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Algorithm and computation of aerosols phase functions

by A.N. Rublev

(Internal Note IAE-5715/16 of Russian Research Center " Kurchatov Institute ", Moscow, 51 pp., 1994).

Extended abstract

Aerosols are known to influence the propagation of the solar radiation in the atmosphere. Aerosols emission sources are numerous: e.g. dust storms, fuel combustion (soot), ocean sprays, etc... Stratospheric aerosols and tropospheric anthropogenic aerosols which play an essential role in climate forcing (Charlson et al.1) can be generated by atmospheric chemical reactions with sulfates, sulfuric acid and nitric acid. The volcanic eruptions are one of the important atmospheric aerosol generators, for example the eruption of the volcano Pinatubo, Philippines, June 1991 resulted in the emission 20 Mts of SO2 (Gregs et al.2) which is a main source of sulfuric acid aerosol fraction.

Despite the large number of different aerosol sources, only some selected basic aerosol components have been considered in the development of various aerosol models (WMO publication3). Principal aerosol models (e.g. continental, urban, maritime, stratospheric, volcanic, upper atmosphere, and cloudy) and their basic components (e.g. dust, water-soluble particles, soot, salt particles (oceanic), sulfuric acid solution droplets, volcanic ash, and water) are listed in Table 1 (from Ref. 3) with the following entries:

  • the aerosol model name (first column) and the related basic aerosol components (second column);
  • the aerosol fraction by volume in % (third column), and the related aerosol relative concentration (fourth column), where Ni is the number of particles of the i-th component, and N is total number of particles in a given aerosol sample.
Main expressions for the aerosol integrated optical properties as given by Deirmendjian4 are:
• the scattering coefficient:                                                                                     (1)

• the extinction coefficient:                                                                                     (2)

       • the scattering phase function corresponding to the scattering angle q:
                                                                                            (3)
 

• the single scattering albedo:                                                                                                                     (4)

• the asymmetry factor:                                                                                                          (5)

• the normalization factor:                                                                                                                 (6)

where:

x=kr is the dimensionless size of the particles with radius r
m=p-iq is the complex index of refraction with the real (p) and imaginary (q) parts
n(x)-is the aerosol particle size distribution function (i.e., n(x)d(x) is the number of particles per cm3 with dimensionless radii x in the interval dx so that is the total number of particles per cm3);
Ksc(x,m)and Kex(x,m) are dimensionless efficiency factors for scattering and extinction, respectively (Ref. 4)
i(x,m,q) is the scattering intensity for non-polarized radiation (Ref. 4): , where S1, S2 are dimensionless complex functions (see Ref. 4, 5 for explicit formulas) which give the complex amplitudes of the scattered wave in terms of the complex amplitudes of the incident radiation resolved along the transverse and parallel directions with respect to the scattering plane, respectively (Ref. 5):

is a linear interpolation of the phase function Iq..
Eqs. (1-5) determine optical properties of the aerosols to be considered in non-polarized radiative transfer problems. In particular, the optical thickness t(l) at the wavelength l of an atmosphere including aerosols is expressed as the sum: t(l)=tgas(l)+taer(l),

where tgas is the atmospheric gases optical thickness calculated using, for example, well-known spectroscopic " line-by-line " methods;

taer is the aerosol optical thickness calculated for an arbitrary non-homogeneous path L:

sex(x;l) is the aerosol extinction coefficient at a point x of the path L.
It should be outlined, that the normalization factor of Eq.(6) () has been calculated to check the reliability of the linear interpolation of the phase function of Eq.(3) used in the calculations of Eq. (5). It is aimed at the determination of a required number of angular mesh points providing an accurate interpolation of the phase function according to the following criterion: the closer Kn is to 1, the better is the interpolation (see last column of Table 2 as an example). In Rublev’s paper 204 angular mesh points (from 0 to 180 degrees) are used in the calculations.
The Mie theory (see, for example, Deirmendjian4, Van de Hulst6) based algorithm has been developed and a related computer code as well, providing a reliable accuracy for computations of the above mentioned aerosol optical properties (estimated relative error £ 0.3%).

Main results presented in the publication are (see Table 2 as an example):

• Tables in the Appendix to the paper provide the computed values of the phase function for the principal aerosol models and their basic components as listed in Table 1. The calculations were made for 8 wavelengths in the UV, visible and IR regions, with an estimated relative error £ 0.3%.

• The principal optical properties of the basic aerosol components (column 2 of Table 1: soot, dust, water-soluble particles, etc...), namely - the extinction coefficient (km-1) for a particle number concentration N=1 particle per 1 cm3; w- the single scattering albedo; g- the asymmetry factor; Kn- the normalization factor and its values at 204 angles.

• The same as above defined optical properties for non-cloudy basic aerosol models (column 1 of Table 1: continental, maritime, urban, etc...). As an example, results of the calculations for the urban aerosol model with basic components from Table 1 (water-soluble, soot, dust) are shown in Table 2.

• The optical properties for a cloudy aerosol model with a particle number concentration N0=353.678 cm-3 corresponding to a typical cloud water content W=0.3 g m-3 (Ref. 7), with the modified Gamma function n(r) (Ref. 6, 7) as a particle size distribution function:

                                                                    (7)
with the following values of parameters (Ref. 4): a=2; r0=1.5 mm.

The software package AERCOMP (FORTRAN code) allowing the determination of the optical properties of more complex aerosol models has been developed. In particular, using optical properties of basic aerosol components, one can calculate (applying linear interpolation on wavelengths and cosines of scattering angels) the optical properties for more complex, composite aerosol models. Table 2 is an example of outputs of this program.

References

  1. Charslon R.J., S.E. Schwartz, J.M. Hales, R.D. Cess, J.A. Coakley, Jr., J.E. Hansen, and D.J. Hofman, " Climate forcing by anthropogenic aerosols ", Science, 255, 423-430 (1992)
  2. Gregs J.S., et al., " Global tracking of the SO2 clouds from the June 1991 month Pinatubo eruptions " Geophys. Res. Letters, 19, 151-154 (1992)
  3. World Meteorology Organization (WMO) publication: "A preliminary cloudless standard atmosphere for radiation computation", WCP-112, WMO/TD-NO. 24 (1986)
  4. Deirmendjian D., Electromagnetic Scattering on Spherical Polydispersions. Elsevier, 290 pp. (1969)
  5. Twomey S. Atmospheric aerosols. Elsevier, 302 pp. (1977)
  6. Van de Hulst, H.C., Light scattering by small particles, 470 pp., New York : Dover Publications, 1981.
  7. Handbook: Clouds and cloudy atmosphere. Leningrad, " Gidrometeoizdat ", 649 p., 1989 (in Russian).
Table 1. Principal aerosol models.

(from Ref. 3)


Aerosol model
Basic aerosol components and their designation
Relative content


volume (%) Ni/N *)
Continental
dust (Dust-Like)

water-soluble (W-S)

soot (Soot)
70

29

1
2.26278E-06

9.37437E-01

6.25607E-02
Urban
water-soluble (W-S)

soot (Soot)

dust (Dust-Like)
61

22

17
5.88931E-01

4.11069E-01

1.64128E-07
Maritime
oceanic (Ocean)

water-soluble (W-S)
95

5
4.29942E-04

9.99573E-01
Stratospheric
sulfuric acid (75% H2SO4)
100
1.0
Volcanic
volcanic ash (V-Ash)
100
1.0
Upper Atmosphere
sulfuric acid (75% H2SO4)
100
1.0
Cloudy 
water
100
1.0
*) Ni- number of particles of i-component; N- total number of particles in an aerosol sample.

Table 2. Integrated optical properties of the urban aerosol model (a non-cloudy model).

 Num
 l(mm)
sex(km-1
w
m
Kn
1
2
3
4
0.200
0.250
0.300
0.337
 0.13889E-05
0.12610E-05
0.11042E-05
0.98538E-06
0.53439E+00
0.59215E+00
0.65632E+00
0.66404E+00
 0.68971E+00
0.64587E+00
0.61527E+00
0.60741E+00
 1.001
1.000
1.000
1.000

  Num- line number;
l - wavelength in micrometers;
sex - extinction coefficient in km-1;
w- single scattering albedo;
m- asymmetry factor;
Ka- normalization factor.