One of the main objectives of the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE) is to provide a better understanding of the physics of upper level clouds. The focus is on just one specific aspect of cirrus physics, namely on characterizing the radiative properties of single, nonspherical ice particles. The basis for further more extensive studies of the radiative transfer through upper level clouds is provided. Radiation provides a potential mechanism for strong feedback between the divergence of in-cloud radiative flux and the cloud microphysics and ultimately on the dynamics of the cloud. Some aspects of ice cloud microphysics that are relevant to the radiation calculations are described. Next, the Discrete Dipole Approximation (DDA) is introduced and some new results of scattering by irregular crystals are presented. The Anomalous Diffraction Theory (ADT) was adopted to investigate the scattering properties of even larger crystals. In this way the scattering properties of nonspherical particles were determined over a range of particle sizes

Draine, Bruce T.

Flatau, Piotr J.

Stephens, Graeme L.

English

NASA Technical Reports Server

APRIL 1988 FLATAU, STEPHENS, DRAINE 
RADIATIVE PROPERTIES OF VISIBLE AND SUBVISIBLE CIRRUS: 
SCATTERING ON HEXAGONAL ICE CRYSTALS 
PIOTR J. FLATAU, GRAEME L. STEPHENS 
Department of Atmoaphedc Science, Colorado State University, Fort Collicrr, CO 80523 
BRUCE T. DRAINE 
Princeton Univertity Ob8ctoatory, Peyton Bolt, Princeton NJ 08644 
(April 1988) 
1. Introduction 
One of the main objectives of the First In- 
ternational Satellite Cloud Climatol y Project 
(ISCCP) Regional Experiment (FIFtg is to pro- 
vide a better understanding of the physics of 
upper level clouds. There are a number of fac- 
tors that have complicated the study of these 
clouds leaving our present understanding of up- 
per level clouds in a less than satisfactory state. 
One factor is simply the distance of the clouds 
from the surface. It is easier to study a low-level 
cloud system because of their proximity to the 
ground. Added to this observational problem is 
the fact that clouds in the upper troposphere 
seem to be governed by an intricate balance of 
not so well understood phenomena. Among the 
factors that have added to the complexity of the 
problem is the fact that particles are no longer 
spherical and both the solid and liquid water 
phases may coexist. Turbulence develops in the 
stable layer environment with its fine scales of 
vertical motion. Horizontal eddies (two dimen- 
sional turbulence) may be important for cloud 
morphology, and a host of interactions between 
gravity waves, turbulence, radiation, and micro- 
physics all seem possible. 
This paper concentrates on just one specific 
aspect of cirrus physics, namely on character- 
izing the radiative properties of single, non- 
spherical ice particles. While focused in this 
way, this study provides the basis for further 
more extensive studies of the radiative transfer 
through upper level clouds. Radiation provides 
a potential mechanism for strong feedback be- 
tween the divergence of in-cloud radiative flux 
and the cloud microphysics and ultimately on 
the dynamics of the cloud. 
We will firstly describe some aspects of ice 
cloud microphysics that are relevant to the ra- 
diation calculations. Next, the Discrete Dipole 
Approximation (DDA) is introduced and some 
new results of scattering by irregular crystals 
are presented. We also adopt the Anomalous 
DBraction Theory (ADT) to investigate the 
scattering properties of even larger crystals. In 
this way we are able to determine the scatter- 
ing properties of non spherical particles over a 
range of particle sizes. The study reported here 
is still preliminary and at the time of writing 
this abstract, the results are incomplete. We 
aim to incorporate the microphysics data col- 
lected during FIRE into these cdculations and 
in a companion study plan to use these scat- 
tering properties to determine daud radiative 
heating rates. 
2. Cloud microphysics 
Characterizing the shape and size of ice crys- 
tals in terms of their environment continues to 
be a subject of extensive research. For envi- 
ronments typical of cirrus clouds and for even 
the colder environments of polar stratospheric 
clouds (PSC’s), ice exists in single- and poly- 
crystalline forms. The more commonly per- 
ceived large composite crystals, of the order of 
500pm, are actually observed more readily at 
lower temperatures. Crystal sizes of the order 
of lOpm and lpm are, respectively, more repre- 
sentative of high cirrus in the tropics (Heyms- 
field, 1986) and PSC’s (Rosen et al., 1988). To 
emphasize this point we present the variation of 
crystal shape drawn in proportion to their size 
in Fig. 1 over the temperature range -20°C to 
-90°C. The crystals at cirrus temperatures, say 
-4O”C, are hexagonal columns although varia- 
tions of this type of crystal exist in the form of 
75 
https://ntrs.nasa.gov/search.jsp?R=19910001149 2020-03-19T20:15:26+00:00Z
SCATTERING ON HEXAGONAL ICE CRYSTALS 
hexagonal bullets and hollow columns. In this 
study we will focus only on single solid hexa 
crystals of more complex shape has been devel- 
oped. 
ond crystals although the foundation to stu f- y 
-60 
CI 
0 
0 
0.10 p m  
"Upper" 
0 Cirrus 
Fig. 1 %ical sizes of ice crystals encountervd 
in the tempemture mnge (-20,-9O)OC. The 
crystal sizes are appmximately dmwn lo scale. 
Notice that the warm cirrua particle (all hatched 
in the low part of the diagrum) w so large that 
only part w presented. 
3. Integral formulation of Maxwell equations 
and its approximation 
Only a very brief outline of the scattering 
methods are described here. We begin with 
Maxwell equations written in the integral form 
(Saxon, 1955) 
E(*) = Eo(x) + 4xk2 
c(x') - 1 (1) 
* E( x') d 'x' 41r 
X 1 G(x,x') 
where Eo(*) is the incident electric field, t(x) 
is a dielectric tensor, and Green's function is 
defined as 
G(x,x') = (1 -E$) - 
4rr 
(2) + (3ii - 1) (& - ') ." 
kr 47rr 
where r = x - d ,  r = Ix - x'l, and i = r/r. 
Boldfaced 1 is a unit matrix, and G is a dyadic 
or tensor) multiplication of two unit vectors i. k he incident electric field Eo(x) is 
~ ~ ( z )  = EoGeik'*x (3) 
where i is a unit vector specifying the wave's 
polarization, EO is the intensity of incident 
wave, & is a unit vector defining the wave 
propagation, k = 27r/X, and X is wavelength. 
We now seek a solution of this integral equa- 
tion to obtain the electric field and all the 
relevant scattering properties follow from this 
solution. Unfortunately, the integral (Fred- 
holm) equation (1) is not easily solvable al- 
though it is amenable to a variety of approx- 
imations commonly used in scattering prob- 
lems. In fact the appraximate methods such 
as the Rayleigh approximation (also known as 
the Rayleigh-Gam-Debye (RGD), the Rayleigh- 
Gans-Rocard or the Kirchoff or the first Born 
approximation), the second and higher Born 
approximations, Anomalous Diffraction Theory 
(ADT also known as High Energy Approxima- 
thy Discrete Dipole Approximation (DDA) and 
the Extended Boundar Method can all be de- 
rived directly from (1) &axon, 1955; Flatau and 
Stephens, 1988b). 
4. Discrete Dipole Approxinuttion 
A US& and general way to solve (1) is to dis- 
cretize the integral term thus reducing the in- 
tegral equation to a linear system of equations. 
In the DDA approximation one assumes that 
each volume element may be replaced by a point 
dipole, whose (tensor) polarizability is related 
to the volume of the element and the dielectric 
tensor using the Clausius-Mossotti relations, 
corrected for the effects of radiative reaction. 
Each of the dipoles acquires a (time-dependent) 
polarization in response to the electric field at 
the position of the dipole, which includes con- 
tributions from all of the other dipoles plus the 
incident wave. This is the approach of the DDA 
as o r i g i d y  formulated by deVoe (1964) and 
Purcell and Pennypacker (1973). Consider OUT 
hexagonal crystal in this instance divided into 
tion ( k EA) or the equivalent WKB method, 
76 
APRIL 1988 FLATAU, STEPHENS, DRAINE 
N discrete domains of volume uj. (Eg. 2) cen- 
tered around point xj, where 3 = 1 ,..., N. 
The volume of such an elementary domain is 
V 
vj = - N (4) 
where V is the volume of the particle, and N 
is the number of domains. Thus we are able to 
form a set of 3N equations from (1 since there 
The resulting equation set can then be repre- 
sented in the form 
are three components of the E fie1 d at each x;. 
as was A = 10.8pm with a corresponding value 
m = 1.089 + i0.182. The wavelengths rep- 
resent near infrared and infrared regions with 
small and relatively large absorption, respec- 
tively and correspond to central wavelengths of 
the AVHRR imager. The 10.8pm wavelength 
is also a relevant wavelength for application to 
COz lidar studies. 
The scattered intensity is presented in Fig. 3 
as a function of scatterin angle for two scat- 
and the 
other parallel to EO. 
tering planes, one perpen 8 'culm to 
dP=& (5) 
where P is the 3N-dimensional vector whose el- 
ements are the dipole moments of the N dipoles. 
For M h e r  details of the matrix structure and 
computational solution procedures the inter- 
ested reader can consult the recent work of 
Draine (1988). The electric field everywhere in 
the three-dimensional space is given in terms 
of the structure matrix A which depends only 
on particle's shape, refictive index and inci- 
dent wavelength but is independent of the direc- 
tion and polarization of the incident wave Eo(x). 
Once P is determined then single scattering 
properties of the particle can be derived from 
the relationships reported by Draine (1988). 
Fig. 2 Hexagonal discrete dipole armys with 
N = 384. Higher resolution was actually wed, 
see text forfirther comments. 
The DDA method has been applied to calcu- 
late the scattering properties of small hexage 
nal ice crystals. Figure 2 presented the discrete 
dipole array for N = 384 dipoles but for the 4- 
culations reported in this paper actually used 
2208 dipoles. The wavelength of A = 3.8pm 
and the corresponding value for the refractive 
index of ice m = 1.38 + i0.0067 was employed 
0 Lo 40 60 m la0 :Lo I40 lea 1m 
8 
. 3 Intensities for hexagonal ice for an in- 
age) and A = 3.8pm. Results for two scattering 
planes are presented. Crystal length L = 8pm, 
crystal mdiw a = 2pm. N = 2208 discrete 
dipoles we= wed. 
Fi% c  ent wave perpendicular to the crystal (aver- 
The incident direction was taken to be normal 
to the L-axis of the hexagon, the length of the 
crystal used in the calculations is specified by 
L, and the aspect ratio is defined as p = L/2a 
where 2a is the width of the crystal. Results are 
shown for two cases: incident electric field par- 
allel to, and perpendicular to, the L-axis of the 
hexagon. Because the hexagon is not rotation- 
ally symmetric, the scattering problem depends 
upon the orientation of the hexagon. Two cases 
have been considered (1) with the hexagon ori- 
ented so that the k vector is normal to one of the 
6 rectangular faces, and (2) with the hexagon 
rotated by 30 degrees, so that the k vector is 
parallel to a line connectin a vertex to the cen- 
has reflection symmetry wllich greatly reduces 
the amount of computing required to obtain a 
solution). The scattering intensities shown in 
Figures 3 and 4 are the averages of the scatter- 
ing intensities for the above two cases. 
ter of the hexagon. (Ea B 1 of these two cases 
77 
SCATTERING ON HEXAGONAL ICE CRYSTALS 
o 10 u w a LOO 110 w ma 1w 
e 
Fig. 4 Same as Fig. 3 but for X = 10.8pm. 
Crystal length L = 8pm, crystal mdiw a = 
2pm. 
Figure 5 shows the results of the M e  calcula- 
tion for a sphere with a volume equal to that of 
the hexagonal crystal. The intensities Ifli and 
I fl: are presented here as a function of scatter- 
ing angle and 
(see e.g. Bohren and Huffman, 1983). There are 
several features worthy of mention in comparing 
between Figs. 3, 4 and 5. For example the 
backscatter linear polarization, defined as 
lfl:(e = 180) - 1 jli(e = 180) 
ifi:(e = 180) + ifii(e = 180) 6(6 = 180) = 
(7) 
is non-vanishing in case of hexagon& as com- 
pared to a zero depolarization for spheres. This 
is a well known characteristic of scattering by 
non-spherical particles but the quantitative re- 
lationship between d and particle shape has 
never been convincingly established. The first 
minimum in intensity doesn't correspond to the 
Mie case, thus indicating that care has to be 
taken when interpreting results from the for- 
ward scattering PMS probes. This again is well 
established experimentally fact but its theoret- 
ical confirmation has to date been scarce. It 
seems that the extinction can be modelled, as 
it usually is, using a sphere of equivalent vol- 
ume or surface, and we plan to perfom more 
detailed calculations for the angularly averaged 
ease. 
A- UOL- @.OUY- 8.@RAD.K& 
20 40 BO 80 100 120 140 160 180 
10-11, a * a ' ' ' ' * * ' ' * * J 
mom 
Fig. 5 Mie results for equivalent volume spheres 
r = 3.41pm, and X = 3.8pm. 
5. Anomalous Diffraction Theory 
The anomalous diffraction theory (ADT) be- 
longs to the broad class of approximations (b 
gether with the Ftayleigh-Gans-Debye or Born 
expansions for example) in which one makes 
some assumption about the electric field under 
the integral of (1). This was noticed by Saxon 
(1955) and derived independently by van de 
Hulst (1957) on the basis of geometrical optics 
and diffraction. In the present context one as- 
sumes that the electric field inside the particle 
is given by 
E(*) = &iexp [ i k i .  (b + &z) - x(b,r)] (8) 
where X(b,z corresponds to the phase change 
particle. The impact vector b is perpendicu- 
lar to the z axis (see Fi . 6) .  The explicit 
other theories such as High Energy Approxi- 
mation and DDA is contained in Flatau and 
Stephens g988t. The anomalous *action 
theory (A T) Stephens, 1984) holds for par- 
ticles with a refractive index close to unity 
(m - 1) << 1 and for a particle with a size to 
due to the ck anged refractive index inside the 
formulation of X(b,z) an 8 relation of ADT to 
APRIL 1888 FLATAU, STEPHENS, DRAINE 
wavelength ratio z > 1. Sice the index of re- 
&action is dose to 1, the problem of total &c- 
tion in the ADT theory reduces to calculations 
of the interference between the almost straight 
transmission and the light diffracted according 
to Huygens’ principle. 
Fig. 6 Geometry for Anomalous D i M t i o n  
Theory calculations. b is an impact pammeter, 
r the position vector, k the dimtion of plane 
wave. The shadow area is in the far field and 
only fornard scattering is consided here. 
3, . . . . . - . . . . . . . . . . . . . ,  
PHASE SHIFT 
Fi 7 Extinction iency ( m u  ‘ng or os- 
c&ting around 2 htge ph~ae’Xj t  d u e s )  
and absorption e@iency for heaagonal ice C~YJ- 
tals for 3 diffetwrt values of cornpkz rejmctive 
index ob a finctwn of phase shijt. 
Thus, in the tmodous difFractian approxi- 
mation, the forward rcattering amplitude S(0) 
is given by 
where qY = &d(m - 1), d is the particle thick- 
ness, and m is the complex refraction: rn = 
n - in’. The quantity qS is the complex phase 
shifl of light passing thro the psrticle rela- 
contributes to absorptive attenuation. 
The extinction and absorption cdc ien t s  are 
defined as 
tive to that passing a ~ ~ u n  “dp it. The kdn’ term 
2 
Cccct = A R C L  (1 - c - ~ )  dA (10) 
1 
and 
C h  = - A A  (1 - e-”’) dA (11) 
and the single scattering albedo is 
(12) 
c, - cob* 
Cat 
w =  
Therefore applicatian of the ADT requires the 
relatively straight forward determination of the 
path length through the particle and evaluation 
of integrals for C, and Cda. Notice that (8) is 
more general because it provides, in principle, 
the full phase function. Calculations of Cc+t, 
Cd,, and w for hexagonal crystds are presented 
in Figs. 7 and 8 using this approximation for the 
geometry depicted in Fig. 6. Nl details of the 
method are planned in a fbrth coming paper and 
we also plan to compare these results, including 
phase functions, with solutions obtained from 
geometric optics and DDA. 
1 
PHASE SHIFT 
Fig. 8 Single scattering a& for hexagonal 
ice crystal in ADT appnnimatwn for 3 digeerent 
d u e s  of complex rejketiue index. 
79 
SCATTERING ON HEXAGONAL ICE CRYSTALS 
0. Summary 
This paper reports on new calculations of the 
scattering of radiation by hexagonal ice crys- 
tals. Results are presented for smal l  crystals 
which are not only the fist  of their type but also 
employ methods, to our knowledge, not previ- 
ously used in atmospheric scattering problems. 
Scattering properties of large hexagonal crystals 
were also modelled using more approximate the- 
ories. While the results presented in this paper 
apply only to hexagonal crystals, the methods 
are general and particles of other geometries wil l  
be considered in fkther studies. 
The work reported in this paper is prelimi- 
nary and we plan to use the micropbysic data 
collected during FIRE into these calculations 
to obtain the optical properties needed in the 
radiative transfer simulations of the cloud flu 
measurements . 
Ackmwledgments. We wish to thank Dr War- 
ren Wiscombe for providing us with his newest 
set of Mie codes. The research presented in 
this paper has been supported in part by the 
grant from the Air Force AFOSR-88-0143 and 
in part by the NSF grants AST-8612013 and 
ATM-8519160. 
REFERENCES 
Bohren, C. F. and D. R. Huffman, 1983: Absorp 
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DeVoe, H., 1964: Optical properties of molec- 
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Draine, B. T., 1988: The discretedipole ap- 
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Flatau, P. J., G. L. Stephens, and B. T. Draine, 
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Flatau, P. J., G. L. Stephens, 1988b: On the 
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80 


Radiative properties of visible and subvisible Cirrus: Scattering on hexagonal ice crystals

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