Abstract

We present diagrams that show how layers in atmospheric thermal structure are related to the altitudes at which they are seen tangentially. These dip diagrams show that the inferior mirage greatly magnifies the apparent angular size of the lowest few centimeters of atmosphere. Conversely, inversion layers below eye level are compressed—even to zero apparent thickness, in ducts. The diagrams show that, even when distant objects are miraged, the ray crossings occur beyond the lowest point on each ray where the line of sight is tangent to a horizontal surface in the atmosphere. Therefore the apparent altitudes of these tangent points are a monotonic function of their actual heights in the atmosphere. This monotonicity explains an apparent paradox in low-Sun images.

© 1998 Optical Society of America

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References

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  1. E. Sang, “On the impossibility of inverted images in the air,” Proc. R. Soc. Edinburgh 12, 129–136 (1884).
  2. R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).
  3. R. Meyer, “Atmosphärische Strahlenbrechung,” in Handbuch der Geophysik, F. Linke, F. Möller, eds., Band 8, Physik der Atmosphäre I (Gebr. Borntraeger, Berlin, 1942–1961), Chap. 13, pp. 769–821.
  4. A. B. Fraser, “The Green Flash and clear air turbulence,” Atmos. 13, 1–10 (1975).
  5. R. White, “A new theory of the green flash,” J. Meteorol. 4, 270–277 (1979).
  6. A. T. Young, G. W. Kattawar, P. Parviainen, “Sunset science. I. The mock mirage,” Appl. Opt. 36, 2689–2700 (1997).
    [CrossRef] [PubMed]
  7. A. Wegener, “Elementare Theorie der atmosphärischen Spiegelungen,” Ann. Phys. (Leipzig) Ser. 4 57, 203–230 (1918).
    [CrossRef]
  8. T. Y. Baker, “On the refraction of electro-magnetic waves in a spherically stratified medium,” Philos. Mag. 4, 955–980 (1927).
  9. O. Haug, “On the theory of superior mirage,” Meteorol. Ann. 3, 295–310 (1953).
  10. H. C. Freiesleben, “Die Berechnung der Kimmtiefe,” Dtsch. Hydrogr. Z. 1, 26–29 (1948).
  11. L. Hasse, “Über den Zusammenhang der Kimmtiefe mit meteorologischen Grössen,” Dtsch. Hydrogr. Z. 13, 181–197 (1960).
    [CrossRef]
  12. L. Hasse, “Temperature-difference corrections for the dip of the horizon,” J. Inst. Navigation 17, 50–56 (1964).
    [CrossRef]
  13. A. Wegener, “Über die Ursache der Zerrbilder bei Sonnenuntergängen,” Beitr. Phys. d. freien Atmos. 4, 26–34 (1912).
  14. R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer, Dordrecht, The Netherlands, 1988).
    [CrossRef]
  15. D. J. K. O’Connell, The Green Flash and Other Low Sun Phenomena (North-Holland, Amsterdam, 1958).

1997

1979

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

1975

A. B. Fraser, “The Green Flash and clear air turbulence,” Atmos. 13, 1–10 (1975).

1964

L. Hasse, “Temperature-difference corrections for the dip of the horizon,” J. Inst. Navigation 17, 50–56 (1964).
[CrossRef]

1960

L. Hasse, “Über den Zusammenhang der Kimmtiefe mit meteorologischen Grössen,” Dtsch. Hydrogr. Z. 13, 181–197 (1960).
[CrossRef]

1953

O. Haug, “On the theory of superior mirage,” Meteorol. Ann. 3, 295–310 (1953).

1948

H. C. Freiesleben, “Die Berechnung der Kimmtiefe,” Dtsch. Hydrogr. Z. 1, 26–29 (1948).

1935

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

1927

T. Y. Baker, “On the refraction of electro-magnetic waves in a spherically stratified medium,” Philos. Mag. 4, 955–980 (1927).

1918

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

1912

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

1884

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

Baker, T. Y.

T. Y. Baker, “On the refraction of electro-magnetic waves in a spherically stratified medium,” Philos. Mag. 4, 955–980 (1927).

Fraser, A. B.

A. B. Fraser, “The Green Flash and clear air turbulence,” Atmos. 13, 1–10 (1975).

Freiesleben, H. C.

H. C. Freiesleben, “Die Berechnung der Kimmtiefe,” Dtsch. Hydrogr. Z. 1, 26–29 (1948).

Hasse, L.

L. Hasse, “Temperature-difference corrections for the dip of the horizon,” J. Inst. Navigation 17, 50–56 (1964).
[CrossRef]

L. Hasse, “Über den Zusammenhang der Kimmtiefe mit meteorologischen Grössen,” Dtsch. Hydrogr. Z. 13, 181–197 (1960).
[CrossRef]

Haug, O.

O. Haug, “On the theory of superior mirage,” Meteorol. Ann. 3, 295–310 (1953).

Kattawar, G. W.

Meyer, R.

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

R. Meyer, “Atmosphärische Strahlenbrechung,” in Handbuch der Geophysik, F. Linke, F. Möller, eds., Band 8, Physik der Atmosphäre I (Gebr. Borntraeger, Berlin, 1942–1961), Chap. 13, pp. 769–821.

O’Connell, D. J. K.

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

Parviainen, P.

Sang, E.

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

Stull, R. B.

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

Wegener, A.

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

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

White, R.

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

Young, A. T.

Ann. Phys. (Leipzig) Ser. 4

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

Appl. Opt.

Atmos.

A. B. Fraser, “The Green Flash and clear air turbulence,” Atmos. 13, 1–10 (1975).

Beitr. Phys. d. freien Atmos.

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

Dtsch. Hydrogr. Z.

H. C. Freiesleben, “Die Berechnung der Kimmtiefe,” Dtsch. Hydrogr. Z. 1, 26–29 (1948).

L. Hasse, “Über den Zusammenhang der Kimmtiefe mit meteorologischen Grössen,” Dtsch. Hydrogr. Z. 13, 181–197 (1960).
[CrossRef]

J. Inst. Navigation

L. Hasse, “Temperature-difference corrections for the dip of the horizon,” J. Inst. Navigation 17, 50–56 (1964).
[CrossRef]

J. Meteorol.

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

Meteorol. Ann.

O. Haug, “On the theory of superior mirage,” Meteorol. Ann. 3, 295–310 (1953).

Meteorol. Z.

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

Philos. Mag.

T. Y. Baker, “On the refraction of electro-magnetic waves in a spherically stratified medium,” Philos. Mag. 4, 955–980 (1927).

Proc. R. Soc. Edinburgh

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

Other

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

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

R. Meyer, “Atmosphärische Strahlenbrechung,” in Handbuch der Geophysik, F. Linke, F. Möller, eds., Band 8, Physik der Atmosphäre I (Gebr. Borntraeger, Berlin, 1942–1961), Chap. 13, pp. 769–821.

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Figures (9)

Fig. 1
Fig. 1

Temperature profile with equally spaced inversions. Each layer begins at an integral multiple of 10 m; inversions are based at even multiples. The temperature gradient dT/dh is +3 K per 100 m in each of the three lowest inversions (from 0 to 10, 20 to 30, and 40 to 50 m) and +5.35 K per 100 m in the top inversion (from 60 to 70 m). The gradient is -0.4 K per 100 m in the bottom region of normal lapse rate (from 10 to 20 m), -1 K per 100 m in the next two (from 30 to 40 and from 50 to 60 m), and has the standard lapse rate of -0.65 K per 100 m above 70 m.

Fig. 2
Fig. 2

Transfer curve produced by the atmospheric structure in Fig. 1 for an observer at 61 m. Note that the inversion bases at 20, 40, and 60 m in Fig. 1 produce characteristic pointed minima here at 2.3, 5.3, and 11.4 arc min above the apparent horizon (at the left end of the curve), which itself is 12.7 min below the astronomical horizon (zero on the abscissa).

Fig. 3
Fig. 3

Geometry of dip. The observer is at O, a height h above the spherical surface SGH with center at C and radius R. The geodetic horizon is at G, where the dashed straight line OG is tangent to the surface; its dip is the angle d g . The apparent (refracted) horizon is at H, with dip angle d below the astronomical horizon (horizontal dashed line). The arc OH, tangent to the surface at H, is the refracted path of the horizon ray.

Fig. 4
Fig. 4

Plot of nR as a function of R for the lowest 22 km of the U.S. Standard Atmosphere. The radius of curvature adopted is the radius of the Earth at 45° latitude, approximately 6357 km. Note the asymptotic approach to the dashed line nR = R, or n = 1. The inset scale of degrees on the right-hand side is explained in the text.

Fig. 5
Fig. 5

Enlarged plot of nR as a function of R for the U.S. Standard Atmosphere (solid line) and for isothermal atmospheres at (top to bottom) 12, 14, 16, and 18 °C (dashed lines). The inset scale is in minutes of arc.

Fig. 6
Fig. 6

Temperature profile producing an inferior mirage. A filled circle is plotted at the surface to emphasize the large temperature change in the lowest meter. The upper part of the curve has nearly the adiabatic lapse rate.

Fig. 7
Fig. 7

Dip diagram for the inferior mirage model. The temperature at the surface is 20.6 °C, but where the temperature falls below 20 °C in the first meter, we did not draw the isothermal model for 20 °C so as to avoid cluttering the diagram. The dip scale is in minutes of arc. The inset shows the hook in the bottom 20 m of the atmosphere magnified three times for clarity; this panel is not quite 30 m high. Its bottom and sides are aligned with ticks on the outer scales, with its lower left corner shifted to (20, 1860) to assist in reading the values. Thus, for example, a height of 10 m in the inset is aligned with 20 + 3 × 10 = 50 m on the outer height scale. The corresponding point on the model curve is level with 1917 m on the outer ordinate scale, corresponding to an actual ordinate of (1917 - 1860)/3 = 19 m above the value of 1720 m at the surface.

Fig. 8
Fig. 8

Temperature profile for duct model. The potential temperature increases by 2.5 °C between 45 and 50 m height. The rest of the atmosphere has the standard lapse rate of 6.5°/km.

Fig. 9
Fig. 9

Dip diagram for the duct model. The solid line is the atmospheric model. Point A marks the bottom of the duct. Points B and C correspond to the left and right corners in Fig. 8. The grid of isothermal models is spaced at 1° intervals from 12 to 15 °C.

Equations (4)

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d = 2 h / R 0 1 - k 1 / 2 ,
d g = 2 h / R 0 1 / 2 .
1 / R 0 - k / R 0 = 1 / R 0 1 - k .
D = 2 hR 0 / 1 - k 1 / 2 ,

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