Abstract

The extinction coefficient, albedo of single scattering, and differential scattering and polarization properties of water clouds and hazes in the visible and infrared have been computed using the complete Mie series. The results with three types of size distributions are presented and compared with observations. These show a strong dependence of angular intensity and polarization patterns on the size distribution, the size range, and the dielectric and absorbing properties of water droplets at each wavelength. A peculiarity of scattering at angles near 45°, observed experimentally and independently by two authors, is corroborated by the numerical results. Prominent observational features characteristic of natural fog, such as an extremely bright and narrow aureole, cloudbows, and glories, are reproduced in a model cloud of spherical water droplets, with a wide distribution in droplet radius and a maximum concentration at a 4-μ radius.

© 1964 Optical Society of America

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References

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  1. D. Deirmendjian, R. Clasen, W. Viezee, J. Opt. Soc. Am. 51, 620 (1961). See also D. Deirmendjian, R. J. Clasen, RAND Reports R-393-PR (1962) and R-407-PR (1963).
    [CrossRef]
  2. R. H. Giese, Z. Astrophys. 51, 119 (1961).
  3. A. Kh. Khrgiana, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 7, 56 (1952).
  4. A. Kh. Khrgian, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 17, 36 (1956).
  5. W. G. Durbin, Tellus 11, 202 (1959).
    [CrossRef]
  6. F. Singleton, D. J. Smith, Quart. J. Roy. Meteorol. Soc. 86, 454 (1960).
    [CrossRef]
  7. D. Deirmendjian, Ann. Geophys. 13, 286 (1957).
  8. D. Deirmendjian, Ann. Geophys. 15, 218 (1959).
  9. D. Deirmendjian, Quart. J. Roy. Meteorol. Soc. 85, 404 (1960).
    [CrossRef]
  10. J. Y. Gilbert, “Condensation Nuclei of the Los Angeles Region,” Univ. of Calif., Department of Meteorology, Los Angeles (1954).
  11. C. Junge, C. W. Chagnon, J. E. Manson, J. Meteorol. 18, 81 (1961).
    [CrossRef]
  12. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).
  13. Z. Sekera, Handbuch Phys. 48, 288 (1957).
  14. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  15. R. Penndorf, J. Opt. Soc. Am. 52, 402 (1962).
    [CrossRef]
  16. M. Centeno, J. Opt. Soc. Am. 31, 244 (1941).
    [CrossRef]
  17. M. G. Gibbons, J. R. Nichols, F. I. Laughridge, R. L. Rudkin, J. Opt. Soc. Am. 51, 633 (1961).
    [CrossRef]
  18. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).
  19. B. S. Pritchard, W. G. Elliott, J. Opt. Soc. Am. 50, 191 (1960).
    [CrossRef]
  20. O. D. Barteneva, Bull. Acad. Sci. USSR, Geophys. Ser. No. 12, 1237 (1960), in AGU translation.
  21. O. D. Barteneva, G. Ya. Bashilov, Bull. Acad. Sci. USSR, Geophys. Ser. No. 4, 395 (1961), in AGU translation.
  22. J. E. Tyler, J. Opt. Soc. Am. 51, 1289 (1961).
    [CrossRef]
  23. G. V. Rozenberg, Soviet Phys.—Uspekhi 3, 346 (1960).
  24. R. S. Fraser, Dissertation, Univ. of Calif., Los Angeles (1959). See also Sci. Rept. No. 2, Contract No. AF 19(604)-2429, AFCRC-TN-60-256 (1959).

1962

1961

1960

F. Singleton, D. J. Smith, Quart. J. Roy. Meteorol. Soc. 86, 454 (1960).
[CrossRef]

D. Deirmendjian, Quart. J. Roy. Meteorol. Soc. 85, 404 (1960).
[CrossRef]

G. V. Rozenberg, Soviet Phys.—Uspekhi 3, 346 (1960).

O. D. Barteneva, Bull. Acad. Sci. USSR, Geophys. Ser. No. 12, 1237 (1960), in AGU translation.

B. S. Pritchard, W. G. Elliott, J. Opt. Soc. Am. 50, 191 (1960).
[CrossRef]

1959

D. Deirmendjian, Ann. Geophys. 15, 218 (1959).

W. G. Durbin, Tellus 11, 202 (1959).
[CrossRef]

1957

D. Deirmendjian, Ann. Geophys. 13, 286 (1957).

Z. Sekera, Handbuch Phys. 48, 288 (1957).

1956

A. Kh. Khrgian, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 17, 36 (1956).

1952

A. Kh. Khrgiana, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 7, 56 (1952).

1941

Barteneva, O. D.

O. D. Barteneva, G. Ya. Bashilov, Bull. Acad. Sci. USSR, Geophys. Ser. No. 4, 395 (1961), in AGU translation.

O. D. Barteneva, Bull. Acad. Sci. USSR, Geophys. Ser. No. 12, 1237 (1960), in AGU translation.

Bashilov, G. Ya.

O. D. Barteneva, G. Ya. Bashilov, Bull. Acad. Sci. USSR, Geophys. Ser. No. 4, 395 (1961), in AGU translation.

Centeno, M.

Chagnon, C. W.

C. Junge, C. W. Chagnon, J. E. Manson, J. Meteorol. 18, 81 (1961).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).

Clasen, R.

Deirmendjian, D.

D. Deirmendjian, R. Clasen, W. Viezee, J. Opt. Soc. Am. 51, 620 (1961). See also D. Deirmendjian, R. J. Clasen, RAND Reports R-393-PR (1962) and R-407-PR (1963).
[CrossRef]

D. Deirmendjian, Quart. J. Roy. Meteorol. Soc. 85, 404 (1960).
[CrossRef]

D. Deirmendjian, Ann. Geophys. 15, 218 (1959).

D. Deirmendjian, Ann. Geophys. 13, 286 (1957).

Durbin, W. G.

W. G. Durbin, Tellus 11, 202 (1959).
[CrossRef]

Elliott, W. G.

Fraser, R. S.

R. S. Fraser, Dissertation, Univ. of Calif., Los Angeles (1959). See also Sci. Rept. No. 2, Contract No. AF 19(604)-2429, AFCRC-TN-60-256 (1959).

Gibbons, M. G.

Giese, R. H.

R. H. Giese, Z. Astrophys. 51, 119 (1961).

Gilbert, J. Y.

J. Y. Gilbert, “Condensation Nuclei of the Los Angeles Region,” Univ. of Calif., Department of Meteorology, Los Angeles (1954).

Junge, C.

C. Junge, C. W. Chagnon, J. E. Manson, J. Meteorol. 18, 81 (1961).
[CrossRef]

Khrgian, A. Kh.

A. Kh. Khrgian, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 17, 36 (1956).

Khrgiana, A. Kh.

A. Kh. Khrgiana, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 7, 56 (1952).

Laughridge, F. I.

Manson, J. E.

C. Junge, C. W. Chagnon, J. E. Manson, J. Meteorol. 18, 81 (1961).
[CrossRef]

Mazin, I. P.

A. Kh. Khrgian, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 17, 36 (1956).

A. Kh. Khrgiana, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 7, 56 (1952).

Minnaert, M.

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

Nichols, J. R.

Penndorf, R.

Pritchard, B. S.

Rozenberg, G. V.

G. V. Rozenberg, Soviet Phys.—Uspekhi 3, 346 (1960).

Rudkin, R. L.

Sekera, Z.

Z. Sekera, Handbuch Phys. 48, 288 (1957).

Singleton, F.

F. Singleton, D. J. Smith, Quart. J. Roy. Meteorol. Soc. 86, 454 (1960).
[CrossRef]

Smith, D. J.

F. Singleton, D. J. Smith, Quart. J. Roy. Meteorol. Soc. 86, 454 (1960).
[CrossRef]

Tyler, J. E.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Viezee, W.

Ann. Geophys.

D. Deirmendjian, Ann. Geophys. 13, 286 (1957).

D. Deirmendjian, Ann. Geophys. 15, 218 (1959).

Bull. Acad. Sci. USSR, Geophys. Ser. No.

O. D. Barteneva, Bull. Acad. Sci. USSR, Geophys. Ser. No. 12, 1237 (1960), in AGU translation.

Bull. Acad. Sci. USSR, Geophys. Ser. No. 4

O. D. Barteneva, G. Ya. Bashilov, Bull. Acad. Sci. USSR, Geophys. Ser. No. 4, 395 (1961), in AGU translation.

Handbuch Phys.

Z. Sekera, Handbuch Phys. 48, 288 (1957).

J. Meteorol.

C. Junge, C. W. Chagnon, J. E. Manson, J. Meteorol. 18, 81 (1961).
[CrossRef]

J. Opt. Soc. Am.

Quart. J. Roy. Meteorol. Soc.

F. Singleton, D. J. Smith, Quart. J. Roy. Meteorol. Soc. 86, 454 (1960).
[CrossRef]

D. Deirmendjian, Quart. J. Roy. Meteorol. Soc. 85, 404 (1960).
[CrossRef]

Soviet Phys.—Uspekhi

G. V. Rozenberg, Soviet Phys.—Uspekhi 3, 346 (1960).

Tellus

W. G. Durbin, Tellus 11, 202 (1959).
[CrossRef]

Tr. Tsentr. Aerolog. Observ.

A. Kh. Khrgiana, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 7, 56 (1952).

A. Kh. Khrgian, I. P. Mazin, Tr. Tsentr. Aerolog. Observ. 17, 36 (1956).

Z. Astrophys.

R. H. Giese, Z. Astrophys. 51, 119 (1961).

Other

J. Y. Gilbert, “Condensation Nuclei of the Los Angeles Region,” Univ. of Calif., Department of Meteorology, Los Angeles (1954).

R. S. Fraser, Dissertation, Univ. of Calif., Los Angeles (1959). See also Sci. Rept. No. 2, Contract No. AF 19(604)-2429, AFCRC-TN-60-256 (1959).

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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

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Figures (10)

Fig. 1
Fig. 1

Three size distribution functions used in the integration of the Mie functions. Total concentration 100 cm−3.

Fig. 2
Fig. 2

Integrated and normalized intensity functions P1/4π (solid line) and P2/4π (dashed line) for haze particles with complex index (see Table I) and infrared illumination. Computed values at θ: 0(5)20(10)150(5)180° joined by straight lines.

Fig. 3
Fig. 3

Same as Fig. 2 but for particles with real index and visible light. Computed values at θ: 0(1)5, 10(10)150(5)170(2)180°.

Fig. 4
Fig. 4

Intensity functions for haze particles at λ0.70 μ and with real index 1.33 but a different distribution than in Fig. 3. Computed values at θ: 0(2.5)20(10)130(2.5)180°.

Fig. 5
Fig. 5

Same as Fig. 4 but with illumination at λ0.45 μ, with real index 1.34.

Fig. 6
Fig. 6

Intensity functions for a water cloud illuminated by λ10.0-μ infrared radiation. Computed values at θ: 0(5)180°.

Fig. 7
Fig. 7

Same as Fig. 6 but at λ5.3 μ. Computed values at θ: 0(2.5)30(10)140(2.5)180°.

Fig. 8
Fig. 8

Same as Fig. 6 but at λ0.70 μ and real index 1.33. Computed values at θ: 0(1)10(5)130(2)180°.

Fig. 9
Fig. 9

Same as Fig. 6 but at λ0.45 μ and real index 1.34. Computed values at θ: 0(1)15(5)120(2)132(1)180°.

Fig. 10
Fig. 10

Integrated and normalized functions P3/4π (solid line) and P4/4π (dashed line) for haze particles at λ0.70 μ. Comparable values, shown by solid and open dots, respectively, for a different size distribution and refractive index, adapted from Fraser (ref. 24, p. 123, “model D”).

Tables (6)

Tables Icon

Table 1 Phase Function P1 (θ), Model C Haze

Tables Icon

Table I Volume Extinction Coefficient and Albedo

Tables Icon

Table II Cloud Phase Functions

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Table III Cloud Phase Functions

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Table IV Cloud Phase Function (P1 + P2)/8π for λ1.61 μ and m = 1.315

Tables Icon

Table V Cloud Phase Function (P1 + P2)/8π for λ3.07 μ and m = 1.525 − 0.0682i

Equations (15)

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n ( r ) = a r α e b r γ
N = 0 n ( r ) d r = a γ b α + 1 γ Γ ( α + 1 γ ) ,
d d r n ( r ) = a r α 1 e b r γ ( α γ b r γ ) ,
b = α γ r γ , r = r c ,
n ( r ) = 2.373 r 6 e 1.5 r cm 3 μ 1 ,
n ( r ) = 5.33 × 10 4 r e 8.944 r .
f ( x , k ) = a k 3 α x α exp { b ( x / k ) γ } ,
β ( m , λ ; x 1 , x 2 ) = π x 1 x 2 x 2 f ( x , k ) K ( m , x ) d x ,
P j 4 π ( m , θ , λ ; x 1 , x 2 ) = 1 β sc x 1 x 2 f ( x , k ) i j ( m , x , θ ) d x , j = 1,2,3,4 ,
d d log r log n ( r ) = α [ ( r r c ) γ 1 ] > 4.
r 2 > r c ( 1 + 4 / α ) 1 / γ .
1 8 π Ω [ P 1 ( θ ) + P 2 ( θ ) ] d ω = 1.
P 1 ( θ ) P 2 ( θ ) P 1 ( θ ) + P 2 ( θ ) ,
P A ( θ ) = β R P R ( θ ) + β M P M ( θ ) β R + β M
1 sin 2 θ 0 x 2 + α e b x [ J 1 ( x sin θ ) ] 2 d x

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