Algorithm and computation of aerosols phase functions
by A.N. Rublev
(Internal Note IAE5715/16 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 SO_{2} (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 publication^{3}). Principal aerosol models
(e.g. continental, urban, maritime, stratospheric, volcanic, upper atmosphere,
and cloudy) and their basic components (e.g. dust, watersoluble
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 N_{i} is the number
of particles of the ith component, and N is total number of particles
in a given aerosol sample.
Main expressions for the aerosol integrated optical properties as given
by Deirmendjian^{4} are:
(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=piq 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 cm^{3} with dimensionless radii x
in the interval dx so that is
the total number of particles per cm^{3});
K_{sc}(x,m)and
K_{ex}(x,m)
are dimensionless efficiency factors for scattering and extinction, respectively
(Ref. 4)
i(x,m,q) is
the scattering intensity for nonpolarized radiation (Ref. 4):
, where
S_{1}, S_{2} 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 I_{q}..
Eqs. (15) determine optical properties of the aerosols to be considered
in nonpolarized 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)=t_{gas}(l)+t_{aer}(l),
where t_{gas}
is the atmospheric gases optical thickness calculated using, for example,
wellknown spectroscopic ” linebyline ” methods;
t_{aer}
is the aerosol optical thickness calculated for an arbitrary nonhomogeneous
path L:
s_{ex}_{(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 K_{n} 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, Deirmendjian^{4}, Van de
Hulst^{6}) 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, watersoluble particles, etc…), namely –
the extinction coefficient (km^{1}) for a particle number concentration
N=1
particle per 1 cm^{3}; w–
the single scattering albedo; g– the asymmetry
factor; K_{n}– the normalization factor
and its values at 204 angles.
• The same as above defined optical properties for noncloudy 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 (watersoluble, soot, dust) are shown
in Table 2.
• The optical properties for a cloudy aerosol model with a particle
number concentration N_{0}=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;
r_{0}=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

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, 423430 (1992) 
Gregs J.S., et al., ” Global tracking of the SO_{2} clouds from
the June 1991 month Pinatubo eruptions ” Geophys. Res. Letters,
19,
151154 (1992) 
World Meteorology Organization (WMO) publication: “A preliminary cloudless
standard atmosphere for radiation computation“, WCP112, WMO/TDNO.
24 (1986) 
Deirmendjian D., Electromagnetic Scattering on Spherical Polydispersions.
Elsevier, 290 pp. (1969)  Twomey S. Atmospheric aerosols. Elsevier, 302 pp. (1977)

Van de Hulst, H.C., Light scattering by small particles, 470 pp.,
New York : Dover Publications, 1981. 
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 (%) N_{i}/N ^{*)} 

Continental 
dust (DustLike)
watersoluble (WS) soot (Soot) 
70
29 1 
2.26278E06
9.37437E01 6.25607E02 
Urban 
watersoluble (WS)
soot (Soot) dust (DustLike) 
61
22 17 
5.88931E01
4.11069E01 1.64128E07 
Maritime 
oceanic (Ocean)
watersoluble (WS) 
95
5 
4.29942E04
9.99573E01 
Stratospheric  sulfuric acid (75% H_{2}SO_{4})  100  1.0 
Volcanic  volcanic ash (VAsh)  100  1.0 
Upper Atmosphere  sulfuric acid (75% H_{2}SO_{4})  100  1.0 
Cloudy  water  100  1.0 
^{*)} N_{i}– number of particles of icomponent; N
total number of particles in an aerosol sample.
Table 2. Integrated optical properties of the urban aerosol model
(a noncloudy model).
Num  l(mm)  s_{ex}(km^{1})  w  m  K_{n} 
1 2 3 4 
0.200 0.250 0.300 0.337 
0.13889E05 0.12610E05 0.11042E05 0.98538E06 
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;
s_{ex }– extinction coefficient
in km^{1};
w– single scattering albedo;
m– asymmetry factor;
K_{a}– normalization factor.