Table Of ContentOptical Propertieso f Eight Water
Cloud Types
GnnEuEL . SrEpsexs
DIVISION OF ATMOSPHERIC PHYSICS TECHMCAL PAPER NO. 36
COMMONWEALTH SCIENTIFIC AND INDUSTRIAL
RESEARCH ORGANIZATION. AUSTRALIA 1979
CSIROA ust. Div. Atmog Phys. Teck. Pap. No. 36, 1-35 (1979)
Optical Propertieso f Eight Water Cloud Types
Grueme L. Stephens
Divisiono f AtmospheriPch ysicsC, SIRO,S tationS treetA, spendaleV,i c. 3195.
Abstract
Mie singles catteringp arameterasr ep rescribefdo r eightd ifferentw aterc loudt ypesu singu pto-date
and consistenvt alueso f liquid water refractivein dex. The main resultsa re presentedin tabular
form for eachc loudt ype for 110 differentw avelengthrasn gingfr om 0.3 prmt o 200 pm. Single
scatteringa lbedo,v olumee xtinctionc oefficienta nd asymmetryf actor are amongt he properties
presented.T heseq uantitiesc an be incorporatedin to multiples catteringra diativetr ansfers chemes
which are usedt o calculateth e grossr adiativep ropertieso f clouds.
The interaction of an electromagneticw ave with an absorbings pherei s described
by Mie theory and is adequatelyd iscussedin Van de Hulst's classicb ook (Van de
Hulst, 1957). Unfortunately, this book was publishedb efore the widespreadu se of
modern computers. Since that time, severaln umerical techniquesh ave been introduced
to effect fast and accurates olutions.
The three important single scatteringp arametersd etermined from Mie theory,
and ultimately used in multiple scatteringm odels, are the single scatteringa lbedo
(ejo), the asymmetry parameter characterizing the angular distribution of scattering
((cos0)) and the Mie extinction efficiency factor (Q"*1) from which the extinction
coefficient and optical depth of the medium is determined. Previousc omputations
of Ois, (cosO) and Q"*1 have been summarizedb y Feigelson( 1964) and Herman
(L962). Yamamoto et al. (197L) listed a number of Mie parametersf or three poly-
disperse( i.e. many sized particle) cloud models for wavelengths) . ) 5 pm. There
have been a large number of computations of Mie single scatteringp arameterso ver
the past decadeo r so. However,m ost of the results listed pertain to monodisperse
(i.e. single sized particle) media. There are few computations of 66 for L ( I pm
and calculationso f (cos0) for all wavelengthsa re lacking in detail. Irvine and
Pollack (1968) presenteda detailed study of 66 for ). ) 1 pm using single particles
of specific radius and presented( cos0) for an ice sphere of radius 10 pm illuminated
by radiationo f \ ) I pm. In this paper singles catteringp arametersa re presented
for a number of different water cloud types pgssessings pecifiedd roplet size distribu-
tions and for wavelengthsr anging from the ultraviolet to the far infrared.
The computations describedi n this paper illustrate the behaviour of the Mie
parametersf or three of the eight water cloud models introduced in Table 1. More
specific and detailed results appeari n the tables at the back of the paper listing all
major single scatteringp arametersf or the eight cloud models for 0.3 pm ( L ( 200 pm.
Mie theory calculationsa re time consuminga nd a table of single scatteringp arameters
for the short- and long-waver egionss hould be valuable in studieso f atmospheric
radiation.
2
Cloud Droplet Distribution
A set of standard cloud droplet distributions for radiation calculationsw as
presentedb y Stephens( 1978a). The representativenesosf these distributions is
difficult to establisha nd the rationale for their selectioni s adequatelyd iscussedin
that paper. The important featureso f the distributions are summarizedi n Table I
and Fig. 1. They representa reasonablyw ide range of water cloud types with the
mode radius (i.e. the radius correspondingt o the maximum droplet number) varying
from2.25 pm and the liquid water content from 0.05 gm-3 to 2.5 gm-3.
Table 1. Cloud modeld roplet distributionp arameteresm ployedin the presents tudy
Cloud type No. of drops Liquid water content Mode radius*
per cm3 (g m-t ) (pm)
Stratus I (St I) 440 0.22
StratusI I (St II) 120 0.05 a a<
Stratocumulus I (Sc I) 3s0 0.r4
StratocumulusI I (Sc II) 150 0.47
/.J
Nimbostratus (Ns) 280 U.5U
Altostratus (As) 430 0.28 4.5
Fair WeatherC umulus( Cu) 300 1.00
Cumulonimbus( Cb) 72 2.s0 5.5,6.s
* Radiusc orrespondintog maximumn umbero f drops.
Single Scattering
The scatteringp ropertieso f a single particle must be specifiedb efore they can be
determined for a volume element containing a distribution of particles (i.e. a poly-
dispersem edium). This is done by specifying three quantities which then determine
OJe and (cosd). Theseq uantities are the scatteringe fficiency factor er.u(r), the
extinction efficiency factot Q"*1(r) and the phasef unction P(r,0) appropriatet o the
individual particle of radius r. TIte latter quantity provides the angular distribution
urd, polarization of the scattered light related to the scattering angle 0.
A scatteringv olume of cloud possesseas distribution of particle sizest ypical of
those presentedi n Fig. l. The scatteringq uantities are determined from the appropriate
integrals over all particle sizes,i .e.
osc=a 7r fe*^e)n(r) d.r,
fi
oext= r Ii ,'gr*r7r1n(r)d ,r, (1)
P(0) = It f.;@ Q_ "" (,r-) P(r, 0) n(r) 12 dr .
,.*
The scatteringo f radiation by such sphericalp articles of radius comparablet o -
or larger than - the wavelengtho f incident radiation was first solvedb y Gustav Mie
in 1908 and thereby bridged the long existing gap between Rayleigh scatteringa nd
geometric optics. The theory is complex and comprehensived iscussionsc an be
found in the books of van de Hulst (195i), Kerker (1969) and Deirmendjian( 1969).
Only an outline of the theory is presentedh ere.
J
The intensity functions Sr and 52 which are the componentsp erpendiculara nd
parallel respectivelyt o the referencep lane (i.e. scatteringp lane), are given as
S1(xm, , 0) = | ; 2!:L {-a n"'( x, m) Iln(cos0 ) +b,(x, m)rn@os0)}12
n=1n ln+l)
(2)
S"(x,m ,q = | {bn(xm, )I l,,(co0s) + an(xm, )rn@os|)}12
f=, #,
E
l
c
o
c
o)
o
o
I
o,
l
o 'iN
ft)
c
Cb -> 6Opcm
5 r o 1 5 2 0 2 5 3 0 3 5
DroPlerto dius (pim)
Fig 1. The dropletd istributionso f eight cloud modelsd escribedin Table 1.
where an(x, m) atd bn@, m) are Mie coefficients. Here x is the Mie size parameter
defined as
x = 2nrfh, (3)
and m is the complex reflactive index of the spherer elative to the surrounding
medium which may be written in terms of its real and imaginary parts as
m = n, t ini. (4)
The angular coefficientsl lrr(cos) and rrr(cos) are defined as
^. dP"(cos0 )
lln(cosrl = and rn(cos0)= cos0l l,(cosg) -sing Ili(cosg) (5)
#
+
where P,r(cos0) is the Iegendre polynomial and the prime denotesd ifferentiation
with respectt o cosd.
The heart of the Mie scatteringp roblem lies in the computation of the coefficients
an and bn. The physicai parametersi nvolved in their determination are the complex
refractive index of the spherer elative to the surroundingm edium, m, and the size
parameter x.
The efficiency factors required to evaluate( l) are determinedb y
Q"*t@*, ) = 3a 62 (2n+1) Re{an(xm, ) +bn@m, )}
e n=7 (6)
Qr"^(xm, ) = 3
L n--1
and the efficiency factor for absorption follows
Q^a"(x,m ) = Q"*t(x, m) -Qr"u@, m) . (7)
The elementso f the phasef unction (normalized to unity) are given by the integrals
= ()r.l2n)2,_" nQ)si(x,m,0d)r (i = r,2) (8)
*o,r*,g,tr) 6
where 7 = 1 and 7 = 2 correspondt o the perpendiculara nd parallel components.
Polarizationi s neglecteda nd the averagep hase function
P(cosd)= %{pr(cos?)+ pr(cos0)l (e)
is adopted. Complete details of the procedure adopted to calculatet he Mie coefficients
an and br, the angularf unctions II,, and rn and the termination of the seriesi n (2)
are discussedb elow.
The phase function (9) can be separatedi nto forward and backward hemispheres
'backward
and the to forward' scatter ratio is related to the asymmetry factor
(cosd) defineda s the integralo ver al1s olid angleso fp(cosg)cosd. That is
(cos9)= Y, It_rP(cosgc)o sd dcosd. (10)
The asymmetry factor is equal to zero for isotropic scatter and +1 and -1 for
complete scatter into the forward and backward directionsr espectivelv and can be
obtained directly from Mie theory by
(cosd)= -x=' !e--"" (r) ,'?= , in* fu+f2 )-Re{ an( x' m) af a 1@,m ) +bn (x, m) b} a /.x, m)}
+ -n?(!n++ l\ Re{ a, (x, m) b}(x, m)} ( 1I )
where afra1 an'db fi'a1aret he complex conjugateso f a,1+t and brar.
'optically
Most water clouds in the atmospherea re thick'; only a very small
proportion of photons incident on a cloud or cloud layer emergew ithout being
scattered. The probability of the photon being multiple scatteredi s very high and
the photons that emergef rom the clouds have many paths within the cloud. con-
sequently the details of the phasef unction for individual particles are 'smoothed'
out by the multiple scatteringp rocessa nd only the grossf eatureso f the phase
function need be specified( usually in telms of (cos0)). The three major single
scatteringp arametersu sed by the multiple scatteringr adiative transfer models are
the asymmetry factor, the extinction coefficient and the single scatteringa lbedo
defined as
d6 = o""ulo"*1 . (r2)
Numerical Aspects
The evaluation of the summation seliesi n (2) involves the computation of the
angularC oefficientsI lr(cos0) and rn(cos?) and the complex Mie coefficientsa n and
b, whiclt are functions of the Mie size parameter (x = Znrl)r) and the complex
refractive index of the particle. The usual expressionsfo r these two functions are
(Van de Hulst, 1957)
,tt - m,lnt 0) {'n@)
an= tl"t('nD(,D1( ,,"@ (x)) - m,tr f'"(x)
'"O)'tt "U) (13 )
mrlr -- ttlltnt U) ** 'n@)
"(y)'lt ""(@x)) "(y) "(x)
winercm is the complex index of refraction and n is a positive integer. In these
definitions, 0 and f are the Ricatti-Besself unctions defined by
*n(z) = zjn(z) = (1/znz)'J/'r *n(z) (14)
(n(z)= ,nf)P1= Y,nt,# Jue)
wlrerc Jray" arrdH fJy" are the Besself unctions of the first and third kind and 7,
and hf;) are the correspondings phericalB esself unctions. The prime denotes
differentiation with respect to the argument of the function and
y = mx- (1s )
The computation schemep resentedb y Kattawar and Plass( 1967) is invoked to
provide a rapid solution to (13). At this point the logarithmic derivativeso f the
Besself unctions are presenteda s
Dn(y)= lln 0,0)]' and Gn@) = [n f"(x)]' (16)
and (13) canb e rewrittena s
tlr"(x) D"$t) -mD"(x)
qnn = - D"(y) -mG"(x)
lr(x)
. {"(x) mD"O) -Dr(x)
uh n = - mD"O) -G"(x)
l,(x)
Kattawar and Plass( 1961) observedt hat D, and Gn satisfy the recursion relation
Dn-{z) = nfz -lDn(z) +nlzl-l
(18)
Gn-{z) = nfz -fG,(z) +nlz)-t.
The upward recursionf ormulaeb ecomesu nstablew hen n) lzl; a regiono f
particular importance for large values of size parameter. On the other hand, the
downward recursion for Dn and G, are always numerically stable and are used in the
6
computations. Kattawar and Plassa lso noted that the calculationsa re insensitivet o
the assumeds tarting value and rapidly converget o the correct value. In practice
zero is a convenient starting value for D, and Go(x) = -i for all x is also useful
(Kattawar and Plass,1 967). Finally, the ratio ,lt"(x)l("@) in (16) can be expressed
in terms of the Besself unctions of the first and secondk ind which can be computed
by well-known recursionr elations (Abramowitz and Segan,1 965).
The angular functions can be determinedf rom (5) using recursion formulae
llral(cos.q =2n! _lftc os9fl,(.c. .- o'^s'0 ) ---=n; fIItn _ 1/(cos0 ) (19)
ill * 1( cos0 ) = (2n- 1) il,,( cos0 ) +fl', -1(cosg )
which are initializeda ssumins
II1 (cos0 ) = 1. f1 = CoS0
(20)
Il2(cos9)= 3 cosd rz=3cos20.
A useful simplification may be employed at this stage. Dave (1969) observedt hat
Il,(-cosd) = (-l)" flr(cosg)
Ql)
Il,(-cosd) = (-l)'rn(cos0)
reducing the number of computationsr equired to determine fln and rn.
Fig. 2. The efficiency factor for
extinction Qext as a function of
size parameter for a single particle.
The refractive index is n; = 0,
n, = I.33 and accordingly
=
Qsca Qext,
5 r0 50 ro0
x=2T(/x
Results and Discussion
Fig. 2, modified from Hansena nd Travis (1974), shows the efficiency factor of
extinction for a single particle plotted as a function of size parameter( and indirectly
wavelength). The calculationsa ssumen o droplet absorption (t.e. ni = 0) and thus
Qe*t = Qs" . The curve is characterizedb y a serieso f major maxima and minima
and superimposed ripples. The maxima and mrnima are due to the interference of
kght diffracted and transmitted by the particle. The smaller ripples superimposedo n
the 6urve result from the last few significant terms in the Mie series (2) and arise
from edge rays (i.e. liglrt rays grazing the sphere) including surface waves. The
diffracted radiation and the radiation arising from these surface waves optically inter-
fere, resulting in the ripples on the curve.
The absorption spectrum of liquid water droplet clouds is most easily studied
from Fig. 3, demonstratingt he expectedd ependenceo f single scatteringa lbedo on
cloud microphysics. The single scatteringa lbedo, for the three cloud types, is less
than 0.5 for tr ) ll pm; that is, the contribution of absorption to the total
extinction is larger than that of scattering. Since the quantity studied in Fig. 3
is a ratio, the oscillationsd iscerniblei n Q"*, (see Fig. 2) are smoothed out. This
smoothing is further accentuatedb ecauset he clouds are polydisperse.
i05 | .2
i Woveleng(lphm )
l3
o O
Eo
,o
o
o
c
o
E U
I
o
o
c
U'
o 2
03 | lo loo
Wovelength( p rm )
Fig. 3. The single scattering albedo 6o for three liquid water cloud
models as a function of wavelength.
It is significant to note that the absorptionb ands of liquid water for ). ( 4 pm
more or less overlap those associatedw ith water vapour absorption. There are weak
liquid water absorptionb ands between l-2 pm with much more intense bands at
about 3 trrma nd extending further in the infrared region. The absorption of short-
wave radiation by water clouds results from the absorption by the liquid water and
the water vapour bands in the near infrared region. For reference,t he major
molecular absorption bands for water vapour in the spectral region encompassedb y
8
the solar spectrum( near infraredr egion)a re centreda t 0.94, 1.1, 1.38, 1.87,2.7
and 3.2 pm.
Figs 4 and 5 illustrate the generalb ehaviour ofo"*1 and (cosd) as a function of
wavelengthf or three of the eight cloud models presentedi n Fig. l. Also superimposed
t60
I
!E -^ A_r---r, /,
9l--.^.r--'-*r\1'-'1i'- !;
_ t2O
c
o
o
o
.Ee o sctr., - -.2 \'t .^.i it .'
r...-Jal ^ l I
r /
o
ot
F= 4 0
o
sfII'
t(J
wovelengqthrm )
C b r , , , " , , , r , " ' , , ,
100 lO xe
g s [ , r . . . . , , r , ' ,
" , , , , , ,
100 lO I xe
srII
t0 X6
Fig. 4. The volume extinction coefficient for the three cloud models of Fig. 3.
on the wavelengtha xis (for o.*1 and (cos0)) is the effective size parumeterx e
defined as
x" = 2nr"flt (19)
wherc r" is the effective radius of distribution
(20)
This radius is a particularly useful parameter. It not only providess ome idea about
the nature of the distribution of particle sizesb ut is also directly related to the
shortwaveo ptical extinction coefficient (and thus optical depth - see Stephens,1 978b).
The extinction coefficient is shown in Fig. 4 as a function of L and is largest
'cumulonimbus' 'stratus
for the cloud model and is smallestf or the optically thin II'
model. The shapeo f oext for the three models is'generally similar and dependso n
9
the effective size parametersw hich are also included on Fig. 4. It is particularly
interesting that o"*1 is approximately constant with wavelength( or x") for shortwave
radiation (). ( 3.0 pm). This feature, reflected somewhato n Fig. 2 in which Q"*1
asymptotically approaches- 2, is indeed extremely usefi.rl,p articularly for the task
of deriving more simple radiative parumeleizations (e.g. Stephens, 1978b).
r.0
08
s
o
o
o
(u
o
E
E
a,
o2
^_\
sill
ro0
Woveleng(tphm )
cb
s c u . '
d d ' , , ' , b ' , . ' . 1 , , , -
sl''
lo I xe
Fig. 5. The asymmetryf actor as a function of wavelengthfo r the
threec loudm odelso f Fig. 3.
In Fig.5, the asymmetryf actor (cos0) for the three samplec loud modelsi s
plotted as a function of wavelength. Again the x" scalei s superimposeda long the
wavelengtha xis. For small x", the asymmetry factor approachesz ero, the value
obtained by Rayleigh scattering. On the other hand, for large x", (cos0) approaches
the result obtained with the geometricalo ptics phasef unction -0.87 (Van de Hulst,
1957). The oscillationsi n (cosd) have the same physical origin as those of Q"*1 and
are likewise smoothed out in a polydispersem edia. An important point is that the
asymmetry factor is somewhatl ess variablew ith changes.i n microphysics. This
feature is often usefully employed by radiation transfer theoristsw ho apply a constant
value of (cos0) for all cloud types. However such simplificationss hould be confined
to wavelengthss horter than 3 gm.
