Abstract

A theory for the brightness of rainbows is presented. The light reaching the observer consists of a beam of singly scattered sunlight, originating from the directly illuminated portion of a rainswath, which, in turn, has suffered depletion by scattering or absorption in its path through the atmosphere. The model incorporates the relevant features of cloud geometry and solar position in relation to the observer appropriate to rainbows. The model helps explain why the bottom (or near-horizon portion) of the rainbow tends to be both brighter and redder than the top (or horizontal portion furthest above the ground) when the sun is near the horizon. The greater brightness of the bottom of the bow derives principally from the greater length of the directly illuminated part of the rainswath near the horizon, while the increased redness of the bow’s bottom is due to the severe depletion of the short-wavelength contribution to the rainbow beam in its passage through the atmosphere.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Airy, Cambridge Philos. Trans. 6, 379 (1849).
  2. F. E. Volz, Physics of Precipitation (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.
  3. A. B. Fraser, J. Atmos. Sci. 29, 211 (1972).
    [CrossRef]
  4. R. S. Greenler, Rainbows, Haloes and Glories (Cambridge U.P., New York, 1980).
  5. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).
  6. S. D. Gedzelman, Appl. Opt. 19, 3068 (1980).
    [CrossRef] [PubMed]
  7. K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
    [CrossRef]
  8. K. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

1980 (1)

1972 (1)

A. B. Fraser, J. Atmos. Sci. 29, 211 (1972).
[CrossRef]

1971 (1)

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

1849 (1)

G. Airy, Cambridge Philos. Trans. 6, 379 (1849).

Airy, G.

G. Airy, Cambridge Philos. Trans. 6, 379 (1849).

Fraser, A. B.

A. B. Fraser, J. Atmos. Sci. 29, 211 (1972).
[CrossRef]

Gedzelman, S. D.

Greenler, R. S.

R. S. Greenler, Rainbows, Haloes and Glories (Cambridge U.P., New York, 1980).

Hansen, J. E.

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Kondratyev, K.

K. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

Liou, K. N.

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).

Volz, F. E.

F. E. Volz, Physics of Precipitation (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.

Appl. Opt. (1)

Cambridge Philos. Trans. (1)

G. Airy, Cambridge Philos. Trans. 6, 379 (1849).

J. Atmos. Sci. (2)

A. B. Fraser, J. Atmos. Sci. 29, 211 (1972).
[CrossRef]

K. N. Liou, J. E. Hansen, J. Atmos. Sci. 28, 995 (1971).
[CrossRef]

Other (4)

K. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

R. S. Greenler, Rainbows, Haloes and Glories (Cambridge U.P., New York, 1980).

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).

F. E. Volz, Physics of Precipitation (American Geophysical Union, Washington, D.C., 1960), pp. 280–286.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Geometry of the rainbow. A cloud, infinite in the y direction with base at z = zo, extends from x = xo to xt while the rainswath extends from xi to xf. The sun is at zenith angle θ and azimuth angle as from the x axis. An observer at O looks up at angle k and at azimuth angle av from the x axis to see the rainbow ray (indicated by arrows) that is illuminated from xi to xm.

Fig. 2
Fig. 2

Spherical trigonometry of the rainbow. Relation between θ, k, the rainbow azimuth angle ar, and the angle h from the top of the rainbow arc.

Fig. 3
Fig. 3

Two-dimensional projection of the rainbow ray. Expressions for all the symbols given in the text.

Fig. 4
Fig. 4

Normalized brightness Bn of the rainbow top (dashed lines) and bottom (solid lines) as a function of σr for θ = 80°, as = 0°, and [0,1,4,1] for a variety of τ values. Dot–dash line shows brightness of bow’s bottom for [−1,0,3,1].

Fig. 5
Fig. 5

Ratio of brightness of rainbow in real atmosphere to that in the absence of atmosphere as a function of k for θ = 80°, as, = 0°, and [0,1,4,1].

Fig. 6
Fig. 6

Normalized brightness of the same rainbow as in Fig. 5 as a function of k for various σr values assuming τ1 = 0.03, τ2 = 0.06.

Fig. 7
Fig. 7

Same as Fig. 6 (solid lines) but with τ1 = 0.119, τ2 = 0.238. Also shown by dashed lines is the same for a cloud 1 km closer, i.e., [−1,0,3,1].

Fig. 8
Fig. 8

Values of length Lb of the directly illuminated part of the rainswath for clouds with 3-km wide rainswaths and θ = 80°, as = 0°.

Fig. 9
Fig. 9

Same as Fig. 8 but with 1-km wide rainswaths.

Fig. 10
Fig. 10

Normalized brightness of the rainbow as a function of h for the same cloud as in Fig. 5 but with as = 30° and σr = 0.64. Plot of av as a function of h is also shown.

Equations (16)

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

a v = a s + a r ,
a r = 42 ° sin h · cos k ,
sin k = cos θ · sin ( - 48 ° ) + sin θ · cos ( - 48 ° ) · cos h .
x m = { z o + x o cot θ cos a s ( tan k cos a v + cot θ cos a s )             x m x f , x f             otherwise .
L b = { ( x m - x i ) [ 1 cos k · cos a v ] , x m > x i , 0 , x m x i .
z m = x m tan k cos a v ,
z t = z m + ( x m - x i ) cot θ cos a s .
z i = x i tan k cos a v .
d B = Φ ( z , λ , θ ) P σ r · csc k · exp ( o z σ r · csc k · d z ) d z ,
σ t = K R ρ 1 ( z ) + K 2 ρ 2 ( z ) + σ r .
Φ ( z , λ , θ ) = Φ o ( λ ) exp { - [ τ 1 · exp ( - z / H 1 ) + τ 2 · exp ( - z / H 2 ) + σ r ( z e - z ) ] · sec θ } ,
z e = ( z t - z m z m - z i ) ( z - z i ) + z ,
exp [ exp ( - z / H ) ] exp ( 1 - z / H )
B = Φ o ( λ ) · P σ r · csc k · exp ( - { ( τ 1 + τ 2 ) · sec θ - σ r z i · [ ( z t - z m z m - z i ) · sec θ + csc k ] } ) · 1 A [ exp ( A z m ) - exp ( A z i ) ] ,
A ( τ 1 H 1 + τ 2 H 2 ) ( sec θ - csc k ) - σ r [ ( z t - z m z m - z i ) sec θ + csc k ] ,
B n B / ( Φ o P ) .

Metrics