Abstract

A ray tracing technique is presented based on the fundamental laws of ray and wave optics; it has been used to calculate the scattering properties of hexagonal ice crystals. These crystals were assumed to be oriented preferably horizontal, and, therefore, the resulting phase functions have been plotted vs direction in 3-D space contrary to earlier calculations of other authors. The anisotropy of the scattered radiation is clearly shown; on the average the phase function varies over ∼2 orders of magnitude. From these single scattering results the multiple scattering between various ice crystals has also been calculated.

© 1989 Optical Society of America

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References

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  1. J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
    [CrossRef] [PubMed]
  2. P. V. Hobbs, Ice Physics (Oxford U. P., London, 1974).
  3. A. J. Heymsfield and R. G. Knollenberg, "Properties of Cirrus Generating Cells," J. Atmos. Sci. 29, 1338–1366 (1972).
    [CrossRef]
  4. A. J. Heymsfield, "Cirrus Uncinus Generating Cells and the Evolution of Cirriform Clouds. Part I. Aircraft Observations of the Growth of the Ice Phase," J. Atmos. Sci. 32, 799–805 (1975).
    [CrossRef]
  5. A. J. Heymsfield, "Precipitation Development in Stratiform Ice Clouds: A Microphysical and Dynamical Study," J. Atmos. Sci. 39, 367–381 (1977).
    [CrossRef]
  6. G. L. Stephens, "Radiative Transfer on a Linear Lattice: Application to Anisotropic Ice Crystal Clouds," J. Atmos. Sci. 37, 2095–2104 (1980).
    [CrossRef]
  7. A. Mugnai and W. J. Wiscombe, "Scattering of Radiation by Moderately Nonspherical Particles," J. Atmos. Sci. 37, 1291–1307 (1980).
    [CrossRef]
  8. W. J. Wiscombe, "Improved Mie Scattering Algorithms," Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  9. Q. Cai and K.-N. Liou, "Polarized Light Scattering by Hexagonal Ice Crystals: Theory," Appl. Opt. 21, 3569–3580 (1982).
    [CrossRef] [PubMed]
  10. S. Asano, "Light Scattering by Horizontally Orientated Spheroidal Particles," Appl. Opt. 22, 1390–1396 (1983).
    [CrossRef] [PubMed]
  11. R. F. Coleman and K.-N. Liou, "Light Scattering by Hexagonal Ice Crystals," J. Atmos. Sci. 38, 1260–1271 (1981).
    [CrossRef]
  12. K. N. Liou, "Light Scattering by Ice Clouds in the Visible Infrared: A Theoretical Study," J. Atmos. Sci. 29, 524–536 (1972).
    [CrossRef]
  13. P. Wendling, R. Wendling, and H. K. Weickmann, "Scattering of Solar Radiation by Hexagonal Ice Crystals," Appl. Opt. 18, 2663–2671 (1979).
    [CrossRef] [PubMed]
  14. Y. Takano and K. Jayaweera, "Scattering Phase Matrix for Hexagonal Ice Crystals Computed from Ray Optics," Appl. Opt. 24, 3254–3262 (1985).
    [CrossRef] [PubMed]
  15. C. M. R. Piatt, "Lidar Backscatter from Horizontal Ice Crystal Patterns," J. Appl. Meterol. 17, 482–488 (1978).
    [CrossRef]
  16. J. E. Hansen and L. D. Travis, "Light Sattering in Planetary Atmospheres," Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  17. M. Born, Optik (Springer-Verlag, Berlin, 1972).
    [CrossRef]
  18. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  19. S. G. Warren, "Optical Constants of Ice from the Ultraviolet to the Microwave," Appl. Opt. 23, 1206–1223 (1984).
    [CrossRef] [PubMed]
  20. K. N. Liou, U. Utah; private communication.

1985 (1)

1984 (1)

1983 (1)

1982 (1)

1981 (2)

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

R. F. Coleman and K.-N. Liou, "Light Scattering by Hexagonal Ice Crystals," J. Atmos. Sci. 38, 1260–1271 (1981).
[CrossRef]

1980 (3)

G. L. Stephens, "Radiative Transfer on a Linear Lattice: Application to Anisotropic Ice Crystal Clouds," J. Atmos. Sci. 37, 2095–2104 (1980).
[CrossRef]

A. Mugnai and W. J. Wiscombe, "Scattering of Radiation by Moderately Nonspherical Particles," J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

W. J. Wiscombe, "Improved Mie Scattering Algorithms," Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

1979 (1)

1978 (1)

C. M. R. Piatt, "Lidar Backscatter from Horizontal Ice Crystal Patterns," J. Appl. Meterol. 17, 482–488 (1978).
[CrossRef]

1977 (1)

A. J. Heymsfield, "Precipitation Development in Stratiform Ice Clouds: A Microphysical and Dynamical Study," J. Atmos. Sci. 39, 367–381 (1977).
[CrossRef]

1975 (1)

A. J. Heymsfield, "Cirrus Uncinus Generating Cells and the Evolution of Cirriform Clouds. Part I. Aircraft Observations of the Growth of the Ice Phase," J. Atmos. Sci. 32, 799–805 (1975).
[CrossRef]

1974 (1)

J. E. Hansen and L. D. Travis, "Light Sattering in Planetary Atmospheres," Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1972 (2)

A. J. Heymsfield and R. G. Knollenberg, "Properties of Cirrus Generating Cells," J. Atmos. Sci. 29, 1338–1366 (1972).
[CrossRef]

K. N. Liou, "Light Scattering by Ice Clouds in the Visible Infrared: A Theoretical Study," J. Atmos. Sci. 29, 524–536 (1972).
[CrossRef]

Asano, S.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born, Optik (Springer-Verlag, Berlin, 1972).
[CrossRef]

Cai, Q.

Coleman, R. F.

R. F. Coleman and K.-N. Liou, "Light Scattering by Hexagonal Ice Crystals," J. Atmos. Sci. 38, 1260–1271 (1981).
[CrossRef]

Hansen, J.

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

Hansen, J. E.

J. E. Hansen and L. D. Travis, "Light Sattering in Planetary Atmospheres," Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Heymsfield, A. J.

A. J. Heymsfield, "Precipitation Development in Stratiform Ice Clouds: A Microphysical and Dynamical Study," J. Atmos. Sci. 39, 367–381 (1977).
[CrossRef]

A. J. Heymsfield, "Cirrus Uncinus Generating Cells and the Evolution of Cirriform Clouds. Part I. Aircraft Observations of the Growth of the Ice Phase," J. Atmos. Sci. 32, 799–805 (1975).
[CrossRef]

A. J. Heymsfield and R. G. Knollenberg, "Properties of Cirrus Generating Cells," J. Atmos. Sci. 29, 1338–1366 (1972).
[CrossRef]

Hobbs, P. V.

P. V. Hobbs, Ice Physics (Oxford U. P., London, 1974).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jayaweera, K.

Johnson, E.

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

Knollenberg, R. G.

A. J. Heymsfield and R. G. Knollenberg, "Properties of Cirrus Generating Cells," J. Atmos. Sci. 29, 1338–1366 (1972).
[CrossRef]

Lacis, A.

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

Lebedeff, S.

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

Lee, P.

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

Liou, K. N.

K. N. Liou, "Light Scattering by Ice Clouds in the Visible Infrared: A Theoretical Study," J. Atmos. Sci. 29, 524–536 (1972).
[CrossRef]

K. N. Liou, U. Utah; private communication.

Liou, K.-N.

Q. Cai and K.-N. Liou, "Polarized Light Scattering by Hexagonal Ice Crystals: Theory," Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

R. F. Coleman and K.-N. Liou, "Light Scattering by Hexagonal Ice Crystals," J. Atmos. Sci. 38, 1260–1271 (1981).
[CrossRef]

Mugnai, A.

A. Mugnai and W. J. Wiscombe, "Scattering of Radiation by Moderately Nonspherical Particles," J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

Piatt, C. M. R.

C. M. R. Piatt, "Lidar Backscatter from Horizontal Ice Crystal Patterns," J. Appl. Meterol. 17, 482–488 (1978).
[CrossRef]

Rind, D.

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

Russell, G.

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

Stephens, G. L.

G. L. Stephens, "Radiative Transfer on a Linear Lattice: Application to Anisotropic Ice Crystal Clouds," J. Atmos. Sci. 37, 2095–2104 (1980).
[CrossRef]

Takano, Y.

Travis, L. D.

J. E. Hansen and L. D. Travis, "Light Sattering in Planetary Atmospheres," Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Warren, S. G.

Weickmann, H. K.

Wendling, P.

Wendling, R.

Wiscombe, W. J.

A. Mugnai and W. J. Wiscombe, "Scattering of Radiation by Moderately Nonspherical Particles," J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

W. J. Wiscombe, "Improved Mie Scattering Algorithms," Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

Appl. Opt. (6)

J. Appl. Meterol. (1)

C. M. R. Piatt, "Lidar Backscatter from Horizontal Ice Crystal Patterns," J. Appl. Meterol. 17, 482–488 (1978).
[CrossRef]

J. Atmos. Sci. (7)

A. J. Heymsfield and R. G. Knollenberg, "Properties of Cirrus Generating Cells," J. Atmos. Sci. 29, 1338–1366 (1972).
[CrossRef]

A. J. Heymsfield, "Cirrus Uncinus Generating Cells and the Evolution of Cirriform Clouds. Part I. Aircraft Observations of the Growth of the Ice Phase," J. Atmos. Sci. 32, 799–805 (1975).
[CrossRef]

A. J. Heymsfield, "Precipitation Development in Stratiform Ice Clouds: A Microphysical and Dynamical Study," J. Atmos. Sci. 39, 367–381 (1977).
[CrossRef]

G. L. Stephens, "Radiative Transfer on a Linear Lattice: Application to Anisotropic Ice Crystal Clouds," J. Atmos. Sci. 37, 2095–2104 (1980).
[CrossRef]

A. Mugnai and W. J. Wiscombe, "Scattering of Radiation by Moderately Nonspherical Particles," J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

R. F. Coleman and K.-N. Liou, "Light Scattering by Hexagonal Ice Crystals," J. Atmos. Sci. 38, 1260–1271 (1981).
[CrossRef]

K. N. Liou, "Light Scattering by Ice Clouds in the Visible Infrared: A Theoretical Study," J. Atmos. Sci. 29, 524–536 (1972).
[CrossRef]

Science (1)

J. Hansen, E. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, "Climate Impact of Increasing Atmospheric Carbon Dioxide," Science 213, 957–966 (1981).
[CrossRef] [PubMed]

Space Sci. Rev. (1)

J. E. Hansen and L. D. Travis, "Light Sattering in Planetary Atmospheres," Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Other (4)

M. Born, Optik (Springer-Verlag, Berlin, 1972).
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

K. N. Liou, U. Utah; private communication.

P. V. Hobbs, Ice Physics (Oxford U. P., London, 1974).

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Figures (11)

Fig. 1
Fig. 1

Hexagonal ice crystal illuminated by a radiation source, e.g., the sun at an elevation angle scatters radiation into a direction described by η and ϕ.

Fig. 2
Fig. 2

Two subsequent planes of incidence rotated against each other require a transformation of the amplitudes ar and al, whichrefer to the unit vectors er and el.

Fig. 3
Fig. 3

P/π vs θ for crystals randomly oriented in 3-D space (solid curve a) yielded by averaging all data for all individual elevation angles compared to results of Liou (dashed curve b). The peaks in curve a correspond to reflections on the crystals and indicate the 6° stepwise increase in elevation angle .

Fig. 4
Fig. 4

P/4π vs θ for elevation angles e of 9, 57, and 87° corresponding to plots (a), (b) and (c).

Fig. 5
Fig. 5

Log(P/4π) vs (η,ϕ) for diffraction at = 45°: (a) downward scattered radiation; (b) upward scattered radiation.

Fig. 6
Fig. 6

Log(P/4π) vs (η,ϕ) for upward scattered radiation; plots (a), (b), and (c) correspond to elevation angles of 21, 51, and 81°.

Fig. 7
Fig. 7

Log(P/4π) vs (η,ϕ) for downward scattered radiation. Plots (a), (b), and (c) correspond to elevation angles of 21, 51 and 81°.

Fig. 8
Fig. 8

Log(P/4π) vs (η,ϕ) for downward scattered radiation for a 51° elevation angle. Particles are rotated around their longest axis and horizontally oriented as in Fig. 7(b); crystals tilted with respect to the horizontal plane at ±15° are included.

Fig. 9
Fig. 9

Log(P/4π) vs (η,ϕ) for a 45° elevation angle resulting from a dual scattering process: (a) downward scattered radiation; (b) upward scattered radiation.

Fig. 10
Fig. 10

Log(P/4π) vs (η,ϕ) for a 45° elevation angle resulting from a triple scattering process: (a) downward scattered radiation; (b) upward scattered radiation.

Fig. 11
Fig. 11

Total amounts of upward scattered energy divided by input energy plotted vs elevation angle for single, dual, and triple scattering processes.

Equations (28)

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[ I Q U V ] = k sca V 4 π r 0 P [ I 0 Q 0 U 0 V 0 ] ,
A 1 = a 1 exp [ i ( ω t kz δ 1 ) ] ;
A r = a r exp [ i ( ω t kz δ r ) ] .
a 1 = e ̂ 1 a 1 ,
a r = e ̂ r a r .
sin Ψ / sin φ = n ,
e P , refl = 2 ( w f + e P , in )
e P , refr = ( n 2 1 + w 2 ) 1 / 2 ( w f + e P , in f
d l = 2 cos φ sin Ψ sin ( φ + Ψ ) cos ( φ Ψ ) a l ;
d r = 2 cos φ sin Ψ sin ( φ + Ψ ) a r ;
r l = sin ( φ Ψ ) cos ( φ + Ψ ) sin ( φ + Ψ ) cos ( φ Ψ ) a l ;
r r = sin ( φ Ψ ) sin ( φ + Ψ ) a r .
a r = [ a l a r sin δ ( 1 / sin 2 ϑ + m 2 sin 2 ϑ ) 1 + m 2 ] 1 / 2 ,
a l = [ a l a r sin δ ( m 2 / sin 2 ϑ + sin 2 ϑ ) 1 + m 2 ] 1 / 2 ,
m = tan ( ρ 0 + ρ ) ,
tan 2 ρ 0 = 2 a l a r a l 2 + a r 2 cos δ ,
sin 2 ϑ = 2 a l a r a l 2 + a r 2 sin δ .
δ = δ + 2 arctan cos φ sin 2 φ n 2 sin 2 φ .
I in , l = c n 1 4 π a l 2 ,
I r , l = c n 1 4 π r l 2 ,
I d , l = c n 2 4 π d l 2 cos Ψ cos φ ,
E in , l = a l 2 ,
E r , l = r 1 2 ,
E d , l = d l 2 n 2 n 1 cos Ψ cos φ = d l 2 tan φ tan Φ .
1 / 4 π P ref ( η , ϕ ) d Ω = 1 .
P dif ( θ , ) = ( 1 + cos θ ) 2 8 π 3 k 2 a 2 R × 0 2 π sin ( k r sin θ sin Φ ) 2 k r sin θ sin Φ × sin ( k r υ sin θ cos Φ ) 2 k r υ sin θ cos Φ d Φ .
υ = cos sin Φ + R 1 cos 2 sin 2 Φ .
P ( η , ϕ ) = P ref + P dif 2 ,

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