Abstract

It is shown that, contrary to classical theory, the circular halos need not be caused by randomly oriented crystals. Furthermore, if Brownian motion is the disorienting mechanism then the circular halos cannot be caused by the randomly oriented crystals, which are too small to produce a reasonably sharp diffraction pattern. However, the circular halos can be caused by crystals that are in the region where there is a transition between randomness and high orientation. These crystals have diameters between about 12 and 40 μm. Larger crystals produce the parhelia and tangent arcs. It is shown that the 46° halo is rare because it can be produced only by solid columns, and then for only a restricted range of sun heights.

© 1979 Optical Society of America

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References

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  1. R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), p. 81.
  2. R. G. Cox, “The steady motion of a particle of the arbitrary shape at small Reynolds numbers,” J. Fluid Mech. 23, 625–643 (1965).
    [Crossref]
  3. K. O. L. F. Jayaweera and R. E. Cottis, “Fall velocities of plate-like and columnar ice crystals,” Q. J. R. Meteorol. Soc. 95, 703–709 (1969).
    [Crossref]
  4. Charles S. Hastings, “A General Theory of Halos,” Monthly Weather Review, 322–330 (June1920).
    [Crossref]
  5. A. H. Auer and D. L. Veal, “The dimensions of ice crystals in natural clouds,” J. Atmos. Sci. 27, 919–926 (1970).
    [Crossref]

1970 (1)

A. H. Auer and D. L. Veal, “The dimensions of ice crystals in natural clouds,” J. Atmos. Sci. 27, 919–926 (1970).
[Crossref]

1969 (1)

K. O. L. F. Jayaweera and R. E. Cottis, “Fall velocities of plate-like and columnar ice crystals,” Q. J. R. Meteorol. Soc. 95, 703–709 (1969).
[Crossref]

1965 (1)

R. G. Cox, “The steady motion of a particle of the arbitrary shape at small Reynolds numbers,” J. Fluid Mech. 23, 625–643 (1965).
[Crossref]

1920 (1)

Charles S. Hastings, “A General Theory of Halos,” Monthly Weather Review, 322–330 (June1920).
[Crossref]

Auer, A. H.

A. H. Auer and D. L. Veal, “The dimensions of ice crystals in natural clouds,” J. Atmos. Sci. 27, 919–926 (1970).
[Crossref]

Cottis, R. E.

K. O. L. F. Jayaweera and R. E. Cottis, “Fall velocities of plate-like and columnar ice crystals,” Q. J. R. Meteorol. Soc. 95, 703–709 (1969).
[Crossref]

Cox, R. G.

R. G. Cox, “The steady motion of a particle of the arbitrary shape at small Reynolds numbers,” J. Fluid Mech. 23, 625–643 (1965).
[Crossref]

Hastings, Charles S.

Charles S. Hastings, “A General Theory of Halos,” Monthly Weather Review, 322–330 (June1920).
[Crossref]

Jayaweera, K. O. L. F.

K. O. L. F. Jayaweera and R. E. Cottis, “Fall velocities of plate-like and columnar ice crystals,” Q. J. R. Meteorol. Soc. 95, 703–709 (1969).
[Crossref]

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), p. 81.

Veal, D. L.

A. H. Auer and D. L. Veal, “The dimensions of ice crystals in natural clouds,” J. Atmos. Sci. 27, 919–926 (1970).
[Crossref]

J. Atmos. Sci. (1)

A. H. Auer and D. L. Veal, “The dimensions of ice crystals in natural clouds,” J. Atmos. Sci. 27, 919–926 (1970).
[Crossref]

J. Fluid Mech. (1)

R. G. Cox, “The steady motion of a particle of the arbitrary shape at small Reynolds numbers,” J. Fluid Mech. 23, 625–643 (1965).
[Crossref]

Monthly Weather Review (1)

Charles S. Hastings, “A General Theory of Halos,” Monthly Weather Review, 322–330 (June1920).
[Crossref]

Q. J. R. Meteorol. Soc. (1)

K. O. L. F. Jayaweera and R. E. Cottis, “Fall velocities of plate-like and columnar ice crystals,” Q. J. R. Meteorol. Soc. 95, 703–709 (1969).
[Crossref]

Other (1)

R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier, New York, 1970), p. 81.

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Figures (6)

FIG. 1
FIG. 1

Cumulative probability distribution for crystal tip angle as a function of plate or column diameter. To the left are crystals that are randomly oriented owing to Brownian rotations. Here the median angle θ0.5 = 45°. To the right are aerodynamically oriented crystals, where for a 40-μm column, θ0.99 < 1/2° or 99% of the crystals are confined below a tip angle of 1/2°.

FIG. 2
FIG. 2

Intensity distribution across the 22° halo based on a diffraction model that includes the effects of the Fresnel coefficients. A 5-μm diameter crystal shows no halo while an 80-μm crystal shows a good one. The calculations, which were done for a point source, show supernumerary halos, which would substantially vanish in nature owing to the 1/2° angular diameter of the sun.

FIG. 3
FIG. 3

Intensity per unit number density versus crystal diameter for various tip angles. Each tip angle has a local maximum.

FIG. 4
FIG. 4

Isopleths of crystal tip angle that are required to produce a spot of light at the angular position, α, on a 22° halo for a particular sun height h. The ordinate is linear in cos α so that the results can easily be transferred to the halo in the center. The dark gray region has crystals that are too small to form halos, the light gray region shows poor halos while the white region shows fair to good halos. The wiggly diagonal line shows the position of the horizon.

FIG. 5
FIG. 5

Same as Fig. 4 only for a 46° halo.

FIG 6
FIG 6

With crystal diameter on the abscissa and length on the ordinate, columns lie above the diagonal while plates lie below it. Shown are the regions where natural crystals occur, based on Auer and Veal, and where various halo phenomena are predicted. Also shown are short isopleths of the tip angle that prefer a particular crystal size. The r marks the boundary of randomly oriented crystals.

Equations (17)

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

r = c { 1 + [ r 3 2 z 2 ( 1 / r ) ] } ,
r = c [ 1 + ( 3 z 2 r 2 1 ) ] ,
G = q sin 2 ψ ,
E ( θ ) = q π / 2 π / 2 + θ sin 2 ψ d ψ = Q sin 2 θ ,
E ( θ ) = q 0 θ sin 2 ψ d ψ = Q sin 2 θ ,
Q = ( 87 / 40 ) π ρ c 3 υ 2 | | .
P ( c , θ ) = A ( c ) exp [ E ( θ ) / k T ] ,
1 = A ( c ) 0 π / 2 exp [ E ( θ ) / k T ] d θ .
0.5 = 0 θ .5 P ( c , θ ) d θ .
I = [ 0.37 D L ( sin x / x ) ] 2 ,
x = π ( 0.37 D / λ ) sin β ,
[ I / n ( D ) ] D 4 P ( D , θ ) .
cos θ = sin α cos h
sin θ = sin α cos h .
sin θ = sin β sin h cos β cos h cos α ,
sin θ = cos β sin h + sin β cos h cos α .
Q / k T c 7 | | L 7 ( 1 + 2 D / L ) 6 | 1 D / L |