Abstract

Model simulations of laboratory-generated and natural crepuscular rays are presented. Rays are created in the laboratory with parallel light beams that pass through artificial fogs and milk–water solutions. Light scattered by 90° in a dilute mixture of whole milk first increases in intensity with distance from the source to a maximum as a result of multiple scattering by mainly small angles before decreasing exponentially due to extinction as distance continues to increase. Crepuscular rays are simulated for three cloud configurations. In case 1, the Sun at the zenith is blocked by a cloud with an overhanging anvil. The rays appear white against blue sky and are brightest when atmospheric turbidity, β11. Shading by the anvil separates maximum brightness from apparent cloud edge. In case 2, a ray passes through a rectangular gap in a cloud layer. The ray is faint blue in a molecular atmosphere but turns pale yellow as β and solar zenith angle, ϕsun, increase. At ϕsun=60° it appears most striking when the cloud is optically thick, β5, and the beam width Δx1000m. In these cases, increasing aerosol radius, raer, to about 1000nm brightens, narrows, and shortens rays. In case 3, the twilight Sun is shaded by a towering cloud or mountain. The shaded rays are deeper blue than the sunlit sky because the light originates higher in the atmosphere, where short waves have suffered less depletion from scattering. The long optical path taken by sunlight at twilight makes color and lighting contrasts of the rays greatest when the air is quite clean, i.e., for β11. In all cases, the brightest rays occur when sunlight passes through an optical thickness of atmosphere, τO(1).

© 2011 Optical Society of America

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References

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  1. M. Minnaert, The Nature of Light and Color in the Open Air, 1938 ed. (Dover, 1954), pp. 1–3, 257–277, reprint of 1938 edition.
  2. R. Greenler, Rainbows, Halos and Glories (Cambridge University, 1980), pp. 129–131, 181–182.
  3. D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University, 2001), pp. 1–19.
  4. M. Vollmer, Lichtspiele in der Luft: Atmosphärische Optik für Einsteiger (Elsevier Spektrum, 2006), pp. 297–309.
  5. D. Lynch, “Optics of sunbeams,” J. Opt. Soc. Am. 4, 609–611(1987).
    [CrossRef]
  6. M. S. Van Den Broeke, W. H. Beasley, and M. B. Richman, “The role of atmospheric conditions in determining intensity of crepuscular and anticrepuscular rays,” Mon. Weather Rev. 138, 2883–2894 (2010).
    [CrossRef]
  7. S. D. Gedzelman, M. Á. López-Álvarez, J. Hernandez-Andrés, and R. Greenler, “Quantifying the “milky sky” experiment,” Appl. Opt. 47, H128–H132, doi:10.1364/AO.47.00H128 (2008).
    [CrossRef] [PubMed]
  8. S. D. Gedzelman and M. Vollmer, “Atmospheric optical phenomena and radiative transfer,” Bull. Am. Meteorol. Soc. 89, 471–485 (2008).
    [CrossRef]
  9. 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]
  10. G. Pretor-Pinney, “The Cloud Appreciation Society,” http://www.cloudappreciationsociety. org. Click on gallery, and then select the category “Optical Effects: Crepuscular Rays & Cloud Shadows.”
  11. S. D. Gedzelman, “Simulating colors of clear and partly cloudy skies,” Appl. Opt. 44, 5723–5736 (2005).
    [CrossRef] [PubMed]
  12. R. L. Lee and J. Hernández-Andrés, “Measuring and modeling twilight’s purple light,” Appl. Opt. 42, 445–457 (2003).
    [CrossRef] [PubMed]
  13. J. L. Monteith, “Crepuscular rays formed by the Western Ghats,” Weather 41, 292–299 (1986).

2010

M. S. Van Den Broeke, W. H. Beasley, and M. B. Richman, “The role of atmospheric conditions in determining intensity of crepuscular and anticrepuscular rays,” Mon. Weather Rev. 138, 2883–2894 (2010).
[CrossRef]

2008

2005

2003

1987

D. Lynch, “Optics of sunbeams,” J. Opt. Soc. Am. 4, 609–611(1987).
[CrossRef]

1986

J. L. Monteith, “Crepuscular rays formed by the Western Ghats,” Weather 41, 292–299 (1986).

1974

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]

Beasley, W. H.

M. S. Van Den Broeke, W. H. Beasley, and M. B. Richman, “The role of atmospheric conditions in determining intensity of crepuscular and anticrepuscular rays,” Mon. Weather Rev. 138, 2883–2894 (2010).
[CrossRef]

Gedzelman, S. D.

Greenler, R.

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]

Hernandez-Andrés, J.

Hernández-Andrés, J.

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, 2001), pp. 1–19.

López-Álvarez, M. Á.

Lynch, D.

D. Lynch, “Optics of sunbeams,” J. Opt. Soc. Am. 4, 609–611(1987).
[CrossRef]

Lynch, D. K.

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University, 2001), pp. 1–19.

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air, 1938 ed. (Dover, 1954), pp. 1–3, 257–277, reprint of 1938 edition.

Monteith, J. L.

J. L. Monteith, “Crepuscular rays formed by the Western Ghats,” Weather 41, 292–299 (1986).

Pretor-Pinney, G.

G. Pretor-Pinney, “The Cloud Appreciation Society,” http://www.cloudappreciationsociety. org. Click on gallery, and then select the category “Optical Effects: Crepuscular Rays & Cloud Shadows.”

Richman, M. B.

M. S. Van Den Broeke, W. H. Beasley, and M. B. Richman, “The role of atmospheric conditions in determining intensity of crepuscular and anticrepuscular rays,” Mon. Weather Rev. 138, 2883–2894 (2010).
[CrossRef]

Van Den Broeke, M. S.

M. S. Van Den Broeke, W. H. Beasley, and M. B. Richman, “The role of atmospheric conditions in determining intensity of crepuscular and anticrepuscular rays,” Mon. Weather Rev. 138, 2883–2894 (2010).
[CrossRef]

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, Lichtspiele in der Luft: Atmosphärische Optik für Einsteiger (Elsevier Spektrum, 2006), pp. 297–309.

Appl. Opt.

Bull. Am. Meteorol. Soc.

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

J. Atmos. Sci.

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.

D. Lynch, “Optics of sunbeams,” J. Opt. Soc. Am. 4, 609–611(1987).
[CrossRef]

Mon. Weather Rev.

M. S. Van Den Broeke, W. H. Beasley, and M. B. Richman, “The role of atmospheric conditions in determining intensity of crepuscular and anticrepuscular rays,” Mon. Weather Rev. 138, 2883–2894 (2010).
[CrossRef]

Weather

J. L. Monteith, “Crepuscular rays formed by the Western Ghats,” Weather 41, 292–299 (1986).

Other

G. Pretor-Pinney, “The Cloud Appreciation Society,” http://www.cloudappreciationsociety. org. Click on gallery, and then select the category “Optical Effects: Crepuscular Rays & Cloud Shadows.”

M. Minnaert, The Nature of Light and Color in the Open Air, 1938 ed. (Dover, 1954), pp. 1–3, 257–277, reprint of 1938 edition.

R. Greenler, Rainbows, Halos and Glories (Cambridge University, 1980), pp. 129–131, 181–182.

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University, 2001), pp. 1–19.

M. Vollmer, Lichtspiele in der Luft: Atmosphärische Optik für Einsteiger (Elsevier Spektrum, 2006), pp. 297–309.

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Figures (15)

Fig. 1
Fig. 1

Photograph of crepuscular rays early in the morning of 22 December 2010 at Fort Lauderdale, Florida.

Fig. 2
Fig. 2

Crepuscular rays created in the laboratory with light from a slide projector passing through artificial fog and passing an opaque obstacle.

Fig. 3
Fig. 3

Scattered light from a green laser pointer ( P = 6.5 mW , λ = 532 nm ) passing through a tank of milky water.

Fig. 4
Fig. 4

Signal of near-monochromatic light (in nanowatts, spectral width 20 nm ) scattered by 90 ° in a tank with a dilute mixture of whole milk as a function of distance (in millimeters) from the source of the light beam.

Fig. 5
Fig. 5

Transmitted light as a function of wavelength (in nanometers) measured 30 and 90 min after adding a dilute mixture of skim milk to a tank of water for conditions given in the text.

Fig. 6
Fig. 6

Multiple scattering Monte Carlo model of the number of light beams scattered by 90 ° as a function of distance (in millimeters) down a cylindrical tube with total optical thickness τ = 7.2 for spherical water drops with radius r aer = 0.3 μm = 300 nm .

Fig. 7
Fig. 7

Photograph of crepuscular rays when the Sun was near the zenith and blocked by a towering cumulus 27 July 2010 in Marietta, Georgia.

Fig. 8
Fig. 8

Geometrical model for crepuscular rays produced by a towering cumulus with a heart-shaped floor plan when the Sun is at the zenith. In this image the observer is x cld from the edge of cloud base at height z cld . Observer looks up at zenith angle, ϕ obs , and the ray is shaded once it passes under the anvil a distance x ovr from the observer.

Fig. 9
Fig. 9

Simulated crepuscular rays produced by a heart-shaped cloud when the observer is x cld = 100 m from the edge of cloud base and x ovr = 600 m from the edge of the anvil. Cloud base pressure, p cld = 500 hPa ; atmospheric turbidity, β = 4 ; aerosol radius, r aer = 400 nm .

Fig. 10
Fig. 10

Relative luminance of crepuscular rays produced by a cloud with an overhanging anvil that is out of sight as a function of observer zenith angle for different values of atmospheric turbidity.

Fig. 11
Fig. 11

Photograph of crepuscular rays due to sunbeams passing through gaps in a cloud layer in Cliffside Park, New Jersey.

Fig. 12
Fig. 12

Geometrical model for crepuscular rays produced by sunlight at zenith angle, ϕ sun , passing through a rectangular gap of width Δ x in a cloud layer with base at height z cld a distance x 1 from the observer.

Fig. 13
Fig. 13

Simulated crepuscular rays Δ x = 400 m wide as a function of scattering angle, ϕ sun ϕ obs produced by sunlight passing through a gap in a cloud layer for β = 1 , 2, and 5 with aerosols of mean radius, r aer = 400 nm , when ϕ sun = 60 ° , x 1 = 3000 m , and p cld = 850 hPa with a bright spot on the water surface below. Optical thickness taken by the sunbeam through the cloud is τ cld = 100 .

Fig. 14
Fig. 14

Relative luminance of crepuscular rays produced by sunlight passing through a gap in a cloud layer as a function of the scattering angle, ϕ sun ϕ obs for β = 5 when ϕ sun = 20 ° , x 1 = 1200 m , p cld = 700 hPa , and other conditions of Fig. 13.

Fig. 15
Fig. 15

Simulated dark crepuscular ray at twilight when ϕ sun = 91.6 ° produced by a cloud at the terminator 8000 m high with surface pressure 880 hPa , atmospheric turbidity β = 1.3 , with aerosols of mean radius r aer = 400 nm . This image contains a triangular wedge of shaded sky grafted onto a clear sky but does not show the barrier.

Equations (1)

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z b 1 , b 2 = z cld tan ( ϕ sun ) x 1 , 2 tan ( ϕ sun ) tan ( ϕ obs ) .

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