Abstract

A theory for the visibility of halos and rainbows is presented. The light reaching the observer’s eye from the direction of the halo or rainbow is assumed to consist of two parts: (1) a beam of singly scattered sunlight (or moonlight) from a cloud of ice crystals or a rainswath, which, in turn, has suffered depletion by scattering or absorption in its passage to the observer, and (2) the general background brightness. The model is able to account for several long-known qualitative observations concerning halos, namely, that the brightest halos are produced by optically thin cirrostratus clouds (i.e., for which the cloud optical depth τc ≤ 1) and that when the sun is low in the sky the top of the halo is visible much more frequently than the bottom. (This is shown to result in good part from extinction by the turbid atmosphere.) With the rainbow the brightness of the beam increases monotonically with the optical depth τR of the sunlit part of the rainswath, but the increase is quite small for τR ≥ 1. On the other hand, the brightness of the background increases more rapidly with τR for τR > 1 so that the rainbow appears most easily visible for τR ≲ 1. This implies that the most easily visible rainbows are produced by light or moderate showers rather than heavy downpours. Finally, suggestions are made for applying the theory to other atmospheric optical phenomena, such as coronas and glories.

© 1980 Optical Society of America

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References

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  1. C. Boyer, Rainbow: From Myth to Mathematics (Yoseloff, New York, 1959).
  2. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).
  3. W. Humphreys, Physics of the Air (Dover, New York, 1940).
  4. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).
  5. R. Tricker, Q. J. R. Meteorol. Soc. 98, (1972)
  6. R. Meyer, Dic Haloerscheinungen Prob. Kosm. Physik 12 (H. Grand, Hamburg, 1929).
  7. R. Meyer, FIAT Review of German Science 1939–1946, Vol. 8 (1948).
  8. H. Neuberger, Compendium of Meteorology (1951), p. 61.
  9. S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
    [CrossRef]
  10. J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
    [CrossRef]
  11. H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
    [CrossRef]
  12. P. Wendling, Meteorologisches Institutder Universitat Munchen; personal communication.
  13. P. Wendling, R. Wendling, H. K. Weickmann, Appl. Opt. 18, 2663 (1979).
    [CrossRef] [PubMed]
  14. A. A. Lacis, J. E. Hansen, J. Atmos. Sci. 31, 118 (1974).
    [CrossRef]
  15. E. Brücke, D. Brücke, Meteorol. Z. 49, 289 (1932).
  16. L. B. McArthur, J. E. Hay, Bull. Am. Meteorol. Soc. 59, 1442 (1978).
  17. R. Scorer, Clouds of the World (Stackpole Books, Harrisburg, Pa.1972).
  18. S. D. Gedzelman, The Science and Wonders of the Atmosphere (Wiley, New York, 1980).
  19. N. K. Nikiforovna, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol. Sci. 8, 243 (1977).
    [CrossRef]
  20. S. Chapman, Proc. Phys. Soc. London 43, 26 (1931).
    [CrossRef]
  21. K. Ya. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).
  22. M. D. Steven, Q. J. R. Meteorol. Soc. 103, 457 (1977).
    [CrossRef]
  23. J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
    [CrossRef]
  24. A. B. Fraser, J. Opt. Soc. Am. 69, 1112 (1979).
    [CrossRef]

1979 (2)

1978 (1)

L. B. McArthur, J. E. Hay, Bull. Am. Meteorol. Soc. 59, 1442 (1978).

1977 (2)

N. K. Nikiforovna, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol. Sci. 8, 243 (1977).
[CrossRef]

M. D. Steven, Q. J. R. Meteorol. Soc. 103, 457 (1977).
[CrossRef]

1974 (2)

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

A. A. Lacis, J. E. Hansen, J. Atmos. Sci. 31, 118 (1974).
[CrossRef]

1972 (1)

R. Tricker, Q. J. R. Meteorol. Soc. 98, (1972)

1971 (1)

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

1970 (1)

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[CrossRef]

1967 (1)

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

1932 (1)

E. Brücke, D. Brücke, Meteorol. Z. 49, 289 (1932).

1931 (1)

S. Chapman, Proc. Phys. Soc. London 43, 26 (1931).
[CrossRef]

Boyer, C.

C. Boyer, Rainbow: From Myth to Mathematics (Yoseloff, New York, 1959).

Brücke, D.

E. Brücke, D. Brücke, Meteorol. Z. 49, 289 (1932).

Brücke, E.

E. Brücke, D. Brücke, Meteorol. Z. 49, 289 (1932).

Chapman, S.

S. Chapman, Proc. Phys. Soc. London 43, 26 (1931).
[CrossRef]

Fraser, A. B.

Gedzelman, S. D.

S. D. Gedzelman, The Science and Wonders of the Atmosphere (Wiley, New York, 1980).

Hansen, J. E.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

A. A. Lacis, J. E. Hansen, J. Atmos. Sci. 31, 118 (1974).
[CrossRef]

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[CrossRef]

Hay, J. E.

L. B. McArthur, J. E. Hay, Bull. Am. Meteorol. Soc. 59, 1442 (1978).

Howell, H. B.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

Humphreys, W.

W. Humphreys, Physics of the Air (Dover, New York, 1940).

Jacobowitz, H.

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

Kondratyev, K. Ya.

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

Lacis, A. A.

A. A. Lacis, J. E. Hansen, J. Atmos. Sci. 31, 118 (1974).
[CrossRef]

McArthur, L. B.

L. B. McArthur, J. E. Hay, Bull. Am. Meteorol. Soc. 59, 1442 (1978).

Meyer, R.

R. Meyer, Dic Haloerscheinungen Prob. Kosm. Physik 12 (H. Grand, Hamburg, 1929).

R. Meyer, FIAT Review of German Science 1939–1946, Vol. 8 (1948).

Minnaert, M.

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

Neuberger, H.

H. Neuberger, Compendium of Meteorology (1951), p. 61.

Nikiforovna, N. K.

N. K. Nikiforovna, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol. Sci. 8, 243 (1977).
[CrossRef]

Pavlova, L. N.

N. K. Nikiforovna, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol. Sci. 8, 243 (1977).
[CrossRef]

Petrushin, A. G.

N. K. Nikiforovna, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol. Sci. 8, 243 (1977).
[CrossRef]

Pollack, J. B.

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[CrossRef]

Scorer, R.

R. Scorer, Clouds of the World (Stackpole Books, Harrisburg, Pa.1972).

Snykov, V. P.

N. K. Nikiforovna, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol. Sci. 8, 243 (1977).
[CrossRef]

Steven, M. D.

M. D. Steven, Q. J. R. Meteorol. Soc. 103, 457 (1977).
[CrossRef]

Travis, L. D.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Tricker, R.

R. Tricker, Q. J. R. Meteorol. Soc. 98, (1972)

R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).

Twomey, S.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

Volkovitsky, O. A.

N. K. Nikiforovna, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol. Sci. 8, 243 (1977).
[CrossRef]

Weickmann, H. K.

Wendling, P.

P. Wendling, R. Wendling, H. K. Weickmann, Appl. Opt. 18, 2663 (1979).
[CrossRef] [PubMed]

P. Wendling, Meteorologisches Institutder Universitat Munchen; personal communication.

Wendling, R.

Appl. Opt. (1)

Bull. Am. Meteorol. Soc. (1)

L. B. McArthur, J. E. Hay, Bull. Am. Meteorol. Soc. 59, 1442 (1978).

J. Aerosol. Sci. (1)

N. K. Nikiforovna, L. N. Pavlova, A. G. Petrushin, V. P. Snykov, O. A. Volkovitsky, J. Aerosol. Sci. 8, 243 (1977).
[CrossRef]

J. Atmos. Sci. (3)

A. A. Lacis, J. E. Hansen, J. Atmos. Sci. 31, 118 (1974).
[CrossRef]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer 11, 691 (1971).
[CrossRef]

Meteorol. Z. (1)

E. Brücke, D. Brücke, Meteorol. Z. 49, 289 (1932).

Proc. Phys. Soc. London (1)

S. Chapman, Proc. Phys. Soc. London 43, 26 (1931).
[CrossRef]

Q. J. R. Meteorol. Soc. (2)

M. D. Steven, Q. J. R. Meteorol. Soc. 103, 457 (1977).
[CrossRef]

R. Tricker, Q. J. R. Meteorol. Soc. 98, (1972)

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Other (11)

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

R. Meyer, Dic Haloerscheinungen Prob. Kosm. Physik 12 (H. Grand, Hamburg, 1929).

R. Meyer, FIAT Review of German Science 1939–1946, Vol. 8 (1948).

H. Neuberger, Compendium of Meteorology (1951), p. 61.

C. Boyer, Rainbow: From Myth to Mathematics (Yoseloff, New York, 1959).

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

W. Humphreys, Physics of the Air (Dover, New York, 1940).

R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).

R. Scorer, Clouds of the World (Stackpole Books, Harrisburg, Pa.1972).

S. D. Gedzelman, The Science and Wonders of the Atmosphere (Wiley, New York, 1980).

P. Wendling, Meteorologisches Institutder Universitat Munchen; personal communication.

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Figures (6)

Fig. 1
Fig. 1

Halo model.

Fig. 2
Fig. 2

Brightness of the halo as a function of cloud optical depth, σch = τc, for solar zenith angle θ = 30. Solid curves give brightness in the absence of an atmosphere, while dashed curves give brightness when optical depth of the atmosphere is 0.357. (This value is used in all subsequent figures.) Light curves give brightness for the top of the halo—heavy curves for the bottom. Dotted curve gives a high estimate for brightness of the background.

Fig. 3
Fig. 3

Same as Fig. 2 except for θ = 60°.

Fig. 4
Fig. 4

Effect of cloud height on brightness of the top (light curves) and bottom (heavy curves) of the halo for θ = 30° (dashed lines) and θ = 60° (solid lines).

Fig. 5
Fig. 5

Rainbow model.

Fig. 6
Fig. 6

Brightness of the rainbow as a function of rainswath optical depth, τR (heavy line). Light lines correspond to background brightness for three widely varying values described in the text.

Equations (17)

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d B = Φ ( z , λ , θ ) P σ c csc k exp ( 0 z σ T csc k d z ) d z ,
σ T = K R ρ 10 exp ( z / H 1 ) + K 1 ρ 20 exp ( z / H 2 ) + σ c .
Φ ( z , λ , θ ) = Φ 0 ( λ ) exp { [ τ 1 exp ( z / H 1 ) + τ 2 exp ( z / H 2 ) + σ c ( h + 1 2 h c z ) ] sec θ } ,
B = Φ 0 P σ c exp { [ τ 1 + τ 2 σ ( h 1 2 h c ) ] × csc k σ c ( h + 1 2 h c ) sec θ } h 1 / 2 h c h + 1 / 2 h c × exp { ( sec θ csc k ) [ τ 1 exp ( z / H 1 ) + τ 2 exp ( z / H 2 ) σ c z ] } csc k d z .
exp ( z / H ) exp ( h / H ) ( 1 z h H ) ·
B = { Φ 0 P τ c sec θ exp [ ( τ 1 + τ 2 + τ c ) sec θ ] ; sec θ = csc k ,
Φ 0 P csc k sec θ csc k exp ( τ c sec θ ) { exp [ τ c ( sec θ csc k ) ] 1 } × exp ( [ τ 1 exp ( h / H 1 ) + τ 2 exp ( h / H 2 ) ] sec θ { τ 1 [ 1 exp ( h / H 1 ) ] + τ 2 [ 1 exp ( h / H 2 ) ] } csc k ) ; sec θ csc k .
B max = { Φ 0 P exp ( 1 ) ; csc k = sec θ ,
Φ 0 P exp ( sec θ csc k sec θ ln csc k sec θ ) ; csc k sec θ ,
τ c = { 1 sec θ ; csc k = sec θ ,
1 csc k sec θ ln csc k sec θ ; csc k sec θ .
a = ( 0.13 τ c ) / ( 1 + 0.13 τ c ) ,
B CLOUD = Φ 0 2 π cos θ [ 1 1 + 0.13 τ c exp ( τ c sec θ ) ] ,
B B = 0.10 exp ( τ c sec θ ) + Φ 0 2 π × cos θ [ 1 1 + 0.13 τ c exp ( τ c sec θ ) ]
d B = Φ 0 exp [ ( τ 1 + τ 2 ) sec θ ] P σ R × exp [ 0 x σ R ( sec ϕ + sec k ) d x ] d x ,
B = Φ 0 P exp [ ( τ 1 + τ 2 ) sec θ ] sec ϕ + sec k { 1 exp [ τ R ( sec ϕ + sec k ) ] } ,
B T = B F + Φ 0 P cos ϕ exp [ ( τ 1 + τ 2 ) sec θ ] ( 0.13 τ R 1 + 0.13 τ R ) + B B ( 1 1 + 0.13 τ R ) ,

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