Abstract

Light and color of geometric optics rainbows are simulated in their atmospheric environment. Sunlight passes through a molecular atmosphere with ozone and an aerosol layer near the ground to strike a cuboidal rain shaft below an overhanging cuboidal cloud. The rainbows are treated as singly scattered sunbeams that are depleted as they pass through the atmosphere and rain shaft. They appear in a setting illuminated by scattered light from behind the observer, from the background beyond the rain shaft, and from the rain shaft. In dark backgrounds the primary and secondary bows first become visible when the optical thickness of rain shafts τR0.0003 and τR0.003, respectively. The bows are brightest and most colorful for 0.1τR3, a range that is typical for most showers. The peaks of the scattering phase function for raindrops that correspond to the geometric optics rainbow are so pronounced that rainbows remain bright and colorful for optically thick rain shafts seen against dark backgrounds, but the bows appear washed out or vanish as the background brightens or where the rain shaft is shaded by an overhanging cloud. Rainbows also redden as the Sun approaches the horizon.

© 2008 Optical Society of America

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References

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  1. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?,” J. Opt. Soc. Am. 73, 1626-1628 (1983).
    [Crossref]
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    [Crossref] [PubMed]
  3. R. L. Lee, “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506-1519 (1998).
    [Crossref]
  4. R. A. R. Tricker, Introduction to Meteorological Optics(Elsevier, 1970), p. 169.
  5. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, 1954), reprint of 1938 edition, p. 169.
  6. R. L. Lee, Jr., “What are 'all the colors of the rainbow?” Appl. Opt. 30, 3401-3407 (1991).
    [Crossref] [PubMed]
  7. S. D. Gedzelman, “Visibility of halos and rainbows,” Appl. Opt. 19, 3068-3074 (1980).
    [Crossref] [PubMed]
  8. S. D. Gedzelman, “Rainbow brightness,” Appl. Opt. 21, 3032-3037 (1982).
    [Crossref] [PubMed]
  9. S. D. Gedzelman, “Simulating rainbows and halos in color,” Appl. Opt. 33, 4607-4613, 4958 (1994).
    [Crossref] [PubMed]
  10. S. D. Gedzelman and M. Vollmer, “Atmospheric optical phenomena and radiative transfer,” Bull. Am. Meteorol. Soc. 89, 471-485, (2008).
    [Crossref]
  11. S. D. Gedzelman, “Atmospheric Optics Programs” http://www.sci.ccny.cuny.edu/~stan.
  12. S. D. Gedzelman, “Simulating halos and coronas in their atmospheric environment,” Appl. Opt. 47, H157-H166 (2008).
    [Crossref]
  13. M. Vollmer and S. D. Gedzelman, “Colours of the Sun and Moon: the role of the optical air mass,” Eur. J. Phys. 27, 299-309 (2006).
    [Crossref]
  14. A. A. Lacis and J. E. Hansen, “A parameterization for the absorption of solar radiation in the Earth's atmosphere,” J. Atmos. Sci. 31, 118-133 (1974).
    [Crossref]
  15. D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University Press, 2001), pp. 109-115.
  16. T. Herd, Kaleidoscope Sky (Abrams Press, 2007), pp. 65-94.
  17. L. Cowley, “Atmospheric Optics,” http://www.atoptics.uk.
  18. R. R. Rogers, A Short Course in Cloud Physics, 2nd ed.(Pergamon, 1979).

2008 (2)

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

S. D. Gedzelman, “Simulating halos and coronas in their atmospheric environment,” Appl. Opt. 47, H157-H166 (2008).
[Crossref]

2006 (1)

M. Vollmer and S. D. Gedzelman, “Colours of the Sun and Moon: the role of the optical air mass,” Eur. J. Phys. 27, 299-309 (2006).
[Crossref]

1998 (1)

1994 (1)

1991 (2)

1983 (1)

1982 (1)

1980 (1)

1974 (1)

A. A. Lacis and J. E. Hansen, “A parameterization for the absorption of solar radiation in the Earth's atmosphere,” J. Atmos. Sci. 31, 118-133 (1974).
[Crossref]

Cowley, L.

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

Fraser, A. B.

Gedzelman, S. D.

S. D. Gedzelman, “Simulating halos and coronas in their atmospheric environment,” Appl. Opt. 47, H157-H166 (2008).
[Crossref]

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, 299-309 (2006).
[Crossref]

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

S. D. Gedzelman, “Rainbow brightness,” Appl. Opt. 21, 3032-3037 (1982).
[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.

Hansen, J. E.

A. A. Lacis and J. E. Hansen, “A parameterization for the absorption of solar radiation in the Earth's atmosphere,” J. Atmos. Sci. 31, 118-133 (1974).
[Crossref]

Herd, T.

T. Herd, Kaleidoscope Sky (Abrams Press, 2007), pp. 65-94.

Lacis, A. A.

A. A. Lacis and J. E. Hansen, “A parameterization for the absorption of solar radiation in the Earth's atmosphere,” J. Atmos. Sci. 31, 118-133 (1974).
[Crossref]

Lee, R. L.

Livingston, W.

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University Press, 2001), pp. 109-115.

Lynch, D. K.

D. K. Lynch and P. Schwartz, “Rainbows and fogbows,” Appl. Opt. 30, 3415-3420 (1991).
[Crossref] [PubMed]

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University Press, 2001), pp. 109-115.

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, 1954), reprint of 1938 edition, p. 169.

Rogers, R. R.

R. R. Rogers, A Short Course in Cloud Physics, 2nd ed.(Pergamon, 1979).

Schwartz, P.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics(Elsevier, 1970), p. 169.

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, 299-309 (2006).
[Crossref]

Appl. Opt. (7)

Bull. Am. Meteorol. Soc. (1)

S. D. Gedzelman and M. Vollmer, “Atmospheric optical phenomena and radiative transfer,” Bull. Am. Meteorol. Soc. 89, 471-485, (2008).
[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, 299-309 (2006).
[Crossref]

J. Atmos. Sci. (1)

A. A. Lacis and J. E. Hansen, “A parameterization for the absorption of solar radiation in the Earth's atmosphere,” J. Atmos. Sci. 31, 118-133 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Other (7)

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University Press, 2001), pp. 109-115.

T. Herd, Kaleidoscope Sky (Abrams Press, 2007), pp. 65-94.

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

R. R. Rogers, A Short Course in Cloud Physics, 2nd ed.(Pergamon, 1979).

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

R. A. R. Tricker, Introduction to Meteorological Optics(Elsevier, 1970), p. 169.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, 1954), reprint of 1938 edition, p. 169.

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Figures (5)

Fig. 1
Fig. 1

Side view (top) and plan view (bottom) of the rainbow cloud and rain shaft model. All the symbols are defined in the text.

Fig. 2
Fig. 2

Angular scattering phase functions for large spherical raindrops (jagged curve), cloud droplets with a radius of 8 μm (thin curve), and air molecules (thick curve).

Fig. 3
Fig. 3

Simulated sky panoramas of double rainbows for six settings: a solar zenith angle of ϕ S u n = 75 ° in all panels but (f), where ϕ S u n = 88 ° ; optical depth of τ R = 1 in all panels but (a), where τ R = 10 ; and (c) where τ R = 1 . The cloud and rain shaft are 500 and 800 m from the observer in all panels but (d), where they are 1300 and 1700 m , respectively.

Fig. 4
Fig. 4

Radiance versus scattering angle ψ measured from the antisolar point for several different values of rain shaft optical depth τ R when B B = 0.01 .

Fig. 5
Fig. 5

CIE chromaticity diagrams showing the impact of background brightness (0.01 in the top panel and 1.0 in the bottom panel) on the color range of the primary (heavy curves) and secondary (thin curves) rainbows for τ R = 1 and other conditions as in Fig. 3b.

Tables (1)

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Table 1 Table 1. User Selected Parameters in the Model

Equations (1)

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A R = 0.13 τ R 1 + 0.13 τ R .

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