A program was developed for computing the scattering and absorption by arbitrarily oriented and randomly oriented prolate and oblate spheroids. This permits examination of the effect of particle shape for cases ranging from needles through spheres to platelets. Applications of this capability to aerosol and cloud problems are discussed. Initial results suggest that the effect of nonspherical particle shape on transfer of radiation through aerosol layers and cirrus clouds, as required for many climate studies, can be readily accounted for by defining an appropriate effective spherical particle radius

Asano, S.

Hansen, J. E.

Sato, M.

English

NASA Technical Reports Server

Paper No. 44
SCATTERING BY RANDOMLY ORIENTED ELLIPSOIDS: APPLICATION
TO AEROSOL AND CLOUD PROBLEMS
Shoji Asano, Makoto Sato and James Hansen, Goddard Institute for Space
Studies, New York, NY 10025
ABSTRACT
A program has been developed for computing the scattering
and absoprtion by arbitrarily oriented and randomly oriented
prolate and oblate spheroids. This permits examination of
the effect of particle shape for cases ranging from needles
through spheres to platelets. We discuss applications of
this capability to aerosol and cloud problems. Initial results
suggest that the effect of nonspheriaal particle shape on trans-
fer of radiation through aerosol layers and cirrus clouds,
as required for many climate studies, can be readily accounted
for by defining an appropriate effective spherical particle radius.
Introduction. Many climate studies involve scattering or absorp-
tion of radiation by nonspherieal particles. Examples include the cli-
matic impact of natural or man-made changes in stratospheric and tro-
pospheric aerosols and cirrus clouds. The nonspherical shape of such
particles has generally been neglected, with the hope that spheres can
adequately represent the results for other particles. With improvement
of climate modeling capabilities and climate observations, it is becoming
increasingly important to obtain a quantitative understanding of the ef-
fects of nonspherical particle shape and a prescription for defining the
spheres which can be used to approximate nonspherical particles.
A general solution for scattering by infinitely long circular cylin-
ders has been obtained and investigated (Kerker, 1969; Lion 1972), but
that case is too special to provide an adequate test of particle shape
effects. The next logical particle shape for analysis, in terms of prac-
ticality for obtaining a general solution analogous to the Mie solution for
spheres, is the spheroid. In fact spheroids, formed by rotation of an
ellipse about its major (prolate spheroid) or minor (oblate spheroid)
axis, provide an excellent opportunity for investigation of the effect of
nonspherical particle shape. As the ratio of the length of the major axis
to length of the minor axis becomes large the particle becomes needle-
like. At the other extreme it approaches the shape of a platelet. In
between it passes through me special case of the sphere.
Asano and Yamamoto (1975) developed the scattering theory for a
homogeneous spheroid. The objective of the work described here is to
develop a program to implement that theory for particles all the way up
to sizes which show the basic features of geometrical optics, and to ap-
ply this theory to climate studies.
Single arLitrarily oriented spheroid. Light scattering by a
265
https://ntrs.nasa.gov/search.jsp?R=19790012506 2020-03-20T18:31:24+00:00Z
spheroidal particle is specified by the following five physical quantities:
1) particle size relative to the wavelength of incident wave, 2) eccentri-
city, 3) complex refractive index _n relative to that of the surrounding
medium, 4) orientation of particle to the incident wave, and 5) the ob-
servation direction.
We define the particle size parameter a by a = 2_a/X , where
a is the semi-major axis of the ellipse and _ is the wavelength of the in-
cident radiation. We specify the shape of the spheroid by the ratio of
the semi-major axis a to the semi-minor axis b, which thus has a value
greater than unity. We use the ratio a/b, rather than the eccentricity,
since it is a more direct measure of the elongation of the spheroid.
Fig. 1 shows the geometry of scattering for a linearly polarized
plane electromagnetic wave incident at anangle _ to the z-axis. The ori-
gin of the coordinate system is taken at the center of the spheroid with
the z-axis as the axis of rotation, i.e., as the major axis for a prolate
spheroid and the minor axis for an oblate one. The spherical coordinate
system ( 0, _ ) is adopted to represent the direction of a far-field obser-
vation point, where 0 is the zenith angle measured from the positive z-
axis and _ is the azimuth angle. The direction of incidence is assumedto
be in the _ = 0 plane (x-z plane) and is thus identified by (_, 0).
The efficiency factor for scattering Qsca is defined here as the
ratio of the scattering cross section Csc a to the area of the
geometrical shadow of the spheroid G(_ ) at the incidence angle _.
In Fig. 2 Qsca for prolate spheroids with ffl = L 50 is plotted, as _ func-
tion of the size parameter a, for various values of the shape parameter
from a/b = L 5 to 5. For small prolate spheroids, a< 4.0, the scattering
efficiency factors are greater for the spheroids which are closer to being
spheres, i.e., for the smaller a/b. Qsca (a) for a/b " 1.5 is very sim-
ilar to that for spheres, with oscillations which damp out for large size
parameters as Qsca approaches the geometrical optics limit of 2. A
striking feature is that with increase of a/b the first reeonance maximum
occurs at large size parameter and the peak value increases.
The major maxima and minima in Fig. 2 are due to interference
of light diffracted with light transmitted by the particle. The phase
shift for a light ray passing through the particle along the axis in the
direction of incidence is p= 2a(r_-l), which equals a or 2_a/rA for the
case ffi = 1.5. Thus successive maxima, caused by constructive inter-
ference, occur at intervals of _2_. The case of a/b = 3 is an exception;
however, the average separation of the maxima would also be_2_ if there
were a_ additional maximum. The results for a/b = 3.5, shown by the
dashed curve, in fact suggest that the expected second and third maxima
have somehow combined into a single maximum for a/b_ 3.
Application of the anomalous diffraction approximation of van de
Hulst (1957) to scattering by prolate spheroids at the parallel incidence
gives extinction efficiency curves identical to those for spheres, without
regard to a/b, but with the phase lag parameter p = 2a(_-1). However,
in fact, the position and peak value of the first maximum of the scattering
efficiency curves in Fig. 2 are strongly dependent on a/b. Furthermore,
the mean periods of the major oscillations of the scattering efficiency
curves are larger for the prolate spheroids than for spheres. These dis-
crepancies from the expectation of the anomalous diffraction approxima-
tion can be attributed to effects of edge phenomena for grazing reflection,
266
Oblate z #
Spheroid _y//_ Observationpoint
/
TE mode / (r;)l-I""_(i,)E
TM mode
Prolote
Spheroid
Fig. 1. Scattering geometry. A spherical coordinate system is
adopted to represent the scattered field in the far-field zone. The
origin of the coordinate system and the z-axis are the center and axis
of revolution of the spheroid, respectively. The angle of incidence _ is
the angle in the plane of incidence (the x-z plane)between the direction
of incidence and the z-axis.
20
O I0
Prolote
o 5
2) °/b=2.0
3) a/b=3.0
1 4) °/b=4.0
q
5) %:5.0
ox :35 4 5
.... I .... I .... I .... I
I0 15 20 25 30
2Fio/X
Fig. 2. Scattering efficiency factors Qsca at _ = 0 as a function of the
size parameter 2_a/X for the prolate spheroids with fh = 1.50, for
several values of the shape parameter a/b.
267
as discussed by Asano (1979).
Randomly oriented spheroids. Fig. 3 compares the scattering by
randomly oriented prolate spheroids with scattering by spheres of'equi-
valen# size. For the spheres integration over a narrow distribution of
particle sizes was performed in order to smooth out the oscillations
which occur for a single size. For the spheroids this was not necessary
since the integration over random orientations served the same function.
We define an equivalent spherical particle radius for any parti-
cle shape to be the radius of the circle which would give the same area
as that of the particle's shadow. Since the complete scattering behavior
of spheres is avatIable in effi etent subroutines (e. g. Hansen and
Travis, 1974) this prescription represents a simple approximation for
scattering by nonspherieal particles.
Parts (a) and (b) of Fig. 3 compare the efficiency factor for scat-
tering (Qsca) and the asymmetry parameter of the scattering diagram
(<cos _ >) for spheres and prolate spheroids. Two principal conclusions
are: (1) spheres provide a good approximation for the nonspherical par-
ticles for these parameters which define the gross radiative properties,
when the equivalent radius we have defined is employed, (2) there is no
evidence that the nouspheres are less forward scattering than spheres.
Parts (c) and (d) of Fig. 3 compare the scattering diagram and
linear polarization for spheres and prolate spheroids. It is apparent that
the scattering at a specific angle is not in general well approximated
bythe simple prescription of using spheres of equivalent cross-sectional
area. There is no evidence in the scattering by spheroids of this size
parameter for special spherical particle geometrical optics features such
as the rainbow and glory.
Discussion. These initial results are encouraging with regard to
climate applications. The principal parameters defining the radiative
transfer characteristics of aerosols and clouds are the efficiency factors
for scattering and absorption and the asymmetry parameter. Our very
limited sample of calculations for randomly oriented nonspherical parti-
cles suggests that it may be possible to accurately approximate these
parameters with results for equivalent spheres. With regard to remote
sensing applications which require knowI_lge of scattering behavior at a
specific angle, the indications are that it will be necessary to take ac-
count of the specific particle shape tuv01ved. The calculations presented
are only a small fraction ofwhat will be needed to draw any firm conclu-
sions; calculations are underway for larger sizes and for randomly orien-
ted oblate spheroids.
Acknowledgements: During the course of this research S.A. held a
NAS-NRC Post-doctoral Research Associateship at the Goddard Institute
for Space Studies. REFERENCES
Asano, S., Appl. Opt., 18 (in press)_ 1979 .
Asauo, S° and G. Yamam-o_, Applo Opt., 14,. 29, i975.
Hansen, J.E, and L.Do Travis, Light Scattering in Plant_ary Atmos-
t)heres, GISS Report, New York, 97 pp., 1976.
Kerker, M., Scattering of Light, Academic Press, NY, 666 pp., 1969.
Llou, K.N., Appl. Opt., 11, 667, 1972.
van de Hulst, H.C., Light Scattering by Small Particles, Wiley, New
York, 470 pp., 1957.
268
//
//
//
5 Jo 15 2o 25 30 35
Size Parameter
(a)
0.9
0.8
(17
O6
°°t// ---_
°,!OD t i I- I I I5 Io e5 20 25 30 35
Size Parameter
(b)
i 1.0 ,,,,.i,,,, , ,,, t..t
e' °-_
o 3o .o ,o _o ,. .o "_'o" ." ,o" ,o" ,,o" ,." .
S¢o.ef_ Ant, S_le,_ A_,
(c) (d)
g. 3. Comparison of scattering by raudomly oriented prolate spher-
es with a/b = 5 with scattering by equivalent spberes. All calculations
e for a real refractive index 1. 33. (a) is the efficiency factor for
attering, (b) is the asymmetry parameter of the phase function,
) is the phase function and (d) is the degree of polarization.
269


Scattering by randomly oriented ellipsoids: Application to aerosol and cloud problems

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