Abstract

A previously unrecognized phenomenon, which we call the mock mirage, produces inverted images of the Sun and Moon near the horizon when the observer looks downward through a thermal inversion. No ducting is involved; the rays can be concave toward the Earth throughout their length, with a radius of curvature larger than the radius of the Earth. Quite mild inversions produce surprisingly large effects, which increase with the height of the observer. Although the phenomenon has frequently been photographed, published pictures have been misinterpreted. Finally, we distinguish between features that are due to waves on inversion layers and the larger features that are due to the inversions themselves.

© 1997 Optical Society of America

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References

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  1. A. Wegener, “Über die Ursache der Zerrbilder bei Sonnenuntergängen,” Beitr. Phys. Freien Atmos. 4, 26–34 (1912).
  2. A. Wegener, “Elementare Theorie der atmosphärischen Spiegelungen,” Ann. Phys. 57(4) , 203–230 (1918).
    [CrossRef]
  3. R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).
  4. W. M. Smart, Text-book in Spherical Astronomy (Cambridge U. Press, Cambridge, 1962), Chap. 3.
  5. E. W. Woolard, G. M. Clemence, Spherical Astronomy (Academic, London, 1966), Chap. 5.
  6. C. A. Murray, Vectorial Astronomy (Hilger, Bristol, 1983), Chap. 7.
  7. P. K. Seidelmann, Explanatory Supplement to the Astronomical Almanac (University Science, Mill Valley, Calif., 1992).
  8. G. Bomford, Geodesy (Clarendon, Oxford, 1971).
  9. E. Sang, “On the impossibility of inverted images in the air,” Proc. R. Soc. Edinburgh 12, 129–136 (1884).
  10. M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).
  11. A. Riccò, “Deformazione del disco solare all’orizzonte,” Mem. Soc. Spett. Ital. 30, 96–110 (1901).
  12. F. Křifka, “Refraktionserscheinungen der aufgehenden Sonne,” Meteorol. Z. 8, 101–102 (1891).
  13. J. M. Pernter, F. M. Exner, Meteorologische Optik (Braumüller, Wien und Leipzig, 1922).
  14. Em. Touchet, “Sur le ‘rayon vert’,” Nature (Paris) 53, 358–366 (1925).
  15. T. Kellen, “Der grüne strahl,” Kosmos (Stuttgart) 23, 13–18 (1926).
  16. D. J. K. O’Connell, The Green Flash and Other Low Sun Phenomena (North-Holland, Amsterdam, 1958).
  17. R. Meyer, “Der ‘Grüne Strahl’ nach niederläandischen Arbeiten,” Meteorol. Z. 56, 342–346 (1939).
  18. P. Feenstra Kuiper, De Groene Straal (de Boer, Helder, 1926).
  19. J. H. Taylor, B. T. Matthias, “Green flash from high altitude,” Nature 222, 157 (1969).
    [CrossRef]
  20. B. E. Schaefer, “The green flash,” Sky Tel. 83, 200–203 (1992).
  21. A. B. Fraser, “The green flash and clear air turbulence,” Atmosphere 13, 1–10 (1975).
  22. R. White, “A new theory of the green flash,” J. Meteorol. 4, 270–277 (1979).
  23. R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer, Dordrecht, 1988).
    [CrossRef]
  24. U. S. Office of Scientific Research and Development, and the National Defense Research Committee, “Radio wave propagation: consolidated summary technical report of the Committee on Propagation of the National Defense Research Committee,” C. R. Burrows, chairman, S. S. Attwood, ed. (Academic, New York, 1949), pp. 41, 155–156, 197–199.

1992 (1)

B. E. Schaefer, “The green flash,” Sky Tel. 83, 200–203 (1992).

1979 (1)

R. White, “A new theory of the green flash,” J. Meteorol. 4, 270–277 (1979).

1975 (1)

A. B. Fraser, “The green flash and clear air turbulence,” Atmosphere 13, 1–10 (1975).

1969 (1)

J. H. Taylor, B. T. Matthias, “Green flash from high altitude,” Nature 222, 157 (1969).
[CrossRef]

1939 (1)

R. Meyer, “Der ‘Grüne Strahl’ nach niederläandischen Arbeiten,” Meteorol. Z. 56, 342–346 (1939).

1935 (1)

R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).

1926 (1)

T. Kellen, “Der grüne strahl,” Kosmos (Stuttgart) 23, 13–18 (1926).

1925 (1)

Em. Touchet, “Sur le ‘rayon vert’,” Nature (Paris) 53, 358–366 (1925).

1918 (1)

A. Wegener, “Elementare Theorie der atmosphärischen Spiegelungen,” Ann. Phys. 57(4) , 203–230 (1918).
[CrossRef]

1912 (1)

A. Wegener, “Über die Ursache der Zerrbilder bei Sonnenuntergängen,” Beitr. Phys. Freien Atmos. 4, 26–34 (1912).

1901 (1)

A. Riccò, “Deformazione del disco solare all’orizzonte,” Mem. Soc. Spett. Ital. 30, 96–110 (1901).

1891 (1)

F. Křifka, “Refraktionserscheinungen der aufgehenden Sonne,” Meteorol. Z. 8, 101–102 (1891).

1884 (1)

E. Sang, “On the impossibility of inverted images in the air,” Proc. R. Soc. Edinburgh 12, 129–136 (1884).

Bomford, G.

G. Bomford, Geodesy (Clarendon, Oxford, 1971).

Burrows, C. R.

U. S. Office of Scientific Research and Development, and the National Defense Research Committee, “Radio wave propagation: consolidated summary technical report of the Committee on Propagation of the National Defense Research Committee,” C. R. Burrows, chairman, S. S. Attwood, ed. (Academic, New York, 1949), pp. 41, 155–156, 197–199.

Clemence, G. M.

E. W. Woolard, G. M. Clemence, Spherical Astronomy (Academic, London, 1966), Chap. 5.

Exner, F. M.

J. M. Pernter, F. M. Exner, Meteorologische Optik (Braumüller, Wien und Leipzig, 1922).

Feenstra Kuiper, P.

P. Feenstra Kuiper, De Groene Straal (de Boer, Helder, 1926).

Fraser, A. B.

A. B. Fraser, “The green flash and clear air turbulence,” Atmosphere 13, 1–10 (1975).

Kellen, T.

T. Kellen, “Der grüne strahl,” Kosmos (Stuttgart) 23, 13–18 (1926).

Krifka, F.

F. Křifka, “Refraktionserscheinungen der aufgehenden Sonne,” Meteorol. Z. 8, 101–102 (1891).

Matthias, B. T.

J. H. Taylor, B. T. Matthias, “Green flash from high altitude,” Nature 222, 157 (1969).
[CrossRef]

Meyer, R.

R. Meyer, “Der ‘Grüne Strahl’ nach niederläandischen Arbeiten,” Meteorol. Z. 56, 342–346 (1939).

R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).

Minnaert, M.

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

Murray, C. A.

C. A. Murray, Vectorial Astronomy (Hilger, Bristol, 1983), Chap. 7.

O’Connell, D. J. K.

D. J. K. O’Connell, The Green Flash and Other Low Sun Phenomena (North-Holland, Amsterdam, 1958).

Pernter, J. M.

J. M. Pernter, F. M. Exner, Meteorologische Optik (Braumüller, Wien und Leipzig, 1922).

Riccò, A.

A. Riccò, “Deformazione del disco solare all’orizzonte,” Mem. Soc. Spett. Ital. 30, 96–110 (1901).

Sang, E.

E. Sang, “On the impossibility of inverted images in the air,” Proc. R. Soc. Edinburgh 12, 129–136 (1884).

Schaefer, B. E.

B. E. Schaefer, “The green flash,” Sky Tel. 83, 200–203 (1992).

Seidelmann, P. K.

P. K. Seidelmann, Explanatory Supplement to the Astronomical Almanac (University Science, Mill Valley, Calif., 1992).

Smart, W. M.

W. M. Smart, Text-book in Spherical Astronomy (Cambridge U. Press, Cambridge, 1962), Chap. 3.

Stull, R. B.

R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer, Dordrecht, 1988).
[CrossRef]

Taylor, J. H.

J. H. Taylor, B. T. Matthias, “Green flash from high altitude,” Nature 222, 157 (1969).
[CrossRef]

Touchet, Em.

Em. Touchet, “Sur le ‘rayon vert’,” Nature (Paris) 53, 358–366 (1925).

Wegener, A.

A. Wegener, “Elementare Theorie der atmosphärischen Spiegelungen,” Ann. Phys. 57(4) , 203–230 (1918).
[CrossRef]

A. Wegener, “Über die Ursache der Zerrbilder bei Sonnenuntergängen,” Beitr. Phys. Freien Atmos. 4, 26–34 (1912).

White, R.

R. White, “A new theory of the green flash,” J. Meteorol. 4, 270–277 (1979).

Woolard, E. W.

E. W. Woolard, G. M. Clemence, Spherical Astronomy (Academic, London, 1966), Chap. 5.

Ann. Phys. (1)

A. Wegener, “Elementare Theorie der atmosphärischen Spiegelungen,” Ann. Phys. 57(4) , 203–230 (1918).
[CrossRef]

Atmosphere (1)

A. B. Fraser, “The green flash and clear air turbulence,” Atmosphere 13, 1–10 (1975).

Beitr. Phys. Freien Atmos. (1)

A. Wegener, “Über die Ursache der Zerrbilder bei Sonnenuntergängen,” Beitr. Phys. Freien Atmos. 4, 26–34 (1912).

J. Meteorol. (1)

R. White, “A new theory of the green flash,” J. Meteorol. 4, 270–277 (1979).

Kosmos (Stuttgart) (1)

T. Kellen, “Der grüne strahl,” Kosmos (Stuttgart) 23, 13–18 (1926).

Mem. Soc. Spett. Ital. (1)

A. Riccò, “Deformazione del disco solare all’orizzonte,” Mem. Soc. Spett. Ital. 30, 96–110 (1901).

Meteorol. Z. (3)

F. Křifka, “Refraktionserscheinungen der aufgehenden Sonne,” Meteorol. Z. 8, 101–102 (1891).

R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).

R. Meyer, “Der ‘Grüne Strahl’ nach niederläandischen Arbeiten,” Meteorol. Z. 56, 342–346 (1939).

Nature (1)

J. H. Taylor, B. T. Matthias, “Green flash from high altitude,” Nature 222, 157 (1969).
[CrossRef]

Nature (Paris) (1)

Em. Touchet, “Sur le ‘rayon vert’,” Nature (Paris) 53, 358–366 (1925).

Proc. R. Soc. Edinburgh (1)

E. Sang, “On the impossibility of inverted images in the air,” Proc. R. Soc. Edinburgh 12, 129–136 (1884).

Sky Tel. (1)

B. E. Schaefer, “The green flash,” Sky Tel. 83, 200–203 (1992).

Other (11)

P. Feenstra Kuiper, De Groene Straal (de Boer, Helder, 1926).

D. J. K. O’Connell, The Green Flash and Other Low Sun Phenomena (North-Holland, Amsterdam, 1958).

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

J. M. Pernter, F. M. Exner, Meteorologische Optik (Braumüller, Wien und Leipzig, 1922).

W. M. Smart, Text-book in Spherical Astronomy (Cambridge U. Press, Cambridge, 1962), Chap. 3.

E. W. Woolard, G. M. Clemence, Spherical Astronomy (Academic, London, 1966), Chap. 5.

C. A. Murray, Vectorial Astronomy (Hilger, Bristol, 1983), Chap. 7.

P. K. Seidelmann, Explanatory Supplement to the Astronomical Almanac (University Science, Mill Valley, Calif., 1992).

G. Bomford, Geodesy (Clarendon, Oxford, 1971).

R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer, Dordrecht, 1988).
[CrossRef]

U. S. Office of Scientific Research and Development, and the National Defense Research Committee, “Radio wave propagation: consolidated summary technical report of the Committee on Propagation of the National Defense Research Committee,” C. R. Burrows, chairman, S. S. Attwood, ed. (Academic, New York, 1949), pp. 41, 155–156, 197–199.

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Figures (8)

Fig. 1
Fig. 1

Atmospheric refraction. OR is the refracted ray of light from a star that reaches the observer at O; OP is the tangent to OR at O and thus the apparent direction to the star. Z, zenith; N, nadir; OT, true direction to the star; OH, astronomical horizon. The apparent altitude of the star is h a = angle HOP; the true altitude is h t = angle HOT; the astronomical refraction is r = h a - h t = angle TOP.

Fig. 2
Fig. 2

Wegener’s calculation of dip. Arc OP is the refracted ray from a point on the visible horizon at P to the observer at O, a height H above the surface of the Earth at S. Q is the center of curvature of the ray OP; C is the center of the Earth, with radius R. The horizontal dashed line is the astronomical horizon. The geodetic dip is d g and the actual refracted dip is d.

Fig. 3
Fig. 3

Formation of the mock mirage: O, observer; S, surface of the Earth; A, B, C, rays reaching the observer’s eye at successively greater zenith distances below the horizon.

Fig. 4
Fig. 4

Wegener’s ray construction: O, observer; S, surface of the Earth, with center at C. Arc OP 1 has its center at Q 1; note that angle COQ 1 is the depression of the ray at the observer. Similarly, angle CP 1Q 1 is the depression of the ray at P 1.

Fig. 5
Fig. 5

Transfer curve: true altitude as a function of apparent altitude (see the text for the labeled points). The observer is 10 m above an inversion layer 30 m thick. Within the inversion layer the temperature gradient is 0.08 K/m, so the top of the inversion is only 2.4 K warmer than the bottom. The rest of the atmosphere has the standard lapse rate of 6.5 K/km.

Fig. 6
Fig. 6

Sunset simulation for the transfer curve of Fig. 5. Below each image is the Sun’s true altitude in minutes of arc; the scales at the sides give apparent altitudes in minutes of arc. The dashed lines represent the astronomical horizon; the dip of the apparent horizon depends on the height of the observer. The atmosphere has the same structure as in Fig. 5; the apparent altitudes of features A, B, and C in that figure are indicated here.

Fig. 7
Fig. 7

Ray diagram for two observers at different heights. Both see grazing ray B, but the higher observer (O′) sees a bigger mock mirage (see text).

Fig. 8
Fig. 8

Two sunsets photographed by Parviainen. The upper six images show a sunset with a mock mirage; the lower six show one with Wegener’s blank strip. Time proceeds from left to right within each row of a sequence (see text for a discussion of the differences between the two sunsets). The last frame in the upper sunset, taken just after the neck of an hourglass shape has broken, shows two obvious images of the upper limb; the lower (erect) image is a direct image of the upper limb, and the upper image (inverted) is its mock mirage. This mock mirage is capped by a third, erect, upper-limb image, very strongly compressed. The faint cloud above the Sun in the upper series should be ignored.

Equations (3)

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sin zapp=1/n,
cos dg=RR+H=11+H/R,
d=21-k.

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