Abstract

The results of a quantitative treatment of the visibility of the circumzenithal arc as a function of solar elevation are compared with records of the arc’s occurrence in France and the Netherlands, 1894–1931. Both calculated and observed frequency distributions peak at a solar elevation of 22° (minimum deviation), and the width of the observed frequency distribution can be closely matched with the assumption that plate ice crystals in quiet air undergo oscillations of around 1° from equilibrium. This result agrees with other estimates based on studies of the subsun and parhelic circle. The circumzenithal arc is shown to be vertically polarized, with Ivert/Ihor ≈ 1.4 in the solar vertical near minimum deviation.

© 1979 Optical Society of America

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References

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  1. R. S. McDowell, “The Formation of Parhelia at Higher Solar Elevations,” J. Atmos. Sci. 31, 1876–1884 (1974).
    [Crossref]
  2. J. M. Pernter and F. M. Exner, Meteorologische Optik, 2nd ed. (Braumülier, Vienna and Leipzig, 1922), pp. 276f, 410–416.
  3. A. Wegener, “Theorie der Haupthalos,” Arch. Deut. Seewarte 43, No. 2, 1–32 (1925), (especially pp 15–17).
  4. A. Wegener, “Optik der Atmosphäre,” Müller-Pouillets Lehrbuch der Physik, 2nd ed., (Viewig & Son, Braunschweig, 1928), Vol. 5, pp. 266–289 (especially pp. 275–278).
  5. R. Meyer, Die Haloerscheinungen. Probleme der Kosmischen Physik (Grand, Hamburg, 1929), Vol. 12, (especially pp. 107–109).
  6. W. J. Humphreys, Physics of the Air, 3rd ed. (McGraw-Hill, NewYork, 1940), pp. 530f.
  7. R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), pp. 132–134.
  8. L. Besson, Sur la Théorie des Halos, Thesis, University of Paris (Gauthier-Villars, Paris, 1909), pp. 53–68;“Sur l’Arc Circumzénithal,” Ann. Soc. Météorol. France 57, 65–72 (1909).
  9. Reference 5, pp. 110–112.
  10. Reference 7, pp. 129–131.
  11. J. M. Pernter, “Sur un Halo Extraordinaire,” C. R. Hebd. Seances Acad. Sci. 140, 1367f (1905).
  12. L. Besson, “Sur l’Arc Tangent Supérieurement au Halo de 46°,” C. R. Hebd. Seances Acad. Sci. 143, 713–715 (1906).
  13. J. M. Pernter, “Zur Theorie der ‘Schönsten der Haloerscheinugen’,” Sitzungsber. Kais. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A,  116, 17–48 (1907).
  14. W. F. J. Evans and R. A. R. Tricker, “Unusual Arcs in the Saskatoon Halo Display,” Weather 27, 234–238 (1972).
    [Crossref]
  15. G. H. Liljequist, Halo-Phenomena and Ice-Crystals.Norw.-Brit.-Swed. Antarctic Exped., 1949–52, Sci. Results, Vol. 2, part 2 (Norsk Polarinstitutt, Oslo, 1956),p. 50.
  16. L. Besson, Thesis (Ref. 8),pp. 25–28. The solid triangles in Fig. 2 represent the number of half minutes that the arc was seen, corrected for cloud cover, divided by the number of half minutes that the sun was visible, from his Table A, p. 26; normalized to a maximum of 100.
  17. E. van Everdingen, “Dagelijksche en Jaarlijksche Gang in het Voorkomen van Circumzenithaalbogen,” Hemel en Dampkring 14, 113–121 (1916);Onweders Opt. Versch. Neder. 35, 84–98 (1916).The solid circles in Fig. 2 represent his relative frequencies (pp. 117f in the Hemel en Dampkring paper, pp. 92f. in Onweders; also quoted by Wegener, Refs. 2 and 3) divided by the solar frequencies given in Ref. 1 (using 148 hr for h= 30–32.2°), and normalized to a maximum of 100.
  18. C. Visser, “De Frequentie van Halowaarnemingen bij de Zon in Nederland, Voornamelijk van 1914–1931,” K. Neder. Meteorol. Inst. Mededeel. Verhandel., No. 37 (1936).The open circles in Fig. 2 are his uncorrected“graadwaarnemingen”, p. 60 (equivalent to the “gradzahlen”, p. 93), divided by solar frequencies of 362, 369, 399, 489, 399, 340, 300, and 278 hr for h= 0°–4°, 4°–8°, …, 28°–32°, respectively, and normalized to a maximum of 100.
  19. K. Stuchtey, “Untersonnen und Lichtsaulen an Sonne und Mond,” Ann. Phys. (Leipzig) 59, 33–55 (1919).
  20. R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The Origin of Sun Pillars,” Amer. Sci. 60, 292–302 (1972).
  21. M. Pinkhof, (1921), cited in Ref. 5, p. 119.
  22. Reference 15, p. 42.
  23. R. Meyer, “Haloerscheinungen: Theoretische Beiträge zur Meteorologischen Optik,” Abhandl. Herder-Inst. Riga 1, No. 5, 1–79 (1925), (especiallypp. 43–49).

1974 (1)

R. S. McDowell, “The Formation of Parhelia at Higher Solar Elevations,” J. Atmos. Sci. 31, 1876–1884 (1974).
[Crossref]

1972 (2)

W. F. J. Evans and R. A. R. Tricker, “Unusual Arcs in the Saskatoon Halo Display,” Weather 27, 234–238 (1972).
[Crossref]

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The Origin of Sun Pillars,” Amer. Sci. 60, 292–302 (1972).

1936 (1)

C. Visser, “De Frequentie van Halowaarnemingen bij de Zon in Nederland, Voornamelijk van 1914–1931,” K. Neder. Meteorol. Inst. Mededeel. Verhandel., No. 37 (1936).The open circles in Fig. 2 are his uncorrected“graadwaarnemingen”, p. 60 (equivalent to the “gradzahlen”, p. 93), divided by solar frequencies of 362, 369, 399, 489, 399, 340, 300, and 278 hr for h= 0°–4°, 4°–8°, …, 28°–32°, respectively, and normalized to a maximum of 100.

1925 (2)

A. Wegener, “Theorie der Haupthalos,” Arch. Deut. Seewarte 43, No. 2, 1–32 (1925), (especially pp 15–17).

R. Meyer, “Haloerscheinungen: Theoretische Beiträge zur Meteorologischen Optik,” Abhandl. Herder-Inst. Riga 1, No. 5, 1–79 (1925), (especiallypp. 43–49).

1919 (1)

K. Stuchtey, “Untersonnen und Lichtsaulen an Sonne und Mond,” Ann. Phys. (Leipzig) 59, 33–55 (1919).

1916 (1)

E. van Everdingen, “Dagelijksche en Jaarlijksche Gang in het Voorkomen van Circumzenithaalbogen,” Hemel en Dampkring 14, 113–121 (1916);Onweders Opt. Versch. Neder. 35, 84–98 (1916).The solid circles in Fig. 2 represent his relative frequencies (pp. 117f in the Hemel en Dampkring paper, pp. 92f. in Onweders; also quoted by Wegener, Refs. 2 and 3) divided by the solar frequencies given in Ref. 1 (using 148 hr for h= 30–32.2°), and normalized to a maximum of 100.

1907 (1)

J. M. Pernter, “Zur Theorie der ‘Schönsten der Haloerscheinugen’,” Sitzungsber. Kais. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A,  116, 17–48 (1907).

1906 (1)

L. Besson, “Sur l’Arc Tangent Supérieurement au Halo de 46°,” C. R. Hebd. Seances Acad. Sci. 143, 713–715 (1906).

1905 (1)

J. M. Pernter, “Sur un Halo Extraordinaire,” C. R. Hebd. Seances Acad. Sci. 140, 1367f (1905).

Besson, L.

L. Besson, “Sur l’Arc Tangent Supérieurement au Halo de 46°,” C. R. Hebd. Seances Acad. Sci. 143, 713–715 (1906).

L. Besson, Sur la Théorie des Halos, Thesis, University of Paris (Gauthier-Villars, Paris, 1909), pp. 53–68;“Sur l’Arc Circumzénithal,” Ann. Soc. Météorol. France 57, 65–72 (1909).

Blumenthal, G.

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The Origin of Sun Pillars,” Amer. Sci. 60, 292–302 (1972).

Drinkwine, M.

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The Origin of Sun Pillars,” Amer. Sci. 60, 292–302 (1972).

Evans, W. F. J.

W. F. J. Evans and R. A. R. Tricker, “Unusual Arcs in the Saskatoon Halo Display,” Weather 27, 234–238 (1972).
[Crossref]

Everdingen, E. van

E. van Everdingen, “Dagelijksche en Jaarlijksche Gang in het Voorkomen van Circumzenithaalbogen,” Hemel en Dampkring 14, 113–121 (1916);Onweders Opt. Versch. Neder. 35, 84–98 (1916).The solid circles in Fig. 2 represent his relative frequencies (pp. 117f in the Hemel en Dampkring paper, pp. 92f. in Onweders; also quoted by Wegener, Refs. 2 and 3) divided by the solar frequencies given in Ref. 1 (using 148 hr for h= 30–32.2°), and normalized to a maximum of 100.

Exner, F. M.

J. M. Pernter and F. M. Exner, Meteorologische Optik, 2nd ed. (Braumülier, Vienna and Leipzig, 1922), pp. 276f, 410–416.

Greenler, R. G.

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The Origin of Sun Pillars,” Amer. Sci. 60, 292–302 (1972).

Humphreys, W. J.

W. J. Humphreys, Physics of the Air, 3rd ed. (McGraw-Hill, NewYork, 1940), pp. 530f.

Liljequist, G. H.

G. H. Liljequist, Halo-Phenomena and Ice-Crystals.Norw.-Brit.-Swed. Antarctic Exped., 1949–52, Sci. Results, Vol. 2, part 2 (Norsk Polarinstitutt, Oslo, 1956),p. 50.

Mallmann, A. J.

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The Origin of Sun Pillars,” Amer. Sci. 60, 292–302 (1972).

McDowell, R. S.

R. S. McDowell, “The Formation of Parhelia at Higher Solar Elevations,” J. Atmos. Sci. 31, 1876–1884 (1974).
[Crossref]

Meyer, R.

R. Meyer, “Haloerscheinungen: Theoretische Beiträge zur Meteorologischen Optik,” Abhandl. Herder-Inst. Riga 1, No. 5, 1–79 (1925), (especiallypp. 43–49).

R. Meyer, Die Haloerscheinungen. Probleme der Kosmischen Physik (Grand, Hamburg, 1929), Vol. 12, (especially pp. 107–109).

Pernter, J. M.

J. M. Pernter, “Zur Theorie der ‘Schönsten der Haloerscheinugen’,” Sitzungsber. Kais. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A,  116, 17–48 (1907).

J. M. Pernter, “Sur un Halo Extraordinaire,” C. R. Hebd. Seances Acad. Sci. 140, 1367f (1905).

J. M. Pernter and F. M. Exner, Meteorologische Optik, 2nd ed. (Braumülier, Vienna and Leipzig, 1922), pp. 276f, 410–416.

Pinkhof, M.

M. Pinkhof, (1921), cited in Ref. 5, p. 119.

Stuchtey, K.

K. Stuchtey, “Untersonnen und Lichtsaulen an Sonne und Mond,” Ann. Phys. (Leipzig) 59, 33–55 (1919).

Tricker, R. A. R.

W. F. J. Evans and R. A. R. Tricker, “Unusual Arcs in the Saskatoon Halo Display,” Weather 27, 234–238 (1972).
[Crossref]

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), pp. 132–134.

Visser, C.

C. Visser, “De Frequentie van Halowaarnemingen bij de Zon in Nederland, Voornamelijk van 1914–1931,” K. Neder. Meteorol. Inst. Mededeel. Verhandel., No. 37 (1936).The open circles in Fig. 2 are his uncorrected“graadwaarnemingen”, p. 60 (equivalent to the “gradzahlen”, p. 93), divided by solar frequencies of 362, 369, 399, 489, 399, 340, 300, and 278 hr for h= 0°–4°, 4°–8°, …, 28°–32°, respectively, and normalized to a maximum of 100.

Wegener, A.

A. Wegener, “Theorie der Haupthalos,” Arch. Deut. Seewarte 43, No. 2, 1–32 (1925), (especially pp 15–17).

A. Wegener, “Optik der Atmosphäre,” Müller-Pouillets Lehrbuch der Physik, 2nd ed., (Viewig & Son, Braunschweig, 1928), Vol. 5, pp. 266–289 (especially pp. 275–278).

Abhandl. Herder-Inst. Riga 1 (1)

R. Meyer, “Haloerscheinungen: Theoretische Beiträge zur Meteorologischen Optik,” Abhandl. Herder-Inst. Riga 1, No. 5, 1–79 (1925), (especiallypp. 43–49).

Amer. Sci. (1)

R. G. Greenler, M. Drinkwine, A. J. Mallmann, and G. Blumenthal, “The Origin of Sun Pillars,” Amer. Sci. 60, 292–302 (1972).

Ann. Phys. (Leipzig) (1)

K. Stuchtey, “Untersonnen und Lichtsaulen an Sonne und Mond,” Ann. Phys. (Leipzig) 59, 33–55 (1919).

Arch. Deut. Seewarte (1)

A. Wegener, “Theorie der Haupthalos,” Arch. Deut. Seewarte 43, No. 2, 1–32 (1925), (especially pp 15–17).

C. R. Hebd. Seances Acad. Sci. (2)

J. M. Pernter, “Sur un Halo Extraordinaire,” C. R. Hebd. Seances Acad. Sci. 140, 1367f (1905).

L. Besson, “Sur l’Arc Tangent Supérieurement au Halo de 46°,” C. R. Hebd. Seances Acad. Sci. 143, 713–715 (1906).

Hemel en Dampkring (1)

E. van Everdingen, “Dagelijksche en Jaarlijksche Gang in het Voorkomen van Circumzenithaalbogen,” Hemel en Dampkring 14, 113–121 (1916);Onweders Opt. Versch. Neder. 35, 84–98 (1916).The solid circles in Fig. 2 represent his relative frequencies (pp. 117f in the Hemel en Dampkring paper, pp. 92f. in Onweders; also quoted by Wegener, Refs. 2 and 3) divided by the solar frequencies given in Ref. 1 (using 148 hr for h= 30–32.2°), and normalized to a maximum of 100.

J. Atmos. Sci. (1)

R. S. McDowell, “The Formation of Parhelia at Higher Solar Elevations,” J. Atmos. Sci. 31, 1876–1884 (1974).
[Crossref]

K. Neder. Meteorol. Inst. Mededeel. Verhandel. (1)

C. Visser, “De Frequentie van Halowaarnemingen bij de Zon in Nederland, Voornamelijk van 1914–1931,” K. Neder. Meteorol. Inst. Mededeel. Verhandel., No. 37 (1936).The open circles in Fig. 2 are his uncorrected“graadwaarnemingen”, p. 60 (equivalent to the “gradzahlen”, p. 93), divided by solar frequencies of 362, 369, 399, 489, 399, 340, 300, and 278 hr for h= 0°–4°, 4°–8°, …, 28°–32°, respectively, and normalized to a maximum of 100.

Sitzungsber. Kais. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A (1)

J. M. Pernter, “Zur Theorie der ‘Schönsten der Haloerscheinugen’,” Sitzungsber. Kais. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A,  116, 17–48 (1907).

Weather (1)

W. F. J. Evans and R. A. R. Tricker, “Unusual Arcs in the Saskatoon Halo Display,” Weather 27, 234–238 (1972).
[Crossref]

Other (12)

G. H. Liljequist, Halo-Phenomena and Ice-Crystals.Norw.-Brit.-Swed. Antarctic Exped., 1949–52, Sci. Results, Vol. 2, part 2 (Norsk Polarinstitutt, Oslo, 1956),p. 50.

L. Besson, Thesis (Ref. 8),pp. 25–28. The solid triangles in Fig. 2 represent the number of half minutes that the arc was seen, corrected for cloud cover, divided by the number of half minutes that the sun was visible, from his Table A, p. 26; normalized to a maximum of 100.

J. M. Pernter and F. M. Exner, Meteorologische Optik, 2nd ed. (Braumülier, Vienna and Leipzig, 1922), pp. 276f, 410–416.

A. Wegener, “Optik der Atmosphäre,” Müller-Pouillets Lehrbuch der Physik, 2nd ed., (Viewig & Son, Braunschweig, 1928), Vol. 5, pp. 266–289 (especially pp. 275–278).

R. Meyer, Die Haloerscheinungen. Probleme der Kosmischen Physik (Grand, Hamburg, 1929), Vol. 12, (especially pp. 107–109).

W. J. Humphreys, Physics of the Air, 3rd ed. (McGraw-Hill, NewYork, 1940), pp. 530f.

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), pp. 132–134.

L. Besson, Sur la Théorie des Halos, Thesis, University of Paris (Gauthier-Villars, Paris, 1909), pp. 53–68;“Sur l’Arc Circumzénithal,” Ann. Soc. Météorol. France 57, 65–72 (1909).

Reference 5, pp. 110–112.

Reference 7, pp. 129–131.

M. Pinkhof, (1921), cited in Ref. 5, p. 119.

Reference 15, p. 42.

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Figures (2)

FIG. 1
FIG. 1

Ray paths through an ice crystal of thickness a, drawn for a solar elevation h = 20°. (a) Crystal horizontal. The two extreme rays are shown, and define the effective aperture A. (b) Crystal tipped by ϕ = 10°.

FIG. 2
FIG. 2

Calculated relative frequency of the circumzenithal arc as a function of solar elevation h for maximum crystal oscillations of 0°, 2°, and 5°. Observed frequencies of van Everdingen and Besson (both at 5° increments in h) and of Visser (4° increments) are shown.

Tables (2)

Tables Icon

TABLE I Properties of the circumzenithal arc for strictly horizontal refracting crystals.a

Tables Icon

TABLE II Maximum oscillation angles of plate crystals, as estimated from halo measurements

Equations (10)

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

cos θ = ( cos h ) / n ,
sin H = sin i = n sin θ = ( n 2 cos 2 h ) 1 / 2 ,
sin δ = ( n 2 1 ) 1 / 2 / cos h ,
T = ( 1 tan 2 ( θ h ) tan 2 ( θ + h ) ) cos i cos r , T = ( 1 tan 2 ( H θ ) tan 2 ( H + θ ) ) cos θ cos H ,
I vert I hor = T T T T = sec 2 ( θ h ) sec 2 ( H θ ) ,
A = a cot θ sin h sin h cos h / ( n 2 cos 2 h ) 1 / 2 .
cos ( δ + α ) = ( 1 n 2 + cos 2 α cos 2 h 1 n 2 + cos 2 h ) 1 / 2 .
Δ r = d δ d α = cos h ( 1 n 2 + cos 2 h ) 1 / 2 1
sin r = ( sin i ) / n = cos ( h ϕ ) / n , sin i = n cos r = [ n 2 cos 2 ( h ϕ ) ] 1 / 2 , H = i + ϕ = sin 1 [ n 2 cos 2 ( h ϕ ) ] 1 / 2 + ϕ ,
Δ H = | 1 sin h cos h [ ( n 2 cos 2 h ) ( 1 n 2 + cos 2 h ) ] 1 / 2 | d ϕ .