To be successful in the development of satellite retrieval methodologies for the determination of cirrus cloud properties, we must have fundamental scattering and absorption data on nonspherical ice crystals that are found in cirrus clouds. Recent aircraft observations (Platt et al. 1989) reveal that there is a large amount of small ice particles, on the order of 10 micron, in cirrus clouds. Thus it is important to explore the potential differences in the scattering and absorption properties of ice crystals with respect to their sizes and shapes. In this study the effects of nonspherical small ice crystals on the infrared radiative properties of cirrus clouds are investigated using light scattering properties of spheroidal particles. In Section 2, using the anomalous diffraction theory for spheres and results from the exact spheroid scattering program, efficient parameterization equations are developed for calculations of the scattering and absorption properties for small ice crystals. Parameterization formulas are also developed for large ice crystals using results computed from the geometric ray-tracing technique and the Fraunhofer diffraction theory for spheroids and hexagonal crystals. This is presented in Section 3. Finally, applications to the satellite remote sensing are described in Section 4

Asano, S.

Heymsfield, A.

Liou, K. N.

Minnis, P.

Takano, Y.

English

NASA Technical Reports Server

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THE EFFECTS OF SMALL ICE CRYSTALS ON
THE INFRARED RADIATIVE PROPERTIES OF CIRRUS CLOUDS
/ " /
•'//
Y. Takano and K. N. Liou
University of Utah, CARSS, Salt Lake City, Utah
S, Asano
84112
Meteorological Research Institute, Tsukuha, Ibaraki 305, Japan
A. Heymsfield
National Center for Atmospheric Research. Boulder, Colorado 80307
F. Minnls
Langley Research Center/NASA, Hampton, Virginia 23665
I. INTRODUCTION
To be successful in the development of
satellite retrieval methodologies for the
determination of cirrus cloud properties, we must
have fundamental scattering and absorption data on
nonspherlcal ice crystals that are found in cirrus
clouds. Recent aircraft observations (Platt et al.
1989) reveal that there is a large amount of small
ice particles, on the order of i0 pm, in cirrus
clouds. Thus it is important to explore the
potential differences in the scattering and
absorption properties of ice crystals with respect
to their sizes and shapes. In this study the
effects of nonspherlcal small ice crystals on the
infrared radiative properties of cirrus clouds are
investigated using light scattering properties of
spheroidal particles.
In Section 2, using the anomalous
diffraction theory for spheres and results from the
exact spheroid scattering program, efficient
parameterlzatatlon equations are developed for
calculations of the scattering and absorption
properties for small ice crystals.
Parameterlzation formulas are also developed for
large ice crystals using results computed from the
geometric ray-traclng technique and the Fraunhofer
diffraction theory for spheroids and hexagonal
crystals. This is presented in Section 3.
Finally, applications to the satellite remote
sensing are described in Section 4.
2. PARAMETERIZATION FOR SMALL ICE CRYSTALS
Because the effects of nonsphericity would
be small in the infrared region, small ice crystals
may be approximated as spheroidal particles. For
small size parameters of spheroidal particles,
scattering parameters computed from the Hie theory
are adjusted to match with those computed exactly
by the method of Asano and Sato (1980) based on the
anomalous diffraction theory (Latimer 1980).
For a spheroid of seml-axes a e and va,, the
equivalent spherical radius is given by
a, - a, [ v/(cose_ + v 2 aln 2 ()_/2]kl, (i)
where ( is the angle between the incident direction
and the rotation axis of the spheroid, a, the
radius of the revolved circle, and v the axial
ratio. The extinction or scattering cross section
of the spheroid is given by
C(() - QI_ _a:[ (cOs2( ÷ v2 sin2()1/2] k2 , (2)
where Q,_ is the optical efficiency factor from
the Hie theory for a sphere of radius, as, and
refractive index, m (- m r - i mi). If k 2 iS I in
Eq. (2), the rlght-hand side becomes Qs_ times the
geometric shadow area of the spheroid.-- The cross
section for randomly oriented spheroids is expres-
sed by
C, s = [,/2 Ce,' (()sin ( dE • (3)
• JO
The unknown constants, k I and k2,_in Eqs. (I) and
(2) are determined by matching Ce/A , where A is the
average geometric cross section of the spheroid,
with that computed exactly by Asano and Sato (1980)
as follows:
kl " 1.O, k 2 - 0.98, for val
k I • 0.96, k 2 - 1.08, for v<l .'
(4)
https://ntrs.nasa.gov/search.jsp?R=19930010857 2020-03-17T08:19:50+00:00Z
Further, by comparing with values computed from the
method of Asano end Sato, the extinction cross
section for randomly oriented spheroids may be
expressed by
Cs + Ca, vts, for_ < 0.07 ,
Ce = for m i _ 0.07p
(5)
where Ca yes is the absorption cross section for the
volume equivalent sphere. The single scattering
albedo is given by
(6)
C5 + CI, yes
When the spheroid shape becomes flat or elongated,
scattered light rays are inclined toward forward
directions. This effect on the asymmetry factor is
mimicked by adjusting the refractive index and
size.
With this procedure, the single scattering
properties of randomly oriented spheroids can be
efficiently calculated. Figures 1 (a) and I (b)
show the single scattering properties of randomly
oriented oblate and prolate spheroids with the
axial ratio of 2 as a function of xG - 2_rG/X for
II and 12 _m wavelengths, where rE is the radius of
area-equlvalent spheres. The solid and dash-dotted
lines are calculated by means of the preceding
parameterization. Circles and crosses are values
computed exactly from the method of Asano and Sato.
The accuracy of the present parameterization is
within about 3%.
2.40
tomde - 11 /.in Lomdm - 12 #m
2.00
,_ | .SO|
,t
z
s.2o
.SO
.,40
3. PARAMETERIZATION FOR LARGE ICE CRYSTALS
For large size parameters we use the
geometric optics approximation to develop pars-
meterlzatlon equations. The single scattering
properties of spheroids are computed from the ray-
tracing method and Fraunhofer diffraction theory.
Using these computed values, parameterlzed formulas
for the single scattering properties of large ice
crystals are derived.
Figure 2 shows the scattering phase
functions for randomly oriented oblate spheroids
wlth the axial ratio, I/v - 2, and m - (1.33, 0)
computed from the ray-optics approximation and the
exact method. The size parameter used is 30, which
is twice the value used in Fig. 9 of Asano and Sato
(1980). The peaks at the 0" scattering angle that
are produced by the Fraunhofer diffracted light
have about the same values for the two methods.
The peak appeared at @ - 76", computed from the
ray-optlcs approximation, corresponds to the
maximum created by one internal reflection (primary
rainbow feature). Since the rainbow angle is
dependent on the incident direction on the
spheroid, the rainbow peak is smaller than that of
spheres. Based on the exact method the rainbow
angle is located at about @ - 94". The peak is
smaller and broader and shifts toward backward
direction. This behavior is the same as that in
the case of spheres (Liou and Hansen 1971) and
circular cylinders (Takano and Tanaka 1980).
Glory-llke peaks appear at the backward direction
(8 - 180") on both methods. If the size parameter
is larger, these two methods give closer values.
This comparison verifies the ray-traclng program
developed in the present research and the
Fraunhofer diffraction approach developed by Takano
and Assno (1983). Hence, it is possible to pars-
m.--
Ler,'_a " 11 pm Lemsa " 12
Prolotl Spheroid (v - 2)
81nole $cettering Albe@o
.00 .00
O 4 O L2 $6 20 0 4 8 i2 IS
Size PerameteP, xg Size Pe,emeter. Xg
2O
Fig. i: Comparison of single scatterln 8 properties for randomly oriented (a) oblate and (b) prolate
spheroids between the parameterlzatlon (solld and dash-dotted lines) and the exact computation
(circles and crosses). The complex refractive indexes of ice used are m - 1.0925-i0.2_80 at I -
11 pm and m - 1.280-10.4133 ar I - 12 #m.
AOj
10 -I
10.2
_ Ray Opt iCl
I i i , i I , .... I .... ,
gO 120 180
Scetterln O AnQle (_e@,}
Fi E . 2: Comparison of the phase function for
randomly oriented oblate spheroids usin E the ray-
optics approximaclon (solid line) and the exact
theory (dash-dotted llne). The axial ratio, the
complex refractive index, and the size parameter
used are 2, (1.33, 0) and 30, respectlvely.
meterlze the single scattering albedo and asymmetry
factor of large spheroids using these programs.
&/hen absorption in a spheroidal particle is
weak, the absorption cross section can be expressed
by
4_mi r ]
c.--q- v , (7)
where m i is the imaginary index of refraction and
r v the radius of the volume equivalent sphere.
Thus, the absorption efficiency is given by
q, . C, 4=mir_ • z (8)
--J " --r--_ "
•rG r_.
The single scattering co-albedo may then be expres-
sed by
N
1 (9)
Using values of _ computed by the ray optics ap-
proximation for a number of z, the coefficients,
an, are determined. Figure 3 (a) shows the single
scattering co-albedo for randomly oriented oblate
spheroids as a function of z. The solid llne is
calculated from Eq. (9), whereas symbols (o, +, and
x) are computed from the ray-optlcs approximation.
The fitting is sufficiently accurate in the range
of z _ 0.4. On the other hand, when absorption
dominates, the single scattering co-albedo should
approach to a constant value, which cannot be
expressed by Eq. (9). Using the anomalous
diffraction formula for spheres, the single
scattering albedo for intensively absorbing
spheroids may be written in the form
&- 1 - (I - $) 2 X(cz) , (i0)
where
I e-W e-W_lK(w) - ¢ + __ + (Ii)
Z w w 2
As a result, _ - I when z - 0, and _ - _ when z
--, -. To be consistent with Eq. (9), z 13 chosen
as the independent variable. The constant, c, in
Eq. (I0) should be determined by fitting this
formula with ray-optlcs computations, e in Eq.
(i0) is given by Qsc,./2. Qsc, ® is 1 ;lus the
external reflected energy. Figu{e 3 (b) shows the
single scattering co-albedo as a function of z (>
0.4). The constant in Eq. (i0) depends on the
axial ratio of spheroid.
The extinction efficiency is given by
qe(x) = 2 + --xc {Qe(Xc) - 2] .
x (12)
With this functional form, Qe - 2 when x -, - and
Qe - Qe(Xc ) at the crossover point, x c. The cross-
over point is the size parameter which separates
small and large particle regions. The crossover
point is determined in the following manner.
10 e
IO'J
z
?
I0"*
[
i
SO "l |0 _
(i)
e I/v - 2
• 1Iv - 3
v-
, * i i
t@ -| 10 -a |O-I
1
10'
o
?
c
e
• Sly - 2
• Sly - 3
• t/v - 5
, , i I , I I , i
2 4 8 @ Io
z
Fig. 3: Single scattering co-albedo for randomly
oriented oblate spheroids as a function of z for
(a) weak and (b) stron E absorption. The assumed
axial ratios are 2, 3, and 5, which are represented
by o, +, and x, respectively.
Figures4 (a) and 4 (b) show the composite single
scattering albedo for randomly oriented oblate and
prolate spheroids with the axial ratio of 2 as a
function of the size parameter, x , at a wavelength
of iI Mm. The solid lines are t_e large particle
parameterlzatlon using Eqs. (9) and (i0), whereas
the dot-dashed lines are the small particle
parameterizatlon using Eq. (6). For comparison
purposes, the values computed by the exact method
are represented by circles. From these figures the
crossover point, x¢, is about 22 In units of the
area equivalent size parameter. A similar
procedure can be applied to the asymmetry factor.
For size parameters larger than about 30,
parameterlzation equations have been derived using
the results computed from the geometric ray-tracing
technique for hexagonal crystals (Takano and Liou
1989; Liou et al. 1990). The extinction cross
section for randomly oriented hexagonal crystals of
the same size can be expressed by
-_- ' (13)
where w is the width of the basal plane of a hexag-
onal crystal and D the crystal maximum length.
Based on numerical calculations involving a number
of wavelengths, particle sizes, D, and aspect ra-
tio, D/W, we find that the single scattering albedo
can be parameterlzed in the form:
.s0
•4O
.30
.iO
{al Omla%eSDhef'ol_ {S/v * 2)
/
Y
/
7 °
+
2" i l , l , , i ......
• s i2 16 _0 24
Sl:t( l%irlilter. Xll
-- LIPII! PInnacle
Sibe}lPlPt_¢le
"_(ect"
o
++
o
o
[
.50
.40
.SO
.00
m+ Pro|lie m_hero_+ (v. 2}
.-+.___
t
/
f
T
-- LePg+ PlP_Scll
----- Sil_ Pirti¢l,
o "Eject"
.......... , . . , . . , , .
• a [2 iS _0 24 Zll 32
Size Perll_ttr. nO
Fig. 4: Single scatterin E albedo for randomly
oriented (a) oblate and (b) prolate spheroids w_th
the axial ratlo of 2 at a wavelength of 11 _m
computed from the large partlcle parameterlzatJon
(solld llne), the small particle parameterlzatlon
(dash-dotted llne), and the exact method (clrcles).
4
I - Z f,z n , forz <0.4
n-i
& - (14)
1 - 0.47 [I - exp(-l.5051z 0-6_9 )] , for z k 0.4 ,
where z is a physical parameter defined by
4_m i r 3
v 4smi w 3 3_-- (D/w)
(15)
The coefficients in Eq. (14) are: fl - 1.1128, f2
- -2.5576, f3 - 5.6257, and f_ - -5.9498.
4. APPLICATION TO SATELLITE REMOTE SENSING
Using these parameterlzatlons, the scatter-
ing and absorption properties for ice crystal size
distributions obtained during the FIRE Intensive
Field Observation and the effects of small ice
crystals on the infrared radiative properties are
examined. As shown in Fig. I, the absorption
efficiencles of ice crystals for the wavelengths of
Ii and 12 _m can differ by as much as a factor of
two. This could provide a means for the detection
of small ice crystals using the radiances measured
from AVHRR channels 4 and 5.
5. ACKNOWLEDGEMENTS
This research has been supported by the
National Science Foundation under Grant ATM88-15712
and NASA Grant 1-1048.
6. REFERENCES
Asano, S. and 14. $ato, 1980: Light scattering by randomly oriented
spheroidal particles. AI_I. _t., 19, 962-974.
Latimer, P., 1980: Predicted scattering by spheroids: Comparison of
Iq_proximte lind exact methods. Ko_L. _t., 19, 3039-3041.
L_ou, K.M. end J.E. Nansen, 1971: Intensity and polarization for
alngie lCltter|ng by poLydilperle spheres: A compar|son of
ray-optics and Nte theory, d. Atlnos. _;fl+, _8, 995-1004.
iiOU, K.M., ¥. lakano, S.C. Ou, A. He_fietd+ and W. Krelss, 1990:
Infrared trlm¢.mlsslen through cirrus clouds: A radiative modet
for target ck_tection. _ (|n press).
Ptatt, C.N.R., J.D. $pin_Irne, and W.D, Hart, 1989: Optical and
microphysicai properties of a cold cirrus cloud: Evidence for
regions of ml| ice partlc[es. J. _o[_%v. Res., ?4, 11151-
11164.
Takan_, Y. lad S. Asano, 1983: Fraur_ofer diffraction by ice crystals
Suspended in the atmosphere. J. Meteor. +oc+ @span, 61, 289-
300.
Takano, Y. and K.M. L_OU, 1989: So[|r redietlve tra_fer in cirrus;
clouds. Part I: $|ngie-lCStter|ng Ind optical properties of
hexagonal ice crystals. J. Atmos. $ci., 46, 3-19.
Takano, Y. lind 14. Tanaka, 1980: Phlse Imtrtx lind cross sections for
Single scattering by circular cylinders: A comparison of ray
optics end wave theory. +4_o[. _t., 19, 2781-2793.


The effects of small ice crystals on the infrared radiative properties of cirrus clouds

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