ( ϕ obs = 88.3 ° ) for a molecular atmosphere (a) and a hazy atmosphere (b) with β = 2 and α = 1 .

Fig. 10
Fig. 10

Simulated sky panoramas for solar zenith angle ϕ Sun = 40 ° in a molecular atmosphere (a) and a hazy atmosphere (b) with β = 2 and α = 1 and for ϕ Sun = 90 ° in a molecular atmosphere with 300 DU (c) and 0 DU (d) of ozone.

Fig. 11
Fig. 11

Comparison of observed spectra of radiance of clear skies for solar zenith angle ϕ Sun = 90 ° in the vertical plane of the Sun and the observer on 07 March 2008 with calculated spectra just above the horizon (a)  ϕ obs = 90 ° and near the midpoint in the sky and (b)  ϕ obs = 60 ° with 0 DU (hollow) and 300 DU (solid) of ozone.

Fig. 12
Fig. 12

Simulated sky panoramas of the 22 ° and 46 ° halos for ϕ Sun = 60 ° , β = 1.2 , p cld = 300 hPa , and cloud optical depths τ cld = 0.03 (top), 0.3 (center), and 3.0 (bottom).

Fig. 13
Fig. 13

Calculated radiance, I tot , of the sky as a function of viewer zenith angle, ϕ obs , for ϕ Sun = 60 ° , β = 1.2 , p cld = 300 hPa , with τ cld = 0.1 , 1.0, and 3.0. The halo is most pronounced at the top when τ cld = 1.0 and at the bottom when τ cld = 0.1 . It is barely visible at bottom when τ cld = 3.0 .

Fig. 14
Fig. 14

Maximum and minimum calculated radiance of top and bottom of the 22 ° halo as a function of the width to height ratio of a cloud element with τ cld = 1.0 for ϕ Sun = 60 ° , β = 1.2 and α = 1 .

Fig. 15
Fig. 15

Simulated sky panorama of a corona produced by a cloud of droplets with radius, r d = 6 μm , τ cld = 1.0 at p cld = 600 hPa , for ϕ Sun = 80 ° , β = 1.2 , and α = 1 .

Tables (1)

Tables Icon

Table 1 User-Selected Parameters in the Models

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

M H = r E H cos ( ϕ ) + ( r E H cos ( ϕ ) ) 2 + 2 r E H + 1 .
M H , Δ H = ( H + Δ H 2 ) M H + Δ H 2 , Δ H ( H Δ H 2 ) M H Δ H 2 Δ H .
I ( λ ) = I 0 ( λ ) e τ ,
RGB = 255 ( I RGB ( ϕ ) I max ) ε .

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