It is shown how the radiative transfer equation reduces to the diffusion equation. To keep the mathematics as simple as possible, the approximation is applied to a cylindrical cloud of radius R and height h. The diffusion equation separates in cylindrical coordinates and, in a sample calculation, the solution is evaluated for a range of cloud radii with cloud heights of 0.5 km and 1.0 km. The simplicity of the method and the speed with which solutions are obtained give it potential as a tool with which to study the effects of finite-sized clouds on the albedo of the earth-atmosphere system

Arduini, R. F.

Barkstrom, B. R.

English

NASA Technical Reports Server

THE DIFFUSION APPROXIMATION - AN APPLICATION 
TO RADIATIVE TRANSFER IN CLOUDS 
Robert F. Arduini and Bruce R. Barkstrom 
The George Washington University 
Joint Institute for Advancement of Flight Sciences 
INTRODUCTION 
Water droplets and ice crystals, the constituents of clouds, are very 
nearly transparent (i.e. they absorb almost no radiation in the visual wave- 
lengths (400 nm to 700 nm)). Clouds are also optically thick with optical 
depths for a one kilometer path ranging from 10 to 50, depending upon droplet 
size and number density of droplets. Therefore, visible light which enters 
a cloud is scattered many times before being absorbed or exiting the cloud. 
This type of process is well described by a diffusion model. 
In this paper it is shown how the radiative transfer equation reduces 
To keep the mathematics as simple as possible, to the diffusion equation. 
the approximation is applied to a cylindrical cloud o f  radius R and height h. 
The diffusion equation separates in cylindrical coordinates and, in a sample 
calculation, the solution is evaluated for a range of cloud radii with cloud 
heights of 0.5 km and 1.0 km. 
The simplicity of the method and the speed with which solutions are 
obtained give it potential as a tool with which to study the effects of 
finite-sized clouds on the albedo of the earth-atmosphere system. 
THE DIFFUSION APPROXIMATION 
The diffusion approximation has long been used in nuclear reactor theory 
(refs. 1-3) and has recently been applied to the transfer of visual radiation 
in snow (ref. 4). In the diffusion approximation, the radiation is assumed 
to have an almost isotropic angular distribution, so that the specific 
intensity at space point y for radiation traveling in direction is 
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https://ntrs.nasa.gov/search.jsp?R=19770003386 2020-03-20T06:46:34+00:00Z
is the  mean in tens i ty ,  D is the diffusion coef f ic ien t ,  and - V is  the  gradient 
operator. Because the ne t  vector f lux 
@ E 1 QI dQ - - -  
is  proportional t o  the  gradient of J: 
it follows t h a t  J satisfies the  diffusion equation 
2 -2 V J - L  J = O  
where the diffusion length, L ,  is  re la ted  t o  D according t o  
2 
L = D/K 
K is  the absorption coeff ic ient .  
The net  f lux  through a surface with normal A i s  
(3 )  
(5) 
where a/an denotes the  direct ional  derivative.  The f lux  i n  the d i rec t ion  of 
+A i s  
and the f lux i n  d i rec t ion  -fi is 
= nJ + 2 n D  aJ/an n @ 
The plus and minus s igns indicate  the  pos i t ive  and negative senses of fi. 
1086 
The diffusion coefficient and the diffusion length are related to the 
extinction coefficient x and the phase function of the cloud droplets 
according to 
and 
In these equations, w is the single scattering albedo (or fraction of light 
scattered in a single interaction with a cloud droplet) and g is the mean 
cosine of single scattering. Equations (9) and (10) are essentially approxi- 
mations to the dispersion relation derived by Mika (ref. 5) for the largest 
singular eigenvalue in the singular eigenfunction solution for a plane- 
parallel atmosphere and expanded by van de Hulst (refs. 6 and 7) in his 
scaling laws. It may be noted that the diffusion equation (5) is identical 
to the Eddington approximation in a plane-parallel medium, in that for this 
geometry, the second moment of the radiation field, K, in a conservative 
atmosphere is equal to one-third of the mean intensity 
deep in the medium. Equation (11) uses the assumption of the plane-parallel 
atmosphere and describes the direction of propagation il in terms of a polar 
coordinate system with a polar angle 8 = cos-11.1 such that 8 = 0 is perpendi- 
cular to the plane of symmetry. 
THE CYLINDRICAL CLOUD 
Clouds in the atmosphere vary markedly in shape and size. 
mathematics as simple as possible, we have applied the diffusion approximation 
to a cylindrical cloud of radius R and height h. The cloud is illuminated 
from the top by a diffuse source, which is normalized to unit flux. .It is 
assumed that none of the radiation that escapes the cloud returns. 
To keep the 
With this choice of geometry and boundary conditions, the diffusion 
equation (5) separates. The solution for the mean intensity may be found 
as an expansion of standard mathematical functions in the form 
m 
J(r, z )  = C [A exp(amz) + Bm exp(-cimz)1Jo(B,r) m m=O 
1087 
with Jo(B,r) being the zeroth order Bessel function of the first kind. 
separation constants am and Bm are related through the diffusion length in 
the following manner: 
The 
2 2 -2 
m a = B m + L  (13) 
@m is the solution of the transcendental equation that results from the appli- 
cation of the boundary condition at the sides: 
where 
R c = - -  
2D 
Equation (14) is solved numerically using a Newton-Raphson method of root 
finding. The application of the boundary conditions at the top and bottom 
yields expressions for the expansion coefficients 
Ill - 
2 2 "'mR J1 ('mR) [ ( 1+2Dam) - (1-2Dam) exp (-2amh) 1 Bm 
and 
( 2Dam- 1 ) 
Am - (2Dam+l) exp (-2amh) Bm (17) 
Our primary interest in this study is in the fate of the energy incident 
on the cloud. As one might expect, the energy may be reflected back out the 
top, may be transmitted out the bottom, escape out the sides, or be absorbed 
within the cloud. The power incident on the cloud top is 
2Tr d$ J R r@L-) (z=O)dr = RR 2 
0 = ro 
+ 
TOP 
1088 
If we normalize the reflected and absorbed power by this amount, we find that 
they may be expressed in terms of the expansion coefficients and geometrical 
properties of the cloud in the form 
m 
E ' = 27r C [A m (1+2Da m 1 + B m (1-2Dam)1J1(BmR)/BmR (19) 
m=O TOP 
m 
= 2n 2 [Am(exp(amh)-l) + Bm(l-exp(-amh))] 
m=O E~~~~ 
m 
= 27-r C [A exp(a h) (1-2Dam) + B exp(-a h) J. E m m m m m=O BOTTOM 
and 
to 
= 8 n ~  C [Am (exp(amh)-l) + Bm(l-exp(-a m h))]. 
m=O ABS 
E 
A SAMPLE CALCULATION 
In a sample calculation the above expressions were evaluated for clouds 
of heights 0.5 and 1.0 km and a range of cloud radii from 0.5 to 10.0 km. 
The cloud droplet radius was assumed to be 10 microns with a number density 
of 100 The mean cosine g was assumed to be 0.8516 and the single 
scattering albedo w was chosen to represent nearly conservative scattering 
(l-w = and non-conservative scattering (1-61 = loB2). 
typical of cumulus clouds as seen in Deirmendjian's model C1 (ref. 8). 
These values are 
The effect of the finite radius is marked, as can be seen in Figs. I and 
2. The albedo of an isolated cloud, 1 km thick, may be reduced by 5 percent 
or more if the radius is less than 5 km. This reduction in albedo is due,to 
the leakage of energy out the sides of the cloud. 
smaller, the escape out the sides becomes more and more important. Clouds 
As the radius becomes .' 
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whose radii are about equal to their height lose nearly as much energy out the 
sides as is reflected back out the top. 
As the radius increases we expect the results to closely approach those 
from a plane-parallel treatment. 
results using the diffusion approximation for a cylindrical cloud and those 
using the Eddington approximation for a plane-parallel layer. 
is quite small for clouds whose horizontal extent is much larger than the 
vertical. 
Table 1 shows the difference between the 
The difference 
It is generally known that Monte Carlo techniques, which are currently 
being used to study the effects of finite-sized clouds (refs. 9 and lo), 
consume great amounts of computer time. 
approximation required less than 30 seconds execution time on a CDC CYBER 175 
computer. The simplicity of the approximation and the speed with which 
results are obtained give the diffusion approximation potential as a tool 
to study the effects of finite-sized clouds on the earth-atmosphere system. 
The above results using the diffusion 
CONCLUDING ITEMARKS 
Clouds represent an optically thick medium for visible radiation in which 
the internal radiation field is very nearly isotropic. Such a medium is well- 
suited to a description by a diffusion model. 
approximation to a cloud of cylindrical geometry, the fraction of the incident 
energy emerging from each of the cloud's surfaces has been calculated. The 
amount of radiation escaping from the sides becomes significant when the 
cloud's horizontal extent is less than ten times its vertical extent. The 
speed and simplicity of the method argue for its use to study the effects 
of finite-sized clouds on the earth's albedo. 
Applying the diffusion 
RFCFEmNCES 
1. Glasstone, S. and Edlund, M. C.: The Elements of Nuclear Reactor Theory. 
D. Van Nostrand, 1952, 416 pp. 
2. Meghreblian, R. V. and Holmes, D. K.: Reactor Analysis. McGraw-Hill, 
1960, 808 pp. 
3. Case, K. M. and Zweifel, P. F.: Linear Transport Theory. Addison-Wesley, 
1967, 342 pp. 
4. Bohren, C. F. and Barkstrom, B. R.: Theory of the Optical Properties of 
Snow. J. Geophys. Res., 1974, 4527-4535. 
5. Mika, J. R.: Neutron Transport With Anisotropic Scattering. Nucl. Sci. 
Eng., 1961, 415-427. 
1090 
6. van de Hulst, H. C.: The Spectrum of the Anisotropic Transfer Equation. 
Astron. Astrophys., 1970, 366-373. 
Albedo 0.875 
Transmission 0.125 
Albedo 0.549 
Transmission 0.009 
-a l-w = 10 
-2 l-w = 10 
7. van de Hulst, H. C.: High Order Scattering in Diffuse Reflection From A 
Semi-Infinite Atmosphere. Astron. Astrophys., 1970, 374-379. 
R = l O k m  5 k m  2 km 1 k m  0.5 km 
0.848 0.821 0.744 0.631 0.464 
0.114 0.104 0.076 0.040 0.009 
0.538 0.528 0.498 0.450 0.361 
0.009 0.008 0.007 0.005 0.002 
8. Deirmendjian, D.: Electromagnetic Scattering on Spherical Polydispersions. 
American Elsevier, 1969, 290 pp. 
9. McKee, T. B. and Cox, S. K.: Scattering of Visible Radiation By Finite 
Clouds. J. Atmos. Sci., 1974, 1885-1892. 
10. Davies, R.: Three Dimensional Transfer of Solar hdiation In Terrestrial 
Clouds. Am. Meteor. SOC., Second Conference on Atmospheric Radiation, 
Oct. 29-31, 1975. 
TABLE 1.-COMPARISON BETWEEN EDDINGTON (PLANE-PARALLEL) AND 
DIFFUSION (CYLINDRICAL) APPROXIMATIONS 
(Cloud Height = 1 km) 
109 1 
ABSO_R_BE& - -_ __ 
CI--------- 
0 2 4 6 8 10 
CLOUD RADIUS, R, km 
Figure 1.- Fa te  of energy inc ident  upon a c y l i n d r i c a l  cloud of 
height 0.5 km as a func t ion  of rad ius  ( so l id  l i n e :  
broken l i n e :  l-w = 
l-w = 10-8; 
FRACTION 
OF .6 
INCl DENT 
RA DIAT ION 
.4 
.2 
0 2 4 6 8 IO 
0 
CLOUD RADIUS, R, km 
Figure 2.- Fa te  of energy inc ident  upon a c y l i n d r i c a l  cloud of 
height 1.0 km as a func t ion  of r ad ius  ( so l id  l i n e :  
broken l i n e :  l - W  = 10-2). 
1-w = 10-8; 
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The diffusion approximation. An application to radiative transfer in clouds

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